Learning the shape of protein microenvironments with a holographic convolutional neural network.

Proteins play a central role in biology from immune recognition to brain activity. While major advances in machine learning have improved our ability to predict protein structure from sequence, determining protein function from its sequence or structure remains a major challenge. Here, we introduce holographic convolutional neural network (H-CNN) for proteins, which is a physically motivated machine learning approach to model amino acid preferences in protein structures. H-CNN reflects physical interactions in a protein structure and recapitulates the functional information stored in evolutionary data. H-CNN accurately predicts the impact of mutations on protein stability and binding of protein complexes. Our interpretable computational model for protein structure-function maps could guide design of novel proteins with desired function.

Inferring the shape of global epistasis.

Genotype-phenotype relationships are notoriously complicated. Idiosyncratic interactions between specific combinations of mutations occur and are difficult to predict. Yet it is increasingly clear that many interactions can be understood in terms of global epistasis. That is, mutations may act additively on some underlying, unobserved trait, and this trait is then transformed via a nonlinear function to the observed phenotype as a result of subsequent biophysical and cellular processes. Here we infer the shape of such global epistasis in three proteins, based on published high-throughput mutagenesis data. To do so, we develop a maximum-likelihood inference procedure using a flexible family of monotonic nonlinear functions spanned by an I-spline basis. Our analysis uncovers dramatic nonlinearities in all three proteins; in some proteins a model with global epistasis accounts for virtually all of the measured variation, whereas in others we find substantial local epistasis as well. This method allows us to test hypotheses about the form of global epistasis and to distinguish variance components attributable to global epistasis, local epistasis, and measurement error.

Fierce Selection and Interference in B-Cell Repertoire Response to Chronic HIV-1.

During chronic infection, HIV-1 engages in a rapid coevolutionary arms race with the host's adaptive immune system. While it is clear that HIV exerts strong selection on the adaptive immune system, the characteristics of the somatic evolution that shape the immune response are still unknown. Traditional population genetics methods fail to distinguish chronic immune response from healthy repertoire evolution. Here, we infer the evolutionary modes of B-cell repertoires and identify complex dynamics with a constant production of better B-cell receptor (BCR) mutants that compete, maintaining large clonal diversity and potentially slowing down adaptation. A substantial fraction of mutations that rise to high frequencies in pathogen-engaging CDRs of BCRs are beneficial, in contrast to many such changes in structurally relevant frameworks that are deleterious and circulate by hitchhiking. We identify a pattern where BCRs in patients who experience larger viral expansions undergo stronger selection with a rapid turnover of beneficial mutations due to clonal interference in their CDR3 regions. Using population genetics modeling, we show that the extinction of these beneficial mutations can be attributed to the rise of competing beneficial alleles and clonal interference. The picture is of a dynamic repertoire, where better clones may be outcompeted by new mutants before they fix.

Information-Geometric Optimization with Natural Selection.

Evolutionary algorithms, inspired by natural evolution, aim to optimize difficult objective functions without computing derivatives. Here we detail the relationship between classical population genetics of quantitative traits and evolutionary optimization, and formulate a new evolutionary algorithm. Optimization of a continuous objective function is analogous to searching for high fitness phenotypes on a fitness landscape. We describe how natural selection moves a population along the non-Euclidean gradient that is induced by the population on the fitness landscape (the natural gradient). We show how selection is related to Newton's method in optimization under quadratic fitness landscapes, and how selection increases fitness at the cost of reducing diversity. We describe the generation of new phenotypes and introduce an operator that recombines the whole population to generate variants. Finally, we introduce a proof-of-principle algorithm that combines natural selection, our recombination operator, and an adaptive method to increase selection and find the optimum. The algorithm is extremely simple in implementation; it has no matrix inversion or factorization, does not require storing a covariance matrix, and may form the basis of more general model-based optimization algorithms with natural gradient updates.

Biophysical Inference of Epistasis and the Effects of Mutations on Protein Stability and Function.

Understanding the relationship between protein sequence, function, and stability is a fundamental problem in biology. The essential function of many proteins that fold into a specific structure is their ability to bind to a ligand, which can be assayed for thousands of mutated variants. However, binding assays do not distinguish whether mutations affect the stability of the binding interface or the overall fold. Here, we introduce a statistical method to infer a detailed energy landscape of how a protein folds and binds to a ligand by combining information from many mutated variants. We fit a thermodynamic model describing the bound, unbound, and unfolded states to high quality data of protein G domain B1 binding to IgG-Fc. We infer distinct folding and binding energies for each mutation providing a detailed view of how mutations affect binding and stability across the protein. We accurately infer the folding energy of each variant in physical units, validated by independent data, whereas previous high-throughput methods could only measure indirect changes in stability. While we assume an additive sequence-energy relationship, the binding fraction is epistatic due its nonlinear relation to energy. Despite having no epistasis in energy, our model explains much of the observed epistasis in binding fraction, with the remaining epistasis identifying conformationally dynamic regions.

Host-Pathogen Coevolution and the Emergence of Broadly Neutralizing Antibodies in Chronic Infections.

The vertebrate adaptive immune system provides a flexible and diverse set of molecules to neutralize pathogens. Yet, viruses such as HIV can cause chronic infections by evolving as quickly as the adaptive immune system, forming an evolutionary arms race. Here we introduce a mathematical framework to study the coevolutionary dynamics between antibodies and antigens within a host. We focus on changes in the binding interactions between the antibody and antigen populations, which result from the underlying stochastic evolution of genotype frequencies driven by mutation, selection, and drift. We identify the critical viral and immune parameters that determine the distribution of antibody-antigen binding affinities. We also identify definitive signatures of coevolution that measure the reciprocal response between antibodies and viruses, and we introduce experimentally measurable quantities that quantify the extent of adaptation during continual coevolution of the two opposing populations. Using this analytical framework, we infer rates of viral and immune adaptation based on time-shifted neutralization assays in two HIV-infected patients. Finally, we analyze competition between clonal lineages of antibodies and characterize the fate of a given lineage in terms of the state of the antibody and viral populations. In particular, we derive the conditions that favor the emergence of broadly neutralizing antibodies, which may have relevance to vaccine design against HIV.

Inferring fitness landscapes by regression produces biased estimates of epistasis.

The genotype-fitness map plays a fundamental role in shaping the dynamics of evolution. However, it is difficult to directly measure a fitness landscape in practice, because the number of possible genotypes is astronomical. One approach is to sample as many genotypes as possible, measure their fitnesses, and fit a statistical model of the landscape that includes additive and pairwise interactive effects between loci. Here, we elucidate the pitfalls of using such regressions by studying artificial but mathematically convenient fitness landscapes. We identify two sources of bias inherent in these regression procedures, each of which tends to underestimate high fitnesses and overestimate low fitnesses. We characterize these biases for random sampling of genotypes as well as samples drawn from a population under selection in the Wright-Fisher model of evolutionary dynamics. We show that common measures of epistasis, such as the number of monotonically increasing paths between ancestral and derived genotypes, the prevalence of sign epistasis, and the number of local fitness maxima, are distorted in the inferred landscape. As a result, the inferred landscape will provide systematically biased predictions for the dynamics of adaptation. We identify the same biases in a computational RNA-folding landscape as well as regulatory sequence binding data treated with the same fitting procedure. Finally, we present a method to ameliorate these biases in some cases.

Clonal interference and Muller's ratchet in spatial habitats.

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller's ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio [Formula: see text] between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using, again, an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.

Design of an optimal combination therapy with broadly neutralizing antibodies to suppress HIV-1.

Infusion of broadly neutralizing antibodies (bNAbs) has shown promise as an alternative to anti-retroviral therapy against HIV. A key challenge is to suppress viral escape, which is more effectively achieved with a combination of bNAbs. Here, we propose a computational approach to predict the efficacy of a bNAb therapy based on the population genetics of HIV escape, which we parametrize using high-throughput HIV sequence data from bNAb-naive patients. By quantifying the mutational target size and the fitness cost of HIV-1 escape from bNAbs, we predict the distribution of rebound times in three clinical trials. We show that a cocktail of three bNAbs is necessary to effectively suppress viral escape, and predict the optimal composition of such bNAb cocktail. Our results offer a rational therapy design for HIV, and show how genetic data can be used to predict treatment outcomes and design new approaches to pathogenic control.

Dynamics of B cell repertoires and emergence of cross-reactive responses in patients with different severities of COVID-19.

Individuals with the 2019 coronavirus disease (COVID-19) show varying severity of the disease, ranging from asymptomatic to requiring intensive care. Although monoclonal antibodies specific to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have been identified, we still lack an understanding of the overall landscape of B cell receptor (BCR) repertoires in individuals with COVID-19. We use high-throughput sequencing of bulk and plasma B cells collected at multiple time points during infection to characterize signatures of the B cell response to SARS-CoV-2 in 19 individuals. Using principled statistical approaches, we associate differential features of BCRs with different disease severity. We identify 38 significantly expanded clonal lineages shared among individuals as candidates for responses specific to SARS-CoV-2. Using single-cell sequencing, we verify the reactivity of BCRs shared among individuals to SARS-CoV-2 epitopes. Moreover, we identify the natural emergence of a BCR with cross-reactivity to SARS-CoV-1 and SARS-CoV-2 in some individuals. Our results provide insights important for development of rational therapies and vaccines against COVID-19.

Dynamics of B-cell repertoires and emergence of cross-reactive responses in COVID-19 patients with different disease severity.

COVID-19 patients show varying severity of the disease ranging from asymptomatic to requiring intensive care. Although a number of SARS-CoV-2 specific monoclonal antibodies have been identified, we still lack an understanding of the overall landscape of B-cell receptor (BCR) repertoires in COVID-19 patients. Here, we used high-throughput sequencing of bulk and plasma B-cells collected over multiple time points during infection to characterize signatures of B-cell response to SARS-CoV-2 in 19 patients. Using principled statistical approaches, we determined differential features of BCRs associated with different disease severity. We identified 38 significantly expanded clonal lineages shared among patients as candidates for specific responses to SARS-CoV-2. Using single-cell sequencing, we verified reactivity of BCRs shared among individuals to SARS-CoV-2 epitopes. Moreover, we identified natural emergence of a BCR with cross-reactivity to SARS-CoV-1 and SARS-CoV-2 in a number of patients. Our results provide important insights for development of rational therapies and vaccines against COVID-19.

Dynamics of B-cell repertoires and emergence of cross-reactive responses in COVID-19 patients with different disease severity.

COVID-19 patients show varying severity of the disease ranging from asymptomatic to requiring intensive care. Although a number of SARS-CoV-2 specific monoclonal antibodies have been identified, we still lack an understanding of the overall landscape of B-cell receptor (BCR) repertoires in COVID-19 patients. Here, we used high-throughput sequencing of bulk and plasma B-cells collected over multiple time points during infection to characterize signatures of B-cell response to SARS-CoV-2 in 19 patients. Using principled statistical approaches, we determined differential features of BCRs associated with different disease severity. We identified 38 significantly expanded clonal lineages shared among patients as candidates for specific responses to SARS-CoV-2. Using single-cell sequencing, we verified reactivity of BCRs shared among individuals to SARS-CoV-2 epitopes. Moreover, we identified natural emergence of a BCR with cross-reactivity to SARS-CoV-1 and SARS-CoV-2 in a number of patients. Our results provide important insights for development of rational therapies and vaccines against COVID-19.

Major antigenic site B of human influenza H3N2 viruses has an evolving local fitness landscape.

Antigenic drift of influenza virus hemagglutinin (HA) is enabled by facile evolvability. However, HA antigenic site B, which has become immunodominant in recent human H3N2 influenza viruses, is also evolutionarily constrained by its involvement in receptor binding. Here, we employ deep mutational scanning to probe the local fitness landscape of HA antigenic site B in six different human H3N2 strains spanning from 1968 to 2016. We observe that the fitness landscape of HA antigenic site B can be very different between strains. Sequence variants that exhibit high fitness in one strain can be deleterious in another, indicating that the evolutionary constraints of antigenic site B have changed over time. Structural analysis suggests that the local fitness landscape of antigenic site B can be reshaped by natural mutations via modulation of the receptor-binding mode. Overall, these findings elucidate how influenza virus continues to explore new antigenic space despite strong functional constraints.

Direct observation of colloidal aggregation by critical Casimir forces.

We present a refractive-index-matched colloidal system that allows direct observation of critical Casimir induced aggregation with a confocal microscope. We show that in this system, in which van der Waals forces are negligible, a simple competition between repulsive screened Coulomb and attractive critical Casimir forces can account quantitatively for the reversible aggregation. Above the temperature T(a), the critical Casimir force drives aggregation of the particles into fractal clusters, while below T(a), the electrostatic repulsion between the particles breaks up the clusters, and the particles resuspend by thermal diffusion. The aggregation is observed in a remarkably wide temperature range of as much as 15 degrees. We derive a simple expression for the particle pair potential that accounts quantitatively for the temperature-dependent aggregation and aggregate breakup.

Detecting epistasis from an ensemble of adapting populations.

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite-site fitness landscapes. First, we analyze the mean fitness trajectory-that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non-epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics-the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions-can detect certain forms of epistasis in the underlying fitness landscape.

Genotype to phenotype mapping and the fitness landscape of the E. coli lac promoter.

Genotype-to-phenotype maps and the related fitness landscapes that include epistatic interactions are difficult to measure because of their high dimensional structure. Here we construct such a map using the recently collected corpora of high-throughput sequence data from the 75 base pairs long mutagenized E. coli lac promoter region, where each sequence is associated with its phenotype, the induced transcriptional activity measured by a fluorescent reporter. We find that the additive (non-epistatic) contributions of individual mutations account for about two-thirds of the explainable phenotype variance, while pairwise epistasis explains about 7% of the variance for the full mutagenized sequence and about 15% for the subsequence associated with protein binding sites. Surprisingly, there is no evidence for third order epistatic contributions, and our inferred fitness landscape is essentially single peaked, with a small amount of antagonistic epistasis. There is a significant selective pressure on the wild type, which we deduce to be multi-objective optimal for gene expression in environments with different nutrient sources. We identify transcription factor (CRP) and RNA polymerase binding sites in the promotor region and their interactions without difficult optimization steps. In particular, we observe evidence for previously unexplored genetic regulatory mechanisms, possibly kinetic in nature. We conclude with a cautionary note that inferred properties of fitness landscapes may be severely influenced by biases in the sequence data.

Accumulation of beneficial mutations in one dimension.

When beneficial mutations are relatively common, competition between multiple unfixed mutations can reduce the rate of fixation in well-mixed asexual populations. We introduce a one-dimensional model with a steady accumulation of beneficial mutations. We find a transition between periodic selection and multiple-mutation regimes. In the multiple-mutation regime, the increase of fitness along the lattice bears a striking similarity to surface growth phenomena, with power-law growth and saturation of the interface width. We also find significant differences compared to the well-mixed model. In our lattice model, the transition between regimes happens at a much lower mutation rate due to slower fixation times in one dimension. Also, the rate of fixation is reduced with increasing mutation rate due to the more intense competition, and it saturates with large population size.

Speeding up Evolutionary Search by Small Fitness Fluctuations.

We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models.