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![Page 1: Outline](https://reader030.fdocuments.net/reader030/viewer/2022033105/56812def550346895d934e03/html5/thumbnails/1.jpg)
Aggregation-Fragmentation Modelsfor Protein Polymerization
Marie Doumic
Journées non linéaires – Journées Venakides
Institut de Mathématiques de JussieuJune 24th, 2010
with: V. Calvez, P. Gabriel, T. Goudon, N. Lenuzza, T. Lepoutre, B. Perthame and biologists: F. Mouthon, H. Rezaei
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
OutlineA (very) little bit about the biology of Prion and Alzheimer
Review on existing mathematical models for prion disease
& previous results
Some recent results:
Link between discrete and continuous modelling
Eigenvalue problem
Parameters dependency
Perspectives
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
What is a common point of
• Prion (Madcow, Kuru, Creutzfeldt-Jakob disease)
• Alzheimer• Huntington • … and some others:
Neurodegenerative diseases characterized by abnormal accumulation ofinsoluble fibrous protein aggregates
called AMYLOIDS
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Where do these fibrils come from ?
Healthy state: the considered protein is present(APP for Alzheimer, PrP for Prion, HTT for Huntington…)
BUT is monomeric (N.B.: mono=1, poly = several)
Disease : Misfolding of the protein, leading to its
polymerization onto long fibrils (monomers attach to one another)
Specificity of Prion disease: transmissible
PRION: PRoteinasceous Infectious Only(Prusiner 1982, Griffith 1967, Nobel Prize)
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Wednesday 19 April 2023
Marie DOUMIC - BONN October, 6th, 2009
+
i+1
Polymerization
i-mermonomer
(i+1)-mer
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Wednesday 19 April 2023
+
Fragmentation
(i-j)-mer
i-mer
j-mer
j monomers if j < i0
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Wednesday 19 April 2023
+
Coagulationj-mer
(i+j)-mer
i-mer
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Wednesday 19 April 2023
Depolymerization
+
(i-1)-mermonomer
i-mer
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Discrete Prion model
Masel, Jansen, Nowak, 1999:
Monomer density
i-polymerdensity
synthesis
degradation
polymerization
fragmentation
Size of a nucleus = minimal size of stable polymers
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Remarks on discrete polymerization models
In Biology and Biophysics
• Huge literature
• Simulation tools e.g. softwares “PREDICI”, “POLYRED”
(e.g. see articles by Wulkow et al., Choi et al.)
• Main difficulty: large (infinite…) number of parameters
often drastic simplifying assumptions
In Mathematics
• Family of coagulation-fragmentation models / Becker-Döring
See e.g. work by Aizenman-Back, 79; Ball, Carr, Penrose1990’s;
Laurençot 02; Jabin-Niethammer, 03; Fournier-Mischler, 04 ;
Dubovski – Stewart ; …
• Specificity of this model: particular role of monomers V
Coupling v / ui in the “advection” term (replaces coalescence)
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Discrete model for prion proliferation: Nowak et al, 1998 and Masel et al., 1999
Continuous version: Prüss, Pujo-Menjouet, Webb, Zacher, 2002
Question: is the continuous framework valid ? (Investigated in D, Goudon, Lepoutre, Comm. Math. Sc., 2009)
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Previous works: Laurençot-Mischler, 2007 (coagulation-fragmentation) Collet, Goudon, Poupaud, Vasseur, 2002 (Lifshitz-Slyozov)
A- Rescale the equations
Dimensionless parameters appear in the equation:
Mass conservation:
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And mass conservation:
Biological interpretation: if and iM typical size
of a polymer
B- Define a small parameter
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C- Assumptions on the coefficients
Under the following assumption
We set
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C- Assumptions on the coefficients
Assumption on n0 :
Biological interpretation: x0=0 iff n0 << iM
Lemma. Under the previous assumptions + some
compactness, we can define continuous functions k
and extract a subsequence εn such that
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Theorem (D, Goudon, Lepoutre). If moreover
We can extract a subsequence εn such that
weak solution of :
D- Asymptotic result (based on moment estimates)
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Theorem (D, Goudon, Lepoutre). If moreover
We can extract a subsequence εn such that
weak solution of :
D- Asymptotic result (based on moment estimates)
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Discrete model: no boundary condition.
Continuous model: needs one to be well-posed. If:
-> obtained rigorously if
-> obtained only (by formal division by x0) otherwise
E- What to choose for a boundary condition ?
We set (in a weak sense):
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
- Several different possible continuous equations as limits of the discrete one, according to assumptions on the coefficients:
e.g. fragmentation term
Can give depolymerization, a boundary condition or a fragmentation kernel
Necessity to obtain orders of magnitude
- Coalescence has been neglected but may be added
Some comments
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Study of the continuous model
Constant coefficient case (reduces to 3 coupled ODE) studied by Engl, Prüss, Pujo-Menjouet, Webb, Zacher
Motivation for studying the general case:- Silveira et al: 2-peak size distributions of polymers- Biophysical considerations (Radford et al, PNAS 09)
Size of particles
Density of polymers
“infectious efficiency”:
≈ Polymerization rate
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Continuous model – Well-PosednessA key feature : Mass balance equation:
(Simonnett, Walker, JMAP, 2006 & Laurençot, Walker, J. Evol. Equ. 2007): proved
For positive polymerization rate - Existence and uniqueness of classical solutions for regular
initial condition and bounded coefficients
- Existence of weak solutions for L1 initial condition
- Stability of the disease-free steady state when it is unique
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1. Disease-free steady state:
2. Disease steady state:
For x0=0: only possible if
Continuous model – Steady States
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Linked to the following eigenvalue problem:
If : the steady state is for
If the eigenvalue increases with V :
Stability of the disease steady state is possible
Continuous model – Disease Steady State
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Continuous model – Asymptotic behaviour
(Calvez, Lenuzza, Mouthon, Perthame et al, BMB 08 & JBD 08)
For a polymerization rate possibly vanishing toward zero:
- Stability or unstability of the zero steady state under 2 conditions:- Solution for the eigenvalue problem- The eigenvalue increases with V(t)Tool: General Relative Entropy (see Perthame, Michel, Mischler, etc)
- Boundedness of the solution
- Necessary assumptions on to obtain an eigenvector with two peaks
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Shape of the eigenvector (figures from Calvez, Lenuzza et al, 2008)
Constant coefficient caseVariable polymerization rate
Question: what conditions to ensure existence and uniqueness of a solution to the eigenvalue problem ?
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Study of the eigenvalue problem
(D. Gabriel, M3AS, 2010)The polymerization rate possibly vanishes toward zero
Rejoin general problems of cell division (see Michel, M3AS, 2006 and Perthame, 2007), and general aggregation/fragmentation models.
Many other possible applications
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Theorem (D., Gabriel). Under some technical assumptions, there exists a unique solution to the previous eigenvalue problem, and
Application. by General Relative Entropy principle (work by Michel, Mischler, Perthame, Ryzhik,…) definethen
Study of the eigenvalue problem
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Some necessary assumptions for the previous theorem
- For the kernel k: ,
- To avoid shattering:
- Fragmentation and polymerization rates:
Assumptions for well-posedness of the eigenvalue problem
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
- Truncated & regularized problem-> existence and uniqueness (Krein-Rutman)
- A priori estimates (by integration of the equation against proper weights,…)
-> weak compactness in L1 (Dunford Pettis) & convergence of the regularized problem
Well-posedness for the Eigenvalue problem – sketch of the proof
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
(Calvez, D., Gabriel, in progress)
Question: how does the eigenvalue depend on the parameters ?
Application: - Asymptotic behaviour (see above: monotonous
dependency on V is required)- Better understanding of paradoxical biological behaviour(e.g. : what happens if fragmentation increases ? See
Knowles et al, Science 09)- Improve “PMCA” protocole
Eigenvalue problem – Fitness dependency on the coefficients
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Marie DOUMIC Journées Venakides
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
Sonication phase: increase of β
Incubation phase: natural coefficients
Some questions:
- Is there an optimal finite λ when β increases ?
- How to choose an optimal strategy for incubation/sonication times in order to increase the polymer density as fast as possible ?
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
(Calvez, D., Gabriel, in progress)
Fragmentation dependence:
Proposition. i) If then
Eigenvalue problem – Fitness dependency on the coefficients
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Marie DOUMIC Journées Venakides
(Calvez, D., Gabriel, in progress)
Fragmentation dependence:
Proposition. ii) If then
Eigenvalue problem – Fitness dependency on the coefficients
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Idea: use a self-similar transformation: if(in the spirit of Escobedo, Mischler, Rodriguez Ricard, 04)
Define
For one has
And
Proving gives the result.
Eigenvalue problem – Fitness dependency on the coefficients
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
(Calvez, D., Gabriel, in progress)
Polymerization dependency: similarly
Proposition.
Eigenvalue problem – Fitness dependency on the coefficients
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Marie DOUMIC Journées Venakides
Eigenvalue problem – Numerical simulations (P. Gabriel)
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Perspectives • Many open questions concerning the eigenvalue
problem / asymptotic behaviour
• Optimization of the PMCA protocole(Calvez, Gabriel)
• Generalize the original model, and justify its use by asymptotic analysis and experiences
(in strong interaction with H. Rezaei)
• Apply inverse problem techniques to recover the equation parameters from INRA experiments
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Wednesday 19 April 2023
Marie DOUMIC Journées Venakides
To be continued…
Thanks to French ANR project grant
TOPPAZTheory and Observation of Protein polymerization
in Prion and Alzheimer diseases