Fitness value of information with delayed phenotype switching: Optimal performance with imperfect sensing

Alexander S. Moffett, Nigel Wallbridge, Carrol Plummer, and Andrew W. Eckford

Phys. Rev. E 102, 052403 – Published 6 November 2020

Abstract

The ability of organisms to accurately sense their environment and respond accordingly is critical for evolutionary success. However, exactly how the sensory ability influences fitness is a topic of active research, while the necessity of a time delay between when unreliable environmental cues are sensed and when organisms can mount a response has yet to be explored at any length. Accounting for this delay in phenotype response in models of population growth, we find that a critical error probability can exist under certain environmental conditions: An organism with a sensory system with any error probability less than the critical value can achieve the same long-term growth rate as an organism with a perfect sensing system. We also observe a tradeoff between the evolutionary value of sensory information and robustness to error, mediated by the rate at which the phenotype distribution relaxes to steady state. The existence of the critical error probability could have several important evolutionary consequences, primarily that sensory systems operating at the nonzero critical error probability may be evolutionarily optimal.

Received 4 June 2020
Revised 2 September 2020
Accepted 12 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052403

Physics Subject Headings (PhySH)

Gene expression Survival strategies

Populations

Information theory Markovian processes

Physics of Living Systems

Authors & Affiliations

Alexander S. Moffett ¹, Nigel Wallbridge ², Carrol Plummer ², and Andrew W. Eckford ^1,*

¹Department of Electrical Engineering and Computer Science, York University, Toronto, Ontario M3J 1P3, Canada
²Vivent SaRL, Crans-près-Céligny 1299, Switzerland

^*aeckford@yorku.ca

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Vol. 102, Iss. 5 — November 2020

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Figure 1
Model schematic. This schematic illustrates an example of the process an individual in our model can follow. In this case, the environment is in state $e_{1}$ and the phenotype of the individual is in state $ϕ_{2}$ at time $t$ . These two variables alone determine the fitness of our individual at time $t$ , which is $w_{ϕ_{2} e_{1}}$ . The environment perceived by our example individual is chosen according to the error probability $ε$ . Our example individual correctly perceives the environment, so the perceived environment is ${\tilde{e}}_{1}$ , chosen with probability $1 - ε$ . The individual then expresses phenotype switching probabilities $χ_{{\tilde{e}}_{1}}$ and $ω_{{\tilde{e}}_{1}}$ accordingly. However, because the phenotype at time $t$ is $ϕ_{2}$ , only $ω_{{\tilde{e}}_{1}}$ is relevant at the moment. The environmental state for $t + 1$ is then chosen according to the environmental switching probabilities $α$ and $β$ , where $α$ is the relevant value because the environment is in $e_{1}$ at time $t$ . Here, state $e_{2}$ is chosen with probability $α$ . The organism chooses phenotype $ϕ_{2}$ for time $t + 1$ according to the probability $1 - ω_{{\tilde{e}}_{1}}$ . The fitness of our individual is now $w_{ϕ_{2} e_{2}}$ . Then the perceived state can again be chosen and the process repeated. This figure was created using Inkscape 0.92 [22].
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Figure 2
The effects of environment sensing error probability on long-term growth rate. (a) The fitness value of information ( ${\hat{Λ}}_{k \to \infty}^{(a, ε)} - {\hat{Λ}}^{(η)}$ ) for $k \to \infty$ is shown as a black curve as a function of $ε$ . This is the value of information achieved by the optimal strategy, as found through numerical optimization. The value of perfect information, ${\hat{Λ}}_{k \to \infty}^{(a, δ)} - {\hat{Λ}}^{(η)} = I (E_{m} | E_{m + 1})$ , is shown in blue, representing an upper bound on the value of information. The value of information when a pure strategy is used ( $π_{ϕ_{1} e_{1}} = π_{ϕ_{2} e_{2}} = 1$ ) is shown in red as a function of $ε$ . Finally, the value of information for the effective strategy resembling the optimal strategy when $ε = 0$ as closely as possible is shown in purple. This corresponds to the optimal strategy when $ε \leq ε_{c_{1}}$ , but is no longer optimal for $ε > ε_{c_{1}}$ . (b) Optimal and suboptimal phenotype expression probabilities as a function of $ε$ . The dashed lines follow the strategy ( $π_{ϕ_{1} e_{1}, δ}$ and $π_{ϕ_{2} e_{2}, δ}$ ) yielding the effective strategy that most closely matches that for when $ε = 0$ . These probabilities yield the purple curve in (a). The thin solid lines follow the optimal strategy, corresponding to the black curve in (a). The thick solid lines follow the effective strategy resulting from the optimal strategy. Model parameters are fixed at $w_{ϕ_{1} e_{1}} = 5.0$ , $w_{ϕ_{2} e_{1}} = 0.01$ , $w_{ϕ_{1} e_{2}} = 1.0$ , $w_{ϕ_{2} e_{2}} = 10.0$ , $α = 0.2$ , and $β = 0.25$ . This figure was created in a Python 3.7 Jupyter Notebook [25] with Matplotlib 3.1.1 [26].
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Figure 3
The critical error probability in different conditions. The value of the critical error probability $ε_{c_{1}}$ in statistically different environments and with different fitness values when $k \to \infty$ . Dotted white lines indicate the values of $α$ and $β$ above which $ε_{c_{1}}$ is greater than zero. This figure was created in a Python 3.7 Jupyter Notebook [25] with Matplotlib 3.1.1 [26].
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Figure 4
The value of information in statistically different environments with different sensing error probabilities. As in the text, $ε$ is the environmental sensing error probability, and $α$ and $β$ are environmental transition probabilities. Fitness values are fixed at $w_{ϕ_{1} e_{1}} = 5.0$ , $w_{ϕ_{2} e_{1}} = 0.01$ , $w_{ϕ_{1} e_{2}} = 1.0$ , $w_{ϕ_{2} e_{2}} = 10.0$ . This figure was created in a Python 3.7 Jupyter Notebook [25] with Matplotlib 3.1.1 [26].
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Figure 5
The fitness value and cost of imperfect information with finite phenotype switching rate. The effects of phenotype switching rate and environment sensing error probability on the fitness value of information and fitness cost of imperfect sensing. Model parameters are fixed at $w_{ϕ_{1} e_{1}} = 5.0$ , $w_{ϕ_{2} e_{1}} = 0.01$ , $w_{ϕ_{1} e_{2}} = 1.0$ , $w_{ϕ_{2} e_{2}} = 10.0$ , $α = 0.2$ , and $β = 0.25$ . (a) The fitness value of information, ${\hat{Λ}}_{k \to \infty}^{(a, ε)} - {\hat{Λ}}^{(η)}$ , is shown as a function of $ε$ at several values of the phenotype switching rate $k$ , denoted by the color bar. The fitness value of information with perfect environmental sensing and infinite phenotype switching rate, $I (E_{m}; E_{m + 1})$ , is shown in light blue. As $k$ decreases, the interval of $ε$ values for which the fitness value is the same as if $ε = 0$ increases. (b) The fitness cost of imperfect information and finite phenotype switching rates. There is a large region below $ε \approx 0.2$ and above $k \approx 3.5$ where the fitness cost is close to zero and largely insensitive to changes in either $ε$ or $k$ . Outside this region of parameter space, the fitness cost increases far more rapidly as a function of $k$ than of $ε$ . Note the dependence of $ε_{c_{1}}$ and $ε_{c_{2}}$ on $k$ . This figure was created in a Python 3.7 Jupyter Notebook [25] with Matplotlib 3.1.1 [26].
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Figure 6
Optimal Lyapunov exponents when $k_{{\tilde{e}}_{1}} \neq k_{{\tilde{e}}_{2}}$ . We numerically optimized Lyapunov exponents for a number of cases in which $k_{{\tilde{e}}_{1}}$ and $k_{{\tilde{e}}_{2}}$ are not necessarily equal. The purple curve with $k_{{\tilde{e}}_{1}} = k_{{\tilde{e}}_{2}} = 4$ is the special case examined in the main text. In some other cases, a critical error probability $ε_{c_{1}}$ exists, such as when $k_{{\tilde{e}}_{1}} = 2$ and $k_{{\tilde{e}}_{2}} = 6$ or when $k_{{\tilde{e}}_{1}} = 6$ and $k_{{\tilde{e}}_{2}} = 2$ . Interestingly, when $k_{{\tilde{e}}_{1}} = 1$ and $k_{{\tilde{e}}_{2}} = 8$ or $k_{{\tilde{e}}_{1}} = 8$ and $k_{{\tilde{e}}_{2}} = 1$ , the Lyapunov exponent increases with $ε$ and reaches a maximum before decreasing. Model parameters are fixed at $w_{ϕ_{1} e_{1}} = 5.0$ , $w_{ϕ_{2} e_{1}} = 0.01$ , $w_{ϕ_{1} e_{2}} = 1.0$ , $w_{ϕ_{2} e_{2}} = 10.0$ , $α = 0.2$ , and $β = 0.25$ . This figure was created in a Python 3.7 Jupyter Notebook [25] with Matplotlib 3.1.1 [26].
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