How Good Are Statistical Models at Approximating Complex Fitness Landscapes?

Genotype sampling for deep-learning assisted experimental mapping of a combinatorially complete fitness landscape

MOTIVATION

Experimental characterization of fitness landscapes, which map genotypes onto fitness, is important for both evolutionary biology and protein engineering. It faces a fundamental obstacle in the astronomical number of genotypes whose fitness needs to be measured for any one protein. Deep learning may help to predict the fitness of many genotypes from a smaller neural network training sample of genotypes with experimentally measured fitness. Here I use a recently published experimentally mapped fitness landscape of more than 260 000 protein genotypes to ask how such sampling is best performed.

RESULTS

I show that multilayer perceptrons, recurrent neural networks, convolutional networks, and transformers, can explain more than 90% of fitness variance in the data. In addition, 90% of this performance is reached with a training sample comprising merely ≈103 sequences. Generalization to unseen test data is best when training data is sampled randomly and uniformly, or sampled to minimize the number of synonymous sequences. In contrast, sampling to maximize sequence diversity or codon usage bias reduces performance substantially. These observations hold for more than one network architecture. Simple sampling strategies may perform best when training deep learning neural networks to map fitness landscapes from experimental data.

AVAILABILITY AND IMPLEMENTATION

The fitness landscape data analyzed here is publicly available as described previously (Papkou et al. 2023). All code used to analyze this landscape is publicly available at https://github.com/andreas-wagner-uzh/fitness_landscape_sampling.

Rapid adaptation of recombining populations on tunable fitness landscapes

How does standing genetic variation affect polygenic adaptation in recombining populations? Despite a large body of work in quantitative genetics, epistatic and weak additive fitness effects among simultaneously segregating genetic variants are difficult to capture experimentally or to predict theoretically. In this study, we simulated adaptation on fitness landscapes with tunable ruggedness driven by standing genetic variation in recombining populations. We confirmed that recombination hinders the movement of a population through a rugged fitness landscape. When surveying the effect of epistasis on the fixation of alleles, we found that the combined effects of high ruggedness and high recombination probabilities lead to preferential fixation of alleles that had a high initial frequency. This indicates that positive epistatic alleles escape from being broken down by recombination when they start at high frequency. We further extract direct selection coefficients and pairwise epistasis along the adaptive path. When taking the final fixed genotype as the reference genetic background, we observe that, along the adaptive path, beneficial direct selection appears stronger and pairwise epistasis weaker than in the underlying fitness landscape. Quantitatively, the ratio of epistasis and direct selection is smaller along the adaptive path (≈ 1 ) than expected. Thus, adaptation on a rugged fitness landscape may lead to spurious signals of direct selection generated through epistasis. Our study highlights how the interplay of epistasis and recombination constrains the adaptation of a diverse population to a new environment.

Reinforcement learning informs optimal treatment strategies to limit antibiotic resistance

Antimicrobial resistance was estimated to be associated with 4.95 million deaths worldwide in 2019. It is possible to frame the antimicrobial resistance problem as a feedback-control problem. If we could optimize this feedback-control problem and translate our findings to the clinic, we could slow, prevent, or reverse the development of high-level drug resistance. Prior work on this topic has relied on systems where the exact dynamics and parameters were known a priori. In this study, we extend this work using a reinforcement learning (RL) approach capable of learning effective drug cycling policies in a system defined by empirically measured fitness landscapes. Crucially, we show that it is possible to learn effective drug cycling policies despite the problems of noisy, limited, or delayed measurement. Given access to a panel of 15 [Formula: see text]-lactam antibiotics with which to treat the simulated Escherichia coli population, we demonstrate that RL agents outperform two naive treatment paradigms at minimizing the population fitness over time. We also show that RL agents approach the performance of the optimal drug cycling policy. Even when stochastic noise is introduced to the measurements of population fitness, we show that RL agents are capable of maintaining evolving populations at lower growth rates compared to controls. We further tested our approach in arbitrary fitness landscapes of up to 1,024 genotypes. We show that minimization of population fitness using drug cycles is not limited by increasing genome size. Our work represents a proof-of-concept for using AI to control complex evolutionary processes.

The community-function landscape of microbial consortia

Quantitatively linking the composition and function of microbial communities is a major aspiration of microbial ecology. Microbial community functions emerge from a complex web of molecular interactions between cells, which give rise to population-level interactions among strains and species. Incorporating this complexity into predictive models is highly challenging. Inspired by a similar problem in genetics of predicting quantitative phenotypes from genotypes, an ecological community-function (or structure-function) landscape could be defined that maps community composition and function. In this piece, we present an overview of our current understanding of these community landscapes, their uses, limitations, and open questions. We argue that exploiting the parallels between both landscapes could bring powerful predictive methodologies from evolution and genetics into ecology, providing a boost to our ability to engineer and optimize microbial consortia.

On the sparsity of fitness functions and implications for learning

The properties of proteins and other biological molecules are encoded in large part in the sequence of amino acids or nucleotides that defines them. Increasingly, researchers estimate functions that map sequences to a particular property using machine learning and related statistical approaches. However, an important question remains unanswered: How many experimental measurements are needed in order to accurately learn these “fitness” functions? We leverage perspectives from the fields of biophysics, evolutionary biology, and signal processing to develop a theoretical framework that enables us to make progress on answering this question. We demonstrate that this framework can be used to make useful calculations on real-world data and suggest how these calculations may be used to guide experiments.

Fitness functions map biological sequences to a scalar property of interest. Accurate estimation of these functions yields biological insight and sets the foundation for model-based sequence design. However, the fitness datasets available to learn these functions are typically small relative to the large combinatorial space of sequences; characterizing how much data are needed for accurate estimation remains an open problem. There is a growing body of evidence demonstrating that empirical fitness functions display substantial sparsity when represented in terms of epistatic interactions. Moreover, the theory of Compressed Sensing provides scaling laws for the number of samples required to exactly recover a sparse function. Motivated by these results, we develop a framework to study the sparsity of fitness functions sampled from a generalization of the NK model, a widely used random field model of fitness functions. In particular, we present results that allow us to test the effect of the Generalized NK (GNK) model’s interpretable parameters—sequence length, alphabet size, and assumed interactions between sequence positions—on the sparsity of fitness functions sampled from the model and, consequently, the number of measurements required to exactly recover these functions. We validate our framework by demonstrating that GNK models with parameters set according to structural considerations can be used to accurately approximate the number of samples required to recover two empirical protein fitness functions and an RNA fitness function. In addition, we show that these GNK models identify important higher-order epistatic interactions in the empirical fitness functions using only structural information.

Lanza V, Rodríguez-Beltrán J, Galán JC, San Millán A, Cantón R, Coque TM. Evolutionary Pathways and Trajectories in Antibiotic Resistance

Evolution is the hallmark of life. Descriptions of the evolution of microorganisms have provided a wealth of information, but knowledge regarding "what happened" has precluded a deeper understanding of "how" evolution has proceeded, as in the case of antimicrobial resistance. The difficulty in answering the "how" question lies in the multihierarchical dimensions of evolutionary processes, nested in complex networks, encompassing all units of selection, from genes to communities and ecosystems. At the simplest ontological level (as resistance genes), evolution proceeds by random (mutation and drift) and directional (natural selection) processes; however, sequential pathways of adaptive variation can occasionally be observed, and under fixed circumstances (particular fitness landscapes), evolution is predictable. At the highest level (such as that of plasmids, clones, species, microbiotas), the systems' degrees of freedom increase dramatically, related to the variable dispersal, fragmentation, relatedness, or coalescence of bacterial populations, depending on heterogeneous and changing niches and selective gradients in complex environments. Evolutionary trajectories of antibiotic resistance find their way in these changing landscapes subjected to random variations, becoming highly entropic and therefore unpredictable. However, experimental, phylogenetic, and ecogenetic analyses reveal preferential frequented paths (highways) where antibiotic resistance flows and propagates, allowing some understanding of evolutionary dynamics, modeling and designing interventions. Studies on antibiotic resistance have an applied aspect in improving individual health, One Health, and Global Health, as well as an academic value for understanding evolution. Most importantly, they have a heuristic significance as a model to reduce the negative influence of anthropogenic effects on the environment.

Towards an engineering theory of evolution

Biological technologies are fundamentally unlike any other because biology evolves. Bioengineering therefore requires novel design methodologies with evolution at their core. Knowledge about evolution is currently applied to the design of biosystems ad hoc. Unless we have an engineering theory of evolution, we will neither be able to meet evolution's potential as an engineering tool, nor understand or limit its unintended consequences for our biological designs. Here, we propose the evotype as a helpful concept for engineering the evolutionary potential of biosystems, or other self-adaptive technologies, potentially beyond the realm of biology.

From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics

Understanding how genotypes map onto phenotypes, fitness, and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution. We refer to this generally as the problem of the genotype-phenotype map. Though we are still far from achieving a complete picture of these relationships, our current understanding of simpler questions, such as the structure induced in the space of genotypes by sequences mapped to molecular structures, has revealed important facts that deeply affect the dynamical description of evolutionary processes. Empirical evidence supporting the fundamental relevance of features such as phenotypic bias is mounting as well, while the synthesis of conceptual and experimental progress leads to questioning current assumptions on the nature of evolutionary dynamics-cancer progression models or synthetic biology approaches being notable examples. This work delves with a critical and constructive attitude into our current knowledge of how genotypes map onto molecular phenotypes and organismal functions, and discusses theoretical and empirical avenues to broaden and improve this comprehension. As a final goal, this community should aim at deriving an updated picture of evolutionary processes soundly relying on the structural properties of genotype spaces, as revealed by modern techniques of molecular and functional analysis.

Statistical Design of Experiments for Synthetic Biology

The design and optimization of biological systems is an inherently complex undertaking that requires careful balancing of myriad synergistic and antagonistic variables. However, despite this complexity, much synthetic biology research is predicated on One Factor at A Time (OFAT) experimentation; the genetic and environmental variables affecting the activity of a system of interest are sequentially altered while all other variables are held constant. Beyond being time and resource intensive, OFAT experimentation crucially ignores the effect of interactions between factors. Given the ubiquity of interacting genetic and environmental factors in biology this failure to account for interaction effects in OFAT experimentation can result in the development of suboptimal systems. To address these limitations, an increasing number of studies have turned to Design of Experiments (DoE), a suite of methods that enable efficient, systematic exploration and exploitation of complex design spaces. This review provides an overview of DoE for synthetic biologists. Key concepts and commonly used experimental designs are introduced, and we discuss the advantages of DoE as compared to OFAT experimentation. We dissect the applicability of DoE in the context of synthetic biology and review studies which have successfully employed these methods, illustrating the potential of statistical experimental design to guide the design, characterization, and optimization of biological protocols, pathways, and processes.

Inferring a complete genotype-phenotype map from a small number of measured phenotypes

Understanding evolution requires detailed knowledge of genotype-phenotype maps; however, it can be a herculean task to measure every phenotype in a combinatorial map. We have developed a computational strategy to predict the missing phenotypes from an incomplete, combinatorial genotype-phenotype map. As a test case, we used an incomplete genotype-phenotype dataset previously generated for the malaria parasite’s ‘chloroquine resistance transporter’ (PfCRT). Wild-type PfCRT (PfCRT^3D7) lacks significant chloroquine (CQ) transport activity, but the introduction of the eight mutations present in the ‘Dd2’ isoform of PfCRT (PfCRT^Dd2) enables the protein to transport CQ away from its site of antimalarial action. This gain of a transport function imparts CQ resistance to the parasite. A combinatorial map between PfCRT^3D7 and PfCRT^Dd2 consists of 256 genotypes, of which only 52 have had their CQ transport activities measured through expression in the Xenopus laevis oocyte. We trained a statistical model with these 52 measurements to infer the CQ transport activity for the remaining 204 combinatorial genotypes between PfCRT^3D7 and PfCRT^Dd2. Our best-performing model incorporated a binary classifier, a nonlinear scale, and additive effects for each mutation. The addition of specific pairwise- and high-order-epistatic coefficients decreased the predictive power of the model. We evaluated our predictions by experimentally measuring the CQ transport activities of 24 additional PfCRT genotypes. The R² value between our predicted and newly-measured phenotypes was 0.90. We then used the model to probe the accessibility of evolutionary trajectories through the map. Approximately 1% of the possible trajectories between PfCRT^3D7 and PfCRT^Dd2 are accessible; however, none of the trajectories entailed eight successive increases in CQ transport activity. These results demonstrate that phenotypes can be inferred with known uncertainty from a partial genotype-phenotype dataset. We also validated our approach against a collection of previously published genotype-phenotype maps. The model therefore appears general and should be applicable to a large number of genotype-phenotype maps.

Biological macromolecules are built from chains of building blocks. The function of a macromolecule depends on the specific chemical properties of the building blocks that make it up. Macromolecules evolve through mutations that swap one building block for another. Understanding how biomolecules work and evolve therefore requires knowledge of the effects of mutations. The effects of mutations can be measured experimentally; however, because there are a vast number of possible combinations of mutations, it is often difficult to make enough measurements to understand biomolecular function and evolution. In this paper, we describe a simple method to predict the effects of mutations on biomolecules from a small number of measurements. This method works by appropriately averaging the effects of mutations seen in different contexts. We test the method by predicting the effects of mutations on a PfCRT—a macromolecule from the malarial parasite that confers drug resistance. We find that our method is fast and effective. Using a small number of measurements, we were able to gain insight into the evolutionary steps by which this macromolecule conferred drug resistance. To make this method accessible to other researchers, we have released it as an open-source software package: https://gpseer.readthedocs.io.

Deconvolving mutational patterns of poliovirus outbreaks reveals its intrinsic fitness landscape

Vaccination has essentially eradicated poliovirus. Yet, its mutation rate is higher than that of viruses like HIV, for which no effective vaccine exists. To investigate this, we infer a fitness model for the poliovirus viral protein 1 (vp1), which successfully predicts in vitro fitness measurements. This is achieved by first developing a probabilistic model for the prevalence of vp1 sequences that enables us to isolate and remove data that are subject to strong vaccine-derived biases. The intrinsic fitness constraints derived for vp1, a capsid protein subject to antibody responses, are compared with those of analogous HIV proteins. We find that vp1 evolution is subject to tighter constraints, limiting its ability to evade vaccine-induced immune responses. Our analysis also indicates that circulating poliovirus strains in unimmunized populations serve as a reservoir that can seed outbreaks in spatio-temporally localized sub-optimally immunized populations.

Poliovirus has a higher mutation rate than HIV, yet has been almost eradicated by vaccination while an effective vaccine against HIV does not exist. Here, the authors develop a fitness model for poliovirus viral protein 1 to show that it is subject to stringent evolutionary constraints that limit its ability to avoid vaccine-induced immune responses.

The causes of evolvability and their evolution

Evolvability is the ability of a biological system to produce phenotypic variation that is both heritable and adaptive. It has long been the subject of anecdotal observations and theoretical work. In recent years, however, the molecular causes of evolvability have been an increasing focus of experimental work. Here, we review recent experimental progress in areas as different as the evolution of drug resistance in cancer cells and the rewiring of transcriptional regulation circuits in vertebrates. This research reveals the importance of three major themes: multiple genetic and non-genetic mechanisms to generate phenotypic diversity, robustness in genetic systems, and adaptive landscape topography. We also discuss the mounting evidence that evolvability can evolve and the question of whether it evolves adaptively.

The Causes and Consequences of Genetic Interactions (Epistasis)

The same mutation can have different effects in different individuals. One important reason for this is that the outcome of a mutation can depend on the genetic context in which it occurs. This dependency is known as epistasis. In recent years, there has been a concerted effort to quantify the extent of pairwise and higher-order genetic interactions between mutations through deep mutagenesis of proteins and RNAs. This research has revealed two major components of epistasis: nonspecific genetic interactions caused by nonlinearities in genotype-to-phenotype maps, and specific interactions between particular mutations. Here, we provide an overview of our current understanding of the mechanisms causing epistasis at the molecular level, the consequences of genetic interactions for evolution and genetic prediction, and the applications of epistasis for understanding biology and determining macromolecular structures.

An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape

Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible.

An intuitive understanding of protein evolution dictates that, with the exception of adaptive substitutions, amino acid states should be freely exchangeable between the same gene from different species. However, the extent to which this assertion holds true has not been tested in a controlled experiment. Here, we show that whether an amino acid state can be exchanged between orthologues depends on other amino acid states in the same protein. Furthermore, we show that the mode of interaction of amino acid states is multidimensional. Assuming that amino acid replacements influence the protein in several independent ways substantially improves our ability to predict the effect of an amino acid state in a protein sequence that has not been observed in nature.

Sex: The power of randomization

In evolutionary biology, randomness has been perceived as a force that, in and of itself, is capable of inventing: mutation creates new genetic information at random across the genome which leads to phenotypic change, which is then subject to selection. However, in science in general and in computer science in particular, the widespread use of randomness takes a different form. Here, randomization allows for the breaking of pattern, as seen for example in its removal of biases (patterns) by random sampling or random assignment to conditions. Combined with various forms of evaluation, this breaking of pattern becomes an extraordinarily powerful tool, as also seen in many randomized algorithms in computer science. Here we show that this power of randomness is harnessed in nature by sex and recombination. In a finite population, and under the assumption of interactions between genetic variants, sex and recombination allow selection to test how well an allele will perform in a sample of combinations of interacting genetic partners drawn at random from all possible such combinations; consequently, even a small number of tests of genotypes such as takes place in a finite population favors alleles that will most likely perform well in a vast number of yet unrealized genetic combinations. This power of randomization is not manifest in asexual populations.

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.

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.

Key issues review: evolution on rugged adaptive landscapes

Adaptive landscapes represent a mapping between genotype and fitness. Rugged adaptive landscapes contain two or more adaptive peaks: allele combinations with higher fitness than any of their neighbors in the genetic space. How do populations evolve on such rugged landscapes? Evolutionary biologists have struggled with this question since it was first introduced in the 1930s by Sewall Wright. Discoveries in the fields of genetics and biochemistry inspired various mathematical models of adaptive landscapes. The development of landscape models led to numerous theoretical studies analyzing evolution on rugged landscapes under different biological conditions. The large body of theoretical work suggests that adaptive landscapes are major determinants of the progress and outcome of evolutionary processes. Recent technological advances in molecular biology and microbiology allow experimenters to measure adaptive values of large sets of allele combinations and construct empirical adaptive landscapes for the first time. Such empirical landscapes have already been generated in bacteria, yeast, viruses, and fungi, and are contributing to new insights about evolution on adaptive landscapes. In this Key Issues Review we will: (i) introduce the concept of adaptive landscapes; (ii) review the major theoretical studies of evolution on rugged landscapes; (iii) review some of the recently obtained empirical adaptive landscapes; (iv) discuss recent mathematical and statistical analyses motivated by empirical adaptive landscapes, as well as provide the reader with instructions and source code to implement simulations of evolution on adaptive landscapes; and (v) discuss possible future directions for this exciting field.

Within-Epitope Interactions Can Bias CTL Escape Estimation in Early HIV Infection

As human immunodeficiency virus (HIV) begins to replicate within hosts, immune responses are elicited against it. Escape mutations in viral epitopes—immunogenic peptide parts presented on the surface of infected cells—allow HIV to partially evade these responses, and thus rapidly go to fixation. The faster they go to fixation, i.e., the higher their escape rate, the larger the selective pressure exerted by the immune system is assumed to be. This relation underpins the rationale for using escapes to assess the strength of immune responses. However, escape rate estimates are often obtained by employing an aggregation procedure, where several mutations that affect the same epitope are aggregated into a single, composite epitope mutation. The aggregation procedure thus rests upon the assumption that all within-epitope mutations have indistinguishable effects on immune recognition. In this study, we investigate how violation of this assumption affects escape rate estimates. To this end, we extend a previously developed simulation model of HIV that accounts for mutation, selection, and recombination to include different distributions of fitness effects (DFEs) and inter-mutational genomic distances. We use this discrete time Wright–Fisher based model to simulate early within-host evolution of HIV for DFEs and apply standard estimation methods to infer the escape rates. We then compare true with estimated escape rate values. We also compare escape rate values obtained by applying the aggregation procedure with values estimated without use of that procedure. We find that across the DFEs analyzed, the aggregation procedure alters the detectability of escape mutations: large-effect mutations are overrepresented while small-effect mutations are concealed. The effect of the aggregation procedure is similar to extracting the largest-effect mutation appearing within an epitope. Furthermore, the more pronounced the over-exponential decay of the DFEs, the more severely true escape rates are underestimated. We conclude that the aggregation procedure has two main consequences. On the one hand, it leads to a misrepresentation of the DFE of fixed mutations. On the other hand, it conceals within-epitope interactions that may generate irregularities in mutation frequency trajectories that are thus left unexplained.