On diffusion posterior sampling via sequential Monte Carlo for zero-shot scaffolding of protein motifs
Authors: James Matthew Young
O. Deniz Akyildiz
Venue: Accepted by TMLR ()

Abstract: With the advent of diffusion models, new proteins can be generated at an unprecedented rate. The motif scaffolding problem requires steering this generative process to yield proteins with a desirable functional substructure called a motif. While models have been trained to take the motif as conditional input, recent techniques in diffusion posterior sampling can be leveraged as zero-shot alternatives whose approximations can be corrected with sequential Monte Carlo (SMC) algorithms. In this work, we introduce a new set of guidance potentials for describing scaffolding tasks and solve them by adapting SMC-aided diffusion posterior samplers with an unconditional model, Genie, as a prior. In single motif problems, we find that (i) the proposed potentials perform comparably, if not better, than the conventional masking approach, (ii) samplers based on reconstruction guidance outperform their replacement method counterparts, and (iii) measurement tilted proposals and twisted targets improve performance substantially. Furthermore, as a demonstration, we provide solutions to two multi-motif problems by pairing reconstruction guidance with an SE(3)-invariant potential. We also produce designable internally symmetric monomers with a guidance potential for point symmetry constraints. Our code is available at: https://github.com/matsagad/mres-project.

Published in Transactions on Machine Learning Research (09/2025)
On diffusion posterior sampling via sequential Monte Carlo for zero-shot scaffolding of protein motifs
James Matthew Young Deportment of Mothematics Imperial College London
uk
O. Deniz Akyildiz Department Mathemalics Imperial College London
deniz. attt::kkkyiiiiimmeaaalll...o...
Reviewed on OpenReview: https: openreview. net/ net/forum? id= KXRYYTsugh
Abstract
With the advent of diffusion modols, now proteins can be generated at an umprocedented rate. The motif scaffolding problem requires steering this generative process to yield proteins with a desirable functional substructure called motif. While models have been trained to take the motif as conditional input, recent techniques in diffusion posterior sampling can be leveraged zero-shot alternatives whose approximations (SMC) algorithms. In this work, we introduce new set of guidance potentials for describing scaffolding tasks and solve them by adapting SMC-aided diffusion posterior samplers with an unconditional model, Genie, as a prior. In single motif problems, we find that (i) the proposed potentials perform comparably, if not better, than the conventional masking approach, (ii) samplers based on reconstruction guidance outperform their replacement method counterparts, and (iii) measurement tilted proposals and twisted targets improve performance substantially. Furthermore, demonstration, we provide solutions to two multi-motif problems by pairing reconstruction guidance with (BB(3)-invarit) potential. We also produce designable internally symmetric monomers with guidance potential for point symmetry constraints. Our code available at: https: //ggtt//bb.ttt.. httpaa/..oo//tttaaaaa/eeeepppppp
be corrected with sequential Monte Carlo
1 Introduction
Proteins are fundamental to many biological systems. Naturally occurring and defined by an amino acid sequence, they fold to structural conformations that determine their function. Nature, however, has explored but tiny fraction of the entire protein universe, furthering its reach through evolution at the scale of millions of years. De novo protein design aims to accelerate this process to days or hours. Recent works (Watson et al. 2023 Trippe et al., 2022: Yim et al. 2023 Lin and Alquraishi 2023) use diffusion models (Sohl-Dicksste ct al. 2015) to learn the diverse distribution of protein structures and permit the sampling of new, potentially novel proteins. Consequently, an important design task imposing the existence of a functional substructure called motif. While works have demonstrated their ability on this task, the motif scaffolding problem, along with its variants, remains significant challenge.
Amortisation has been the most successful approach, whereby the diffusion model is trained to condition upon the motif as label (Watson et al. 2023 Lin et al. 2024 Didi et al. 2023). However, with the growing list of design requirements, e.g. secondary structure or binding aflinity, training for each may be too expensive, especially when requirements are composed or the underlying model is replaced, prompting the need for retraining. Independent of this, problem-agnostic posterior sampling techniques have been used solve numerous inverse problems with diffusion model acting as prior :Chung al.) [0023] [Snn et al. 2023). These methods can further guarantee asymptotically exact sampling when paired with sequential Monte

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By formalising motif scaffolding as an inverse problem, it becomes compatible with posterior samplers and solvable in wero-shot fashion. In particular, WW et al. (2023) and Trippe et al. (0022) have done this for the single-motif case by conditioning on fixed partial view of the protein backbone. However, while they have laid the foundation, their methods are not directly applicable the event of multiple motifs, which is an important but challenging design task. Moreover, SMC-aided diffusion posterior samplers have not been sufficiently compared in the protein domain, where scaffolding tasks are severely ill-posed and high-dimensional but with solutions that can be quantitatively evaluated. We aim to address these concerns.
Carlo (SMC) algorithms (Cardoso et al. 2023 Dou and Song 2023 Wu et al. 2023
In this work, we make the following contributions:
We propose a set of guidance polentials for various scaffolding tasks. Notably, we explore alternatives to masking for guiding the denoising process. These potential functions permit a natural extension to scaffolding multiple motifs while having comparablc, if not better, single-motif in-silico performance Ihan masking. Additionally, we provide potential function for developing monomers with possibly loose internal symmetries that achieves designability in the cyclic and dihedral case.
We provide thorough comparison of recent SMC-aided diffusion posterior samplers on motif scaffolding. We identify their design choices, report their empirical costs, and compare them to a baseline in the bootstrap particle filter.
We demonstrate that it is possible to jointly scaffold multiple motifs in zero-shot by pairing a reconstruction guidance-basee SMC sampler with one of our proposed potentials. We also point towards possible improvements sampler efficiency for future research.
The rest of this paper is organised as follows. In Section 2 we introduce some background to our methods, namely inverse problems, diffusion posterior sampling, and motif scaffolding. We also briefly review relevant literature and existing methods in relation to our algorithms. In Section 3 we present our proposed potentials across several scaffolding tasks while highlighting the intuition behind them. In Section 4 we describe our experimental selup and elucidate our findings. Finally, we highlight limitations and future work in Section 5
2 Background
Inverse Problems. Commonplace in many scientific disciplines, inverse problems require estimating a latent quantity x RDD given its measurement, y E TRd, an often ill-posed problem. Formally, we define it
y A(x) + n,
(1)
for some possibly nonlinear measurement function A: : TR - Rd and noise n ~ NO, II). Eq. 1 analogously
defines a likelihood
(-
20? (-203 lly A(x))I)),
1
g x) x exp
that we generally express in this work as
gly x) exp Lx(x:))),
for some > O and potential LA IR. - R. We adopt this notation to define arbitrary potentials but remark that our formulations largely remain Gaussian. Conveniently, the composition of several inverse problem constraints can then be represented by summing together individual potentials. Because inverse problems are often underdeterminee,, we may assume access to a prior distribution on the latent variable qxx) and aim to
sample from the posterior distribution q(x y), which given under Bayes' rule as q(x y) x (xx)g(y x).
Diffusion models (Sohl-Dickstein et al. 2015 Song et al. 2020 Ho et al. 2020 are class of generative models that Icarn the time reversal of diffusion process applied to target distribution g(Xn), the end result of which is a Gaussian q(xr) (0 I). This involves learning the score function Vx, log (xx;,, an otherwise intractable quantity, by parameterising it with a neural network. The time reversal forms a bridge between simple Gaussian and the target distribution, which permits sampling from the target by progressively moving Gaussian samples in the direction of the score. As such, trained diffusion models can be used priors for complex data distributions and their associated inverse problems.
Diffusion Models.
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Diffusion Posterior Samplers. Suppose wish to solve the noisy inverse problem y= A + with diffusion (model) prior Xo- To frame the posterior our new target, we can modify the reverse process, so it is instead steered by the conditional score Vx, lx,loggu ((x, y). This quantity can be decomposed into a
sum of two terms using the Bayes' rule as (Chung et al. 2023 Song et al. 2023 Boys et al. 2024
lx, Iy) - Xx, lx,g(x;) + Vx, logggy xx),
where Xx, log (Xx) is score of the prior (unconditional model) and Vx, xx) is called the guidance lerm. However, the guidance lerm is lypically intractable, as it involves a direct comparison between a clean measurement and a noisy latent variable. Existing works bridge this gap through approximations that essentially noise the measurement or denoise the latent variable.
Song et al. 2020) define a measurement noising process {yel that parallels the latent's forward diffusion process, where Yo = y represents the clean measurement and p(y, yo) gives the noisy measurement distribution at time t. They subsequently project the latent variables towards or orthogonally onto the constraint manifold (x, : A(xx) ye to maintain sample consistency with the measurement throughout the entire denoising process. They make the approximation
V x, log q(x, yy) = Xx, log / qxx, ye. v) ly)dy, = Vx, log / g(x; Iy) Iy)dy: x Vx, log (x: 8c) = Fx.log + Vx,logg(1 xt);
(3)
where it is assumed g(x, y) R q(x, y.) and v(y: y) oy, for ÿr vy: y)- This measurement
process is available under a linear structure to the inverse problem (see App. A.2 When A(-) is masking transformation, e.g. in image in-painting problems, the sequence ($r)7-1 is elfectively constructed by forward diffusing the measurement y. Here, the guidance term's contribution is equivalent to replacing or interpolating the masked segment of X2 with ý: (Song et al., (202).. This is known as the replacement method (Trippe) et al. 2022 Ho et al. 2022). More generally, we will refer to approaches that rely on the score estimate in eq. 13 as replacement methods in our experiments.
The replacement method becomes unsuitable, however, when the mcasurement procoss is intractable, with non-linear inverse problems. Replacement in the linear masking case has also been empirically shown to lose coherence between the masked and unmasked regions (Ho et al. 2022 A Alternatively, Chung et al. 2023 propose the diffusion posterior sampling (DPS) method, using the approximation
= I s(yix:) =
~ g(yl 8o (x),,) f)),
where it is assumed (xx))xx)) xe (xo(xx,)), and so(xx, t) = E[xo xx] is the reconstructed (clean) sample. This leads to the score estimate
Vx. logg ly) RO Vx, lx. + Vx, gy (8((x,,)) f)),
where g(y (x,, t)) = Ny:; t)),, III), whose logarithm's gradient computed via backpropagation. We refer to this method as reconstruction guidance.
The above methods rely single-sample approximations. One can instead devise Gaussian approximations for q(xolx;) (Song et al. 2023 Boys et al. 2024), but this too can be strong assumption. Taking a different approach, one can adopt sequential Monte Carlo (SMC) algorithms (Del Moral et al. 2006) to correct such errors, given that it permits asymptotically exact posterior sampling with added liberties in the algorithm's 's construction, namely the proposal and intermediate target choices. SMC methods are often used in the context of state space models (SSMs), with their Markovian structure and sequential measurements. A diffusion model's reverse process naturally fits this framework, with the exception of measurements taken at intermediate steps. After all, the inverse problem is only defined samples in the final target. A workaround is to adopt probabilistic models with single terminal measurement yo or artificially construct sequence of measurements [ye)T- However, the former produces low likelihood samples if filtering is only done at the last step, and the latter imposes a realisation of measurements that may not be easily satisfied. We discuss two ways adopted by SMC methods to mitigate these issues.
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First, twisting Pitt and Shephard 1999 Jo-
Algorithm 1: Recipe for an SMC Diffusion Posterior Sampler
hansen and Doucet 2008) is an SMC tech- nique that improves sampler efficiency by accounting for measurements in future it- ions.ons. It involves tilting the intermedi- ate targets by some twist function tt, with th - vT- = 1, maintaining the initial and final targets. Twisting allows the terminal measurement to be accounted for earlier in the process and for smoother transitions belween sequential measurements. Second, the reverse diffusion kernel proposal can be further tilted by the likelihood to obtaín the (locally) optimal proposal (x X Vt); choice that minimises the variance of im- portance weights. This permits the condi- tional signal to be present beyond just the resampling steps. These building blocks are especially important when working in high dimensions, where a correct but inefficient sampler may face frequent weight degenera- cies. Alg. provides a general recipe for SMC diffusion posterior samplers.
input measurement ye, proposal (t, likelihood It.
reverse diffusion transition kernel po, no. of particles K, no. of time steps T', (if replacement method) noisy measurement dist. V, (if twisting) twist functions de (otherwise set 1/: = 1)
output: approximate posterior samples K
# Create a sequence of measurements
for recoustruction guidance
Sample yr (v(- for replacement method
for tort=1m 1=1,...,?
# Define transition betueen targets Set fe-1(Z-)) 2i)
pa p (x) p 2i (yo x(-1) if ==t=) measurement
if 1 for terminal
po Zi ) (yc-1 Z(-1) for sequential measurements
# Generate samples guided by measurements Sample x ~ N(O.T) Set vß + (yr x7)) for i I....,K K
for t= TT,,..., do
Set w/ t wi// اØ= w/ for i l,,...; K Resample x; 1:K ~ Multinomial(ue!" x, x
The replacement method precisely creates the sequence of measurements that can be used together with SMC. Trippe et al. 2022 apply filtering with the replacement method as part of their SMCDiff algorithm to solve in-painting problems. They maintain the original reverse kernel as their proposal, but end replace the masked segment with a noisy version of the measurement. For linear inverse problems in general, Dou and Song 2023 employ the optimal proposal and a noise-sharing technique in their method, FPS-SMC. In the in in-painting case, their proposal smoothly interpolates the masked segments with the noisy measurements, which share the same noise applied to their latent counterparts. The consistency of both methods, however, relies on the forward and reverse processes having the same joint distribution, which is often untrue in practice (see (Trippe et al.) 2022 App.
for i= L......, K do
Sample xi_ 9t-1 (: (.1 xx)
Set w t- J (8) xi) 2he (8(-1)
g (2 x!)
(x{)
end
D.2) and (Dou and Song] 2023 Prop. 4.1)). Maintaining consistency without this assumption, Cardoso ct al. (2023) use the same optimal proposal but adopt terminal mcasurement model with targets tilted by the likelihood of sequential measurements. Their MCGDiff algorithm solves general linear inverse problems. While they do not explicitly noise the measurements, they use the mean of the measurement distribution as a single sample approximation in eq. (3). In parallel, Wu et al. (2023) use reconstruction guidance together with filtering for solving general inverse problems. They similarly adopt terminal measurement model but have twist functions and an optimal proposal approximation based on eq. 4.. Their method, TDS, provides state-of teate-of-the-art performance in motif scaffolding among training- free methods. Specific choices in proposals and twist functions of cach sampler, together with their weight updates, are laid out in App. A.2
Other works that fit. within this framework include SVDD (Li et al., 2024 an instance of the bootstrap particle filter (BPF) equipped with the likelihood in eq. 4 which avoids backpropagating through the score model, and PDDS (Phillips et al. 2024), & sampler that reframes the sampling problem as diffusion posterior sampling, leveraging
diffusion paths, and uses SMC with twisting to correct for errors. A unique feature of these SMC methods is their increasingly accurate posterior sampling at the controlled expense of larger number of particles.
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Motif Scaffolding. A protein's function is characterised by several of its structural domains. As result, motils from known domains can act as foundation that, when scaffolded, can give rise to proteins with specific functional or conformational properties. The process of fixing a substructure is non-trivial, however, as the protein's folding dynamics must be respected. Generative models on protein structure can be leveraged, given their inherent understanding of well-folded structures. RFDiffusion (Watson al. 2023 and Genie2 [Li et al., 2024) are current stato-of-the-art diffusion models for this task, capable of scaffolding several benchmark motifs (Watson et al., 2023), from small molecule binding sites viral epitopes. More generally, one can also scaffold multiple motifs simultaneously, e.g., to map out different functional domains. Multi-motif seaffolding is more challenging as interactions between motifs must be accommodated by the scaffold. This differs from scaffolding a single motif with fixed discontiguous segments, as each motif can be oriented independently.
Another useful task is the generation of oligomers embodying some point symmetry, i.c. through the self- assembly of multiple identical monomers (Watson et al. 2023 Ingraham et al. 2023). We additionally bring altention to the design of internally symmetric monomers which fits with current monomerie generative efforts. Symmetric motif scaffolding demands symmetric protein structure with each asymmetric subunit containing a copy of the motif. For example, the SARS-CoV-2 spike protein, a C3 symmetric trimer, may be inhibited by attaching a similarly symmetric binder- feat RFDiffusion demonstrated in-silico.
In these scaffolding tasks, it is common to sample from a range of positions in the backbone and fix the motif place. WWu et al. (2023) explored allowing multiple motif placements to jointly considered during the denoising process, We refer to this as scaffolding with degrees of freedom. All the scaffolding tasks are illustrated in Fig.
LAM DAL FIAIR
MAM
&
Scaffold
NMN-NNN
PPP
MNM?NNN?
?PPP?
HKN?PPР?NNN Multi-Motif
?PPP ?PPP? PPP?
Motifs
Single-Motif
Symmetrie
With DoF
Figure 1: Different motif scaffolding tasks. discontiguous. Scaffolds are illustratdd in white.
The motif in blue is contiguous, and the other in red is
Related Work. We focus on solving motif scaffolding problems using an unconditional protein backbone diffusion model without additional training. Several works have previously explored the single-motif case in a similar vein. SMC-based methods include SMCDiff, a replacement method paired with ProtDiff (Trippe et al, 2022), and TDS, reconstruction guidance method paired with FrameDiff (Yim al. 2023). similar work, Chroma (Ingraham al.) 2023), provides a suite of conditioners built around an unconditional model and addresses substructure motifs through reconstruction guidance and root mean square deviation (RMSD) metric. We believe the latter idea can be incorporated without additional scaling hyperparameters.
On scaffolding multiple motifs, diffusson models have been trained to condition on pairwise residue Co distances, allowing motifs to be represented independently of each other. Genie2, an extension of Genie [Lin and Alquraishi 2023) under this scheme, achieves in-silico success on some of its curated multi-motif benchmarks. On the other hand, [astro et al. (2024) demonstrate experimental success in jointly scaffolding three epitope motifs using RFJoint2 (Wang et al. 2021), modified
network for in-painting. Taking inspiration from these efforts while side-stepping the requirement for training, explore using distances as part of the inverse problem.
Lastly, on diffusion posterior sampling, replacement methods such as FPS-SMC (Dou and Song (2023) and MCGDiff (Cardoso et al. (2023) have not been explored in the motif scaffolding setting. SMC methods in general have also yet to be compared with a bascline such as the bootstrap particle filter (BPF). Formalising the scaffolding inverse problems, we aim investigate which samplers and formalisations perform best for the tasks.
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3 Guidance Potentials for Motif Scaffolding
The motif scaffolding problem has structure and sequence requirements, only the former of which is accossible to a typical backbone diffusion model. Hence, our focus is on the structural aspect of the problem. In this section, we provide different ways (in particular a number potential functions) to encode the scaffolding problems to be solved by diffusion posterior samplers.
For convenience, we work with the flattened representation each represented by its three-dimensional Ca coordinates. From these coordinates, we may represent the protein as a set of rigid body frames, where the ith residue is given by T,(x) = (x) t,()) E SE(3), a pair of rotation matrix and a translation vector with respect to global reference frame. For a proper rigid transformation T E SE(3), we denote by T o x the result of applying the transformation to each of the residue coordinates of the protein.
RR&,, protein backbone with N residues,
3.1 Single-Motif Scaffolding
We define the motif and scaffold index sets (M,S) as a partition over all the backbone coordinate values, with each residue's coordinates belonging the same set. Furthermore, we assume ordering on (Mi)) and {S:),-o first according to residue number and then coordinate axis. Given the Co coordinates of a molif m E R MM,, we set our measurement to be y = m. The (single~) motif scaffolding problem requires sampling from the distribution p(x XM = y), where the conditioned cquality holds when the motif, oriented in whichever way, appears in the backbone region specified by the motif index set, i.c. there exists a T SE(3) where XM T y.
This task is commonly framed a linear inverse problem by fixing the motif's orientation and conditioning on a partial view of the backbone. Here, we use a linear transformation =
= Am E RlM]xaN, given by (AM)i = SM to mask the backbone and allow a direct comparison with the motif, which is translated to share the same centre-of-mass (CoM). The corresponding potential is thus given by
Lmuask (x; y. lly AMx
(5)
Throughout, we equivalently use X.M in place of AMx. This approach is taken in several posterior sampling works (Trippe et al. 2022 Wu et al. 2023), drawing parallels to image inpainting. However, unlike images, a
protein motil can further be oriented in 3D space, and fixing it disrupts the equivariance of the denoising
process while posing questions on how multiple motifs may be handled. To address these concerns, we now introduce guidance potentials we develop in this work as alternatives the masking potential (5)
Distance. Notably, Wu et al. (2023) improved masking performance by parameterising over finite set of possible motif orientations. This is likely because we impose narrower path towards the posterior distribution without the additional degree of freedom. Motivaled by this, we consider conditioning on the full pairwise Ca distances of the unmasked backbone, Denote the pairwise Euclidean distance matrix derived from two flattened 3D coordinate vectors u, V E R as Done Ruxn, where (Duye); = [t,(() tz(y)]l. We define the potential as
Ldist (x; y, M) D
Doue D
where F| is the Frobenius norm. Contrary to masking, this approach yields an orientation-free motif representation. Its indifference to reflections, however, violates the chirality of proteins, and we explore this drawback our experiments. The above is akin an unnormalised dRMSD metric (AlQuraishi 2019
Distance We propose to break the ieleatiion-invaraance of the distance potential ( by further conditioning the backbone's pairwise orientation deviations within its frame representation. Similarly, we define distances between the rotation matrices [x).). First, we remove the dependence on the protein's 's current orientation by computing the relative rotations R, (R) use the result. measuring the cosine of the angular difference between two rolation matrices
R& for every residuc pair k). Then, we
1 (eo, R2) = (Tr (R,R,) 1), 2
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to compare the sample's relative rotations with those of the motif. We define
Lihiral (x: yx y, M) |11 - D
y.x li:
where (D yx ) dens (y) (R, RR
We keep the angle in its cosine form due to the instability of arccosine gradients near one. Summing with the previous formulation, we have the potential
Liramddist y. = Laist ( M) + Iehhirll (x.y.
(7)
where Vchiral is a hyperparameter to scale the magnitude of the chiral contribution.
Frame-Aligned Point Error (FAPE). A similar idea of an 3E(3)-inva measure with chiral properties is the frame-aligned point error (FAPF) the main component of the AlphaFold2 loss function (Jumper ct al. 2021 FAPE defines distance metric between two sets of rigid body frames of the same dimension and allains value of zero when the two sets of frames are identical up to proper rigid transformations. To match the magnitude of the masking approach, we modify the original formulation to yield the potential
M
(xa (,
Lmaak (T.(xM) o T.
M
where T-1 is the inverse of the transformation T. For each motil residue, essentially set its Ca atom the global reference, allowing both the specified backbone region and the motif to coincide at the residue and their deviation to be computed.
Root Mean Squared Deviation (RMSD). A common measure for structural similarity between protein structures is through their root mean squared deviation (RMSD). For protein backbones u, g each with n residues, we have
RMSD(u, v) min 1 T*ES((3 vn
||u T vll
Similarly, we define potential matching the form and magnitude of masking
Leanddx; M) = M RMSD(xM.,) min T ESE(3)
Lxmak(x; T y,
Here, we can obtain the minimiser transformation T (R", t²) using the Kabsch algorithm (Kbschh 1976 or directly compute the RMSD through quaternions (Coutsias et al. 2004). Both methods are differentiable.
A consequence of the proposed potentials' -E(3)iinvari is their gradients, used reconstruction guidance, maintain the equivariance of the denoising process -i.e. the global orientation of noisy samples do not affect. how they are denoised.
3.2 Multi-Motif Scaffolding
Now, suppose the index sets {M'... (M'...,, MM S form partition over the backbone coordinates and are each ordered according to residue number and coordinate axis. Given motifs m m., we define the measurement y E RRN as ys = 0 and yM = m; for i E [1, MJ. The multi-motif scaffolding problem requires sampling from the distribution p(x XMI = YMM..... XMM = = MM),, with the conditioned equalities similar as before, i.e. there exists T E SE(3) where X. = T y. for [1, M].
Unlike in the single motif case, the masking approach fixes the motif-to-moit orientations and severely underrepresents the posterior distribution. Instead, may the other proposed approaches to keep each motif's orientation independent of others. This can be achieved by summing the polentials
M
I M.M
(X.,,M) MM.,,.MMMM
MM) = Σ LM MX))
1=1
where L'M is the chosen potential corresponding the ith motif. While is possible use different potentials for each motif, we choose to keep it fixed in analyses.
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3.3 Symmetric Generation
For some point symmetry group in R2, define g = {gx )n-1 as the set of all its symmetry operations. We consider designing internally symmetric monomers as we focus on diffusion models that produce a single chain. However, our formulation can analogously be applied to models supporting multiple chains to design symmetric oligomers by treating each subunit as monomer. Suppose the chain composed of n identical subunits with N divisible by In dealing with 3D atom coordinates, & is a set of transformation matrices in R3x3, Without loss of generality, select go the identity matrix. We can then construct the potential
Lg(x) = l1(Ag LIN)x|1
where our mcasurement is implicitly set to y = and Ag E R3Nx3N is given by
diag(go..... ............................ gu)
Ag diag(gn Bn-1)
composed block diagonals diag(g...... ....g................ g+) ) E R3N/nx3N/n, Effectively, this constrains the generated protein to be identical to several symmetric projections of its first subunit. This partitions the chain into n contiguous segments representing each subunit. However, one can shuffle the block diagonals between group operations to render the subunits discontiguous. We demonstrate this process for cyclic and dihedral symmetries.
3.3.1 Cyclic and Dihedral Symmetries
Proteins with cyclic symmetry Cn are invariant to any integer multiple rotations of 2 /n with respect to given axis. Without loss of generality, we choose to work with the z-axis and accordingly translate so its CoM lies on it. Denote R 3333 as the rotation matrix that rotates vector anti-clockwise about the d-axis by an angle of 0. As such, we have the set of rotations 9c., = 2. 1n-1 While any ordering of the set G is conducive to producing a cyclic protein, it may be favourable to have adjacent angles for adjacent sub-sequences, c.g. gx = R2,2nk/n.
Proteins with dihedral symmetry D, similarly have Cn symmetry in one axis but have C2 symmetry in another axis orthogonal to the first. We choose the and y-axes as primary and secondary axes of symmetry and translate x to have its CoM at the origin. Thus, we have Gp. = Gc. U (R 1n-1 k== We may then substitute Agc, and Ago, into our symmetry potential to impose cyclic and dihedral symmetries.
3.3.2 Symmetric Motif Scaffolding
In addition to symmetric constraints, we may also condition the existence of a motif. Note that we can assume the motif is local to exactly one subunit and n copics cxist in the others. Otherwise, if the motif lies on the boundary between subunits, we can redefine the motif to be the residues entirely situated in one subunit. Suppose then that the motif is in the first subunit, ie. M; S 3N/n for all
We may adapt the masking approach with motif measurement, y and define the potential
(x:mass(xx y, [|[em, eMMM 1y (Ag LsN + diag(1 x||
where e; E N is the ith standard basis vector. The additional diagonal term unmasks the motif indices and asserts it equal to the chosen motif y. We remark that this preserves the linearity of the inverse problem. More generally, e.g. for the distance approaches, we can simply add the symmetry potential Lg onto the motif potential. For masking, however, we choose to combine the linear measurement functions so a single matrix may be supplied for supporting posterior samplers.
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4 Results
4.1 Experimental Setup
we use an in silico self-consislency pipeline for measuring the quality of protein backbones. The procedure involves an inverig-folding network and a structure prediction network. We use ProteinMPNN (Dauparas et al. 2022 and ESMFold (Lin al. 2022p, respectively. First, the inverse-folding network prodicts cach generated backbone's representative amino-acid sequences. Then, the structure prediction network folds these sequences into structures. The celf-consistency root mean squared devialion (scRMSD) between the generated and predicted backbones are then computed, and the smallest is reported. The premise of this technique is that generated backbones possessing natural structures are likely to be represented consistently across orthogonal methods. We use the available in-silico design pipeline provided by the authors of Genie The set-up is summarised in Fig.
Similar to cxisting works (Tripee ct al. 2022 Watson et al., 2023 Lin and Alquraishi 2023
Designability and Diversity Metrics. We adopt a similar designability criterion as Lin al. (2024) We consider a protein backbone designable if it deviates with the most similar predicted design by most two Angstroms (scRMSD < 2A) and if the designs are confidently predicted (pLDDT 2 70). For motif scaffolding tasks, we consider scaffold to be successful if it is designable as above, the motif is present in the predicted backbone within one Angstrom in alignment deviation (motif RMSD S 1A), and there is low predicted alignment error (pAE) between residues (pAE < 5). For multi-mouif scaffolding, every motif must be within one Angstrom. We remark that scRMSD in motif scaffolding differs from the unconditional setting, as inverse-folded sequences are conditioned to fix the motif's sequence. Furthermore, as success rates can be mislcading with identically designed structures, we also mcasure sample diversity. Designs are grouped using single-linkage hierarchical clustering with a distance threshold given by a TM-score of 0.6. We report the number of unique successful scaffolds for motif scaffolding tasks.
Conditional Setup. In our analyses, we use Genie Lin and Alquraishi 2023 an unconditional protein backbone diffusion model. Specifically, we use SENNE-SCOPER 128 and GENIE SCOPE-255 256,, models trained on proteins from the SCOPe dataset (Fox et al., 2014), capable generating proteins of up to 128 and 256 residues, respectively. For single-motif problems, where the overall length is at most 128 residues, we use GRNIR-SCOPR-122 for quicker inference times. For the multi-motif case, with problems requiring longer samples, we use GENNE-CCOPE-256 In each model, samples are denoised for T 1000 steps. We parallelise our setup across two NVIDIA GeForce RTX-3090 GPUs to accelerate the process but limit each motif problem to 32 backbones per posterior sampler.
motif RMSD
Motif
BEEDRFKLSFOR
AHASDFKI.SPPIC MAWEFFKLSFHNG
Unconditional Diffusion Model
Sample
Inverse-
Fold
Fold
p(x|x y))
Posterior
Generated Structure
Predicted Structure
Sampler
scRMSD
An overview of the motif scaffolding experimental setup. Protein backbones are first sampled from the conditional setup with the motif as an observation. These generated structures are then folded with the motif sequence fixed and folded back into structures. Finally, metrics such self-consistency RMSD and motif RMSD computed between the predicted structure and both the generated structure and the motif.
Figure 2:
The repository is available httpper//gthhh..oolaaaaaarttteee http://gaaaaaaaaoeeeleelooaaaaap

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Published in Transactions on Machine Learning Research (09/2025)
We examine the performance of posterior samplers across different motif scaffolding benchmarks. Under the replacement, method, we use FPS-SMC (FPSSMC) and the noiseless version of MCGDiff (MCGDIFF) and compare them with the bootstrap particle filter (BPF) as baseline. In particular, we define the BPF likelihoods by constructing a sequence of measurements {yr) either via forward noising the motif as in SMCDifT (BPF-rw) or using the reverse process as in FPS (BPF-BW). These are limited to linear inverse problems and thereby use the masking approach. Under reconstruction guidance, we use TDS paired with masking (TDS-MASK), distance (TDS-DIST), frame-distance (TDS-PRAMM-DDSTT),, FAPE (TDS-FAPE), and RMSD (TDS- RMMD)) guidance potentials.
We set the potential scales to be тι = 1/ for replacement methods and 1/: = 1/((11 - 5.) + oz) for reconstruction guidance, under a fixed likelihood standard deviation o, = 0.05. We also fix the number of particles to be K = and perform adaptive (residual) resampling, with resampling taking place when the effective sample size becomes less than half the particles. We additionally perform hyperparameter searches (sec App. D.I) jointly for each sampler and corresponding guidance potential based on two motif problems: TIXT and 1PRW contiguous and discontiguous problem, respectively.
4.2 Single-Motif Scaffolding
We sampled proteins conditioned on motifs from the RFDiffusion benchmark WWtson et al. 2023 Twelve of the 24 problems had at least one successful solution among the 32 designs generated for each problem. Fig. B summarises the performance of samplers across the benchmarks. Among the methods, TDS-RMSD solved the most, with (en problems. The non-linear potentials paired with TDS showed a comparable, if not higher, number of unique successes over TDS- TDS-MASK for several problems. On the one hand, this shows the viability of masking for singlo-motif scaffolding, maintaining the lincarity of the inverse problem and being broadly applicable to many posterior samplers. On the other hand, the non-linear potentials, with their comparable performance and generalisability to multiple motifs, present a case for being a drop-in replacement.
We tested our BPF samplers as baseline to see if resampling provides sufficient conditional signal to warrant keeping the proposal unchanged. Results of the replacement- based BPFs and TDS hyperparameter search (App. D.I). in which a zero guidance scale yields a twisted reconstruction guidance-based BPF, indicate that tilting the proposal substantially improves performance and may even be necessary for complex motifs. Intuitively, when weights are degenerate and all particles are identical, BPF chooses the most favourable noise leading to the motif's formation. But as motifs grow in size or deviate from common substructures, even crude approximation of the conditional score is crucial to move in the right direction in such high-dimensional space.
Between the replacement methods, we found MCGDIFF to be overall more successful than FPSSMC. The asymmetry between Genic's forward and reverse processes was likely incompatible with the lattor's consistency requirements. However, both methods were outperformed by TDS paired with various guidance potentials. This comes despite reconstruction guidance viewing the motif's formation as an optimisation problem, uncertain to be solved at the end of sampling, unlike in replacement methods.
Given the diversity of motifs, we provide additional context to elucidate our results. Motif problems 1BCF, 1PRW, and 2KT8 are discontiguous, restricting the possible conformations the scaffold can take, and have scaffolds substantially shorter than the motif. Because of this, valid scaffolds have minimal variation, resulting in low diversity among successful designs. Hence, we are more interested in whether success is achieved rather than its frequency. State-of-the-art methods yield one unique success for each of the three problems, alboit with the low diversity due to one thousand backbones being sampled (Lin et al., 2024). In 32 samples, TDS-MASK, TDS-FAPE, and TDS-RMSD were successful in all three. The distance approaches had mixed successes on this front but had several designs that missed out only one of the four success criteria. In contrast, more successos were achieved for contiguous motif problems 1YCR, 3IXT, and 6EXZ, with TDS-FRAME-DIST and TDS-RMSD attaining up to six unique successes in the 6EXZ problems.
Notably, problems such as 1QJG, 5TRV, and 7MRX had several designs with low scRMSD but few with low motif RMSD Consider, for example, the 1QJG motif made up of three residues separated by some scaffolding. Many protein backbones can satisfy three residues being at these distances away from each other, but only a subset of those backbones can be achieved by folding protein with a particular trio of amino acids in its
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(A)
1
BPF-BW
1
FPSSMC
Unique Successful Scadfolds
MOCDIFF
1 2
1
1
TDS-MASK
1
8
23
0
TIS-NIST
TDS-FRAME-DIST
11
1
2
TDS-FAPE
23
1
IDS-RMES
16
1VO
1
83
28
15
V
15
29
RPF-PW
8
1
80
1
6
8
9
Unique scRMSD
5
8
1
1
8
4
20
MOGDIPP
6
TDS-MASK
24
R
16
11
83
5L
57
101
10
17
10
4
TDS-DIST
16
18
11
8
11
13
18
06
20
12
12
22
TIDS-FRAME.DE
1
19
B
X
8
8
10
13
12
4
TDS-RMSD
1
1.7
30
BPF-PW
1
2
2
2
BPT-RW
Unique Motif RMSD
83
1
26
8
MOGDIFF
123
11
22
8
8
22
9
28
TDS-MASK
TDS-DIST
LA
19
1
4
10
10
8
TDS-FRAME-DDEY
18
11
8
11
15
2
TIS-PAPE
1
0
8
TDS-RMSD
1BCF
1PRW
1QJG
IYCR
2618
aXX
4JHW
4ZYP
sIUs
ETPN
long
mee
SYU1
GEen, long
GEAR. med
GEaR short
dEXZ. long
GENZ short
und
TMIX 128
TMIX 60
7MHXX 85
STRV
STRY
STIY
GEXZ.
(B)
Successful
motif RMSD IA
scRMSD 2A
1PRW
2KL8
3IXT
4JHW
5WN9
Figure 3: (A) Performance of sampling methods on the 24 motif scaffolding benchmarks. Thirty- two backbones are sampled from each method across all the motif problems. Scaffolds that are successful and those which meet at least one of the main success criteria are reported according to their unique count. (B) Examples of the designed scaffolds. The motif, in grey, is aligned with the scaffold, in white. Most unsuccessful scaffolds either do not possess the motif full or have poor self-consistency..
sequence. In other words, the motif's sequence critical to defining the support of our target distribution, which is currently underspecified. We believe that our setup already successful in sampling from the much larger support of natural structures possessing the backbone folds of the motif. However, more samples are required to cover the subset of structures with the motif's sequence. To test this hypothesis, modify the self-consistency pipeline to remove the conditioning of the motif sequence in App. D.2 Indeed, our setup had more successes overall and had solutions for all seven of the IQJG, 5TRV, and 7MRX problems, indicating that the backbones generated could be designed with an alternate sequence but were incompatible with the motif sequence. Given our setup only conditions on structure, we believe incorporating sequence information can substantially improve performance select motifs.
We remark that limiting setup to single motif placement in each problem may have resulted in too restrictive motif placements. Common approaches that could be employed are uniformly sampling valid placements or allowing multiple placements to be considered during denoising, as done by Wu et al. (2023). We exclude these in the study to minimise the variance of our experiments, given our choice of relatively smaller sample sizes. With more samples, the diversity of successes is expected to diminish. It may then be preferable to use methods that generally produce more diverse samples, as they have a higher ceiling for unique successes. We hypothesise that methods that do not fix the motif's orientation allow for more flexibility during denoising, enabling designs to be more diverse.
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(A)
I'SSDIIT
TDS- IDS-FRAME-DIST RAME-DIST
TDS-FAPE
MDS-RMS
(B)
Average
10
scRMSD (4)
scRMSD S
10
IPRW two
scRMSD 1.822A motif RMSD 0.872A
PLDDD 82.877
Average Max Motif
10
pAE 4.252
RMSD (4)
Max Motif
25
RMSD & 14
33IK+3BP5
Correct Helix
100
Handedness
scRMSD 0.901A motif RMSD 0.926A pLDDT pAE 2.939
(%) (f)
1PRW
1PRW
3NTN
3BIK-3RP5
2A51
4JHW+55N3
Figure 4: (A) Success metrics of sampling methods on the six multi-motif scaffolding benchmarks. Thirty-two scaffolds are sampled for each problem. Values for pass in each criterion denoted by the dashed line. Error bars shown are one standard deviation from the mean. Only samples with correct handedness were considered. (B) Examples of successful designs from the 512 samples generated via TDS-rmsd. The motifs, in colour, are aligned with the scaffold, in white.
4.3 Multi-Motif Scaffolding
We tested the non-linear guidance potentials paired with TDS on the six multi-motif problems from the Genic2 benchmark. In our initial assessment, where 32 designs were produced for each problem, none of the methods succeeded, but some designs narrowly missed the success criteria. Our reference, conditionally trained model Genie2, solved four problems with less than 20 unique successes among 1000 samples for each problem. To match the same order of magnitude in samples, we chose the most promising method, EDSS-MMS. and produced 512 samples for problems 1PRW __wo and 3BIK+3BP5. Our choice is motivated by the observation that some TDS-RMSD designs in the two problems successful when alternate motif sequence was used. In our larger batch of samples, TDS-RMSD found one unique success in cach problem. Fig. f summarises our findings.
First, we review our initial assessment, where there is an apparent dynamic between the distance approaches. We found that, independent of the number of motifs, TDS-DIST was more likely to produce reflections when motifs had discontiguities, as indicated by its lower proportion of samples with correct handedness. TDS-FRAME-DESS had fixed this issue in virtually all samples, only behind TDS-FAPE and TDS-RMSD, which maintained the correct handedness throughout. Unlike the latter two, TDS-FRAME-DDE hyperparameter Mchiral that scales the chiral contribution of the likelihood. We hypothesise that a good setting should account for the contiguity of motifs, but principled choice is difficult to make priori. For these reasons, TDS-FAPE and TDS-RMSD may be more suitable, as they work right out of the box.
has an additional
In our attempt to scaffold two multi-motif problems with more samples, TDS-RMSD could only produce one unique success out of 512 generated backbones. Fehoing our findings in the single-motif case, we attribute this to our fixed motif placements and the lack of sequence information in our inverse problem formulation, both of which are addressed by baseline in Genie2. To put this in perspective, there around one million possible motif placements for 1PRW two, varying between 120 and 200 residues in total length. With the goal of demonstrating the fcasibility of our setup, we used the same placements as our initial assessment, which, from our metrics, had hinted at its suitability. A variable motif placement is especially crucial for 3BIK+3BP5, as nearly all designs were grouped in the same cluster. Moreover, without accounting for sequence information, we may need to sample far more cover the subset of backbones that can be attained by folding an amino acid chain containing all the motifs' sequences. This is evidenced by the higher number of successes among methods when the motif sequence is not fixed in the evaluation process (see App. D.2)
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Despite our low success yield, remain optimistic that future setups addressing our concerns even- tually narrow the gap between zero-shot and amortised scaffolding of multiple motifs, with our result as a demonstration of what is possible.
4.4 Symmetric Generation
Using replacement and reconstruction guidance methods, we tested our sym- metric formulation by generating inter- nally symmetric monomers for cyclic and dihedral point symmetries. Several met the designability criteria shown in Fig. 5 Structures were less des- ignable at higher orders of symmetry, as the subunits became increasingly short. Several desigus were toroidal, with some resembling TIM barrels.
(A)
FPSSMC-128
TDS MASK-128
PPSSMC-256
TDS TDS-MASK 256
Unique 10 Designabie Scaffolds
20
Average scRMSD (4)
10
C-8
(B)
While, method implicitly imposes symmetry, the generated backbones were all found to be symmetric. We attribute this to the tight inverse prob- lem variance and the samplers guiding the backbone in sufficiently meeting the inverse problem formulation. This implicit approach has some advantages over explicitly symmetrising backbones at each step. It allows for control over looser symmetrics by increasing the variance o. This widens the target space to include commonly observed monomeric proteins with non-cxact in- ternal symmetries. It is also compos- able with other constraints without having to orient the asymmetric sub- units at fixed distances at each step.
L S 128:
C-3
D-2
D-3
D.4
L. S 256:
C-3
C-6
C-8
C-10
D-2
Figure 5: (A) Designability of symmetric designs across several point symmetries. Sixteen scaffolds with maximum of 128 and 256 residues were sampled for each symmetry through F PSSMC and TDS-MASK. The total number of designable scaffolds dashed atop the unique count. The success threshold for scRMSD indicated by the dashed line. (B) Examples of the successfully designed scaffolds. The first and second rows show designs with a maximum of 128 and 256 residues, respectively. The primary axis of symmetry points directly outwards of the page.
4.5 Computational Costs
The samplers and guidance potentials used throughout our experiments incurred different computational costs. We outline their differences to inform future motif scaffolding experiments.
Table summaariss the cost and estimated runtime of each sampler. We report the number forward and backward passes to the score model, as was the main bottleneck of the methods. TDS was the most expensive, having to backpropagate through the entire score model each time, regardless of its guidance potential pairing. Empirically, this resulted in runtimes up to three times longer than those of the next most expensive sampler in MCGDiff, with the backward passes taking about twice as long as the forward pass. While the rest of the samplers rely solely on forward passes to the model, we found that subtle design choice in their weight computations resulted in significant speed-ups between them. FPS-SMC and BPF with the noised measurements do not use the score model (i.e. to get E[xo xx] or E[x;-1 xx]) when calculating the weights. This enables the simple optimisation of only evaluating the score of the unique particles after resampling. Noting that the ESS (estimator) can be interpreted as rough estimate of the unique particle count, found that this optimisation evaluates the score of only fraction of the K particles each time, roughly equal to the average ESS across all time steps ESSavg- This led to speed-ups of up to K times when the weights were consistently degenerate, fairly common occurrence in high-dimensional problem such
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motif scaffolding. For completeness, also report the cost of BPF using the likelihood in 4 We did not evaluate it on the full benchmark, as its twisted version is special case of TDS with a guidance scale of zero, and we found this to be sub-par to larger scale values in our hyperparameter search.
# Model Passes
Est. Speedup TDS
Sampler
FW
BW
Runtime Complexity
Potential
BPF (with noised (yr) BPF (with approx. in 4
ESS
3K X BSS 3x
Mask
O(M)
KT
Distance
O(IM)
FPS-SMC
ESSavg T
0
ESS
3K X
Trame-Distanee
O(IMI)
MCGDiff
KT
3x
FAPE
O(MP))
TDS
KT
KT
1x
RMSD
O(LM)
Table 1: The number of score model passes and estimated runtime speedup over TDS for each sampler. Here, K is the number of particles, T is the number of denoising time steps, and avg is the average ESS taken across all time steps.
Table 2:
The runtime complexity of guidance potential calculations in terms motif length.
The calculation of guidance potentials also contributes to the overall runtime, notably going through automatic differentiation, albeit considerably less than the score model invocations. How the runtime complexity scales with the motif length is listed in Table 2 Depending on the problem, the motif may have a total length more than half the entire protein (e-g- 1PRW). However, we found the quadratic complexity to be non-issue given the relatively small maximum number of residues. In any case, masking, which maintains the problem's 's linearity, and RMSD, which has the most promising results, both scale linearly, With multiple motifs, the complexity is in terms of the longest motif.
5 Discussion
We proposed set of guidance potentials and evaluated various SMC samplers for the motif scaffolding problem. We produced successful scaffolds for several single-motif and multi-motif problems in zero-shot. On single-motif problems, our proposed potentials perform comparably to masking while generalising to the multi-motif case, and reconstruction guidance outperformed replacement methods when aided by SMC. Internally symmetric monomers were also successfully designed by our setup, and their composition with motifs left for subsequent work.
Our assessments were limited in the number of backbones sampled and motif placements considered. We welcome further exploration of the performance of various potentials with larger sample sizes and uniform sampling of valid motif placements. With scaffolding performance being only as good as the underlying unconditional model, the model-agnostic quality of the methods allows future, more performant models to be swapped in. Another major point of future work is incorporating sequence information in our sctup specify the correct target distribution and thereby improve the efficiency of samplers. Unlike structure, where bespoke potentials be crafted, sequence information must likely come from pre-trained models with inherent understanding of amino acid interactions. We have shown that it is possible to address multi-motif scaffolding without sequence information but believe it is essential to make zero-shot practices viable. Finally, several guidance potentials were motivated by work within the backbone modelling space and were rooted in the rigid body assumption involving idcalised bond lengths and angles. With all-atom diffusion models on the horizon, additional work can done to support these more general protein representations.
Acknowledgements
J.M.Y. would like to thank Joshua Southern for insightful discussions and a warm welcome to the world of protein design.
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A Additional Background
A.1 Diffusion Models and Protein Structure
Denoising Diffusion Probabilistic Models (DDPMs). Recall that diffusion models learn to reverse a diffusion process applicd to a target distribution ((xo). Here, we outline them in more detail under the framework of denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020). In general, their dynamics are governed by a forward (diffusion) process and reverse (denoising) process both of which are Markovian. Adopting the terminology of Ho et al. (2020), we start with the forward process with transition kernel
qxx x1-1) N (xL: V1 Bix 911).
(8)
for some decreasing variance schedule , Br E (0, 1). Equivalently, denoting == Br and ā = IT. Og the noising process can be done in single step via
(xx xo) = N (x:: Vaixo, 11-aj a))).
(9)
For large enough T, we have q(x xo) R N (xT; O, I). As a result, the forward process constructs a bridge from the data distribution (Xxs) to the Gaussian distribution N I). Naturally, the time reversal this process maps Gaussian samples back to the data distribution. DDPMs do this by learning mcasure the total noise accumulated from the forward process. Manipulating eq. 9 above, we can write
Xo
Xx - V1 - vi-a, where & ~ N(0, I), Voxz
(10)
which enables reconstructing the clean sample upon predicting the total noise. The reverse process, however, is done gradually rather than in single step through the transition kernel
po Xxx) = - (x--1 no(xi, Et))))
parameterised by
no(x,, 1)= = Va,(1 ax-1)
vot-1B 13
X1 +
& &x(xe,
Eo(() = BeI,
Ct
1 ar
0tt
cØ (xxx (x)
X;
Vat
v1 Oe.
where tx,, is the total noise as predicted by denoising network and *o(,,, is the predicted clean sample found by substituting eœ(x,, t) into eq. 10). The procedure to generate samples xo g(-) then involves the reverse process
po(Xo.T) p(xT) II po (x (xx),
first sampling from XT NO, I) then gradually denoising the samples for T steps.
A key connection be made with score-based generative models (SBGMs) (Song and Ermon 2019 in
their learning objectives. SBGMs predict the score function s(x) Vzlog (xx), vector that points in the
direction of areas with high density in the data distribution. Taking the gradient of the log density of eq. (9)
we precisely have
Ex(X,, t)
s0 5x(,,
VI a
When working with denoising networks, we this relationship to compute the score.
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(A) Generic Protein Backbone
(B) Rigid Body Frame Representation
Rest-1
Res1+1
N
Caiia
Cas
N
Cas
R
H
Res:
H
Figure 6: The protein backbone illustrated. (A) The backbone is linear, with a repeating N Ca - C atomic structure. Each side chain residue is anchored at a Ca atom, varying structurally depending on its amino acid type. (B) Fixing the N - Ca C substructures as rigid bodies, the ith residue can be represented as (triangular) frame, defined by a rotation matrix R, and translation vector t, with respect global reference frame.
Primer Protein Structure. A protein's tertiary structure its three-dimension arrangement in space-is often the modality of choice for generative modelling. A key reason is that applications are centred around designing proteins with some function, and while the sequence defines the protein, its function is more evident in its structure. Fig. G illustrates this structure, which comprises the backbone and side chains. Like most generative efforts, we limit our scope to monomeric (i.e., single-chain) protein backbones. Only after the backbone is modelled are the side chains predicted.
Since protein backbones high-dimensional, an efficient reprosentation incorporating their symmetrics is ideal for making their associated learning problems more tractable. First, we may assume idealised bond lengths and angles, allowing the Ca coordinates to be representative of the entire backbone. Next, we note that rotations and translations in threo-dimensional space do not change the inherent structure of the protein. These symmetrics, as well as the chiral propertics of proteins, associate them with the three-dimensional special Euclidean group, SE(3), where each transformation T (R, t) SE(3) a pair R 33x3 and translation vector E R3, Given the Ca coordinates x E gxx of protein backbone with N residucs, denote T x the backbone resulting from the transformation, where (T x)); Rx, + t.
rotation matrix
The frame representation (Jumper et al. 2021) incorporates the above symmetry group by treating each - Ca sub- structure as rigid body, whose orien- tation and position are indicated by transformation in SE(3) with respect to global reference frame. Here, we de- note the äth residue of the backbone as (x(x) = (R,(x), (x))) E SE(3), derived from the Ca coordinates x via the Gram- Schmidt process (Alg. 2 This formula- tion is used by several diffusion models Watsson et al. 2023 Yim et al.) 2023 Lin et al. 2024] to learn the distribution of protein backbones efficiently.
Algorithm 2: Frame Construction from Atom Coordinates (Adapted from Supplementary Material Algorithm 21 of Jumper et al. (202))
input :coordinates ith N, C-o, and C atoms x X.C. output: frame representation of ith residue (R, t.) # vectors pointing from C-a to and Set Va,1 + X1,0 XI,C
Set Va,2 <- Xi,x Xi,C # Do Gram-Schmidt process Set e,,1 + Vi.i/Ilvi.ll
Set u,22 v.2 0,,1 (e
Set e,2 +- uu,//||luu.
Set e.3 0,, 0,2
# Construct frame components Set R, (e ee, es,3)
Set t; +- X,CA
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A.2 SMC-Aided Diffusion Posterior Samplers
With SMC, we reiterate that two main design choices are available: the proposals dr and the intermediate targets Yt. In this section, we detail the specific choices made by the SMC samplers considered in our experiments. We first outline how a lincar structure to the inverse problem can define a noisy mcasurement process. This utilised by two existing samplers, FPS-SMC and MCGDiff, that operate on linear problems. Then, we discuss how each sampler fits the recipe in Alg. 1 Seeing as the weight updates can be a source of confusion, show their derivations. Where there are distributions that we are free design, time-index them.
Linear Measurement Process. We follow the construction by Dou and Song (2023) for producing measurement process under linear inverse problem
Axo + J.ZO,
where E R2, E RR, A E RaxD and Zo ~ I). Recall that the DDPM's forward process written as
S can be
X /1 3,x,1 + V B1z--1,
for Z;-1 2 ~1.1~~11 N(0,1) Applying the matrix A this recursion, we can define an analogous measurement process
yr = Biy--1 B.AZ-I,
where 2 is randomily sampled or chosen Lo be equal to the noise applied in the latent forward process. The intermediate noisy measurements can then be constructed by
T
v(yLT (yo) = II vly IYc-1);
(11)
where v(y, yc-1) Nyy VI- BLy--1, B,ATT).
Using the above recursions and the inverse problem statement, the likelihood has the form
glye xx) Nyy Axi, o?aaI).
(12)
and its logarithm's gradient is precisely the guidance term for replacement methods. From here, we develop simple particle filter by considering BPF, the baseline sampler for SSMs.
Bootstrap Particle Filter. For now, let us assume a (filtering) model with sequential measurements and the likelihood in eq. [2]. BPF, the proposal qi matches the DDPM's reverse kernel py and the target % is the joint distribution between the latent and measurement. variables. Under this choice, the weight update becomes
Px(x: X (ge xx) ú (X;
(v x2)
xx-1)
and the likelihood values essentially become the fitness criteria for filtering samples.
To improve the efficiency of BPF, we can choose to be the locally optimal proposal oi, better estimat- ing the target with fewer particles. This notion is formalised by minimising the KT-divergence between 7441 (x xx+1) and (x(x)) For SSMs, this is a known result with
de (x: Xx) x 7((xTT) = Po (x: Xe+ (4+11g(yy (x))
(x4411))
fe (x+41)
where fr is what we refer as the transition between targets. Note that this can be different for terminal measurement model, where do not need to explain each intermediate measurement, only the last one.
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Filtering Posterior Sampling. Dou and Song 2023
preciscly leverage the linear measurement process
and the corresponding optimal proposal. As the likelihood in eq. (12) and the reverse diffusion kernel are both Gaussian, the optimal proposal of is analytically available and is given by
qi (Xe. X+) = Nxxi n² Vt. + 1), E1 1)),
(13)
(+) (20(t)-1+ oGār-1
ATA
1 (xt, Yc-1 x Ee(t) (x(x).
1
A (1-1)
obal-1
With the optimal proposal, the new weight update is simply
(x X I (ye x1) Le
(x, x1)
However, instead of heward-nnising Yo, they find improved performance through a noise-sharing technique by setting yI = AxT and building the sequence backwards with
vyrs(y yo) N (y--1: vār- V (1-c)(1-ā--1) (ye Voryo), c(1 - 1-a:
for some tunable parameter E [0, 1]. We choose c= 11/(1 à=-1 to match our DDPM variance Eg 3,I.
Monte Carlo Guided Diffusion. Cardoso et al. (023) use the same optimal proposal of eq. 13 (for sequential measurements) but adopt terminal measurement model. To improve efficiency, they tilt the targets with the twist function the (y: xx)- Because of their algorithm's construction, where the final samples are not reweighted (i.. wh = 1), the weight updates shifted one time step in advance. For t >> 1,
(x x,)
9 9:-1 (y,-1 x(-1)
Po Px x2)
9t (v--1 x,-1)
0 of (x Xc-)
9: (ve xx)
Pe px( lec- (Mi-1 (
9: x²)
po (xt_1 1 xe)0
1 (ye-1 Xx- Xx- 1xx1)
X2 xx)99-1 (yr-1 X, x
9t- (yc-1 81-1)
(14)
(y xx)
9r (y xt)
where we used &,-1 E[[x-1 xe] as single-sample approximation of the integral. Note that, at ( 1, we get the same weight computation, the but the target transition contains an extra term go xx) to explain the terminal measurement. To maintain the correct targets in the midst of the shifted weight computations, the first weight. is set to be (T = 1 -) where & E[xr-1 xr].
Furthermore, instead noising Yo, they set y: = varyo, the conditional expectation of the measurement process given the terminal measurement. For experiments, we use the noiseless inpainting version of MCGDiff (i.e. o. = and A is masking matrix), which involves a similar proposal, but we omit the details here.
Twisted Diffusion Sampler. Wu et al. (2023) similarly adopt terminal mcasurement model, with the twist function vi = ge(yo &o(x,, ()) and proposal ,DS relying on the likelihood approximation in eq. Their proposal given by
TDS (x, Xx) = N dt
Xx;
(x++1 (1 o4) 1)), Eo(t))
(15)
Va,
where we have the conditional score approximation
iy Yo. t) s0(x,, t) Xx, logg(yo (x((x))) t)).
For t > 0, this leads to the weight update
Po (x: x(+1)
(y {x)( f))
W
q6
TDS (x: Xo41. Ic+ (yo 50(xx11,++0
which also coincides with the update for 0, as vho 1 and the terminal measurement has to be explained.
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Sampler
Proposal (1:)
Twist. Function (vi)
Measurement Model
BPF
Po(x Xx+1)
Any
FPS-SMC
(x, X(+1+
(eq.
Sequential
MCGDif
(7 Xx)
(eq- 13
Ir. (yx xx)
Terminal
TDS
DS (X: Xo)
(eq. 15
Gr (x),,tt)
Terminal
Table 3: The design choices of each SMC sampler for diffiusion posterior sampling. To define their intermediate measurements, FPS-SMC samples from the measurement noising process (11), and MCGDiff takes the expectation of the same process conditioned on the terminal mcasurement.
We have shown that these existing works merely differentiated by their design choices. Table 8 gives overview. While they all target the same distribution under the right assumptions, their efficiencies, or lack thereof, are more prominent in the finite particle setting. Even more, the high cost of the score model can impose significantly smaller particle counts. The high-dimensional nature of inverse problems, such as motif scaffolding, compounds the effect so that the weights become almost always degencrate. As a consequence, virtually all samplers will return multiple copies the same sample. This because particles have less room diversify towards the end of the reverse process, and resampling in the later stages becomes detrimental diversity. Some workarounds include resampling only until a certain point and running parallel instances of samplers. In any case, utilising the optimal proposal and twisted targets has become commonplace in the most performant samplers.
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B Likelihood Formulation
B.1 SEv3)-Invare and Chirality of Frame-Distance Likelihood
The potential in remains to be shown that the chiral component is invariant to rotations but not reflections.
has distance and chiral components- of which are translation invariant. It.
Suppose we have a protein's shree-dimensional C-a coordinates x. can be converted into a set frames {R,, t:) via Alg. 2 Note that the positions for the N and atoms are fixed, given the rigid body assumption, and are determined based the adjacent C-a coordinates. We consider the case of reflected protein Xxef and rotated protein X == Rax. Under the Gram-Schmidt process, we get the frame reprosentations
Xrot,i 1 (RoR,, Rot;). X + R (8 -t)),
First, the pairwise deviations between each residue are computed. For pair (i,j), we have
0 R. 0 1) R ( 0
0 R R2 ( -0-0)
R(xref,)) ((xrof.s)
0 R (Xxot,,) RR (Xrot..)) RR) RoR, R. R,R, R. R3:
which shows that such a deviation, and therefore the chiral component as whole, invariant rotations. We remark that this expression is different from R,R,,. the rotation matrix that transforms the jth frame's orientation to that of the ith frame. Next, the cosine of the deviations' angles computed. In the reflection case, have
deos RR.R,, Rj. R(xeet,) R(xxeet)))
(Tr R R
-)) o 0 R
-1).
Setting V = R.RR, R2 and expanding the expression, we have
(Tr(VVT) - + 123 + U/1 + v220) - 1)
2
= =1- (vīs + uz2 + U2g + 222)
where we have used the fact that Tr(VV Tr(II= = for rotation matrix V. For the above to be equal to one, i.c. the cosine of zero, must have V13 123 131 U32 0 which precisely requires R.R,R to be rotation matrix strictly about the z-axis which is not true in general. Hence, the orientation component is not generally inlectioniivvariat.
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C Experimental Setup
C.1 Benchmark Problems
In the following benchmarks, we fix the motif placement by taking the median of scaffold length ranges in their specifications like Wu et al. (2023). We isolate this variability to provide less stochastic comparison of the methods over small sample sizes.
Motif Scaffolding Benchmark. Problems in the RFDiffusion motif scaffolding benchmark listed in Table 4 Excluding 6VW1, which involves multiple chains, there are 24 problems involving different motif types, lengths, and contiguity. Where the length and configuration define a range, we can design any length scalfold that fits those specifications but choose to fix the configuration in our experiments.
Name
Description
Configuration
Length
IPRW 1BCF
Double EF-hand motif Di-iron binding motif
5-20, A16-35, 10-25, A52-71, 5-20
60-105
8-15, A92-99, 16-30, A123-130, 16-30,
96-152
A47-54, 16-30, A18-25, 8-15
5TPN
RSV F-protein Site V
10-40, A163-181, 10-40
50-75
STUS
PD-L1 binding interface on
0-30, A119-140, 15-40, A63-82, 0-30
57-142
1
3IXT
RSV F-protein Site II Carbonic anhydrase active site
10-40, P254-277, 10-10
50-75
5YUI
5-30, A98-97, 5-20, A118-120, 10-35,
50-100
A198-200, 10-30
1QJG
Dellab-3-ketosterrodd isomerase
10-20, A38, 15-30, A14, 15-30, A99, 10-20
53-103
active site
1YCR
P53 helix that binds to Mdm2
10-40, B19-27, 10-40
40-100
2KL8
De novo designed protein
A1-7, 20, A28-79
7MRX
60
Barnase ribonuclease inhibitor
0-38, B25-46, 0-38
60
7MRX
85
Barnase ribomuclease inhibitor
0-68,
85
7MRX
128
Barnase ribonuclease inhibitor
0-122, B25-46, 01-22
128
4JHW
RSV F-protein Site 0
10-25, F196-212, 15-30, F63-69, 10:25
60-90
4ZYP
RSV F-protein Site 4
10-40, A422-436, 10-40
30-50
5WN9
RSV G-protein 2D10 site
10-40, A170-189, 10-10
35-50
STRV_she short
De novo designed protein
0-35, A45-65, 0-35
5TRV med
De novo designed prolein
0-63, A45-65, 0-65
5TRV long
De novo designed protein
0-95, A45-65, 0-95
116
6E6R short
Ferridoxin Protein
0-35, A23-35, 0-35
48
6E6R med
Ferridoxin Protein
0-63, A23-35, 0-65
78
6E6R long
Ferridoxin Protein
0-95, 233-35, 0-95
108
6EXZ short
RNA export factor
0-35, A28-42, 0-35
50
6EXZ med
RNA export factor
0-65, A28-42, 0-65
6EXZ long
RNA export factor
0-95, A28-42, 0-95
110
Table 4: RFDiffusion motif scaffolding benchmark details. The specification for each scaffolding problem is under 'Configuratoon' with the motif structures in bold. For example, in motif 2KL8, the problem requires protein that contains residues 1-7 and 28-79 from chain A of the motif, joined together by a scaffold of 20 residues. Furthermore, the total length of the generated protein must fall in the range specified in the "Length" column.
Multi-Motif Scaffolding Benchmark. Problems in the Genie2 multi-motif scaffolding benchmark are listed in Table 5 In the Genie2 pro-print, problem 3NTN had a configuration with ranges in reverse, but the specification differed from their GitHub repository. We chose work with the ranges specified in their repository and made the correction in Table 5
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Name
Description
Configuration
Length
4JHW +5WN9
Two epitopes
10.40.
4JHW
(F25
20-50,
85-175
SWN9/ A (17011991))8 10-40
1PRW two
Two 4-helix bundles
5-20. HPRW/A16 (11//1166351) 10-25, 1PRW IPRW/A52 /A52-
120-200
71(1): 10-30, 1PRW/A80-108(2), 10-25,
IPRW/A125-144/2. 5-29
1PRW four
Four EF-hands
5-20, 1PRW/A21-2(1)), 10-25, 1PRW/A55
88-163
68(2). 10-25, (B))))
10-25,
IPRW/A125-144/9 5-20
3BIK+3BP5
Two PD-1 binding motifs
5-15, 311K/A121-155(1)) 10-20, ABP5/B110-
30-60
114(2), 5-15
3NTN
Two 3-helix bundles
GNTN/A342-328(1], 372(2). 10-20,
3N37/A367-
89-109
3NTN (TTN/B34-3448(2) 10.
ENTN/Ba67-3722 372(1), 10, 8NTN/C312-348(2)
10-20, 3NTN/C367
2B51
Two binding siles
5-15, 2B61/A11-23(2), 10-20. 2B51/A35- 116-166
45(1), 10-20, 2B61/A61-72(1), 5-13,
2851/A81-95(2); 20-30, 2851/A119-133(2)
Table 5: Genie2 multi-motif scaffolding benchmark details. The specification for each scaffolding problem is under 'Configuration', with the motif structures bold. Here, the motif structures are formatted as <MOTIF <MOTIF_NAME>//E NAME>/<CHAIN VEGMENT><<MOTH) [ where structures belonging the same motif group (BSE:CCCAATASEEEEEE><(CCCC
are fixed in their orientations relative to each other. Furthermore, the total length of the generated protein must fall in the range specified in the 'Length' column.
C.2 Diffusion Posterior Samplers
By defining the likelihoods appropriately, we can use the guidance potentials together with the SMC samplers. For FPS-SMC and MCGDifI, which operate on linear inverse problems only, they are confined to using the masking potential, and thereby the likelihood
ge xx) x exp (- L-muk (xt: (x:: Yc.M..
with 7), = a, for FPS-SMC and, for MCGDiff, N) = 1/(1-a;) when x, is known and n) = 1/( 14 On the other hand, TDS uses the DPS likelihood
when using & = E[x; xt+1] in the numerator eq.
approximation and adopts a general likelihood
gy xx) exp (-L (8o(xx (xx.))) ( 1/t
where n/: = 1/07. Wu al. (2023) recommend setting o? = Var(x; xo), leniently filtering early in the reverse process, when the reconstructed sample &o is still unreliable, and gradually tightening the filter towards the end. To keep it non-zero set o? ā,) + of to match inverse problem formulation at t 0.
In our implementation of the samplers, we make several optimisations. By partitioning particles into groups, we can run multiple trials simultaneously, making the most of throughput gains with increased model batch sizes. This is achieved by computing weights and resampling on a per-group basis. We further perform multiprocessing across several GPUs. Beyond parallelism, we minimise the number calls made to the diffusion modcl-the main bottlenock. We cache the predicted noise (or score) between computing weights and sampling from the proposal. Moreover, we only compute the predicted noise of unique particles for BPF and FPS-SMC, where duplicate particles may be fed to the proposal. Different batches of Gaussian noise are added afterwards to differentiate the duplicate particles. Computationally, it is if we set the number of particles equal to the mean of the ESS (estimator).
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D Additional Results
D.1 Hyperparameter Search
The hyperparameters of each method were chosen to maximise performance on two selected motif problems, 1PRW and 3IXT. In discretised grid-search fashion, 16 backbones are sampled and evaluated for each paramcter combination. A key hyperparameter we consider is the diffusion model's noise or temperature scale E (0, 1] that scales the noise added to each step in the reverse process. A high Ç leads to higher sample diversity, and a low Ç typically yields higher sample quality. The Genie models have been shown to attain the best F1 designaablity-diversstt score for unconditional generation at around Ç < 0.4 (Lin and Alquraishi, 2023 However, the effect of this setting on the conditional samplers wrapped around the model unclear. We tested three values Ç = 0.4, 0.7, 1.0 to gauge the impact on sampler performance. The proposals of the samplers, all of which were Gaussian, had their standard deviation scaled by this value.
Fig. 7 shows that, for the replacement methods, lower values indeed improved the scRMSD but had different effects on motif RMSD depending on the motif. We suspect that with minimal noise, motifs common to backbones lying in the modes of the data distribution are easily scaffolded, whereas those outside are not. We chose Ç = 0.4 for both replacement methods to prioritise a lower scRMSD, given the motif RMSD was already near the success threshold.
PPSSMC
MOCDIFF
1PRW
3IXT
FI(IPRW, 3IXT)
0.05
Unique Seaflolding Success Rate (%)
20
0.05
Average
7.5
7.5
seRMSD
5.0
0.85
1.5
Average
1.0
Motif
0.50
RMSD
0.75
1.0
0.8
Average ESS
5.0
6.0
2.5
2.5
2.5
0.4
0.7
1.0
0.4
0.7
1.0
0.4
0.7
1.0
Noise Seale Ç
Figure 7: Noise scale grid search for FPSSMC and MCGDIFF. The unique success rate and average values for two of the four designability criteria are shown. The right column depicts the F1 score between the values in the left and middle columns.
In reconstruction guidance, an additional hyperparameter we consider is the guidance scale y that controls the strength of the conditional signal. For high 7, the target distribution becomes more peaky, and the adherence to the conditional label is higher at the cost of diversity. We consider six values in 7 = 0, 0.25, 0.5, 1.0, 2.0, 4.0. Notably, 7 0 is identical to BPF under likelihood involving the reconstructed sample and the label. Fig. [ shows results for TDS-MASE and TDS-DIST. For TDS-MASK, with the best unique success rate but with the lower average scRMSD. Owing to the reficction-invariance of TDS-DIST, higher guidance scales made it produce reflected motifs up to 50% of the time. We found this issue less pervasive with a smaller guidance scale, where the unconditional model, informed on handedness, makes bigger contribution to the do-noising process. We therefore configured TDS-DIST at y = 0.25 and 0.4. 0.4. As for TDS-FAPE and TDS- RMSD, there were no issues whatsoever with reflected motifs. Fig. g shows their grid search metrics. We chose 7 0.5 and Ç = 1.0 for TTDS-FAPR as it had the highest unique
chose 7 1.0 and ( ( 1.0, one of the two
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success rate and the lowest average scRMSD and motif RMSD between the two motifs. For TDS-RMSD, several had high unique success rates, but we chose 7 = 2.0 and
0.4, which yielded the lowest scRMSD.
TDS DIST, moise ( 0.4
TDS- MASK, muise ( 0.4
TDS DIST, moise ( 0.7
TDS MASK, mnise ( 0.7
TOS-DIST; molen C 1.0
TDS-MASK, mnioe C 1.0
IPRW
3XXT
FI(IPRW, SIXT)
Unique Seaffolding Succrss Hate (%)
20
10
10
100
100
Correct Helix Handedness Rate (%6
90
7.5
Average seRMSD
2.5
1.6
1.25
Average Motif RMSD
1.0
1.0
1.00
0.75
5.0
Average ESS
2.5
0.0
0.25
0.5
1.0
2.0
4.0
0.0
0.25
0.5
1.0
2.0
4.0
0.0
0.25
1.0 20
4.0
Guidance Scale 7
Figure Hyperparameter grid search for TDS-dist and TDS-mask. The rates at which unique success and right- right-handed helices were achieved are reported. Average values for two of the four designa bility criteria also shown. The right column depicts the F1 score between the values in the left and middle columns. TDS-DIST with 4.0 is not shown in problem 1PRW, as it had numerical overflows in its backbone coordinates.
With TDS-PRAMR-DISTT, an additional parameter Mehiral is available to scale the chiral contribution of the rotation deviations to the likelihood. When 7ehiral O, we retrieve back TDS-DIST. Due to the large parameter space, we limit our search to fixed temperature value of Ç 0.4 to match TDS-DIST. As shown in Fig. 10 a non-zero rotation scale indeed corrected for reflections from as little as Vchhral = 0.5. However, as with 1PRW, there was a range of values between 0.5 and 4.0 where the rotation contribution was too weak, increasing the motif RMSD as it attempted to steer the trajectory away from solutions that meet the distance constraints those with the correct handedness. When the scale was too large, solutions had the correct handedness but incorrect distances, leading to malformed backbones. Here, the scRMSD was at its highest. We found that the values Mchiral 32.0 and 0.25 balance this tradc-off but remark that the optimal value differed in the case of both motifs. For example, zero chiral contribution was necessary for motif problem 3IXT.
In summary, however, we acknowledge that our approach to hyperparameter tuning is limited. In particular, optimising for two select motifs not. guaranteed to generalise to other motif problems with different sizes, contiguity, and frequency in the data distribution. Where we can, we highlight possible reasons for our observations to help make more informed choices in the future. Furthermore, the computational expense of the procedure could not permit larger sample sizes in tuning the multi-motif case. Our results can certainly be improved, and the demonstrated albeit limited success. in select multi-motif problems points to the feasibility of zero-shot approaches.
separate search for hyperparameters in
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TDS FAPE, HoGse 5 0.4
TDS RASD, moise = 0.4
PIS- moiee (= 0.7
TDS- RMSD, noiee C=
T1S PAPE, HOose C 1.0
TDS- RMSD, voler
IPRW
3IXT
PI(PRW 3IXT)
Unique
5.0
Scaffolding
10
Success Rate
(%6)
750
750
Average
6000
scRMSD
250
1:25
Average Motif RMSD
1.25
1.0
1.00
1.00
0.75
Average
ESS
2
0.0 0.25
05
1.0
2.
0:25 0.5 1.0 2.0 400 0.0
0.0 0.25 05
1.0 20
4.0
Guidance Scale 7
Figure 9: Hyperparameter grid search for TDS-fape and TDS-rmsd. The unique success rate and average values for two of the four designability criteria are shown. The right column depicts the FI score between the values in the left and middle columns.
Guidsnce ; 0.25
Guidane y
Guidiace 1.0
IPRW
3IXT
FI(LPRW, 3IXT)
Unique Scaffolding
Sucuess Rate (%)
Correct Helix Handodness Rate (%)
100
50
95
10
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Figure 10: Effect of chiral scale Mhira on the handedness and designability of generated structures by PDS-frame-dist. The right column depicts the F1 score between the values the left and middle columns. A noise scale value of Ç 0.4 is used throughout
27

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Published in Transactions on Machine Learning Research (09/2025)
D.2 Exclusion of Sequence Requirements
The motif scalfolding problem, in full, requires both the motil's structure and sequence to be present in the designed proteins. The self-consistency pipeline described in Figure 2 precisely evaluates samples in this regard. The scRMSD mcasures the cxtent to which the generated structures can be producod by folding a protein sequence that contains s as a subsequence. In contrast, the motif RMSD measures the existence of m as substructure of the generated structures.
However, zero-shot setups like ours only condition on the (backbone) structural information of the motif. Without providing sequence information to posterior samplers, effectively sample from the distribution of protein backbones that contain m as substructure, despite them possibly having sequences that do not contain S. To quantify the success of our samplers at the structural aspect of the motif scaffolding task, we remove the fixing of the motif sequence in the self-consistency pipeline, allowing alternate sequences to be considered place the motif sequence. Fig. [1 shows that several single-motif problems that our samplers previously could not solve can be successfully scaffolded without the sequence requirements. As for problems that had decrease in success, particularly in motif RMSD, we suspect this due to combination of stochasticity in the evaluation procedure and the loss of stability of the motif substructure that could have been achieved by fixing the sequence.
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Figure 11: Performance of sampling methods on the 24 motif scaffolding benchmarks when motif sequence is not fixed in the sclf-consistency pipclinc. Thirty-two backbones are sampled from cach method across all the motif problems. Scaffolds that successful and those which meet at least of the main success criteria are reported according to their unique count.
Similarly, Fig. 12 shows the results for the multi-mott problems where the sequence requirements were omitted in the evaluation. TDS-RMSD solved the most problems under this looser success criteria. The difficulty with scaffolding multiple motifs is that the issues with sequence information are presumably compounded.
28

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Published in Transactions on Machine Learning Research (09/202)
TUS-DIST
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Figure 12: Performance of sampling methods on the six multi-motif scaffolding benchmarks when motif sequences are not fixed in the self-consistency pipeline. Thirty-two backbones are sampled from cach method across all the motif problems. Scaffolds that are successful and those which meet. at least of the main success criteria reported according to their unique count.
29


Reviews:
[{"review_id": "h25m44HcIx", "reviewer": "Reviewer_6Enk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written and tackles a fairly complex problem with elegance. The empirical results are generally enough to back up the paper's claims, although in the multi-motif experiments, the results are more \"signs of life\" than strong quantitative results. Generally, the paper does offer some insights into why their method works, but these insights are in a way \"negative\" (e.g. handedness results); they show that a series of steps are needed to get SE(3) guidance right and they point out some challenges of the problem, but there is no very deep comparison with prior work.\n\nIn terms of TMLR's second criteria, audience, I think the paper is very timely and will find many readers.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a method for protein scaffold design using pretrained general models $q(x)$ while enforcing specific motifs. This is done by fixing the motifs and using diffusion guidance + SMC to solve the conditional posterior sampling problem $\\propto q(x)g(y|x)$. Specifically, the authors argue that prior work defines $g$ in counterproductive ways because they work directly in coordinate space, $x$, and its gradients perturb the motifs that we are trying to keep fixed thus creating a suboptimal guidance path when masking; instead, they propose to define $g$ in internal coordinates/in an SE(3)-invariant way so that the diffusion guidance keeps the motif intact. \n\nThis new guidance is defined by a combination of RMSD, frame-aligned point error (as in AF2), and internal pairwise distances.\n\nThe paper then shows how this enables sampling, as well as doing more complex motif conditioning, in fairly hard settings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I think the paper is good as is, more investigation-type results as I've alluded to above could help but are not strictly necessary.\n\nSome other suggestions:\n- There's no clear analysis of the computational costs of the methods involved here. Are there trade-offs?\n- It seems like there is high hyperparameter sensitivity; while that analysis is welcome, it would be good to understand why, or perhaps to suggest a guide to the reader.\n- If possible, improving the multi-motif experiments with more evidence points.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "slXAOSxqqx", "reviewer": "Reviewer_a1ij", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Weaknesses\n\n1. The paper is sometimes too informally written examples:\n    a. \"One idea is \"\n    b. \"complicated algorithms\" (what makes them complicated ???)\n2. The notation lacks defining things, for example, in equation (2), what is p(y_t|y) ? \n\nTypically when doing conditional genreative modelling and inverse problems we have:\n\n$$\np(y|x_0) = \\mathcal{N}(y| A(x_0) , \\tau I)\n$$\n\nAnd the conditional score (not an approximation ) which is guaranteed (under certain assumptions on the noise distribution/noise schedule [1,2 ]) is given by :\n\n$$\n\\nabla_{x_t} \\ln p_{t}(x_t|y) = \\nabla_{x_t} \\ln  \\int p(x_t, y, x_0) dx_0 =   \\nabla_{x_t} \\ln  \\int p(y| x_t, x_0) p(x_t|x_0) dx_0 \n$$\nWhere $ p(y| x_t, x_0) =  p(y| x_0)$ is not an approximation, however, the decomposition of the conditional score you have written of the score is different, so this is not an issue, but just poiting out why what you wrote seemed confusing at first. I assume is applying the product rule of prob in a different direction, I dont think its wrong but its difficult to read with the level of provided detail. What is $y_t$ ?  you seemed to have defined a full state space model / filtering process in a single sentence :\n\n```\nOne idea is to project the intermediate latent variables xt onto a measurement subspace at each denoising\n```\n\nThis is just not enough detail. \n\n\n[1] Denker, A., Vargas, F., Padhy, S., Didi, K., Mathis, S., Barbano, R., Dutordoir, V., Mathieu, E., Komorowska, U.J. and Lio, P., 2024. DEFT: Efficient Fine-tuning of Diffusion Models by Learning the Generalised $ h $-transform. Advances in Neural Information Processing Systems, 37, pp.19636-19682.\n\n[2] Domingo-Enrich, C., Drozdzal, M., Karrer, B. and Chen, R.T., 2024. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. arXiv preprint arXiv:2409.08861.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "At a highlevel this paper applies SMC based methodologies for diffusion such as TDS (Wu et al 2023) to a newly defined likelihood for motif scaffolding that combining different guidance potentials each of which incorporate better inductive biases / probabilistic modelling assumptions for the mottif scaffolding task.\n\nThe empirical results seem compelling and the proposed guidance potentials are novel and well motivated however the authors need to work harder to present the overall method.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please start by including an algorithm with pseudocode of your overall scheme in the main, well summarised, and possibly colour-coded to highlight differences between TDS and SMCDiff as much as possible (what adaptations were necessery to incorporate these new guidances).  I can see there are a lot  of algorithms in the Appendix but it has not been structured / layed out in a way that is easy to review.\n2. Define all core terms of equations in the main such that the main equations are self-contained. I shouldn't have to go and read the Song inverse problem paper to figure out your notation. See weaknesses in particular if the filtering model implied by equation (2) is important to your method (eq 2 takes up a very central location in the paper ...) please define it in more detail and motivate why its relevant.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ZUvbjHWVwx", "reviewer": "Reviewer_CqDk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n1. First, introduce SE(3)-invariant potentials to generalize beyond masking, enabling symmetric scaffolding.\n2. Extensive evaluations across 30+ tasks with multiple SMC-based samplers.\n\nWeakness:\n1. Evaluation scaling is quite limited. It seems to be restricted only to fixed motif placements and small sample sizes.\n2. Can the authors comment on the multi-motif experimental results? It does not look too impressive to me.\n3. One major drawback to me is the lack of essential coverage of the literature. Several Sequential Monte Carlo (SMC) variants are not discussed [1-2]. Compared to the more standardized SMC methods, many variants are reported to have less computational costs and can have even better performances in terms of reward values. These variants can probably also work with your novel potential, and it can be interesting to see it working. The authors could consider referring to [3] for a complete review of relevant inference-time techniques (i.e., SMC methods and variants).\n\n[1] https://arxiv.org/pdf/2402.06320 [2] https://arxiv.org/pdf/2408.08252 [3] https://arxiv.org/abs/2501.09685", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a zero-shot approach to protein motif scaffolding using diffusion posterior sampling combined with Sequential Monte Carlo (SMC). The authors propose several new SE(3)-invariant guidance and frame-distance. The authors have conducted extensive experiments benchmarking different SMC-based samplers, showing that reconstruction-guidance methods with the proposed potentials match or outperform standard masking. \nThe major contribution of this work is generalizing posterior sampling to complex scaffolding tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. It might be important to add variable motif placement or placement sampling to improve robustness of your results.\n2. I am curious to see the computational costs for all these methods. Consider reporting them in the next version.\n3. The details of how potential parameters are tuned are missing. It is advised to provide instructions on how those are selected.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)