Steady-state thermodynamics for population growth in fluctuating environments

Information geometric bound on general chemical reaction networks

We investigate the convergence of chemical reaction networks (CRNs), aiming to establish an upper bound on their reaction rates. The nonlinear characteristics and discrete composition of CRNs pose significant challenges in this endeavor. To circumvent these complexities, we adopt an information geometric perspective, utilizing the natural gradient to formulate a nonlinear system. This system effectively determines an upper bound for the dynamics of CRNs. We corroborate our methodology through numerical simulations, which reveal that our constructed system converges more rapidly than CRNs within a particular class of reactions. This class is defined by the count of chemicals, the highest stoichiometric coefficients in the reactions, and the total number of reactions involved. Further, we juxtapose our approach with traditional methods, illustrating that the latter falls short in providing an upper bound for CRN reaction rates. Although our investigation centers on CRNs, the widespread presence of hypergraphs across various disciplines, ranging from natural sciences to engineering, indicates potential wider applications of our method, including in the realm of information science.

Derivation of evolutionary entropy in the steady-state thermodynamics of evolutionary dynamics

We investigate the parallel mutation-selection model with varying population size, which is formulated in terms of individuals undergoing the evolution processes of reproduction and mutation, to derive evolutionary entropy. Under the framework of the steady-state thermodynamics for evolutionary dynamics, the excess growth (the difference between the maximum growth rate and the total growth rate) can be interpreted as the evolutionary entropy defined in terms of the probability distributions characteristic of evolutionary dynamics. The Clausius inequality states that the excess growth is always less than or equal to the entropy difference in evolutionary dynamics. Analytically, by using the genome sequence length L=3, we derive the growth after evolutionary dynamics with the finite number of environmental changes and calculate the entropy difference during this evolutionary dynamics, and we verify the Clausius inequality. Furthermore, by taking the infinite limit of the number of environmental changes, we verify that the equality holds for the quasistatic environmental change. By using the derived evolutionary entropy, we propose the thermodynamic relation between the free fitness and evolutionary entropy, where the free fitness is the maximum growth rate possible. Numerically, we use the Gillespie-type simulations, which provides direct realizations of the master equation governing evolutionary dynamics, to verify the Clausius inequality and we find that the simulation results are in good agreement with the analytic results.

Large Deviation Principle Linking Lineage Statistics to Fitness in Microbial Populations

In exponentially proliferating populations of microbes, the population doubles at a rate less than the average doubling time of a single-cell due to variability at the single-cell level. It is known that the distribution of generation times obtained from a single lineage is, in general, insufficient to determine a population's growth rate. Is there an explicit relationship between observables obtained from a single lineage and the population growth rate? We show that a population's growth rate can be represented in terms of averages over isolated lineages. This lineage representation is related to a large deviation principle that is a generic feature of exponentially proliferating populations. Due to the large deviation structure of growing populations, the number of lineages needed to obtain an accurate estimate of the growth rate depends exponentially on the duration of the lineages, leading to a nonmonotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.

The interplay of phenotypic variability and fitness in finite microbial populations

In isogenic microbial populations, phenotypic variability is generated by a combination of stochastic mechanisms, such as gene expression, and deterministic factors, such as asymmetric segregation of cell volume. Here we address the question: how does phenotypic variability of a microbial population affect its fitness? While this question has previously been studied for exponentially growing populations, the situation when the population size is kept fixed has received much less attention, despite its relevance to many natural scenarios. We show that the outcome of competition between multiple microbial species can be determined from the distribution of phenotypes in the culture using a generalization of the well-known Euler-Lotka equation, which relates the steady-state distribution of phenotypes to the population growth rate. We derive a generalization of the Euler-Lotka equation for finite cultures, which relates the distribution of phenotypes among cells in the culture to the exponential growth rate. Our analysis reveals that in order to predict fitness from phenotypes, it is important to understand how distributions of phenotypes obtained from different subsets of the genealogical history of a population are related. To this end, we derive a mapping between the various ways of sampling phenotypes in a finite population and show how to obtain the equivalent distributions from an exponentially growing culture. Finally, we use this mapping to show that species with higher growth rates in exponential growth conditions will have a competitive advantage in the finite culture.

Fitness response relation of a multitype age-structured population dynamics

We construct a pathwise formulation for a multitype age-structured population dynamics, which involves an age-dependent cell replication and transition of gene- or phenotypes. By employing the formulation, we derive a variational representation of the stationary population growth rate; the representation comprises a tradeoff relation between growth effects and a single-cell intrinsic dynamics described by a semi-Markov process. This variational representation leads to a response relation of the stationary population growth rate, in which statistics on a retrospective history work as the response coefficients. These results contribute to predicting and controlling growing populations based on experimentally observed cell-lineage information.

Making sense of snapshot data: ergodic principle for clonal cell populations

Population growth is often ignored when quantifying gene expression levels across clonal cell populations. We develop a framework for obtaining the molecule number distributions in an exponentially growing cell population taking into account its age structure. In the presence of generation time variability, the average acquired across a population snapshot does not obey the average of a dividing cell over time, apparently contradicting ergodicity between single cells and the population. Instead, we show that the variation observed across snapshots with known cell age is captured by cell histories, a single-cell measure obtained from tracking an arbitrary cell of the population back to the ancestor from which it originated. The correspondence between cells of known age in a population with their histories represents an ergodic principle that provides a new interpretation of population snapshot data. We illustrate the principle using analytical solutions of stochastic gene expression models in cell populations with arbitrary generation time distributions. We further elucidate that the principle breaks down for biochemical reactions that are under selection, such as the expression of genes conveying antibiotic resistance, which gives rise to an experimental criterion with which to probe selection on gene expression fluctuations.

Delayed bet-hedging resilience strategies under environmental fluctuations

Many biological populations, such as bacterial colonies, have developed through evolution a protection mechanism, called bet hedging, to increase their probability of survival under stressful environmental fluctuation. In this context, the concept of preadaptation refers to a common type of bet-hedging protection strategy in which a relatively small number of individuals in a population stochastically switch their phenotypes to a dormant metabolic state in which they increase their probability of survival against potential environmental shocks. Hence, if an environmental shock took place at some point in time, preadapted organisms would be better adapted to survive and proliferate once the shock is over. In many biological populations, the mechanisms of preadaptation and proliferation present delays whose influence in the fitness of the population are not well understood. In this paper, we propose a rigorous mathematical framework to analyze the role of delays in both preadaptation and proliferation mechanisms in the survival of biological populations, with an emphasis on bacterial colonies. Our theoretical framework allows us to analytically quantify the average growth rate of a bet-hedging bacterial colony with stochastically delayed reactions with arbitrary precision. We verify the accuracy of the proposed method by numerical simulations and conclude that the growth rate of a bet-hedging population shows a nontrivial dependency on their preadaptation and proliferation delays. Contrary to the current belief, our results show that faster reactions do not, in general, increase the overall fitness of a biological population.