Large deviationsfor models in

Systems Biology

Lea PopovicConcordia University, Montreal

(with Giovanni Zorrodu)

IMA, Ecological and Biological SystemsJune 2018

Stochastic processes in systems biology

• chemical reaction networks describing basic cellular processes

• stochasticity arises from molecular interactions andenvironmental noise

• different amounts of molecular abundances and interactionrate magnitudes implies some stochastic features are essentialand persistent in the long-term dynamics

Rare events in systems biology

? departure from typical behaviour (± st.dev. fluctuations)

? in bistable systems leads to transitions to new stable state

? deviations can arise from intrinsic stochasticity of the system(rather than from external perturbations)

Bistable examples in systems biology

? competing positive-negative feedback, enzymatic futile cycle,de/phosphorylation cycle, bistable repressible switch

yeast cells switch between expressing and nonexpressing states(Kaufmann, Yang, Mettetal, and van Oudenaarden, PLOS Bio. 2007)

I cell growth+division: rare events have many trials to occurnot rare on the level of the cell population

Stochastic models for reaction network system

• continuous-time Markov chains and their rescaled versions

• multi-scale properties can lead to various limiting processes:

jump diffusions with state-dependent coefficients

Xt = X0 +

∫ t

0µ(Xs)ds +

∫ t

0σ(Xs)dWs +

∑s≤t

∆Xs

jumps have rate λ(X ) and jump measure∫|y |ν(X , dy) <∞

• reflection of Xs ∈ [0, b] at the boundaries 0, b ≤ ∞

Vt = V0 + Xt + Lt − Ut

Lt and Ut are local times at the lower and upper boundary

• intrinsic constraints in reaction systems give processes with reflection

(Leite and Williams, www.math.ucsd.edu 2017)

Examples

I self-regulated gene expression

w/ protein bursts (due to protein translation+transcription):

Xt = X0 +

∫ t

0(c±0 +c±0?Xs

c±1 +c±2 Xs− c3Xs)ds

+ ε

∫ t

0(c±0 +c±0?Xs

c±1 +c±2 Xs

[ c±4 +c±4?x

(c±1 +c±2 Xs)2+c±5

]+ c3Xs)dWs + δ

∑s≤t

∆Xs

I enzymatic (Michaelis-Menten) kinetics

w/ substrate bursts (due to cell division):

Xt = X0 −∫ t

0

Xsc1+c2Xs

ds + ε

∫ t

0

c3Xs

(c4+c5Xs)2dWs + δ∑s≤t

∆Xs

Short-term behaviour

on a finite time interval t ∈ [0,T ]

when noise contribution (ε, δ) is small

? FLLN: (Xt)[0,T ] → (xt)[0,T ] with dxt = µ(xt)dt

? FCLT: 1ε (Xt − xt)[0,T ] ⇒ (Ut)[0,T ] with U Gaussian

? PLDP: P( supt∈[0,T ]

|Xt − x̃t | ≤ ε) ≈ e−εI ((x̃t)[0,T ])

(Freidlin and Wentzell; Anderson and Orey - for reflected processes)

calculating the rate function I (x̃)

• from limiting non-linear exponential operator:

Hf = limε→0

log E[e1εf ((Xt)[0,T ])], I (x̃) = sup

f ∈Cb([0,T ])[f (x̃)− Hf ]

• operator H has additional term on the boundary for reflection

(Feng and Kurtz ‘Large Deviations for Stochastic Processes’ 2006)

0.25 0.75

sample path and occupation measure for δ � ε2

sample path and switching times for δ � ε2 (same system)

(McSweeney and Popovic, Ann.Appl.Prob. 2014)

Long-term behaviour

on long time intervals t →∞with sizeable contribution from diffusion and jumps

? assuming Vt is ergodic, stationary measure π, gives average:

1

t

∫ t

0f (Vs)ds →

t→∞Eπ[f (x)]

? fluctuations for f with Eπ[f (x)] = 0 can be calculated usingsolution g of the Poisson equation:

f (x) = µ(x)g ′(x) +1

2σ2(x)g ′′(x)

+ λ(x)

∫(g(δ(x , y))− g(x))ν(x , dy), g ′(0) = g ′(b) = 0

where Vs = δ(Vs−,∆Xs) if ∆X 6= 0

for δ(x , y) = 0 1x+y≤0 + (x + y)1x+y∈(0,b) + b 1x+y≥b

Generalized occupation time

• consider the Additive process - generalization of local time

Λt =

∫ t

0f (Vs)ds +

∑s≤t

f̃ (Vs−,∆Xs) + f0Lct + fbU

ct

f bdd, f̃ (x , 0) = 0,∫eθf̃ (x ,y)ν(x , dy) < C ∀x , f0, fb ∈ R+

Lc ,Uc are the continuous parts of local times at 0, b

I e.g. Λt = Lt with f = 0, f̃ = −[x + y ]−, f0 = 1, fb = 0

this gives the fraction of time a protein is not expressed in, orfraction of time enzymatic substrate is at extremely low level

Gaertner-Ellis theorem

for stochastic process on Rd

I Suppose1

tE[eθΛt ] →

t→∞ψ(θ)

exists for ψ such that

I 0 ∈ int(Dψ), Dψ = {θ : ψ(θ) <∞}I ψ is lower semi-conts, differentiable on int(Dψ)

I Dψ = R or limθ→∂Dψ |ψ′(θ)| =∞

ThenΛtt satisfies the Large Deviation Principle

with ‘speed’ t and ‘good’ convex rate function

ψ∗(a) = supθ∈R

[aθ − ψ(θ)]

Integro-differential equation

each fixed θ → pair (uθ(x), ψθ)

I Suppose there exist uθ(x) ∈ C2≥0 and ψθ ∈ R satisfying the

integro-differential equation

0 = µ(x)u′θ(x) +1

2σ2(x)u′′θ (x) + (θf (x)− ψθ)uθ(x)

+ λ(x)

∫(eθf̃ (x ,y)u(θ, δ(x , y))− uθ(x))ν(x , dy)

subject to boundary conditions

uθ(0) = 1, u′θ(0) = −f0θ, uθ(b) = 1, u′θ(b) = fbθ

? the integral term makes it somewhat more difficult to solvethe differential equation boundary problem numerically

Martingale argument

I Then (under assumptions on µ, σ, λ, ν)

Mt(θ) = eθΛtuθ(Vt)

is a martingale (uθ is bounded) and for each θ

1

tlog E[eθΛt ] →

t→∞ψθ

I Assuming ψ(θ) := (ψθ)θ∈Dψ satisfies Gaertner-Ellis theorem

1

tlog P[Λt ≥ at] →

t→∞ψ(θ∗)− θ∗a,

for any θ∗ > 0 such that ψ ∈ C 1 around θ∗, a = ∂θψ(θ∗)

? numerically solving for ψ ⇒ ‘verifying’ needed assumptionsonly numerically ⇒ estimating probabilities of rare events

Simulations

I Vt is jump-diffusion reflected at: 0 and b = 2.5 withdrift and diffusion: µ(x) = 2.62(1.61− x), σ(x) = 0.62

√x

jump rate and measure: λ(x) = x , ν(x , dy) = 1210.72 + 1

212.87

U t = local time at b and Lt = local time at 0

I Λt = Lt is local time at 0

1tE[Lt ] ∼ mina ψ

∗(a) LDP rate function ψ∗