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Improved contact prediction in proteins: Using pseudolikelihoods to infer Potts models

Magnus Ekeberg, Cecilia Lövkvist, Yueheng Lan, Martin Weigt, Erik Aurell

arXiv:1211.1281v2q-bio.QMcond-mat.dis-nncond-mat.stat-mechphysics.data-an

TL;DR

Protein contact prediction must distinguish direct residue interactions from correlations propagated through other residues. The paper applies pseudolikelihood maximization to 21-state Potts models for evolutionary protein families and compares it with mean-field DCA. Across protein-domain families, plmDCA consistently outperforms mfDCA, with part of the advantage attributable to a sampling-corrected coupling score.

  • Problem

    Correlations in protein sequences can arise from direct or indirect interactions, so correlation alone does not reliably identify spatially proximate residues.

  • Method

    The paper applies pseudolikelihood maximization to 21-state Potts models describing statistical properties of homologous protein-sequence families.

  • Results

    plmDCA consistently outperforms mfDCA across large protein-domain families, with the improvement partly intrinsic and partly dependent on a sampling-corrected Frobenius-norm score.

  • Takeaways & Limitations

    Pseudolikelihood-based direct-coupling analysis improves protein-contact prediction relative to the previously described mean-field approach in the evaluated families.

  • Takeaways & Limitations

    The comparison changes both scoring and regularization, so some of the observed improvement may come from the new score rather than PLM alone.

Abstract

from arXiv · show

Spatially proximate amino acids in a protein tend to coevolve. A protein's three-dimensional (3D) structure hence leaves an echo of correlations in the evolutionary record. Reverse engineering 3D structures from such correlations is an open problem in structural biology, pursued with increasing vigor as more and more protein sequences continue to fill the data banks. Within this task lies a statistical inference problem, rooted in the following: correlation between two sites in a protein sequence can arise from firsthand interaction but can also be network-propagated via intermediate sites; observed correlation is not enough to guarantee proximity. To separate direct from indirect interactions is an instance of the general problem of inverse statistical mechanics, where the task is to learn model parameters (fields, couplings) from observables (magnetizations, correlations, samples) in large systems. In the context of protein sequences, the approach has been referred to as direct-coupling analysis. Here we show that the pseudolikelihood method, applied to 21-state Potts models describing the statistical properties of families of evolutionarily related proteins, significantly outperforms existing approaches to the direct-coupling analysis, the latter being based on standard mean-field techniques. This improved performance also relies on a modified score for the coupling strength. The results are verified using known crystal structures of specific sequence instances of various protein families. Code implementing the new method can be found at http://plmdca.csc.kth.se/.

I. INTRODUCTION

Protein structure prediction can exploit evolutionary correlations in homologous sequences, but direct spatial contacts must be separated from correlations propagated through intermediate residues. This paper frames that challenge as inverse statistical mechanics and evaluates improved inference methods for direct-coupling analysis.

  • The inference problem: Direct-coupling analysis seeks to infer hidden direct interactions from observable correlations in evolutionarily related protein sequences.The task is an instance of inverse statistical mechanics, in which model parameters are reconstructed from observables.
  • Paper scope: The paper compares pseudolikelihood maximization with mean-field-based inference for 21-state Potts models and assesses predictions against known crystal structures.The paper also examines interaction scoring and discusses computational practicality for realistic protein families.
  • Motivation: Protein folds connect amino-acid sequence positions to biological function, motivating computational prediction of three-dimensional structure.Experimental structure determination is costly and time-consuming, while sequence databases continue to expand.
  • Motivation: Large protein-domain families provide sequence variability that can encode statistical information about the three-dimensional structures shared by homologous domains.Pfam families contain aligned sequences from common evolutionary origins, making their variability useful for structure prediction.
  • The inference problem: Correlations between aligned amino-acid positions may reflect either direct contact or indirect interactions mediated through another residue.Observed correlations alone therefore do not reliably identify spatial proximity.
  • Paper scope: Earlier comparisons indicate that DCA and PSICOV can outperform Bayesian-network approaches, while their relative efficiency remains unassessed here.This leaves the comparative efficiency of these methods as an open issue within the paper’s scope.

III. METHOD DEVELOPMENT

The method models aligned homologous protein sequences with a 21-state Potts distribution whose fields and pairwise couplings reproduce empirical single-site and pairwise statistics. Parameters can be inferred through maximum likelihood, although exact optimization is computationally infeasible for realistic systems.

  • Data representation: Each aligned sequence is represented as a length-N array whose 21 states encode 20 amino acids plus gaps.An MSA contains B aligned sequences, one row per sequence and one column per chain position.
  • Empirical statistics: Empirical single-site and pairwise frequencies measure amino-acid occupancy at individual positions and amino-acid combinations across position pairs.Gaps are counted as a 21st amino-acid state when calculating single-site frequencies.
  • Potts model: A generalized Potts model uses fields and pairwise couplings to reproduce the empirical frequencies and correlations of the sequence family.The model is also characterized as the maximum-entropy distribution satisfying those statistical constraints.
  • Maximum likelihood: The maximum-likelihood approach adjusts fields and couplings to maximize the probability of the input MSA under the Potts model.Because exponential-family means and correlations are sufficient statistics, exact likelihood inference would use the same empirical statistics if the partition function were available.
  • Computational challenge: Exact likelihood optimization is computationally intractable for realistic system sizes because it requires calculating the Potts partition function for arbitrary parameters.The paper therefore turns to approximate inference schemes for practical PSP applications.

B. Model parameters and gauge invariance

Potts-model parameters are redundant under gauge transformations, so the paper removes this overparameterization through gauge choices. Exact maximum-likelihood equations match model and empirical statistics, but evaluating the partition function makes the procedure impractical at realistic sizes.

  • Parameterization: The full Potts parameterization contains redundant field and coupling values that can represent the same probability distribution.The number of nonredundant parameters is smaller than the total parameter count.
  • Gauge invariance: Gauge choices eliminate this redundancy by imposing constraints on fields and couplings.The paper presents both a zero-sum gauge and a gauge that treats one Potts state as a reference.
  • Gauge invariance: Reconstruction is well-defined in either gauge, and results can be translated between the two parameterizations.The reference-state gauge corresponds to a lattice-gas representation with q −1 particle types.
  • Maximum likelihood: Maximum-likelihood estimates satisfy equations equating model-derived fields and couplings to empirical single-site and pairwise frequencies.These conditions arise by setting derivatives of the likelihood objective to zero.
  • Computational limitation: A realistic 21-state Potts model requires evaluating a partition function whose computation is intractable for reasonable system sizes.Approximate minimization is therefore essential for protein-structure prediction applications.

D. Naive mean-field inversion

Naive mean-field inversion provides a fast approximation by inverting a covariance matrix, while pseudolikelihood replaces the full likelihood with conditional probabilities that avoid the intractable partition function. Both approaches require regularization for high-dimensional protein families.

  • Naive mean-field inversion: Naive mean-field inversion reconstructs Potts couplings from a covariance matrix using linear response in a reduced lattice-gas gauge.The covariance matrix has dimension N(q −1) × N(q −1), excluding the final Potts state.
  • Pseudolikelihood maximization: Pseudolikelihood conditions each variable on all other variables and optimizes the resulting conditional likelihoods rather than the full likelihood.The objective is formed by combining node-specific conditional objectives across all positions.
  • Pseudolikelihood maximization: Replacing likelihood with pseudolikelihood changes the optimizer but removes computational intractability from parameter optimization.Each conditional normalization sums over only the q = 21 states, while averages over conditioning variables use the observed data.
  • Regularization: Protein families with sequences of 50-300 amino acids require approximately 5 · 10^5 to 2·10^7 parameters, making regularization necessary to avoid overfitting.For mean-field inversion, pseudocounts promote covariance-matrix invertibility; their useful value is reported as being on the order of B.
  • Regularization: PLM uses a penalty term, commonly an l2 regularizer, whereas NMFI uses pseudocounts under gauge constraints.The regularizer also removes the need to fix a gauge in the full parameterization by selecting one representative among gauge-equivalent parameters.

G. Sequence reweighting

Maximum-likelihood inference assumes independently generated samples, but biological sequences exhibit phylogenetic bias. Sequence reweighting reduces redundancy by weighting each sequence according to its similarity neighborhood, and the same weights enter pseudolikelihood calculations.

  • Maximum-likelihood inference assumes independently generated sample configurations, an assumption violated by phylogenetic bias in biological sequence databases.Related species may not have evolved independently long enough to reach statistical independence.
  • Sequence similarity is measured as the fraction of conserved positions and compared with a threshold x satisfying 0 < x < 1.The weight of sequence σ(b) is set from the number of sequences similar to it.
  • Using x = 0.9 makes the reweighted frequency counts reflect effectively nonredundant sequences.Earlier work found x = 0.8 suitable, with weak dependence across 0.7 < x < 0.9.
  • Pseudolikelihood uses the direct analog of the reweighted frequency-count construction.Each sequence contributes weight w_b rather than the unit weight used for i.i.d. samples.

H. Interaction scores

Because Potts couplings are q×q matrices rather than scalar interactions, the paper evaluates position pairs with scalar coupling scores. It compares direct information with gauge-aware norm-based scores and applies an average-product correction to the preferred corrected norm.

  • Each Potts position pair has a q×q coupling matrix, so a scalar score S_ij is required to rank coupling strength.This differs from the inverse Ising case, where each interaction has one scalar coupling.
  • Direct information scores mutual information from a restricted two-position distribution constructed from the inferred coupling.The associated fields are chosen to match empirical single-site frequencies, and the score is gauge-invariant.
  • Pseudolikelihood avoids direct information because its frequency regularization would introduce a third regularization variable alongside λ_h and λ_J.The paper instead considers the Frobenius norm, which requires the zero-sum gauge because it is gauge-dependent.
  • The corrected norm combines a norm-based score with an average-product correction intended to suppress phylogenetic-bias and insufficient-sampling effects.The resulting score is called CN, or corrected norm.

IV. EVALUATING THE PERFORMANCE OF MFDCA AND PLMDCA ACROSS PROTEIN FAMILIES

The study evaluates mfDCA and plmDCA on 17 Pfam protein families using representative high-resolution crystal structures as reference contact maps. Contact-prediction accuracy is measured by the true-positive rate among the strongest-scored pairs.

  • Numerical experiments compare mfDCA and plmDCA across 17 Pfam domain families.The smaller collection was selected because PLM is more computationally demanding than NMFI.
  • Each family is evaluated against one representative high-resolution X-ray crystal structure with resolution below 3 Å.The structures provide a reference for assessing predicted contacts.
  • Native position distances are measured as the minimum distance between heavy atoms belonging to the amino acids at each pair of sequence positions.The pooled distance distributions contain peaks below 1.5 Å, around 3–5 Å, and around 7–8 Å.
  • An 8.5 Å distance cutoff defines true contacts, chosen to include the full second peak in the distance distribution.The evaluation considers structural distances and distinguishes contacts from noncontacts using this cutoff.
  • The true-positive rate for p is the fraction of the p strongest-scored pairs that are contacts in the crystal structure.This is the primary accuracy measure reported in the comparison.

B. Parameter settings

The comparison uses reweighted sequences, fixed field regularization, scanned coupling regularization, and method-specific optimal settings. Across families, PLM generally achieves higher true-positive rates than NMFI, while its score-versus-distance pattern and limitations are also examined.

  • B. Parameter settings: The reported experiments use x = 0.9, an 8.5 Å contact cutoff, λ_h = 0.01, and scanned λ and λ_J values.These settings define the evaluation configuration.
  • C. Main comparison of mfDCA and plmDCA: PLM achieves consistently higher true-positive rates than NMFI, especially for families with large B_eff.Changing NMFI-DI to true frequencies avoids a strong PF00084 failure but lowers performance for several other families.
  • B. Parameter settings: λ = B_eff is optimal for NMFI, whereas λ_J = 0.01 is optimal for PLM across essentially all families.The PLM optimum remains the same despite wide ranges of N and B_eff.
  • C. Main comparison of mfDCA and plmDCA: Both PLM and NMFI-DI assign high scores predominantly to pairs below the 8.5 Å contact cutoff and show progressively weaker interactions across the distance peaks.Most spatially close pairs do not have elevated scores, but almost all strongly coupled pairs are close.
  • C. Main comparison of mfDCA and plmDCA: PLM and NMFI-DI agree on many high-scoring pairs, although their score distributions differ because they use different scoring schemes.Both methods capture clusters of contacts with long sequence separation.
  • C. Main comparison of mfDCA and plmDCA: Only a relatively small fraction of contacts is recovered before false positives appear, and many native contacts remain missed.DCA does not aim to reproduce a complete contact map because it uses sequence correlations alone and omits several structural constraints.
  • C. Main comparison of mfDCA and plmDCA: The PLM improvement over NMFI is partly attributable to the new coupling score, because the comparison changes both scoring and regularization styles.The authors therefore evaluate NMFI-CN, an updated NMFI version using the corrected norm score.

D. Run times

PLM-based DCA is feasible but substantially slower than matrix-inversion NMFI, with runtime increasing with protein-family size. Parallelization and improved initialization are identified as possible ways to reduce this cost.

  • Runtime comparison: NMFI runs in seconds for most families, whereas PLM runtimes range from under two minutes for small families to over an hour for larger domains.Reported PLM runtimes include 50, 160, and 90 seconds for PF00014, PF00017, and PF00018, versus 80 and 65 minutes for PF00026 and PF00006.
  • Runtime scaling: PLM execution time depends on both the number of family sequences B and the aligned-sequence length N.
  • Possible acceleration: An asymmetric pseudolikelihood variant could parallelize across N independent logistic-regression problems, potentially reducing runtime by a factor as large as 10N.
  • Initialization: The reported runtimes used zero-valued fields and couplings at initialization, while initialization from NMFI was suggested but not implemented.
  • Accuracy context: Across large protein-domain families, plmDCA consistently outperformed mean-field mfDCA, with the advantage partly intrinsic and partly attributable to the sampling-corrected Frobenius score.

Appendix A: Circle plots

Circle plots compare structural contacts with the strongest predicted interactions from PLM and NMFI-DI across four protein families. They highlight where false positives cluster and how the methods differ.

  • Error pattern: False positives show some loopiness, especially for NMFI-DI, with relatively few biased nodes apparently responsible for many erroneous predictions.
  • Plot layout: The left column shows crystal-structure contacts, while the PLM and NMFI-DI columns show the top 1.5N non-neighbor predictions.
  • Contact encoding: Structural contacts use dark gray for distances below 5 Å and light gray for distances from 5 Å up to 8.5 Å.
  • Prediction encoding: Predicted connections are gray for true positives and black for false positives, enabling direct comparison of prediction errors between PLM and NMFI-DI.

Appendix B: Other scores for naive mean-field inversion

Alternative coupling scores improve naive mean-field contact prediction, narrowing the gap with pseudolikelihood while revealing sensitivity to neighboring and gap-rich alignment regions.

  • APC in SDI_ij scoring increases NMFI TP rates slightly, while SCN_ij scoring can improve TP rates overall.
  • Using SCN_ij makes NMFI-CN and PLM prediction quality about the same for several families, although PLM retains somewhat higher TP rates overall.
  • NMFI assigns more attention than PLM to neighboring position pairs with small |j − i|, even when SCN_ij makes their score scatter more linear.
  • NMFI reacts more strongly than PLM to gap-gap couplings, which are among the largest parameters, especially in the gap-rich family PF00043.
  • Removing gap contributions only affects pairs with small |j − i| and does not significantly improve either method, leaving gap handling unresolved.
  • With x = 0.75 or after removing sequences containing more than 20% gaps, PLM maintains its advantage over NMFI-CN; deleting gappy sequences can reduce TP rates.

Appendix C: Extension to 28 protein families

An extended survey of 28 protein families tests whether the reported scoring and pseudolikelihood advantages persist beyond the shorter families used initially.

  • The extended survey covers 28 families with lengths ranging from 50 to 400 residues, using an 8.5 Å contact cutoff.
  • Across the larger family set, TP rates increase partly from changing SDI_ij to SCN_ij and partly from changing NMFI to PLM.
  • The positive results do not appear specific to short-length protein families.
  • The PLM–NMFI difference primarily appears at the high end of the ranking, according to the distribution of p values with perfect TP rates.
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