Modeling Chemical Reactions with Single Reactant Specie

Transcription

1 Modeling Cheical Reactions with Single Reactant Specie Abhyudai Singh and João edro Hespanha Abstract A procedure for constructing approxiate stochastic odels for cheical reactions involving a single reactant specie is presented This is done by representing the population of various species involved in a cheical reaction as the continuous state of a polynoial Stochastic Hybrid Syste (pshs) An iportant property of pshss is that the dynaics of all the statistical oents of its continuous states, evolves according to a infinite-diensional linear ordinary differential equation (ODE) Under appropriate conditions, this infinite-diensional ODE can be accurately approxiated by a finite-diensional nonlinear ODE, the state of which typically contains soe oents of interest In this paper we provide existence and uniqueness conditions for these finite-diensional nonlinear ODEs Furtherore, explicit forulas to construct the are also provided I INTRODUCTION The tie evolution of a spatially hoogeneous ixture of cheically reacting olecules is often odeled using a stochastic forulation, which takes into account the inherent randoness of theral olecular otion This forulation is superior to the traditional deterinistic forulation of cheical kinetics and is otivated by coplex reactions inside living cells, where sall populations of key reactants can set the stage for significant stochastic effects [1], [2], [3] Although ost of the atheatical odeling of genetic networks represents gene expression and regulation as deterinistic processes, recent observations of gene expression in individual cells illustrate the stochastic nature of transcription [4], [5] Furtherore, studies of engineered genetic circuits designed to act as toggle switches or oscillators have revealed large stochastic effects [2], [6] Stochasticity is therefore an inherent feature of biological dynaics and developing odels which capture this have becoe increasingly iportant In the stochastic forulation of cheical reactions, the tie evolution of the syste is described by a single equation for a grand probability function, where tie and species populations appear as independent variables, called the Master equation [7] However, this equation can only be solved for relatively few, highly idealized cases and generally Monte Carlo siulation techniques [8]-[9], are used Since one is often interested in only the first and This aterial is based upon work supported by the Institute for Collaborative Biotechnologies through grant DAAD19-3-D-4 fro the US Ary Research Office and by the National Science Foundation under Grant No CCR ASingh and JHespanha are with the Center for Control Engineering and Coputation University of California, Santa Barbara, CA 9311 abhi@engineeringucsbedu, hespanha@eceucsbedu second order oents for the nuber of olecules of the different species involved, uch effort can be saved by applying approxiate ethods to estiate the evolution of these low-order oents, without actually having to solve for the probability density function Various such approxiate ethods have been developed, for exaple, using the Fokker-lank approxiation or expanding the Master equation [7] In [1], an alternative approxiate ethod for estiating lower-order oents was introduced This was done by representing the dynaics of a cheical reaction as a Stochastic Hybrid Syste (SHS) whose state x, is the population of all the species involved in the reaction In order to fit the fraework of our proble, this class of SHS had trivial continuous dynaics ẋ, with reset aps and transitional intensities defined by the stoichioetry and the reaction rates of the reactions, respectively In essence, if no reaction takes place, the population of species reain constant and whenever the reaction takes place, the reset ap is activated and the population is reset according to the stoichioetry of the reaction Details for the stochastic odeling of cheical reactions are presented in Sec II It was also shown in [1] that these SHSs used to odel cheical reactions are actually polynoial Stochastic Hybrid Systes (pshs) An iportant property of pshss is that, if one creates an infinite vector containing all the statistical oents of x, the dynaics of this vector is governed by an infinite-diensional linear ordinary differential equation (ODE) which we call the infinite-diensional oent dynaics In general, the infinite-diensional oent dynaics cannot be solved analytically, however, as shown in Sec III, under appropriate conditions, they can be approxiated by a finite-diensional nonlinear ODE, which we call the truncated oent dynaics The state of the truncated oent dynaics µ, typically contains soe lower-order oents of interest A procedure to construct these truncated oent dynaics was outlined in [1] The procedure proposed in [1] is general but not systeatic Moreover, [1] provided no conditions under which the truncated oent dynaics could be found In this paper we resolve these issues for a general class of cheical reactions involving one specie More specifically, we show in Sec IV, that given a vector µ, containing the first n oents of x, one can find truncated oent dynaics, if one drops soe lower order oents fro the tie derivatives of µ These dropped oents are at least of

2 order n less than the highest order oent appearing in that tie derivative Thus when n is taken sufficiently large these lower order oents will be doinated by higher order ones and one can drop the without incurring too uch error We also give specific conditions on the stoichioetry of the reactions, for which, this truncated oent dynaics is unique Explicit forulas to construct the truncated oent dynaics are also provided The striking features of these forulas is that they are independent of the nuber of reactions, the stoichioetry of the reactions and the reaction rates To illustrate our results we consider a siple bi-olecular reaction for which the exact solution for the first and second order oents can be obtained fro the Master equation Truncated oent dynaics odel are built for n 2 and 3 and oent estiates obtained are copared with the exact solution II STOCHASTIC MODEING OF CHEMICAY REACTING SYSTEMS Consider the general set of eleentary reactions 1 involving a single reactant specie, which without loss of any generality can be expressed as follows: R 1 : 2X R : 2X c 1 v1 X, c v X, c 1 R 1 : X A 1 v 1 X, R : X A (1) c v X, R 1 : X c 1 v 1 X, R : X c v X, where,, N 1, species A 1,,A have a constant nuber of olecules a 1,,a respectively and represents products that are other than the specie X The reaction paraeter c i characterizes the reaction R i and, together with the stoichioetry, defines the probability that a particular reaction takes place in an infinitesial tie interval (t, t dt] This probability is given by the product of the following two ters: 1) the nuber h i of distinct olecular reactant cobinations present in V at tie t for the reaction R i, 2) the probability c i dt that a particular cobination of R i reactant olecules will actually react on (t,t dt] The nuber, h i depends both on the reactants stoichioetry in R i and on the nuber of reactant olecules in V Table I shows the value of h i for different reaction types [8] In this table and in the sequel, we denote by x(t) R, the nuber of olecules of the specie X in the volue V at tie t 1 Reactions involving two or less reactant olecules [11] TABE I h i (x) AND c i FOR DIFFERENT REACTION TYES Reaction R i X X A i h i (x) x xa i 2X 1 2 x(x 1) To odel the tie evolution of the nuber of olecules x, a special class of Stochastic Hybrid Systes (SHS) were introduced in [1] More specifically, to fit the fraework of our proble, these syste are characterized by trivial dynaics a faily of reset aps ẋ, (2) x φ i (x,t), φ i : R [, ) R, (3) and a corresponding faily of transition intensities λ i (x,t), λ i : R [, ) [, ) (4) for all i {1,,} Each of the reset aps φ i (x), and corresponding transition intensities λ i (x) are uniquely defined by the i th reaction and given by x φ 1 (x) x 2 v 1, λ 1 (x) c 1 x(x 1), 2 x φ (x) x 2 v, λ (x) c x(x 1), 2 x φ 1 (x) x 1 v 1, λ 1 (x) c 1 xa 1, (5) x φ (x) x 1 v, λ (x) c xa, x φ 1 (x) x 1 v 1, λ 1 (x) c 1 x, x φ (x) x 1 v, λ (x) c x, respectively In essence, if no reaction takes place, the state reains constant and whenever the i th reaction takes place, φ i (x) is activated and the state x is reset according to (5), furtherore, the probability of the activation taking place in an infinitesial tie interval (t,t dt] is λ i (x)dt III MOMENT DYNAMICS To fully characterize the dynaics of a cheical reaction one would like to deterine the evolution of the probability distribution for x(t) In general, this is difficult and a ore reasonable goal is to deterine the evolution of a few loworder oents Given N, we define the th order (uncentered) oent of x to be µ (t) E [ x(t) ], t (6)

3 The tie evolution of oents is given by the following result, which is a straightforward application of Theore 1 in [12] to the SHS (2)-(4) Theore 1: that where x R For every function ψ : R R we have de[ψ(x)] dt (ψ)(x) E[(ψ)(x)], (7) (ψ(φ i (x)) ψ(x))λ i (x) (8) The operator ψ ψ defined by (8) is called the extended generator of the SHS Since the reset aps φ i (x) and transitional intensities λ i (x) are finite polynoials in x, the extended generator (ψ)(x, t) is a finite-polynoial in x for every finitepolynoial ψ(x, t) in x Such SHSs whose extended generator is closed on the set of finite-polynoials in x are called polynoial Stochastic Hybrid Syste (pshs) With ψ(x) x in (7), we have fro (5), (6) and (8) that µ E[(ψ)(x)] (9) (ψ)(x) c i 2 x(x 1)[(x 2 v i) x ] i1 i1 c i a i x[(x 1 v i ) x ] c i x[(x 1 v i ) x ] (1) Doing a binoial expansion of (1) we have (ψ)(x) where C j c i 2 x(x 1) C j x j (v i 2) j i1 i1 c i a i x c i x C j x 2 j f ( j) C j x j (v i 1) j C j x j (v i 1) j is defined as follows 2 j, N C j 2 n! denotes the factorial of n! ( j)! j!, j,, < j, C j x 1 j g( j) (11) and j {1,,n 1} f ( j) g( j) 1 2 c i(v i 2) j, (12) i1 1 2 c i(v i 2) j i1 c i a i (v i 1) j c i (v i 1) j (13) As j only takes values greater than equal to one we define f (), g() Using (9) and (11), the evolution of µ can be written as µ 1 r1 C j µ 2 j f ( j) C j µ 1 j g( j) [C 2 r f (2 r) C 1 r g(1 r)]µ r (14) One can see fro the right-hand-side of (14) that the tie derivative of µ is a linear cobinations of oents of x upto order 1 Hence, if one stacks all oents in an infinite vector µ [µ 1, µ 2, ] T, its dynaics can be written as µ A µ, (15) for soe infinite atrix A We refer to (15) as the infinitediensional oent dynaics et µ [µ 1, µ 2,, µ n ] T R n contains the top n eleents of µ, ie, the first n order oents of x which correspond to the lower-order oents of interest Then, using (14) the evolution of µ is given by µ Aµ B µ, µ µ n1, (16) for soe n n and n 1 atrices A and B having the following structure A (17) and B (18) where * represents non-zero entries Our goal is to approxiate (16) by a finite-diensional nonlinear ODE of the for ν Aν Bϕ(ν,t), ν [ν 1,ν 2,,ν n ] T (19) where the ap ϕ : R [, ) R should be chosen so as to keep ν(t) close to µ(t) We call (19) the truncated oent dynaics and ϕ the truncation function

4 et Ω µ be a set of initial conditions for which solutions to (15) exists globally and their first n eleents are uniforly bounded by the sae constant When a sufficiently large but finite nuber of derivatives of µ(t) and ν(t) atch point-wise, then, the difference between solutions to (16) and (19) reains close on a given copact tie interval This follows fro a Taylor series approxiation arguent To be ore precise, for each δ > and T R, there exists an integer N, sufficiently large, for which the following result holds: Assuing that for every t, µ (t ) Ω µ µ(t ) ν(t ) di µ(t) dt i tt di ν(t) dt i tt, i {1,,N} (2) where di µ(t) dt i and di ν(t) dt i represent the i th tie derivative of µ(t) and ν(t) along the trajectories of syste (15) and (19) respectively Then, µ(t ) ν(t ) µ(t) ν(t) δ, t [t,t ], (21) along solutions of (15) and (19), where µ denote the first n eleents of µ It has been shown in [13] that under appropriate asyptotic stability conditions on (15), the inequality (21) can actually be extended t [t, ) In the next section we will use equality (2) to construct truncation functions ϕ IV CONSTRUCTION OF AROXIMATE TRUNCATIONS In this section we construct approxiate truncated oent dynaics (19) for the general set of reactions introduced in Section II using (2) After replacing (16) and (19) in (2), equality (2) becoes a DE on ϕ and we look for solutions for ϕ having the following separable for ϕ(ν) n 1 ν γ, ν [ν 1,,ν n ] (22) for appropriately chosen constants γ R In general, it is not possible to find ϕ(ν) of the for (22) for which (2) holds We will therefore relax this condition and siply deand the following µ(t ) ν(t ) di µ(t) dt i tt di ν(t) dt i tt E[ε i (x(t ))], (23) i {1, 2,, N}, where each eleent of the vector ε i (x(t )) is a polynoial in x(t ) One can think of (23) as an approxiation to (2) which will be valid as long as the oents in di µ (t) dt i tt doinate E[ε i (x(t ))] We ake the following assuption on the stoichioetry of the reactions in (1), which as we will see, is needed for the uniqueness for solution of γ Assuption 1 (Uniqueness): in (12) satisfies The constant f ( j) defined f ( j), j {1,,n} (24) The following theore suarizes the ain result TABE II SOUTION FOR TRUNCATION FUNCTION ϕ FOR DIFFERENT VAUES OF n n ϕ(ν) ν 3 3 ν 3 2 ν 4 1 ν4 3 ν 6 2 ν 1 2 ν5 4 ν 5 1 ν1 3 ν1 6ν6 5 ν2 3 ν2 15ν15 4 Theore 2: For every deterinistic initial condition µ (t ) [x,x 2,]T where x(t ) x with probability one, there exists γ for which µ(t ) ν(t ) and di ν(t) dt i dµ(t) dt tt dν(t) dt tt d i µ(t) dt i tt di ν(t) dt i tt ε i (x ) i 2 (25) where di µ(t) dt i represent the i th tie derivative of µ(t) and ν(t) along the trajectories of syste (15) and (19) respectively and the th eleent of ε i (x ) is a polynoial in x of order n i, i 2 n and zero otherwise Furtherore, if Assuption 1 holds then this γ is uniquely deterined d Reark 1 It can be shown that i µ (t) dt i is a linear cobination of oents of x upto order i Thus, d i µ (t) dt i tt is a polynoial in x of order i, and, for d a given x, by choosing n large enough, i µ (t) dt i tt can be ade to doinates the corresponding eleent in ε i (x ) which is a polynoial in x of order n i, upto any orders of agnitude Hence, with increasing n the truncated oent dynaics odel should provides ore accurate approxiations for the lower order oents Reark 2 If Assuptions 1 holds, then the unique γ for which (25) holds is given by γ ( 1) n C n1, (26) which is independent of the reaction paraeter c i, the nuber of reactions or even the kind of reaction occurring in (1) Truncation functions for different values of n as given by (26) are listed in Table II Moreover, one can check that the approxiations for µ as given by ϕ(ν) in Table II is as if x(t) is lognorally distributed V EXAMES Consider the following bi-olecular irreversible reaction: 2X c (27) Owing to the siplicity of this reaction, one can obtain the exact solution for µ 1 (t) E[x(t)] and µ 2 (t) E[x(t) 2 ] fro the Master equation as has been done in [14] We will develop a truncated oent dynaics odel for the above cheical reaction and copare it with the exact solutions

5 The nuber of olecules x of X, can be generated by a SHS with continuous dynaics ẋ and a reset ap x φ(x) x 2 with intensity λ(x) c 2 x(x 1) For ψ(x) x, N we have (ψ)(x) c 2 x(x 1)[(x 2) x ] (28) Fro (12) we have f ( j) c( 1) j 2 j 1 and thus Assuption 1 holds n N 1 We first build the truncated oent dynaics odel for n 2 Towards that end, taking µ [µ 1, µ 2 ] T, equation (16) can be written as follows [ µ1 µ 2 ] [ c 2c c 4c ][ µ1 µ 2 ] [ 2c ] µ 3 Using Table II, the truncated oent dynaics for which (25) holds is given by [ ] [ ][ ] [ ]( ) 3 ν1 c c ν1 ν2 (29) ν 2 2c 4c 2c ν 2 Table III and Figure 1 copare our estiates for the ean and the coefficient of variation respectively, with that of the exact solution obtained fro [14] and the approxiate truncated odel (29) for c 1 and initial condition x() 1 Our estiates of E[x] are alost identical to those of the exact solution, while, those of CV [x] are close We now build the truncated oent dynaics odel for n 3 Taking µ [µ 1, µ 2, µ 3 ] T, equation (16) can be written as µ 1 µ 2 µ 3 c c 2c 4c 2c 4c 1c 9c µ 1 µ 2 µ 3 ν 1 3c µ 4 Fro Table II the truncated oent dynaics is given by ν 1 c c ν 2 2c 4c 2c ν 1 ν 2 ν 3 4c 1c 9c ν 3 3c ν4 1 ν4 3 ν 6 2 (3) One can see fro Table III and Figure 1 that the truncated odels (3) provides alost identical estiates for E[x] and CV [x] to those of the exact solution VI CONCUSION AND FUTURE WOR An approxiate stochastic odel for cheically reacting systes with single reactant specie was presented in this paper This was done by representing the population of various species involved in a set of cheical reactions as the continuous state of a pshs With such a representation, the dynaics of the infinite vector containing all the statistical oents of the continuous state are governed by an infinite-diensional linear syste of ODEs, which under appropriate conditions can be approxiated by finitediensional nonlinear ODEs Explicit conditions under CV[x] TABE III E[x] FOR THE EXACT AND AROXIMATE TRUNCATED MODES AT DIFFERENT TIMES t FOR THE CHEMICA REACTION (27) E[x] REFERS TO THE EXACT SOUTION, ν 1 (n 2) REFERS TO THE TRUNCATED MODE (29) AND ν 1 (n 3) REFERS TO THE TRUNCATED MODE (3) t E[x] ν 1 (n 2) ν 1 (n 3) t E[x Fig 1 CV [x] 2 ] E[x] 2 E[x] for the exact and approxiate truncated odels at different ties t for the cheical reaction (27) refers to the exact solution, to the truncated odel (29) and o to the truncated odel (3) which these finite-diensional nonlinear ODEs exist and are unique were given along with forulas to construct the Using these forulas, for a siple bi-olecular reaction, we constructed approxiate stochastic odel for the evolution of the ean and standard deviation and copared the with the exact results obtained fro the Master equation Iportant observations that point towards directions of future work are as follows 1) In previous work [15], a siilar version of Theore 2 was proven for ulti-specie reactions but for a vector µ containing only the first and second order oents of the populations of species involved in the reaction Also in [15] equality (25) was proven only for i 1 and 2 A direction for future research would be to extend these results to that of this paper 2) Another direction of future work is to use equality (25) to copute explicit error bounds between the solution of the truncated oent dynaics odel (19) and the actual odel (16)

6 ACNOWEDGMENTS We would like to thank Mustafa haash and Hana El Saad for several discussions that led us to consider pshs as a odeling tool for cheical reactions REFERENCES [1] H H McAdas and A Arkin, Stochastic echaniss in gene expression, roceedings of the National Acadey of Sciences USA, vol 94, pp , 1997 [2] A Arkin, J Ross, and H H McAdas, Stochastic kinetic analysis of developental pathway bifurcation in phage λ-infected Escherichia coli cells, Genetics, vol 149, pp , 1998 [3] D Endy and R Brent, Modelling cellular behaviour, Nature (ondon), vol 49, pp , 21 [4] M Elowitz and S eibler, A synthetic oscillatory netwrok of transcriptional regulators, Nature (ondon), vol 43, pp , 2 [5] T S Gardner, C R Cantor, and J J Collings, Construction of a genetic toggle switch in Escherichia coli, Nature (ondon), vol 43, pp , 2 [6] M C Walters, S Fiering, J Eideiller, W Magis, M Groudine, and D I Martin, Enhancers increase the probability but not the level of gene expression, roceedings of the National Acadey of Sciences USA, vol 92, pp , 1995 [7] N G V apen, Stochastic rocesses in hysics and Cheistry Asterda, The Netherlands: Elsevier Science, 21 [8] D T Gillespie, A general ethod for nuerically siulating the stochastic tie evolution of coupled cheical reactions, J of Coputational hysics, vol 22, pp , 1976 [9], Approxiate accelerated stochastic siulation of cheically reacting systes, J of Cheical hysics, vol 115, no 4, pp , 21 [1] J Hespanha and A Singh, Stochastic odels for cheically reacting systes using polynoial stochastic hybrid systes, Int J of Robust and Nonlinear Control, 25, to appear [11] F Wilkinson, Cheical inetics and Reaction Mechaniss New York: Van Nostrand Reinhold Co, 198 [12] J Hespanha, Stochastic hybrid systes: Applications to counication networks, in Hybrid Systes: Coputation and Control, ser ect Notes in Coput Science, R Alur and G J appas, Eds Berlin: Springer-Verlag, Mar 24, no 2993, pp [13], olynoial stochastic hybrid systes, in Hybrid Systes : Coputation and Control (HSCC) 25, Zurich, Switzerland, 25 [14] D A McQuarrie, Stochastic approach to cheical kinetics, J of Applied robability, vol 4, pp , 1967 [15] A Singh and J Hespanha, Models for gene regulatory networks using polynoial stochastic hybrid systes, 25, subitted