Stochastic coagulation-fragmentation models for the study of protein aggregation phenomena

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Stochastic coagulation-fragmentation models for the

study of protein aggregation phenomena

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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Stochastic coagulation-fragmentation models for

the study of protein aggregation phenomena

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.

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Numerical results

Coarse-graining

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Outline

Amyloid diseases and Becker-D¨oring model

Numerical results Coarse-graining

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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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Nucleation time in the Stochastic Becker-D¨

oring model

Reversible one-step agregation

Ci ` C1 pi ÝÝá âÝÝ qi `1 Ci `1 (1)

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Nucleation time in the Stochastic Becker-D¨

oring model

The nucleation time is given by the following First Passage Time, TN,M :“ inftt ě 0 : CNptq “ 1 | Cipt “ 0q “ Mδi “1u . (2)

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

the model parameters ?

total mass : M ; nucleus size : N

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Nucleation time in the Stochastic Becker-D¨

oring model

The nucleation time is given by the following First Passage Time, TN,M :“ inftt ě 0 : CNptq “ 1 | Cipt “ 0q “ Mδi “1u . (2)

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

nucleus size N ?

lim

M,NÑ8T N,M

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Nucleation time in the Stochastic Becker-D¨

oring model

Ci ` C1 pi ÝÝá âÝÝ qi `1 Ci `1 (8,0,0) (6,1,0) (4,2,0) (2,3,0) (5,0,1) (3,1,1) (1,2,1) (2,0,2) (0,4,0) (0,1,2) (7,0,0) (5,1,0) (3,2,0) (1,3,0) (4,0,1) (2,1,1) (0,2,1) (1,0,2) (a) (b)

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Outline

Amyloid diseases and Becker-D¨oring model

Numerical results

Coarse-graining

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§ Non-monotonous w.r.t

reaction rate

§ Bimodal for ’small’

fragmentation rate

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§ _{For M Ñ 8 : deterministic trajectory.} § _{Metastable behavior : ’pure-aggregation’.}

§ Medium-large polymer formed only after a longer time

pk “ 1 and qk ” 1 for k ě 2, M “ 105 (we plot M´1CkptM´1q).

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Outline

Amyloid diseases and Becker-D¨oring model Numerical results

Coarse-graining

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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

Amyloid diseases and Becker-D¨oring model Numerical results

Coarse-graining

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Quantifying the large deviation in the SBD 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 FPT are given by E“T1,0N ‰ “ N´1 ÿ i “1 i ÿ j “1 śi k“j`1qk śi k“jpkpm ´ εkq .

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Quantifying the large deviation in the SBD 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 dt “ ppy qpm ´ x ´ y q ´ qpy q 0.0 0.2 0.4 0.6 0.8 Size cluster 1 0.0 0.2 0.4 0.6 0.8 S iz e cl us te r2

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Quantifying the large deviation in the SBD model

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Quantifying the large deviation in the SBD model

which converges (with time rescaling) to dx dt “ ppxqpm ´ x ´ y q ´ qpxq dy dt “ ppy qpm ´ x ´ y q ´ qpy q 0.0 0.2 0.4 0.6 0.8 1.0 Size cluster 1 0.0 0.2 0.4 0.6 0.8 1.0 S iz e cl us te r2

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Quantifying the large deviation in the SBD model

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Quantifying the large deviation in the SBD model

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§ Extension to spatial models (diffusion) ?

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Toy model with time-scale separation

$ ’ ’ & ’ ’ % X ÝÝÝá_âÝÝÝγ γ˚_{εr X ˚_, X˚` X˚ ÝÑεα 2Y , X ` Y ÝÑεβ 2Y ,