Joachim Krug

Origins of scale invariance in growth processes

Abstract This review describes recent progress in the understanding of the emergence of scale invariance in far-from-equilibrium growth. The first section is devoted to ‘solvable’ needle models which illustrate the relationship between long-range competition mediated, for example, through shadowing or a Laplacian field, and scale invariance. The following three sections, which comprise the bulk of the article, develop the theory of kinetic surface roughening in a comprehensive manner. The two large classes of kinetic roughening processes, characterized by non-conserved (Kardar-Parisi-Zhang) and conserved (ideal molecular beam epitaxy (MBE)) surface relaxation, respectively, are treated separately. For the former case, which has been extensively reviewed elsewhere, the focus is on recent developments. For the case of ideal MBE we give a systematic derivation of the various universality classes in terms of microscopic processes, and compare the predictions of continuum theory to computer simulations and exp...

Empirical fitness landscapes and the predictability of evolution

The genotype–fitness map (that is, the fitness landscape) is a key determinant of evolution, yet it has mostly been used as a superficial metaphor because we know little about its structure. This is now changing, as real fitness landscapes are being analysed by constructing genotypes with all possible combinations of small sets of mutations observed in phylogenies or in evolution experiments. In turn, these first glimpses of empirical fitness landscapes inspire theoretical analyses of the predictability of evolution. Here, we review these recent empirical and theoretical developments, identify methodological issues and organizing principles, and discuss possibilities to develop more realistic fitness landscape models.

Quantitative analyses of empirical fitness landscapes

The concept of a fitness landscape is a powerful metaphor that offers insight into various aspects of evolutionary processes and guidance for the study of evolution. Until recently, empirical evidence on the ruggedness of these landscapes was lacking, but since it became feasible to construct all possible genotypes containing combinations of a limited set of mutations, the number of studies has grown to a point where a classification of landscapes becomes possible. The aim of this review is to identify measures of epistasis that allow a meaningful comparison of fitness landscapes and then apply them to the empirical landscapes to discern factors that affect ruggedness. The various measures of epistasis that have been proposed in the literature appear to be equivalent. Our comparison shows that the ruggedness of the empirical landscape is affected by whether the included mutations are beneficial or deleterious and by whether intra- or intergenic epistasis is involved. Finally, the empirical landscapes are compared to landscapes generated with the Rough Mt.\ Fuji model. Despite the simplicity of this model, it captures the features of the experimental landscapes remarkably well.

Evolutionary Accessibility of Mutational Pathways

Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection (‘fitness’), measured values over all possible combinations of a set of mutations yield a fitness landscape that determines which mutational states can be reached from a given initial genotype. Understanding the accessibility properties of fitness landscapes is conceptually important in answering questions about the predictability and repeatability of evolutionary adaptation. Here we theoretically investigate accessibility of the globally optimal state on a wide variety of model landscapes, including landscapes with tunable ruggedness as well as neutral ‘holey’ landscapes. We define a mutational pathway to be accessible if it contains the minimal number of mutations required to reach the target genotype, and if fitness increases in each mutational step. Under this definition accessibility is high, in the sense that at least one accessible pathway exists with a substantial probability that approaches unity as the dimensionality of the fitness landscape (set by the number of mutational loci) becomes large. At the same time the number of alternative accessible pathways grows without bounds. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus Aspergillus niger. By analyzing subgraphs of the full landscape containing different subsets of mutations, we are able to probe the mutational distance scale in the empirical data. The predicted effect of high accessibility is supported by the empirical data and is very robust, which we argue reflects the generic topology of sequence spaces. Together with the restrictive assumptions that lie in our definition of accessibility, this implies that the globally optimal configuration should be accessible to genome wide evolution, but the repeatability of evolutionary trajectories is limited owing to the presence of a large number of alternative mutational pathways.

Clonal interference in large populations

Clonal interference, the competition between lineages arising from different beneficial mutations in an asexually reproducing population, is an important factor determining the tempo and mode of microbial adaptation. The standard theory of this phenomenon neglects the occurrence of multiple mutations as well as the correlation between loss by genetic drift and clonal competition, which is questionable in large populations. Working within the Wright–Fisher model with multiplicative fitness (no epistasis), we determine the rate of adaptation asymptotically for very large population sizes and show that the standard theory fails in this regime. Our study also explains the success of the standard theory in predicting the rate of adaptation for moderately large populations. Furthermore, we show that the nature of the substitution process changes qualitatively when multiple mutations are allowed for, because several mutations can be fixed in a single fixation event. As a consequence, the index of dispersion for counts of the fixation process displays a minimum as a function of population size, whereas the origination process of fixed mutations becomes completely regular for very large populations. We find that the number of mutations fixed in a single event is geometrically distributed as in the neutral case. These conclusions are based on extensive simulations combined with analytic results for the limit of infinite population size.

Exploring the Effect of Sex on Empirical Fitness Landscapes

The nature of epistasis has important consequences for the evolutionary significance of sex and recombination. Recent efforts to find negative epistasis as a source of negative linkage disequilibrium and associated long‐term advantage to sex have yielded little support. Sign epistasis, where the sign of the fitness effects of alleles varies across genetic backgrounds, is responsible for the ruggedness of the fitness landscape, with several unexplored implications for the evolution of sex. Here, we describe fitness landscapes for two sets of strains of the asexual fungus Aspergillus niger involving all combinations of five mutations. We find that ∼30% of the single‐mutation fitness effects are positive despite their negative effect in the wild‐type strain and that several local fitness maxima and minima are present. We then compare adaptation of sexual and asexual populations on these empirical fitness landscapes by using simulations. The results show a general disadvantage of sex on these rugged landscapes, caused by the breakdown by recombination of genotypes on fitness peaks. Sex facilitates movement to the global peak only for some parameter values on one landscape, indicating its dependence on the landscape’s topography. We discuss possible reasons for the discrepancy between our results and the reports of faster adaptation of sexual populations.

Island nucleation in the presence of step-edge barriers: Theory and applications

We develop a theory of nucleation on top of two-dimensional islands bordered by steps with an additional energy barrier

The Speed of Evolution in Large Asexual Populations

\ensuremath{\Delta}{E}_{S}

Patterns of Epistasis between Beneficial Mutations in an Antibiotic Resistance Gene

for descending atoms. The theory is based on the concept of the residence time of an adatom on the island, and yields an expression for the nucleation rate which becomes exact in the limit of strong step-edge barriers. This expression differs qualitatively and quantitatively from that obtained using the conventional rate-equation approach to nucleation [J. Tersoff et al., Phys. Rev. Lett.

Unstable epitaxy on vicinal surfaces

72,