Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Abstract

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie’s algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

Appendices

Appendices

Scaling of the Deterministic and Stochastic Models

1.1 Dimensionalization of Deterministic Model

Note that our deterministic model (Xu and Jilkine 2018) used dimensionless units including all the concentrations and the cell size parameter L. Hence, for the stochastic simulation we need to reconstruct the dimensional model with parameters that have appropriate units. Recall that there are three types of variables with different units: length, time, and concentration. For concentrations, we have both membrane and cytosolic concentrations. The choices for these two concentrations are proteins per unit area ($\upmu \hbox {m}^{-2}$) and protein copy numbers per unit volume ($\upmu \hbox {m}^{-3}$).

Denote $\tilde{x}$ and $\tilde{t}$ by the length and time variables in dimensions. Denote $\tilde{C}(\tilde{G})$ and $\tilde{c}(\tilde{g})$ by the cytosolic and membrane concentrations of Cdc42(GEF) in dimensions. Consider the change of variables

$$\begin{aligned} \tilde{x}=\alpha x, \quad \tilde{t}=\beta t,\quad \tilde{C}(\tilde{x},\tilde{t})=\gamma C(x,t), \quad \tilde{c}_{1,2}(\tilde{t})=\gamma _m c_{1,2}(t). \end{aligned}$$

(34)

Here, $\alpha$ represents the units of length. Since $L=1$ corresponds to $9\,\upmu \hbox {m}$, we set $\alpha =9\,\upmu \hbox {m}$. Also, the Cdc42 dissociation rate $k^-=1$ corresponds to $4\min ^{-1}$ (Das et al. 2012); hence, we take $\beta =1/4\min =15\,s$. We will discuss the units of concentrations (i.e., values of $\gamma , \gamma _m$) at a later time.

Using Eq. (34), we have

$$\begin{aligned} \frac{\partial C}{\partial t}=\frac{\beta }{\gamma }\frac{\partial \tilde{C}}{\partial \tilde{t}},\quad \frac{\partial C}{\partial x}=\frac{\alpha }{\gamma }\frac{\partial \tilde{C}}{\partial \tilde{x}},\quad \frac{\partial ^2C}{\partial x^2}=\frac{\alpha ^2}{\gamma }\frac{\partial ^2\tilde{C}}{\partial \tilde{x}^2}, \quad \frac{\mathrm{d}c_i}{\mathrm{d}t}=\frac{\beta }{\gamma _m}\frac{\mathrm{d}\tilde{c_i}}{\mathrm{d}\tilde{t}}. \end{aligned}$$

(35)

Hence, from Eq. (1), the diffusion equation and ODEs with dimensions are given by

$$\begin{aligned} \frac{\partial \tilde{C}}{\partial \tilde{t}}(\tilde{x},\tilde{t})&=\frac{D_C\alpha ^2}{\beta }\frac{\partial ^2\tilde{C}}{\partial \tilde{x}^2}, \end{aligned}$$

(36a)

$$\begin{aligned} \frac{\mathrm{d}\tilde{c}_1}{\mathrm{d}\tilde{t}}(\tilde{t})&=\left( \frac{k_0}{\gamma \beta }+\frac{k_{\mathrm{cat}}}{\gamma \gamma _m^2\beta }\tilde{c}_1^2\right) \tilde{g}_1\tilde{C}(0,\tilde{t})-\frac{k^-}{\beta }\tilde{c}_1(\tilde{t}),\end{aligned}$$

(36b)

$$\begin{aligned} \frac{\mathrm{d}\tilde{g}_1}{\mathrm{d}\tilde{t}}(\tilde{t})&=\frac{k^\mathrm{on_\mathrm{max}}/\beta }{1+\kappa \tilde{c}_1^2/\gamma _m^2}\frac{\gamma _m}{\gamma }\tilde{G}(0,\tilde{t})-\frac{k^\mathrm{off}}{\beta }\tilde{g}_1(\tilde{t}). \end{aligned}$$

(36c)

We list the dimensions of rates below.

$$\begin{aligned}&\tilde{k}_0=\frac{k_0}{\gamma \beta } \,\frac{1}{{\mathrm{[conc.]}}\times \sec }, \quad \tilde{k}_{\mathrm{cat}}=\frac{k_{\mathrm{cat}}}{\gamma \gamma _m^2 \beta } \frac{1}{{\mathrm{[conc.]}}_{\mathrm{m}}^2\times {\mathrm{[conc.]}}\times \sec }, \end{aligned}$$

(37a)

$$\begin{aligned}&\tilde{k}^\mathrm{on}_\mathrm{max}=\frac{k^{\mathrm{on}}_{\mathrm{max}}}{\beta }\frac{\gamma _m}{\gamma }\,\frac{{\mathrm{[conc.]}}_{{\mathrm{m}}}}{{\mathrm{[conc.]}}\times \sec }, \quad \tilde{\kappa }= \frac{\kappa }{\gamma _m^2}\,\frac{1}{{\mathrm{[ conc.]}}_{\mathrm{m}}^2}, \quad \tilde{k}^{{\mathrm{off}}}=\frac{k^{{\mathrm{off}}}}{\beta } \frac{1}{\sec }. \end{aligned}$$

(37b)

Here, [conc.] represents cytosolic concentration and $[\mathrm{conc.]}_m$ represents membrane concentration. It remains to determine the units of concentrations $\gamma$ and $\gamma _m$ and the corresponding order of magnitude.

1.2 Units of Concentration and Model Geometry

For the stochastic model geometry, we model the rod-shaped cell as a cylinder. In this case, each cell tip is a small disk with a radius r and a small thickness. This is the model geometry that is consistent with our one-dimensional deterministic model. For molecules attached to the cell membrane, we measure the concentration using molecular copy number per cross-sectional area since the thickness is fixed. For cytosolic concentration, we use the units of molecular copy number per volume ($\upmu \hbox {m}^3$).

Let $\gamma _m=\frac{\#}{\mathrm{area}}=n_1 \upmu \hbox {m}^{-2}$ and $\gamma =\frac{\#}{\mathrm{volume}}=n_c \upmu \hbox {m}^{-3}$. Here, $n_1$ and $n_c$ are numbers that represent the order of magnitude. The number of molecules in each compartment is given by

$$\begin{aligned} n_1= \tilde{c}_1 A,\quad m_1=\tilde{g}_1A,\quad n_c(\tilde{t})=\tilde{C}(\tilde{t})V,\quad m_c(\tilde{t})=\tilde{G}(\tilde{t})V. \end{aligned}$$

(38)

Note that in the last two equations, we drop the spatial dependence. This holds when diffusion in the cytosol is fast. The conservation equation is then given by

$$\begin{aligned} N_\mathrm{cdc}&=n_1+n_2+n_c \Rightarrow \tilde{c}_1 A+ \tilde{c}_2 A+ \tilde{C}(\tilde{t})V=\tilde{C}_\mathrm{tot}V,\nonumber \\&\Rightarrow \tilde{C}(\tilde{t})=\frac{\tilde{C}_\mathrm{tot}V/A-\tilde{c}_1 - \tilde{c}_2}{V/A}. \end{aligned}$$

(39)

Since we model the cell as a cylinder, the ratio V / A is given by

$$\begin{aligned} V\approx A\times (\alpha L), \end{aligned}$$

(40)

where $\alpha =9\,\upmu \hbox {m}$. Equation (39) becomes

$$\begin{aligned} \tilde{C}(\tilde{t})=\frac{\tilde{C}_\mathrm{tot}\alpha L-\tilde{c}_1 - \tilde{c}_2}{\alpha L}. \end{aligned}$$

(41)

We now rewrite the above equation in terms of the dimensionless variables. Using the change of variables (34) to remove the tilde notations gives

$$\begin{aligned} C(t)\gamma&=\frac{C_{\mathrm{tot}}\gamma \alpha {L}-c_1\gamma _m - c_2\gamma _m}{\alpha L},\nonumber \\ \Rightarrow C(t)\frac{\alpha \gamma }{\gamma _m}&=\frac{C_{\mathrm{tot}} L(\alpha \gamma /\gamma _m)-c_1 - c_2}{L} . \end{aligned}$$

(42)

Suppose we choose $\gamma$ and $\gamma _m$ such that

$$\begin{aligned} \frac{\alpha \gamma }{\gamma _m}=1. \end{aligned}$$

(43)

Then, we can recover the conservation equation (6).

1.3 Derivation of Stochastic Rates

We now write down the equation for the number of molecules in each compartment using Eq. (38) and the differential equations (36). The ODEs for the number of Cdc42 at tip1 ($n_1$) and the number of GEF at tip1 ($m_1$) are given by

$$\begin{aligned} \frac{\mathrm{d}n_1}{\mathrm{d}\tilde{t}}(\tilde{t})&=\left( \frac{k_0}{\gamma \beta }+\frac{k_{\mathrm{cat}}}{\gamma \gamma _m^2\beta }\frac{n_1^2}{A^2}\right) m_1\frac{n_c(\tilde{t})}{V}-\frac{k^-}{\beta }n_1(\tilde{t}), \end{aligned}$$

(44a)

$$\begin{aligned} \frac{\mathrm{d}m_1}{\mathrm{d}\tilde{t}}(\tilde{t})&=\frac{k^\mathrm{on}_\mathrm{max}/\beta }{1+\kappa n_1^2/(\gamma _m^2A^2)}\frac{\gamma _m}{\gamma }\frac{A}{V}m_c(\tilde{t})-\frac{k^\mathrm{off}}{\beta }m_1(\tilde{t}). \end{aligned}$$

(44b)

Our goal is to derive the stochastic rates. The above ODEs imply the rescaling relations

$$\begin{aligned}&k'_0=\frac{\tilde{k}_0}{V}=\frac{k_0}{\gamma \beta V} , \end{aligned}$$

(45a)

$$\begin{aligned}&k'_\mathrm{cat}=\frac{\tilde{k}_{\mathrm{cat}}}{A^2V}=\frac{k_{\mathrm{cat}}}{\gamma \gamma _m^2 \beta }\frac{1}{A^2V} , \end{aligned}$$

(45b)

$$\begin{aligned}&k'_\mathrm{on}=\frac{\tilde{k}^\mathrm{on}A}{V}=\frac{k^\mathrm{on}_\mathrm{max}\gamma _m}{\gamma \beta }\frac{A}{ V},\end{aligned}$$

(45c)

$$\begin{aligned}&\kappa '=\frac{\tilde{\kappa }}{A^2}=\frac{\kappa }{\gamma _m^2A^2}, \end{aligned}$$

(45d)

$$\begin{aligned}&k'_{-}=\tilde{k}_{-}=\frac{k_{-}}{\beta } ,\quad k'_\mathrm{off}=\tilde{k}_\mathrm{off}=\frac{k^\mathrm{off}}{\beta }. \end{aligned}$$

(45e)

Here, the rates with the $'$ notation represent the rates we use in stochastic simulation, the rates with the tilde notation represent the rates in physical units, and the rates without any notation are dimensionless.

Using the relations $V=A\alpha L$ and $\alpha \gamma =\gamma _m$, we simplify the rescaling relation as

$$\begin{aligned}&k'_0=\frac{k_0}{\gamma \beta \alpha LA} =\frac{k_0}{\gamma _m\beta LA},\end{aligned}$$

(46a)

$$\begin{aligned}&k'_\mathrm{cat}=\frac{k_{\mathrm{cat}}}{\gamma _m^3A^3L\beta } , \end{aligned}$$

(46b)

$$\begin{aligned}&k'_\mathrm{on}=\frac{k^\mathrm{on}_\mathrm{max}\gamma _m}{\gamma \beta }\frac{A}{V}=\frac{k^\mathrm{on}_\mathrm{max}}{\beta L},\end{aligned}$$

(46c)

$$\begin{aligned}&\kappa '=\frac{\kappa }{\gamma _m^2A^2}, \end{aligned}$$

(46d)

$$\begin{aligned}&k'_{-}=\frac{k_{-}}{\beta } ,\quad k'_\mathrm{off}=\frac{k^\mathrm{off}}{\beta }. \end{aligned}$$

(46e)

Setting $\gamma _m=602$, we recover Eq. (10). Note that the unit of all stochastic rate constants except for $\kappa '$ is per $1/4\min$ ($15\,$s) due to the factor $1/\beta$ in (46).

1.4 Derivation of the Reduced ODE Model From the Stochastic Model

Remind that the numbers of molecules of Cdc42 at tip1, at tip2, and in the cytosol at time $\tilde{t}$ are $n_1(\tilde{t})$, $n_2(\tilde{t})$, and $n_c(\tilde{t})$. Similarly, the numbers of molecules of GEF at tip1, tip2, and in the cytosol at time $\tilde{t}$ are $m_1(\tilde{t})$, $m_2(\tilde{t})$, and $m_c(\tilde{t})$. Then, the species copy numbers are governed by the following stochastic equations

$$\begin{aligned} n_i(\tilde{t})= & {} n_i(0) + Y_{i,1}\left( \int _0^{\tilde{t}} \left( k_0'+k_\mathrm{cat}'n_i(\tilde{s})^2\right) m_i(\tilde{s})n_c(\tilde{s}) \,\hbox {d}\tilde{s}\right) - Y_{i,2}\left( \int _0^{\tilde{t}} k_{-}'n_i(\tilde{s})\,\hbox {d}\tilde{s}\right) ,\quad i=1,2, \nonumber \\ n_c(\tilde{t})= & {} n_c(0) +\sum _{i=1}^{2}\left[ - Y_{i,1}\left( \int _0^{\tilde{t}} \left( k_0'+k_\mathrm{cat}'n_i(\tilde{s})^2\right) m_i(\tilde{s})n_c(\tilde{s}) \,\hbox {d}\tilde{s}\right) + Y_{i,2}\left( \int _0^{\tilde{t}} k_{-}'n_i(\tilde{s})\,\hbox {d}\tilde{s}\right) \right] ,\nonumber \\ m_i(\tilde{t})= & {} m_i(0) + Y_{i,3}\left( \int _0^{\tilde{t}} \frac{k_\mathrm{on}'}{1+\kappa ' n_i(\tilde{s})^2} m_c(\tilde{s}) \,\hbox {d}\tilde{s}\right) - Y_{i,4}\left( \int _0^{\tilde{t}} k_\mathrm{off}' m_i(\tilde{s})\,\hbox {d}\tilde{s}\right) ,\quad i=1,2, \nonumber \\ m_c(\tilde{t})= & {} m_c(0) + \sum _{i=1}^{2}\left[ - Y_{i,3}\left( \int _0^{\tilde{t}} \frac{k_\mathrm{on}'}{1+\kappa ' n_i(\tilde{s})^2} m_c(\tilde{s}) \,\hbox {d}\tilde{s}\right) + Y_{i,4}\left( \int _0^{\tilde{t}} k_\mathrm{off}' m_i(\tilde{s})\,\hbox {d}\tilde{s}\right) \right] ,\nonumber \\ \end{aligned}$$

(47)

where $Y_k$’s are independent Poisson processes. Adding the equations for CDC42 (or GEF) in (47), the total number of molecules of Cdc42 (or GEF) is conserved as

$$\begin{aligned} \begin{aligned}&n_1(\tilde{t})+n_2(\tilde{t})+n_c(\tilde{t})=n_1(0)+n_2(0)+n_c(0) \equiv N_{\mathrm{cdc}}, \\&m_1(\tilde{t})+m_2(\tilde{t})+m_c(\tilde{t})=m_1(0)+m_2(0)+m_c(0) \equiv N_{gef}. \\ \end{aligned} \end{aligned}$$

(48)

Plugging in $\tilde{t}=\beta t$ and using the change of variables ($\tilde{s}=\beta s$), (47) become

$$\begin{aligned} n_i(\beta t)= & {} n_i(0) + Y_{i,1}\left( \int _0^{t} \left( k_0'+k_\mathrm{cat}'n_i(\beta s)^2\right) m_i(\beta s)n_c(\beta s) \beta \,\hbox {d}s\right) \nonumber \\&- Y_{i,2}\left( \int _0^{t} k_{-}'n_i(\beta s)\beta \,\hbox {d}s\right) , \nonumber \\ n_c(\beta t)= & {} n_c(0) +\sum _{i=1}^{2}\left[ - Y_{i,1}\left( \int _0^{t} \left( k_0'+k_\mathrm{cat}'n_i(\beta s)^2\right) m_i(\beta s)n_c(\beta s) \beta \,\hbox {d}s\right) \right. \nonumber \\&\left. + Y_{i,2}\left( \int _0^{t} k_{-}'n_i(\beta s)\beta \,\hbox {d}s\right) \right] ,\nonumber \\ m_i(\beta t)= & {} m_i(0) + Y_{i,3}\left( \int _0^{t} \frac{k_\mathrm{on}'}{1+\kappa ' n_i(\beta s)^2} m_c(\beta s) \beta \,\hbox {d}s\right) - Y_{i,4}\left( \int _0^{t} k_\mathrm{off}' m_i(\beta s) \beta \,\hbox {d}s\right) ,\nonumber \\ m_c(\beta t)= & {} m_c(0) + \sum _{i=1}^{2}\left[ - Y_{i,3}\left( \int _0^{t} \frac{k_\mathrm{on}'}{1+\kappa ' n_i(\beta s)^2} m_c(\beta s) \beta \,\hbox {d}s\right) + Y_{i,4}\left( \int _0^{t} k_\mathrm{off}' m_i(\beta s) \beta \,\hbox {d}s\right) \right] ,\nonumber \\ \end{aligned}$$

(49)

for $i=1,2$. Assuming all species copy numbers are of the same order, we normalize the species copy numbers by a scaling parameter $N=\gamma _m A=602A$. Define normalized variables as

$$\begin{aligned} c_i^N(t)=\frac{n_i(\beta t)}{N},&\quad&C^N(t)=\frac{n_c(\beta t)}{NL},\qquad g_i^N(t)=\frac{m_i(\beta t)}{N},\qquad G^N(t)=\frac{m_c(\beta t)}{NL}, \nonumber \\ \end{aligned}$$

(50)

for $i=1,2$.

We express the stochastic rate constants in (46) using N as the following:

$$\begin{aligned} \begin{aligned} k_0'&= \frac{k_0}{N \beta L},\quad k_\mathrm{cat}' = \frac{k_{\mathrm{cat}}}{N^3 \beta L},\quad k_\mathrm{on}' = \frac{k^\mathrm{on}_\mathrm{max}}{\beta L}, \quad \kappa ' = \frac{\kappa }{N^2},\quad k_{-}'=\frac{k_{-}}{\beta },\quad k_\mathrm{off}'=\frac{k^\mathrm{off}}{\beta }. \end{aligned} \end{aligned}$$

(51)

Then, plugging in the normalized variables and scaled rate constants in (50) and (51), (49) becomes

$$\begin{aligned} c_i^N(t)= & {} c_i^N(0) + \frac{1}{N} Y_{i,1}\left( \int _0^t N\left( k_0+k_{\mathrm{cat}}c_i^N(s)^2\right) g_i^N(s)C^N(s) \,\hbox {d}s\right) \nonumber \\&- \frac{1}{N} Y_{i,2}\left( \int _0^t N k_{-}c_i^N(s)\,\hbox {d}s\right) ,\nonumber \\ C^N(t)= & {} C^N(0) +\sum _{i=1}^2 \frac{1}{NL} \left[ -Y_{i,1}\left( \int _0^t N\left( k_0+k_{\mathrm{cat}}c_i^N(s)^2\right) g_i^N(s)C^N(s) \,\hbox {d}s\right) \right. \nonumber \\&\left. + Y_{i,2}\left( \int _0^t N k_{-}c_i^N(s)\,\hbox {d}s\right) \right] ,\nonumber \\ g_i^N(t)= & {} g_i^N(0) + \frac{1}{N} Y_{i,3}\left( \int _0^t \frac{Nk^\mathrm{on}_\mathrm{max}}{1+\kappa c_i^N(s)^2} G^N(s) \,\hbox {d}s\right) - \frac{1}{N} Y_{i,4}\left( \int _0^t N k^\mathrm{off} g_i^N(s)\,\hbox {d}s\right) ,\nonumber \\ G^N(t)= & {} G^N(0) + \sum _{i=1}^2 \frac{1}{NL} \left[ -Y_{i,3}\left( \int _0^t \frac{Nk^\mathrm{on}_\mathrm{max}}{1+\kappa c_i^N(s)^2} G^N(s) \,\hbox {d}s\right) + Y_{i,4}\left( \int _0^t N k^\mathrm{off} g_i^N(s)\,\hbox {d}s\right) \right] ,\nonumber \\ \end{aligned}$$

(52)

for $i=1,2$. The strong law of large numbers states that, for a unit Poisson process Y, $\frac{1}{N}Y(Nu)\rightarrow u$ almost surely as $N\rightarrow \infty$. Therefore, assuming that $c_i^N(0)\rightarrow c_i(0)$, $C^N(0)\rightarrow C(0)$, $g_i^N(0)\rightarrow g_i(0)$, and $G^N(0)\rightarrow G(0)$ as $N\rightarrow \infty$, $X^N\equiv \left( c_1^N,c_2^N,C^N, g_1^N,g_2^N,G^N\right)$ converges to the limit which is a solution of

$$\begin{aligned} \begin{aligned} c_i(t)&= c_i(0) + \int _0^t \left[ \left( k_0+k_{\mathrm{cat}}c_i(s)^2\right) g_i(s) C(s) - k_{-}c_i(s) \right] \,\hbox {d}s, \\ C(t)&= C(0) + \frac{1}{L} \int _0^t \sum _{i=1}^{2} \left[ -\left( k_0+k_{\mathrm{cat}}c_i(s)^2\right) g_i(s) C(s) + k_{-}c_i(s) \right] \,\hbox {d}s, \\ g_i(t)&= g_i(0) + \int _0^t \left[ \frac{k^\mathrm{on}_\mathrm{max}}{1+\kappa c_i(s)^2} G(s) - k^\mathrm{off} g_i(s) \right] \,\hbox {d}s, \\ G(t)&= G(0) + \frac{1}{L} \int _0^t \sum _{i=1}^{2} \left[ -\frac{k^\mathrm{on}_\mathrm{max}}{1+\kappa c_i(s)^2} G(s) + k^\mathrm{off} g_i(s) \right] \,\hbox {d}s,\\ \end{aligned} \end{aligned}$$

(53)

for $i=1,2$ as $N\rightarrow \infty$. Note that $c_i$ and $g_i$ are the solution of the reduced ODE model for Cdc42 and GEF given in (7) and the total concentrations of Cdc42 and GEF are conserved as

$$\begin{aligned} \begin{aligned} \frac{c_1(t)+c_2(t)}{L}+C(t) \equiv C_{\mathrm{tot}}, \\ \frac{g_1(t)+g_2(t)}{L}+G(t) \equiv G_{\mathrm{tot}}, \end{aligned} \end{aligned}$$

(54)

which is consistent with (6). Setting $N=602 A$ and $t=0$ in (50), we obtain that

$$\begin{aligned} n_i(0)= & {} 602A\, c_i(0),\quad n_c(0) = 602AL\, C(0),\quad m_i(0) = 602A\, g_i(0),\nonumber \\ m_c(0)= & {} 602AL\, G(0), \end{aligned}$$

(55)

for $i=1,2$. Then, (55), (48), and (54) provide how $C_{\mathrm{tot}}$ and $N_{\mathrm{cdc}}$ ($G_{\mathrm{tot}}$ and $N_{gef}$) are related as follows:

$$\begin{aligned} \begin{aligned} N_{\mathrm{cdc}} = 602A(c_1(t)+c_2(t)+LC(t)) = 602AL\, C_{\mathrm{tot}}, \\ N_{gef} = 602A(g_1(t)+g_2(t)+LG(t)) = 602AL\, G_{\mathrm{tot}}. \end{aligned} \end{aligned}$$

(56)

Power Spectrum Analysis

1.1 Discrete Fourier Transform

Here, we use the fast Fourier transform to estimate the temporal power spectrum density of a discrete time series $\{n_1(t_i)\}_{i=1}^{N_t}$ generated by Gillespie’s algorithm. We subtract the steady state from the time series and replace $n_1(t_i)$ by $n_1(t_i)-n_1^*$. The default time interval is given by $t=0:\Delta t:t_\mathrm{end}$ with $t_\mathrm{end}=200$ and a uniform time step $\Delta t=0.001$. For the frequency interval, we set the sample frequency $f_s=1/\Delta t=1000$ and define the frequency domain by

$$\begin{aligned} ff=[0:N_t-1]*\Delta f,\quad \Delta f=\frac{f_s}{N_t}. \end{aligned}$$

(57)

Here, $N_t$ is the length of the time series. For each frequency $f_k$, the discrete Fourier transform of $n_1(t)$ is given by

$$\begin{aligned} \tilde{n}_1(k):=\sum _{j=0}^{N_t-1}n_1(t_j)\mathrm{e}^{-2\pi ijk/N}. \end{aligned}$$

(58)

Define the one-sided power spectrum P(k) by Press et al. (1996); Toner and Grima (2013)

$$\begin{aligned} P(k) = {\left\{ \begin{array}{ll} \displaystyle \frac{|\tilde{n}_1(k)|^2}{N_t^2}, &{} k=0,\,N_t/2,\\ \displaystyle \frac{|\tilde{n}_1(k)|^2+|\tilde{n}_1({N_t-k})|^2}{N_t^2}, &{} 1\le k\le N_t/2-1. \end{array}\right. } \end{aligned}$$

(59)

The power spectral density (PSD) over the frequency range $(f_k-f_s/(2N_t),f_k+f_s/(2N_t))$ is given by

$$\begin{aligned} S(f_k\pm \frac{1}{2}f_s)=\frac{P(k)}{\Delta f}= \frac{|\tilde{n}_1(k)|^2+|\tilde{n}_1({N_t-k})|^2}{N_tf_s}. \end{aligned}$$

(60)

The corresponding PSD over the angular frequency range $(\omega _k-\Delta \omega /2, \omega _k+\Delta \omega /2)$ is

$$\begin{aligned} S(\omega _k\pm \frac{1}{2}\Delta \omega )=\frac{P(k)}{\Delta \omega }. \end{aligned}$$

(61)

Here, the angular frequency increment is $\Delta \omega =2\pi f_s/N_t$. Finally, we average the PSD of a single realization r over R realizations to obtain a numerical approximation for the PSD. Denote the PSD of a single realization r by $S^r(\omega _k)$. The averaged PSD is given by

$$\begin{aligned} S(\omega _k)=\frac{1}{R}\sum _{r=1}^RS^r(\omega _k). \end{aligned}$$

(62)

1.2 MATLAB Code