Jakub Otwinowski

Inferring fitness landscapes by regression produces biased estimates of epistasis

Significance The dynamics of evolution depend on an organism’s fitness landscape: the mapping from genotypes to reproductive capacity. Knowledge of the fitness landscape can help resolve questions, such as how quickly a pathogen will acquire drug resistance or by what pattern of mutations. However, direct measurement of a fitness landscape is impossible because of the vast number of genotypes. Here, we critically examine regression techniques used to approximate fitness landscapes from data. We find that such regressions are subject to two inherent biases that distort the biological quantities of greatest interest, often making evolution seem less predictable than it actually is. We discuss methods that may mitigate these biases in some cases. 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.

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.

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.

A totally asymmetric exclusion process with hierarchical long range connections

A non-equilibrium particle transport model, the totally asymmetric exclusion process, is studied on a one-dimensional lattice with a hierarchy of fixed long range connections. This model breaks the particle?hole symmetry observed on an ordinary one-dimensional lattice and results in a surprisingly simple phase diagram, without a maximum current phase. Numerical simulations of the model with open boundary conditions reveal a number of dynamic features and suggest possible applications.

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.

Clonal interference and Mullerʼs 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 Mullers 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.

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.

Clonal competition in B-cell repertoires during chronic HIV-1 infection

Abstract 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 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 B-cell receptors (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.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 modes of immune response are still unknown. Traditional population genetics methods fail to distinguish a chronic immune response from natural repertoire evolution in healthy individuals. Here, we infer the evolutionary modes of B-cell repertoire response and identify a complex dynamics where, instead of one winning clone, there is a constant production of new better mutants that compete with each other. A substantial fraction of mutations in pathogen-engaging CDRs of B-cell receptors are beneficial, in contrast to the many deleterious changes in structurally relevant framework regions. The picture is of a dynamic repertoire, where better clones may be outcompeted by new mutants before they fix, challenging current vaccine design and therapy ideas.

Inferring the shape of global epistasis

Significance How does an organism’s genetic sequence govern its measurable characteristics? New technologies provide libraries of randomized sequences to study this relationship in unprecedented detail for proteins and other molecules. Deriving insight from these data is difficult, though, because the space of possible sequences is enormous, so even the largest experiments sample a tiny minority of sequences. Moreover, the effects of mutations may combine in unexpected ways. We present a statistical framework to analyze such mutagenesis data. The key assumption is that mutations contribute in a simple way to some unobserved trait, which is related to the observed trait by a nonlinear mapping. Analyzing three proteins, we show that this model is easily interpretable and yet fits the data remarkably well. 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.