macroscopic descriptions of biological systems in rn and in …hg94/singapore.pdf · 2020. 1....

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



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).


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

Chemotaxis



Absent/sparse attractant ⇒ change in τ distribution:

Figure 1: Movement ofDictyostelium cells. From L. Li etal., PLoS one (2008).


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

Chemotaxis



In this talk: τ with long tail fractional diffusion orchemotactic equations that involve non-local, fractionaldifferential operators.


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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 .


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

General strategy


∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0

∫S


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 .


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

General strategy


∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0

∫S



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.


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

General strategy


∂τσ+∂tσ+cθ·∇σ = −βσ, σ(·, η, 0) =∫ t0

∫S



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.


Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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


Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



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)

).


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



Ct D



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.


Introduction

Chemotaxis

Superdiffusion



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


Introduction

Chemotaxis

Superdiffusion



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:


Introduction

Chemotaxis

Superdiffusion



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)]


Introduction

Chemotaxis

Superdiffusion



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?


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

Metaplex


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).


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions

Metaplex


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).


Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



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 .



Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



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)

.



Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



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)

.



Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



∂tu = Du


H =

divJ1 · · · 0... . . . ...0 · · · divJN

.I Jj is the flux of particles.I divJj is the generator of the diffusion process in Ωj .


Introduction

Chemotaxis

Superdiffusion



Result I

T cells

Result II

Hitting times

From Rn tonetworks

Metaplexdiffusion

Numerics

Real-worldexamples

Conclusions



∂tu = Du


T =

−∑

j αj1Tj1 · · · α1NT1N...

. . ....

αN1TN1 · · · −∑

j αjNTjN

I Tij transition operators between Ωi , Ωj .I αij are the transition probabilities.


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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).


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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(+).


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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 (◦).


Introduction

Chemotaxis

Superdiffusion



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


Introduction

Chemotaxis

Superdiffusion



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.


Introduction

Chemotaxis

Superdiffusion



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.


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.


References

Figure 7: Distribution of tumble (gray) and run (black)intervals from the E. coli. From E. Korobkova et al.,Nature (2004).


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.


References


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)).


References


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 .


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 · ∇)

.


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.


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.


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


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) .


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)


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

).


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)


References



Ct D



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 .


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

fd@rm@0:

macroscopic descriptions of biological systems in rn and in …hg94/singapore.pdf · 2020. 1....

Documents