Asymptotic decoupling of population growth rate and cell size distribution

The rate at which individual bacterial cells grow depends on the concentrations of cellular components such as ribosomes and proteins. These concentrations continuously fluctuate over time and are inherited from mother to daughter cells leading to correlations between the growth rates of cells across generations. Division sizes of cells are also stochastic and correlated across generations due to a phenomenon known as cell size regulation. Fluctuations and correlations from both growth and division processes affect the population dynamics of an exponentially growing culture. Here, we provide analytic solutions for the population dynamics of cells with continuously fluctuating growth rates coupled with a generic model of cell-size regulation. We show that in balanced growth, the effects of growth and division processes decouple; the population growth rate only depends on the single-cell growth rate process, and the population cell size distribution only depends on the model of division and cell size regulation. The population growth rate is always higher than the average single-cell growth rate, and the difference increases with growth rate variability and its correlation time. This difference also sets the timescale for the population to reach its steady state. We provide analytical solutions for oscillations in population growth rate and traveling waves in size distribution during this approach to the steady state

Stochastic dynamics in spatially extended physical and biological systems

In this thesis, I discuss three different problems of stochastic nature in spatially extended systems: (1) a noise induced mechanism for the emergence of biological homochirality in early life self-replicators, (2) the amplification effect of nonnormality on stochastic Turing patterns in reaction diffusion systems, and (3) the velocity statistics of edge dislocations in plastic deformation of crystalline material. In Part I, I present a new model for the origin of homochirality, the observed single-handedness of biological amino acids and sugars, in prebiotic self-replicator. Homochirality has long been attributed to autocatalysis, a frequently assumed precursor for self-replication. However, the stability of homochiral states in deterministic autocatalytic systems relies on cross inhibition of the two chiral states, an unlikely scenario for early life self-replicators. Here, I present a theory for a stochastic individual-level model of autocatalysis due to early life self-replicators. Without chiral inhibition, the racemic state is the global attractor of the deterministic dynamics, but intrinsic multiplicative noise stabilizes the homochiral states, in both well-mixed and spatially-extended systems. I conclude that autocatalysis is a viable mechanism for homochirality, without imposing additional nonlinearities such as chiral inhibition. In Part II, I study the amplification effect of nonnormality on the steady state amplitude of fluctuation-induced Turing patterns. The phenomenon occurs generally in Turing-like pattern forming systems such as reaction-diffusion systems, does not require a large separation of diffusion constant, and yields pattern whose amplitude can be orders of magnitude larger than the fluctuations that cause the patterns. The analytical treatment shows that patterns are amplified due to an interplay between noise, non-orthogonality of eigenvectors of the linear stability matrix, and a separation of time scales, all built-in feature of stochastic pattern forming systems. I conclude that many examples of biological pattern formations are nonnormal stochastic patterns. In Part III, I study the dynamics of edge dislocations with parallel Burgers vectors, moving in the same slip plane, by mapping the problem onto Dyson's model of a two-dimensional Coulomb gas confined in one dimension. I show that the tail distribution of the velocity of dislocations is power-law in form, as a consequence of the pair interaction of nearest neighbors in one dimension. In two dimensions, I show the presence of a pairing phase transition in a system of interacting dislocations with parallel Burgers vectors. The scaling exponent of the velocity distribution at effective temperatures well below this pairing transition temperature can be derived from the nearest-neighbor interaction, while near the transition temperature, the distribution deviates from the form predicted by the nearest-neighbor interaction, suggesting the presence of collective effects

Noise-induced symmetry breaking far from equilibrium and the emergence of biological homochirality

The origin of homochirality, the observed single-handedness of biological amino acids and sugars, has long been attributed to autocatalysis, a frequently assumed precursor for early life self-replication. However, the stability of homochiral states in deterministic autocatalytic systems relies on cross-inhibition of the two chiral states, an unlikely scenario for early life self-replicators. Here we present a theory for a stochastic individual-level model of autocatalytic prebiotic self-replicators that are maintained out of thermal equilibrium. Without chiral inhibition, the racemic state is the global attractor of the deterministic dynamics, but intrinsic multiplicative noise stabilizes the homochiral states. Moreover, we show that this noise-induced bistability is robust with respect to diffusion of molecules of opposite chirality, and systems of diffusively coupled autocatalytic chemical reactions synchronize their final homochiral states when the self-replication is the dominant production mechanism for the chiral molecules. We conclude that nonequilibrium autocatalysis is a viable mechanism for homochirality, without imposing additional nonlinearities such as chiral inhibition.United States. National Aeronautics and Space Administration (NNA13AA91A

Evolutionary dynamics in non-Markovian models of microbial populations

In the past decade, great strides have been made to quantify the dynamics of single-cell growth and division in microbes. In order to make sense of the evolutionary history of these organisms, we must understand how features of single-cell growth and division influence evolutionary dynamics. This requires us to connect processes on the single-cell scale to population dynamics. Here, we consider a model of microbial growth in finite populations which explicitly incorporates the single-cell dynamics. We study the behavior of a mutant population in such a model and ask: can the evolutionary dynamics be coarse-grained so that the forces of natural selection and genetic drift can be expressed in terms of the long-term fitness We show that it is in fact not possible, as there is no way to define a single fitness parameter (or reproductive rate) that defines the fate of an organism even in a constant environment. This is due to fluctuations in the population averaged division rate. As a result, various details of the single-cell dynamics affect the fate of a new mutant independently from how they affect the long-term growth rate of the mutant population. In particular, we show that in the case of neutral mutations, variability in generation times increases the rate of genetic drift, and in the case of beneficial mutations, variability decreases its fixation probability. Furthermore, we explain the source of the persistent division rate fluctuations and provide analytic solutions for the fixation probability as a multispecies generalization of the Euler-Lotka equation