Instabilities and Turing patterns in stochastic spatiotemporal systems in which a fraction of an evolving population, after undergoing a series of dynamic transitions, returns to its original state, remain largely unexplored. Adopting an epidemic model incorporating reinfections as an exemplar of such a system, we present stability and pattern-formation analyses of the stochastic reaction-diffusion equations that represent the model. Saturation effects in epidemic spread lead to nonlinear considerations, while random environmental effects motivate a stochastic term. Turing bifurcation and the emergence of equilibrium patterns are analysed with respect to three fundamental parameters - reinfection, saturation, and noise intensity. Using higher-order stability analysis and stochastic averaging, we find the Turing instability and also uncover self–organized, distinct equilibrium patterns of infection spread. Additionally, results elucidating the effects of stochastic excitation and its intensity, as well as the competing influence of saturation and reinfection on stability and pattern formation, are presented. The results are also expected to be broadly significant beyond epidemic modelling, for studies of noise-induced instabilities and morphogenesis in spatiotemporal nonlinear dynamical systems.

While RNA folding prediction remains challenging, even with machine and deep learning methods, it can also be approached from a topological mathematics perspective. The purpose of the present paper is to elucidate this problem for students and researchers in both the mathematical physics and biology fields, fostering interest in developing novel theoretical and applied solutions that could propel RNA research forward. With this intention, the mathematical method, based on matrix field theory, to compute the topological classification of RNA structures is reviewed. Similarly, McGenus, a computational software that exploits matrix field theory for topological and folding predictions, is examined. To further illustrate the outcomes of this mathematical approach, two types of analyses are performed: the prediction results from McGenus are compared with topological information extracted from experimentally-determined RNA structures, and the topology of RNA structures is investigated for biological significance, both in evolutionary and functional terms. Lastly, we advocate for more research efforts to be conducted at the intersection between physics, mathematics and biology, with a particular focus on the potential contributions that topology can make to the study of RNA folding and structure.

Coarse-grain models are essential to understand the biological function of DNA molecules because the length and time scales of the sequence-dependent physical properties of DNA are often beyond the reach of experimental and all-atom computational methods. Simulating coarse-grain models of DNA, e.g. using Langevin dynamics, requires the parametrization of both potential and kinetic energy functions. Many studies have shown that the flexibility (i.e., potential energy) of a DNA molecule depends on its sequence. In contrast, little is known about the sequence-dependence of DNA mass parameters required to model its kinetic energy. In this paper, an algebraic expression is derived for the kinetic energy as a function of linear and angular velocities of each DNA base parameterized by its mass, center of mass, and rotational inertia tensor. The parameters of this function are then approximated from a set of fine-grain molecular dynamics simulations representing all combinations of the four DNA base pairs AT, TA, GC, and CG, in different sequence contexts. Compatibility conditions associated with the assumption of each base being modeled as a rigid body were verified to be good approximations. The kinetic parameters were found to be significantly different between the four G, C, A, and T bases, and to not be dependent on the sequence context. This suggests that the effective kinetic parameters of a DNA base may depend only on the base itself, not on its neighbors.

The emergence of an autocatalytic network from an available set of elements is a fundamental step in early evolutionary processes, such as the origin of metabolism. Given the set of elements, the reactions between them (chemical or otherwise), and with various elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set is a subset R' of reactions that is self-generating from a given food set, and with each reaction in R' being catalysed from within R'. RAF theory has been applied to various phenomena in theoretical biology, and a key feature of the approach is that it is possible to efficiently identify and classify RAFs within large systems. This is possible because RAFs can be described as the (nonempty) subsets of the reactions that are the fixed points of an (efficiently computable) interior map that operates on subsets of reactions. Although the main generic results concerning RAFs can be derived using just this property, we show that for systems with at least 12 reactions there are generic results concerning RAFs that cannot be proven using the interior operator property alone.

The variance effective population size (N_{eV}) is frequently used to quantify the expected rate at which a population’s allele frequencies change over time. The purpose of this paper is to find expressions for the global N_{eV} of a spatially structured population that are of interest for conservation of species. Since N_{eV} depends on allele frequency change, we start by dividing the cause of allele frequency change into genetic drift within subpopulations (I) and a second component mainly due to migration between subpopulations (II). We investigate in detail how these two components depend on the way in which subpopulations are weighted as well as their dependence on parameters of the model such a migration rates, and local effective and census sizes. It is shown that under certain conditions the impact of II is eliminated, and N_{eV} of the metapopulation is maximized, when subpopulations are weighted proportionally to their long term reproductive contributions. This maximal N_{eV} is the sought for global effective size, since it approximates the gene diversity effective size N_{eGD}, a quantifier of the rate of loss of genetic diversity that is relevant for conservation of species and populations. We also propose two novel versions of N_{eV}, one of which (the backward version of N_{eV}) is most stable, exists for most populations, and is closer to N_{eGD} than the classical notion of N_{eV}. Expressions for the optimal length of the time interval for measuring genetic change are developed, that make it possible to estimate any version of N_{eV} with maximal accuracy.

Mutations are often described as being "random with respect to fitness." Here we show that the experiments used to establish randomness with respect to fitness are only capable of showing that mutations are random with respect to current external selection. Current debates about whether or not mutations are directed may be at least partially resolved by making use of this distinction. Additionally, this distinction has important mathematical, experimental, and inferential implications.

We propose a framework for the description of the effects of vaccinations on the spreading of an epidemic disease. Different vaccines can be dosed, each providing different immunization times and immunization levels. Differences due to individuals’ ages are accounted for through the introduction of either a continuous age structure or a discrete set of age classes. Extensions to gender differences or to distinguish fragile individuals can also be considered. Within this setting, vaccination strategies can be simulated, tested and compared, as is explicitly described through numerical integrations.

We propose a novel model to explain the mechanisms underlying dominance hierarchical structures. Guided by a predetermined social convention, agents with limited cognitive abilities optimize their strategies in a Hawk-Dove game. We find that several commonly observed hierarchical structures in nature such as linear hierarchy and despotism, emerge as the total fitness-maximizing social structures given different levels of cognitive abilities.