Fluctuation Relations of Fitness and Information in Population Dynamics

Abstract

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...

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