MATHICSE Mathematics Institute of Computational Science and Engineering School of Basic Sciences - Section of Mathematics

Transcription

1 MATHICSE Mathematis Institute of Computational Siene and Engineering Shool of Basi Sienes - Setion of Mathematis MATHICSE Tehnial Report Nr April 211 Boosted hybrid method for solving hemial reation systems with multiple sales in time and population size Yuheng Hu, Assyr Abdulle, Tiejun Li Address: Phone: EPFL - SB - MATHICSE (Bâtiment MA) Station 8 - CH Lausanne - Switzerland Fax:

2 Boosted Hybrid Method for Solving Chemial Reation Systems with Multiple Sales in Time and Population Size Yuheng Hu 1,, Assyr Abdulle 2, and Tiejun Li 1 1 Laboratory of Mathematis and Applied Mathematis and Shool of Mathematial Sienes, Peking University, Beijing 1871, China. 2 Mathematis Setion, Eole Polytehnique Fédérale de Lausanne, CH-115 Lausanne, Switzerland. Abstrat. A new algorithm, alled boosted hybrid method, is proposed for the simulation of hemial reation systems with sale-separation in time and disparity in speies population. For suh stiff systems, the algorithm an automatially identify sale-separation in time and slow down the fast reations while maintaining a good approximation to the original effetive dynamis. This tehnique is alled boosting. As disparity in speies population may still exist in the boosted system, we propose a hybrid strategy based on oarse-graining methods, suh as the tau-leaping method, to aelerate the reations among large population speies. The ombination of the boosting strategy and the hybrid method allow for an effiient and adaptive simulation of omplex hemial reations. The new method does not need a priori knowledge of the system and an also be used for systems with hierarhial multiple time sales. Numerial experiments illustrate the versatility and effiieny of the method. Key words: hemial reation, multisale, boosting, hybrid method 1 Introdution Advanes in experimental and omputational methods over the last deades have made a quantitative, systemati understanding of ellular proesses in moleular level possible [1 6]. For miro-sale biohemial systems, suh as one single living ell, onsiderable evidene indiates that stohastiity plays an important role, espeially when lowmoleular-number reatant speies are being onsidered [2, 3]. The usefulness of the traditional deterministi approah, based on reation rate equations, is limited in suh situation. In turn, many stohasti biohemial reation networks have been built to take into Corresponding author. addresses: huy@pku.edu.n(y. Hu), assyr.abdulle@epfl.h(a. Abdulle), tieli@pku.edu.n(t. Li)

3 aount the randomness in biologial proesses. Beause of the disparity of time sales and speies population, the simulation of suh systems is often hallenging and the development of new numerial tehniques for multisale hemial reations has beome an ative researh field [4, 7, 8]. One fundamental method in simulating hemial reation systems is Gillespie s Stohasti Simulating Algorithm (SSA) [9,1]. It an generate statistially exat trajetories of the system state by randomly sampling eah reation event. In priniple, SSA applies to any hemial reation system, but the method beome omputationally ostly when reation events our very frequently in the system. This often happens beause of the o-existene of fast and slow dynamis in a system, or reations involving speies with very large populations, or both. On one hand, the o-existene of fast and slow dynamis in a system leads often to severe step-size restrition for standard methods. Suh systems are alled stiff and need a speial numerial treatments. For stiff ordinary or stohasti differential equations, impliit methods or stabilized expliit methods (alled Chebyshev methods) an be effiient [11 13]. But fast variables in hemial reation system often flutuate quikly around a slow manifold, and impliit or stabilized method usually fail to apture the right stationary distribution of the fast variables [12, 14, 15]. Rao and Arkin [16] first formalized the quasi-equilibrium approximation in hemial reation system and implemented it in SSA. Their idea was further extended by Cao et al. in developing the slowsale SSA [17]. Both methods require expliit form of the stationary distribution for the fast variables whih in general is diffiult to get. To remove this restrition, E et al. developed the nested-ssa [18]. In this method, the averaged rates of the slow reations are sampled by inner SSA, as the miro-solver ating on fast reations only, during a period of time that is muh larger than the fast time sale and at the same time muh smaller than the slow time sale. Then the average rates of the slow reations will be used by the outer SSA, as the maro-solver ating on slow reations only, to marh the system forward. On the other hand, reations involving speies with large population size fire very frequently whih also make the SSA omputationally ineffiient. To overome this diffiulty, Haseltine and Rawlings proposed a hybrid method for solving hemial reation systems with disparity in speies population [8]. The main idea is to apply a oarse-graining approximation (based on stohasti or ordinary differential equations) for speies with large population size and SSA for speies with small population size. Many variants of hybrid method have been proposed [6, 19 25]. Based on the system size, the τ-leaping method [7], hemial Langevin equations or reation rate equations [26] are often used as the oarse solver. Numerial algorithm that an handle both the multiple time sales and the disparity in speies population is urrently a hallenging researh diretion. Cao et. al [27] demonstrate that hybrid method an be ombined with slow-sale SSA and impliit τ-leaping to handle stiff separations. Samant et. al [28] designed a more general algorithm that an identify sale-separation and partition the reations on-the-fly. Another improvement in 2

4 this work is the generalization of the nested-ssa to nest-tau-leaping to handle fast reations involving large population speies. Compared with the nested-ssa by taking the heterogeneous multi-sale method (HMM) approah [18] and slow-sale SSA based on taking the expliit quasi-equilibrium approximation [17], another nie idea for multi-sale methods whih is alled boosting has never been investigated in the hemial reation literature [29]. The idea of boosting for aelerating the multi-sale simulation is simply to resale the sale separation parameter λ 1 to an artifiial onstant λ suh that λ λ 1, whih an speed up the simulation of fast reations and keep the omputational auray at the same time. The idea an be dated to Carr-Parrinello s moleular dynamis and Chorin s artifiial ompressibility method in fluid mehanis [29]. In this paper we present a new general algorithm, alled boosted hybrid method, that an effiiently simulate omplex hemial reation systems with time-sale separation and disparity in speies population. The algorithm ombines the idea of boosting and hybrid simulations. It ollets the system state information on-the-fly and automatially partition the reations into fast and slow groups if ertain riteria is satisfied. Then the original system is approximated by a boosted system, in whih the fast reations are slowed-down by a hange of time-unit. The new system, whih poses less saleseparation in time, is simulated using a hybrid method. The shemati overview of our method is shown in Fig. 1. The hybrid solver is built in Blok B, whose details will be given in Setion 2. It is similar to that of Haseltine et al. s approah [8] but has several ruial improvements. Blok C is the monitor blok. It does not diretly update the system but only ollets information. With the olleted statistial information over a short time window, the monitor will generate an approximate system as a replaement of the original one. The main tool used in maintaining a reasonable approximation of the system is boosting [29, 3]. It relies on the following idea: in presene of large time-sale separation, it is possible to slow down the fast reations of a system while maintaining a good approximation to the original effetive dynamis. A boosted system beomes less stiff without destroying the effetive dynamis. It an then be solved by a hybrid method, so that the reations among speies with large population size an be effiiently simulated. Basially speaking, the time-sale separation is dealt with boosting, and the disparity in the speies population is taken are by hybrid method. The paper is organized as follows. In Se. 2 we disuss the riteria for partitioning the reations and present a improved hybrid method. The boosting strategy is introdued in Se. 3. Its ombination with the boosting method is presented in Se. 4. Various numerial experiments are given in Se. 5 illustrating the versatility and effiieny of the method. Finally in Se. 6 we summarize our findings and disuss unsolved issues and future works. 3

5 Boosted Hybrid Method C. Monitor A. Initialization Reset to the original system. B. Hybrid Solver Partition the reations: ritial reations Use Gillespie's exat simulation method. oarse-grainable reations Use tau-leaping approximation method. Collet run-time data Keep the urrent boosted system or further boost the system if it will not break the quasi-equilibrium and sale separation. Y Quasi-equilibrium? Y Sale separation? N N D. Exit Figure 1: Shemati overview of the boosted hybrid method. It onsists of two major omponents: the monitor and the hybrid solver. The monitor ollets information, inluding the average rates and system state hange. With this statistial information, it will deide if quasi-equilibrium is reahed. If so, a boosted system will be proposed to replae the original system to be simulated by the hybrid method. 2 Adaptive Hybrid Method In this setion we start by desribing how to adaptively partition the reations in a omplex hemial reation systems. A hybrid method is then introdued whih allows to use different numerial strategies for hemial speies with different population size. 2.1 Partition of the system Consider a well stirred hemial reation system onsisting of N different speies interating through M reation hannels. Let X i (t) be the population number of the i-th speie at time t, and the system state vetor be X(t) = (X 1 (t),x 2 (t),,x N (t)) N N. Eah reation hannel j has a propensity funtion, or reation rate, a j (x),x N N. The probability that the j-th reation fires during an infinitesimal time dt is a j (X(t))dt, independent of the other reations. If the j-th reation fires, the system is updated as X(t) X(t)+ν j, where ν j = (ν 1j,...,ν Nj ) T N N is the state-hange vetor orresponding to this reation. Given the initial state X() and the aforementioned evolutionary law, we an use Gillespie s SSA [9] to simulate trajetories of X(t) and study its statistial properties. But for systems with multiple well separated time-sales and disparity in speies population, this proedure often beomes too ostly. The aim of this work is to provide a more effiient simulation strategy for suh systems. The first step is to partition the system. We divide the reations into different groups aording to their property (see Fig. 2). Eah group will then be treated with different algorithms. As shown in Fig. 2, we partition all the reations into four groups based on 4

6 population size I fast oarse-grainable III II slow oarse-grainable IV fast ritial slow ritial time sale Figure 2: After partition, the reations are divided into four non-overlapping groups. (I) Fast oarse-grainable reations, suitable for boosting and oarse-graining approximations. (II) Slow oarse-grainable reations, suitable for oarse-graining approximation. (III) Fast ritial reations, suitable for boosting approximation. (IV) Slow ritial reations, suitable for SSA. To apply quasi-equilibrium approximation, we need time-sale separation between fast and slow reations, whih is represented by the gray zone in the figure. the speies population numbers and the reation rates. Reations with fast dynamis are simply alled fast reations, the remaining reations are alled slow reations. We also distinguish reations involving speies with large population size, alled oarse-grainable reations, from reations involving speies with small population size, alled ritial reations. We give below the implementation details for realizing suh a partition. Adaptive seletion of ritial and non-ritial reations. Given the system state X, we define the bottlenek speie number z j for eah reation j as, z j (X) = min i=1,...,n;ν ij = { } Xi. ν ij Note that if reation j fires one, at least one speie will be hanged by a proportion of 1/z j. If z j is large, this will be just a small hange to the system. For these reations oarse-graining approximation suh as tau-leaping an apply. The idea of tau-leaping is to assume the system state to be fixed for a small period of time so that many reations an be updated in one time. But if z j is small, the system will experiene a signifiant hange, and exat method like the SSA need to be used. We hoose a small value ɛ (say, ɛ=.1, the same ɛ will be used later in the tau-leaping method). If z j <1/ɛ, then reation j will be alled as ritial reation, otherwise as oarse-grainable reation. Adaptive seletion of fast and slow reations. We further partition all the reations as being either fast or slow reations. Define the harateristi rate for reation j as A j (X) = a j (X) max{ɛz j (X),1}, (2.1) 5

7 where ɛ is a small positive parameter. Note that for ritial reations A j (X)=a j (X), while for oarse-grainable reations A j (X) < a j (X). To apply quasi-equilibrium approximations, time-sale separation between the averaged harateristi rates is required. Then we an partition all the reations into four non-overlapping groups as shown in Fig Hybrid method The hybrid method allows effiient simulation of hemial reation systems with disparity in speies population. It is also a major building blok of the boosted hybrid method relying on the use of boosting to handle the time-sale separation and a hybrid treatment of the resulting reations groups (II) and (IV). The idea is to use SSA for ritial reations and at the same time use tau-leaping method for oarse-grainable reation. The diffiulty is the oupling of time-steps between those used by tau-leaping and SSA. There are a lot of works in this diretion [6, 19 25]. The oupling strategy that we use here relies on Ref. [8]. We however propose some improvements of the aforementioned work related to an adaptive partitions of the reations as explained below. Remark 2.1. In Ref. [8], the authors use SSA ombined with stohasti differential equation integrator or ordinary differential equation integrator. Here we will disuss the hybridization of SSA and tau-leaping but the framework also apply to other oarse-graining solvers. Consider a system with only slow reations. The ritial reations (group (IV)) are labeled as 1,...,q and the oarse-grainable reations (group (II)) are labeled as q+1,...,m. Define a Λ (X) as the sum of rates of all the ritial reations, a Λ (X) q j=1 a j (X). The waiting time for the next ritial reation to fire is a random variable τ with probability density funtion ( t+τ ) p(τ) = a Λ (X(t+τ))exp a Λ (X(s))ds. (2.2) t In order to generate suh a random variable, one an first generate a standard uniformly distributed random variable u and then solve t +τ q t j=1 a j (X(s))ds+lnu 1 =. (2.3) During (t,t +τ) the oarse-grainable reations may hange the system state X(t). But under the leaping ondition [31] X j (t +t leap ) X j (t ) ɛx j, j = q+1,,m, (2.4) 6

8 we an find t leap > so that before time t +t leap the system-hange aused by the oarsegrainable reations an be negleted. Assuming the system state remains unhanged to be X(t ), Eq. (2.3) predits dt 1 = lnu 1 /a Λ (X(t )). If we have dt 1 < t leap, the leaping ondition is not violated and we hoose a ritial reation to fire at time t +dt 1. We also use this dt 1 as the leaping step-size to update the oarse-grainable reations in the tau-leaping method. If, however, we have dt 1 > t leap, then we need to update the oarsegrainable reations at the time t +dt 1. So we update the oarse-grainable reations using the tau-leaping method with step-size t leap and define dt 1 =t leap instead. Then we update the time to t t +dt 1. Sine no ritial reation has fired, we update Eq. (2.3) as t +τ q t j=1 a j (X(s))ds = lnu 1 q j=1 a j (X(t ))dt 1 >. Now we let t t and repeat the above proedures to get dt i (i = 2,...) in eah step. As the left-hand-side of the above equation will derease eah time, eventually a ritial reation will fire and Eq. (2.3) an be solved. Thus the final τ = i dt i. Then, using SSA, the ritial reation that will fire an be randomly hosen by finding a l that satisfies l 1 j=1 a j (X(t +τ)) < u 2 a Λ (X(t +τ)) where u 2 is a random variable uniformly distributed in [,1]. l j=1 a j (X(t +τ)), (2.5) Remark 2.2. Note that after several steps of the numerial method for the oarse-grainable reations, the lassifiation of the ritial and non-ritial reations may no longer be valid. In this ase, we have to partition the system again. Remark 2.3. Eqs. (2.2) and (2.3) are the same as Eqs. (17) and (2a) in Ref. [8]. But the way of solving them is different here. Instead of using no-reation, whih does not hange the system state but only makes ritial reations to happen at a more frequent basis in order to sale stohasti time step, we searh for the next zero-rossing time point for Eq. (2.3) under the onstrain of the leaping ondition. The no-reation approah is easy to use and still appliable here, but it may result in omputational overhead beause more ritial reations (inluding the no-reations) need to be handled. Lastly, we point out that that our approah is similar with the work in Ref. [2], but the later is based on the first reation variant of Gillespie s SSA method. Generally speaking, there is no signifiant differene between the applied priniples. Finally we disuss how to generate t leap. There already exist a ouple of proedures to selet a reasonable t leap that meets the leaping ondition Eq. (2.4), suh as those in Ref. [7,31,32]. They an be used diretly in our hybrid method. Here we propose another step-size seletion strategy, whih is simpler to be implemented in our algorithm. We let t leap = min j=q+1,...,m 7 1 A j (X),

9 where A j (X) is the harateristi rate of the reation j as given in Eq. (2.1). Note that within a time-step 1/A j (X), reation j fires on average a j (X)/A j (X)=ɛz j (X) times, and ause a hange of ɛz j (X)ν j ɛx i (i =1,...,N), whih is onsistent with the leaping ondition (2.4). The hybrid method is given in Algorithm 1. Note that we need to do a new partition (step 3 below) eah time after we update the system state beause the ritial reation set may have hanged. Algorithm 1. Hybrid method. Initialization: Let t =, give X(), and set ɛ =.1, γ =. 1. If γ, generate a standard uniform random variables u 1 and let γ = logu 1 <. Otherwise ontinue. 2. Compute the rates a j (X(t)), j =1,...,M. 3. For eah reation j, ompute its bottlenek speie number as z j (X(t))=min νij ={X i / ν ij }, j=1,...,m. If ɛz j (X(t)) 1, label it as ritial reation, otherwise as oarse-grainable reation and ompute its harateristi rate A j (X(t)) = a j (X(t))/ɛz j (X(t)). 4. Choose the leaping step-size t leap =1/max(A j (X(t))), for all oarse-grainable reation j. Compute τ = γ/a Λ (X(t)). 5. If t leap > τ, fire one ritial reation using SSA as in Eq. (2.5). Apply tau-leaping with step-size τ for all oarse-grainable reations. Update time t=t+τ. Otherwise, apply tau-leaping with step-size t leap for all oarse-grainable reations, and do not fire ritial reations. 6. Update time t t+t leap and γ γ+a Λ (X(t))t leap. Repeat from step 1. 3 Boosting method for hemial reation systems In hemial reation systems, it is very ommon to have large time-sale separations. A widely used strategy is quasi-equilibrium approximation. There exist a number of ways of implementing this idea in hemial reation systems, suh as slow-sale SSA, nested SSA. Here we onsider another approah alled boosting, first introdued by Vanden- Eijnden [29]. This approah, as we will see, has the advantage of allowing for an easy implementation. The idea of boosting has been originally proposed in dynamial systems with sale-separation in time, suh as stiff ODEs and SDEs. We will see here that it is also a good strategy for stohasti systems arising in hemial reations. 8

10 A simple example. First let us onsider a simple hemial reation system 1 S 1 S 2, 2 3 S 1 S 3. 4 Suppose 1 =2, 2 =1 3 = 4 =.5, and initially X()=(1,,). One an think of S 1, S 2 and S 3 as being different forms of a protein. This protein will swith between state S 1 and S 2 at a very fast rate, and it an also swith between state S 1 and S 3 at a muh slower rate. Suppose that we are interested in the slow variable S 3. In fat the rate of the third reation S 1 S 3 only depends on the amount of time that the protein stays at state S 1, whih is determined by the ratio of 1 and 2. The idea of boosting for this example is to slow down the fast reation while maintaining orret behavior of the slow dynamis. We an simply dividing 1 and 2 both by 1, so that the protein still osillate faster between S 1 and S 2. But the system is less stiff sine we have dereased the sale separation and it is easier to solve with a standard solver. 3.1 Boosted-SSA Consider a hemial reation system with only ritial reations. The slow reations in group (III) are labeled as 1,2,...,q, and the fast reations in group (IV) are labeled as q+ 1,q+2,...,M. Sale separation between fast and slow reations requires min(a q+1,...,a M ) max(a 1,...,a q ). If we simulate this system with SSA, fast reations will fire frequently and involve a high omputational ost. But it is only the slow dynamis that we want to apture. If we restrit our attention to slow reations, the waiting time for the next slow reation in the system, τ, has the probability density funtion ( ) p(τ) = q j=1 a j (X(τ))exp τ q j=1 a j (X(s))ds. (3.1) The above equation is exat, but X(t) ontains the information of the fast reations whih we do not want to resolve. If the fast reations quikly drive X(t) into quasi-equilibrium, then we an approximate the effetive rate of slow reation j b j as b j 1 τ τ a j (X(s))ds 1 h h a j (X(s))ds, j =1,...,q. (3.2) Here h a parameter hosen so that it is large in mirosopi sense, meaning quasi-equilibrium is well established for X(t), and small in marosopi sense, meaning h is far less than 9

11 the waiting time of the next slow reation τ. Assume h = κτ, where κ 1. In (3.2) we make the transformation b j 1 h h a j (X(s))ds = 1 τ τ a j (X(κs))ds, (3.3) for j=1,...,q. The above equation implies that we an approximate the effetive rate b j by slowing down the fast reations by a fator of κ, whih an be done simply by setting the rate onstants as j = κ j,j = 1,...,q to form a modified system. The boosted-ssa as given by algorithm 2 is nothing but to solve a modified system with SSA. Algorithm 2. Boosted-SSA. 1. Divide all reations into slow and fast reations. 2. Choose parameter κ 1 (we will disuss how to hoose κ in Setion 4). Modify the system by resalling the rate onstants of all the fast reations as j = κ j. 3. Simulate the modified system using SSA. 3.2 Comparison with slow-sale SSA and nested SSA The slow-sale SSA [17] and nested SSA [18] are also based on quasi-equilibrium approximation of the fast reations in the system. In the slow-sale SSA, the system state is divided into fast and slow variables as X =(Z,Y). Conditioned on the urrent state of the slow variable Y = y, it alulates the onditional probability distribution funtion of the fast variables p(z y), analytially or approximately. Then it omputes the effetive rate b j (y) of the slow reation j by using b j (y) = a j (y,z)p(z y). (3.4) z This amount to integrate out the fast variable z in the system, whih leads to a redued system with state Y and slow reations with rates b j (y). Then Gillespie s SSA is applied to the redued slow-sale system and Y gets updated. The slow-sale SSA is very effiient if one an obtain the onditional distribution p(z y). However, for omplex system there is no systemati way to do so. The nested SSA does not require an expliit form of p(z y), not even the partition of slow and fast variables. It only partitions the reations into slow and fast reations, whih is relatively easier. It then uses an inner-ssa subroutine to simulate only the fast reations for a small amount of time h to ompute the effetive rate of slow reations j as b j = 1 h h a j (X(s))ds. 1

12 The above equation is exatly (3.2). With the effetive rates, an outer-ssa subroutine is alled to simulate one single event among the slow reations and update the system state and the time. The nested SSA and boosted SSA share similar theoretial basis: both need to partition the reations, and share the same quasi-equilibration time h. The major differene is in the implementation: in the former, the fast and slow reations are simulated separately and only the slow reations get updated, but in the latter, the fast reations are first slowed down, then all the reations are updated together. Boosting is easier to implement, espeially for multisale systems with a hierarhy of time sales. In suh a situation, if using nested-ssa, we need to identify and order the hierarhy of time sales and implement suessively a hierarhy of fast solvers for the different sales involved with the averaged rates for one level oming from the quasi-equilibrium of the fast reation at the previous level. In the boosting framework, one does not need any additional subroutine to handle the time sale separations, we just boost the reation rates of the fast dynamis, and use some single sale hybrid solver in hand to perform the time integration. 3.3 Boosted tau-leaping Boosting an also be used in oarse-grainable reations, so to get the boosted tau-leaping. Beause in tau-leaping the step-size τ is deterministi, the onventional proedure of boosting that is used in solving stiff ODEs and SDEs an be applied here. The following algorithm an be obtained by following Ref. [3]. Algorithm 3. Seemless tau-leaping with fixed step-size. 1. Divide all reations into slow and fast reations. 2. Selet the step-size δt for the fast reations and t for the slow reations. Choose a parameter K, whih is the estimated number of steps that the fast reations require to reah quasi-equilibrium. Beause of the time-sale separation between fast and slow reations, we have Kδt τ. Repeat the following proedure K times: (a) Integrate the fast reations with step-size δt using tau-leaping. (b) Integrate the slow reations with step-size t = t/k using tau-leaping. 3. Repeat from step 1. While the above sheme works for tau-leaping, we do not know how to generalize it into SSA beause of the randomness of the step-size. The boosted tau-leaping algorithm is as follows: Algorithm 4. Boosted tau-leaping with modified rates. 1. Choose κ = Kδt/ t and modify the rate onstants of the fast reations as = κ. 11

13 2. Integrate the whole system with step-size t/k. In fat, Algorithms 3 and 4 are theoretially equivalent, but the latter is more flexible, and more importantly, it works for both SSA and tau-leaping. 4 Boosted hybrid method In this setion we introdue the boosted hybrid method as an adaptive solver for multisale hemial reation system. As Fig. 1 shows, it has two major omponents. The monitor will maintain an approximating system. It will monitor the system state during the simulation to see if there exists quasi-equilibrium for fast reations and deide whether it is possible to boost the system to redue stiffness. The approximating system is simulated using the hybrid method. 4.1 The monitor We use the hybrid method to simulate the hemial system for a time of length t, whih is alled a monitoring window. Information suh as reation rates and system state is olleted during the monitoring window, with whih the monitor an identify timesale separation between fast and slow reations, and if so, hek if quasi-equilibrium is reahed for the fast reations. Based on these judgments, the method is able to maintain a reasonable approximation of the original system. We say that time-sale separation between the fast reations group Ω and its omplementary, the slow reations group, Ω, exist if min j Ω A j >1 q max j Ω A j > ω, (4.1) where A j is the harateristi rates in Eq. (2.1). There are two parameters in the above formula, q and ω. The parameter q represents the degree of sale separation, for example, q = 2 orresponds to a 1 times sale differene between fast and slow reations. The parameter ω indiates the time-sale we are interested in. In other words, we only apply boosting to reations whose rates are greater than ω. It may vary from system to system, based on the partiular dynamis we want to learn. Note that in out algorithm we use the averaged harateristi rates in Eq. (4.1). This is beause for some fast reations their rates may vary rapidly. For simpliity, in the rest of the paper we still denote the averaged rates as A j. Time-sale separation between the fast reations Ω and slow reations Ω does not guarantee quasi-equilibrium. We laim quasi-equilibrium for reations in Ω is reahed if, during the last monitoring window, the total flux due to reations in Ω is muh larger than the net-hange of the system state (a similar idea as used in Ref. [27]). More speifially, in a monitoring window, we ompute the total flux as X f lux = r j ν j, j Ω, j 12

14 where r j is the total number that reation j fired, and reord Xi min and Xi max, the minimum and the maximum state reahed during the monitoring window, respetively. We then ompute the so-alled redundany oeffiient defined by, ζ = min i=1,...,n max(x f lux i,1) max(x max i X min i,1). (4.2) We take max(x f lux i,1) and max(xi max Xi min,1) to avoid having vanishing terms. We will assume that the fast reations have reahed their quasi-equilibrium if ζ 1, and not if ζ <1. In the algorithm, we keep a vetor alled boosting vetor κ = (κ 1,...,κ M ). The approximating system has rate onstant j = jκ j. If κ = 1, we just have the original system. If some κ j < 1, it means that in the boosted system reation j has been slowed down. Now we introdue two parameters ζ rit,1,ζ rit,2 related to ritial values of ζ. Above the value ζ rit,2 the reations will be onsidered fast enough to be slowed down. We implement the following strategy: if ζ < ζ rit,1 we reset the approximating system to the original one by setting κ = 1; if ζ > ζ rit,2 we boost the system by let κ j =.75κ j, j Ω. if ζ rit,1 ζ ζ rit,2, we just hold κ unhanged. The parameters ζ rit,1,ζ rit,2 are hosen in an empirial way and different values leads to different performane of the algorithm. Clearly more investigation is needed to set these values, whih is left as a future work. In the numerial experiments we set ζ rit,1 = 1,ζ rit,2 =3. Note that the boosting is done gradually based on the urrent system state. When boosting is turned on, we only multiply κ j by.75 for the fast reations. Keep in mind the rates of all the fast reations should be multiplied by the same onstant other wise the effetive dynamis will be altered. At the next monitoring window, we may keep reduing some κ j, hold the urrent κ unhanged, or even reset κ =1. The above adaptive proess an thus automatially handle systems with multiple sale separations hierarhially. For example, if a system onsists of three groups of reations with disparate rates, the algorithm will first slow down the fastest reations. Then, it will onsider the two fastest group of reations as long as quasi-equilibrium approximation is valid. The length of the monitoring window should be large enough to let the fast reations reah equilibrium, but not too large, otherwise both the effiieny and the auray of the system may be ompromised. We hoose t to be 1 times the minimum harateristi rate among all the fast reations: t =1/min j Ω A j. 13

15 4.2 The overall algorithm A. Initialization Set the initial moleule number x i = x i () (i = 1,...,N), the boosting oeffiient κ j = 1 (j = 1,...,M), the initial time t =, γ =, the leaping parameter ɛ =.1, the initial window length t =.2, and the fast reation set Ω to empty set. Choose q = 2 and a slow sale ω for the system. Invoke hybrid solver Subroutine B. B. Hybrid Solver (κ j, t, Ω as input) 1. Let t monitor = t+ t. 2. If γ, generate a uniformly distributed random variables u 1 and let γ=logu 1 <. 3. Compute propensity funtions a j and do the boosting a j = κ j a j (j =1,...,M). 4. For eah reation j, ompute its bottlenek-speie number z j (X)=min νij ={X i / ν ij }, j = 1,...,M. If ɛz j (X) 1, label it as ritial reation, otherwise oarse-grainable reation with oarse-grained reation rate A j (X(t)) = a j (X(t))/ɛz j (X(t)). 5. Choose step-size: first hoose the leaping step-size t leap =1/max(A j (X(t))), where j runs over all oarse-grainable reations. Then ompute the waiting time for the next ritial reation τ=γ/a Λ using a Λ = a j, where j runs over all ritial reations. The step-size is hosen as dt =min(t leap,τ). 6. Apply τ-leaping with step-size dt for all oarse-grainable reations. Let γ=γ+a Λ dt, update time t = t+dt. If t leap > τ, simulate one reation events among the ritial reations using SSA. 7. Invoke Subroutine D if exit onditions are met, otherwise ollet information: X max =max(x(t),x max ), X min =min(x(t),x min ), X f lux = X f lux + r j ν j,j Ω. j 8. If t > t monitor, invoke Subroutine C. Otherwise go bak to Step 2. C. Monitor 1. Compute the redundany oeffiient ζ =min i max(x f lux i,1) max(x max i 14 X min i,1).

16 s.s. ζ t new <5 t Ω new = Ω ation <1 κ j =1,j =1,...,M, keep the old t and Ω 1 keep the old κ j, t and Ω keep the old κ j for this and the next monitoring windows, use the old t, use Ω new keep the old κ j, use Ω new and t new >3 κ j =.75κ j,j Ω, use Ω new and t new [1,3] keep the old κ j, use Ω new and t new <1 κ j =1,j =1,...,M, use Ω new and t new If there exists κ j <1 and min j) <5max j) κ j <1 κ j =1 κ j =1,j =1,...,M, use Ω new and t new Table 1: Rules that deide whether to apply boosting or not in the boosted hybrid method. The symbol,, or indiates whether a speifi ondition is satisfied, not satisfied, or does not matter. s.s. is short for sale separation. ζ is the redundany oeffiient given by Eq. (4.2). t and t new is the time length of the previous and next monitor window, respetively. Ω and Ω new is the fast reations set in the previous and next monitor window, respetively. Under different onditions, we will take different ations, with the method of hoosing the new window length t, the boosting oeffiient κ and the fast reation table Ω for the next monitor window. Some rational of doing this is given in the appendix. 2. If there exists a non empty set Ω new suh that min A j >1 q max A j > ω, (4.3) j Ω new j Ω new then there exists a sale separation in reation rates between Ω new and Ω new and we define t new =1/ max A j. j Ω new 3. Set the window length t, the boosting oeffiient κ and the fast reation set Ω aording to the rules given by Table 1. Invoke Subroutine B. D. Exit 5 Numerial Examples In this setion we test the effiieny and the versatility of our method on five numerial examples. We will see that for the first three examples, our method is effiient and an handle numerially the hemial systems in robust and adaptive way. For the last two examples, the method shows limited appliability. The omparison of our method with other algorithms will be disussed in a future work. 15

17 5.1 System 1 This system is a toy model to test the method. It has five speies and seven reations. 1 1,2 : S 1 +L 1 S 2 +L ,4 : S 1 S : L 1 6 6,7 : L 1 L 2. 7 The initial ondition is L 1 = L 2 = 1,S 1 = 1,S 2 = S 3 =. The rate onstants are 1 = 1, 2 =1, 3 =1, 4 =.2, 5 =.5, 6 =1, 7 =1. The system is simulated in time [,1]. Fig. 3 shows the numerial results of the boosted hybrid method on System 1. When S 3 =, κ 1 and κ 2 are less than 1. This means that the algorithm identifies quasi-equilibrium of the fast reations 1 and 2, and applies boosting to them. More interestingly, as L 1,2 derease, this time-sale separation gets weaker and the algorithm automatially adjusts the boosting oeffiients κ. Eventually, the fast reations 1 and 2 are no longer fast enough to support any boosting. When this happens, κ 1 and κ 2 are equal to 1 (even though S 3 = ), whih demonstrates the adaptivity of the algorithm. Conerning the auray of the method, we an see from the histograms of L 1 and S 3 that the results of the boosted hybrid method approximate those of the SSA very well. Next, we make some modifiations to the system 1 by letting 3 =.2, 4 =.1, 6 = 1, 7 = 1, and simulate the modified system in the time interval [,4]. The remaining parameters are unhanged. The modified system ontains hierarhial sale separation in time, namely, reations 6, 7 are very fast, reations 1, 2 are fast, and the remaining ones are slow. One trajetory of the boosting oeffiient κ is given in Fig. 4. The algorithm first reognizes reations 6 and 7 as fast group and boosts the system by reduing κ 6 and κ 7. After the first round of boosting, reations 6, 7, 1, 2 are treated as fast group and the system an be further boosted. Note also that when t is lose to 15, L 1 drops below 1, and reations 6, 7 beome ritial and must be simulated using SSA. κ is set to 1 here when some reation hange from oarse-grainable to ritial type. However, ritial reations 6 and 7 are still fast enough for new boosting and κ 6 and κ 7 derease again. We an see from this example that the new algorithm is apable of treating system with hierarhial time sales whose hierarhy an hange over time. 5.2 System 2 This system is adapted from Ref. [28] and was originally proposed in Ref. [33], in whih it was used to model stohasti gene regulation. It has four reatant speies θ,θ 1,M,D 16

18 1.9 S SSA Boosted Hybrid.8 S relative frequeny t (a) L 1 (b) SSA BtHyMC.8.6 κ relative frequeny t () 1 S 3 (d) Figure 3: Numerial results for System 1. (a) A trajetory of S 3 obtained by the boosted hybrid method (when S 3 = the reations 1 and 2 are fast); (b) Histogram of L 1 sampled from SSA (irle, solid line) and the boosted hybrid method (star, dashed line) at final time, with sample size 1 6 ; () Evolution of κ 1 and κ 2 in the boosted hybrid method (the two are idential in the figure), the other omponents of κ are all equal to 1; (d) Histogram of S 3 sampled from SSA and the boosted hybrid method at final time, with sample size

19 κ 1 κ 2 κ 6 κ κ t Figure 4: One sampled trajetory of κ for the modified system 1. κ 1 (solid line) and κ 6 (dashed line) oinide with κ 2 and κ 7. The other entries of κ equal to 1 all the time (log sale for the y axis). and seven reations. 1,2 : 2M 1 D 2 3,4 : θ +D 3 θ : θ θ +M 6 6 : θ 1 θ1 +M 7 : M 7. Initially we have θ = 1,θ 1 =,M = 5,D = 1. The rate onstants are 1 = 1, 2 = 1, 3 =.2, 4 = 1.5, 5 = 5, 6 = 1, 7 = 1. Fig. 5 shows a trajetory of M and κ 1 in time interval [2,4]. In this system, M appears to be have a bistable pattern. When M is relatively large, reations 1, 2 are very fast, and hene κ 1 and κ 2 are very small. When M is small, reations 1and 2 are not as fast as before, but still fast enough to be boosted (with a larger κ). Also note that M and D sometimes get very unstable, for example around t = 334, at this time the approximating system is reset to the original system by the algorithm. This is exatly what we need, beause suh transition period are usually very important and interesting and quasi-equilibrium approximation is not valid there. In order to ompare the auray of the boosted hybrid method with SSA, we first obtain ensemble histograms for speie M using 1 5 samples at time t = 1 ( (b) in Fig. 5). We also simulate a trajetory up-to time 2, and draw a sample of speie M at eah integer value of time starting from t = 1, from whih we get the time histograms ( (d) in Fig. 5). It shows the result of the boosted hybrid method mathes with SSA quite well. To test effiieny, we let 1 = 1k, 2 = 1k, with k =.1,.1,.1,1 to see the performane of the boosted hybrid method for varying time-sale separation (the larger the k is, the larger the time-sale separation will be in this system). The time taken for eah simulation of one trajetory during [,2] using SSA and boosted hybrid method, respetively, is listed in Table 2. SSA beomes dramatially slow as k inreases, but the boosted hybrid method is still fast beause of its adaptivity. 18

20 Table 2: Simulation time using SSA and boosted hybrid for system 2 with different k. As k inreases, the system beomes more stiff. k SSA boosted hybrid.1 22s 5s.1 147s 3s s 35s s 44s SSA Boosted Hybrid moleular number of M relative frequeny t (a) M (b) 1.9 κ 1.35 SSA Boosted Hybrid κ relative frequeny () t M (d) Figure 5: Numerial results for System 2. (a) A typial trajetory of M obtained by the boosted hybrid method; (b) Histograms of M at time t=1 obtained by using 1 5 samples with the boosted hybrid method (star, dashed line) and SSA (irle, solid line), respetively. () A typial trajetory of κ 1 in the boosted hybrid method. κ 2 is idential with κ 1 and others are equal to 1; (d) Histograms of M over time [1,2] obtained by using the boosted hybrid method (star, dashed line) and SSA (irle, solid line), respetively. 19

21 5.3 System 3 This model is adapted from Ref. [27]. There are seven speies and ten reations. 1 : D D+M+R 2 : M M+P 3 : M 4 : P 5 : D+R D 6 : D D+R 7 : P+P P2 8 : P2 P+P 9 : D+P2 Q 1 : Q D+P2. Here D and D represents the ativated and deativated states of a DNA moleule, respetively. M is the mrna, R is the RNA, P is a protein, and P 2 a protein dimmer. Initially we have D = 1,R = 3, the other speie populations are all zeros. The rate onstants are 1 =.78, 2 =.43, 3 =.39, 4 =.7, 5 =.38, 6 =3, 7 =.5, 8 =5, 9 =.12, 1 =9. The simulation time is [,5]. In Ref. [27], reations 5, 6, 9, 1 are approximated by slow-sale SSA. Reations 7, 8 are first solved by expliit τ-leaping then impliit τ-leaping when the system beomes more stiff. For this system, we hoose different sale-separation threshold q = 1,2 (see Eq.(4.1)). q = 1 means there must be at least one order of magnitude sale-separation between the fast and slow reation rates to try boosting, and q = 2 means two orders of magnitude sale-separation. So q = 1 is a more aggressive boosting strategy. Fig. 6 (a) and (b) show a typial trajetory of κ using different q. () and (d) show the histograms sampled from 1 5 trajetories simulated by using SSA and the boosted hybrid method to t = 5. We an see by hoosing more aggressive boosting strategy, q = 1, κ is smaller and hene the algorithm is faster (about.7 seond for one trajetory for q = 1, 16.2 seonds for q = 2). But doing so also inreases numerial errors as shown in (d). We also modified other parameters in the algorithm, suh as ɛ that ontrols the oarsegrained time-step. We hange ɛ from.1 to.5. So now ritial reation is more frequent and the leaping step-size is smaller. The algorithm will be muh slower but no appreiable improvement in auray is observed for this system (results are not shown). 5.4 System 4 This system desribes the heat shok response of the E. Coli bateria [34, 35]. It onsists of 28 speies and 61 reations as given in the Appendix B. 2

22 1 κ κ 8 κ κ 6 κ κ 1.7 κ 7 κ 8 κ 5 κ 6 κ 9 κ κ κ t (a) x t (b) x SSA Boosted Hybrid.14 SSA Boosted Hybrid relative frequeny relative frequeny P 2 () P 2 (d) Figure 6: Numerial results for system 3. (a) A typial trajetory of κ in the boosted hybrid method for q = 2. In this ase, there are always two elements of κ are idential; (b) A typial trajetory of κ in the boosted hybrid method for q = 1; () Histograms of P 2 obtained by using the boosted hybrid method (irle, solid line) and SSA (star, dashed line) at t=5, q=2, sample size 1 5 ; (d) Histograms of P 2 obtained by using the boosted hybrid method (irle, solid line) and SSA (star, dashed line) at t =5, q =1, sample size

23 1 1 4 κ 5.9 κ 6 κ κ κ 8 κ 46 κ 47 boosted average rate t (a) sorted reation index (b) Figure 7: Numerial results for system 4. (a) A typial trajetory of κ. The entries of κ not shown in the piture are equal to one; (b) Boosted average oarse-grain rates κ j A j (log sale in the y-axis). It shows that, after boosting, the rates still spread over a large interval without obvious sale-separation. The initial amount of speies are s 1 = s 2 = s 3 = s 4 =,s 5 = 1,s 6 = ,s 7 = 1324,s 8 = 8,s 9 = 16,s 1 = 3413,s 11 = 29,s 12 = 584,s 13 = 1,s 14 = 22,s 15 =,s 16 = 17144,s 17 = 915,s 18 = 228,s 19 =6,s 2 =596,s 21 =,s 22 =13,s 23 =3,s 24 =3,s 25 =7,s 26 =,s 27 =26,s 28 =. The numerial results are shown in Fig. 7. During the simulation, six omponents of κ are less than 1, but they are not very small (see Fig. 7 (a)). This means that the effet of boosting is limited. Moreover, even after boosting, Fig. 7 (b) shows that the rates A j still oupy quite a large sope, whih means the system is still very stiff. Sine here the sale separation is not large enough we an not boost the system further. For this system, the boosted hybrid method is even slower than SSA beause of the widely and ontinuously distributed reation rates. 5.5 System 5 Another limitation of the new method (and most of the urrent simulation method) is illustrated by the following simple example, 1 : A 2 : A+E EA 3 : EA B+E 4 : B. A is a transription fator, E is DNA, EA is the protein-dna binding omplex, and B is protein. Usually the speies A and B have large populations and speie E is only one in moleular number. Assume the rate onstant of reations 2 and 3 are muh faster than reations 1 and 4. The overall effet is that A is onstantly onverted into B. In this system, 22

24 reations 1 and 4 are suitable for oarse-grained approximation while reations 2 and 3 are ritial. Moreover, the number of E will quikly reah to a stationary distribution. However, we an not apply boosting here beause no partial equilibrium holds for the fast reations 2 and 3 as there is a non-zero flux from A to B. A possible approximation to simulate the above system fast may be done by ombining the two fast ritial reations into one slow oarse-grained reations. This gives a modified system 1 : A 2 : A B, or B 3 : B. Note that the reation 2 in the above system does not exist in the original system. This amounts to modify the reation network itself, rather then treating different reations differently. One should be very areful when using the above approximation beause the model itself has been hanged. Some progress for handling this kind of problems are reported in Ref. [36] but they remain, in general, diffiult to simulate due to the diffiulty of dealing with these systems in a systemati way. 6 Disussion We proposed a new numerial method to simulate hemial reation systems with timesale separation and disparity in population speies. The method relies on boosting, whih is a strategy to derease the stiffness of a system in presene of time-sale separation. After stiffness redution, the system may still exhibit disparity in speies and we suggest the use of a hybrid method for its simulation in whih ertain reations are oarse-grained. We showed that both stiffness redution and oarse-graining approximation an be adapted in an automati way in time. This adaptivity is suitable in pratie. We notie that works remain to be done to optimize some parameters of the algorithm. Even so, by ombining boosting and hybrid methods it is possible to save substantial omputational ost for many omplex systems, when ompared to SSA. This has been illustrated numerially for various hemial systems. We showed how the main soure of error in the algorithm, the error from boosting and the error from oarse-graining, an be ontrolled by tuning the sale separation threshold at whih the boosting is turned on or by tuning the parameter threshold to set the number of reations hosen for oarsegraining. The new algorithm has many advantages. First, for hemial systems with multiple time sales, it an hierarhially slow down (boost) the system whih an then be numerially integrated by a single-sale solver. It avoids the use of multisale solver suh as the slow-sale SSA or nested SSA solver whih makes its oding easier for omplex systems. Seond, it does not need an a priori knowledge of the features of the hemial system (suh as reation rates and population size). Third, it allows for a systemati ontrol of 23

25 the main soures of the numerial errors oming from the boosting in time and the oarse graining proedure for speies with large population size. Numerial experiments show good performane for the long-time simulation of systems with multiple sales in time and population size. We found two situations in whih the boosted hybrid method is not very effiient. First, when the reation rates are spread over a large interval without obvious sale separation, boosting strategy is diffiult to be applied. In this situation we do not know of any other method apable of handling effiiently this kind of system. We note that thanks to the adaptivity in the boosted hybrid method, the algorithm remains robust (i.e. gives a reasonable approximation of the original system) even though not it is not effiient when ompared to SSA. The seond situation, illustrated in Se. 5.5, is onerned with a system where an effiient simulation requires to hange the original model. This remains a hallenging and interesting task for future work. Refinement and further appliations of the boosted hybrid method is also urrently under investigation. Aknowledgement Hu and Li are supported by the National Siene Foundation of China under grant The authors also thank the referees for their onstrutive omments and suggestions to improve the urrent paper. 7 Appendix 7.1 Some implementation details We ollet here a few details for a pratial implementation of the boosted hybrid algorithm. 1. Initially, we just let the fast reation set Ω to be empty, and pik up a small window length for monitoring, for example, t = After boosting, sine the fast reation rates tend to derease, the new fast reations set Ω new may beome empty. We do not reset κ to 1 here, but keep using the old Ω oming from the last step. But as soon as ζ <1 (see Setion 4.2), we rest κ to If min (A j) < 5max (A j) we reset κ to 1. This guarantees enough time-sale separation between boosted reations and non-boosted κ j <1 κ j =1 reations. 4. Some are has to be taken whether or not to aept the new set a fast reation Ω new. In our implementation, when t new given by the algorithm has hanged by magnitude from the old t, we will keep the urrent partition for the system and do another monitoring step with time-step t new to see if Ω new have reahed quasiequilibrium state. 24