Unraveling the causes of adaptive benefits of synonymous mutations in TEM-1 ß-lactamase.

While synonymous mutations were long thought to be without phenotypic consequences, there is growing evidence they can affect gene expression, protein folding, and ultimately the fitness of an organism. In only a few cases have the mechanisms by which synonymous mutations affect the phenotype been elucidated. We previously identified 48 mutations in TEM-1 ß-lactamase that increased resistance of Escherichia coli to cefotaxime, 10 of which were synonymous. To better understand the molecular mechanisms underlying the beneficial effect of these synonymous mutations, we made a series of measurements for a panel containing the 10 synonymous together with 10 non-synonymous mutations as a reference. Whereas messenger levels were unaffected, we found that total and functional TEM protein levels were higher for 5 out of 10 synonymous mutations. These observations suggest that some of these mutations act on translation or a downstream process. Similar effects were observed for some small-benefit non-synonymous mutations, suggesting a similar causal mechanism. For the synonymous mutations, we found that the cost of resistance scales with TEM protein levels. A resistance landscape for four synonymous mutations revealed strong epistasis: none of the combinations of mutations exceeded the resistance of the largest-effect mutation and there were synthetically neutral combinations. By considering combined effects of these mutations, we could infer that functional TEM protein level is a multi-dimensional phenotype. These results suggest that synonymous mutations may have beneficial effects by increasing the expression of an enzyme with low substrate activity, which may be realized via multiple, yet unknown, post-transcriptional mechanisms.

Interaction Analysis of Longevity Interventions Using Survival Curves.

A long-standing problem in ageing research is to understand how different factors contributing to longevity should be expected to act in combination under the assumption that they are independent. Standard interaction analysis compares the extension of mean lifespan achieved by a combination of interventions to the prediction under an additive or multiplicative null model, but neither model is fundamentally justified. Moreover, the target of longevity interventions is not mean life span but the entire survival curve. Here we formulate a mathematical approach for predicting the survival curve resulting from a combination of two independent interventions based on the survival curves of the individual treatments, and quantify interaction between interventions as the deviation from this prediction. We test the method on a published data set comprising survival curves for all combinations of four different longevity interventions in Caenorhabditis elegans. We find that interactions are generally weak even when the standard analysis indicates otherwise.

Phase transition in random adaptive walks on correlated fitness landscapes.

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength c. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size L. When the distribution of the random fitness component has an exponential tail, we find a phase transition of the walk length D between a phase at small c, where walks are short (D∼lnL), and a phase at large c, where walks are long (D∼L). For all other distributions only a single phase exists for any c>0. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.

Multidimensional epistasis and the transitory advantage of sex.

Identifying and quantifying the benefits of sex and recombination is a long-standing problem in evolutionary theory. In particular, contradictory claims have been made about the existence of a benefit of recombination on high dimensional fitness landscapes in the presence of sign epistasis. Here we present a comparative numerical study of sexual and asexual evolutionary dynamics of haploids on tunably rugged model landscapes under strong selection, paying special attention to the temporal development of the evolutionary advantage of recombination and the link between population diversity and the rate of adaptation. We show that the adaptive advantage of recombination on static rugged landscapes is strictly transitory. At early times, an advantage of recombination arises through the possibility to combine individually occurring beneficial mutations, but this effect is reversed at longer times by the much more efficient trapping of recombining populations at local fitness peaks. These findings are explained by means of well-established results for a setup with only two loci. In accordance with the Red Queen hypothesis the transitory advantage can be prolonged indefinitely in fluctuating environments, and it is maximal when the environment fluctuates on the same time scale on which trapping at local optima typically occurs.

Adaptation in tunably rugged fitness landscapes: the rough Mount Fuji model.

Much of the current theory of adaptation is based on Gillespie's mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyze a simple fitness landscape model with tunable ruggedness based on the rough Mount Fuji (RMF) model originally introduced by Aita et al. in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulas for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly fit first-step background in a recent evolution experiment with a microvirid bacteriophage.

Patterns of Epistasis between beneficial mutations in an antibiotic resistance gene.

Understanding epistasis is central to biology. For instance, epistatic interactions determine the topography of the fitness landscape and affect the dynamics and determinism of adaptation. However, few empirical data are available, and comparing results is complicated by confounding variation in the system and the type of mutations used. Here, we take a systematic approach by quantifying epistasis in two sets of four beneficial mutations in the antibiotic resistance enzyme TEM-1 ß-lactamase. Mutations in these sets have either large or small effects on cefotaxime resistance when present as single mutations. By quantifying the epistasis and ruggedness in both landscapes, we find two general patterns. First, resistance is maximal for combinations of two mutations in both fitness landscapes and declines when more mutations are added due to abundant sign epistasis and a pattern of diminishing returns with genotype resistance. Second, large-effect mutations interact more strongly than small-effect mutations, suggesting that the effect size of mutations may be an organizing principle in understanding patterns of epistasis. By fitting the data to simple phenotype resistance models, we show that this pattern may be explained by the nonlinear dependence of resistance on enzyme stability and an unknown phenotype when mutations have antagonistically pleiotropic effects. The comparison to a previously published set of mutations in the same gene with a joint benefit further shows that the enzyme's fitness landscape is locally rugged but does contain adaptive pathways that lead to high resistance.

Exact results for amplitude spectra of fitness landscapes.

Starting from fitness correlation functions, we calculate exact expressions for the amplitude spectra of fitness landscapes as defined by Stadler [1996. Landscapes and their correlation functions. J. Math. Chem. 20, 1] for common landscape models, including Kauffman's NK-model, rough Mount Fuji landscapes and general linear superpositions of such landscapes. We further show that correlations decaying exponentially with the Hamming distance yield exponentially decaying spectra similar to those reported recently for a model of molecular signal transduction. Finally, we compare our results for the model systems to the spectra of various experimentally measured fitness landscapes. We claim that our analytical results should be helpful when trying to interpret empirical data and guide the search for improved fitness landscape models.

Predictability of evolution depends nonmonotonically on population size.

To gauge the relative importance of contingency and determinism in evolution is a fundamental problem that continues to motivate much theoretical and empirical research. In recent evolution experiments with microbes, this question has been explored by monitoring the repeatability of adaptive changes in replicate populations. Here, we present the results of an extensive computational study of evolutionary predictability based on an experimentally measured eight-locus fitness landscape for the filamentous fungus Aspergillus niger. To quantify predictability, we define entropy measures on observed mutational trajectories and endpoints. In contrast to the common expectation of increasingly deterministic evolution in large populations, we find that these entropies display an initial decrease and a subsequent increase with population size N, governed, respectively, by the scales Nµ and Nµ(2), corresponding to the supply rates of single and double mutations, where µ denotes the mutation rate. The amplitude of this pattern is determined by µ. We show that these observations are generic by comparing our findings for the experimental fitness landscape to simulations on simple model landscapes.

Quantifying the adaptive potential of an antibiotic resistance enzyme.

For a quantitative understanding of the process of adaptation, we need to understand its "raw material," that is, the frequency and fitness effects of beneficial mutations. At present, most empirical evidence suggests an exponential distribution of fitness effects of beneficial mutations, as predicted for Gumbel-domain distributions by extreme value theory. Here, we study the distribution of mutation effects on cefotaxime (Ctx) resistance and fitness of 48 unique beneficial mutations in the bacterial enzyme TEM-1 ß-lactamase, which were obtained by screening the products of random mutagenesis for increased Ctx resistance. Our contributions are threefold. First, based on the frequency of unique mutations among more than 300 sequenced isolates and correcting for mutation bias, we conservatively estimate that the total number of first-step mutations that increase Ctx resistance in this enzyme is 87 [95% CI 75-189], or 3.4% of all 2,583 possible base-pair substitutions. Of the 48 mutations, 10 are synonymous and the majority of the 38 non-synonymous mutations occur in the pocket surrounding the catalytic site. Second, we estimate the effects of the mutations on Ctx resistance by determining survival at various Ctx concentrations, and we derive their fitness effects by modeling reproduction and survival as a branching process. Third, we find that the distribution of both measures follows a Fréchet-type distribution characterized by a broad tail of a few exceptionally fit mutants. Such distributions have fundamental evolutionary implications, including an increased predictability of evolution, and may provide a partial explanation for recent observations of striking parallel evolution of antibiotic resistance.

Rare events in population genetics: stochastic tunneling in a two-locus model with recombination.

We study the evolution of a population in a two-locus genotype space, in which the negative effects of two single mutations are overcompensated in a high-fitness double mutant. We discuss how the interplay of finite population size N and sexual recombination at rate r affects the escape times t(esc) to the double mutant. For small populations demographic noise generates massive fluctuations in t(esc). The mean escape time varies nonmonotonically with r, and grows exponentially as lnt(esc)∼N(r-r(*))(3/2) beyond a critical value r(*).

Dynamics of perturbations in disordered chaotic systems.

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatiotemporal behavior of typical perturbations deltau(x,t) is analyzed in terms of the Hopf-Cole transform h(x,t) identical withlnmid R:deltau(x,t)mid R: . Our analysis shows that the associated surface h(x,t) self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.

Structure of characteristic Lyapunov vectors in spatiotemporal chaos.

We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz '96 model exhibit the same features in quantitative and qualitative terms. Additionally, we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro, Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.

Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices.

The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called characteristic LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs (obtained through Gram-Schmidt orthogonalization) spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction.

Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces.

We study a surface growth model related to the Kardar-Parisi-Zhang equation for nonequilibrium kinetic roughening, but where the thermal noise is replaced by a static columnar disorder eta(x) . This model is one of the many representations of the problem of particle diffusion in trapping or amplifying disordered media. We find that probability localization in the latter translates into facet formation in the equivalent surface growth problem. Coarsening of the pattern can therefore be identified with the diffusion of the localization center. The emergent faceted structure gives rise to nontrivial scaling properties, including anomalous surface roughening in excellent agreement with an existing conjecture for kinetic roughening of faceted surfaces. In a wider context, our study sheds light onto the scaling properties in other systems displaying this kind of patterned surface.

Universal critical behavior of the synchronization transition in delayed chaotic systems.

We numerically investigate the critical behavior of the synchronization transition of two unidirectionally coupled delayed chaotic systems. We map the problem to a spatially extended system to show that the synchronization transition in delayed systems exhibits universal critical properties. We find that the synchronization transition is absorbing and generically belongs to the universality class of the bounded Kardar-Parisi-Zhang equation, as occurs in the case of extended systems. We also argue that directed percolation critical behavior may emerge for systems with strong nonlinearities.

Scaling properties of growing noninfinitesimal perturbations in space-time chaos.

We study the spatiotemporal dynamics of random spatially distributed noninfinitesimal perturbations in one-dimensional chaotic extended systems. We find that an initial perturbation of finite size epsilon0 grows in time obeying the tangent space dynamic equations (Lyapunov vectors) up to a characteristic time tx(epsilon0) approximately b-(1/lambda(max))ln(epsilon0), where lambda(max) is the largest Lyapunov exponent and b is a constant. For times t