We study the evolution of asexual microorganisms with small mutation rate in fluctuating environments, and develop techniques that allow us to expand the formal solution of the evolution equations to first order in the mutation rate. Our method can be applied to both discrete time and continuous time systems. While the behavior of continuous time systems is dominated by the average fitness landscape for small mutation rates, in discrete time systems it is instead the geometric mean fitness that determines the system's properties. In both cases, we find that in situations in which the arithmetic (resp. geometric) mean of the fitness landscape is degenerate, regions in which the fitness fluctuates around the mean value present a selective advantage over regions in which the fitness stays at the mean. This effect is caused by the vanishing genetic diffusion at low mutation rates. In the absence of strong diffusion, a population can stay close to a fluctuating peak when the peak's height is below average, and take advantage of the peak when its height is above average.
Molecular evolution is often conceptualised as adaptive walks on rugged fitness landscapes, driven by mutations and constrained by incremental fitness selection. It is well known that epistasis shapes the ruggedness of the landscape's surface, outlining their topography (with high-fitness peaks separated by valleys of lower fitness genotypes). However, within the strong selection weak mutation (SSWM) limit, once an adaptive walk reaches a local peak, natural selection restricts passage through downstream paths and hampers any possibility of reaching higher fitness values. Here, in addition to …
Background: The rate at which fitness declines as an organism's genome accumulates random mutations is an important variable in several evolutionary theories. At an intuitive level, it might seem natural that random mutations should tend to interact synergistically, such that the rate of mean fitness decline accelerates as the number of random mutations is increased. However, in a number of recent studies, a prevalence of antagonistic epistasis (the tendency of multiple mutations to have a mitigating rather than reinforcing effect) has been observed. Results: We studied in silico the net amoun…
We investigate the evolutionary dynamics of a finite population of sequences adapting to NK fitness landscapes. We find that, unlike in the case of an infinite population, the average fitness in a finite population is maximized at a small but finite, rather than vanishing, mutation rate. The highest local maxima in the landscape are visited for even larger mutation rates, close to a transition point at which the population delocalizes (i.e., leaves the fitness peak at which it was localized) and starts traversing the sequence space. If the mutation rate is increased even further, the populatio…
We study the evolution of mutation rates for an asexual population living on a static fitness landscape, consisting of multiple peaks forming an evolutionary staircase. The optimal mutation rate is found by maximizing the diffusion towards higher fitness. Surprisingly the optimal genomic copying fidelity is given by Q = e^(-1/ln(n)) (where n is the genome length), independent of all other parameters in the model. Simulations confirm this theoretical result. We also discuss the relation between the optimal mutation rate on static and dynamic fitness landscapes.
We investigate the relationship between the average fitness decay due to single mutations and the strength of epistatic interactions in genetic sequences. We observe that epistatic interactions between mutations are correlated to the average fitness decay, both in RNA secondary structure prediction as well as in digital organisms replicating in silico. This correlation implies that during adaptation, epistasis and average mutational effect cannot be optimized independently. In experiments with RNA sequences evolving on a neutral network, the selective pressure to decrease the mutational load t…
The idea of adaptive walks on fitness landscapes as a means of studying evolutionary processes on large time scales is extended to fitness landscapes that are slowly changing over time. The influence of ruggedness and of the amount of static fitness contributions are investigated for model landscapes derived from Kauffman's $NK$ landscapes. Depending on the amount of static fitness contributions in the landscape, the evolutionary dynamics can be divided into a percolating and a non-percolating phase. In the percolating phase, the walker performs a random walk over the regions of the landscape …
We investigate the competition between two quasispecies residing on two disparate neutral networks. Under the assumption that the two neutral networks have different topologies and fitness levels, it is the mutation rate that determines which quasispecies will eventually be driven to extinction. For small mutation rates, we find that the quasispecies residing on the neutral network with the lower replication rate will disappear. For higher mutation rates, however, the faster replicating sequences may be outcompeted by the slower replicating ones in case the connection density on the second neu…
We considered a {multi-block} molecular model of biological evolution, in which fitness is a function of the mean types of alleles located at different parts (blocks) of the genome. We formulated an infinite population model with selection and mutation, and calculated the mean fitness. For the case of recombination, we formulated a model with a multidimensional fitness landscape (the dimension of the space is equal to the number of blocks) and derived a theorem about the dynamics of initially narrow distribution. We also considered the case of lethal mutations. We also formulated the finite po…
We show how concepts from statistical physics, such as order parameter, thermodynamic limit, and quantum phase transition, translate into biological concepts in mutation-selection models for sequence evolution and can be used there. The article takes a biological point of view within a population genetics framework, but contains an appendix for physicists, which makes this correspondence clear. We analyze the equilibrium behavior of deterministic haploid mutation-selection models. Both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is c…
Mutations create the genetic diversity on which selective pressures can act, yet also create structural instability in proteins. How, then, is it possible for organisms to ameliorate mutation-induced perturbations of protein stability while maintaining biological fitness and gaining a selective advantage? Here we used a new technique of site-specific chromosomal mutagenesis to introduce a selected set of mostly destabilizing mutations into folA - an essential chromosomal gene of E. coli encoding dihydrofolate reductase (DHFR) - to determine how changes in protein stability, activity and abunda…
We study self-replicating molecules under externally varying conditions. Changing conditions such as temperature variations and/or alterations in the environment's resource composition lead to both non-constant replication and decay rates of the molecules. In general, therefore, molecular evolution takes place in a dynamic rather than a static fitness landscape. We incorporate dynamic replication and decay rates into the standard quasispecies theory of molecular evolution, and show that for periodic time-dependencies, a system of evolving molecules enters a limit cycle for $t\to\infty$. For fa…
Evolution in changing environments is an important, but little studied aspect of the theory of evolution. The idea of adaptive walks in fitness landscapes has triggered a vast amount of research and has led to many important insights about the progress of evolution. Nevertheless, the small step to time-dependent fitness landscapes has most of the time not been taken. In this work, some elements of a theory of adaptive walks on changing fitness landscapes are proposed, and are subsequently applied to and tested on a simple family of time-dependent fitness landscapes, the oscillating NK landscap…
Mutation rate is a key determinant of the pace as well as outcome of evolution, and variability in this rate has been shown in different scenarios to play a key role in evolutionary adaptation and resistance evolution under stress caused by selective pressure. Here we investigate the dynamics of resistance fixation in a bacterial population with variable mutation rates and show that evolutionary outcomes are most sensitive to mutation rate variations when the population is subject to environmental and demographic conditions that suppress the evolutionary advantage of high-fitness subpopulation…
Mutational escape from vaccine induced immune responses has thwarted the development of a successful vaccine against AIDS, whose causative agent is HIV, a highly mutable virus. Knowing the virus' fitness as a function of its proteomic sequence can enable rational design of potent vaccines, as this information can focus vaccine induced immune responses to target mutational vulnerabilities of the virus. Spin models have been proposed as a means to infer intrinsic fitness landscapes of HIV proteins from patient-derived viral protein sequences. These sequences are the product of non-equilibrium vi…
In this paper we study the evolution of the mutation rate for simple organisms in dynamic environments. A model with multiple fitness coding loci tracking a moving fitness peak is developed and an analytical expression for the optimal mutation rate is derived. Surprisingly it turns out that the optimal mutation rate per genome is approximately independent of genome length, something that also has been observed in nature. Simulations confirm the theoretical predictions. We also suggest an explanation for the difference in mutation frequency between RNA and DNA based organisms.
We present an extension of Eigen's model for quasi-species including the competition among individuals, proposed as the simplest mechanism for the formation of new species in a smooth fitness landscape. We are able to obtain analytically the critical threshold for species formation. The comparison with numerical simulations is very good.
A quasispecies evolving on a fitness landscape with a single peak of fluctuating height is studied. In the approximation that back mutations can be ignored, the rate equations can be solved analytically. It is shown that the error threshold on this class of dynamic landscapes is defined by the time average of the selection pressure. In the case of a periodically fluctuating fitness peak we also study the phase-shift and response amplitude of the previously documented low-pass filter effect. The special case of a small harmonic fluctuation is treated analytically.
Eigen's quasi-species model describes viruses as ensembles of different mutants of a high fitness "master" genotype. Mutants are assumed to have lower fitness than the master type, yet they coexist with it forming the quasi-species. When the mutation rate is sufficiently high, the master type no longer survives and gets replaced by a wide range of mutant types, thus destroying the quasi-species. It is the so-called "error catastrophe". But natural selection acts on phenotypes, not genotypes, and huge amounts of genotypes yield the same phenotype. An important consequence of this is the appeara…
In a co-evolutionary context, the survive probability of individual elements of a system depends on their relation with their neighbors. The natural selection process depends on the whole population, which is determined by local events between individuals. Particular characteristics assigned to each individual, as larger memory, usually improve the individual fitness, but an agent possess also endogenous characteristics that induce to re-evaluate her fitness landscape and choose the best-suited kind of interaction, inducing a non absolute value of the outcomes of the interaction. In this work,…
We consider a continuum version of a previously introduced and numerically studied model of macroevolution (PRL 75, 2055, (1995)) in which agents evolve by an optimization process in a rugged fitness landscape and die due to their competitive interactions. We first formulate dynamical equations for the fitness distribution and the survival probability. Secondly we analytically derive the $t^{-2}$ law which characterizes the life time distribution of biological genera. Thirdly we discuss other dynamical properties of the model such as the rate of extinction and conclude with a brief discussion.
In this paper we investigate error-thresholds on dynamics fitness-landscapes. We show that there exists both lower and an upper threshold, representing limits to the copying fidelity of simple replicators. The lower bound can be expressed as a correction term to the error-threshold present on a static landscape. The upper error-threshold is a new limit that only exists on dynamic fitness-landscapes. We also show that for long genomes on highly dynamic fitness-landscapes there exists a lower bound on the selection pressure needed to enable effective selection of genomes with superior fitness in…
