Punctuated equilibrium and shock waves in molecular models of biological evolution

Cryptic selection forces and dynamic heritability in generalized phenotypic evolution

Individuals with different phenotypes can have widely-varying responses to natural selection, yet many classical approaches to evolutionary dynamics emphasize only how a population's average phenotype increases in fitness over time. However, recent experimental results have produced examples of populations that have multiple fitness peaks, or that experience frequency-dependence that affects the direction and strength of selection on certain individuals. Here, we extend classical fitness gradient formulations of natural selection in order to describe the dynamics of a phenotype distribution in terms of its moments-such as the mean, variance, and skewness. The number of governing equations in our model can be adjusted in order to capture different degrees of detail about the population. We compare our simplified model to direct Wright-Fisher simulations of evolution in several canonical fitness landscapes, and we find that our model provides a low-dimensional description of complex dynamics not typically explained by classical theory, such as cryptic selection forces due to selection on trait ranges, time-variation of the heritability, and nonlinear responses to stabilizing or disruptive selection due to asymmetric trait distributions. In addition to providing a framework for extending general understanding of common qualitative concepts in phenotypic evolution - such as fitness gradients, selection pressures, and heritability - our approach has practical importance for studying evolution in contexts in which genetic analysis is infeasible.

Accurate analytic solution of chemical master equations for gene regulation networks in a single cell

Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.078101] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.

An optimal strategy to solve the Prisoner's Dilemma

Cooperation is a central mechanism for evolution. It consists of an individual paying a cost in order to benefit another individual. However, natural selection describes individuals as being selfish and in competition among themselves. Therefore explaining the origin of cooperation within the context of natural selection is a problem that has been puzzling researchers for a long time. In the paradigmatic case of the Prisoner's Dilemma (PD), several schemes for the evolution of cooperation have been proposed. Here we introduce an extension of the Replicator Equation (RE), called the Optimal Replicator Equation (ORE), motivated by the fact that evolution acts not only at the level of individuals of a population, but also among competing populations, and we show that this new model for natural selection directly leads to a simple and natural rule for the emergence of cooperation in the most basic version of the PD. Contrary to common belief, our results reveal that cooperation can emerge among selfish individuals because of selfishness itself: if the final reward for being part of a society is sufficiently appealing, players spontaneously decide to cooperate.

Crossing fitness canyons by a finite population

We consider the Wright-Fisher model of the finite population evolution on a fitness landscape defined in the sequence space by a path of nearly neutral mutations. We study a specific structure of the fitness landscape: One of the intermediate mutations on the mutation path results in either a large fitness value (climbing up a fitness hill) or a low fitness value (crossing a fitness canyon), the rest of the mutations besides the last one are neutral, and the last sequence has much higher fitness than any intermediate sequence. We derive analytical formulas for the first arrival time of the mutant with two point mutations. For the first arrival problem for the further mutants in the case of canyon crossing, we analytically deduce how the mean first arrival time scales with the population size and fitness difference. The location of the canyon on the path of sequences has a crucial role. If the canyon is at the beginning of the path, then it significantly prolongs the first arrival time; otherwise it just slightly changes it. Furthermore, the fitness hill at the beginning of the path strongly prolongs the arrival time period; however, the hill located near the end of the path shortens it. We optimize the first arrival time by applying a nonzero selection to the intermediate sequences. We extend our results and provide a scaling for the valley crossing time via the depth of the canyon and population size in the case of a fitness canyon at the first position. Our approach is useful for understanding some complex evolution systems, e.g., the evolution of cancer.

Solution of classical evolutionary models in the limit when the diffusion approximation breaks down

The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.