# Fitness Gain of Individually Sensed Information by Cells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Sensing and Adaptation Processes

#### Fitness of a Population with Individual and Common Sensing

## 3. Stochastic Trajectories of Individual and Common Sensing

## 4. Value of Individual Sensing is Always Greater than that of Common Sensing

#### 4.1. The Gain of Fitness by Individual Sensing

#### 4.2. Connection with Other Information Measures

## 5. Discussion and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Equations

#### Appendix A.1. Derivation of Equations (7) and (8)

#### Appendix A.2. Derivation of Equation (15)

#### Appendix A.3. Derivation of Equations (16) and (17)

#### Appendix A.4. Derivation of Equation (21)

#### Appendix A.5. Derivation of Equation (30)

#### Appendix A.6. Derivation of Equation (35)

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**Figure 1.**Schematic diagrams of population dynamics of cells with individual (

**a**) and common (

**b**) sensing. The colors of cells and molecules on the cells represent phenotypic states and sensing signal, respectively. Bars on the diagrams indicate the histories of environmental states and common sensing. In (

**a**), the sensing singal of each cell is correlated with the environmental state but has an intercellular variation due to the stochasticity of individual sensing. In (

**b**), on the other hand, all the cells at certain time points share the same sensing signal, which is shown by the background colors in the diagram.

**Figure 2.**(

**a**) A diagrammatic representation of state transitions of the environment used for simulation in Figure 3, Figure 4 and Figure 5. Three states are assumed for the environment; (

**b**) Replication rates of cells with two different phenotypic states under different environmental states; (

**c**) Environment-dependence of the sensing signal, and the probabilities to obtain a certain sensing signal under each environmental state; (

**d**) Signal-dependent phenotype switching; The thickness of arrows represent relative probabilities and rates of replications. The values of the parameters used for the simulation are given by Equations (11)–(14).

**Figure 3.**(

**a**,

**b**) Trajectories of population sizes with individual sensing under two different realizations of the environment. The history of the environment, ${\mathcal{Y}}_{t}$, is shown by the colored bar on each panel. The color represents the state of the environment, which was defined in Figure 2a. Each colored line corresponds to the population size of the cells with phenotypic state x and sensing signal z; the actual value of $(x,z)$ is designated in the panels. The gray lines in (

**a**,

**b**) are the replicate of the trajectories in (

**c**,

**d**), respectively, for comparison. (

**c**,

**d**) Trajectories of population sizes with common sensing under the same realizations of the environment as in (

**a**) and (

**b**), respectively. On each panel, the history of the common signal, ${\mathcal{Z}}_{t}$, is additionally shown by the colored bar. Each colored line corresponds to the population size of the cells with phenotypic state x, with the actual value of x designated in the panels. (

**e**,

**f**) Fitnesses of the populations with the individual and the common sensing, ${\Psi}^{i}\left[{\mathcal{Y}}_{t}\right]$ (red solid curve) and ${\Psi}^{c}[{\mathcal{Y}}_{t},{\mathcal{Z}}_{t}]$ (blue solid curve) under the same realizations of the environment and common signal as in (

**a**,

**c**) and (

**b**,

**d**). Related quantities are also shown for comparison.

**Figure 4.**(

**a**) Average values of fitnesses and related quantities; (

**b**,

**c**) Fluctuation of the fitness with individual sensing ${\Psi}^{i}\left[{\mathcal{Y}}_{t}\right]$ (

**b**); and that with common sensing ${\Psi}^{c}\left[{\mathcal{Y}}_{t}\right]$ (

**c**); (

**d**–

**f**) Fluctuation of ${\Psi}_{0}\left[{\mathcal{Y}}_{t}\right]$ (

**d**); ${\Psi}_{0}\left[{\mathcal{Y}}_{t}\right]+i[{\mathcal{Z}}_{t}\to {\mathcal{Y}}_{t}]+g[{\mathcal{Y}}_{t},{\mathcal{Z}}_{t}]$ (

**e**); and ${\Psi}_{0}\left[{\mathcal{Y}}_{t}\right]+i[{\mathcal{Z}}_{t}\to {\mathcal{Y}}_{t}]$ (

**f**).

**Figure 5.**Numerical verification of IFRs for $g\left[{\mathcal{Y}}_{t}\right]$ (

**a**,

**b**); ${\gamma}_{t}-\sigma \left[{\mathcal{Y}}_{t}\right]$ (

**c**,

**d**); and ${\Psi}_{0}\left[{\mathcal{Y}}_{t}\right]+i[{\mathcal{Z}}_{t}\to {\mathcal{Y}}_{t}]+g[{\mathcal{Y}}_{t},{\mathcal{Z}}_{t}]-{\Psi}^{i}\left[{\mathcal{Y}}_{t}\right]$ (

**e**,

**f**). Left panels are behaviors of the integrands of the IFRs for 100 different realizations of the environmental and common signal histories. Right panels are the sample averages of the integrands of the IFRs. Thin colored curves are obtained by averaging ${10}^{5}$ different samples, and the thick black curves are obtained by the average of $1.2\times {10}^{8}$ samples.

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**MDPI and ACS Style**

Kobayashi, T.J.; Sughiyama, Y.
Fitness Gain of Individually Sensed Information by Cells. *Entropy* **2019**, *21*, 1002.
https://doi.org/10.3390/e21101002

**AMA Style**

Kobayashi TJ, Sughiyama Y.
Fitness Gain of Individually Sensed Information by Cells. *Entropy*. 2019; 21(10):1002.
https://doi.org/10.3390/e21101002

**Chicago/Turabian Style**

Kobayashi, Tetsuya J., and Yuki Sughiyama.
2019. "Fitness Gain of Individually Sensed Information by Cells" *Entropy* 21, no. 10: 1002.
https://doi.org/10.3390/e21101002