On Volume-Surface Reaction-Diffusion systems Klemens Fellner - - PowerPoint PPT Presentation

On Volume-Surface Reaction-Diffusion systems

Klemens Fellner

Institute of Mathematics and Scientific Computing, University of Graz joint works with L. Desvillettes, H. Egger, J.-F. Pietschmann,

• E. Latos, B.Q. Tang

Marrakech 18.04.2018 – p. 1/13

Complex-Balanced Volume-Surface RD Network

Protein-localisation before asymmetric stem-cell division Asymmetric stem-cell division: Cell-diversity by localisation of cell-fate determinants into one side of the cell cortex and into one of two daughter cells.a

aGFP-Pon in SOP precursor cells in living Drosophila larvae [Meyer, Emery,

Berdnik, Wirtz-Peitz, Knoblich, Current Biology, 2005]

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Complex-Balanced Volume-Surface RD Network

Protein-localisation before asymmetric stem-cell division Mathematical model: “high” concentrations, insignificant stochastic effects system of (reversible) reaction-diffusion equations volume(cytoplasm)-surface(membran) dynamics

Marrakech 18.04.2018 – p. 2/13

A Volume-Surface Reaction-Diffusion Model

Model Assumptions and Quantities Key protein: Lgl in cytoplasm (Ω) and cell cortex (Γ = ∂Ω). Key kinase: aPKC phosphorylates Lgl on a part Γ2 of cortex. L(Ω) P(Ω) ℓ(Γ) p(Γ2)

α β γ λ σ(aPKC) ξ

L(t, x) cytoplasmic Lgl ↔ l(t, x) cortical Lgl → activation of aPKC → p(t, x) cortical p-Lgl → P(t, x) cytoplasmic p-Lgl ↔ L(t, x)

Complex-balanced reaction-diffusion network Bio: qualitative interplay reaction/surface/volume diffusion

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model

Detailed versus Complex Balance Equilibria A detailed balance equilibrium balances the forward and backward reactions between all species/complexes. A complex balance equilibrium balances the total inflow and total outflow from and into all species/complexes.

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model

A prototypical model I Volume equations with diffusion coefficients dL, dP > 0 (V)        Lt − dL∆L = αP − βL, x ∈ Ω, t > 0, Pt − dP∆P = −αP + βL, x ∈ Ω, t > 0, L(0, x) = L0(x), P(0, x) = P0(x), x ∈ Ω Boundary conditions on ∂Ω = Γ =Γ1 ∪ Γ2 and Γ1 ∩ Γ2 = ∅ (BC)        dL

∂L ∂ν = γl − λL,

x ∈ Γ, t > 0, dP

∂P ∂ν = 0,

x ∈ Γ1, t > 0, dP

∂P ∂ν = ξp,

x ∈ Γ2, t > 0, Reaction rates α, β, γ, λ, σ, ξ are positive constants

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A Volume-Surface Reaction-Diffusion Model

A prototypical model II Boundary dynamics (BD)                    lt − dl∆Γl = λL − γl − σχΓ2l, x ∈ Γ, t > 0 pt − dp∆Γ2p = σl − ξp, x ∈ Γ2, t > 0, dp

∂p ∂νΓ2 = 0,

x ∈ ∂Γ2, l(0, x) = l0(x), x ∈ Γ, p(0, x) = p0(x), x ∈ Γ2, ∆ is the usual Laplacian in the domain Ω ∆Γ and ∆Γ2 are Laplace-Beltrami operator on Γ and Γ2 χΓ2 is the characteristic function of Γ2

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model

Properties and Local well-posedness Conservation law: total Lgl mass d dt

• Ω

(L(t, x) + P(t, x)) +

• Γ

l(t, x) +

• Γ2

p(t, x)

• = 0.

Local well-posedness: There exists of a unique weak/strong local solution (L, P, l, p)

• n (0, T), which is non-negative if the intital data are so.a

a[K.F

., S. Rosenberger, B.Q. Tang, Comm. Math. Sciences 2016]

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model

Complex balance reaction network

Figure 1: l-Lgl(Γ) with and without surface diffusion

Numerical analysis of VSRD models including discrete entropy structure/estimates: a

a[Egger, F

., Pietschmann, Tang, to appear in Applied Math & Computation]

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model Figure 2: p-Lgl(Γ) with and without surface diffusion

Surface diffusion O(10−2) : indirect surface diffusion effect via weakly reversible reaction O(1) and volume diffusion O(10−2)

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A Volume-Surface Reaction-Diffusion Model Figure 3: L-Lgl(Ω) with and without surface diffusion

Surface diffusion and weakly reversible reaction lead to stationary hump in L within Ω.

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A Volume-Surface Reaction-Diffusion Model Figure 4: P-Lgl(Ω) with and without surface diffusion

Stationary hump in L as consequence of inflow from p into P → L and shape of Ω.

Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model

Global existence and large time behaviour Theorem: Unique global-in-time weak solution (L, P, l, p). Proof: L2-type energy estimate and Gronwall. Question: Convergence to complex balance equilibrium for all initial data and parameter? L2-Entropy?

Marrakech 18.04.2018 – p. 3/13

Another (Volume-Surface) RD Model

Lipolysis Lipolysis: Breakdown of lipids and hydrolysis of triglycerides into glycerol and fatty acids.

Marrakech 18.04.2018 – p. 4/13

Systems of Reaction-Diffusion Equations

Nonlinear Complex Balance Networks Substances: S = {S1, . . . , SN}, Complexes: C = {y1, . . . , y|C|} with yi ∈ ({0} ∪ [1, ∞))N, Reactions: R = {y → y′} from source y into product y′ ∈ C. Mass action law reaction rate for yr → y′

r:

cyr = N

i=1 c yr,i i

Reaction rate constant kr of the reaction yr → y′

r.

Reaction vector: R(c) = |R|

r=1 krcyr(y′ r − yr)

Marrakech 18.04.2018 – p. 5/13

Systems of Reaction-Diffusion Equations

Nonlinear Complex Balance Networks Nonlinear reaction-diffusion network ∂ ∂tc − D∆c = R(c) for (x, t) ∈ Ω × (0, +∞), with D = diag(d1, . . . , dN). Homogeneous Neumann BCs on Lipschitz domain Ω. [JH72]: A complex balanced network has a unique positive equilibrium, which balances the total outflow and inflow for all complexes y ∈ C:

• {r: yr=y}

krcyr

∞ =

• {s: y′

s=y}

kscys

∞.

Marrakech 18.04.2018 – p. 5/13

Systems of Reaction-Diffusion Equations

Nonlinear Complex Balance Networks Relative (free energy) entropy functional E(c|c∞) =

N

• i=1
• Ω
• ci log ci

ci,∞ − ci + ci,∞

• dx

Explicit (nontrivial) entropy dissipation functional with e(x, y) = x log (x/y) − x + y D(c) = − d dtE(c|c∞) =

N

• i=1

di

• Ω

|∇ci|2 ci dx +

|R|

• r=1

krcyr

∞e

cyr cyr

∞, cy′

r

cy′

r

∞

• ≥ 0

Marrakech 18.04.2018 – p. 5/13

Systems of Reaction-Diffusion Equations

Nonlinear Complex Balance Networks Theorem:a For complex balanced RD networks without boundary equilibria, any renormalised (Fisher [2015]) solution c(x, t) converges exponentially to c∞ in L1 with a rate λ/2:

N

• i=1

ci(t) − ci,∞2

L1(Ω) ≤ C−1 CKP E(c0|c∞) e−λt

for a.a. t > 0, where CCKP is the constant in a Csiszár-Kullback-Pinsker type inequality. Renormalised solutions satisfy all mass/charge conservation laws and a weak entropy-dissipation law, Fisher [2017]

a[K.F

. B.Q.Tang, ZAMP 2018]

Marrakech 18.04.2018 – p. 5/13

The Entropy Method

Quantitative large-time behaviour E(f) non-increasing convex entropy functional D(f) entropy production, f∞ entropy minimising equilibrium d dtE(f) = d dtE(f) − E(f∞)) = −D(f) ≤ 0 provided conservation laws: D(f) = 0 ⇐ ⇒ f = f∞ D ≥ Φ(E(f) − E(f∞)), Φ(0) = 0, Φ ≥ 0 ⇒ explicit convergence in entropy, exponential if Φ′(0) > 0 ⇒ convergence in L1 : f − f∞2

1 ≤ C(E(f) − E(f∞))

Cziszár-Kullback-Pinsker inequalities for convex entropies

Marrakech 18.04.2018 – p. 6/13

The Entropy Method

Entropy Method Advantages: based on functional inequalities → "robust" avoids linearisation → "global" results allows for explicit constants nonlinear diffusion: [T], [CJMTU], [AMTU], [DV]. . . inhomogeneous kinetic equations: [DV], ... reaction-diffusion systems: [Grö83], [Grö92], [DF06], [DF08], [DF14], [MMH15], [FL16], [PSZ17], [DFT17], [FT17], [HHMM18], [FT18] no Bakry-Emery strategy

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Systems of Reaction-Diffusion Equations

Entropy Method for Complex Balance Networks Theorem:a For any complex balanced reaction networks without boundary equilibria, there exists a constant λ > 0 and the “exponential” entropy entropy-dissipation estimate D(c(t)) ≥ λ E(c(t)|c∞), Proof via convexification: [MMH15], [PSZ17] (detailed b.) Proof via explicit estimates using conservation laws Q c = M: [DFT17], [FT17], [FLT18] Method applies also to volume-surface RD systems Proof via reduction to finite-dimensional inequality: [FT18]

a[L. Desvillettes, K.F

., B.Q. Tang, SIMA 2017], [K.F ., B.Q. Tang, Nonlinear Anal- ysis 2017.], [K.F . E.Latos B.Q.Tang, Annales IHP (C) 2018]

Marrakech 18.04.2018 – p. 8/13

Systems of Reaction-Diffusion Equations

Entropy Method for Complex Balance Networks Lemma:a For all states c ∈ RN

>0 satisfying E(c|c∞) < +∞ and

the conservation laws Q c = M, there exists a positive constant H1 = H1(Q, M, y ∈ C, E(c|c∞)) such that

|R|

• r=1
• c

c∞

yr

−

• c

c∞

y′

r2

≥ H1

N

• i=1
• ci

ci,∞ − 1 2 . Here,

• c

c∞ =

• c1

c1,∞, . . . ,

• cN

cN,∞

• .

This finite-dimensional inequality implies D(c(t)) ≥ λ(H1) E(c(t)|c∞),

a[K.F

., B.Q. Tang, ZAMP 2018]

Marrakech 18.04.2018 – p. 8/13

Systems of Reaction-Diffusion Equations

Weakly Reversible Network of Linear Reactions Explicit exponential convergence to equilibrium state for weakly reversible volume-surface reaction-diffusion system:a L P ℓ p

α β γ λ σ(aPKC) ξ

aK.F

., Bao Q. Tang Springer Proc. Math & Stats 2017

Marrakech 18.04.2018 – p. 9/13

Systems of Reaction-Diffusion Equations

Weakly Reversible Network of Linear Reactions Similar: Friedmann-Neumann-Rannacher modela u1 u3 u4 u5 u6 u7 u2 u0

rimp2 rdelay rdelay rdelay rdelay rdelay ractpJAK rexp rimp

• aE. Friedmann, R. Neumann, R. Rannacher, Well-posedness of a linear spatio-

temporal model of the JAK2/STAT5 signaling pathway, Comm. Math. Anal. 15 (2013) 76-102.

Marrakech 18.04.2018 – p. 9/13

Systems of Reaction-Diffusion Equations

Boundary Equilibria Example: A B + C 2B

k1 k3 k2

       at − da∆a = −k1a + k3b2, bt − db∆b = k1a + k2bc − 2k3b2, ct − dc∆c = k1a − k2bc, Boundary equilibrium (a∗, b∗, c∗) = (0, 0, M). Problem: D(a∗, b∗, c∗)) = 0, but E(c∗|c∞) > 0 No global entropy-entropy dissipation estimate possible!

Marrakech 18.04.2018 – p. 10/13

Systems of Reaction-Diffusion Equations

Boundary Equilibria Our approach: weaker entropy-entropy dissipation estimate along solution trajectories D(c(t)) ≥ λ(t) E(c(t)|c∞) Difficulty: λ(t) → 0 near boundary equilibria. However, if λ(t) satisfies +∞ λ(s)ds = +∞, then E(c(t)|c∞) ≤ E(c0|c∞)e−

R t

0 λ(s)ds → 0

as t → ∞. ⇒ (algebraic) instabilty of boundary equilibria ⇒ Exponential convergence to positive equilibrium

Marrakech 18.04.2018 – p. 10/13

Systems of Reaction-Diffusion Equations

Boundary Equilibria Corresponding RD system              at − da∆a = −k1a + k3b2, x ∈ Ω, t > 0, bt − db∆b = k1a + k2bc − 2k3b2, x ∈ Ω, t > 0, ct − dc∆c = k1a − k2bc, x ∈ Ω, t > 0, ∇a · ν = ∇b · ν = ∇c · ν = 0, x ∈ ∂Ω, t > 0, inf

x∈Ω b(x, t) ≥

1

• 1

b0

• L∞ + 2k3t,

for all t ≥ 0. Solutions would need infinite initial entropy to remain close to boundary equilibria for an unbounded time interval. a

a[L. Desvillettes, K.F

., B.Q. Tang, SIMA 2017]

Marrakech 18.04.2018 – p. 10/13

Systems of Reaction-Diffusion Equations

Boundary Equilibria Theorem:a Let c(t) be a renormalised solution of an arbitray complex balanced network. Assume that there exists H1 : [0, ∞) → [0, ∞) such that ∞

0 H1(s)ds = +∞ and for a.a.

t ≥ 0

|R|

• r=1

 

• c(t)

c∞

yr

−

• c(t)

c∞

y′

r



2

≥ H1(t)

N

• i=1
• ci(t)

ci,∞ − 1 2 . Then, the renormalised solution c(t) converges exponentially to the positive equilibrium c∞.

a[K.F

., B.Q. Tang, ZAMP 2018], [M. Pierre, T. Suzuki, Umakoshi]

Marrakech 18.04.2018 – p. 10/13

Systems of Reaction-Diffusion Equations

Boundary Equilibria Global Attractor Conjecture: For any complex balanced mass action law reaction networks, all solution trajectory subject to positive initial data are conjectured to converge to the positive equilibrium c∞. Proof for ODE systems by Gheorghe Craciun in 2015? Above finite-dimensional inequality has ODE structure!? But ODE system and averaged PDE concentrations: d dtu = R(u) = R(c) = d dtc(t)

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Systems of Reaction-Diffusion Equations

Boundary Equilibria Boundary equilibria for complex balanced reaction networks: S1 + S2 3S1 2S1 + S3 2S2 Open problem!

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Systems of Reaction-Diffusion Equations

Nonlinear Diffusion        ∂tci − di∆(cmi

i ) = fi(c),

x ∈ Ω, t > 0, i = 1, . . . , N, di∇(cmi

i ) · −

→ n = 0, x ∈ ∂Ω, t > 0, i = 1, . . . , N, ci(x, 0) = ci,0(x), x ∈ Ω, i = 1, . . . , N, (i) |fi(c)| ≤ C(1+|c|ν), ∀c = (c1, . . . , cN) ∈ RN, ∀i = 1, . . . , N (ii) Mass dissipation: There exist positive constants λ1, . . . , λN > 0 such that: S

i=1 λifi(u) ≤ 0,

∀c ∈ RS (iii) Quasi-positivity ⇒ Propagation of non-negativity

Marrakech 18.04.2018 – p. 11/13

Systems of Reaction-Diffusion Equations

Nonlinear Diffusion Assume mi > max{ν − 1; 1} and mi > ν −

4 d+2 if d ≥ 3.

⇒ Existence of global weak nonnegative solutions ci ∈ C([0, ∞); L1(Ω)), cmi

i

∈ L1(0, T; W 1,1(Ω)), fi(c) ∈ L1(Ω × [0, T]) and ciL∞(QT ) ≤ CT for all T > 0 and i = 1, . . . , N, Single reaction α1A1 + · · · + αMAM

kb

⇌

kf β1B1 + · · · + βNBN.

⇒ Exponential convergence to equilibrium ∀1 ≤ p < ∞,

M

• i=1

ai(t) − ai∞Lp(Ω) +

N

• j=1

bj(t) − bj∞Lp(Ω) ≤ C e−λpt

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Systems of Reaction-Diffusion Equations

Nonlinear Diffusion Proof of existence theory extends [LP17] Duality estimates Specific bootstrap

Marrakech 18.04.2018 – p. 11/13

Systems of Reaction-Diffusion Equations

Nonlinear Diffusion A generalised version of Logarithmic Sobolev Inequality:

• Ω

|∇ai|2 a2−mi

i

dx ≥ C(Ω, mi) a mi−1

i

• Ω

ai log ai ai dx. Degeneracy for ai ∼ 0 is control by functional inequalities for indirect diffusion effect and conservation law, since not all ai ∼ 0 can be small at the same time. Setting of “slowly growing” apriori estimates: First algebraic convergence, then exponential convergence! Indirect diffusion effect ∼ “coercive hypocoercivity”

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Quasi-steady-state approximation

QSSA as ξ → +∞ Limit of fast cortical release of Lgl: ξ → +∞ L P ℓ p

α β γ λ σ ξ

ξ→+∞

− − − − → L P l

α β σ γ λ

Goal: convergence towards reduced QSSA model. The reduced model is still a complex balance system.

Marrakech 18.04.2018 – p. 12/13

Quasi-steady-state approximation

QSSA as ξ → +∞ Theorem:a For any (L0, P0, l0, p0) ∈ L2(Ω) × L2(Ω) × L2(Γ) × L2(Γ2) Lξ

ξ→+∞

− − − − → L in L2([0, T] × L2(Ω), P ξ

ξ→+∞

− − − − → P in L1([0, T] × Ω), lξ

ξ→+∞

− − − − → l in L2([0, T] × Γ) pξ

ξ→+∞

− − − − → 0 in L2([0, T] × Γ2) for any T > 0 and up to a subsequence. Proof: duality method [M. Pierre, D. Bothe] and entropy.

a[T.Q.Bao, K.F

., S. Rosenberger Commun. Math Sciences 2016]

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Models for amyloids and protein aggregation

with Marie Doumic, Mathieu Mézache, Human Rezaei Model for transient oscillations in coagulation-fragmentation experiments of PrP fibrils        V + W

k

− → 2W, W + Ci

ai

− → Ci+1, 1 ≤ i ≤ n, Ci + V

bi

− → Ci−1 + 2V, 2 ≤ i ≤ n. Simplest two-polymer model with normalised coefficients dv

dt = v [−kw + c2] , dw dt = w [kv − c1] ,

dc1

dt = −wc1 + vc2, dc2 dt = wc1 − vc2,

Marrakech 18.04.2018 – p. 13/13

Models for amyloids and protein aggregation

with Marie Doumic, Mathieu Mézache, Human Rezaei small parameter ε = 1

k

  

dv dt = v [w∞ − w] + εv [v∞ + w∞ − v − w] , dw dt = w [v − v∞] + εw [v∞ + w∞ − v − w] .

Zero-order Hamiltonian H = v0 − v∞ ln v0 + w0 − w∞ ln w0 Full model entropy d dtH(v(t), w(t)) = −ε [(v − v∞) + (w − w∞)]2 . ⇒ Equilibration and transient oscillations for k large.

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Models for amyloids and protein aggregation

with Marie Doumic, Mathieu Mézache, Human Rezaei THANK YOU VERY MUCH!!

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