Accuracy Analysis of Hybrid Stochastic Simulation Algorithm on Linear Chain Reaction Systems

Abstract

Noise in cellular systems is often modeled and simulated with Gillespie’s stochastic simulation algorithm (SSA), but the low efficiency of the SSA limits its application to large biochemical networks. To improve the efficiency of stochastic simulations, Haseltine and Rawlings (HR) proposed a hybrid algorithm, which combines ordinary differential equations for traditional deterministic models and the SSA for stochastic models. In this paper, accuracy of the HR hybrid method is studied based on a linear chain reaction system. Mathematical analysis and numerical results both show that the HR hybrid method is accurate if either the quantity of reactant molecules in fast reactions is above a certain threshold, or the reaction rates of fast reactions are much larger than those of slow reactions. This analysis also shows that the HR hybrid method approximates the chemical master equation well for a much greater region in system parameter space than the slow-scale SSA and the stochastic quasi-steady-state assumption methods.

Appendices

Appendix A: Solution of NSRFT for the Simple Case ($n=2$)

1.1 A.1: Solution to CME

Matrix A for the simple system (24) is

$$\begin{aligned} A = \left[ \begin{array}{cc} 1 &{}\quad -1\\ -1 &{}\quad 1+k_c \end{array} \right] . \end{aligned}$$

A has two eigenvalues

$$\begin{aligned} \lambda _{1,2}=\frac{2+k_c \mp \sqrt{4+k_c^2}}{2}. \end{aligned}$$

Let $\mathbb {P}(0)=(1$,$0)^T$. The solution is given by

$$\begin{aligned} p_{11}(t, k_c)= & {} \frac{\lambda _2-1}{\lambda _2-\lambda _1}e^{-\lambda _1t}+\frac{1-\lambda _1}{\lambda _2-\lambda _1}e^{-\lambda _2t},\\ p_{12}(t, k_c)= & {} \frac{1}{\lambda _2-\lambda _1}e^{-\lambda _1t}-\frac{1}{\lambda _2-\lambda _1}e^{-\lambda _2t}. \end{aligned}$$

Let ${\mathbb {P}}(0)=(0$,$1)^T$. The solution is then

$$\begin{aligned} p_{21}(t, k_c)= & {} \frac{1}{\lambda _2-\lambda _1}e^{-\lambda _1t}-\frac{1}{\lambda _2-\lambda _1}e^{-\lambda _2t},\\ p_{22}(t, k_c)= & {} \frac{1-\lambda _1}{\lambda _2-\lambda _1}e^{-\lambda _1t}+\frac{\lambda _2-1}{\lambda _2-\lambda _1}e^{-\lambda _2t}. \end{aligned}$$

Define $q_{1}(t,k_c)$ and $q_{2}(t,k_c)$ as

$$\begin{aligned} q_{1}(t,k_c)= & {} p_{11}(t,k_c) + p_{12}(t,k_c),\\ q_{2}(t,k_c)= & {} p_{21}(t,k_c) + p_{22}(t,k_c). \end{aligned}$$

According to Eq. (14), $T_c$ satisfies

$$\begin{aligned} \int ^{T_c}_0f_c(t,k_c)\hbox {d}t= & {} \int ^{T_c}_0 \left( \theta _1 \frac{p_{12}(t,k_c)}{q_{1}(t,k_c)} +\theta _2 \frac{p_{22}(t,k_c)}{q_{2}(t,k_c)}\right) \hbox {d}t \nonumber \\= & {} \frac{r}{mk_c}. \end{aligned}$$

(28)

1.2 A.2: Solution to the HR Hybrid Method

The ODE system for the fast subsystem is

$$\begin{aligned} \mathbf {x}'(t) = -\tilde{A}{\mathbf {x}}(t), \end{aligned}$$

with

$$\begin{aligned} \tilde{A} = \left[ \begin{array}{cc} 1 &{}\quad -1\\ -1 &{}\quad 1 \end{array} \right] . \end{aligned}$$

Given the initial condition $(x_1(0)$, $x_2(0))^T=(m_1$,$m_2)^T$, the solution is

$$\begin{aligned} x_1(t)= & {} \frac{m}{2} + \frac{m_1-m_2}{2}e^{-2t},\\ x_2(t)= & {} \frac{m}{2} + \frac{m_2-m_1}{2}e^{-2t}. \end{aligned}$$

$T_h$ satisfies the equation

$$\begin{aligned} \int ^{T_h}_0f_h(t)\hbox {d}t = \int ^{T_h}_0 \left( \frac{1}{2} + \frac{\theta _2-\theta _1}{2} e^{-2t}\right) \hbox {d}t = \frac{r}{mk_c}. \end{aligned}$$

(29)

1.3 A.3: Solutions to ssSSA and SQSSA

By applying the ssSSA, the fast subsystem is assumed to stay at steady state without the impact of $k_c$, which is $x_1^*=x_2^*=m/2$. Thus, $f_w=1/2$ and $T_w$ satisfies the equation

$$\begin{aligned} \int ^{T_w}_0f_w\hbox {d}t = \int ^{T_w}_0 \frac{1}{2}\hbox {d}t = \frac{r}{mk_c}. \end{aligned}$$

(30)

In contrast, under the SQSSA assumption, only species ${S}_2$ stays at steady state. $x_2(t)$ satisfies

$$\begin{aligned} \frac{\hbox {d}x_2(t)}{\hbox {d}t} = x_1(t)-(1+k_c)x_2(t) = 0, \end{aligned}$$

and the solution is $x_2(t)= m/(2+k_c)$. So $f_q(k_c)=1/(2+k_c)$ and $T_q$ as a result satisfies

$$\begin{aligned} \int ^{T_q}_0f_q(k_c)\hbox {d}t = \int ^{T_q}_0 \frac{1}{2+k_c}\hbox {d}t = \frac{r}{mk_c}. \end{aligned}$$

(31)

Appendix B: Accuracy Analysis of General Cases

In Sect. 4.2, the accuracy analysis is applied to a general linear model with ${S}_n$ as the exit state. For a model with ${S}_v$ ($1\le v \le n$) as the exit state, we have a similar analysis process for the HR hybrid method.

Consider a general linear model

(32)

For the HR hybrid method, the linear chain reactions are taken as the fast subsystem and the irreversible reaction is the slow subsystem. The HR hybrid method shares the same matrix $\tilde{A}$ in Eq. (15) regardless of the value of v. Factor v only affects the computation of $T_h$. $T_h$ is implicitly given by

$$\begin{aligned} \int ^{T_h}_0 \mathbf {e}_v^Te^{-\tilde{A}t}{\Theta } \hbox {d}t = \int ^{T_h}_0 f^{(v)}_{h}(t)\hbox {d}t = \frac{r}{mk_c}. \end{aligned}$$

(33)

Comparing with Eq. (17), we find $\mathbf {e}^T_n$ is replaced with $\mathbf {e}^T_v$ in Eq. (33).

As to the CME for this general system, a new matrix $E_v$ is introduced. $E_v$ has an element $E_{v}(v,v)$ equal to $k_c$ and all the others equal to 0. Matrix $A_v$ is constructed as

$$\begin{aligned} A_v=\tilde{A}+E_v. \end{aligned}$$

Following the same derivation in Sect. 3.1, we have

$$\begin{aligned} \int ^{T_c}_0 \sum ^{n}_{j=1}\theta _i\frac{\mathbf {e}^T_ve^{-A_vt}\mathbf {e}_j}{\mathbf {e}_n^Te^{-A_vt}\mathbf {e}_j}\hbox {d}t = \int ^{T_c}_0 f^{(v)}_{c}(t,k_c)\hbox {d}t = \frac{r}{mk_c}. \end{aligned}$$

(34)

In Sect. 4.2, the effects of parameter $k_c$, population m as well as the length of chain n on accuracy have been discussed. Here, we will show how the parameter v impacts the accuracy.

Consider a general chain reaction system of length n. Reaction rates satisfy $f_i=b_i$ for $1\le i\le n-1$. At the initial time, the probability that one particle stays in state ${S}_i$ is 1 / n. To trigger a slow reaction event, a particle must reach state ${S}_v$ first. The length one particle jumps from ${S}_i$ to ${S}_v$ is defined by $|i-v|$. The average jumping length is computed as

$$\begin{aligned} d_v = \frac{1}{n}\sum ^n_{i=1}|i-v|. \end{aligned}$$

We can verify that when $v=1$ or $v=n$, $d_v$ reaches its maximum value $(n-1)/2$. When $v=\lceil n/2\rceil$, the minimum value of $d_v$ is about n / 4, half the value of the maximum one. From intuition, $d_v$ can be regarded as a measurement of the average next slow reaction time T. The larger the value of $d_v$ is, the longer time one particle may spend to reach the exit state ${S}_v$.

On the other hand, we have defined two functions $f^{(v)}_{h}(t)$ and $f^{(v)}_{c}(t,k_c)$ in Eqs. (33) and (34), respectively. These two functions have the same properties as their counterparts in Sect. 4.2. We illustrate that the error of the HR hybrid method decreases as $T_c$ and $T_h$ move toward 0 from the analysis in Sect. 4.2, Thus the smaller $d_v$ is, the more accurate the HR hybrid method will be.

We set $n=10$ and vary v from 6 to 10. Average relative errors $|T_c-T_h|/T_c$ were calculated with different m and $k_c$. Values of m and $k_c$ which lead to the relative error of 0.01 are listed in Table 3. From the results, we can see parameter v has a crucial impact on the threshold of $k_c$. The value of $k_c$ with respect to $v=6$ is almost four times larger than the one with respect to $v=10$. Whereas, when reducing v from 10 to 6, the threshold of m only decreases by 10%.

Table 3 Thresholds of m and $k_c$ corresponding to different v values

Appendix C: Calculation of $\beta _1$

The value of $\beta _1$ in Sect. 4.3 is calculated below. For $\tilde{\lambda }_1=0$, its left eigenvector is $\mathbf {e}$. Its right eigenvector ${\varvec{\eta }}_1$ is

$$\begin{aligned} {\varvec{\eta }}_1 = \left( 1, \frac{f_1}{b_1}, \frac{f_1f_2}{b_1b_2},\ldots ,\prod ^{n-1}_{i=1}\frac{f_i}{b_i} \right) ^T. \end{aligned}$$

There is a symmetric tridiagonal matrix $\tilde{A}_\mathrm{s}$ similar to $\tilde{A}$, given by

$$\begin{aligned} \tilde{A}_\mathrm{s} = D^{-1}\tilde{A}D, \end{aligned}$$

where D is a diagonal matrix that

$$\begin{aligned} D = \text {diag}\left( 1, \sqrt{\frac{f_1}{b_1}},\ldots , \prod ^{n-1}_{i=1} \sqrt{\frac{f_i}{b_i}}\right) . \end{aligned}$$

Since $\tilde{A}_\mathrm{s}$ is symmetric, there is a unitary matrix $Q=[\mathbf {q}_1,\mathbf {q}_2,\ldots ,\mathbf {q}_n]$, where $\mathbf {q}_i$ are eigenvectors of ${\tilde{A}}_\mathrm{s}$. Q satisfies

$$\begin{aligned} \tilde{A}_\mathrm{s} = Q\tilde{\varLambda } Q^T \text { and } QQ^T=I, \end{aligned}$$

where ${\tilde{\varLambda }} = \text {diag}(0,{\tilde{\lambda }}_2,\ldots ,{\tilde{\lambda }}_{n})$. Let $P=[{\varvec{\eta }}_1,\mathbf {\eta }_2,\ldots ,{\varvec{\eta }}_n]$ and we have

$$\begin{aligned} \tilde{A} =P{\tilde{\varLambda }} P^{-1}= DQ{\tilde{\varLambda }} Q^TD^{-1}. \end{aligned}$$

As a consequence, ${\varvec{\eta }}_i = a_iD\mathbf {q}_i$, where $a_i$ is a real number. By applying the property $\mathbf {q}_i^T\mathbf {q}_i=1$, $a_i$ is computed as

$$\begin{aligned} a_i = \sqrt{{\varvec{\eta }}_i^TD^{-2}{\varvec{\eta }}_i}. \end{aligned}$$

On the other hand, $e^{-\tilde{A}t}$ can be written as

$$\begin{aligned} e^{-\tilde{A}t} = DQe^{-\tilde{\varLambda } t}Q^TD^{-1}= & {} \sum ^n_{i=1}D\mathbf {q}_i\mathbf {q}_i^TD^{-1}e^{-\tilde{\lambda }_it}\\= & {} \sum ^n_{i=1}{\varvec{\eta }}_i \frac{{\varvec{\eta }}_i^TD^{-2}}{{\varvec{\eta }}_i^TD^{-2}{\varvec{\eta }}_i} e^{-\tilde{\lambda }_it}. \end{aligned}$$

So we get

$$\begin{aligned} \beta _1 = \mathbf {e}^T_n {\varvec{\eta }}_1\frac{{\varvec{\eta }}_1^TD^{-2}\mathbf {e}_i}{{\varvec{\eta }}_1^TD^{-2}{\varvec{\eta }}_1}=\frac{\mathbf {e}_n^T {\varvec{\eta }}_1}{\mathbf {e}^T {\varvec{\eta }}_1}. \end{aligned}$$

Appendix D: Formulation of Bistable Switch Model

Section 5.1 studied the bistable switch model as a system (26). In this appendix, we will now show how multiple linear chain systems like (32) can formulate system (26). In system (26), proteins $\text {Cdh1P}_i$ ($i=0,1,\dots ,9$), form a linear chain which is the fast subsystem. At the same time, each $\text {Cdh1P}_i$ acts as an enzyme activating the Clb2 degradation reaction. Notice that the synthesis and degradation of Clb2 are taken as slow reactions. Between two consecutive slow reactions, the population of Clb2 does not change. By ignoring the synthesis and degradation of Clb2, for each $\text {Cdh1P}_i$ we have an equivalent system which reads as

(35)

where the slow reaction is the irreversible reaction which consumes a $\text {Cdh1P}_i$ and generates a $\text {Cdh1P}_i$. Clb2 in this reaction is considered as part of the slow reaction rate. Therefore, system (35) is exactly a concrete instance of system (32).