Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness

Addressing epistasis in the design of protein function

Mutations in protein active sites can dramatically improve function. The active site, however, is densely packed and extremely sensitive to mutations. Therefore, some mutations may only be tolerated in combination with others in a phenomenon known as epistasis. Epistasis reduces the likelihood of obtaining improved functional variants and dramatically slows natural and lab evolutionary processes. Research has shed light on the molecular origins of epistasis and its role in shaping evolutionary trajectories and outcomes. In addition, sequence- and AI-based strategies that infer epistatic relationships from mutational patterns in natural or experimental evolution data have been used to design functional protein variants. In recent years, combinations of such approaches and atomistic design calculations have successfully predicted highly functional combinatorial mutations in active sites. These were used to design thousands of functional active-site variants, demonstrating that, while our understanding of epistasis remains incomplete, some of the determinants that are critical for accurate design are now sufficiently understood. We conclude that the space of active-site variants that has been explored by evolution may be expanded dramatically to enhance natural activities or discover new ones. Furthermore, design opens the way to systematically exploring sequence and structure space and mutational impacts on function, deepening our understanding and control over protein activity.

Evolutionary sequence and structural basis for the distinct conformational landscapes of Tyr and Ser/Thr kinases

Protein kinases are molecular machines with rich sequence variation that distinguishes the two main evolutionary branches - tyrosine kinases (TKs) from serine/threonine kinases (STKs). Using a sequence co-variation Potts statistical energy model we previously concluded that TK catalytic domains are more likely than STKs to adopt an inactive conformation with the activation loop in an autoinhibitory folded conformation, due to intrinsic sequence effects. Here we investigate the structural basis for this phenomenon by integrating the sequence-based model with structure-based molecular dynamics (MD) to determine the effects of mutations on the free energy difference between active and inactive conformations, using a thermodynamic cycle involving many (n = 108) protein-mutation free energy perturbation (FEP) simulations in the active and inactive conformations. The sequence and structure-based results are consistent and support the hypothesis that the inactive conformation DFG-out Activation Loop Folded, is a functional regulatory state that has been stabilized in TKs relative to STKs over the course of their evolution via the accumulation of residue substitutions in the activation loop and catalytic loop that facilitate distinct substrate binding modes in trans and additional modes of regulation in cis for TKs.

Activation of Polycystin-1 Signaling by Binding of Stalk-derived Peptide Agonists

Polycystin-1 (PC1) is the membrane protein product of the PKD1 gene whose mutation is responsible for 85% of the cases of autosomal dominant polycystic kidney disease (ADPKD). ADPKD is primarily characterized by the formation of renal cysts and potential kidney failure. PC1 is an atypical G protein-coupled receptor (GPCR) consisting of 11 transmembrane helices and an autocatalytic GAIN domain that cleaves PC1 into extracellular N-terminal (NTF) and membrane-embedded C-terminal (CTF) fragments. Recently, signaling activation of the PC1 CTF was shown to be regulated by a stalk tethered agonist (TA), a distinct mechanism observed in the adhesion GPCR family. A novel allosteric activation pathway was elucidated for the PC1 CTF through a combination of Gaussian accelerated molecular dynamics (GaMD), mutagenesis and cellular signaling experiments. Here, we show that synthetic, soluble peptides with 7 to 21 residues derived from the stalk TA, in particular, peptides including the first 9 residues (p9), 17 residues (p17) and 21 residues (p21) exhibited the ability to re-activate signaling by a stalkless PC1 CTF mutant in cellular assays. To reveal molecular mechanisms of stalk peptide-mediated signaling activation, we have applied a novel Peptide GaMD (Pep-GaMD) algorithm to elucidate binding conformations of selected stalk peptide agonists p9, p17 and p21 to the stalkless PC1 CTF. The simulations revealed multiple specific binding regions of the stalk peptide agonists to the PC1 protein including an "intermediate" bound yet inactive state. Our Pep-GaMD simulation findings were consistent with the cellular assay experimental data. Binding of peptide agonists to the TOP domain of PC1 induced close TOP-putative pore loop interactions, a characteristic feature of the PC1 CTF signaling activation mechanism. Using sequence covariation analysis of PC1 homologs, we further showed that the peptide binding regions were consistent with covarying residue pairs identified between the TOP domain and the stalk TA. Therefore, structural dynamic insights into the mechanisms of PC1 activation by stalk-derived peptide agonists have enabled an in-depth understanding of PC1 signaling. They will form a foundation for development of PC1 as a therapeutic target for the treatment of ADPKD.

Machine learning in biological physics: From biomolecular prediction to design

Machine learning has been proposed as an alternative to theoretical modeling when dealing with complex problems in biological physics. However, in this perspective, we argue that a more successful approach is a proper combination of these two methodologies. We discuss how ideas coming from physical modeling neuronal processing led to early formulations of computational neural networks, e.g., Hopfield networks. We then show how modern learning approaches like Potts models, Boltzmann machines, and the transformer architecture are related to each other, specifically, through a shared energy representation. We summarize recent efforts to establish these connections and provide examples on how each of these formulations integrating physical modeling and machine learning have been successful in tackling recent problems in biomolecular structure, dynamics, function, evolution, and design. Instances include protein structure prediction; improvement in computational complexity and accuracy of molecular dynamics simulations; better inference of the effects of mutations in proteins leading to improved evolutionary modeling and finally how machine learning is revolutionizing protein engineering and design. Going beyond naturally existing protein sequences, a connection to protein design is discussed where synthetic sequences are able to fold to naturally occurring motifs driven by a model rooted in physical principles. We show that this model is "learnable" and propose its future use in the generation of unique sequences that can fold into a target structure.

Towards parsimonious generative modeling of RNA families

Generative probabilistic models emerge as a new paradigm in data-driven, evolution-informed design of biomolecular sequences. This paper introduces a novel approach, called Edge Activation Direct Coupling Analysis (eaDCA), tailored to the characteristics of RNA sequences, with a strong emphasis on simplicity, efficiency, and interpretability. eaDCA explicitly constructs sparse coevolutionary models for RNA families, achieving performance levels comparable to more complex methods while utilizing a significantly lower number of parameters. Our approach demonstrates efficiency in generating artificial RNA sequences that closely resemble their natural counterparts in both statistical analyses and SHAPE-MaP experiments, and in predicting the effect of mutations. Notably, eaDCA provides a unique feature: estimating the number of potential functional sequences within a given RNA family. For example, in the case of cyclic di-AMP riboswitches (RF00379), our analysis suggests the existence of approximately 1039 functional nucleotide sequences. While huge compared to the known <4000 natural sequences, this number represents only a tiny fraction of the vast pool of nearly 1082 possible nucleotide sequences of the same length (136 nucleotides). These results underscore the promise of sparse and interpretable generative models, such as eaDCA, in enhancing our understanding of the expansive RNA sequence space.

Some mechanistic underpinnings of molecular adaptations of SARS-COV-2 spike protein by integrating candidate adaptive polymorphisms with protein dynamics

We integrate evolutionary predictions based on the neutral theory of molecular evolution with protein dynamics to generate mechanistic insight into the molecular adaptations of the SARS-COV-2 spike (S) protein. With this approach, we first identified candidate adaptive polymorphisms (CAPs) of the SARS-CoV-2 S protein and assessed the impact of these CAPs through dynamics analysis. Not only have we found that CAPs frequently overlap with well-known functional sites, but also, using several different dynamics-based metrics, we reveal the critical allosteric interplay between SARS-CoV-2 CAPs and the S protein binding sites with the human ACE2 (hACE2) protein. CAPs interact far differently with the hACE2 binding site residues in the open conformation of the S protein compared to the closed form. In particular, the CAP sites control the dynamics of binding residues in the open state, suggesting an allosteric control of hACE2 binding. We also explored the characteristic mutations of different SARS-CoV-2 strains to find dynamic hallmarks and potential effects of future mutations. Our analyses reveal that Delta strain-specific variants have non-additive (i.e., epistatic) interactions with CAP sites, whereas the less pathogenic Omicron strains have mostly additive mutations. Finally, our dynamics-based analysis suggests that the novel mutations observed in the Omicron strain epistatically interact with the CAP sites to help escape antibody binding.

Evolutionary sequence and structural basis for the distinct conformational landscapes of Tyr and Ser/Thr kinases

Protein kinases are molecular machines with rich sequence variation that distinguishes the two main evolutionary branches - tyrosine kinases (TKs) from serine/threonine kinases (STKs). Using a sequence co-variation Potts statistical energy model we previously concluded that TK catalytic domains are more likely than STKs to adopt an inactive conformation with the activation loop in an autoinhibitory "folded" conformation, due to intrinsic sequence effects. Here we investigated the structural basis for this phenomenon by integrating the sequence-based model with structure-based molecular dynamics (MD) to determine the effects of mutations on the free energy difference between active and inactive conformations, using a novel thermodynamic cycle involving many (n=108) protein-mutation free energy perturbation (FEP) simulations in the active and inactive conformations. The sequence and structure-based results are consistent and support the hypothesis that the inactive conformation "DFG-out Activation Loop Folded", is a functional regulatory state that has been stabilized in TKs relative to STKs over the course of their evolution via the accumulation of residue substitutions in the activation loop and catalytic loop that facilitate distinct substrate binding modes in trans and additional modes of regulation in cis for TKs.

Evolutionary sequence and structural basis for the distinct conformational landscapes of Tyr and Ser/Thr kinases

Protein kinases are molecular machines with rich sequence variation that distinguishes the two main evolutionary branches - tyrosine kinases (TKs) from serine/threonine kinases (STKs). Using a sequence co-variation Potts statistical energy model we previously concluded that TK catalytic domains are more likely than STKs to adopt an inactive conformation with the activation loop in an autoinhibitory "folded" conformation, due to intrinsic sequence effects. Here we investigated the structural basis for this phenomenon by integrating the sequence-based model with structure-based molecular dynamics (MD) to determine the effects of mutations on the free energy difference between active and inactive conformations, using a novel thermodynamic cycle involving many (n=108) protein-mutation free energy perturbation (FEP) simulations in the active and inactive conformations. The sequence and structure-based results are consistent and support the hypothesis that the inactive conformation "DFG-out Activation Loop Folded", is a functional regulatory state that has been stabilized in TKs relative to STKs over the course of their evolution via the accumulation of residue substitutions in the activation loop and catalytic loop that facilitate distinct substrate binding modes in trans and additional modes of regulation in cis for TKs.

Kinetic coevolutionary models predict the temporal emergence of HIV-1 resistance mutations under drug selection pressure

Drug resistance in HIV type 1 (HIV-1) is a pervasive problem that affects the lives of millions of people worldwide. Although records of drug-resistant mutations (DRMs) have been extensively tabulated within public repositories, our understanding of the evolutionary kinetics of DRMs and how they evolve together remains limited. Epistasis, the interaction between a DRM and other residues in HIV-1 protein sequences, is key to the temporal evolution of drug resistance. We use a Potts sequence-covariation statistical-energy model of HIV-1 protein fitness under drug selection pressure, which captures epistatic interactions between all positions, combined with kinetic Monte-Carlo simulations of sequence evolutionary trajectories, to explore the acquisition of DRMs as they arise in an ensemble of drug-naive patient protein sequences. We follow the time course of 52 DRMs in the enzymes protease, RT, and integrase, the primary targets of antiretroviral therapy. The rates at which DRMs emerge are highly correlated with their observed acquisition rates reported in the literature when drug pressure is applied. This result highlights the central role of epistasis in determining the kinetics governing DRM emergence. Whereas rapidly acquired DRMs begin to accumulate as soon as drug pressure is applied, slowly acquired DRMs are contingent on accessory mutations that appear only after prolonged drug pressure. We provide a foundation for using computational methods to determine the temporal evolution of drug resistance using Potts statistical potentials, which can be used to gain mechanistic insights into drug resistance pathways in HIV-1 and other infectious agents.

Potts Hamiltonian Models and Molecular Dynamics Free Energy Simulations for Predicting the Impact of Mutations on Protein Kinase Stability

Single-point mutations in kinase proteins can affect their stability and fitness, and computational analysis of these effects can provide insights into the relationships among protein sequence, structure, and function for this enzyme family. To assess the impact of mutations on protein stability, we used a sequence-based Potts Hamiltonian model trained on a kinase family multiple-sequence alignment (MSA) to calculate the statistical energy (fitness) effects of mutations and compared these against relative folding free energies (ΔΔGs) calculated from all-atom molecular dynamics free energy perturbation (FEP) simulations in explicit solvent. The fitness effects of mutations in the Potts model (ΔEs) showed good agreement with experimental thermostability data (Pearson r = 0.68), similar to the correlation we observed with ΔΔGs predicted from structure-based relative FEP simulations. Recognizing the possible advantages of using Potts models to rapidly estimate protein stability effects of kinase mutations seen in cancer genomics data, we used the Potts statistical energy model to estimate the stability effects of 65 conservative and nonconservative mutations across three distinct kinases (Wee1, Abl1, and Cdc7) with somatic mutations reported in the Genomic Data Commons (GDC) database. The ΔEs of these mutations calculated from the Potts model are consistent with the corresponding ΔΔGs from FEP simulations (Pearson ratio of 0.72). The agreement between these methods suggests that the Potts model may be used as a sequence-based tool for high-throughput screening of mutational effects as part of a computational pipeline for predicting the stability effects of mutations. We also demonstrate how the scalability of the fitness-based Potts model calculations permits analyses that are not easily accessed using FEP simulations. To this end, we employed site-saturation mutagenesis in the Potts model in order to investigate the relative stability effects of mutations seen in different cancer evolutionary scenarios. We used this approach to analyze the effects of drug pressure in Abl kinase by contrasting the relative fitness penalties of somatic mutations seen in miscellaneous cancer types with those calculated for mutations associated with cancer drug resistance. We observed that, in contrast to somatic mutations of Abl seen in various tumors that appear to have evolved neutrally, cancer mutations that evolved under drug pressure in Abl-targeted therapies tend to preserve enzyme stability.

Coevolutionary Information Captures Catalytic Functions and Reveals Divergent Roles of Terpene Synthase Interdomain Connections

Inferring the historical and biophysical causes of diversity within protein families is a complex puzzle. A key to unraveling this problem is characterizing the rugged topography of sequence-function adaptive landscapes. Using biochemical data from a 29 = 512 combinatorial library of tobacco 5-epi-aristolochene synthase (TEAS) mutants engineered to make the native major product of Egyptian henbane premnaspirodiene synthase (HPS) and a complementary 512 mutant HPS library, we address the question of how product specificity is controlled. These data sets reveal that HPS is far more robust and resistant to mutations than TEAS, where most mutants are promiscuous. We also combine experimental data with a sequence Potts Hamiltonian model and direct coupling analysis to quantify mutant fitness. Our results demonstrate that the Hamiltonian captures variation in product outputs across both libraries, clusters native family members based on their substrate specificities, and exposes the divergent catalytic roles of couplings between the catalytic and noncatalytic domains of TEAS versus HPS. Specifically, we found that the role of the interdomain connectivities in specifying product output is more important in TEAS than connectivities within the catalytic domain. Despite being 75% identical, this property is not shared by HPS, where connectivities within the catalytic domain are more important for specificity. By solving the X-ray crystal structure of HPS, we assessed structural bases for their interdomain network differences. Last, we calculate the product profile Shannon entropies of the two libraries, which showcases that site-site connectivities also play divergent roles in catalytic accuracy.

In vivo functional phenotypes from a computational epistatic model of evolution

Computational models of evolution are valuable for understanding the dynamics of sequence variation, to infer phylogenetic relationships or potential evolutionary pathways and for biomedical and industrial applications. Despite these benefits, few have validated their propensities to generate outputs with in vivo functionality, which would enhance their value as accurate and interpretable evolutionary algorithms. We demonstrate the power of epistasis inferred from natural protein families to evolve sequence variants in an algorithm we developed called sequence evolution with epistatic contributions (SEEC). Utilizing the Hamiltonian of the joint probability of sequences in the family as fitness metric, we sampled and experimentally tested for in vivo [Formula: see text]-lactamase activity in Escherichia coli TEM-1 variants. These evolved proteins can have dozens of mutations dispersed across the structure while preserving sites essential for both catalysis and interactions. Remarkably, these variants retain family-like functionality while being more active than their wild-type predecessor. We found that depending on the inference method used to generate the epistatic constraints, different parameters simulate diverse selection strengths. Under weaker selection, local Hamiltonian fluctuations reliably predict relative changes to variant fitness, recapitulating neutral evolution. SEEC has the potential to explore the dynamics of neofunctionalization, characterize viral fitness landscapes, and facilitate vaccine development.

Some mechanistic underpinnings of molecular adaptations of SARS-COV-2 spike protein by integrating candidate adaptive polymorphisms with protein dynamics

We integrate evolutionary predictions based on the neutral theory of molecular evolution with protein dynamics to generate mechanistic insight into the molecular adaptations of the SARS-COV-2 Spike (S) protein. With this approach, we first identified Candidate Adaptive Polymorphisms (CAPs) of the SARS-CoV-2 Spike protein and assessed the impact of these CAPs through dynamics analysis. Not only have we found that CAPs frequently overlap with well-known functional sites, but also, using several different dynamics-based metrics, we reveal the critical allosteric interplay between SARS-CoV-2 CAPs and the S protein binding sites with the human ACE2 (hACE2) protein. CAPs interact far differently with the hACE2 binding site residues in the open conformation of the S protein compared to the closed form. In particular, the CAP sites control the dynamics of binding residues in the open state, suggesting an allosteric control of hACE2 binding. We also explored the characteristic mutations of different SARS-CoV-2 strains to find dynamic hallmarks and potential effects of future mutations. Our analyses reveal that Delta strain-specific variants have non-additive (i.e., epistatic) interactions with CAP sites, whereas the less pathogenic Omicron strains have mostly additive mutations. Finally, our dynamics-based analysis suggests that the novel mutations observed in the Omicron strain epistatically interact with the CAP sites to help escape antibody binding.

GENERALIST: A latent space based generative model for protein sequence families

Generative models of protein sequence families are an important tool in the repertoire of protein scientists and engineers alike. However, state-of-the-art generative approaches face inference, accuracy, and overfitting- related obstacles when modeling moderately sized to large proteins and/or protein families with low sequence coverage. Here, we present a simple to learn, tunable, and accurate generative model, GENERALIST: GENERAtive nonLInear tenSor-factorizaTion for protein sequences. GENERALIST accurately captures several high order summary statistics of amino acid covariation. GENERALIST also predicts conservative local optimal sequences which are likely to fold in stable 3D structure. Importantly, unlike current methods, the density of sequences in GENERALIST-modeled sequence ensembles closely resembles the corresponding natural ensembles. Finally, GENERALIST embeds protein sequences in an informative latent space. GENERALIST will be an important tool to study protein sequence variability.

Protein dynamics provide mechanistic insights about epistasis among common missense polymorphisms

Sequencing of the protein coding genome has revealed many different missense mutations of human proteins and different population frequencies of corresponding haplotypes, which consist of different sets of those mutations. Here, we present evidence for pairwise intramolecular epistasis (i.e., nonadditive interactions) between many such mutations through an analysis of protein dynamics. We suggest that functional compensation for conserving protein dynamics is a likely evolutionary mechanism that maintains high-frequency mutations that are individually nonneutral but epistatically compensating within proteins. This analysis is the first of its type to look at human proteins with specific high population frequency mutations and examine the relationship between mutations that make up that observed high-frequency protein haplotype. Importantly, protein dynamics revealed a separation between high and low frequency haplotypes within a target protein cytochrome P450 2A7, with the high-frequency haplotypes showing behavior closer to the wild-type protein. Common protein haplotypes containing two mutations display dynamic compensation in which one mutation can correct for the dynamic effects of the other. We also utilize a dynamics-based metric, EpiScore, that evaluates the epistatic interactions and allows us to see dynamic compensation within many other proteins.

Mechanisms of HIV-1 integrase resistance to dolutegravir and potent inhibition of drug-resistant variants

HIV-1 infection depends on the integration of viral DNA into host chromatin. Integration is mediated by the viral enzyme integrase and is blocked by integrase strand transfer inhibitors (INSTIs), first-line antiretroviral therapeutics widely used in the clinic. Resistance to even the best INSTIs is a problem, and the mechanisms of resistance are poorly understood. Here, we analyze combinations of the mutations E138K, G140A/S, and Q148H/K/R, which confer resistance to INSTIs. The investigational drug 4d more effectively inhibited the mutants compared with the approved drug Dolutegravir (DTG). We present 11 new cryo-EM structures of drug-resistant HIV-1 intasomes bound to DTG or 4d, with better than 3-Å resolution. These structures, complemented with free energy simulations, virology, and enzymology, explain the mechanisms of DTG resistance involving E138K + G140A/S + Q148H/K/R and show why 4d maintains potency better than DTG. These data establish a foundation for further development of INSTIs that potently inhibit resistant forms in integrase.

In vivo functional phenotypes from a computational epistatic model of evolution

Computational models of evolution are valuable for understanding the dynamics of sequence variation, to infer phylogenetic relationships or potential evolutionary pathways and for biomedical and industrial applications. Despite these benefits, few have validated their propensities to generate outputs with in vivo functionality, which would enhance their value as accurate and interpretable evolutionary algorithms. We demonstrate the power of epistasis inferred from natural protein families to evolve sequence variants in an algorithm we developed called Sequence Evolution with Epistatic Contributions. Utilizing the Hamiltonian of the joint probability of sequences in the family as fitness metric, we sampled and experimentally tested for in vivo β -lactamase activity in E. coli TEM-1 variants. These evolved proteins can have dozens of mutations dispersed across the structure while preserving sites essential for both catalysis and interactions. Remarkably, these variants retain family-like functionality while being more active than their WT predecessor. We found that depending on the inference method used to generate the epistatic constraints, different parameters simulate diverse selection strengths. Under weaker selection, local Hamiltonian fluctuations reliably predict relative changes to variant fitness, recapitulating neutral evolution. SEEC has the potential to explore the dynamics of neofunctionalization, characterize viral fitness landscapes and facilitate vaccine development.

Latent generative landscapes as maps of functional diversity in protein sequence space

Variational autoencoders are unsupervised learning models with generative capabilities, when applied to protein data, they classify sequences by phylogeny and generate de novo sequences which preserve statistical properties of protein composition. While previous studies focus on clustering and generative features, here, we evaluate the underlying latent manifold in which sequence information is embedded. To investigate properties of the latent manifold, we utilize direct coupling analysis and a Potts Hamiltonian model to construct a latent generative landscape. We showcase how this landscape captures phylogenetic groupings, functional and fitness properties of several systems including Globins, β-lactamases, ion channels, and transcription factors. We provide support on how the landscape helps us understand the effects of sequence variability observed in experimental data and provides insights on directed and natural protein evolution. We propose that combining generative properties and functional predictive power of variational autoencoders and coevolutionary analysis could be beneficial in applications for protein engineering and design.

Prediction of designer-recombinases for DNA editing with generative deep learning

Site-specific tyrosine-type recombinases are effective tools for genome engineering, with the first engineered variants having demonstrated therapeutic potential. So far, adaptation to new DNA target site selectivity of designer-recombinases has been achieved mostly through iterative cycles of directed molecular evolution. While effective, directed molecular evolution methods are laborious and time consuming. Here we present RecGen (Recombinase Generator), an algorithm for the intelligent generation of designer-recombinases. We gather the sequence information of over one million Cre-like recombinase sequences evolved for 89 different target sites with which we train Conditional Variational Autoencoders for recombinase generation. Experimental validation demonstrates that the algorithm can predict recombinase sequences with activity on novel target-sites, indicating that RecGen is useful to accelerate the development of future designer-recombinases.

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.

From sequence to function through structure: Deep learning for protein design

The process of designing biomolecules, in particular proteins, is witnessing a rapid change in available tooling and approaches, moving from design through physicochemical force fields, to producing plausible, complex sequences fast via end-to-end differentiable statistical models. To achieve conditional and controllable protein design, researchers at the interface of artificial intelligence and biology leverage advances in natural language processing (NLP) and computer vision techniques, coupled with advances in computing hardware to learn patterns from growing biological databases, curated annotations thereof, or both. Once learned, these patterns can be used to provide novel insights into mechanistic biology and the design of biomolecules. However, navigating and understanding the practical applications for the many recent protein design tools is complex. To facilitate this, we 1) document recent advances in deep learning (DL) assisted protein design from the last three years, 2) present a practical pipeline that allows to go from de novo-generated sequences to their predicted properties and web-powered visualization within minutes, and 3) leverage it to suggest a generated protein sequence which might be used to engineer a biosynthetic gene cluster to produce a molecular glue-like compound. Lastly, we discuss challenges and highlight opportunities for the protein design field.

Molecular Dynamics Simulations Establish the Molecular Basis for the Broad Allostery Hotspot Distributions in the Tetracycline Repressor

It is imperative to identify the network of residues essential to the allosteric coupling for the purpose of rationally engineering allostery in proteins. Deep mutational scanning analysis has emerged as a function-centric approach for identifying such allostery hotspots in a comprehensive and unbiased fashion, leading to observations that challenge our understanding of allostery at the molecular level. Specifically, a recent deep mutational scanning study of the tetracycline repressor (TetR) revealed an unexpectedly broad distribution of allostery hotspots throughout the protein structure. Using extensive molecular dynamics simulations (up to 50 μs) and free energy computations, we establish the molecular and energetic basis for the strong anticooperativity between the ligand and DNA binding sites. The computed free energy landscapes in different ligation states illustrate that allostery in TetR is well described by a conformational selection model, in which the apo state samples a broad set of conformations, and specific ones are selectively stabilized by either ligand or DNA binding. By examining a range of structural and dynamic properties of residues at both local and global scales, we observe that various analyses capture different subsets of experimentally identified hotspots, suggesting that these residues modulate allostery in distinct ways. These results motivate the development of a thermodynamic model that qualitatively explains the broad distribution of hotspot residues and their distinct features in molecular dynamics simulations. The multifaceted strategy that we establish here for hotspot evaluations and our insights into their mechanistic contributions are useful for modulating protein allostery in mechanistic and engineering studies.

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.

Unique features of different classes of G-protein-coupled receptors revealed from sequence coevolutionary and structural analysis

G-protein-coupled receptors (GPCRs) are the largest family of human membrane proteins and represent the primary targets of about one third of currently marketed drugs. Despite the critical importance, experimental structures have been determined for only a limited portion of GPCRs and functional mechanisms of GPCRs remain poorly understood. Here, we have constructed novel sequence coevolutionary models of the A and B classes of GPCRs and compared them with residue contact frequency maps generated with available experimental structures. Significant portions of structural residue contacts were successfully detected in the sequence-based covariational models. "Exception" residue contacts predicted from sequence coevolutionary models but not available structures added missing links that were important for GPCR activation and allosteric modulation. Moreover, we identified distinct residue contacts involving different sets of functional motifs for GPCR activation, such as the Na⁺ pocket, CWxP, DRY, PIF, and NPxxY motifs in the class A and the HETx and PxxG motifs in the class B. Finally, we systematically uncovered critical residue contacts tuned by allosteric modulation in the two classes of GPCRs, including those from the activation motifs and particularly the extracellular and intracellular loops in class A GPCRs. These findings provide a promising framework for rational design of ligands to regulate GPCR activation and allosteric modulation.

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.

Field-theoretic density estimation for biological sequence space with applications to 5' splice site diversity and aneuploidy in cancer

Density estimation in sequence space is a fundamental problem in machine learning that is also of great importance in computational biology. Due to the discrete nature and large dimensionality of sequence space, how best to estimate such probability distributions from a sample of observed sequences remains unclear. One common strategy for addressing this problem is to estimate the probability distribution using maximum entropy (i.e., calculating point estimates for some set of correlations based on the observed sequences and predicting the probability distribution that is as uniform as possible while still matching these point estimates). Building on recent advances in Bayesian field-theoretic density estimation, we present a generalization of this maximum entropy approach that provides greater expressivity in regions of sequence space where data are plentiful while still maintaining a conservative maximum entropy character in regions of sequence space where data are sparse or absent. In particular, we define a family of priors for probability distributions over sequence space with a single hyperparameter that controls the expected magnitude of higher-order correlations. This family of priors then results in a corresponding one-dimensional family of maximum a posteriori estimates that interpolate smoothly between the maximum entropy estimate and the observed sample frequencies. To demonstrate the power of this method, we use it to explore the high-dimensional geometry of the distribution of 5' splice sites found in the human genome and to understand patterns of chromosomal abnormalities across human cancers.

Efficient generative modeling of protein sequences using simple autoregressive models

Generative models emerge as promising candidates for novel sequence-data driven approaches to protein design, and for the extraction of structural and functional information about proteins deeply hidden in rapidly growing sequence databases. Here we propose simple autoregressive models as highly accurate but computationally efficient generative sequence models. We show that they perform similarly to existing approaches based on Boltzmann machines or deep generative models, but at a substantially lower computational cost (by a factor between 10² and 10³). Furthermore, the simple structure of our models has distinctive mathematical advantages, which translate into an improved applicability in sequence generation and evaluation. Within these models, we can easily estimate both the probability of a given sequence, and, using the model's entropy, the size of the functional sequence space related to a specific protein family. In the example of response regulators, we find a huge number of ca. 10⁶⁸ possible sequences, which nevertheless constitute only the astronomically small fraction 10^-80 of all amino-acid sequences of the same length. These findings illustrate the potential and the difficulty in exploring sequence space via generative sequence models.

Sparse generative modeling via parameter reduction of Boltzmann machines: Application to protein-sequence families

Boltzmann machines (BMs) are widely used as generative models. For example, pairwise Potts models (PMs), which are instances of the BM class, provide accurate statistical models of families of evolutionarily related protein sequences. Their parameters are the local fields, which describe site-specific patterns of amino acid conservation, and the two-site couplings, which mirror the coevolution between pairs of sites. This coevolution reflects structural and functional constraints acting on protein sequences during evolution. The most conservative choice to describe the coevolution signal is to include all possible two-site couplings into the PM. This choice, typical of what is known as Direct Coupling Analysis, has been successful for predicting residue contacts in the three-dimensional structure, mutational effects, and generating new functional sequences. However, the resulting PM suffers from important overfitting effects: many couplings are small, noisy, and hardly interpretable; the PM is close to a critical point, meaning that it is highly sensitive to small parameter perturbations. In this work, we introduce a general parameter-reduction procedure for BMs, via a controlled iterative decimation of the less statistically significant couplings, identified by an information-based criterion that selects either weak or statistically unsupported couplings. For several protein families, our procedure allows one to remove more than 90% of the PM couplings, while preserving the predictive and generative properties of the original dense PM, and the resulting model is far away from criticality, hence more robust to noise.

Deep Mutational Scanning of Viral Glycoproteins and Their Host Receptors

Deep mutational scanning or deep mutagenesis is a powerful tool for understanding the sequence diversity available to viruses for adaptation in a laboratory setting. It generally involves tracking an in vitro selection of protein sequence variants with deep sequencing to map mutational effects based on changes in sequence abundance. Coupled with any of a number of selection strategies, deep mutagenesis can explore the mutational diversity available to viral glycoproteins, which mediate critical roles in cell entry and are exposed to the humoral arm of the host immune response. Mutational landscapes of viral glycoproteins for host cell attachment and membrane fusion reveal extensive epistasis and potential escape mutations to neutralizing antibodies or other therapeutics, as well as aiding in the design of optimized immunogens for eliciting broadly protective immunity. While less explored, deep mutational scans of host receptors further assist in understanding virus-host protein interactions. Critical residues on the host receptors for engaging with viral spikes are readily identified and may help with structural modeling. Furthermore, mutations may be found for engineering soluble decoy receptors as neutralizing agents that specifically bind viral targets with tight affinity and limited potential for viral escape. By untangling the complexities of how sequence contributes to viral glycoprotein and host receptor interactions, deep mutational scanning is impacting ideas and strategies at multiple levels for combatting circulating and emergent virus strains.

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.

ELIHKSIR Web Server: Evolutionary Links Inferred for Histidine Kinase Sensors Interacting with Response Regulators

Two-component systems (TCS) are signaling machinery that consist of a histidine kinases (HK) and response regulator (RR). When an environmental change is detected, the HK phosphorylates its cognate response regulator (RR). While cognate interactions were considered orthogonal, experimental evidence shows the prevalence of crosstalk interactions between non-cognate HK–RR pairs. Currently, crosstalk interactions have been demonstrated for TCS proteins in a limited number of organisms. By providing specificity predictions across entire TCS networks for a large variety of organisms, the ELIHKSIR web server assists users in identifying interactions for TCS proteins and their mutants. To generate specificity scores, a global probabilistic model was used to identify interfacial couplings and local fields from sequence information. These couplings and local fields were then used to construct Hamiltonian scores for positions with encoded specificity, resulting in the specificity score. These methods were applied to 6676 organisms available on the ELIHKSIR web server. Due to the ability to mutate proteins and display the resulting network changes, there are nearly endless combinations of TCS networks to analyze using ELIHKSIR. The functionality of ELIHKSIR allows users to perform a variety of TCS network analyses and visualizations to support TCS research efforts.

Episodic evolution of coadapted sets of amino acid sites in mitochondrial proteins

The rate of evolution differs between protein sites and changes with time. However, the link between these two phenomena remains poorly understood. Here, we design a phylogenetic approach for distinguishing pairs of amino acid sites that evolve concordantly, i.e., such that substitutions at one site trigger subsequent substitutions at the other; and also pairs of sites that evolve discordantly, so that substitutions at one site impede subsequent substitutions at the other. We distinguish groups of amino acid sites that undergo coordinated evolution and evolve discordantly from other such groups. In mitochondrion-encoded proteins of metazoans and fungi, we show that concordantly evolving sites are clustered in protein structures. By analysing the phylogenetic patterns of substitutions at concordantly and discordantly evolving site pairs, we find that concordant evolution has two distinct causes: epistatic interactions between amino acid substitutions and episodes of selection independently affecting substitutions at different sites. The rate of substitutions at concordantly evolving groups of protein sites changes in the course of evolution, indicating episodes of selection limited to some of the lineages. The phylogenetic positions of these changes are consistent between proteins, suggesting common selective forces underlying them.

The mode and rate of evolution of a protein site depends on the effect of its mutations on protein fitness. The fitness effect of a mutation itself can change in the course of evolution for at least two reasons. First, it can be modulated by substitutions occurring at other sites, a phenomenon called epistasis. Second, changes in selection can be non-epistatic, affecting sites independently of one another. Here, we analyse substitutions accumulated by the evolving lineages of the five proteins encoded by the mitochondrial genomes of thousands of species of metazoans and fungi. We show that substitutions at different amino acid sites occur in a coordinated fashion, and this coordination is caused both by epistasis and by episodes of selection affecting groups of sites. We partition each protein into several groups of concordantly evolving sites such that evolution of sites from different groups is discordant, and show that the proteins encoded by the mitochondrial genome consist of coevolving structural blocks. Some of these blocks have a clear functional specialization, e.g. are associated with interfaces between proteins composing respiratory complexes. Together, our results reveal a previously unrecognized complexity in the causes of variation in evolutionary rates between protein sites.

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.

Predominance of positive epistasis among drug resistance-associated mutations in HIV-1 protease

Drug-resistant mutations often have deleterious impacts on replication fitness, posing a fitness cost that can only be overcome by compensatory mutations. However, the role of fitness cost in the evolution of drug resistance has often been overlooked in clinical studies or in vitro selection experiments, as these observations only capture the outcome of drug selection. In this study, we systematically profile the fitness landscape of resistance-associated sites in HIV-1 protease using deep mutational scanning. We construct a mutant library covering combinations of mutations at 11 sites in HIV-1 protease, all of which are associated with resistance to protease inhibitors in clinic. Using deep sequencing, we quantify the fitness of thousands of HIV-1 protease mutants after multiple cycles of replication in human T cells. Although the majority of resistance-associated mutations have deleterious effects on viral replication, we find that epistasis among resistance-associated mutations is predominantly positive. Furthermore, our fitness data are consistent with genetic interactions inferred directly from HIV sequence data of patients. Fitness valleys formed by strong positive epistasis reduce the likelihood of reversal of drug resistance mutations. Overall, our results support the view that strong compensatory effects are involved in the emergence of clinically observed resistance mutations and provide insights to understanding fitness barriers in the evolution and reversion of drug resistance.

An evolution-based model for designing chorismate mutase enzymes

The rational design of enzymes is an important goal for both fundamental and practical reasons. Here, we describe a process to learn the constraints for specifying proteins purely from evolutionary sequence data, design and build libraries of synthetic genes, and test them for activity in vivo using a quantitative complementation assay. For chorismate mutase, a key enzyme in the biosynthesis of aromatic amino acids, we demonstrate the design of natural-like catalytic function with substantial sequence diversity. Further optimization focuses the generative model toward function in a specific genomic context. The data show that sequence-based statistical models suffice to specify proteins and provide access to an enormous space of functional sequences. This result provides a foundation for a general process for evolution-based design of artificial proteins.

TLmutation: Predicting the Effects of Mutations Using Transfer Learning

A reccurring challenge in bioinformatics is predicting the phenotypic consequence of amino acid variation in proteins. With the recent advancements in sequencing techniques, sufficient genomic data has become available to train models that predict the evolutionary statistical energies, but there is still inadequate experimental data to directly predict functional effects. One approach to overcome this data scarcity is to apply transfer learning and train more models with available data sets. In this study, we propose a set of transfer learning algorithms we call TLmutation, which implements a supervised transfer learning algorithm that transfers knowledge from survival data of a protein to a particular function of that protein. This is followed by an unsupervised transfer learning algorithm that extends the knowledge to a homologous protein. We explore the application of our algorithms in three cases. First, we test the supervised transfer on 17 previously published deep mutagenesis data sets to complete and refine missing data points. We further investigate these data sets to identify which mutations build better predictors of variant functions. In the second case, we apply the algorithm to predict higher-order mutations solely from single point mutagenesis data. Finally, we perform the unsupervised transfer learning algorithm to predict mutational effects of homologous proteins from experimental data sets. These algorithms are generalized to transfer knowledge between Markov random field models. We show the benefit of our transfer learning algorithms to utilize informative deep mutational data and provide new insights into protein variant functions. As these algorithms are generalized to transfer knowledge between Markov random field models, we expect these algorithms to be applicable to other disciplines.

Statistical physics of interacting proteins: Impact of dataset size and quality assessed in synthetic sequences

Identifying protein-protein interactions is crucial for a systems-level understanding of the cell. Recently, algorithms based on inverse statistical physics, e.g., direct coupling analysis (DCA), have allowed to use evolutionarily related sequences to address two conceptually related inference tasks: finding pairs of interacting proteins and identifying pairs of residues which form contacts between interacting proteins. Here we address two underlying questions: How are the performances of both inference tasks related? How does performance depend on dataset size and the quality? To this end, we formalize both tasks using Ising models defined over stochastic block models, with individual blocks representing single proteins and interblock couplings protein-protein interactions; controlled synthetic sequence data are generated by Monte Carlo simulations. We show that DCA is able to address both inference tasks accurately when sufficiently large training sets of known interaction partners are available and that an iterative pairing algorithm allows to make predictions even without a training set. Noise in the training data deteriorates performance. In both tasks we find a quadratic scaling relating dataset quality and size that is consistent with noise adding in square-root fashion and signal adding linearly when increasing the dataset. This implies that it is generally good to incorporate more data even if their quality are imperfect, thereby shedding light on the empirically observed performance of DCA applied to natural protein sequences.

Computationally Aided Discovery of LysEFm5 Variants with Improved Catalytic Activity and Stability

Bacteriophage-derived lysin proteins are potentially effective antimicrobials that would benefit from engineered improvements to their bioavailability and specific activity. Here, the catalytic domain of LysEFm5, a lysin with activity against vancomycin-resistant Enterococcus faecium (VRE), was subjected to site-saturation mutagenesis at positions whose selection was guided by sequence and structural information from homologous proteins. A second-order Potts model with parameters inferred from large sets of homologous sequence information was used to predict the average change in the statistical fitness for mutant libraries with diversity at pairs of sites within the secondary catalytic shell. Guided by the statistical fitness, nine double mutant saturation libraries were created and plated on agar containing autoclaved VRE to quickly identify and segregate catalytically active (halo-forming) and inactive (non-halo-forming) variants. High-throughput DNA sequencing of 873 unique variants showed that the statistical fitness was predictive of the retention or loss of catalytic activity (area under the curve [AUC], 0.840 to 0.894), with the inclusion of more diverse sequences in the starting multiple-sequence alignment improving the classification accuracy when pairwise amino acid couplings (epistasis) were considered. Of eight random halo-forming variants selected for more sensitive testing, one showed a 1.8 (±0.4)-fold improvement in specific activity and an 11.5 ± 0.8°C increase in melting temperature compared to those of the wild type. Our results demonstrate that a computationally informed approach employing homologous protein information coupled with a mid-throughput screening assay allows for the expedited discovery of lysin variants with improved properties.IMPORTANCE Broad-spectrum antibiotics can indiscriminately kill most bacteria, including commensal species that are a part of the normal human flora. This can potentially lead to the proliferation of drug-resistant bacteria upon elimination of competing species and to unwanted autoimmune effects in patients. Bacteriophage-derived lysin proteins are an alternative to conventional antibiotics that have coevolved alongside specific bacterial hosts. Lysins are capable of targeting conserved substrates in the bacterial cell wall essential for its viability. To engineer these proteins to exhibit improved therapeutically relevant properties, homology-guided statistical approaches can be used to identify compelling sites for mutation and to quantify the functional constraints acting on these sites to direct mutagenic library creation. The platform described herein couples this informed approach with a visual plate assay that can be used to simultaneously screen hundreds of mutants for catalytic activity, allowing for the streamlined identification of improved lysin variants.

Deciphering protein evolution and fitness landscapes with latent space models

Protein sequences contain rich information about protein evolution, fitness landscapes, and stability. Here we investigate how latent space models trained using variational auto-encoders can infer these properties from sequences. Using both simulated and real sequences, we show that the low dimensional latent space representation of sequences, calculated using the encoder model, captures both evolutionary and ancestral relationships between sequences. Together with experimental fitness data and Gaussian process regression, the latent space representation also enables learning the protein fitness landscape in a continuous low dimensional space. Moreover, the model is also useful in predicting protein mutational stability landscapes and quantifying the importance of stability in shaping protein evolution. Overall, we illustrate that the latent space models learned using variational auto-encoders provide a mechanism for exploration of the rich data contained in protein sequences regarding evolution, fitness and stability and hence are well-suited to help guide protein engineering efforts.

Toward Inferring Potts Models for Phylogenetically Correlated Sequence Data

Global coevolutionary models of protein families have become increasingly popular due to their capacity to predict residue–residue contacts from sequence information, but also to predict fitness effects of amino acid substitutions or to infer protein–protein interactions. The central idea in these models is to construct a probability distribution, a Potts model, that reproduces single and pairwise frequencies of amino acids found in natural sequences of the protein family. This approach treats sequences from the family as independent samples, completely ignoring phylogenetic relations between them. This simplification is known to lead to potentially biased estimates of the parameters of the model, decreasing their biological relevance. Current workarounds for this problem, such as reweighting sequences, are poorly understood and not principled. Here, we propose an inference scheme that takes the phylogeny of a protein family into account in order to correct biases in estimating the frequencies of amino acids. Using artificial data, we show that a Potts model inferred using these corrected frequencies performs better in predicting contacts and fitness effect of mutations. First, only partially successful tests on real protein data are presented, too.

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.

Effect of Correlated Pair Mutations in Protein Misfolding

A Monte Carlo simulation based sequence design method is proposed to explore the effect of correlated pair mutations in proteins. In the designed sequences, the most correlated residue pairs are identified and mutated with all possible amino acid pairs except those already present. The cumulative correlated pair mutations generated an array of mutated sequences. Results show a significant increase in the probability of misfolding for correlated pair mutations as compared to that of the random pair mutations. The pair mutations of correlated residues that are in contact record a higher probability of misfolding as compared to the correlated residues that are not in contact. The probability of misfolding increases on pair mutation of nonlocally correlated residue pairs as compared to that of the locally correlated residue pairs. The choice of a compact or expanded conformation does not depend on the type of correlated pair mutations. Pair mutation of the most correlated residue pairs at the surface with hydrophobic amino acids results in higher misfolding probability as compared to that in the core. An exactly opposite behavior is observed on pair mutation with hydrophilic and charged amino acid pairs. The neutral amino acid pairs do not differentiate between core and surface sites. This study may be used for targeted mutation experiments to predict complex mutation patterns, reengineer the existing proteins, and design new proteins with reduced misfolding propensity.

Massively Parallel Assays and Quantitative Sequence-Function Relationships

Over the last decade, a rich variety of massively parallel assays have revolutionized our understanding of how biological sequences encode quantitative molecular phenotypes. These assays include deep mutational scanning, high-throughput SELEX, and massively parallel reporter assays. Here, we review these experimental methods and how the data they produce can be used to quantitatively model sequence-function relationships. In doing so, we touch on a diverse range of topics, including the identification of clinically relevant genomic variants, the modeling of transcription factor binding to DNA, the functional and evolutionary landscapes of proteins, and cis-regulatory mechanisms in both transcription and mRNA splicing. We further describe a unified conceptual framework and a core set of mathematical modeling strategies that studies in these diverse areas can make use of. Finally, we highlight key aspects of experimental design and mathematical modeling that are important for the results of such studies to be interpretable and reproducible.

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

Potts statistical models have become a popular and promising way to analyze mutational covariation in protein multiple sequence alignments (MSAs) in order to understand protein structure, function, and fitness. But the statistical limitations of these models, which can have millions of parameters and are fit to MSAs of only thousands or hundreds of effective sequences using a procedure known as inverse Ising inference, are incompletely understood. In this work we predict how model quality degrades as a function of the number of sequences N, sequence length L, amino-acid alphabet size q, and the degree of conservation of the MSA, in different applications of the Potts models: in "fitness" predictions of individual protein sequences, in predictions of the effects of single-point mutations, in "double mutant cycle" predictions of epistasis, and in 3D contact prediction in protein structure. We show how as MSA depth N decreases an "overfitting" effect occurs such that sequences in the training MSA have overestimated fitness, and we predict the magnitude of this effect and discuss how regularization can help correct for it, using a regularization procedure motivated by statistical analysis of the effects of finite sampling. We find that as N decreases the quality of point-mutation effect predictions degrade least, fitness and epistasis predictions degrade more rapidly, and contact predictions are most affected. However, overfitting becomes negligible for MSA depths of more than a few thousand effective sequences, as often used in practice, and regularization becomes less necessary. We discuss the implications of these results for users of Potts covariation analysis.

How Pairwise Coevolutionary Models Capture the Collective Residue Variability in Proteins?

Global coevolutionary models of homologous protein families, as constructed by direct coupling analysis (DCA), have recently gained popularity in particular due to their capacity to accurately predict residue-residue contacts from sequence information alone, and thereby to facilitate tertiary and quaternary protein structure prediction. More recently, they have also been used to predict fitness effects of amino-acid substitutions in proteins, and to predict evolutionary conserved protein-protein interactions. These models are based on two currently unjustified hypotheses: 1) correlations in the amino-acid usage of different positions are resulting collectively from networks of direct couplings; and 2) pairwise couplings are sufficient to capture the amino-acid variability. Here, we propose a highly precise inference scheme based on Boltzmann-machine learning, which allows us to systematically address these hypotheses. We show how correlations are built up in a highly collective way by a large number of coupling paths, which are based on the proteins three-dimensional structure. We further find that pairwise coevolutionary models capture the collective residue variability across homologous proteins even for quantities which are not imposed by the inference procedure, like three-residue correlations, the clustered structure of protein families in sequence space or the sequence distances between homologs. These findings strongly suggest that pairwise coevolutionary models are actually sufficient to accurately capture the residue variability in homologous protein families.

Inference of epistatic effects in a key mitochondrial protein

We use Potts model inference to predict pair epistatic effects in a key mitochondrial protein-cytochrome c oxidase subunit 2-for ray-finned fishes. We examine the effect of phylogenetic correlations on our predictions using a simple exact fitness model, and we find that, although epistatic effects are underpredicted, they maintain a roughly linear relationship to their true (model) values. After accounting for this correction, epistatic effects in the protein are still relatively weak, leading to fitness valleys of depth 2Ns≃-5 in compensatory double mutants. Interestingly, positive epistasis is more pronounced than negative epistasis, and the strongest positive effects capture nearly all sites subject to positive selection in fishes, similar to virus proteins evolving under selection pressure in the context of drug therapy.

Contact Potential for Structure Prediction of Proteins and Protein Complexes from Potts Model

The energy function is the key component of protein modeling methodology. This work presents a semianalytical approach to the development of contact potentials for protein structure modeling. Residue-residue and atom-atom contact energies were derived by maximizing the probability of observing native sequences in a nonredundant set of protein structures. The optimization task was formulated as an inverse statistical mechanics problem applied to the Potts model. Its solution by pseudolikelihood maximization provides consistent estimates of coupling constants at atomic and residue levels. The best performance was achieved when interacting atoms were grouped according to their physicochemical properties. For individual protein structures, the performance of the contact potentials in distinguishing near-native structures from the decoys is similar to the top-performing scoring functions. The potentials also yielded significant improvement in the protein docking success rates. The potentials recapitulated experimentally determined protein stability changes upon point mutations and protein-protein binding affinities. The approach offers a different perspective on knowledge-based potentials and may serve as the basis for their further development.