Developmental Partial Differential Equations Nastassia Pouradier - PowerPoint PPT Presentation

Developmental Partial Differential Equations Nastassia Pouradier Duteil Rutgers University - Camden Kinet Young Researchers’ Workshop November 30, 2016 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 1 / 32

Outline Motivation: A description of oogenesis 1 The heat equation on time-varying manifolds 2 A “Lie bracket” between transport and heat 3 Control of growth via a signal 4 Future Directions 5 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 2 / 32

Motivation: A description of oogenesis Drosophila oogenesis Figure: Drosophila melanogaster oogenesis Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 3 / 32

Motivation: A description of oogenesis Morphogens Morphogen Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

Motivation: A description of oogenesis Morphogens Morphogen Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

Motivation: A description of oogenesis Morphogens Morphogen Figure: “French flag model” Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

Motivation: A description of oogenesis Morphogens Morphogen Figure: “French flag model” Figure: Morphlogies of Drosophila eggshells and Gurken patterning Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

Motivation: A description of oogenesis Mechanism of Gurken diffusion and internalization Figure: Gurken diffusion from oocyte nucleus in the perivitelline space and internalization into the follicle cells Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 5 / 32

Motivation: A description of oogenesis Mechanism of Gurken diffusion and internalization Figure: Gurken in Drosophila willistoni Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 6 / 32

The heat equation on time-varying manifolds Motivation: A description of oogenesis 1 The heat equation on time-varying manifolds 2 A “Lie bracket” between transport and heat 3 Control of growth via a signal 4 Future Directions 5 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 7 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t M 0 u x w Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t v [ · ] : P c ( R d ) → Lip ( R d , R d ) M 0 u Growth vector field x w Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t v [ · ] : P c ( R d ) → Lip ( R d , R d ) M 0 u Growth vector field x w Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t v [ · ] : P c ( R d ) → Lip ( R d , R d ) M 0 u Growth vector field x w Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t v [ · ] : P c ( R d ) → Lip ( R d , R d ) M 0 u Growth vector field x w Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds General model M t : time-varying compact manifold of dimension n M t embedded in R d = R n +1 φ # u Organism’s membrane φ φ # w x t v [ · ] : P c ( R d ) → Lip ( R d , R d ) M 0 u Growth vector field x µ t ∈ P ( M t ): probability w measure on M t (also, µ t ∈ P c ( R d )) Morphogen diffusing in intercellular space Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 ∆ t : Laplace-Beltrami operator on M t Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 ∆ t : Laplace-Beltrami operator on M t Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

The heat equation on time-varying manifolds Wasserstein distance: Monge transportation problem How do you best move a pile of sand to fill up a given hole of the same total volume? Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

The heat equation on time-varying manifolds Wasserstein distance: Monge transportation problem Monge’s problem (1781) Given µ, ν ∈ P ( X ) and c : X × X → R + a Borel-measurable function, � Minimize c ( x , T ( x )) d µ ( x ) X among all transport maps T : X → X s.t. T # µ = ν . Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

The heat equation on time-varying manifolds Wasserstein distance: Monge transportation problem Kantorovich’s formulation (1940’s) Given µ, ν ∈ P ( X ) and c : X × X → R + a Borel-measurable function, � Minimize c ( x , y ) d γ ( x , y ) X × X where γ ∈ Π( µ, ν ) := { ρ ∈ P ( X × X ) | π 1 # ρ = µ, π 2 # ρ = ν } . Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

The heat equation on time-varying manifolds p -Wasserstein distance ��� � 1 / p � R n × R n | x − y | p d γ ( x , y ) W p ( µ, ν ) = inf γ ∈ Π( µ,ν ) Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 11 / 32

The heat equation on time-varying manifolds p -Wasserstein distance ��� � 1 / p � R n × R n | x − y | p d γ ( x , y ) W p ( µ, ν ) = inf γ ∈ Π( µ,ν ) Figure: Two measures with different L 1 and W 1 distances (respectively O (1) and O ( δ )). Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 11 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 ∆ t : Laplace-Beltrami operator on M t Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 12 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 ∆ t : Laplace-Beltrami operator on M t Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 12 / 32

The heat equation on time-varying manifolds Laplace-Beltrami operator Laplace-Beltrami operator: generalization of the Laplacian on Riemannian manifolds. ∆ f := ∇ · ∇ f Let ( x i ) i ∈{ 1 ,..., n } be a coordinate system on M t and g t be the metric tensor of M t . Let f ∈ C ∞ ( M t ). n n 1 ∂ ∂ � � � g ij ∆ t f = ( | g t | f ) t � ∂ x i ∂ x j | g t | i =1 j =1 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 13 / 32

The heat equation on time-varying manifolds Coupling of diffusion and manifold evolution Evolution of µ t by the combined transport and diffusion: Transport-diffusion PDE ∂ t µ t + ∇ · ( v [ µ t ] µ t ) = ∆ t µ t (1) where v : a Lipschitz function w.r.t. the Wasserstein distance W 2 ∆ t : Laplace-Beltrami operator on M t Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 14 / 32