Modélisation probabiliste en biologie cellulaire et mol ... fileBursting Nucleation Modélisation probabiliste en biologie cellulaire et moléculaire Thèse sous la direction de

Bursting Nucleation

Modelisation probabiliste en biologie cellulaire etmoleculaire

These sous la direction de

M. Adimy, M.C. Mackey & L. Pujo-Menjouet

Romain Yvinec

Institut Camille Jordan - Universite Claude Bernard Lyon 1

Vendredi 05 octobre 2012

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Outline

Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model

Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time

Outline



Bursting Nucleation Experiments Reduction Limiting model

Central Dogma

◮ Expression of a gene through transcription/translationprocesses.

◮ Non-linear Feedback regulation.

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◮ Bifurcation analysis in Ordinary Differential Equation.

◮ Application to synthetic biology.

[Goodwin, 1965],[Hasty et al., 2001].

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Stochasticity in molecular biology

[Eldar and Elowitz, 2010].

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Much more accurate measurements

◮ Bifurcation can be studied on probability distributions.

[Song et al., 2010].

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Much more accurate measurements

◮ Trajectories can be analyzed on single cells.

[Yu et al., 2006].

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New Central dogma

◮ Take into account gene state switching. Interpretation asstochastic processes.

[Berg, 1978],[Peccoud and Ycart, 1995],[Kepler and Elston, 2001],[Paulsson, 2005],[Lipniacki et al., 2006],[Paszek, 2007],

[Shahrezaei and Swain, 2008].

The bursting phenomena

Question 1) When does the stochastic model predict burstphenomenon ?Question 2) What can we say in such cases ?

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Outline




We consider the following 2d stochastic kinetic chemical reactionmodel (X=’mRNA’, Y=’Protein’)

∅λ1(X ,Y )−−−−−→ X , Production of X at rate λ1(X ,Y )

XNγ1(X ,Y )−−−−−−→ ∅, Destruction of X at rate Nγ1(X ,Y )

∅Nλ2(X ,Y )−−−−−−→ Y , Production of Y at rate Nλ2(X ,Y )

Yγ2(X ,Y )−−−−−→ ∅, Destruction of Y at rate γ2(X ,Y )

with γ1(0,Y ) = γ2(X , 0) = 0 to ensure non-negativity.

BN f (x , y) =λ1(x , y)[

f (x + 1, y)− f (x , y)]

+ Nγ1(x , y)[

f (x − 1, y) − f (x , y)]

+ Nλ2(x , y)[

f (x , y + 1)− f (x , y)]

+ γ2(x , y)[

f (x , y − 1)− f (x , y)]

.

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Theorem (R.Y.)If

◮ The degradation function on X satisfies

infx≥1,y≥0

γ1(x , y) = γ > 0.

◮ The production rate of Y satisfies λ2(0, y) = 0, for all y ≥ 0.

◮ λ1 and λ2 are linearly bounded by x + y , and either λ1 or λ2 isbounded.

Then, for all T > 0, (XN(t),Y N(t))t≥0 converges in L1(0,T ) to(0,Y (t)), whose generator is given by

B∞ϕ(y) = λ1(0, y)( ∫ ∞

0

Pt(γ1(1, · )ϕ(· ))(y)dt − ϕ(y))

+ γ2(0, y)[

ϕ(y − 1)− ϕ(y)]

,

Ptg(y) = E[g(Z (t, y)e−

∫t

0γ1(1,Z (s,y))ds

],

Ag(z) = λ2(1, z)(g(z + 1)− g(z)

).

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Sketch of the proof

◮ We first show tightness and convergence of X based on

NγE[∫ t

01{XN(s)≥1}ds

]≤ E

[XN(0)

]+E

[∫ t

0λ1(X

N(s),Y N(s))ds].

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Sketch of the proof


NγE[∫ t

01{XN(s)≥1}ds

]≤ E

[XN(0)

]+E

[∫ t

0λ1(X

N(s),Y N(s))ds].

◮ We identify the limiting martingale problem

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Sketch of the proof


NγE[∫ t

01{XN(s)≥1}ds

]≤ E

[XN(0)

]+E

[∫ t

0λ1(X

N(s),Y N(s))ds].


λ2(x , y)[

f (x , y+1)−f (x , y)]

+γ1(x , y)[

f (x−1, y)−f (x , y)]

= 0.9/40


Sketch of the proof


NγE[∫ t

01{XN(s)≥1}ds

]≤ E

[XN(0)

]+E

[∫ t

0λ1(X

N(s),Y N(s))ds].


Axg(y) = λ2(x , y)[

g(y + 1)− g(y)]

,

for any x ≥ 1. and we introduce the semigroup Pxt

Pxt g(y) = E

[g(Z x ,y

t )e−∫ t0 γ1(x ,Z

x,ys )ds

].

Now for any bounded function g , define f (0, y) = g(y) and

f (x , y) =

∫ ∞

0Pxt (γ1(x , .)f (x − 1, .))(y)dt.

Then

λ2(x , y)[

f (x , y+1)−f (x , y)]

+γ1(x , y)[

f (x−1, y)−f (x , y)]

= 0.9/40


◮ A similar proof for a (continuous state) PDMP model, ofgenerator

Bf (x , y) =− Nγ1(x , y)∂f

∂x+ (Nλ2(x , y)− γ2(x , y))

∂f

∂y

+ λ1(x , y)

∫ ∞

0(f (x + z , y)− f (x , y))h(z)dz .

◮ These proofs are based on a simple idea([Debussche et al., 2011],[Kang and Kurtz, 2011]).

◮ Other proof : reduction on the Fokker-Planck equation.

◮ Different scalings lead to different models.

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Outline




We look at the stochastic process

dx = −γ(x)dt + dN(λ(x), h(x , ·)),

whose generator is

Af = −γ(x)f ′(x) + λ(x)( ∫ ∞

0f (x + y)h(x , y)dy − f (x)

)

,

and evolution equation on densities

∂u(t, x)

∂t=

∂γ(x)u(t, x)

∂x−λ(x)u(t, x)+

∫ x

0u(t, y)λ(y)h(y , x−y)dy ,

and with

∫ ∞

0h(x , y)dy = 1, for all x .

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Probabilistic techniques

If jumps are independent of positions, i.e. h(x , y) = h(y), we have :

Proposition

Suppose x 7→ λ(x) is continuous on (0,∞), λ(0) > 0, γ(x) = γx,E[h]< ∞, and

limx→∞

λ(x)E[h]

γx< 1,

then there exist β < 1, B < ∞ and π (invariant measure) such that

‖P(t, x , ·) − π‖V ≤ BV (x)βt , x ∈ E , t > 0,

where ‖µ‖f = sup|g |≤f | µ(g) | and V (x) = x + 1.

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Probabilistic techniques

If jumps are independent of positions, i.e. h(x , y) = h(y), we have :

Proposition

Suppose x 7→ λ(x) is continuous on (0,∞), λ(0) > 0, γ(x) = γx,E[h]< ∞, and

limx→∞

λ(x)E[h]

γx< 1,

then there exist β < 1, B < ∞ and π (invariant measure) such that

‖P(t, x , ·) − π‖V ≤ BV (x)βt , x ∈ E , t > 0,

where ‖µ‖f = sup|g |≤f | µ(g) | and V (x) = x + 1.

◮ Ax = −γx + λ(x)( ∫∞

0 (x + y)h(y)dy − x)

= −(1−λ(x)E

[h

]

γx)γx

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Semigroup techniques

∂u(t, x)

∂t︸︷︷︸

dudt

=∂γ(x)u(t, x)

∂x− λ(x)u(t, x)

︸︷︷︸

Au=(A0−λ)u

+

∫ x

0u(t, y)λ(y)h(y , x − y)dy

︸︷︷︸

Bu=J(λu)

(A,D(A)) ⇒ S(t)u(x) = P0(t)u(x)e−

∫t

0λ(φr x)dr

Let C = A+ B. Denote the resolvent RSs u =

∫∞0 e−stS(t)udt.

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Semigroup techniques

∂u(t, x)

∂t︸︷︷︸

dudt

=∂γ(x)u(t, x)

∂x− λ(x)u(t, x)

︸︷︷︸

Au=(A0−λ)u

+

∫ x

0u(t, y)λ(y)h(y , x − y)dy

︸︷︷︸

Bu=J(λu)

(A,D(A)) ⇒ S(t)u(x) = P0(t)u(x)e−

∫t

0λ(φr x)dr

Let C = A+ B. Denote the resolvent RSs u =

∫∞0 e−stS(t)udt.

Theorem ([Tyran-Kaminska, 2009])There is a minimal substochastic semigroup P generated by anextension of (C ,D(A)), and which resolvent is given by

RPs u = lim

n→∞

RSs

n∑

k=0

(J(λRSs ))

ku,

and if K = limσ→0 J(λRSσ ) has a unique invariant density, then so

does for P (and P is stochastic).

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◮ Under good conditions, K is the transition operator for thediscrete Markov chain“post-jump”, and has for kernel

k(x , y) =

∫ x

0

1{(0,y)}(z)h(z , y − z)λ(z)

γ(z)eQ(x)−Q(z)dz ,

Q(x) =

∫ x

x

λ(z)

γ(z)dz .

◮ Modulo integrability conditions, invariant density v∗ for K andinvariant density u∗ for P are related through

γ(x)u∗(x) =

∫ x

0

H(z , x − z)λ(z)u∗(z)dz , H(z , x) =

∫ ∞

x

h(z , y)dy ,

v∗(x) =

∫ x

0

h(z , x − z)λ(z)u∗(z)dz ,

u∗(x) =1

γ(x)

∫ ∞

x

eQ(y)−Q(x)v∗(y)dy ,

v∗(x) =

∫ x

0

h(z , x − z)λ(z)

γ(z)e−Q(z)

∫ ∞

z

v∗(y)eQ(y)dydz .

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Condition for ergodicity in the exponential case

If jumps are independent of positions, i.e. h(x , y) = h(y) andexponentially distributed, of mean b, i.e. h(y) = 1

be−y/b, then

Theorem (M. Tyran-Kaminska, M. Mackey, R.Y.)

Under technical assumptions (for integrability), and if

limx→∞

λ(x)

γ(x)<

1

b,

Q(0) :=

∫ x

0

λ(z)

γ(z)dz = ∞,

then P is ergodic with unique invariant density

u∗(x) =1

cγ(x)e−x/b−Q(x).

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Bifurcation

This analytical approach allows us to deduce that the number ofmodes of the stationary state is linked to the solution of

λ(x) =γ(x)

b+ γ′(x).

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Further results (not developed here)

◮ This can be used to find λ(x) and b from observations of(u∗, γ).

◮ The convergence rate can be estimated from couplingtechniques.

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Further results (not developed here)

◮ This can be used to find λ(x) and b from observations of(u∗, γ).

◮ The convergence rate can be estimated from couplingtechniques.

Perspectives

◮ Other jump size kernel h.

◮ Waiting time properties.

◮ Switch and bursting model.

◮ Include cell division and study population dynamics.

◮ Characterize oscillations in two-dimensional model.

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Outline



Bursting Nucleation Prion Model Experiments Nucleation time

Prion Diseases

◮ Creutzfeldt-Jakob : Firsthuman prion diseasedescribed (1929).

◮ Mad cow disease, Scrapie.

◮ Kuru (New Guinea)

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Prion Diseases

Epidemiology

◮ All prion diseases aretransmissible, i.e. infectious.

◮ Some prion diseases aresporadic, they appearspontaneously, withoutcause.

◮ Some prion diseases aregenetic.

Symptoms

◮ Affect the structure of thebrain ;

◮ Convulsion, Dementia, Lossof balance ;

◮ Always fatal.

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Outline




What is prion ?

A protein

◮ A protein called PRION isthe cause of this disease

◮ It is neither a bacteria, noran viroid like agent !

◮ Stanley Prusiner wasawarded Nobel price inPhysiology and Medicine in1997 for his discovery.

Histopathology

◮ Accumulation of a protein inthe amyloid form.

◮ Spongiosis.

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Nucleation-Polymerization

Conformation change

Normal ⇔ Misfolded

Lansbury’s model

[Lansbury and Caughey, 1995].

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Outline




◮ In vitro spontaneous polymerization experiments.

◮ Time series of polymer mass.

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Statistics of nucleation time

◮ Relation between the nucleation time and the initialconcentration in log plots.

◮ Full distribution of the nucleation time.

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Questions

◮ Can a probabilistic model reproduce the observed variability ?

◮ Can it help to identify parameters ?

◮ Can a model include different strain structures ?

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Outline




Reversible aggregation model

Ci + C1

p−⇀↽−qCi+1

where

Ci = ♯{molecules of size i}.

The nucleation time is given by a waiting time problem,

Tlag = inf{t ≥ 0 : CN(t) = 1},

with initial condition C1(0) = M, Ci(0) = 0, i ≥ 2.N is the nucleus size.

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Constant monomer formulation

If we supposeC1(t) ≡ M

We can solve exactly the probability distributions (Poisson) and wededuce

S(t) := P{CN(s) = 0, s ≤ t

}= P

{CN(t) = 0

}= e−cN (t),

where (ci )i=2..N are solution of the linear deterministic system (cnis absorbing) :

c2 = pM(12M − c2)− q(c2 − c3),ci = pM(ci−1 − ci )− q(ci − ci+1), 3 ≤ i ≤ N − 2,cN−1 = pM(cN−2 − cN−1)− qcN−1,

cN = pMcN−1.

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Nucleation time distribution (C1(t) ≡ M)

M → ∞ : Weibull MN

2(N−2)!tN−2 exp(− MN

2(N−1)! tN−1)

q → ∞ : exponential MN

2qN−2 exp(−MN

2qN−2 t)

N = 6,M = 1000,

q =

102,

103,

4.103,

104.

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Mean nucleation time versus initial monomer quantity inlog scale (C1(t) ≡ M)

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Nucleation time distribution in log scale (C1(t) ≡ M)

M → ∞ : Weibull MN

2(N−2)!tN−2 exp(− MN

2(N−1)! tN−1)

q → ∞ : exponential MN

2qN−2 exp(−MN

2qN−2 t)

N = 6,M = 1000,

q =

102,

103,

4.103,

104.

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Mass conservative formulation

We now supposeN∑

i=1

iCi(t) ≡ M.

Kolmogorov backward equations ⇒ Linear system on

S(t, {C 0}) = P{CN(t) = 0 | Ci(0) = C 0

i

}

Problem : Dimension of the system

♯{configuration {C 0},N∑

i=1

iC 0i = M,C 0

N = 0} ≈MN

N!

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In general, we look for approximate solution for extreme parametervalues : q ≫ M and q ≪ M. We use

◮ known deterministic solution ;

◮ time scale separation ;

◮ scaling laws ;

◮ phase space dimension reduction ;

◮ linear model ;

◮ numerical simulation.

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Trajectories in the unfavorable case q ≫ M

Pre-equilibrium hypothesis (ex : M = 200,N = 8,q = 1000).

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Nucleation time distribution in the unfavorable caseq ≫ M : exponential law

Tlag ∼ exponential law, of parameter

< C1CN−1 >t→∞ (M) ≈ c1(t → ∞)cN−1(t → ∞) ≈MN

2qN−2

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Trajectories in the favorable case M >> q and N large

“Metastable“ trajectory (ex :M = 30000,N = 10,q = 1).

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Trajectories in the favorable case M >> q and N large

“Metastable“ trajectory. Known phenomenon for the deterministicmodel ([Penrose, 1989],[Wattis, 2006])

1. irreversibleaggregation(up to c∗i )

2. slow ”diffusion”with constantmonomerC1(t) ≡ c∗1

3. convergence toequilibrium

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Nucleation time distribution in the favorable case M >> q

and N large, c∗N < 1 : bimodal distribution

c∗N < 1 : Linear model with C1 ≡ c∗1 , Ci (0) = c∗i (solid line).

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Nucleation time distribution in the favorable case M >> q

and N small, c∗N > 1 : Weibull law

c∗N > 1 : Linear model with C1 ≡ M, Weibull law (dashed line).

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Mean nucleation time versus initial monomer quantity inlog scale : 2 or 3 phases according nucleus size N

1. exponential

2. Linearmodel(startingfrom c∗i )

3. Weibull

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Mean nucleation time versus initial monomer quantity inlog scale : 2 or 3 phases according nucleus size N

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Conclusion/Perspectives

◮ Different behavior of the nucleation time

◮ Parameter Identifiability depending on parameter region

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Conclusion/Perspectives

◮ Different behavior of the nucleation time

◮ Parameter Identifiability depending on parameter region

Perspectives

◮ Different nucleation regime ⇒ Different polymerization regime

◮ Possibility to take into account different polymer structures

◮ Study the nucleation time for size-dependent rates

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