Adaptive cluster expansion for inferring boltzmann machines with noisy data

Functional effects of mutations in proteins can be predicted and interpreted by guided selection of sequence covariation information

Predicting the effects of one or more mutations to the in vivo or in vitro properties of a wild-type protein is a major computational challenge, due to the presence of epistasis, that is, of interactions between amino acids in the sequence. We introduce a computationally efficient procedure to build minimal epistatic models to predict mutational effects by combining evolutionary (homologous sequence) and few mutational-scan data. Mutagenesis measurements guide the selection of links in a sparse graphical model, while the parameters on the nodes and the edges are inferred from sequence data. We show, on 10 mutational scans, that our pipeline exhibits performances comparable to state-of-the-art deep networks trained on many more data, while requiring much less parameters and being hence more interpretable. In particular, the identified interactions adapt to the wild-type protein and to the fitness or biochemical property experimentally measured, mostly focus on key functional sites, and are not necessarily related to structural contacts. Therefore, our method is able to extract information relevant for one mutational experiment from homologous sequence data reflecting the multitude of structural and functional constraints acting on proteins throughout evolution.

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.

Small-correlation expansion to quantify information in noisy sensory systems

Neural networks encode information through their collective spiking activity in response to external stimuli. This population response is noisy and strongly correlated, with a complex interplay between correlations induced by the stimulus, and correlations caused by shared noise. Understanding how these correlations affect information transmission has so far been limited to pairs or small groups of neurons, because the curse of dimensionality impedes the evaluation of mutual information in larger populations. Here, we develop a small-correlation expansion to compute the stimulus information carried by a large population of neurons, yielding interpretable analytical expressions in terms of the neurons' firing rates and pairwise correlations. We validate the approximation on synthetic data and demonstrate its applicability to electrophysiological recordings in the vertebrate retina, allowing us to quantify the effects of noise correlations between neurons and of memory in single neurons.

Thermodynamics of the Ising Model Encoded in Restricted Boltzmann Machines

The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.

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.

Inferring the intrinsic mutational fitness landscape of influenzalike evolving antigens from temporally ordered sequence data

There still are no effective long-term protective vaccines against viruses that continuously evolve under immune pressure such as seasonal influenza, which has caused, and can cause, devastating epidemics in the human population. To find such a broadly protective immunization strategy, it is useful to know how easily the virus can escape via mutation from specific antibody responses. This information is encoded in the fitness landscape of the viral proteins (i.e., knowledge of the viral fitness as a function of sequence). Here we present a computational method to infer the intrinsic mutational fitness landscape of influenzalike evolving antigens from yearly sequence data. We test inference performance with computer-generated sequence data that are based on stochastic simulations mimicking basic features of immune-driven viral evolution. Although the numerically simulated model does create a phylogeny based on the allowed mutations, the inference scheme does not use this information. This provides a contrast to other methods that rely on reconstruction of phylogenetic trees. Our method just needs a sufficient number of samples over multiple years. With our method, we are able to infer single as well as pairwise mutational fitness effects from the simulated sequence time series for short antigenic proteins. Our fitness inference approach may have potential future use for the design of immunization protocols by identifying intrinsically vulnerable immune target combinations on antigens that evolve under immune-driven selection. In the future, this approach may be applied to influenza and other novel viruses such as SARS-CoV-2, which evolves and, like influenza, might continue to escape the natural and vaccine-mediated immune pressures.

Inverse problems for structured datasets using parallel TAP equations and restricted Boltzmann machines

We propose an efficient algorithm to solve inverse problems in the presence of binary clustered datasets. We consider the paradigmatic Hopfield model in a teacher student scenario, where this situation is found in the retrieval phase. This problem has been widely analyzed through various methods such as mean-field approaches or the pseudo-likelihood optimization. Our approach is based on the estimation of the posterior using the Thouless-Anderson-Palmer (TAP) equations in a parallel updating scheme. Unlike other methods, it allows to retrieve the original patterns of the teacher dataset and thanks to the parallel update it can be applied to large system sizes. We tackle the same problem using a restricted Boltzmann machine (RBM) and discuss analogies and differences between our algorithm and RBM learning.

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.

Direct coupling analysis of epistasis in allosteric materials

In allosteric proteins, the binding of a ligand modifies function at a distant active site. Such allosteric pathways can be used as target for drug design, generating considerable interest in inferring them from sequence alignment data. Currently, different methods lead to conflicting results, in particular on the existence of long-range evolutionary couplings between distant amino-acids mediating allostery. Here we propose a resolution of this conundrum, by studying epistasis and its inference in models where an allosteric material is evolved in silico to perform a mechanical task. We find in our model the four types of epistasis (Synergistic, Sign, Antagonistic, Saturation), which can be both short or long-range and have a simple mechanical interpretation. We perform a Direct Coupling Analysis (DCA) and find that DCA predicts well the cost of point mutations but is a rather poor generative model. Strikingly, it can predict short-range epistasis but fails to capture long-range epistasis, in consistence with empirical findings. We propose that such failure is generic when function requires subparts to work in concert. We illustrate this idea with a simple model, which suggests that other methods may be better suited to capture long-range effects.

Allostery in proteins is the property of highly specific responses to ligand binding at a distant site. To inform protocols of de novo drug design, it is fundamental to understand the impact of mutations on allosteric regulation and whether it can be predicted from evolutionary correlations. In this work we consider allosteric architectures artificially evolved to optimize the cooperativity of binding at allosteric and active site. We first characterize the emergent pattern of epistasis as well as the underlying mechanical phenomena, finding the four types of epistasis (Synergistic, Sign, Antagonistic, Saturation), which can be both short or long-range. The numerical evolution of these allosteric architectures allows us to benchmark Direct Coupling Analysis, a method which relies on co-evolution in sequence data to infer direct evolutionary couplings, in connection to allostery. We show that Direct Coupling Analysis predicts quantitatively point mutation costs but underestimates strong long-range epistasis. We provide an argument, based on a simplified model, illustrating the reasons for this discrepancy. Our analysis suggests neural networks as more promising tool to measure epistasis.

Erasure machine: Inverse Ising inference from reweighting of observation frequencies

Maximum likelihood estimation (MLE) is fundamental to system inference for stochastic systems. In some generality, MLE will converge to the correct model in the infinite data limit. In the context of physical approaches to system inference, such as Boltzmann machines, MLE requires the arduous computation of partition functions summing over all configurations, both observed and unobserved. We present a conceptually transparent data-driven inference computation based on a reweighting of observed configuration frequencies that allows us to recast the inference problem as a simpler calculation. Modeling our approach on the high-temperature limit of statistical physics, we reweight the frequencies of observed configurations by multiplying with reciprocals of Boltzmann weights and update the Boltzmann weights iteratively to make these products close to the high-temperature limit of the Boltzmann weights. This converts the required partition function computation in the reweighted MLE to a tractable leading-order high-temperature term. We show that this is a convex optimization at each step. Then, for systems with a large number of degrees of freedom where other approaches are intractable, we demonstrate that this data-driven algorithm gives accurate inference with both synthetic data and two real-world examples.

Inference of compressed Potts graphical models

We consider the problem of inferring a graphical Potts model on a population of variables. This inverse Potts problem generally involves the inference of a large number of parameters, often larger than the number of available data, and, hence, requires the introduction of regularization. We study here a double regularization scheme, in which the number of Potts states (colors) available to each variable is reduced and interaction networks are made sparse. To achieve the color compression, only Potts states with large empirical frequency (exceeding some threshold) are explicitly modeled on each site, while the others are grouped into a single state. We benchmark the performances of this mixed regularization approach, with two inference algorithms, adaptive cluster expansion (ACE) and pseudolikelihood maximization (PLM), on synthetic data obtained by sampling disordered Potts models on Erdős-Rényi random graphs. We show in particular that color compression does not affect the quality of reconstruction of the parameters corresponding to high-frequency symbols, while drastically reducing the number of the other parameters and thus the computational time. Our procedure is also applied to multisequence alignments of protein families, with similar results.

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.

Modelling and in vitro testing of the HIV-1 Nef fitness landscape

An effective vaccine is urgently required to curb the HIV-1 epidemic. We have previously described an approach to model the fitness landscape of several HIV-1 proteins, and have validated the results against experimental and clinical data. The fitness landscape may be used to identify mutation patterns harmful to virus viability, and consequently inform the design of immunogens that can target such regions for immunological control. Here we apply such an analysis and complementary experiments to HIV-1 Nef, a multifunctional protein which plays a key role in HIV-1 pathogenesis. We measured Nef-driven replication capacities as well as Nef-mediated CD4 and HLA-I down-modulation capacities of thirty-two different Nef mutants, and tested model predictions against these results. Furthermore, we evaluated the models using 448 patient-derived Nef sequences for which several Nef activities were previously measured. Model predictions correlated significantly with Nef-driven replication and CD4 down-modulation capacities, but not HLA-I down-modulation capacities, of the various Nef mutants. Similarly, in our analysis of patient-derived Nef sequences, CD4 down-modulation capacity correlated the most significantly with model predictions, suggesting that of the tested Nef functions, this is the most important in vivo. Overall, our results highlight how the fitness landscape inferred from patient-derived sequences captures, at least in part, the in vivo functional effects of mutations to Nef. However, the correlation between predictions of the fitness landscape and measured parameters of Nef function is not as accurate as the correlation observed in past studies for other proteins. This may be because of the additional complexity associated with inferring the cost of mutations on the diverse functions of Nef.

Coevolutionary Landscape of Kinase Family Proteins: Sequence Probabilities and Functional Motifs

The protein kinase catalytic domain is one of the most abundant domains across all branches of life. Although kinases share a common core function of phosphoryl-transfer, they also have wide functional diversity and play varied roles in cell signaling networks, and for this reason are implicated in a number of human diseases. This functional diversity is primarily achieved through sequence variation, and uncovering the sequence-function relationships for the kinase family is a major challenge. In this study we use a statistical inference technique inspired by statistical physics, which builds a coevolutionary "Potts" Hamiltonian model of sequence variation in a protein family. We show how this model has sufficient power to predict the probability of specific subsequences in the highly diverged kinase family, which we verify by comparing the model's predictions with experimental observations in the Uniprot database. We show that the pairwise (residue-residue) interaction terms of the statistical model are necessary and sufficient to capture higher-than-pairwise mutation patterns of natural kinase sequences. We observe that previously identified functional sets of residues have much stronger correlated interaction scores than are typical.

Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties

The pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80-95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations.

Ising distribution as a latent variable model

During the past decades, the Ising distribution has attracted interest in many applied disciplines, as the maximum entropy distribution associated to any set of correlated binary ("spin") variables with observed means and covariances. However, numerically speaking, the Ising distribution is unpractical, so alternative models are often preferred to handle correlated binary data. One popular alternative, especially in life sciences, is the Cox distribution (or the closely related dichotomized Gaussian distribution and log-normal Cox point process), where the spins are generated independently conditioned on the drawing of a latent variable with a multivariate normal distribution. This article explores the conditions for a principled replacement of the Ising distribution by a Cox distribution. It shows that the Ising distribution itself can be treated as a latent variable model, and it explores when this latent variable has a quasi-normal distribution. A variational approach to this question reveals a formal link with classic mean-field methods, especially Opper and Winther's adaptive TAP approximation. This link is confirmed by weak coupling (Plefka) expansions of the different approximations and then by numerical tests. Overall, this study suggests that an Ising distribution can be replaced by a Cox distribution in practical applications, precisely when its parameters lie in the "mean-field domain."

The intrinsic dimension of protein sequence evolution

It is well known that, in order to preserve its structure and function, a protein cannot change its sequence at random, but only by mutations occurring preferentially at specific locations. We here investigate quantitatively the amount of variability that is allowed in protein sequence evolution, by computing the intrinsic dimension (ID) of the sequences belonging to a selection of protein families. The ID is a measure of the number of independent directions that evolution can take starting from a given sequence. We find that the ID is practically constant for sequences belonging to the same family, and moreover it is very similar in different families, with values ranging between 6 and 12. These values are significantly smaller than the raw number of amino acids, confirming the importance of correlations between mutations in different sites. However, we demonstrate that correlations are not sufficient to explain the small value of the ID we observe in protein families. Indeed, we show that the ID of a set of protein sequences generated by maximum entropy models, an approach in which correlations are accounted for, is typically significantly larger than the value observed in natural protein families. We further prove that a critical factor to reproduce the natural ID is to take into consideration the phylogeny of sequences.

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.

Integration and multiplexing of positional and contextual information by the hippocampal network

The hippocampus is known to store cognitive representations, or maps, that encode both positional and contextual information, critical for episodic memories and functional behavior. How path integration and contextual cues are dynamically combined and processed by the hippocampus to maintain these representations accurate over time remains unclear. To answer this question, we propose a two-way data analysis and modeling approach to CA3 multi-electrode recordings of a moving rat submitted to rapid changes of contextual (light) cues, triggering back-and-forth instabitilies between two cognitive representations (“teleportation” experiment of Jezek et al). We develop a dual neural activity decoder, capable of independently identifying the recalled cognitive map at high temporal resolution (comparable to theta cycle) and the position of the rodent given a map. Remarkably, position can be reconstructed at any time with an accuracy comparable to fixed-context periods, even during highly unstable periods. These findings provide evidence for the capability of the hippocampal neural activity to maintain an accurate encoding of spatial and contextual variables, while one of these variables undergoes rapid changes independently of the other. To explain this result we introduce an attractor neural network model for the hippocampal activity that process inputs from external cues and the path integrator. Our model allows us to make predictions on the frequency of the cognitive map instability, its duration, and the detailed nature of the place-cell population activity, which are validated by a further analysis of the data. Our work therefore sheds light on the mechanisms by which the hippocampal network achieves and updates multi-dimensional neural representations from various input streams.

As an animal moves in space and receives external sensory inputs, it must dynamically maintain the representations of its position and environment at all times. How the hippocampus, the brain area crucial for spatial representations, achieves this task, and manages possible conflicts between different inputs remains unclear. We propose here a comprehensive attractor neural network-based model of the hippocampus and of its multiple input streams (including self-motion). We show that this model is capable of maintaining faithful representations of positional and contextual information, and resolves conflicts by adapting internal representations to match external cues. Model predictions are confirmed by the detailed analysis of hippocampal recordings of a rat submitted to quickly varying and conflicting contextual inputs.

Optimal structure and parameter learning of Ising models

Reconstruction of the structure and parameters of an Ising model from binary samples is a problem of practical importance in a variety of disciplines, ranging from statistical physics and computational biology to image processing and machine learning. The focus of the research community shifted toward developing universal reconstruction algorithms that are both computationally efficient and require the minimal amount of expensive data. We introduce a new method, interaction screening, which accurately estimates model parameters using local optimization problems. The algorithm provably achieves perfect graph structure recovery with an information-theoretically optimal number of samples, notably in the low-temperature regime, which is known to be the hardest for learning. The efficacy of interaction screening is assessed through extensive numerical tests on synthetic Ising models of various topologies with different types of interactions, as well as on real data produced by a D-Wave quantum computer. This study shows that the interaction screening method is an exact, tractable, and optimal technique that universally solves the inverse Ising problem.

Inverse statistical physics of protein sequences: a key issues review

In the course of evolution, proteins undergo important changes in their amino acid sequences, while their three-dimensional folded structure and their biological function remain remarkably conserved. Thanks to modern sequencing techniques, sequence data accumulate at unprecedented pace. This provides large sets of so-called homologous, i.e. evolutionarily related protein sequences, to which methods of inverse statistical physics can be applied. Using sequence data as the basis for the inference of Boltzmann distributions from samples of microscopic configurations or observables, it is possible to extract information about evolutionary constraints and thus protein function and structure. Here we give an overview over some biologically important questions, and how statistical-mechanics inspired modeling approaches can help to answer them. Finally, we discuss some open questions, which we expect to be addressed over the next years.

Sparse dynamical Boltzmann machine for reconstructing complex networks with binary dynamics

Revealing the structure and dynamics of complex networked systems from observed data is a problem of current interest. Is it possible to develop a completely data-driven framework to decipher the network structure and different types of dynamical processes on complex networks? We develop a model named sparse dynamical Boltzmann machine (SDBM) as a structural estimator for complex networks that host binary dynamical processes. The SDBM attains its topology according to that of the original system and is capable of simulating the original binary dynamical process. We develop a fully automated method based on compressive sensing and a clustering algorithm to construct the SDBM. We demonstrate, for a variety of representative dynamical processes on model and real world complex networks, that the equivalent SDBM can recover the network structure of the original system and simulates its dynamical behavior with high precision.

Fitness landscape of the human immunodeficiency virus envelope protein that is targeted by antibodies

HIV is a highly mutable virus, and over 30 years after its discovery, a vaccine or cure is still not available. The isolation of broadly neutralizing antibodies (bnAbs) from HIV-infected patients has led to renewed hope for a prophylactic vaccine capable of combating the scourge of HIV. A major challenge is the design of immunogens and vaccination protocols that can elicit bnAbs that target regions of the virus's spike proteins where the likelihood of mutational escape is low due to the high fitness cost of mutations. Related challenges include the choice of combinations of bnAbs for therapy. An accurate representation of viral fitness as a function of its protein sequences (a fitness landscape), with explicit accounting of the effects of coupling between mutations, could help address these challenges. We describe a computational approach that has allowed us to infer a fitness landscape for gp160, the HIV polyprotein that comprises the viral spike that is targeted by antibodies. We validate the inferred landscape through comparisons with experimental fitness measurements, and various other metrics. We show that an effective antibody that prevents immune escape must selectively bind to high escape cost residues that are surrounded by those where mutations incur a low fitness cost, motivating future applications of our landscape for immunogen design.

Biomolecular coevolution and its applications: Going from structure prediction toward signaling, epistasis, and function

Evolution leads to considerable changes in the sequence of biomolecules, while their overall structure and function remain quite conserved. The wealth of genomic sequences, the ‘Biological Big Data’, modern sequencing techniques provide allows us to investigate biomolecular evolution with unprecedented detail. Sophisticated statistical models can infer residue pair mutations resulting from spatial proximity. The introduction of predicted spatial adjacencies as constraints in biomolecular structure prediction workflows has transformed the field of protein and RNA structure prediction toward accuracies approaching the experimental resolution limit. Going beyond structure prediction, the same mathematical framework allows mimicking evolutionary fitness landscapes to infer signaling interactions, epistasis, or mutational landscapes.

Inverse problem for multispecies ferromagneticlike mean-field models in phase space with many states

In this paper we solve the inverse problem for the Curie-Weiss model and its multispecies version when multiple thermodynamic states are present as in the low temperature phase where the phase space is clustered. The inverse problem consists of reconstructing the model parameters starting from configuration data generated according to the distribution of the model. We demonstrate that, without taking into account the presence of many states, the application of the inversion procedure produces very poor inference results. To overcome this problem, we use the clustering algorithm. When the system has two symmetric states of positive and negative magnetizations, the parameter reconstruction can also be obtained with smaller computational effort simply by flipping the sign of the magnetizations from positive to negative (or vice versa). The parameter reconstruction fails when the system undergoes a phase transition: In that case we give the correct inversion formulas for the Curie-Weiss model and we show that they can be used to measure how close the system gets to being critical.

Functional coupling networks inferred from prefrontal cortex activity show experience-related effective plasticity

Functional coupling networks are widely used to characterize collective patterns of activity in neural populations. Here, we ask whether functional couplings reflect the subtle changes, such as in physiological interactions, believed to take place during learning. We infer functional network models reproducing the spiking activity of simultaneously recorded neurons in prefrontal cortex (PFC) of rats, during the performance of a cross-modal rule shift task (task epoch), and during preceding and following sleep epochs. A large-scale study of the 96 recorded sessions allows us to detect, in about 20% of sessions, effective plasticity between the sleep epochs. These coupling modifications are correlated with the coupling values in the task epoch, and are supported by a small subset of the recorded neurons, which we identify by means of an automatized procedure. These potentiated groups increase their coativation frequency in the spiking data between the two sleep epochs, and, hence, participate to putative experience-related cell assemblies. Study of the reactivation dynamics of the potentiated groups suggests a possible connection with behavioral learning. Reactivation is largely driven by hippocampal ripple events when the rule is not yet learned, and may be much more autonomous, and presumably sustained by the potentiated PFC network, when learning is consolidated.

Cell assemblies coding for memories are widely believed to emerge through synaptic modification resulting from learning, yet their identification from activity is very arduous. We propose a functional-connectivity-based approach to identify experience-related cell assemblies from multielectrode recordings in vivo, and apply it to the prefrontal cortex activity of rats recorded during a task epoch and the preceding and following sleep epochs. We infer functional couplings between the recorded cells in each epoch. Comparisons of the functional coupling networks across the epochs allow us to identify effective potentiation between the two sleep epochs. The neurons supporting these potentiated interactions strongly coactivate during the task and subsequent sleep epochs, but not in the preceding sleep, and, hence, presumably belong to an experience-related cell assembly. Study of the reactivation of this assembly in response to hippocampal ripple inputs suggests possible relations between the stage of behavorial learning and memory consolidation mechanisms.

Inferring structural connectivity using Ising couplings in models of neuronal networks

Functional connectivity metrics have been widely used to infer the underlying structural connectivity in neuronal networks. Maximum entropy based Ising models have been suggested to discount the effect of indirect interactions and give good results in inferring the true anatomical connections. However, no benchmarking is currently available to assess the performance of Ising couplings against other functional connectivity metrics in the microscopic scale of neuronal networks through a wide set of network conditions and network structures. In this paper, we study the performance of the Ising model couplings to infer the synaptic connectivity in in silico networks of neurons and compare its performance against partial and cross-correlations for different correlation levels, firing rates, network sizes, network densities, and topologies. Our results show that the relative performance amongst the three functional connectivity metrics depends primarily on the network correlation levels. Ising couplings detected the most structural links at very weak network correlation levels, and partial correlations outperformed Ising couplings and cross-correlations at strong correlation levels. The result was consistent across varying firing rates, network sizes, and topologies. The findings of this paper serve as a guide in choosing the right functional connectivity tool to reconstruct the structural connectivity.

Inference of Epistatic Effects Leading to Entrenchment and Drug Resistance in HIV-1 Protease

Understanding the complex mutation patterns that give rise to drug resistant viral strains provides a foundation for developing more effective treatment strategies for HIV/AIDS. Multiple sequence alignments of drug-experienced HIV-1 protease sequences contain networks of many pair correlations which can be used to build a (Potts) Hamiltonian model of these mutation patterns. Using this Hamiltonian model, we translate HIV-1 protease sequence covariation data into quantitative predictions for the probability of observing specific mutation patterns which are in agreement with the observed sequence statistics. We find that the statistical energies of the Potts model are correlated with the fitness of individual proteins containing therapy-associated mutations as estimated by in vitro measurements of protein stability and viral infectivity. We show that the penalty for acquiring primary resistance mutations depends on the epistatic interactions with the sequence background. Primary mutations which lead to drug resistance can become highly advantageous (or entrenched) by the complex mutation patterns which arise in response to drug therapy despite being destabilizing in the wildtype background. Anticipating epistatic effects is important for the design of future protease inhibitor therapies.

Functional connectivity models for decoding of spatial representations from hippocampal CA1 recordings

Hippocampus stores spatial representations, or maps, which are recalled each time a subject is placed in the corresponding environment. Across different environments of similar geometry, these representations show strong orthogonality in CA3 of hippocampus, whereas in the CA1 subfield a considerable overlap between the maps can be seen. The lower orthogonality decreases reliability of various decoders developed in an attempt to identify which of the stored maps is active at the moment. Especially, the problem with decoding emerges with a need to analyze data at high temporal resolution. Here, we introduce a functional-connectivity-based decoder, which accounts for the pairwise correlations between the spiking activities of neurons in each map and does not require any positional information, i.e. any knowledge about place fields. We first show, on recordings of hippocampal activity in constant environmental conditions, that our decoder outperforms existing decoding methods in CA1. Our decoder is then applied to data from teleportation experiments, in which an instantaneous switch between the environment identity triggers a recall of the corresponding spatial representation . We test the sensitivity of our approach on the transition dynamics between the respective memory states (maps). We find that the rate of spontaneous state shifts (flickering) after a teleportation event is increased not only within the first few seconds as already reported, but this instability is sustained across much longer (> 1 min.) periods.

Rational design of vaccine targets and strategies for HIV: a crossroad of statistical physics, biology, and medicine

Vaccination has saved more lives than any other medical procedure. Pathogens have now evolved that have not succumbed to vaccination using the empirical paradigms pioneered by Pasteur and Jenner. Vaccine design strategies that are based on a mechanistic understanding of the pertinent immunology and virology are required to confront and eliminate these scourges. In this perspective, we describe just a few examples of work aimed to achieve this goal by bringing together approaches from statistical physics with biology and clinical research.

A Perspective on the Role of Computational Models in Immunology

This is an exciting time for immunology because the future promises to be replete with exciting new discoveries that can be translated to improve health and treat disease in novel ways. Immunologists are attempting to answer increasingly complex questions concerning phenomena that range from the genetic, molecular, and cellular scales to that of organs, whole animals or humans, and populations of humans and pathogens. An important goal is to understand how the many different components involved interact with each other within and across these scales for immune responses to emerge, and how aberrant regulation of these processes causes disease. To aid this quest, large amounts of data can be collected using high-throughput instrumentation. The nonlinear, cooperative, and stochastic character of the interactions between components of the immune system as well as the overwhelming amounts of data can make it difficult to intuit patterns in the data or a mechanistic understanding of the phenomena being studied. Computational models are increasingly important in confronting and overcoming these challenges. I first describe an iterative paradigm of research that integrates laboratory experiments, clinical data, computational inference, and mechanistic computational models. I then illustrate this paradigm with a few examples from the recent literature that make vivid the power of bringing together diverse types of computational models with experimental and clinical studies to fruitfully interrogate the immune system.

Improving landscape inference by integrating heterogeneous data in the inverse Ising problem

The inverse Ising problem and its generalizations to Potts and continuous spin models have recently attracted much attention thanks to their successful applications in the statistical modeling of biological data. In the standard setting, the parameters of an Ising model (couplings and fields) are inferred using a sample of equilibrium configurations drawn from the Boltzmann distribution. However, in the context of biological applications, quantitative information for a limited number of microscopic spins configurations has recently become available. In this paper, we extend the usual setting of the inverse Ising model by developing an integrative approach combining the equilibrium sample with (possibly noisy) measurements of the energy performed for a number of arbitrary configurations. Using simulated data, we show that our integrative approach outperforms standard inference based only on the equilibrium sample or the energy measurements, including error correction of noisy energy measurements. As a biological proof-of-concept application, we show that mutational fitness landscapes in proteins can be better described when combining evolutionary sequence data with complementary structural information about mutant sequences.

Resummed mean-field inference for strongly coupled data

We present a resummed mean-field approximation for inferring the parameters of an Ising or a Potts model from empirical, noisy, one- and two-point correlation functions. Based on a resummation of a class of diagrams of the small correlation expansion of the log-likelihood, the method outperforms standard mean-field inference methods, even when they are regularized. The inference is stable with respect to sampling noise, contrarily to previous works based either on the small correlation expansion, on the Bethe free energy, or on the mean-field and Gaussian models. Because it is mostly analytic, its complexity is still very low, requiring an iterative algorithm to solve for N auxiliary variables, that resorts only to matrix inversions and multiplications. We test our algorithm on the Sherrington-Kirkpatrick model submitted to a random external field and large random couplings, and demonstrate that even without regularization, the inference is stable across the whole phase diagram. In addition, the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution.

Network inference in the nonequilibrium steady state

Nonequilibrium systems lack an explicit characterization of their steady state like the Boltzmann distribution for equilibrium systems. This has drastic consequences for the inference of the parameters of a model when its dynamics lacks detailed balance. Such nonequilibrium systems occur naturally in applications like neural networks and gene regulatory networks. Here, we focus on the paradigmatic asymmetric Ising model and show that we can learn its parameters from independent samples of the nonequilibrium steady state. We present both an exact inference algorithm and a computationally more efficient, approximate algorithm for weak interactions based on a systematic expansion around mean-field theory. Obtaining expressions for magnetizations and two- and three-point spin correlations, we establish that these observables are sufficient to infer the model parameters. Further, we discuss the symmetries characterizing the different orders of the expansion around the mean field and show how different types of dynamics can be distinguished on the basis of samples from the nonequilibrium steady state.

A Tractable Method for Describing Complex Couplings between Neurons and Population Rate

Neurons within a population are strongly correlated, but how to simply capture these correlations is still a matter of debate. Recent studies have shown that the activity of each cell is influenced by the population rate, defined as the summed activity of all neurons in the population. However, an explicit, tractable model for these interactions is still lacking. Here we build a probabilistic model of population activity that reproduces the firing rate of each cell, the distribution of the population rate, and the linear coupling between them. This model is tractable, meaning that its parameters can be learned in a few seconds on a standard computer even for large population recordings. We inferred our model for a population of 160 neurons in the salamander retina. In this population, single-cell firing rates depended in unexpected ways on the population rate. In particular, some cells had a preferred population rate at which they were most likely to fire. These complex dependencies could not be explained by a linear coupling between the cell and the population rate. We designed a more general, still tractable model that could fully account for these nonlinear dependencies. We thus provide a simple and computationally tractable way to learn models that reproduce the dependence of each neuron on the population rate.

Learning maximum entropy models from finite-size data sets: A fast data-driven algorithm allows sampling from the posterior distribution

Maximum entropy models provide the least constrained probability distributions that reproduce statistical properties of experimental datasets. In this work we characterize the learning dynamics that maximizes the log-likelihood in the case of large but finite datasets. We first show how the steepest descent dynamics is not optimal as it is slowed down by the inhomogeneous curvature of the model parameters' space. We then provide a way for rectifying this space which relies only on dataset properties and does not require large computational efforts. We conclude by solving the long-time limit of the parameters' dynamics including the randomness generated by the systematic use of Gibbs sampling. In this stochastic framework, rather than converging to a fixed point, the dynamics reaches a stationary distribution, which for the rectified dynamics reproduces the posterior distribution of the parameters. We sum up all these insights in a "rectified" data-driven algorithm that is fast and by sampling from the parameters' posterior avoids both under- and overfitting along all the directions of the parameters' space. Through the learning of pairwise Ising models from the recording of a large population of retina neurons, we show how our algorithm outperforms the steepest descent method.

Neural assemblies revealed by inferred connectivity-based models of prefrontal cortex recordings

We present two graphical model-based approaches to analyse the distribution of neural activities in the prefrontal cortex of behaving rats. The first method aims at identifying cell assemblies, groups of synchronously activating neurons possibly representing the units of neural coding and memory. A graphical (Ising) model distribution of snapshots of the neural activities, with an effective connectivity matrix reproducing the correlation statistics, is inferred from multi-electrode recordings, and then simulated in the presence of a virtual external drive, favoring high activity (multi-neuron) configurations. As the drive increases groups of neurons may activate together, and reveal the existence of cell assemblies. The identified groups are then showed to strongly coactivate in the neural spiking data and to be highly specific of the inferred connectivity network, which offers a sparse representation of the correlation pattern across neural cells. The second method relies on the inference of a Generalized Linear Model, in which spiking events are integrated over time by neurons through an effective connectivity matrix. The functional connectivity matrices inferred with the two approaches are compared. Sampling of the inferred GLM distribution allows us to study the spatio-temporal patterns of activation of neurons within the identified cell assemblies, particularly their activation order: the prevalence of one order with respect to the others is weak and reflects the neuron average firing rates and the strength of the largest effective connections. Other properties of the identified cell assemblies (spatial distribution of coactivation events and firing rates of coactivating neurons) are discussed.

Structural propensities of kinase family proteins from a Potts model of residue co-variation

Understanding the conformational propensities of proteins is key to solving many problems in structural biology and biophysics. The co-variation of pairs of mutations contained in multiple sequence alignments of protein families can be used to build a Potts Hamiltonian model of the sequence patterns which accurately predicts structural contacts. This observation paves the way to develop deeper connections between evolutionary fitness landscapes of entire protein families and the corresponding free energy landscapes which determine the conformational propensities of individual proteins. Using statistical energies determined from the Potts model and an alignment of 2896 PDB structures, we predict the propensity for particular kinase family proteins to assume a "DFG-out" conformation implicated in the susceptibility of some kinases to type-II inhibitors, and validate the predictions by comparison with the observed structural propensities of the corresponding proteins and experimental binding affinity data. We decompose the statistical energies to investigate which interactions contribute the most to the conformational preference for particular sequences and the corresponding proteins. We find that interactions involving the activation loop and the C-helix and HRD motif are primarily responsible for stabilizing the DFG-in state. This work illustrates how structural free energy landscapes and fitness landscapes of proteins can be used in an integrated way, and in the context of kinase family proteins, can potentially impact therapeutic design strategies.

ACE: adaptive cluster expansion for maximum entropy graphical model inference

MOTIVATION

Graphical models are often employed to interpret patterns of correlations observed in data through a network of interactions between the variables. Recently, Ising/Potts models, also known as Markov random fields, have been productively applied to diverse problems in biology, including the prediction of structural contacts from protein sequence data and the description of neural activity patterns. However, inference of such models is a challenging computational problem that cannot be solved exactly. Here, we describe the adaptive cluster expansion (ACE) method to quickly and accurately infer Ising or Potts models based on correlation data. ACE avoids overfitting by constructing a sparse network of interactions sufficient to reproduce the observed correlation data within the statistical error expected due to finite sampling. When convergence of the ACE algorithm is slow, we combine it with a Boltzmann Machine Learning algorithm (BML). We illustrate this method on a variety of biological and artificial datasets and compare it to state-of-the-art approximate methods such as Gaussian and pseudo-likelihood inference.

RESULTS

We show that ACE accurately reproduces the true parameters of the underlying model when they are known, and yields accurate statistical descriptions of both biological and artificial data. Models inferred by ACE more accurately describe the statistics of the data, including both the constrained low-order correlations and unconstrained higher-order correlations, compared to those obtained by faster Gaussian and pseudo-likelihood methods. These alternative approaches can recover the structure of the interaction network but typically not the correct strength of interactions, resulting in less accurate generative models.

AVAILABILITY AND IMPLEMENTATION

The ACE source code, user manual and tutorials with the example data and filtered correlations described herein are freely available on GitHub at https://github.com/johnbarton/ACE CONTACTS: jpbarton@mit.edu, cocco@lps.ens.frSupplementary information: Supplementary data are available at Bioinformatics online.

Relative rate and location of intra-host HIV evolution to evade cellular immunity are predictable

Human immunodeficiency virus (HIV) evolves within infected persons to escape being destroyed by the host immune system, thereby preventing effective immune control of infection. Here, we combine methods from evolutionary dynamics and statistical physics to simulate in vivo HIV sequence evolution, predicting the relative rate of escape and the location of escape mutations in response to T-cell-mediated immune pressure in a cohort of 17 persons with acute HIV infection. Predicted and clinically observed times to escape immune responses agree well, and we show that the mutational pathways to escape depend on the viral sequence background due to epistatic interactions. The ability to predict escape pathways and the duration over which control is maintained by specific immune responses open the door to rational design of immunotherapeutic strategies that might enable long-term control of HIV infection. Our approach enables intra-host evolution of a human pathogen to be predicted in a probabilistic framework.

Benchmarking Inverse Statistical Approaches for Protein Structure and Design with Exactly Solvable Models

Inverse statistical approaches to determine protein structure and function from Multiple Sequence Alignments (MSA) are emerging as powerful tools in computational biology. However the underlying assumptions of the relationship between the inferred effective Potts Hamiltonian and real protein structure and energetics remain untested so far. Here we use lattice protein model (LP) to benchmark those inverse statistical approaches. We build MSA of highly stable sequences in target LP structures, and infer the effective pairwise Potts Hamiltonians from those MSA. We find that inferred Potts Hamiltonians reproduce many important aspects of ‘true’ LP structures and energetics. Careful analysis reveals that effective pairwise couplings in inferred Potts Hamiltonians depend not only on the energetics of the native structure but also on competing folds; in particular, the coupling values reflect both positive design (stabilization of native conformation) and negative design (destabilization of competing folds). In addition to providing detailed structural information, the inferred Potts models used as protein Hamiltonian for design of new sequences are able to generate with high probability completely new sequences with the desired folds, which is not possible using independent-site models. Those are remarkable results as the effective LP Hamiltonians used to generate MSA are not simple pairwise models due to the competition between the folds. Our findings elucidate the reasons for the success of inverse approaches to the modelling of proteins from sequence data, and their limitations.

Inverse statistical approaches, modeling pairwise correlations between amino acids in the sequences of homologous proteins across many different organisms, can successfully extract protein structure (contact) information. Here, we benchmark those statistical approaches on exactly solvable models of proteins, folding on a 3D lattice, to assess the reasons underlying their success and their limitations. We show that the inferred parameters (effective pairwise interactions) of the statistical models have clear and quantitative interpretations in terms of positive (favoring the native fold) and negative (disfavoring competing folds) protein sequence design. New sequences randomly drawn from the statistical models are likely to fold into the native structures when effective pairwise interactions are accurately inferred, a performance which cannot be achieved with independent-site models.

Identification of drug resistance mutations in HIV from constraints on natural evolution

Human immunodeficiency virus (HIV) evolves with extraordinary rapidity. However, its evolution is constrained by interactions between mutations in its fitness landscape. Here we show that an Ising model describing these interactions, inferred from sequence data obtained prior to the use of antiretroviral drugs, can be used to identify clinically significant sites of resistance mutations. Successful predictions of the resistance sites indicate progress in the development of successful models of real viral evolution at the single residue level and suggest that our approach may be applied to help design new therapies that are less prone to failure even where resistance data are not yet available.

Reconstruction of a Real World Social Network using the Potts Model and Loopy Belief Propagation

The scope of this paper is to test the adoption of a statistical model derived from Condensed Matter Physics, for the reconstruction of the structure of a social network. The inverse Potts model, traditionally applied to recursive observations of quantum states in an ensemble of particles, is here addressed to observations of the members' states in an organization and their (anti)correlations, thus inferring interactions as links among the members. Adopting proper (Bethe) approximations, such an inverse problem is showed to be tractable. Within an operational framework, this network-reconstruction method is tested for a small real-world social network, the Italian parliament. In this study case, it is easy to track statuses of the parliament members, using (co)sponsorships of law proposals as the initial dataset. In previous studies of similar activity-based networks, the graph structure was inferred directly from activity co-occurrences: here we compare our statistical reconstruction with such standard methods, outlining discrepancies and advantages.

Inference of the sparse kinetic Ising model using the decimation method

In this paper we study the inference of the kinetic Ising model on sparse graphs by the decimation method. The decimation method, which was first proposed in Decelle and Ricci-Tersenghi [Phys. Rev. Lett. 112, 070603 (2014)] for the static inverse Ising problem, tries to recover the topology of the inferred system by setting the weakest couplings to zero iteratively. During the decimation process the likelihood function is maximized over the remaining couplings. Unlike the ℓ(1)-optimization-based methods, the decimation method does not use the Laplace distribution as a heuristic choice of prior to select a sparse solution. In our case, the whole process can be done auto-matically without fixing any parameters by hand. We show that in the dynamical inference problem, where the task is to reconstruct the couplings of an Ising model given the data, the decimation process can be applied naturally into a maximum-likelihood optimization algorithm, as opposed to the static case where pseudolikelihood method needs to be adopted. We also use extensive numerical studies to validate the accuracy of our methods in dynamical inference problems. Our results illustrate that, on various topologies and with different distribution of couplings, the decimation method outperforms the widely used ℓ(1)-optimization-based methods.

Scaling laws describe memories of host-pathogen riposte in the HIV population

The enormous genetic diversity and mutability of HIV has prevented effective control of this virus by natural immune responses or vaccination. Evolution of the circulating HIV population has thus occurred in response to diverse, ultimately ineffective, immune selection pressures that randomly change from host to host. We show that the interplay between the diversity of human immune responses and the ways that HIV mutates to evade them results in distinct sets of sequences defined by similar collectively coupled mutations. Scaling laws that relate these sets of sequences resemble those observed in linguistics and other branches of inquiry, and dynamics reminiscent of neural networks are observed. Like neural networks that store memories of past stimulation, the circulating HIV population stores memories of host-pathogen combat won by the virus. We describe an exactly solvable model that captures the main qualitative features of the sets of sequences and a simple mechanistic model for the origin of the observed scaling laws. Our results define collective mutational pathways used by HIV to evade human immune responses, which could guide vaccine design.

Inverse spin glass and related maximum entropy problems

If we have a system of binary variables and we measure the pairwise correlations among these variables, then the least structured or maximum entropy model for their joint distribution is an Ising model with pairwise interactions among the spins. Here we consider inhomogeneous systems in which we constrain, for example, not the full matrix of correlations, but only the distribution from which these correlations are drawn. In this sense, what we have constructed is an inverse spin glass: rather than choosing coupling constants at random from a distribution and calculating correlations, we choose the correlations from a distribution and infer the coupling constants. We argue that such models generate a block structure in the space of couplings, which provides an explicit solution of the inverse problem. This allows us to generate a phase diagram in the space of (measurable) moments of the distribution of correlations. We expect that these ideas will be most useful in building models for systems that are nonequilibrium statistical mechanics problems, such as networks of real neurons.