Control Of Metabolic Systems Modeled with Timed Continuous Petri Nets

Transcription

1 Control Of Metabolic Systems Modeled with Timed Continuous Petri Nets Roberto Ross-León Antonio Ramirez-Treviño José Alejandro Morales Javier Ruiz-León June 14, 2010

2 Outline 1 Introduction Introduction 2 Basic Definitions Timed Continuous Petri Nets Controllability 3 Modelling The Metabolome Molecular Interpretation Obtaining The Metabolome Module 4 Control Law Regulation Control Problem Extended TCPN Solution to the RCP 5 Illustrative Controlling Metabolic System Example 6 Conclusions 2 / 20

3 Introduction Introduction T CP N are amenable to model biochemical reactions and cell metabolism. A Modelling methodology. Problem of reaching a required state (marking) representing a certain metabolite concentration. 3 / 20

4 Basic Definitions Timed Continuous Petri Nets T CP N = (N, λ, m 0 ). N = (P, T, P re, P ost), m 0 {R + 0} P. P re, P ost {N 0} P T. λ : T {R + } T 4 / 20

5 Basic Definitions Timed Continuous Petri Nets The controlled state equation of a T CP N system is: m = C [ΛΠ (m) m u] (1) 0 u i [ΛΠ (m) m] i (2) f = ΛΠ (m) m. (3) The controlled state equation can be rewritten as: With 0 I ci 1. m = CI c ΛΠ (m) m (4) 5 / 20

6 Basic Definitions Controllability 6 / 20

7 Basic Definitions Equilibrium Points It is said that (m r, I cr ) is an equilibrium point if m = CI cr ΛΠ (m r ) m r = 0, it is and 0 I cri (τ) 1 τ. m(τ) Icr m r (5) 7 / 20

8 Modelling The Metabolome Representing Reactions A reaction E + S E + Q (6) is represented by the next Petri net: 8 / 20

9 Modelling The Metabolome Merging Elementary Modules After a merging of elementary modules is made, pathway modules are obtained. 9 / 20

10 Modelling The Metabolome Metabolome Module 10 / 20

11 Control Law Regulation Control Problem Computation of I c and I cr such that with 0 I ci (τ) 1 for 0 τ < τ f, and with 0 I cri (τ) 1 for τ τ f. m(0) Ic m(τ f ) = m r (7) m(τ ) Icr m r (8) 11 / 20

12 Control Law Extended TCPN m x = [ m m a ] = [ CIc ΛΠ (m) m I c ΛΠ (m) m ] (9) m r = m 0 + Cσ r (10) 12 / 20

13 Control Law Solution to the RCP e = m r m (11) e a = σ r m a (12) I ci = { 1 if ma [i] < σ r [i] 0 otherwhise (13) 13 / 20

14 Control Law Solution to the RCP Suppose that it occurs m a (τ 1 ) = σ 1 and σ 1 [i] < σ r [i] i, now from m 1 = m 0 + Cσ 1 we have m r = m 1 + C(σ r σ 1 ) = m 1 + C(σ r2 ) (14) Since σ r2 > 0 then it is feasible and the same procedure can be applied in succession. [ m ] [ ] CIc ΛΠ (m) m m x = = (15) m a I c ΛΠ (m) m I ci = { 1 if ma [i] < σ r [i] 0 otherwhise (16) 14 / 20

15 Illustrative Controlling Metabolic System Example Example 15 / 20

16 Illustrative Controlling Metabolic System Example Example Example Let a metabolome model be the system T CP N = (N, λ, m 0 ) with Λ = diag (2, 3, 4, 1) and m 0 = [ ] T. Let mr = [ ] T be a required marking. σ r = [ ] T Notice that from τ = 0 to τ = τ f 4.5 occurs the transitory dynamics, and for τ > τ f the steady state is reached. 16 / 20

17 Illustrative Controlling Metabolic System Example Example 17 / 20

18 Illustrative Controlling Metabolic System Example Example 18 / 20

19 Conclusions Conclusions This work presented a model methodology to capture the metabolome behavior. It uses a bottom-up approach where each individual biochemical reaction is modeled by elementary T CP N modules and, afterwards, all the modules are merged into a single one to capture the whole metabolome behavior. This work also presented the problem of reaching a required metabolome state. The solution to this problem are the instantaneous reaction velocities that are realizable in biological system. 19 / 20

20 Conclusions Actual Work and Future Perspective Present results are being applied to optimize metabolome fermentation in the production of tequila and to biofuels generation. Future perspective involves introduction of stochastic modelling and merging the metabolome with the signaling and genetic networks. 20 / 20