Computational Biology and Chemistry 32 (2008) 240–242

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Computational Biology and Chemistry

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Accelerated stochastic simulation algorithm for coupled chemical

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reactions with delays

Wen Zhoua,b, Xinjun Penga, Zhenglou Yana, Yifei Wa Department of Mathematics, Shanghai University, Shanghai 200444, PR Chinab College of Mathematics and Computer Science, Anhui Normal University, Wuhu 24100

a r t i c l e i n f o

Article history:Received 11 October 2007Received in revised form 17 March 2008Accepted 19 March 2008

Keywords:Delay stochastic simulation algorithmDelay final all possible steps approachChemically reacting system

a b s t r a c t

Some biochemical procesthem. In the existed metha great deal of simulationapproach, called the “Delof stochastic simulation, bthat the proposed methodobtain a significant impro

1. Introduction

The stochastic simulation of chemically reacting systemsattracts a great deal of interest, since discreteness and stochastic-ity are important in systems formed by living cells where some

key reactant molecules may be present in small numbers (Arkinet al., 1998; Fedoroff and Fontana, 2002; McAdams and Arkin,1997; Van Kampen, 1981). The stochastic formulation of chemi-cal kinetics, described by the chemical master equation (CME), hasa rigorous microphysical basis for well-stirred chemically react-ing systems. While the analytical solution of the CME is rarelyavailable, the stochastic simulation algorithm (SSA) can simulatechemically reacting systems with exact statistical properties stipu-lated by the CME (Gillespie, 1977). To accelerate the SSA, Gillespieproposed an approximate simulation method called the �-leapmethod (Gillespie, 2001). After then, many improvements have alsobeen proposed (Cai and Xu, 2007; Chatterjee et al., 2005; Gillespieand Petzold, 2003; Peng et al., 2007).

In the above algorithms, a necessary condition for reliable statis-tical properties is averaging over a large number of runs. To reducethe simulation times, Lipshtat (2007) proposed the “All PossibleSteps” (APS) approach to obtain the reliable statistical characters ofevery species during the whole time course.

The SSA and the accelerated stochastic simulation algorithms allassume that all reactions occur instantly. However, some biologi-

∗ Corresponding author. Tel.: +86 21 66134331.E-mail address: yifei wang@staff.shu.edu.cn (Y. Wang).

1476-9271/$ – see front matter © 2008 Published by Elsevier Ltd.doi:10.1016/j.compbiolchem.2008.03.007

ga,∗

China

not occur instantaneously but have considerably delays associated withhich solve these chemically reacting systems with delays, averaging overeeded for reliable statistical characters. Here we present an acceleratingal All Possible Steps” (DFAPS) approach, which does not alter the courseduces the required running times. Numerical simulation results indicatebe applied to a wide range of chemically reacting systems with delays andnt on efficiency and accuracy over the existed methods.

© 2008 Published by Elsevier Ltd.

cal and chemical processes such as transcription, translation anddegradation do not occur instantaneously but have considerabledelays associated with them. So Bratsun et al. (2005) and Barrioet al. (2006), respectively, proposed the versions of delay stochas-tic simulation algorithms (DSSAs). Recently, Cai (2007) developedan exact DSSA which is rigorously based upon the fundamentalpremise of stochastic chemical kinetics (Gillespie, 1977). In thesealgorithms, averaging over a great deal of simulations is still nec-essary for reliable statistical properties.

In the present work, we develop the “Delay Final All PossibleSteps” (DFAPS) approach to produce significant gains in the effi-ciency of the statistical properties for the simulation of biochemicalreactions with delays. The DFAPS approach focuses on efficientanalysis of output data and reduces significantly the computationalresources needed in stochastic simulations. This paper is organizedas follows. In Section 2 we develop the DFAPS approach for chem-ically reacting systems with delays. Two numerical examples arepresented to illustrate the performance of the approach in Section3. Finally, summary is presented in Section 4.

2. The “Delay Final All Possible Steps” Approach

We see that the estimation of the various moments is the typicalaim of a stochastic simulation method. In all the DSSAs (Barrio etal., 2006; Bratsun et al., 2005; Cai, 2007), the selection procedurefor the next reaction and choice of data structure are describedin detail. However, there is no mention of the translation of theresulting trajectory into moments. We can easily find the momentsfrom the probability distribution. Here we suggest a novel approach

logy a

3. Results and Discussion

To demonstrate the accuracy and efficiency of the DFAPSmethod, we present simulation results for two specific examples.We run enough repeated DSSA simulations to make the histogramnearly smooth and regard this histogram as the real distribution.Then we compute the histogram distance between the DSSA, DFAPSalgorithm and the real distribution after a series of run times. Sincethe histogram distance provides a measure of the error, we com-pare the run times and the CPU time of these two algorithms withthe same histogram distance. All simulations are run in MATLAB ona personal computer with a 1.99 GHz CPU and 256 Mbyte memoryrunning WINDOWS XP.

3.1. A System With Two Reaction Channels

This simple example was used by Cai (2007). It is

R1 : S1 + S2c1→S3, R2 : S3

c2→∅. (5)

W. Zhou et al. / Computational Bio

for calculating the probability distribution at a certain time, calledthe “Delay Final All Possible Steps” approach.

We consider a well-stirred mixture of N ≥ 1 molecular species{S1, . . . , SN} that chemically interact through M ≥ 1 reaction chan-nels {R1, . . . , RM}. The state of this chemical system is described bythe state vector X(t) = (X1(t), . . . , XN(t)), where Xn(t), n = 1, . . . , N,is the molecular number of species Sn at time t. The molecular popu-lations Xi(t) are random variables whose dynamics are determinedby the reactions. The dynamic of reaction channel Rj is charac-terized by its propensity function aj and the state-change vectorvj ≡ (v1j, . . . , vNj), where vj ≡ vrj + vpj , vrj is the state-change vec-tor of reactant species, and vpj is that of production species inducedby the Rj reaction.

Suppose that a subset of, or, all reaction channels, incurs somea delay, respectively. Following Barrio et al. (2006), we classifyreactions with delays RD into two categories: (1) nonconsumingreaction, where the reactants of an unfinished reaction can par-ticipate in a new reaction, and (2) consuming reaction, where thereactants of an unfinished reaction cannot participate in a newreaction. We denote the set of nonconsuming reactions as RD1and the set of consuming reactions as RD2 . When a reaction Roccurs, if R∈RD1 , the population of the reactants does not change,however, if R∈RD2 , the population of the reactants change imme-diately.

If there are Nd ongoing reactions at time t, which will finishat t + T1, . . . , t + TNd

, respectively. Without loss of generality, T1 ≤T2 ≤ . . . TNd

is assumed. In Cai’s DSSA, after generating two standarduniform random variables u1 and u2, � and � can be calculatedas

� = Ti +− ln(1− u1)−

∑i−1

j=0a0(t + Tj)(Tj+1 − Tj)

a0(t + Ti),

� ∈ [Ti, Ti+1), i = 0, . . . , Nd. (1)

where T0 = 0 and TNd+1 = ∞, a0(t) =∑M

�=1a�(t). Ti is generated

from F�(Ti) ≤ u2 < F�(Ti+1), where F�(�) = 1− exp(−∑i−1

j=0a0(t +Tj)(Tj+1 − Tj)− a0(t + Ti)(� − Ti)), � ∈ [Ti, Ti+1), i = 0, . . . , Nd,and

�−1∑

j=1

aj(t + Ti) < u1a0(t + Ti) ≤�∑

j=1

aj(t + Ti). (2)

Assuming at the final step before reaching the acquiredtime the system’s state is X(t) = x. Without loss of generality,we assume there are M possible reactions R1, . . . , RM to occurand R1, . . . , RM1 ∈RD1 , RM1+1, . . . , RM2 ∈RD2 , (M1 ≤M2 ≤M). TheDSSA chooses randomly the next reaction Rj according to the pointprobability a�(t + Ti)/a0(t + Ti), i = 0, . . . , Nd, and only updates theprobability due to the reaction Rj as

q(x)→ q(x)+ 1, if Rj ∈RD1 ,q(x+ vrj)→ q(x+ vrj)+ 1, if Rj ∈RD2 ,

q(x+ vj)→ q(x+ vj)+ 1, if Rj ∈̄RD.(3)

However, in the DFAPS algorithm, we consider all the possiblereactions and update the probabilities:

q(x)→ q(x)+ a�/a0, � = 1, . . . , M1,q(x+ vr�)→ q(x+ vr�)+ a�/a0, � =M1 + 1, . . . , M2,

q(x+ v�)→ q(x+ v�)+ a�/a0, � =M2, . . . , M.(4)

After all the runs are over, the probabilities are normalized.The DFAPS approach is summarized as follows:

(1) Initialize t0 = 0, tfinal = T, x = x(t0), p(x) = 0, q(x) = 0;

nd Chemistry 32 (2008) 240–242 241

(2) Repeat steps 3–5 until t + � ≥ T;(DSSA)

(3) Calculate the propensity functions, am(x), m = 1, . . . , M, anda0(x) =

∑M�=1a�(x);

(4) Generate � from Eq. (1);(5) Select the next reaction to be R� with point probability a�(t +

Ti)/a0(t + Ti), where Ti ≤ � < Ti+1. If R� ∈̄RD, we update thestate vector as x← x+ v�; If R� ∈RD2 , then x← x+ vr�;

(DFAPS)(6) Update the probabilities of each reaction R�, � = 1, . . . , M,

according to Eq. (4);(7) After l runs, normalize q(x) as p(x) = q(x)/

∑xq(x).

In every run, steps 1–6 are done. The DFAPS algorithm onlyrequires more probabilities computation which are simply arith-metic operations than the DSSA and the DFAPS algorithm does notneed to run the last step. So it does not need more time in everyrun. The running times l is decided by the required accuracy. Withthe same accuracy, l in the DFAPS algorithm is much less than in theDSSA. It is well-known that in general systems the time spent oneach run is long. Therefore, the DFAPS algorithm is more efficient.

Fig. 1. The histogram distance of X3(1)vs. run times for two reaction channel (5).

242 W. Zhou et al. / Computational Biology a

Table 1Run times and CPU time (in seconds) to obtain the corresponding histogram distance(hist.dist.) of X3(1) for two independent reactions (5)

Run times CPU time Hist.dist. Run times CPU time Hist.dist.

DSSA 964 133 0.10 1314 182 0.08DFAPS 598 78 0.10 866 113 0.08

pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics

Fig. 2. The histogram distance of X2(80)vs. run times for gene expression with neg-ative feedback reactions (6).

Table 2Run times and the average needed CPU time (in seconds) to obtain the correspondinghistogram distance (hist.dist.) of X2(80) for the example of gene expression withnegative feedback

Run times CPU time Hist.dist. Run times CPU time Hist.dist.

DSSA 855 11 701 0.10 1312 17 955 0.08DFAPS 422 5 916 0.10 580 8 130 0.08

The first reaction R1 incurs a delay and R1 ∈RD2 , but the secondreaction R2 reacts without delay. We also choose c1 = 0.001, c2 =0.001 and X1(0) = 1000, X2(0) = 1000, X3(0) = 0, set the delay ofR1 to be d = 0.1. The simulation starts at t0 = 0 and ends at tfinal =

1.

We first obtain the smooth histogram of X3(1) from 800 000runs using the DSSA as the level distribution. Then Fig. 1 depictsthe histogram distances of X3(1) versus run times using theDSSA and DFAPS approach, respectively. It shows that withthe same histogram distance the running times of the DSSAis nearly twice more than that of the DFAPS approach. Thisis because that the number x3 of species S3 has two possi-ble states: x3 − 1 and x3, and their occurrence probability maybe different. The DFAPS approach considers all the two pos-sible states without virtually visit them. However, the DSSArandomly chooses one reaction to occur and obtain the correspond-ing state with possibility 1. So the DFAPS approach outperformsthe DSSA. We can further see that from Table 1, which liststhe average run times, CPU time and the histogram distance ofX3(1).

3.2. Gene Expression With Negative Feedback

Bratsun et al. (2005) and Cai (2007) used this example in theirmethods. This model has the following four reaction channels:

nd Chemistry 32 (2008) 240–242

R1 : S1 + S2c1→S3, R2 : S3

c2→S1 + S2, R3 : S1c3→S1 + S2,

R4 : S2c4→∅. (6)

There is no delay in reactions R1, R2 and R4. R3 is a nonconsum-ing delay reaction. The delay of R3 is d = 50. The initial molecularnumbers of S1, S2 and S3 are x1 = 1, x2 = 0 and x3 = 0. The proba-bility rate constants are c1 = 1, c2 = 500, c3 = 100 and c4 = 1. Thesimulation starts at t = 0 and ends at tfinal = 80.

After running the DSSA 200 000 times, we obtain the nearlysmooth histogram of X2(80). Regarding it as the level result, Fig. 2depicts the histogram distance of X2(80) versus run times using theDSSA and DFAPS approach, respectively. Further, Table 2 lists theaverage run times, CPU time and the histogram distances of X2(80).From Fig. 2 and Table 2, we can see that the DFAPS approach is moreefficient than the DSSA. With the same histogram distance, the runtimes and the CPU time of DSSA are both about twice more thanthose of the DFAPS approach.

4. Summary

The present DSSAs have to run a great deal of simulations toobtain a high precision level. We here propose a novel approach,namely the DFAPS approach, which significantly decreases the runtimes. It computes the possibilities that all reactions will possiblyoccur at the last step before reaching the final time in the DSSA.Test results on two different reaction sets support the conclusionthat the DFAPS approach significantly improves the simulation effi-ciency with better accuracy. When a species in the reaction sethas many possible states, the efficiency gain afforded by the DFAPSapproach will be most significant. Further work is to build the the-ory of the DFAPS method.

Acknowledgements

This work is partly supported by the National Natural ScienceFoundation of China (General Program No. 30571059), and theNational High-Tech Research and Development Program of China(Grant No. 2006AA02Z190).

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