Patent Publication Number: US-2021183465-A1

Title: Computational Design of Self-Assembling Cyclic Protein Homo-oligomers

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 15/813,872 filed Nov. 15, 2017, which claims priority to U.S. Provisional Patent Application Ser. No. 62/422,872 filed Nov. 16, 2016, incorporated by reference herein in its entirety 
    
    
     BACKGROUND 
     Cyclic homo-oligomers assembled from multiple identical protein subunits symmetrically arranged around a central axis play key roles in many biological processes including catalysis, signaling and allostery. Despite their prevalence in natural systems, currently there is no systematic approach to design cyclic homo-oligomers starting from a monomeric protein structure. 
     SUMMARY OF THE INVENTION 
     In one aspect are provided polypeptides polypeptide comprising the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1,   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein X2 possesses changes from SEQ ID NO:1 at least at residues 16, 20, and 24;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein X2 possesses changes from SEQ ID NO:2 at least at residues 18, 22, and 26; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein X2 possesses changes from SEQ ID NO:3 at least at residues 18, 22, and 26; and       

     (c) X3, X4, and X5 are independently absent, or comprise the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1,   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NO: 5-7. 
     In another aspect, the invention provides polypeptides comprising the general formula X1-X2-X3-X4, wherein: 
     X1 is at least 50% identical along its length to residues 1-34 of the amino acid sequence of SEQ ID NO: 8, wherein the amino acid sequence of X1 differs from the amino acid sequence of residues 1-34 of SEQ ID NO: 8 at least at residues 6, 8, 13, 21, 25, and 28; 
     X2 is absent, or is at least 50% identical along its length to residues 36-68 of the amino acid sequence of SEQ ID NO: 8; 
     X3 is absent, or is at least 50% identical along its length to residues 69-102 of the amino acid sequence of SEQ ID NO: 8; and 
     X4 is absent, or is at least 50% identical along its length to residues 103-119 of the amino acid sequence of SEQ ID NO: 8. 
     In another aspect are provided polypeptides comprising the amino acid sequence at least 50% identical to the amino acid sequence of SEQ ID NO: 10, wherein all oligomerizing positions in SEQ ID NO: 10 have the amino acid residue shown in SEQ ID NO: 10, or conservative substitutions thereof, and wherein the polypeptide does not comprise acid sequence of SEQ ID NO: 9. 
     In a further aspect are provided polypeptides comprising the amino acid sequence at least 50% identical to SEQ ID NO: 11, wherein the polypeptide amino acid sequence differs from SEQ ID NO: 11 at least at residues 7, 8, 10, 14, 17, 118, 122, 146, 149, and 150. 
     In a still further aspect are provided polypeptides comprising the amino acid sequence that is at least 50% identical over its length to the amino acid sequence of a polypeptide selected from the group consisting of SEQ ID NOS: 10 and 12-40. 
     In a further aspect, a method is provided. A computing device determines a cycle of monomeric proteins. The computing device determines a docking score for the cycle of monomeric proteins. The docking score represents interaction between two or more monomeric proteins in the cycle of monomeric proteins with respect to a multi-dimensional rigid body transformation between three or more backbone atoms of the two or more monomeric proteins. The computing device determines whether the docking score for the cycle of monomeric proteins is a relatively-low docking score. After determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score, the computing device determines one or more interfaces between the two or more monomeric proteins in the cycle of monomeric proteins. An output is generated related to the cycle of monomeric proteins. 
     In another aspect, a computing device is provided. The computing device includes one or more processors; and non-transitory data storage that is configured to store at least computer-readable instructions that, when executed by the one or more processors, cause the computing device to perform functions. The functions include: determining a cycle of monomeric proteins; determining a docking score for the cycle of monomeric proteins, the docking score representing interaction between two or more monomeric proteins in the cycle of monomeric proteins with respect to a multi-dimensional rigid body transformation between three or more backbone atoms of the two or more monomeric proteins; determining whether the docking score for the cycle of monomeric proteins is a relatively-low docking score; after determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score, determining one or more interfaces between the two or more monomeric proteins in the cycle of monomeric proteins; and generating an output related to the cycle of monomeric proteins. 
     In another aspect, a non-transitory computer-readable medium is provided. The non-transitory computer-readable medium is configured to store at least computer-readable instructions that, when executed by one or more processors of a computing device, cause the computing device to perform functions. The functions include: determining a cycle of monomeric proteins; determining a docking score for the cycle of monomeric proteins, the docking score representing interaction between two or more monomeric proteins in the cycle of monomeric proteins with respect to a multi-dimensional rigid body transformation between three or more backbone atoms of the two or more monomeric proteins; determining whether the docking score for the cycle of monomeric proteins is a relatively-low docking score; after determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score, determining one or more interfaces between the two or more monomeric proteins in the cycle of monomeric proteins; and generating an output related to the cycle of monomeric proteins. 
     In another aspect, a device is provided. The device includes: means for determining a cycle of monomeric proteins; determining a docking score for the cycle of monomeric proteins, the docking score representing interaction between two or more monomeric proteins in the cycle of monomeric proteins with respect to a multi-dimensional rigid body transformation between three or more backbone atoms of the two or more monomeric proteins; means for determining whether the docking score for the cycle of monomeric proteins is a relatively-low docking score; means for, after determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score, determining one or more interfaces between the two or more monomeric proteins in the cycle of monomeric proteins; and means for generating an output related to the cycle of monomeric proteins. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1 : A block diagram of an example computing network. 
         FIG. 2A : A block diagram of an example computing device. 
         FIG. 2B : A block diagram of an example network of computing devices arranged as a cloud-based server system. 
         FIG. 3 : A flowchart of a method. 
         FIG. 4A-G : Alignment of 1na0 or tpr repeat designed peptides with the 1na0 reference sequence (1naRS, SEQ ID NO: 8) or the tpr reference sequence (tpRS, SEQ ID NO: 41). Column 1: 1na0C3_1 (SEQ ID NO: 16); Column 2: 1na0C3_int2 (SEQ ID NO: 17); Column 3: 1na0C3_3 (SEQ ID NO: 18); Column 4: 1na0C3_5 (SEQ ID NO: 19); Column 5: 1na0C3_7 (SEQ ID NO: 20); Column 6: 1na0C4_1 (SEQ ID NO: 21); Column 7: tpr1C4_2 (SEQ ID NO: 10); Column 8: T33_dn2A (SEQ ID NO: 33); Column 9: T33 dn2B (SEQ ID NO: 34); Column 10: T33_dn5A (SEQ ID NO: 35); Column 11: T33_dn10A (SEQ ID NO: 37); Column 12: 153_dn5B (SEQ ID NO: 40). 
         FIG. 5 : Computational design protocol. Left, starting with a monomeric protein we exhaustively sample cyclic docked configurations, score them using the RPX model and generate sequences to drive the complex formation using a full atom RosettaDesign™ calculation. Right, schematic representation of the RPX model scoring procedure. 
         FIG. 6 : Comparison between the experimentally determined crystal structures and corresponding design models. Crystal structures are shown in cyan and models in gray. Left column, full model and crystal structure superposition; Right column, superposition showing hydrophobic side chains at the designed interface. a, ank3C2_1 (r.ms.d. to model 1 Å) b, ank1C2_1 (r.ms.d. to model 0.9 Å) c, 1na0C3_3 (r.ms.d. to model 1 Å) d, HR00C3_2 (r.ms.d. to model 0.9 Å) e, ank1C4_2 pair of chains (r.ms.d. to model 1.1 Å)  FIG. 7 : Robustness of designs to subunit extension by repeat addition. From left to right: computational model of the original design, computational model of the extended design, SEC-MALS chromatogram used for molecular weight determination (n represents number of repeat modules in each monomer; original design: solid line; extended design: dotted line), SAXS scattering profiles (original design: experimental data in black circles, computed profile in red; extended design: experimental data open circles, computed profile in cyan). a, ank1C2_1. B, HR04C4_1. 
     
    
    
     DETAILED DESCRIPTION 
     All references cited are herein incorporated by reference in their entirety. Within this application, unless otherwise stated, the techniques utilized may be found in any of several well-known references such as:  Molecular Cloning: A Laboratory Manual  (Sambrook, et al., 1989, Cold Spring Harbor Laboratory Press),  Gene Expression Technology  (Methods in Enzymology, Vol. 185, edited by D. Goeddel, 1991. Academic Press, San Diego, Calif.), “Guide to Protein Purification” in  Methods in Enzymology  (M. P. Deutshcer, ed., (1990) Academic Press, Inc.);  PCR Protocols: A Guide to Methods and Applications  (Innis, et al. 1990. Academic Press, San Diego, Calif.),  Culture of Animal Cells: A Manual of Basic Technique,  2 nd  Ed. (R. I. Freshney. 1987. Liss, Inc. New York, N.Y.),  Gene Transfer and Expression Protocols , pp. 109-128, ed. E. J. Murray, The Humana Press Inc., Clifton, N.J.), and the Ambion 1998 Catalog (Ambion, Austin, Tex.). 
     As used herein, the singular forms “a”, “an” and “the” include plural referents unless the context clearly dictates otherwise. “And” as used herein is interchangeably used with “or” unless expressly stated otherwise. 
     As used herein, the amino acid residues are abbreviated as follows: alanine (Ala; A), asparagine (Asn; N), aspartic acid (Asp; D), arginine (Arg; R), cysteine (Cys; C), glutamic acid (Glu; E), glutamine (Gln; Q), glycine (Gly; G), histidine (His; H), isoleucine (Ile; I), leucine (Leu; L), lysine (Lys; K), methionine (Met; M), phenylalanine (Phe; F), proline (Pro; P), serine (Ser; S), threonine (Thr; T), tryptophan (Trp; W), tyrosine (Tyr; Y), and valine (Val; V). 
     All embodiments of any aspect of the invention can be used in combination, unless the context clearly dictates otherwise. 
     In one aspect are provided isolated polypeptides comprising the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1 [[[residues 1-31 of SEQ ID NO: 4]]   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2 [[[residues 32-64 of SEQ ID NO: 4]]; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3 [[[residues 131-157 of SEQ ID NO:4]];       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein X2 possesses changes from SEQ ID NO:1 at least at residues 16, 20, and 24;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein X2 possesses changes from SEQ ID NO:2 at least at residues 18, 22, and 26; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein X2 possesses changes from SEQ ID NO:3 at least at residues 18, 22, and 26; and       

     (c) X3, X4, and X5 are independently absent, or comprise the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1,   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NO: 5-7. 
     
       
         
           
               
            
               
                 Ank reference sequence (i.e.: includes SEQ ID 
               
               
                 NOS: 1, 2, and 3) 
               
               
                 (SEQ ID NO: 4) 
               
               
                 SELGKRLIEAAENGNKDRVKDLLENGADVNASDSDGKTPLHLAAENGHKE 
               
               
                   
               
               
                 VVKLL(I/L)S(Q/K)G(D/A)(P/D)(V/P/N)(N/T)(A/T/S) 
               
               
                   
               
               
                 (S/K)DSDG(K/R)TPLH(H/L/Y)AAENGHKE(V/I)VKLL(I/L)S 
               
               
                   
               
               
                 (Q/K)G(D/A)(P/D)(V/P/N)(N/T)(A/T/S)(S/K)DSDG(K/R) 
               
               
                   
               
               
                 TPLH(H/L/Y)AAENGHKE(V/I)VKLL(I/L)S(Q/K)G(D/A)(P/D) 
               
               
                   
               
               
                 (V/P/N)(N/T)(A/T/S)SDSDGRTPLDLAREHGNEE(V/I)VKLLEKQ 
               
               
                   
               
               
                 Wild type ankyrins 
               
               
                 &gt; ank1 
               
               
                 (SEQ ID NO: 5) 
               
               
                 SELGKRLIEAAENGNKDRVKDLIENGADVNASDSDGRTPLHHAAENGHKE 
               
               
                   
               
               
                 VVKLLISKGADVNAKDSDGRTPLHHAAENGHKEVVKLLISKGADVNAKDS 
               
               
                   
               
               
                 DGRTPLHHAAENGHKEVVKLLISKGADVNTSDSDGRTPLDLAREHGNEEV 
               
               
                   
               
               
                 VKLLEKQ 
               
               
                   
               
               
                 &gt; ank3 
               
               
                 (SEQ ID NO: 6) 
               
               
                 SELGKRLIEAAENGNKDRVKDLLENGADVNASDSDGKTPLHLAAENGHKE 
               
               
                   
               
               
                 VVKLLLSQGADPNAKDSDGKTPLHLAAENGHKEVVKLLLSQGADPNAKDS 
               
               
                   
               
               
                 DGKTPLHLAAENGHKEVVKLLLSQGADPNTSDSDGRTPLDLAREHGNEEV 
               
               
                   
               
               
                 VKLLEKQ 
               
               
                   
               
               
                 &gt; ank4 
               
               
                 (SEQ ID NO: 7) 
               
               
                 SELGKRLIEAAENGNKDRVKDLLENGADPNASDSDGRTPLHYAAENGHKE 
               
               
                   
               
               
                 IVKLLLSKGADPNAKDSDGRTPLHYAAENGHKEIVKLLLSKGADPNAKDS 
               
               
                   
               
               
                 DGRTPLHYAAENGHKEIVKLLLSKGADPNTSDSDGRTPLDLAREHGNEEI 
               
               
                   
               
               
                 VKLLEKQ 
               
            
           
         
       
     
     As described in the examples that follow, the polypeptides of all aspects and embodiments of the invention were designed for their ability to self-assemble to cyclic homoligomers with tunable shape, size, and symmetry enables rigid display of binding domains at arbitrary orientations and distances for a range of biological applications. 
     In this aspect, design interfaces were grafted onto ankyrin repeat scaffolds, permitting the designed polypeptides to direct their assembly into homo-oligomeric complexes (such as dimers, trimers, tetramers, and pentamers). 
     The ankyrin-derived repeat polypeptides of the invention include between 2-5 repeat domains (X1-X5), depending on where the design interfaces (referred to herein as “oligomerizing positions”) are located. Thus, if all of the oligomerizing positions are located in domains X1 and X2, then X3-X5 may be absent, or may be present. The repeat domains in the ankyrin-derived polypeptides are interchangeable, and thus each domain X1-X5 may be selected from modified regions of a scaffold ankyrin domain. 
     Oligomerizing positions for a variety of polypeptides of this aspect of the invention are shown Table 1 below, aligned with the wild-type ankyrin repeat consensus sequences. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
               
                   
                 Repeat domain 
                 ank reference 
                   
                   
                   
                   
               
               
                 Position 
                 designation 
                 sequence 
                 ank1C2_1 
                 ank1C4_2 
                 ank3C2_1 
                 ank4D2 
               
               
                   
               
             
            
               
                  1 
                 N-terminal Cap 
                 S 
                   
                   
                   
                   
               
               
                  2 
                 N-terminal Cap 
                 E 
                   
                 E 
                   
                 T 
               
               
                  3 
                 N-terminal Cap 
                 L 
                   
                 D 
                   
                 E 
               
               
                  4 
                 N-terminal Cap 
                 G 
                   
                   
                   
                   
               
               
                  5 
                 N-terminal Cap 
                 K 
                   
                 E 
                   
                 K 
               
               
                  6 
                 N-terminal Cap 
                 R 
                   
                 L 
                   
                 M 
               
               
                  7 
                 N-terminal Cap 
                 L 
                   
                   
                   
                   
               
               
                  8 
                 N-terminal Cap 
                 I 
                   
                   
                   
                   
               
               
                  9 
                 N-terminal Cap 
                 E 
                   
                 L 
                   
                 I 
               
               
                  10 
                 N-terminal Cap 
                 A 
                   
                 A 
                   
                 A 
               
               
                  11 
                 N-terminal Cap 
                 A 
                   
                   
                   
                   
               
               
                  12 
                 N-terminal Cap 
                 E 
                   
                   
                   
                 R 
               
               
                  13 
                 N-terminal Cap 
                 N 
                   
                 L 
                   
                 E 
               
               
                  14 
                 N-terminal Cap 
                 G 
                   
                   
                   
                   
               
               
                  15 
                 N-terminal Cap 
                 N 
                   
                 I 
                   
                 M 
               
               
                  16 
                 N-terminal Cap 
                 K 
                 K 
                 A 
                 K 
                 I 
               
               
                  17 
                 N-terminal Cap 
                 D 
                   
                 E 
                 D 
                 I 
               
               
                  18 
                 N-terminal Cap 
                 R 
                   
                 A 
                   
                 V 
               
               
                  19 
                 N-terminal Cap 
                 V 
                   
                   
                   
                   
               
               
                  20 
                 N-terminal Cap 
                 K 
                   
                 R 
                 K 
                 I 
               
               
                  21 
                 N-terminal Cap 
                 D 
                   
                 M 
                   
                 V 
               
               
                  22 
                 N-terminal Cap 
                 L 
                   
                 L 
                   
                 L 
               
               
                  23 
                 N-terminal Cap 
                 I/L 
                   
                   
                   
                   
               
               
                  24 
                 N-terminal Cap 
                 E 
                   
                   
                   
                 E 
               
               
                  25 
                 N-terminal Cap 
                 N 
                   
                 Q 
                   
                 K 
               
               
                  26 
                 N-terminal Cap 
                 G 
                   
                   
                   
                   
               
               
                  27 
                 N-terminal Cap 
                 A 
                   
                   
                   
                   
               
               
                  28 
                 N-terminal Cap 
                 D 
                   
                   
                   
                   
               
               
                  29 
                 N-terminal Cap 
                 V/P 
                   
                   
                   
                   
               
               
                  30 
                 N-terminal Cap 
                 N 
                   
                   
                   
                   
               
               
                  31 
                 N-terminal Cap 
                 A 
                   
                   
                   
                   
               
               
                  32 
                 Internal Repeat 1 
                 S/K 
                   
                   
                   
                   
               
               
                  33 
                 Internal Repeat 1 
                 D 
                   
                   
                   
                   
               
               
                  34 
                 Internal Repeat 1 
                 S 
                   
                 D 
                   
                   
               
               
                  35 
                 Internal Repeat 1 
                 D 
                   
                   
                   
                   
               
               
                  36 
                 Internal Repeat 1 
                 G 
                   
                   
                   
                   
               
               
                  37 
                 Internal Repeat 1 
                 K/R 
                   
                   
                   
                   
               
               
                  38 
                 Internal Repeat 1 
                 T 
                   
                   
                   
                   
               
               
                  39 
                 Internal Repeat 1 
                 P 
                   
                   
                   
                   
               
               
                  40 
                 Internal Repeat 1 
                 L 
                   
                   
                   
                   
               
               
                  41 
                 Internal Repeat 1 
                 H 
                   
                   
                   
                   
               
               
                  42 
                 Internal Repeat 1 
                 H/L/Y 
                   
                   
                   
                   
               
               
                  43 
                 Internal Repeat 1 
                 A 
                   
                   
                   
                   
               
               
                  44 
                 Internal Repeat 1 
                 A 
                   
                   
                   
                   
               
               
                  45 
                 Internal Repeat 1 
                 E 
                   
                   
                   
                   
               
               
                  46 
                 Internal Repeat 1 
                 N 
                   
                   
                   
                   
               
               
                  47 
                 Internal Repeat 1 
                 G 
                   
                   
                   
                   
               
               
                  48 
                 Internal Repeat 1 
                 H 
                   
                   
                   
                   
               
               
                  49 
                 Internal Repeat 1 
                 K 
                 A 
                 L 
                 A 
                 L 
               
               
                  50 
                 Internal Repeat 1 
                 E 
                 E 
                 A 
                 K 
                 I 
               
               
                  51 
                 Internal Repeat 1 
                 V/I 
                   
                   
                   
                   
               
               
                  52 
                 Internal Repeat 1 
                 V 
                   
                   
                   
                   
               
               
                  53 
                 Internal Repeat 1 
                 K 
                 A 
                 L 
                 L 
                 L 
               
               
                  54 
                 Internal Repeat 1 
                 L 
                   
                 L 
                 L 
                 L 
               
               
                  55 
                 Internal Repeat 1 
                 L 
                   
                   
                   
                   
               
               
                  56 
                 Internal Repeat 1  
                 I/L 
                   
                   
                   
                   
               
               
                  57 
                 Internal Repeat 1 
                 S 
                 E 
                 L 
                 E 
                 E 
               
               
                  58 
                 Internal Repeat 1 
                 Q/K 
                   
                   
                   
                   
               
               
                  59 
                 Internal Repeat 1 
                 G 
                   
                   
                   
                   
               
               
                  60 
                 Internal Repeat 1 
                 D/A 
                   
                   
                   
                   
               
               
                  61 
                 Internal Repeat 1 
                 P/D 
                   
                   
                   
                   
               
               
                  62 
                 Internal Repeat 1 
                 V/P/N 
                   
                   
                   
                   
               
               
                  63 
                 Internal Repeat 1 
                 N/T 
                   
                   
                   
                   
               
               
                  64 
                 Internal Repeat 1 
                 A/T/S 
                   
                   
                   
                   
               
               
                  65 
                 Internal Repeat 2 
                 S/K 
                   
                   
                   
                   
               
               
                  66 
                 Internal Repeat 2 
                 D 
                   
                   
                   
                   
               
               
                  67 
                 Internal Repeat 2 
                 S 
                   
                   
                   
                   
               
               
                  68 
                 Internal Repeat 2 
                 D 
                   
                   
                   
                   
               
               
                  69 
                 Internal Repeat 2 
                 G 
                   
                   
                   
                   
               
               
                  70 
                 Internal Repeat 2 
                 K/R 
                   
                   
                   
                   
               
               
                  71 
                 Internal Repeat 2 
                 T 
                   
                   
                   
                   
               
               
                  72 
                 Internal Repeat 2 
                 P 
                   
                   
                   
                   
               
               
                  73 
                 Internal Repeat 2 
                 L 
                   
                   
                   
                   
               
               
                  74 
                 Internal Repeat 2 
                 H 
                   
                   
                   
                   
               
               
                  75 
                 Internal Repeat 2 
                 H/L/Y 
                   
                   
                   
                   
               
               
                  76 
                 Internal Repeat 2 
                 A 
                   
                   
                   
                   
               
               
                  77 
                 Internal Repeat 2 
                 A 
                   
                   
                   
                   
               
               
                  78 
                 Internal Repeat 2 
                 E 
                   
                   
                   
                   
               
               
                  79 
                 Internal Repeat 2 
                 N 
                   
                   
                   
                   
               
               
                  80 
                 Internal Repeat 2 
                 G 
                   
                   
                   
                   
               
               
                  81 
                 Internal Repeat 2 
                 H 
                   
                   
                   
                   
               
               
                  82 
                 Internal Repeat 2 
                 K 
                 D 
                   
                 A 
                   
               
               
                  83 
                 Internal Repeat 2 
                 E 
                 E 
                 T 
                 V 
                   
               
               
                  84 
                 Internal Repeat 2 
                 V/I 
                   
                   
                   
                   
               
               
                  85 
                 Internal Repeat 2 
                 V 
                   
                   
                   
                   
               
               
                  86 
                 Internal Repeat 2 
                 K 
                 L 
                 L 
                 A 
                   
               
               
                  87 
                 Internal Repeat 2 
                 L 
                 I 
                 L 
                 L 
                   
               
               
                  88 
                 Internal Repeat 2 
                 L 
                   
                   
                   
                   
               
               
                  89 
                 Internal Repeat 2 
                 I/L 
                 L 
                   
                 K 
                   
               
               
                  90 
                 Internal Repeat 2 
                 S 
                 L 
                 L 
                 M 
                   
               
               
                  91 
                 Internal Repeat 2 
                 Q/K 
                 K 
                 M 
                 H 
                   
               
               
                  92 
                 Internal Repeat 2 
                 G 
                   
                   
                   
                   
               
               
                  93 
                 Internal Repeat 2 
                 D/A 
                   
                   
                   
                   
               
               
                  94 
                 Internal Repeat 2 
                 P/D 
                   
                   
                   
                   
               
               
                  95 
                 Internal Repeat 2 
                 V/P/N 
                   
                   
                   
                   
               
               
                  96 
                 Internal Repeat 2 
                 N/T 
                   
                   
                   
                   
               
               
                  97 
                 Internal Repeat 2 
                 A/T/S 
                   
                   
                   
                   
               
               
                  98 
                 Internal Repeat 3 
                 S/K 
                   
                   
                   
                   
               
               
                  99 
                 Internal Repeat 3 
                 D 
                   
                   
                   
                   
               
               
                 100 
                 Internal Repeat 3 
                 S 
                   
                   
                   
                   
               
               
                 101 
                 Internal Repeat 3 
                 D 
                   
                   
                   
                   
               
               
                 102 
                 Internal Repeat 3 
                 G 
                   
                   
                   
                   
               
               
                 103 
                 Internal Repeat 3 
                 K/R 
                   
                   
                   
                   
               
               
                 104 
                 Internal Repeat 3 
                 T 
                   
                   
                   
                   
               
               
                 105 
                 Internal Repeat 3 
                 P 
                   
                   
                   
                   
               
               
                 106 
                 Internal Repeat 3 
                 L 
                   
                   
                   
                   
               
               
                 107 
                 Internal Repeat 3 
                 H 
                   
                   
                   
                   
               
               
                 108 
                 Internal Repeat 3 
                 H/L/Y 
                   
                   
                   
                   
               
               
                 109 
                 Internal Repeat 3 
                 A 
                   
                   
                   
                   
               
               
                 110 
                 Internal Repeat 3 
                 A 
                   
                   
                   
                   
               
               
                 111 
                 Internal Repeat 3 
                 E 
                   
                   
                   
                   
               
               
                 112 
                 Internal Repeat 3 
                 N 
                   
                   
                   
                   
               
               
                 113 
                 Internal Repeat 3 
                 G 
                   
                   
                   
                   
               
               
                 114 
                 Internal Repeat 3 
                 H 
                   
                   
                   
                   
               
               
                 115 
                 Internal Repeat 3 
                 K 
                 K 
                   
                 E 
                   
               
               
                 116 
                 Internal Repeat 3 
                 E 
                 R 
                   
                 E 
                   
               
               
                 117 
                 Internal Repeat 3 
                 V/I 
                   
                   
                   
                   
               
               
                 118 
                 Internal Repeat 3 
                 V 
                   
                   
                   
                   
               
               
                 119 
                 Internal Repeat 3 
                 K 
                 L 
                   
                 I 
                   
               
               
                 120 
                 Internal Repeat 3 
                 L 
                 V 
                   
                 L 
                   
               
               
                 121 
                 Internal Repeat 3 
                 L 
                   
                   
                   
                   
               
               
                 122 
                 Internal Repeat 3  
                 I/L 
                 I 
                   
                   
                   
               
               
                 123 
                 Internal Repeat 3 
                 S 
                 L 
                   
                 A 
                   
               
               
                 124 
                 Internal Repeat 3 
                 Q/K 
                 A 
                 K 
                 M 
                   
               
               
                 125 
                 Internal Repeat 3 
                 G 
                   
                   
                   
                   
               
               
                 126 
                 Internal Repeat 3 
                 D/A 
                   
                   
                   
                   
               
               
                 127 
                 Internal Repeat 3 
                 P/D 
                   
                   
                   
                   
               
               
                 128 
                 Internal Repeat 3 
                 V/P/N 
                   
                   
                   
                   
               
               
                 129 
                 Internal Repeat 3 
                 N/T 
                   
                   
                   
                   
               
               
                 130 
                 Internal Repeat 3 
                 A/T/S 
                   
                   
                   
                   
               
               
                 131 
                 C-terminal Cap 
                 S 
                   
                   
                   
                   
               
               
                 132 
                 C-terminal Cap 
                 D 
                   
                   
                   
                   
               
               
                 133 
                 C-terminal Cap 
                 S 
                   
                   
                   
                   
               
               
                 134 
                 C-terminal Cap 
                 D 
                   
                   
                   
                   
               
               
                 135 
                 C-terminal Cap 
                 G 
                   
                   
                   
                   
               
               
                 136 
                 C-terminal Cap 
                 R 
                   
                   
                   
                   
               
               
                 137 
                 C-terminal Cap 
                 T 
                   
                   
                   
                   
               
               
                 138 
                 C-terminal Cap 
                 P 
                   
                   
                   
                   
               
               
                 139 
                 C-terminal Cap 
                 L 
                   
                   
                   
                   
               
               
                 140 
                 C-terminal Cap 
                 D 
                   
                   
                   
                   
               
               
                 141 
                 C-terminal Cap 
                 L 
                   
                   
                   
                   
               
               
                 142 
                 C-terminal Cap 
                 A 
                   
                   
                   
                   
               
               
                 143 
                 C-terminal Cap 
                 R 
                   
                   
                   
                   
               
               
                 144 
                 C-terminal Cap 
                 E 
                   
                   
                   
                   
               
               
                 145 
                 C-terminal Cap 
                 H 
                   
                   
                   
                   
               
               
                 146 
                 C-terminal Cap 
                 G 
                   
                   
                   
                   
               
               
                 147 
                 C-terminal Cap 
                 N 
                   
                   
                   
                   
               
               
                 148 
                 C-terminal Cap 
                 E 
                   
                   
                   
                   
               
               
                 149 
                 C-terminal Cap 
                 E 
                 E 
                   
                 E 
                   
               
               
                 150 
                 C-terminal Cap 
                 V/I 
                   
                   
                   
                   
               
               
                 151 
                 C-terminal Cap 
                 V 
                   
                   
                   
                   
               
               
                 152 
                 C-terminal Cap 
                 K 
                 K 
                   
                 K 
                   
               
               
                 153 
                 C-terminal Cap 
                 L 
                 A 
                   
                 V 
                   
               
               
                 154 
                 C-terminal Cap 
                 L 
                   
                   
                   
                   
               
               
                 155 
                 C-terminal Cap 
                 E 
                   
                   
                 E 
                   
               
               
                 156 
                 C-terminal Cap 
                 K 
                 K 
                   
                 D 
                   
               
               
                 157 
                 C-terminal Cap 
                 Q 
                 Q 
                   
                 H 
               
               
                   
               
            
           
         
       
     
     Modifications to the ankyrin-repeat domain proteins (as well as the other starting scaffold repeat domains discussed herein) were all made at potential oligomerizing positions. These are the residues that drive homo-oligomerization; residues outside of these regions can be significantly modified without affecting oligomerization of the polypeptides. 
     In this first aspect, the recited alternative positions in X2 are modified in all designed peptides. In one further embodiment, X1 comprises the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein (a) X1 possesses changes from SEQ ID NO:1 at least at one or more of residues 2, 3, 5, 6, 9, 12, 13, 15, 16, 17, 17, 20, 21, and 25, or (b) X1 possesses changes from SEQ ID NO:2 at least at one or more of residues 4, 5, 7, 8, 11, 14, 15, 17, 18, 19, 20, 22, 23, 26, and 27; or (c) X1 possesses changes from SEQ ID NO:3 at least at one or more of residues 4, 5, 7, 8, 11, 14, 15, 17, 18, 19, 20, 22, 23, 26, and 27. These are oligomerizing positions in a number of the designed peptides exemplified in Table 1. 
     In another embodiment, X3 is present, and wherein X3 has the amino acid sequence selected from the group consisting of
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein X2 possesses changes from SEQ ID NO:1 at least at residues 20, 24, and 25;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:2, wherein X2 possesses changes from SEQ ID NO:2 at least at residues 22, 26, and 27; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein X2 possesses changes from SEQ ID NO:3 at least at residues 22, 26, and 27.       

     In a further embodiment, X4 is present, and wherein X4 has the amino acid sequence selected from the group consisting of
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein X2 possesses changes from SEQ ID NO:1 at least at residue 25;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:2, wherein X2 possesses changes from SEQ ID NO:2 at least at residue 27; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein X2 possesses changes from SEQ ID NO:3 at least at residue 27.       

     In a still further embodiment, X5 is present, and wherein X5 has the amino acid sequence selected from the group consisting of
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein X2 possesses changes from SEQ ID NO:1 at least at residue 23;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:2, wherein X2 possesses changes from SEQ ID NO:2 at least at residue 25; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein X2 possesses changes from SEQ ID NO:3 at least at residue 25.       

     In another aspect, the polypeptides are based on the ank1C2_1 scaffold (see Table 1). In this embodiment, the polypeptide comprises the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 16 is K or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 18 is K or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 18 is K or a conservative substitution thereof;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A or a conservative substitution thereof, residue 17 is E or a conservative substitution thereof, residue 20 is A or a conservative substitution thereof, and residue 24 is E or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is A or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is A or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is D or a conservative substitution thereof, residue 17 is E or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is I or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, and residue 25 is K or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is D or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is I or a conservative substitution thereof, residue 25 is L or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is K or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is D or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is I or a conservative substitution thereof, residue 25 is L or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is K or a conservative substitution thereof;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is K or a conservative substitution thereof, residue 17 is R or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is V or a conservative substitution thereof, residue 23 is I or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, and residue 25 is A or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is K or a conservative substitution thereof, residue 19 is R or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 23 is V or a conservative substitution thereof, residue 25 is I or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is A or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is K or a conservative substitution thereof, residue 19 is R or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 23 is V or a conservative substitution thereof, residue 25 is I or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is A or a conservative substitution thereof; and       

     (e) X5 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is E or a conservative substitution thereof, residue 20 is K or a conservative substitution thereof, residue 21 is A or a conservative substitution thereof, residue 24 is K or a conservative substitution thereof, and residue 25 is Q or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is E or a conservative substitution thereof, residue 21 is K or a conservative substitution thereof, residue 23 is A or a conservative substitution thereof, residue 26 is K or a conservative substitution thereof, and residue 27 is Q or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is E or a conservative substitution thereof, residue 21 is K or a conservative substitution thereof, residue 23 is A or a conservative substitution thereof, residue 26 is K or a conservative substitution thereof, and residue 27 is Q or a conservative substitution thereof;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NO: 5-7. 
     In one embodiment of the polypeptides based on the ank1C2_1 scaffold: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 16 is K;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 18 is K; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 18 is K;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A, residue 17 is E, residue 20 is A, and residue 24 is E;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A, residue 19 is E, residue 22 is A, and residue 26 is E; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A, residue 19 is E, residue 22 is A, and residue 26 is E;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is D, residue 17 is E, residue 20 is L, residue 21 is I, residue 23 is L, residue 24 is L, and residue 25 is K;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is D, residue 19 is E, residue 22 is L, residue 23 is I, residue 25 is L, residue 26 is L, and residue 27 is K; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is D, residue 19 is E, residue 22 is L, residue 23 is I, residue 25 is L, residue 26 is L, and residue 27 is K;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is K, residue 17 is R, residue 20 is L, residue 21 is V, residue 23 is I, residue 24 is L, and residue 25 is A;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is K, residue 19 is R, residue 21 is L, residue 23 is V, residue 25 is I, residue 26 is L, and residue 27 is A; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is K, residue 19 is R, residue 21 is L, residue 23 is V, residue 25 is I, residue 26 is L, and residue 27 is A; and       

     (e) X5 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is E, residue 20 is K, residue 21 is A, residue 24 is K, and residue 25 is Q;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is E, residue 21 is K, residue 23 is A, residue 26 is K, and residue 27 is Q; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is E, residue 21 is K, residue 23 is A, residue 26 is K, and residue 27 is Q.       

     In another aspect, the polypeptides are based on the ank1C4_2 scaffold (see Table 1). In this embodiment, the polypeptide comprises the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 2 is E or a conservative substitution thereof, residue 3 is D or a conservative substitution thereof, residue 5 is E or a conservative substitution thereof, residue 6 is L or a conservative substitution thereof, residue 9 is L or a conservative substitution thereof, residue 10 is A or a conservative substitution thereof, residue 13 is L or a conservative substitution thereof residue 15 is I or a conservative substitution thereof. residue 16 is A or a conservative substitution thereof, residue 17 is E or a conservative substitution thereof, residue 18 is A or a conservative substitution thereof, residue 20 is R or a conservative substitution thereof, residue 21 is M or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, and residue 25 is Q or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 4 is E or a conservative substitution thereof, residue 5 is D or a conservative substitution thereof, residue 7 is E or a conservative substitution thereof, residue 8 is L or a conservative substitution thereof, residue 11 is L or a conservative substitution thereof, residue 12 is A or a conservative substitution thereof, residue 15 is L or a conservative substitution thereof residue 17 is I or a conservative substitution thereof. residue 18 is A or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 20 is A or a conservative substitution thereof, residue 22 is R or a conservative substitution thereof, residue 23 is M or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, and residue 27 is Q or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 4 is E or a conservative substitution thereof, residue 5 is D or a conservative substitution thereof, residue 7 is E or a conservative substitution thereof, residue 8 is L or a conservative substitution thereof, residue 11 is L or a conservative substitution thereof, residue 12 is A or a conservative substitution thereof, residue 15 is L or a conservative substitution thereof residue 17 is I or a conservative substitution thereof. residue 18 is A or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 20 is A or a conservative substitution thereof, residue 22 is R or a conservative substitution thereof, residue 23 is M or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, and residue 27 is Q or a conservative substitution thereof;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is L or a conservative substitution thereof, residue 17 is A or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, and residue 24 is L or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is L or a conservative substitution thereof, residue 19 is A or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is L or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is L or a conservative substitution thereof, residue 19 is A or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is L or a conservative substitution thereof;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is T or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, and residue 25 is M or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is T or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is M or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is T or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 26 is L or a conservative substitution thereof, and residue 27 is M or a conservative substitution thereof;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 25 is K or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 27 is K or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 27 or a conservative substitution thereof; and       

     (e) X5 is absent, or comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NOs: 5-7. 
     In one embodiment of the polypeptides based on the ank1C4_2 scaffold: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 2 is E, residue 3 is D, residue 5 is E, residue 6 is L, residue 9 is L, residue 10 is A, residue 13 is L. residue 15 is I, residue 16 is A, residue 17 is E, residue 18 is A, residue 20 is R, residue 21 is M, residue 22 is L, and residue 25 is Q;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 4 is E, residue 5 is D, residue 7 is E, residue 8 is L, residue 11 is L, residue 12 is A, residue 15 is L. residue 17 is I, residue 18 is A, residue 19 is E, residue 20 is A, residue 22 is R, residue 23 is M, residue 24 is L, and residue 27 is Q; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 4 is E, residue 5 is D, residue 7 is E, residue 8 is L, residue 11 is L, residue 12 is A, residue 15 is L. residue 17 is I, residue 18 is A, residue 19 is E, residue 20 is A, residue 22 is R, residue 23 is M, residue 24 is L, and residue 27 is Q;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is L, residue 17 is A, residue 20 is L, residue 21 is L, and residue 24 is L;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is L, residue 19 is A, residue 22 is L, residue 23 is L, and residue 26 is L; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is L, residue 19 is A, residue 22 is L, residue 23 is L, and residue 26 is L;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is T, residue 20 is L, residue 21 is L, residue 24 is L, and residue 25 is M;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is T, residue 21 is L, residue 23 is L, residue 26 is L, and residue 27 is M; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is T, residue 21 is L, residue 23 is L, residue 26 is L, and residue 27 is M;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 25 is K;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 27 is K; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 27; and       

     (e) X5 is absent, or comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NOs: 5-7. 
     In another aspect, the polypeptides are based on the ank3C2_1 scaffold (see Table 1). In this embodiment, the polypeptide comprises the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 16 is K or a conservative substitution thereof, residue 17 is D or a conservative substitution thereof, and residue 20 is K or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 18 is K or a conservative substitution thereof, residue 19 is D or a conservative substitution thereof, and residue 22 is K or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 18 is K or a conservative substitution thereof, residue 19 is D or a conservative substitution thereof, and residue 22 is K or a conservative substitution thereof;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A or a conservative substitution thereof, residue 17 is K or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, and residue 24 is E or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A or a conservative substitution thereof, residue 19 is K or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A or a conservative substitution thereof, residue 19 is K or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A or a conservative substitution thereof; 17 is V or a conservative substitution thereof, residue 20 is A or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 24 is M or a conservative substitution thereof, and residue 25 is H or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A or a conservative substitution thereof; 19 is V or a conservative substitution thereof, residue 22 is A or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 25 is L or a conservative substitution thereof, residue 26 is M or a conservative substitution thereof, and residue 27 is H or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A or a conservative substitution thereof; 19 is V or a conservative substitution thereof, residue 22 is A or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 25 is L or a conservative substitution thereof, residue 26 is M or a conservative substitution thereof, and residue 27 is H or a conservative substitution thereof;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is E or a conservative substitution thereof, residue 17 is E or a conservative substitution thereof, residue 20 is I or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, residue 24 is A or a conservative substitution thereof, and residue 25 is M or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is E or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is I or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 26 is A or a conservative substitution thereof, and residue 27 is M or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is E or a conservative substitution thereof, residue 19 is E or a conservative substitution thereof, residue 22 is I or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, residue 26 is A or a conservative substitution thereof, and residue 27 is M or a conservative substitution thereof; and       

     (e) X5 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is E or a conservative substitution thereof, residue 20 is K or a conservative substitution thereof, residue 21 is V or a conservative substitution thereof, residue 23 is E or a conservative substitution thereof, residue 24 is D or a conservative substitution thereof, and residue 25 is H or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is E or a conservative substitution thereof, residue 22 is K or a conservative substitution thereof, residue 23 is V or a conservative substitution thereof, residue 25 is E or a conservative substitution thereof, residue 26 is D or a conservative substitution thereof, and residue 27 is H or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is E or a conservative substitution thereof, residue 22 is K or a conservative substitution thereof, residue 23 is V or a conservative substitution thereof, residue 25 is E or a conservative substitution thereof, residue 26 is D or a conservative substitution thereof, and residue 27 is H or a conservative substitution thereof;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NOs: 5-7. 
     In one embodiment of the polypeptides based on the ank3C2_1 scaffold 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 16 is K, residue 17 is D, and residue 20 is K;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 18 is K, residue 19 is D, and residue 22 is K; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 18 is K, residue 19 is D, and residue 22 is K;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A, residue 17 is K, residue 20 is L, residue 21 is L, and residue 24 is E;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A, residue 19 is K, residue 22 is L, residue 23 is L, and residue 26 is E; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A, residue 19 is K, residue 22 is L, residue 23 is L, and residue 26 is E;       

     (c) X3 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is A; 17 is V, residue 20 is A, residue 21 is L, residue 23 is L, residue 24 is M, and residue 25 is H;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is A; 19 is V, residue 22 is A, residue 23 is L, residue 25 is L, residue 26 is M, and residue 27 is H; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is A; 19 is V, residue 22 is A, residue 23 is L, residue 25 is L, residue 26 is M, and residue 27 is H;       

     (d) X4 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is E, residue 17 is E, residue 20 is I, residue 21 is L, residue 24 is A, and residue 25 is M;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is E, residue 19 is E, residue 22 is I, residue 23 is L, residue 26 is A, and residue 27 is M; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is E, residue 19 is E, residue 22 is I, residue 23 is L, residue 26 is A, and residue 27 is M; and       

     (e) X5 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 17 is E, residue 20 is K, residue 21 is V, residue 23 is E, residue 24 is D, and residue 25 is H;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 19 is E, residue 22 is K, residue 23 is V, residue 25 is E, residue 26 is D, and residue 27 is H; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 19 is E, residue 22 is K, residue 23 is V, residue 25 is E, residue 26 is D, and residue 27 is H;       

     In another aspect, the polypeptides are based on the ank4D2 scaffold (see Table 1). In this embodiment, the polypeptide comprises the general formula X1-X2-X3-X4-X5, wherein: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 2 is T or a conservative substitution thereof, residue 3 is E or a conservative substitution thereof, residue 5 is K or a conservative substitution thereof, residue 6 is M or a conservative substitution thereof, residue 9 is I or a conservative substitution thereof, residue 10 is A or a conservative substitution thereof, residue 12 is R or a conservative substitution thereof, residue 13 is E or a conservative substitution thereof. residue 15 is M or a conservative substitution thereof. residue 16 is I or a conservative substitution thereof, residue 17 is I or a conservative substitution thereof, residue 18 is V or a conservative substitution thereof, residue 20 is R or a conservative substitution thereof, residue 21 is M or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 24 is E or a conservative substitution thereof, and residue 25 is K or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 4 is T or a conservative substitution thereof, residue 5 is E or a conservative substitution thereof, residue 7 is K or a conservative substitution thereof, residue 8 is M or a conservative substitution thereof, residue 10 is I or a conservative substitution thereof, residue 12 is A or a conservative substitution thereof, residue 14 is R or a conservative substitution thereof, residue 15 is E or a conservative substitution thereof. residue 17 is M or a conservative substitution thereof. residue 18 is I or a conservative substitution thereof, residue 19 is I or a conservative substitution thereof, residue 20 is V or a conservative substitution thereof, residue 22 is R or a conservative substitution thereof, residue 23 is M or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, residue 26 is E or a conservative substitution thereof, and residue 27 is K or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 4 is T or a conservative substitution thereof, residue 5 is E or a conservative substitution thereof, residue 7 is K or a conservative substitution thereof, residue 8 is M or a conservative substitution thereof, residue 10 is I or a conservative substitution thereof, residue 12 is A or a conservative substitution thereof, residue 14 is R or a conservative substitution thereof, residue 15 is E or a conservative substitution thereof. residue 17 is M or a conservative substitution thereof. residue 18 is I or a conservative substitution thereof, residue 19 is I or a conservative substitution thereof, residue 20 is V or a conservative substitution thereof, residue 22 is R or a conservative substitution thereof, residue 23 is M or a conservative substitution thereof, residue 24 is L or a conservative substitution thereof, residue 26 is E or a conservative substitution thereof, and residue 27 is K or a conservative substitution thereof;       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is L or a conservative substitution thereof, residue 17 is I or a conservative substitution thereof, residue 20 is L or a conservative substitution thereof, residue 21 is L or a conservative substitution thereof, and residue 24 is E or a conservative substitution thereof;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is L or a conservative substitution thereof, residue 19 is I or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is L or a conservative substitution thereof, residue 19 is I or a conservative substitution thereof, residue 22 is L or a conservative substitution thereof, residue 23 is L or a conservative substitution thereof, and residue 26 is E or a conservative substitution thereof;       

     (c) X3 is absent or comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     (d) X4 is absent or comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3; and       

     (e) X5 is absent, or comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3;       

     wherein the polypeptide does not comprise the amino acid sequence of SEQ ID NOs: 5-7. 
     In one embodiment of the polypeptides based on the ank4D2 scaffold: 
     (a) X1 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to SEQ ID NO:1, wherein residue 2 is T, residue 3 is E, residue 5 is K, residue 6 is M, residue 9 is I, residue 10 is A, residue 12 is R, residue 13 is E. residue 15 is M. residue 16 is I, residue 17 is I, residue 18 is V, residue 20 is R, residue 21 is M, residue 22 is L, residue 24 is E, and residue 25 is K;   (ii) the amino acid sequence at least 50% identical along its length to SEQ ID NO: 2, wherein residue 4 is T, residue 5 is E, residue 7 is K, residue 8 is M, residue 10 is I, residue 12 is A, residue 14 is R, residue 15 is E. residue 17 is M. residue 18 is I, residue 19 is I, residue 20 is V, residue 22 is R, residue 23 is M, residue 24 is L, residue 26 is E, and residue 27 is K; and   (iii) the amino acid sequence at least 50% identical along its length to SEQ ID NO:3, wherein residue 4 is T, residue 5 is E, residue 7 is K, residue 8 is M, residue 10 is I, residue 12 is A, residue 14 is R, residue 15 is E. residue 17 is M. residue 18 is I, residue 19 is I, residue 20 is V, residue 22 is R, residue 23 is M, residue 24 is L, residue 26 is E, and residue 27 is K; and       

     (b) X2 comprises the amino acid sequence that is selected from the group consisting of:
         (i) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:1, wherein residue 16 is L, residue 17 is I, residue 20 is L, residue 21 is L, and residue 24 is E;   (ii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:2, wherein residue 18 is L, residue 19 is I, residue 22 is L, residue 23 is L, and residue 26 is E; and   (iii) the amino acid sequence at least 50% identical along its length to residues SEQ ID NO:3, wherein residue 18 is L, residue 19 is I, residue 22 is L, residue 23 is L, and residue 26 is E.       

     In one embodiment of all of the various ankyrin-derived polypeptides of the invention, all oligomerizing positions in the amino acid sequence selected from the group consisting of SEQ ID NO: 12-15 have the amino acid residue shown in the amino acid sequence selected from the group consisting of SEQ ID NO: 12-15 or conservative substitutions thereof. 
     In another aspect, the invention provides isolated polypeptides comprising the general formula X1-X2-X3-X4, wherein: 
     X1 is at least 50% identical along its length to residues 1-34 of the amino acid sequence of SEQ ID NO: 8, wherein the amino acid sequence of X1 differs from the amino acid sequence of residues 1-34 of SEQ ID NO: 8 at least at residues 6, 8, 13, 21, 25, and 28; 
     X2 is absent, or is at least 50% identical along its length to residues 36-68 of the amino acid sequence of SEQ ID NO: 8; 
     X3 is absent, or is at least 50% identical along its length to residues 69-102 of the amino acid sequence of SEQ ID NO: 8; and 
     X4 is absent, or is at least 50% identical along its length to residues 103-119 of the amino acid sequence of SEQ ID NO: 8; 
     wherein the polypeptide is not identical to SEQ ID NO:8 or SEQ ID NO:41 (the tpr reference sequence). 
     
       
         
           
               
            
               
                 SEQ ID NO: 8: &gt;1na0 
               
               
                 NSAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAWYNLGNAYYKQ 
               
               
                   
               
               
                 GDYDEAIEYYQKALELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALEL 
               
               
                   
               
               
                 DPNNAEAKQNLGNAKQKQG 
               
            
           
         
       
     
     In this aspect, design interfaces were grafted onto 1na0 or tpr repeat scaffolds which are examples of tetratricopeptide repeat proteins, permitting the designed polypeptides to direct their assembly into homo-oligomeric complexes (such as dimers, trimers, tetramers, and pentamers). 
     Oligomerizing positions for a variety of polypeptides of this aspect of the invention are shown  FIG. 4 , aligned with the wild-type reference sequences. These are the residues that drive homo-oligomerization of the polypeptides of this aspect; residues outside of these regions can be significantly modified without affecting oligomerization of the polypeptides. 
     In one embodiment, the amino acid sequence of X1 differs from the amino acid sequence of residues 1-34 of SEQ ID NO: 8 at least as follows: 
     W6 is substituted with A, M or conservative substitutions thereof; 
     N8 is substituted with I, L, K, D or conservative substitutions thereof; 
     Y13 is substituted with I, A, M or conservative substitutions thereof; 
     E21 is substituted with I, L or conservative substitutions thereof; 
     Y25 is substituted with M, A or conservative substitutions thereof; and 
     K28 is substituted with I, L, V or conservative substitutions thereof. 
     In another embodiment, the amino acid sequence of X1 further differs from the amino acid sequence of residues 1-34 of SEQ ID NO: 8 at least at residues 2, 5, 18, and 27. 
     In a further embodiment, the amino acid sequence of X1 differs from the amino acid sequence of residues 1-34 of SEQ ID NO: 8 at least as follows: 
     S2 is substituted with R, E, L or conservative substitutions thereof; 
     A5 is substituted with M, L, K, V or conservative substitutions thereof; 
     D18 is substituted with L, E, Q or conservative substitutions thereof; and 
     Q27 is substituted with L, R, T, V or conservative substitutions thereof. 
     In a further embodiment, X1 is at least 50% identical to residues 1-34 of the amino acid sequence of one of SEQ ID NOS:16-21. In another embodiment, all oligomerizing positions in residues 1-34 of the amino acid sequence selected from the group consisting of SEQ ID NO: 16-21 have the amino acid residue shown in the amino acid sequence selected from the group consisting of SEQ ID NO: 16-21 or conservative substitutions thereof. 
     In one embodiment, X2 is present; in one non-limiting embodiment, the amino acid sequence of X2 differs from the amino acid sequence of residues 35-68 of SEQ ID NO: 8 at one or more of residues 47, 50, 55, 58, and 59. In another embodiment, X2, X3, and X4 are present. In one such embodiment, the amino acid sequence of X3 differs from the amino acid sequence of residues 69-102 of SEQ ID NO: 8 at least at residue 95. In another embodiment, residue Q95 is substituted with K, E, R, or conservative substitutions thereof. 
     In another embodiment, the amino acid sequence of X4 differs from the amino acid sequence of residues 103-119 of SEQ ID NO: 8 at least at residues 108, 112, and 116. In one such embodiment, residue K108 is substituted with I, L, M, or conservative substitutions thereof; residue G112 is substituted with I, L or conservative substitutions thereof; and residue Q116 is substituted with E, A, D, M or conservative substitutions thereof. 
     In another aspect is provided polypeptides comprising the amino acid sequence at least 50% identical to the amino acid sequence of SEQ ID NO: 10, wherein all oligomerizing positions in SEQ ID NO: −10 have the amino acid residue shown in SEQ ID NO: 10 or conservative substitutions thereof, and wherein the polypeptide does not comprise acid sequence of SEQ ID NO: 9. 
     
       
         
           
               
            
               
                 &gt;tpr1C4_2 (tpr1C4_pm3) 
               
               
                 (SEQ ID NO: 10) 
               
               
                 A SSWVML G LLLSLLNR L SL AA EA YK K AIELDPNDA L AW LL LG S VLE K L 
               
               
                   
               
               
                   KR LDEAAEAYKKAIEL K PN D A S AW KE LGKVLEKLGRLDEAAEAYKKAI 
               
               
                   
               
               
                 EL D P E D A EAWKELGKVLEKLGRLDEAAEAYKKAIEL D PND 
               
               
                   
               
               
                 &gt;tpr1 
               
               
                 (SEQ ID NO: 9) 
               
               
                 AEAWKELGKVLEKLGRLDEAAEAYKKAIELDPNDAEAWKELGKVLEKL 
               
               
                   
               
               
                 GRLDEAAEAYKKAIELDPNDAEAWKELGKVLEKLGRLDEAAEAYKKAI 
               
               
                   
               
               
                 ELDPNDAEAWKELGKVLEKLGRLDEAAEAYKKAIELDPND 
               
            
           
         
       
     
     In another aspect are provided polypeptides comprising the amino acid sequence at least 50% identical to SEQ ID NO: 11, wherein the polypeptide amino acid sequence differs from SEQ ID NO: 11 at least at residues 7, 8, 10, 14, 17, 118, 122, 146, 149, and 150. 
     
       
         
           
               
            
               
                 &gt;3ltj 
               
               
                 (SEQ ID NO: 11) 
               
               
                 TDPEKVEMYIKNLQDDSYYVRRAAAYALGKIGDERAVEPLIKALKDEDAW 
               
               
                   
               
               
                 VRRAAADALGQIGDERAVEPLIKALKDEDGWVRQSAAVALGQIGDERAVE 
               
               
                   
               
               
                 PLIKALKDEDWFVRIAAAFALGEIGDERAVEPLIKALKDEDGWVRQSAAD 
               
               
                   
               
               
                 ALGEIGGERVRAAMEKLAETGTGFARKVAVNYLETHK 
               
            
           
         
       
     
     In this aspect, design interfaces were grafted onto HEAT repeat scaffolds, permitting the designed polypeptides to direct their assembly into homo-oligomeric complexes (such as dimers, trimers, tetramers, and pentamers). “HEAT” is an acronym for four proteins in which this repeat structure is found: Huntingtin, elongation factor 3 (EF3), protein phosphatase 2A (PP2A), and the yeast kinase TOR1. 
     Oligomerizing positions for a variety of polypeptides of this aspect of the invention are shown Table 2, aligned with the wild-type reference sequences. These are the residues that drive homo-oligomerization of the polypeptides of this aspect; residues outside of these regions can be significantly modified without affecting oligomerization of the polypeptides. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                 Heat reference 
                   
                   
                   
               
               
                 Position 
                 sequence 
                 3ltjC3_int1 
                 3ltjC3_int11 
                 T33_dn5B 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 1 
                 — 
                 — 
                 
                   R 
                 
                 — 
               
               
                 2 
                 — 
                 — 
                 
                   R 
                 
                 — 
               
               
                 3 
                 — 
                 — 
                 
                   E 
                 
                 — 
               
               
                 4 
                 T 
                   
                 
                   E 
                 
               
               
                 5 
                 D 
                 
                   D 
                 
                 
                   D 
                 
                 
                   D 
                 
               
               
                 6 
                 P 
                 
                   P 
                 
                 
                   P 
                 
                 
                   P 
                 
               
               
                 7 
                 E 
                 
                   L 
                 
                 
                   L 
                 
                 
                   L 
                 
               
               
                 8 
                 K 
                 
                   A 
                 
                 
                   A 
                 
                 
                   A 
                 
               
               
                 9 
                 V 
               
               
                 10 
                 E 
                 
                   I 
                 
                 
                   V 
                 
                 
                   I 
                 
               
               
                 11 
                 M 
                 
                   L 
                 
                 
                   M 
                 
                 
                   L 
                 
               
               
                 12 
                 Y 
                 
                   Y 
                 
                   
                 
                   Y 
                 
               
               
                 13 
                 I 
                 
                   I 
                 
                 
                   R 
                 
                 
                   I 
                 
               
               
                 14 
                 K 
                 
                   A 
                 
                 
                   L 
                 
                 
                   A 
                 
               
               
                 15 
                 N 
                 
                   I 
                 
                 
                   N 
                 
                 
                   I 
                 
               
               
                 16 
                 L 
               
               
                 17 
                 Q 
                 
                   K 
                 
                 
                   R 
                 
                 
                   K 
                 
               
               
                 18 
                 D 
                 
                   A 
                 
                   
                 
                   A 
                 
               
               
                 19 
                 D 
                 
                   E 
                 
                   
                 
                   E 
                 
               
               
                 20 
                 S 
                 
                   K 
                 
                   
                 
                   K 
                 
               
               
                 21 
                 Y 
                 
                   S 
                 
                   
                 
                   S 
                 
               
               
                 22 
                 Y 
                 
                   I 
                 
                   
                 
                   I 
                 
               
               
                 23 
                 V 
                 
                   A 
                 
                   
                 
                   A 
                 
               
               
                 24 
                 R 
               
               
                 25 
                 R 
                 
                   A 
                 
                   
                 
                   A 
                 
               
               
                 26 
                 A 
                 
                   K 
                 
                   
                 
                   K 
                 
               
               
                 27 
                 A 
               
               
                 28 
                 A 
               
               
                 29 
                 Y 
                 
                   E 
                 
                   
                 
                   E 
                 
               
               
                 30 
                 A 
               
               
                 31 
                 L 
               
               
                 32 
                 G 
               
               
                 33 
                 K 
                 
                   K 
                 
                   
                 
                   K 
                 
               
               
                 34 
                 I 
               
               
                 35 
                 G 
               
               
                 36 
                 D 
               
               
                 37 
                 E 
               
               
                 38 
                 R 
                 
                   R 
                 
                   
                 
                   R 
                 
               
               
                 39 
                 A 
               
               
                 40 
                 V 
               
               
                 41 
                 E 
               
               
                 42 
                 P 
               
               
                 43 
                 L 
               
               
                 44 
                 I 
               
               
                 45 
                 K 
               
               
                 46 
                 A 
                 
                   A 
                 
                   
                 
                   A 
                 
               
               
                 47 
                 L 
                 
                   L 
                 
                   
                 
                   L 
                 
               
               
                 48 
                 K 
               
               
                 49 
                 D 
               
               
                 50 
                 E 
               
               
                 51 
                 D 
               
               
                 52 
                 A 
               
               
                 53 
                 W 
                 
                   L 
                 
                   
                 
                   L 
                 
               
               
                 54 
                 V 
               
               
                 55 
                 R 
               
               
                 56 
                 R 
                 
                   L 
                 
                   
                 
                   L 
                 
               
               
                 57 
                 A 
               
               
                 58 
                 A 
               
               
                 59 
                 A 
               
               
                 60 
                 D 
               
               
                 61 
                 A 
               
               
                 62 
                 L 
               
               
                 63 
                 G 
               
               
                 64 
                 Q 
               
               
                 65 
                 I 
               
               
                 66 
                 G 
               
               
                 67 
                 D 
               
               
                 68 
                 E 
               
               
                 69 
                 R 
               
               
                 70 
                 A 
               
               
                 71 
                 V 
               
               
                 72 
                 E 
               
               
                 73 
                 P 
               
               
                 74 
                 L 
               
               
                 75 
                 I 
               
               
                 76 
                 K 
               
               
                 77 
                 A 
               
               
                 78 
                 L 
               
               
                 79 
                 K 
               
               
                 80 
                 D 
               
               
                 81 
                 E 
               
               
                 82 
                 D 
                 
                   E 
                 
                   
                 
                   E 
                 
               
               
                 83 
                 G 
               
               
                 84 
                 W 
                 
                   L 
                 
                   
                 
                   L 
                 
               
               
                 85 
                 V 
               
               
                 86 
                 R 
               
               
                 87 
                 Q 
                 
                   A 
                 
                   
                 
                   A 
                 
               
               
                 88 
                 S 
               
               
                 89 
                 A 
               
               
                 90 
                 A 
               
               
                 91 
                 V 
                 
                   I 
                 
                   
                 
                   I 
                 
               
               
                 92 
                 A 
               
               
                 93 
                 L 
               
               
                 94 
                 G 
               
               
                 95 
                 Q 
               
               
                 96 
                 I 
               
               
                 97 
                 G 
               
               
                 98 
                 D 
               
               
                 99 
                 E 
               
               
                 100 
                 R 
               
               
                 101 
                 A 
               
               
                 102 
                 V 
                   
                   
                 
                   V 
                 
               
               
                 103 
                 E 
                   
                   
                 
                   Q 
                 
               
               
                 104 
                 P 
               
               
                 105 
                 L 
               
               
                 106 
                 I 
                   
                   
                 
                   I 
                 
               
               
                 107 
                 K 
                   
                   
                 
                   K 
                 
               
               
                 108 
                 A 
               
               
                 109 
                 L 
                   
                   
                 
                   L 
                 
               
               
                 110 
                 K 
                   
                   
                 
                   T 
                 
               
               
                 111 
                 D 
               
               
                 112 
                 E 
               
               
                 113 
                 D 
                 
                   R 
                 
                   
                 
                   R 
                 
               
               
                 114 
                 W 
                 
                   D 
                 
                   
                 
                   D 
                 
               
               
                 115 
                 F 
                 
                   L 
                 
                   
                 
                   L 
                 
               
               
                 116 
                 V 
               
               
                 117 
                 R 
                   
                   
                 
                   R 
                 
               
               
                 118 
                 I 
                 
                   V 
                 
                 
                   A 
                 
                 
                   V 
                 
               
               
                 119 
                 A 
                 
                   A 
                 
                 
                   A 
                 
                 
                   A 
                 
               
               
                 120 
                 A 
               
               
                 121 
                 A 
               
               
                 122 
                 F 
                 
                   V 
                 
                 
                   A 
                 
                 
                   V 
                 
               
               
                 123 
                 A 
               
               
                 124 
                 L 
               
               
                 125 
                 G 
               
               
                 126 
                 E 
                 
                   R 
                 
                 
                   R 
                 
                 
                   R 
                 
               
               
                 127 
                 I 
               
               
                 128 
                 G 
               
               
                 129 
                 D 
               
               
                 130 
                 E 
               
               
                 131 
                 R 
                   
                   
                 
                   K 
                 
               
               
                 132 
                 A 
               
               
                 133 
                 V 
               
               
                 134 
                 E 
                   
                   
                 
                   R 
                 
               
               
                 135 
                 P 
                   
                   
                 
                   P 
                 
               
               
                 136 
                 L 
               
               
                 137 
                 I 
               
               
                 138 
                 K 
                   
                   
                 
                   I 
                 
               
               
                 139 
                 A 
                   
                   
                 
                   V 
                 
               
               
                 140 
                 L 
               
               
                 141 
                 K 
                   
                   
                 
                   K 
                 
               
               
                 142 
                 D 
               
               
                 143 
                 E 
                   
                   
                 
                   E 
                 
               
               
                 144 
                 D 
                 
                   E 
                 
                   
                 
                   E 
                 
               
               
                 145 
                 G 
                   
                 
                   E 
                 
               
               
                 146 
                 W 
                 
                   E 
                 
                 
                   M 
                 
                 
                   E 
                 
               
               
                 147 
                 V 
               
               
                 148 
                 R 
               
               
                 149 
                 Q 
                 
                   E 
                 
                 
                   E 
                 
                 
                   E 
                 
               
               
                 150 
                 S 
                 
                   A 
                 
                 
                   I 
                 
                 
                   A 
                 
               
               
                 151 
                 A 
               
               
                 152 
                 A 
               
               
                 153 
                 D 
                 
                   I 
                 
                 
                   L 
                 
                 
                   I 
                 
               
               
                 154 
                 A 
               
               
                 155 
                 L 
               
               
                 156 
                 G 
               
               
                 157 
                 E 
                 
                   S 
                 
                 
                   M 
                 
                 
                   S 
                 
               
               
                 158 
                 I 
               
               
                 159 
                 G 
               
               
                 160 
                 G 
               
               
                 161 
                 E 
               
               
                 162 
                 R 
               
               
                 163 
                 V 
               
               
                 164 
                 R 
               
               
                 165 
                 A 
               
               
                 166 
                 A 
               
               
                 167 
                 M 
               
               
                 168 
                 E 
               
               
                 169 
                 K 
               
               
                 170 
                 L 
               
               
                 171 
                 A 
               
               
                 172 
                 E 
               
               
                 173 
                 T 
                   
                   
                 
                   R 
                 
               
               
                 174 
                 G 
               
               
                 175 
                 T 
               
               
                 176 
                 G 
               
               
                 177 
                 F 
               
               
                 178 
                 A 
               
               
                 179 
                 R 
               
               
                 180 
                 K 
               
               
                 181 
                 V 
               
               
                 182 
                 A 
               
               
                 183 
                 V 
               
               
                 184 
                 N 
               
               
                 185 
                 Y 
               
               
                 186 
                 L 
               
               
                 187 
                 E 
               
               
                 188 
                 T 
               
               
                 189 
                 H 
               
               
                 190 
                 K 
               
               
                   
               
            
           
         
       
     
     In one embodiment of this aspect, the polypeptide amino acid sequence differs from SEQ ID NO: 11 at least as follows: 
     E7 is substituted with L or a conservative substitution thereof; 
     K8 is substituted with A or a conservative substitution thereof; 
     E10 is substituted with I, V or conservative substitutions thereof; 
     K14 is substituted with A, L or conservative substitutions thereof; 
     Q17 is substituted with R, K or conservative substitutions thereof; 
     1118 is substituted with A, V or conservative substitutions thereof; 
     F122 is substituted with A, V or conservative substitutions thereof; 
     W146 is substituted with E, M or conservative substitutions thereof; 
     Q149 is substituted with E or a conservative substitution thereof; and 
     S150 is substituted with I, A, or conservative substitutions thereof. 
     In another embodiment, the polypeptide amino acid sequence further differs from the amino acid sequence of SEQ ID NO: 11 at least at residues 11, 15, 18, 19, 20, 22, 23, 25, 26, 29, 82, 84, 87, 91, 113, 114, 115, and 144. In one such embodiment, the polypeptide amino acid sequence differs from SEQ ID NO: 11 at least as follows: 
     M11 is substituted with L or a conservative substitution thereof; 
     N15 is substituted with I or a conservative substitution thereof; 
     D18 is substituted with A or a conservative substitution thereof; 
     D19 is substituted with E or a conservative substitution thereof; 
     S20 is substituted with K or a conservative substitution thereof; 
     Y22 is substituted with I or a conservative substitution thereof; 
     V23 is substituted with A or a conservative substitution thereof; 
     R25 is substituted with A or a conservative substitution thereof; 
     A26 is substituted with K or a conservative substitution thereof; 
     Y29 is substituted with E or a conservative substitution thereof; 
     D82 is substituted with E or a conservative substitution thereof; 
     W84 is substituted with L or a conservative substitution thereof; 
     Q87 is substituted with A or a conservative substitution thereof; 
     V91 is substituted with I or a conservative substitution thereof; 
     D113 is substituted with R or a conservative substitution thereof; 
     W114 is substituted with D or a conservative substitution thereof; 
     F115 is substituted with L or a conservative substitution thereof; and 
     D144 is substituted with E or a conservative substitution thereof. 
     In another aspect, the invention provides polypeptides comprising or consisting of a polypeptide having at least 50% identity over its length with a polypeptide having the amino acid sequence selected from the group consisting of SEQ ID NO:10 and 12-40. As described in the examples that follow, the polypeptides of the invention were designed for their ability to self-assemble to cyclic homoligomers with tunable shape, size, and symmetry enables rigid display of binding domains at arbitrary orientations and distances for a range of biological applications. 
     In various embodiments, the polypeptides comprise or consist of a polypeptide having at least 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, 95%, 96%, 97%, 98%, 99%, or 100% identity over its length with a polypeptide having the amino acid sequence selected from the group consisting of SEQ ID NO: 10 and 12-40. Oligomerizing positions are in bold font. 
     
       
         
           
               
               
            
               
                 Ankyrin based oligomers 
                   
               
               
                 &gt;ank1C2_1 (ank1C2_G3) 
               
               
                 (SEQ ID NO: 12) 
                   
               
               
                 SELGKRLIEAAENGN K DRVKDLIENGADVNASDSDGRTPLHHAAENGH AE VV A LLI E KGADVN 
                   
               
               
                   
               
               
                 AKDSDGRTPLHHAAENGHD E VV LI L LLK GADVNAKDSDGRTPLHHAAENGH KR VV LV L ILA GA 
               
               
                   
               
               
                 DVNTSDSDGRTPLDLAREHGNE E VV KA LE KQ   
               
               
                   
               
               
                 &gt;ank1C4_2 (ank1C4_7) 
               
               
                 (SEQ ID NO: 13) 
                   
               
               
                 S ED G EL LI LA AE L G IAEA V RML IE Q GADVNASD D DGRTPLHHAAENGH LA VV LL LL L KGADVN 
                   
               
               
                   
               
               
                 AKDSDGRTPLHHAAENGHK T VV LL LI LM GADVNAKDSDGRTPLHHAAENGHKEVVKLLIR K GA 
               
               
                   
               
               
                 DVNTSDSDGRTPLDLAREHGNEEVVKLLEKQ 
               
               
                   
               
               
                 &gt;ank3C2_1 
               
               
                 (SEQ ID NO: 14) 
                   
               
               
                 SELGKRLIEAAENGN KD RV K DLLENGADVNASDSDGKTPLHLAAENGH AK VV LL LL E QGADPN 
                   
               
               
                   
               
               
                 AKDSDGKTPLHLAAENGH AV VV AL L LMH GADPNAKDSDGKTPLHLAAENGH EE VV IL LL AM GA 
               
               
                   
               
               
                 DPNTSDSDGRTPLDLAREHGNE E VV KV L EDH   
               
               
                   
               
               
                 &gt;ank4C4 (ank4C4.2, D2 residues selected here) 
               
               
                 (SEQ ID NO: 15) 
                   
               
               
                 S TE G KM LI IA A RE G MIIV V IVL L EK GADPNASDKDGRTPLHYAAENGH LI IV LL LL E KGADPN 
                   
               
               
                   
               
               
                 AKDSDGRTPLHYAAENGHKEIVEALLEHGADPNAKDSDGRTPLHYAAENGHKEIVKLLLSKGA 
               
               
                   
               
               
                 DPNTSDSDGRTPLDLAREHGNEEIVKLLEKQLE 
               
               
                   
               
               
                 TPR based oligomers 
               
               
                 &gt;1na0C3_1 
               
               
                 (SEQ ID NO: 16) 
                   
               
               
                   ER A EMAAIV G DAI YI M G L Y RL AI KM Y LI AL KL DPNNAEAWYNLGNAYYKQGDYDEAIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;1na0C3_int2* 
               
               
                 (SEQ ID NO: 17) 
                   
               
               
                   EE A ELA Y LL G ELA Y KL G E Y RI AI RA Y RI AL KR DPNNAEAWYNLGNAYYKQGDYDEAIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;1na0C3_3 
               
               
                 (SEQ ID NO: 18) 
                   
               
               
                   NL A EKM Y KA GN AM Y RK G Q Y TI A IIA Y TL A LLK DPNNAEA W YN L GNA A YK K G E YD E AI EA YQ K A 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;1na0C3_5 (1na0C3_6) 
               
               
                 (SEQ ID NO: 19) 
                   
               
               
                 N E AE LA Y DL GN EA YK D G E Y RL A AVA Y VL A LAV DPNNAEA W YN L GNAYYK Q G R YD K AI K YYQ 
                   
               
               
                   
               
               
                   K ALELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;1na0C3_7 (1na0C3_G1) 
               
               
                 (SEQ ID NO: 20) 
                   
               
               
                 N S AE AM Y KM GN AA YKQG D Y IL A IIA Y LL AL EK DPNNAEAWYNLGNAAYK Q G D YD E AIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;1na0C4_1 (1na0C4_G1) 
               
               
                 (SEQ ID NO: 21) 
                   
               
               
                   TL A RVA Y IL G AIA Y AQ G E YD I AI TA YQ V ALD L DPNNAEAWYNLGNAYYKQGDYDEAIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYYQKALELDPNNAEAKQNLGNAKQKQG 
               
               
                   
               
               
                 &gt;tpr1C4_2 (tpr1C4_pm3) 
               
               
                 (SEQ ID NO: 10) 
                   
               
               
                 A SSWVML G LLLSLLNR L SL AA EA YK K AIELDPNDA L AW LL LG S VLE K L KR LDEAAEAYKKAIE 
                   
               
               
                   
               
               
                 L K PN D A S AW KE LGKVLEKLGRLDEAAEAYKKAIEL D P E D A EAWKELGKVLEKLGRLDEAAEAY 
               
               
                   
               
               
                 KKAIELDPND 
               
               
                   
               
               
                 HEAT based oligomers 
               
               
                 &gt;3ltjC3_int1* 
               
               
                 (SEQ ID NO: 22) 
                   
               
               
                 T DPLA V ILYIAI L KAEKSIA R AK AA E ALG K IGDE R AVEPLIK AL KDEDA L VR A AAADALGQIG 
                   
               
               
                   
               
               
                 DERAVEPLIKALKDE E G L VR A SAA I ALGQIGDERAVEPLIKALKDE RDL VR VA AA V ALG R IGD 
               
               
                   
               
               
                 ERAVEPLIKALKDE E G E VR EA AA I ALG S IGGERVRAAMEKLAETGTGFARKVAVNYLETHK 
               
               
                   
               
               
                 &gt;3ltjC3_int11* 
               
               
                 (SEQ ID NO: 23) 
                   
               
               
                   RREEDPLA V VM Y RLN L R DDSYYVRRAAAYALGKIGDERAVEPLIKALKDEDAWVRRAAADALG 
                   
               
               
                   
               
               
                 QIGDERAVEPLIKALKDEDGWVRQSAAVALGQIGDERAVEPLIKALKDEDWFVR AA AA A ALG R   
               
               
                   
               
               
                 IGDERAVEPLIKALKDED EM VR EI AA L ALG M IGGERVRAAMEKLAETGTGFARKVAVNYLETH 
               
               
                   
               
               
                 K 
               
               
                   
               
               
                 HR based oligomers 
               
               
                 &gt;HR00C3_2 
               
               
                 (SEQ ID NO: 24) 
                   
               
               
                   I EE VV A EMI D IL AE S SK K SIE EL AR A ADN K TTE KA VA E AI EE IA RL ATAAI QL IE AL A KN LAS 
                   
               
               
                   
               
               
                 EEFMARAISAIAELAKKAIEAIYRLADNHTTD T FMA R AI A AIA N LA VT AI L AIA A LA S NHTTE 
               
               
                   
               
               
                 EFMARAISAIAELAKKAIEAIYRLADNHTTD KF MAAAI E AIA L LA TL AI LA I AL LA S NHTTEE 
               
               
                   
               
               
                 FMAKAISAIAELAKKAIEAIYRLADNHTSPTYIE K AIEAIEKIAR K AIKAIE M LAK N ITTEEY 
               
               
                   
               
               
                 KEKAKSAIDEIREKAKEAIKRLEDNRT 
               
               
                   
               
               
                 &gt;HR04C3_int2 (tj04C3_int5v2)* 
               
               
                 (SEQ ID NO: 25) 
                   
               
               
                 DECEEKARRVAEKVERLKRSGTSEDEIAEEVAREISEVIRTLKESGS E Y KV IC R CVARIVAEI 
                   
               
               
                   
               
               
                 VEALKRSGTSEDEIAEIVARVISEVIRTLKESGS D Y LI IC V CV AI IV AE IV E ALKRSGTS ED E 
               
               
                   
               
               
                 IAEIVARVISEVIRTLKESGSSYEVIK E CVQ I IV LA I ILA L MK SG TEVEEILL I LLR V KT EV R   
               
               
                   
               
               
                 RTLKESGS 
               
               
                   
               
               
                 &gt;HR04C4_1 (tj10C4_G1) 
               
               
                 (SEQ ID NO: 26) 
                   
               
               
                 DECEEKARRVAEKVERLKRSGTSEDEIAEEVAREISEVIRTLKESGSSYEVICECVARIVAEI 
                   
               
               
                   
               
               
                 VEALKRSGT SA VEIA K IVARVIS E VIRTLKESGSSYEVICECVARIVAEIVEALKRSGT SAAI   
               
               
                   
               
               
                 I AL IVA L VISEVIRTLKESGSS F EVI LE CV I RI VL EI I EAL K RSGT SEQDVMLIVMAVLLVVL   
               
               
                   
               
               
                   A TLQLSG S   
               
               
                   
               
               
                 &gt;HR08C3 (tj08C3_V13) 
               
               
                 (SEQ ID NO: 27) 
                   
               
               
                 DEM R KV ML AL AI A LVR A LL N E D IE V AKE I AR A AD EI E EAL R ENNSDEMAKVMLALAKAVLLA 
                   
               
               
                   
               
               
                 AKNNDD R VAE V IA L AAA E IV K ALR R NNSDEMAKVMLALAKAVLLAAKNNDDEVA K EIA I AA 
               
               
                   
               
               
                   MIIVI AL RAE NSDEMAKKMLELAKRVLDAAKNNDDETA R EI AE QA AE E LEA   
               
               
                   
               
               
                 &gt;HR10C2_2 (bex2C2_G2) 
               
               
                 (SEQ ID NO: 28) 
                   
               
               
                   
               
               
                 SSEKEELRERLVKIVVENAKRKGDDTE E AR EA AR A AFE I VR AA A KLA GIDSSEVLELAIRLIK 
                   
               
               
                   
               
               
                 EVVENAQREGYDIA V AA IA AA V AF AV VA VA A AA ADITSSEVLELAIRLIKEVVENAQREGY V I 
               
               
                   
               
               
                   LL AA L AAA A AF VV VA A AAK R AGITSSETLKRAIEEIRKRVEEAQREGNDISEAARQAAEEFRK 
               
               
                   
               
               
                 KAEELK 
               
               
                   
               
               
                 &gt;HR10C5_2 (bex2C5_G2) 
               
               
                 (SEQ ID NO: 29) 
                   
               
               
                   SAE K LM L MAK L II IV AE NA K RKG DDTLI A IM AA KL AFE I VR IA AE E AGIDSSEVLELAIRLIK 
                   
               
               
                   
               
               
                 EVVENAQREGY D IS I AA LA AA M AF AL VA I AAK R AGITSSEVLELAIRLIKEVVENAQREGYDI 
               
               
                   
               
               
                 AEAARAAAEAFKRVAEAAKRAGITSSETLKRAIEEIRKRVEEAQREGNDISEAARQAAEEFRK 
               
               
                   
               
               
                 KAEELK 
               
               
                   
               
               
                 &gt;HR18C2 (tj18C2_V03)* 
               
               
                 (SEQ ID NO: 30) 
                   
               
               
                   RI E KL C RI AE AL AR E ARS K AEELRQRHPDSQAARDAQKLASQAEEAVKLACELAQEHPNA I IA 
                   
               
               
                   
               
               
                   IL C IV AA IA AA I AA SM AAALAQ R HPDSQAARDAIKLASQAAEAVKLACELAQEHPNA K IA VL C 
               
               
                   
               
               
                 I L AAALAAIAAALAALLAQLHPDSQAARDAIKLASQAAEAVKLACELAQEHPNADIAEKCILL 
               
               
                   
               
               
                 AILAA LL A IL AA LL A MLH PDSQ L A RDL I DL AS EL AEEVKERCER 
               
               
                   
               
               
                 &gt;HR79C2 (tj79C2_V01) 
               
               
                 (SEQ ID NO: 31) 
                   
               
               
                   RE DE LR A RL LI LL A EL AA ER A DI AA E RTGDPRVRELARELKRLAQEAAEEVKRDPSSSDVNEA 
                   
               
               
                   
               
               
                 L K LIVEA IE AAVRALEAAERTGDPEVRELARELVRLAVEAAEEVQRNPSSSDVNEALKLIV E A 
               
               
                   
               
               
                 IEAAV R ALEAAERTGDPEVRELARELVRLAVEAAEEVQRNPSS K EVD M AL LL I LI AI LE A VL S 
               
               
                   
               
               
                 L LR AE R SGDPEKREKARERVREAVE R AEEVQRDPS 
               
               
                   
               
               
                 &gt;HR81C2 (tj81C2_V39) 
               
               
                 (SEQ ID NO: 32) 
                   
               
               
                   ER E EL SE L AER I LQ K ARK L SEEARERGDL KE LA L AL IL EA L AV LL LA I AA LL RGNSEEAERAS 
                   
               
               
                   
               
               
                 EKAQRVLEEARKVSEEAREQGDDEVLA L ALIAIA L AV L ALA L VACCRGNSEEAERASEKAQRV 
               
               
                   
               
               
                 LEEARKVSEEAREQGDD E VLA L ALIAIA L AV L ALA I VACCRGNKEEAERAAEDA IK VA M EA LE   
               
               
                   
               
               
                 V LL SA VE QGD LK V ALA A VI A ILL A IA A LLMVIM C K G 
               
               
                   
               
               
                 &gt;T33_dn2A* (Based on 1na0C3_3) 
               
               
                 (SEQ ID NO: 33) 
                   
               
               
                   NL A EKM Y KA GN AM Y RK G Q Y TI A IIA Y TL A LLK DPNNAEA W YN L GNA A YK K GEYD E AI EA YQ K A 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYY K KA LR LD PR N VD A IE N LIE A EE KQG 
               
               
                   
               
               
                 &gt;T33_dn2B* (Based on 1na0C3_int2) 
               
               
                 (SEQ ID NO: 34) 
                   
               
               
                   EE A ELA Y LL G ELA Y KL G E Y RI AI RA Y RI AL KR DPNNAEAWYNLGNAYYKQGDY R EA IR YY LR A 
                   
               
               
                   
               
               
                   LK LD PE NAEA W YN L GNA L YKQG K Y DL A IIA Y QA AL EE DPNNAEAKQNLGNAKQK Q G 
               
               
                   
               
               
                 &gt;T33_dn5A* (Based on 1na0C3_G1) 
               
               
                 (SEQ ID NO: 35) 
                   
               
               
                 N S AE AM Y KM GN AA YKQG D Y IL A IIA YLLAL EK DPNNAEAWYNLGNAAYK Q G D YD E AIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYY E KA LE LDPNN A EA LK N LLE A IAE Q D   
               
               
                   
               
               
                 &gt;T33_dn5B* (Based on 3ltjC3_int1) 
               
               
                 (SEQ ID NO: 36) 
                   
               
               
                 T DPLA V ILYIAI L KAEKSIA R AK AA E ALG K IGDE R AVEPLIK AL KDEDA L VR A AAADALGQIG 
                   
               
               
                   
               
               
                 DERAVEPLIKALKDE E G L VR A SAA I ALGQIGDERA VQ PL IK A LT DE RDL V RVA AA V ALG R IGD 
               
               
                   
               
               
                 E K AV RP LI IV L K D EE G E VR EA AA I ALG S IGGERVRAAMEKLAE R GTGFARKVAVNYLETHK 
               
               
                   
               
               
                 &gt;T33_dn10A* (Based on 1na0C3_int2) 
               
               
                 (SEQ ID NO: 37) 
                   
               
               
                   EE A EL AY LL G ELA Y KL G E Y RI AI RA Y RI AL KR DPNNAEAWYNLGNAYYKQGDYDEAIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYKQGDYDEAIEYY E KALELDP ENL EA LQNLLN A MD KQ G   
               
               
                   
               
               
                 &gt;T33_dn10B* (Based on HR00C3_2) 
               
               
                 (SEQ ID NO: 38) 
                   
               
               
                   I EE VV A EM ID IL AE S SK K SIE EL AR A ADN K TTE KA VA E AI EE IA RL ATAAI QL IE AL A KN LAS 
                   
               
               
                   
               
               
                 EEFMARAISAIAELAKKAIEAIYRLADNHTTD T FMA R AI A AIA N LA VT AI L AIA A LA S NHTTE 
               
               
                   
               
               
                 EFMARAISAIAELAKKAIEAIYRLADNHTTD KF MAAAI E AIA L LA TL AI LA I AL LA S NHTTE K   
               
               
                   
               
               
                 FMA R AI M AI AI LA A KAI E AIYRLADNHTSPTYIE K AIEAIEKIARKAI K AIE M LAK N ITTEEY 
               
               
                   
               
               
                 KE K AK KI ID I IRK L AK MA I KK LEDNRT 
               
               
                   
               
               
                 &gt;I53_dn5A* 
               
               
                 (SEQ ID NO: 39) 
                   
               
               
                 KYDGSKLRIGIL H A R W NAE II L AL VL GA L KRL Q EFGVK R ENII IET VPGSFELPYGSKLFVEK 
                   
               
               
                   
               
               
                 QKRLGKPLDAIIPIGVLIKGSTMHFEYICDSTTHQLMKLNFELGIPVIFGVLTCLTDEQAEAR 
               
               
                   
               
               
                 AGLIEGKMHNHGEDWGAAAVEMATKFN 
               
               
                   
               
               
                 &gt;I53_dn5B* (Based on 1na0C3_int2) 
               
               
                 (SEQ ID NO: 40) 
                   
               
               
                   EE A ELA Y LL G ELA Y K LG E Y RI AI RA Y RI AL KR DPNNAEAWYNLGNAYYKQGRYREAIEYYQKA 
                   
               
               
                   
               
               
                 LELDPNNAEAWYNLGNAYYERGEYEEAIEYY R KA LR LDP N N AD A MQ N LLN A KMREE   
               
            
           
         
       
     
     In another embodiment, all oligomerizing positions in the amino acid sequence selected from the group consisting of SEQ ID NO: 10 and 12-40 have the amino acid residue shown in the amino acid sequence selected from the group consisting of SEQ ID NO: 10 and 12-40, or conservative substitutions thereof. 
     As used throughout the present application, the term “polypeptide” is used in its broadest sense to refer to a sequence of subunit amino acids. The polypeptides of the invention may comprise L-amino acids, D-amino acids (which are resistant to L-amino acid-specific proteases in vivo), or a combination of D- and L-amino acids. The polypeptides described herein may be chemically synthesized or recombinantly expressed. The polypeptides may be linked to other compounds to promote an increased half-life in vivo, such as by PEGylation, HESylation, PASylation, glycosylation, or may be produced as an Fc-fusion or in deimmunized variants. Such linkage can be covalent or non-covalent as is understood by those of skill in the art. 
     In various embodiments of all aspects and embodiments of the invention, the polypeptides and individual domains thereof may comprise or consist of the amino acid sequence having at least 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, 95%, 96%, 97%, 98%, 99%, or 100% identity over its length with the reference sequence. 
     As will be understood by those of skill in the art, the polypeptides of the invention may include additional residues at the N-terminus, C-terminus, or both that are not present in the polypeptides disclosed herein; these additional residues are not included in determining the percent identity of the polypeptides of the invention relative to the reference polypeptide. 
     As used herein, “conservative amino acid substitution” means amino acid or nucleic acid substitutions that do not alter or substantially alter polypeptide or polynucleotide function or other characteristics. Amino acids can be divided into groups based on common side-chain properties: (1) hydrophobic: Norleucine, Met, Ala, Val, Leu, Ile; (2) neutral hydrophilic: Cys, Ser, Thr, Asn, Gln; (3) acidic: Asp, Glu; (4) basic: His, Lys, Arg; (5) residues that influence chain orientation: Gly, Pro; (6) aromatic: Trp, Tyr, Phe. Non-conservative substitutions will entail exchanging a member of one of these classes for another class. Particular conservative substitutions include, for example; Ala into Gly or into Ser; Arg into Lys; Asn into Gln or into H is; Asp into Glu; Cys into Ser; Gln into Asn; Glu into Asp; Gly into Ala or into Pro; His into Asn or into Gln; Ile into Leu or into Val; Leu into Ile or into Val; Lys into Arg, into Gln or into Glu; Met into Leu, into Tyr or into Ile; Phe into Met, into Leu or into Tyr; Ser into Thr; Thr into Ser; Trp into Tyr; Tyr into Trp; and/or Phe into Val, into Ile or into Leu. 
     As noted above, the polypeptides of the invention may include additional residues at the N-terminus, C-terminus, or both. Such residues may be any residues suitable for an intended use, including but not limited to detection tags (i.e.: fluorescent proteins, antibody epitope tags, etc.), linkers, ligands suitable for purposes of purification (His tags, etc.), and peptide domains that add functionality to the polypeptides, such as a polypeptide to be displayed on the surface of the homo-oligomers formed from the polypeptides of the invention (i.e.: a “cargo”). 
     In another embodiment, the invention provides homo-oligomeric protein assemblies, comprising a plurality of polypeptides of the present invention having the same amino acid sequence. Such homo-oligomeric assemblies may comprise, for example, dimers, trimers, tetramers, and pentamers. 
     In a further aspect, the present invention provides isolated nucleic acids encoding a polypeptide of the present invention. The isolated nucleic acid sequence may comprise RNA or DNA. As used herein, “isolated nucleic acids” are those that have been removed from their normal surrounding nucleic acid sequences in the genome or in cDNA sequences. Such isolated nucleic acid sequences may comprise additional sequences useful for promoting expression and/or purification of the encoded protein, including but not limited to polyA sequences, modified Kozak sequences, and sequences encoding epitope tags, export signals, and secretory signals, nuclear localization signals, and plasma membrane localization signals. It will be apparent to those of skill in the art, based on the teachings herein, what nucleic acid sequences will encode the polypeptides of the invention. 
     In another aspect, the present invention provides recombinant expression vectors comprising the isolated nucleic acid of any aspect of the invention operatively linked to a suitable control sequence. “Recombinant expression vector” includes vectors that operatively link a nucleic acid coding region or gene to any control sequences capable of effecting expression of the gene product. “Control sequences” operably linked to the nucleic acid sequences of the invention are nucleic acid sequences capable of effecting the expression of the nucleic acid molecules. The control sequences need not be contiguous with the nucleic acid sequences, so long as they function to direct the expression thereof. Thus, for example, intervening untranslated yet transcribed sequences can be present between a promoter sequence and the nucleic acid sequences and the promoter sequence can still be considered “operably linked” to the coding sequence. Other such control sequences include, but are not limited to, polyadenylation signals, termination signals, and ribosome binding sites. Such expression vectors can be of any type known in the art, including but not limited plasmid and viral-based expression vectors. The control sequence used to drive expression of the disclosed nucleic acid sequences in a mammalian system may be constitutive (driven by any of a variety of promoters, including but not limited to, CMV, SV40, RSV, actin, EF) or inducible (driven by any of a number of inducible promoters including, but not limited to, tetracycline, ecdysone, steroid-responsive). The construction of expression vectors for use in transfecting host cells is well known in the art, and thus can be accomplished via standard techniques. (See, for example, Sambrook, Fritsch, and Maniatis, in: Molecular Cloning, A Laboratory Manual, Cold Spring Harbor Laboratory Press, 1989 ; Gene Transfer and Expression Protocols , pp. 109-128, ed. E. J. Murray, The Humana Press Inc., Clifton, N.J.), and the Ambion 1998 Catalog (Ambion, Austin, Tex.). The expression vector must be replicable in the host organisms either as an episome or by integration into host chromosomal DNA. In various embodiments, the expression vector may comprise a plasmid, viral-based vector, or any other suitable expression vector. 
     In a further aspect, the present invention provides host cells that comprise the recombinant expression vectors disclosed herein, wherein the host cells can be either prokaryotic or eukaryotic. The cells can be transiently or stably engineered to incorporate the expression vector of the invention, using standard techniques in the art, including but not limited to standard bacterial transformations, calcium phosphate co-precipitation, electroporation, or liposome mediated-, DEAE dextran mediated-, polycationic mediated-, or viral mediated transfection. (See, for example,  Molecular Cloning: A Laboratory Manual  (Sambrook, et al., 1989, Cold Spring Harbor Laboratory Press;  Culture of Animal Cells: A Manual of Basic Technique,  2 nd  Ed. (R. I. Freshney. 1987. Liss, Inc. New York, N.Y.). A method of producing a polypeptide according to the invention is an additional part of the invention. The method comprises the steps of (a) culturing a host according to this aspect of the invention under conditions conducive to the expression of the polypeptide, and (b) optionally, recovering the expressed polypeptide. The expressed polypeptide can be recovered from the cell free extract, but preferably they are recovered from the culture medium. Methods to recover polypeptide from cell free extracts or culture medium are well known to the person skilled in the art. 
     In another aspect, the invention provides computational methods for designing polypeptides that can self-assemble into homoligomers. 
     Example Computing Environment 
       FIG. 1  is a block diagram of an example computing network. Some or all of the above-mentioned techniques disclosed herein, such as but not limited to techniques disclosed as part of and/or being performed by software, the Rosetta software suite, RosettaDesign, Rosetta applications, and/or other herein-described computer software and computer hardware, can be part of and/or performed by a computing device. For example,  FIG. 1  shows protein design system  102  configured to communicate, via network  106 , with client devices  104   a ,  104   b , and  104   c  and protein database  108 . In some embodiments, protein design system  102  and/or protein database  108  can be a computing device configured to perform some or all of the herein described methods and techniques, such as but not limited to, method  300  and functionality described as being part of or related to Rosetta. Protein database  108  can, in some embodiments, store information related to and/or used by Rosetta. 
     Network  106  may correspond to a LAN, a wide area network (WAN), a corporate intranet, the public Internet, or any other type of network configured to provide a communications path between networked computing devices. Network  106  may also correspond to a combination of one or more LANs, WANs, corporate intranets, and/or the public Internet. 
     Although  FIG. 1  only shows three client devices  104   a ,  104   b ,  104   c , distributed application architectures may serve tens, hundreds, or thousands of client devices. Moreover, client devices  104   a ,  104   b ,  104   c  (or any additional client devices) may be any sort of computing device, such as an ordinary laptop computer, desktop computer, network terminal, wireless communication device (e.g., a cell phone or smart phone), and so on. In some embodiments, client devices  104   a ,  104   b ,  104   c  can be dedicated to problem solving/using the Rosetta software suite. In other embodiments, client devices  104   a ,  104   b ,  104   c  can be used as general purpose computers that are configured to perform a number of tasks and need not be dedicated to problem solving/using the Rosetta software suite. In still other embodiments, part or all of the functionality of protein design system  102  and/or protein database  108  can be incorporated in a client device, such as client device  104   a ,  104   b , and/or  104   c.    
     Computing Environment Architecture 
       FIG. 2A  is a block diagram of an example computing device (e.g., system) In particular, computing device  200  shown in  FIG. 2A  can be configured to: include components of and/or perform one or more functions of some or all of the herein described methods and techniques, such as but not limited to, method  300  and functionality described as being part of or related to Rosetta. Computing device  200  may include a user interface module  201 , a network-communication interface module  202 , one or more processors  203 , and data storage  204 , all of which may be linked together via a system bus, network, or other connection mechanism  205 . 
     User interface module  201  can be operable to send data to and/or receive data from external user input/output devices. For example, user interface module  201  can be configured to send and/or receive data to and/or from user input devices such as a keyboard, a keypad, a touch screen, a computer mouse, a track ball, a joystick, a camera, a voice recognition module, and/or other similar devices. User interface module  201  can also be configured to provide output to user display devices, such as one or more cathode ray tubes (CRT), liquid crystal displays (LCD), light emitting diodes (LEDs), displays using digital light processing (DLP) technology, printers, light bulbs, and/or other similar devices, either now known or later developed. User interface module  201  can also be configured to generate audible output(s), such as a speaker, speaker jack, audio output port, audio output device, earphones, and/or other similar devices. 
     Network-communications interface module  202  can include one or more wireless interfaces  207  and/or one or more wireline interfaces  208  that are configurable to communicate via a network, such as network  106  shown in  FIG. 1 . Wireless interfaces  207  can include one or more wireless transmitters, receivers, and/or transceivers, such as a Bluetooth transceiver, a Zigbee transceiver, a Wi-Fi transceiver, a WiMAX transceiver, and/or other similar type of wireless transceiver configurable to communicate via a wireless network. Wireline interfaces  208  can include one or more wireline transmitters, receivers, and/or transceivers, such as an Ethernet transceiver, a Universal Serial Bus (USB) transceiver, or similar transceiver configurable to communicate via a twisted pair, one or more wires, a coaxial cable, a fiber-optic link, or a similar physical connection to a wireline network. 
     In some embodiments, network communications interface module  202  can be configured to provide reliable, secured, and/or authenticated communications. For each communication described herein, information for ensuring reliable communications (i.e., guaranteed message delivery) can be provided, perhaps as part of a message header and/or footer (e.g., packet/message sequencing information, encapsulation header(s) and/or footer(s), size/time information, and transmission verification information such as CRC and/or parity check values). Communications can be made secure (e.g., be encoded or encrypted) and/or decrypted/decoded using one or more cryptographic protocols and/or algorithms, such as, but not limited to, DES, AES, RSA, Diffie-Hellman, and/or DSA. Other cryptographic protocols and/or algorithms can be used as well or in addition to those listed herein to secure (and then decrypt/decode) communications. 
     Processors  203  can include one or more general purpose processors and/or one or more special purpose processors (e.g., digital signal processors, application specific integrated circuits, etc.). Processors  203  can be configured to execute computer-readable program instructions  206  contained in data storage  204  and/or other instructions as described herein. Data storage  204  can include one or more computer-readable storage media that can be read and/or accessed by at least one of processors  203 . The one or more computer-readable storage media can include volatile and/or non-volatile storage components, such as optical, magnetic, organic or other memory or disc storage, which can be integrated in whole or in part with at least one of processors  203 . In some embodiments, data storage  204  can be implemented using a single physical device (e.g., one optical, magnetic, organic or other memory or disc storage unit), while in other embodiments, data storage  204  can be implemented using two or more physical devices. 
     Data storage  204  can include computer-readable program instructions  206  and perhaps additional data. For example, in some embodiments, data storage  204  can store part or all of data utilized by a protein design system and/or a protein database; e.g., protein designs system  102 , protein database  108 . In some embodiments, data storage  204  can additionally include storage required to perform at least part of the herein-described methods and techniques and/or at least part of the functionality of the herein-described devices and networks. 
       FIG. 2B  depicts a network  106  of computing clusters  209   a ,  209   b ,  209   c  arranged as a cloud-based server system in accordance with an example embodiment. Data and/or software for protein design system  102  can be stored on one or more cloud-based devices that store program logic and/or data of cloud-based applications and/or services. In some embodiments, protein design system  102  can be a single computing device residing in a single computing center. In other embodiments, protein design system  102  can include multiple computing devices in a single computing center, or even multiple computing devices located in multiple computing centers located in diverse geographic locations. 
     In some embodiments, data and/or software for protein design system  102  can be encoded as computer readable information stored in tangible computer readable media (or computer readable storage media) and accessible by client devices  104   a ,  104   b , and  104   c , and/or other computing devices. In some embodiments, data and/or software for protein design system  102  can be stored on a single disk drive or other tangible storage media, or can be implemented on multiple disk drives or other tangible storage media located at one or more diverse geographic locations. 
       FIG. 2B  depicts a cloud-based server system in accordance with an example embodiment. In  FIG. 2B , the functions of protein design system  102  can be distributed among three computing clusters  209   a ,  209   b , and  209   c . Computing cluster  209   a  can include one or more computing devices  200   a , cluster storage arrays  210   a , and cluster routers  211   a  connected by a local cluster network  212   a . Similarly, computing cluster  209   b  can include one or more computing devices  200   b , cluster storage arrays  210   b , and cluster routers  211   b  connected by a local cluster network  212   b . Likewise, computing cluster  209   c  can include one or more computing devices  200   c , cluster storage arrays  210   c , and cluster routers  211   c  connected by a local cluster network  212   c.    
     In some embodiments, each of the computing clusters  209   a ,  209   b , and  209   c  can have an equal number of computing devices, an equal number of cluster storage arrays, and an equal number of cluster routers. In other embodiments, however, each computing cluster can have different numbers of computing devices, different numbers of cluster storage arrays, and different numbers of cluster routers. The number of computing devices, cluster storage arrays, and cluster routers in each computing cluster can depend on the computing task or tasks assigned to each computing cluster. 
     In computing cluster  209   a , for example, computing devices  200   a  can be configured to perform various computing tasks of protein design system  102 . In one embodiment, the various functionalities of protein design system  102  can be distributed among one or more of computing devices  200   a ,  200   b , and  200   c . Computing devices  200   b  and  200   c  in computing clusters  209   b  and  209   c  can be configured similarly to computing devices  200   a  in computing cluster  209   a . On the other hand, in some embodiments, computing devices  200   a ,  200   b , and  200   c  can be configured to perform different functions. 
     In some embodiments, computing tasks and stored data associated with protein design system  102  can be distributed across computing devices  200   a ,  200   b , and  200   c  based at least in part on the processing requirements of protein design system  102 , the processing capabilities of computing devices  200   a ,  200   b , and  200   c , the latency of the network links between the computing devices in each computing cluster and between the computing clusters themselves, and/or other factors that can contribute to the cost, speed, fault-tolerance, resiliency, efficiency, and/or other design goals of the overall system architecture. 
     The cluster storage arrays  210   a ,  210   b , and  210   c  of the computing clusters  209   a ,  209   b , and  209   c  can be data storage arrays that include disk array controllers configured to manage read and write access to groups of hard disk drives. The disk array controllers, alone or in conjunction with their respective computing devices, can also be configured to manage backup or redundant copies of the data stored in the cluster storage arrays to protect against disk drive or other cluster storage array failures and/or network failures that prevent one or more computing devices from accessing one or more cluster storage arrays. 
     Similar to the manner in which the functions of protein design system  102  can be distributed across computing devices  200   a ,  200   b , and  200   c  of computing clusters  209   a ,  209   b , and  209   c , various active portions and/or backup portions of these components can be distributed across cluster storage arrays  210   a ,  210   b , and  210   c . For example, some cluster storage arrays can be configured to store one portion of the data and/or software of protein design system  102 , while other cluster storage arrays can store a separate portion of the data and/or software of protein design system  102 . Additionally, some cluster storage arrays can be configured to store backup versions of data stored in other cluster storage arrays. 
     The cluster routers  211   a ,  211   b , and  211   c  in computing clusters  209   a ,  209   b , and  209   c  can include networking equipment configured to provide internal and external communications for the computing clusters. For example, the cluster routers  211   a  in computing cluster  209   a  can include one or more internet switching and routing devices configured to provide (i) local area network communications between the computing devices  200   a  and the cluster storage arrays  201   a  via the local cluster network  212   a , and (ii) wide area network communications between the computing cluster  209   a  and the computing clusters  209   b  and  209   c  via the wide area network connection  213   a  to network  106 . Cluster routers  211   b  and  211   c  can include network equipment similar to the cluster routers  211   a , and cluster routers  211   b  and  211   c  can perform similar networking functions for computing clusters  209   b  and  209   b  that cluster routers  211   a  perform for computing cluster  209   a.    
     In some embodiments, the configuration of the cluster routers  211   a ,  211   b , and  211   c  can be based at least in part on the data communication requirements of the computing devices and cluster storage arrays, the data communications capabilities of the network equipment in the cluster routers  211   a ,  211   b , and  211   c , the latency and throughput of local networks  212   a ,  212   b ,  212   c , the latency, throughput, and cost of wide area network links  213   a ,  213   b , and  213   c , and/or other factors that can contribute to the cost, speed, fault-tolerance, resiliency, efficiency and/or other design goals of the moderation system architecture. 
     Example Methods of Operation 
       FIG. 3  is a flow chart of an example method  300 . Method  300  can be carried out by a computing device, such as computing device  200  described in the context of at least  FIG. 2A . At least the embodiments of method  300  mentioned below are discussed herein. 
     Method  300  can begin at block  310 , where the computing device can determine a cycle of monomeric proteins. 
     At block  320 , the computing device can determine a docking score for the cycle of monomeric proteins using the computing device, the docking score representing interaction between two or more monomeric proteins in the cycle of monomeric proteins with respect to a multi-dimensional rigid body transformation between one or more backbone atoms of the two or more monomeric proteins. In some embodiments, determining a docking score can include determining a docking score representing interaction between the two or more monomeric proteins in the cycle of monomeric proteins with respect to a six-or-more-dimensional rigid body transformation. In particular of these embodiments, determining the docking score representing interaction between the two or more monomeric proteins in the cycle of monomeric proteins with respect to the six-or-more-dimensional rigid body transformation can include: determining that the two or more monomeric proteins come into contact; and after determining that the two or more monomeric proteins come into contact, reducing the six-or-more-dimensional rigid body transformation to a three-or-more-dimensional rigid body transformation. In other of these embodiments, determining that the two or more monomeric proteins come into contact can include determining that the two or more monomeric proteins come into contact based on a slide vector that brings the two or more monomeric proteins into contact. In even other of these embodiments, determining that the two or more monomeric proteins come into contact based on a slide vector that brings the two or more monomeric proteins into contact can include determining a slide distance between the two or more monomeric proteins using the slide vector. In still other of these embodiments, determining that the two or more monomeric proteins come into contact can include determining a slide distance between the two or more monomeric proteins using an octree. 
     At block  330 , the computing device can determine whether the docking score for the cycle of monomeric proteins is a relatively-low docking score. In some embodiments, determining the docking score can include determining a plurality of bins for two or more monomeric proteins, each bin representing a particular position and a particular orientation of the two or more monomeric proteins. In particular of these embodiments, each bin of the plurality of bins can include a bin index determined using a hash transform computed for the particular position and the particular orientation of the two or more monomeric proteins. In some of these embodiments, the hash transform can receive backbone atom positions of the two or more monomeric proteins as inputs. In other of these embodiments, the hash transform can compute a rigid body transformation between at least three backbone atom positions of the two or more monomeric proteins. In still other of these embodiments, the at least three backbone atom positions comprise positions for at least one nitrogen atom, at least one alpha-carbon atom, and at least one carbon atom. 
     At block  340 , after determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score, the computing device can determine one or more interfaces between the two or more monomeric proteins in the cycle of monomeric proteins. In some embodiments, the cycle of monomeric proteins can include a particular monomer; then, determining whether the docking score for the cycle of monomeric proteins is a relatively-low docking score can include: determining a plurality of docking scores for a plurality of cycles of monomeric proteins that each include the particular monomer; determining a predetermined number of lowest docking scores of the plurality of docking scores; determining whether the docking score for the cycle of monomeric proteins is a docking score of the predetermined number of lowest docking scores; and after determining that the docking score for the cycle of monomeric proteins is a docking score of the predetermined number of lowest docking scores, determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score. In other embodiments, the cycle of monomeric proteins includes a particular monomer; then, determining whether the docking score for the cycle of monomeric proteins is a relatively-low docking score can include: determining a plurality of docking scores for a plurality of cycles of monomeric proteins that each include the particular monomer; determining a threshold docking score value; determining whether the docking score for the cycle of monomeric proteins is less than the threshold docking score value; and after determining that the docking score for the cycle of monomeric proteins is less than the threshold docking score value, determining that the docking score for the cycle of monomeric proteins is a relatively-low docking score. 
     At block  350 , the computing device and/or one or more other entities can generate an output related to the cycle of monomeric proteins. In some embodiments, generating the output related to the cycle of monomeric proteins can include designing one or more molecules based on the cycle of monomeric proteins. In other embodiments, generating the output related to the cycle of monomeric proteins can include: generating a synthetic gene that is based the cycle of monomeric proteins; expressing a particular protein in vivo using the synthetic gene; and purifying the particular protein. In some of these embodiments, expressing the particular protein sequence in vivo using the synthetic gene can include expressing the particular protein sequence in one or more  Escherichia coli  that include the synthetic gene. In even other embodiments, generating the output related to the cycle of monomeric proteins comprises generating one or more images that include at least part of the cycle of monomeric proteins. 
     In some examples, at least a portion of method  300  is performed by a computing device that includes: one or more processors; and non-transitory data storage, configured to store at least computer-readable instructions that, when executed, cause the computing device to perform the at least a portion of method  300 . 
     In other examples, a non-transitory computer-readable medium is provided, where the computer-readable medium is configured to store at least computer-readable instructions that, when executed by one or more processors of a computing device, cause the computing device to perform at least a portion of method  300 . 
     In still other examples, an apparatus is provided, where the apparatus can include means to perform at least a portion of method  300 . 
     The particulars shown herein are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of various embodiments of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for the fundamental understanding of the invention, the description taken with the drawings and/or examples making apparent to those skilled in the art how the several forms of the invention may be embodied in practice. 
     The above definitions and explanations are meant and intended to be controlling in any future construction unless clearly and unambiguously modified in the following examples or when application of the meaning renders any construction meaningless or essentially meaningless. In cases where the construction of the term would render it meaningless or essentially meaningless, the definition should be taken from Webster&#39;s Dictionary, 3rd Edition or a dictionary known to those of skill in the art, such as the Oxford Dictionary of Biochemistry and Molecular Biology (Ed. Anthony Smith, Oxford University Press, Oxford, 2004). 
     As used herein and unless otherwise indicated, the terms “a” and “an” are taken to mean “one”, “at least one” or “one or more”. Unless otherwise required by context, singular terms used herein shall include pluralities and plural terms shall include the singular. 
     Unless the context clearly requires otherwise, throughout the description and the claims, the words ‘comprise’, ‘comprising’, and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to”. Words using the singular or plural number also include the plural or singular number, respectively. Additionally, the words “herein,” “above” and “below” and words of similar import, when used in this application, shall refer to this application as a whole and not to any particular portions of this application. 
     The above description provides specific details for a thorough understanding of, and enabling description for, embodiments of the disclosure. However, one skilled in the art will understand that the disclosure may be practiced without these details. In other instances, well-known structures and functions have not been shown or described in detail to avoid unnecessarily obscuring the description of the embodiments of the disclosure. The description of embodiments of the disclosure is not intended to be exhaustive or to limit the disclosure to the precise form disclosed. While specific embodiments of, and examples for, the disclosure are described herein for illustrative purposes, various equivalent modifications are possible within the scope of the disclosure, as those skilled in the relevant art will recognize. 
     Specific elements of any of the foregoing embodiments can be combined or substituted for elements in other embodiments. Furthermore, while advantages associated with certain embodiments of the disclosure have been described in the context of these embodiments, other embodiments may also exhibit such advantages, and not all embodiments need necessarily exhibit such advantages to fall within the scope of the disclosure. 
     The above detailed description describes various features and functions of the disclosed systems, devices, and methods with reference to the accompanying figures. In the figures, similar symbols typically identify similar components, unless context dictates otherwise. The illustrative embodiments described in the detailed description, figures, and claims are not meant to be limiting. Other embodiments can be utilized, and other changes can be made, without departing from the spirit or scope of the subject matter presented herein. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations, all of which are explicitly contemplated herein. 
     With respect to any or all of the ladder diagrams, scenarios, and flow charts in the figures and as discussed herein, each block and/or communication may represent a processing of information and/or a transmission of information in accordance with example embodiments. Alternative embodiments are included within the scope of these example embodiments. In these alternative embodiments, for example, functions described as blocks, transmissions, communications, requests, responses, and/or messages may be executed out of order from that shown or discussed, including substantially concurrent or in reverse order, depending on the functionality involved. Further, more or fewer blocks and/or functions may be used with any of the ladder diagrams, scenarios, and flow charts discussed herein, and these ladder diagrams, scenarios, and flow charts may be combined with one another, in part or in whole. 
     A block that represents a processing of information may correspond to circuitry that can be configured to perform the specific logical functions of a herein-described method or technique. Alternatively or additionally, a block that represents a processing of information may correspond to a module, a segment, or a portion of program code (including related data). The program code may include one or more instructions executable by a processor for implementing specific logical functions or actions in the method or technique. The program code and/or related data may be stored on any type of computer readable medium such as a storage device including a disk or hard drive or other storage medium. 
     The computer readable medium may also include non-transitory computer readable media such as computer-readable media that stores data for short periods of time like register memory, processor cache, and random access memory (RAM). The computer readable media may also include non-transitory computer readable media that stores program code and/or data for longer periods of time, such as secondary or persistent long term storage, like read only memory (ROM), optical or magnetic disks, compact-disc read only memory (CD-ROM), for example. The computer readable media may also be any other volatile or non-volatile storage systems. A computer readable medium may be considered a computer readable storage medium, for example, or a tangible storage device. Moreover, a block that represents one or more information transmissions may correspond to information transmissions between software and/or hardware modules in the same physical device. However, other information transmissions may be between software modules and/or hardware modules in different physical devices. 
     Numerous modifications and variations of the present disclosure are possible in light of the above teachings. 
     Examples 
     Self-assembling cyclic protein homo-oligomers play important roles in biology and the ability to generate custom homo-oligomeric structures could enable new approaches to probe biological function. Here we report a. general approach to design cyclic homo-oligomers that employs a new residue pair transform method for assessing the designability of a protein-protein interface. This method is sufficiently rapid to allow systematic enumeration of cyclically docked arrangements of a monomer followed by sequence design of the newly formed interfaces. We use this method to design interfaces onto idealized repeat proteins that direct their assembly into complexes that possess cyclic symmetry. Of 96 designs that were experimentally characterized, 21 were found to form stable monodisperse homo-oligomers in solution, and 15 (4 homodimers, 6 homotrimers, 6 homotetramers and 1 homopentamer) had solution small angle X-ray scattering data consistent with the design models. X-ray crystal structures were obtained for five of the designs and each of these were shown to be very close to their design model. 
     Cyclic homo-oligomers assembled from multiple identical protein subunits symmetrically arranged around a central axis play key roles in many biological processes including catalysis, signaling and allostery. Despite their prevalence in natural systems, currently there is no systematic approach to design cyclic homo-oligomers starting from a monomeric protein structure. 
     Here we present a general method for designing cyclic homo-oligomers in silico and use it to design interfaces onto recently developed repeat proteins that direct their assembly into dimeric, trimeric, tetrameric and pentameric complexes. Structural characterization shows that many of the designs adopt the target oligomerization state and structure, demonstrating that we have a basic understanding of the determinants of oligomerization state. The capability of designing proteins with tunable shape, size, and symmetry enables rigid display of binding domains at arbitrary orientations and distances for a range of biological applications. 
     Results 
     The self-assembly of naturally occurring complexes is driven by chemical and shape complementarity. Protein-protein interfaces are generally comprised of a hydrophobic core that is buried upon binding and surrounded by a rim of polar residues that prevent non-specific aggregation. We developed a design strategy to generate such interfaces between protein monomers docked in a range of cyclic geometries. The strategy has two steps ( FIG. 5 ): first, low resolution docking to sample and rank symmetric arrangements of a given scaffold protein based on their designability (the likelihood of finding an amino acid sequence that can stabilize a given rigid body conformation), and second, full atom RosettaDesign™ calculations to optimize the sequence at the protein-protein interfaces fix high affinity binding. To explore the generality of the method, symmetries ranging from C2 through C6 were designed. 96 designs were selected for experimental characterized, and 4 homodimers, 6 homotrimers, 6 homotetramers and 1 homopentamer were found to form stable monodisperse homooligomers in solution. 
     Computational Design 
     Existing methods for protein-protein docking fall into three general categories: (1) voxelized rigid representations with Fast Fourier Transform (FFT)-based docking, (2) docking based on patches of high-resolution local shape complementarity, and (3) Monte Carlo sampling with soft centroid models. The first two categories are not ideal for the protein design problem because the precise shape and chemical detail of the docked surfaces are unavailable, as the interface residues are not known in advance. The approach we take is one in which docked backbones are generated and then scored using a low-resolution representation of the proteins (requiring only the backbone coordinates and secondary structure assignments) but with two notable improvements. First, we employ a six-dimensional implicit side chain scoring methodology, which better predicts the result of subsequent full atom design calculation than a traditional coarse-grained model, and second, we use an enumerative strategy to generate docked backbones, which samples more robustly the low-dimensional docking space than a Monte Carlo search. 
     In past efforts, scoring at the docking stage has been accomplished using coarse-grained models in which the absent side chains are represented by one or two points in space, and the interaction potential between two amino acids is evaluated as a function of the distance or distances between these points, and in some cases an associated angle. These representations are incomplete since they do not capture the full six-dimensional rigid body relationship between pairs of side chains. To avoid loss of information, we have developed a Residue Pair Transform (RPX) model that represents the interaction between two residues by the full six dimensional rigid body transformation between their respective backbone N, Cα and C atoms. We employ a precompiled database of all favorable residue pair interactions found in structures from the Protein Data Bank involving alanine, isoleucine, leucine, valine and methionine, binning these data based on the rigid body transform between amino acids. The score of a given docked configuration is the sum, over each pair of residues across the interface, of the lowest Rosetta full atom energy found in the associated spatial transformation bin of the database. This approach predicts the interface energy resulting from full atom sequence design calculation better than the Rosetta™ centroid energy function. As the residue-pair-transform database is compiled offline, arbitrary data selection (different subsets of amino acid identities) and processing (alternative smoothing and scoring schemes) can be employed with no impact on runtime of the docking calculations. 
     To best leverage the RPX scoring methodology described above, we employ deterministic sampling of the complete docking space. The configurational space for cyclic docking is four dimensional: the usual six degrees of freedom required for orienting a rigid body, minus translations along and rotations around the symmetry axis of the oligomer (to which the structure is invariant). These four degrees of freedom can be reduced effectively to three by the requirement that the subunits must be roughly in contact. We realize this dimensionality reduction using a fast slide-into-contact algorithm. To rapidly compute the translational distance along a slide vector, which will bring two rigid clouds of atoms into contact, we create a pair of two-dimensional arrays containing the leading face of each cloud along the slide vector. Corresponding cells of each array are checked, and the pair of atoms with least separation along the slide vectors defines an upper bound on the slide distance. The final slide distance is calculated using a local octree-like data structure (Methods). This results in a significant savings in the total number of samples that must be evaluated compared to a simpler brute force search. 
     In some examples, the slide vectors can be determined using a singular value decomposition (SVD) and/or the octree-like data structure can be a recursive structure used to divide 3-D spaces. In other examples, other data structures than slide vectors and/or octree-like data structures can be used in connection with other algorithms; e.g., stochastic Monte-Carlo based algorithms. 
     For the ten best RPX scoring docked arrangements of each monomer, low energy and shape complementary interfaces between protomers were generated using Rosetta™ sequence design calculations employing a Monte-Carlo simulated annealing protocol. Designs were filtered on number of mutations, buried surface area, shape complementarity and computed interaction energy, and 96 were selected for experimental characterization. The 11 dimers, 34 trimers, 19 tetramers, 17 pentamers and 15 hexamers are named according to the following nomenclature: the first 4 letters refer to the scaffold protein (as described in the supplementary information), the symmetry is denoted as Cn, and finally an integer is added to differentiate oligomers of identical symmetry and scaffold identity. 
     Protein Expression and Oligomerization State Screening 
     Synthetic genes encoding each of the 96 designs were synthesized and cloned into a vector with a T7 promoter system and either an N- or C-terminal (His) 6  tag, and the corresponding proteins expressed in  E. coli . The proteins were purified by immobilized nickel-affinity chromatography (Ni 2+  IMAC) and size-exclusion chromatography (SEC). 64 designs were soluble and amenable to purification. The oligomerization states for 44 designs that eluted from SEC with a single predominant species were determined by size-exclusion chromatography in tandem with multi-angle light scattering (SEC-MALS). For 21 of the designs, the molecular weights determined by light scattering agreed with the designed oligomerization state. 
     Structural Characterization 
     To further assess the configuration of the designed proteins in solution, small-angle X-ray scattering (SAXS) measurements were performed on designs that had predominantly monodisperse traces in the SEC screen. A total of 26 designs (the 21 with consistent SEC-MALS data and 5 additional designs that had monodisperse SEC profiles) were characterized with this technique and the measured scattering profile was compared to that expected from the computational model. Designs with a deviation of less than or equal to 3.1 a.u. using the χ measure and a deviation of less than 11% between the computed and experimental radius of gyration were considered to be in the designed supramolecular arrangement (these thresholds were chosen based on the deviations between computed and measured values for designs with crystal structures consistent with the corresponding models; see below). 
     Of the 26 designs, 15 fulfill these criteria; 5 dimers, 6 trimers, 3 tetramers, and 1 pentamer. The docked configurations and designed interfaces of 13 of these are unique (three of the trimers have similar geometries with pairwise r.m.s.d. values between 1.9-2.5 Å; the lowest pairwise r.ms.d. among the remaining designs is 5.3 Å with no similarity in designed interface). Computational models, in silico symmetric docking energy landscapes, SEC-MALS chromatograms and SAXS experimental and computed profiles for the 30 designs are. 
     Crystal structures that contain the designed interface were obtained for five of the designed proteins: two dimers, two trimers and one tetramer, and are compared to the design models in  FIG. 6 . For each of the five cases the side chain rotamers of the hydrophobic residues are similar to those in the design model. The two dimers, ank3C2_1 and ank1C2_1, are both built from idealized ankyrin repeat proteins and are shown in  FIGS. 6 a  and 6 b   . The ank3C2_1 design has a large hydrophobic patch (1100 Å 2 ) that is buried upon binding; all interface hydrophobic side chains are in the same rotameric state in the design model and the crystal structure with the exception of methionine  90  ( FIG. 6 a   , right panel). The backbone r.m.s.d. between the design model and the crystal structure is 1.0 Å. The agreement between the model and the structure of ank1C2_1 ( FIG. 6 b   ) is even closer: both polar and hydrophobic side chain rotamers were correct and the r.m.s.d. to the model is only 0.9 Å. 
     The two trimeric designs with solved structures are 1na0C3_3 ( FIG. 6 c   ) built from a consensus designed tpr protein 16 , and HR00C3_2 ( FIG. 6 d   ) built from a de novo designed repeat protein. 1na0C3_3 has a hydrophobic core that lies on the 3-fold axis formed by residues in all subunits. The r.m.s.d. between the crystal structure and design model is 1.0 Å. HR00C3_2 contains a pore on the symmetry axis and is stabilized by three separate heterologous interfaces. This trimer was designed using the computational model of a designed repeat protein whose structure had not previously been confirmed by X-ray crystallography. Thus the crystal structure, which has a backbone r.m.s.d. to the model of 0.9 Å, validates the design of both the monomer and oligomer simultaneously. This ability to accurately design higher order structures based on design models of monomers will considerably streamline future computational design of nanomaterials using monomers with custom designed properties. 
     For the two dimers and the two trimers, the χ values between the measured SAXS scattering profiles and the profiles computed from either the corresponding design models or crystal structures are less than 3.1. In contrast, the experimental SAXS data for the designed tetramer, ank1C4_2 ( FIG. 6 e   ), deviates considerably from that computed using the crystal structure (data not shown). The ank1C4_2 crystal structure adopts a C2 symmetric tetrameric structure in which 2 pairs of chains accurately match the design model (r.m.s.d. of 1.1 Å), but exhibit clear overall distortion relative to the C4 symmetric design model (r.m.s.d. of 4.5 Å). There are two distinct interfaces present in the structure, one of which corresponds to the designed interface. The experimental SAXS profile is closer to the design model of the tetramer than the crystal structure, and hence it seems likely that the symmetry breaking in the crystal is due to lattice contacts. 
     A sixth structure was solved for design ank4C4, which shows a single symmetric peak by SEC and forms a tetrameric complex in solution as determined by MALS. The SAXS profile of this design does not match that computed from the design model (χ=3.8), and the crystal structure exhibits D2 symmetry rather than the target C4 symmetry. The SAXS profile computed from the D2 oligomer matches the measured scattering curve better than the target C4 model (χ=1.2) indicating that the D2 state corresponds to the conformation of the design in solution (data not shown). 
     Subunit Extensions 
     To explore the modularity of the designs and the robustness of the designed interfaces, we extended two of the designed oligomers by appending two additional repeats to the original constructs. Extended versions of ank1C2_1 and HR04C4_1 were expressed and characterized as described above. SEC-MALS traces of the long constructs show the expected shifts to larger apparent sizes compared to the original constructs ( FIG. 7 , third column), and the calculated molecular weights are close to those expected. Experimental SAXS profiles of the extended designs are in good agreement with the extended computational models suggesting that the supramolecular arrangement of the subunits is maintained upon extending the scaffold protein. This ability to maintain oligomer geometry while extending the length of the monomers will be very useful for systematically varying the distance between binding moieties and for nanomaterial design. 
     Resilience to Guanidine Denaturation 
     The repeat protein scaffolds used to construct the designed oligomers are very stable proteins, and thus guanidine denaturation can be used to probe the stability of the designed interfaces independent of effects on the monomers. Four designed oligomers (one selected from each symmetry C2-05) were purified in an initial round of IMAC and SEC, and subsequently run through SEC-MALS in TBS supplemented with 1M or 2M GuHCl. In both conditions, all four designs remained in their designed oligomeric state (as determined by MALS) without indications of smaller assembly formation (data not shown). 
     Discussion 
     Our results show that homo-oligomeric protein complexes with cyclic symmetry can be generated from repeat protein building blocks by computationally designing geometrically complementary, low-energy interfaces. A key advance is the new fast method for assessing designability that provides a reasonable estimate of the energy obtained after a full atom combinatorial sequence design calculation with roughly six orders of magnitude less computational cost. This allows exhaustive evaluation of the possible cyclically docked configurations of a monomer, which would not be possible with a combinatorial, all-atom sequence design calculation. The broad applicability of the computational pipeline developed here is highlighted by the number of successful designs (15) and symmetries (C2-05). We have experimentally validated dimers, trimers, tetramers and pentamer—the broad range of structures and the variety of interface geometries and architectures far exceeds that reported in any previous study. The combination of RPX search for designable interfaces followed by Rosetta all atom design calculations can clearly generate a wide range of new interfaces involving three to five alpha helices; the ability of the approach to design new beta sheet and loop containing interfaces is an area for future investigation. 
     Progress in protein design will require study not only of the successes but also the failures. The results reported in this paper provide a valuable resource for understanding failure modes as the input scaffolds are all very stable designed proteins (in previous design studies, the often unknown stability of the starting native scaffolds and the robustness to amino acid substitutions were potentially confounding factors). We are able to distinguish distinct failure modes for the designs reported: 32 were not expressed solubly in  E. Coli,  24 adopt multiple oligomerization states, 4 were monomeric, 15 were monodisperse but had an oligomerization state different from that designed, and 6 occupied the designed oligomerization state but had unanticipated configurations based on SAXS data. Analysis of the properties of the design models revealed that designs with (1) a high total charge (greater than −50), (2) small (under 750 Å 2 ) interfaces, (3) poor shape complementarity (sc&lt;0.625), or (4) for which asymmetric pairwise docking calculations found much lower energy alternative arrangements than the two body interactions in the design model were generally unsuccessful. Furthermore, despite the success with HROOC3_2, designs based on monomers with crystal structures had higher success rates (19%) than those based on monomers validated only by SAXS (4%). The fraction of designs experimentally confirmed to be in the designed state increases from 15/96 in the overall population to 14/45 restricting to models that satisfy the above criteria (low electrostatic repulsion, larger shape complementary interfaces, absence of much lower energy competing dimeric states, and crystallographically validated monomer structures). 
     Our robust design pipeline can be combined with the modularity of computationally designed repeat proteins to control the three-dimensional arrangement of the protomers at multiple length scales. While the designed interfaces control the nanoscale three-dimensional arrangement, extensions of the repeat proteins allow for the placement of functional motifs with sub-nanometer resolution in each of the interacting proteins. Designed proteins can remain folded under strongly denaturing conditions, and the design process provides unparalleled control over their geometry and amino acid composition allowing for reactive chemical moieties, such as thiols or aromatic rings, to be reserved to engineer function in downstream applications. An immediate use for these designed oligomers is to probe how the geometry and valency of tethered signaling molecules affects the clustering of receptors and the cellular response. The relationship between ligand valency, spatial orientation, and signaling outcome is not well understood, and designed homoligomerization with systematically tunable lengths should be very well suited for investigating this and other basic biochemical questions. 
     Methods 
     Scaffold Set. A set of 17 monomeric designed repeat proteins with high-resolution crystal structures as well as 6 computational models that were validated by SAXS were used as a scaffold set for our design protocol. 
     Motif Database and Scoring. We construct Cartesian frames given two N—Cα-C backbone segments across the symmetric interface. The relative position and orientation of the two N—Cα-C segments form a six dimensional space that can be divided into bins, assigning to any possible position/orientation a bin index. The best-scoring, superimposable residue-pair available in a large database of candidates can then be found with a single memory lookup keyed on the bin index. The residue pair-motif database was constructed from residue pairs observed in a set of high quality structures from the Protein Data Bank (PDB), filtered for energetic favorability, separation by at least 10 residues in sequence, and residue composition of only alanine, isoleucine, leucine, valine, and methionine. To compute an aggregate score for each conformation, we consider all pairs of N—Cα-C backbone segments across the newly formed symmetric interface within 9 Å of one another. For each such pair, the score of the best superimposable residue pair motif is looked up, and the results are summed. 
     The bin index is based on the concept that a space of all rigid body transformations, as long as objects are closer than some maximum distance, is actually a finite 6D manifold. In one example, six coordinates are chosen that can be used to unequivocally map any point in the manifold, where the six coordinates include a 3D vector (x y z) for the relative translation and a pseudo-vector (a b g) of 3 Euler angles for the relative rotation. Other coordinates/parameters are possible as well. A grid is then placed on the 6D space to produce the bins. Each bin is assigned a unique index in the form of a 64 bit integer. When evaluating a pose, or a pair of proteins with a symmetric configuration, for docking, a relative transformation between pairs of amino acids that are interacting across the interface of the monomer pair and the respective 6d pseudo-vector (x1 y1 z1 a1 b1 g1) along with its bin index is calculated. This bin index allows us to check the content of a hash value that is filled prior to the docking calculation by collecting favorable interactions that are observed in natural proteins that have the same rigid body transformation and therefore the same bin index. 
     The above-mentioned hash value is determined using a hash function or hash transform. The hash function/hash transform can receive backbone atom positions of both residues as inputs, computes their relative rigid body transformation and returns the bin index as mentioned above. The keys are integer values (e.g., 16 bit integers, 32 bit integers, 64 bit integers, 128 bit integers, 256 bit integers, etc.) that are assigned when the hash is initially constructed prior to the docking calculation. In the hash function, data for at least three atoms for each residue are used (e.g., N, Ca, C) to construct a local orthonormal frame which encodes the geometric information regarding the position and orientation of each amino acid in the pair. 
     Cyclic Docking. To generate cyclic homooligomeric arrangements of n copies of a protein monomer, we center it at the origin, finely sample the 3 rotational degrees of freedom, generate a symmetric copy by (360/n°) rotation around the Z-axis, and slide the two bodies into contact along the X-axis allowing a small range of X offsets close to the contact value. For each of these, the axis of symmetry is determined from the relative orientation of the two subunits, and the full oligomer is generated and evaluated using the residue pair motif database. A rapid slide into contact operation is required for this sampling strategy. Computing the slide distance along a given slide vector is accomplished using two two-dimensional arrays perpendicular to the slide direction into which the atoms along the leading face of each body are placed. Corresponding cells are checked, and the pair with the least separation provides an estimate of the slide distance. The bodies are placed according to this estimate, but may still have clashes. All contacting pairs of atoms across the bodies are checked using an octree-like data structure, and the bodies are backed off so as to relieve the largest clash found. This process is repeated until no dashes are found. In practice, only one or two iterations through the fast clash check are required in most cases, making the slide move rapid. 
     Interface Design. An interface design protocol was implemented in RosettaScripts™ and is described briefly here. In each design trajectory, the protomer was initially perturbed by a small translation perpendicular to the axis of symmetry, as well as a random rotation around its center of mass. An oligomer with the specified cyclic symmetry was then generated using the information stored in the symmetry definition file. Amino acids at the interface were optimized using the Monte-Carlo simulated annealing protocol available in the Rosetta™ Macromolecular Modeling suite. An initial optimization step was executed with a modified score function with a soft repulsive term. Once a sequence was converged upon, designable positions were allowed to minimize side chain torsion angles using the same reduced repulsive term weight. A subsequent round of design and minimization was conducted, but with the standard score function in order to obtain a sequence that corresponds to a local minimum of the energy function. Initially, the extended rotamer library available in Rosetta™ was utilized but in later design rounds it was augmented with the rotamers available in the residue pair motif database. Individual design trajectories were filtered by the following criteria: difference between Rosetta energy of bound (oligomeric) and unbound (monomeric) states less than −20.0 Rosetta energy units, interface surface area greater than 700 Å 2 , Rosetta shape complementarity greater than 0.65, and less than 45 mutations made from the respective native scaffold. Designs that passed these criteria were manually inspected and refined by single point reversions for mutations that were deemed as not contributing to stabilizing the bound state of the interface. The design with the best overall scores for each docked configuration was then added to a set of finalized proteins to be experimentally validated. 
     Size exclusion chromatography. Elution samples for each designed protein were concentrated down using a 10,000 MWCO protein concentrator (Novagen) and fractionated by size on an AKTA pure chromatography system using a Superdex™ 200 10/300 GL gel filtration column (GE Life Sciences) in 25 mM Tris 150 mM Nacl pH 8 (TBS) unless otherwise. Sizing profiles were noted based on absorption at 220 nm and 280 nm wavelength light for each fraction. Molecular weights for predominant species in each protein trace were estimated by comparison to the corresponding monomeric profile. 
     Protein Expression and Purification. 
     Synthetic genes for these designed proteins were optimized for  E. coli  expression and assembled from purchased genes (Genscript) ligated into the pET21-NESG vector at restriction sites NdeI and XhoI. These plasmids were cloned into BL21 (DE3)  E. coli  competent cells. Transformants were inoculated and grown in either LB or TB medium with either 100 mg L −1  carbenicillin or 150 mg L −1  ampicillin at 37° C. until an OD 600  of 0.7. Isopropyl-thio-β-D-galactopyranoside was then added at a concentration of 1 mM to induce protein expression. Expression proceeded for 20 hours at 18° C. until the cell cultures were harvested by centrifugation. Cell pellets were resuspended in TBS and lysed by sonication. Each filtered lysate was then purified by Ni 2+  immobilized metal affinity chromatography with Ni-NTA Superflow™ resin (Qiagen). Resin with bound cell lysate was washed with five column volumes of 25 mM imidazole and five column volumes of 50 mM imidazole. The desired proteins were then eluted with five column volumes of 400 mM imidazole and further purified by size exclusion chromatography. 
     Size Exclusion Chromatography with Multi-Angle Light Scattering. 
     Fractions containing single predominant species from the initial round of size exclusion chromatography were concentrated down with 10,000 MWCO protein concentrators (Novagen) to a concentration of 1.0-2.0 mg mL −1 . 100 uL of each sample was then run through a high-performance liquid chromatography system (Agilent) using (unless otherwise noted) a Superdex 200 10/300 GL gel filtration column (GE Life Sciences) at an elution rate of 0.50 mL min −1  in TBS. These fractionation runs were coupled to a multi-angle light scattering detector (Wyatt) in order to determine the absolute molecular weights for each designed protein. The following equation 1  derived from the Rayleigh-Debye-Gans theory of light scattering 2  was used in the ASTRA software to calculate the molecular weight of the major species present in each sample: 
     
       
         
           
             
               
                 
                   K 
                   * 
                 
                  
                 c 
               
               
                 R 
                  
                 
                   ( 
                   
                     θ 
                     , 
                     c 
                   
                   ) 
                 
               
             
             = 
             
               
                 1 
                 
                   
                     M 
                     w 
                   
                    
                   
                     P 
                      
                     
                       ( 
                       θ 
                       ) 
                     
                   
                 
               
               + 
               
                 2 
                  
                 
                   A 
                   2 
                 
                  
                 c 
               
             
           
         
       
     
     where:
         R(θ,c) is the excess Rayleigh ratio of the solution as a function of scattering angle θ and concentration c. It is directly proportional to the intensity of the scattered light in excess of the light scattered by the pure solvent.   c is the solute concentration.   M w  is the weight-averaged solute molar mass.   A 2  is the second virial coefficient in the virial expansion of the osmotic pressure.   K* is the constant 4π 2 (dn/dc) 2 n 0   2 /N a λ 0   4 .   N a  is Avogadro&#39;s number. This number always appears when concentration is measured in g/mL and molar mass in g/mol.   P(θ) describes the angular dependence of the scattered light, and can be related to the rms radius.   n 0  is the index of refraction of the solvent   λ 0  is the vacuum wavelength of the laser
 
Accounting for error in light scattering data acquisition, species with calculated molecular weights within 13% of the expected target molecular weight for each design were considered to be forming the anticipated oligomeric state.
       

     Small-Angle X-Ray Scattering. 
     Designed proteins that predominantly formed the target oligomeric species were re-expressed and purified for low-resolution structure determination while in solution by small-angle X-ray scattering (SAXS). A purified elution sample and concentrated sample of each protein were sent for data collection at the SIBYLS High Throughput SAXS Advanced Light Source in Berkeley, Calif. A beam exposure time of between 0.5-2.0 seconds was used to obtain diffraction data, which we represent in plots of log intensity (I) vs. q. 
     where:
         q=(4π sin θ)/λ.   2θ is angle of diffraction from detector origin   λ is wavelength of the incident X-ray beam       

     Experimental diffraction data was then analyzed with the java-based application, Scatter. Minimum q values (q min ) and experimental radii of gyration (Rg) were determined by Guinier analysis. Data resolution, reflected by maximum q value (q max ), was determined by a characteristic asymptote in signal intensity described by Porod&#39;s Law. Refined data sets and corresponding designed model .pdb files were input to the FoXS web server to compute the agreement (evaluated as X) between the experimental and model-computed profiles 6 . 
     Generation of Extension Ensemble and Determination of SAXS-Suggested Model. 
     A set of designed homooligomers, one each of C2 and C4 symmetry, that had been structurally validated by X-ray diffraction crystallography and/or SAXS were selected as candidates for extension. Because the repeating units of the initial scaffolds were not perfectly superimposable, each unique repeat unit (aside from N- and C-capping repeats) was propagated to generate several models with two additional repeat units (three for C2 oligomer, two for C4 oligomer). 100 trajectories of a Rosetta™ protocol that previously showed to conformationally sample the local energy landscape was then performed on each extended model. The total extension set was then input to FoXS with an experimentally-obtained profile to determine an ensemble of models that agreed within a threshold to the data. 
     Crystallography, Data Acquisition, Structure Determination and Refinement. 
     Selected designs were expressed as above and purified by IMAC and SEC on a Superdex™ 200 10/300 GL gel filtration column using a buffer containing 25 mM Tris pH 8.0 and 50 mM NaCl. Fractions corresponding to the designed oligomers were combined and concentrated for screening. 
     Crystallization trials for ank3C2_1 were performed at 16.5 mg/ml. The protein crystallized readily in a variety of conditions and optimization was performed using 100 mM Tris pH 8.5, 200 mM magnesium chloride and 30% (v/v) PEG 400. Initial crystallization for 1na0C3_3 trials were performed at 15 mg/ml and produced crystals in 2.4 M sodium malonate pH 7.0 that did not yield a diffraction pattern. Upon concentration crystals that diffracted up to 2.1 Angstroms grew in 2 months. Crystallization trials for ank1C4_2 were performed at 12 mg/ml and pyramidal crystals were observed within 2 weeks in 100 mM sodium acetate pH 4.6 and 2.0 M ammonium sulfate. Diffraction data were collected at Advanced Photon Source at Argonne National Laboratory in Lemont, Ill. Data reduction was carried out using XDS/SCALE™. Molecular Replacement was performed in the program PHASER™ using the design models as search models. Solutions were refined using the program PHENIX™ or BUSTER™. MR solutions were initially subjected to rigid body refinement and subsequently coordinate refinement. Individual atomic displacement parameter (ADP) refinement and automated water picking were also performed. Refinement protocols were run iteratively while the quality of the model was assessed by the R/R-free values. Finally, alternating cycles of refinement and model building in COOT were performed using the using the 2mFo-DFc map to obtain the final coordinates 12 . 
     HR00C3_2 and ank1C2_1 were dialyzed against 25 mM Tris buffer pH 8.0 and 150 mM NaCl. The final concentration of HR00C3_2 and ank1C2_1 used for crystallization trials were 12 mg ml −1 . The HR00C3_2 and ank1C2_1 protein were screened with a Phoenix Robot (Art Robbins Instruments) using the following crystallization screens: Crystal Screen, Natrix, PEG/Ion, Index and PEGRx (Hampton Research, Aliso Viejo, Calif.) and Berkeley Screen (Lawrence Berkeley National Laboratory). Crystals of HR00C3_2 and ank1C2_1 were found in Berkeley Screen conditions. HR00C3_2 was found in condition of 0.3 M Sodium Citrate, 0.1 M Hepes pH 7.5 and 15% PEG 3,350 and ank1C2_1 was found in 0.4 M Sodium Chloride, 0.1 M Tris-HCl pH 8.5 and 30% PEG 3,350. HR00C3_2 and ank1C2_1 crystals were obtained after 4 days by the sitting-drop vapor-diffusion method with the drops consisting of a mixture of 0.2 μl of protein solution and 0.2 μl of reservoir solution. Crystallization trials for ank4C4 were performed with a stock protein concentration of 15 mg/ml with three sample to condition ratios in the following crystallization screens: PEG/Ion, Index (Hampton Research, Aliso Viejo, Calif.), Morpheus (Molecular Dimensions). Hanging-drop optimization was performed with an evenly distributed pH and concentration gradient, and the protein produced crystals within 3 days in a mixture of 1 μl protein solution and 1 μl reservoir solution of 2.1 M DL-Malic Acid pH 7.0. Diffraction data were collected at Advanced Light Source (at Beamline 8.2.1) at Lawrence Berkeley National Laboratory in Berkeley, Calif. Integration, scaling and merging of the X-ray diffraction data were carried out with the HKL2000 package 13 . An analysis of the intensity statistics carried out on HR00C3_2 by Phenix xtriage program indicated that the data was merohedrally twinned with twin law (−h, −k, l) with an estimated twin fraction of 46%. Molecular replacement was carried out using PHASER™ in PHENIX™ suite (using a monomer predicted by Rosetta ab initio structure prediction as the initial search model. Refinement was carried out with phenix.refine, using a twin-based target for HR00C3_2 and a maximum likelihood target for ANK1C2-G3ank1C2_1. Reciprocal space refinement was complemented by rounds of manual model adjustment in COOT™. Root-mean-square deviation differences from ideal geometries for bond lengths, angles and dihedrals were calculated with Phenix™. The overall stereochemical quality of all final models was assessed using the program MOLPROBITYPRO™. 
     REFERENCES 
     
         
         1 Ali, M. H. &amp; Imperiali, B. Protein oligomerization: How and why.  Bioorganic  &amp;  Medicinal Chemistry  13, 5013-5020, doi:10.1016/j.bmc.2005.05.037 (2005). 
         2 Goodsell, D. S. &amp; Olson, A. J. Structural symmetry and protein function.  Annual Review of Biophysics and Biomolecular Structure  29, 105-153, doi:10.1146/annurev.biophys.29.1.105 (2000). 
         3 Nishi, H., Hashimoto, K., Madej, T. &amp; Panchenko, A. R. Evolutionary, Physicochemical, and Functional Mechanisms of Protein Homooligomerization.  Oligomerization in Health and Disease  117, 3-24, doi:10.1016/b978-0-12-386931-9.00001-5 (2013). 
         4 Fletcher, J. M. et al. A Basis Set of de Novo Coiled-Coil Peptide Oligomers for Rational Protein Design and Synthetic Biology.  Acs Synthetic Biology  1, 240-250, doi:10.1021/sb300028q (2012). 
         5 Smock, R. G., Yadid, I., Dym, O., Clarke, J. &amp; Tawfik, D. S. De Novo Evolutionary Emergence of a Symmetrical Protein Is Shaped by Folding Constraints.  Cell  164, 476-486, doi:10.1016/j.cell.2015.12.024 (2016). 
         6 Voet, A. R. D., Nogushi, H., Addy, C., Simoncini, D., Terada, D., Unzai, S., Park, S.-Y., Zhang, K. Y. J. and Tame, J. R. H. Computational Design of a Self-assembling Symmetrical β-propeller protein.  PNAS  111, 15102-15107, doi:10.1073/pnas.1412768111 (2014). 
         7 Stranges, P. B., Machius, M., Miley, M. J., Tripathy, A. &amp; Kuhlman, B. Computational design of a Symmetric Homodimer Using Beta-strand Assembly.  Proceedings of the National Academy of Sciences of the United States of America  108, 20562-20567, doi:10.1073/pnas.1115124108 (2011). 
         8 Der, B. S. et al. Metal-Mediated Affinity and Orientation Specificity in a Computationally Designed Protein Homodimer.  Journal of the American Chemical Society  134, 375-385, doi:10.1021/ja208015j (2012). 
         9 Mou, Y., Huang, P. S., Hsu, F. C., Huang, S. J. &amp; Mayo, S. L. Computational Design and Experimental Verification of a Symmetric Protein Homodimer.  Proceedings of the National Academy of Sciences of the United States of America  112, 10714-10719, doi:10.1073/pnas.1505072112 (2015). 
         10 Huang, P. S. et al. De novo Design of an Ideal TIM-barrel Scaffold.  Protein Science  24, 186-186 (2015). 
         11 Lin, Y.-R. et al. Control Over Overall Shape and Size in de novo Designed Proteins.  Proceedings of the National Academy of Sciences of the United States of America  112, E5478-E5485, doi:10.1073/pnas.1509508112 (2015). 
         12 Koga, N. et al. Principles for Designing Ideal Protein Structures.  Nature  491, 222-227, doi:10.1038/nature11600 (2012). 
         13 Parmeggiani, F. et al. A General Computational Approach for Repeat Protein Design.  Journal of Molecular Biology  427, 563-575, doi:10.1016/j.jmb.2014.11.005 (2015). 
         14 Huang, P.-S. et al. High Thermodynamic Stability of Parametrically Designed Helical Bundles.  Science  346, 481-485, doi:10.1126/science.1257481 (2014). 
         15 Brunette, T. J. et al. Exploring the Repeat Protein Universe through Computational Protein Design.  Nature, doi: 10.1038/nature16162 (2015). 
         16 Main, E. R. G., Xiong, Y., Cocco, M. J., D&#39;Andrea, L. &amp; Regan, L. Design of Stable Alpha-helical arrays from an Idealized TPR Motif.  Structure  11, 497-508, doi:10.1016/s0969-2126(03)00076-5 (2003). 
         17 Pechmann, S., Levy, E. D., Tartaglia, G. G. &amp; Vendruscolo, M. Physicochemical Principles that Regulate the Competition between Functional and Dysfunctional Association of P roteins.  Proceedings of the National Academy of Sciences of the United States of America  106, 10159-10164, doi:10.1073/pnas.0812414106 (2009). 
         18 Levy, E. D. &amp; Teichmann, S. Structural, Evolutionary, and Assembly Principles of Protein Oligomerization.  Oligomerization in Health and Disease  117, 25-51, doi:10.1016/b978-0-12-386931-9.00002-7 (2013). 
         19 Bahadur, R. P., Chakrabarti, P., Rodier, F. &amp; Janin, J. Dissecting Subunit Interfaces in Homodimeric Proteins.  Proteins - Structure Function and Bioinformatics  53, 708-719, doi:10.1002/prot.10461 (2003). 
         20 Bahadur, R. P., Chakrabarti, P., Rodier, F. &amp; Janin, J. A Dissection of Specific and Non-specific Protein—Protein Interfaces.  Journal of Molecular Biology  336, 943-955, doi:10.1016/j.jmb.2003.12.073 (2004). 
         21 Janin, J., Bahadur, R. P. &amp; Chakrabarti, P. Protein-protein Interaction and Quaternary Structure.  Quarterly Reviews of Biophysics  41, 133-180, doi:10.1017/s0033583508004708 (2008). 
         22 Leaver-Fay, A. et al. in  Methods in Enzymology, Vol  487:  Computer Methods, Pt C Methods in Enzymology  545-574 (Elsevier Academic Press Inc, 2011). 
         23 Chen, R. &amp; Weng, Z. P. Docking Unbound Proteins Using Shape Complementarity, Desolvation, and Electrostatics.  Proteins - Structure Function and Genetics  47, 281-294, doi:10.1002/prot.10092 (2002). 
         24 Pierce, B., Tong, W. W. &amp; Weng, Z. P. M-ZDOCK: a Grid-based Approach for C-n Symmetric Multimer Docking.  Bioinformatics  21, 1472-1478, doi:10.1093/bioinformatics/bti229 (2005). 
         25 Schneidman-Duhovny, D., Inbar, Y., Nussinov, R. &amp; Wolfson, H. J. PatchDock and SymmDock: Servers for Rigid and Symmetric docking.  Nucleic Acids Research  33, W363-W367, doi:10.1093/nar/gki481 (2005). 
         26 Gray, J. J. &amp; Baker, D. Protein-protein Docking Predictions with RosettaDock. Biophysical Journal 86, 306A-306A (2004). 
         27 Gray, J. J. et al. Protein-protein Docking with Simultaneous Optimization of Rigid-body Displacement and Side-chain Conformations.  Journal of Molecular Biology  331, 281-299, doi:10.1016/s0022-2836(03)00670-3 (2003). 
         28 Wei, H. et al. Lysozyme-stabilized gold fluorescent cluster: Synthesis and Application as Hg2+ Sensor.  Analyst  135, 1406-1410, doi:10.1039/c0an00046a (2010). 
         29 Lu, M., Dousis, A. D. &amp; Ma, J. OPUS-PSP: An Orientation-dependent Statistical All-atom Potential Derived from Side-chain Packing.  Journal of Molecular Biology  376, 288-301, doi:10.1016/j.jmb.2007.11.033 (2008). 
         30 Zheng, W. H., Schafer, N. P., Davtyan, A., Papoian, G. A. &amp; Wolynes, P. G. Predictive Energy Landscapes for Protein-protein Association.  Proceedings of the National Academy of Sciences of the United States of America  109, 19244-19249, doi:10.1073/pnas.1216215109 (2012). 
         31 DeBartolo, J., Dutta, S., Reich, L. &amp; Keating, A. E. Predictive Bcl-2 Family Binding Models Rooted in Experiment or Structure.  Journal of Molecular Biology  422, 124-144, doi:10.1016/j.jmb.2012.05.022 (2012). 
         32 Fleishman, S. J. et al. RosettaScripts: A Scripting Language Interface to the Rosetta Macromolecular Modeling Suite.  Plos One  6, doi:10.1371/journal.pone.0020161 (2011). 
         33 Schneidman-Duhovny, D., Hammel, M. &amp; Sali, A. FoXS: a Web server for Rapid Computation and Fitting of SAXS Profiles.  Nucleic Acids Research  38, W540-W544, doi:10.1093/nar/gkq461 (2010). 
         34 Boyken, S. E. et al. De novo Design of Protein Homo-oligomers with Modular Hydrogen-Bond Network-mediated Specificity.  Science  352, 680-687, doi:10.1126/science.aad8865 (2016). 
         35 Park, K. et al. Control of Repeat-protein Curvature by Computational Protein Design.  Nature Structural  &amp;  Molecular Biology  22, 167-174, doi:10.1038/nsmb.2938 (2015).