Time dependent multivariate distributions for piecewise-deterministic models of gene networks Ovidiu Radulescu, Guilherme D.C.P. Innocentini University of Montpellier
Stochastic gene expression Heterogeneity of clone cells populations Dynamics, individual cell Dynamics of stochastic gene expression, population Montpellier-BIOSS 2017, 2 / 15
Applications Adaptation of heterogeneous Gaines/Poppelbaum populations: stochastic computing stochastic switching Montpellier-BIOSS 2017, 3 / 15
Markovian models of stochastic promoters Montpellier-BIOSS 2017, 4 / 15
Multi-scaleness of stochastic gene expression Switching and discreteness timescales (a) The partition of species and of the reactions; dotted lines mean that reaction rates depend on the corresponding species. (b) Typical trajectories of continuous and discrete variables: switched processes. Radulescu et al Phys.Rev.E 85 (2012) 041919. Montpellier-BIOSS 2017, 5 / 15
Continuous and hybrid approximations Montpellier-BIOSS 2017, 6 / 15
Continuous and hybrid approximations Montpellier-BIOSS 2017, 7 / 15
Continuous and hybrid approximations Montpellier-BIOSS 2017, 8 / 15
Partial omega expansion Ω : size variable, x c = X c / Ω . ∂ p [ V i ( X D − γ D i , x c ; µ ) p ( X D − γ D ∑ ∂ t ( X D , x c , X , t ) = i , x c , t ) − V i ( X ; µ ) p ( X D , x c , t )]+ i ∈ R D ∪ R DC Ω [ v i ( X D , x c − γ c i / Ω ; µ ) p ( X D , x c − γ C ∑ i / Ω , t ) − v i ( X ; µ ) p ( X D , x c , t )] + i ∈ R C ∪ R CD Master equation ∂ p − ∂ [ Φ ( X D , x c ; µ ) p ( X D , x c , t )] [ V i ( X D − γ D i , x c ; µ ) p ( X D − γ D ∑ ∂ t ( X D , x c , t ) = + i , x c , t ) − ∂ x c i ∈ R D ∪ R DC − V i ( X D , x c ; µ ) p ( X D , x c , t )] , where ∑ γ C Φ ( X D , x c ; µ ) = i v i ( X D , x c ; µ ) i ∈ R C ∪ R CD Liouville-master equation Montpellier-BIOSS 2017, 9 / 15
Single gene ON/OFF promoter: chemical master equation X : P , Y : R . ∂ p ∂ t ( 1 , X , Y , t ) = k Ω ( p ( 1 , X , Y − 1 , t ) − p ( 1 , X , Y , t ))+ + ρ (( Y + 1 ) p ( 1 , X , Y + 1 , t ) − Yp ( 1 , X , Y , t ))+ bY ( p ( 1 , X − 1 , Y , t ) − p ( 1 , X , Y , t ))+ + a (( X + 1 ) p ( 1 , X + 1 , Y , t ) − Xp ( 1 , X , Y , t ))+ fp ( 0 , X , Y , t ) − hp ( 1 , X , Y , t ) ∂ p ∂ t ( 0 , X , Y , t ) = ρ (( Y + 1 ) p ( 0 , X , Y + 1 , t ) − Yp ( 0 , X , Y , t ))+ bY ( p ( 0 , X − 1 , Y , t ) − p ( 0 , X , Y , t ))+ + a (( X + 1 ) p ( 0 , X + 1 , Y , t ) − Xp ( 0 , X , Y , t ))+ hp ( 1 , X , Y , t ) − fp ( 0 , X , Y , t ) Montpellier-BIOSS 2017, 10 / 15
Single gene ON/OFF promoter: Liouville-master equation ∂ p − ∂ [( by − ax ) p ( 1 , x , y , t )] − ∂ [( k − ρ y ) p ( 1 , x , y , t )] ∂ t ( 1 , x , t ) = + fp ( 0 , x , y , t ) − hp ( 1 , x , y , t ) ∂ x ∂ y ∂ p − ∂ [( by − ax ) p ( 0 , x , y , t )] − ∂ [ − ρ yp ( 0 , x , y , t )] ∂ t ( 0 , x , y , t ) = + hp ( 1 , x , y , t ) − fp ( 0 , x , y , t ) ∂ x ∂ y x : X / Ω , y : Y / Ω . Montpellier-BIOSS 2017, 11 / 15
Single gene ON/OFF promoter: Monte-Carlo (1) Set s = s ( 0 ) , x = x ( 0 ) , y = y ( 0 ) , t = t 0 , i = 0. (2) Generate u ∼ U [ 0 , 1 ] , (3) Integrate the system of ODEs dx dt = by − ax dy dt = k δ s , 1 − ρ y , dF dt = − ( f + h ) F , x ( t i ) = x ( i ) , y ( t i ) = y ( i ) , F ( t i ) = 1 , between t i and t i + τ i with the stopping condition F ( t i + τ i ) = u . (4) Generate v ∼ U [ 0 , 1 ] use it to pick s ( i + 1 ) . (the decision is made in the same way as in the Gillespie algorithm). (5) Change the system state ( s ( i ) , x , y ) to ( s ( i + 1 ) , x , y ) , and the time t i to t i + 1 = t i + τ i . (6) Reiterate the system from 2) with the new state until a time t max previously defined is reached. Montpellier-BIOSS 2017, 12 / 15
Single gene ON/OFF promoter: push forward (1) Consider fixed partition t 0 = 0 , t 1 ,..., t N = T fine enough such that s is constant on most subintervals. (2) For each possible instance s 0 , s 1 ,..., s N − 1 compute its probability ∑ P [ s 0 ,..., s N − 1 ] = P [ s 0 ] P [ s 1 | s 0 ] ... P [ s N − 1 | s N − 2 ] s 0 ,..., s N − 1 where P [ s k + 1 | s k ] is the exact solution of dp 0 dt = − fp 0 + h ( 1 − p 0 ) , p 1 = 1 − p 0 . (3) Compute x ( t ) and y ( t ) as exact solutions of dx dy dt = by − ax , dt = k δ s , 1 − ρ y (4) Gather all x ( t ) , y ( t ) leading to the same distribution bin in ( x , y ) plane and sum the probabilities. NB: we use the exact distribution of s to obtain the one of x , y . Possible to use the exact distribution of ( s , y ) to obtain the one of x Montpellier-BIOSS 2017, 13 / 15 Innocentini et al, Bull Math Biol (2016) 78:110-131.
Single gene ON/OFF promoter: illustration of the methods Dynamical evolution of protein probability density for slow switch ( ε = ( h + f ) / ρ = 0.1) in a) and fast switch ( ε = 5) in b). Innocentini et al, Bull Math Biol (2016) 78:110-131. Montpellier-BIOSS 2017, 14 / 15
Gene networks: work in progress ◮ The methods can be applied to any combination of promoters with or without feed-back. ◮ Limitations imposed by the number of distinct genes N g . ◮ Push-forward is better than solving the 2 N g Liouville-master PDEs. ◮ Push-forward is better than Monte-Carlo for small to medium N g (circuits). Montpellier-BIOSS 2017, 15 / 15
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