Phenotype-switching with and without sensing environment is a ubiquitous strategy of organisms to survive in fluctuating environment. Fitness of a population of organisms with phenotype-switching may be constrained and restricted by hidden relations as the entropy production in a thermal system with and without sensing and feedback is well-characterized via fluctuation relations (FRs) . In this work, we derive such FRs of fitness together with an underlying information-theoretic structure in selection. By using path-integral formulation of a multi-phenotype population dynamics, we clarify that the optimal switching strategy is characterized as a consistency condition for time-forward and backward path probabilities. Within the formulation, the selection is regarded as passive information compression, and the loss of fitness from the optimal strategy is shown to satisfy various FRs that constrain the average and fluctuation of the loss. These results are naturally extended to the situation that organisms can use an environmental signal by actively sensing the environment. FRs of 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 fitness gain. Submitted to PRL on 25/Jul/2014; resubmitted to PRL for revision on 10/Apr/2015. PACS numbers: Valid PACS appear herePhenotype-switching is a strategy of living systems to survive in stochastically changing environment [1]. Even if no environmental information is available, diversification of phenotypes by stochastic switching (known as bethedging) can lead to gain of fitness when a subpopulation with a resistant phenotype can survive in a harsh environment to feed the next generation [2][3][4]. If environmental signal that conveys information of the environment is exploitable, further gain of fitness is possible by switching into the phenotypes adapted to the future environment (known as decision-making) [5][6][7]. Ubiquitous observations of phenotype-switching and environmental sensing in living systems from higher organisms down to bacteria implies its actual fitness advantage over the diversification loss and the metabolic load of switching and sensing mechanisms [8][9][10][11].The fitness gain enjoyed by switching and sensing, however, must be constrained by the environmental statistics and the sensed information. On the one hand, previous investigations clarified such constrains for average fitness gain at least in specific situations [2,3,7,[12][13][14][15]. On the other hand, the similarity between evolutionary dynamics and statistical physics [16][17][18] suggests that more general relations may exist as the series of fluctuation relations (FRs) characterize not only the average but also the fluctuation of entropy production in a thermal system with and without sensing and feedback [19,20]. Finding such relations is crucial to understand the constraints and predicability of adaptive dynamics of organisms in ever cha...

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.

We establish a Hessian geometric structure in chemical thermodynamics which describes chemical reaction networks (CRNs) with equilibrium states. In our setup, the assumptions of ideal gas and mass-action kinetics are not required. The existence and uniqueness condition of the equilibrium state is derived by using the Legendre duality inherent to the Hessian structure. The entropy production during a relaxation to the equilibrium state can be evaluated by the Bregman divergence. Furthermore, the equilibrium state is characterized by four distinct minimization problems of the divergence, which are yielded by the generalized Pythagorean theorem originating from the dual flatness. For the ideal-gas case, we confirm that our existence and uniqueness condition implies Birch's theorem, and the entropy production represented by the divergence coincides with the generalized Kullback-Leibler divergence. In addition, under mass-action kinetics, our general framework reproduces the local detailed balance condition.

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 informationthermodynamic 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 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.

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.