Abstract
Background
Genome sequencing projects have expanded the gap between the amount of known protein sequences and structures. The limitations of current high resolution structure determination methods make it unlikely that this gap will disappear in the near future. Small angle Xray scattering (SAXS) is an established low resolution method for routinely determining the structure of proteins in solution. The purpose of this study is to develop a method for the efficient calculation of accurate SAXS curves from coarsegrained protein models. Such a method can for example be used to construct a likelihood function, which is paramount for structure determination based on statistical inference.
Results
We present a method for the efficient calculation of accurate SAXS curves based on the Debye formula and a set of scattering form factors for dummy atom representations of amino acids. Such a method avoids the computationally costly iteration over all atoms. We estimated the form factors using generated data from a set of high quality protein structures. No ad hoc scaling or correction factors are applied in the calculation of the curves. Two coarsegrained representations of protein structure were investigated; two scattering bodies per amino acid led to significantly better results than a single scattering body.
Conclusion
We show that the obtained point estimates allow the calculation of accurate SAXS curves from coarsegrained protein models. The resulting curves are on par with the current stateoftheart program CRYSOL, which requires full atomic detail. Our method was also comparable to CRYSOL in recognizing native structures among nativelike decoys. As a proofofconcept, we combined the coarsegrained Debye calculation with a previously described probabilistic model of protein structure, TorusDBN. This resulted in a significant improvement in the decoy recognition performance. In conclusion, the presented method shows great promise for use in statistical inference of protein structures from SAXS data.
Background
The fast progress of large scale gene sequencing projects has lead to a rapid increase in the amount of known protein sequences, extending the gap between known sequences and known structures [1]. Highresolution methods have successfully been applied to resolve the structure of many proteins at the atomic level but the class of experimental conditions to which they can be applied is limited by the crystallization process for Xray crystallography and protein size for Nuclear Magnetic Resonance spectroscopy (NMR).
These limitations can be overcome by turning to different low resolution structure determination methods. Small Angle Xray Scattering (SAXS) [24] is a well established low resolution method that relies on an isotropical 1D description of the excess electron density of the sample versus the surrounding environment. Recently, automated methods for highthroughput SAXS analysis of biomolecules have been developed [5,6], opening the prospect of structure determination on a genomic scale from SAXS experiments. SAXS data provide information on the structure of a protein in solution, but the information content is small compared to Xray crystallography or NMR data due to the inherent ambiguity arising from spherical averaging. This means that SAXS data only provides structural information at low resolution; additional model constraints are therefore typically needed to assist the structural interpretation.
Early SAXS structure determination methods were based on ab initio shape determination using spherical harmonics expansions [7]. These methods provide good computational efficiency, at the cost of limitations in accuracy for complex shapes; for instance for proteins with internal cavities [8]. Another modeling approach has been to fit the scattering curve using a gas of "dummy beads". This is done using conformational searches by a genetic algorithm in DALAI_GA [9] or simulated annealing in DAMMIN [10] and its optimized implementation DAMMIF [11]. A higher resolution approach is found in the GASBOR program [12], where a SAXS curve is fitted using a packed assembly of spheres in a pseudoCα chain. This program does not use amino acid sequence information, but does enforce a realistic packing density for the Cα atoms. In GASBOR, the scattering intensity is calculated using the Debye formula while simulated annealing is used for searching the conformational space. Other recent structure prediction methods, such as the ORNL [13] and IMP [14] programs, utilize the SAXS curve in the form of an extra energy term. Since these methods are nonprobabilistic, the weight that scales the SAXS energy with respect to the other energy terms must be chosen heuristically [15,16].
According to the Bayesian probability calculus, the conditional probability of an event given some data depends on the likelihood (which brings in the data), multiplied by the prior distribution (which brings in the knowledge regarding the event prior to observing the data) [17]. If experimental data D is used to infer the structure X of a protein with a known primary sequence A, this results in the following expression:
Such an approach was used by Rieping et al. [15] for inferential structure determination using NMR data. The likelihood function P(DX, A) accounts for the experimental data and quantifies the probability of observing data D given a protein structure X with sequence A. The prior, P(XA), on the other hand accounts for general knowledge about protein structures with a given amino acid sequence [15,1821]. In our case, the data D is the experimentally measured scattering curve I resulting from a SAXS experiment.
For the evaluation of the likelihood, it is necessary to compute the probability of the scattering profile, I, given a proposal structure, X. This work therefore focuses specifically on the calculation of the theoretical SAXS scattering curve I '(X, A). The likelihood can then be calculated by evaluating the discrepancy  in a probabilistic way  between the experimental curve I and the calculated curve I '(X, A). Both the curve calculation and evaluation of discrepancy must be reasonably fast in order to be useful for macromolecular structure determination. We used the wellknown Debye formula [22] for calculating the scattering curve, combined with a coarsegrained representation of protein structure in order to comply with the speed requirement. In such a coarsegrained representation, certain groups of atoms are represented by one scattering body or dummy atom [9,12,23]. The coarsegraining thus avoids a costly iteration over all atoms (see Equation 2 below for details). The main goal of this study is therefore to obtain good point estimates of the form factors for these dummy atoms.
To illustrate and evaluate the potential of this approach in statistical inference of protein structure from SAXS data, we also perform two decoy recognition experiments (see Methods). In both cases, we use SAXS curves calculated from the native structure by the program CRYSOL as "experimental" data; the goal is to identify the native structure among a set of decoys by using the experimental data. In the first experiment, we use a simple likelihood function based on the SAXS curves calculated by our coarsegrained Debye method combined with a uniform prior. In a second experiment, we instead incorporate a probabilistic model of local protein structure as the prior distribution.
Results and Discussion
Coarsegrained protein models
We used two coarsegrained models of protein structure, in which the amino acids were represented by one and two scattering bodies (here called dummy atoms), respectively. In the twobody model, the amino acids  with the exception of glycine and alanine  were represented by two dummy atoms; one representing the backbone, and the other representing the side chain. The dummy atoms were placed at the respective centers of mass (see Figure 1). Exceptions were made for the representation of glycine and alanine due to their small size; in both cases, one dummy atom represents the whole amino acid. For the other 18 amino acids, a side chain specific dummy atom was combined with the generic, backbone dummy atom. As a result, a total of 21 form factors needed to be estimated for the twobody model: one for alanine, one for glycine, one for the backbone and 18 for the remaining side chains. For the onebody model, we used a straightforward approach with one dummy atom that is placed at the center of mass. Hence, 20 form factors need to be estimated; one for each amino acid type. The onebody model results in roughly half the number of scattering bodies as compared to the twobody model for a given protein.
Figure 1. Coarsegrained model of protein structure. Example of a protein backbone stretch (dark gray) with side chain atoms (light gray), and a corresponding coarsegrained model. The positions of the corresponding dummy atoms for the backbone (red spheres) and side chains (blue spheres) are shown. This representation is used in the twobody model in the article. The figure was made using PyMOL [40].
Calculation of the SAXS curves
The observed data in a SAXS experiment is a onedimensional intensity curve, I(q), where q = 4π sin(θ)/λ is the scattering momentum, λ is the wavelength and 2θ is the scattering angle. The calculation of a theoretical SAXS scattering curve from structure is based on the wellestablished Debye formula [22]:
where F_{i }and F_{j }are the scattering form factors of the individual particles i and j, and r_{ij }is the Euclidean distance between them. The summations run over all M scattering particles.
Since an average amino acid has around eight heavy atoms, and considering the pairwise character of the summation in Equation 2, it can be expected that a coarsegrained protein model with one or two scattering bodies per amino acid can lead to a computational speedup of more than an order of magnitude (see Methods).
Estimation of scattering form factors
The estimation of the form factors was carried out using artificial SAXS curves generated by the stateoftheart program CRYSOL [24]. We used a set of 297 high resolution crystal structures from the Protein Data Bank (PDB) [25].
Atomic scattering form factors are continuous functions of the scattering momentum q; the same can be expected for the coarsegrained form factors. In order to render the estimation of the 20 (for the one body model) or 21 (for the two body model) form factors tractable, we discretized the problem by considering resolution bins. We divided the relevant scattering momentum interval  ranging from 0 to 0.75 Å^{1 } into 51 discrete bins, with a width equal to 0.015 Å^{1}. Our strategy was to obtain a point estimate for each of the 20 or 21 form factors in each of the qbins, resulting in a total of 1010 and 1071 parameters for the one and two body models, respectively. We will denote the vector of form factor values for a specific qbin as . Our scheme is to sample form factor values from a suitable posterior distribution for each bin, then calculate a point estimate from the obtained samples. We start with the classic Bayesian approach, and consider the following posterior distribution:
where is the 20 or 21dimensional form factor vector for bin q and I_{q }is the intensity calculated by CRYSOL at a given qbin for a certain structure X. The approach will be generalized to multiple structures below. We assume conditional independence between the structure X and the form factor vector , that is, P(X ) = P(X), and a uniform density for the prior P(). To evaluate the likelihood P(I_{q}, X)  the probability of the data I_{q }given the form factor vector  we use the following strategy. Applying , we calculate the scattering intensity for the given structure X using the Debye formula (Equation 2) and evaluate the difference between the two intensities. The likelihood is thus expressed as:
In the following, will be used as a short notation for . In order to calculate the likelihood, I_{q }was interpreted as a sample from a Gaussian distribution where the mean is given by :
with σ_{q }being the standard deviation. The standard deviation σ_{q }was set to a value that is typically observed in real experiments (see Methods). For multiple structures, the likelihood function simply becomes a product of Gaussian distributions:
where N is the number of structures in the training set, is the calculated scattering intensity curve for structure X_{i }using and I_{q,i }is the intensity as calculated by CRYSOL from structure X_{i}. Using this probabilistic model, it becomes possible to sample form factor vectors from the posterior distribution for a given bin.
The form factor vectors are sampled from the posterior distribution for each qbin using a generalized Markov chain Monte Carlo (MCMC) method as implemented in the Muninn program [26].
Form factor estimates
The resulting distributions for two qbins are shown in Figures 2 and 3 (twobody model). From these distributions, it is clear that some side chains are less determining for the scattering curve than others; the hydrophobic side chains of leucine, isoleucine and valine only contribute marginally at low resolution. These amino acids are most often buried in the hydrophobic core; as a result, their contributions to the scattering intensity for low q values  which is mostly determined by the protein's external shape  are rather small. The final step in the estimation is to obtain the point estimates for the form factor vectors from the samples; this is done by computing the centroid of these samples (see Methods), which is an attractive point estimator for highdimensional problems [27]. The resulting form factor curves as a function of q for the different amino acids and the generic backbone are shown in Figure 4. In all cases, the resulting curves are smooth in q. Since the estimation of the form factors has been carried out independently for each qbin, the observed continuity testifies to the efficiency and consistency of the MCMC procedure. The 20 form factors for the onebody model are shown in Figure 5. Although using only one dummy atom per amino acid is computationally attractive, it comes at the cost of a significantly lower accuracy, except for very low resolutions (see Figure 6). The difference is particularly significant in the central part of the qrange, which is of the highest interest for structure prediction [28]. Therefore, we focus on the two body model in the rest of this article.
Figure 2. Distributions of the components of the form factor vector for q = 0. An asterisk indicates that this form factor describes both the backbone and side chain atoms of the residue. Note that different scales are used on the Xaxes. The scale on the Yaxis shows the percentage of observations.
Figure 3. Distributions of the components of the form factor vector for q = 0.3. An asterisk indicates that this form factor describes both the backbone and side chain atoms of the residue. Note that different scales are used on the Xaxes. The scale on the Y axis shows the percentage of observations.
Figure 4. Form factors for the twobody model. Mean (dark blue curve) and standard deviations (blue areas) for the backbone (top left) and the 20 sidechain form factors (Yaxis) as a function of q (Xaxis). An asterisk indicates that this form factor describes both the backbone and side chain atoms of the residue.
Figure 5. Form factors for the onebody model. Mean (dark blue curve) and standard deviation (blue area) of the 20 single dummy atom form factors (Yaxis) as a function of q (Xaxis).
Figure 6. Overall accuracy of the SAXS curve calculations. The deviations between the SAXS curves calculated by CRYSOL and by the coarsegrained Debye methods as as function of the resolution, q, are shown in units of standard deviations. Mean and standard deviations are plotted in red for the onebody model and in green for the twobody model.
Evaluation using a test set
The estimated form factors were assessed by calculating scattering curves for fifty proteins that have low sequence similarity with the proteins in the training set (sequence similarity below 25%). The calculated curves were compared to curves generated by CRYSOL, which uses full atomic detail. The results are shown in Table 1. The dissimilarities between the curves are quantified by a χ^{2 }measure, S in the table, which is scaled by the standard deviations that were used in the training of the form factors (see Methods for details). Since the errors are within the usual magnitude of the experimental errors [9,10,14,29], our results are in excellent agreement with the CRYSOL predictions. Scattering curves for six proteins of various sizes are shown in Figure 7.
Table 1. Accuracy of the SAXS curve calculation for the individual structures.
Figure 7. SAXS curves for six example proteins. Comparison of I(q) calculated by CRYSOL from the all atom structure (blue) and by the twobody model (green). Error bars indicate the simulated "experimental" error for each bin. The example proteins are: (A) 1ENH 54 residues, (B) 2CRO 71 residues, (C) 2PTH 193 residues, (D) 1PTA 327 residues, (E) 1A12 (chain A) 401 residues and (F) 1JET 520 residues.
Protein decoy recognition
In order to investigate the utility of the coarsegrained model in inferential structure determination [15,16], we carried out a decoy recognition experiment (see Methods). As previously discussed, the Bayesian approach to this problem employs the posterior probability distribution. The posterior probability distribution P(XI) is proportional to the product of the likelihood P(IX) and the prior probability P(X). Below, we first test the model by combining the likelihood function with a uniform prior and subsequently with a suitable prior probability distribution, P(XA). The performance of decoy recognition experiments is commonly evaluated using the Zscore. The Zscore is defined as the difference between the score of the native conformation and the average score of all conformations belonging to that decoy set, divided by the standard deviation [30]. Ideally, the native structures have the lowest energy. For this experiment we used a decoy set from TASSER [31].
In the first test, the likelihood was used to assign an energy to the decoys, and a corresponding Zscore was calculated. The results were compared to Zscores obtained using CRYSOL (see Table 2). Strikingly, the coarsegrained Debye method is generally as good as CRYSOL in identifying the native structure among the decoys. In some cases our method even performs better than CRYSOL; the coarse grained approach is quite likely less sensitive to differences on the atomic scale.
Table 2. Decoy recognition Zscores for CRYSOL and the twobody model.
In the second part of this experiment, we also incorporated a probabilistic model of the structure of proteins as a prior. This model, called TorusDBN, was previously developed in our group [21] and evaluates the probability of observing a certain sequence of ϕ and ψ angles for a given amino acid sequence. TorusDBN is a model of the local structure of proteins; nonlocal interactions such as hydrogen bonds or the formation of a hydrophobic core are not part of this model.
Including the TorusDBN prior in the definition of the posterior probabilities leads to a clear improvement in decoy recognition; the average Zscore was enhanced by 16%. As illustrated in Figure 8, there is no correlation between protein size and Zscores; the performance stays constant over a wide range of protein sizes.
Figure 8. Decoy recognition Zscores for CRYSOL and the twobody model versus protein length. For each of the proteins in the set, the Zscores (Yaxis) are plotted against the length of the protein (Xaxis). Red spheres: using CRYSOL. Blue squares: using the likelihood according to the twobody model. Green triangles: using the likelihood according to the twobody model and the TorusDBN prior. More detailed information is presented in Table 2.
Conclusions
We have demonstrated that it is possible to obtain accurate SAXS curves from coarsegrained protein structures and matching estimated form factors without the use of ad hoc correction factors. We obtained point estimates of the form factors and assessed their performance for a diverse set of proteins; the resulting SAXS curves are on par with the current stateoftheart program CRYSOL, at least up to scattering vector lengths of 0.75 Å^{1 }(see Methods). As a further validation of this model, we used a comparison of the Zscores for a set of protein decoys based on the SAXS curves generated by CRYSOL and by our coarsegrained Debye method, respectively. Again, the performance was excellent. Including prior information from a probabilistic model of local protein structure [21] further improved the decoy recognition.
Before a rigorous incorporation of SAXS information in a fully probabilistic model for data driven structure prediction is possible, two additional developments are needed: a proper description of the hydration layer that surrounds the protein [29] and a probabilistic description of the experimental errors associated with a SAXS data acquisition experiment [15,16]. We are currently implementing such an approach in the PHAISTOS software package [32,33].
Methods
Protein data sets
Three protein data sets were used throughout this work: one for training of the form factors, another for validating the model, and finally a set of native structures and corresponding artificial decoys for the Zscore calculations. The training data set consists of 297 structures with lengths between 50 and 400 from the Top500 data set of high quality protein structures [34]. To ensure that CRYSOL and our program processed these structures in the same manner, structures with conflicting atoms or nonstandard amino acids were excluded (selected structures can be found in additional file 1).
Additional file 1. Data set of selected protein structures from the Top500 data set [34] used in the form factor estimation. Each column contains PDB identifier, primary sequence length and radius of gyration calculated from the atomic structure.
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The estimated form factors were validated by calculating scattering curves for 50 proteins (see Table 1) extracted using the PISCES server [35]. These were randomly selected among an initial group of 81 proteins with low sequence similarity (below 25%) with those in the training set, a resolution better than 3 Å and an Rfactor below 30%.
The decoy set used in the Zscore evaluation was generated by the structure prediction program TASSER [31]. This decoy set consists of 47 proteins, each with 1040 proteinlike decoy structures with varying similarity to the native structure. The decoys are constructed from energyminimized snapshots from molecular dynamics simulations using the AMBER force field [36]. A single protein from the set, [1CBP], was excluded from our Zscore evaluation, as this structure resulted in an input error when evaluated by the CRYSOL program. Six proteins were left out of the evaluation as they were also present in our training data set.
In all cases, the backbone and side chain centroids were calculated using all nonhydrogen atoms. The Cβ atoms were only included in the calculation of the backbone centroid.
SAXS training data
Due to the lack of publicly available highquality experimental data needed for the estimation of the form factors, artificial data curves were generated for highresolution protein structures using the stateoftheart program CRYSOL [24]. This program calculates the theoretical scattering curve from a given fullatom resolution structure using spherical harmonics expansions. CRYSOL was used to compute a simulated scattering curve including the vacuum and excluded volume scattering components, but without hydration layer contribution; the electron density of the solvent layer was set equal to that of the bulk solvent (i.e. CRYSOL was run with the command line "crysol/dro 0.0 inputfile.pdb"). For the scattering curve evaluations, an upper qlimit of 0.75 Å^{1 }was chosen in order to be well within the expected valid resolution range for CRYSOL.
Error model and qbinning
The range of the scattering momentum ranges from 0 to 0.75 Å^{1}, divided in 51 discretized bins. For each bin we sampled form factors from the posterior distribution to obtain the 21dimensional form factor distributions. The intensity at a given bin was evaluated using the lefthand side qvalue, starting at q = 0 in the first bin.
To account for the "experimental" error for the SAXS curves, a standard deviation σ_{q }= I_{q}β (with β = 0.3) has previously been used in the literature as a realistic estimate [14]. Aiming to be more precise in the portion of the curve of primary interest in a structure prediction application  approximately between q = 0.1 and q = 0.5 Å^{1 }[37]  we introduced a scaling factor (q + α):
with α = 0.15 and β = 0.3. This is significantly stricter at mid qrange than the reference parameters.
Posterior sampling
In order to explore the large parameter space efficiently, we used an optimized, maximumlikelihood based MCMC method implemented in the Muninn program [26]. In the MCMC method, we used the negative of the logarithm of the posterior probability as an energy. We employed a sampling scheme where the density of states is weighted according to the inverse cumulative density of states (1/k ensemble) [38]. Compared to standard MCMC methods, this ensures a more frequent generation of samples in the lower energy regions. Avoiding slow relaxation in the Markov chain also minimizes the risk of the chain getting trapped at a local minimum [26]; a problem often encountered in rough energy landscapes when using Metropolis techniques, as for example in simulated annealing.
An initial proposal value for the vector was obtained by uniformly sampling its components from the interval [0, f_{max}]. The value of f_{max }was set to 40; well beyond the limit of the form factor components described in the literature [12]. The MCMC proposed new form factor vectors, , with a transition in which a randomly chosen component f of the vector was resampled uniformly from the interval:
where the width m was equal to 1.5. Border effects were taken into account in order to respect the detailed balance condition, which is
The transition probabilities, , can be expressed as:
where represents the probability of selecting given and is the acceptance probability of this transition. The selection probability implied by the local uniform proposal scheme is:
and according to the MetropolisHastings criterion [39], the following acceptance probability satisfies the detailed balance condition:
where is given below by Equations 7 and 8 for the TorusDBN independent and dependent cases, respectively.
Point estimation of form factors
The MCMC procedure results in a set of samples from the posterior distribution. In order to obtain a point estimate, we calculated the centroid vector obtained from the set of sampled form factor vectors . The centroid vector was defined as the one with the lowest Euclidean distance to all the other vectors in the set:
where T is the number of samples.
SAXS curve distance measures
A χ^{2 }measure was used to quantify the difference between two scattering curves. S was scaled by the "experimental" error that was used in the estimation of the form factors, thus evaluating the statistical quality of our reconstruction:
where Q is the number of qbins and σ_{q }is the experimental error as described previously.
Decoy recognition
The first decoy recognition experiment  which only uses the likelihood  is performed as follows. First, a SAXS curve I is calculated from the native structure using CRYSOL. For each decoy structure, a corresponding SAXS curve I' is calculated with the Debye formula using the twobody model. In this case, the posterior  which is simply equal to the likelihood P(I  X)  of the decoy X is calculated as:
where the product runs over all qbins. For the comparison with CRYSOL, scattering curves were calculated from the decoys using CRYSOL instead of the twobody model.
In the second decoy recognition experiment, TorusDBN is included as a prior distribution, and amino acid information is explicitly included. In this case, the posterior becomes:
where and are the backbone angles of the decoy and A is the amino acid sequence.
The Zscore for a given decoy set was calculated as the difference between the energy (defined as the negative of the logarithm of the posterior) of the native structure E(X_{N}) and the average energy of all structures in the decoy set, divided by the standard deviation σ of the energies:
TorusDBN prior
TorusDBN is a probabilistic model of the local structure of proteins, formulated as a dynamic Bayesian network. It can be considered as a probabilistic alternative to the well known fragment libraries, It allows sampling of plausible protein conformations in continuous space, and it can assign a probability density value to a given sequence of ϕ and ψ angles. For the prior, we used:
where and are the ϕ and ψ angle sequences, respectively. The calculation of from TorusDBN is straightforward using the forward algorithm [21]. We used the model that is described in [21] with default parameters.
Computational efficiency
The naive implementation of the Debye formula (Equation 2) leads to a computational complexity of O(M^{2}), where M is the number of scatterers in the structure under evaluation. Our coarsegrained approach reduces M by representing several atoms by one scattering body (a dummy atom). Each of the dummy atoms contains an average of k atoms, thus lowering the execution time by a constant factor of k^{2}, replacing O(M^{2}) with .
The exact value of k is obviously dependent on the primary sequence of the protein. For both training and validation sets, employing a dummy atom for the backbone and one for the side chain leads to an average k of 4.24. This means that each dummy atom contains on average 4.24 nonhydrogen atoms, leading to an increase in speed of k^{2}≃18. Using only one dummy atom to describe a complete amino acid results in k≃7.8 atoms, allowing for a k^{2}≃60 times faster execution. The absolute running time for a single scattering evaluation is approximately 30 ms for a 129 residues protein ([6LYZ]), using the twobody model and a standard desktop computer (AMD Athlon X2 5200+). Absolute running time comparisons with other programs are obviously unfair, since for a single evaluation the overhead introduced by the file system and the operative system is considerable. This said, our approach is significantly faster than CRYSOL [24](~786 ms).
Availability
The point estimates of the form factor vectors for all bins are available as supplementary information (for the onebody model in additional file 2 and the twobody model in additional file 3).
Additional file 2. Scattering form factor centroids for the onebody model. Form factor centroids for each amino acid in the qrange [0, 0.750] Å^{1}.
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Additional file 3. Scattering form factors for the twobody model. Form factor centroids for the generic backbone component and each amino acid in the qrange [0, 0.750] Å^{1}. An asterisk is used to mark the residues where the form factor includes both the backbone and side chain scatterers.
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Authors' contributions
CA and KS contributed equally to this work. JFB provided the Muninn method, and assisted in its application. JFB and TH conceived the study. All authors read and approved the final manuscript.
Acknowledgements
The authors thank our colleagues at the Bioinformatics Centre for valuable comments and suggestions. We acknowledge funding from the Danish Council for Strategic Research (Program Commission on Nanoscience, Biotechnology and IT; project: Simulating proteins on a millisecond time scale, 2106060009) and the Danish Council for Independent Research (Danish Research Council for Technology and Production Sciences; project: Data driven protein structure prediction, 274060380).
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