Queuing Models of Gene Expression: Analytical Distributions and Beyond

Abstract

Activation of a gene is a multistep biochemical process, involving recruitments of transcription factors and histone kinases as well as modification of histones. Many of these intermediate reaction steps would have been unspecified by experiments. Therefore, classical two-state models of gene expression established based on the memoryless (or Markovian) assumption would not well describe the reality in gene expression. Recent experimental data have indicated that the inactive phases of gene promoters are differently distributed, showing strong memory. Here, we use a nonexponential waiting-time distribution to model the complex activation process of a gene, and then analyze a queuing model of stochastic transcription. We successfully derive the analytical expression of the stationary mRNA distribution, which provides insight into the effect of molecular memory created by complex activating events on the mRNA expression. We find that the reduction in the waiting-time noise may result in the increase in the mRNA noise, contrary to the previous conclusion. Based on the derived distribution, we also develop a method to infer the waiting-time distribution from a known mRNA distribution. Data analysis on a realistic example verifies the validity of this method.

Significance

Activation of a gene is a complex biochemical process and involves several intermediate reaction steps, many of which have been unspecified by experiments. Stochastic models of gene expression that were previously established based on the constant reaction rates would not well reflect the reality in gene expression. To this end, we study a queuing model of stochastic transcription, which assumes that the reaction waiting time for the gene activation follows a general distribution. Importantly, we derive the analytical expression of stationary mRNA distribution. Our results provide insight into the role of molecular memory in fine-tuning the gene expression noise, and can be used in the inference of the underlying molecular mechanism.

Introduction

Gene expression is a complex biochemical process, inevitably leading to stochastic fluctuations in mRNA and further protein (1, 2, 3, 4). Although this inherent noise would be important for the maintenance of cellular functioning and the generation of cell phenotypic variability, mathematical models are a strong tool to quantify the contributions of different noisy sources of gene expression.

Single-cell studies on gene expression have indicated that most genes in prokaryotic and eukaryotic cells are expressed in a bursty fashion (5, 6, 7). In theory, this kind of expression manner may be modeled by two-state models in which the gene switches between two states: one transcriptionally active (on) state and one transcriptionally inactive (off) state. Classical two-state models of gene expression (8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) assume that all the involved reaction rates are constants, implying that the reaction waiting times, particularly including those from the on-to-off states and vice versa, follow exponential distributions. This Markovian assumption has led to important successes in the modeling of many gene expression processes (22,23). With the Markovian assumption, analytical mRNA or protein distributions have also been derived (8,11,12,24, 25, 26), which provide the best description of the stochastic properties of the underlying gene systems. These studies have revealed that the characteristic parameters in the reaction waiting-time distributions can have a significant impact on the mRNA and protein levels.

However, biochemical events involved in gene expression occur not necessarily in a Markovian manner but may take place in a non-Markovian manner. First, the complex control process of transcription initiation can generate nonexponential time intervals between transcription windows (27, 28, 29). Second, the synthesis of an mRNA would involve multiple intermediate reaction steps that have been unspecified because of experimental technologies, creating a memory between individual reaction events (30,31). For example, the inactive phases of promoter involving the prolactin gene in a mammalian cell are differently distributed, showing strong memory (27). Indeed, the increasing availability of time-resolved data on different kinds of interactions has verified the extensive existence of molecular memory in biological systems (27, 28, 29,32, 33, 34). How molecular memory impacts gene expression remains elusive even though non-Markovian models of gene expression have been established in terms of chemical master equations (21,35).

There have been studies on non-Markovian models of stochastic gene expression, which can be divided into two classes from the viewpoint of continuous-time random walk (CTRW) (35,36): active CTRWs in which the waiting time needs to be reset and passive CTRWs in which the waiting time cannot be reset. Qiu et al. (37) studied a specific active CTRW model of gene expression and derived the analytical expression for stationary gene-product distribution. In contrast, more studies focused on queuing models of gene expression (belonging to the passive CTRW class). Pedraza and Paulsson (31) studied gene expression from the viewpoint of queue theory, showing that common types of molecular mechanisms can produce gestation and senescence periods that reduce noise without requiring higher abundances, shorter lifetimes, or any concentration-dependent control loops. Kulkarni et al. (38,39) analyzed a stochastic model of bursty gene expression, which considers general waiting-time distributions governing arrival and decay of proteins. By mapping gene systems to standard queuing models, they also derived analytical expressions for the steady-state protein noise (defined as the ratio of the variance over the square mean). In addition, previous studies (11,25,40,41) also used multistate models to analyze some special class of non-Markovian processes. Nevertheless, for queuing models of gene expression with general waiting-time distributions, analytical mRNA or protein distributions have not been derived so far.

In this article, we also study a queuing model of stochastic gene expression with a nonexponential waiting-time distribution, focusing on gene-product distribution rather than statistical quantities such as the mean and variance studied in (27), (39), (42), and (43). We derive the analytical expression for stationary gene-product distributions, which provide insight into the role of molecular memory in fine-tuning the mRNA or protein noise and properties of the steady states. We also present estimations on the bounds of the expression noise constrained by the mean and variance of the waiting-time distribution. In addition, we develop an effective method to infer a waiting-time distribution and further the promoter structure from a known mRNA or protein distribution that can be obtained by experimental methods such as mRNA-FISH labeling or RT-qPCR (44,45). Our inference method is different from previous inference methods (28,42,44, 45, 46, 47), which used qualitative properties of the gene expression noise to infer microscopic kinetic parameters, such as transcription and translation rates. Data analysis on a realistic example verifies the validity of our inference method.

Materials and Methods

The classical two-state model of stochastic transcription assumes that the gene promoter has two activity states: one active (on) state in which the gene is transcribed and the other inactive (off) state in which transcription is prohibited. These transcriptionally active and inactive states may switch to each other. The mRNA molecules transcribed from DNA degrade in a linear manner. All involved reaction rates are assumed as constants, implying that the reaction waiting times follow exponential distributions or the reaction events take place in a Markovian (memoryless) manner (22,23). Different from the classical on-off model, our model assumes that the waiting time from the off state to the on state follows a general distribution denoted by $f\left(t\right)$. This assumption is reasonable because the turning on of transcription is a complex process with intermediate reaction steps that would have been unspecified by experiments. We point out that although this model has been studied in terms of statistical quantities (27,42,43), the analytical mRNA distribution has not been derived so far.

For convenience, we define $h\left(t\right)=f\left(t\right)/F\left(t\right)$, which is referred to as a hazard function (43,48, 49, 50), where $F\left(t\right)={\int }_t^{\infty }f\left(x\right)dx$ is the survival function of off state. Note that if $f\left(t\right)$ is an exponential distribution of the form $f\left(t\right)=\alpha e^{-\alpha t}$ (where $\alpha$ is a positive constant, representing the mean transition rate from off-to-on states), then we have $h\left(t\right)\equiv \alpha$. In other words, hazard function $h\left(t\right)$ is an extension of the constant transition rate from off-to-on states.

For the modeling convenience, we list all the involved biochemical reactions as follows

$$\begin{array}{l}{\text{R}}_1:off\stackrel{f\left(t\right)}{\to }on,{\text{R}}_2:on\stackrel{k}{\to }off,\\ {\text{R}}_3:on\stackrel{\lambda }{\to }on+X,{\text{R}}_4:X\stackrel{\mu }{\to }\varnothing ,\end{array}$$ (1)

where $X$ may represent mRNA or protein (a random variable; without loss of generality, we will assume that it represents mRNA), $k$ is the mean switching rate from on-to-off states, $\lambda$ represents the mean transcription rate, and $\mu$ represents the mean degradation rate of mRNA. These reaction rates are assumed to be constants, implying that the corresponding reaction waiting-time distributions are exponential.

Let $S\left(t\right)$ record the elapsed time since the gene enters the off state at time $t$, and $U\left(t\right)$ denote the promoter state at time $t$. Assume that the number of mRNA molecules is $n$ at time $t$. Then, the state of the system at time $t$ can be described by the following time-evolutional probabilities:

$$\begin{array}{l}{\tilde{P}}_1\left(n,t\right)=\mathrm{Pr}\left\{X\left(t\right)=n,U\left(t\right)=on\right\},\\ {\tilde{P}}_0\left(n,\tau ,t\right)\Delta t=\mathrm{Pr}\left\{X\left(t\right)=n,\tau <S\left(t\right)<\tau +\Delta t,U\left(t\right)=off\right\},\end{array}$$ (2)

where $\tau$ represents the elapsed time, ${\tilde{P}}_0(\cdot )$ and ${\tilde{P}}_1(\cdot )$ represent the probability density functions in off and on states at time $t$, respectively. To derive the renewal equations for ${\tilde{P}}_0\left(n,t\right)$ and ${\tilde{P}}_1\left(n,\tau ,t\right)$, we consider a small time increment $\Delta t$. Then, $F\left(\tau +\Delta t\right)/F\left(\tau \right)˜1-\left[f\left(\tau \right)/F\left(\tau \right)\right]\Delta t$ represents the conditional probability that reaction ${\text{R}}_1$ in Eq. 1 does not occur within time interval $\left(\tau ,\tau +\Delta t\right)$ if the elapsed time is $\tau$, whereas hazard function $h\left(\tau \right)$ represents the instantaneous probability that reaction ${\text{R}}_1$ happens also if the elapsed time is $\tau$. According to the total probability principle, we can establish the following renewal equations for ${\tilde{P}}_1\left(n,\tau ,t\right)$ and ${\tilde{P}}_0\left(n,t\right)$

$${\tilde{P}}_1\left(n,t+\Delta t\right)={\tilde{P}}_1\left(n,t\right)\left(1-\lambda \Delta t\right)\left(1-n\mu \Delta t\right)\left(1-k\Delta t\right)+{\tilde{P}}_1\left(n-1,t\right)\lambda \Delta t\left(1-\left(n-1\right)\mu \Delta t\right)\left(1-k\Delta t\right),+{\tilde{P}}_1\left(n+1,t\right)\left(1-\lambda \Delta t\right)\left(n+1\right)\mu \Delta t\left(1-k\Delta t\right)+\left(1-\lambda \Delta t\right)\left(1-n\mu \Delta t\right)\Delta t{\int }_0^t{\tilde{P}}_0\left(n,\tau ,t\right)h\left(\tau \right)d\tau$$ (3a)

$$\begin{array}{c}{\tilde{P}}_0\left(n,\tau +\Delta t,t+\Delta t\right)={\tilde{P}}_0\left(n,\tau ,t\right)\left(1-n\mu \Delta t\right)\left[1-\frac{f\left(\tau \right)}{F\left(\tau \right)}\Delta t\right]\\ +{\tilde{P}}_0\left(n+1,\tau ,t\right)\left(n+1\right)\mu \Delta t\left[1-\frac{f\left(\tau \right)}{F\left(\tau \right)}\Delta t\right].\end{array}$$ (3b)

In Eqs. 3a and 3b, ${\tilde{P}}_i(\cdot )$ should be understood as the probability set $\mathrm{Pr}\{\cdot \}$ defined in Eq. 2. The following boundary condition at $\tau =0$ needs to be imposed because consideration of the probability equilibrium

$${\tilde{P}}_0\left(n,\Delta t,t+\Delta t\right)\Delta t=k\Delta t{\tilde{P}}_1\left(n,t\right)\left(1-\lambda \Delta t\right)\left(1-n\mu \Delta t\right).$$ (3c)

Note that in the limit of small $\Delta t$, Eqs. 3a and 3b will become differential equations. If the stationary distributions of ${\tilde{P}}_1\left(n,t\right)$ and ${\tilde{P}}_0\left(n,\tau ,t\right)$ with regard to $t$ exist, and are denoted by $P_1\left(n\right)$ and $P_0\left(n,\tau \right)$, respectively, Eqs. 3a and 3b at steady state become

$$P_1\left(n\right)\left(\lambda +n\mu +k\right)=\lambda P_1\left(n-1\right)+\left(n+1\right)\mu P_1\left(n+1\right)+{\int }_0^{\infty }{P}_0\left(n,\tau \right)h\left(\tau \right)d\tau ,$$ (4a)

$$\frac{\partial P_0\left(n,\tau \right)}{\partial \tau }=-\left(n\mu +h\left(\tau \right)\right)P_0\left(n,\tau \right)+\left(n+1\right)\mu P_0\left(n+1,\tau \right),$$ (4b)

whereas Eq. 3c becomes

$$P_0\left(n,0\right)=k{P}_1\left(n\right).$$ (4c)

To solve (4a), (4b) with Eq. 4c, we introduce the probability-generating functions, as done previously (8,11),

$$G_1\left(z\right)=\sum _{n=0}^{\infty }{z}^n{P}_1\left(n\right),G_0\left(z,\tau \right)=\sum _{n=0}^{\infty }{z}^n{P}_0\left(n,\tau \right),$$ (5)

and set $W_0\left(z,\tau \right)=G_0\left(z,\tau \right)/F\left(\tau \right)$ for convenience. Then, we can obtain the following differential equations

$$\mu \left(z-1\right)\frac{d{G}_1\left(z\right)}{dz}=\left[\lambda \left(z-1\right)-k\right]G_1\left(z\right)+{\int }_0^{\infty }{W}_0\left(z,\tau \right)f\left(\tau \right)d\tau ,$$ (6a)

$$\frac{\partial W_0\left(z,\tau \right)}{\partial \tau }=-\mu \left(z-1\right)\frac{\partial W_0\left(z,\tau \right)}{\partial z},$$ (6b)

with the constraint condition

$$G_0\left(z,0\right)=k{G}_1\left(z\right).$$ (6c)

One will see that Eqs. 6a, 6b, and 6c are analytically solvable. As such, we can obtain the total probability-generating function $G\left(z\right)=G_0\left(z\right)+G_1\left(z\right)$ with $G_0\left(z\right)={\int }_0^{\infty }{G}_0\left(z,\tau \right)d\tau$ and further the total stationary probability distribution $P\left(m\right)=P_0\left(m\right)+P_1\left(m\right)$ according to the relationship between generating function and probability distribution.

Results

Analytical distributions

Using the characteristic method, we can derive the formal expression of $W_0\left(z,\tau \right)$ from Eq. 6b with Eq. 6c

$$W_0\left(z,\tau \right)=k{G}_1\left(\left(z-1\right)e^{-\mu \tau }+1\right).$$ (7)

If we denote $G_0\left(z\right)={\int }_0^{\infty }{G}_0\left(z,\tau \right)d\tau$, then Eq. 7 becomes

$$G_0\left(s+1\right)=k{\int }_0^{\infty }{G}_1\left(s{e}^{-\mu \tau }+1\right)F\left(\tau \right)d\tau ,$$ (8a)

whereas Eq. 6a becomes

$$s\mu \frac{d{G}_1\left(s+1\right)}{ds}=\left(\lambda s-k\right)G_1\left(s+1\right)+{\int }_0^{\infty }{G}_1\left(s{e}^{-\mu \tau }+1\right)f\left(\tau \right)d\tau ,$$ (8b)

where $s=z-1$. If we make Taylor expansions $G_j\left(s+1\right)={\sum }_{n=1}^{\infty }{a}_n^{\left(j\right)}{s}^n$ ($j=\mathrm{0,1}$), then coefficients $a_n^{\left(j\right)}$ are just binomial moments and $b_n=a_n^{\left(0\right)}+a_n^{\left(1\right)}$ is the total binomial moment of order $n$ (51,52). Equivalently,$b_n$ can be expressed as $b_n=\sum _{m=n}^{\infty }\left(\begin{array}{c}m\\ n\end{array}\right)P\left(m\right)$ (51,52), where $P\left(m\right)$ represents the stationary mRNA distribution, and the symbol $\left(\begin{array}{c}n\\ m\end{array}\right)$ represents a combinatorial number. According to Appendix A, we obtain the following expression of $b_n$,

$$b_n=\frac{{\lambda }^n{\tau }_1^n}{n!{\mu }^n{\prod }_{i=0}^{n-1}\left({\tau }_1+\tilde{F}\left(i\mu \right)\right)},$$ (9)

where ${\tau }_1=1/k$ represents the average waiting time in the on state.

Note that the binomial moments mentioned above have good properties, e.g., they converge to zero as their orders tend to infinity, and can be used in the reconstruction of the underlying distribution, with a reconstruction formula

$$P\left(m\right)=\sum _{n=m}^{\infty }{\left(-1\right)}^{n-m}\left(\begin{array}{c}n\\ m\end{array}\right)b_n,$$ (10)

where $m=\mathrm{0,1,2},\mathrm{...}$. We finally obtain the following formal expression of the stationary mRNA distribution

$$P\left(m\right)=\frac{{\left(\lambda /\mu \right)}^m}{m!{\prod }_{i=0}^{m-1}\left(1+k\tilde{F}\left(i\mu \right)\right)}\sum _{n=0}^{\infty }\frac{{\left(-\lambda /\mu \right)}^n}{n!{\prod }_{i=m}^{m+n-1}\left(1+k\tilde{F}\left(i\mu \right)\right)},$$ (11)

where we define ${\prod }_{i=0}^{n-1}\left(1+k\tilde{F}\left(i\mu \right)\right)=1$ if $n=0$, and function $\tilde{F}\left(s\right)$ represents the Laplace transform of function $F\left(t\right)$, i.e., $\tilde{F}\left(s\right)={\int }_0^{\infty }{e}^{-s\tau }F\left(\tau \right)d\tau$. Equation 11 shows how molecular memory characterized by waiting-time distribution $f\left(t\right)$ impacts the mRNA distribution through $\tilde{F}\left(i\mu \right)$. To more clearly see the effect of molecular memory on gene expression, we will analyze the mRNA noise in the next subsection.

In particular, if the waiting time from off-to-on states follows an exponential distribution of the form $f\left(t\right)=r{e}^{-rt}$, where $r$ is a mean transition rate from off-to-on states, then we have $\tilde{F}\left(s\right)=1/\left(s+r\right)$. In this case, the binomial moment of order $n$ can be expressed as $b_n=\left({\left(i+r/\mu \right)}_n/{\left(i+\left(r+k\right)/\mu \right)}_n\right)\left({\left(\lambda /\mu \right)}^n/n!\right)$, where symbol ${\left(c\right)}_n=c\left(c-1\right)\cdots \left(c-n+1\right)$ and $n=\mathrm{0,1},\mathrm{...}$. According to the above reconstruction formula, we thus obtain the analytical expression of the mRNA distribution

$$P\left(m\right)=\frac{\mathrm{\Gamma }\left(\frac{r}{\mu }+m\right)\mathrm{\Gamma }\left(\frac{k+r}{\mu }\right)}{\mathrm{\Gamma }\left(\frac{r}{\mu }\right)\mathrm{\Gamma }\left(\frac{k+r}{\mu }+m\right)}{\frac{{\left(\lambda /\mu \right)}^m}{m!}}_1^{}{F}_1\left(\frac{r}{\mu }+m,\frac{k+r}{\mu }+m,-\frac{\lambda }{\mu }\right),$$ (12)

where ${}_{1}F_1$ denotes the confluent hypergeometric function (53), and $\mathrm{\Gamma }(\cdot )$ is the common Γ-function. This distribution has been derived in previous works (11,54,55).

Next, we use Eq. 11 to calculate the mRNA distribution for a two-state model with arbitrarily many inactive pathways with each consisting of several off states (Fig. 1 A). The probability density function for the waiting time takes the form $f\left(t\right)={\mathbf{a}}_0\exp \left(\mathbf{A}t\right){\mathbf{u}}_0$ (seeing Appendix B or referring to (11) and (42)), where $\mathbf{A}$ (a matrix) describes transitions among off states, ${\mathbf{u}}_0$ (a column vector) describes the transitions from the off states to the on state, and ${\mathbf{a}}_0$ (a row vector) denotes the initial condition. The Laplace transform of probability density function $f\left(t\right)$ is given by $\tilde{f}\left(s\right)={\mathbf{a}}_0{\left(s\mathbf{E}-\mathbf{A}\right)}^{-1}{\mathbf{u}}_0$, and the Laplace transform of survival probability function $F\left(t\right)$ is given by

$$\tilde{F}\left(s\right)={\mathbf{a}}_0{\left(s\mathbf{E}-\mathbf{A}\right)}^{-1}{\mathbf{u}}_1,$$ (13)

where ${\mathbf{u}}_1={\left(\mathrm{1,1},\cdots ,1\right)}^{\text{T}}$ is a column vector. Substituting Eq. 13 into Eq. 11, we can obtain the mRNA distribution for the two-state model with arbitrarily many inactive pathways with cross talk.

Figure 1.

(A) A two-state model of gene expression with arbitrarily many inactive pathways; (B) two parallel inactive pathways; (C) waiting-time distribution, where the red line represents an approximate solution obtained using a mixed Γ-distribution, empty circles represent the exact solution obtained by the Gillespie algorithm, the parameter values are set as $k_{10}=k_{20}=50$, $k_{11}=k_{12}=\mathrm{...}=k_{15}=1$, $k_{21}=k_{22}=\mathrm{...}=k_{26}=6$; (D) mRNA distribution, where the red line represents the analytical solution, empty circles represent the numeric solution obtained by the Gillespie algorithm, parameter values are set as $k=1/3$, $\lambda =15$, and$\mu =1$, and the other parameters are same as (C). To see this figure in color, go online.

For clarity, let us consider the special case that there is only one inactive pathway consisting of two off states:

$$of{f}_1\stackrel{k_1}{\to }of{f}_2\stackrel{k_2}{\to }on,$$ (14)

then the Laplace transform of the waiting-time distribution is given by

$$\tilde{f}\left(s\right)=\frac{s+k_1+k_2}{\left(s+k_1\right)\left(s+k_2\right)}.$$ (15)

Substituting this expression into Eq. 11, we obtain the mRNA distribution of the form

$$P\left(m\right)=\frac{\prod _{i=1}^2\left({\alpha }_i+m\right)\prod _{i=1}^2\left({\beta }_i+m\right)}{\prod _{i=1}^2{\alpha }_i\prod _{i=1}^2{\beta }_i}\frac{{\left(\lambda /\mu \right)}^m}{m}{}_2{F}_2\left({\alpha }_1+m,{\alpha }_2+m;{\beta }_1+m,{\beta }_2+m;-\frac{\lambda }{\mu }\right),$$ (16)

where ${\alpha }_1=k_1$, ${\alpha }_2=k_2$, ${\beta }_{\mathrm{1,2}}=\left(k_1+k_2-k\pm \sqrt{{\left(k_1+k_2-k\right)}^2-4k_1{k}_2}\right)/2$. This distribution was also derived in a previous work (11,25).

If there are two parallel inactive pathways with each consisting of several off states (referring to Fig. 1 B), then the Laplace transform of waiting-time distribution is given by

$$\tilde{f}\left(s\right)=\frac{k_{10}}{k_{10}+k_{20}}\frac{k_{10}+k_{20}}{s+k_{10}+k_{20}}\prod _{i=1}^n\frac{k_{1i}}{s+k_{1i}}+\frac{k_{20}}{k_{10}+k_{20}}\frac{k_{10}+k_{20}}{s+k_{10}+k_{20}}\prod _{i=1}^m\frac{k_{2i}}{s+k_{2i}},$$ (17)

Under the assumption of $k_{11}=k_{12}=\cdots =k_{1n}\equiv k_1$, $k_{21}=k_{22}=\cdots =k_{2m}\equiv k_2$, and $k_{10}+k_{20}\gg \max \left\{k_1,k_2\right\}$, we find that the waiting-time distribution can be approximated by a mixed Γ-distribution of the form (referring to Fig. 1 C)

$$f\left(t\right)=\frac{k_{10}}{k_{10}+k_{20}}\gamma \left(t;n,k_1\right)+\frac{k_{20}}{k_{10}+k_{20}}\gamma \left(t;m,k_2\right),$$ (18)

where $\gamma \left(t;\alpha ,r\right)=\left(t^{\alpha -1}{r}^{\alpha }{e}^{-rt}/\mathrm{\Gamma }\left(\alpha \right)\right)$ is a common Γ-distribution, which is an extension of exponential distribution. Parameter $\alpha$ will be called memory index because $\alpha >1$ corresponds to the non-Markovian case whereas $\alpha =1$ to the Markovian case. Moreover, the larger the $\alpha$ is, the stronger is the molecular memory. In particular, if $\alpha$ is a positive integer, then $\gamma \left(t;\alpha ,r\right)$ is an Erlang distribution, which can model a multistep process. The validity of the approximation given by Eq. 18 has numerically been verified, referring to Fig. 1 D, where empty circles represents the “exact” mRNA distribution obtained by the Gillespie algorithm (56), whereas the line represents the “approximate” mRNA distribution obtained using Eq. 11.

Effect of molecular memory on the mRNA noise

Using the first two binomial mRNA moments derived for the gene model with a general waiting-time distribution, we can show that the mRNA noise (defined as the ratio of the variance over the square mean) takes the following form

$${\eta }_m^2=\frac{{\sigma }_m^2}{{\langle m\rangle }^2}=\frac{\mu }{\lambda }\frac{{\tau }_1+{\tau }_0}{{\tau }_1}+\frac{{\tau }_0-\tilde{F}\left(\mu \right)}{{\tau }_1+\tilde{F}\left(\mu \right)},$$ (19)

where ${\tau }_1=1/k$ represents the average waiting time in the on state whereas ${\tau }_0=\tilde{F}\left(0\right)$represents the average waiting time in the off state. Different from the approximate result obtained in (31), this formula is exact because the corresponding mRNA distribution is exact. Moreover, the mean mRNA is given by $⟨m⟩=\left(\lambda /\mu \right)\left({\tau }_1/\left({\tau }_1+{\tau }_0\right)\right)$. If the waiting time from off-to-on states follows a Γ-distribution of the form $\gamma \left(t;\alpha ,r\right)$, then we have $\tilde{F}\left(\mu \right)=\left(1/\mu \right)\left[1-{\left(r/\left(\mu +r\right)\right)}^{\alpha }\right]$. Furthermore, if $\alpha =1$, then the model reduces to the case of exponential distribution and the corresponding formula for the mRNA noise given by Eq. 19 becomes

$${\eta }_m^2=\frac{\mu }{\lambda }\frac{{\tau }_1+{\tau }_0}{{\tau }_1}+\frac{\mu {\tau }_0^2}{{\tau }_0+{\tau }_1+\mu {\tau }_0{\tau }_1},$$ (20)

which is also a known result (11,54,57).

For the Gamma distribution of waiting time as set above, the mRNA noise given by Eq. 19 can be rewritten as

$${\eta }_m^2=\frac{\langle b\rangle }{\langle m\rangle }\left[\frac{1}{\mu {\tau }_1+1-{\left(r/\left(r+\mu \right)\right)}^{\alpha }}-\frac{1}{\mu \left({\tau }_0+{\tau }_1\right)}+\frac{1}{\langle b\rangle }\right],$$ (21)

where $\langle b\rangle =\lambda /k$ represents the mean burst size. If the degradation of mRNA is slower or the waiting time in the on state is shorter (i.e., $\mu {\tau }_1\ll 1$), the above calculation formula can be approximated as

$${\eta }_m^2\approx \frac{\langle b\rangle }{2\langle m\rangle }\left[\frac{{\left(r+\mu \right)}^{\alpha }+r^{\alpha }}{{\left(r+\mu \right)}^{\alpha }-r^{\alpha }}-\frac{2r}{\alpha \mu }+{\eta }_b^2+\frac{1}{\langle b\rangle }\right],$$ (22)

where ${\eta }_b^2=1+1/\langle b\rangle$ represents the burst noise. Equation 22 is the same as the result derived by Schwabe et al. (42), which can reduce to the formula derived by Pedraza and Paulsson (31) in the absence of degradation.

Previous studies showed that if the waiting time from off-to-on states follows a Γ-distribution, a reduction in the noise of the waiting-time distribution leads to a reduction in the noise of the mRNA distribution (11,25,31). A different waiting-time distribution can result in a different result. As seen above, the waiting-time distribution can be well approximated by a mixed Γ-distribution if the inactive process consists of two parallel pathways. Now, we assume that this mixed Γ-distribution takes the form

$$f\left(t\right)=\frac{{\rho }_1{\rho }_2-{\rho }_1}{{\rho }_1{\rho }_2-1}\gamma \left(t;{\alpha }_1,\frac{{\alpha }_1{\rho }_1}{{\tau }_0}\right)+\frac{{\rho }_1-1}{{\rho }_1{\rho }_2-1}\gamma \left(t;{\alpha }_2,\frac{{\alpha }_2}{{\tau }_0{\rho }_2}\right),$$ (23)

where ${\rho }_1>1$ and ${\rho }_2>1$ are constants. Then, it is not difficult to show that the mean of the waiting time is given by $\langle T\rangle ={\tau }_0$, whereas the variance of the waiting time by

$${\sigma }_T^2={\tau }_0^2\left[\frac{{\rho }_2-1}{{\rho }_1\left({\rho }_1{\rho }_2-1\right)}\frac{1+{\alpha }_1}{{\alpha }_1}+\frac{{\rho }_2^2\left({\rho }_1-1\right)}{{\rho }_1{\rho }_2-1}\frac{1+{\alpha }_2}{{\alpha }_2}-1\right].$$ (24)

Changing the parameter ${\rho }_1$ and ${\rho }_2$ allows to tune the level of the waiting-time noise defined by ${\sigma }_T^2/{\langle T\rangle }^2$, while keeping the mean waiting time fixed. For clarity, we set ${\rho }_1=\left(c+1\right)/c$ and ${\rho }_2=\left(2-c\right)/\left(1-c\right)$, where $0<c<1$. With this setting, we find that with the increase in parameter $c$, the waiting-time noise increases (Fig. 2 A), but the mRNA noise decreases (referring to Fig. 2 B). Alternatively, a reduction in the waiting-time noise leads to an increase in the mRNA noise (Fig. 2 C), which is contrary to the previous conclusion. The possible reason is that the mRNA noise can vary in a wide range even if the mean and noise of the waiting time are fixed (Fig. 2 C).

Figure 2.

The mRNA noise adjusted by molecular memory. (A) Waiting-time distributions for different values of $c$, where ${\tau }_0=3$, $k=1/3$, ${\alpha }_1={\alpha }_2=4$, $\lambda =15$, $\mu =1$; (B) mRNA distribution for different values of $c$, where the parameter values are set as the same as (A). Here, the probability of the zero mRNA is not shown; (C) mRNA noise versus waiting-time noise for different values of $c$, where ${\alpha }_1={\alpha }_2=\alpha =\mathrm{3,4,5}$ are set, and other parameter values are the same as (A); (D) mRNA noise versus waiting-time noise, where $b_1,b_2,{\alpha }_1,{\alpha }_2$ are uniformly sampled from the interval (1,200), parameter values are set as ${\tau }_0=2$, $k=0.5$, $\lambda =15$, and $\mu =1$, and the red and blue lines respectively represent the upper bound and lower bounds of the mRNA noise obtained by Eq. 26 and Eq. 19. To see this figure in color, go online.

Next, we estimate the bounds of the mRNA noise in the case that the mean and noise of waiting time are fixed. For this, we first need to estimate the bounds of $\tilde{f}\left(s\right)$. By calculation, we can obtain the following estimation (seeing (58))

$$\frac{{\sigma }^2}{{\sigma }^2+{\left(B-W\right)}^2}{e}^{-sB}+\frac{{\left(B-W\right)}^2}{{\sigma }^2+{\left(B-W\right)}^2}{e}^{-s\left(W-{\sigma }^2/\left(B-W\right)\right)}\le \tilde{f}\left(s\right)\le \frac{{\sigma }^2}{{\sigma }^2+W^2}+\frac{W^2}{{\sigma }^2+W^2}{e}^{-s\left(W+{\sigma }^2/W\right)},$$ (25)

where $B$ is the upper bound of waiting-time $T$, $W$ and ${\sigma }^2$ are the mean and variance of the waiting time, respectively. The upper bound of $\tilde{f}\left(s\right)$ corresponds to a two-point mixed distribution with mass ${\sigma }^2/\left({\sigma }^2+W^2\right)$ at position 0 and mass $W^2/\left({\sigma }^2+W^2\right)$ at position $W+{\sigma }^2/W$. The lower bound of $\tilde{f}\left(s\right)$ corresponds also to a two-point mixed distribution with mass ${\sigma }^2/\left({\sigma }^2+{\left(B-W\right)}^2\right)$ at position $B$ and mass ${\left(B-W\right)}^2/\left({\sigma }^2+{\left(B-W\right)}^2\right)$ at position $W-{\sigma }^2/\left(B-W\right)$. When $B\to \infty$, the estimation given by Eq. 25 then becomes

$$e^{-sW}\le \tilde{f}\left(s\right)\le \frac{{\sigma }^2}{{\sigma }^2+W^2}+\frac{W^2}{{\sigma }^2+W^2}{e}^{-s\left(W+{\sigma }^2/W\right)},$$ (26)

which depends only on the mean and variance of the waiting time from off-to-on states, independent of the detail of the waiting-time distribution. Furthermore, we can obtain first the estimation for the bounds of $\tilde{F}\left(s\right)$ and then the estimation for the bounds of the mRNA noise, each depending only on the mean and variance of the waiting time from off-to-on states, independent of the detail of the waiting-time distribution. These estimations imply that the lower and upper bounds of the mRNA noise are independent of the way (e.g., many inactive pathways) of transition from off-to-on states, where a different way corresponds to a different waiting-time distribution.

To numerically verify the resulting estimations on the lower and upper bounds of the mRNA noise, we randomly sample four parameters ${\rho }_1,{\rho }_2,{\alpha }_1,{\alpha }_2$ in the $\left({\rho }_1,{\rho }_2,{\alpha }_1,{\alpha }_2\right)$ space while keeping the other parameter values in the mixed distribution fixed. Numerical results on the bounds of the mRNA noise are shown in Fig. 2 D. From this panel, we observe that for a fixed value of mean waiting time, there are two sets of parameter values such that the waiting-time noise increases, but the mRNA noise decreases. Note that Eq. 26 holds for any waiting-time distribution. Therefore, this result still holds for any waiting-time distribution.

Inferring waiting-time distribution from mRNA distribution

The mRNA distribution can be obtained through experimental techniques such as mRNA-FISH labeling or RT-qPCR on individual cells. An interesting question is whether we can infer the underlying transcription mechanism from experimental data. In this subsection, we will provide a method to infer the waiting-time distribution from the known mRNA distribution. Without loss of generality, we set the mRNA degradation rate to be unit, i.e., $\mu =1$. Therefore, the key to infer the gene model described by Eq. 1 from a known mRNA distribution is to infer two parameters $k$ and $\lambda$ as well as function $f\left(t\right)$. We point out that in our inference process, it is important to make use of binomial moments as a bridge of inference.

Based on binomial moments $b_n$ introduced above, we further introduce a new index (see a reason given in Appendix C for this introduction)

$$\tilde{G}\left(n\right)=\left(1+\gamma \right)\prod _{i=1}^n{R}_i-1,$$ (27)

where we define $R_i=\left(i{b}_i^2\right)/\left[\left(i+1\right)b_{i-1}{b}_{i+1}\right]$, and $\gamma$ is a positive constant. Distribution $P\left(n\right)$ can be easily obtained from experimental data, so can binomial moments $b_n$ ($n=\mathrm{0,1},\cdots ,N+1$), all $R_i$ ($i=\mathrm{1,2},\cdots ,n$), and all $\tilde{G}\left(n\right)$ ($n=\mathrm{1,2},\cdots ,N$), where $N$ is some large positive integer (which practically represents the highest order of binomial moment). Furthermore, we can obtain first the continuous-variable function, $\tilde{G}\left(s\right)$ (representing the Laplace transform of the discrete-variable $G\left(n\right)$) using the Burlisch-Stoer rational interpolation in the R package “pracma” (59), and then $G\left(t\right)$ by the numerical inverse Laplace transform of $\tilde{G}\left(s\right)$ using the function “invlap” also in the R package “pracma.” For $t\ge {\delta }_0$ (where ${\delta }_0$ is a very small number), we can prove that $F\left(t\right)=c_{0}G\left(t\right)$, where $c_0$ is a constant ($\approx 1/G\left({\delta }_0\right)$) (seeing Appendix C for detail). Therefore, we can reconstruct an approximate waiting-time distribution according to $f\left(t\right)=-dF\left(t\right)/dt$. We point out that parameter $\gamma$ does not influence the shape of the inferred waiting-time distribution, so it can be set as a positive constant (we may set $\gamma =2$ in simulation). In addition, we can determine two parameter values: first parameter $k$ (representing the mean switching rate from on-to-off states) according to $k=\left(1-R_1/R_1\tilde{F}\left(0\right)-\tilde{F}\left(1\right)\right)$ and then parameter $\lambda$ (representing the mean transcription rate) according to $\lambda =b_1\left(1+k\tilde{F}\left(0\right)\right)$. To that end, we finish the inference of parameter values in our model. We emphasize that in contrast to previous inference methods, our inference method do not need to presuppose the type of waiting-time distribution.

To verify the effectiveness of the above inference method, we generate an mRNA distribution using the two-state model with a mixed Γ-waiting-time distribution of the form $f\left(t\right)=p_1\gamma \left(t;{\alpha }_1,r_1\right)+p_2\gamma \left(t;{\alpha }_2,r_2\right)$. Then, we define

$$P^{\ast }\left(n\right)=\{\begin{array}{l}P\left(n\right)+\xi ifP\left(n\right)\ge \omega ,\\ 0ifP\left(n\right)<\omega ,\end{array}$$ (28)

where $\omega$ is a parameter, $P\left(n\right)$ represents the exact distribution, whereas $P^{\ast }\left(n\right)$ represents the observed distribution, and $\xi$ is a random variable which follows the uniform distribution $\xi ˜U\left(-\left(\omega /2\right),\left(\omega /2\right)\right)$. If there is no error (i.e., $\omega =0$), the predicted waiting-time distribution gradually tends to the exact one with the increase of $N$ (Fig. 3 A), implying that the accurate waiting-time distribution can be inferred from the precisely known mRNA distribution. However, if an error exists (i.e., $\omega ={10}^{-12}$), the predicted waiting-time distribution cannot converge to the known waiting-time distribution with the increase of $N$ (Fig. 3 B), implying that it is difficult to infer the exact waiting-time distribution when there is an error in the mRNA distribution. Nevertheless, the inferred “approximate waiting-time distributions” (Fig. 3 B) always have similar characteristics as the known waiting-time distribution.

Figure 3.

Inferring waiting-time distribution from mRNA distribution. (A) The parameter values corresponding to the exact distribution (red line) are as follows: $p_1=0.3$, $p_2=0.7$, $r_1=4$,$r_2=1$, ${\alpha }_1={\alpha }_2=4$,$k=2$,$\lambda =5$, and$\mu =1$, the parameter of sample error is $\omega =0$; (B) the parameter values corresponding to the exact distribution (red line) are the same as in (A), and the parameter of sample error is$\omega ={10}^{-12}$; (C) the waiting-time distribution inferred from the experimental mRNA distribution for PDR5 gene is shown, where the red line denotes the distribution fitted using Γ-distribution. (D) A comparison is shown between observed data (44) (blue histogram) and the mRNA distribution obtained by stochastic simulation (red line), where the parameter values are set as $\alpha =18$, $r=17.75$, $k=2$,$\lambda =41.5$, and$\mu =1$. To see this figure in color, go online.

Finally, we apply the above inference method to a realistic example. We try to infer the waiting-time distribution from the mRNA distribution for gene PDR5 measured by Zenklusen et al. (44). The reconstructed waiting-time distribution is demonstrated in Fig. 3 C. Here, we fit the predicted waiting-time distribution ($N=5$) with a Γ-distribution, and determine the values of parameters in the model (see Appendix C for detail). The mRNA distribution obtained through the stochastic simulation can well fit to the experimental data (Fig. 3 D). We point out that although the exact waiting-time distribution is difficult to infer, our numerical result indicates that the waiting-time distribution is definitely unimodal (referring to Fig. 3 C).

Conclusions

In this article, we have derived the analytical stationary mRNA distribution for the two-state model of gene expression in which the waiting time for the activation process of the gene is assumed to follow a general distribution whereas the waiting time from on-to-off states is assumed to follow an exponential distribution. This derived distribution can provide insight into the role of molecular memory characterized by nonexponential waiting-time distribution in fine-tuning the gene expression noise. In contrast to previous methods (8,11,12,24,25), our method is not limited to a specific Markov chain description of various possible promoters containing transcriptionally active and inactive states. Actually, our derivation method can be applied to gene models with arbitrarily many parallel or cross talk inactive pathways.

Previous studies showed that a reduction in the waiting-time noise can lead to a reduction in the mRNA noise (25,31), under the assumption that the waiting time follows a Γ-distribution. In this article, however, we have shown that a reduction in the waiting-time noise may lead to the increase in the mRNA noise, under the assumption that the waiting-time distribution follows a mixed Γ-distribution, which can model the case that the inactive process consists of two parallel inactive pathways. The plausible reason is that in our case, the mRNA noise may vary in a wide range even if the mean and noise of waiting time are fixed. For the two-state model of gene expression, we only considered that the waiting-time distribution for the off-to-on reaction is general. Actually, waiting-time distributions for on-to-off reaction, transcription reaction, and degradation reaction may also be nonexponential (31,32,60). For these cases, how mRNA or protein distributions are derived as well as how molecular memory affects the expression level and noise needs further studies.

Based on the derived distribution, we have also developed an effective method to infer the waiting-time distribution from a known mRNA distribution. Compared to previous methods (28,42,44), our method do not need to presuppose the type of distribution. Our results indicated that an accurate waiting-time distribution can be inferred provided that the mRNA distribution is precisely known. However, only “approximate distributions” can be inferred if there is an error in the mRNA distribution. Our method can also be used in inference of the waiting-time distribution from the mRNA distribution obtained by experimental data. For example, when our inference method is applied to the PDR5 gene (44), we found that the inferred waiting-time distribution is unimodal, but this qualitative result needs experimental verification. It is worth pointing out that intermediate processes of gene expression, e.g., the partitioning of mRNA/protein at cell division (10,41), would additionally contribute the noise of gene expression and would skew the inferred results, but they are neglected in our study. How details of intermediate processes affect the inferred results is worth further study.

Editor: Alexander Berezhkovskii.

Appendix A: Derivation of Binomial Moments

By Taylor expanding the probability-generating function $G_j\left(s+1\right)={\sum }_{n=1}^{\infty }{a}_n^{\left(j\right)}{s}^n$ ($j=\mathrm{0,1}$), we can derive the following relations

$$a_n^{\left(0\right)}=k{a}_n^{\left(1\right)}\tilde{F}\left(\mu n\right),$$ (A1)

$$\left[n\mu +k\left(1-\tilde{f}\left(\mu n\right)\right)\right]a_n^{\left(1\right)}=\lambda a_{n-1}^{\left(1\right)},$$ (A2)

where $\tilde{F}\left(s\right)={\int }_0^{\infty }{e}^{-sx}F\left(x\right)dx$ and $\tilde{f}\left(s\right)={\int }_0^{\infty }{e}^{-sx}f\left(x\right)dx=1-s\tilde{F}\left(s\right)$. Note that $a_0^{\left(0\right)}+a_0^{\left(1\right)}=1$ because of the conservative condition of probability. We get the expression of $a_0^{\left(1\right)}$ from Eq. A1 as follows

$$a_0^{\left(1\right)}=\frac{1}{1+k\tilde{F}\left(0\right)},$$ (A3)

Because the total probability distribution of mRNA is $P\left(m\right)=P_0\left(m\right)+P_1\left(m\right)$, the corresponding total binomial moments are $b_n=a_n^{\left(0\right)}+a_n^{\left(1\right)}$. Combining Eqs. A1, A2, and A3, we obtain the expression of $b_n$ given by Eq. 9 in the main text.

Appendix B: Calculation of Waiting-Time Distribution

For the inactive pathway shown in Fig. 1 B, the probability distribution of waiting time is calculated according to $f\left(t\right)={\mathbf{a}}_0\exp \left(\mathbf{A}t\right){\mathbf{u}}_0$, where matrix $\mathbf{A}$ is

$$\mathbf{A}=\left(\begin{array}{ccc}-\left(k_{10}+k_{20}\right)& {\mathbf{v}}_1& {\mathbf{v}}_2\\ \mathbf{0}& {\mathbf{A}}_1& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{A}}_2\end{array}\right)\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\mathbf{A}}_1=\left(\begin{array}{ccccc}-k_{11}& k_{11}& 0& \cdots & 0\\ 0& -k_{12}& k_{12}& \cdots & 0\\ ⋮& \ddots & \ddots & & ⋮\\ 0& 0& 0& -k_{1,n-1}& k_{1,n-1}\\ 0& 0& 0& 0& -k_{1n}\end{array}\right),$$

$${\mathbf{A}}_2=\left(\begin{array}{ccccc}-k_{21}& k_{21}& 0& \cdots & 0\\ 0& -k_{22}& k_{22}& \cdots & 0\\ ⋮& \ddots & \ddots & & ⋮\\ 0& 0& 0& -k_{2,m-1}& k_{2,m-1}\\ 0& 0& 0& 0& -k_{2m}\end{array}\right),$$

and ${\mathbf{v}}_1=\left(k_{10},0,\cdots ,0\right)$, ${\mathbf{v}}_2=\left(k_{20},0,\cdots ,0\right)$. The column vector ${\mathbf{u}}_0$ takes the form ${\mathbf{u}}_0={\left(0,\cdots ,0,k_{1n},0,\cdots ,0,k_{2m}\right)}^{\text{T}}$, the row vector ${\mathbf{a}}_0$ takes the form ${\mathbf{a}}_0=\left(\mathrm{1,0},\cdots ,0\right)$. According to $F\left(t\right)={\int }_t^{\infty }f\left(x\right)dx$, we thus have $F\left(t\right)={\mathbf{a}}_0\exp \left(\mathbf{A}t\right){\mathbf{A}}^{-1}{\mathbf{u}}_0$. The Laplace transform of $f\left(t\right)$ is calculated according to $\tilde{f}\left(s\right)={\int }_0^{\infty }{e}^{-sx}f\left(x\right)dx$. Thus, we have $\tilde{f}\left(s\right)={\mathbf{a}}_0{\left(s\mathbf{E}-\mathbf{A}\right)}^{-1}{\mathbf{u}}_0$, where $\mathbf{E}$ is the unit matrix. Similarly, we can have

$\tilde{F}\left(s\right)={\mathbf{a}}_0{\left(s\mathbf{E}-\mathbf{A}\right)}^{-1}{\mathbf{A}}^{-1}{\mathbf{u}}_0={\mathbf{a}}_0{\left(s\mathbf{E}-\mathbf{A}\right)}^{-1}{\mathbf{u}}_1$, where ${\mathbf{u}}_1={\left(\mathrm{1,1},\cdots ,1\right)}^{\text{T}}$.

Appendix C: A Method to Infer the Waiting-Time Distribution from Experimental Data

Note that binomial moments $b_i$ ($i=\mathrm{1,2},\cdots ,N$) can be estimated from experimental data. Thus, $R_i=\left(i{b}_i^2\right)/\left[\left(i+1\right)b_{i-1}{b}_{i+1}\right]$ are known. Setting $\mu =1$ and substituting these expressions of$R_i$ into Eq. 9 in the main text, we obtain the following relationship

$$\tilde{F}\left(n\right)=\left({\tau }_0+{\tau }_1\right)\prod _{i=1}^n{R}_i-{\tau }_1,$$ (C1)

where ${\tau }_0$ and ${\tau }_1$ represent the mean waiting times of the OFF and ON states, which are experimentally measurable quantities and are therefore known. By smoothing discrete-variable function $\tilde{F}\left(n\right)$ through interpolation, we can obtain continuous-variable function $\tilde{F}\left(s\right)$. Furthermore, we can obtain survival function $F\left(t\right)$ through the inverse Laplace transform.

If we set ${\tau }_0=\gamma {\tau }_1$ (where $\gamma$ is a positive constant of more than) and ${\tau }_1=1$, then

$$\tilde{G}\left(n\right)=\left(1+\gamma \right)\prod _{i=1}^n{R}_i-1,$$ (C2)

Also by smoothing discrete-variable function $\tilde{G}\left(n\right)$ through interpolation, we can obtain continuous-variable function $\tilde{G}\left(s\right)$. Furthermore, we can obtain the function $G\left(t\right)$ through the inverse Laplace transform.

Combining Eqs. C1 and C2, we obtain the following relation:

$$\tilde{F}\left(s\right)=\frac{{\tau }_0+{\tau }_1}{1+\gamma }\tilde{G}\left(s\right)+\frac{{\tau }_0-\gamma {\tau }_1}{1+\gamma }.$$ (C3)

Applying inverse Laplace transform to Eq. C3, we have:

$$F\left(t\right)=\frac{{\tau }_0+{\tau }_1}{1+\gamma }G\left(t\right)+\frac{{\tau }_0-\gamma {\tau }_1}{1+\gamma }\delta \left(t\right),$$ (C4)

where $\delta \left(t\right)$ denotes the Dirac δ-function. Therefore, for $t\ge {\delta }_0$ (where ${\delta }_0$ is a very small number), we obtain $F\left(t\right)=c_{0}G\left(t\right)$. Because $F\left({\delta }_0\right)\approx 1$, we have $c_0\approx 1/G\left({\delta }_0\right)$.

Once the survival function $F\left(t\right)$ is obtained in the above manner, we can further determine the values of other parameters in the model. From Eq. C1, we can determine the value of $k$ as $k=\left((1-R_1)/(R_1\tilde{F}\left(0\right)-\tilde{F}\left(1\right))\right)$. Thus, the value of $\lambda$ can be determined according to $\lambda =b_1\left(1+k\tilde{F}\left(0\right)\right)$.

Author Contributions

C.S., Y.J., and T.Z. designed the research. C.S. performed the research and analyzed the data. C.S. and T.Z. wrote the article.