Lineage EM algorithm for inferring latent states from cellular lineage trees.

SUMMARY: Phenotypic variability in a population of cells can work as the bet-hedging of the cells under an unpredictably changing environment, the typical example of which is the bacterial persistence. To understand the strategy to control such phenomena, it is indispensable to identify the phenotype of each cell and its inheritance. Although recent advancements in microfluidic technology offer us useful lineage data, they are insufficient to directly identify the phenotypes of the cells. An alternative approach is to infer the phenotype from the lineage data by latent-variable estimation. To this end, however, we must resolve the bias problem in the inference from lineage called survivorship bias. In this work, we clarify how the survivorship bias distorts statistical estimations. We then propose a latent-variable estimation algorithm without the survivorship bias from lineage trees based on an expectation-maximization (EM) algorithm, which we call lineage EM algorithm (LEM). LEM provides a statistical method to identify the traits of the cells applicable to various kinds of lineage data. AVAILABILITY AND IMPLEMENTATION: An implementation of LEM is available at https://github.com/so-nakashima/Lineage-EM-algorithm. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

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.

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.

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.

Discreteness-induced transitions in multibody reaction systems.

A decrease in system size can induce qualitatively different behavior compared to the macroscopic behavior of the corresponding large-size system. The mechanisms of this transition, which is known as the small-size transition, can be attributed to either a relative increase in the noise intensity or to the discreteness of the state space due to the small system size. The former mechanism has been intensively investigated using several toy and realistic models. However, the latter has rarely been analyzed and is sometimes confused with the former, because a toy model that extracts the essence of the discreteness-induced transition mechanism is lacking. In this work, we propose a one- and three-body reaction system as a minimal model of the discreteness-induced transition and derive the conditions under which this transition occurs in more complex systems. This work enriches our understanding of the influence of small system size on system behavior.

Motif analysis for small-number effects in chemical reaction dynamics.

The number of molecules involved in a cell or subcellular structure is sometimes rather small. In this situation, ordinary macroscopic-level fluctuations can be overwhelmed by non-negligible large fluctuations, which results in drastic changes in chemical-reaction dynamics and statistics compared to those observed under a macroscopic system (i.e., with a large number of molecules). In order to understand how salient changes emerge from fluctuations in molecular number, we here quantitatively define small-number effect by focusing on a "mesoscopic" level, in which the concentration distribution is distinguishable both from micro- and macroscopic ones and propose a criterion for determining whether or not such an effect can emerge in a given chemical reaction network. Using the proposed criterion, we systematically derive a list of motifs of chemical reaction networks that can show small-number effects, which includes motifs showing emergence of the power law and the bimodal distribution observable in a mesoscopic regime with respect to molecule number. The list of motifs provided herein is helpful in the search for candidates of biochemical reactions with a small-number effect for possible biological functions, as well as for designing a reaction system whose behavior can change drastically depending on molecule number, rather than concentration.

Fluctuation Relations of Fitness and Information in Population Dynamics.

Phenotype switching with and without sensing environment is a common strategy of organisms to survive in a fluctuating environment. Understanding the evolutionary advantages of switching and sensing requires a quantitative evaluation of their fitness gain and its fluctuation together with the conditions for the switching and sensing strategies being adapted to a given environment. In this work, by using a pathwise formulation of the population dynamics, we show that the optimal switching strategy is characterized by a consistency condition for time-forward and backward path probabilities. The formulation also clarifies the underlying information-theoretic aspect of selection as a passive information compression. The loss of fitness by a suboptimal strategy is also shown to satisfy a fluctuation relation, which provides us with the information on how environmental fluctuation impacts the advantages of the optimal strategy. These results are naturally extended to the situation that organisms can use an environmental signal by actively sensing the environment. The fluctuation relations of the fitness gain by sensing are derived in which the multivariate mutual information among the phenotype, the environment, and the signal plays the role to quantify the relevant information in the signal for the fitness gain.

Pathwise thermodynamic structure in population dynamics.

We reveal thermodynamic structure in population dynamics with phenotype switching. Mean fitness for a population of organisms is determined by a thermodynamic variational principle described by the large deviation of phenotype-switching dynamics. Owing to this variational principle, a response relation of the mean fitness with respect to changes of environments and phenotype-switching dynamics is represented as a thermodynamic differential form. Furthermore, we discuss the strength of the selection by using the difference between time-forward and time-backward (retrospective) processes.

Fluctuation theorem for the renormalized entropy change in the strongly nonlinear nonequilibrium regime.

A nonlinear relaxation process is considered for a macroscopic thermodynamic quantity, generalizing recent work by Taniguchi and Cohen [J. Stat. Phys. 126, 1 (2006)] that was based on the Onsager-Machlup theory. It is found that the fluctuation theorem holds in the nonlinear nonequilibrium regime if the change of entropy characterized by local equilibria is appropriately renormalized. The fluctuation theorem for the ordinary entropy change is recovered in the linear near-equilibrium case.