A Novel Dynamical Regulation of mRNA Distribution by Cross-Talking Pathways

Abstract

In this paper, we use a similar approach to the one proposed by Chen and Jiao to calculate the mathematical formulas of the generating function $V(z,t)$ and the mass function $P_m\left(t\right)$ of a cross-talking pathways model in large parameter regions. Together with kinetic rates from yeast and mouse genes, our numerical examples reveal novel bimodal mRNA distributions for intermediate times, whereby the mode of distribution $P_m\left(t\right)$ displays unimodality with the peak at $m=0$ for initial and long times, which has not been obtained in previous works. Such regulation of mRNA distribution exactly matches the transcriptional dynamics for the osmosensitive genes in Saccharomyces cerevisiae, which has not been generated by those models with one single pathway or feedback loops. This paper may provide us with a novel observation on transcriptional distribution dynamics regulated by multiple signaling pathways in response to environmental changes and genetic perturbations.

1. Introduction

Gene transcription is a central and stochastic process in cells, through which, the genetic information stored in DNA is transcribed into mRNA molecules, which are then translated into proteins. The stochasticity and bursting fashion of transcription were widely studied both in experiments and theoretical research in the last two decades [1,2,3,4]. The stochasticity is manifested by the heterogeneous distribution of mRNA copy numbers in the population of isogenic cells [4,5,6,7]. The probability mass function $P_m\left(t\right)$, which is the probability that there are exactly m mRNA copies of the gene at time t in one cell, is often used to quantify fluctuations in mRNA levels. By using recent single-cell measurements [8,9] and parameter estimation methods [10,11,12,13], we can obtain massive data on the histogram of the probability mass function $P_m\left(t\right)$ [9,14,15]. The distribution profiles of $P_m\left(t\right)$ have been studied in [16,17,18,19,20]. They found that there are three types of distribution: decaying distribution, for which, $P_m\left(t\right)$ decreases in m; unimodal distribution, for which, $P_m\left(t\right)$ peaks uniquely at $m>0$; biomodal distribution, for which, $P_m\left(t\right)$ has two peaks: one is at $m=0$ and the other is at some $m>0$.

The two-state model is considered as one of the most classic models to understand how varying external signals influence the profile of $P_m\left(t\right)$. As shown in the diagram

$$\mathrm{gene\; off}\underset{\gamma }{\stackrel{\lambda }{\rightleftarrows }}\mathrm{gene\; on}\stackrel{v}{⟶}\mathrm{mRNA}\stackrel{\delta }{⟶}\varnothing$$

(1)

the gene is suggested to switch randomly between off (inactive) and on (active) states [2,4], where $\lambda$ is the activation rate and $\gamma$ is the inactivation rate. The processes of the birth and death of mRNAs are determined by the synthesis rate $v>0$ and the degradation rate $\delta >0$, respectively. Some indices of this model, such as the mean, noise, fano factor and probability distribution $P_m\left(t\right)$, have been applied to fit the experimental data obtained in yeast [21], bacteria [22] and mammals [23]. However, some inducible genes in the important physiological processes, such as immunity, development and stem cell renewal, are activated by multiple pathways [24,25,26]. Modeling the transcription by a single signal pathway cannot be sufficient in capturing the multiple biochemical reactions. Therefore, some authors introduced transcription models with two or more competitive cross-talking pathways responding to environmental changes [6,27,28,29,30,31]. The cross-talking model can be generalized as the following diagram,

(2)

The two competitive cross-talking pathways can activate the transcription alternatively. Here, we use ${\lambda }_1$ to denote the induction strength of the first pathway and ${\lambda }_2$ to denote the induction strength of the second one, where

$$0<{\lambda }_1\le {\lambda }_2<\infty .$$

Let $O_i(i=1,2)$ denote the gene off state if the gene is transferred to the gene on state by the ith pathway. The pathway selection probabilities are denoted by

$$\begin{array}{c}\hfill q_1=\mathrm{Prob}(O=O_1)\mathrm{and}{q}_2=\mathrm{Prob}(O=O_2),\end{array}$$

and satisfy

$$0<q_1,q_2<1,q_1+q_2=1.$$

Other parameters $\gamma$, v and $\delta$ are in accordance with the two-state model (1). Then, the gene switches randomly between off (inactive) and on (active) states, and the respective probabilities of m mRNA molecules existing at time t with the j-th pathway being chosen are defined as $P_{m,j}\left(t\right),j=1,2$. We let $P_{m,e}\left(t\right)$ denote the probability that there are m mRNA molecules and the gene resides at an active state. Then, the total probability mass function is

$$P_m\left(t\right)=P_{m,1}\left(t\right)+P_{m,2}\left(t\right)+P_{m,e}\left(t\right).$$

According to the standard procedure in the theory of stochastic processes [31], the three partial mass functions satisfy the following system of master equations

$$\begin{array}{cc}\hfill P_{m,1}^{\prime }\left(t\right)=& q_1\gamma P_{m,e}\left(t\right)-(m\delta +{\lambda }_1)P_{m,1}\left(t\right)+(m+1)\delta P_{m+1,1}\left(t\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill P_{m,2}^{\prime }\left(t\right)=& q_2\gamma P_{m,e}\left(t\right)-(m\delta +{\lambda }_2)P_{m,2}\left(t\right)+(m+1)\delta P_{m+1,2}\left(t\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill P_{m,e}^{\prime }\left(t\right)=& {\lambda }_1{P}_{m,1}\left(t\right)+{\lambda }_2{P}_{m,2}\left(t\right)-(v+m\delta +\gamma )P_{m,e}\left(t\right)\hfill \\ \hfill & +(m+1)\delta P_{m+1,e}\left(t\right)+v{P}_{m-1,e}\left(t\right),\hfill \end{array}$$

(5)

where $P_{-1,e}\left(t\right)=0$. Without a loss of generality, we assume that the gene is inactive and the number of transcripts is zero at $t=0$. Then, the initial condition is

$$\begin{array}{c}\hfill P_{0,1}\left(0\right)=q_1,P_{0,2}\left(0\right)=q_2,P_{0,e}\left(0\right)=P_{m,i}\left(0\right)=0\mathrm{for}i=1,2,e\mathrm{and}m\ge 1.\end{array}$$

(6)

We define the probability generating functions [19,32] as

$$V_i(z,t)=\sum _{m=0}^{\infty }{(z+1)}^m{P}_{m,i}\left(t\right),i=1,2,e.$$

(7)

Then, the initial value problem of the master Equations (3)–(5) is transformed into a system of first-order partial differential equations

$$\begin{array}{cc}\hfill \frac{\partial V_1}{\partial t}(z,t)& =-{\lambda }_1{V}_1(z,t)+q_1\gamma V_e(z,t)-\delta z\frac{\partial V_1}{\partial z}(z,t),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \frac{\partial V_2}{\partial t}(z,t)& =-{\lambda }_2{V}_2(z,t)+q_2\gamma V_e(z,t)-\delta z\frac{\partial V_2}{\partial z}(z,t),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \frac{\partial V_e}{\partial t}(z,t)& ={\lambda }_1{V}_1(z,t)+{\lambda }_2{V}_2(z,t)-\gamma V_e(z,t)+vz{V}_e(z,t)-\delta z\frac{\partial V_e}{\partial z}(z,t)\hfill \end{array}$$

(10)

with the initial condition

$$\begin{array}{ccc}& & V_1(z,0)=q_1,V_2(z,0)=q_2\mathrm{and}{V}_e(z,0)=0.\hfill \end{array}$$

(11)

By adding the three solutions of the initial value problem (8)–(11) together and then applying the conversion formula, we obtain $P_m\left(t\right)$ as

$$P_m\left(t\right)=\frac{1}{m!}\frac{{\partial }^{z}V(z,t)}{\partial z^m}{|}_{z=-1},$$

(12)

where $V(z,t)=V_1(z,t)+V_2(z,t)+V_e(z,t)$ and $m\ge 0$. We introduce non-dimensionalized system parameters

$$\begin{array}{c}\hfill {\overline{\lambda }}_1={\lambda }_1/\delta ,{\overline{\lambda }}_2={\lambda }_2/\delta ,\overline{\gamma }=\gamma /\delta ,\overline{v}=v/\delta ,\end{array}$$

(13)

and two real numbers $\overline{\alpha }>\overline{\beta }>0$ determined by algebra equation

$$\overline{\alpha }+\overline{\beta }={\overline{\lambda }}_1+{\overline{\lambda }}_2+\overline{\gamma }\mathrm{and}\overline{\alpha }\overline{\beta }={\overline{\lambda }}_1{\overline{\lambda }}_2+q_1{\overline{\lambda }}_2\overline{\gamma }+q_2{\overline{\lambda }}_1\overline{\gamma }.$$

Recently, Zhu, Han and Jiao [20] studied the dynamical regulation of mRNA distribution produced in a transcription system activated by two competitive cross-talking pathways. They expressed distribution in mathematical dynamical formula under the moderate regulation, and displayed the dynamical transitions from mRNA decaying distribution to unimodal distribution undergoing the significant intermediate bimodal stage for stress genes in yeast. In this work, we continue to study the transcription model activated by two competitive cross-talking pathways via introducing a similar approach to the one in [16] to derive the mathematical formulas of the generating function $V(z,t)$ and mass function $P_m\left(t\right)$. Meanwhile, we get a dynamical bimodal distribution that is different from the results in [20].

The rest of this paper is organized as follows. In Section 2 and Section 3, we express mathematical formulas of the generating function $V(z,t)$ and the distribution $P_m\left(t\right)$, respectively. We then give some numerical simulations to illustrate the dynamical bimodal distribution in Section 4. Finally, conclusions and discussion are given in Section 5.

2. Expressing Generating Function $V(z,t)$

In this section, we use a novel approach to calculate the analytical formula of the generating function $V(z,t)$ in a special parameter region.

Theorem 1.

If $\gamma =2\delta$, $q_1>q_2$ and ${\lambda }_2={\lambda }_1+\delta /(q_1-q_2)$, the generating function $V(z,t)$ can be expressed as

$$\begin{array}{cc}\hfill V(z,t)=& \left[{\overline{\lambda }}_1{\overline{\lambda }}_2+a{\left(\overline{v}z{e}^{-\delta t}\right)}^2\right]{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{\overline{v}z(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{\overline{vz}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t\left[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}\right]e^{\overline{v}z{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& \frac{{\left(\overline{v}z\right)}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}[({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^t{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}ds\hfill \\ \hfill -& {\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^t{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}ds],\hfill \end{array}$$

(14)

where $a=(q_1{\overline{\lambda }}_1+q_2{\overline{\lambda }}_2)/\left(\overline{v}{z}_0\right).$

Proof. 

For $\gamma =2\delta$, $q_1>q_2$ and ${\lambda }_2={\lambda }_1+\delta /(q_1-q_2)$, without a loss of generality, we suppose $\overline{\alpha }={\overline{\lambda }}_2+1$ and $\overline{\beta }={\overline{\lambda }}_1+1$. Moreover, we let ${}_2{F}_2(x_1,x_2;y_1,y_2;x)$ be the generalized hypergeometric function [33], which is denoted by

$$\begin{array}{c}\hfill {}_2{F}_2(x_1,x_2;y_1,y_2;x)=\sum _{k=0}^{\infty }\frac{1}{k!}\frac{{\left(x_1\right)}_k{\left(x_2\right)}_k}{{\left(y_1\right)}_k{\left(y_2\right)}_k}{x}^k.\end{array}$$

(15)

Thus, ${}_2{F}_2$ is equal to 1 if it contains $1+{\overline{\lambda }}_2-\overline{\alpha }$ or $1+{\overline{\lambda }}_1-\overline{\beta }.$

Let $z_0$ be a parameter and

$$\begin{array}{c}\hfill u_i\left(t\right)=V_i\left(z_0{e}^{\delta t},t\right)\mathrm{for}i=1,2,e,\mathrm{and}u\left(t\right)=u_1\left(t\right)+u_2\left(t\right)+u_e\left(t\right).\end{array}$$

(16)

Then, we transform Equations (8)–(10) into ordinary differential equations [20,34],

$$\begin{array}{c}\hspace{1em}\hspace{1em}{u}_1^{\prime }\left(t\right)=-{\lambda }_1{u}_1\left(t\right)+q_1\gamma u_e\left(t\right),u_2^{\prime }\left(t\right)=-{\lambda }_2{u}_2\left(t\right)+q_2\gamma u_e\left(t\right),\hfill \end{array}$$

(17)

$$\begin{array}{c}\hspace{1em}{u}_e^{\prime }\left(t\right)={\lambda }_1{u}_1\left(t\right)+{\lambda }_2{u}_2\left(t\right)-\gamma u_e\left(t\right)+v{z}_0{e}^{\delta t}{u}_e\left(t\right),\hfill \end{array}$$

(18)

$$\begin{array}{c}{\displaystyle u_1\left(0\right)=q_1,u_2\left(0\right)=q_2\mathrm{and}{u}_e\left(0\right)=0.}\hfill \end{array}$$

(19)

By introducing a further transformation

$$\begin{array}{c}\hfill w_i\left(x\right)=u_i\left(t\right)\mathrm{for}i=1,2,e,\mathrm{and}w\left(x\right)=u\left(t\right),\mathrm{where}x=\overline{v}{z}_0{e}^{\delta t},\end{array}$$

(20)

we obtain $u_i^{\prime }\left(t\right)=\delta x{w}_i^{\prime }\left(x\right)$. Then, Equations (17)–(19) can be transformed into

$$\begin{array}{c}x{w}_1^{\prime }\left(x\right)=-{\overline{\lambda }}_1{w}_1\left(x\right)+q_1\overline{\gamma }{w}_e\left(x\right),x{w}_2^{\prime }\left(x\right)=-{\overline{\lambda }}_2{w}_2\left(x\right)+q_2\overline{\gamma }{w}_e\left(x\right),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & x{w}_e^{\prime }\left(x\right)={\overline{\lambda }}_1{w}_1\left(x\right)+{\overline{\lambda }}_2{w}_2\left(x\right)-\overline{\gamma }{w}_e\left(x\right)+x{w}_e\left(x\right),\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& & {\displaystyle w_1\left(\overline{v}{z}_0\right)=q_1,w_2\left(\overline{v}{z}_0\right)=q_2\mathrm{and}{w}_e\left(\overline{v}{z}_0\right)=0.}\hfill \end{array}$$

(23)

We give a linear operator as

$$\begin{array}{c}\hfill {\mathcal{L}}_{c,d}^{a,b}\left(f\right)=x^2{f}^{‴}\left(x\right)+x(1-x+c+d)f^{″}\left(x\right).+(cd-x(a+b+1))f^{\prime }\left(x\right)-abf\left(x\right),\end{array}$$

(24)

and then prove that $w\left(x\right)$ is the unique solution of the following initial value problem

$$\begin{array}{c}\hfill {\mathcal{L}}_{\overline{\alpha },\overline{\beta }}^{{\overline{\lambda }}_1,{\overline{\lambda }}_2}\left(w\right)=0,w\left(\overline{v}{z}_0\right)=1,w^{\prime }\left(\overline{v}{z}_0\right)=0\mathrm{and}{w}^{″}\left(\overline{v}{z}_0\right)=(q_1{\overline{\lambda }}_1+q_2{\overline{\lambda }}_2)/\overline{v}{z}_0,\end{array}$$

(25)

by using a similar calculation to the one in [6], where a, b, c and d are real constants and $f=f\left(x\right)$ is a smooth function of x.

Since ${\mathcal{L}}_{\overline{\alpha },\overline{\beta }}^{{\overline{\lambda }}_1,{\overline{\lambda }}_2}\left(w\right)=0$ in (25) is the generalized hypergeometric equation, then, if $\overline{\alpha }-\overline{\beta }\ne 1,2,\cdots$, there are three independent particular solutions of ${\mathcal{L}}_{\overline{\alpha },\overline{\beta }}^{{\overline{\lambda }}_1,{\overline{\lambda }}_2}\left(w\right)=0$, that is,

$$\begin{array}{cc}\hfill {\omega }_1\left(t\right)=& {}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2;{\overline{\lambda }}_2+1,{\overline{\lambda }}_1+1;x),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\omega }_2\left(t\right)=& x^{1-\overline{\alpha }}{}_2{F}_2(1+{\overline{\lambda }}_1-\overline{\alpha },1+{\overline{\lambda }}_2-\overline{\alpha };2-\overline{\alpha },1+\overline{\beta }-\overline{\alpha };x)=x^{-{\overline{\lambda }}_2},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\omega }_3\left(t\right)=& x^{1-\overline{\beta }}{}_2{F}_2(1+{\overline{\lambda }}_1-\overline{\beta },1+{\overline{\lambda }}_2-\overline{\beta };2-\overline{\beta },1+\overline{\alpha }-\overline{\beta };x)=x^{-{\overline{\lambda }}_1}.\hfill \end{array}$$

(28)

Then, the general solution for ${\mathcal{L}}_{\overline{\alpha },\overline{\beta }}^{{\overline{\lambda }}_1,{\overline{\lambda }}_2}\left(w\right)=0$ is derived and given as

$$\begin{array}{c}\hfill \omega \left(x\right)=A{\omega }_1\left(x\right)+B{\omega }_2\left(x\right)+C{\omega }_3\left(x\right),\end{array}$$

(29)

where A, B and C are undetermined constants. By substituting the inial value condition $w\left(\overline{v}{z}_0\right)=1,w^{\prime }\left(\overline{v}{z}_0\right)=0$ and $w^{″}\left(\overline{v}{z}_0\right)=(q_1{\overline{\lambda }}_1+q_2{\overline{\lambda }}_2)/\overline{v}{z}_0$ into (29), we can obtain the following system of equations:

$$\begin{array}{ccc}& & A{\omega }_1\left(\overline{v}{z}_0\right)+B{\omega }_2\left(\overline{v}{z}_0\right)+C{\omega }_3\left(\overline{v}{z}_0\right)=1,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & A{\omega }_1^{\prime }\left(\overline{v}{z}_0\right)+B{\omega }_2^{\prime }\left(\overline{v}{z}_0\right)+C{\omega }_3^{\prime }\left(\overline{v}{z}_0\right)=0,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & A{\omega }_1^{″}\left(\overline{v}{z}_0\right)+B{\omega }_2^{″}\left(\overline{v}{z}_0\right)+C{\omega }_3^{″}\left(\overline{v}{z}_0\right)=a,\hfill \end{array}$$

(32)

where $a=(q_1{\overline{\lambda }}_1+q_2{\overline{\lambda }}_2)/\left(\overline{v}{z}_0\right).$

In order to solve Equations (30)–(32), multiplying (32) by $-{\omega }_3\left(\overline{v}{z}_0\right)/{\omega }_3^{″}\left(\overline{v}{z}_0\right)$ and adding (30) lead to

$$\begin{array}{cc}\hfill & A[{\omega }_1\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_1^{″}\left(\overline{v}{z}_0\right){\omega }_3\left(\overline{v}{z}_0\right)]\hfill \\ \hfill +& B[{\omega }_2\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_2^{″}\left(\overline{v}{z}_0\right){\omega }_3\left(\overline{v}{z}_0\right)]={\omega }_3^{″}\left(\overline{v}{z}_0\right)-a{\omega }_3^{\prime }\left(\overline{v}{z}_0\right).\hfill \end{array}$$

(33)

Using the same method, we multiply (32) by $-{\omega }_3^{\prime }\left(\overline{v}{z}_0\right)/{\omega }_3^{″}\left(\overline{v}{z}_0\right)$ and add (31), then obtain

$$\begin{array}{cc}\hfill & A[{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_1^{″}\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)]\hfill \\ \hfill +& B[{\omega }_2^{\prime }\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_2^{″}\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)]=-a{\omega }_3^{\prime }\left(\overline{v}{z}_0\right).\hfill \end{array}$$

(34)

By a straightforward calculation, we solve (33) and (34) and obtain

$$\begin{array}{ccc}& & A=\frac{{\omega }_2^{\prime }\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_2^{″}\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)+a({\omega }_2\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_2^{\prime }\left(\overline{v}{z}_0\right){\omega }_3\left(\overline{v}{z}_0\right))}{\tilde{W}\left(\overline{v}{z}_0\right)},\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & B=\frac{{\omega }_1^{″}\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)+a({\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_3\left(\overline{v}{z}_0\right)-{\omega }_1\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right))}{\tilde{W}\left(\overline{v}{z}_0\right)},\hfill \end{array}$$

(36)

where $\tilde{W}\left(s\right)$ is the $Wronskian$ of ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$, namely,

$$\begin{array}{cc}\hfill \tilde{W}\left(s\right)=& {\omega }_1\left(s\right)[{\omega }_2^{\prime }\left(s\right){\omega }_3^{″}\left(s\right)-{\omega }_2^{″}\left(s\right){\omega }_3^{\prime }\left(s\right)]-{\omega }_1^{\prime }\left(s\right)[{\omega }_2\left(s\right){\omega }_3^{″}\left(s\right)-{\omega }_2^{″}\left(s\right){\omega }_3\left(s\right)]\hfill \\ \hfill & +{\omega }_1^{″}\left(s\right)[{\omega }_2\left(s\right){\omega }_3^{\prime }\left(s\right)-{\omega }_2^{\prime }\left(s\right){\omega }_3\left(s\right)].\hfill \end{array}$$

(37)

We substitute (35) and (36) into (32) and obtain

$$\begin{array}{c}\hfill C=\frac{{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_2^{″}\left(\overline{v}{z}_0\right)-{\omega }_1^{″}\left(\overline{v}{z}_0\right){\omega }_2^{\prime }\left(\overline{v}{z}_0\right)+a({\omega }_1\left(\overline{v}{z}_0\right){\omega }_2^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_2\left(\overline{v}{z}_0\right))}{\tilde{W}\left(\overline{v}{z}_0\right)}.\end{array}$$

(38)

Differentiating the generalized hypergeometric function, we obtain

$$\begin{array}{c}\hfill \frac{d^k{}_2{F}_2(x_1,x_2;y_1,y_2;x)}{d{x}^k}=\frac{{\left(x_1\right)}_k{\left(x_2\right)}_k}{{\left(y_1\right)}_k{\left(y_2\right)}_k}{}_2{F}_2(x_1+k,x_2+k;y_1+k,y_2+k;x),k=0,1,\cdots \end{array}$$

(39)

and

$$\begin{array}{c}\hfill \frac{d^k{x}^{y_1-1}{}_2{F}_2(x_1,x_2;y_1,y_2;x)}{d{x}^k}={(y_1-k)}_k{x}^{y_1-k-1}{}_2{F}_2(x_1,x_2;y_1-k,y_2;x),k=0,1,\cdots \end{array}$$

(40)

Applying (26)–(28), (39) and (40) leads to

$$\begin{array}{c}{\omega }_1^{\prime }\left(s\right)=\frac{{\overline{\lambda }}_1{\overline{\lambda }}_2}{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s),\hfill \end{array}$$

(41)

$$\begin{array}{c}{\omega }_1^{″}\left(s\right)=\frac{{\overline{\lambda }}_1{\overline{\lambda }}_2}{({\overline{\lambda }}_1+2)({\overline{\lambda }}_2+2)}{}_2{F}_2({\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;{\overline{\lambda }}_1+3,{\overline{\lambda }}_2+3;s),\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & {\omega }_2^{\prime }\left(s\right)=-\frac{{\overline{\lambda }}_2}{s^{{\overline{\lambda }}_2+1}},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & {\omega }_2^{″}\left(s\right)=\frac{{\overline{\lambda }}_2({\overline{\lambda }}_2+1)}{s^{{\overline{\lambda }}_2+2}},\hfill \end{array}$$

(44)

$$\begin{array}{ccc}& & {\omega }_3^{\prime }\left(s\right)=-\frac{{\overline{\lambda }}_1}{s^{{\overline{\lambda }}_1+1}},\hfill \end{array}$$

(45)

$$\begin{array}{ccc}& & {\omega }_3^{″}\left(s\right)=\frac{{\overline{\lambda }}_1({\overline{\lambda }}_1+1)}{s^{{\overline{\lambda }}_1+2}}.\hfill \end{array}$$

(46)

We set

$$\begin{array}{cc}\hfill \tilde{W}\left(s\right)=& (\overline{\alpha }-\overline{\beta })(\overline{\alpha }-1)(\overline{\beta }-1)e^s{s}^{-(1+\overline{\alpha }+\overline{\beta })}\hfill \\ \hfill =& ({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2{e}^s{s}^{-({\overline{\lambda }}_1+{\overline{\lambda }}_2+3)}.\hfill \end{array}$$

(47)

Then, we verify (47). Due to the fact that ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ are three independent solutions of the third order ordinary differential equation ${\mathcal{L}}_{\overline{\alpha },\overline{\beta }}^{{\overline{\lambda }}_1,{\overline{\lambda }}_2}\left(w\right)=0$, for any given real number $s_0$, $\tilde{W}\left(s\right)$ has the following form:

$$\begin{array}{cc}\hfill \tilde{W}\left(s\right)=& \tilde{W}\left(s_0\right)exp\left\{-{\int }_{s_0}^s\frac{\overline{\alpha }+\overline{\beta }+1-s}{s}ds\right\}\hfill \\ \hfill =& \tilde{W}\left(s_0\right)e^{s-s_0}{(s_0/s)}^{\overline{\alpha }+\overline{\beta }+1}\hfill \\ \hfill =& \tilde{W}\left(s_0\right)e^{s-s_0}{(s_0/s)}^{{\overline{\lambda }}_1+{\overline{\lambda }}_2+3}.\hfill \end{array}$$

(48)

Since $s_0$ is arbitrary, we could let $s_0=0$. Moreover, we have ${}_2{F}_2(x_1,x_2;y_1,y_2;0)=1$ by calculation. Substituting (26)–(28), (37) and (41)–(46) into (48), we obtain (47) immediately.

We substitute (35), (36), (38) and (47) into (29), and obtain

$$\begin{array}{cc}\hfill \omega \left(x\right)=& \frac{e^{-\overline{v}{z}_0}{\left(\overline{v}{z}_0\right)}^{{\overline{\lambda }}_1+{\overline{\lambda }}_2+3}{\omega }_1\left(x\right)}{({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2}[{\omega }_2^{\prime }\left(\overline{v}{z}_0\right){\omega }_3^{″}\left(\overline{v}{z}_0\right)-{\omega }_2^{″}\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)\hfill \\ \hfill & +a({\omega }_2\left(\overline{v}{z}_0\right){\omega }_3^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_2^{\prime }\left(\overline{v}{z}_0\right){\omega }_3\left(\overline{v}{z}_0\right))]\hfill \\ \hfill & +\frac{e^{-\overline{v}{z}_0}{\left(\overline{v}{z}_0\right)}^{{\overline{\lambda }}_1+{\overline{\lambda }}_2+3}{\omega }_2\left(x\right)}{({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2}[{\omega }_3^{\prime }\left(\overline{v}{z}_0\right){\omega }_1^{″}\left(\overline{v}{z}_0\right)-{\omega }_3^{″}\left(\overline{v}{z}_0\right){\omega }_1^{\prime }\left(\overline{v}{z}_0\right)\hfill \\ \hfill & +a({\omega }_3\left(\overline{v}{z}_0\right){\omega }_1^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_3^{\prime }\left(\overline{v}{z}_0\right){\omega }_1\left(\overline{v}{z}_0\right))]\hfill \\ \hfill & +\frac{e^{-\overline{v}{z}_0}{\left(\overline{v}{z}_0\right)}^{{\overline{\lambda }}_1+{\overline{\lambda }}_2+3}{\omega }_3\left(x\right)}{({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2}[{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_2^{″}\left(\overline{v}{z}_0\right)-{\omega }_1^{″}\left(\overline{v}{z}_0\right){\omega }_2^{\prime }\left(\overline{v}{z}_0\right)\hfill \\ \hfill & +a({\omega }_1\left(\overline{v}{z}_0\right){\omega }_2^{\prime }\left(\overline{v}{z}_0\right)-{\omega }_1^{\prime }\left(\overline{v}{z}_0\right){\omega }_2\left(\overline{v}{z}_0\right))].\hfill \end{array}$$

(49)

Substituting (41)–(46) into (49), we have

$$\begin{array}{cc}\hfill \omega \left(x\right)=& \frac{\mathcal{A}\left(\overline{v}{z}_0\right)}{({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{\overline{v}{z}_0}}{\omega }_1\left(x\right)+\frac{{\overline{\lambda }}_1{\overline{\lambda }}_2{\left(\overline{v}{z}_0\right)}^{\overline{\alpha }}\mathcal{B}\left(\overline{v}{z}_0\right)}{\overline{\alpha }\overline{\beta }({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{\overline{v}{z}_0}}{\omega }_2\left(x\right)\hfill \\ \hfill & +\frac{{\overline{\lambda }}_1{\overline{\lambda }}_2{\left(\overline{v}{z}_0\right)}^{\overline{\beta }}\mathcal{C}\left(\overline{v}{z}_0\right)}{\overline{\alpha }\overline{\beta }({\overline{\lambda }}_2-{\overline{\lambda }}_1){\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{\overline{v}{z}_0}}{\omega }_3\left(x\right),\hfill \end{array}$$

(50)

here

$$\begin{array}{cc}\hfill \mathcal{A}\left(s\right)=& {\overline{\lambda }}_1{\overline{\lambda }}_2({\overline{\lambda }}_2-{\overline{\lambda }}_1)+s^{2}a({\overline{\lambda }}_2-{\overline{\lambda }}_1),\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill \mathcal{B}\left(s\right)=& (-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)[\frac{({\overline{\lambda }}_2+1)s}{({\overline{\lambda }}_1+2)({\overline{\lambda }}_2+2)}{}_2{F}_2({\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;{\overline{\lambda }}_1+3,{\overline{\lambda }}_2+3;s)\hfill \\ \hfill & +{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)]\hfill \\ \hfill & +\frac{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)as}{{\overline{\lambda }}_2}[\frac{{\overline{\lambda }}_{2}s}{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)\hfill \\ \hfill & +{}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;s)],\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill \mathcal{C}\left(s\right)=& {\overline{\lambda }}_2({\overline{\lambda }}_2+1)[\frac{({\overline{\lambda }}_1+1)s}{({\overline{\lambda }}_1+2)({\overline{\lambda }}_2+2)}{}_2{F}_2({\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;{\overline{\lambda }}_1+3,{\overline{\lambda }}_2+3;s)\hfill \\ \hfill & +{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)]\hfill \\ \hfill & -{\overline{\lambda }}_{2}as[\frac{{\overline{\lambda }}_{1}s}{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)\hfill \\ \hfill & +{}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;s)].\hfill \end{array}$$

(53)

Noting that [16]

$$\begin{array}{c}\hfill \frac{b_{1}s}{c_1{c}_2}{}_2{F}_2(b_1+1,b_2+1;c_1+1,c_2+1;s)+{}_2{F}_2(b_1,b_2;c_1,c_2;s)={}_2{F}_2(b_1,b_2+1;c_1,c_2;s)\end{array}$$

and

$$\begin{array}{c}\hfill \frac{b_{2}s}{c_1{c}_2}{}_2{F}_2(b_1+1,b_2+1;c_1+1,c_2+1;s)+{}_2{F}_2(b_1,b_2;c_1,c_2;s)={}_2{F}_2(b_1+1,b_2;c_1,c_2;s).\end{array}$$

We can further simplify $\mathcal{B}\left(s\right)$ and $\mathcal{C}\left(s\right)$ as

$$\begin{array}{cc}\hfill \mathcal{B}\left(s\right)=& -{\overline{\lambda }}_1({\overline{\lambda }}_1+1){}_2{F}_2({\overline{\lambda }}_1+2,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)\hfill \\ \hfill & +\frac{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)as}{{\overline{\lambda }}_2}{}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;s)\hfill \\ \hfill =& -{\overline{\lambda }}_1({\overline{\lambda }}_1+1){}_1{F}_1({\overline{\lambda }}_2+1;{\overline{\lambda }}_2+2;s)+\frac{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)as}{{\overline{\lambda }}_2}{}_1{F}_1({\overline{\lambda }}_2;{\overline{\lambda }}_2+1;s),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \mathcal{C}\left(s\right)=& {\overline{\lambda }}_2({\overline{\lambda }}_2+1){}_2{F}_2({\overline{\lambda }}_1+1,{\overline{\lambda }}_2+2;{\overline{\lambda }}_1+2,{\overline{\lambda }}_2+2;s)\hfill \\ \hfill & -{\overline{\lambda }}_{2}as{}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2+1;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;s)\hfill \\ \hfill =& {\overline{\lambda }}_2({\overline{\lambda }}_2+1){}_1{F}_1({\overline{\lambda }}_1+1;{\overline{\lambda }}_1+2;s)-{\overline{\lambda }}_{2}as{}_1{F}_1({\overline{\lambda }}_1;{\overline{\lambda }}_1+1;s),\hfill \end{array}$$

(55)

where ${}_1{F}_1$ is the confluent hypergeometric function, which is the following:

$$\begin{array}{c}\hfill {}_1{F}_1(b_1;c_1;x)=\sum _{k=0}^{\infty }\frac{1}{k!}\frac{{\left(b_1\right)}_k}{{\left(c_1\right)}_k}{x}^k.\end{array}$$

(56)

By substituting (26)–(28) into (29) and replacing x with $\overline{v}z$, and $z_0$ with $z{e}^{-\delta t}$, we obtain

$$\begin{array}{cc}\hfill V(z,t)=& \left[e^{-\overline{v}z{e}^{-\delta t}}+\frac{{\left(\overline{v}z{e}^{-\delta t}\right)}^{2}a}{{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{\overline{v}z{e}^{-\delta t}}}\right]{}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;\overline{v}z)\hfill \\ \hfill & +\frac{\overline{v}z{e}^{-({\overline{\lambda }}_2+1)\delta t}}{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)({\overline{\lambda }}_2-{\overline{\lambda }}_1)e^{\overline{v}z{e}^{-\delta t}}}[-{\overline{\lambda }}_1({\overline{\lambda }}_1+1){}_1{F}_1({\overline{\lambda }}_2+1;{\overline{\lambda }}_2+2;\overline{v}z{e}^{-\delta t})\hfill \\ \hfill & +\frac{a({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)\overline{v}z{e}^{-\delta t}}{{\overline{\lambda }}_2}{}_1{F}_1({\overline{\lambda }}_2;{\overline{\lambda }}_2+1;\overline{v}z{e}^{-\delta t})]\hfill \\ \hfill & +\frac{\overline{v}z{e}^{-({\overline{\lambda }}_1+1)\delta t}}{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)({\overline{\lambda }}_2-{\overline{\lambda }}_1)e^{\overline{v}z{e}^{-\delta t}}}[{\overline{\lambda }}_2({\overline{\lambda }}_2+1){}_1{F}_1({\overline{\lambda }}_1+1;{\overline{\lambda }}_1+2;\overline{v}z{e}^{-\delta t})\hfill \\ \hfill & -{\overline{\lambda }}_{2}a\overline{v}z{e}^{-\delta t}{}_1{F}_1({\overline{\lambda }}_1;{\overline{\lambda }}_1+1;\overline{v}z{e}^{-\delta t})]\hfill \end{array}$$

(57)

We now denote $\mathsf{\Gamma }(·)$ as a gamma function, and rewrite the Euler’s integral for ${}_1{F}_1$ and ${}_2{F}_2$ as follows [35]:

$$\begin{array}{cc}\hfill {}_1{F}_1(b_1;c_1;x)=& \frac{\mathsf{\Gamma }\left(c_1\right)}{\mathsf{\Gamma }\left(b_1\right)\mathsf{\Gamma }(c_1-b_1)}{\int }_0^1{e}^{xs}{s}^{b_1-1}{(1-s)}^{c_1-b_1-1}ds,\mathrm{for}{\mathrm{c}}_1>{\mathrm{b}}_1>0,\end{array}$$

(58)

$$\begin{array}{cc}\hfill {}_2{F}_2(b_1,b_2;c_1,c_2;x)=& M{\int }_0^1{\int }_0^1{s}^{b_1-1}{(1-s)}^{c_1-b_1-1}{e}^{xs\sigma }{\sigma }^{b_2-1}{(1-\sigma )}^{c_2-b_2-1}d\sigma ds,\end{array}$$

(59)

when $c_1>b_1>0$ and $c_2>b_2>0$, where

$$M=\frac{\mathsf{\Gamma }\left(c_1\right)\mathsf{\Gamma }\left(c_2\right)}{\mathsf{\Gamma }\left(b_1\right)\mathsf{\Gamma }\left(b_2\right)\mathsf{\Gamma }(c_1-b_1)\mathsf{\Gamma }(c_2-b_2)}.$$

Moreover, we note that $\mathsf{\Gamma }(s+1)=s\mathsf{\Gamma }\left(s\right)$. Together with (58) and (59), it is easy to calculate that

$$\begin{array}{cc}\hfill {}_1{F}_1({\overline{\lambda }}_1;{\overline{\lambda }}_1+1;\overline{v}z{e}^{-\delta t})=& {\overline{\lambda }}_1{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}s}{s}^{{\overline{\lambda }}_1-1}ds,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {}_1{F}_1({\overline{\lambda }}_2;{\overline{\lambda }}_2+1;\overline{v}z{e}^{-\delta t})=& {\overline{\lambda }}_2{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}s}{s}^{{\overline{\lambda }}_2-1}ds,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {}_1{F}_1({\overline{\lambda }}_1+1;{\overline{\lambda }}_1+2;\overline{v}z{e}^{-\delta t})=& ({\overline{\lambda }}_1+1){\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}s}{s}^{{\overline{\lambda }}_1}ds,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {}_1{F}_1({\overline{\lambda }}_2+1;{\overline{\lambda }}_2+2;\overline{v}z{e}^{-\delta t})=& ({\overline{\lambda }}_2+1){\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}s}{s}^{{\overline{\lambda }}_2}ds,\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {}_2{F}_2({\overline{\lambda }}_1,{\overline{\lambda }}_2;{\overline{\lambda }}_1+1,{\overline{\lambda }}_2+1;\overline{v}z)=& {\overline{\lambda }}_1{\overline{\lambda }}_2{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{\overline{v}zs\sigma }{\sigma }^{{\overline{\lambda }}_2-1}.\hfill \end{array}$$

(64)

We can obtain the integral expression of $V(z,t)$ immediately by substituting (60)–(64) into (69). The proof of Theorem 1 is completed. □

3. Expressing Mathematical Formulas of the Distribution

In this section, we express mathematical formulas of the distribution in the same parameter region that was given in the last section.

Theorem 2.

If $\gamma =2\delta$, $q_1>q_2$ and ${\lambda }_2={\lambda }_1+\delta /(q_1-q_2)$, the probability mass function $P_m\left(t\right)$ can be expressed as

$$\begin{array}{cc}\hfill P_0\left(t\right)=& \left[{\overline{\lambda }}_1{\overline{\lambda }}_2+a{\left(\overline{v}{e}^{-\delta t}\right)}^2\right]{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{-\overline{v}(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill -& \frac{\overline{v}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_2}]e^{-\overline{v}{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& \frac{{\left(\overline{v}\right)}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}[({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^t{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}ds\hfill \\ \hfill & -{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^t{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}ds],\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill P_1\left(t\right)=& {\overline{\lambda }}_1{\overline{\lambda }}_2\overline{v}(s\sigma -e^{-\delta t}){\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{-\overline{v}(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{\overline{v}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}][1-\overline{v}{e}^{-\delta t}(s-1)]e^{-\overline{v}{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& a{\overline{v}}^2{e}^{-2\delta t}{\int }_0^1{\int }_0^1\left[-2+\overline{v}(s\sigma -e^{-\delta t})\right]e^{-\overline{v}(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[-2+\overline{v}{e}^{-\delta t}(s-1)\right]ds-{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}\left[-2+\overline{v}{e}^{-\delta t}(s-1)\right]ds\},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill P_2\left(t\right)=& \frac{{\overline{\lambda }}_1{\overline{\lambda }}_2{\overline{v}}^2{(s\sigma -e^{-\delta t})}^2}{2}{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{-\overline{v}(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^2{e}^{-\delta t}}{2({\overline{\lambda }}_2-{\overline{\lambda }}_1)}{\int }_0^t\left[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}\right](s-1)\left[2-\overline{v}{e}^{-\delta t}(s-1)\right]e^{-\overline{v}{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& \frac{a{\overline{v}}^2{e}^{-2\delta t}}{2}{\int }_0^1{\int }_0^1\left[-2-4\overline{v}(s\sigma -e^{-\delta t})\right]e^{-\overline{v}(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^{2}a{e}^{-\delta t}}{2({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[-2-4\overline{v}(s\sigma -e^{-\delta t})\right]ds-{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{-\overline{v}{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}\left[-2-4\overline{v}(s\sigma -e^{-\delta t})\right]ds\},\hfill \end{array}$$

(67)

and when $m\ge 3$,

$$\begin{array}{cc}\hfill P_m\left(t\right)=& \frac{1}{m!}{\overline{\lambda }}_1{\overline{\lambda }}_2{\overline{v}}^m{(s\sigma -e^{-\delta t})}^m{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{-\overline{v}(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill & +\frac{\overline{v}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)m!}{\int }_0^t[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}]e^{\overline{v}{e}^{-\delta t}(s-1)}[m+\overline{v}{e}^{-\delta t}(1-s)].\hfill \\ \hfill & {\left[\overline{v}{e}^{-\delta t}(s-1)\right]}^{m-1}{e}^{\overline{v}(-1)e^{-\delta t}(s-1)}ds+\frac{a{\overline{v}}^2{e}^{-2\delta t}}{m!}{\int }_0^1{\int }_0^1\{2{\overline{v}}^{m-2}{(s\sigma -e^{-\delta t})}^{m-2}\hfill \\ \hfill & +4\overline{v}(s\sigma -e^{-\delta t})\left[m-2+\overline{v}(e^{-\delta t}-s\sigma )\right]{\overline{v}}^{m-3}{(s\sigma -e^{-\delta t})}^{m-3}\}{e}^{\overline{v}(-1)(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill & +\frac{{\overline{v}}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)m!}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{\overline{v}(-1)e^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[2{\overline{v}}^{m-2}{\left(e^{-\delta t}(s-1)\right)}^{m-2}+\overline{v}{e}^{-\delta t}(s-1)[m-2+\overline{v}{e}^{-\delta t}(1-s)]{\overline{v}}^{m-3}{\left[e^{-\delta t}(s-1)\right]}^{m-3}\right]ds\hfill \\ \hfill & -{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{\overline{v}(-1)e^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}[2{\overline{v}}^{m-2}{\left(e^{-\delta t}(s-1)\right)}^{m-2}\hfill \\ \hfill & +4\overline{v}{e}^{-\delta t}(s-1)[m-2+\overline{v}{e}^{-\delta t}(1-s)]{\overline{v}}^{m-3}{\left[e^{-\delta t}(s-1)\right]}^{m-3}\left]ds\right\}.\hfill \end{array}$$

(68)

Proof. 

Noting the identities

$$\begin{array}{c}\hfill \frac{{\partial }^m\left(z{e}^{az}\right)}{\partial z^m}=(m+az)a^{m-1}{e}^{az},\end{array}$$

and

$$\frac{{\partial }^m\left(z^2{e}^{az}\right)}{\partial z^m}=\left\{\begin{array}{c}2z{e}^{az}+z^{2}a{e}^{az},m=1,\hfill \\ 2z{e}^{az}+4az{e}^{az},m=2,\hfill \\ 2a^{m-2}{e}^{az}+4a(m-2+az)a^{m-3}{e}^{az},m\ge 3.\hfill \end{array}\right.$$

$P_m\left(t\right)$ can be extracted from (14) through the conversion formula (12). By calculation, we obtain

$$\begin{array}{cc}\hfill \frac{\partial V(z,t)}{\partial z}=& {\overline{\lambda }}_1{\overline{\lambda }}_2\overline{v}(s\sigma -e^{-\delta t}){\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{\overline{v}z(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{\overline{v}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t\left[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}\right]\left[1+\overline{v}{e}^{-\delta t}(s-1)z\right]e^{\overline{v}z{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& a{\overline{v}}^2{e}^{-2\delta t}{\int }_0^1{\int }_0^1\left[2z+z^2\overline{v}(s\sigma -e^{-\delta t})\right]e^{\overline{v}z(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[2z+z^2\overline{v}{e}^{-\delta t}(s-1)\right]ds-{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}\left[2z+z^2\overline{v}{e}^{-\delta t}(s-1)\right]ds\},\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill \frac{{\partial }^{2}V(z,t)}{\partial z^2}=& {\overline{\lambda }}_1{\overline{\lambda }}_2{\overline{v}}^2{(s\sigma -e^{-\delta t})}^2{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{\overline{v}z(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^2{e}^{-\delta t}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}][2+\overline{v}{e}^{-\delta t}(s-1)z](s-1)e^{\overline{v}z{e}^{-\delta t}(s-1)}ds\hfill \\ \hfill +& a{\overline{v}}^2{e}^{-2\delta t}{\int }_0^1{\int }_0^1\left[2z+4\overline{v}z(s\sigma -e^{-\delta t})\right]e^{\overline{v}z(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[2z+4\overline{v}z(s\sigma -e^{-\delta t})\right]ds-{\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}\left[2z+4\overline{v}z(s\sigma -e^{-\delta t})\right]ds\},\hfill \end{array}$$

and when $m=3,4,\cdots$,

$$\begin{array}{cc}\hfill \frac{{\partial }^{m}V(z,t)}{\partial z^m}=& {\overline{\lambda }}_1{\overline{\lambda }}_2{\overline{v}}^m{(s\sigma -e^{-\delta t})}^m{\int }_0^1{\int }_0^1{s}^{{\overline{\lambda }}_1-1}{e}^{\overline{v}z(s\sigma -e^{-\delta t})}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{\overline{v}}{{\overline{\lambda }}_2-{\overline{\lambda }}_1}{\int }_0^t[{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{s}^{{\overline{\lambda }}_1}-{\overline{\lambda }}_1{e}^{-({\overline{\lambda }}_2+1)\delta t}{s}^{{\overline{\lambda }}_2}][m+\overline{v}{e}^{-\delta t}(s-1)z]·\hfill \\ \hfill & {\left[\overline{v}{e}^{-\delta t}(s-1)\right]}^{m-1}{e}^{\overline{v}z{e}^{-\delta t}(s-1)}ds+a{\overline{v}}^2{e}^{-2\delta t}{\int }_0^1{\int }_0^1\{2{\overline{v}}^{m-2}{(s\sigma -e^{-\delta t})}^{m-2}\hfill \\ \hfill +& 4\overline{v}(s\sigma -e^{-\delta t})[m-2+\overline{v}(s\sigma -e^{-\delta t})z]{\overline{v}}^{m-3}{(s\sigma -e^{-\delta t})}^{m-3}\}{e}^{\overline{v}z(s\sigma -e^{-\delta t})}{s}^{{\overline{\lambda }}_1-1}{\sigma }^{{\overline{\lambda }}_2-1}d\sigma ds\hfill \\ \hfill +& \frac{{\overline{v}}^{2}a{e}^{-\delta t}}{({\overline{\lambda }}_2-{\overline{\lambda }}_1)({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)}\{({\overline{\lambda }}_1+1)({\overline{\lambda }}_2+1)e^{-({\overline{\lambda }}_2+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_2-1}·\hfill \\ \hfill & \left[2{\overline{v}}^{m-2}{\left(e^{-\delta t}(s-1)\right)}^{m-2}+4\overline{v}{e}^{-\delta t}(s-1)[m-2+\overline{v}{e}^{-\delta t}(s-1)z]{\overline{v}}^{m-3}{\left[e^{-\delta t}(s-1)\right]}^{m-3}\right]ds\hfill \\ \hfill -& {\overline{\lambda }}_1{\overline{\lambda }}_2{e}^{-({\overline{\lambda }}_1+1)\delta t}{\int }_0^1{e}^{\overline{v}z{e}^{-\delta t}(s-1)}{s}^{{\overline{\lambda }}_1-1}[2{\overline{v}}^{m-2}{\left(e^{-\delta t}(s-1)\right)}^{m-2}\hfill \\ \hfill +& 4{\overline{v}}^{m-2}{e}^{-\delta t}(s-1)[m-2+\overline{v}{e}^{-\delta t}(s-1)z]{\left[e^{-\delta t}(s-1)\right]}^{m-3}\left]ds\right\}.\hfill \end{array}$$

(70)

Substituting $z=-1$ into the expressions (14), (69) and (70) and then dividing them by $m!$, we obtain (65)–(68). The proof of Theorem 2 is completed. □

4. Dynamical Bimodal Distribution

In this section, we will demonstrate through numerical examples that competitive cross-talking pathways can generate new dynamics within several parameter regions compared with [20].

In growing yeast cells, transcripts are turned over rapidly, with a median mRNA half-life of ${\tau }_{1/2}=11$ min [36], so the degradation rate $\delta =ln2/{\tau }_{1/2}=0.063$ min${}^{-1}$. The upper bound of total transcripts ${\nu }_m/\delta =10$ gives the transcription rate, which is ${\nu }_m=0.63$ min${}^{-1}$. From the burst size ${\nu }_m/\delta =5$, we obtain $\gamma =0.126$ min${}^{-1}$. The selection probabilities of the two pathways are given as $q1=0.51$ and $q2=0.49$. Set a relative small strength rate ${\lambda }_1=0.001$ min${}^{-1}$ for the first pathway and a relative large strength rate ${\lambda }_2=3.151$ min${}^{-1}$ for the second one. The observed mRNA average level behaves as up-and-down dynamics, while the distribution transits from the decaying mode to the significant bimodality during a mediate time interval, and finally returns to decaying distribution; see Figure 1. We cannot find this dynamical regulation in the two-state [19], three-state [16] or even multi-state [37] models with a single pathway.

Therefore, we guess that cross-talking regulation can generate these special dynamics compared with single-pathway regulation for some special parameters when the dynamic of the mRNA average level is up-and-down. What about the dynamical distribution by cross-talking pathways in the general condition?

Cross-talking pathways can modulate transcription dynamics of mean in two groups of fibroblast genes after TNF stimulation, and the dynamics of the two groups are both up-and-down [38]. In [39], they used the following two groups of data to fit the two groups dynamics in [38]:

Group I: $\delta =1.22{\mathrm{h}}^{-1},v_m=400{\mathrm{h}}^{-1},\gamma =90{\mathrm{h}}^{-1},q1=0.025,q2=0.975,{\lambda }_1=0.03{\mathrm{h}}^{-1},{\lambda }_2=300{\mathrm{h}}^{-1}.$

Group II: $\delta =0.14{\mathrm{h}}^{-1},v_m=45{\mathrm{h}}^{-1},\gamma =1.9{\mathrm{h}}^{-1},q1=0.35,q2=0.65,{\lambda }_1=0.03{\mathrm{h}}^{-1},{\lambda }_2=15{\mathrm{h}}^{-1}.$

Using these data to simulate the dynamics of distribution in the cross-talking pathways model, it is interesting to find that it generates similar dynamics to those in the special parameter region; see Figure 2.

5. Conclusions and Discussion

In this study, we calculated the expressions of the generating function $V(z,t)$ and the probability mass function $P_m\left(t\right)$ of the model for which the gene is activated by competitive cross-talking pathways. In a special parameter region, we successfully obtained the dynamic expression $V(z,t)$ by simplifying some expressions firstly (Theorem 1) and deriving the expression $P_m\left(t\right)$ in integral forms (Theorem 2). Together with transcriptional kinetic rates from yeast and mouse fibroblast genes, we found that, when the transcription average level regulated by the cross-talking pathways displays up-and-down dynamics, the corresponding mRNA distribution transits from the decaying mode to the intermediate bimodality as time develops, and finally returns back to the decaying mode (Figure 1 and Figure 2), which is not obtained [20].

In this work, the approach we present for calculating the $P_m\left(t\right)$ of a cross-talking pathways model in a special parameter region only needs all parameters to be positive and finite. However, other current approaches to calculating $P_m\left(t\right)$ require several parameters to be zero or infinity. This approach may be further developed for the general parameters condition. The dynamical scenario of decaying–bimodal–decaying distribution transition is different from the transcription of the mammalian c-Fos gene, which exhibits up-and-down dynamics of an average level, whereby the intermediate distribution takes unimodality [40]. However, such decaying–bimodal–decaying dynamical transitions matches exactly with the experimental data of yeast stress-response genes, which are regulated by a weak basal pathway under normal growth conditions, whereas they are strongly activated by the HOG-MAPK signaling pathway under acute stresses [15]. The theoretical result of this work is in good agreement with the experimental observations. Note that our observed mRNA distribution dynamics have not been generated by the models in which the gene is activated by a single pathway with multiple steps [37] or regulated by feedback loops [14]. Our work thus provides novel dynamics of mRNA distribution modulated by multiple signaling pathways.