Fluctuation Relations of Fitness and Information in Population Dynamics

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.

Kobayashi T. Population dynamics models for various forms of adaptation

Adaptability to changing environments is one of the universal characteristics of living organisms. Because individual modes of adaptation are diverse, a unified understanding of these diverse modes is essential to comprehend adaptation. Adaptations can be categorized from at least two perspectives with respect to information. One is the passivity and activity of adaptation and the other is the type of information transmission. In Darwinian natural selection, organisms are selected among randomly generated traits under which individual organisms are passive in the sense that they do not process any environmental information. On the other hand, organisms can also adapt by sensing their environment and changing their traits. This is an active adaptation in that it makes use of environmental information. In terms of information transfer, adaptation through phenotypic heterogeneity, such as bacterial bet-hedging, is intragenerational in which traits are not passed on to the next generation. In contrast, adaptation through genetic diversity is intergenerational. The theory of population dynamics enables us to unify these various modes of adaptations and their properties can be analyzed qualitatively and quantitatively using techniques from quantitative genetics and information thermodynamics. In addition, such methods can be applied to situations where organisms can learn from past experiences and pass them on from generation to generation. In this work, we introduce the unified theory of biological adaptation based on population dynamics and show its potential applications to evaluate the fitness value of information and to analyze experimental lineage tree data. Finally, we discuss future perspectives for its development. This review article is an extended version of the Japanese article in SEIBUTSU BUTSURI Vol. 57, p. 287-290 (2017).

A unified framework for measuring selection on cellular lineages and traits

Intracellular states probed by gene expression profiles and metabolic activities are intrinsically noisy, causing phenotypic variations among cellular lineages. Understanding the adaptive and evolutionary roles of such variations requires clarifying their linkage to population growth rates. Extending a cell lineage statistics framework, here we show that a population's growth rate can be expanded by the cumulants of a fitness landscape that characterize how fitness distributes in a population. The expansion enables quantifying the contribution of each cumulant, such as variance and skewness, to population growth. We introduce a function that contains all the essential information of cell lineage statistics, including mean lineage fitness and selection strength. We reveal a relation between fitness heterogeneity and population growth rate response to perturbation. We apply the framework to experimental cell lineage data from bacteria to mammalian cells, revealing distinct levels of growth rate gain from fitness heterogeneity across environments and organisms. Furthermore, third or higher order cumulants' contributions are negligible under constant growth conditions but could be significant in regrowing processes from growth-arrested conditions. We identify cellular populations in which selection leads to an increase of fitness variance among lineages in retrospective statistics compared to chronological statistics. The framework assumes no particular growth models or environmental conditions, and is thus applicable to various biological phenomena for which phenotypic heterogeneity and cellular proliferation are important.

Generalized Euler-Lotka equation for correlated cell divisions

Cell division times in microbial populations display significant fluctuations that impact the population growth rate in a nontrivial way. If fluctuations are uncorrelated among different cells, the population growth rate is predicted by the Euler-Lotka equation, which is a classic result in mathematical biology. However, cell division times can be significantly correlated, due to physical properties of cells that are passed through generations. In this Letter, we derive an equation remarkably similar to the Euler-Lotka equation which is valid in the presence of correlations. Our exact result is based on large deviation theory and does not require particularly strong assumptions on the underlying dynamics. We apply our theory to a phenomenological model of bacterial cell division in E. coli and to experimental data. We find that the discrepancy between the growth rate predicted by the Euler-Lotka equation and our generalized version is relatively small, but large enough to be measurable by our approach.

Sex as information processing: Optimality and evolution

The long-term growth rate of populations in varying environments quantifies the evolutionary value of processing the information that biological individuals inherit from their ancestors and acquire from their environment. Previous models were limited to asexual reproduction with inherited information coming from a single parent with no recombination. We present a general extension to sexual reproduction and an analytical solution for a particular but important case, the infinitesimal model of quantitative genetics which assumes traits to be normally distributed. We study with this model the conditions under which sexual reproduction is advantageous and can evolve in the context of autocorrelated or directionally varying environments, mutational biases, spatial heterogeneities, and phenotypic plasticity. Our results generalize and unify previous analyses. We also examine the proposal made by Geodakyan that the presence of two phenotypically distinct sexes permits an optimal adaptation to varying environments. We verify that conditions exists where sexual dimorphism is adaptive but find that its evolutionary value does not generally compensate for the twofold cost of males.

Fitness Gain of Individually Sensed Information by Cells

Mutual information and its causal variant, directed information, have been widely used to quantitatively characterize the performance of biological sensing and information transduction. However, once coupled with selection in response to decision-making, the sensing signal could have more or less evolutionary value than its mutual or directed information. In this work, we show that an individually sensed signal always has a better fitness value, on average, than its mutual or directed information. The fitness gain, which satisfies fluctuation relations (FRs), is attributed to the selection of organisms in a population that obtain a better sensing signal by chance. A new quantity, similar to the coarse-grained entropy production in information thermodynamics, is introduced to quantify the total fitness gain from individual sensing, which also satisfies FRs. Using this quantity, the optimizing fitness gain of individual sensing is shown to be related to fidelity allocations for individual environmental histories. Our results are supplemented by numerical verifications of FRs, and a discussion on how this problem is linked to information encoding and decoding.

Linking lineage and population observables in biological branching processes

Using a population dynamics inspired by an ensemble of growing cells, a set of fluctuation theorems linking observables measured at the lineage and population levels is derived. One of these relations implies specific inequalities comparing the population doubling time with the mean generation time at the lineage or population levels. While these inequalities have been derived before for age-controlled models with negligible mother-daughter correlations, we show that they also hold for a broad class of size-controlled models. We discuss the implications of this result for the interpretation of a recent experiment in which the growth of bacteria strains has been probed at the single-cell level.

Many-body perturbation theory and fluctuation relations for interacting population dynamics

Population dynamics deals with the collective phenomena of living organisms, and it has attracted much attention since it is expected to explain how not only living organisms but also human beings have been adapted to varying environments. However, it is quite difficult to insist on a general statement on living organisms since mathematical models heavily depend on phenomena that we focus on. Recently, it was reported that the fluctuation relations on the fitness of living organisms held for a quite general problem setting. But, interactions between organisms were not incorporated in the problem setting, though interaction plays critical roles in collective phenomena in physics and population dynamics. In this paper, we propose interacting models for population dynamics and provide the perturbative theory of population dynamics. Then, we derive the variational principle and fluctuation relations for interacting population dynamics.

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.

Information-theoretic analysis of the directional influence between cellular processes

Inferring the directionality of interactions between cellular processes is a major challenge in systems biology. Time-lagged correlations allow to discriminate between alternative models, but they still rely on assumed underlying interactions. Here, we use the transfer entropy (TE), an information-theoretic quantity that quantifies the directional influence between fluctuating variables in a model-free way. We present a theoretical approach to compute the transfer entropy, even when the noise has an extrinsic component or in the presence of feedback. We re-analyze the experimental data from Kiviet et al. (2014) where fluctuations in gene expression of metabolic enzymes and growth rate have been measured in single cells of E. coli. We confirm the formerly detected modes between growth and gene expression, while prescribing more stringent conditions on the structure of noise sources. We furthermore point out practical requirements in terms of length of time series and sampling time which must be satisfied in order to infer optimally transfer entropy from times series of fluctuations.

Information Theory Broadens the Spectrum of Molecular Ecology and Evolution

Information or entropy analysis of diversity is used extensively in community ecology, and has recently been exploited for prediction and analysis in molecular ecology and evolution. Information measures belong to a spectrum (or q profile) of measures whose contrasting properties provide a rich summary of diversity, including allelic richness (q=0), Shannon information (q=1), and heterozygosity (q=2). We present the merits of information measures for describing and forecasting molecular variation within and among groups, comparing forecasts with data, and evaluating underlying processes such as dispersal. Importantly, information measures directly link causal processes and divergence outcomes, have straightforward relationship to allele frequency differences (including monotonicity that q=2 lacks), and show additivity across hierarchical layers such as ecology, behaviour, cellular processes, and nongenetic inheritance.

Transitions in optimal adaptive strategies for populations in fluctuating environments

Biological populations are subject to fluctuating environmental conditions. Different adaptive strategies can allow them to cope with these fluctuations: specialization to one particular environmental condition, adoption of a generalist phenotype that compromises between conditions, or population-wise diversification (bet hedging). Which strategy provides the largest selective advantage in the long run depends on the range of accessible phenotypes and the statistics of the environmental fluctuations. Here, we analyze this problem in a simple mathematical model of population growth. First, we review and extend a graphical method to identify the nature of the optimal strategy when the environmental fluctuations are uncorrelated. Temporal correlations in environmental fluctuations open up new strategies that rely on memory but are mathematically challenging to study: We present analytical results to address this challenge. We illustrate our general approach by analyzing optimal adaptive strategies in the presence of trade-offs that constrain the range of accessible phenotypes. Our results extend several previous studies and have applications to a variety of biological phenomena, from antibiotic resistance in bacteria to immune responses in vertebrates.

Stochastic and information-thermodynamic structures of population dynamics in a fluctuating environment

Adaptation in a fluctuating environment is a process of fueling environmental information to gain fitness. Living systems have gradually developed strategies for adaptation from random and passive diversification of the phenotype to more proactive decision making, in which environmental information is sensed and exploited more actively and effectively. Understanding the fundamental relation between fitness and information is therefore crucial to clarify the limits and universal properties of adaptation. In this work, we elucidate the underlying stochastic and information-thermodynamic structure in this process, by deriving causal fluctuation relations (FRs) of fitness and information. Combined with a duality between phenotypic and environmental dynamics, the FRs reveal the limit of fitness gain, the relation of time reversibility with the achievability of the limit, and the possibility and condition for gaining excess fitness due to environmental fluctuation. The loss of fitness due to causal constraints and the limited capacity of real organisms is shown to be the difference between time-forward and time-backward path probabilities of phenotypic and environmental dynamics. Furthermore, the FRs generalize the concept of the evolutionary stable state (ESS) for fluctuating environment by giving the probability that the optimal strategy on average can be invaded by a suboptimal one owing to rare environmental fluctuation. These results clarify the information-thermodynamic structures in adaptation and evolution.

The genealogical decomposition of a matrix population model with applications to the aggregation of stages

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

Inferring fitness landscapes and selection on phenotypic states from single-cell genealogical data

Recent advances in single-cell time-lapse microscopy have revealed non-genetic heterogeneity and temporal fluctuations of cellular phenotypes. While different phenotypic traits such as abundance of growth-related proteins in single cells may have differential effects on the reproductive success of cells, rigorous experimental quantification of this process has remained elusive due to the complexity of single cell physiology within the context of a proliferating population. We introduce and apply a practical empirical method to quantify the fitness landscapes of arbitrary phenotypic traits, using genealogical data in the form of population lineage trees which can include phenotypic data of various kinds. Our inference methodology for fitness landscapes determines how reproductivity is correlated to cellular phenotypes, and provides a natural generalization of bulk growth rate measures for single-cell histories. Using this technique, we quantify the strength of selection acting on different cellular phenotypic traits within populations, which allows us to determine whether a change in population growth is caused by individual cells’ response, selection within a population, or by a mixture of these two processes. By applying these methods to single-cell time-lapse data of growing bacterial populations that express a resistance-conferring protein under antibiotic stress, we show how the distributions, fitness landscapes, and selection strength of single-cell phenotypes are affected by the drug. Our work provides a unified and practical framework for quantitative measurements of fitness landscapes and selection strength for any statistical quantities definable on lineages, and thus elucidates the adaptive significance of phenotypic states in time series data. The method is applicable in diverse fields, from single cell biology to stem cell differentiation and viral evolution.

Selection is a ubiquitous process in biological populations in which individuals are endowed with heterogeneous reproductive abilities, and it occurs even among genetically homogeneous cells due to the existence of phenotypic noise. Unlike genotypes, which can remain stable for many generations, phenotypic fluctuations at the single cell level are often comparable to cellular generation times. For this reason, quantifying the contribution of specific phenotypic states to cellular fitness remains a major challenge. Here, we develop a method to measure the fitness landscape and selection strength acting on diverse cellular phenotypes by employing a novel conceptual framework in which cellular histories are regarded as a basic unit of selection. With this framework, one can tell quantitatively whether a population adapts to environmental changes by selection or through individual responses. This new analytical approach to genetics reveals the roles of heterogeneous expression patterns and dynamics without directly perturbing genes. Applications in diverse fields including stem cell differentiation and viral evolution are discussed.

Steady-state thermodynamics for population growth in fluctuating environments

We report that population dynamics in fluctuating environments is characterized by a mathematically equivalent structure to steady-state thermodynamics. By employing the structure, population growth in fluctuating environments is decomposed into housekeeping and excess parts. The housekeeping part represents the integral of the stationary growth rate for each condition during a history of the environmental change. The excess part accounts for the excess growth induced by environmental fluctuations. Focusing on the excess growth, we obtain a Clausius inequality, which gives the upper bound of the excess growth. The equality is shown to be achieved in quasistatic environmental changes. We also clarify that this bound can be evaluated by the "lineage fitness", which is an experimentally observable quantity.