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Motivation: Many important aspects of evolutionary dynamics can only be addressed through simulations. However, accurate simulations of realistically large populations over long periods of time needed for evolution to proceed are computationally expensive. Mutants can be present in very small numbers and yet (if they are more fit than others) be the key part of the evolutionary process. This leads to significant stochasticity that needs to be accounted for. Different evolutionary events occur at very different time scales: mutations are typically much rarer than reproduction and deaths., Results: We introduce a new exact algorithm for fast fully stochastic simulations of evolutionary dynamics that include birth, death and mutation events. It produces a significant speedup compared to direct stochastic simulations in a typical case when the population size is large and the mutation rates are much smaller than birth and death rates. The algorithm performance is illustrated by several examples that include evolution on a smooth and rugged fitness landscape. We also show how this algorithm can be adapted for approximate simulations of more complex evolutionary problems and illustrate it by simulations of a stochastic competitive growth model.

The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.

We introduce a public key encryption scheme that is based on additive mixing of a message with chaotic nonlinear dynamics. A high-dimensional dissipative nonlinear dynamical system is distributed between transmitter and receiver. The transmitter dynamics is public (known to all) and the receiver dynamics is private (known only to the authorized receiver). Bidirectional signals that couple transmitter and receiver are transmitted over a public channel. Once the chaotic dynamics which is initialized with a random state converges to the attractor, a message is mixed with the chaotic dynamics at the transmitter. The authorized receiver who knows the entire dynamics can use a simple algorithm to decode the message. An unauthorized receiver does not know the receiver dynamics and needs to use computationally unfeasible algorithms in order to decode the message. Security is maintained by altering the private receiver dynamics during transmission. We show that using additive mixing modulation is more efficient than the attractor position modulation distributed dynamics encryption scheme. We demonstrate the concept of this new scheme by simulating a simple coupled map lattice. Index Terms--Chaos, nonlinear dynamics, public key encryption.