Moment closure for biochemical networks

Transcription

1 Momet closure for biochemical etworks João Hespaha Departmet of Electrical a Computer Egieerig Uiversity of Califoria, Sata Barbara hespaha@ece.ucsb.eu Abstract Momet closure is a techique use to costruct systems of ifferetial equatios to approximately compute meas, staar eviatios, a correlatios betwee molecule couts of species ivolve i chemical reactios. These techiques are especially useful whe the umber of molecules exhibit large stochasticity, which is ot ucommo i bio-chemical reactios. We iscuss several approaches to momet closure that have bee propose i the literature a that have bee recetly implemete i a Matlab toolbox. I. INTRODUCTION TO MOMENT CLOSURE Cosier a set of chemical species X,X,...,X ivolve i a set of chemical reactios a let us eote by x := x,x,...,x a vector cotaiig their molecule couts. Give a vector of itegers m := m,m,...,m, we use the otatio µ m to eote the followig ucetere momet of x: µ m := Ex m xm xm. Such momet if sai to be of orer i m i. With species there are exactly first orer momets Ex i, i {,,...,}, which are just the meas; / seco orer momets Ex i, i a Ex ix j, i j, which ca be use to compute variaces a covariace; / thir orer momets; a so o. It was show i, that if we costruct a vector µ cotaiig all the ucetere momets of x up to some orer k, the evolutio of µ is etermie by a ifferetial equatio of the form µ = Aµ + B µ, µ R K, µ R K where A a B are appropriately efie matrices a µ is a vector cotaiig momets of orer larger tha k. The equatio is exact a we call it the exact k-orer momet yamics a the iteger k is calle the orer of trucatio. Note that the imesio K of is always larger tha k sice there are may momets of each orer. I fact, i geeral K is of orer k. Whe all chemical reactios have oly oe reactat, the term B µ oes ot appear i a we say that the exact momet yamics are close. However, whe at least oe chemical reactio has or more reactats, the the term B µ appears a we say that the momet yamics are ope sice This research was supporte by the NSF. Whe oe oes ot iclue i µ the zero-orer momet µ =, this term will appear i µ. epes o the momets i µ, which are ot part of the state µ. Whe all chemical reactios are elemetary i.e., with at most reactats, the all momets i µ are exactly of orer k+. Momet closure is a proceure by which oe approximates the exact but ope momet yamics by a approximate but ow close equatio of the form ν = Aν + Bϕν, ν R K where ϕν is a colum vector that approximates the momets i µ. The fuctio ϕν is calle the momet closure fuctio a is calle the approximate kth orer momet yamics. The goal of ay momet closure metho is to costruct ϕν so that the solutio ν to is close to the solutio µ to. Some momet closure methos approximate the exact momets yamics by a close equatio of larger orer, such as i φ = ψφ, φ R N, a ν = Aν + Bϕφ,ν, ν R K, b where oe ow approximates µ by the fuctio ϕφ, ν that is allowe to also epe o the state φ of a aitioal yamic system. Ofte ϕφ,ν ca be mae liear i ν. I this case, oce φ reaches a steay state, the ν yamics became liear a time-ivariat. There are three mai approaches to costruct the momet closure fuctio ϕ : Matchig-base methos irectly attempt to match the solutios to a or. Distributio-base methos costruct ϕ by makig reasoable assumptios o the statistical istributio of the molecule couts vector x. Large volume methos costruct ϕ by assumig that reactios take place o a large volume. It is importat to emphasize that this classificatio is about methos to costruct momet closure. It turs out that sometimes ifferet methos lea to the same momet closure fuctio ϕ. II. METHODS FOR MOMENT CLOSURE I this sectio we iscuss several methos to costruct the momet closure fuctio. We shall see that the choice of which metho to use epes o the type of system

2 istributios have low variability i.e., low staar eviatios whe compare to the mea or are fairly symmetric TABLE I WHICH MOMENT CLOSURE TO USE? istributios have large staar eviatios whe compare to the mea, but populatios o ot become zero with high probability populatios ca become zero with high probability accuracy zero cumulat closure erivative matchig closure o goo solutio yet simple yamics quasi-etermiistic closure or a Kampe s erivative matchig closure but will ot be very o goo solutio yet liear oise approximatio simple e.g., how populatio meas compare with staar eviatios for the system cosiere a also o the primary goal i costructig the approximate momet yamics e.g., how importat is accuracy versus simplicity of the equatios. Table I summarizes some rules of thumb o the choice of which approximatio to use. A. Derivative Matchig Derivative matchig is a matchig-base metho for momet closure escribe i. It uses momet closure fuctios ϕ i whose etries are separable, i.e., of the form ν γ νγ νγ. The coefficiets γ i R are the compute to make the relative error l ν l µ t l t l l µ t l as small as possible for molecule couts larger tha oe. Somewhat surprisigly, this miimizatio leas to explicit formulas for the momet closure fuctios ϕ that o ot epe o the reactio parameters. B. Zero Cumulats Zero cumulats is a istributio-base metho for momet closure that fis the kth orer momet closure fuctio ϕ i by assumig that all multi-variable cumulats of the populatio x with orer larger tha k are egligible. This makes the istributio of x as close as possible to a Gaussia istributio, which has all cumulats of orer higher tha two equal to zero. To costruct zero cumulat closures, oe uses the fact that the cumulat κ m ca be expresse as κ m = µ m + α m µ m, m i < m i where the summatio is over momets µ m of orer strictly smaller tha m i a the α m are appropriately selecte costats. This shows that the cumulat κ m epes oly o the momet µ m a lower-orer momets µ m, so settig κ m = oe obtais a expressio for µ m as a fuctio of lower-orer momets. The proceure to compute the zero-cumulats momet closure fuctio ϕ cosists of settig to zero all cumulats correspoig to the momets that o ot appear i µ a the solvig the equatios for the momets i µ. C. Low Dispersio Low ispersio is a istributio-base metho for momet closure that fis the momet closure fuctio ϕ i by assumig that the istributios of the populatios are tightly clustere arou their meas, with staar eviatios much smaller tha the meas. Specifically, for the kth orer momet closure oe assumes that the ormalize cetere momets of orer larger tha k are much smaller tha oe. We recall that give a vector of itegers m := m,m,...,m, the correspoig ormalize cetere momet is efie by η m := E x Ex Ex m x Ex Ex m. Such momet if sai to be of orer i m i. For fairly symmetric istributios the o-orer momets ca be quite small a therefore this techique is especially useful for eve-orer momet closures for which the o-orer higher momets ca be safely eglecte. To costruct low ispersio closures, oe uses the fact that a ucetere momet µ m ca be expresse i terms of the ormalize cetere momet as follows µ m = Ex m Ex m Ex m +η m + β m η m, m i < m i where the summatio is over momets η m of orer two or larger a strictly smaller tha m i, a the β m are appropriately selecte oegative costats. Whe a particular ormalize cetere momet η m is much smaller tha oe, we have that µ m Ex m Ex m Ex m + β m η m, m i < m i which allows oe to express the ucetere momet µ m solely i terms of ormalize cetere momets η m of orer strictly smaller tha m i. O the other ha, we ca express all these ormalize cetere momets as liear combiatios of the ucetere momets of orer strictly smaller tha m i as follows µ m η m = Ex m Ex m Ex m + γ µ m m m i < m i Ex m Ex m Ex m,

3 where the summatio is over ucetere momets µ m of orer strictly smaller tha m i a the γ m are appropriately selecte costats. The proceure to compute the low ispersios momet closure fuctio ϕ i thus cosists of usig a to approximate ay momet that oes ot appear i µ as a liear combiatio of the momets i µ. Note however that the coefficiets of these liear combiatios will epe o moomials of the form Ex ˆm Ex ˆm Ex ˆm, with all the ˆm i a therefore the momet closure fuctio φ will be polyomial but oliear o µ. Relatioship with zero-cumulats closure: For seco orer momet closure k = oe sets to zero th-orer ormalize cetere momets, which is equivalet to settig to zero the th-orer cumulats. Therefore for -orer closures, zero cumulat a low ispersio coicie. D. Quasi Determiistic Quasi-etermiistic is a istributio-base metho for momet closure that fis the momet closure fuctio ϕ i by assumig that the istributios of the populatios are tightly clustere arou the solutio φ to the etermiistic yamics φ = A et φ + B et ψφ, φ := φ,φ,...,φ R, 7 which are obtaie by assumig that each φ i := x i is etermiistic a therefore Eφ i φ j = Eφ i Eφ j = φ i φ j. Specifically, for the kth orer momet closure oe assumes that the quasi-etermiistic ormalize cetere momets of orer larger tha k are much smaller tha oe. Give a vector of itegers m := m,m,...,m, the correspoig quasietermiistic ormalize cetere momet is efie by ˆη m := E x φ φ m x φ φ Such momet if sai to be of orer i m i. m x φ φ m. To costruct quasi-etermiistic closures, oe uses the fact that a ucetere momet µ m ca be expresse i terms of the quasi-etermiistic ormalize cetere momet as follows µ m = φ m φ m φ m + ˆη m + β m ˆη m, m i < m i where the summatio is over momets η m of orer oe or larger a strictly smaller tha m i, a the β m are appropriately selecte oegative costats. Whe a particular quasi-etermiistic ormalize cetere momet ˆη m is much smaller tha oe, we have that µ m φ m φ m φ m + m i < m i β m ˆη m, which allows oe to express the ucetere momet µ m solely i terms of quasi-etermiistic ormalize cetere momets ˆη m of orer strictly smaller tha m i. O the other ha, we ca express all these quasi-etermiistic ormalize cetere momets as liear combiatios of the ucetere momets of orer strictly smaller tha m i as follows ˆη m = µ m φ m φ m φ m + m i < m i γ m µ m φ m φ m φ m, 9 where the summatio is over ucetere momets µ m of orer strictly smaller tha m i a the γ m are appropriately selecte costats. The proceure to compute the quasi-etermiistic momet closure fuctio ϕ i thus cosists of usig a to approximate ay momet that oes ot appear i µ as a liear combiatio of the momets i µ. The coefficiets of these liear combiatios will epe o moomials of the form φ ˆm φ ˆm φ ˆm, with all the ˆm i a therefore the momet closure fuctio will be liear o µ for a fixe φ. This meas that the approximate yamics i are of the form φ = A et φ + B et ψφ, φ R, a ν = A ν φν + c ν φ, ν R K, b a, whe φ reaches a steay state value, the ν yamics become liear. Relatioship with low ispersio closure: I geeral the ormalize cetere momet are smaller tha their quasi-etermiistic versio a therefore wheever quasietermiistic momet closure provies a goo approximatio, oe shoul expect low-ispersio momet closure to o at least as well. However, quasi-etermiistic momet closure has the avatage that it results i momet yamics that are almost liear a therefore geerally easier to aalyze. E. a Kampe s Liear Noise Approximatio a Kampe s Liear Noise Approximatio is evelope i, Chapter X a ca be applie whe the matrices A,B i epe o some parameter that ca be assume large, i.e., whe we have µ = Aµ + B µ, µ R K, with large. This form of momet closure results i a system of the form a is exact i the limit as. Typically, is the volume o which the chemical reactios take place. To costruct, oe starts by choosig φ to satisfy the etermiistic large-volume yamics φ = A et φ + B et ψ φ, φ := φ, φ,..., φ R, which are obtaie by assumig that each φ i := x i / is etermiistic a therefore a also by makig. E φ i φ j = E φ i E φ j = φ i φ j Regarig the vector φ i as a etermiistic approximatio to the stochastic vector x/, motivates efiig the

4 followig stochastic perturbatio vector χ := χ, χ,..., χ, with χ i := x i φ i x i = φ i + χi, where the ormalizatio by will be eee to keep the momets of χ boue as. Give a vector of itegers m := m,m,...,m, we use the otatio ξ m to eote the followig ucetere momet of χ: ξ m := Eχ m χm χ m. The momets of x a χ are relate by µ m = E φ + χ m φ + χ m = i m i α m ξ m m i m i m where the summatio is over momets ξ m of orer up to m i a the α m are appropriately selecte costats. Computig the exact momet yamics for ξ, oe obtais ξ = A ξ, φξ + B ξ, φ ξ, ξ R K, where ξ a ξ cotai the momets of χ correspoig to the momets of x i µ a µ, respectively. For elemetary reactios with reactio rates that epe o the volume as follows: / X rate =c rate =c XX X X +Y rate =cx rate =c XY the ope system coverges as to a close system of the form ξ = A ξ, φξ + B ξ, φ ξ A ξ, φξ, ξ R K. Sice the momets µ a ξ are relate through, oe ca use to obtai a close equatio for µ as i. Moreover, this equatio will be liear i µ, leaig to approximate yamics similar to. Relatioship with quasi-etermiistic closure: The etermiistic equatios 7 a iffer by two facts: i the state i was ormalize through a ivisio by the volume, a ii i we took the limit as. For elemetary reactios with molecule couts much larger tha oe, takig the limit as has almost o effect a we essetially have φ = φ. I this case, ξ m φ m φ m φ m ˆη m. i m i I view of this, settig a quasi-etermiistic momet ˆη m to zero is equivalet to settig to zero the correspoig ucetere momet ξ m of χ. This meas that we ca view the quasi-etermiistic closure as takig the a Kampe equatios a a simply settig ξ i to zero, without igorig other terms that woul also isappear as. Sice we are keepig more terms of the exact equatios, with quasi-etermiistic closure oe ofte obtais more accurate results the with a Kampe s liear oise approximatio. III. ST O C HDY NTO O L S TOOLBOX I this sectio we illustrate how to use the StochDyTools Matlab toolbox to compute ifferet momet closure yamics for the etwork of chemical reactios cosiere i, p. : A αφ A X, X γxx / Y, Y βy B, where the populatio of A is assume costat with a cocetratio φ A, is the volume o which the reactios take place, a the expressios above the arrows i correspo to the rates at which the ifferet reactios take place. With some abuse of otatio we will use the symbols X a Y to eote both the ames of the species a their molecular couts. Withi StochDyTools, this etwork of chemical reactios is escribe by the followig.et file: species: X stochastic; % umber of X molecules Y stochastic; % umber of Y molecules parameters: = ; % volume phia "\phi_a"= ; % cocetratio of A fixe al "\alpha" = ; be "\beta" = ; ga "\gamma" = ; reactios: rate = al*phia*; {X} >{X+}; % A > X rate = ga*x*x-/; {X,Y}>{X-,Y+};%X > Y rate = be*y; {Y} >{Y-}; % Y > B Proviig a etaile iscussio of.et files sytax is beyo the scope of this paper a the reaer is referre to for etails. Assumig that the above file is calle Kp.et, the exact -orer momet yamics for this system ca be compute usig the followig Matlab commas: et=reanet Kp.et ; my=mometdyamicset,; from which oe obtais after coversio to LATEX: t EX γ γ EX EY γ γ EX β EY = αφ A γ γ EX EXY γ EY αφ A γ β+ γ EXY γ γ β γ β EY αφ A + αφ A γ γ γ γ EX EX Y The ifferet momet closure approximatios coul the be obtaie usig the followig Matlab commas: my_m =mometclosureet,my, erivativematchig ; my_zc =mometclosureet,my, zerocumulats ; my_l =mometclosureet,my, lowdispersio ; my_q =mometclosureet,my, quasidetermiistic ; my_la=mometclosureet,my,{ akampe, };.

5 From these commas we woul obtai the ifferet approximate momet yamics liste below. Derivative matchig: EX EX Y EX EX Y EX EX EX EXY EX EY Zero cumulats a low ispersio: Quasi etermiistic: t φx φ Y = EX EX Y γφ X EX EX EX EX EY+EXEXY EX EY γφ X +αφ A γφ X β φ Y + γφ X φ X φ X EX+φ X EX φ X φ Y +φ Y EX φ X φ Y EX φ X EY+φ X EXY a Kampe s liear oise approximatio: φx t φ Y = γφx +αφ A βφ Y +γφ X t EX EY EX EXY EY γφ X γφ X β γ φ X +αφ A γφ X γ φ X γ φ X +αφ A γφ X β γφ X γ φ X γφ X β + γ φ X +αφ A γ φ X γ φ X +αφ A γ φ X γ φ X +β φ Y EX EY EX EXY EY Figure compares the accuracy of the ifferet momet closure methos for a low volume = a a high volume =. For the larger volume all momet closure techiques provie a very goo match with Mote Carlo results, but for the smaller volume erivative matchig prouces the most accurate results eve with oly a seco orer trucatio. These results are fairly typical. ACKNOWLEDGMENT The author woul like to thaks Mustafa Khammash a Abhyuai Sigh for several iscussios that motivate this paper. REFERENCES Y. Cao, A. Hall, H. Li, a S. Lampoui. User s Guie for STOCHKIT. Uiversity of Califoria, Sata Barbara. Available at cse/stochkit. C. A. Gomez-Uribe a G. C. erghese. Mass fluctuatio kietics: Capturig stochastic effects i systems of chemical reactios through couple mea-variace computatios. J. of Chemical Physics, :9 9, 7. J. P. Hespaha. StochDyTools a MATLAB toolbox to compute momet yamics for stochastic etworks of bio-chemical reactios. Available at hespaha/software, Dec.. A. Sigh a J. P. Hespaha. Logormal momet closures for biochemical reactios. I Proc. of the th Cof. o Decisio a Cotr., Dec.. N. G. a Kampe. Stochastic Processes i Physics a Chemistry. Elsevier Sciece,.

6 erivative matchig X MC X.9±. m, X.±.7 m, zero cumulats X.9 zc, X.7±.97 zc, X.±. zc, low ispersio X.9 l, X.7±.97 l, X.7±.9 l, quasi etermiistic X.9 q, X.±.9 q, X.±.99 q, a Kampe X. la, X.±.7 la, X.±.7 la, X.97±. MC, erivative matchig zero cumulats low ispersio quasi etermiistic a Kampe Y.±.9 MC, Y MC Y.9±.9 m, Y.97±.9 m, Y. zc, Y.±.9 zc, Y.9±.9 zc, Y. l, Y.±.9 l, Y.±.9 l, Y. q, Y.9±.9 q, Y.9±.9 q, Y. la, Y.±.9 la, Y.±.9 la, a volume = erivative matchig zero cumulats low ispersio quasi etermiistic a Kampe X MC X.±.9 m, X.±.7 m, X.7 zc, X.±.7 zc, X.±.79 zc, X.7 q, X.9±. q, X.±. q, X.7 l, X.±.7 l, X.±. l, X. la, X.±.7 la, X.±.7 la, X.±.7 MC, erivative matchig Y MC Y.±. m, Y.±. m, zero cumulats Y.7 zc, Y.±. zc, Y.±. zc, low ispersio Y.7 l, Y.±. l, Y.±. l, quasi etermiistic Y.7 q, Y.±. q, Y.±. q, a Kampe Y. la, Y.±. la, Y.±. la, Y.7±. MC, b volume = Fig.. Compariso of accuracy betwee ifferet momet closure methos for the example escribe i Sectio III with two ifferet volumes. The leges show i the values of the mea ± oe staar eviatio at the fial time ii a two-character strig iicatig the momet closure metho, a iii a iteger iicatig the orer of trucatio. The istributios, meas a staar eviatios i the right-most plots were obtaie usig, Mote Carlo simulatios prouce by. The left-most plots iclue a typical Mote Carlo ru.