A stochastic programming approach to the antibiotics time machine problem

Antibiotics Time Machine is an important problem to understand antibiotic resistance and how it can be reversed. Mathematically, it can be modeled as follows: Consider a set of genotypes, each of which contain a set of mutated and unmutated genes. Suppose that a set of growth rate measurements of each genotype under a set of antibiotics is given. The transition probabilities of a 'realization' of a Markov chain associated with each arc under each antibiotic are computable via a predefined function given the growth rate realizations. The aim is to maximize the expected probability of reaching to the genotype with all unmutated genes given the initial genotype in a predetermined number of transitions, considering the following two sources of uncertainties: (i) the randomness in growth rates, (ii) the randomness in transition probabilities, which are functions of growth rates. We develop stochastic mixed-integer linear programming and dynamic programming approaches to solve static and dynamic versions of the Antibiotics Time Machine Problem under the aforementioned uncertainties. We adapt a Sample Average Approximation approach that exploits the special structure of the problem and provide accurate solutions that perform very well in an out-of-sample analysis.

Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

Geometry of fitness landscapes: peaks, shapes and universal positive epistasis

Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length L, an overview of possible evolutionary trajectories is provided by the L-cube graph with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. The fitness landscape is defined by the fitness values of all genotypes in the system. Some notion of curvature is necessary for a more complete analysis of the landscapes, including the effect of recombination. The shape approach uses triangulations (shapes) induced by fitness landscapes. The main topic for this work is the interplay between peak patterns and shapes. Because of constraints on the shapes for [Formula: see text] imposed by peaks, there are in total 25 possible combinations of peak patterns and shapes. Similar constraints exist for higher L. Specifically, we show that the constraints induced by the staircase triangulation can be formulated as a condition of universal positive epistasis, an order relation on the fitness effects of arbitrary sets of mutations that respects the inclusion relation between the corresponding genetic backgrounds. We apply the concept to a large protein fitness landscape for an immunoglobulin-binding protein expressed in Streptococcal bacteria.

Accessibility percolation on Cartesian power graphs

A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph [Formula: see text], where [Formula: see text] is the number of genetic loci and the allele graph [Formula: see text] encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in [Formula: see text] displays a transition from 0 to a positive value at a threshold [Formula: see text] for the fitness difference between the initial and final genotype. We derive a lower bound on [Formula: see text] for general [Formula: see text] and show that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs.

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

Population dynamics and evolutionary genetics underly the structure of ecosystems, changing on the same timescale for interacting species with rapid turnover, such as virus (e.g. HIV) and immune response. Thus, an important problem in mathematical modeling is to connect ecology, evolution and genetics, which often have been treated separately. Here, extending analysis of multiple virus and immune response populations in a resource-prey (consumer)-predator model from Browne and Smith (2018), we show that long term dynamics of viral mutants evolving resistance at distinct epitopes (viral proteins targeted by immune responses) are governed by epistasis in the virus fitness landscape. In particular, the stability of persistent equilibrium virus-immune (prey-predator) network structures, such as nested and one-to-one, and bifurcations are determined by a collection of circuits defined by combinations of viral fitnesses that are minimally additive within a hypercube of binary sequences representing all possible viral epitope sequences ordered according to immunodominance hierarchy. Numerical solutions of our ordinary differential equation system, along with an extended stochastic version including random mutation, demonstrate how pairwise or multiplicative epistatic interactions shape viral evolution against concurrent immune responses and convergence to the multi-variant steady state predicted by theoretical results. Furthermore, simulations illustrate how periodic infusions of subdominant immune responses can induce a bifurcation in the persistent viral strains, offering superior host outcome over an alternative strategy of immunotherapy with strongest immune response.

Mechanisms of antibiotic action shape the fitness landscapes of resistance mutations

Antibiotic-resistant pathogens are a major public health threat. A deeper understanding of how an antibiotic's mechanism of action influences the emergence of resistance would aid in the design of new drugs and help to preserve the effectiveness of existing ones. To this end, we developed a model that links bacterial population dynamics with antibiotic-target binding kinetics. Our approach allows us to derive mechanistic insights on drug activity from population-scale experimental data and to quantify the interplay between drug mechanism and resistance selection. We find that both bacteriostatic and bactericidal agents can be equally effective at suppressing the selection of resistant mutants, but that key determinants of resistance selection are the relationships between the number of drug-inactivated targets within a cell and the rates of cellular growth and death. We also show that heterogeneous drug-target binding within a population enables resistant bacteria to evolve fitness-improving secondary mutations even when drug doses remain above the resistant strain's minimum inhibitory concentration. Our work suggests that antibiotic doses beyond this "secondary mutation selection window" could safeguard against the emergence of high-fitness resistant strains during treatment.

Modelling the fitness landscapes of a SCRaMbLEd yeast genome

The use of microorganisms for the production of industrially important compounds and enzymes is becoming increasingly important. Eukaryotes have been less widely used than prokaryotes in biotechnology, because of the complexity of their genomic structure and biology. The Yeast2.0 project is an international effort to engineer the yeast Saccharomyces cerevisiae to make it easy to manipulate, and to generate random variants using a system called SCRaMbLE. SCRaMbLE relies on artificial evolution in vitro to identify useful variants, an approach which is time consuming and expensive. We developed an in silico simulator for the SCRaMbLE system, using an evolutionary computing approach, which can be used to investigate and optimize the fitness landscape of the system. We applied the system to the investigation of the fitness landscape of one of the S. saccharomyces chromosomes, and found that our results fitted well with those previously published. We then simulated directed evolution with or without manipulation of SCRaMbLE, and revealed that controlling the SCRaMbLE process could effectively impact directed evolution. Our simulator can be applied to the analysis of the fitness landscapes of any organism for which SCRaMbLE has been implemented.

Redundancy-selection trade-off in phenotype-structured populations

Realistic fitness landscapes generally display a redundancy-fitness trade-off: highly fit trait configurations are inevitably rare, while less fit trait configurations are expected to be more redundant. The resulting sub-optimal patterns in the fitness distribution are typically described by means of effective formulations, where redundancy provided by the presence of neutral contributions is modelled implicitly, e.g. with a bias of the mutation process. However, the extent to which effective formulations are compatible with explicitly redundant landscapes is yet to be understood, as well as the consequences of a potential miss-match. Here we investigate the effects of such trade-off on the evolution of phenotype-structured populations, characterised by continuous quantitative traits. We consider a typical replication-mutation dynamics, and we model redundancy by means of two dimensional landscapes displaying both selective and neutral traits. We show that asymmetries of the landscapes will generate neutral contributions to the marginalised fitness-level description, that cannot be described by effective formulations, nor disentangled by the full trait distribution. Rather, they appear as effective sources, whose magnitude depends on the geometry of the landscape. Our results highlight new important aspects on the nature of sub-optimality. We discuss practical implications for rapidly mutant populations such as pathogens and cancer cells, where the qualitative knowledge of their trait and fitness distributions can drive disease management and intervention policies.

Predator-prey population dynamics may induce the evolutionary dynamics of behavioral unpredictability

Behavioral unpredictability (within-individual behavioral variability that cannot be explained by extrinsic and intrinsic factors) has been observed in a wide variety of species, but its adaptive significance is not well understood. This study examines the possibility that behavioral unpredictability is maintained through predator-prey population dynamics. An individual-based model was constructed to track the status of individual predators. The handling time of a predator is characterized by its expected value and unpredictability, which are heritable, and each predator features a unique combination of these characteristics. A discrete generation model in which one prey species and one predator species interact was constructed. The model showed that the evolutionarily stable strategy (ESS) of handling time is associated with no behavioral unpredictability and that the ESS handling time depends on the densities of the prey and predators. When populations exhibit cyclic dynamics, the ESS also changes along with the population dynamics, thereby creating mismatches between the traits of predators and the ESS because the dynamics of the ESS and population are faster than the evolution of the handling time traits. This mismatch can generate conditions in which individuals with behavioral unpredictability are at least transiently selected because of the topology of the fitness landscape. However, the model also showed that the selection strength of behavioral unpredictability is weak and can be overruled by inherent stochastic processes.

Fitness landscapes for coupled map lattices

Our goal is to match some dynamical aspects of biological systems with that of networks of coupled logistic maps. With these networks we generate sequences of iterates, convert them to symbol sequences by coarse-graining, and count the number of times combinations of symbols occur. Comparison of this with the number of times these combinations occur in experimental data-a sequence of interbeat intervals for example-is a measure of the fitness of each network to describe the target data. The most fit networks provide a cartoon that suggests a decomposition of the experimental data into a component that may be produced by a simple dynamical subsystem, and a residual component, the result of detailed, particular characteristics of the system that generated the target data. In the space of all network parameters, each point corresponds to a particular network. We construct a fitness landscape when we assign a fitness to each point. Because the parameters are distributed continuously over their ranges, and because fitnesses are estimated numerically, any plot of the landscape involves a finite sample of parameter values. We'll investigate how the local landscape geometry changes when the array of sample parameters is refined, and use the landscape geometry to explore complex relations between local fitness maxima.

On coevolution: Asymmetry in the NKCS model

The NKCS model was introduced to explore coevolutionary systems, that is, systems in which multiple species are closely interconnected. The fitness landscapes of the species are coupled to a controllable amount, where the underlying properties of the individual landscapes are also controllable. Previous work has assumed symmetry with respect to the controlling parameters. This paper explores the effects of reducing that symmetry on the behaviour of the coevolutionary system, including varying genome complexity, the degree of landscape coupling, and the use of local learning. Significant changes in behaviour from the traditional model are seen across the parameter space. These findings are suggested as particularly pertinent to symbiotic relationships.

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.

Quasi-species evolution maximizes genotypic reproductive value (not fitness or flatness)

Growing efforts to measure fitness landscapes in molecular and microbial systems are motivated by a longstanding goal to predict future evolutionary trajectories. Sometimes under-appreciated, however, is that the fitness landscape and its topography do not by themselves determine the direction of evolution: under sufficiently high mutation rates, populations can climb the closest fitness peak (survival of the fittest), settle in lower regions with higher mutational robustness (survival of the flattest), or even fail to adapt altogether (error catastrophes). I show that another measure of reproductive success, Fisher's reproductive value, resolves the trade-off between fitness and robustness in the quasi-species regime of evolution: to forecast the motion of a population in genotype space, one should look for peaks in the (mutation-rate dependent) landscape of genotypic reproductive values-whether or not these peaks correspond to local fitness maxima or flat fitness plateaus. This new landscape picture turns quasi-species dynamics into an instance of non-equilibrium dynamics, in the physical sense of Markovian processes, potential landscapes, entropy production, etc.

Fitness optimization and evolution of permanent replicator systems

In this paper, we discuss fitness landscape evolution of permanent replicator systems applying the hypothesis that the specific time of evolutionary adaptation of system parameters is much slower than the time of internal evolutionary dynamics. In other words, we suppose that the extremal principle of Darwinian evolution based on Fisher's fundamental theorem of natural selection is valid for the steady-states of permanent replicator systems. Various cases illustrating this concept are considered.

Biological fitness landscapes by deep mutational scanning

Knowledge of the distribution of fitness effects (DFE) of mutations is critical to the understanding of protein evolution. Here, we describe methods for large-scale, systematic measurements of the DFE using growth competition and deep mutational scanning. We discuss techniques for producing comprehensive libraries of gene variants as well as provide necessary considerations for designing these experiments. Using these methods, we have constructed libraries containing over 18,000 variants, measured fitness effects of these mutations by deep mutational scanning, and verified the presence of fitness effects in individual variants. Our methods provide a high-throughput protocol for measuring biological fitness effects of mutations and the dependence of fitness effects on the environment.

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes the phenotypes over many genetic loci, and the shape of this landscape can be used to predict evolutionary trajectories. Epistasis in a fitness landscape makes prediction of evolutionary trajectories more complex because the interactions between loci can produce local fitness peaks or troughs, which changes the likelihood of different paths. While various mathematical frameworks have been proposed to investigate properties of fitness landscapes, Beerenwinkel et al. (Stat Sin 17(4):1317-1342, 2007a) suggested studying regular subdivisions of convex polytopes. In this sense, each locus provides one dimension, so that the genotypes form a cube with the number of dimensions equal to the number of genetic loci considered. The fitness landscape is a height function on the coordinates of the cube. Here, we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool, which provides a concise combinatorial way of processing metric information from epistatic interactions. Furthermore, we extend the calculation of genetic interactions to consider interactions between microbial taxa in the gut microbiome of Drosophila fruit flies. We demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information on the fitness landscape where the previous approach is less conclusive.

Mathematical modeling of movement on fitness landscapes

Background

Movement of populations on fitness landscapes has been a problem of interest for a long time. While the subject has been extensively developed theoretically, reconciliation of the theoretical work with recent experimental data has not yet happened. In this work, we develop a computational framework and study evolution of the simplest transcription network between a single regulator, R and a single target protein, T.

Results

Through our simulations, we track evolution of this transcription network and comment on its dynamics and statistics of this movement. Significantly, we report that there exists a critical parameter which controls the ability of a network to reach the global fitness peak on the landscape. This parameter is the fraction of all permissible values of a biochemical parameter that can be accessed from its current value via a single mutation.

Conclusions

Overall, through this work, we aim to present a general framework for analysis of movement of populations (and particularly regulatory networks) on landscapes.

Electronic supplementary material

The online version of this article (10.1186/s12918-019-0704-0) contains supplementary material, which is available to authorized users.

Applications of high-throughput sequencing to analyze and engineer ribozymes

A large number of catalytic RNAs, or ribozymes, have been identified in the genomes of various organisms and viruses. Ribozymes are involved in biological processes such as regulation of gene expression and viral replication, but biological roles of many ribozymes still remain unknown. Ribozymes have also inspired researchers to engineer synthetic ribozymes that function as sensors or gene switches. To gain deeper understanding of the sequence-function relationship of ribozymes and to efficiently engineer synthetic ribozymes, a large number of ribozyme variants need to be examined which was limited to hundreds of sequences by Sanger sequencing. The advent of high-throughput sequencing technologies, however, has allowed us to sequence millions of ribozyme sequences at low cost. This review focuses on the recent applications of high-throughput sequencing to both characterize and engineer ribozymes, to highlight how the large-scale sequence data can advance ribozyme research and engineering.

Visualising the global structure of search landscapes: genetic improvement as a case study

The search landscape is a common metaphor to describe the structure of computational search spaces. Different landscape metrics can be computed and used to predict search difficulty. Yet, the metaphor falls short in visualisation terms because it is hard to represent complex landscapes, both in terms of size and dimensionality. This paper combines local optima networks, as a compact representation of the global structure of a search space, and dimensionality reduction, using the t-distributed stochastic neighbour embedding algorithm, in order to both bring the metaphor to life and convey new insight into the search process. As a case study, two benchmark programs, under a genetic improvement bug-fixing scenario, are analysed and visualised using the proposed method. Local optima networks for both iterated local search and a hybrid genetic algorithm, across different neighbourhoods, are compared, highlighting the differences in how the landscape is explored.

Substitutions into amino acids that are pathogenic in human mitochondrial proteins are more frequent in lineages closely related to human than in distant lineages

Propensities for different amino acids within a protein site change in the course of evolution, so that an amino acid deleterious in a particular species may be acceptable at the same site in a different species. Here, we study the amino acid-changing variants in human mitochondrial genes, and analyze their occurrence in non-human species. We show that substitutions giving rise to such variants tend to occur in lineages closely related to human more frequently than in more distantly related lineages, indicating that a human variant is more likely to be deleterious in more distant species. Unexpectedly, substitutions giving rise to amino acids that correspond to alleles pathogenic in humans also more frequently occur in more closely related lineages. Therefore, a pathogenic variant still tends to be more acceptable in human mitochondria than a variant that may only be fit after a substantial perturbation of the protein structure.

Selecting among three basic fitness landscape models: Additive, multiplicative and stickbreaking

Fitness landscapes map genotypes to organismal fitness. Their topographies depend on how mutational effects interact - epistasis - andare important for understanding evolutionary processes such as speciation, the rate of adaptation, the advantage of recombination, and the predictability versus stochasticity of evolution. The growing amount of data has made it possible to better test landscape models empirically. We argue that this endeavor will benefit from the development and use of meaningful basic models against which to compare more complex models. Here we develop statistical and computational methods for fitting fitness data from mutation combinatorial networks to three simple models: additive, multiplicative and stickbreaking. We employ a Bayesian framework for doing model selection. Using simulations, we demonstrate that our methods work and we explore their statistical performance: bias, error, and the power to discriminate among models. We then illustrate our approach and its flexibility by analyzing several previously published datasets. An R-package that implements our methods is available in the CRAN repository under the name Stickbreaker.

Reconstructing the genotype-to-fitness map for the bacterial chemotaxis network and its emergent behavioural phenotypes

The signal-transduction network responsible for chemotaxis in Escherichia coli has been characterised in extraordinary detail. Yet, relatively little is known about eco-evolutionary aspects of chemotaxis, such as how the network has been shaped by selection and to what extent natural populations may fine-tune their chemotactic behaviour to the ecological conditions. To address these questions, we here develop an evolutionary-systems-biology model of the chemotaxis network of E. coli, which we apply to estimate the resource accumulation rate (here used as a proxy for fitness) of wildtype and a large number of potential mutant genotypes. Mutant genotypes differ from the wildtype in the concentrations of one or more constituent proteins of the chemotaxis signalling network or in one or more of its kinetic parameters. To guarantee model consistency across the genotype space, we explicitly incorporated biochemical constraints that underly observed phenotypic trade-offs. The model was validated by reconstructing the phenotypic properties of several known mutant genotypes. We also characterised differences in the fitness distribution between genotypes, and reconstructed adaptive walks in genotype space for populations exposed to different environmental conditions. We found that the local fitness landscape is rugged, due to non-additive interactions between mutations. When selection has a consistent direction, just a few adaptive mutations are required to reach a local peak, and different local peaks can be reached by adaptive walks starting from the same initial genotype. However, when the direction of selection is fluctuating, evolutionary paths are much longer and genotype space is explored further. Longer adaptive walks were also observed when evolution was started from a low-fitness genotype such as a CheZ knockout mutant. In line with empirical observations, the initial ΔcheZ mutant did not respond to a step-down stimulus, but a dynamic response similar to the wildtype was recovered following the fixation of compensatory mutations.

Monotonicity of fitness landscapes and mutation rate control

A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher’s work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. First, we consider the geometric case of fitness being isomorphic to distance from an optimum, and show how problems of optimal mutation rate control can be solved exactly or approximately depending on additional constraints of the problem. Then we consider the general case of fitness communicating only partial information about the distance. We define weak monotonicity of fitness landscapes and prove that this property holds in all landscapes that are continuous and open at the optimum. This theoretical result motivates our hypothesis that optimal mutation rate functions in such landscapes will increase when fitness decreases in some neighbourhood of an optimum, resembling the control functions derived in the geometric case. We test this hypothesis experimentally by analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. Our findings support the hypothesis and find that the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We discuss the relevance of these findings to living organisms.

Fitness landscapes among many options under social influence

Cultural learning represents a novel problem in that an optimal decision depends not only on intrinsic utility of the decision/behavior but also on transparency of costs and benefits, the degree of social versus individual learning, and the relative popularity of each possible choice in a population. In terms of a fitness-landscape function, this recursive relationship means that multiple equilibria can exist. Here we use discrete-choice theory to construct a fitness-landscape function for a bi-axial decision-making map that plots the magnitude of social influence in the learning process against the costs and payoffs of decisions. Specifically, we use econometric and statistical methods to estimate not only the fitness function but also movements along the map axes. To search for these equilibria, we employ a hill-climbing algorithm that leads to the expected values of optimal decisions, which we define as peaks on the fitness landscape. We illustrate how estimation of a measure of transparency, a measure of social influence, and the associated fitness landscape can be accomplished using panel data sets.

Master equation and relative species abundance distribution for Lotka-Volterra models of interacting ecological communities

DESCRIPTION

Understanding the factors that control the dynamics of interacting species is a fundamental problem in ecology. The nature of the interactions among different species is usually not completely understood, but it is assumed that the species interaction plays an important role in the ecosystem properties as predicted by the niches models for an ecosystem. However, recent studies point out as the neutral hypothesis proposed by Hubbell of non-interacting species with an external source from the surrounding environment, allows to explain the relative species abundance distribution when the ecosystem has reached a stationary situation. In this paper we use the concept of fitness landscape to introduce a random dynamical model that describes the evolution of different communities near a stationary situation. The average dynamics can be associated to a system of Lotka-Volterra equations with mutualistic interactions. Then we derive a Master equation that satisfies the detailed balance condition of thermodynamical equilibria and allows to analytically compute the relative species abundance distribution near the stationary state as a multinomial negative distribution. These results suggest a possible approach to a synthetic theory that joins the niche theories and the Hubbell's neutral theory for RSA distribution.

Evolutionary dynamics of a polymorphic self-replicator population with a finite population size and hyper mutation rate

Self-replicating biomolecules, subject to experimental evolution, exhibit hyper mutation rates where the genotypes of most offspring have at least a one point mutation. Thus, we formulated the evolutionary dynamics of an asexual self-replicator population with a finite population size and hyper mutation rate, based on the probability density of fitnesses (fitness distribution) for the evolving population. As a case study, we used a Kauffman's "NK fitness landscape". We deduced recurrence relations for the first three cumulants of the fitness distribution and compared them with the results of computer simulations. We found that the evolutionary dynamics is classified in terms of two modes of selection: the "radical mode" and the "gentle mode". In the radical mode, only a small number of genotypes with the highest or near highest fitness values can leave offspring. In the gentle mode, genotypes with moderate fitness values can leave offspring. We clarified how the evolutionary equilibrium and climbing rate depend on given parameters such as gradient and ruggedness of the landscape, mutation rate and population size, in terms of the two modes of selection. Roughly, the radical mode conducts the fast climbing but attains to the stationary states with low fitness, while the gentle mode conducts the slow climbing but attains to the stationary states with high fitness.

Evolution of RNA-Based Networks

RNA molecules have served for decades as a paradigmatic example of molecular evolution that is tractable both in in vitro experiments and in detailed computer simulation. The adaptation of RNA sequences to external selection pressures is well studied and well understood. The de novo innovation or optimization of RNA aptamers and riboswitches in SELEX experiments serves as a case in point. Likewise, fitness landscapes building upon the efficiently computable RNA secondary structures have been a key toward understanding realistic fitness landscapes. Much less is known, however, on models in which multiple RNAs interact with each other, thus actively influencing the selection pressures acting on them. From a computational perspective, RNA-RNA interactions can be dealt with by same basic methods as the folding of a single RNA molecule, although many details become more complicated. RNA-RNA interactions are frequently employed in cellular regulation networks, e.g., as miRNA bases mRNA silencing or in the modulation of bacterial mRNAs by small, often highly structured sRNAs. In this chapter, we summarize the key features of networks of replicators. We highlight the differences between quasispecies-like models describing templates copied by an external replicase and hypercycle similar to autocatalytic replicators. Two aspects are of importance: the dynamics of selection within a population, usually described by conventional dynamical systems, and the evolution of replicating species in the space of chemical types. Product inhibition plays a key role in modulating selection dynamics from survival of the fittest to extinction of unfittest. The sequence evolution of replicators is rather well understood as approximate optimization in a fitness landscape for templates that is shaped by the sequence-structure map of RNA. Some of the properties of this map, in particular shape space covering and extensive neutral networks, give rise to evolutionary patterns such as drift-like motion in sequence space, akin to the behavior of RNA quasispecies. In contrast, very little is known about the influence of sequence-structure maps on autocatalytic replication systems.

Empirically founded genotype-phenotype maps from mammalian cyclic nucleotide-gated ion channels

A major barrier between evolutionary and functional biology is the difficulty of determining appropriate genotype-phenotype-fitness maps, particularly in metazoans. Concrete perspectives towards unifying these approaches are offered by studies on the physiological systems that depend on ion channel dynamics. I focus on the cyclic nucleotide-gated (CNG) channels implicated in the photoreceptor's response to light. From an evolutionary standpoint, sensory systems offers interpretative advantages, as the relation between the sensory response and environment is relatively straightforward. For CNG and other ion channels, extensive data are available about the physiological consequences of scanning mutagenesis on sensitive protein domains, such as the conduction pore. Mutant ion channels can be easily studied in living cells, so that the relation between genotypes and phenotypes is less speculative than usual. By relying on relatively simple theoretical frameworks, I used these data to relate the sequence space with phenotypes at increasing hierarchical levels. These empirical genotype-phenotype and phenotype-phenotype landscapes became smoother at higher integration levels, especially in heterozygous condition. The epistatic interaction between sites was analyzed from double mutant constructs. Magnitude epistasis was common. Moreover, evidence of reciprocal sign epistasis and the presence of permissive mutations were also observed, which suggest how adaptive regions can be connected across maladaptive valleys. The approach I describe suggests a way to better relate the evolutionary dynamics with the underlying physiology.

The probability of improvement in Fisher's geometric model: a probabilistic approach

Fisher developed his geometric model to support the micro-mutationalism hypothesis which claims that small mutations are more likely to be beneficial and therefore to contribute to evolution and adaptation. While others have provided a general solution to the model using geometric approaches, we derive an equivalent general solution using a probabilistic approach. Our approach to Fisher's geometric model provides alternative intuition and interpretation of the solution in terms of the model's parameters: for mutation to improve a phenotype, its relative beneficial effect must be larger than the ratio of its total effect and twice the difference between the current phenotype and the optimal one. Our approach provides new insight into this classical model of adaptive evolution.

Dynamic landscapes: a model of context and contingency in evolution

Although the basic mechanics of evolution have been understood since Darwin, debate continues over whether macroevolutionary phenomena are driven by the fitness structure of genotype space or by ecological interaction. In this paper we propose a simple model capturing key features of fitness-landscape and ecological models of evolution. Our model describes evolutionary dynamics in a high-dimensional, structured genotype space with interspecies interaction. We find promising qualitative similarity with the empirical facts about macroevolution, including broadly distributed extinction sizes and realistic exploration of the genotype space. The abstraction of our model permits numerous applications beyond macroevolution, including protein and RNA evolution.

Quasispecies dynamics with network constraints

A quasispecies is a set of interrelated genotypes that have reached a stationary state while evolving according to the usual Darwinian principles of selection and mutation. Quasispecies studies invariably assume that it is possible for any genotype to mutate into any other, but recent finds indicate that this assumption is not necessarily true. Here we revisit the traditional quasispecies theory by adopting a network structure to constrain the occurrence of mutations. Such structure is governed by a random-graph model, whose single parameter (a probability p) controls both the graph's density and the dynamics of mutation. We contribute two further modifications to the theory, one to account for the fact that different loci in a genotype may be differently susceptible to the occurrence of mutations, the other to allow for a more plausible description of the transition from adaptation to degeneracy of the quasispecies as p is increased. We give analytical and simulation results for the usual case of binary genotypes, assuming the fitness landscape in which a genotype's fitness decays exponentially with its Hamming distance to the wild type. These results support the theory's assertions regarding the adaptation of the quasispecies to the fitness landscape and also its possible demise as a function of p.