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Macroscopic descriptions of biological systemsin Rn and in networks
Gissell Estrada-Rodríguez 1
(H. Gimperlein 2, K. J. Painter 2, J. Štoček 2 and E. Estrada 3)
1LJLL Sorbonne Université, (France). 2Maxwell Institute and Heriot-WattUniversity, (U.K.). 3IUMA, Universidad de Zaragoza, (Spain).
Workshop on Mathematical Biology: Modelling, Analysisand Simulations
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Results of this talk
I From biological systems:
I Fractional Patlak-Keller-Segel equation:∂tu = c0∇ · (Cα∇α−1u − χu∇ρ) (bacteria E. coli).
I Space-time fractional diffusion:Ct D
κutot = ∇ ·(Cα,κ∇α−1utot
)(T cells in the brain).
I Study of hitting times for immune cell search strategies.
I Analysis of diffusion and superdiffusion in complexgeometries:
I We introduce a network of subdomains, corresponding tothe nodes of a graph.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 2 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Chemotaxis
Classical case of chemotaxis: the individual runs for sometime τ , it stops at (x, t) and chooses a new direction at random.τ follows a Poisson process Patlak-Keller-Segel equations:
∂tu = ∇ · (C (u, ρ)∇u − χ(u, ρ)∇ρ),∂tρ = Dρ∆ρ+ f (u, ρ).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 3 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Chemotaxis
Classical case of chemotaxis: the individual runs for sometime τ , it stops at (x, t) and chooses a new direction at random.τ follows a Poisson process Patlak-Keller-Segel equations:
∂tu = ∇ · (C (u, ρ)∇u − χ(u, ρ)∇ρ),∂tρ = Dρ∆ρ+ f (u, ρ).
Absent/sparse attractant ⇒ change in τ distribution:
Figure 1: Movement ofDictyostelium cells. From L. Li etal., PLoS one (2008).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 3 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Chemotaxis
Classical case of chemotaxis: the individual runs for sometime τ , it stops at (x, t) and chooses a new direction at random.τ follows a Poisson process Patlak-Keller-Segel equations:
∂tu = ∇ · (C (u, ρ)∇u − χ(u, ρ)∇ρ),∂tρ = Dρ∆ρ+ f (u, ρ).
In this talk: τ with long tail fractional diffusion orchemotactic equations that involve non-local, fractionaldifferential operators.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 3 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Superdiffusion
Diffusion (Brownianmotion):
〈x2〉 ∝ t
Nonlocal diffusion (Lévymotion):
〈x2〉 ∝ t2/α, 1 ≤ α ≤ 2
Note: The systems we study don’t directly follow a Lévyprocess in space.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 4 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Experimental evidence for superdiffusion
Figure 1: MSD ofDictyostelium cells in absenceof a chemotactic signal. FromL. Li et al., PLoS one (2008).
Figure 2: Distribution oftumble (gray) and run (black)intervals from the E. coli.From E. Korobkova et al.,Nature (2004).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 5 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Microscopic description
Consider an individual in a medium in Rn with an attractantconcentration ρ(x, t):
I Running probability: ψ(·, θ, τ) =(S(ρ,Dθρ)S(ρ,Dθρ)+τ
)αwhere
Dθρ = ∂tρ+ cθ · ∇ρ, θ ∈ S = {|x| = 1} and 1 < α < 2.
I The stopping frequency during a run phase is
β(·, θ, τ) = −∂τψ(·, θ, τ)ψ(·, θ, τ)
.
I The turn angle distribution is given byk(·, θ; η) = `(·, |η − θ|) where
∫S k(·, θ; η)dθ = 1.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 6 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Microscopic description
Consider an individual in a medium in Rn with an attractantconcentration ρ(x, t):
I Running probability: ψ(·, θ, τ) =(S(ρ,Dθρ)S(ρ,Dθρ)+τ
)αwhere
Dθρ = ∂tρ+ cθ · ∇ρ, θ ∈ S = {|x| = 1} and 1 < α < 2.
I The stopping frequency during a run phase is
β(·, θ, τ) = −∂τψ(·, θ, τ)ψ(·, θ, τ)
.
I The turn angle distribution is given byk(·, θ; η) = `(·, |η − θ|) where
∫S k(·, θ; η)dθ = 1.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 6 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Microscopic description
Consider an individual in a medium in Rn with an attractantconcentration ρ(x, t):
I Running probability: ψ(·, θ, τ) =(S(ρ,Dθρ)S(ρ,Dθρ)+τ
)αwhere
Dθρ = ∂tρ+ cθ · ∇ρ, θ ∈ S = {|x| = 1} and 1 < α < 2.
I The stopping frequency during a run phase is
β(·, θ, τ) = −∂τψ(·, θ, τ)ψ(·, θ, τ)
.
I The turn angle distribution is given byk(·, θ; η) = `(·, |η − θ|) where
∫S k(·, θ; η)dθ = 1.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 6 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
General strategy
I The density σ(x, t, θ, τ) satisfies
∂τσ + ∂tσ + cθ · ∇σ︸ ︷︷ ︸Run phase
= −βσ︸︷︷︸Tumble phase
σ(·, η, 0)︸ ︷︷ ︸Particles immediatelystarting a new run
=
∫ t0
∫S
(βσ)k(·, θ;η)dθdτ .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 7 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
General strategy
I The density σ(x, t, θ, τ) satisfies
∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0
∫S
(βσ)k(·, θ;η)dθdτ .
I Scaling: (x, t, c , τ) 7→ (x/ε, t/ε, c0/εγ , τ/εµ) for µ > 0and 0 < γ < 1, where ε = τ̄ /T .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 7 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
General strategy
I The density σ(x, t, θ, τ) satisfies
∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0
∫S
(βσ)k(·, θ;η)dθdτ .
I Scaling: (x, t, c , τ) 7→ (x/ε, t/ε, c0/εγ , τ/εµ) for µ > 0and 0 < γ < 1, where ε = τ̄ /T .
I Integrate out τε∂t σ̄ + ε
1−γc0θ · ∇σ̄ = −(1− T )A,
σ̄(·, θ) =∫ t0σ(·, θ, τ)dτ, Tφ(η) =
∫Sk(·, θ; η)φ(θ)dθ,
A =∫ t0Bε(·, θ, t − s)σ̄(x− cθ(t − s), s, θ)ds.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 7 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
General strategy
I The density σ(x, t, θ, τ) satisfies
∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0
∫S
(βσ)k(·, θ;η)dθdτ .
I Scaling: (x, t, c , τ) 7→ (x/ε, t/ε, c0/εγ , τ/εµ) for µ > 0and 0 < γ < 1, where ε = τ̄ /T .
I Integrate out τε∂t σ̄ + ε
1−γc0θ · ∇σ̄ = −(1− T )A.
A =∫ t0Bε(·, θ, t − s)σ̄(x− cθ(t − s), s, θ)ds.
I Conservation equation: ∂tu + c0∇ · w = 0 where
u := 1|S|∫S σ̄dθ and w :=
εγ
|S |∫S θσ̄dθ.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 7 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Nonlocal Patlak-Keller-Segel equation
Assuming that τ is distributed according to a power law withexponent α then, the motion of the individuals is described by1
∂tu = c0∇ · (Cα∇α−1u − χu∇ρ) ,
Cα = −π(τ0c0)
α−1(α− 1)sin(πα)Γ(α)
(|S | − n2ν1)|S |(1− ν1)
> 0 and χ =τ1c0τ0
.
I Cα can be determined from the microscopic parameters.I For s ∈ (0, 2) the fractional gradient is defined
[Meerschaert et al. (2006)]
∇su(x, t) = 1|S |
∫Sθ(θ · ∇)su(x, t)dθ.
1ER-Gimperlein-Painter. “Fractional PKS equations for chemotacticsuperdiffusion”. In: SIAP 78 (2018).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 8 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Nonlocal T cell movement: Experimental evidence
Immune cells in thebrain.
T cells:
From T. Harris et.al., Nature(2012).
I Further experiments show longwaiting times which combinedwith the long runs give rise to ageneralized Lévy walk.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 9 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
T cells: Long runs + Long waiting times
New ingredients: The running and waiting time probabilities,for α ∈ (1, 2) and κ ∈ (0, 1) are
ψ(x, τ) =( τ0(x)τ0(x) + τ
)α, and ψr (r) =
( r0r0 + r
)κ.
Kinetic system:
∂t σ̄(·, θ) + cθ · ∇σ̄(·, θ)︸ ︷︷ ︸Moving particles
= σ(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
−∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸
Particles enteringa tumble phase
∂t σ̄0(·, θ)︸ ︷︷ ︸Resting particles
= T
∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸Particles enteringthe tumble phase
−σ0(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 10 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
T cells: Long runs + Long waiting times
New ingredients: The running and waiting time probabilities,for α ∈ (1, 2) and κ ∈ (0, 1) are
ψ(x, τ) =( τ0(x)τ0(x) + τ
)α, and ψr (r) =
( r0r0 + r
)κ.
Kinetic system:
∂t σ̄(·, θ) + cθ · ∇σ̄(·, θ)︸ ︷︷ ︸Moving particles
= σ(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
−∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸
Particles enteringa tumble phase
∂t σ̄0(·, θ)︸ ︷︷ ︸Resting particles
= T
∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸Particles enteringthe tumble phase
−σ0(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 10 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Main result
If the run time and pauses during re-orientations followapproximate Lévy distributions, then for utot = umoving + uresting,the movement of the individuals is described by2
Ct D
κutot = ∇ ·(Cα,κ∇α−1utot
).
Remarks:I No chemotaxis (according to experiments).I Two populations: moving and resting individuals.I Fractional derivative in time is introduced by solving the
resting density equation.
2ER-Gimperlein-Painter-Štoček. “Space-time fractional diffusion in cellmovement models with delay”. In: M3AS (2019).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 11 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Fundamental solution
Assuming Cα,κ independent of x, the fundamental solution of
Ct D
κ utot = Cα,κ∇ ·(∇α−1utot
)= C̃α,κ(−∆)α/2utot,
is given in terms of Fox H functions,
G (x, t) =1
πn/2|x|nH2,12,3
(|x|α
2αC̃α,κtκ
∣∣∣(1,1);(1,κ)(n/2,α/2);(1,1);(1,α/2)
).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 12 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Fundamental solution
Assuming Cα,κ independent of x, the fundamental solution of
Ct D
κ utot = Cα,κ∇ ·(∇α−1utot
)= C̃α,κ(−∆)α/2utot,
is given in terms of Fox H functions,
G (x, t) =1
πn/2|x|nH2,12,3
(|x|α
2αC̃α,κtκ
∣∣∣(1,1);(1,κ)(n/2,α/2);(1,1);(1,α/2)
).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 12 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Fundamental solution
Assuming Cα,κ independent of x, the fundamental solution of
Ct D
κ utot = Cα,κ∇ ·(∇α−1utot
)= C̃α,κ(−∆)α/2utot,
is given in terms of Fox H functions,
G (x, t) =1
πn/2|x|nH2,12,3
(|x|α
2αC̃α,κtκ
∣∣∣(1,1);(1,κ)(n/2,α/2);(1,1);(1,α/2)
).
In the limit |x|α
C̃α,κtκ� 1 the asymptotic behaviour of H is known,
G (x, t) ' 1|x|n
(|x|α
2αC̃α,κtκ
)−1.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 12 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Hitting times
We seek the first time at which the density of the solution inthe target position T , reaches a certain threshold δ, i.e., weseek t0 such that
δ =
∫T
∫Rn
G (x− y, t0)u0(y)dydx.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.810
0
101
102
103
104
105
106
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 13 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
From Rn to networks
Diffusion in complex geometries such as the brain:
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 14 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
From Rn to networks
Diffusion in complex geometries such as the brain:
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 14 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
From Rn to networks
Diffusion in complex geometries such as the brain:
Other examples:
• Long range human movement[Brockmann et. al (2006)]
• Metapopulations• Nonlocal foraging strategy ofanimal [Ramos-Fernandez et. al(2004)]
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 14 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Metaplex
Main questions: How do we study global diffusion in thesecomplex geometries? How does the internal structure of thenodes in the network affect the global phenomenon?
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 15 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Metaplex
Main questions: How do we study global diffusion in thesecomplex geometries? How does the internal structure of thenodes in the network affect the global phenomenon?
Metaplexa: A metaplex is a 4-tuple Υ = (V ,E , I, ω), where:• (V ,E ) is a graph• ω = {Ωj}kj=1 = {�, ©, 4, ...} with Borel measure µj .• I : V → ω
aEstrada-ER-Gimperlein. “Metaplex networks: influence of the exo-endostructure of complex systems on diffusion”. In: SIAM Review, ResearchSpotlight (2019).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 15 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Metaplex
Main questions: How do we study global diffusion in thesecomplex geometries? How does the internal structure of thenodes in the network affect the global phenomenon?
In this talk:• ω = {B(0, r) ⊂ R2}• Lebesgue measure on the domain• I is constant.
a
aEstrada-ER-Gimperlein. “Metaplex networks: influence of the exo-endostructure of complex systems on diffusion”. In: SIAM Review, ResearchSpotlight (2019).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 15 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (external dynamics)3
Diffusion process in the network: The diffusion among thenodes of the graph is controlled by
du (t)dt
= −
(dmax∑d=1
cd∆d
)u (t) , u (0) = u0,
where ∆d is the d-path Laplacian operator of the graph:
3Estrada. “Path Laplacian matrices: introduction and application to theanalysis of consensus in networks”. In: LAA (2012).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 16 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (external dynamics)3
Diffusion process in the network: The diffusion among thenodes of the graph is controlled by
du (t)dt
= −
(dmax∑d=1
cd∆d
)u (t) , u (0) = u0,
where ∆d is the d-path Laplacian operator of the graph:
∆d f (v) =∑
w∈V ,dist(v ,w)=d
(f (v)− f (w)) , v ∈ V .
3Estrada. “Path Laplacian matrices: introduction and application to theanalysis of consensus in networks”. In: LAA (2012).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 16 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (external dynamics)3
Diffusion process in the network: The diffusion among thenodes of the graph is controlled by
du (t)dt
= −
(dmax∑d=1
cd∆d
)u (t) , u (0) = u0,
where ∆d is the d-path Laplacian operator of the graph:cd tunes the hopping of the diffusive particle between
∆M f :=dmax∑d=1
d−s∆d f︸ ︷︷ ︸long- ranged (Mellin transform)
, ∆e f :=dmax∑d=1
e−ds∆d f︸ ︷︷ ︸short-ranged (Laplace transform)
.
3Estrada. “Path Laplacian matrices: introduction and application to theanalysis of consensus in networks”. In: LAA (2012).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 16 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (external dynamics)3
Diffusion process in the network: The diffusion among thenodes of the graph is controlled by
du (t)dt
= −
(dmax∑d=1
cd∆d
)u (t) , u (0) = u0,
where ∆d is the d-path Laplacian operator of the graph:cd tunes the hopping of the diffusive particle between
∆M f :=dmax∑d=1
d−snet ∆d f︸ ︷︷ ︸long- ranged (Mellin transform), snet ∈ (1, 3)
, ∆e f :=dmax∑d=1
e−dsnet ∆d f︸ ︷︷ ︸short-ranged (Laplace transform)
.
3Estrada. “Path Laplacian matrices: introduction and application to theanalysis of consensus in networks”. In: LAA (2012).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 16 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (internal dynamics)
Diffusion process in the continuous space: uj (x, t) in thenode vj ∈ V , for x ∈ Ω ⊂ Rn evolves according to
∂tuj (x, t) = (−∆)snod uj (x, t) .
For snod ∈ (0, 1] and x ∈ Rn, (−∆)snod is defined as
(−∆)snoduj(x) = cn,s P.V .∫
Rn
uj(x)− uj(y)|x− y|n+2snod
dy .
We consider this operator in a bounded domain withNeumann boundary conditions.4
4ER-Gimperlein-Painter-Štoček. “Space-time fractional diffusion in cellmovement models with delay”. In: M3AS (2019).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 17 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Preliminaries (internal dynamics)
Diffusion process in the continuous space: uj (x, t) in thenode vj ∈ V , for x ∈ Ω ⊂ Rn evolves according to
∂tuj (x, t) = (−∆)snod uj (x, t) .
For snod ∈ (0, 1] and x ∈ Rn, (−∆)snod is defined as
(−∆)snoduj(x) = cn,s P.V .∫
Rn
uj(x)− uj(y)|x− y|n+2snod
dy .
We consider this operator in a bounded domain withNeumann boundary conditions.4
4ER-Gimperlein-Painter-Štoček. “Space-time fractional diffusion in cellmovement models with delay”. In: M3AS (2019).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 17 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Diffusion in a Metaplex: Mathematical Description
For u = (u1, u2, ..., uN)T , where uj(x, t) is a density in thenode vj ∈ V and for (x, t) ∈ Ωj × (0,∞), we have
∂tu = Du
where D = H + T is an N × N block operator matrix ofunbounded operators with N the number of nodes.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 18 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Diffusion in a Metaplex: Mathematical Description
For u = (u1, u2, ..., uN)T , where uj(x, t) is a density in thenode vj ∈ V and for (x, t) ∈ Ωj × (0,∞), we have
∂tu = Du
where D = H + T is an N × N block operator matrix ofunbounded operators with N the number of nodes.
H =
divJ1 · · · 0... . . . ...0 · · · divJN
.I Jj is the flux of particles.I divJj is the generator of the diffusion process in Ωj .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 18 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Diffusion in a Metaplex: Mathematical Description
For u = (u1, u2, ..., uN)T , where uj(x, t) is a density in thenode vj ∈ V and for (x, t) ∈ Ωj × (0,∞), we have
∂tu = Du
where D = H + T is an N × N block operator matrix ofunbounded operators with N the number of nodes.
T =
−∑
j αj1Tj1 · · · α1NT1N...
. . ....
αN1TN1 · · · −∑
j αjNTjN
I Tij transition operators between Ωi , Ωj .I αij are the transition probabilities.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 18 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Numerical Set Up
Metaplex of 51 circular domains Ω = Ωj ⊂ R2 in the form of apath graph. We start from a uniform distribution in node 1and we study the effect of:
I snet (diffusion in network) and snod (diffusion in nodes).
I Size of the nodes Ωs = B(0, 1) and Ωb = B(0, 100).
I Coupling points: central and disjoint sinks and sources.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 19 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Numerical Set Up
Metaplex of 51 circular domains Ω = Ωj ⊂ R2 in the form of apath graph. We start from a uniform distribution in node 1and we study the effect of:
I snet (diffusion in network) and snod (diffusion in nodes).
I Size of the nodes Ωs = B(0, 1) and Ωb = B(0, 100).
I Coupling points: central and disjoint sinks and sources.
I Nature of the coupling:• Short-ranged: 2−snetdist(i ,j) (Laplace-transformed ∆d).• Long-ranged: dist(i , j)−snet (Mellin-transformed ∆d).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 19 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Influence of the node’s size
I Disjoint sinks and sources and short range coupling2−snetdist(i ,j).
I Equilibration: |∫
Ω1u1(x, t)dx− 1N
∫Ω1
u0(x)dx|
0 2000 4000 6000 8000 1000
Time
10-3
10-2
10-1
100
101
102
Eq
uili
bra
tio
n o
f th
e d
en
sity
snet=0.4, snod=0.2
snet=0.6, snod=0.2
snet=0.8, snod=0.2
snet=0.4, snod=0.8
snet=0.6, snod=0.8
snet=0.8, snod=0.8
0 200 400 600 800 1000
Time
40
50
60
70
80
90
Equili
bra
tion o
f th
e d
ensity
snet=0.4, snod=0.2
snet=0.6, snod=0.2
snet=0.8, snod=0.2
snet=0.4, snod=0.8
snet=0.6, snod=0.8
snet=0.8, snod=0.8
Figure 3: Density equilibration in node 1 for Ωs (left) and Ωb (right).
I Supported by analytical results by looking at the spectralproperties of the operators.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 20 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Superdiffusion in a Metaplex (exo-dynamics)
100
101
102
Nodes
10-10
10-5
100
Density
snode=0.8
snode=0.2
snode=0.8
snode=0.2
t=10
100
101
102
Nodes
10-6
10-4
10-2
100
102
Density
snode=0.8
snode=0.2
snode=0.8
snode=0.2
t=100
100
101
102
Nodes
100
101
102
Density
snode=0.8
snode=0.2
snode=0.8
snode=0.2
t=1000
100
101
102
Nodes
10-4
10-2
100
102
Density
t=10
t=100
t=1000
t=10
t=100
t=1000
snet=1.5
100
101
102
Nodes
10-4
10-2
100
102
Density
t=10
t=100
t=1000
t=10
t=100
t=1000
snet=2
100
101
102
Nodes
10-6
10-4
10-2
100
102
Density
t=10
t=100
t=1000
t=10
t=100
t=1000
snet=2.5
Figure 4: Top: short range coupling, snet = 0.8 (◦) resp. 0.4(×).Bottom: long range coupling, snod = 0.2 (◦) resp. 0.8(+).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 21 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Real-World Metaplexes
Figure 5: Landscape (left)and macaque visual cortexmetaplex (right).
Landscape (183 nodes): ≈ linear metaplexI Average shortest path distance is ∼ 11.88 and dmax = 32.I The average degree of the patches is ∼ 5.78.
Macaque visual cortex (30 nodes): small-world metaplexI Average shortest path distance is ∼ 1.54 and dmax = 3.I The average degree is ∼ 12.67.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 22 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Real-World Metaplexes
Figure 5: Landscape (left)and macaque visual cortexmetaplex (right).
Landscape (183 nodes): ≈ linear metaplexI Average shortest path distance is ∼ 11.88 and dmax = 32.I The average degree of the patches is ∼ 5.78.
Macaque visual cortex (30 nodes): small-world metaplexI Average shortest path distance is ∼ 1.54 and dmax = 3.I The average degree is ∼ 12.67.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 22 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Landscape metaplex
I There are no significant differences between low and highdegree nodes (only for the case snod = snet = 0.8).
I A trade-off between the endo and exo structure (as for thelinear metaplex).
Figure 6: Density as a function of distance for long range coupling(a) and short range coupling (b). snod = 0.8 (+), resp. 0.2 (◦).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 23 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Macaque metaplex
I The internal dynamics dominate the global diffusion in themetaplex:
10 1000 2000 3000 4000 5000
Time
10-4
10-3
10-2
10-1
100
De
nsity
snet=0.4, snod=0.2
snet=4, snod=0.2
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 24 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Results
I The geometry of the nodes and their coupling playcrucial roles for the global dynamics.
I Superdiffusion in the nodes accelerates diffusion in themetaplex, but it cannot lead to superdiffusion.
I For small-world metaplexes, the external dynamicsdoes not play any significant role.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 25 / 26
-
Introduction
Chemotaxis
Superdiffusion
Experimentalevidence
Microscopicdescription
Result I
T cells
Result II
Hitting times
From Rn tonetworks
Metaplexdiffusion
Numerics
Real-worldexamples
Conclusions
Outlook
I Include internal biochemical pathways into thechemotactic system.
I Combine the models derived from biological systems andthe notion of metaplex to find efficient search strategiesfor robots swarm in complex environments.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 26 / 26
-
References
Thanks! Any questions?ljll.math.upmc.fr/estradarodriguez/
Estrada. “Path Laplacian matrices: introduction and application to theanalysis of consensus in networks”. In: LAA (2012).
Estrada-ER-Gimperlein. “Metaplex networks: influence of the exo-endostructure of complex systems on diffusion”. In: SIAM Review, ResearchSpotlight (2019).
ER-Gimperlein-Painter. “Fractional PKS equations for chemotacticsuperdiffusion”. In: SIAP 78 (2018).
ER-Gimperlein-Painter-Štoček. “Space-time fractional diffusion in cellmovement models with delay”. In: M3AS (2019).
Tajie Harris et al. “Generalized Lévy walks and the role of chemokines inmigration of effector CD8+ T cells”. In: Nature 486 (2012).
E Korobkova et al. “From molecular noise to behavioural variability in asingle bacterium”. In: Nature 428 (2004).
L Li, S F Nørrelykke, and E C Cox. “Persistent cell motion in the absenceof external signals: a search strategy for eukaryotic cells”. In: PLoS one 3.5(2008), e2093.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 27 / 26
-
References
Figure 7: Distribution of tumble (gray) and run (black)intervals from the E. coli. From E. Korobkova et al.,Nature (2004).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 28 / 26
-
References
From long jump RW to the fractional Laplacian
Normal diffusion:
u(x , t + dt) =12u(x − dx , t) + 1
2u(x + dx , t)
u(x , t + dt)−u(x , t)dt
=dx2
2dtu(x − dx , t)−2u(x , t) + u(x + dx , t)
dx2
For dx → 0 and dt → 0 and dx2/2dt = D, we obtain ut = D∆u.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 29 / 26
-
References
From long jump RW to the fractional Laplacian
Normal diffusion:
ut = D∆u
Anomalous diffusion: [E. Valdinoci (2009)]
p(jump of length d) = C(β)/dβ,
u(x , t + dt)− u(x , t)dt
=∑y
1dt
C (β)
|y |β(u(x + y , t)− u(x , t)).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 29 / 26
-
References
From long jump RW to the fractional Laplacian
Normal diffusion:
ut = D∆u
Anomalous diffusion: [E. Valdinoci (2009)]
u(x , t + dt)− u(x , t)dt
=∑y
C̃ (dy)β
(dy)α(u(x + y , t)− u(x , t))
|y |β.
Passing to the limit: ut = C̃ (−∆)α/2u for β = n + α inn-dimensions where
(−∆)α/2 = c∫Rn
u(x + y)− u(x)|y |n+α
dy .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 29 / 26
-
References
Derivation of A
I Integrate out τ
ε∂t σ̄ + ε1−γc0θ · ∇σ̄ = −(1− T )
∫ t0β(·, θ, τ)σ(·, θ, τ)dτ ,∫ t
0 β(·, θ, τ)σ(·, θ, τ)dτ =∫ t0 Bε(·, θ, t−s)σ̄(x−cθ(t−s), s, θ)ds.
In Laplace space B̂ε =ϕ̂(·, θ, ελ+ ε1−γc0θ · ∇)ψ̂(·, θ, ελ+ ε1−γc0θ · ∇)
.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 30 / 26
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References
Data for T cells
Figure 8: Nonlinear growth of them.s.d. in time.
Figure 9: ζ(t), which is a rescaleddisplacement, increasesapproximately as a power law.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 31 / 26
-
References
New ingredients in the modelling
I Running + waiting probabilities, for α ∈ (1, 2) andκ ∈ (0, 1)
ψ(x, τ) =(
τ0(x)τ0(x) + τ
)α, ψr (r) =
(r0
r0 + r
)κ.
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 32 / 26
-
References
New ingredients in the modelling
I Running + waiting probabilities, for α ∈ (1, 2) andκ ∈ (0, 1)
ψ(x, τ) =(
τ0(x)τ0(x) + τ
)α, ψr (r) =
(r0
r0 + r
)κ.
I Density of moving + resting populations:
∂t σ̄(·, θ) + cθ · ∇σ̄(·, θ)︸ ︷︷ ︸Moving particles
= σ(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
−∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸
Particles enteringa tumble phase
∂t σ̄0(·, θ)︸ ︷︷ ︸Resting particles
= T
∫ t0β(x, τ)σ(·, θ, τ)dτ︸ ︷︷ ︸Particles enteringthe tumble phase
−σ0(·, θ, τ = 0)︸ ︷︷ ︸Particles starting
a new run
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 32 / 26
-
References
System for resting and moving particles
I Density of moving + resting populations:
∂t σ̄ + cθ · ∇σ̄ = σ(·, θ, 0)−∫ t0 β(x, τ)σ(·, θ, τ)dτ ,
∂t σ̄0 = T∫ t0 β(x, τ)σ(·, θ, τ)dτ − σ0(·, θ, 0) ,
σ(·, θ, 0) = σ0(·, θ, 0) ,
σ(·, θ, 0) =∫ t0drφ(r)
∫ t−r0
dτ
∫Sdηβ(x, τ)σ(x, t−r , η, τ)k(·, η; θ) .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 33 / 26
-
References
Solving resting particles’equation
The particles at rest satisfy the following equation
εν∂t σ̄0(·, θ) = T∫ t0B(x, t − s)σ̄(x− cθ(t − s), s, θ)ds
− T∫ t0φε(t − s)
(∫ s0B(x, t − s ′)σ̄(x− cθ(t − s ′), s ′, θ)ds ′
)ds .
(1)
The Laplace transform of this expression is
ενλˆ̄σ0(x, λ, θ)− εν σ̄00(x, θ) = rκ0 ε(ν+%)κλκT B̂ε(x, ε1−γc0θ · ∇)ˆ̄σ(x, λ, θ) ,(2)
and if we assume that ν > 1− γ as before we get
εν∂t σ̄0(·, θ) = rκ0 ε(ν+%)κT tDκBε(x, ε1−γc0θ · ∇)σ̄(·, θ) . (3)
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 34 / 26
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References
General strategy
I For µ, γ > 0 and %� ν > 0, we use the scaling(x, t, c, τ, r) 7→ (x/ε, t/ε, c0/εγ , τ/εµ, r/ε%).
I For σtot = 1|S |(ū + εϑnθ · w̄
)+ 1|S | ū0 the conservation
equation is
∂t(ū + ū0) + nc0∇ · w̄ = ∂tutot +∇ · (Cα∇α−1(utot − ū0))where
ū0(x, t) =1|S |
∫Sσ̄0(·, θ)dθ .
I w̄ is obtained from the equation of the moving particles.
I Integrating the equation of the resting particles we obtainū0 and the fractional operator in time tD1−κ is introduced,hence
∂tutot = tD1−κ∇ ·
(Cα,κ∇α−1utot
).
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 35 / 26
-
References
Back up
⇒ ∂tutot = ∇ ·(Cα∇α−1ū
)= ∇ · (Cα∇α−1(utot − ū0)) .
Expanding the right hand side of (3) and choosing only theleading terms we obtain, in the Laplace space,
λˆ̄σ0(x, λ, θ)−σ̄00(x, 0, θ) = rκ0 λκ(α− 1)τ0
ˆ̄u+O(ε(ν+%)κ+µ(α−2)+(α−1)(1−γ)
).
Substituting ū = utot− ū0 into the right hand side and groupingterms we obtain
ˆ̄u0(x, λ)−1λū00(x, 0) =
ûtot1 + τ0rκ0 (α−1)λ
1−κ . (4)
Since λ→ 0 then, applying a Taylor expansion and assuming allparticles are moving at t = 0, i.e. ū00 = 0, we have
ˆ̄u0 =
(1− τ0λ
1−κ
rκ0 (α− 1)+O
(λ2(1−κ)
))ûtot . (5)
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 36 / 26
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References
Fundamental solution
Assuming Cα,κ independent of x, the fundamental solution of
Ct D
κ utot = Cα,κ∇ ·(∇α−1utot
)= C̃α,κ(−∆)α/2utot,
is given in terms of Fox H functions.
G (t, x) =1
πn/2|x|nH2,12,3
(|x|α
2αC̃α,κtκ
∣∣∣(1,1);(1,κ)(n/2,α/2);(1,1);(1,α/2)
).
In the limit |x|α
C̃α,κtκ� 1 the asymptotic behaviour of H is known,
G (t, x) ' 1|x|n
(|x|α
2αC̃α,κtκ
)qfor q = −1 .
Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 37 / 26
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References
Hitting time
δ =
∫T
∫Rn
G (x− y, t0)u0(y)dydx. (6)
Assuming that the initial positions of the particles are given byxi , so that u0(x) =
∑i δxi (x), we obtain
δ =∑i
∫TG (x− xi , t0)dx (7)
If all initial positions are at distance � (C̃α,κτκ)1/α from thetarget T , we may use the asymptotic expansion of theH-function from the previous subsection to obtain
δ ' 2αC̃α,κt
κ0
πn/2
∑i
∫T|x− xi |−α−ndx
' 2αC̃α,κt
κ0
πn/2vol(T )
∑i
|x0 − xi |−α−n,(8)
where x0 is a centre of the target T .Gissell Estrada-Rodríguez Nonlocal macroscopic PDEs from kinetic equations 20/01/2020 38 / 26
IntroductionChemotaxisSuperdiffusionExperimental evidenceMicroscopic descriptionResult IT cellsResult IIHitting timesFrom Rn to networksMetaplex diffusionNumericsReal-world examplesConclusionsAppendixReferences
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