Nucleation time in stochastic Becker-Doring model

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Nucleation time in stochastic Becker-Doring model

Romain Yvinec, Samuel Bernard, Tom Chou, Julien Deschamps, Maria R.

d’Orsogna, Erwan Hingant, Laurent Pujot-Menjouet

To cite this version:

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Nucleation Time in Stochastic Becker-D¨

oring

Model

Romain Yvinec1, Samuel Bernard2,3, Tom Chou4, Julien

Deschamps5, Maria R. D’Orsogna6, Erwan Hingant7 and

Laurent Pujo-Menjouet2,3

1

BIOS group INRA Tours, France.

2_{Institut Camille Jordan, Universit´}_{e Lyon 1, Villeurbanne, France.} 3

INRIA Team Dracula, Inria Center Grenoble Rhˆone-Alpes, France. 4_{Depts. of Biomathematics and Mathematics, UCLA, Los Angeles, USA.}

5

DIMA, Universita degli Studi di Genova, Italy. 6_{Dept. of Mathematics, CSUN, Los Angeles, USA.} 7

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Numerical results

Coarse-graining

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Outline

Amyloid diseases and Becker-D¨oring model

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Protein accumulation in amyloid by

nucleation-polymerization

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Protein accumulation in amyloid by

nucleation-polymerization

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Times series of in-vitro spontaneous polymerization

Xue et al. PNAS (2008)

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Quantification of experiment

ThT fluorescence « Number of polymerized protein

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Becker-D¨

oring model

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Becker-D¨

oring model

Reversible one-step agregation Ci ` C1 pi ÝÝá âÝÝ qi `1 Ci `1 (1)

§ _{Purely dynamic model (law of mass-action) : no space, no polymer}

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Becker-D¨

oring model

§ _{Indirect interaction between polymer C}_i_{, i ě 2 via the available}

number of monomers C1.

C1ptq `

ÿ

i ě2

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Becker-D¨

oring model

§ _{In spontaneous polymerization experiment,}

§ Initial condition given by C_i_{pt “ 0q “ 0 @i ě 2.} § Measured variable :ř

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Becker-D¨

oring model

§ _{The (observed) nucleation time is given by}

inftt ě 0 : ÿ

i ěN

iCiptq ě ρM | Cipt “ 0q “ Mδi “1u .

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Becker-D¨

oring model

Reversible one-step agregation Ci ` C1 pi ÝÝá âÝÝ qi `1 Ci `1

§ _{What are the dependencies of the nucleation time with}

respect to the model parameters ?

total mass : M ; nucleus size : N

aggregation rates : pi, i ě 1 fragmentation rates : qi, i ě 2

§ _{What is the nucleation time for very large initial quantity M}

and nucleus size N ?

§ In experiment, M « 1010_{´ 10}15,

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Deterministic BD model and Classical Nucleation Theory

$ ’ ’ ’ & ’ ’ ’ % dci dt “ Ji ´1´ Ji, i ě 2 , Ji “ pic1ci´ qi `1ci `1, i ě 1 , dc1 dt “ ´J1´ ř8 i “1Ji.

Equilibrium is given by Ji ” J “ 0, which implies

ci “ Qic1i , Qi “

p1p2¨ ¨ ¨ pi ´1

q2q3¨ ¨ ¨ qi

The mass at equilibrium is given by ρpc1q “

ÿ

i ě1

iQic1i

If this series has a finite radius of convergence, zs, then there is a

critical mass

ρs “

ÿ

i ě1

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Deterministic BD model and Classical Nucleation Theory

$ ’ ’ ’ & ’ ’ ’ % dci dt “ Ji ´1´ Ji, i ě 2 , Ji “ pic1ci´ qi `1ci `1, i ě 1 , dc1 dt “ ´J1´ ř8 i “1Ji.

[Ball, Carr, Penrose, CMP (1986)]

If M ď ρs, then (with strong convergence)

lim

tÑ8ciptq “ Qiz

i_, _{ρpzq “ M}

If M ą ρs, then (with weak convergence)

lim

tÑ8ciptq “ Qiz i

s, M ´ ρpzsq ““loss of mass”

As M Œ ρs, there is a solution for which Ji « J˚ is exponentially

small, and

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Deterministic BD model – Some remarks

§ _{For constant or linear kinetic rates p}_i_{, q}_i_{, one can reduce the}

system to 1 or 2 ODEs on ÿ i ě1 ci, ÿ i ě1 ici.

§ Based on scaling arguments, one can show that for q_i _{“ 0}

(irreversible nucleation),

inftt ě 0 : cNptq ě ρM | cipt “ 0q “ Mδi “1u »

1

M .

while for “qi Ñ 82 (pre-equilibrium nucleation),

inftt ě 0 : cNptq ě ρM | cipt “ 0q “ Mδi “1u »

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Stochastic Becker-D¨

oring model

We define a continuous time Markov Chain on

tpCiqi ě 1 : Ci P N,

ÿ

i ě1

iCi “ Mu

Transitions are given by

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Stochastic Becker-D¨

oring model

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Outline

Numerical results

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§ _{Law of large numbers as}

M Ñ 8 [Jeon. CMP (1998)]

§ _{Any macroscopic quantity like.}

inftt ě 0 : ÿ

i ěN

iCiptq ě ρM

| Cipt “ 0q “ Mδi “1u .

converges (if reachable) to a finite deterministic value as M Ñ 8.

§ _{This may not be true for}

microscopic quantity, for instance.

inftt ě 0 : CNptq ě 1

| Cipt “ 0q “ Mδi “1u .

[Y., D’Orsogna, Chou JCP (2012)]

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§ Non-monotonous w.r.t

reaction rate

§ Bimodal for ’small’

fragmentation rate

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Outline

Numerical results

Coarse-graining

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When N Ñ 8?

We start from a rescaled model (ε “ 1{N, ε2“ 1{M)

$ ’ & ’ % dc_iε dt “ 1 ε“J ε i ´1´ Jiε‰ , i ě 2 , mε “ c₁εptq ` ε2ÿ i ě2 ic_iεptq . With fεpt, x q “ř_{i ě2}c_iεptq1rpi ´1{2qε,pi `1{2qεqpx q,

Scaling idea : excess of monomer c₁εptq “ ε2c1ptq ,

Compensated aggregation / fragmentation pεpx q “ ÿ i ě2 ppi{ε2q1rεi ,εpi `1qr qεpx q “ ÿ i ě3 qi1rεi ,εpi `1qr

and slow first step :

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When N Ñ 8?

From the polymer point of view, we have accelerated fluxes, all of the same order :

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When N Ñ 8?

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d dt ż`8 0 fεpt, x qϕpx q dx ““p1εc1εptq2´ q2c2εptq ‰ ˜ 1 ε ż5{2ε 3{2ε ϕpxq dx ¸ ` ż`8 0 rpεpx qc1εptqfεpt, x q∆εϕpxq ´ qεpx qfεpt, x q∆´εϕpxqs dx ,

Theorem (Deschamps, Hingant, Y. (2016))

we have fε Ñ f (weakly in X “ tν P Mbpr0, 8q : ş xνpdxq ă 8u)

solution of d dt ż`8 0 f pt, xqϕpxq dx “ ż`8 0 rppx qc1ptq ´ qpx qs ϕ1px qf pt, x q dx ,

and c1ptq `ş xf pt, xq “ m, for all ϕ P Ccp0, 8q.

This is the weak form of Bf

Bt `

BpJpx , tqf pt, x qq

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Theorem (Deschamps, Hingant, Y. (2016)) When c1p0q ą limx Ñ0 qpxq_ppxq, d dt ż`8 0 f pt, xqϕpxq dx “ Nptqϕp0q ` ż`8 0 rppx qc1ptq ´ qpx qs ϕ1px qf pt, x q dx ,

for all ϕ P Cbp0, 8q, which gives the boundary condition

lim

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Theorem (Deschamps, Hingant, Y. (2016))

Nptq is an explicit function of c1ptq, and is given by a quasi

steady-state approximation of c₂ε“ fεpt, 2εq, given by the solution of #

0 “ rJi ´1pc1q ´ Jipc1qs , i ě 2 ,

c1ptq “ c1.

When c1 ą limx Ñ0qpxq_ppxq, the solution of Ji ” J ‰ 0 is linked to the

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Large nucleus N „

?

M

§ _{First case (pp0qm ą qp0q) : Convergence towards a}

deterministic value.

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Large nucleus N „

?

M

§ _{Second case (pp0qM ă qp0q) : Exponentially large time and}

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Outline

Numerical results Coarse-graining

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Quantifying the large deviation in toy model

A much simpler version of this model consider that a single aggregate may be formed at a time :

k ÝÝÝÝÝÝá_{âÝÝÝÝÝÝ}pkpm´kεq

qk`1

k ` 1 , which converges (with time rescaling) to

dx

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Quantifying the large deviation in toy model

A much simpler version of this model consider that a single aggregate may be formed at a time :

k ÝÝÝÝÝÝá_{âÝÝÝÝÝÝ}pkpm´kεq

qk`1

k ` 1 , which converges (with time rescaling) to

dx

dt “ ppx qpm ´ x q ´ qpx q

§ To leading order the stationary

prob. density is u˚ px q “ Ce ´1_εşxlog ´ qpy q ppy qpm´y q ¯ dy a ppxqpm ´ xqqpxq . § _{MFPT is explicit and is} exponentially large in ε

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Quantifying the large deviation in toy model

Can we perform similar calculations with n clusters ?

pk0, k1q p_k0pm´pk0`k1qεq ÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝ q_k0`1 pk0` 1, k1q , pk0, k1q p_k1pm´pk0`k1qεq ÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝ q_k1`1 pk0 , k1` 1q ,

which converges (with time rescaling) to

dx

dt “ ppxqpm ´ x ´ y q ´ qpxq

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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Quantifying the large deviation in toy model

dx

dy

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§ _{Quantifying the MFPT}

§ _{Data fitting in spontaneous polymerization experiment}