Solving a Chemical Reaction System by a PDE

Transcription

1 Solving a Chemical Reaction System by a PDE François Lemaire, Michel Petitot, Adrien Poteaux, Rosanne Ushi Robira, Samuel Vidal,... University Lille 1 (Symbolic Computation Team) FELIM 2012 Functional Equations in LIMoges 2012 Faculty of Sciences and Techniques of the University of Limoges 6 Mars 2012 Work supported by the ANR LEDA F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

2 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

3 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

4 Prospective work Initiated by Michel Petitot around Keywords modelization based on chemical reactions systems stochastic simulations statistics, probabilities Application modelization in Biology exact and approximate computation of probabilities and statistics F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

5 Presentation Modelization in Biology one possible approach : chemical reactions system Reaction : list of products list of reactants Example: G + P α β H (1) G k G + P (2) P d 0 (3) Species are G: free gene, H: gene with protein bound, P: protein Difficulties in biology, there are not many molecules reactions do not arise deterministically reactions can arise in (almost) any order randomness is important for understanding cells in biology F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

6 A simple example A k 0 similar to a nuclear desintegration A A Figure: Two different simulations with A(0) = 10 and k=1 Note: the simulations are obtained following the Gillespie algorithm, which repeats the following steps: compute when the next reaction will occur compute which reaction will occur update the numbers of molecules according the reaction occurring F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

7 A simple example A k 0 similar to a nuclear desintegration A A A A A A A Figure: Ten superposed simulations with A(0) = 10 et k=1 F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

8 A simple example A k 0 similar to a nuclear desintegration A Figure: Experimental mean value of 1000 simulations F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

9 Probabilities, statistics Notation/properties one denotes X(t) the (integer) number of molecules of X at time t X(t) is a random variable (i.e. at each time t, X(t) is a random variable) Denote π i (t) = Prob(X(t) = i), i.e. probability that there are i molecules of X at time t. Fundamental questions compute π i (t) compute the mean value, variance, moments... of X(t). avoid Gillespie algorithm (time consuming, and missing rare events) try to get exact or approximate formulaes, or numerical solutions if not possible Models of Stochastic Gene Expression and Weyl Algebra, Vidal et al., ANB2010. F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

10 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

11 Complete solution of the degradation A k 0 Initial condition: A(0) = 10 similar to a nuclear desintegration Formal series and the PDE Let us introduce Φ(t, z) = π i (t)z i the chemical master equation (demonstrations skipped) yields : Φ t i=0 = k(1 z) Φ z Maple pdsolve (!) : Φ(t, z) = F ((z 1)e kt ) where F is arbitrary if A(0) = 10, then π 10 (0) = 1, so Φ(0, z) = z 10 easy computation: F(z) = (1 + (z 1)e kt ) 10. One can compute π i (t), and also mean value, variance, moments,... F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

12 Complete solution of the degradation (2) A k 0 similar to a nuclear desintegration Φ(t, z) = π i (t)z i i=0 Mean value (i.e. first moment) by definition: E(X(t)) = i=0 iπ i(t) Φ = z i=1 iπ i(t)z i 1 one obtains the mean value by setting z = 1 E(X(t)) = Φ (t, 1) =... = 10e kt z Second moment similar computations yields (using a second derivative) the second moment where E 2 (X(t)) = i=0 i 2 π i (t) E 2 (X(t)) = 10e t (1 e t ) F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

13 Complete solution of the degradation (2) A k 0 similar to a nuclear desintegration Φ(t, z) = π i (t)z i i= t mean second moment Figure: Mean and second moment, for k = 1 F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

14 Complete solution of the degradation (3) Transformation into system of ODEs: E(X(t)) = ke(x(t)) E 2 (X(t)) = 2kE 2 (X(t)) + ke(t) Infinite If a reaction involves two or more reactants (always the case except in toy examples), an infinite cascad occurs: E(X(t)) = depends on E(X(t)), E 2 (X(t)) E 2 (X(t)) = depends on E(X(t)), E 2 (X(t)), E 3 (X(t))... One cannot truncate the system easily F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

15 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

16 General framework Ideal case introduce an indeterminate z i for each species denote Φ(t, z) = π α(t)z α where z = (z 1,..., z n) and α = (α 1,..., α n) transform the system of reactions into Φ t = HΦ where H is a differential operator in z with polynomial coefficients in z (similarities with an Hamiltonian) solve the PDE! How to solve the PDE? symbolically (as in the degradation example) numerically convert the PDE into a ODE system, where unknown functions are mean values, variances,... F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

17 Difficulties Symbolically for exact solving, special functions arise even for easy examples for approximate solving, truncations are necessary where to truncate? Numerically except in toy systems, the operator H always involves second order derivatives the initial condition is given at t = 0 in the form Φ(0, z) = z α 0 (α 0 is the vector of number of species at t = 0) method of characteristics and usual PDE tricks fail Conversion to an ODE system except in toy systems, the ODE system is infinite, and cannot be truncated exactly (infinite cascad) F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

18 A subproblem Stationary distribution when t, the distribution might stabilize, if so, this distribution is time independant, and satisfies HΦ(z) = 0 (no more t) it gives information on the behaviour at t = Degradation example Φ t one looks fo k(1 z) Φ(z) z = 0 yields, Φ(z) = c, where c is a constant = HΦ where H = k(1 z) z since Φ(z) is a distribution, Φ(1) = 1, so Φ(z) = 1 this means that the system stabilizes with π 0 = 1 (no molecules) F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

19 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

20 Double degradation with creation 2A µ 0 0 λ A Φ t = HΦ where H = 1 2 µ(1 z2 ) 2 + λ(z 1) z2 For the moment, we have no solution for Φ(t, z), but we could solve the stationary distribution Stationary distribution equations HΦ(z) = 0 where Φ(z) = π i z i, π i 0, Φ(1) = i=0 π i = 1 F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

21 A dimension problem? After simplication (1 + z)φ (z) = 2a Φ(z) where a = λ/µ Φ has nonnegative coefficients, whose sum is 1 Uniqueness of the expected solution the physics tells us the expected solution is unique the ODE yield two independent solutions as formal power series around the origin Looking for the right solution the expected f.p.series solution converges in 1, so in -1 also the expected solution goes "through" the singularity in -1 F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

22 Two solvings Hypergeometric (Petitot) ( ) Φ(z) = K 1 (1 + z) 0 F 1 2 2a(1 + z) where ( ) K = 2 0 F 1 2 4a Expansion at z=-1 By a change of variable y = 1 + z, and φ(y) = b i y i, and using gfun, b 0 = 0 (4) b 1 arbitrary (5) b i+1 = 2a b i for i 1 (6) i(i + 1) Remark: Φ has an infinite radius of convergence, and is extremely convergent (I believe b i = o((i!) 2+ɛ ). F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

23 A generic example p times {}}{ A + A + + A µ 0 0 λ A Φ t Stationary distribution equations = HΦ where H = 1 p! µ(1 zp ) p + λ(z 1) zp (1 + z + + z p 1 )Φ (p) (z) = p!a Φ(z) where a = λ/µ Φ has nonnegative coefficients, whose sum is 1 finding the right solution p-dimensional space of solutions for the PDE the wanted solution goes through the p 1 singularities (p 1 roots of the unity) F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

24 Thoughts / conjectures Stationary distribution equations (1 + z + + z p 1 )Φ (p) (z) = p!a Φ(z) where a = λ/µ Φ has nonnegative coefficients, whose sum is 1 Difficulties "easy" to solve at each singularity, hard to deal with several singularities at the same time a Picard approach does not always converge (by solving the BVP satisfying Φ(w) = 0 for each singularity w, and Φ(1) = 1) Conjectures there is only one solution with an infinite radius of convergence (and the general term is o((i!) p+ɛ ) the expected solution is that one! F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

25 Plan 1 Background/Motivation 2 Complete treatment of the degradation 3 General framework 4 Stationary distribution for different examples 5 Conclusion F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

26 Towards the real examples real examples are in several variables one needs to treat the complete system (not just the stationary distribution) G + P α β H (7) G k G + P (8) P d 0 (9) Φ t = α(h GP) 2 Φ Φ Φ Φ + β(gp H) + k(gp G) + d(1 P) G H H G P where Φ = Φ(t, G, H, P) Remark: the term H GP is the "singularity" for the stationary distribution. F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24

27 Software See for a prototype in Maple New version in preparation (Michel Petitot, Kyrill Batmanov, Christian Versari,... ) enough work for a PhD student (proposal made this year in the Computer Algebra team) F. Lemaire (Lille 1) Chemical Reactions and PDE FELIM / 24