Time scales in a coagulation-fragmentation model, Nucleation Time in Stochastic Becker-Döring Model

(1)

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Time scales in a coagulation-fragmentation model, Nucleation Time in Stochastic Becker-Döring Model

Romain Yvinec

To cite this version:

Romain Yvinec. Time scales in a coagulation-fragmentation model, Nucleation Time in Stochastic

Becker-Döring Model. Workshop on Protein Aggregation: Biophysics and Mathematics, Jun 2017,

vienna, Austria. �hal-02785742�

(2)

Time scales in a coagulation-fragmentation model

Nucleation Time in Stochastic Becker-D¨oring Model

Romain Yvinec

¹

, Samuel Bernard

^2,3

, Tom Chou

⁴

, Julien Deschamps

⁵

, Maria R. D’Orsogna

⁶

, Erwan Hingant

⁷

and

Laurent Pujo-Menjouet

^2,3

1BIOS group INRA Tours, France.

2Institut Camille Jordan, Universit´e Lyon 1, Villeurbanne, France.

3INRIA Team Dracula, Inria Center Grenoble Rhˆone-Alpes, France.

4Depts. of Biomathematics and Mathematics, UCLA, Los Angeles, USA.

5DIMA, Universita degli Studi di Genova, Italy.

6Dept. of Mathematics, CSUN, Los Angeles, USA.

7Departamento de Matem´atica, U. Federal de Campina Grande, PB, Brasil.

(3)

Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

Re-scaling reaction rates with M

Re-scaling nucleus size with M

Back to classical nucleation theory

(4)

Outline

Amyloid diseases and nucleation Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

(5)

Protein accumulation in amyloid by nucleation-dependent polymerization

Misfolding Prusiner model for prion

The early aggregation formation requires a series of association steps that are thermodynamically unfavorable (with an dissociation constant K

d

" 1).

These aggregation steps are unfavorable up to a given size (that is

not currently known), which is referred to the nucleus size.

(6)

Motivation BD Coarse-graining SBD Variability

Protein accumulation in amyloid by nucleation-dependent polymerization

Misfolding Prusiner model for prion

The early aggregation formation requires a series of association steps that are thermodynamically unfavorable (with an dissociation constant K

_d

" 1).

These aggregation steps are unfavorable up to a given size (that is

not currently known), which is referred to the nucleus size.

(7)

Key questions

We want to study nucleation mechanism for in-vitro spontaneous polymerization experiments of rPrP (kinetics monitored by fluorescence intensity)

§

How to include nucleation in (macroscopic) model of protein polymerization ?

§

How to explain large variability in nucleation lag time, despite

the large number of proteins ?

(8)

Outline

Amyloid diseases and nucleation Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

(9)

Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation

C

i

`C

1 pi

ÝÝá âÝÝ

qi`1

C

i`1

, i “ 2 , , 3 , ¨ ¨ ¨

§

First used in the work Kinetic treatment of nucleation in supersaturated vapors by physicists Becker and D¨ oring (1935).

§

Traditionally used as an infinite set of Ordinary Differential

Equations. More recently used as a finite state-space Markov

Chain.

(10)

Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

i

` C

1

pi

ÝÝá âÝÝ

qi`1

C

i`1

Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

§

Purely kinetic model (law of mass-action) : no space, no polymer structure (but size-dependent kinetic rates).

§

Indirect interaction between polymer C

i

, i ě 2 via the available number of monomers C

1

.

C

1

ptq ` ÿ

iě2

iC

i

ptq “ constant

(11)

Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

i

` C

1

pi

ÝÝá âÝÝ

qi`1

C

i`1

§

Purely kinetic model (law of mass-action) : no space, no polymer structure (but size-dependent kinetic rates).

§

Indirect interaction between polymer C

i

, i ě 2 via the available number of monomers C

1

.

C

1

ptq ` ÿ

iě2

iC

i

ptq “ constant

(12)

Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

§

Typical coefficient are derived from physical principles p

_i

“ i

^α

, q

_i

“ p

_i

´ z

_s

` q

i

^γ

¯

.

(13)

Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

§

Typical coefficient are derived from physical principles p

_i

“ i

^α

, q

_i

“ p

_i

´ z

_s

` q

i

^γ

¯

.

(14)

Becker-D¨ oring model

Reversible one-step coagulation-fragmentation Set of kinetic reactions :

C

i

` C

1 p_i

ÝÝá âÝÝ

qi`1

C

i`1

, i ě 1 .

§

In spontaneous polymerization experiment,

§ Initial condition given bycipt“0q “0 @iě2.

§ Measured variable :ř

iěniCi (nis an unknown parameter)

§

The (observed) nucleation time is given by inftt ě 0 : ÿ

iěn

iC

i

ptq ě δm | C

i

pt “ 0q “ mδ

i“1

u . Another quantity of interest is the following First Passage Time,

inf tt ě 0 : C

_N

ptq ě 1 | C

_i

pt “ 0q “ mδ

_i“1

u .

(15)

Deterministic Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i`1

, i ě 1 , dc

1

dt “ ´J

1

´ ř

₈

i“1

J

i

.

§

Deterministic version : infinite system of ODEs.

X

“

#

pc

_i

q

iě1

P

R^N`

: ÿ

iě1

ic

_i

ă 8 +

§

Preserves mass for all times

8

ÿ

i“1

ic

i

ptq “

8

ÿ

i“1

ic

i

p0q “: m .

(16)

Deterministic Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i`1

, i ě 1 , dc

1

dt “ ´J

1

´ ř

₈

i“1

J

i

.

§

Deterministic version : infinite system of ODEs.

§

Well-posedness theory for sublinear coefficients in

X

“

#

pc

_i

q

iě1

P

R^N`

: ÿ

iě1

ic

_i

ă 8 +

§

Preserves mass for all times

8

ÿ

i“1

ic

i

ptq “

8

ÿ

i“1

ic

i

p0q “: m .

(17)

Deterministic Becker-D¨ oring model

Reversible one-step

coagulation-fragmentation C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i`1

, i ě 1 , dc

1

dt “ ´J

1

´ ř

₈

i“1

J

i

.

§

Deterministic version : infinite system of ODEs.

§

Well-posedness theory for sublinear coefficients in

X

“

#

pc

_i

q

iě1

P

R^N`

: ÿ

iě1

ic

_i

ă 8 +

§

Preserves mass for all times

8

ÿ

i“1

ic

i

ptq “

8

ÿ

i“1

ic

i

p0q “: m .

(18)

Equilibrium of the BD model

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i_`1

, i ě 1 , dc

1

dt “ ´J

1

´ ř

₈

i“1

J

i

.

Ball, Carr, Penrose, Comm.

Math. Phys 104(4), 1986

Equilibrium is given by J

_i

” J “ 0, which implies c

i

“ Q

i

z

ⁱ

, Q

i

“ p

₁

p

₂

¨ ¨ ¨ p

_i_´1

q

2

q

3

¨ ¨ ¨ q

i

z is given by the mass at equilibrium, mpzq :“ ÿ

iě1

iQ

_i

z

ⁱ Is there a solution of

mpz q “

^?

mp“ ÿ

iě1

ic

i

ptqq

(19)

Equilibrium of the BD model

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i_`1

, i ě 1 , dc

₁

dt “ ´J

₁

´ ř

₈

i“1

J

_i

.

Math. Phys 104(4), 1986

If the serie mpzq “ ř

iě1

iQ

i

z

ⁱ

has a finite radius of convergence z

s

and if

suptmpz q , z ă z

_s

u “: m

_s

ă 8 ,

then there is a critical mass such that there is no equilibrium with

mass m ą m

s

.

(20)

Deterministic BD model and Classical Nucleation Theory

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i_`1

, i ě 1 , dc

1

dt “ ´J

1

´ ř

₈

i“1

J

i

.

Math. Phys 104(4), 1986 Slemrod, Nonlinearity 2(3), 1989

Ca˜nizo, Lods, J. Diff. Eqs.

255(5), 2013

If m ď m

_s

, then (with strong convergence)

tÑ8

lim c

i

ptq “ Q

i

z

ⁱ

, mpz q “ m If m ą m

_s

, then (with weak convergence)

tÑ8

lim c

_i

ptq “ Q

_i

z

_sⁱ

, m ´ m

_s

“ ”loss of mass to 8”

Remark

There is a Lyapounov function, given by Hpc q “

ÿ

iě1

c

i

ˆ ln

ˆ c

i

Q

_i

˙

´ 1

˙

.

(21)

Deterministic BD model and Classical Nucleation Theory

$

’ ’

’ &

’ ’

’ % dc

i

dt “ J

i´1

´ J

i

, i ě 2 ,

J

_i

“ p

_i

c

₁

c

_i

´ q

_i`1

c

_i_`1

, i ě 1 , dc

₁

dt “ ´J

₁

´ ř

₈

i“1

J

_i

.

Penrose, Comm. Math. Phys 124, 1989

There exist ”almost steady-states”, for which J

_i

” J

^˚

pmq ‰ 0. As m Œ m

_s

, if such steady-states are used as initial condition, then the solution

§

(for finite t) c

_i

ptq ´ c

_i

p0q is exponentially small

§

lim

_tÑ8

c

_i

ptq ´ c

_i

p0q is not exponentially small Moreover J

^˚

pmq is exponentially small

§

The new phase is being formed extremely slowly, after a

long metastable period.

(22)

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

c

₁

, ÿ

iě2

c

_i

, ÿ

iě2

ic

_i

.

§

Based on scaling arguments, one can show that for q

_i

“ 0 (irreversible nucleation),

inf tt ě 0 : c

n

ptq ě δm | c

i

pt “ 0q “ mδ

i“1

u » 1 m . while for “q

i

Ñ 8

²

(pre-equilibrium nucleation),

inftt ě 0 : c

n

ptq ě δm | c

i

pt “ 0q “ mδ

i“1

u » 1

m

ⁿ

.

(23)

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

c

₁

, ÿ

iě2

c

_i

, ÿ

iě2

ic

_i

.

§

Based on scaling arguments, one can show that for q

_i

“ 0 (irreversible nucleation),

inf tt ě 0 : c

n

ptq ě δm | c

i

pt “ 0q “ mδ

i“1

u » 1 m . while for “q

i

Ñ 8

²

(pre-equilibrium nucleation),

inftt ě 0 : c

n

ptq ě δm | c

i

pt “ 0q “ mδ

i“1

u » 1

m

ⁿ

.

(24)

Use of BD-like model in protein polymerization models

Irreversible nucleation step, ”Heaviside” rates

Powers & Powers, Biophys. J. 91, 2006 For b"c :

pre-equilibrium hypothesis.

(25)

Use of BD-like model in protein polymerization models

pre-equilibrium nucleation step, constant rates

$

’ ’

&

’ ’

% dc

1

dt “ ´ppc

1

´ qqy, . dy

dt “ Kc

₁ⁿ

p`Qpm ´ c

₁

pt qqq , dz

dt “ ppc

1

´ qqy , .

Ferrone et al., Bophys. J.

32, 1980 y “ř

iěnc_i z “ř

iěnic_i ppiq “p,qpiq “q Q “secondary nucleation mechanism (fragmentation, heterogeneous nucleation...)

(26)

Use of BD-like model in protein polymerization models

pre-equilibrium nucleation,

polymerization-fragmentation, ”oligomers at 0”

$

’ ’

&

’ ’

% dc

₁

dt “ ´pc

₁

ÿ

iěn

c

_i

` 2q

n´1

ÿ

i“1

ÿ

jěi`1

ic

_j

´ nKc

₁ⁿ

. dc

_i

dt “ pc

₁

pc

_i´1

´ c

_i

q ´ qpi ´ 1qc

_i

` 2q ÿ

jěi`1

c

_j

` Kc

₁ⁿ

δ

_i,n

, i ě n ,

Knowles et al., Science 326, 2009

Approximate analytical solution.

$

’ ’

&

’ ’

% dc

₁

dt “ ´pc

₁

y ` npn ´ 1qqy ´ nKc

₁ⁿ

. dy

dt “ qz ´ p2n ´ 1qqy ` Kc

₁ⁿ

, y “ ÿ

iěn

c

_i

, dz

dt “ pc

₁

y ´ npn ´ 1qqy ` nKc

₁ⁿ

, z “ ÿ

iěn

ic

_i

.

(27)

Use of BD-like model in protein polymerization models

Continuous approximation, nucleation as a boundary condition

$

’ ’

’ &

’ ’

’ %

Bf px, tq

Bt ` c

₁

ptq Bppxqf px, t q

Bx “ r¨ ¨ ¨ s . c

₁

ptqppx

₀

qf px

₀

, tq “ Npc

₁

ptqq ,

dc

1

dt “ λ ´ γc

1

´ nNpc

1

q ´ c

1

ż

₈

x0

ppxqf px, tqdx ,

Helal et al., J. Math.

Biol., 2013 fpt,xq=number of polymer sizex Npc1q “αc₁ⁿ

(28)

Use of BD-like model in protein polymerization models

Continuous approximation, nucleation as a boundary condition

$

’ &

’ %

Bf px, tq

Bt ` Bpppxqc

₁

ptq ´ qpx qqf px, t q

Bx “ r¨ ¨ ¨ s .

ppx

0

qf px

0

, tq “ ppx

0

q p

_N

c

1

ptq

ⁿ

q

_N

` ppx

₀

qc

₁

ptq ,

Prigent et al., Plos One, 7, 2012

Banks et al., J. Math.

Biol., 74, 2017

(29)

Outline

Amyloid diseases and nucleation Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

(30)

Large Size, Excess of monomer

We start from a rescaled model (ε “ 1{n, ε

²

“ 1{m)

$

’ &

’ % dc

_i^ε

dt “ 1 ε

“ J

_i´1^ε

´ J

_i^ε

‰

, i ě 2 , m

^ε

“ c

₁^ε

ptq ` ε

²

ÿ

iě2

ic

_i^ε

pt q . Scaling idea : excess of monomer, time scale “ 1{ε

c

₁^ε

ptq :“ ε

²

c

1

pt{εq , c

_i^ε

ptq :“ c

_i

pt{εq Compensated aggregation / fragmentation

p

_i^ε

:“ p

_i

ε

²

, q

^ε_i

:“ q

_i

, J

_i^ε

“ p

_i^ε

c

₁^ε

c

_i^ε

´ q

_i`1^ε

c

_i`1^ε

and slow first step :

p

₁^ε

:“ p

1

ε

⁴

,

(31)

Large Size, Excess of monomer

We start from a rescaled model (ε “ 1{n, ε

²

“ 1{m)

$

’ &

’ % dc

_i^ε

dt “ 1 ε

“ J

_i´1^ε

´ J

_i^ε

‰

, i ě 2 , m

^ε

“ c

₁^ε

ptq ` ε

²

ÿ

iě2

ic

_i^ε

pt q .

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

1 εp^ε₁C₁^εC₁^ε

Ý ÝÝÝÝÝ á â ÝÝÝÝÝ Ý

1 εq^ε₂C₂^ε

C

₂^ε

C

_i´1^ε

1

εp^εpεpi´1qqC₁^εC_i´1^ε

Ý ÝÝÝÝÝÝÝÝÝÝÝ á â ÝÝÝÝÝÝÝÝÝÝÝ Ý

1 εq^εpεiqC_i^ε

C

_i^ε

1

εp^εpεiqC₁^εC_i^ε

ÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝ

1

εq^εpεpi`1qqC_i`1^ε

C

_i`1^ε

,

(32)

Large Size, Excess of monomer

We start from a rescaled model (ε “ 1{n, ε

²

“ 1{m)

$

’ &

’ % dc

_i^ε

dt “ 1 ε

“ J

_i´1^ε

´ J

_i^ε

‰

, i ě 2 , m

^ε

“ c

₁^ε

ptq ` ε

²

ÿ

iě2

ic

_i^ε

pt q . Weak form : for any test function (ϕ

_i

),

d dt

ÿ

iě2

c

_i^ε

ϕ

_i

“ 1

ε J

₂^ε

ϕ

2

` ÿ

iě3

J

_i^ε

„ ϕ

i`1

´ ϕ

i

ε



.

(33)

Large Size, Excess of monomer

We start from a rescaled model (ε “ 1{n, ε

²

“ 1{m)

$

’ &

’ % dc

_i^ε

dt “ 1 ε

“ J

_i´1^ε

´ J

_i^ε

‰

, i ě 2 , m

^ε

“ c

₁^ε

ptq ` ε

²

ÿ

iě2

ic

_i^ε

pt q .

f^εpt,xq “ř

iě2c_i^εptq1rpi´1{2qε,pi`1{2qεqpxq,ϕ_i “şpi`1{2qε

pi´1{2qεϕpxqdx,

$

’’

’&

’’

’% d dt

ż`8 0

f^εpt,xqϕpxqdx “ “

p^ε₁c₁^εptq²´q₂^εc₂^εptq‰

˜ 1 ε

ż5{2ε 3{2ε

ϕpxqdx

¸

` ż`8

0

J^εpt,xq∆_εϕpxqdx,

m^ε “ c₁^εptq ` ż`8

0

xf^εpt,xqdx.

where ∆_εϕpxq “ϕpx`εq´ϕpxq

ε andJ^εpt,xq “.p^εpxqc₁^εptqf^εpt,xq ´q^εpx` εqf^εpt,x`εq

(34)

d dt

ż

_`8

0

f

^ε

pt, xqϕpxq dx “ “

p

₁^ε

c

₁^ε

ptq

²

´ q

^ε₂

c

₂^ε

ptq ‰

˜ 1 ε

ż

5{2ε 3{2ε

ϕpxq dx

¸

` ż

_`8

0

rp

^ε

pxqc

₁^ε

ptqf

^ε

pt, xq∆

_ε

ϕpxq ´ q

^ε

pxqf

^ε

pt, xq∆

_´ε

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

We suppose :

§

Control and convergence of rate functions

§

Control and convergence of initial condition

§

ppxq „ px

^r^p

, qpxq „ qx

^r^q

near x “ 0, and r

_q

ě r

_p

.

§

c

1

p0q ą ρ :“ lim

xÑ0

qpxq{ppxq

(35)

d dt

ż

_`8

0

f

^ε

pt, xqϕpxq dx “ “

p

₁^ε

c

₁^ε

ptq

²

´ q

^ε₂

c

₂^ε

ptq ‰

˜ 1 ε

ż

5{2ε 3{2ε

ϕpxq dx

¸

` ż

_`8

0

rp

^ε

pxqc

₁^ε

ptqf

^ε

pt, xq∆

_ε

ϕpxq ´ q

^ε

pxqf

^ε

pt, xq∆

_´ε

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

we have f

^ε

Ñ f (in

C

pr0, T s; w ´ ˚ ´

M

pr0, 8qqq) solution of d

dt ż

_`8

0

f pt, xqϕpxq dx “ Npt qϕp0q

` ż

_`8

0

rppxqc

₁

ptq ´ qpxqs ϕ

¹

pxqf pt, xq dx , for all ϕ P C

₀

r0, 8q, which is the weak form of

Bf

Bt ` BpJpx, tqf pt, xqq

Bx “ 0 , lim

xÑ0

Jpx, tqf pt, xq “ Nptq .

(36)

d dt

ż

_`8

0

f

^ε

pt, xqϕpxq dx “ “

p

₁^ε

c

₁^ε

ptq

²

´ q

^ε₂

c

₂^ε

ptq ‰

˜ 1 ε

ż

5{2ε 3{2ε

ϕpxq dx

¸

` ż

_`8

0

rp

^ε

pxqc

₁^ε

ptqf

^ε

pt, xq∆

_ε

ϕpxq ´ q

^ε

pxqf

^ε

pt, xq∆

_´ε

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

Nptq is an explicit function of c

1

ptq, and is given by a quasi steady- state approximation of c

₂^ε

“ f

^ε

pt, 2εq, given by the solution of

$

’ &

’ %

0 “ rJ

i´1

pc

1

q ´ J

i

pc

1

qs , i ě 2 , c

₁

ptq “ c

₁

.

J

_i

pc

1

q “ pi

^r^p

c

1

´ qpi ` 1q

^r^q

1

rp“rq

.

When c

₁

ą lim

_xÑ0 ^qpxq_ppxq

, the solution of J

_i

” J ‰ 0 is linked to the

loss of mass in the classical BD theory.

(37)

Exemples

§

For r

p

ă r

q

, we get Npc

1

q “ αc

₁²

, and

xÑ0

lim

^`

x

^r^p

f pt, xq “ α p c

1

ptq .

For r

_p

“ r

_q

, we get Npc

₁

q “

_p

c

₁

ppc

₁

´ qq, and lim

xÑ0^`

x

^r^p

f pt, xq “ α p c

₁

ptq .

§

For faster fragmentation rate q

^ε₂

, we may get Npc

₁

q “ αc

₁²_pc^pc¹

1`q2

and lim

xÑ0^`

x

^r^p

f pt, xq “ αc

₁

ptq c

₁

ptq pc

1

ptq ` q

2

, or Npc

1

q “ 0, and

lim

xÑ0^`

x

^r^p

f pt , xq “ 0 .

(38)

Exemples

§

For r

p

ă r

q

, we get Npc

1

q “ αc

₁²

, and

xÑ0

lim

^`

x

^r^p

f pt, xq “ α p c

1

ptq .

§

For r

_p

“ r

_q

, we get Npc

₁

q “

^α_p

c

₁

ppc

₁

´ qq, and lim

xÑ0^`

x

^r^p

f pt, xq “ α p c

₁

ptq .

§

For faster fragmentation rate q

^ε₂

, we may get Npc

₁

q “ αc

₁²_pc^pc¹

1`q2

and lim

xÑ0^`

x

^r^p

f pt, xq “ αc

₁

ptq c

₁

ptq pc

1

ptq ` q

2

, or Npc

1

q “ 0, and

lim

xÑ0^`

x

^r^p

f pt , xq “ 0 .

(39)

Exemples

§

For r

p

ă r

q

, we get Npc

1

q “ αc

₁²

, and

xÑ0

lim

^`

x

^r^p

f pt, xq “ α p c

1

ptq .

§

For r

_p

“ r

_q

, we get Npc

₁

q “

^α_p

c

₁

ppc

₁

´ qq, and lim

xÑ0^`

x

^r^p

f pt, xq “ α p c

₁

ptq .

§

For faster fragmentation rate q

^ε₂

, we may get Npc

₁

q “ αc

₁²_pc^pc¹

1`q2

and lim

xÑ0^`

x

^r^p

f pt, xq “ αc

₁

ptq c

₁

ptq pc

1

ptq ` q

2

, or Npc

₁

q “ 0, and

lim

xÑ0^`

x

^r^p

f pt, xq “ 0 .

(40)

Outline

Amyloid diseases and nucleation Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

(41)

Stochastic Becker D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

$

’ ’

&

’ ’

%

C

i

ptq “ C

_iⁱⁿ

` J

i´1

ptq ´ J

i

ptq , i ě 2 , J

_i

ptq “ Y

_i^`

´ ż

_t

0

p

_i

C

₁

psqC

_i

ps qds

¯

´Y

_i`1^´

´ ż

_t

0

q

_i`1

C

_i`1

ps qds

¯

C

1

ptq “ C

₁ⁱⁿ

´ 2J

1

pt q ´ ÿ

iě2

J

i

ptq ,

§

Stochastic version : Finite-state space Markov Chain, in X

M

:“

#

C “ pC

i

q

iě1

P

N^N

:

8

ÿ

i“1

iC

i

“ M +

.

8

ÿ

i“1

iC

i

ptq “

8

ÿ

i“1

iC

i

p0q “: M .

(42)

Stochastic Becker D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

$

’ ’

&

’ ’

%

C

i

ptq “ C

_iⁱⁿ

` J

i´1

ptq ´ J

i

ptq , i ě 2 , J

_i

ptq “ Y

_i^`

´ ż

_t

0

p

_i

C

₁

psqC

_i

ps qds

¯

´Y

_i`1^´

´ ż

_t

0

q

_i`1

C

_i`1

ps qds

¯

C

1

ptq “ C

₁ⁱⁿ

´ 2J

1

pt q ´ ÿ

iě2

J

i

ptq ,

§

Stochastic version : Finite-state space Markov Chain, in X

M

:“

#

C “ pC

i

q

iě1

P

N^N

:

8

ÿ

i“1

iC

i

“ M +

.

§

Preserves mass for all times

8

ÿ

i“1

iC

i

ptq “

8

ÿ

i“1

iC

i

p0q “: M .

(43)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Transitions are given by

P

"

C

1

pt ` dtq “ C

1

ptq ´ 2 C

2

pt ` dtq “ C

2

ptq ` 1

*

“ p

1

C

1

ptqpC

1

ptq ´ 1qdt ` o pdtq

(44)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Transitions are given by

P

$

&

%

C

1

pt ` dt q “ C

1

ptq ´ 1 C

i

pt ` dt q “ C

i

ptq ´ 1 C

_i`1

pt ` dt q “ C

_i_`1

ptq ` 1

, . -

“ p

i

C

1

ptqC

i

ptqdt ` opdtq

(45)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Transitions are given by

P

$

&

%

C

1

pt ` dtq “ C

1

ptq ` 1 C

i

pt ` dtq “ C

i

ptq ` 1 C

_i`1

pt ` dtq “ C

_i`1

pt q ´ 1

, . -

“ q

i`1

C

i`1

pt qdt ` o pdt q

(46)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(47)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(48)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(49)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(50)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(51)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

Time interval between transition T

i`1

´ T

i

„

E

˜

p

1

C

1

pC

1

´ 1q ` ÿ

iě2

p

i

C

1

C

i

` q

i

C

i

¸

A given transition is selected at random according to its weight.

(52)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i`1

XM:“

#

CPN^N :

8

ÿ

i“1

iCi“M +

.

Drawback : exponential increase of the size of the state-space ! M | X

M

|“

M

ÿ

i“1

σpiq | X

M´i

| , | X

M

| 9 1 4M ?

3 exp

˜ π

c 2M 3

¸ ,

where σpiq is the sum of the divisors of i

(53)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

i

` C

1 pi

ÝÝá âÝÝ

qi`1

C

i`1

Due to detailed-balance, the asymptotic prob. distribution is ΠpC q “ B

_M

M

ź

i“1

pQ

_i

q

^Cⁱ

C

_i

! , Q

_i

“ p

₁

p

₂

¨ ¨ ¨ p

_i´1

q

₂

q

₃

¨ ¨ ¨ q

_i

.

(54)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

i

` C

1 pi

ÝÝá âÝÝ

qi`1

C

i`1

Due to detailed-balance, the asymptotic prob. distribution is ΠpC q “ B

_M

M

ź

i“1

pQ

_i

q

^Cⁱ

C

_i

! , Q

_i

“ p

1

p

2

¨ ¨ ¨ p

_i´1

q

₂

q

₃

¨ ¨ ¨ q

_i

. The expected number of clusters of size i is

E

Π

C

i

“ Q

i

B

M

{B

M´i

, and MB

_M^´1

“

M

ÿ

i“1

iQ

i

B

_M´i^´1

.

(55)

Stochastic Becker-D¨ oring model

Reversible one-step coag.-frag.

C

_i

` C

₁

ÝÝá âÝÝ

^pⁱ

qi`1

C

_i_`1

Due to detailed-balance, the asymptotic prob. distribution is ΠpC q “ B

M

ź

i“1

pQ

_i

q

^Cⁱ

C

i

! , Q

i

“ p

₁

p

₂

¨ ¨ ¨ p

_i´1

q

2

q

3

¨ ¨ ¨ q

i

.

The expected number of clusters of size i is E

_Π

C

_i

“ Q

_i

B

_M

{B

_M´i

, and MB

_M^´1

“

ÿ

M

i“1

iQ

_i

B

_M´i^´1

. Moreover, analogy with supercritical case in BD holds :

ˆ

iÑ8

lim p

i

q

_i`1

“ z

s

ą 0

˙ ñ

ˆ

MÑ8

lim E

_Π

C

_i

“ Q

_i

z

_sⁱ

˙

(56)

§

With the large volume scaling : c

_i^ε

“ εC

_i

, and p

_i

“ εp

_i

,

q

_i

“ q

_i

: Law of large numbers as M Ñ 8 [Jeon, CMP (1998)]

§

Any macroscopic quantity like inftt ě 0 : ÿ

iěN

iC

i

ptq ě ρM

| C

_i

pt “ 0q “ M δ

_i“1

u .

converges (in standard scaling)

to a finite deterministic value as

M Ñ 8.

(57)

§

With the large volume scaling : c

_i^ε

“ εC

_i

, and p

_i

“ εp

_i

,

q

_i

“ q

_i

: Law of large numbers as M Ñ 8 [Jeon, CMP (1998)]

§

Any macroscopic quantity like inftt ě 0 : ÿ

iěN

iC

i

ptq ě ρM

| C

_i

pt “ 0q “ M δ

_i“1

u . converges (in standard scaling) to a finite deterministic value as M Ñ 8.

§ This may not be true for

microscopic quantity, for instance.

inftt ě0 :CNptq ě1

|C_ipt “0q “Mδ_i“1u.

[Y., D’Orsogna, Chou JCP (2012)]

[Y., Bernard, Hingant, Pujo-Menjouet JCP (2016)]

(58)

Outline

Amyloid diseases and nucleation Becker-D¨ oring model

Coarse-graining : to include nucleation in continuous model Stochastic Becker D¨ oring model

Variability in nucleation time

Re-scaling reaction rates with M

Re-scaling nucleus size with M

Back to classical nucleation theory

(59)

How to explain large variability in M Ñ 8 ?

Roughly speaking, due to the law of large number (+CLT), in order to obtain a positive variance in a continuous settings, one needs to avoid that the nucleation occurs in finite time in the limit M Ñ 8.

§

We seek situations (model, scaling) where the nucleation is a

rare event, that do not occurs in the deterministic limit

M Ñ 8.

(60)

Motivation BD Coarse-graining SBD Variability Rate scaling Size scaling CNT

Coarse-Grained model

C

₁

, Y , Z ÞÑ

$

&

%

C

₁

´ n, Y ` 1, Z ` n at rate αpC

₁

q , C

₁

´ 1, Y , Z ` 1 at rate pC

₁

Y ,

C

1

, Y ` 1, Z at rate qZ . Then, for ”small α”, and large

volume, the lag time is composed of the convolution of an Exponential variable of rate α and a

deterministic time given by the ODE

Szavits-Nossan et al., PRL 113, 2014

Y “ř

iě2Ci,Z “ř

iě2iCi

$

’ ’

&

’ ’

% dc

₁

dt “ ´pc

1

y

p´nαpc1qq

. dy

dt “ qz

p`αpc₁qq

, y “ ÿ

iěn

c

_i

, dz

dt “ pc

₁

y

`pnαpc₁qq

, z “ ÿ

iěn

ic

_i