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Time dependent multivariate distributions for piecewise-deterministic models of gene networks

Ovidiu Radulescu, Guilherme D.C.P. Innocentini University of Montpellier

Montpellier-BIOSS 2017, 2 / 15

Stochastic gene expression

Heterogeneity of clone cells populations Dynamics, individual cell Dynamics of stochastic gene expression, population

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Applications

Adaptation of heterogeneous populations: stochastic switching Gaines/Poppelbaum stochastic computing

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Markovian models of stochastic promoters

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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.

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Continuous and hybrid approximations

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Continuous and hybrid approximations

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Continuous and hybrid approximations

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Partial omega expansion Ω: size variable, xc = Xc/Ω.

∂p ∂t (XD,xc,X,t) =

∑

i∈RD∪RDC

[Vi(XD −γD

i ,xc;µ)p(XD −γD i ,xc,t)− Vi(X;µ)p(XD,xc,t)]+

+

∑

i∈RC∪RCD

Ω[vi(XD,xc −γc

i /Ω;µ)p(XD,xc −γC i /Ω,t)− vi(X;µ)p(XD,xc,t)]

Master equation

∂p ∂t (XD,xc,t) = − ∂[Φ(XD,xc;µ)p(XD,xc,t)] ∂xc +

∑

i∈RD∪RDC

[Vi(XD −γD

i ,xc;µ)p(XD −γD i ,xc,t)−

−

Vi(XD,xc;µ)p(XD,xc,t)],where

Φ(XD,xc;µ) =

∑

i∈RC∪RCD

γC

i vi(XD,xc;µ)

Liouville-master equation

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Single gene ON/OFF promoter: chemical master equation

X : P, Y : R.

∂p ∂t (1,X,Y,t) =

kΩ(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)

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Single gene ON/OFF promoter: Liouville-master equation

∂p ∂t (1,x,t) = −∂[(by − ax)p(1,x,y,t)] ∂x − ∂[(k −ρy)p(1,x,y,t)] ∂y + fp(0,x,y,t)− hp(1,x,y,t) ∂p ∂t (0,x,y,t) = −∂[(by − ax)p(0,x,y,t)] ∂x − ∂[−ρyp(0,x,y,t)] ∂y + hp(1,x,y,t)− fp(0,x,y,t)

x : X/Ω, y : Y/Ω.

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Single gene ON/OFF promoter: Monte-Carlo

(1) Set s = s(0), x = x(0), y = y(0), t = t0, i = 0. (2) Generate u ∼ U [0,1], (3) Integrate the system of ODEs

      

dx dt = by − ax dy dt = kδs,1 −ρy, dF dt = −(f + h)F,

x(ti) = x(i),y(ti) = y(i),F(ti) = 1, between ti and ti +τi with the stopping condition F(ti +τi) = 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 ti to ti+1 = ti +τi. (6) Reiterate the system from 2) with the new state until a time tmax previously defined is reached.

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Single gene ON/OFF promoter: push forward

(1) Consider fixed partition t0 = 0,t1,...,tN = T fine enough such that s is constant on most subintervals. (2) For each possible instance s0,s1,...,sN−1 compute its probability

P[s0,...,sN−1] =

∑

s0,...,sN−1

P[s0]P[s1|s0]...P[sN−1|sN−2]

where P[sk+1|sk] is the exact solution of

dp0 dt = −fp0 + h(1− p0),

p1 = 1− p0. (3) Compute x(t) and y(t) as exact solutions of

dx dt = by − ax, dy dt = kδs,1 −ρy

(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

Innocentini et al, Bull Math Biol (2016) 78:110-131.

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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.

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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

Ng.

◮ Push-forward is better than solving the 2Ng

Liouville-master PDEs.

◮ Push-forward is better than Monte-Carlo for small to

medium Ng (circuits).