arXiv subjects

Joachim Krug

Publications and source records attributed to Joachim Krug.

At least 19 recordsLinked to original sources

arXiv2026

Evolution as fitness landscape navigation: concepts, measures, and emerging questions

Fitness landscapes are mappings between genotypes, phenotypes, and fitness that shape evolution. In recent years, empirical work and theoretical models have greatly advanced our understanding of how populations navigate rugged fitness landscapes. Here, we provide a timely review of the theoretical aspects of this field. Its rapidly growing literature employs a wide range of terms, which are sometimes used ambiguously or inconsistently. We therefore begin by defining the major concepts and the field's vocabulary, highlighting our own terminology choices wherever needed. We then review key results on the relationships between epistasis, ruggedness, accessibility, and navigability for genotype-fitness maps, highlighting several complex and sometimes counterintuitive connections that have emerged. Further, we review how the conserved structural properties of the underlying genotype-phenotype map, which can lead to the formation of large connected neutral networks of genotypes, influence dynamics on fitness landscapes. We then compare the two levels to study landscape navigation: the level of genotype-phenotype maps and the level of genotype-fitness maps. Our review leads us to propose a new measure of navigability, based on evolutionary outcomes, that is broadly applicable and overcomes limitations of existing measures. Finally, we highlight examples from the smaller body of work that relaxes the common assumption of fitness-monotonic paths on static landscapes, and discuss how this can fundamentally change the nature of fitness landscape navigation. Throughout the review, we identify directions for future work to fill existing gaps and to synthesize the disparate strands of research within the field.

q-bio.PE↗

arXiv2026

Epistatic strength, modularity, and locus heterogeneity shape the number of local optima in fitness landscapes

Fitness landscapes provide a quantitative framework for understanding how natural selection shapes evolutionary trajectories. A central feature of these landscapes is their number of local optima, which determines whether fitness-increasing evolution can proceed towards a global optimum or become trapped on suboptimal peaks. Although multiple peaks are known to require reciprocal sign epistasis, the quantitative relationship between epistasis and number of peaks remains incompletely understood. Here, we show that for a broad class of unstructured fitness landscapes, i.e. isotropic Gaussian random fields, the expected number of local optima is determined by a single local measure of epistasis: the correlation of fitness effects. This provides a baseline prediction for the number of peaks in typical unstructured landscapes and links peak density directly to the amount of reciprocal sign epistasis. This baseline changes when epistatic interactions are structured. We show that clustering interactions within blocks of loci slightly increases the number of local optima. In contrast, strong heterogeneity between loci, where only a small subset of loci participate in epistatic interactions, causes the number of peaks to collapse. These results show that the number of local optima is governed not only by the overall strength of epistasis, but also by how epistatic interactions are distributed across the genotype space. Our framework therefore reconciles the central role of reciprocal sign epistasis with the observation that landscapes with similar amounts of epistasis can differ substantially in ruggedness, and provides a guide to the range of peak numbers expected in typical landscapes.

q-bio.PE↗

arXiv2026

Simple sign epistasis and evolutionary detours in fitness landscapes

In epistatic fitness landscapes, the fitness effect of a mutation depends on the genetic background and may even switch between deleterious and beneficial depending on the presence of another mutation. Epistatic interactions may cause both mutations to change the sign of each other's fitness effects (reciprocal sign epistasis) or only one mutation to do so (simple sign epistasis). Both these forms of epistasis influence evolutionary trajectories. While reciprocal sign epistasis has been associated with multi-peaked landscapes and their ruggedness, the role and relative frequency of simple sign epistasis in fitness landscapes have not been systematically investigated. Here, we prove that the presence of simple sign epistasis is associated with evolutionary detours, i.e., indirect, longer fitness-increasing paths to fitness peaks that include back-mutations. We also show that in experimentally resolved, weakly epistatic landscapes, simple sign epistasis occurs much more frequently than reciprocal sign epistasis. This result is consistent with the theoretical predictions we derive for most landscape models, with the exception of the block model and of landscapes dominated by pairwise allelic incompatibilities, such as RNA stability landscapes. Our results suggest that detours represent a general feature of evolutionary trajectories in weakly epistatic landscapes.

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arXiv2025

Complexity and accessibility of random landscapes

These notes introduce probabilistic landscape models defined on high-dimensional discrete sequence spaces. The models are motivated primarily by fitness landscapes in evolutionary biology, but links to statistical physics and computer science are mentioned where appropriate. Elementary and advanced results on the structure of landscapes are described with a focus on features that are relevant to evolutionary searches, such as the number of local maxima and the existence of fitness-monotonic paths. The recent discovery of submodularity as a biologically meaningful property of fitness landscapes and its consequences for their accessibility is discussed in detail.

q-bio.PE↗

arXiv2025

Branching with selection and mutation II: Mutant fitness of Gumbel type

We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave.

math.PR↗

arXiv2024

Evolutionary accessibility of random and structured fitness landscapes

Biological evolution can be conceptualized as a search process in the space of gene sequences guided by the fitness landscape, a mapping that assigns a measure of reproductive value to each genotype. Here we discuss probabilistic models of fitness landscapes with a focus on their evolutionary accessibility, where a path in a fitness landscape is said to be accessible if the fitness values encountered along the path increase monotonically. For uncorrelated (random) landscapes with independent and identically distributed fitness values, the probability of existence of accessible paths between genotypes at a distance linear in the sequence length $L$ becomes nonzero at a nontrivial threshold value of the fitness difference between the initial and final genotype, which can be explicitly computed for large classes of genotype graphs. The behaviour in uncorrelated random landscapes is contrasted with landscape models that display additional, biologically motivated structural features. In particular, landscapes defined by a tradeoff between adaptation to environmental extremes have been found to display a combinatorially large number of accessible paths to all local fitness maxima. We show that this property is characteristic of a broad class of models that satisfy a certain global constraint, and provide further examples from this class.

q-bio.PE↗

arXiv2023

Exclusion model of mRNA translation with collision-induced ribosome drop-off

The translation of messenger RNA transcripts to proteins is commonly modeled as a one-dimensional totally asymmetric exclusion process with extended particles. Here we focus on the effects of premature termination of translation through the irreversible detachment of ribosomes. We consider a model where the detachment is induced by the unsuccessful attempt to move to an occupied site. The model is exactly solvable in a simplified geometry consisting of the translation initiation region followed by a single slow site representing a translation bottleneck. In agreement with recent experimental and computational studies we find a non-monotonic dependence of the ribosome current on the initiation rate, but only if the leading particle in a colliding pair detaches. Simulations show that the effect persists for larger lattices and extended bottlenecks. In the homogeneous system the ribosome density decays asymptotically as the inverse square root of the distance to the initiation site.

q-bio.SC↗

arXiv2023

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 $L=3$ 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 {\emph{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.

q-bio.PE↗

arXiv2023

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 $\mathcal{A}^L$, where $L$ is the number of genetic loci and the allele graph $\mathcal{A}$ 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 $L$ displays a transition from 0 to a positive value at a threshold $β_c$ for the fitness difference between the initial and final genotype. We derive a lower bound on $β_c$ for general $\mathcal{A}$ 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.

q-bio.PE↗

arXiv2022

Branching with selection and mutation I: Mutant fitness of Fréchet type

We investigate two stochastic models of a growing population subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent's fitness, but some are mutants and obtain a fitness randomly sampled from a distribution in the domain of attraction of the Fréchet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth.

math.PR↗

arXiv2022

Nonlinear dynamics of an epidemic compartment model with asymptomatic infections and mitigation

A significant proportion of the infections driving the current {SARS-CoV-2} pandemic are transmitted asymptomatically. Here we introduce and study a simple epidemic model with separate compartments comprising asymptomatic and symptomatic infected individuals. The linear dynamics determining the outbreak condition of the model is equivalent to a renewal theory approach with exponential waiting time distributions. Exploiting a nontrivial conservation law of the full nonlinear dynamics, we derive analytic bounds on the peak number of infections in the absence and presence of mitigation through isolation and testing. The bounds are compared to numerical solutions of the differential equations.

q-bio.PE↗

arXiv2022

Unpredictable repeatability in molecular evolution

The extent of parallel evolution at the genotypic level is quantitatively linked to the distribution of beneficial fitness effects (DBFE) of mutations. The standard view, based on light-tailed distributions (i.e. distributions with finite moments), is that the probability of parallel evolution in duplicate populations is inversely proportional to the number of available mutations, and moreover that the DBFE is sufficient to determine the probability when the number of available mutations is large. Here we show that when the DBFE is heavy-tailed, as found in several recent experiments, these expectations are defied. The probability of parallel evolution decays anomalously slowly in the number of mutations or even becomes independent of it, implying higher repeatability of evolution. At the same time, the probability of parallel evolution is non-self-averaging, that is, it does not converge to its mean value even when a large number of mutations are involved. This behavior arises because the evolutionary process is dominated by only a few mutations of high weight. Consequently, the probability varies widely across systems with the same DBFE. Contrary to the standard view, the DBFE is no longer sufficient to determine the extent of parallel evolution, making it much less predictable. We illustrate these ideas theoretically and through analysis of empirical data on antibiotic resistance evolution.

q-bio.PE↗

arXiv2022

A Driven Disordered Systems Approach to Biological Evolution in Changing Environments

Biological evolution of a population is governed by the fitness landscape, which is a map from genotype to fitness. However, a fitness landscape depends on the organisms environment, and evolution in changing environments is still poorly understood. We study a particular model of antibiotic resistance evolution in bacteria where the antibiotic concentration is an environmental parameter and the fitness landscapes incorporate tradeoffs between adaptation to low and high antibiotic concentration. With evolutionary dynamics that follow fitness gradients, the evolution of the system under slowly changing antibiotic concentration resembles the athermal dynamics of disordered physical systems under external drives. Exploiting this resemblance, we show that our model can be described as a system with interacting hysteretic elements. As in the case of the driven disordered systems, adaptive evolution under antibiotic concentration cycling is found to exhibit hysteresis loops and memory formation. We derive a number of analytical results for quasistatic concentration changes. We also perform numerical simulations to study how these effects are modified under driving protocols in which the concentration is changed in discrete steps. Our approach provides a general framework for studying motifs of evolutionary dynamics in biological systems in a changing environment.

q-bio.PE↗

arXiv2021

Accessibility percolation in random fitness landscapes

The fitness landscape encodes the mapping of genotypes to fitness and provides a succinct representation of possible trajectories followed by an evolving population. Evolutionary accessibility is quantified by the existence of fitness-monotonic paths connecting far away genotypes. Studies of accessibility percolation use probabilistic fitness landscape models to explore the emergence of such paths as a function of the initial fitness, the parameters of the landscape or the structure of the genotype graph. This chapter reviews these studies and discusses their implications for the predictability of evolutionary processes.

q-bio.PE↗

arXiv2021

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 into a critical and constructive attitude in 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.

q-bio.PE↗

arXiv2021

Non-equilibrium Roughness Evolution of Small Molecule Mixed Films Reflecting Equilibrium Phase Behavior

Understanding non-equilibrium phenomena, such as growth, and connecting them to equilibrium phase behavior is a major challenge, in particular for complex multicomponent materials. We use X-ray reflectivity to determine the surface roughness of binary mixtures of several prototypical organic compounds. By analyzing the roughness as a function of composition, we find a systematic behavior depending on the bulk phase behavior in terms of intermixing, co-crystallization or phase separation. Supported by kinetic Monte Carlo simulations, we provide evidence that the growth behavior can be rationalized by a lowered step edge barrier in the mixed films which is induced by reduced in-plane crystallinity.

physics.app-ph↗

arXiv2020

The fitness landscapes of translation

Motivated by recent experiments on an antibiotic resistance gene, we investigate genetic interactions between synonymous mutations in the framework of exclusion models of translation. We show that the range of possible interactions is markedly different depending on whether translation efficiency is assumed to be proportional to ribosome current or ribosome speed. In the first case every mutational effect has a definite sign that is independent of genetic background, whereas in the second case the effect-sign can vary depending on the presence of other mutations. The latter result is demonstrated using configurations of multiple translational bottlenecks induced by slow codons.

q-bio.SC↗

arXiv2020

Distribution of the number of fitness maxima in Fisher's Geometric Model

Fisher's geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an $n$-dimensional Euclidean phenotype space with a nonlinear, single-peaked phenotype-fitness map. Genotypes are represented by binary sequences of length $L$, and the phenotypic effects of mutations at different sites are represented by $L$ random vectors drawn from an isotropic Gaussian distribution. Recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima. Extending our previous study of the mean number of local maxima, here we focus on the distribution of the number of maxima when the limit $L \to \infty$ is taken at finite $n$. We identify the typical scale of the number of maxima for general $n$, and determine the full scaled probability density and two point correlation function of maxima for the one-dimensional case. We also elaborate on the close relation of the model to the anti-ferromagnetic Hopfield model with $n$ random continuous pattern vectors, and show that many of our results carry over to this setting. More generally, we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems.

q-bio.PE↗