# Information Theoretical Study of Cross-Talk Mediated Signal Transduction in MAPK Pathways

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## Abstract

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## 1. Introduction

## 2. Results and Discussion

#### 2.1. Two Variable Mutual Information

#### 2.2. Three Variable Mutual Information

## 3. Materials and Methods

#### 3.1. Two Variable Mutual Information

#### 3.2. Three Variable Mutual Information

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MAPK | Mitogen activated protein kinase |

LNA | Linear noise approximation |

PID | Partial information decomposition |

## Appendix A. Calculation of variance and covariance

## References

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**Figure 1.**(color online) Two variable mutual information and net information transduction as a function of cross-talk parameter. (

**A–F**) Two variable mutual information profiles $\mathcal{I}({x}_{p1};{x}_{p2})$, $\mathcal{I}({x}_{p1};{y}_{p2})$, $\mathcal{I}({y}_{p1};{y}_{p2})$, $\mathcal{I}({y}_{p1};{x}_{p2})$, $\mathcal{I}({x}_{p2};{y}_{p2})$ and $\mathcal{I}({x}_{p3};{y}_{p3})$ as a function of cross-interaction parameter ${\epsilon}_{2}$ for a fixed value of ${\epsilon}_{1}=0.5\times {10}^{-4}$. In all figures, red (with open circle) and green (with open diamond) lines are generated using faster (Table 1) and slower (Table 2) relaxation rate parameters, respectively. The symbols are generated using stochastic simulation algorithm [43] and the lines are due to theoretical calculation; (

**G,H**) Bar diagram of two variable mutual information of three parallel cascade kinases under an equivalent cross-talk condition (${\epsilon}_{1}={\epsilon}_{2}=0.5\times {10}^{-4}$) for faster (Table 1) and slower (Table 2) relaxation rate parameters, respectively; (

**I**) Net information transduction D as a function of cross-interaction parameter ${\epsilon}_{2}$ for a fixed value of ${\epsilon}_{1}=0.5\times {10}^{-4}$. The red (with open circle) and the green (with open diamond) lines are due to faster (Table 1) and slower (Table 2) relaxation rate parameters, respectively. The figure indicates data collapse for two relaxation rate parameters. The symbols are generated using stochastic simulation algorithm [43] and the lines are obtained from theoretical calculation. All the simulation data (open circles and open diamonds) are ensemble average of ${10}^{7}$ independent trajectories; (

**J**) 2d-surface plot of net information transduction D as a function of two cross-talk parameters ${\epsilon}_{1}$ and ${\epsilon}_{2}$ for faster (Table 1) relaxation rate parameters.

**Figure 2.**(color online) 2d-surface plots of two variable mutual information, stochastic time trajectories and scatter plots. (

**A,B**) 2d-surface plot of two variable mutual information $\mathcal{I}({x}_{p2};{y}_{p2})$ and $\mathcal{I}({x}_{p3};{y}_{p3})$ as a function of two cross-talk parameters ${\epsilon}_{1}$ and ${\epsilon}_{2}$ for faster relaxation rate parameters (Table 1). In both figures, I, II, III and IV correspond to four different values of ${\epsilon}_{1}$ and ${\epsilon}_{2}$; (

**C**) Stochastic time trajectories and steady state population of two parallel kinases for four different sets of ${\epsilon}_{1}$ and ${\epsilon}_{2}$. For CI, CII, CIII and CIV we have used ${\epsilon}_{1}={\epsilon}_{2}=0.1\times {10}^{-4},$ ${\epsilon}_{1}=0.1\times {10}^{-4}$ and ${\epsilon}_{2}=0.9\times {10}^{-4}$, ${\epsilon}_{1}={\epsilon}_{2}=0.9\times {10}^{-4}$ and ${\epsilon}_{1}=0.9\times {10}^{-4}$ and ${\epsilon}_{2}=0.1\times {10}^{-4}$, respectively. In each scatter plot, ${\rho}_{ij}(i=j)$ represents analytical value of Pearson’s correlation coefficient. The stochastic trajectories and the scatter plots are generated using stochastic simulation algorithm [43] and the surface plots are due to theoretical calculation.

**Figure 3.**(color online) 2d-surface plots of two variable mutual information, stochastic time trajectories and scatter plots. (

**A,B**) 2d-surface plot of two variable mutual information $\mathcal{I}({x}_{p2};{y}_{p2})$ and $\mathcal{I}({x}_{p3};{y}_{p3})$ as a function of two cross-talk parameters ${\epsilon}_{1}$ and ${\epsilon}_{2}$ for slower relaxation rate parameters (Table 2). In both figures, I, II, III and IV correspond to four different values of ${\epsilon}_{1}$ and ${\epsilon}_{2}$; (

**C**) Stochastic time trajectories and steady state population of two parallel kinases for four different sets of ${\epsilon}_{1}$ and ${\epsilon}_{2}$. For CI, CII, CIII and CIV we have used ${\epsilon}_{1}={\epsilon}_{2}=0.1\times {10}^{-4},$ ${\epsilon}_{1}=0.1\times {10}^{-4}$ and ${\epsilon}_{2}=0.9\times {10}^{-4}$, ${\epsilon}_{1}={\epsilon}_{2}=0.9\times {10}^{-4}$ and ${\epsilon}_{1}=0.9\times {10}^{-4}$ and ${\epsilon}_{2}=0.1\times {10}^{-4}$, respectively. In each scatter plot, ${\rho}_{ij}(i=j)$ represents analytical value of Pearson’s correlation coefficient. The stochastic trajectories and the scatter plots are generated using stochastic simulation algorithm [43] and the surface plots are due to theoretical calculation.

**Figure 4.**(color online) Three variable mutual information as a function of cross-talk parameter. (

**A,B**) Three variable mutual information $\mathcal{I}({x}_{p1},{y}_{p1};{x}_{p2})$ (

**A**) and net synergy $\Delta \mathcal{I}({x}_{p1},{y}_{p1};{x}_{p2})$ ((

**B**) for signal integration motif. Schematic diagram of signal integration motif in composite MAPK network (see inset in (

**A**). (

**C,D**) - Three variable mutual information $\mathcal{I}({x}_{p1};{x}_{p2},{y}_{p2})$ (

**C**) and net synergy $\Delta \mathcal{I}({x}_{p1};{x}_{p2},{y}_{p2})$ (

**D**) for signal bifurcation motif. Schematic diagram of signal bifurcation motif in composite MAPK network (see inset in (

**C**). All the figures are drawn as a function of cross-interaction parameter ${\epsilon}_{2}$ for a fixed value of ${\epsilon}_{1}=0.5\times {10}^{-4}$. Here red (with open circle) and green (with open diamond) lines are drawn for faster (Table 1) and slower (Table 2) relaxation rate parameters, respectively. The symbols are generated using stochastic simulation algorithm [43] and the lines are obtained from theoretical calculation. All the simulation data (open circles and open diamonds) are ensemble average of ${10}^{7}$ independent trajectories.

**Figure 5.**(color online) Schematic diagram of two parallel MAPK (equivalent and identical) signalling pathways (X and Y). Each pathway consists of three successively connected cascade kinases, MAPKKK (red), MAPKK (green) and MAPK (blue). The first activated kinase facilitates the activation of the second one and then the second kinase regulates the activation of the last one. Both signalling pathways are exposed to two different signals (${S}_{x}$ and ${S}_{y}$). Cross-talk is developed due to inter-pathway interactions. ${\epsilon}_{1}$ and ${\epsilon}_{2}$ are the cross-interaction parameters and the directionality of these interactions are ${x}_{p1}\to {y}_{p2}$ and ${y}_{p1}\to {x}_{p2}$, respectively.

**Table 1.**Reactions and corresponding parameter values for the MAPK network motif of S. cerevisiae [6,9,66], related to faster relaxation rate. Other Parameters are ${s}_{x}={s}_{y}=10$ molecules/cell, ${x}_{T1}={x}_{1}+{x}_{p1}=250$ molecules/cell, ${x}_{T2}={x}_{2}+{x}_{p2}=1700$ molecules/cell, ${x}_{T3}={x}_{3}+{x}_{p3}=5000$ molecules/cell, ${y}_{T1}={y}_{1}+{y}_{p1}=250$ molecules/cell, ${y}_{T2}={y}_{2}+{y}_{p2}=1700$ molecules/cell and ${y}_{T3}={y}_{3}+{y}_{p3}=5000$ molecules/cell. The kinetic schemes adopted in the present work follows the model of Heinrich et al. [63].

Description | Reaction | Propensity Function | Rate Constant | |
---|---|---|---|---|

Activation of ${x}_{1}$ | ${x}_{1}+{s}_{x}\stackrel{{k}_{x}}{\u27f6}{x}_{p1}+{s}_{x}$ | ${k}_{x}{s}_{x}{x}_{1}$ | ${k}_{x}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p1}$ | ${x}_{p1}\stackrel{{\alpha}_{1}}{\u27f6}{x}_{1}$ | ${\alpha}_{1}{x}_{p1}$ | ${\alpha}_{1}=0.01$ s${}^{-1}$ | |

Activation of ${y}_{1}$ | ${y}_{1}+{s}_{y}\stackrel{{k}_{y}}{\u27f6}{y}_{p1}+{s}_{y}$ | ${k}_{y}{s}_{y}{y}_{1}$ | ${k}_{y}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p1}$ | ${y}_{p1}\stackrel{{\beta}_{1}}{\u27f6}{y}_{1}$ | ${\beta}_{1}{y}_{p1}$ | ${\beta}_{1}=0.01$ s${}^{-1}$ | |

Activation of ${x}_{2}$ | ${x}_{2}+{x}_{p1}\stackrel{{k}_{12x}}{\u27f6}{x}_{p2}+{x}_{p1}$ | ${k}_{12x}{x}_{p1}{x}_{2}$ | ${k}_{12x}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Activation of ${x}_{2}$ | ${x}_{2}+{y}_{p1}\stackrel{{\epsilon}_{2}}{\u27f6}{x}_{p2}+{y}_{p1}$ | ${\epsilon}_{2}{y}_{p1}{x}_{2}$ | ${\epsilon}_{2}=(0-1)\times {10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p2}$ | ${x}_{p2}\stackrel{{\alpha}_{2}}{\u27f6}{x}_{2}$ | ${\alpha}_{2}{x}_{p2}$ | ${\alpha}_{2}=0.05$ s${}^{-1}$ | |

Activation of ${y}_{2}$ | ${y}_{2}+{y}_{p1}\stackrel{{k}_{12y}}{\u27f6}{y}_{p2}+{y}_{p1}$ | ${k}_{12y}{y}_{p1}{y}_{2}$ | ${k}_{12y}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Activation of ${y}_{2}$ | ${y}_{2}+{x}_{p1}\stackrel{{\epsilon}_{1}}{\u27f6}{y}_{p2}+{x}_{p1}$ | ${\epsilon}_{1}{x}_{p1}{y}_{2}$ | ${\epsilon}_{1}=(0-1)\times {10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p2}$ | ${y}_{p2}\stackrel{{\beta}_{2}}{\u27f6}{y}_{2}$ | ${\beta}_{2}{y}_{p2}$ | ${\beta}_{2}=0.05$ s${}^{-1}$ | |

Activation of ${x}_{3}$ | ${x}_{3}+{x}_{p2}\stackrel{{k}_{23x}}{\u27f6}{x}_{p3}+{x}_{p2}$ | ${k}_{23x}{x}_{p2}{x}_{3}$ | ${k}_{23x}=5\times {10}^{-5}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p3}$ | ${x}_{p3}\stackrel{{\alpha}_{3}}{\u27f6}{x}_{3}$ | ${\alpha}_{3}{x}_{p3}$ | ${\alpha}_{3}=0.05$ s${}^{-1}$ | |

Activation of ${y}_{3}$ | ${y}_{3}+{y}_{p2}\stackrel{{k}_{23y}}{\u27f6}{y}_{p3}+{y}_{p2}$ | ${k}_{23y}{y}_{p2}{y}_{3}$ | ${k}_{23y}=5\times {10}^{-5}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p3}$ | ${y}_{p3}\stackrel{{\beta}_{3}}{\u27f6}{y}_{3}$ | ${\beta}_{3}{y}_{p3}$ | ${\beta}_{3}=0.05$ s${}^{-1}$ |

**Table 2.**Reactions and corresponding parameter values for the MAPK network motif of S. cerevisiae [6,9,66], related to slower relaxation rate. Other Parameters are ${s}_{x}={s}_{y}=10$ molecules/cell, ${x}_{T1}={x}_{1}+{x}_{p1}=250$ molecules/cell, ${x}_{T2}={x}_{2}+{x}_{p2}=1700$ molecules/cell, ${x}_{T3}={x}_{3}+{x}_{p3}=5000$ molecules/cell, ${y}_{T1}={y}_{1}+{y}_{p1}=250$ molecules/cell, ${y}_{T2}={y}_{2}+{y}_{p2}=1700$ molecules/cell and ${y}_{T3}={y}_{3}+{y}_{p3}=5000$ molecules/cell. The kinetic schemes adopted in the present work follows the model of Heinrich et al. [63].

Description | Reaction | Propensity Function | Rate Constant | |
---|---|---|---|---|

Activation of ${x}_{1}$ | ${x}_{1}+{s}_{x}\stackrel{{k}_{x}}{\u27f6}{x}_{p1}+{s}_{x}$ | ${k}_{x}{s}_{x}{x}_{1}$ | ${k}_{x}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p1}$ | ${x}_{p1}\stackrel{{\alpha}_{1}}{\u27f6}{x}_{1}$ | ${\alpha}_{1}{x}_{p1}$ | ${\alpha}_{1}=0.01$ s${}^{-1}$ | |

Activation of ${y}_{1}$ | ${y}_{1}+{s}_{y}\stackrel{{k}_{y}}{\u27f6}{y}_{p1}+{s}_{y}$ | ${k}_{y}{s}_{y}{y}_{1}$ | ${k}_{y}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p1}$ | ${y}_{p1}\stackrel{{\beta}_{1}}{\u27f6}{y}_{1}$ | ${\beta}_{1}{y}_{p1}$ | ${\beta}_{1}=0.01$ s${}^{-1}$ | |

Activation of ${x}_{2}$ | ${x}_{2}+{x}_{p1}\stackrel{{k}_{12x}}{\u27f6}{x}_{p2}+{x}_{p1}$ | ${k}_{12x}{x}_{p1}{x}_{2}$ | ${k}_{12x}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Activation of ${x}_{2}$ | ${x}_{2}+{y}_{p1}\stackrel{{\epsilon}_{2}}{\u27f6}{x}_{p2}+{y}_{p1}$ | ${\epsilon}_{2}{y}_{p1}{x}_{2}$ | ${\epsilon}_{2}=(0-1)\times {10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p2}$ | ${x}_{p2}\stackrel{{\alpha}_{2}}{\u27f6}{x}_{2}$ | ${\alpha}_{2}{x}_{p2}$ | ${\alpha}_{2}=0.01$ s${}^{-1}$ | |

Activation of ${y}_{2}$ | ${y}_{2}+{y}_{p1}\stackrel{{k}_{12y}}{\u27f6}{y}_{p2}+{y}_{p1}$ | ${k}_{12y}{y}_{p1}{y}_{2}$ | ${k}_{12y}={10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Activation of ${y}_{2}$ | ${y}_{2}+{x}_{p1}\stackrel{{\epsilon}_{1}}{\u27f6}{y}_{p2}+{x}_{p1}$ | ${\epsilon}_{1}{x}_{p1}{y}_{2}$ | ${\epsilon}_{1}=(0-1)\times {10}^{-4}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p2}$ | ${y}_{p2}\stackrel{{\beta}_{2}}{\u27f6}{y}_{2}$ | ${\beta}_{2}{y}_{p2}$ | ${\beta}_{2}=0.01$ s${}^{-1}$ | |

Activation of ${x}_{3}$ | ${x}_{3}+{x}_{p2}\stackrel{{k}_{23x}}{\u27f6}{x}_{p3}+{x}_{p2}$ | ${k}_{23x}{x}_{p2}{x}_{3}$ | ${k}_{23x}={10}^{-5}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${x}_{p3}$ | ${x}_{p3}\stackrel{{\alpha}_{3}}{\u27f6}{x}_{3}$ | ${\alpha}_{3}{x}_{p3}$ | ${\alpha}_{3}=0.01$ s${}^{-1}$ | |

Activation of ${y}_{3}$ | ${y}_{3}+{y}_{p2}\stackrel{{k}_{23y}}{\u27f6}{y}_{p3}+{y}_{p2}$ | ${k}_{23y}{y}_{p2}{y}_{3}$ | ${k}_{23y}={10}^{-5}$ molecules${}^{-1}$ s${}^{-1}$ | |

Deactivation of ${y}_{p3}$ | ${y}_{p3}\stackrel{{\beta}_{3}}{\u27f6}{y}_{3}$ | ${\beta}_{3}{y}_{p3}$ | ${\beta}_{3}=0.01$ s${}^{-1}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Maity, A.K.; Chaudhury, P.; Banik, S.K.
Information Theoretical Study of Cross-Talk Mediated Signal Transduction in MAPK Pathways. *Entropy* **2017**, *19*, 469.
https://doi.org/10.3390/e19090469

**AMA Style**

Maity AK, Chaudhury P, Banik SK.
Information Theoretical Study of Cross-Talk Mediated Signal Transduction in MAPK Pathways. *Entropy*. 2017; 19(9):469.
https://doi.org/10.3390/e19090469

**Chicago/Turabian Style**

Maity, Alok Kumar, Pinaki Chaudhury, and Suman K. Banik.
2017. "Information Theoretical Study of Cross-Talk Mediated Signal Transduction in MAPK Pathways" *Entropy* 19, no. 9: 469.
https://doi.org/10.3390/e19090469