Stochastic Simulations of Cellular Biological Processes

Transcription

1 Proceedngs of Department of Defense Hgh Performance Computng Modernzaton Program Users Group Conference 2007, June 18-22, 2007, Pttsburgh, PA Stochastc Smulatons of Cellular Bologcal Processes Yaroslav Chushak Botechnology HPC Software Applcatons Insttute, HPCMP, Wrght-Patterson AFB, OH Brent Foy Wrght State Unversty, Dayton, OH John Frazer Ar Force Research Laboratory, Wrght-Patterson AFB, OH Introducton From a systems engneerng pont of vew, cells consst of a complex set of nested, nonlnear control systems domnated by feed-back and feed-forward loops. The more complex the system s, the more mportant are the ssues concernng the robustness and parameter optmzaton, therefore modelng and smulatons are mportant for both engneerng and reverse-engneerng of bosystems. Obectve At the functonal level, all bologcal processes n cells can be represented as a seres of bochemcal reactons. Snce such reactons are stochastc n nature, the user must run thousands of smulatons to characterze the ensemble behavor of bologcal systems. We developed a software package called Bomolecular Network Smulator (BNS) to model and smulate complex bomolecular reacton networks usng Hgh Performance Computng (HPC). Methodology The Bomolecular Network Smulator uses the Gllespe stochastc algorthm to smulate the evoluton of a system of bochemcal reactons. The BNS code s a combnaton of MATLAB and C-coded modules. Ths combnaton allows one to use the nteractve features and vsualzaton tools of MATLAB, whle achevng hgh speed for the computatonally ntensve part of the software. The software s parallelzed wth the MPI lbrary to run on multprocessor archtectures. Result The Bomolecular Network Smulator conssts of two sets of tools: for smulatons of the system and for the analyss of smulaton results. The Graphcal User Interface of BNS allows users to easly set parameters for the model and smulatons and to select analyss method. Multple types of post-smulaton analyses are avalable. Sgnfcance to DoD The Developed software allows DoD scentsts to buld, smulate and analyze complex cellular bomolecular networks utlzng the capactes of HPC. It provdes the foundatonal capablty to desgn and ntegrate bologcal constructs nto a new generaton of botechnology products.

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 1204, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subect to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Stochastc Smulatons of Cellular Bologcal Processes 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Ar Force Research Laboratory,Wrght Patterson AFB,OH, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES Proceedngs of Department of Defense Hgh Performance Computng Modernzaton Program Users Group Conference 2007, June 18-22, 2007, Pttsburgh, PA 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed Same as Report (SAR) 18. NUMBER OF PAGES 9 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

3 1 INTRODUCTION All chemcal reactons requre the nteracton of the reactants wth suffcent energy to overcome the actvaton energy barrer and, fundamentally, they are stochastc n nature. The best way to model knetcs of a system of chemcal reactons s to use a stochastc approach n terms of the Chemcal Master Equaton (CME), wth the number of molecules of each molecular speces as varables. The CME descrbes transtons of the system from one state to another state based on probablstc methods. Gllespe proposed a method to smulate probablstcally-correct traectores based on the CME through the use of Monte Carlo methods [1]. Although the Gllespe stochastc algorthm s a method for exact smulatons, as t explctly counts each reacton event that occurred n the system, the accuracy of the method comes at a hgh computatonal cost. Ths s especally true for systems wth a large number of molecular speces where reactons occur numerous tmes n a short perod of tme. To accelerate dscrete stochastc smulaton, Gllespe proposed the tau-leapng method as an approxmate smulaton strategy [2]. By usng Posson random numbers, the tau-leapng method can leap over many reactons wthout a sgnfcant loss of accuracy. We developed a software package the Bomolecular Network Smulator (BNS) that can use the Gllespe stochastc algorthm or the tau-leapng algorthm to smulate the behavor of a system of bochemcal reactons. It allows scentsts to buld a synthetc bomolecular network and explore ts performance utlzng the capactes of Hgh Performance Computng. BNS contans tools for both smulatng the system and for analyzng the results of the smulaton. Snce the smulatons are stochastc n nature, the user must run thousands of smulatons to characterze the ensemble behavor of bologcal systems. The parallelzed BNS code allows users to run multple smulatons and store results on multprocessor platforms. In ths paper, we present a bref descrpton of the Bomolecular Network Smulator software along wth some examples. 2 STOCHASTIC SIMULATION ALGORITHM Let us consder a system composed of N well mxed chemcal speces, S ( = 1, N), n a fxed volume V, whch are nvolved n M reactons, R ( = 1, M). The dynamcal state of the system can be specfed by the state vector X(t) = (X 1 (t), X 2 (t), X N (t)), where X (t) s the number of molecules of speces S at tme t. The probablty for each reacton R s defned by a propensty functon a (X)= c H (X), where c s the stochastc probablty constant and H (X) represents the number of possble combnatons of reactants. The probablty that reacton R wll occur n the tme nterval dt s defned as a (X)dt. Each reacton s also characterzed by ts state-change vector v = (v 1, v 2, v N ), where v s the change n the number of molecules S caused by one R reacton. To study the evoluton of the state vector X(t), Gllespe proposed an algorthm for Monte Carlo generaton of stochastc traectores [1]. The drect smulaton algorthm, whch s mplemented n the Bomolecular Network Smulator, answers two questons: (1) whch reacton wll occur next, and (2) what s the watng tme for the next reacton to occur. To answer these questons, two random numbers unformly dstrbuted over the nterval (0,1) r 1 and r 2 are generated. The frst random number s used to determne the next reacton R, such that 1 = 1 a < r 1 a 0 < = 1 a, (1)

4 where M a = 1 a0 =. (2) The dstrbuton of the watng tme s gven by followng probablty densty functon: P( τ, ) = a exp( a0τ ). (3) Here, P( τ, ) s the probablty that the watng tme for the reacton s τ and that t wll be an R reacton. The watng tme for the next reacton s calculated as [1] 1 1 τ = log. (4) a0 r2 After the next reacton and ts watng tme are determned, the reacton s executed and the state of the system s changed accordng to the state-change vector v. The smulaton tme s ncreased then by τ and the next smulaton step s generated. The Gllespe stochastc algorthm s exact for the elementary reactons (un-un, un-b and b-un types of reactons) wth any number of molecules n the system. For the system wth large numbers of molecules, the traectores generated by stochastc Monte Carlo smulatons converge to the traectory generated by determnstc dfferental equatons [1]. 3 TAU-LEAPING ALGORITHM The tau-leapng method calculates a tme nterval τ whch encompasses more than one reacton event and satsfes the Leap Condton,.e., the expected state change nduced by the leap must be suffcently small that no propensty functon changes ts value by a sgnfcant amount. Several methods have been proposed recently to choose the sze of the tme nterval for tau-leapng [3-5]. We mplemented the tauleapng method proposed by Cao et al. [6]. In that method a tau selecton formula s gven by 2 ' max{ εx / g,1} max{ εx / g,1} τ = mn, 2, (5) I rs µ ( x) σ ( x) where g s the hghest order of reacton n whch speces S appears as reactant, ε s an error control parameter, I rs s the set of ndces of all reactant speces, and µ ( x ) ν a ( x), (6) 2 = = 2 σ ( x ) ν a ( x). (7) After the tme nterval τ has been selected, the number of frngs of each reacton channel durng ths tme nterval s approxmated as a Posson random varable. The Posson random varable can have arbtrary large sample value and t s possble that the populaton of some of the molecular speces can run negatve. Therefore, a crtcal number of molecules n c (typcally n the range of 5-20) was ntroduced. If the number of molecular speces gets less then n c, all reacton n whch that speces appears as reactant are defned as crtcal. These reactons are smulated by the stochastc smulaton algorthm. We c calculate the sum of propensty functons of all the crtcal reactons a 0 and generate a second canddate tme τ accordng to '' 1 1 τ = log. (8) r a c o

5 The actual tme leap τ s selected as the smaller of τ and τ. If τ = τ, a number of frngs k s generated as a Posson random varable wth mean a (x) τ for all of the noncrtcal reacton and reactons are executed; no crtcal reacton s executed n ths case. If τ=τ, we generate k and fre noncrtcal reactons as n the prevous case; also, another random number wth unform dstrbuton s generated to fnd whch crtcal reacton needs to be fred accordng to Eq. (1). Evaluatons of the performance of the tau-leapng algorthm shows a 2-3 fold speed up of smulatons compared to the exact stochastc algorthm whle mantanng excellent accuracy wth regard to both rare and frequent events. 4 BIOMOLECULAR NETWORK SIMULATOR The Bomolecular Network Smulator uses a combnaton of MATLAB and C-coded modules. The front-end, graphcal user nterface (GUI) and analyss tools of BNS are wrtten n MATLAB, whle the smulaton core engne s wrtten n C. Such a combnaton allows one to use the nteractve features and vsualzaton tools of MATLAB, whle achevng hgh speed for the computatonally ntensve part of the software. The parallelzaton of the code s done wth the help of the MPI lbrary. The BNS can be run on any computer platform where MATLAB 6.5 or newer s nstalled. 4.1 Input Data A model s a set of mathematcal relatonshps that descrbe the behavor of bochemcal reactons that control cellular bologcal processes. Each of the Model drectores contans one or more subdrectores wth model descrpton fles and/or dfferent sets of parameters for the same model and an Output drectory where the results of smulatons are stored. There are two types of model drectores: one for models defned n the Systems Bology Markup Language (SBML) format [5] and one for models defned as a set of MATLAB m-fles, referred to as the BNS format. In addton, BNS allows one to perform smulatons wth multple parameter sets, wth each parameter set beng run multple tmes. Smulatons wth multple parameter sets can be used for optmzaton and senstvty analyss of the model. 4.2 Output Data Fgure 1. Scalng of BNS wth the number of processors. There are two types of output fles: snapshot data and event log data. Snapshot data fles contan the state of the system (number of molecules of each molecular speces) at user specfed tme ntervals. The nformaton stored n the snapshot fles are used to create runtme nteractve graphcs and for post hoc

6 analyss of the data. The second type of output fles the event log fles contan the record of every dscrete event that occurs durng the smulaton. The user should be aware that event log fles may requre consderable memory or hard dsk space and, therefore, may create memory management problems for smulatons nvolvng a large number of long runs or for large reacton networks. 4.3 Parallelzaton The parallelzaton of the BNS code s accomplshed usng the MPI lbrary. In our parallelzaton scheme, the master processor dvdes the total number of user specfed runs between the avalable processors, sends a set of obs to each of the workers and performs some of the smulaton runs tself. In ths approach to parallelzaton, sometmes called embarrassngly parallel we reduce the communcaton between the nodes and ncrease the speedup of multple smulatons. To test BNS scalng wth the number of processors, we ran 1000 smulatons on an SGI Orgn 3900 machne at ASC MSRC, whch has shared memory archtecture. A smulaton here s defned as a sngle run of the mathematcal model of a partcular bochemcal reacton network. A 92-fold speedup was observed by runnng BNS on 100 processors (Fgure 1). 4.4 Runnng the Smulatons The BNS can be run ether n command lne mode or va a GUI. The GUI allows the user to modfy model parameters at runtme and to execute smulatons n the nteractve or batch mode on HPC resources. (a) (b) Fgure 2. The screen shots of the BNS GUI dalog wndows. (a) The man BNS dalog wndow. (b) The parameters dalog wndow whch allows users to modfy the model parameters and to set smulaton parameters. The man dalog wndow of the BNS GUI s shown n Fgure 2. It allows the user to select the approprate Model and Parameters drectores and set the Run mode. A clck on the Detals button

7 next to the Parameters drectory opens a new wndow, shown n Fgure 2(b). Ths dalog wndow allows the user to modfy model parameters and to set parameters for the smulaton. If smulatons are run n the nteractve mode, partal results of the smulatons wll appear on the screen durng the run. Usually, HPC centers allocate lmted resources (n number of processors and runnng tme) for nteractve smulatons, therefore BNS can be run n Batch mode. In ths mode all output data are stored on the hard drve for further analyss. 4.5 Analyss The Bomolecular Network Smulator has a comprehensve set of tools for post-smulaton analyss. A GUI for the analyss tools allows the user to easly select the data and to set condtons for the analyss. Multple types of post-smulaton analyses are avalable Plots of number of molecules vs. tme The most frequently used type of analyss s a plot of the number of molecules vs. tme. Such plots are avalable n the nteractve mode or as post-smulaton analyss. There are two ways to create plots: each compound s plotted on a separate fgure or multple compounds are plotted on the same fgure wndow (groupng mode). The number of molecules vs. tme plots can be created wth both types of output fles: snapshot data or event log data. Fgure 3. The averaged number of compounds S1 and S2 n the smulaton nterval (0, 2000) obtaned usng the exact Gllespe stochastc algorthm and approxmate tau-leapng algorthm (ε=0.1, n c =5).The tme-weghted average was calculated usng a 50-sec tme-averagng nterval and the ndvdual tmeweghted averages were averaged over the 200 smulaton runs. Data for the mean ± SD are shown Tme-weghted average analyss A tme-weghted average analyss refers to the calculatons of the average number of molecules of a partcular molecular speces durng a user selected tme-averagng nterval. The average s weghted accordng to the amount of tme the compound exsts n each state durng the selected tme-averagng nterval. The tme weghted average s then plotted versus tme. The averagng analyss can be performed for a

8 sngle run or for a selected set of runs. When the analyss s appled to multple runs, the plot shows the between run average (the average of each ndvdual tme-weghted average) and standard devaton. The graphs n Fgure 3 show the behavor of two molecular speces, S1 and S2, over the tme nterval of 2000 seconds for a bomolecular reacton network contanng transcrpton, translaton and metabolc reactons. The smulatons were carred out usng two dfferent algorthms mplemented n BNS: Gllespe stochastc algorthm and tau-leapng algorthm wth parameters ε = 0.1 and n c = 5. Fgure 3 shows the between run average of the tme-weghted average number of molecules for 200 runs usng a tme-averagng nterval of 50 sec. The average number of molecules of speces S1 changed n the range of ~0-20 molecules, whle the average number of molecules of speces S2 changed from 0 to ~2500 molecules. Results obtaned usng an approxmate tau-leapng algorthm are almost ndstngushable from the result of exact Gllespe algorthm for both types of speces. On the other hand, usng the tau-leapng algorthm speeds up smulatons more than 3-fold compared to the exact stochastc algorthm Reacton frequency analyss Complex bomolecular reacton networks usually contan reactons that occur on dfferent tme scales: some reactons have a low propensty and occur rarely; other reactons are very fast and occur frequently. The data stored n the event log fles allow the user to perform varous reacton frequency analyses of the smulaton data to learn more about the basc nature of the system. One type of analyss creates plots of the total number of tmes each reacton n the network occurred durng the smulaton. Fgure 4. The averaged number of reacton occurrences per tme-averagng nterval for the reacton channels rxn21 and rxn24 obtaned usng the exact Gllespe stochastc algorthm and approxmate tauleapng algorthm (ε = 0.1, n c = 5). The tme-averaged rate was calculated usng a 5 sec tme-averagng nterval and the ndvdual tme-averaged rates were averaged over the 200 smulaton runs. Data for the mean ± SD are shown. A second type of analyss s the average and standard devaton of the tme-averaged reacton frequency n each reacton channel. Fgure 4 shows an example of the tme-averaged reacton frequency for two reacton channels averaged over the 200 runs obtaned by runnng smulatons wth the two algorthms. The reacton rxn21 belongs to the group of slow reactons wth the frequency of occurrences n

9 the range of 0-20 frngs per sec. The reacton rxn24 s a fast reacton and reached occurrences per sec. As n the case of average number of molecules, the tau-leapng algorthm shows an excellent agreement wth the results from exact stochastc smulatons for ths partcular bomolecular reacton network. 5 CONCLUSIONS The Bomolecular Network Smulator allows the users to smulate the behavor of complex bologcal processes utlzng the capactes of hgh performance computers. Some of the features that dstngush BNS from smlar tools are: usage of MATLAB and C-coded functons allows the user to combne ntensve vsualzaton of data wth hgh speed computatons; parallelzed code for multple smultaneous smulatons allows the user to run BNS on multprocessor machnes; optons to run the code n the nteractve or batch mode; user frendly graphcal user nterface allows the user to easly set and modfy parameters of the model, smulaton and analyss; and comprehensve tool sets provde for post-smulaton analyss of results. ACKNOWLEDGMENT The work of Yaroslav Chushak was sponsored by the US Department of Defense Hgh Performance Computng Modernzaton Program (HPCMP), under the Hgh Performance Computng Software Applcatons Insttutes (HSAI) ntatve. The work of Brent Foy was made possble by a grant from the Ar Force Offce of Scentfc Research (AFOSR) and by the Ar Force Summer Faculty Fellowshp Program. DISCLAIMER The opnons or assertons contaned heren are the prvate vews of the authors and are not to be construed as offcal or as reflectng the vews of the US Army or the US Department of Defense. Ths paper has been approved for publc release; dstrbuton s unlmted. REFERENCES [1] Gllespe, D., Exact Stochastc Smulaton of Coupled Chemcal Reactons. J. Phys. Chem. v. 81, p (1977). [2] Gllespe, D., 2001, Approxmate accelerated stochastc smulatons of chemcally reactng systems. J. Chem. Phys. v. 115, p (2001). [3] Gllespe, D. and Petzold, L., Improved leap-sze selecton for accelerated stochastc smulatons. J. Chem. Phys. v. 119, p.8229 (2003). [4] Tan, T. and Burrage, K. Bnomal leap methods for smulaton chemcal knetcs. J. Chem. Phys. v. 121, p (2004). [5] Chatteree, A., Vlachos, D. and Katsoulaks, M. Bnomal dstrbuton based τ-leap accelerated stochastc smulaton. J. Chem. Phys. v. 122, (2005). [6] Cao, Y., Gllespe, D. and Petzold, L. Effcent step sze selecton for the tau-leapng smulaton method. J. Chem. Phys. v.124, (2006).

10 [5] Huska, M., Fnney, A., Sauro H. M., Bolour, H. et al., 2003, The Systems Bology Markup Language (SBML): A medum for representaton and exchange of bochemcal network models. Bonformatcs, v. 19, no. 4: