Model Order Reduction For Linear Noise Approximation
Model order reduction for Linear Noise Approximation using time-scaleseparationNarmada Herath1 and Domitilla Del Vecchio2Abstract— In this paper, we focus on model reduction ofbiomolecular systems with multiple time-scales, modeled usingthe Linear Noise Approximation. Considering systems wherethe Linear Noise Approximation can be written in singular perturbation form, with as the singular perturbation parameter,we obtain a reduced order model that approximates the slowvariable dynamics of the original system. In particular, we showthat, on a finite time-interval, the first and second moments ofthe reduced system are within an O( )-neighborhood of thefirst and second moments of the slow variable dynamics of theoriginal system. The approach is illustrated on an example ofa biomolecular system that exhibits time-scale separation.I. I NTRODUCTIONTime-scale separation is a ubiquitous feature in biomolecular systems, which enables the separation of the system dynamics into ‘slow’ and ‘fast’. This property is widely used inbiological applications to reduce the complexity in dynamicalmodels. In deterministic systems, where the dynamics aremodeled using ordinary differential equations, the processof obtaining a reduced model is well defined by singularperturbation and averaging techniques [1], [2]. However,employing time-scale separation to obtain a reduced ordermodel remains an ongoing area of research for stochasticmodels of biological systems [3].Biological systems are inherently stochastic due to randomness in chemical reactions [4], [5]. Thus, differentstochastic models have been developed to capture the randomness in the system dynamics, especially at low populations numbers. The chemical Master equation is a prominentstochastic model which considers the species counts as aset of discrete states and provides a description for thetime-evolution of their probability density functions [6], [7].However, analyzing the chemical Master equation directlyproves to be a challenge due to the lack of analytical tools toanalyze its behavior. Therefore, several approximations of theMaster equation have been developed, which provide gooddescriptions of the system dynamics under certain assumptions. The chemical Langevin equation (CLE) is one suchapproximation, where the dynamics of the chemical speciesare described as a set of stochastic differential equations [8].The Fokker-Plank equation is another method equivalent tothe CLE, which considers the species counts as continuousvariables and provides a description of the time evolution*This work was funded by AFOSR grant # FA9550-14-1-0060.1 Narmada Herath is with the Department of Electrical Engineering andComputer Science, Massachusetts Institute of Technology, 77 Mass. Ave,Cambridge MA nherath@mit.edu2 Domitilla Del Vecchio is with the Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Mass. Ave, Cambridge MAddv@mit.eduof their probability density functions [6]. The Linear NoiseApproximation (LNA) is another approximation, where thesystem dynamics are portrayed as stochastic fluctuationsabout a deterministic trajectory, assuming that the systemvolume is sufficiently large such that the fluctuations aresmall relative to the average species counts [7], [9].In our previous work, we considered a class of stochasticdifferential equations in singular perturbation form, whichcaptures the case of multiple scale chemical Langevin equation with linear propensity functions. We obtained a reducedorder model for which the error between the moment dynamics were of O( ), where is the singular perturbationparameter [10], [11]. In this work, we consider systems withnonlinear propensity functions, modeled using the LinearNoise Approximation.There have been several works that obtain reduced ordermodels for systems modeled using LNA, under differentapproaches for time-scale separation. One such model isderived by Pahlajani et. al, in [12], where the slow and fastspecies are identified by categorizing the chemical reactionsas slow and fast. In [13], [14], Thomas et. al, derive a reducedorder model by considering the case where the species areseparated using the decay rate of their transients, accordingto the quasi-steady-state approximation for chemical kinetics.It is also shown that, imposing the time-scale separationconditions arising from slow and fast reactions, on theirmodel, leads to the same reduced model obtained in [12]. Inthese previous works, the error between the original systemand the reduced system has been studied numerically andhas not been analytically quantified. The work by Sootlaand Anderson in [15] gives a projection-based model orderreduction method for systems modeled by Linear NoiseApproximation. The authors extend this work in [16], wherethey also provide an error quantification in mean square sensefor the reduced order model derived in [13] under quasisteady state assumptions. However, to provide an error boundthe authors explicitly use the Lipschitz continuity of thediffusion term, which is not Lipschitz continuous in general.In this paper, we consider biomolecular systems modeled using the Linear Noise Approximation where systemdynamics are represented by a set of ordinary differentialequations that give the deterministic trajectory and a set ofstochastic differential equations that describe the stochasticfluctuations about the deterministic trajectory. We considerthe case where the system dynamics evolve on well separatedtime-scales with slow and fast reactions, and the LNA can bewritten in singular perturbation form with as the singularperturbation parameter, as in [12]. We define a reduced order
model and prove that the first and second moments of thereduced system are within an O( )-neighborhood of the firstand second moments of the original system. Our results donot rely on Lipschitz continuity assumptions on the diffusionterm of the LNA.This paper is organized as follows. In Section II, wedescribe the model considered. In Section III, we define thereduced system and derive the moment dynamics for theoriginal and reduced systems. In Section IV, we prove themain convergence results. Section V illustrates our approachwith an example and Section VI includes the concludingremarks.II. S YSTEM M ODELA. Linear Noise ApproximationConsider a biomolecular system with n species interactingthrough m reactions in a given volume . The Chemical Master Equation (CME) describes the evolution of theprobability distribution for the species counts to be in stateY (Y1 , . . . , YN ), by the partial differential equationm@P (Y, t) X [ai (Y@ti 1vi , t)P (Yvi , t)ai (Y, t)P (Y, t)],(1)where ai (Y, t) is the microscopic reaction rate withai (Y, t)dt being the probability that a reaction i will takeplace in an infinitesimal time step dt and vi the change instate produced by reaction i for i 1, . . . , m [17].The Linear Noise Approximation (LNA) is an approximation to the CME obtained under the assumption thatthe system volume and the number of molecules in thesystem areplarge [7]. To derive the LNA it is assumed thatY y , where y is a deterministic quantity and is astochastic variable accounting for the stochastic fluctuations.Then by expanding the chemical Master equation in a Taylorseries and equating the terms of order 1/2 and 0 , it isshown that y is the macroscopic concentration and is aGaussian process whose dynamics are given by [7], [9]ẏ f (y, t), A(y, t) (y, t) ,(2)(3)wherenoise process,Pism an m-dimensional white(y,t)f (y, t) i 1 vi ãi (y, t), A(y, t) @f@yand (y, t) pp[v1 ã1 (y, t), . . . , vm ãm (y, t)]. ãi (y, t) is the macroscopicreaction rate which can be approximated by ãi (y, t) 1 ai ( y, t) at the limit of ! 1 and Y ! 1 such thatthe concentration y Y / remains constant [18].B. Singularly Perturbed SystemWe consider the case where the biomolecular system in (2)- (3) exhibits time-scale separation, with ms slow reactionsand mf fast reactions where ms mf m. This allowsthe use of a small parameter to decompose the reactionrate vector as ã(y, t) [âs (y, t), (1/ )âf (y, t)]T whereâs (y, t) 2 Rms represents the reaction rates for the slowreactions and (1/ )âf (y, t) 2 Rmf represents the reactionrates for the fast reactions. The corresponding vi vectorsrepresenting the change of state by each reaction i couldbe represented as v [v1 , . . . , vms , vms 1 , . . . , vms mf ] forms slow and mf fast reactions. However, such a decomposition does not guarantee that the individual species in thesystem will evolve on well-separated time-scales. Therefore,a coordinate transformation may be necessary to identify theslow and fast variables in the system as seen in deterministicsystems [19] and chemical Langevin models [20]. Thus, wemake the following claim.Claim 1: Assume there is an invertible matrix A [Ax , Az ]T with Ax 2 Rns n and Az 2 Rnf n , such thatthe change of variables x Ax y, z Az y, allows thedeterministic dynamics in (2) to be written in the singularperturbation formẋ fx (x, z, t),(4) ż fz (x, z, t, ),(5)then, the change of variables x Ax , z Az takesthe dynamics of the stochastic fluctuations given in (3), into the singular perturbation form x A1 (x, z, t) x A2 (x, z, t) z x (x, z, t) x , (6) z B1 (x, z, t, ) x B2 (x, z, t, ) z z (x, z, t, ) z ,(7)where x is an ms -dimensional white noise process, z [ x , f ]T , where f is an mf -dimensional white noiseprocess and@fx (x, z, t),@x@fx (x, z, t)A2 (x, z, t) ,@z@fz (x, z, t, )B1 (x, z, t, ) ,@x@fz (x, z, t, )B2 (x, z, t, ) ,@zx (x, z, t) qpAx v1 âs1 (A 1 [x, z]T , t), . . . , vms âsms (AA1 (x, z, t) 1 [x, z]T , t), h pi3Tp Az v1 âs1 (A 1 [x, z]T , t), . . . , vms âsms (A 1 [x, z]T , t)76 67p6 Az vms 1 âf (A 1 [x, z]T , t), . . . ,7167 .67q45vms mf âf mf (A 1 [x, z]T , t)2z (x, z, t, )Proof: See Appendix A-1.Based on the result of Claim 1, in this work, we considerthe Linear Noise Approximation represented in the singularperturbation form:x(0) x0 , (8)ẋ fx (x, z, t), ż fz (x, z, t, ), x A1 (x, z, t) x A2 (x, z, t)z(0) z0 , (9)z x (x, z, t) x ,x (0) x0,(10)
z B1 (x, z, t, )x B2 (x, z, t, )z z (x, z, t, ) z ,z (0) z 0,(11)where x 2 Dx Rns , x 2 D x Rns are the slowvariables and z 2 Dz Rnf , z 2 D z Rnf are thefast variables. x is an ms -dimensional white noise process.Then, z [ x , f ]T , where f to be a mf -dimensionalwhite noise process. We assume that the system (8) - (9) hasa unique solution on a finite-time interval t 2 [0, t1 ].We refer to the system (8) - (11) as the original systemand obtain a reduced order model when 0. To this end,we make the following assumptions on system (8) - (11) forx 2 Dx Rns , z 2 Dz Rnf and t 2 [0, t1 ].Assumption 1: The functions fx (x, z, t), fz (x, z, t, ) aretwice continuously differentiable. The Jacobian @fz (x,z,t,0)@zhas continuous first and second partial derivatives withrespect to its arguments.Assumption 2: Thematrix-valuedfunctionsTT(x,z,t)(x,z,t),(x,z,t, )[(x,z,t)0]andxxzxT(x,z,t, )(x,z,t, )arecontinuouslydifferentiable.zzFurthermore, we have that z (x, z, t, 0) 0 andTlim !0 z (x,z,t, ) z (x,z,t, ) (x, z, t) where (x, z, t) isbounded for given x, z, t and @ (x,z,t)is continuous.@zAssumption 3: There exists an isolated real root z 1 (x, t), for the equation fz (x, z, t, 0) 0, for which, thematrix @fz (x,z,t,0)is Hurwitz, uniformly in x and@zz 1 (x,t)t. Furthermore, we have that the first partial derivative of1 (x, t) is continuous with respect to its arguments. Also,the initial condition z0 is in the region of attraction ofdzthe equilibrium point z 1 (x0 , 0) for the system d fz (x0 , z, 0, 0).Assumption 4: The system ẋ fx (x, 1 (x, t), t) has aunique solution for t 2 [0, t1 ].III. P RELIMINARY R ESULTSA. Reduced SystemThe reduced system is defined by setting 0 in theoriginal system (8) - (11), which yieldsfz (x, z, t, 0) 0,B1 (x, z, t, 0)x B2 (x, z, t, )z 0.(12)(13)Let z 1 (x, t) be an isolated root of equation (12), which satisfies Assumption 3. Then, we havethat z B2 (x, 1 (x, t), t, 0) 1 B1 (x, 1 (x, t), t, 0) xis the unique solution of equation (13). Let 2 (x, t) B2 (x, 1 (x, t), t, 0) 1 B1 (x, 1 (x, t), t, 0). Then, substituting z 1 (x, t) and z 2 (x, t) x in equations (8) and(10), we obtain the reduced systemx̄ fx (x̄, 1 (x̄, t), t), A(x̄, t) (x̄,xxwhereA(x̄, t) A1 (x̄,x1 (x̄, t), t) x , A2 (x̄,1 (x̄, t), t) xx̄(0) x0 , (14) x (0) x ,0(15)1 (x̄, t), t) 2 (x̄, t).Next, we derive the first and second moment dynamics ofthe variable x in the reduced system. To this end, we makethe following claim:Claim 2: The first and second moment dynamics for thevariable x of the reduced system (14) - (15) can be writtenin the formdE[ x ] A(x̄, t)E[ x ],E[ x (0)] x 0 , (16)dtdE[ x xT ] A(x̄, t)E[ x xT ] E[ x xT ]A(x̄, t)Tdt x (x̄, 1 (x̄, t), t), t) x (x̄, 1 (x̄, t), t), t)T ,E[ x (0) x (0)T ] x 0 x T0 . (17)Proof: Similar to [21], the first and second momentdynamics of x in (15) can be written asdE[ x ] E[A(x̄, t) x ],dtdE[ x xT ] E[A(x̄, t) x xT ] E[ x ( xT A(x̄, t)T )]dt x (x̄, 1 (x̄, t), t), t) x (x̄, 1 (x̄, t), t), t)T .Since the dynamics of x̄ given by (14) is deterministic, usingthe linearity of the expectation operator we can write themoment dynamics of the reduced system as (16) - (17).Next, we proceed to derive the moment dynamics forx andz in the original system (8) - (11) given by thefollowing claim.Claim 3: The first and second moment dynamics for thevariables x and z of the original system (8) - (11) can bewritten in the formdE[ x ] A1 (x, z, t)E[ x ] A2 (x, z, t)E[ z ],(18)dtTdE[ x x ] A1 (x, z, t)E[ x xT ] A2 (x, z, t)E[ z xT ]dt E[ x xT ]A1 (x, z, t)T (E[ z xT ])T A2 (x, z, t)T x (x, z, t) x (x, z, t)T ,(19)dE[ z ] B1 (x, z, t, )E[ x ] B2 (x, z, t, )E[ z ], (20)dtdE[ z xT ] E[ z xT ]A1 (x, z, t)Tdt E[ z zT ]A2 (x, z, t)T B1 (x, z, t, )E[ x xT ] B2 (x, z, t, )E[ dE[zTz ]zTx] z (x, z, t, )[ x (x, z, t) B1 (x, z, t, )E[dt B2 (x, z, t, )E[ z zT ] E[1 E[ z zT ]B2 (x, z, t, )T xTz ]zTTx ]B1 (x, z, t, )0]T ,(21)z (x, z, t, ) z (x, z, t, )T,(22)where x and z are the solutions of the equations (8) - (9),and the initial conditions are given by E[ x (0)] x 0 ,E[ x xT (0)] x 0 x T0 , E[ z (0)] z 0 , E[ z xT (0)] TTTz 0 x 0 , E[ z z (0)] z 0 z 0 .
Proof: The equations (10) - (11) can be written in theform x A1 (x, z, t) x A2 (x, z, t) z [ x (x, z, t) 0] z , z B1 (x, z, t, ) x B2 (x, z, t, ) z z (x, z, t, ) z ,where [ x (x, z, t) 0 ] 2 R. Then, using thefact that the x and z are deterministic and the linearity ofthe expectation operator, the dynamics for the first momentscan be written asdE[ x ] A1 (x, z, t)E[ x ] A2 (x, z, t)E[ z ],(23)dtdE[ z ]11 B1 (x, z, t, )E[ x ] B2 (x, z, t, )E[ z ].dt (24)n (ms mf )Similarly, using Proposition III.1 in [21], the second momentdynamics can be written as TTdx xx zE TTdtz xz z Tx (A1 (x, z, t) x A2 (x, z, t) z )Tz (A1 (x, z, t) x A2 (x, z, t) z ) 1 1 x (B1 (x, z, t, ) x B2 (x, z, t, )z (B1 (x, z, t, ) x B2 (x, z, t, )(A1 (x, z, t)1 (B1 (x, z, t, ) A2 (x, z, t) z ) xTx B2 (x, z, t, ) z )Tz)Tz)x Employing the linearity of the expectation operator, we cansum the corresponding entries of the matrices in equation(25), and multiply by to write the moment equations(23) - (25) in the form of (18) - (22). Note that, sinceE[ x zT ] (E[ z xT ])T , we have eliminated the dynamicsof the variable E[ x zT ].Claim 4: Setting 0 in the system of moment dynamics(18) - (22) and the dynamics of x and z given by (8) - (9),yields the moment dynamics of the reduced system (16) (17) where the dynamics of x̄ are given by (14).Proof: Setting 0 in the equations (8) - (9) and (19)- (20), yields(26)0 fz (x, z, t, 0),0 B1 (x, z, t, 0)E[x] B2 (x, z, t, 0)E[xTx](27)z ], B2 (x, z, t, 0)E[zTx ].(28)By definition of the reduced system, we have that z 1 (x, t) is an isolated root for equation (26). Then, underthe Assumption 3, we have that the unique solutions for theequations (27) and (28) are given byE[z] B2 (x,1 (x, t), t, 0)1(B1 (x, zB2 (x,1 (x, t), t, 0)T2 (x, t)E[ x x ].1(B1 (x,1 (x, t), t, 0)E[ xTx ])(30)Substituting z 1 (x, t) and equations (29) - (30), in (8)and (18) - (22) results inẋ fx (x, 1 (x, t), t),dE[ x ] A1 (x, 1 (x, t), t)E[ x ]dt A2 (x, 1 (x, t), t) 2 (x, t)E[dE[xdtTx] A1 (x,1 (x, t), t)E[ x(31)x ],(32)Tx]T1 (x, t), t) 2 (x, t)E[ x x ]E[ x xT ]A1 (x, z, t)T( 2 (x, t)E[ x xT ])T A2 (x, 1 (x, t), t)TTx (x, 1 (x, t), t) x (x, 1 (x, t), t) . A2 (x, (33)It follows that equation (31) is equivalent to the reducedsystem given by (14) and since we have that A(x, t) A1 (x, 1 (x, t), t) x A2 (x, 1 (x, t), t) 2 (x, t), the system(32) - (33) is equivalent to the moment dynamics of thereduced system given by (16) - (17).IV. M AIN R ESULTS(A1 (x, z, t) x A2 (x, z, t) z )1T (B1 (x, z, t, ) x B2 (x, z, t, ) z ) zTx (x, z, t) x (x, z, t)1T z (x, z, t, )[ x (x, z, t) 0 ]1T [ x (x, z, t) 0 ] z (x, z, t, ).(25)1T 2 z (x, z, t, ) z (x, z, t, )0 B1 (x, z, t, 0)E[E[(29)2 (x, t)E[ x ],Tx]TxTz 1 (x, t), t, 0)E[x ])Lemma 1: Consider the original system in (8) - (11), thereduced system in (14) - (15), and the moment dynamicsfor the original and reduced systems in (18) - (22), (16) (17) respectively. We have that, under Assumptions 1 - 3,the commutative diagram in Fig. 1 holds.Proof: The proof follows from Claim 1, Claim 2 andClaim 3.Theorem 1: Consider the original system (8) - (11), thereduced system in (14) - (15) and the moment dynamics forthe original and reduced systems in (18) - (22), (16) - (17)respectively. Then, under Assumptions 1 - 4, there exists 0 such that for 0 , we havex̄(t)k O( ), t 2 [0, t1 ],(34)kE[ x (t)] E[ x (t)]k O( ),(35)TT kE[ x (t) x (t) ] E[ x (t) x (t) ]k O( ).(36)Proof: From Lemma 1, we see that setting 0 inthe moment dynamics of the original system (18) - (22) andin the dynamics of x and z given by (8) - (9), yields themoment dynamics of the reduced system (16) - (17) wherethe dynamics of x̄ is given by (14). Therefore to proveTheorem 1, we apply Tikhonov’s theorem [1] to the systemof moment dynamics of the original system given by (18) (22) together the dynamics of x and z given by (8) - (9). Inorder to apply Tikhonov’s theorem, we first prove that theassumptions of the Tikhonov’s theorem are satisfied. To thisend, let us define the boundary layer variableskx(t)b1 zb2 E[(37)1 (x, t),z]2 (x, t)E[x ],(38)
Original SystemReduced Systemẋ fx (x, z, t), ż fz (x, z, t, ), x A1 (x, z, t) x A2 (x, z, t) z B1 (x, z, t, )xx̄ fx (x̄,1 (x̄, t), t), x A(x̄, t) x x (x̄,z B2 (x, z, t, )z z (x, z, t, ) z . !0Moments of the Original SystemzTx]2 (x, t)E[ xMoments of the Reduced Systemx̄ fx (x̄, 1 (x̄, t), t), dE[ x ] .T E[ x x ]dtẋ fx (x, z, t), ż fz (x, z, t, ),01E[ x ]T]E[BCx xd BCB E[ z ]C .dt @ E[T Az x] E[ z zT ]b3 E[1 (x̄, t), t) x .x (x, z, t) x ,Fig. 1.Tx ].Commutative Diagram.(39)The dynamics of the boundary layer variables are given bydb1dzd 1 (x, t) ,dtdtdtdb2dE[ z ] d 2 (x, t)E[ x ] ,dtdtdtdb3dE[ z x ] d 2 (x, t)E[ x xT ] .dtdtdtDenote by t/ the time variable in the fast time-scale.Then, expanding using the chain rule, we havedb1dz@ 1 (x, t)@ 1 (x, t) dx ,d dt@t@xdtdb2dE[ z ]@ 2 (x, t)@ 2 (x, t) dx E[ x ] E[ x ]d dt@t@xdtdE[ x ] 2 (x, t),dtTdb3dE[ z x ]@ 2 (x, t) E[ x xT ]d dt@tdE[ x xT ]T @ 2 (x, t) dx E[ x x ] 2 (x, t).@xdtdtSubstituting from equations (9), (20) and (22) yieldsdb1@ 1 (x, t)@ 1 (x, t) dx fz (x, z, t, ) ,(40)d @t@xdtdb2 B1 (x, z, t, )E[ x ] B2 (x, z, t, )E[ z ]d @ 2 (x, t)@ 2 (x, t) dx E[ x ] E[ x ]@t@xdtdE[ x ] 2 (x, t),(41)dtdb3 E[ z xT ]A1 (x, z, t)T E[ z zT ]A2 (x, z, t)Td B1 (x, z, t, )E[ x xT ] B2 (x, z, t, )E[ z xT ]@ 2 (x, t) z (x, z, t, )[ x (x, z, t) 0]T E[ x xT ]@tT@(x,t)dxdE[2xx] E[ x xT ] 2 (x, t). (42)@xdtdtwhere we take z b1 1 (x, t) and E[ z ] b2 TT2 (x, t)E[ x ], and E[ z x ] b3 2 (x, t)E[ x x ]. Since,from Assumption 3, 1 (x, t) is a continuously differentiable(x,t) @ 1 (x,t)functions in its arguments, we have that @ 1@t,dxare bounded in a finite time interval t 2 [0, t1 ]. SinceB2 (x, 1 (x, t), t, 0) 1 B1 (x, 1 (x, t), t, 0), and2 (x, t) B1 and B2 are continuously differentiable from Assumption(x,t)(x,t)1, we have that @ 2@xand @ 2@tare bounded in a finitetime interval t 2 [0, t1 ]. Then, the boundary layer systemobtained by setting 0 in (40) - (42) is given bydb1 fz (x, b1 1 (x, t), t, 0),d db2 B1 (x, b1 1 (x, t), t, 0)E[ x ]d B2 (x, b1 1 (x, t), t, 0)(b2 2 (x, t)E[ : g1 (b1 , b2 , x, t),db3 B1 (x, b1 1 (x, t), t, 0)E[ xd B2 (x, b1 1 (x, t), t, 0)(b3 (43)x ])(44)Tx]2 (x, t)E[ xTx ])(45) : g2 (b1 , b2 , b3 , x, t).To prove that the origin of the boundary layer system isexponentially stable, we consider the dynamics of the vectorb [b1 , b2 , b3 ]. Linearizing the system (43) - (45) around theorigin, we obtain the dynamics for b̃ b 0 as23J1100db̃0 5 b̃ 4 J21 J22(46)d J31 J32 J33where J11 @fz (x,b1 1 (x,t),t,0),@b1b1 0@g1 (b1 ,b2 ,x,t),@b1@g2 (b1 ,b2 ,b3 ,x,t),@b1J21 1 (x, t), t, 0) b 0 , J31 1@g2 (b1 ,b2 ,b3 ,x,t),J B(x,b1 1 (x, t), t, 0) b 0 .332@b21J22 B2 (x, b1 J32 Since the eigenvalues of a block triangular matrix aregiven by the union of eigenvalues of the diagonal blocks,we consider the eigenvalues of @fz (x,b1 @b11 (x,t),t,0) b1 0and B2 (x, b1 1 (x, t), t, 0) b 0 . Under Assumption 3,1
@fz (x,b1 1 (x,t),t,0) @b1b1 0@fz (x,z,t,0)is Hurwitz.@zz 1 (x,t)we have that the matrix@fz (x,z,t,0) dz@zdb1 z 1 (x,t)From the definition of the original system (8) - (11), wehave that B2 (x, z, t, ) @fz (x,z,t, ). Therefore, B2 (x, b1 @z@fz (x,z,t,0)(x,t),t,0) . Under Assump1@zb1 0z 1 (x,t)tion 3 we have that the matrix @fz (x,z,t,0)is Hur@zz 1 (x,t)witz, and thus, the boundary layer system is exponentiallystable.From Assumptions 1 and 2 we have that thefunctions fx (x, z, t), fz (x, z, t, ), A1 (x, z, t), A2 (x, z, t),TB1 (x, z, t, ),B2 (x, z, t, ),x (x, z, t) x (x, z, t) ,Tand z (x, z, t, ) z (x, z, t, )Tz (x, z, t, )[ x (x, z, t) 0]and their first partial derivatives are continuouslydifferentiable. From Assumption 1 we have that the@fz (x,z,t,0), @B1 (x,z,t,0), @B2 (x,z,t,0)have continuous@z@z@zfirst partial derivatives with respect to their arguments.From Assumptions 1 and 3 we have that the 1 (x, t),Thave continuous first2 (x, t)E[ x ],2 (x, t)E[ x x ]partial derivatives with respect to their arguments. FromAssumption 4 we have that the reduced system (14) has aunique bounded solution for t 2 [0, t1 ]. Since the momentequations (16) - (17), are linear in E[ x ] and E[ x xT ] thereexists a unique solution to (16) - (17) in for t 2 [0, t1 ].From Assumption 3 we have that the initial conditionz0 is in the region of attraction of the equilibrium point1 (x0 , 0), and thus the initial condition z01 (x0 , 0) forthe boundary layer system b1 is in the region of attractionof the equilibrium point b1 0. Since the system (18) (22) is linear in the moments, the system (20) - (22) hasa unique equilibrium point for given x and thus the initialconditions z 0 and z 0 x T0 are in the region of attractionof the equilibrium point for (20) - (22). Therefore, the initialTTconditions z 02 (x0 , 0) x 0 , z 0 x 02 (x0 , 0) x 0 x 0for the boundary layer variables b2 and b3 are in the regionof attraction of the equilibrium point at b2 0 and b3 0.Thus, the assumptions of the Tikhonov’s theorem on a finitetime-interval [1] are satisfied and applying the theorem tothe moment dynamics of the original system in (18) - (22)and the dynamics of x and z given by (8) - (9), we obtainthe result (34) - (36).Remark: From [7], we have that the x (t) and x (t) aremultivariate Gaussian processes. Since a Gaussian distribution is fully characterized by their mean and the covariance,and from Theorem 1, we have that the E[ x (t)],lim !0 E[ x (t) x (t) ] E[ x (t) x (t)T ].lim !0 E[x (t)]Twe have that for given t 2 [0, t1 ], the vectorin distribution to the vector x (t) as ! 0.x (t)(47)(48)convergesV. E XAMPLEIn this section we demostrate the application of the modelreduction approach on an example of a biolomelcular system.Consider the system in Fig. 2, where a phosphorylatedprotein X , binds to a downstream promoter site p whichproduces the protein G. Such a setup can be seen commonlyoccurring in natural biological systems, an example beingthe two component signalling systems in bacteria [22].Moreover, similar setups are also used in synthetic biology todesign biological circuits that are robust to the loading effectsthat appear due to the presence of downstream components[23], [24].Protein X is phosphorylated by kinase Z and dephosphorylated by phosphatase Y. Phosphorylated protein X binds to thedownstream promoter p.Fig. 2.The chemical reactions for the system are as follows: X konk1k2Z !X Z, X Y !X Y, X p )* C, C ! C koffG, G ! . The protein X is phosphorylated by kinase Z anddephosphorylated by phosphatase Y with the rate constantsk1 and k2 , respectively. The binding between phosphorylatedprotein X and promoter p produces a complex C, wherekon and ko are the binding and unbinding rate constants.Protein G is produced at rate , which encapsulates bothtranscription and translation processes and decayed at rate, which includes both degradation and dilution. We assumethat the total concentration of protein X and promoter p areconserved, giving Xtot X X C and ptot p C.Then, the dynamics for the macroscopic concentrations ofX , C and G denoted by x , c and g, respectively, can bewritten as dx dt k1 Z(t)(Xtotkon x (ptotdc kon x (ptotdtdg cg.dtx c)k2 Y x c) ko c,(49)c)(50)ko c,(51)Binding and unbinding reactions are much faster than phosphorylation/dephosphorylation, and therefore, we can writek2 Y /ko 1. Taking kd ko /kon , we havedx k1 Z(t)(Xtotdtk2 Y x (ptot kddck2 Y x (ptotdt kddg cg.dtx c) c)c)k2 Y x k2 Yc, k2 Yc, (52)(53)(54)The system (52) - (54) is in the form of system (2), with y [x , c, g]T . To take the system in the singular perturbationform given in (8) - (9), we consider the change of variablev x c, which yieldsdv k1 Z(t)(Xtotdtdg cg,dtv)k2 Y (vc),(55)(56)
dck2 Y (vdtkdc)(ptotc)(57)k2 Y c.This change of coordinates corresponds to having Ax [1 1 0, 0 0 1]T , Az [0 1 0], x [v, g]T and z c inClaim 1. Then, the dynamics for the stochastic fluctuationscan be written asd v ( k1 Z(t) k2 Y ) v k2 Y cdtpp k1 Z(t)(Xtot v) 1k2 Y (v c) 2 ,ppd g c 5g 6,cg dtd ck2 Y (ptot c) vdt kd k2 Yk2 Yk2 Y vptot 2c k2 Yckdkdkdrpk2 Y (v c)(ptot c) 3 k2 Y c 4 .kd(58)(59)Errors in the second moments decreases as decreases.The parameters used are Z(t) 1, k1 0.01, k2 0.01, kd 100, Xtot 200, Y 200, ptot 100, 0.1, 0.1, v(0) 0, c(0) 0, g(0) 0, v (0) 0, g (0) 0.Fig. 3.(60)with x [ v , g ]T and x [ v , g ]T . Therefore, theequations (55) - (60) are in the form of the original systemin (8) - (9) with x [v, g]T and z c. It follows thatAssumptions 1 and 2 are satisfied since the system functionsof (55) - (57) are polynomials of the state variables. Weevaluate fz kk2dY (v z)(ptot z) k2 Y z 0, which1yieldsp the unique solution z(v) 2 (v ptot kd )12(v ptot kd )4vptot , feasible under the physical2constraints 0 c ptot . We have that Assumption 3 iszsatisfied since @f@z is negative. Thus, we obtain the reducedsystemwheredv̄ k1 Z(t)(Xtot v̄) k2 Y (v̄ c̄),dtdḡ c̄ḡ,dtd v ( k1 Z(t) k2 Y ) v k2 Y cdtpp k1 Z(t)(Xtot v̄) 1k2 Y (v̄ppd g g cc̄ 5ḡ 6 ,dt11pc̄ (v̄ ptot kd )(v̄ ptot kd )222(ptot c̄) v c .(v̄ ptot 2c̄ kd Y )c̄)2,system. This result can be used to approximate the slowvariable dynamics of the LNA with a system of reduceddimensions, which will be useful in analysis and simulationsof biomolecular system especially when the system size islarge. The reduced model that we obtain is equivalent to thereduced order model derived in [12]. Our results are alsoconsistent with the error analysis that they have performednumerically, where it is approximated that the maximumerrors in the mean and the variance over time are of O( ).In future work, we aim to extend this analysis to obtainan approximation for the fast variable dynamics.A PPENDIXA-1:Applyingthecoordinatetransformationx Ax y, z Az y to equation (2), withã(y, t) [âs (y, t), (1/ )âf (y, t)]T and v [v1 , . . . , vms , vms 1 , . . . , vms mf ], with y A 1 [x, z]T wehave1Tẋ Ax f (A [x, z] , t)msX Axvi âsi (A 1 [x, z]T , t)i 1ms mf4v̄ptot ,Fig. 3 includes the simulation results for the error insecond moments of the stochastic fluctuations of v and g.We use zero initial conditions for all variables and thus thefirst moment of the stochastic fluctuations remains zero at alltimes. The simulations are carried out with Euler-Maruyamamethod and the sample means are calculated using 3 106realizations.VI. C ONCLUSIONIn this work, we obtained a reduced order model for theLinear Noise Approximation of biomolecular systems withseparation in time-scales. It was shown that, for a finite timeinterval the first and second moments of the reduced systemare within an O( )-neighborhood of the first and secondmoments of the slow variable dynamics of the original AxXvi (1/ )âf i (A1[x, z]T , t)i vms 1(61) fx (x, z, t),ż Az f (A 1 [x, z]T , t)msX Azvi âsi (A 1 [x, z]T , t)i 1ms mf AzXvi (1/ )âf i (A1[x, z]T , t)i vms 1 1fz (x, z, t, ). (62)Thus, from equation (61), if follows that Ax vi 0 for i ms 1, . . . , ms mf .Applying the coordinate transformation x Ax , z Az , to equation (3), we have that x Ax [A(y, t) ] Ax (y, t) , z Az
Model order reduction for Linear Noise Approximation using time-scale separation Narmada Herath1 and Domitilla Del Vecchio2 Abstract—In this paper, we focus on model reduction of biomolecular systems with multiple time-scales, modeled using
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