Influence of multiple-sequence-alignment depth on Potts statistical models of protein covariation

Gauge fixing for sequence-function relationships

Quantitative models of sequence-function relationships are ubiquitous in computational biology, e.g., for modeling the DNA binding of transcription factors or the fitness landscapes of proteins. Interpreting these models, however, is complicated by the fact that the values of model parameters can often be changed without affecting model predictions. Before the values of model parameters can be meaningfully interpreted, one must remove these degrees of freedom (called "gauge freedoms" in physics) by imposing additional constraints (a process called "fixing the gauge"). However, strategies for fixing the gauge of sequence-function relationships have received little attention. Here we derive an analytically tractable family of gauges for a large class of sequence-function relationships. These gauges are derived in the context of models with all-order interactions, but an important subset of these gauges can be applied to diverse types of models, including additive models, pairwise-interaction models, and models with higher-order interactions. Many commonly used gauges are special cases of gauges within this family. We demonstrate the utility of this family of gauges by showing how different choices of gauge can be used both to explore complex activity landscapes and to reveal simplified models that are approximately correct within localized regions of sequence space. The results provide practical gauge-fixing strategies and demonstrate the utility of gauge-fixing for model exploration and interpretation.

Symmetry, gauge freedoms, and the interpretability of sequence-function relationships

Quantitative models that describe how biological sequences encode functional activities are ubiquitous in modern biology. One important aspect of these models is that they commonly exhibit gauge freedoms, i.e., directions in parameter space that do not affect model predictions. In physics, gauge freedoms arise when physical theories are formulated in ways that respect fundamental symmetries. However, the connections that gauge freedoms in models of sequence-function relationships have to the symmetries of sequence space have yet to be systematically studied. Here we study the gauge freedoms of models that respect a specific symmetry of sequence space: the group of position-specific character permutations. We find that gauge freedoms arise when model parameters transform under redundant irreducible matrix representations of this group. Based on this finding, we describe an "embedding distillation" procedure that enables analytic calculation of the number of independent gauge freedoms, as well as efficient computation of a sparse basis for the space of gauge freedoms. We also study how parameter transformation behavior affects parameter interpretability. We find that in many (and possibly all) nontrivial models, the ability to interpret individual model parameters as quantifying intrinsic allelic effects requires that gauge freedoms be present. This finding establishes an incompatibility between two distinct notions of parameter interpretability. Our work thus advances the understanding of symmetries, gauge freedoms, and parameter interpretability in sequence-function relationships.

ZEPPI: Proteome-scale sequence-based evaluation of protein-protein interaction models

We introduce ZEPPI (Z-score Evaluation of Protein-Protein Interfaces), a framework to evaluate structural models of a complex based on sequence coevolution and conservation involving residues in protein-protein interfaces. The ZEPPI score is calculated by comparing metrics for an interface to those obtained from randomly chosen residues. Since contacting residues are defined by the structural model, this obviates the need to account for indirect interactions. Further, although ZEPPI relies on species-paired multiple sequence alignments, its focus on interfacial residues allows it to leverage quite shallow alignments. ZEPPI can be implemented on a proteome-wide scale and is applied here to millions of structural models of dimeric complexes in the Escherichia coli and human interactomes found in the PrePPI database. PrePPI's scoring function is based primarily on the evaluation of protein-protein interfaces, and ZEPPI adds a new feature to this analysis through the incorporation of evolutionary information. ZEPPI performance is evaluated through applications to experimentally determined complexes and to decoys from the CASP-CAPRI experiment. As we discuss, the standard CAPRI scores used to evaluate docking models are based on model quality and not on the ability to give yes/no answers as to whether two proteins interact. ZEPPI is able to detect weak signals from PPI models that the CAPRI scores define as incorrect and, similarly, to identify potential PPIs defined as low confidence by the current PrePPI scoring function. A number of examples that illustrate how the combination of PrePPI and ZEPPI can yield functional hypotheses are provided.

A Perspective on Protein Structure Prediction Using Quantum Computers

Despite the recent advancements by deep learning methods such as AlphaFold2, in silico protein structure prediction remains a challenging problem in biomedical research. With the rapid evolution of quantum computing, it is natural to ask whether quantum computers can offer some meaningful benefits for approaching this problem. Yet, identifying specific problem instances amenable to quantum advantage and estimating the quantum resources required are equally challenging tasks. Here, we share our perspective on how to create a framework for systematically selecting protein structure prediction problems that are amenable for quantum advantage, and estimate quantum resources for such problems on a utility-scale quantum computer. As a proof-of-concept, we validate our problem selection framework by accurately predicting the structure of a catalytic loop of the Zika Virus NS3 Helicase, on quantum hardware.

Undersampling and the inference of coevolution in proteins

Protein structure, function, and evolution depend on local and collective epistatic interactions between amino acids. A powerful approach to defining these interactions is to construct models of couplings between amino acids that reproduce the empirical statistics (frequencies and correlations) observed in sequences comprising a protein family. The top couplings are then interpreted. Here, we show that as currently implemented, this inference unequally represents epistatic interactions, a problem that fundamentally arises from limited sampling of sequences in the context of distinct scales at which epistasis occurs in proteins. We show that these issues explain the ability of current approaches to predict tertiary contacts between amino acids and the inability to obviously expose larger networks of functionally relevant, collectively evolving residues called sectors. This work provides a necessary foundation for more deeply understanding and improving evolution-based models of proteins.

Contingency and Entrenchment of Drug-Resistance Mutations in HIV Viral Proteins

The ability of HIV-1 to rapidly mutate leads to antiretroviral therapy (ART) failure among infected patients. Drug-resistance mutations (DRMs), which cause a fitness penalty to intrinsic viral fitness, are compensated by accessory mutations with favorable epistatic interactions which cause an evolutionary trapping effect, but the kinetics of this overall process has not been well characterized. Here, using a Potts Hamiltonian model describing epistasis combined with kinetic Monte Carlo simulations of evolutionary trajectories, we explore how epistasis modulates the evolutionary dynamics of HIV DRMs. We show how the occurrence of a drug-resistance mutation is contingent on favorable epistatic interactions with many other residues of the sequence background and that subsequent mutations entrench DRMs. We measure the time-autocorrelation of fluctuations in the likelihood of DRMs due to epistatic coupling with the sequence background, which reveals the presence of two evolutionary processes controlling DRM kinetics with two distinct time scales. Further analysis of waiting times for the evolutionary trapping effect to reverse reveals that the sequences which entrench (trap) a DRM are responsible for the slower time scale. We also quantify the overall strength of epistatic effects on the evolutionary kinetics for different mutations and show these are much larger for DRM positions than polymorphic positions, and we also show that trapping of a DRM is often caused by the collective effect of many accessory mutations, rather than a few strongly coupled ones, suggesting the importance of multiresidue sequence variations in HIV evolution. The analysis presented here provides a framework to explore the kinetic pathways through which viral proteins like HIV evolve under drug-selection pressure.

HDIContact: a novel predictor of residue-residue contacts on hetero-dimer interfaces via sequential information and transfer learning strategy

Proteins maintain the functional order of cell in life by interacting with other proteins. Determination of protein complex structural information gives biological insights for the research of diseases and drugs. Recently, a breakthrough has been made in protein monomer structure prediction. However, due to the limited number of the known protein structure and homologous sequences of complexes, the prediction of residue-residue contacts on hetero-dimer interfaces is still a challenge. In this study, we have developed a deep learning framework for inferring inter-protein residue contacts from sequential information, called HDIContact. We utilized transfer learning strategy to produce Multiple Sequence Alignment (MSA) two-dimensional (2D) embedding based on patterns of concatenated MSA, which could reduce the influence of noise on MSA caused by mismatched sequences or less homology. For MSA 2D embedding, HDIContact took advantage of Bi-directional Long Short-Term Memory (BiLSTM) with two-channel to capture 2D context of residue pairs. Our comprehensive assessment on the Escherichia coli (E. coli) test dataset showed that HDIContact outperformed other state-of-the-art methods, with top precision of 65.96%, the Area Under the Receiver Operating Characteristic curve (AUROC) of 83.08% and the Area Under the Precision Recall curve (AUPR) of 25.02%. In addition, we analyzed the potential of HDIContact for human-virus protein-protein complexes, by achieving top five precision of 80% on O75475-P04584 related to Human Immunodeficiency Virus. All experiments indicated that our method was a valuable technical tool for predicting inter-protein residue contacts, which would be helpful for understanding protein-protein interaction mechanisms.

Limits to detecting epistasis in the fitness landscape of HIV

The rapid evolution of HIV is constrained by interactions between mutations which affect viral fitness. In this work, we explore the role of epistasis in determining the mutational fitness landscape of HIV for multiple drug target proteins, including Protease, Reverse Transcriptase, and Integrase. Epistatic interactions between residues modulate the mutation patterns involved in drug resistance, with unambiguous signatures of epistasis best seen in the comparison of the Potts model predicted and experimental HIV sequence “prevalences” expressed as higher-order marginals (beyond triplets) of the sequence probability distribution. In contrast, experimental measures of fitness such as viral replicative capacities generally probe fitness effects of point mutations in a single background, providing weak evidence for epistasis in viral systems. The detectable effects of epistasis are obscured by higher evolutionary conservation at sites. While double mutant cycles in principle, provide one of the best ways to probe epistatic interactions experimentally without reference to a particular background, we show that the analysis is complicated by the small dynamic range of measurements. Overall, we show that global pairwise interaction Potts models are necessary for predicting the mutational landscape of viral proteins.

Modeling sequence-space exploration and emergence of epistatic signals in protein evolution

During their evolution, proteins explore sequence space via an interplay between random mutations and phenotypic selection. Here, we build upon recent progress in reconstructing data-driven fitness landscapes for families of homologous proteins, to propose stochastic models of experimental protein evolution. These models predict quantitatively important features of experimentally evolved sequence libraries, like fitness distributions and position-specific mutational spectra. They also allow us to efficiently simulate sequence libraries for a vast array of combinations of experimental parameters like sequence divergence, selection strength, and library size. We showcase the potential of the approach in reanalyzing two recent experiments to determine protein structure from signals of epistasis emerging in experimental sequence libraries. To be detectable, these signals require sufficiently large and sufficiently diverged libraries. Our modeling framework offers a quantitative explanation for different outcomes of recently published experiments. Furthermore, we can forecast the outcome of time- and resource-intensive evolution experiments, opening thereby a way to computationally optimize experimental protocols.

The generative capacity of probabilistic protein sequence models

Potts models and variational autoencoders (VAEs) have recently gained popularity as generative protein sequence models (GPSMs) to explore fitness landscapes and predict mutation effects. Despite encouraging results, current model evaluation metrics leave unclear whether GPSMs faithfully reproduce the complex multi-residue mutational patterns observed in natural sequences due to epistasis. Here, we develop a set of sequence statistics to assess the "generative capacity" of three current GPSMs: the pairwise Potts Hamiltonian, the VAE, and the site-independent model. We show that the Potts model's generative capacity is largest, as the higher-order mutational statistics generated by the model agree with those observed for natural sequences, while the VAE's lies between the Potts and site-independent models. Importantly, our work provides a new framework for evaluating and interpreting GPSM accuracy which emphasizes the role of higher-order covariation and epistasis, with broader implications for probabilistic sequence models in general.

Mi3-GPU: MCMC-based Inverse Ising Inference on GPUs for protein covariation analysis

Inverse Ising inference is a method for inferring the coupling parameters of a Potts/Ising model based on observed site-covariation, which has found important applications in protein physics for detecting interactions between residues in protein families. We introduce Mi3-GPU ("mee-three", for MCMC Inverse Ising Inference) software for solving the inverse Ising problem for protein-sequence datasets with few analytic approximations, by parallel Markov-Chain Monte-Carlo sampling on GPUs. We also provide tools for analysis and preparation of protein-family Multiple Sequence Alignments (MSAs) to account for finite-sampling issues, which are a major source of error or bias in inverse Ising inference. Our method is "generative" in the sense that the inferred model can be used to generate synthetic MSAs whose mutational statistics (marginals) can be verified to match the dataset MSA statistics up to the limits imposed by the effects of finite sampling. Our GPU implementation enables the construction of models which reproduce the covariation patterns of the observed MSA with a precision that is not possible with more approximate methods. The main components of our method are a GPU-optimized algorithm to greatly accelerate MCMC sampling, combined with a multi-step Quasi-Newton parameter-update scheme using a "Zwanzig reweighting" technique. We demonstrate the ability of this software to produce generative models on typical protein family datasets for sequence lengths L ~ 300 with 21 residue types with tens of millions of inferred parameters in short running times.

Remote homology search with hidden Potts models

Most methods for biological sequence homology search and alignment work with primary sequence alone, neglecting higher-order correlations. Recently, statistical physics models called Potts models have been used to infer all-by-all pairwise correlations between sites in deep multiple sequence alignments, and these pairwise couplings have improved 3D structure predictions. Here we extend the use of Potts models from structure prediction to sequence alignment and homology search by developing what we call a hidden Potts model (HPM) that merges a Potts emission process to a generative probability model of insertion and deletion. Because an HPM is incompatible with efficient dynamic programming alignment algorithms, we develop an approximate algorithm based on importance sampling, using simpler probabilistic models as proposal distributions. We test an HPM implementation on RNA structure homology search benchmarks, where we can compare directly to exact alignment methods that capture nested RNA base-pairing correlations (stochastic context-free grammars). HPMs perform promisingly in these proof of principle experiments.

Epistasis and entrenchment of drug resistance in HIV-1 subtype B

The development of drug resistance in HIV is the result of primary mutations whose effects on viral fitness depend on the entire genetic background, a phenomenon called 'epistasis'. Based on protein sequences derived from drug-experienced patients in the Stanford HIV database, we use a co-evolutionary (Potts) Hamiltonian model to provide direct confirmation of epistasis involving many simultaneous mutations. Building on earlier work, we show that primary mutations leading to drug resistance can become highly favored (or entrenched) by the complex mutation patterns arising in response to drug therapy despite being disfavored in the wild-type background, and provide the first confirmation of entrenchment for all three drug-target proteins: protease, reverse transcriptase, and integrase; a comparative analysis reveals that NNRTI-induced mutations behave differently from the others. We further show that the likelihood of resistance mutations can vary widely in patient populations, and from the population average compared to specific molecular clones.

GEMME: a simple and fast global epistatic model predicting mutational effects

The systematic and accurate description of protein mutational landscapes is a question of utmost importance in biology, bioengineering, and medicine. Recent progress has been achieved by leveraging on the increasing wealth of genomic data and by modeling intersite dependencies within biological sequences. However, state-of-the-art methods remain time consuming. Here, we present Global Epistatic Model for predicting Mutational Effects (GEMME) (www.lcqb.upmc.fr/GEMME), an original and fast method that predicts mutational outcomes by explicitly modeling the evolutionary history of natural sequences. This allows accounting for all positions in a sequence when estimating the effect of a given mutation. GEMME uses only a few biologically meaningful and interpretable parameters. Assessed against 50 high- and low-throughput mutational experiments, it overall performs similarly or better than existing methods. It accurately predicts the mutational landscapes of a wide range of protein families, including viral ones and, more generally, of much conserved families. Given an input alignment, it generates the full mutational landscape of a protein in a matter of minutes. It is freely available as a package and a webserver at www.lcqb.upmc.fr/GEMME/.