Evolutionary graph theory is the study of birthdeath processes that are constrained by population structure. A principal problem in evolutionary graph theory is to obtain the probability that some initial population of mutants will fixate on a graph, and to determine how that fixation probability depends on the structure of that graph. A fluctuating mutant population on a graph can be considered as a random walk. Martingales exploit symmetry in the steps of a random walk to yield exact analytical expressions for fixation probabilities. They do not require simplifying assumptions such as large population sizes or weak selection. In this paper, we show how martingales can be used to obtain fixation probabilities for symmetric evolutionary graphs. We obtain simpler expressions for the fixation probabilities of star graphs and complete bipartite graphs than have been previously reported and show that these graphs do not amplify selection for advantageous mutations under all conditions.

A model is presented for predicting the capture of magnetic micro/nano-particles in a bioseparation microsystem. This bioseparator consists of an array of conductive elements embedded beneath a rectangular microfluidic channel. The magnetic particles are introduced into the microchannel in solution, and are attracted and held by the magnetic force produced by the energized elements. Analytical expressions are obtained for the dominant magnetic and fluidic forces on the particles as they move through the microchannel. These expressions are included in the equations of motion, which are solved numerically to predict particle trajectories and capture time. This model is well-suited for parametric analysis of particle capture taking into account variations in particle size, material properties, applied current, microchannel dimensions, fluid properties, and flow velocity.

The core design of spiking neurones is remarkably similar throughout the animal kingdom. Their basic function as fast-signalling thresholding cells might have been established very early in their evolutionary history. Identifying the selection pressures that drove animals to evolve spiking neurones could help us interpret their design and function today. We review fossil, ecological and molecular evidence to investigate when and why animals evolved spiking neurones. Fossils suggest that animals evolved nervous systems soon after the advent of animal-on-animal predation, 550 million years ago (MYa). Between 550 and 525 MYa, we see the first fossil appearances of many animal innovations, including eyes. Animal behavioural complexity increased during this period as well, as evidenced by their traces, suggesting that nervous systems were an innovation of that time. Fossils further suggest that, before 550 MYa, animals were either filter feeders or microbial mat grazers. Extant sponges and Trichoplax perform these tasks using energetically cheaper alternatives than spiking neurones. Genetic evidence testifies that nervous systems evolved before the protostome-deuterostome split. It is less clear whether nervous systems evolved before the cnidarian-bilaterian split, so cnidarians and bilaterians might have evolved their nervous systems independently. The fossil record indicates that the advent of predation could fit into the window of time between those two splits, though molecular clock studies dispute this claim. Collectively, these lines of evidence indicate that animals evolved spiking neurones soon after they started eating each other. The first sensory neurones could have been threshold detectors that spiked in response to other animals in their proximity, alerting them to perform precisely timed actions, such as striking or fleeing.

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

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