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 a GFP-Pon in SOP precursor cells in living Drosophila larvae [Meyer, Emery, Berdnik, Wirtz-Peitz, Knoblich, Current Biology, 2005] Marrakech 18.04.2018 – p. 2/13

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 Γ 2 of cortex. β L (Ω) P (Ω) α γ ξ λ ℓ (Γ) p (Γ 2 ) σ ( aPKC ) 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 d L , d P > 0  L t − d L ∆ L = αP − βL, x ∈ Ω , t > 0 ,    ( V ) P t − d P ∆ P = − αP + βL, x ∈ Ω , t > 0 ,   x ∈ Ω L (0 , x ) = L 0 ( x ) , P (0 , x ) = P 0 ( x ) ,  Boundary conditions on ∂ Ω = Γ =Γ 1 ∪ Γ 2 and Γ 1 ∩ Γ 2 = ∅  ∂L d L ∂ν = γl − λL, x ∈ Γ , t > 0 ,    ( BC ) ∂P x ∈ Γ 1 , t > 0 , d P ∂ν = 0 ,   ∂P d P ∂ν = ξp, x ∈ Γ 2 , t > 0 ,  Reaction rates α, β, γ, λ, σ, ξ are positive constants Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model A prototypical model II Boundary dynamics  l t − d l ∆ Γ l = λL − γl − σχ Γ 2 l, x ∈ Γ , t > 0     p t − d p ∆ Γ 2 p = σl − ξp, x ∈ Γ 2 , t > 0 ,      ∂p ( BD ) x ∈ ∂ Γ 2 , d p ∂ν Γ2 = 0 ,   x ∈ Γ , l (0 , x ) = l 0 ( x ) ,       p (0 , x ) = p 0 ( x ) , x ∈ Γ 2 ,  ∆ is the usual Laplacian in the domain Ω ∆ Γ and ∆ Γ 2 are Laplace-Beltrami operator on Γ and Γ 2 χ Γ 2 is the characteristic function of Γ 2 Marrakech 18.04.2018 – p. 3/13

A Volume-Surface Reaction-Diffusion Model Properties and Local well-posedness Conservation law: total Lgl mass d �� � � � ( L ( t, x ) + P ( t, x )) + l ( t, x ) + p ( t, x ) = 0 . dt Ω Γ Γ 2 Local well-posedness: There exists of a unique weak/strong local solution ( L, P, l, p ) on (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 ) Marrakech 18.04.2018 – p. 3/13

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 . Marrakech 18.04.2018 – p. 3/13

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: L 2 -type energy estimate and Gronwall. Question: Convergence to complex balance equilibrium for all initial data and parameter? L 2 -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 = { S 1 , . . . , S N } , Complexes: C = { y 1 , . . . , y |C| } with y i ∈ ( { 0 } ∪ [1 , ∞ )) N , Reactions: R = { y → y ′ } from source y into product y ′ ∈ C . y r,i c y r = � N Mass action law reaction rate for y r → y ′ r : i =1 c i Reaction rate constant k r of the reaction y r → y ′ r . R ( c ) = � |R| r =1 k r c y r ( y ′ r − y r ) Reaction vector: Marrakech 18.04.2018 – p. 5/13

Systems of Reaction-Diffusion Equations Nonlinear Complex Balance Networks Nonlinear reaction-diffusion network ∂ ∂t c − D ∆ c = R ( c ) for ( x, t ) ∈ Ω × (0 , + ∞ ) , with D = diag( d 1 , . . . , d N ) . 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 : � � k r c y r ∞ = k s c y s ∞ . { r : y r = y } { s : y ′ s = y } Marrakech 18.04.2018 – p. 5/13

Systems of Reaction-Diffusion Equations Nonlinear Complex Balance Networks Relative (free energy) entropy functional N c i log c i � � � � E ( c | c ∞ ) = − c i + c i, ∞ dx c i, ∞ Ω i =1 Explicit (nontrivial) entropy dissipation functional with e ( x, y ) = x log ( x/y ) − x + y D ( c ) = − d dt E ( c | c ∞ ) |R| N |∇ c i | 2 ∞ , c y ′ � c y r � � r � � k r c y r ≥ 0 = d i dx + ∞ e c y r c y ′ c i r Ω ∞ i =1 r =1 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 L 1 with a rate λ/ 2 : N � � c i ( t ) − c i, ∞ � 2 L 1 (Ω) ≤ C − 1 CKP E ( c 0 | c ∞ ) e − λt for a.a. t > 0 , i =1 where C CKP 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 dt E ( f ) = d d dt E ( f ) − E ( f ∞ )) = −D ( f ) ≤ 0 provided conservation laws: D ( f ) = 0 ⇐ ⇒ f = f ∞ D ≥ Φ( E ( f ) − E ( f ∞ )) , Φ ≥ 0 Φ(0) = 0 , ⇒ explicit convergence in entropy, exponential if Φ ′ (0) > 0 ⇒ convergence in L 1 : � 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 Marrakech 18.04.2018 – p. 7/13

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