STUDY OF INTERACTION OF THE CYCLIN-DEPENDENT KINASE 5 WITH ITS 

ACTIVATOR, P25 AND WITH THE P25-DERIVED INHIBITOR, CIP 

Antonio Cardone1

, R. Wayne Albers, Ram D. Sriram, Harish C. Pant 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 

National Institutes of Health 

Bethesda, Maryland 20892

ABSTRACT 

A high-affinity inhibitor protein called CIP , which can be produced by small truncations of p35, 

was earlier identified by Amin, Albers, and Pant. P35 is one of the physiological activators of 

cdk5, a member of the cyclin-dependent kinase family. P25 is an activator derived from the 

truncation of the physiological cdk5 activator p35 upon exposure to Aβ peptides, and it is known 

to be associated with the hyperphosphorylation of specific neuronal proteins. This typically 

occurs in the case of neurodegenerative diseases such as Alzheimer’s. In this paper we study in 

silico the binding mechanism of cdk5-p25 and cdk5-CIP complexes more in detail. This provides 

a better understanding of the inhibitory activity of the protein CIP. We use a geometry-based 

technique to verify the following hypothesis: p25’s truncation provides increased flexibility to 

CIP, and hence CIP is able to conform better to cdk5 interface than p25. Therefore CIP is 

expected to bind to cdk5 more easily than p25 and prevent it from reaching its active 

conformation. Our in silico study is based on a geometry-based alignment algorithm. The 

algorithm is capable of efficiently aligning two protein conformations with respect to their 

interfaces, which are represented as point sets. The algorithm is based on biochemical criteria as 

well as geometrical ones. Our results indicate the validity of the flexibility hypothesis. They 

could be used, along with some observations we made on cdk5 activation, as a basis for a set of 

very targeted and therefore more efficient molecular dynamics simulations. 

Keywords: Cyclin-dependent kinase; neurodegenerative diseases; protein phosphorylation; 

computational geometry; feature-based shape similarity assessment; shape signature; point 

alignment algorithm.

1 INTRODUCTION 

 

Cell functions are produced by chains of temporally interacting proteins that form complex 

networks. Major components of such networks are regulatory proteins that switch other 

components “on” or “off”. Very commonly the regulator is a protein phosphokinase that transfers 

phosphate from ATP to a serine, threonine or tyrosine residue of the target protein. More than 

500 protein phosphokinases are specified in the human genome. In their active states these 

1 Corresponding Author 

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proteins all have similar kinase domains, whereas their inactive conformations are quite variable: 

in many cases the active state is only formed after combination with an activator protein (i.e. the 

cyclin-dependent kinase family that regulates cell division). Other phosphokinases normally 

exist combined with an inhibitor protein that must be displaced by a regulatory event such as 

combination with a ‘second messenger’ molecule, such as cyclic AMP in the case of protein 

kinase A [Hus02]. Frequently kinases are subject to multiple regulatory mechanisms involving 

both conformational and covalent modifications [Hus02]. 

Cdk5, a member of the cyclin-dependent kinase family, differs from other members of 

this family in several respects: it has no known role in cell division; its activity is expressed 

primarily in mature (non-dividing) neurons; it is not activated by cyclins, but rather by two 

different physiological activators, p35 and p39, which appear to have evolved independently of 

the cyclins. As the interface of p35 and p39 to the active conformation of cdk5 is very similar to 

those of the other cdk family members, it is reasonable to assume that the kinase is activated by a 

mechanism similar to that proposed for other cyclin-dependent kinases: an active conformation 

of the comparatively flexible kinase molecule is attained upon binding it to a more rigid interface 

provided by activator protein [Jef95]. 

Interest in the above binding mechanism arose from a study designed to determine the 

minimal part of p35 required for cdk5 activation [Ami02]. In the course of this study it was 

observed that the 301 residues of p35 when reduced to 154 (p16 = Δ138-291) will fully activate 

recombinant cdk5. However, further small truncations produce a 126 residue, high-affinity 

inhibitor protein called CIP (CIP = Δ154-279) [Ami02]. Several neurodegenerative diseases are 

associated with hyperphosphorylation of specific neuronal proteins including, in the case of 

Alzheimer’s (AD), the microtubule-associated protein tau which then aggregates within neurons 

to form neurofibrillary tangles. For instance, cdk5 phosphorylates tau in correspondence of ADspecific phospho-epitopes when it associates with p25. p25 is an activator derived from the 

truncation of the physiological cdk5 activator p35 upon exposure to Aβ peptides. Therefore 

hyperphosphorylation and formation of “neurofibrillary tangles” can be initiated by transfecting 

p25 into neuronal cultures whereas co-transfection of CIP can prevent both [Zhe05]. 

Questions about the above-described binding mechanism might be addressed by directly 

simulating cdk5-p25 and cdk5-CIP interactions. However, simulation of protein interactions is a 

computationally challenging task due to the complexity of protein structure and properties. On 

the other hand, advances in areas like computational geometry suggest that the above-described 

cdk5-p25 and cdk5-CIP interactions might be initially addressed by studying their binding 

mechanism. In fact protein interactions need to be studied with reference to both their 

geometrical and their biochemical characteristics. 

This paper is organized as follows. In the next section some of the most popular 

techniques that are available to model and simulate protein interactions are described. As 

geometrical shape characteristics and conformational transitions play a major role in protein 

interactions, dimensional shape similarity assessment techniques will also be covered in the 

literature survey. In Section 3 the problem in the current study is formally defined and the 

working hypotheses are stated. The working hypotheses focus on the geometrical aspects of the 

problem as well as on the protein biochemical characteristics. In Section 4 the geometry-based 

techniques to study cdk5-p25 and cdk5-CIP interactions are described. In Section 5 the results of 

the study are presented and discussed. Finally, in Section 6 the conclusions are presented, along 

with our future plans. 

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2 LITERATURE SURVEY

Advances in bioinformatics in the past two decades have facilitated the structural identification 

of single protein three dimensional structures, obtained through X-ray diffraction and NMR 

(Nuclear Magnetic Resonance) techniques. However, only a fraction of the complexes consisting 

of more than one protein have been experimentally characterized. Therefore a number of 

computational techniques have been developed for characterizing protein structures and for 

simulating their interactions. Protein docking represents one of the most promising 

computational techniques to deal with this problem. 

Protein docking can be defined as the prediction of the structure of a protein complex 

based on the independently solved structures of the components. The main challenge is to 

account for protein flexibility, which greatly increases the complexity of protein docking. Protein 

docking can be viewed as a process involving interaction and binding of two molecules where 

both geometric and chemical characteristics play a major role. Because of its nature docking is 

significantly more complex than an assembly problem of two rigid parts. For example the 

internal flexibility due to internal bond rotations needs to be considered for protein docking. 

Therefore a large number of variables are involved. Protein docking techniques can be classified 

based on the criteria used to predict the resulting protein complex. In this review we will cover 

purely geometric docking techniques and interaction energy-based docking techniques. 

Purely geometric docking techniques [Nor99, Yue90] rely exclusively on protein 

geometric complementarity. Therefore, interaction energy and protein flexibility are accounted 

for indirectly. For instance, soft docking techniques account for protein internal flexibility by 

partially allowing overlapping of the interacting surfaces. Examples are BiGGER [Pal00] and 

FTDOCK [Gab97]. The latter relies on Fourier correlation theory. However, the accuracy of 

purely geometric techniques is not very high, as energetic terms and internal flexibility of 

proteins are not accounted for very accurately. 

Energy-based docking techniques perform an explicit minimization of interaction energy, 

therefore accounting for internal flexibility in detail [Bur03, Kra99, Mor98, Ver03]. Modeling 

and minimizing the interaction energy function can be very complex, especially for large 

proteins. Several energy terms such as electrostatic, Van der Waals, solvation and hydrogen bond 

energies are required for computational accuracy. Each can have a very complex analytical form 

and may explicitly depend on the internal degrees of freedom of the proteins. For large proteins a 

very high number of degrees of freedom might be involved. Several attempts have been made to 

reduce the computational complexity of the problem, e.g., by limiting the flexibility of the 

proteins to only the surface side-chains. Other attempts have ignored the flexibility in certain 

surface areas based on biochemical criteria [Bet99, Con99]. Hence, in general energy-based 

techniques are more accurate than purely geometric ones, but they can be computationally 

intensive. 

Among the energy-based docking techniques, the ICM technique is based on Internal 

Coordinate Mechanics (Figure 1) [Aba94a, Aba94b, Fer02]. It consists of rigid docking followed 

by flexible side-chain refinement based on energy minimization. The refinement is performed 

using a biased Monte Carlo flexible docking, where the internal parameter considered are 

torsion, phase and bond angles and lengths. The ICM Molsoft docking software uses the ICM 

technique. In the annual CAPRI (Critical Assessment of Protein Interactions) docking 

competition [Vaj04], ICM technique was shown to be one of the most accurate protein docking 

 3
techniques. However, its efficiency can become very low if large proteins are involved (e.g. 

proteins with molecular weight higher than 10 kDa). 

An alternative docking technique is based on the Lamarckian model of genetics applied 

to the minimum energy search [Hue07, Mor96, Mor98]. The Lamarckian genetic algorithm has 

been selected by the authors in after comparing its performance, on seven known dockings, to 

the following two search methods: Monte Carlo simulated annealing and a traditional genetic 

algorithm. The Lamarckian genetic algorithm can handle ligands with more degrees of freedom 

than the simulated annealing method and it proved to be the most efficient, reliable, and 

successful of the three algorithms. The docking software AutoDock is based on this technique 

[Hue07, Mor96, Mor98]. 

An additional energy-based docking technique employs a fast algorithm for energy 

minimization, based on an augmented Lagrange-multiplier method. In this case bond lengths, 

angles, and planar groups are constrained, while the potential energy related to torsions, 

hydrogen bonds, Van der Waals and electrostatic interactions is modeled by using elastic 

restraints. Then a local minimum of the total energy that satisfies the constraints is found by 

using the augmented Lagrange-multiplier method. The technique is user-friendly as it is possible 

to define user-specified springs, and its complexity grows linearly with the number of atoms 

involved. However, its performance is very good primarily on small proteins. This technique was 

used to build the software SCULPT, which is capable of modeling protein interactions in 

interactive computer graphics systems [Sur94]. 

Protein conformational changes are very important to protein function. For instance 

kinases are frequently subject to multiple regulatory mechanisms involving both conformational 

and covalent modifications. Protein docking techniques try to address the docking problem by 

accounting for conformational changes. However, accounting for conformational changes is 

computationally expensive because of the high number of variables involved. Therefore it is 

necessary to trade-off between efficiency and accuracy of the results, which is not an easy task. 

An alternative and in some cases more accurate method to study protein conformational changes 

is molecular dynamics (MD). MD techniques explicitly solve the well-known Newton equations 

of motion for all protein atoms and therefore they are able to predict protein conformational 

changes in detail. Most of the techniques that try to efficiently solve the problem are based on 

classical energy terms [And01, Ati01, Cas05, Hum96, Kal99, Lui00, Mou01]. However, in MD 

an acceptable accuracy of trajectories requires significantly short time steps. This makes MD 

simulations computationally expensive, especially for large protein complexes. In order to gain 

insight on protein dynamics without engaging in computationally expensive simulations, hybrid 

methods that couple MD simulations with collective motions derived from normal mode analysis 

(NMA) were developed [Tam00, Tam01]. In some cases the outcome provides insights about 

protein motions that go beyond the range of conventional molecular dynamics simulations. A 

potentially useful insight into the underlying principles governing protein dynamics and its 

relationship to protein functions can be derived from NMA techniques (see [Bah05] for a survey 

of NMA techniques). 

An alternative to MD simulations uses the Lagrangian approach instead of the Newtonian 

approach, which is the primary technique used in the approaches described above. The 

Newtonian approach is fairly computationally expensive. On the other hand, the Lagrangian 

approach provides feasible atom conformations based on sets of constraints. The constraints are 

defined based on the biochemical characteristics of a protein complex. The software 

FIRST/FRODA is mainly based on the Lagrangian approach [Wel05]. It consists of two main 

 4
modules: FIRST and FRODA. The FIRST module performs a rigidity analysis by balancing an 

atom’s degrees of freedom with constraints based on biochemical criteria such as hydrogen 

bonds, covalent bonds and so on. In order to establish final atomic positions based on degrees of 

freedom and constraints, a pebble-game algorithm is used [Wel05]. The outcome is a rigid 

cluster decomposition of the protein (Figure 2). Then FRODA performs the protein dynamics 

simulation based on the rigid cluster decomposition. Interatomic potentials are represented by 

fictitious rigid bodies (referred to as ghost templates) that depict atoms’ ideal positions according 

to the obtained constraints. So atoms are randomly displaced and then their positions are 

iteratively fitted to ghost templates [Wel05]. Due to the nature of the Lagrangian approach, 

FRODA is more efficient than traditional MD methods and it can be fairly accurate depending on 

the biochemical criteria used to define the constraints. However, the interaction energy is not 

explicitly minimized, as in MD simulations, and the outcome is not as accurate as in the MD 

case. 

Protein geometric shape plays a major role in protein interactions. Often proteins with 

similar shapes have similar functions and/or interact similarly with their activators. Therefore, 

the study of proteins and their interactions might involve external shape similarity assessment. 

Over the last few years several algorithms for shape similarity assessment have been developed 

[Car03, Cam01, Iye05, Jay05]. As proteins are three dimensional entities, we will mainly focus 

on three dimensional shape similarity assessment algorithms. This problem can be efficiently 

solved by abstracting three dimensional shapes into shape signatures. Shape signatures are 

abstractions of the actual shape that completely characterize a three dimensional object. A shape 

signature can be represented by a matrix, a set of vectors or a graph. Therefore three dimensional 

similarity assessment involves computing the shape signatures and then comparing the shape 

signatures by a distance function. The distance function should ideally satisfy certain 

mathematical properties, such as identity, positivity and triangle inequality [Car03]. 

Three dimensional shape can be classified based on the shape signature that is used. The 

most common three dimensional shape similarity assessment techniques are based on spatial 

functions, shape histograms, section images, topological graphs and shape statistics [Car03]. In 

particular, feature-based techniques compute the shape signature of an object based on properties 

of the object’s geometric features such as type, size, orientation and number. Once the features 

are extracted and their significant characteristics are determined based on the application, the 

three dimensional shapes are compared by using a distance function. Feature-based techniques 

discriminate the three dimensional models based on the features and their characteristics rather 

than the gross shape of the object. Feature interactions and multiple interpretations still pose 

significant challenges to successful extraction of features [Car04, Car06]. Some of these 

techniques appear to be promising for specific domains such as manufacturing cost estimation, 

while some others can be used as a filter to quickly prune dissimilar objects based on the shape 

similarity assessment criteria [Gup99, Kar05, Ram01]. 

Since feature-based similarity measures are defined using feature-based representations 

of objects, a three dimensional object is represented by a set of feature vectors. A distance 

function will provide the similarity measure between the two sets of feature vectors representing 

two objects. The distance function value depends on the relative positions and orientations of 

features. In particular, in order to compute the distance function, it is necessary to identify, for 

each feature belonging to one three dimensional object, the closest feature belonging to the other 

three dimensional object. The closest feature to a given feature is often referred to in literature as 

closest neighbor. Therefore, the distance value generally changes if a rigid body transformation 

 5
is applied to one of the feature sets. From these considerations it can be inferred that featurebased shape similarity assessment involves finding the alignment between sets of feature vectors 

that minimizes the distance function. Such an alignment can be referred to as optimal alignment. 

This problem is directly related to point pattern alignment problems. 

Considerable work has been done on the point pattern-alignment problem in several 

fields such as computer vision, pattern recognition, and computational geometry [Alt96, Hut90, 

Atk87, Alt88, Spr94, Alt88, Hef94, Mou99]. In the cases of two point sets with different 

cardinalities, the standard Hausdorff distance is used. This is based on the assumption that each 

point in one set has a close match in the other set [Che99, Hut92, Hut93a]. Alternative similarity 

measures have been used when such an assumption might not be valid. For instance the partial 

Hausdorff distance [Hut93b, Hut93c] allows some fraction of the points to be unmatched by 

minimizing the kth largest distance rather than the maximum distance. Other point pattern 

alignment techniques can be found in [Aga03a, Aga03b, Vel01]. In particular geometric hashing 

represents an important class of alignment techniques [Ira96, Wol97]. 

Based on the literature review we believe that the most accurate techniques for modeling 

and simulating protein interactions are computationally intense. On the other hand, based on 

recent advances in computational geometry, we think that it would be more convenient to 

initially study the geometrical characteristics of cdk5-p25 and cdk5-CIP interactions rather than 

directly simulating them without having any insight. In the next section a more formal definition 

of the problem described in the introduction is given, and the working hypotheses are stated. 

 

3 PROBLEM FORMULATION 

One approach for studying our protein-protein binding mechanism would be to use MD 

simulations to compare cdk5-p25 and cdk5-CIP interactions. However, as explained in the 

previous section, molecular dynamic simulations require hours of computer time to simulate 

nanoseconds of real time. Docking algorithms that employ approximations which permit 

mapping possible conformational trajectories independent of time are available [Fer02, Hue07]. 

Although some of these are able to predict the influence of small molecules on protein 

conformations, none appear adequate to predict conformational responses to protein-protein 

interactions. As a first step we have decided to analyze geometric aspects of this problem to 

decide if there are molecular dynamics criteria that may test our hypotheses by involving shorter 

timescale simulations. 

Our work on this project is based on two main hypotheses: 1) the flexibility hypothesis 

and 2) the conservation hypothesis. The flexibility hypothesis is based on the following 

observations. Protein kinases are notable for assuming different conformations in response to 

interactions with different cdk regulatory proteins [Pav99]. In contrast, the cdk activator 

proteins, including cyclins and p25, interface with their kinases by means of a relatively 

compact, rigid and conserved ‘cyclin box’ consisting of 5 α-helices connected by short loops, 

which in the case of cyclin A does not significantly change its conformation upon complexing 

with cdk2 (cyclin A structure can be found in [Bro95]). The N- and C-terminal residues removed 

from p25 to produce CIP are helical in p25 and they contact each other on the surface opposite 

the cdk5/p25 interface. Their removal may relax structural constraints and introduce flexibility 

into CIP relative to p25. We suggest that increased flexibility may allow CIP to retain high 

affinity for the kinase interface while also producing inhibition by disorienting cdk5 substrate 

binding at the kinase active site. Therefore the flexibility hypothesis shows that p25 truncation 

 6
that creates the inhibitory characteristics of CIP does so by increasing its flexibility relative to 

that of p25, such that the cdk5/CIP complex does not align the substrate protein or peptide with 

the catalytic site adequately to produce a significant rate of phosphorylation. This follows 

directly from the experiments described in [Ami02], which show that CIP inhibits cdk5 activity 

with high affinity and competitively with p16 or p25 activation. 

The conservation hypothesis is based on the following observations. P35 is the principal 

physiological activator of cdk5 expressed in neurons and consists of 307 residues. P25 consists 

of p35 residues 99-307 and combines with cdk5 to elicit a higher rate of histone H1

phosphorylation than cdk5/p35. CIP consists of p35 residues 154-279. As this further truncation 

of p25 at both N- and C-termini does not remove any residues that form the normal cdk5-p25

interface (Figure 3), the conservation hypothesis asserts that the high affinity of CIP binding to 

the inhibitory interface with cdk5 consists of some or all of the cdk5-p25 interface residues on 

cdk5. It is important that all the interfacial residues need not be involved at every conformational 

state during the interactions that produce either activation or inhibition. In fact, looking at the 

known active and inactive conformations, it is clear that different parts of the cdk5/p25 interface 

are subject to varying extents of conformational change. However, neither p25 nor CIP

components are expected to change as extensively as the cdk5 component. A relatively small 

part of the interface is likely to interact first, and then, progressively, the rest of the interface will 

interact as cdk5 conforms to more of the available interacting surface. In the case of its 

interaction with CIP, we expect that cdk5 binding does not progress sufficiently to attain an 

active conformation. We refer to this progressive interaction as zipper mechanism. 

We can now consider strategies for testing these hypotheses in more detail. One approach 

would be to obtain a set of geometrically and biochemically feasible cdk5/p25 and cdk5/CIP

intermediate conformations. Given these sets of conformations, the extent to which the two 

interface components, cdk5 and p25 or cdk5 and CIP, remain similar can be compared. This, in 

turn, may be correlated with the conformation of the catalytic site. 

Geometric affinity between protein conformations can be evaluated based on the 

conservation hypothesis: two proteins are geometrically affine if their interface is sufficiently 

close to the active conformation interface. If the resulting geometric affinities verify the 

flexibility hypothesis, then CIP conformations would be expected to adapt geometrically to more 

of the intermediate cdk5 conformations than p25. 

Based on recent advances in computational bioinformatics together with examination of 

the currently available structural data we believe that testing the flexibility hypothesis based on 

the conservation hypothesis is a feasible project from a computational perspective. 

In the next section the problem formulated above is addressed. In particular, in Section 

4.1 the method to obtain a set of geometrically and biochemically feasible cdk5/p25 and 

cdk5/CIP intermediate conformations is described. Then in Section 4.2 the geometry-based 

algorithm to study the obtained protein geometric conformations is described after giving the 

necessary definitions. 

4 METHODS 

In this section our geometry-based technique to study cdk5/p25 and cdk5/CIP intermediate 

conformations is presented. The technique consists of two following steps, which address the 

two tasks that were identified in the previous section. First, different cdk5/p25 and cdk5/CIP

conformations are obtained by using the software FIRST/FRODA. Then, the obtained cdk5/p25

 7
and cdk5/CIP conformations are analyzed by using our geometry-based algorithm. The algorithm 

outputs a distance value for each given protein complex. These distance values are used to 

compare the extent to which cdk5/p25 and cdk5/CIP interfaces remain similar to their active 

conformations. Figure 4 gives an overview of our technique. 

4.1 Obtaining protein conformations 

The first task consists of obtaining sets of geometrically and biochemically feasible cdk5/p25

and p25/CIP conformational trajectories. Given the protein sizes and the large conformational 

transitions involved, we chose to examine the conformational transitions using the 

FIRST/FRODA software. As explained above, FIRST/FRODA employs a constraint-based 

approach that efficiently explores the flexibility of proteins. It is not a purely geometrical 

approach, as the constraints are defined in relation to the conservation of covalent bond lengths 

and angles on existing hydrophobic and hydrogen bonds as well as on the avoidance of steric 

clashes. Therefore the internal flexibility of proteins that is evaluated by FIRST/FRODA 

employs both geometrical and chemical criteria. 

A FIRST/FRODA simulation is typically run by using as starting conformation an initial, 

known protein (or protein complex) and as target conformation the desired final conformation of 

the protein (or complex). Given starting and target conformations, FIRST/FRODA will 

efficiently explore the flexibility of the protein complex and output feasible atom trajectories, if 

any. The obtained atom trajectories are such that the final protein (complex) conformation is 

reached within a user-defined approximation. Obviously there will be cases in which no feasible 

trajectories can be found. 

Initially we tried to use FIRST/FRODA software to examine the behavior of the cdk5/p25

complex directly. In fact our ‘flexibility hypothesis’ implies that p25 is sufficiently rigid to resist 

most of the force of interaction without much distortion by cdk5, or at least more so than CIP. 

Therefore a cdk5/p25 complex could be run through FIRST/FRODA using as target the cdk5/p25

complex itself after replacing the cdk5 active conformation with the inactive one. 

However, a fundamental consequence of the ‘flexibility hypothesis’ is the fact that cdk5

is likely to be activated by progressively combining with p25 through the above-mentioned 

‘zipper’ effect. Unfortunately this phenomenon could not be examined directly by using 

FIRST/FRODA software. In fact we preliminarily verified that if the correct force fields were 

applied between cdk5 and p25 in FIRST/FRODA, then as cdk5 is forced by the program to go 

from the active to inactive direction, cdk5 tends to detach from p25 or from CIP when the 

‘normal’ binding force is overcome by the imposed target force. Successive attempts to force the 

complex to interact resulted in a completely rigid cdk5/p25 interface, which suggested to us that 

FIRST/FRODA is not a viable option for examining cdk5/p25 complex directly. Therefore we 

decided to use FIRST/FRODA software to examine the conformational transitions of each 

protein individually using the procedures described in the following paragraphs. 

The ideal data input for FIRST/FRODA consists of the initial and final coordinates of a 

protein. As mentioned in Section 3, an X-ray structure of the active complex, cdk5/p25, was 

obtained, but the inactive, unconstrained conformation of the cdk5 molecule is yet to be 

synthesized. However, the cyclin-dependent kinases are a well-studied class. In particular cdk2

kinase is a member with a primary sequence very similar to that of cdk5. It has been crystallized 

both as the unliganded, inactive form and also as an active complex, including one with cyclin A, 

Mg, ATP, and a peptide substrate (1qmz_Cdk2 file from [Bro95]). An unconstrained, inactive 

 8
cdk5 conformation can be obtained by homology with the unconstrained cdk2 conformation 

(Figure 5). 

At this stage it was possible to apply the FIRST/FRODA analysis using as the initial 

structure, the p25-constrained active cdk5 conformation and, as the target, a homology structure 

of unliganded cdk5. The outcome of such a FIRST/FRODA analysis is a set of intermediate cdk5

conformations that are geometrically and chemically feasible and are expected to predict the 

spatial trajectory of the inactivating conformational transition. In Figure 6 some of the generated 

conformations are shown. Note that the sampling frequency between the initial and the target 

conformations is a FIRST/FRODA user-defined parameter that determines the level of detail. 

Thus it is possible to estimate intervening conformations by using a pair of cdk5 conformations 

determined by one run as initial and target conformations in a subsequent run of FIRST/FRODA. 

The above-described procedure for obtaining cdk5 conformations cannot be applied to 

p25 or CIP based on currently available data since there are no closely homologous models. An 

alternative way of obtaining accurate p25 and CIP conformations is employed based on the 

conservation hypothesis. The results are shown in Figure 7: on the left the active cdk5/p25

complex is shown, and in particular its interface residues are marked; on the right a cdk5

conformational change is shown with particular focus on an interface amino acid. Based on the 

conservation hypothesis we expect that p25 and CIP interaction with any cdk5 intermediate 

conformation will involve the same interface residues as the active conformation. Therefore, as 

shown in Figure 8, we represent the conformational changes of cdk5 interface residues as 

vectors. Each vector represents a specific cdk5 residue displacement. Then, by using 

FIRST/FRODA, we bias p25 and CIP interfaces to follow cdk5 interface conformational 

changes. As shown in the figure, this is achieved by applying FIRST/FRODA to a given initial 

p25 or CIP conformation by using the above-defined vectors as transformation biases. This way 

new p25 and chemically feasible CIP conformations are obtained in compliance with the 

conservation hypothesis. 

The above-described procedure can be more formally defined as follows: 

For i = 1 to n-1, consider two given cdk5 conformations cdk5i and cdk5i+1 (cdk5o = active 

cdk5 conformation, Cn = inactive cdk5 conformation, n = number of obtained cdk5

conformation) 

1. For each cdk5/p25 interface residue, obtain the set of directional vectors Dv

representing each residue displacement from cdk5i to cdk5i+1. 

2. Apply FIRST/FRODA using the i-th p25 conformation p25i (p25o = active p25

conformation) as input without any target but with the directional vectors Dv as bias. 

The same procedure is used for CIP, and it can be formalized likewise by just using the notation 

CIPi for CIP i-th conformation. 

As expected, based on the above observations p25 and CIP were not able to adjust to all 

cdk5 interface changes from active to inactive conformation. In fact after a few cdk5

conformations the outcome of the above algorithm would not produce a feasible p25 or CIP

structure. As shown in Figure 9, at some point p25 or CIP conformations resulting from the 

application of the above algorithm show unacceptable discontinuities in the backbone. Therefore 

 9
both p25 and CIP are only able to follow cdk5 conformational changes to a certain extent, as 

shown in Figure 10. 

We know that FIRST/FRODA simulation accounts for significant biochemical 

characteristics such as covalent bond lengths and angles, existing hydrophobic and hydrogen 

bonds as well as steric clashes. Therefore several parameters need to be set to determine how the 

above-listed biochemical characteristics will affect the simulation. For a more detailed 

understanding of the parameters meaning and significance, refer to FIRST/FRODA User Guide, 

which can be found at [http://flexweb.asu.edu/software/first/]. Here we note that a trial and error 

procedure was used to fine-tune the parameter set. 

Hence, after applying the above-descried procedures the following conformation sets are 

available: 

• cdk5i with i = 0 to n, where cdk5o = cdk5 active conformation and cdk5n = cdk5 inactive 

conformation. 

• p25i with i = 0 to m, where p25o = p25 conformation from cdk5/p25 active complex and p25m

= p25 last feasible conformation obtained. 

• CIPi with i = 0 to l, where CIPo = CIP conformation from cdk5/p25 active complex and CIPl

= CIP last feasible conformation obtained. 

Observe that n > l > m, as shown in Figure 10. Also note that, for a given i, both p25i and CIPi

conformations are obtained as explained above by referring to the interface conformational 

changes between cdk5i-1 and cdk5i. Therefore any comparison between p25 and CIP

conformations should be performed between p25i or CIPj where i = j. Hence from now on we 

will refer to p25i, CIPj and cdk5k conformations with i = j = k as corresponding conformations.

Based on the obtained sets of geometrically and biochemically feasible conformations, it 

is possible to assess the geometric affinity of p25 and CIP conformations with cdk5. This will 

allow us to verify whether the flexibility hypothesis holds. 

4.2 Studying the geometric affinity of the obtained protein conformations 

In this section the geometric affinity between corresponding conformations of cdk5, p25 and CIP

will be assessed based on the conservation hypothesis. We refer to the geometric affinity of two 

interacting protein conformations as the geometric similarity of their interaction with respect to 

the protein active conformation. The geometric affinity assessment will be used to compare 

cdk5/p25 interaction to cdk5/CIP interaction in order to verify if the flexibility hypothesis holds. 

The above-stated problem of geometric similarity assessment of protein interaction is 

addressed as follows, with the assumption that each protein conformation is considered rigid. 

First we define a unique geometric representation of protein interaction. Then, based on that, we 

define a protein interface shape signature. As mentioned above, shape signatures are defined as 

abstractions of actual shapes that completely characterize a three dimensional object. Then a 

distance function based on protein interface shape signatures is defined. The distance function 

yields the geometric affinity between corresponding conformations of cdk5, p25 and CIP based 

on the conservation hypothesis. Finally we design an efficient algorithm to compute the 

previously defined distance function. 

 10
The next sections will describe the above stages in detail. 

4.2.1 Unique geometric representation of protein interaction 

Protein interaction can be seen as a complex network of single amino acid interactions that are 

arranged in space in such a way as to minimize interaction energy. In general both geometrical 

and biochemical aspects are important for interaction. Therefore protein interaction prediction 

and evaluation are complex tasks, as it is not only necessary to focus on geometric aspects but 

also to evaluate the biochemical characteristics of the amino acids involved. 

In our case based on the conservation hypothesis our task is to evaluate how the interface 

structure varies with respect to that of the reference active conformation as the complex transits 

from the reference active conformation to an inactive state. This simplifies the task, as in 

analyzing protein interaction it is possible to reference the active conformation as a template. 

From this point on we will refer to the cdk5/p25 interaction in this crystallized structure as 

template interaction. 

The crystallized cdk5/p25 complex is shown in Figure 11. A representation for cdk5/p25

complex’s interface information is needed. The criterion on which the representation is based is 

that atoms on amino acid residues from different peptide chains are interacting if their nearest 

approach is within 3.5 Å (see Figure 11). This numeric value is frequently employed for defining 

molecular interaction. By this criterion 22 amino acid pairs are identified. These 22 pairs will be 

our reference in this paper for representing the cdk5/p25 complex’s interface. Based on the 

conservation hypothesis we will evaluate interactions between the different conformations of 

cdk5 and p25 or CIP, providing a measure of how similarly the 22 pairs are interacting relative 

to their template interaction. Hence a unique geometric representation is needed to represent 

such interaction. 

Based on the above observations we choose to represent protein interaction in this case at 

the amino acid level rather than at the protein surface level. In fact the template interaction can 

be exhaustively described by referring to the relative position and orientation of each amino acid 

pair. 

Figure 12 focuses on a pair of interacting amino acids GLU42 and TRP190 from 

cdk5/p25 crystal interface. We choose to represent a single amino acid in R3

 by referring to three 

atoms: α-carbon, backbone nitrogen and the atom that is closest to the interacting amino acid. In 

general we will refer to these as C, N and A, respectively. If the atom closest to the interacting 

amino acid is either the α-carbon, C or the backbone nitrogen, N, the carboxyl group oxygen is 

used instead (for the sake of brevity this case is not shown in the figure). We will refer to it as O. 

Therefore, as shown in the figure, each amino acid is represented by a triangle in R3

. 

In order to uniquely represent the three dimensional position and orientation of an amino 

acid pair we need to define 6 independent parameters. In Figure 13 the chosen parameters are 

shown for the pair consisting of TRP190 from cdk5 and GLU42 from p25. In order to formally 

define these parameters, we introduce the following definitions. 

{ } 22,...,1 , = 21 iii iAAP =∀ (1) 

( )( ) ( ) ( ) jijiji jijijiji ANACANACn jiji

rr r r r r r r r −×−= / f −×− or j = 1, 2 (2) 

 11
π ji = plane identified by , Aji

r N ji

r and Cji

r for j = 1, 2 (3) 

In Equation (1) the set of amino acid pairs that identify the protein interface is defined, in 

Equation (2) the normal vector for amino acid Aji is defined, where Cji

r , and are the 

position vectors of the above-defined C, A and N atoms from amino acid Aji for j = 1,2. 

Aji

r N ji

r

Using equations (1), (2) and (3) it is now possible to formally define 6 parameters that 

uniquely identify the relative position and orientation of an amino acid pair Pi (see Figure 13). 

As it can be inferred from the figure, in order to define the 6 parameters three preliminary rigid 

transformations are applied to the amino acid pair Pi. 

• Transformation T1: amino acid pair Pi is translated so that A1i

r lies at the origin of the 

coordinate system. 

• Transformation T2: amino acid pair Pi is rotated around the origin of the coordinate system 

so that n1i

r is parallel to Z-axis. 

• Transformation T3: amino acid pair Pi is rotated around Z-axis so that C1i lies on X-axis. r

The described transformations do not affect the relative position and orientation of the amino 

acids within pair Pi, as they are applied to both amino acids from the pair. The outcome of 

transformations T1, T2 and T3 is shown in Figure 13 where amino acid A1i is TRP190 and amino 

acid A2i is GLU42; amino acid A2i’s position and orientation has changed according to the 

transformations. With respect to the new positions and orientations that have been obtained both 

for amino acid A1i and for amino acid A2i, the following set of parameters uniquely defines their 

relative orientation. 

• α i angle between n2i

r projection onto coordinate plane XY and X axis 

• β i angle between n2i

r and Z axis 

• i γ angle between π 2i ∩π norm and AC 22 ii

r r − where π 2i is defined in (3) and π norm is the 

plane identified by n1i

r and n2ir

• ) ziyi relative position between A2i ( = xi ,, ddddr r and A1i

r . 

Therefore, for each amino acid pair Pi, the relative position and orientation of amino acid A2i

with respect to amino acid A1i is uniquely defined by a 6 component vector 

( ) ddd ziyixiiii α β γ ,,,,, . Observe that transformations T1, T2 and T3 are applied to the amino 

acids from pair Pi to simplify the parameters definition. However, this does not affect the 

generality of the definition. On the other hand, it is useful to formally define an algorithm to 

align two amino acids A2i and A1i based on a given parameter vector representing their desired 

relative position and orientation. In fact this algorithm will be extensively used later, and it is 

defined as follows. 

 12
Algorithm: ALIGN_RES_PAIR

Input: 

• Amino acid pair Pi randomly positioned and oriented. 

• Vector ( ) ddd ziyixiiii α β γ ,,,,, representing the desired relative position and orientation 

for amino acid pair Pi. 

Output: 

• Amino acid pair Pi positioned and oriented according to vector ( ) ddd ziyixiiii α β γ ,,,,, . 

Steps: 

1. Translate both amino acid A2i and A1i from amino acid pair Pi such that both C1i

r and 

C2i

r are positioned at the origin of coordinate system (not exactly coincident to 

transformation T1 above-mentioned since both C1i

r and C2ir are moved). 

2. Rotate both amino acid A2i and A1i, first around Z-axis and then around X-axis, such 

that n1i

r is aligned to Z-axis (transformation T2 above-mentioned). 

3. Rotate both amino acid A2i and A1i, around Z-axis, such that C1i

r lies on X-axis 

(transformation T3 above-mentioned). 

4. Rotate amino acid A2i, first around Z-axis and then around X-axis, such that n2i

r is 

aligned to Z-axis. 

5. Rotate amino acid A2i around Z-axis such that the angle between AC 22 ii

r r − and X-axis 

(C2i

r and A2ir both lay on XY coordinate plane) is i γ . 

6. Rotate amino acid A2i around X-axis such that the angle between n2i

r and Z-axis (they 

both lay on XZ coordinate plane) is β i . 

7. Rotate amino acid A2i around Z-axis such that the angle between the projection of n2i

r

onto XY coordinate plane and X-axis is α i . 

8. Translate amino acid A2i such that ( ) dddAA ziyixiii ,, 12 =− r r . 

9. Apply to both amino acid A1i and A2i the following transformations in order to restore 

amino acid A1i position and orientation: 

a. reverse of the rotation defined in Step 3; 

b. reverse of the rotation defined in Step 2; 

c. reverse of the translation defined in Step 1. 

The various rotations mentioned in the previous algorithm can be obtained by using 

standard mathematical procedures to minimize a function of one independent variable [Bar94]. 

Figures 14 (a) to 14(h) show an instance of application of steps 1 to 8 of the above-defined 

algorithm. 

The above-defined 6 component vector ( ) ddd ziyixiiii α β γ ,,,,, uniquely defines the 

interaction between a single amino acid pair Pi. In general, referring to all the 22 amino acid 

pairs { } 22,...,1 , , we can define a set of vectors = 21 iii iAAP =∀

 13
=Κ ( ) iddd =∀ 22,...,1 ,,,,, ziyixiiiii α β γ , which uniquely defines the interaction between a given 

pair of proteins. 

In the next section the shape signature for protein interfaces is defined based on their 

unique geometric representation. 

4.2.2 Definition of the shape signature of protein interface 

The shape signature for a three dimensional object can be expressed by a matrix, a set of vectors 

or a graph. Once the object shape signature is defined, it is possible to compare different objects 

by referring to their shape signatures. The geometrical details of the three dimensional shape that 

are included in the shape signature are closely related to the mathematical representation of the 

shape signature itself. Also, the efficiency of the method chosen to compare shape signatures will 

be affected by their mathematical representation. Therefore a shape signature should completely 

and efficiently characterize the three dimensional shape of the object. To efficiently define a 

shape signature it is important to realize that it need not necessarily account for all geometric 

characteristics, but just for the ones that are relevant for object representation. 

A distance function will be used to evaluate the geometric affinity between corresponding 

conformations of cdk5 and p25 or CIP, guided by the conservation hypothesis. Thus, in 

accordance to the conservation hypothesis, the geometric affinity evaluation should be based on 

the interaction taking place in the crystal structure (file 1H4L.pdb from 

http://www.rcsb.org/pdb/home/home.do), which we defined earlier as the template interaction. 

In the previous section we identified 22 amino acid interacting pairs and defined a unique 

geometric representation at the atomic level. Based on this geometric representation, we decided 

to define a protein interface shape signature focusing on the closest atoms from each amino acid 

pair in the template interaction, as defined in the previous section for amino acid pair Pi by Aji

r

with j = 1, 2. 

This choice transforms our protein interface geometric affinity assessment problem into a 

feature-based shape similarity assessment problem, where the features are the atom pairs, 

represented in general as attributed points in R3

. Point attributes are transformation-independent 

characteristics of the atoms and/or amino acid whose location in space is represented by the 

coordinates of atom . Feature-based shape similarity assessment is a problem that has been 

extensively addressed in the literature [Car03, Cam01, Iye05, Jay05]. Later it will be shown that 

our feature-based shape signature has useful mathematical properties. These mathematical 

properties allow us to define an efficient algorithm to compute a distance function that can be 

used to evaluate the geometric affinity between corresponding conformations of cdk5, p25 and 

CIP based on the conservation hypothesis. 

Aji

r

A1i

r

Let us define our shape signatures as follows: consider two interacting proteins, cdk5i and

p25i or CIPi,, and also an amino acid pairs list Pi that is based on the interacting amino acids from 

the template. The shape signature representing protein cdk5i interface is defined as a list of three 

dimensional points for i = 1, 2, ..., 22. Each point represents a particular amino acid from 

the protein conformation cdk5i interface, and any transformation-independent attribute can be 

attached to it. This also holds true for protein conformations, p25i or CIPi: their shape signature 

is defined as a list of three dimensional points A2i

r for i = 1, 2, ..., 22, and any transformation-

 14
independent attribute can be attached. These definitions reduce our problem to a three 

dimensional feature-based shape similarity assessment. 

4.2.3 Definition of the distance function to evaluate protein geometric affinity 

Our task is to evaluate the geometric affinity at the interface between a given protein pair based 

on the conservation hypothesis. As discussed above, the conservation hypothesis implies that the 

inhibitory interactions of CIP with cdk5 mostly involve the same interfacial residue pairs as 

those that activate the cdk5 kinase as a result of p25 binding. We defined as the interactions of 

the 22 amino acid pairs at the interface of the crystallized complex of cdk5 activated by p25 the 

template interaction. Thus evaluating the geometric affinity between a given protein pair cdk5i

and p25i or CIPi involves assessing how close the interface between cdk5i and p25i or CIPi is to 

the template interface, cdk5o/p25o. To accomplish this, we need the following definitions. 

( ) 22,...,1 ,,,,, 0000000 =Κ γβα ziyixiiiii iddd =∀ (4) 

The vectors defined in Equation (4) are the unique geometric representations of the relative 

positions and orientations of the 22 amino acid pairs from the interface between cdk5 active 

conformation, cdk5o, and p25 active conformation, p25o. 

0 Pi

( ) A1ii

0 Κ= , 0

i

0

i ΤΤ (5) 

Equation (5) defines the transformation needed to align amino acid A2i with respect to amino 

acid A1i such that their relative positions and orientation are the same as in the template 

interaction. The transformation defined in Equation (5) is obtained from the transformations 

described in the algorithm ALIGN_RES_PAIR. In particular, using Equation (5) and referring to our 

shape signature, the following definition follows. 

iii

t A i AA 21

0

2 ),( r r Κ= 0 Τi (6) 

For a given p25 or CIP conformation, t A2i

r is the atom A2ir location such that atom would be 

in the same relative position and orientation as in the active conformation with respect to atom 

 from amino acid pair Pi. From now on we will refer to 

A2i

r

A1i

r t A2ir as ideal location of atom A2ir

with respect to atom . The expression ideal location is used because if, for two given 

corresponding protein conformations cdk5i and p25i, 

A1i

r

t A2i A2i

r r = for i = 1, 2,..., 22, then cdk5 and 

p25 would interact exactly as they were interacting in the crystallized form. Based on the 

conservation hypothesis, that would mean that the interacting conformations, cdk5i and p25i, can 

interact with the highest possible geometric affinity. 

From equations (4), (5) and (6) and from what was just stated we can infer that a 

technique to evaluate the geometric affinity between two given corresponding protein 

conformations is to compare how near the interface atoms A2i

r from p25 conformation p25i (or 

CIP conformation CIPi) are to their ideal locations, t A2i

r . In other words, it is necessary to 

 15
evaluate the shape similarity between two three dimensional protein interfaces, namely between 

the ideal interface and the actual interface. This is a three dimensional feature-based shape 

similarity assessment problem, with the protein interface represented by sets of feature points. 

Based on the shape signature selected we will provide a general definition of the distance 

function to evaluate the geometric affinity between a given protein pair interface. 

Observe that three dimensional objects in general and proteins in particular are 

represented in different coordinate systems. Therefore in order to assess similarity between two 

protein interfaces they need to be aligned such that the distance between the two corresponding 

sets of points is minimized. The aligning transformation that minimizes the distance between two 

sets of points is referred to as optimal feature alignment. Observe also that the point sets 

representing protein interfaces in general carry parameters other than their coordinates. In our 

case, point coordinates represent atom positions in the space, while other parameters could 

represent significant chemical characteristics such as atom type and residue type to which the 

atom belongs. We will refer to points carrying parameters other than their coordinates as 

attributed points. 

Most of the available point alignment techniques mentioned in the literature survey 

involve identifying, for each point belonging to one set, the closest point from the other set in 

terms of the chosen distance measure (e.g. Euclidean distance). We will refer, given a point p

from one set, to the closest point to p from the other set as closest neighbor. Therefore in general 

the distance function to compare two attributed point sets is defined as follows. Consider an 

attributed point p in R3

 represented by three transformation-dependent coordinates xp, yp and zp

and a transformation-invariant attribute wp. For the sake of simplicity we define a single 

transformation-invariant attribute, which can be obtained as the combination of any number of 

transformation-invariant attributes without affecting the generality of the problem. The point sets 

A and B in R3

 can be compared using the following general distance function. 

n

qpd

BAD

n

i

i ∑ Bq = ∈

= 1

),(min

),(

Τ

Τ (7) 

In Equation (7) T is the optimal alignment mentioned above and d(Tpi,q) could be defined as 

follows. 

( )( ) ( ) ( ) 2 2 2 2 ),( qp qp qp −+−+−+−= wwzzyyxxqpd qp (8) 

Properties such as positivity, identity, symmetry and triangle inequality may or may not be 

satisfied for a given distance function. In this case positivity and identity properties are satisfied. 

The distance function defined in Equation (7) does not satisfy triangle inequality because it 

consists of a summation of quadratic terms. However, it is easy to differentiate, which is a highly 

desirable property. 

The distance function D(TA,B) defined in Equation (7) changes with the aligning 

transformation T. Hence, as mentioned above, it is necessary to evaluate the optimal alignment T

that minimizes the distance function. This is, in general, a hard problem, as 6 degrees of freedom 

(DOFs) are involved in aligning two sets of points that are rigidly moving in R3

. Furthermore, as 

can be noticed by Equation (7), in order to evaluate D(TA,B) it is necessary to know the closest 

 16
point q from set B for each point pi from set A based on distance function d(Tpi,q). This makes 

the problem even harder. In fact standard minimization techniques cannot be used as, in general, 

the closest point q from set B to a given point pi from set A changes as transformation T is 

applied to set A [Car06]. In particular, among the feasible values of transformation T there will 

be some specific values in correspondence of which the closest point q from set B to a given 

point pi from set A will change [Car06]. For a 2 DOF instance that is simpler to visualize, it can 

be seen that the transformation space for a given point pi of set A is divided into a set of 

subspaces such that its closest neighbor does not change within them (see Figure 15). Therefore 

D(TA,B) should be minimized within each subspace, and then the minimum value over all 

subspaces should be computed [Car06]. In general this is a computationally complex problem to 

address. However, in our case the conservation hypothesis significantly simplifies the problem 

from a computational perspective. In fact the distance function that is needed to evaluate the 

geometric affinity between protein interfaces is slightly different from the one defined in 

Equation (7). Our distance function is formally defined in the following paragraphs. 

Let us consider the following protein conformations: cdk5 k-th conformation cdk5k, p25

k-th conformation p25k and CIP k-th conformation CIPk. For the k-th conformations, equations 

(1), (2) and (3) can be written as follows. 

{ } 22,...,1, 21 iAAP =∀= k

i

k

i

k

i (9) 

( )( ) ( ) ( ) k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji ANACANACn rr r r r r r r r −×−= / f −×− or j = 1, 2 (10) 

In equations (9) and (10) the set of amino acid pairs representing the interface and the normal 

vector for amino acid are defined, where k A1i

k Cji

r , k Ajir and k N ji

r are the position vectors of the 

above-defined C, A and N atoms for amino acid for j = 1,2. Other definitions referring to 

amino acid pairs can be changed accordingly. In particular the definitions from equations (5) and 

(6) can be modified as follows for the k-th conformations. 

k Aji

( ) k A1ii

0 Κ= , 0k

i

0k

i ΤΤ (11) 

k

i

k

ii

tk A i AA 21

0

2 ),( r r Κ= 0k Τi (12) 

Finally, let us define the amino acid sets representing the shape signature of actual and ideal 

interface as { } 22,...,1 2 iAA =∀= k

i

k r and { } 22,...,1 2 iAB =∀= tk

i

k r . Now it is possible to rewrite the 

distance function defined in Equation (7) as follows. 

22

),(

),(

22

1

∑ 22

= = i

tk

i

k

i

kkk

AAd

BAD

rr Τ

Τ (13) 

In Equation (13), ),( is defined as follows. 22

tk

i

k

i AAd

rr Τ

 17
( )( ) ( ) 2 2 2 ),( ba ba ba −+−+−= zzyyxxqpd (14) 

where and . Please note that the distance function defined in 

Equation (13) differs in two ways with respect to the general one defined in Equation (7). One 

difference is that as we are using the conservation hypothesis the distance function does not 

require identifying the closest neighbors anymore. In fact our task is to align as closely as 

possible the actual protein interface to the ideal one. Therefore the amino acid pairs that interact 

(i.e. that are closer to each other) are already known. This makes the distance function easier to 

compute. Furthermore, referring to Equation (14), another difference is that point attributes are 

not explicitly accounted for. This is because the biochemical characteristics of the interface 

amino acids are incorporated in the conservation hypothesis, as both the unique geometric 

representation and the ideal interface are based on amino acid biochemical properties. Therefore 

point attributes, which represent atom biochemical characteristics, are implicitly accounted for in 

the very definition of point sets. 

),,( 2 aaa

k

i = zyxAr

Τ ),,( 2 bbb

tk

i = zyxAr

At this stage the distance function defined in Equation (13) can be 

computed after finding the alignment transformation T. The next section will describe an 

algorithm to do this. 

,( ) kkk Τ BAD

4.2.4 Alignment algorithm to evaluate distance function yielding protein geometric affinity 

In this section an alignment algorithm to compute the distance function defined in Equation (13) 

is presented. The resulting distance value will yield the geometric affinity between cdk5 and p25

or CIP k-th conformation. Observe that, based on the definitions and related observations in the 

previous section, this is a computationally feasible task because of the characteristics of the 

biochemical problem we are addressing. In fact we intend to evaluate the geometric affinity 

between pairs of corresponding protein conformations based on the conservation hypothesis. 

This means that we need to compare the previously defined actual and ideal interfaces of p25 or 

CIP k-th conformation. Therefore the mathematical form of the distance function and the related 

alignment problem are significantly simplified. 

We mentioned that aligning two sets of attributed applied vectors in R3

 is a six degree of 

freedom problem. However, the protein interface characteristics can be used to discriminate 

among the possible alignments and therefore to decrease the computational complexity of the 

problem. Based on the conservation hypothesis and on the zipper-mechanism described in 

Section 3, we expect at least one interface amino acid pair to interact very similarly to the 

template interaction. As our approach involves the comparison between the actual and ideal 

interface of the same protein (either p25 or CIP), the optimal alignment should have at least one 

amino acid pair from the actual interface very closely positioned with respect to the 

corresponding pair from the ideal interface. By applying the described preliminary alignment to 

a given amino acid pair five degrees of freedom are constrained. After the alignment the only 

degree of freedom left is a rotation around the axis identified by the aligned amino acid pair 

(refer to Figure 16). As it will be explained more in detail in the algorithm steps described below, 

each amino acid pair is aligned in two different ways, each corresponding to one amino acid 

from the pair. Therefore the total number of alignments in the case of 22 pairs is equal to the 

permutations P(22,2) = 462. 

 18
In order to evaluate the distance function defined in Equation (13), it is necessary to 

perform P(22,2) = 462 preliminary amino acid pair alignments and for each of them solve the 

corresponding one degree of freedom alignment problem. This is a computationally feasible 

problem because we are able to analytically minimize the distance function from Equation (13). 

The minimization is performed with respect to the one degree of freedom rotation shown in 

Figure 16. As observed above, both the distance function defined in Equation (7) and the one 

defined in Equation (14) do not satisfy the triangle inequality because they consist of a 

summation of quadratic terms. However the derivative is such that the summation of quadratic 

terms can be analytically minimized with respect to the one degree of freedom rotation. The 

algorithm that is used to compute the distance function defined in Equation (13) is presented as 

follows. 

Algorithm: COMPUTEKTHDISTANCEFUNCTION

Input: 

• { } [ ] ( ) [ 22,1 ,, ] = AA 2 ∈ 22,1 = izyx ∈∀ k

i

k

i

k

i i

k

i

k r , which are the actual atom locations from 

corresponding amino acid pairs Pi defined above for k-th protein conformations. 

• { } [ ] ( ) [ 22,1 ,, ] = AB 2 ∈ 22,1 = izyx ∈∀ tk

i

tk

i

tk

i i

tk

i

k r set of the ideal atom locations from 

corresponding amino acid pairs Pi defined above for k-th protein conformations. 

Output: 

• Value k Dmin of the distance function , ) defined in Equation (13) 

corresponding to aligning transformation Tk

 for k-th protein conformations. 

(

k kk BAD k Τ

Steps: 

1. For i = 1 to 22 

For j = 1 to 22, j≠i

i. Apply the translation Tri to set Ak

 such that tk

i

k A i A22

r r = . 

ii. If tk

j

tk

i AA 22

rr ≠

a. Apply the rotation Rij around pivot point tk

i

k A i A22

r r = to set Ak

 such that 

k A2 j

r is located as close as possible to tk A2 jr , which means that k A2 jr lies 

on the line through tk

i

k A i A22

r r = and tk A2 j

r (see Figure 17). 

b. Define the rotation axis r as the line passing through atom locations 

k A2i

r and k A2 jr . 

Else 

c. Define the rotation axis r based on the locations of the atoms k A1i

r and 

k A1 j

r above defined. 

d. Apply the rotation Rij around pivot point tk

i

k A i A22

r r = to set Ak

 such that 

k A2 j

r lies on the line through tk

i

k A i A22

r r = and parallel to rotation axis r 

(see Figure 18). 

 19
iii. Compute the rotation θij and corresponding rotation matrix ) ( R θij around 

axis r defined in Step 1.ii that minimizes the distance function 

),) k (( (see Figure 19). k D iji RRTr θ BA k

iv. Compute the minimum distance )(( ), . kk

ij

kk

ij = DD iji RRTr θ BA

2. Identify the minimum distance value over all alignments from Step 1: 

k

ji

k

ij ij

k Dmin min == DD . The corresponding aligning transformation is 

)( ji

k T RRTr θ jii = . 

Observe that in Step 1.ii of the algorithm COMPUTEKTHDISTANCEFUNCTION the cases in which 

 are specifically addressed. Such cases are possible because both and are ideal 

(i.e. not actual but theoretical) atom locations, and therefore they might be coincident. Hence no 

rotation axis r could be defined based on them. This is the reason why steps 1.ii.c and 1.ii.d 

define the rotation axis r based on atoms 

tk

j

tk

i AA 22

rr = tk A2ir tk A2 jr

k A1i

r and k A1 jr from amino acids A1i and A1j. Amino acids 

A1i and A1j can be used because they are distinct amino acids belonging to cdk5 real interface. 

Therefore we are guaranteed that k A1 j

k A1i

r r , otherwise two distinct atoms would hold the same 

spatial position, which is physically impossible. 

≠

The rotation θij around axis r that minimizes the distance function 

 from Step 1.iii of algorithm COMPUTEKTHDISTANCEFUNCTION is 

formally defined in the next paragraphs. As shown in Figure 19 we can assume, without loss of 

generality, that axis r has been previously aligned to Z-axis. 

),)(( k kk D iji RRTr θ BA

The quantities and in Figure 19 are defined as follows. k

g d g θ0

⎪⎩

⎪

⎨

⎧

=

=

axis-X and XY plane coordinate onto of projection ebetween th angle initial 

constant correspond axisrotation from point of distance axis- Z toing

0 2

2

k

g

g

k

g

k

g

A

d A r

r

θ 

(15) 

Now, the distance function from Equation (13) can be written as follows. 

22

),)((

),)((

22

1

∑ 22

= = g

tk

g

k

g

k kk

AAd

BAD

rr θ

θ

R

R (16) 

In equation (16) θ represents the one degree of freedom rotation around the rotation axis r

corresponding to Z-axis. Equation (14) also gets modified as follows, based on the definitions 

from Equations (15). 

( ) ( )( ) 2 2 2

22 )cos(),)(( )sin( tk

g

k

g

tk

g

g

o

k

g

tk

g

i

o

k

g

tk

g

k

g θ = dAAd θθ +−+ dx θθ −+−+ zzy rr R (17) 

 20
In Equation (17) ),,( are the coordinates of tk

g

tk

g

tk

g zyx tk A2g

r . In order to find the minimum value of 

the distance function from Equation (16) the following condition needs to hold. 

0 ),)(( = ∂

∂

θ

θ k kk R BAD (18) 

The values of the rotation θ that satisfies Equation (18) is computed as follows. 

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎝

⎛

=

∑

∑

=

− =

22

1

22

1 1 tan

g

k

g

k

g

g

k

g

k

g

Md

Nd

θ (19) 

In Equation (19) the quantities and are defined as follows. k Ng

k M g

⎪⎩

⎪

⎨

⎧

= −

= +

)cos()sin(

)sin()cos(

g

o

tk

g

g

o

tk

g

k

g

g

o

tk

g

g

o

tk

g

k

g

xM y

xN y

θ θ

θ θ

 (20) 

A given θ value resulting from Eq. (19) represents a local or global minimum only if the 

corresponding second derivative value from Equation (18) is positive. Therefore we select the θ

value that minimizes the distance function from Equation (16) based on the sign of the second 

derivative. 

Equation (19) yields the value θ corresponding to the best possible aligning rotation 

between the two amino acid sets Ak

 and Bk defined above. The aligning rotation is performed 

around a fixed axis through two amino acids k A2i

r and k A2 j

r from set Ak

. The two amino acids k A2i

r

and defining the fixed rotation axis are aligned to their respective ideal locations. Therefore, 

for each amino acid alignment pair, it is possible to analytically obtain the optimal aligning 

rotation. Hence, as observed above, the algorithm COMPUTEKTHDISTANCEFUNCTION performs all 

possible amino acid pair alignments and for each of them optimally solves the corresponding one 

degree of freedom alignment problem. Therefore the resulting minimum distance value 

k A2 j

r

k

ji

ji

k

ij

k D = = DD,

min min , whose corresponding aligning transformations is )( ji

k T RRTr θ jii = , gives 

an accurate measure of the geometric similarity between the two feature sets Ak

 and Bk . Feature 

set Ak

 represents the amino acids belonging to the actual interface of a given protein 

conformation. Feature set Bk

 represents the ideal position, based on the conservation hypothesis, 

of the amino acids belonging to the interface of a given protein conformation. 

The algorithm COMPUTEKTHDISTANCEFUNCTION for computing the distance function 

defined in Equation (13) is presented in this section. The resulting distance value gives an 

accurate measure of the geometric affinity between the interfaces of corresponding protein 

conformations based on their known active conformations. The algorithm is based on a discrete 

representation of protein interfaces as attributed point sets, where each point represents an amino 

 21
acid belonging to the protein interface. This allows for the characteristics of protein interface at 

amino acid and atomic level to be considered. Furthermore, by using the known active protein 

conformations as reference, the point alignment problem corresponding to the geometric affinity 

assessment is significantly simplified. This simplification is due to the use of previously studied 

amino acid interactions from the active protein complex. This increases the efficiency of the 

feature point set alignment algorithm, maintaining a good accuracy level. 

In the next section the algorithm COMPUTEKTHDISTANCEFUNCTION will be used to 

compare the geometric affinity of the proteins p25 and CIP to the cyclin-dependent kinase cdk5. 

In particular, Section 5.1 provides details about how our algorithm is applied to the available 

protein conformations. Then in Section 5.2 the results are presented and finally in Section 5.3 

they are discussed. 

5 RESULTS AND DISCUSSION 

5.1 Description of algorithm application 

A software system has been implemented based on the above-described 

COMPUTEKTHDISTANCEFUNCTION algorithm in Matlab on a Windows platform. As described in 

Section 4.1 cdk5, p25 and CIP conformations have been obtained using the software 

FIRST/FRODA. Cdk5 conformations have been obtained by using the crystallized cdk5 active 

conformation as initial conformation and the cdk2-like homology inactive conformation as target 

conformation. FIRST/FRODA sampling frequency has been set to obtain a total of n ≈ 14300 

conformations. Then p25 and CIP conformations have been obtained by biasing their interface 

amino acids to follow cdk5 conformational changes in compliance with the conservation 

hypothesis by using FIRST/FRODA. Among the obtained p25 and CIP conformations 

corresponding to the n ≈ 14300 cdk5 conformations, only about 3000 conformations, starting 

from the crystallized one, are geometrically and chemically feasible. In fact the majority of the 

obtained conformations show breaks in the backbone. This means that p25 or CIP were not able 

to follow cdk5 conformational changes all the way to conformation of unliganded cdk5, in 

compliance with the conservation hypothesis. This was not surprising as cdk5 is expected to be 

more flexible than p25 and CIP. So finally only l CIP conformations and m p25 conformations 

are considered, with l > m and both l and m ≈ 3000. 

Therefore, referring to these available data, the following inputs are given to the system. 

• Files in pdb format [www.rcsb.org/pdb] representing the p25 feasible conformations obtained 

as described in Section 4.1. The p25 conformations were defined as p25i for each i = 0 to m, 

where p25o is the active conformation and p25m is last feasible conformation available (refer 

to Section 4.1 for details). 

• Files in pdb format representing some of the feasible CIP conformations obtained as 

described in Section 4.1. The CIP conformations were defined as CIPi for each i = 0 to l, 

where CIPo is the active conformation and CIPl is the last feasible conformation available. 

As l > m, we will focus only on the conformations CIPi for each i = 1 to m (refer to Section 

4.1 for details). 

 22
• Files in pdb format representing some of the cdk5 conformations obtained as described in 

Section 4.1. The cdk5 conformations were defined as cdk5i for each i = 0 to n, where cdk5o is 

the active conformation and cdk5n is the inactive conformation. As n > m, we will focus only 

on the conformations cdk5i for each i = 1 to m (refer to Section 4.1 for details). 

• List of 22 amino acid pairs from active cdk5/p25 complex interface. 

• Vectors ( ) 22,...,1 ,,,,, 0000000 =Κ γβα ziyixiiiii iddd =∀ representing the relative position and 

orientation of the 22 interface amino acid pairs 0 Pi between cdk5 active conformation cdk5o

and the corresponding p25/CIP conformation p25o/CIPo. 

The alignment is performed using the COMPUTEKTHDISTANCEFUNCTION algorithm and it yields 

a measure of the geometric affinity between corresponding cdk5, p25 and CIP conformations 

(cdk5i, p25i and CIPi for the same i). In Figure 20 a schematic view of the software system that 

has been implemented is shown. Then, in Figure 21 an example of the application of 

COMPUTEKTHDISTANCEFUNCTION algorithm to a pair of protein conformations is also shown. 

The geometric affinity measure is based on the distance function described in Section 4.2.3. 

Therefore p25 and CIP corresponding conformations can be compared based on this distance 

function: smaller distance values correspond to higher geometric affinities. 

Our software data structure is consistent with pdb file format, which is one of the most 

commonly used file formats for representing proteins [www.rcsb.org/pdb]. It is also compatible 

with virtually any protein representation and modification software, such as SwissPro, VMD, 

NAMD. 

COMPUTEKTHDISTANCEFUNCTION algorithm is applied to selected sets of corresponding 

p25, CIP and cdk5 conformations to evaluate their geometric affinity based on the conservation 

hypothesis. For instance, two given p25 and cdk5 conformations are geometrically affine if their 

interface is very close to the active conformation interface. Our algorithm application will 

determine if the geometric affinities verify the flexibility hypothesis, i.e. if cdk5/CIP

conformations remain geometrically affine over a greater range of conformations than cdk5/p25. 

Before applying the COMPUTEKTHDISTANCEFUNCTION algorithm to a given protein pair, note 

that the distance function defined in Equation (13) is significant only if sets Ak

and Bk

 actually represent protein interface amino acids. In some cases, two protein 

conformations could be such that the two proteins do not interact anymore and, hence, no protein 

interface is defined. A geometric criterion is needed to identify such cases. 

,( ) k kk BAD k Τ

Observe that sets Ak

 and Bk are defined based on the interface of the active cdk5/p25

complex. Hence, they actually represent protein interface amino acids for the active 

conformation. Intuitively, Ak

 and Bk are very likely to represent actual protein interface amino 

acids for protein conformations other than the active one if such conformations are very close 

geometrically to cdk5/p25 and cdk5/CIP active conformations. However, if larger protein 

conformational changes occur with respect to cdk5/p25 and cdk5/CIP active conformations, then 

the protein pairs might not be able to interact anymore. One way to identify such cases is to 

measure the displacement of the template interface amino acids for a given conformation: if the 

displacement values are significant then the amino acids may not actually represent a protein 

interface. At such conformations the proteins are expected to dissociate, based on the 

 23
conservation hypothesis (see Figure 22). Also the distance function defined in 

Equation (13) will provide a measure of the geometric affinity with no actual biochemical 

meaning, as the given protein pair is not expected to interact. Therefore, this is a useful 

geometric criterion for discarding geometric affinity values when the interface conformations are 

such that no interaction can be expected. 

,( ) k kk BAD k Τ

For a formal definition of the above described geometric criterion, recall that the distance 

function defined in Equation (13) consists of the summation of second powers of 

the distances between 22 interface amino acid pairs and their corresponding ideal locations based 

on conservation hypothesis. Also the interface amino acids are identified as the ones having 

distance less than 3.5 Å from the other protein. Therefore, a threshold for the maximum allowed 

mean displacement of interface amino acids from their ideal position can be used as a 

displacement criterion (refer to Figure 22). It can be shown by simple mathematical formulae 

that the distance function value obtained from Equation (13) has the following relationship with 

the displacement threshold. 

,( ) k kk BAD k Τ

2 npD ∗< ζ k (21) 

where Dk is the distance value obtained from the distance function defined in 

Equation (13) for the k-th protein conformation pair, np is the number of interface amino acid 

pairs (22 in our case), and 

,( ) k kk BAD k Τ

ζ is the chosen threshold value in terms of the mean displacement of 

interface amino acids from their ideal location. Inequality (21) represents the distance value 

above which the mean displacement of interface amino acids from their ideal position is so large 

that no significant protein interaction should occur, based on the conservation hypothesis. In this 

case the geometric affinity value has no meaning and can be discarded. For instance, setting a 

threshold of ζ = 1 Å, from Equation (21) we would have a corresponding distance value of 22. 

Similarly, a threshold of ζ = 5.1 corresponds to a distance value of 88. Considering that we used 

the 3.5 Å threshold to identify protein surfaces, a threshold of ζ ≈ .1 8 Å which corresponds to a 

distance value of ≈ 71 seems reasonable. 

5.2 Results 

The results of the comparison between the geometric affinities to cdk5 of p25 and CIP are 

presented in this section (refer to figures 23 to 29). Figure 23 shows the distance values from 

Equation (13), resulting from the application of the COMPUTEKTHDISTANCEFUNCTION algorithm 

to the cdk5/p25 and cdk5/CIP conformations. The following p25 and CIP conformations were 

investigated: p25i and CIPi with i = 0, 500, 1000, 1500, 2000, 2500, 3000. Note that the distance 

values from Equation (13) for both p25 and CIP go beyond the threshold value of ≈ 71 defined 

above. This means that the mean displacement of interface amino acids from their ideal location 

is greater than ζ ≈ 8.1 . Based on the conservation hypothesis, the interface conformational 

changes are such that significant interactions should not be expected between cdk5 and p25 or 

CIP beyond. So in these cases the geometric affinity values for protein interfaces can be 

discarded. This is not surprising as both p25 and CIP are expected to be significantly less 

flexible than cdk5 and hence not to be able to follow cdk5 conformational changes very far from 

 24
their active conformations. Figure 24 plots the distance values from Equation (13) vs. p25 and 

CIP corresponding conformations on an expanded scale. The following p25 and CIP

conformations have been investigated: p25i and CIPi with i = 0, 50, 100, 150, 200, 250, 300, 350, 

400, 450, 500. Even in this case the distance values from Equation (13) for both p25 and CIP go 

beyond the threshold value above defined of ≈ 71. Only p25i and CIPi with i < 50 identify p25

and CIP conformations that are expected to interact substantially with cdk5. For those 

conformations our technique will be able compare p25 and CIP geometric affinity to cdk5. 

Figure 25 is a plot of distance values from Equation (13) computed for several p25 and 

CIP conformations. In this case the following p25 and CIP conformations have been 

investigated: p25i and CIPi with i = 0, 10, 20, 30, 40, 50. The distance values from Equation (13) 

for both p25 and CIP are higher than the threshold value in correspondence of the 40th

conformations p2540 and CIP40. Therefore we will focus on conformations between the active 

one and the 40th. As it can be noticed from Figure 25, there is a full p25 and CIP conformation 

range between 20th conformation and 40th one such that p25 distance values from Equation (13) 

are higher than CIP ones. This means that CIP geometric affinity is higher than p25 for those 

conformations. Furthermore 10th to 20th conformation range values need to be checked. In fact in 

correspondence of 10th conformation p25 distance value from Equation (13) is lower than CIP, 

but the opposite occurs in correspondence of 20th conformation. Hence, more detailed results 

about the range between active and 40th conformation are needed. 

In Figure 26 the distance values from Equation (13) vs. p25 and CIP corresponding 

conformations are shown for conformations p25i and CIPi with i = 10 to 20. In the whole range 

except for the 10th conformation CIP distance values are clearly lower than for p25. This means 

that CIP geometric affinity is higher than p25 one for those conformations. Same holds for 

Figure 27, where distance values from Equation (13) vs. p25 and CIP corresponding 

conformations are shown for conformations p25i and CIPi with i = 20 to 30. In this case CIP

distance values are lower than for p25 for the whole range. Finally, in Figure 28 distance values 

from Equation (13) vs. p25 and CIP corresponding conformations are shown for conformations 

p25i and CIPi with i = 30 to 40. As observed above, conformations 36th to 40th for both CIP and 

p25 have distance values higher than the threshold. For the remaining distance values we still see 

that CIP distance values are lower than for p25: CIP geometric affinity is higher than p25 for 

those conformations. 

From figures 26 to 28 we observe that, in compliance with the flexibility hypothesis, CIP

conformations seem to be more geometrically affine to cdk5 conformations then p25 ones. 

Geometric affinity has been evaluated by using the distance function defined in Equation (13), 

which is based on the difference between p25/CIP interface amino acid current location and their 

ideal location. The ideal location is the location interface amino acids would have if they were 

interacting exactly as they were doing in the crystallized cdk5/p25 active complex, in 

compliance with the conservation hypothesis. The ideal location has been defined based on the 

closest amino acid atoms, as well as the unique geometric representation of the relative position 

and orientation of the interface amino acid pairs, defined by the vector 

( ) 22,...,1 ,,,,, 0000000 =Κ γβα ziyixiiiii iddd =∀ from Equation (4). Hence, the geometric affinity 

evaluation problem is efficiently addressed as a feature alignment problem with a decreased 

number of degrees of freedom involved. 

Additional insights on protein interface relative orientations are also given. In Figure 

29(a) to (c) plots referring to interface amino acid relative angle values α β and , γ from 

Equation (4) for a few p25 or CIP conformations are shown. These angle values represent p25 or 

 25
CIP interface amino acid orientation with respect to cdk5, as defined in Section 4.2.1. Let us 

define as the angles for the j-th p25 or CIP conformation and 

 as the angles for the active p25 or CIP conformation. Then, the plots 

in Figure 29(a) to (c) show, respectively, for a given conformation j the following values: 

i =∀ 22,...,1 and , j

i

j

i

j

i γβα

i =∀ 22,...,1 and o

i , γ oi

o

i βα

 , and 

22

1

∑=

j =Δ−=Δ

i

o

i

j

j i βααα

i =∀ 22,...,1 and , j

i

j

i

j

i γβα

22

1

22

1

∑ ∑ = =

−=Δ−

i

o

i

j

j i

i

o

i

j

i γγγββ . The angle values 

 are computed after performing the alignment through algorithm 

COMPUTEKTHDISTANCEFUNCTION. In particular p25 and CIP conformations p25i and CIPi with i

= 0, 10, 20, 30, 40, 50 have been considered. Note that for both α and γ the trend is similar to 

the distance values from Equation (13): CIP angle values are closer to the active conformations 

than p25 ones. As for β case, p25 angle values are slightly closer to the active conformations 

than CIP ones, but there is no significant difference as these values are obtained as a summation 

over all the 22 amino acid pairs. Therefore, these plots referring more directly to relative 

interface amino acid orientations seem not to go against the above results. Hence, we can infer 

that a few CIP conformations seem to be more geometrically affine to cdk5 conformations than 

p25 ones, in compliance with our flexibility hypothesis. 

5.3 Discussion of results 

Our results should be interpreted in light of the following issues. 

• Steric clashes between atoms. Observe that in calculating the geometric affinity each amino 

acid residue is treated as a rigid body, and the corresponding alignment is obtained based on 

one set of coordinates per amino acid location. Thus there is no direct control on the 

orientation of non-interface residues or to side chains of interface residues. Therefore an 

alignment yielding an apparently high geometric affinity value between two interfaces may 

actually cause a number of steric clashes (see Figure 30). Hence, steric clashes between 

atoms should be analyzed before accepting the obtained geometric affinity values. 

However, the presence of some clashes after an alignment does not necessarily mean that 

the corresponding interface affinity value needs to be discarded. In fact the rigid side chain 

atoms might just need a little readjusting after the rigid alignment. Therefore, alignments 

with few clashes involving only a few side chain atoms can be still considered significant. 

For example, consider the 20th configuration of cdk5, p25 and CIP in figures 31 and 32. 

As we can see, the protein alignment in the figure causes some atom clashes. If we focus on 

the atom clashes in detail, we realize that they involve side chain amino acids that can easily 

readjust in order not to cause clashes anymore, without significantly affecting the geometric 

affinity measure. In fact, even if the amino acid side chains that are readjusted involved 

atoms representing the protein interface, their displacement resulting from side chain 

adjustment would be usually so small that the overall geometric affinity value would not 

change significantly. In figures 31 and 32 the protein alignment after eliminating the clashes 

is shown, with particular focus on the affected side chains. The readjustment has been 

obtained by using the molecular dynamics software NAMD [Kal99]. Protein atoms were 

kept fixed in space except for the clashing side chains, which were left free to move and 

 26
readjust. This way a minimum energy configuration was reached without significantly 

affecting the obtained alignment. The corresponding distance values are shown: there are no 

significant changes as anticipated. Therefore CIP has still higher geometric affinity than p25

based on the distance values. 

Hence, as long as the small number of clashes can be eliminated as in the abovedescribed case, the corresponding geometric affinity value is still significant. We have 

checked a few of the obtained protein alignments by the above procedure based on the 

software NAMD. We focused on the range between the 10th protein conformations and the 

36th protein conformations, where CIP geometric affinity to cdk5 seems to always be higher 

than p25. For all of them very few or no clashes were identified and they were easily 

eliminated without significantly affecting the geometric affinity values. This could be 

expected just looking at the protein alignments (Figure 33): all of them involve protein 

conformations that are very close to the active complex cdk5/p25. Therefore no major 

clashes are expected upon alignment. On the other hand protein conformations that are very 

different from the active complex cdk5/p25 cannot be handled similarly. In fact in the figure 

an instance of alignment of conformation 2000 is also shown: in this case the protein 

conformations are very different from the active cdk5/p25 complex and therefore there are 

many clashes that cannot be fixed by just allowing simple readjustments of the side chains. 

So in conclusion the geometric affinity values for the protein conformations of our interest 

are significant and can be used to verify the flexibility hypothesis. 

• Assessment of active/inactive state for given cdk5 conformations. A range of cdk5, p25

and CIP conformations has been identified that seems to show that CIP has higher geometric 

affinity to cdk5 than p25 has to cdk5. This suggests that CIP might bind to cdk5 more easily 

than p25 does in correspondence of these conformations. However, in order for CIP to bind 

to cdk5 and therefore prevent p25 from activating cdk5 (i.e. flexibility hypothesis), the 

binding has to involve inactive cdk5 conformations (see Figure 34). This means that there 

must be at least a few cdk5 conformations from the identified range that are still inactive. 

Observe that cdk5 is not expected to suddenly become inactive, but a few conformations, 

very close to the crystallized one, are expected to be still active. Then, as cdk5 main elements 

significantly change location/orientation, cdk5 will become inactive at some point. Therefore 

it would be important to identify the cdk5 conformations that are expected to be active 

among the ones from the above-mentioned range. 

For other cyclin-dependent kinases such as cdk2/cyclin complex the activation 

mechanism has already been studied in detail [Bro95]. Therefore, similarly to when we 

obtained cdk5 inactive conformation, we use cdk2 as model. From [Bro95] it can be inferred 

that very small conformational changes are expected to inactivate cdk2. In particular, a few 

cdk2 and cyclin amino acids are identified, whose relative locations cannot change 

significantly in order for cdk2 to stay active. Hence, in order to evaluate if a given cdk5

conformation is expected to be active, we refer to the corresponding significant amino acid 

positions in cdk5/p25 complex. By homology with cdk2/cyclin complex, the 

hyperphosphorylation of other proteins can only occur if cdk5/p25 complex binds to the 

protein to be hyperphosphorylated (i.e. substrate) as well as to ATP in a very specific way. A 

few significant cdk5/p25 complex amino acids are identified as follows.

Let us consider the active cdk5/p25 complex shown in Figure 35(a). Based on [Bro95] 

and on our knowledge of cdk5 function in the brain, there are two regions of interest in 

 27
cdk5/p25 complex. They are the binding pocket for ATP and the binding site for the substrate 

protein to be phosphorylated. As it is shown in the figure, we chose three amino acids to 

represent the above-mentioned regions. Amino acids GLU 240 from p25 (in purple) and ASP 

126 from cdk5 (in red) represent cdk5 and p25 regions that bind to the substrate. In fact ASP 

126 is supposed to bind to a particular amino acid of the substrate and keep it in the right 

location to be phosphorylated. On the other hand GLU 240 is an amino acid from p25 that is 

likely to interact with the substrate, holding its last amino acid in place. Finally GLU 81 (in 

green) represents cdk5 binding pocket for ATP. So these three selected amino acids, 

represented by Van Der Waals spheres in the figure, are significant for cdk5 activation. In 

Figure 35(b) the relative distances are shown for the active complex. As observed above, in 

order for a given protein conformation to be active the three amino acid relative locations 

need to be conserved very precisely with respect to the fully active protein complex. To 

verify this, we have focused on a few representative cdk5 and p25 conformations, so as to 

check if there were significant changes in the relative locations of these three amino acids. 

The proteins were first aligned by using the above-described 

COMPUTEKTHDISTANCEFUNCTION algorithm. In Figure 36 only the three amino acids of 

interest are shown upon each alignment for the following cdk5 and p25 conformations: cdk5i 

and p25i with i = 10, 20, 30, 40. As we can see from the figure, for the 10th conformation the 

relative distances are still very close to the active conformation ones, suggesting that the 

complex is likely to be active. As can be observed in Figure 36 (b)-(c)-(d), all three distances 

generally increase. In particular, the one between ASP 126 of cdk5 and GLU 240 of p25

significantly increases, suggesting that the substrate might be displaced in such a way that no 

phosphorylation would be possible anymore, as both the amino acids are supposed to interact 

with the substrate. In addition, the distance between GLU 81 and ASP 126 is also slightly 

increasing, suggesting that the relative position between ATP and the substrate might not be 

anymore suitable for phosphorylation. Therefore, if we consider the need of a very precise 

binding in order for cdk5/p25 complex to be active, starting from the 20th cdk5 and p25

conformations we start seeing significant changes in the distance values. Therefore, based on 

these observations, it is reasonable to expect that somewhere in the range between 20th and 

30th conformations cdk5 inactivation may occur. However, we are aware that it will be 

necessary to use molecular dynamics simulations to study more accurately cdk5

activation/inactivation process; we are currently pursuing this. 

Finally, based also on the two items discussed above, the outcome of the comparison 

between p25 and CIP geometric affinity to cdk5 corresponding conformations can be interpreted 

as follows. We identified a range of cdk5, p25 and CIP conformations that seems to show, based 

on conservation hypothesis, that CIP has higher geometric affinity to cdk5 than p25. So CIP is 

expected to bind to cdk5 more easily than p25 for these conformations. We also observed, based 

on [Bro95] and on our knowledge of cdk5 function, that a few cdk5 conformations from the 

identified range might be inactive. Therefore CIP seems to be able to bind to cdk5 inactive 

conformations and therefore prevent p25 from binding and activating them. This complies with 

our flexibility hypothesis. In the next section we draw the conclusions about this work and 

illustrate future directions. 

 28
6 CONCLUSIONS AND FUTURE WORK 

Experimental studies, which are described in [Ami02], provided evidence that small truncations 

of p35 produce a high-affinity inhibitor protein called CIP. These results are of fundamental 

importance for neurodegenerative diseases such as Alzheimer’s, which are known to be 

associated with hyperphosphorylation of specific neuronal proteins. In this paper the binding 

mechanism of cdk5-p25 and cdk5-CIP complexes has been studied in order to gain a better 

insight on the inhibitory activity of the protein CIP. Our in silico study is based on the 

conservation hypothesis, which asserts that at least part of the interface interaction for the 

crystallized cdk5-p25 complex will be conserved for the other cdk5-p25 and cdk5-CIP complex 

conformations. Our aim was to verify the flexibility hypothesis, which asserts that increased 

flexibility is induced by p25’s truncation to CIP which allows it to conform to the cdk5 interface 

more fully and prevents cdk5 from achieving its fully active conformation. 

 We were able to compare the interactions between the proteins at the atomic level by 

using a geometry-based alignment algorithm. This algorithm attempts to fully align two given 

protein conformations based on the conservation hypothesis. The protein interfaces are 

represented as sets of points representing in turn their interacting amino acids. The algorithm 

aligns the proteins such that their interfaces are interacting geometrically as similarly as possible 

to the interface of the crystallized complex. Our alignment algorithm can be potentially 

generalized to the cases in which the protein surfaces to be aligned need to satisfy certain 

geometric conditions following from biochemical considerations. 

 Based on the conservation hypothesis our results seem to verify the flexibility hypothesis. 

Furthermore, we have also studied the protein conformations, derived by the FIRST/FRODA 

software, in order to gain additional insight on the activation state of the cdk5-p25 and cdk5-CIP

complexes. The flexibility and conservation hypotheses as well as our observations on the 

activation of the cdk5-p25 and cdk5-CIP complexes can now be used as the basis for MD 

(molecular dynamics) simulations. In general, simulating cdk5-p25 and cdk5-CIP interactions 

seems to be a computationally challenging task due to the complexity of their structures. 

However, based on our current results, we can efficiently setup our MD simulations by focusing 

on particular aspects rather than trying to fully reproduce in silico what took place in vitro. Some 

potentially interesting MD simulations that we intend to pursue are listed as follows. 

• Simulate cdk5-p25 and cdk5-CIP interaction for conformations that are very close to the 

crystallized cdk5-p25 complex, focusing on the behavior of specific amino acids from the 

interface, from the ATP binding pocket and from the substrate binding pocket, in order to 

infer substantial differences in the behavior of the two complexes. 

• Simulate cdk5-p25-substrate-ATP and cdk5-CIP-substrate-ATP interactions for 

conformations that are very close to the crystallized cdk5-p25 complex focusing on the 

activation status of the complexes. 

• Simulate cdk5-p25-substrate-ATP and cdk5-CIP-substrate-ATP interactions for 

conformations that are very close to the crystallized cdk5-p25 complex after mutating some 

amino acids that we have identified as significant for the cdk5 interaction with p25 and CIP

and for ATP and substrate binding. 

 29
These and additional MD simulations can now be more efficiently setup. In fact we identified 

several amino acids that seem to be more significant than others for cdk5 interaction with p25

and CIP and for cdk5 activation. Also, our results indicates that the phenomena of interest occurs 

very close to the crystalized cdk5-p25 conformation. Therefore we expect MD simulations to 

provide useful insights on cdk5 conformations close to the crystallized forms, which will involve 

much less computational effort. Also, the analysis of MD data can be performed by focusing on 

only a few significant amino acids. Both these will lead to more efficient MD simulations. 

DISCLAIMER 

Certain commercial software systems are identified in this paper. Such identification does not 

imply recommendation or endorsement by the National Institute of Standards and Technology 

(NIST); nor does it imply that the products identified are necessarily the best available for the 

purpose. Further, any opinions, findings, conclusions or recommendations expressed in this 

material are those of the authors and do not necessarily reflect the views of NIST or any other 

supporting US government or corporate organizations.

 30
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 36
Figure 1. ICM technique is based on Internal Coordinate Mechanics 

[Aba94a, Aba94b, Fer02] 

Figure 2. An example of rigid cluster decomposition by FIRST

[Wel05] 

 37
cdk5 active p25 interface

cdk5 interface

p25

active

Figure 3.(a) p25 truncation; (b) interface residues from active cdk5/p25 complex

Truncated amino 

acids from p25 to 

obtain CIP

p25 active

(a) (b)

cdk5 active p25 interface

cdk5 interface

p25

active

Figure 3.(a) p25 truncation; (b) interface residues from active cdk5/p25 complex

Truncated amino 

acids from p25 to 

obtain CIP

p25 active

(a) (b)

cdk5 active

cdk5 inactive

cdk5 intermediate 

conformations

Figure 4. Overview of our geometry-based technique to study cdk5/p25 and cdk5/CIP

intermediate conformations

FIRST/FRODA 

software 

application

FIRST/FRODA 

INPUT: known 

protein 

conformations

FIRST/FRODA 

OUTPUT: 

intermediate protein 

conformations

CIP and p25

active

CIP/p25 intermediate 

conformations

Application of software system based on 

our geometry-based algorithm

OUTPUT: Distance values used to compare 

cdk5/p25 and cdk5/CIP interfaces

cdk5 active

cdk5 inactive

cdk5 intermediate 

conformations

Figure 4. Overview of our geometry-based technique to study cdk5/p25 and cdk5/CIP

intermediate conformations

FIRST/FRODA 

software 

application

FIRST/FRODA 

INPUT: known 

protein 

conformations

FIRST/FRODA 

OUTPUT: 

intermediate protein 

conformations

CIP and p25

active

CIP/p25 intermediate 

conformations

Application of software system based on 

our geometry-based algorithm

OUTPUT: Distance values used to compare 

cdk5/p25 and cdk5/CIP interfaces

 38
cdk5 inactive configuration 

obtained by homology

Crystallized cdk2

inactive conformation

Figure 5. Cdk5 inactive conformation is obtained by homology to the very similar 

crystalized cdk2 inactive conformation

cdk5 inactive configuration 

obtained by homology

Figure 5. Cdk5 inactive conformation is obtained by homology to the very similar 

crystalized cdk2 inactive conformation

Crystallized cdk2

inactive conformation

cdk5 active

cdk5 10th intermediate 

configuration

cdk5 inactive

cdk5 45th intermediate 

configuration

Figure 6. Instances of cdk5 conformations obtained from FIRST/FRODA software

FIRST/FRODA

INPUT

.........

OUTPUT

cdk5 active

cdk5 10th intermediate 

configuration

cdk5 inactive

cdk5 45th intermediate 

configuration

Figure 6. Instances of cdk5 conformations obtained from FIRST/FRODA software

FIRST/FRODA

INPUT

.........

OUTPUT

 39
cdk5 active 

conformation

p25 active 

conformation

p25 interface

cdk5 interface

Interface residues are such that mutual 

distance is < 3.5 A

cdk5 active 

conformation

cdk5 conformation Interface 

residue 37 in 

active 

conformation

Interface 

residue 37 in 

intermediate 

conformation

Interface residue 37 displacement is 

identified

Figure 7. Active cdk5/p25 complex with interface residues and residue 37 displacement 

corresponding to a given cdk5 conformation

cdk5 active 

conformation

p25 active 

conformation

p25 interface

cdk5 interface

Interface residues are such that mutual 

distance is < 3.5 A

cdk5 active 

conformation

cdk5 conformation Interface 

residue 37 in 

active 

conformation

Interface 

residue 37 in 

intermediate 

conformation

Interface residue 37 displacement is 

identified

Figure 7. Active cdk5/p25 complex with interface residues and residue 37 displacement 

corresponding to a given cdk5 conformation

FRODA

For each interface residue 

conformation change a 

vector is obtained

Vectors are used as 

directional bias in 

FRODA

cdk5 active 

conformation

p25 active 

conformation

resulting p25

intermediate 

conformation

Figure 8. Vector representation of the cdk5 interface conformational changes and 

corresponding p25 conformation obtained by using FIRST/FRODA

FRODA

For each interface residue 

conformation change a 

vector is obtained

Vectors are used as 

directional bias in 

FRODA

cdk5 active 

conformation

p25 active 

conformation

resulting p25

intermediate 

conformation

Figure 8. Vector representation of the cdk5 interface conformational changes and 

corresponding p25 conformation obtained by using FIRST/FRODA

 40
The interface of p25 has 

been forced to follow cdk5

interface conformational 

changes from A to B 

through FRODA

Blue arrows show 

discontinuities in 

the resulting p25

backbone

conformation A 

of cdk5

conformation B 

of cdk5

Figure 9. P25 conformation resulting from the application of FRODA shows 

discontinuities in the backbone 

The interface of p25 has 

been forced to follow cdk5

interface conformational 

changes from A to B 

through FRODA

Blue arrows show 

discontinuities in 

the resulting p25

backbone

conformation A 

of cdk5

conformation B 

of cdk5

Figure 9. P25 conformation resulting from the application of FRODA shows 

discontinuities in the backbone 

First conformation 0 

corresponds to cdk5

active conformation

Last conformation m

corresponds to cdk5

inactive conformation

cdk5

p25

CIP

p25 feasible conformation

CIP feasible conformation CIP unfeasible conformation

l

p25 unfeasible conformation

Protein conformations

Figure 10. Plot representing cdk5, p25 and CIP conformation obtained through 

FRODA

n m

First conformation 0 

corresponds to cdk5

active conformation

 41

Last conformation m

corresponds to cdk5

inactive conformation

cdk5

p25

CIP

p25 feasible conformation p25 unfeasible conformation

Protein conformations

Figure 10. Plot representing cdk5, p25 and CIP conformation obtained through 

FRODA

n m

CIP feasible conformation CIP unfeasible conformation

l
cdk5

p25

cdk5 interface

p25 interface

Amino acid pairs with 

mutual distance < 3.5 A 

will be used for 

representing the interface

Crystallized cdk5/p25

complex

VdW sphere 

representation

Figure 11. cdk5/p25 complex interface and its representation

cdk5

p25

cdk5 interface

p25 interface

Amino acid pairs with 

mutual distance < 3.5 A 

will be used for 

representing the interface

Crystallized cdk5/p25

complex

VdW sphere 

representation

Figure 11. cdk5/p25 complex interface and its representation

p25 interface

cdk5 interface

COOH

Cα

N

Side chain

Side chain

COOH

Cα

N

A2

A1

GLU42

TRP190 A2

Cα

N

A1

Cα

N

GLU42 TRP190

GLU42

TRP190

n1

n2

Figure 12. interacting amino acids GLU42 and TRP190 from cdk5/p25 crystal interface

p25 interface

cdk5 interface

COOH

Cα

N

Side chain

Side chain

COOH

Cα

N

A2

A1

GLU42

TRP190 A2

Cα

N

A1

Cα

N

GLU42 TRP190

GLU42

TRP190

n1

n2

Figure 12. interacting amino acids GLU42 and TRP190 from cdk5/p25 crystal interface

 42
Relative position/orientation between GLU42 and TRP190 is function of α, β, γ and 

d=(dx, dy, dz

)

d

n1

n2

x

z

y

α

β

A2

Cα

N

A1

Cα

N

GLU42

TRP190

n1

n2

Cα

A2

N

A1

Cα

N

γ

Plane π Plane π

n1

Figure 13. The six independent parameters that univocally represent the relative 

position and orientation of amino acid pair GLU42/TRP190 from cdk5/p25

crystallized complex

TRP190

GLU42

Relative position/orientation between GLU42 and TRP190 is function of α, β, γ and 

d=(dx, dy, dz

)

d

n1

n2

x

z

y

α

β

A2

Cα

N

A1

Cα

N

GLU42

TRP190

n1

n2

Cα

A2

N

A1

Cα

N

γ

Plane π Plane π

n1

Figure 13. The six independent parameters that univocally represent the relative 

position and orientation of amino acid pair GLU42/TRP190 from cdk5/p25

crystallized complex

TRP190

GLU42

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

Non active conformation

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(a) Translation of both amino acids such that A2 lies on the origin

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

Non active conformation

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(a) Translation of both amino acids such that A2 lies on the origin

 43
x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(b) Rotation of both amino acids such that n2 is aligned to Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(b) Rotation of both amino acids such that n2 is aligned to Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(c) Rotation of both amino acids such that Cα lies on X-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(c) Rotation of both amino acids such that Cα lies on X-axis

 44
x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(d) Rotation of amino acid Res1 such that n1 is aligned to Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

(d) Rotation of amino acid Res1 such that n1 is aligned to Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

γ

(e) Rotation of amino acid Res1 such that CαA1 has an angle equal to γ with X-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

γ

(e) Rotation of amino acid Res1 such that CαA1 has an angle equal to γ with X-axis

 45
x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

γ

β

(f) Rotation of amino acid Res1 such that n1 has an angle equal to β with Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

γ

β

(f) Rotation of amino acid Res1 such that n1 has an angle equal to β with Z-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

β A1

Cα

N

Res1

z

A y 2

Cα N

n1

n2

Res2

α

x

x’

z’

(g) Rotation of amino acid Res1 such that n1 projection onto XY-plane has an angle 

equal to α with X-axis

x

z

y

A2

Cα

N

A1

Cα

N

n1

n2

Res1

Res2

β A1

Cα

N

Res1

z

A y 2

Cα N

n1

n2

Res2

α

x

x’

z’

(g) Rotation of amino acid Res1 such that n1 projection onto XY-plane has an angle 

equal to α with X-axis

 46
A1

Cα

N

Res1

z

A y 2

Cα N

n1

n2

Res2

x

d A2

A1 Cα

Cα

N

Res1

Res2

n1

n2

Active 

conformation 

restored

z

y

x

N

(h) Translation of amino acid Res1 such that A1 is at distance d 

from the origin

Figure 14. Example of transformations described in steps 1 to 8 from 

ALIGN_RES_PAIR algorithm

A1

Cα

N

Res1

z

A y 2

Cα N

n1

n2

Res2

x

d A2

A1 Cα

Cα

N

Res1

Res2

n1

n2

Active 

conformation 

restored

z

y

x

N

(h) Translation of amino acid Res1 such that A1 is at distance d 

from the origin

Figure 14. Example of transformations described in steps 1 to 8 from 

ALIGN_RES_PAIR algorithm

y

pi x

Closest neighbor = point q4

Closest neighbor = point q5

qj points of 

Set B

Closest neighbor 

= point q1

Closest neighbor 

= point q2

Closest neighbor 

= point q3

2 DOF transformation space for point pi: any point (x,y) represents a 

transformation T=(x,y) applied to point pi

Figure 15. For each transformation T=(x,y) belonging to the same cell the 

resulting closest neighbor to point pi from set B does not change 

y

pi x

2 DOF transformation space for point pi: any point (x,y) represents a 

transformation T=(x,y) applied to point pi

Figure 15. For each transformation T=(x,y) belonging to the same cell the 

resulting closest neighbor to point pi from set B does not change 

Closest neighbor = point q4

Closest neighbor = point q5

qj points of 

Set B

Closest neighbor 

= point q1

Closest neighbor 

= point q2

Closest neighbor 

= point q3

 47
Aligned amino acid pair identifying rotation axis r

rotation axis r

1 DOF rotation θ

Figure 16. One degree of freedom rotation of protein p25 around r

axis which is identified by the aligned amino acid pair shown

Aligned amino acid pair identifying rotation axis r

rotation axis r

1 DOF rotation θ

Figure 16. One degree of freedom rotation of protein p25 around r

axis which is identified by the aligned amino acid pair shown

actual 

LYS254 

location

actual 

GLU255 

location

LEU37

HIS71

LYS254

GLU255

ideal 

GLU255 

location

ideal 

LYS254 

HIS71 location

LEU37

GLU255 ideally 

aligned

LYS254 aligned as 

close as possible to 

ideal location

rotation axis r

Figure 17. Instance of amino acid pair alignment in the case of non coincident 

ideal locations

(a) Amino acid pairs (b) Ideal and actual amino acid locations

(c) Translation Tri to ideally align GLU255 (d) Rotation Rij to ideally align LYS254

Tri

Rij

p25

cdk5 actual 

LYS254 

location

actual 

GLU255 

location

LEU37

HIS71

LYS254

GLU255

ideal 

GLU255 

location

ideal 

LYS254 

HIS71 location

LEU37

GLU255 ideally 

aligned

LYS254 aligned as 

close as possible to 

ideal location

rotation axis r

Figure 17. Instance of amino acid pair alignment in the case of non coincident 

ideal locations

(a) Amino acid pairs (b) Ideal and actual amino acid locations

(c) Translation Tri to ideally align GLU255 (d) Rotation Rij to ideally align LYS254

Tri

Rij

p25

cdk5

 48
ideal location of amino acid represented 

by feature point 

x

z

y

θ

⎪⎩

⎪

⎨

⎧

=

=

initial angle between th e projection of onto coordinate plane XY and -X axis

constant distance of point from rotation axis correspond ing to Z - axis

0 2

2

k

g

g

k

g

k

g

A

d A r

r

θ

k A2 g

r

g θ0

k A2i

r

k A2 j

r

tk A2 g

r

Rotation axis Z identified by amino acids 

represented by feature points and k A2 j

r k A2ir

k dg

k dg

k A2g

r tk A2g

r

index k refers to k-th protein conformation

Figure 19. Instance of geometric parameters used in distance function evaluation

ideal location of amino acid represented 

by feature point 

x

z

y

θ

⎪⎩

⎪

⎨

⎧

=

=

initial angle between th e projection of onto coordinate plane XY and -X axis

constant distance of point from rotation axis correspond ing to Z - axis

0 2

2

k

g

g

k

g

k

g

A

d A r

r

θ

k A2 g

r

g θ0

k A2i

r

k A2 j

r

tk A2 g

r

Rotation axis Z identified by amino acids 

represented by feature points and k A2 j

r k A2ir

k dg

k dg

k A2g

r tk A2g

r

index k refers to k-th protein conformation

Figure 19. Instance of geometric parameters used in distance function evaluation

input

CIP conformations 

obtained by using 

FIRST/FRODA 

software

Software system implemented based on 

geometric alignment algorithm 

COMPUTEKTHDISTANCEFUNCTION 

p25 conformations 

obtained by using 

FIRST/FRODA 

software

input

output

Geometric affinity between 

corresponding cdk5, p25

and CIP conformations 

input

CIP conformations 

obtained by using 

FIRST/FRODA 

software

Software system implemented based on 

geometric alignment algorithm 

COMPUTEKTHDISTANCEFUNCTION 

p25 conformations 

obtained by using 

FIRST/FRODA 

software

input

output

Geometric affinity between 

corresponding cdk5, p25

and CIP conformations 

Figure 20. Schematic view of the software system implemented in this work

 49
(a) Initial cdk5 and p25 position and orientation

p25 cdk5

PRO45

LYS56

TRP258

ASN266

Ideal positions of TRP258 and ASN266 

determined based on LYS56 and PRO45 

positions

(b) TRP258 and ASN266 are aligned to 

their ideal positions (c) p25 is optimally aligned to cdk5 by 

rotation around axis r

Rotation axis r identified by 

TRP258 and ASN 266 ideal 

positions

Figure 21. Instance of application of COMPUTEKTHDISTANCEFUNCTION

algorithm for a given amino acid pair alignment

(a) Initial cdk5 and p25 position and orientation

p25 cdk5

TRP258

ASN266

Ideal positions of TRP258 and ASN266 

determined based on LYS56 and PRO45 

positions

(b) TRP258 and ASN266 are aligned to 

their ideal positions

PRO45

LYS56

(c) p25 is optimally aligned to cdk5 by 

rotation around axis r

Rotation axis r identified by 

TRP258 and ASN 266 ideal 

positions

Figure 21. Instance of application of COMPUTEKTHDISTANCEFUNCTION

algorithm for a given amino acid pair alignment

(a) aligned cdk5 and p25

conformations

(b) cdk5 and p25 actual 

interface amino acids

(c) p25 ideal interface 

amino acids obtained 

based on cdk5 interface 

and conservation 

hypothesis

(d) p25 ideal and 

actual interface (e) Displacement vectors 

representing p25 amino 

acid displacements between 

actual and ideal interface

Figure 22. cdk5 and p25 are not interacting if the mean norm of displacement 

vectors shown in (e) exceeds a threshold value

p25

cdk5 cdk5

interface

cdk5

interface

p25 actual 

interface

p25 ideal 

interface

p25 actual 

interface

p25 ideal 

interface

(a) aligned cdk5 and p25

conformations

(b) cdk5 and p25 actual 

interface amino acids

(c) p25 ideal interface 

amino acids obtained 

based on cdk5 interface 

and conservation 

hypothesis

(d) p25 ideal and 

actual interface (e) Displacement vectors 

representing p25 amino 

acid displacements between 

actual and ideal interface

Figure 22. cdk5 and p25 are not interacting if the mean norm of displacement 

vectors shown in (e) exceeds a threshold value

p25

cdk5

p25 actual 

interface

p25 ideal 

interface

p25 actual 

interface

p25 ideal 

interface

cdk5

interface

cdk5

interface

 50
Figure 23. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

Distance value

Protein k-th conformation

0 500 1000 1500 2000 2500 3000

distance 

threshold

CIP distance

p25 distance

Figure 23. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

Distance value

Protein k-th conformation

0 500 1000 1500 2000 2500 3000

CIP distance

p25 distance distance 

threshold

Distance value

Protein k-th conformation

0 50 100 150 200 250 300 350 400 450 500

distance 

threshold

Figure 24. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP distance

p25 distance

Distance value

Protein k-th conformation

0 50 100 150 200 250 300 350 400 450 500

CIP distance

distance 

threshold

p25 distance

Figure 24. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

 51
Distance value

Protein k-th conformation

0 5 10 15 20 25 30 35 40 45 50

distance 

threshold

Figure 25. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP distance

p25 distance

Distance value

Protein k-th conformation

0 5 10 15 20 25 30 35 40 45 50

CIP distance

p25 distance

distance 

threshold

Figure 25. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP higher geometric affinity range

Distance value

Protein k-th conformation

10 11 12 13 14 15 16 17 18 19 20

Figure 26. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP distance

p25 distance

CIP higher geometric affinity range

Distance value

Protein k-th conformation

10 11 12 13 14 15 16 17 18 19 20

Figure 26. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

p25 distance

 52
CIP higher geometric affinity range

Distance value

Protein k-th conformation

20 21 22 23 24 25 26 27 28 29 30

Figure 27. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP distance

 53

p25 distance

CIP distance

CIP higher geometric affinity range

Distance value

Protein k-th conformation

20 21 22 23 24 25 26 27 28 29 30

Figure 27. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

p25 distance

Discarded range (p25 and CIP 

are not expected to interact 

with cdk5)

CIP higher geometric affinity 

range

Distance value

Protein k-th conformation

30 31 32 33 34 35 36 37 38 39 40

Figure 28. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

distance 

threshold

CIP distance

p25 distance

Discarded range (p25 and CIP 

are not expected to interact 

with cdk5)

CIP higher geometric affinity 

range

Distance value

Protein k-th conformation

30 31 32 33 34 35 36 37 38 39 40

Figure 28. Plot of distance values from Equation (13) vs. p25 and CIP corresponding 

conformations

CIP distance

p25 distance

distance 

threshold
(a) Alpha angle (b) Beta angle

Figure 29. Plot of relative angle values representing interface amino acid relative 

orientations

CIP distance

p25 distance

Distance 

value

Distance 

value

Distance 

value

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

(c) gamma angle

(a) Alpha angle (b) Beta angle

CIP distance

p25 distance

Distance 

value

Distance 

value

Distance 

value

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

(c) gamma angle

Figure 29. Plot of relative angle values representing interface amino acid relative 

orientations

2500th conformation 

of p25

2500th conformation 

of cdk5

Clashing atoms represented 

as sticks and as spheres

Figure 30. Instance of a protein alignment obtained by using our technique that 

causes a major number of atom clashes

2500th conformation 

of p25

2500th conformation 

of cdk5

Clashing atoms represented 

as sticks and as spheres

Figure 30. Instance of a protein alignment obtained by using our technique that 

causes a major number of atom clashes

 54
p25 20th

conformation

cdk5 20th

conformation

 55

cdk5 clashing 

amino acids

p25 clashing 

amino acids

cdk5 clashing amino 

acids before fixing 

clashes

cdk5 clashing amino 

acids after fixing 

clashes

p25 clashing amino 

acids before fixing 

clashes

p25 clashing amino 

acids after fixing 

clashes

Figure 31. Instance of identification and elimination of clashes for cdk5/p25

aligned 20th conformations

Only little changes in correspondence of amino acid side chains occur

cdk5/p25 distance value yielding geometric affinity before fixing clashes = 52.26

cdk5/p25 distance value yielding geometric affinity before fixing clashes = 54.57

SER46

LYS56

ARG149

SER46

LYS56

ARG149

ASN266

SER242

MET237

MET237

SER242

MET237

p25 20th

conformation

cdk5 20th

conformation p25 clashing 

amino acids

cdk5 clashing 

amino acids

cdk5 clashing amino 

acids before fixing 

clashes

cdk5 clashing amino 

acids after fixing 

clashes

p25 clashing amino 

acids before fixing 

clashes

p25 clashing amino 

acids after fixing 

clashes

Figure 31. Instance of identification and elimination of clashes for cdk5/p25

aligned 20th conformations

Only little changes in correspondence of amino acid side chains occur

cdk5/p25 distance value yielding geometric affinity before fixing clashes = 52.26

cdk5/p25 distance value yielding geometric affinity before fixing clashes = 54.57

SER46

LYS56

ARG149

SER46

LYS56

ARG149

ASN266

SER242

MET237

MET237

SER242

MET237
CIP 20th

conformation

cdk5 20th

conformation

 56

cdk5 clashing 

amino acids

CIP clashing 

amino acids

cdk5 clashing amino 

acids before fixing 

clashes

cdk5 clashing amino 

acids after fixing 

clashes

CIP clashing amino 

acids before fixing 

clashes

CIP clashing amino 

acids after fixing 

clashes

Figure 32. Instance of identification and elimination of clashes for cdk5/CIP

aligned 20th conformations

Only little changes in correspondence of amino acid side chains occur

cdk5/CIP distance value yielding geometric affinity before fixing clashes = 47.18

cdk5/CIP distance value yielding geometric affinity before fixing clashes = 48.08

ARG50

ALA150

ARG50

ALA150

ILE241

ASN276

ILE241

ASN276

CIP 20th

conformation

cdk5 20th

conformation CIP clashing 

amino acids

cdk5 clashing 

amino acids

cdk5 clashing amino 

acids before fixing 

clashes

cdk5 clashing amino 

acids after fixing 

clashes

CIP clashing amino 

acids before fixing 

clashes

CIP clashing amino 

acids after fixing 

clashes

Figure 32. Instance of identification and elimination of clashes for cdk5/CIP

aligned 20th conformations

Only little changes in correspondence of amino acid side chains occur

cdk5/CIP distance value yielding geometric affinity before fixing clashes = 47.18

cdk5/CIP distance value yielding geometric affinity before fixing clashes = 48.08

ARG50

ALA150

ARG50

ALA150

ILE241

ASN276

ILE241

ASN276
p25 cdk5

Active conformation

p25 cdk5

5th conformation

p25 cdk5

10th conformation

p25 cdk5

15th conformation

p25 cdk5

20th conformation

cdk5

p25

2000th conformation

(a) Major clashes are expected 

for this alignment, which is very 

different from active complex

p25 cdk5

Active conformation

cdk5

5th conformation

p25 cdk5

10th conformation

p25 cdk5

15th conformation

p25 cdk5

20th conformation

cdk5

p25

2000th conformation

(b) No major clashes are expected for these alignments, 

as they are very similar to active complex

Figure 33. No major clashes are expected for protein alignments very similar 

to the active complex cdk5/p25 and vice versa

p25

(a) Major clashes are expected 

for this alignment, which is very 

different from active complex

(b) No major clashes are expected for these alignments, 

as they are very similar to active complex

Figure 33. No major clashes are expected for protein alignments very similar 

to the active complex cdk5/p25 and vice versa

 57
 58

Distance value

Protein k-th conformation

0 5 10 15 20 25 30 35 40 45 50

CIP distance

p25 distance

Conformations 

p2535 and CIP35

Conformations 

p2511 and CIP11

CIP might bind to an inactive cdk5

conformation in the identified range, 

preventing cdk5 from activating it

p25 Active 

cdk5

Inactive 

cdk5 CIP

(a) Identified range show CIP higher 

geometric affinity to cdk5 than p25

(b) Identified range interpretation

Figure 34. Identified range showing CIP higher geometric affinity and our interpretation 

Distance value

Protein k-th conformation

0 5 10 15 20 25 30 35 40 45 50

CIP distance

p25 distance

Conformations 

p2535 and CIP35

Conformations 

p2511 and CIP11

CIP might bind to an inactive cdk5

conformation in the identified range, 

preventing cdk5 from activating it

p25 Active 

cdk5

Inactive 

cdk5 CIP

(a) Identified range show CIP higher 

geometric affinity to cdk5 than p25

(b) Identified range interpretation

Figure 34. Identified range showing CIP higher geometric affinity and our interpretation 
(a) cdk5/p25 active complex 

and significant amino acids 

from activation perspective

(b) Significant amino 

acids relative distances

GLU81

GLU240

ASP126

 59

Figure 35. Identified amino acids from cdk5/p25 active complex that are 

important from activation perspective

(a) cdk5/p25 active complex 

and significant amino acids 

from activation perspective

(b) Significant amino 

acids relative distances

GLU81

GLU240

Figure 35. Identified amino acids from cdk5/p25 active complex that are 

important from activation perspective

ASP126
(a) 10th conformation

Figure 36. Significant amino acid relative distances from activation perspective for 

several cdk5 and p25 conformations

(b) 20th conformation

(c) 30th conformation (d) 40th conformation

(a) 10th conformation

Figure 36. Significant amino acid relative distances from activation perspective for 

several cdk5 and p25 conformations

(b) 20th conformation

(c) 30th conformation (d) 40th conformation

 60