## 1. Introduction

_{j}, induces the modification of X

_{j}

_{+1}into X

_{j}

_{+1}*. In individual steps, the dephosphorylation of X

_{j}

_{+1}* into X

_{j}

_{+1}occurs by phosphatase Ph

_{j}by the release ability of inorganic phosphate Pi from X

_{j}

_{+1}*, and the pre-stimulation steady state is subsequently recovered. A signaling step in the above cascades may be described as follows:

_{j}and k

_{−j}are the kinetic coefficients. ATP, ADP, and Pi represent adenosine triphosphate, adenosine diphosphate, and inorganic phosphate, respectively. Consider all the possible distinct signal transduction events that correspond to all the possible combinations of the signal molecule X

_{j}, whose transduction length is ${\tau}_{j}$. For instance, an event is described as follows:

_{1}X

_{3}X

_{2}X

_{3}X

_{1}X

_{2}X

_{3}X

_{5}X

_{3}X

_{4}X

_{3}

_{j}with numbers ${N}_{j}\left(1\le j\le n\right)$, will correspond to all the possible combinations of X

_{j}. Therefore, N

_{1}= 2, N

_{2}= 2, N

_{3}= 5, N

_{4}= 1, N

_{5}= 1 and n = 5 in the signal event (2). In actuality, signaling cascades have been studied extensively using models of Mitogen-activated Protein Kinase (MAPK) pathways, in which the epidermal growth factor receptor, $c-\mathrm{Raf}$, MAP kinase-extracellular signal-regulated kinase [17], and kinase-extracellular signal-regulated kinase (ERK) are phosphorylated following treatment with growth factors. The Ras-c-Raf-ERK cascade (RRE) is a ubiquitous signaling pathway that conveys mitogenic and differentiation signals from the cell membrane to the nucleus.

_{j}or X

_{j}*, respectively:

_{j}*, and non-active j molecules, X

_{j}, participating in signaling cascades is regarded as constant, the protein production process is relatively slower than the signal transduction step:

## 2. Mixing Entropy in Signal Transduction

_{j}* and X

_{j}difference between the steps, the mixing entropy change of the j-th step, $\mathrm{d}{S}_{j}^{\mathrm{mix}}$, with a minimal concentration difference in X

_{j}*, $\mathrm{d}{{p}_{j}}^{\ast}$, and in X

_{j}, $\mathrm{d}{p}_{j}=-\mathrm{d}{{p}_{j}}^{\ast}$, is described:

## 3. Entropy Current and Signal Transduction

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Schematic of a reaction cascade in cell signal transduction. The receptor mediates the cellular response to the presence of the ligand in the extracellular medium. $A$ is a messenger, ATP, of signal transduction. Individual signaling molecules ${X}_{j}\left(1\le j\le n\right)$ relay the modification of individual steps, and the last species ${X}_{n}$ is translocated to the nucleus, where it controls gene expression by the transcription of mRNA. Ph denotes a phosphatase.

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