Zero-shot protein stability prediction by inverse folding models: a free energy interpretation
Authors: Jes Frellsen
Maher M. Kassem
Tone Bengtsen
Lars Olsen
Kresten Lindorff-Larsen
Jesper Ferkinghoff-Borg
Wouter Boomsma
Venue: NeurIPS 2025 poster ()
Keywords: protein stability
inverse folding models
zero-shot prediction
free energy
interpretability
Primary area: machine_learning_for_sciences
TL;DR: revealing This paper analyses the free energy foundations of inverse folding models for protein stability prediction, revealing limitations in current practice and proposing simple improvements for better zero-shot prediction.

Abstract: Inverse folding models have proven to be highly effective zero-shot predictors of protein stability. Despite this success, the link between the amino acid preferences of an inverse folding model and the free-energy considerations underlying thermodynamic stability remains incompletely understood. A better understanding would be of interest not only from a theoretical perspective, but also potentially provide the basis for stronger zero-shot stability prediction. In this paper, we take steps to clarify the free-energy foundations of inverse folding models. Our derivation reveals the standard practice of likelihood ratios as a simplistic approximation and suggests several paths towards better estimates of the relative stability. We empirically assess these approaches and demonstrate that considerable gains in zero-shot performance can be achieved with fairly simple means.

Zero-shot protein stability prediction by inverse folding models: a free energy interpretation
Jes Frellsen' Technical University of Denmark
Maher M. Kassem" Novonesis
Tone Bengtsen Novonesis?
Lars Olsen Novonesis
Kresten Lindorff-Laree
Jesper Ferkinghoff-Bort Novo Nordisk
Wouter Boomsma! i University of Copenhagen
University of Copenhagen
Abstract
Inverse folding models have proven to be highly effective zero-shot predictors of protein stability. Despite this success, the link between the amino acid preferences of an inverse folding model and the free-energy considerations underlying thermo- dynamic stability remains incompletely understood. A better understanding would be of interest not only from theoretical perspective, but also potentially provide the basis for stronger zero-shot stability prediction. In this paper, we take steps clarify the free-energy foundations of inverse folding models. Our derivation reveals the standard practice of likelihood ratios as a simplistic approximation and suggests several paths towards better estimates of the relative stability. We empirically assess these approaches and demonstrate that considerable gains zero-shot performance can be achieved with fairly simple means.
Introduction
Quantifying how amino acid substitutions affect protein structural stability is of fundamental impor- tance for understanding human genetic diseases (Stein et al., 2019), and for our ability design and optimise industrial enzymes and protein therapeutics (Listov et al., 2024). In recent years, multiplexed assays of variant effects (MAVE; see, e.g., Starita al., 2017) experiments have greatly enhanced our ability characterize variants experimentally, but still only scratch the surface compared to the astro- nomically large number of possible variants (we use the term variant to refer to a protein that differs from reference, 'wild-type' protein by one more amino acid substitutions). In silico prediction, therefore, remains an important tool, as a low-cost and fast alternative to experimental characterisation, but also extrapolate mcaningfully from experimental results to uncharacterized variants.
Despite the progress in -hghuhhooughput experimental characterisation, variant-effect data still falls in the low-data regime, and supervised learning in this domain has been observed to be prone to overfitting, exemplified by several large-scale studies on human variants (Livesey and Marsh, 2020). As consequence, unsupervised or weakly supervised methods are often the most attractive approach to the problem. For protein stability prediction, inverse folding models, which provide a probability distribution over amino acid sequences given fixed 3D structure, have emerged as particularly useful. particular, variations log-ratios based on the form
p( variant sequence wild-type structure) In
(1)
p( (wild-type sequence wild-type structure)
'Equal contribution. Work performed while employed at Novonesis. CCoresppnding authors: jefr dis & wbëdi. ku. dk
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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have shown strong empirical correlation with experimental measurements of stability (Boomsma and Frellsen, 2017: Meier et al., 2021; Hsu et al., 2022; Dutton et al., 2024; Cagiada al., 2025).
An inverse folding model describes amino acid preferences conditioned on their structural environ- ment. While it is intuitively appealing to assume that likely amino acids correspond to stable protein structures, the formal connection between these probabilities and folding free energies remains incompletely understood. For instance, since thermodynamic stability is an ensemble property (a property of the entire structural distribution), it is not clear why it would be sufficient to condition on a single structure. Likewise, since thermostabilityy is balance between a folded and an unfolded state, one might reasonably wonder whether one should explicitly model the unfolded-stee in addition to the current practice of generally considering the folded state alone. This raises the question: Can we interpret these inverse folding models in terms of thermodynamic stability?
propensities,
In this paper, we derive theoretical connection between inverse folding probabilities and thermody- namic stability and use this to elucidate current practices. We also suggest a number of improvements the current protocols. Our contributions are:
We derive a formal relationship between changes in thermodynamic stability (AAG) and changes in inverse folding probabilities, and describe the approximations necessary to explain the current practice of simple probability ratios, cf. eq. (1).
We show that the current practice corresponds to single-sample Monte Carlo estimates, and demonstrate empirical performance gains by using multiple samples from molecular dynamics (MD) simulations or the BioEmu generative model (Lewis al., 2025).
We show that the unfolded state can be disregarded. ef. eq. (1), but this introduces an additional factor in the probability ratio. We also propose and evaluate several strategies for explicitly modelling the unfolded state and show that these improve empirical performance.
2 Background
Assuming isothermal-isobaric conditions (i.e., an NPT ensemble), the stability of a protein is deter- mined the difference in Gibbs free energy, AG, between its folded and unfolded states. These two states are macro states in the sense that they each correspond to a set of structural conformations. How this quantity changes as a result of changing one or several amino acid residues with another (substitutions) is referred to as a AAG. Denoting the original sequence as wild type (WT) with amino sequence a and the new sequence as variant (MT) with sequence a', we have that
AAGa-a = AGU-F AGU-F = (G G.d) - (G GH))
(2)
2.1 Gibbs free energy
The Gibbs free energy is calculated from the Boltzmann distribution, expressing the probability of a protein microstate with structural degrees of freedom X E E and solvent w, given by
p(x. w|a, 3) = --Ha-_ Za / (H.(x.)) dx dw
where Ha(x, w) Ua(x, w) (V w) is the enthalpy of the microstate for amino acid sequence a, (x, w) is the internal energy (Hamiltonian), V(x, w) is volume of the microstate, P is the pressure, and 3 T is the inverse of the thermodynamic temperature. The Gibbs free energy associated with the amino acid sequence is defined as
G. 3 log Z.
Since our focus is on the degrees of freedom of the protein, we integrate out the solvent degrees of freedom. Furthermore, we split the structure degrees of freedom X = (xx, s) into backbone degrees of freedom æ and side-chain degrees of freedom s, and integrate out the side chains. The probability of backbone configuration z for given amino acid sequence is then given by
p a, 3) - (H)(w) Za =
(a) dx
(5)
where (x) is the free energy associated with implicit treatment of the solvent and side-chain freedom defined by e (0) = fe- - (x,w) ds dw.

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2.2 ceermodynamicc stability
To calculate the stability of a protein, we usually partition' the space of structural degrees of freedom I into a folded subset 3. and an unfolded subset xú (Brandts, 1969; Lindorff-Larsen and Teilum, 2021). We will use S E {F, U} to denote either of the two states. We will assume that the states can be fully characterized by the backbone degrees of freedom, such that the space of backbone degrees of freedom Xa can be partitioned into a folded subset X and an unfolded subset X For generality, we use a soft partitioning given by the probability p(S|æ, a), where the hard partitioning above is special case specified through the indicator function. We can then write the partition function (normalisation constant) for the state as
Z 5 f e (--8aal)) p(S| a) dz.
(6)
Similarly eq. (4), the free energy of state S is then given by
G - log Záa
and the folding stability of the protein with amino acid sequence a is given by
U z
AG = GO Gt or equivalently BAG = In
(8)
If we write the stability in terms of integrals over the Boltzmann distributions, we that the stability can be expressed as a ratio of probabilities
BAGU-F_ =In S BHH (z) p(U [., a)
-In p(xla, (a, a)dæ: I p(araa, (F [a, a)dæ p(S= aa B)
= In p(S= U a. 3)
(9)
ʃ e -H p(F ar, a)dæ
where p(S|a, 9) p(z|a, 3)p(S x, a) dæ denotes the probability of finding the protein with sequence a in state S. Since (FF a. 8) + p(U |a, 3) = 1, it follows from eq. (9) that
pS F a. B) In ( (
BAGU-F = p(S= F a, 3)
(10)
which means that knowing P la, B) is sufficient for calculating the stability BAG
2.3 Databases of structure-sequence pairs
Consider dataset of structure-sequence pairs D = {(xi, a:)} that is sampled from the marginal of some data- ceata-generating distribution PD a, 3). The dataset could, for instance, be the Protein Data Bank (PDB; Berman et al., 2000) or some subset of it. We make the fairly strong assumption that all structures in the database are sampled approximately from their respective Boltzmann distribution with potentially different and unknown (latent) B. That is, for any and (3, we assume
(11) Note that thermodynamics do not, per se, make any statements about the marginal distribution over the amino sequences. Furthermore, the data marginal over sequences, PD (a). may not reflect the true biological sequence prevalence due to data collection biases, as documented for the PDB (Gerstein, 1998; Orlando et al., 2016).
pD( 3) R p(xla, 3))
2.4 Inverse folding models
We assume a joint model pa(a, x, 33 over the sequence a, backbone degrees of freedom æ, and inverse temperature 3. Since available datasets typically do not include the inverse temperature, this limits what can be learned in practice. From dataset of structure-sequence pairs, we learn the conditional distribution (a(xx)) referred to as the inverse folding model, as well the marginal distribution pe (a) over sequences. For simplicity. we let 9 denote the combined parameters of all components, although in practice the parameter sets may be disjoint and estimated separately.
Building on the assumption in eq. (11), we further assume that the model provides a good estimate of the conditional distribution over structures given sequence and inverse temperature, such that
pe (3) As PD (z) (,,3) Rs p(a)a ())
(12)
This reflects the idea that the learned posterior approximates the Boltzmann distribution well.
Thii means that Xa x U x and x!x xxx

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3 Methods
Consider a wild-type sequence a and a variant sequence a' that differ by one or more amino acid substitutions. The question now, can utilise inverse folding models calculate the change in stability between the two proteins? The definition of the change in stability is given by the difference in folding free energy between the mutant and wild-type. We can reformulate this
BAAGa-o In M(S-Ulak
p(S U 3) In p(S= U l' In p(S- F a B) p Fa,
In
PSS= 3)
p U aI,3) P(S a. 3)
SAG
BAGU-P
BAGU
BAGE a'
(13) where the second form expresses the mutation effects on the folded and unfolded states in terms of the two pseudo change in free energy terms BAG which we define for analysis but are not physical quantities. In the following, we show how these terms can be estimated using importance sampling.
3.1 Change in stability using inverse folding models
To calculate the pscudo change in free energy AGS a'a we begin by expressing p(S]a', a ()) using importance sampling, similarly to free energy perturbation (Zwanzig, 1954). That is,
P(S)a', 3) f p(æ: a', ()p(S) x a²) dæ = Ba-
p(a:|aa, ,,,,) () a') a(a 3)
(14)
Using Bayes' theorem, we can express the structure posterior as p(æ|S, a. 3) - p( x a. () / p(S)a, B), which allows us rewrite the importance sampling expression from eq. (14) as
p(Sla', 9) E (e
p(azla', (S(,, p( a, ()
(15)
p(S a, (3) To evaluate the ratio p(
p(æ a. we use the approximation assumption from eq. (12) and apply Bayes' theorem in both numerator and denominator, i.c.
p(zla', (3)
Po( (3) Po(a'Iz (2pp(()))) Po(a'(3) po(a'|æ, 3) pa(a
(16)
p(z|a, B) pa (a a,
pa pa(a| B)pp x B) (a] po(alæ, B) pa(a'
Substituting this into eq. (15), we arrive at
p(S)a' (a=pp(=(S p(S)a, 3)
po(a'l [x, 3) p( x. a') po(a 3) p(S) (SSx a)
po (a
(17)
pe (a' (3)
Assumptions Recall that the inverse folding model introduced in section 2.4 does not depend explicitly on B. To evaluate eq. (17), we assume that the considered 33 is equally representative of both sequences, a and a', and of the structure under our model, i.e., pe a) ~ po a') and
po(3 æ. a) = (3 x, a'). This assumption may be reasonable when the two sequences differ by only few substitutions. It then follows that
po Po(a' (x. ()) ~
po (xx) and (o (a) po(a')
(18)
Pe (z, ())
(o(a)z)
(a(a pe (a)
Furthermore, we assume that small number of sequence substitutions does not alter the backbone- to-state mapping, that is, p(S) (S|e,, a) & p(5 x a'), and we can thus assume that the ratio of these terms 1 in eq. (17). Together, these assumptions yield
p(Sla', B)
pa)))
(a)
xe Emm-p(a
(19)
(a(
Pe (a')'
providing a means to approximate BAGS aa using an inverse folding model.
Note that it would be mathematically tempting to write the importance sampler in eqs. (14) and (15) using the unconditional Boltzmann distribution p(z|a, (3) as the proposal distribution. However, we note that it is much more difficult to sample from pl x/a, (3) than from the conditional p(z) SS, a, 3) as it requires sampling both the folded and unfolded states. Unfolding events are typically rare and difficult to sample in, e.g., MD simulations, and we would need to sample multiple folding-unfolding events to obtain low-variance estimate. no unfolded structures are sampled, would (erroneously) imply that p(F a, 3) = 1, see appendix A.I for further discussion.

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3.2
Change in stability from folded and unfolded ensembles
By combining eqs. (13) and (19), we can express the change in stability as
pa(a' [x) In Be-pol F.a.8) Pa x)]
( (a' x Po(a x))
3AAGa-aa In E Ba (a v.a.9)
(20)
which follows from the important observation that the marginal sequence probabilities pe (a(a) and pe(a') cancel between the folded and unfolded terms.
Equation (20) represents key result: it shows that the change in thermodynamic stability can be estimated using an inverse folding model. The terms inside the expectations can be computed using an inverse folding model, and the full expression becomes tractable through Monte Carlo estimation, provided can sample structures from the conditional structure distributions po (æ) S, a, () for both the unfolded and folded ensembles.
For the folded state, (ar F, a, a,, we can approximate this distribution using structural data available in the dataset. Usually, the dataset will only contain very few structures for each sequence, and only single structure aa is available for sequence a, one-sample approximation yields
Po a [=) Po a ax) Po a
po (aaa
[x--po(a)) F,a,9)
(21)
Presumably, more accurate estimate could be obtained by sampling local structural variations around Va through molecular simulation. Similarly, simulations could be used to approximate the expectation for the unfolded state. We explore these strategies empirically in section 4.
3.3 Change of stability from a folded ensemble or structure
Recall from eq. (10) that knowing p(F (a, 3) is sufficient for determining the stability of protein. In this section, we investigate the extent to which we can estimate the change in stability from folded ensemble alone. We consider two approaches leading to the same expression: the first case, we assume that BAGU aa 0, and in the second case we only consider ranking mutations.
3.3.1 Simplified change of stability estimation via unfolded-state invariance
If we consider stable wild-type sequence, i.e., p(F a. 3) is close to 1, and variant with a similar or higher folding probability pF la', 3), then the folded term BA a' is close to zero. In this case, the dominant contribution to the stability change 3AAG arises from the unfolded term BAČU -a- cf. eq. (20). Thus, for relatively stable variants, the unfolded pseudo-free-energy change, BAGU a' the key term to evaluate. Conversely, for strongly destabilising mutations, where P(F |a', 3) < p(U (a, 3), the dominant term in BAAG is the folded term /A a' These two cases are illustrated in fig. Consequently, if our main interest lies in quantifying strongly destabilising mutations, we may neglect the unfolded state and approximate the stability change as
Po (a:)
(a(a) po(a')
3AAGa-aa BBAGE a -n (a F.a.9)
In
(22)
Po (a (x)
where the second term accounts for the conditional sequence probabilities under the model. If only single structure Fa po(x) F, a) is available, we can approximate the expectation with one-sample estimator
BAAGa-a ~ - BBAGE Rs - In (a'aa(aa) In (a(a) (o(a)aa))
(23)
(a
The expression in eq. (23) closely resembles standard practice in the field, ef. eq. (1), and thus provides an explanation for zero-shot prediction of inverse-folding models. However, we note that the expression includes an additional correction term that accounts for the frequency the substituted amino acid under the model (or in the underlying dataset). The fact that the raw log-odds scores work well in practice suggests that this is not dominating term, but we would expect performance to improve when including it. We investigate this empirically in section 4.
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3.3.2 Ranking changes of stability
When comparing the stability change of multiple variants a d(()) to a'n), one would ideally compute and compare their respective values a However, note that the term BAGH is constant across all variants and thus cancels out when comparing values. As a result, ranking variants by their BAGU values preserves the same ordering. Moreover, since p(F |a, 33 is constant, AG is a monotonic function of p(F la', (3) and thus also monotonic function of -BAG a'= See fig. 3 for illustration and appendix A.2 for detailed derivation. Therefore, ranking set of variants a'(1) ..... a((n) by -BAGE -a yields the same ordering ranking them by their full stability changes BAAG n---- ('(h)
ar(i)
implies that we are only interested in ranking variants, rather than computing exact stability changes, we can ignore the unfolded ensemble and instead use -BAC am0) given by eqs. (22) and (23).
Importantly, this ranking argument does not rely on the approximation B -a & O, but still leads to the same practical expression. This helps explain why strong Spearman correlations have been observed between the simple log-ratio expression - In(po(a'|zaa //pp (l/pp(aaaaaa and experimentally measured values of stability changes (Meier et al., 2021), the Spearman coefficient is purely rank-based metric.
3.4
Change in stability with sequence models
In the previous sections, we derived expressions for estimating the change in thermodynamic stability using inverse folding models. These derivations relied on the ability to sample structural ensembles for both folded and unfolded states. However, in practical setting, it may be preferable more convenient to estimate free energy changes using only sequence-based models, without requiring structure-conditionee models.
We assume joint probabilistic model over amino acid sequences and structural states of the form
Py (a. S) (a (S),
(24)
where y denotes the model parameters. Using Bayes' theorem, the pseudo free energy change for given structural state S can be expressed as
p(S a', a' p(S a)
In (S a') PS a) P- S) + In (a (a))
In pa 8)
(25)
where we assume access both the marginal and conditional probabilities under the model, and that P(S 3) = (S a) for all a. This means that the change in thermodynamic stability can be estimated using only
condiiionii sequence model; see appendix A.3 for further details and section 4 and table 2 for empirical results.
This framework also enables to combine estimates from different sources by using sequence- based model for one state and a structure- based model for the other. A practical special case arises when we have good characterisation of p. (a U) from data on, e.g., intrinsically disordered proteins or regions, which predominantly represent the unfolded ensemble. By combining this with an inverse folding model for the folded state, and assuming that the marginal sequence probabilities agree across models, i.e., (a) pa(a), the change in stability can be expressed as
3AAGa-a R In (a' In E Bee F.a.9) (a U)
Pa (a(a(()) po(a ar)
(26)
This hybrid approach is particularly useful when only folded structures are available, as a state- conditional sequence model can provide an estimate of the unfolded pseudo-change in stability. We evaluate this hybrid approach empirically in section 4 and include ablations across different combinations of sequence- and structure-based models in table
Experiments
To evaluate the effects of the different assumptions and approximations discussed the previous sec- tion, we conclude the paper with series of experiments in which the individual terms estimated

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Single sample for folded state
Multiple samples (MD) for folded state
Ignoring unfolded
In pu [Xx] Pa [xa)
In
po (xa) Po(a (xa)
state
state.
Ignoring unfolded
In po(a' Xx) Pe (a Xxa)
In Po(a)
In
Pola (xx)
In Po(a) pola")
p(a) correction
po(aa)
Po(a (xa)
Unfolded state using
In
polal (xa) pola (xa)
In
pola ia ixx) palaa (all (xa))
In po(aa (xa) Po(a (xx))
In
Po(a lxx) po(al(xa))
Monte Carlo
Unfolded state using
minimal--ontext
In po(allxa>)
In (o (o0 pu(aqixny 1
In
pol xxa) Pala (Xa)
1n1 Po pola
Pio (au 1502
inverse-folding
pola (xx)
Unfolded
pa(al [xa) In
Pala (U In
In Paole (x)) Po (xx)
In (a (U Fo (a' (Uppe)
state using IDP statistics
pa (xx))
po(a' [U
F
0.6
Pearson r
0.4
0.2
0.0
VAMP-seq
Guerois
Protein G
Figure 1: Correlation coefficients obtained using the different expressions discussed in the paper, all involving the inverse-folls foersse model ESM-IF. The top-left expression is the approach typically employed as zero-shot predictor of protein stability. The left column contains methods that consider only a single folded structure, while the right column considers a structural ensemble from an MD simulation. The different rows represent increasingly accurate approximations to the 3AAG (see text for details). The error bars represent the standard error of the mean calculated using 100 bootstrap samples. For simplicity, we used the bracket notation (-)s to denote the expectation (a)
using available computational methods on a representative selection of protein datasets. In the follow- ing sections, we discuss these choices in turn. We will initially conduct our experiments using the pre- trained ESM inverse folding (ESM-IF) model (Hsu et al., 2022), as has been shown to perform well in zero-shot setting (Notin et al., 2023). An ablation with ProteinMPNN is discussed subsequently.
4.1 Data
Our primary analysis considers three different data sets. The first is a high-quality data set measuring the thermodynamic stability of nearly all variants of a single 56-residue protein, the B domain of Protein G (hereafter called Protein G; Nisthal et al., 2019). The second an older benchmark set, compiled for the FoldX prediction method by Guerois et al. (2002), mostly consisting of entries from the ProTherm database (Gromiha et al., 1999). This set also directly provides experimental changes in thermodynami stability. but is heterogeneous in terms of experimental conditions and is known to be biased towards substitutions in which a large amino acid residue is replaced by smaller one and, in particular, mutations to Alanine (Stein et al., 2019). Finally, we include data generated using variant abundance by massively parallel sequencing (VAMP-seq) experiments that probe stability only indi- rectly, by quantifying the variant abundance in cultured cells using a combination of fluorescent tags and sequencing (Matreyek et al., 2018). The data generated by VAMP-seq have previously been shown to correlate with both biophysical measurements (Matreyek et al., 2018 8) and computational predictions (Cagiada et al., 2021) for protein stability, despite the assay reflecting cellular factors beyond ther- modynamic stability. We will refer to these sets as Protein G, Guerois, and VAMP-seg, seq, respectively.
7

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We will conclude the experiments section with brief discussion of scaling. For this purpose, we employ a subset of the mega-scale stability dataset (Tsuboyama al., 2023) as included in Prote- inGym (Notin et al., 2023). Both VAMP-seq and the mega-scale set represent recent developments in high-throughput assays based on deep sequencing. which are becoming increasingly common for variant characterisation. Combined, the chosen data sets are selected to reflect the different quality/noise regimes in experimental stability data (see appendix B.1 for details).
4.2 The folded ensemble
We approximate the expectation over pa(2 F, a. 3) using unbiased molecular dynamics (MD) simulations. Using the OpenMM framework (Eastman et al., 2017), 20 simulations were conducted at 300 K using 2 femtosecond time steps with the Langevin integrator, combined with the Amber 14 force field with a TIP3P water model, adding counter ions assure overall neutrality. See appendix B.2 for details on the choice of simulation ensemble.
When considering only the folded state, sequence correction factor arises in eq. (23). For simplicity, we will in our experiments use a position independent sequence model estimated from PD. meaning that for single mutation amino acid i the factor becomes (ppo(aapo(aa))) = (pp((a.) (c)))p(aa)). We also did the analysis using the ESM2 language model (Lin et al., 2022), but found this choice to be detrimental (see section 4.5 and table 2).
4.3 The unfolded ensemble
For the unfolded ensemble, it is less clear what the best approach is, and we therefore try differ- ent strategies. In the first approach, we conduct Metropolis- Hastinggs simulations in the Phaistos framework (Boomsma et al., 2013), using the TorusDBN (Boomsma et al., 2008, 2014) and Basilisk (Harder et al., 2010) statistical models obtain reasonable backbone and side-chain conformations, but otherwise keep the chain an unfolded state. We simulated segments with five flanking amino acids each side of the position of interest, rumning for 10,000 iterations, where each 100th structure was saved. In the second approach, we again evaluate the ESM-IF model on segments, but unlike the first approach, we extract single fixed segment from the crystal structure (i.c., the folded state). The fragment length is kept short (1 flanking amino acid to each side) to approximate an unfolded state, and no structural averaging done. This approach is similar to that introduced by Dutton et al. (2024), but using segments of length 3 instead of 1. The third approach differs from the first two by not considering the unfolded structural ensemble all, and instead using a sequence model detailed in eq. (26). Specifically, we consider protein disorder as a proxy for the unfolded state, and estimate p-, (a UIPD) using amino acid frequencies obtained from disordered regions according to the 'curated- disorder-uniprot annotation in the MobiDB (Piovesan et al., 2021) database (extracted Jan 21, 2021).
4.4 Ablations and scaling
Ablations were performed on the Protein G dataset. First, we repeated all analyses using the ProteinMPNN model (Dauparas et al., 2022). Second, we probed whether the expensive MD simulations could be replaced by predicted ensembles using the BioEmu model (Lewis et al., 2025). After observing promising results, the BioEmu approach was repeated on the larger mega-scale stability dataset (Tsuboyama et al., 2023).
4.5 Results
Results for folded/unfolded strategies are reported in fig. I and appendix B. with additional results on sequence models in table 2. The ProteinMPNN and BioEmu results are found in table 3, while the mega-scale results are displayed in fig. 2.
Folded state: single-sample vs multi-sample approximation The left column in the legend in fig. corresponds methods that use only single native structure to approximate the expectation, while the right column approximates the ensemble average using multiple sampled structures. For the VAMP-seq and Protein-G dataset, this choice consistently improved performance. On the Guerois dataset, no general trends can be observed, and we observe extensive fluctuations among the 40 different structures in the dataset, reflecting the heterogeneous nature of this older dataset, and perhaps indicating issues with our MD simulations for some of these systems (see also fig. 4).
8

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foldlddssingle
folded:single
unfolded:IDP
foldedmulti
foldedmmlli
unfolded:IDP
0.8
Peerson
0.7
0..6
0.4
2M16
11.00
1OSK
FRoM
IPV0
2IVD
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TWCL
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2HFV
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2KWH
TWIG HLRR
aL
200y
2017
17G0
2ARU
2014
Zulk
2817
SUBS 2AMI 3DKM
2YS8
20FF
Figure 2: Scaling using BioEmu (Lewis et al., 2025). When replacing MD simulations of the folded state with structural ensembles generated using BioEmu (20 samples; filtered to include only folded states), we still observe consistent improvements over the single-structure performance this subset of the mega-scale data set (Tsuboyymaa et al., 2023). This suggests that learned generators of molecular ensembles offer a promising route to scaling our ensemble-based approach in practice.
Considering only the folded state: the pe (a) correction For the folded-only case, the experiments generally show an improvement of including the (n(p(()) (a)/p((a)) correction term, but the effect is fairly minor as anticipated. We note that in light of eq. (26), the correction term in eq. (23) can be interpreted as specific choice of model of the unfolded state, namely the one where P- (a U) is assumed to factorize over positions and follow the general amino acid propensities. This perspective provides another reason why this simplistic model might not provide very accurate results.
Three models for the unfolded state Estimating the contribution from the unfolded state using Monte Carlo simulation worked less well than expected, generally performing worse than the simple log-odds baseline. One explanation could be that ESM-IF has been trained on structures generated by AlphaFold, and thus has learnt specific geometric features that may not be present in the structures generated by our Monte Carlo simulations. Another explanation could be that the sequence- and local structure signal in ESM-IF dominates when no structural environment is present. Since our Monte Carlo sampler uses proposal distribution that guarantees native-like local structure, ESM-IF IF apparently displays folded-like preferences when evaluated on unfolded fragments with native local structure. Replacing the Monte Carlo simulation with single short structural fragment extracted from the folded state yields some improvements, in line with previous reporting (Dutton et al., 2024). Remarkably, the best approach was to approximate the unfolded state using amino acid frequency statistics from disordered regions. Despite its simplicity and ease of implementation, generally outperforms the other models of the unfolded state on the considered datasets.
Ablation and scaling The ablation results in table confirm earlier reports that the likelihoods from ProteinMPNN are less informative for zero-shot stability prediction than those from ESM-IF (Notin et al., 2023). It has been shown that the autoregressive order of evaluation can improve likelihoods Dutton al. (2024), but this was not implemented here. Despite the different baseline performance levels, both models improve similarly across the different strategies. The BioEmu ablation shows only a very minor drop performance relative the MD ensembles, suggesting that BioEmu can serve as an efficient proxy when averaging structural ensembles. When applied to range of proteins from the mega-scale dataset, the BioEmu protocol indeed yields consistent improvements over the baseline at fraction of the computational cost of full MD) analysis (fig. 2).
5 Related work
The idea of connecting free energies to the statistical properties of experimentally determined protein structures has long history (Tanaka and Scheraga, 1976; Lifson and Levitt, 1979; Sippl, 1990; Miyazawa and Jernigan, 1996; Simons et al., 1999). Before the advent of deep models, these ap- proaches have essentially been based on deriving approximate free-energy stability energies from single and pairwise amino-acid statistics (Borg et al., 2012). Commonly used physico-chemical based modelling tools for calculating AAG include the FoldX (Schymkowitz et al., 2005) and Rosetta

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force-fields (Alford et al., 2017). While providing reasonable estimates for conservative mutations, these force-fields are known to be sensitive to the specific choice of the backbone template in a given application and, as such, do not yield consistent AAG estimates when applied to the full native ensemble of backbone structures. The connection between AAG and inverse-folding likelihoods was initially explored by Boomsma and Frellsen (2017) in the context of a 3D convolutional model. This study introduced a correction term compensating for the base frequencies of amino acids similar to eq. (23), but argued for it in terms of the unfolded state, while we show here that it follows more natu- rally as a consequence of assuming zero contribution from the unfolded state. More recently, a study demonstrated performance gains by including correction term by evaluating inverse -oodel only on the coordinates of the amino acid in question, motivating it as a representation of the unfolded state (Dutton et al., 2024). Our paper provides the theoretical basis for this argument, and we include very similar strategy in our experiments (using fragments of length 3). Finally, contemporaneously with our work, a recent study on binding affinity prediction reported substantial performance gains by explicitly incorporating the unfolded state (Jiao et al., 2024), using Bayes' theorem in a similar way as we do in eq. (15), but without considering the full structural ensembles subsequently extended this approach by fine-tuning the inverse folding model on folding stability data.
do here. Deng et al. (2025)
6 Discussion
Log-odds scores from protein inverse-folding models correlate remarkably well with changes in protein stability, but the underlying reasons for this correspondence have remained incompletely understood. In this paper, we take steps to establish a formal connection between the two. We demon- strate that the standard log-odds practice arises as consequence of a specific set of assumptions, and explore how these assumptions can be relaxed to improve zero-shot prediction further.
Based on experiments, two choices appear to have the most significant impact over the simple log-odds baseline: 1) including a contribution from the unfolded ensemble, and 2) approximating the structural ensemble of the folded state with more than single sample. From a practical perspective, both are potentially inconvenient in that they involve molecular simulation. Fortunately, our experiments indicate that the unfolded state can be approximated by a simple static distribution extracted from disordered regions, and we show that computationnaly--cnveeeiee found for the folded ensemble, based recent generative models of molecular ensembles (Lewis et al., 2025). We therefore anticipate that these improvements can be readily implemented on top of any existing pre-trained free energy model. Finally, we note that while our study has focused on protein stability, it should extend directly to the analysis of binding affinity, generalizing the approach derived by Jiao et al. (2024).
proxies can also be
Limitations While our derivations are general, the experiments section necessitates choices regard- ing practical implementations of the individual terms. We believe we have made reasonable choices, but have not exhaustively explored the space of possible models. We consider our experimental section as proof-of-coneee
exemplifying that better theoretical treatment can lead to gains in per- formance. The relative size of these performance gains will depend on the protein and the models used to approximate the terms, and cannot be conclusively established from our limited set of experiments. Another outstanding issue is that our analysis does not explain the recent observation that inverse- folding likelihoods also correlate surprisingly well with absolute stabilities (Cagiada et al., 2025).
Broader impact As machine learning models play increasing role in science, it is important to understand how such models work and interact with existing interpretable models. By establishing a link between pre-trained protein models and the free-energy considerations that drive our physical understanding of protein stability, we hope to make these models more broadly applicable to the sci- entific community. Although we acknowiedge that fnverse-folding models can be considered dual-use technologies, we believe any such risks are mitigated by the fact that our work focuses on a theoretical understanding of an existing model class, rather than the development of new predictive capabilities.
Availability Code for reproducing the experiments is accessible at: MachineisannnggLieeeeeeeeeeeeeee frreinnnnnfeeeeeeeeeeeeeeeeeeeee energies
com/
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Acknowledgments and Disclosure of Funding
JF and WB are supported by the Center for Basic Machine Learning Research in Life Science (MLLS) through the Novo Nordisk Foundation (NNF200C00626066, and the Pioneer Centre for AI (DNRF grant PI). KL-L is supported by the PRISM (Protein Interactions and Stability in Medicine and Genomies) centre funded by the Novo Nordisk Foundation (NNFI80C0033950),
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A Details on evaluating the change in stability using inverse folding models
A.1 Using the unconditional Boltzmann distribution as the proposal
In eq. (15), we use the conditional Boltzmann distribution p(a) (S, a, 3) as the proposal, since is easier to sample from than p( 3,) 3). Here, we will investigate the unconditional proposal distribution.
We can write the importance sampler from eq. (14) using the unconditional Boltzmann distribution as
p(Sla', = / p(z (((e)aa)p(Saaa) (p(S) a', B) dæ Bm-p(=/a,,9)
(3))p()xx.. p( a 3)
(27)
In the extreme case, when we try to sample from p(a/a, B), efficiently only sample the folded state, i.e,, we use p(a) F, a, 3) as the proposal distribution. This corresponds to assuming that
p(Sla', B) Eawploo F.a.9)
p/xzla...)p(( (3)(()mm.a)) Using Bayes' rule on this assumption gives us that
0)
p 3)p (S) ae, a') p(F Le, a) F(F la, () p( alzlaa, ())((( x a')
p(Sla'3) Eemp(aaa,
(28)
p(a a, 3)
B ppppla),
pp(F [ a.)
(29)
P(F Ja, (3)
p(æ: a, (3)
p( 3)p(( a')
p(S)a')
E p(
(30)
P(F la, B)
p(æ: a, 3)
p(F aa)
which is only true for p(F (a, 3) So the assumption implies that p(F aa. ()
A.2
Monotonicity of stability
We aim to show that AGU is a monotone increasing function of -BAGa In the following. all distributions are conditioned on 5, but we omit it to simplify notation. Starting from eq. (10), we can write
p(S = F a') -1)
BAG = In
(31)
= In
pSs Fla) p(S= p(S= (aa) -1
(32)
In
1
(33)
p(S= F (a) exp
In
P(S=Fla') p(8=Fla)
In p(S=Fla')
exp
p(S=F(a)
ln
(34)
p(S= F F(a)
exp(y)
-1)= =1 f()
In
(35)
P(S=F a)
where y - -BA a' = In paS=Fla')
aa)=F We note that y exp(y) is monotonically increasing
and + In(z/c 1) is monotonically increasing for e > 0. Since p(S= F (a) is positive constant and function composition preserves monotonicity, we have that f(y) is a monotonically increasing function of y.
A.3 Details on using sequence models for change in stability
By combining eq. (13) with eq. (25), the change in thermodynamic stability can be approximated as
B BAAGa-a In (a U)
In (a F)
(36)
(a U) (a F)
14

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Stability change as a function p(Fla', for p(Fla, 33 0.95
12.5
BAGY
10.0
BAGG
BAAGn = BAG 8'-a
-BAG
7.5
a'-ta
5.0
2.5
0.0
2.5
p(Fla', ((U),3)
55.0
0.0
0.2
0.4
0.6
0.8 1.0
p(Fla',
Figure 3: Stability change calculated from the two pseudo-fire-energy terms, 3 plotted as a function of p(F (a, for p(F 3) 0.95. As discussed in section 3.3.1, we observe that when p(F |a', (3) is around 0.95, the unfolded-state term, BAGU a' -a provides the dominant contribution the stability change, BAAGa-aa' whereas for p(F |a (3 pU la, 33), the folded- state term, BAGE aa aaa dominates. We also observe that the stability change, 3AAGa is monotonic function of the variant's folding probability, p(F la', 3), as discussed in section 3.3.2 and appendix A.2.
where the marginal sequence probabilities (a) and (a') cancel between the folded and unfolded terms, analogous to the cancellation in eq. (20). This expression shows that we can estimate the change in stability using only a sequence model that provides conditional likelihoods given the structural state, i.e., a model capable of computing (a S) for S (F, U3.
B
Experimental details
B.1 Dataset preprocessing details
The Guerois data set (Guerois al., 2002) contains 988 entries that we filtered contain only single amino acid substitutions (i.e. no double and triple substitutions). 911 entries remained after filtering and are associated to PDB structures.
The Protein G data set (Nisthal et al., 2019) contains 907 entries associated with a single PDB (PDBID:1PGA) We used the values labelled "ddG(mAvg) mean" which are associated with the lowest median uncertainty reported to be 0.1 kcal/mol. For 107 of the very destabilising entries, only lower bound of 4.0 kcal/mol were reported. Note that these values the AAG of unfolding, therefore, we inverted the sign to obtain A AAG values for folding.
The VAMP-seq data (Matreyck et al., 2018) contains 8096 entries for two proteins, TPMT and PTEN (associated with two structures with PDBID: 2H11 and ID5R, respectively). After filtering to include only amino acid residues that are resolved in the protein structures, 6909 entries remain. We use the values labelled as "score" with negative sign.
For our last experiment, we considered subset of the mega-scale thermodynamic folding stability dataset (Tsuboyama et al., 2023), with a total 42253 variants across 40 proteins (PDB-IDs can be found in Figure 2). The experimental values were retrieved through the ProteinGymm (Notin et al., 2023) interface without further processing. The structural ensembles for the wild-type variants of these proteins were retrieved from the supporting material of the BioEmu paper (Lewis et al., 2025).
15

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B.2 Choice of simulation ensemble
For the molecular dynamics simulations used in our study, we initially conducted simulations in an NVT ensemble. Over the course of the study, we refined this protocol and switched to an NPT simulation setup, which we used for the simulations for the Protein G, TPMT and PTEN in the VAMP-seq and Protein G datasets. Since the change in protocol turned out to have very minor effect on the prediction accuracy we decided not to rerun the simulations for the 40 proteins the Guerois data set.
B.3 BioEmu samples
The BioEmu model produces both folded and unfolded conformations. Since we only use BioEmu as proxy for the folded state in our analysis, we filter out conformations with fraction of native contacts (FNC) below 0.5. Since BioEmu aims to sample states proportionally to their Boltzmann weights, and the considered proteins are stable, less than 1% of samples were discarded by this procedure.
B.4 Resources
The experiments in this paper comprise running Monte Carlo and molecular dynamics simulations for 40 proteins, in addition model evaluation of pretrained models on all samples. Since no training was involved, no large scale GPU-resources were necessary for this study.
C Licenses and references for used assets
Below we list the external software and dataset assets used in this study, along with their licenses and access information:
ESM-IF
The ESM-IF inverse folding model (Hsu al., 2022) is released under the MIT li- cense. Source code and pretrained models are available at: https://githan com/ facebookresearchh/
Phaistos Framework
The Phaistos framework (Boomsma et al., 2013), used for Monte Carlo simulations of unfolded structures, is available under the LGPLv2 or GPLv3 license. The source code is available at: httt::://sourefffoi
OpenMM The OpenMM molecular dynamics engine (Eastman et al., 2017) was used for MD simula- tions. It released under a mix of licenses, including MIT, LGPLv3, CC BY 3.0 and several other linces for specific parts (see details at https: //githb.. cww/ coam///ppmm/////mm oppemm// blob/master/docs
txt) and available at: https://
github. com hwtpm//penmmm/mmge /openmm
Protein G Dataset The thermodynamic stability measurements for the B1 domain of Protein G by Nisthal al. (2019) available with license from: https://wwe www.abank.aaaa org/ h
Guerois Benchmark Set This benchmark set of experimental AAG values was compiled by Guerois et al. (2002), with data sourced from the ProTherm database (Gromiha al., 1999). The dataset is available through the ProTherm database at: https: h/tppp:///bbb.e iitm. ac. in/bioonfo22/poohemmmd
VAMP-seq Dataset Stability data for the TPMT and PTEN proteins were taken from Ma- treyek et al. (2018) and are available for non-profit, non-commercial use at: https: //github. com httmm//oooerL.aaa/MPaeee...
MobiDB Disorder Statistics Amino acid frequencies for disordered regions were extracted from the suratedddiisordderuuiproo entries in the MobiDB database (Piovesan et al., 2021). The database is available under the CC BY 4.0 licence at: attps://mobidd: bio. inipd.it
Mega-scale thermodynamic folding stability dataset The mega-scale thermodynamic folding sta- bility dataset (Tsuboyama et al., 2023) was accessed through the ProteinGym interface (Notin et al., 2023). It is licenced under the MIT licence.
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Table I: Tabular overview of the Pearson correlation coefficients presented in the fig, 1, where the numbers parentheses are the standard error of the mean.
Strategy
Guerois
Protein G
VAMP-seq
folded-only, single-sane sample
In Po (xx) (a(a (xa)
0.63 (0.02) 0.66 (0.02)
0.51 (0.01)
folded-only, (a) correction
In pa(a')aa
In (o
0.63 (0.02)
0.67 (0.02)
0.51 (0.01)
Po (m (Xa)
folded
In Pa (a(laa))) (xx) (a (xa)
In pa Ixa) Po (o/(a//))))
0.59 (0.02)
0.66 (0.02)
0.51 (0.01)
sample,
unfolded
multi sample (MC)
folded
Pa(a [xsa)
In Pa Do, (otn Meg 1:11
0.62 (0.02)
0.67 (0.02)
0.51 (0.01)
Ps (a (xxx)
single-sample,
inv-fold single-aa unfolded
folded
In pa(a'(xa)
In Pa(a (Ume) Po (
0.64 (0.02)
0.69 (0.02)
0.52 (0.01)
Po (a (xa)
single-sample, IDP aa-stats as unfolded
folded-only,
In (e(aa(/x))
0.6 (0.03)
0.7 (0.03)
0.53 (0.01)
pola (xa)
multi-sample
folded only,
Po(a' po(a/(xa)) (xa))
In Pu(a
0.61 (0.03)
0.71 (0.02)
(0.01)
multi-sample (MD), (a(a) correction
Po(a (xa)
F
(ax')
folded multi-sample (MD), unfolded multi-sample (MC)
Pe(a' (xa) In
In
Pa(a (xa) Pio (onn (Ma)
0.55 (0.03)
0.69 (0.02)
0.53 (0.01)
Mola (xm)
folded multi sample, inv-fold single-aa unfolded
In
/ po (xa) no(a (xa))
In (aay 2x
0.59 (0.02)
0.71 (0.02)
0.53 (0.01)
folded multi multi-sample (MD), IDP na-stats unfolded
pala' Ixa) Pola
In Pala [Upe]] (ax Uwr]
0.62 (0.03)
0.72 (0.02)
0.54 (0.01)
17

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In pa (a' xxa)
In po poo(a//x.) (xaa)
tho (a (xa))
Po(a ( (xa) F
foldeed-only single-samplo
foldec--nn only multi-sample
In pa [xa Pa (x.))
In Po(a
In /Po(al/xa)) (xa))
In Po(a)
pu(a()
PPlla (xx))
pu(a()
foided only, multi- saulti-sample (MD), po (sx) correction
folded-only. pa(a] correction
In pa(a' Xxa) Pa (x,))
In Pala xx. Pola' xxa) TT
In ( ( Pola (xa) F
In 2ala Xx.
pula' Xa.
U
folded single-sample, unfolded mulli-sample (MC)
lolded multi-sample (MD), unfolded multi sample (MC)
In pa [xa)
In pulau 1911
In (oo(alixa) Pola (xa) / F pu(af] Xoy 1it1 folded multi-sample (MD), inv-fold single-aa as unfoltied
Pe(ay 111 >
pa(a (xa) pola Day 1.s1 foldeed single-samper fol- single-as na unfolded
In Da (a' [x))
(a [xa)
In
Po(a (ine) pppa)
In /o(aa(xxa)
Po(a (xa) /
F po(a"
In
Po(a UUDe)
folded single-sample, IDP aa stats as unfolded
folded multisemmle (MD), IDP aa-stats as unfolded
Pearson
0.8
0.6
0.4
0.2
0.0
1D5R 2H11 1A2P 1BPI 1FKB 1FMK 1HZ6 1663 1PGA 1REX 1STN 1TEN 2012 2LZM 1PGA
VAMP-seq
Guerois
Protein G
Figure 4: A breakdown of the performance for the individual proteins within the three original datasets. Since correlations are computed, only proteins with at least 20 variant observations are included. The top-left variant is the approach typically employed as zero-shot predictor for protein stability prediction. The left column are methods based that consider only single folded structure, while the right column considers a structural ensemble from an MD simulation. Note the considerable variation among the proteins in the Guerois set.
18

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folded:single
folded:multi
folded:single, compensainn
folded:multi, compensatnnn
folded:single
unfolded::ulti
folded:multi unfolded:muiti
folded:single
unfolded:invfold
folded:multi unfolded:invfold
folded:single
unfolded:IDP
folded:multi
unfolded:IDP
0.6
0.4
0.2
0.0
VAMP-seq
Guerois Pearson
Protein
VAMP-seq
Guerois Spearman
Protein
Figure 5: Pearson and Spearman correlations behave similarly. As expected from our derivations, the relationship between zero-shot scores and stability linear, and employing a rank-based procedure like Spearman rho is therefore not necessary.
folded:single, (oldedsiiggle,pcc)ccommees compentminoitiiooonn
folded:multi, poom-compnnsttion
foldedssingle, (
folded:multi, compenaioooo
0.6
Pearson
0.4
0.2
0.0
VAMP-seq
Guerois
Protein G
Figure Compensating with a richer model for p(a). A comparison between using the simple p(a) and a full language model, ESM2 (Lin et al., 2022). The language model does not provide a benefit over the simpler model. One potential explanation could be that the ESM2 model itself captures considerable structural signal, as illustrated by its use for structure prediction ESMFold (Lin al., 2022).
19

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Table 2: Pearson correlation coefficients for different strategies, focusing on the sequence-only strategies (as compensation terms as model for the folded state). The first 10 rows in this table also present in appendix B.I. but are repeated here for ease comparison. Numbers in parentheses represent standard errors of the mean.
Strategy
Guerois
Protein G
VAMP-seq
songed:single
0.63 (0.02)
0.66 (0.02)
0,51 (0.01)
unfolded:mulii unfoldedinvfold
0.59 (0.02)
0.66 (0.02)
0.51 (0.01)
0.62 (0.03)
0.67 (0.02)
0.51 (0.01)
unfolded:IDP
0.64 (0.02)
0.69 (0.02)
0.52 (0.01)
folded:multi
0.60 (0.02)
0.70 (0.02)
0.53 (0.01)
unfolded:multi
0.55 (0.03)
0.69 (0.02)
0.53 (0.01)
unfolded:invfold
0.59 (0.03)
0.71 (0.02)
0.53 (0.01)
unfolded:IDP
0.62 (0.03)
0.72 (0.02)
0.54 (0.01)
fonded:single, colpeencomnntttionn folded:multi, p(a) compeensatin folded:single, Pasar ( compensatnnn folded:multi, (aSMr(a) (ot(oa)))cmmmiiiiooo
0.63 (0.02)
0.67 (0.02)
0.51 (0.01)
0.61 (0.03)
0.71 (0.02)
0.54 (0.01)
0.59 (0.03)
0.64 (0.02)
0.59 (0.01)
0.55 (0.03)
0.63 (0.02)
0.60
folded:p(a)
0.05 (0.04)
0.12 (0.04)
(0.01)
unfolded:mulii
-0.04 (0.03)
-0.04 (0.04)
(0.01)
unfoldedinnfoll
-0.08 (0.04)
-0.01 (0.04)
0.01 (0.01)
unfolded:1DP
0.14 (0.04)
0.21 (0.03)
0.12 (0.01)
folded:pessr24)
0.38 (0.04)
0.41 (0.03)
0.49 (0.01)
unfoldedmultii
0.34 (0.03)
0.37 (0.03)
0.48 (0.01)
unfolded:invfold
0.35 (0.03)
0.39 (0.03)
0.46 (0.01)
unfolded:IDP
(0.03)
0.45 (0.03)
0.50 (0.01)
Table 3: Ablation the Protein G dataset of 1) the inverse-folding model employed and 2) the procedure for generating structural ensembles. As has been observed previously (Notin et al., 2023), ProteinMPNN generally produces lower correlation scores than ESM-IF, but displays relatively larger performance boost with the more advanced strategies. Remarkably, the MD and BioEmu results are almost identical on these datasets, suggesting that BioEmu could provide reasonable approximation to the expensive MD structural ensembles. Values are Pearson correlation scores and standard errors of the mean are provided in parenthesis.
ESM-IF
MPNN
Strategy
Ensemble type
folded:single
0.66 (0.02)
0.4 (0.03)
folded:single, col-peoomnnntioon
0.67 (0.02)
0.43 (0.03)
folded:single unfolded:mulii
0.66 (0.02)
0.39 (0.03)
folded:single unfolded:nnyoold
0.67 (0.02)
0.35 (0.03)
folded:single unfolded:1DP
0.69 (0.02)
0.48 (0.02)
folded:multi
MD
0.70 (0.02)
0.50 (0.02)
BioEmu
0.69 (0.02)
folded:multi, pla)-compensation
MD
0.71 (0.02)
0.53 (0.02)
BioEmu
0.70 (0.02)
folded:multi unfolded:multi
MD
0.69 (0.02)
0.49 (0.02)
BioEmu
0.68 (0.02)
folded:multi unfoldedinvfold
MD
0.71 (0.02)
0.43 (0.03)
BioEmu
0.70 (0.02)
folded:multi unfolded:IDP
MD
0.72 (0.02)
0.58 (0.02)
BioEmu
0.72 (0.02)
20

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27


Reviews:
[{"review_id": "8OaLD7nkFv", "reviewer": "Reviewer_u4zq", "summary": "This paper presents a new theoretical perspective on the connection between the likelihood learned by inverse folding models and thermostability. Based on thermodynamic equations, this paper clarifies the free-energy foundations of inverse folding models and explore new ways to estimate relative stability other than the standard log likelihood ratio methods. The experiments on three protein stability datasets show that considerable gains in zero-shot performance can be achieved with fairly simple alternatives.", "questions": "1. If you run the same experiments with proteinMPNN, would the performance of different prediction methods change?\n2. Can you discuss how to extend your theoretical analysis to binding energy prediction. People typically use the SKEMPI database to evaluate different methods and they found that protein inverse folding models is also predictive. Can we adopt the same extensions you developed here?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. This paper is one of the few papers that presents theoretical perspective on connection between the likelihood learned by inverse folding models and thermostability. It's a very interesting paper. The thermodynamics equations / derivations presented in this paper may inspire other researchers to design better methods for stability prediction\n2. Based on the thermodynamic perspective, it proposes new alternatives for stability prediction other than standard log-likelihood ratio. The new alternative metrics actually perform better and more principled.\n\nWeakness\n1. The test sets used in this paper is relatively small. Additional datasets can be included, such as Tsuboyama et al. paper listed below.\n2. This paper only use ESM-IF. How about proteinMPNN? It's unclear whether the same observation holds for ProteinMPNN.\n\nTsuboyama K, Dauparas J, Chen J, Laine E, Mohseni Behbahani Y, Weinstein JJ, Mangan NM, Ovchinnikov S, Rocklin GJ. Mega-scale experimental analysis of protein folding stability in biology and design. Nature. 2023\nJul; p. 1–11.", "was_revised": false, "final_justification": "", "rebuttal": "## Summary of responses to all reviewers\nWe thank all the reviewers for their constructive reviews and relevant feedback. The comments regarding the theoretical contribution were generally positive, but several of the reviewers raised concerns regarding the scope of our empirical evaluations. Although we believe that the primary contribution of our paper is the theoretical connection between inverse folding likelihoods and thermodynamic stability, we agree that expanding the empirical evaluations would further strengthen the paper. We have therefore initiated the following experiments:\n\n1. Repeat experiments with ProteinMPNN, a leading inverse-folding model, \n2. Replace simplistic site-independent model for $p(\\mathbf{a})$ with a pLM, \n3. Rerun our protocol using ensembles obtained from BioEmu, thereby avoiding the computational bottleneck of molecular dynamics – which would allow us to scale to larger datasets. \n\nWe hope to have preliminary results for 1) ready within the rebuttal period, and 2) and 3) will be added before camera-ready. Finally, we will change the paper title to \"Zero-shot protein stability prediction by inverse folding models: a free energy interpretation\" – to more specifically reflect its contents.\n\n## Response to Reviewer u4zq\n\nWe thank Reviewer u4zq for their positive remarks. We agree that the test sets used in our paper were not as extensive as one might have wished for. The primary reason for focusing on a limited set of systems was the molecular dynamics simulations: not only the computational cost, but also making sure that the molecular systems were set up correctly, relaxed properly in the forcefield, etc. We believe the current set of experiments is sufficient as a sanity check of the presented theoretical contribution. However, we do agree that it would be valuable to see whether the trends hold across a larger dataset, and to develop a protocol that would make the approach more robustly applicable in practice. We are therefore now in the process of running our experiments using ensembles sampled from BioEmu, and will attempt to scale this to the Tsuboyama et al dataset. We believe we can have this full evaluation completed in time for camera-ready. If the BioEmu ensemble-approach fails, we would be happy to at least extend our current analysis to several more systems using the full molecular dynamics protocol.\n\nWe also agree that it would be interesting to see if MPNN shows similar trends. We are setting up these experiments now. This does not involve new molecular dynamics simulations, we expect that we should have preliminary results ready before the discussion period ends, and hope to post the results when they are ready. A priori, it is unclear to us what to expect from MPNN. The ProteinGym zero-shot benchmark for the stability category reports ESM-IF likelihoods to have somewhat higher correlation to experimental values (0.624) than ProteinMPNN (0.565), suggesting that it might perform slightly worse, but the relative trends of the ablation might show very similar behavior.\n\nIt would indeed be interesting to investigate our approach in the context of binding energy. Theoretically, it is a simple extension, since the standard way to calculate mutational effects on binding is to look at the difference in mutational effects on the stability of the complex ($A:B$) and the monomers ($A$ & $B$) is $\\Delta\\Delta G = \\Delta G_{A:B}^{\\text{wt} \\to \\text{mut}} - \\Delta G_{A}^{\\text{wt} \\to \\text{mut}} - \\Delta G_{B}^{\\text{wt} \\to \\text{mut}}$ . In practice, it would require some work to do the analysis, since structural ensembles would need to be acquired for these different structural states, over a range of systems for which we have experimental results (e.g. from SKEMPI as you suggest). Because of the work involved in these practical matters, we defined this to be out of scope of the current paper, and focused our attention on stability. In the new version of the manuscript, we will, however, add a comment to the discussion to make it clearer how our approach applies to the binding case."}, {"review_id": "00FfoPWR4B", "reviewer": "Reviewer_PxEM", "summary": "In this paper, the authors investigate the relationship between thermodynamic stability of proteins and log-odds scores from inverse folding models. They derive corrections to the typical simple zero-shot predictor and comprehensively evaluate incorporating additional terms related to the unfolded and folded ensembles. They consider three datasets and conclude that it is possible to improve on the zero-shot predictor by including corrections from the unfolded ensemble and additional samples from the folded ensemble.", "questions": "Are there additional datasets or experimental settings, such as subsets of ProteinGym, where the authors think the effects of the corrective terms may be more pronounced?\nCan the authors highlight any specific cases where the ranking of different mutants within a dataset via the different methods is substantially different between the different terms considered in the paper?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strengths\nThe paper provides a clear and well-reasoned theoretical basis for exploring the connection between thermodynamic stability and likelihoods under an inverse folding model. \nThe theory naturally suggests two distinct categories of correction terms, and many specific implementations to ablate.\nThe finding that the unfolded state is best approximated by simple amino acid frequency statistics from disordered regions is interesting and could be of interest for future work on modeling and incorporating information from disordered regions in proteins, which current generation structure-based models largely do not account for.\n\nWeaknesses\nThe experiments, while thorough and comprehensive across the three datasets considered and the many additional terms, do not suggest any improvements to the folded-only, single-sample standard approach that are robust enough and worth the additional complexity of computing the correction terms.\nThe paper considers only ESM-IF, whereas most protein design literature relies on ProteinMPNN and related models.\nThe experiments do not consider more widely used benchmarks like ProteinGym.\nFigure 1 is difficult to parse. It may be useful to indicate, via text within the figure and another visual element like hash marks, that the left column considers single-structure approaches and the right column considers ensemble approaches. Highlighting the standard zero-shot predictor would also be helpful.\nThe abstract and introduction do not mention any specific quantitative results, conclusions, or prescriptions for future work.", "was_revised": true, "final_justification": "The authors addressed both my major comments during the rebuttal period, and further clarified the scope and motivation of their work. Hence, I have raised my score.", "rebuttal": "## Summary of responses to all reviewers\nWe thank all the reviewers for their constructive reviews and relevant feedback. The comments regarding the theoretical contribution were generally positive, but several of the reviewers raised concerns regarding the scope of our empirical evaluations. Although we believe that the primary contribution of our paper is the theoretical connection between inverse folding likelihoods and thermodynamic stability, we agree that expanding the empirical evaluations would further strengthen the paper. We have therefore initiated the following experiments:\n\n1. Repeat experiments with ProteinMPNN, a leading inverse-folding model, \n2. Replace simplistic site-independent model for $p(\\mathbf{a})$ with a pLM, \n3. Rerun our protocol using ensembles obtained from BioEmu, thereby avoiding the computational bottleneck of molecular dynamics – which would allow us to scale to larger datasets. \n\nWe hope to have preliminary results for 1) ready within the rebuttal period, and 2) and 3) will be added before camera-ready. Finally, we will change the paper title to \"Zero-shot protein stability prediction by inverse folding models: a free energy interpretation\" – to more specifically reflect its contents.\n\n## Response to Reviewer PxEM\n\nWe thank the Reviewer PxEM for their positive evaluation of our theoretical contribution and their critical assessment of our paper. Our main motivation with this paper was to attempt to understand why log-odds score from inverse-folding models correlate so well with stability changes. From this perspective, we did not consider it a problem that not all correction terms provide substantial improvements, and considered the fact that these correction terms generally had a positive effect mostly as a sanity-check of the underlying derivations. \n\nOf course, we agree with the reviewer that the paper would benefit from an expanded set of experiments – both to get some more statistically significant results and because the assumptions we make might hold to a different degree for different protein systems. As we have discussed above with reviewer u4zq, the main bottleneck was conducting properly converged molecular dynamics simulations. We agree with the reviewer that this computational complexity makes our approach less interesting for practical application. In an effort to overcome this, we are now repeating our experiments using ensembles from the BioEmu model, with the hope that the structural ensembles sampled from this model are of sufficient quality to make it an alternative to molecular dynamics. If this proves successful, it allows us to scale our analysis to a larger stability subset of ProteinGym (e.g., the Tsuboyama subset), and also suggests a path towards making our approach more useful for practical use.\n\nWe agree that it makes sense to compare to MPNN as well. We chose ESM-IF because it generally performs slightly better on zero-shot stability prediction than ProteinMPNN (according to ProteinGym), but the reviewer is right that the ubiquitous use of ProteinMPNN makes it a relevant comparison. We are currently running these experiments and hope to have them done before the rebuttal period ends.\n\nWe thank the reviewer for their comments regarding Figure 1 and the abstract/introduction, which we will incorporate in the new version of the manuscript.\n\nIt is a good question whether subsets of ProteinGym – or differences in experimental settings – would display different tendencies. We could clearly expect the most meaningful results for subsets categorized under the \"Stability\" label. In particular, the Tsuboyama megascale experiments are direct probes for protein stability and should thus behave well. The VAMP-seq datasets – like the one we use in our paper – measure abundance, but have previously been shown to correlate well with stability, and should thus be another meaningful subset of ProteinGym to consider (although likely displaying somewhat higher levels of noise).\n\nWe do see differences in the ranking of mutants between different ablations of terms, as reflected by the Spearman rank correlation scores in Figure 1 and Table 1. It would indeed be interesting to investigate whether there are any patterns to these effects – e.g., whether there is a clear difference between surface and core residues."}, {"review_id": "uH8WYjXHyE", "reviewer": "Reviewer_sDYD", "summary": "The authors investigate the relation between free-energy consideration, from a chemistry perspective, and recent inverse folding models, improving the understanding of good results obtained by these models for stability prediction and testing different formulations (based on different assumptions and data availability) in empirical experiments. The results show that inverse folding models represent a simplistic approximation for the free-energy modeling and different considerations improving the modeling have the potential to improve the performance of free-energy predictors.", "questions": "Comments:\n\n1. Equation 22: Should the second term be calculated with the same model as the first term or can different models be used?\n2. Can the authors clarify how the first term in equation 25 can be calculated or how this conditioning can be trained?\n3. (line 220) The authors mention that they use predicted structures for UR50. Does this mean that some of the structures used for the datasets analyzed are predicted structures?\n4. What is the meaning of the sentence in lines 232-235: “While data generated… for protein stability.”\n5. (lines 246-249) Why a position-independent sequence model? Why did the authors not test a masked protein LM in this case?\n6. The writing and explanation of section 4.3 are confusing. It needs additional explanation, intuition, and, if possible, a visualization. As there is no clear way to do that, readers should be clear about the reasoning of the authors for generating or modeling these unfolded ensembles.\n7. Can the authors clarify the differences between this work and the work done by Jiao et al, 2024? (currently being discussed in lines 301-304 and lines 319-321)\n8. How can the foundations presented in the work be applied to generate better objectives for self-supervised losses when training models using datasets like the PDB? Do the authors think these can also help model more efficient losses for using datasets with augmented predicted structures?\n\nMinor Comments:\n\n1. Typo: (line 99) “which has by”\n2. Typo: (line 120) “change-in-free”\n3. Expression: (line 217) “on a representative selection of protein dataset”\n4. Typo: Missing a dot in line 271", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The attempt to improve the understanding and the relation between free-energy and inverse folding models is timely, given the results of recent references, and has the potential to improve the pre-training objectives of protein-related foundation models.\n2. The authors show the potential of including an unfolded ensemble and approximating the structural ensemble of the folded state with multiple samples. \n3. The theoretical derivation is well explained and clear giving meaningful insights for the reader.\n\nWeaknesses:\n\n1. Given the popularity of protein language models, it would improve the strength of the paper using these to model correction terms or terms needing to model p(a).\n2. The empirical results hint that a better modeling can improve the performance but a stronger discussion and analysis regarding these empirical results would improve the manuscript. The empirical results are limited in the current version of the manuscript.\n3. The clarity of the sections for the folded ensemble and unfolded ensemble could be improved.", "was_revised": true, "final_justification": "During the rebuttal phase, the authors clarified my concerns and added more empirical results, which addressed a weakness of the manuscript. My recommended score is based on the theoretical contributions of the proposed manuscript, that is timely in this research field.", "rebuttal": "## Summary of responses to all reviewers\nWe thank all the reviewers for their constructive reviews and relevant feedback. The comments regarding the theoretical contribution were generally positive, but several of the reviewers raised concerns regarding the scope of our empirical evaluations. Although we believe that the primary contribution of our paper is the theoretical connection between inverse folding likelihoods and thermodynamic stability, we agree that expanding the empirical evaluations would further strengthen the paper. We have therefore initiated the following experiments:\n\n1. Repeat experiments with ProteinMPNN, a leading inverse-folding model, \n2. Replace simplistic site-independent model for $p(\\mathbf{a})$ with a pLM, \n3. Rerun our protocol using ensembles obtained from BioEmu, thereby avoiding the computational bottleneck of molecular dynamics – which would allow us to scale to larger datasets. \n\nWe hope to have preliminary results for 1) ready within the rebuttal period, and 2) and 3) will be added before camera-ready. Finally, we will change the paper title to \"Zero-shot protein stability prediction by inverse folding models: a free energy interpretation\" – to more specifically reflect its contents.\n\n## Response to Reviewer sDYD\n\nWe thank the Reviewer sDYD for their positive remarks and detailed feedback. We agree that it would be natural to use a language model for calculating $p(\\mathbf{a})$, and will add this result to the paper. As stated above in response to reviewers u4zq and PxEM, we also agree that the paper would benefit from a more elaborate empirical evaluation, and we are currently setting up experiments using ProteinMPNN and BioEmu (see details above). Thanks for pointing out the difficulties reading sections 4.2 and 4.3. We will make an effort to rephrase these. And thanks for pointing out the list of typos, which we will incorporate immediately.\n\nPlease see the point-by-point answers to the questions below:\n\n1. *Equation 22: Should the second term be calculated with the same model as the first term or can different models be used?*\nIn principle, it should be calculated using the same model. So ideally we would want a joint model that supported marginalizing out the structural context, and we could use the same model for both terms. In practice, inverse-folding models do not support this, and we therefore typically employ different models for these two terms: an inverse-folding model for the first and a sequence-only model for the 2nd. \n\n2. *Can the authors clarify how the first term in equation 25 can be calculated or how this conditioning can be trained?*\nThe first term in equation 25 is the probability of observing an amino acid sequence, conditioned on being in an unfolded state. It is indeed not trivial how to evaluate this in practice – and we therefore dedicate section 4.3 to discussing various alternatives. The most simple solution that we found is to use the amino acid frequencies obtained from intrinsically disordered proteins for this quantity. Given your comment above about section 4.3, we acknowledge that this was not sufficiently clear in the paper, and will elaborate on this explanation in the new version of the paper.\n\n3. *(line 220) The authors mention that they use predicted structures for UR50. Does this mean that some of the structures used for the datasets analyzed are predicted structures?*\nApologies for the confusing sentence structure. The intention was to communicate that the original authors of ESM-IF trained their model on both on CATH AND on predicted structures for UR50. When evaluating the models, we only used crystal structures and structures from molecular dynamics simulations. We have rephrased the sentence to avoid the confusion.\n\n4. *What is the meaning of the sentence in lines 232-235: \"While data generated … for protein stability.\"*\nWe apologize for this poorly formulated sentence. We meant to say that the abundance signal measured by VAMP-seq is a convoluted signal of thermodynamic stability, effects on local unfolding stability and interactions with the cell's quality control system, and we therefore do not expect to get as clear a signal on protein stability from this type of experiment. We will rephrase to make this clearer.\n\n5. *(lines 246-249) Why a position-independent sequence model? Why did the authors not test a masked protein LM in this case?*\nThe choice of the position-independent sequence model was made as a simple proxy which had similar sequence statistics as the training data of the ESM-IF model (which was trained on UniRef50 sequences). Protein language models like ESM2 are trained on similar sequence datasets, and should therefore indeed be directly applicable as well. We will do so in the new version of the paper.\n\n6. *The writing and explanation of section 4.3 are confusing. It needs additional explanation, intuition, and, if possible, a visualization. As there is no clear way to do that, readers should be clear about the reasoning of the authors for generating or modeling these unfolded ensembles.*\nWe acknowledge that this was not sufficiently clear. One of the main problems seems to be that we do not explicitly relate the discussion in 4.3 back to the terms we wish to approximate (e.g. the first term in eq 25). We will rework these sections to make this clearer, possibly adding a figure if necessary.\n\n7. *Can the authors clarify the differences between this work and the work done by Jiao et al, 2024? (currently being discussed in lines 301-304 and lines 319-321)*\nOur work considers the more general case of the full structural ensembles, while Jiao et al. represent structural states by single conformations. In addition, Jiao et al. consider protein binding while we consider protein stability, but as discussed in our response to reviewer u4zq above, these tasks are closely related, and it would be possible to use our approach in the setting introduced by Jiao et al., by considering the full structural ensembles.\n\n8. *How can the foundations presented in the work be applied to generate better objectives for self-supervised losses when training models using datasets like the PDB? Do the authors think these can also help model more efficient losses for using datasets with augmented predicted structures?*\nIndeed, it would be possible to construct an self-supervised objective, in a very similar way as Jiao et al. (2024), but in our case the loss would be evaluated over a full structural ensemble. It would be interesting to explore whether this leads to improved binding energy prediction. However, due to the considerable experimentation required, we consider it outside the scope of the current manuscript."}, {"review_id": "t6zYYgq8P2", "reviewer": "Reviewer_r63Z", "summary": "In this paper, the authors lay out a series of assumptions that are needed in order to show that the log-odds based on the inverse-folding model (log p(structure|wild-type seq)/p(structure|mutated seq)) and a small number of (even just one) folded structures is a good approximation to ΔΔG (which is the change in the change of the energy from unfolded to folded states of a sequence after a small mutation.) I enjoyed reading the whole series of approximations, although it raises a question whether those approximations naturally arise from our knowledge of molecular biology and reasonable intuition or were selected to fit this picture that the inverse-folding-based log-odds was found earlier by others as a reasonable approximation to the stability of a protein. \n\nPerhaps, it’s okay as long as we can arrive at a usual practice (the log-odds) from ΔΔG via this chain of assumptions, since this chain of assumptions itself may be what was missing. But, it does raise a question whether we could arrive at this conclusion from a different angle and from a different set of explanations and assumptions. I am curious about such an alternative, since the experimental results show that a supposedly better approximation scheme (under the proposed scheme) to ΔΔG (e.g. by using more folded structures, by drawing unfolded structures, etc.) does not really lead to a significantly better approximation of ΔΔG. the correlation does seem to go up generally, but if you look at the bars carefully in Fig. 1, the improvement is at best modest, and sometimes better approximation schemes result in worse correlations.\n\nSo, do I have any potential alternatives in my mind? Unfortunately no.", "questions": "See my summary comment above.", "limitations": "See my summary comment above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "See my summary comment above.", "was_revised": false, "final_justification": "", "rebuttal": "## Summary of responses to all reviewers\nWe thank all the reviewers for their constructive reviews and relevant feedback. The comments regarding the theoretical contribution were generally positive, but several of the reviewers raised concerns regarding the scope of our empirical evaluations. Although we believe that the primary contribution of our paper is the theoretical connection between inverse folding likelihoods and thermodynamic stability, we agree that expanding the empirical evaluations would further strengthen the paper. We have therefore initiated the following experiments:\n\n1. Repeat experiments with ProteinMPNN, a leading inverse-folding model, \n2. Replace simplistic site-independent model for $p(\\mathbf{a})$ with a pLM, \n3. Rerun our protocol using ensembles obtained from BioEmu, thereby avoiding the computational bottleneck of molecular dynamics – which would allow us to scale to larger datasets. \n\nWe hope to have preliminary results for 1) ready within the rebuttal period, and 2) and 3) will be added before camera-ready. Finally, we will change the paper title to \"Zero-shot protein stability prediction by inverse folding models: a free energy interpretation\" – to more specifically reflect its contents.\n\n## Response to Reviewer r63Z\n\nWe thank the Reviewer r63Z for their feedback on our paper. It is a valid point that we cannot rule out that a similar outcome could have been obtained through a different line of reasoning. However, the assumptions we make do not arise out of the blue. They are based on approximations of the underlying biophysical systems. In our view, the value of our work lies in the fact that our derivations make it possible to have very specific discussions about the types of assumptions we make implicitly when using inverse folding models – and potentially develop better models in the future by revisiting these assumptions.\n\nAs also discussed in our response to PxEM, we considered our empirical studies primarily as a sanity check on the underlying theory, and were comforted to see that the correction terms generally contribute positively to performance. It is true that many of the correction terms contribute only marginally to the $\\Delta\\Delta G$ prediction performance. However, apart from the noisy Guerois dataset, we consider the total performance gains as quite substantial (0.66 vs 0.72 for ProteinG and 0.51 vs 0.54 for VAMP-seq), in particular when compared to the small variations typically reported between the top performing methods on ProteinGym. \n\nWe acknowledge that stronger conclusions about the relative merits of the individual correction terms would require a more elaborate set of experiments. Scaling up to more systems has been hindered by the fact that we need to conduct full molecular dynamics simulations for each considered system. In an attempt to solve this issue, we are currently investigating the use of BioEmu as a replacement for full molecular dynamics simulations (for details, see the general post above)."}, {"review_id": "DFjTIp01BR", "reviewer": "Reviewer_NjCR", "summary": "This paper presents a mathematical framework that ties the ddG of mutation to inverse folding probabilities. More specifically, they show under some assumptions that the ddG can be approximated by the ratio of the inverse folding model's predictions given folded and unfolded structures (Eqn 19).", "questions": "Questions\n- How does equation 10 connect to the rest of the technical section?\n- How does section 3.4 Change in Stability with sequence models tie into estimating protein stability using inverse folding models?\n- Why does the inverse folding model compute a meaningful $p_\\theta(a|x)$ for unfolded structures, as it was never trained on unfolded structures?\n- Could the authors elaborate on the position independent sequence model estimated from $p_D$ described in L248?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n- The technical section is crisp and extremely easy to follow.\n- The assumptions made are presented forthright and are largely intuitive and reasonable.\n\nWeaknesses\n- Assume the same $\\beta$ (and thus temperature) across the data source (PDB).\n- The inverse folding model estimates  $p_\\theta(a|x)$, however is only trained on folded CATH structures.\n- The related work is short and narrowly scoped. It does not discuss other avenues of estimating protein stability beyond inverse folding methods.\n\t- Tsuboyama, Kotaro, et al. \"Mega-scale experimental analysis of protein folding stability in biology and design.\" Nature 620.7973 (2023).\n\t- Schymkowitz, Joost, et al. \"The FoldX web server: an online force field.\" Nucleic acids research. (2005).", "was_revised": true, "final_justification": "my concerns have largely been addressed", "rebuttal": "## Summary of responses to all reviewers\nWe thank all the reviewers for their constructive reviews and relevant feedback. The comments regarding the theoretical contribution were generally positive, but several of the reviewers raised concerns regarding the scope of our empirical evaluations. Although we believe that the primary contribution of our paper is the theoretical connection between inverse folding likelihoods and thermodynamic stability, we agree that expanding the empirical evaluations would further strengthen the paper. We have therefore initiated the following experiments:\n\n1. Repeat experiments with ProteinMPNN, a leading inverse-folding model, \n2. Replace simplistic site-independent model for $p(\\mathbf{a})$ with a pLM, \n3. Rerun our protocol using ensembles obtained from BioEmu, thereby avoiding the computational bottleneck of molecular dynamics – which would allow us to scale to larger datasets. \n\nWe hope to have preliminary results for 1) ready within the rebuttal period, and 2) and 3) will be added before camera-ready. Finally, we will change the paper title to \"Zero-shot protein stability prediction by inverse folding models: a free energy interpretation\" – to more specifically reflect its contents.\n\n## Response to Reviewer NjCR\n\n\nWe thank Reviewer NjCR for their detailed feedback to our paper. We give a point-by-point response to the identified weaknesses and questions below:\n\n**Assume the same beta (and thus temperature) across the data source (PDB)**\n\nWe agree that the assumption about constant $\\beta$ (inverse temperature) across all of PDB (eq. 13) is quite strong.  However, we have since realized that this assumption can be substantially relaxed. It is sufficient to assume (cf. eq. 11) that each structure in the database is sampled from its own Boltzmann distribution with a potentially different (and unknown) $\\beta$, that is\n$$\np_{\\operatorname{D}}(\\boldsymbol{\\chi} | \\boldsymbol{a}, \\beta) \\approx p(\\boldsymbol{\\chi} | \\boldsymbol{a}, \\beta).\n$$\nCrucially, the downstream analysis does not require access to the individual $\\beta$-values. In fact, we can also include the inverse temperature $\\beta$ in our model from eq. 12, meaning that we assume a joint model $p_\\theta(\\mathbf{a}, \\mathbf{x}, \\beta)$ over sequence, structure and inverse temperature. This weaker assumption above (i.e., allowing for variation in $\\beta$ across the database) then implies (cf. eq. 13) that\n$$\n   p_{\\theta}(\\mathbf{x} | \\mathbf{a}, \\beta) \\approx p_{\\operatorname{D}}(\\mathbf{x} | \\mathbf{a}, \\beta) \\approx p(\\mathbf{x} | \\mathbf{a}, \\beta) ,\n$$\nUsing this assumption, we can now explicitly condition all factors in eq. 17 on the specific $\\beta$ beta that the structure is observed and simulated at, and we get\n$$\n\\frac{p(\\mathbf{x} | \\mathbf{a}', \\beta)}{p(\\mathbf{x} | \\mathbf{a}, \\beta)} \\approx \\frac{p_\\theta(\\mathbf{x} | \\mathbf{a}', \\beta)}{p_\\theta(\\mathbf{x} | \\mathbf{a}, \\beta)}\n= \\frac{\\frac{p_\\theta(\\mathbf{a}' | \\mathbf{x}, \\beta) p_\\theta(\\mathbf{x}| \\beta) } {p_\\theta(\\mathbf{a}'| \\beta)}}{\\frac{p_\\theta(\\mathbf{a} | \\mathbf{x}, \\beta) p_\\theta(\\mathbf{x}| \\beta) } {p_\\theta(\\mathbf{a}|\\beta)}}\n= \\frac{p_\\theta(\\mathbf{a}' | \\mathbf{x}, \\beta)}{p_\\theta(\\mathbf{a} | \\mathbf{x}, \\beta )} \\frac{p_\\theta(\\mathbf{a}|\\beta)}{p_\\theta(\\mathbf{a}'|\\beta)}\n$$\nWe now make the same weak assumption that is implicitly made when training an inverse folding model: that, conditioned on a given structure, the distribution over amino acid sequences is independent of the temperature at which the structure was recorded. In other words, we treat the structure as a complete description of the structural state, which means that $p_\\theta(\\mathbf{a} | \\mathbf{x}, \\beta) = p_\\theta(\\mathbf{a} | \\mathbf{x})$. We make the similar weak assumption that the sequence distribution is independent of the temperature, i.e., $p_\\theta(\\mathbf{a} | \\beta) = p_\\theta(\\mathbf{a})$. This gives us that\n$$\n\\frac{p(\\mathbf{x} | \\mathbf{a}', \\beta)}{p(\\mathbf{x} | \\mathbf{a}, \\beta)} \\approx \\frac{p_\\theta(\\mathbf{a}' | \\mathbf{x})}{p_\\theta(\\mathbf{a} | \\mathbf{x})} \\frac{p_\\theta(\\mathbf{a})}{p_\\theta(\\mathbf{a}')}.\n$$\nThis can now be substituted into eq. 16, yielding\n$$\n\\beta \\Delta \\tilde{G}\\_{\\mathbf{a}' \\to \\mathbf{a}}^{S} = \\ln\n\\frac{p(S | \\mathbf{a}', \\beta)}{p(S | \\mathbf{a}, \\beta)} \\approx \\ln \\mathbb{E}\\_{\\mathbf{x} \\sim p\\_\\theta(\\mathbf{x} | S, \\mathbf{a}, \\beta)} \\left[ \\frac{p\\_\\theta(\\mathbf{a}' | \\mathbf{x})}{p\\_\\theta(\\mathbf{a} | \\mathbf{x} )} \\right] \\frac{p\\_\\theta(\\mathbf{a})}{p\\_\\theta(\\mathbf{a}')}  .\n$$\nThis provides a temperature-dependent estimate of $\\beta \\Delta \\tilde{G}\\_{\\mathbf{a}' \\to \\mathbf{a}}^{S}$, where the temperature is determined by that of the MD simulation, i.e., it is specified in the distribution $p_\\theta(\\mathbf{x} | S, \\mathbf{a}, \\beta)$. We have updated the paper to reflect this relaxed assumption.\n\n**The inverse folding model estimates $p(a|x)$, however is only trained on folded CATH structures**\n\nIt is indeed not obvious that this is reasonable. Empirically, we found that inverse folding models do seem to provide somewhat reasonable amino acid preferences for unfolded states. A possible explanation is that in the autoregressive factorization of an inverse-folding model, the distribution of the $i$'th amino acid is typically conditioned only on the local structural environment surrounding it. Some of these local structural environments (e.g., coil or surface regions) will be similar to those of unfolded proteins, providing some degree of generalization to the unfolded state. We rely on similar generalizations even within the folded state: if $x$ actually represented the full structural state, CATH would provide us only with a single sequence for $\\mathbf{a}$ given $\\mathbf{x}$, and our model would predict a delta function for each amino acid, providing no information about alternative amino acids (or $\\Delta\\Delta G$). \n\n**The related work is short and narrowly scoped. It does not discuss other avenues of estimating protein stability beyond inverse folding methods.**\n\nDue to space constraints, we indeed ended up with a quite narrowly scoped related work section. With the extra page provided for camera-ready, we will broaden it to include a broader overview of the developments over the last decades, including methods such as FoldX and Rosetta.\n\n**How does equation 10 connect to the rest of the technical section?**\n\nEquation 10 is used in 3.3.2. We acknowledge that this was not very clear, since 3.3.2 refers to appendix A.2, which in turn refers to eq 10. We will add a direct reference to eq 10 from 3.3.2 to make the link clearer.\n\n**How does section 3.4 Change in Stability with sequence models tie into estimating protein stability using inverse folding models?**\n\nSection 3.4 was introduced to motivate the way we deal with the unfolded state (using amino acid frequencies from the unfolded state). However, the section indeed suggests a protocol relying entirely on sequence-only models, which we did not find time to test empirically. Since we will be evaluating pLMs for $p(a)$ anyway (see response to reviewer sDYD), we will add the results of this protocol as well. This should integrate section 3.4 more closely with the remaining story.\n\n**Why does the inverse folding model compute a meaningful $p_\\theta(a|x)$ for unfolded structures, as it was never trained on unfolded structures?**\n\nSee discussion above.\n\n**Could the authors elaborate on the position independent sequence model estimated from  described in L248?**\n\nThis is a simple amino acid frequency model estimated from the human proteome (`UP000005640_9606.fasta`). As was also pointed out by reviewer sDYD, it would be meaningful to estimate $p(a)$ using a pLM, and we will include this result in the new version of the manuscript."}]

Author rebuttal:

Meta-review: The reviewers and I are in unanimous agreement on accepting the paper: it makes an insightful connection between thermodynamic stability and inverse folding probabilities and experimentally explores how this results in corrections to the zero-shot predictor using unfolded and folded ensembles. Initially multiple reviewers found only using ESM-IF insufficient, but the rebuttal adding ProteinMPNN convincingly addresses this.
Decision: Accept (poster)