Kinetic equations and cell motion Beno^ t Perthame

Kinetic equations and cell motion

Benoıt Perthame

Outline

I. Bacterial motion

II. Kinetic models of bacterial motion

III. Diffusion and hyperbolic limits (macroscopic equations)

IV. Traveling bands

V. Molecular pathways

Cell movement

Eukaryotic cell movement (from Vic Small, Wien)

Cell movement

E Coli swimming (from H. Berg, Boston)

Cell movement

.

B. Subtilis. Top S. Serror (CNRS). Bottom K. Ben Jacob (Tel Aviv Univ.)

Cell movement

Yo Bisschops (displayed in Lille)

Cell movement

Gray-Scott model from chemical reaction

∂∂tu(t, x)− d1∆u = r u

(uv − µ

),

∂∂tv(t, x)− d2∆v = −r u2v,

∂∂tw(t, x) = −r µ u.

The dynamics is driven by the chemical reaction. It is an exampleof ’diffusion enhanced’ instability :The cells in the front have an advantage for nutients availability

Survey : I. Golding, Y. Kozlovsky, I. Cohen, E. Ben-Jacob. Phys. A,260 (1998).

Cell movement

Solution to the Gray-Scott model : u+ w. By A. Marrocco, INRIA Bang.

Cell movement

MIMURA’s model for B. Subtilis

∂∂tn(t, x)− d1∆n = r n

(S − µ n

(n0+n)(S0+S)

),

∂∂tS(t, x)− d2∆S = −r nS,

∂∂tf(t, x) = r n µ n

(n0+n)(S0+S)

The dynamics is driven by the source terms, i.e., by bacterial growth.

Cell movement

Cell movementPattern formation based on nutrient depletion

Questioned by micro-biologists for nutrient rich experiments

Courtesy of B. Holland, S. Serror

Cell movement

A force based model with branching

∂∂tn(t, x) + div

[n(1− n)∇c− n∇S

]= 0,

−dc∆c+ c = αcn,

∂∂tf(t, x)− df∆f = αfn+ f(1− f)

∂∂tS(t, x)− dS∆S + S = αS

(nmother colony + f + n

).

• n = swarmer (leader) cells,

• f = supporter (follower) cells,

• c = short range ’attractant’

• S = long range signal (surfactant ?)

Cell movement

F. Cerreti, B. P., C. Schmeiser, M. Tang, N. Vauchlet

Cell movement

Claim. Branching instabilities occur in the reduced systems for

swarmer cells∂∂tn(t, x) + div

[n(1− n)∇c− n∇S

]= 0, x ∈ Rd, t ≥ 0,

−dc∆c+ c = αcn, (short range attraction)

∂∂tS(t, x)− dS∆S + τSS = αS n ( long range repulsion )

Several approaches on 1-dimensional traveling pulses.

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Solution n n

Solution c c

Cell movement

Special case 2. L� 0, dS ≥ dc

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Short range attraction, Long range repulsion

Cell movement

So far we have two explanations for instability

Nurient depletion


Cell movement

Chemotaxis (E. Coli)∂∂tn(t, x)−∆n(t, x) + div(nχ∇c) = 0, x ∈ Rd,

−∆c(t, x) = n(t, x),

Has a deep mathematical structure Jager-Luckhaus, Biler et al,

Herrero- Velazquez, Suzuki-Nagai, Brenner et al, Laurencot, Corrias,

Horstman, Winkler.

Cell movement

Theorem (dimensions d ≥ 2) - (method of Sobolev inequalities)

(i) for ‖n0‖Ld/2(Rd) small, then there are global weak solutions,

(ia) they gain Lp regularity,

(ib) and ‖n(t)‖L∞(Rd) → 0 with the rate of the heat equation,

(ii) for( ∫|x|2n0

)(d−2)< C‖n0‖d

L1(Rd)with C small, there is blow-up

in a finite time T ∗.

Cell movementThe existence proof relies on Jager-Luckhaus argument

ddt

∫n(t, x)p = −4

p

∫|∇ np/2|2 +

∫p∇np−1 n χ ∇c︸︷︷︸

χ∫∇np·∇c=−χ

∫np∆c

= −4

p

∫|∇np/2|2︸︷︷︸

parabolic dissipation

+ χ∫np+1︸︷︷︸

hyperbolic effect

Using Gagliardo-Nirenberg-Sobolev ineq. on the quantity

u(x) = np/2, we obtain∫np+1 ≤ Cgns(d, p)‖∇np/2‖2

L2 ‖n‖Ld2

Cell movementIn dimension 2, for Keller and Segel model :

{∂∂tn(t, x)−∆n(t, x) + div(nχ∇c) = 0, x ∈ R2,−∆c(t, x) = n(t, x),

Theorem (d=2) (Method of energy) (Blanchet, Dolbeault, BP)

(i) for ‖n0‖L1(R2) <8πχ , there are smooth solutions,

(ii) for ‖n0‖L1(R2) >8πχ , there is creation of a singular measure

(blow-up) in finite time.

(iii) For radially symmetric solutions, blow-up means

n(t) ≈ 8 πχ δ(x = 0) +Rem (M. Herrero, J.-L. Velazquez)

Cell movementExistence uses (i) energy

d

dt

[∫R2n logn dx−

χ

2

∫R2n c dx

]= −

∫R2

∣∣∣∇√n− χ∇c∣∣∣2 dx ≤ 0

(ii) a limiting Hardy-Littlewood-Sobolev ineq. (Beckner, Carlen-Loss, 96)

for n > 0, M =∫n(x)dx,∫

R2n logn dx+

2

M

∫ ∫R2×R2

n(x)n(y) log |x− y| dx dy ≥M(1 + logπ + logM)

Notice that in d = 2 we have

−∆c = n, c(t, x) =−1

2π

∫n(t, y) log |x− y| dy

n ∈ L1log =⇒

∫nc <∞.

Cell movementBlow-up, critical mass, continuation after blow-up, see

Hererro, Velazquez...

Blanchet, Carrillo, Masmoudi, Carlen

Dolbeault, Schmeiser

Cell movement

. From A. Marrocco (INRIA, BANG)

Cell movement

So far we have 3 explanations for instabilities/patterns

Nurient depletion


Chemotaxis

We now consider only E. Coli (chemotactic bacteria)

Cell movement

We now consider only E. Coli (chemotactic bacteria)

Go back to the individual cell

E Coli swimming (from H. Berg, Boston)

Bacterial movement

E. Coli is known (since the 80’s) to move by run and tumble

depending on the coordination of motors that control the flagella

See Alt, Dunbar, Othmer, Stevens.

Bacterial movement

Denote by f(t, x, ξ) the density of cells moving with the velocity ξ

∂

∂tf(t, x, ξ) + ξ · ∇xf︸︷︷︸

run

= K[c, f ]︸︷︷︸tumble

,

K[c, f ] =∫BK(c; ξ, ξ′)f(ξ′)dξ′︸︷︷︸

cells of velocity ξ′ turning to ξ

−∫BK(c; ξ′, ξ)dξ′ f,

−∆c(t, x) = n(t, x) :=∫Bf(t, x, ξ)dξ,

Properties : Nonnegativity, Mass conservation

Bacterial movement

What is the correct description of the tumbling rate

K[c, f ] =∫BK(c; ξ, ξ′)f(ξ′)dξ′ −

∫BK(c; ξ′, ξ)dξ′ f ?

General principle

K(c; ξ, ξ′) = K(c; ξ′)

Cells can measure along their path,

they cannot forsee

Bacterial movement

What is the correct description of the tumbling rate

K[c, f ] =∫BK(c; ξ, ξ′)f(ξ′)dξ′ −

∫BK(c; ξ′, ξ)dξ′ f ?

Rule No 1

K(c; ξ, ξ′) = k−(c(x− εξ′)

)And ε represents a memory effect, measuring the chemoattractant

concentration along a path.

Bacterial movementExer For K(c; ξ, ξ′) = k−

(c(x− εξ′)

), f ≥ 0, assume

A := sup0≤t≤T

‖k−(c)‖Lq(Rd) <∞.

Use the method of characteristics and Young’s inequality to provethat for 1

p + 1q = 1,

‖f(t)‖Lp(Rd×V ) ≤ ‖f0‖Lp(Rd×V ) + C(T )A

∫ t0‖f(s)‖Lp(Rd×V )ds.

Apply the method when, for some constant B > 0, k−(c) ≤ Bc and csolves

−∆c+ c = n,

Chalub, Markowich, BP, Schmeiser

Bacterial movement

More realistic

When c increases, jumps are longer

Bacterial movement

What is K[c, f ] =∫BK(c; ξ, ξ′)f(ξ′)dξ′ −

∫BK(c; ξ′, ξ)dξ′ f?

Rule No 2. This leads Dolak and Schmeiser to choose

K(c; ξ, ξ′) = k(∂c∂t

+ ξ′.∇c).

With (stiff response)

k(z) =

k− for z < 0,

k+ < k− for z > 0.

More generally k(·) a (smooth) decreasing function

Bacterial movement

Conclusion of this section

We have two rules

Find a way to assert which is better

Diffusion limitAssume that

M(ξ) ≥ 0,∫RdM(ξ)dξ = 1,

∫Rd|ξ|2M(ξ)dξ <∞,∫

RdξM(ξ)dξ = 0, 0 < k− ≤ k(x) ≤ k+,

ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

k(x)

ε

[nε(t, x)M(ξ)− fε]

Theorem (usual limit) The limit is

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)

∂

∂tn(t, x)− div

[ A

k(x)∇n(t, x)

]= 0

Diffusion limitProof. (i) Because∫

V

k(x)

εnε(t, x)M(ξ)dξ =

∫V

k(x)

εfεdξ

we conclude that∂

∂tnε + divJε = 0

Jε(t, x) =∫V

ξ

εfε(t, x, ξ)dξ

Diffusion limitProof. (ii) Prove that∫

V×Rdfε(t, x, ξ)

2dxdξ ≤ C

∫ ∞0

∫V×Rd

|fε(t, x, ξ)− nεM(ξ)|2dxdξdt ≤ C

∫ ∞0

∫V×Rd

|Jε(t, x)|2dxdt ≤ C

Diffusion limitProof. (iii) Extract from the equation

ξ

εfε =

ξ

εnε(t, x)M(ξ)− ε

ξ

k(x)

∂

∂tfε(t, x, ξ)−

ξ

k(x)ξ · ∇xfε


ξ

εfε =

ξ

εnε(t, x)M(ξ)− ε

ξ

k(x)

∂


ξ

k(x)ξ · ∇xfε

Jε =∫V

ξ

εfε = −ε

∫V

ξ

k(x)

∂


∫V

ξ

k(x)ξ · ∇xfε

−→ε→0−∫Vξ ⊗ ξM(ξ)dξ

∇n(t, x)

k(x):= −A.

∇n(t, x)

k(x)

∂

∂tn+ divJ = 0 =

∂

∂tn− div

[A.∇n(t, x)

k(x)

]

Diffusion limit

Come back to rule No 1

It has the advantage that a small time scale is given

ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε = M(ξ)

∫V

k(x− εξ′)ε

fε(t, x, ξ′)dξ′ −

k(x− εξ)ε

fε

Diffusion limit

Come back to rule No 1

ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε = M(ξ)

∫V

k(x− εξ′)ε


k(x− εξ)ε

fε

Theorem (rule No 1) The limit is

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)

∂

∂tn(t, x)− div

[ A

k(x)∇n(t, x)

]+ div

[nA.∇k(x)

k(x)

]= 0

(Keller-Segel, Drift-diffusion)


ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε = M(ξ)

∫V

k(x− εξ′)ε


k(x− εξ)ε

fε

ξ

εfε =

ξM(ξ)

ε

∫V

k(x− εξ′)k(x− εξ)

fε(t, x, ξ′)dξ′ − ε

ξ

k(x+ εξ)

∂

∂tfε(t, x, ξ)

−ξ

k(x+ εξ)ξ · ∇xfε


ξ

εfε =

ξM(ξ)

ε

∫V



ξ

k(x+ εξ)

∂

∂tfε(t, x, ξ)

−ξ


Jε =∫V

ξ

εfε =

∫V

∫V

ξM(ξ)

ε

k(x)

k(x)− εξ.∇k(x)fε(t, x, ξ

′)dξ′

+O(ε)−∫V

ξ

k(x)ξ · ∇xfε


ξ

εfε =

ξM(ξ)

ε

∫V



ξ

k(x+ εξ)

∂

∂tfε(t, x, ξ)

−ξ


Jε =∫V

ξ

εfε =

∫V

∫V

ξM(ξ)

ε

k(x)

k(x)− εξ.∇k(x)fε(t, x, ξ

′)dξ′

+O(ε)−∫V

ξ

k(x)ξ · ∇xfε

Jε =∫V

ξM(ξ)

ε

[1 +

εξ.∇k(x)

k(x)

]nε(t, x) +O(ε)−

∫V

ξ

k(x)ξ · ∇xfε

Diffusion limit

Jε =∫V

ξM(ξ)

ε

[1 +

εξ.∇k(x)

k(x)

]nε(t, x) +O(ε)−

∫V

ξ

k(x)ξ · ∇xfε

Jε =∫Vξ ⊗ ξM(ξ).

∇k(x)

k(x)n(t, x) +O(ε)−

∫Vξ ⊗ ξM(ξ)

∇xnk(x)

Diffusion limit

Jε =∫V

ξM(ξ)

ε

[1 +

εξ.∇k(x)

k(x)

]nε(t, x) +O(ε)−

∫V

ξ

k(x)ξ · ∇xfε

Jε =∫Vξ ⊗ ξM(ξ).

∇k(x)

k(x)n(t, x) +O(ε)−

∫Vξ ⊗ ξM(ξ)

∇xnk(x)

Jε −→ε→0

A.∇k(x)

k(x)n(t, x)−A

∇xnk(x)

Diffusion limit

For rule No 2 there is no identified small parameter so far

V = Sphere

∂

∂tf + ξ · ∇xf =

∫VK(c; ξ′)f(ξ′)− |V |K(c; ξ)f(ξ)

K(c; ξ′) = k(∂c∂t

+ ξ′.∇c).

Diffusion limit

Exer (Large asymmetry) Consider the scales

∂

∂tfε(t, x, ξ) + ξ · ∇xfε = M(ξ)

∫V

K(x, ξ′)

εfε(t, x, ξ

′)dξ′ −K(x, ξ)

εfε

Show that the limit reads

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)k0(x)

K(x, ξ)

∂

∂tn(t, x) + div[n U(t, x)] = 0

and identify the velocity field U

See F. James and N. Vauchelet for the stiff case

Diffusion limit

For Rule No 2., we can also write

K(c; ξ′) = k(∂c∂t

+ ξ′.∇c)

:= k0 + εk1

(∂c∂t

+ ξ′.∇c)

Diffusion limit

K(c; ξ′) = k(∂c∂t

+ ξ′.∇c)

:= k0 + εk1

(∂c∂t

+ ξ′.∇c)

Theorem (rule No 2)

∂

∂tn(t, x)− div

[ ak0∇n(t, x)

]+ div

[nU ] = 0

U(∂c∂t

,∇c)

=1

k0

∫ξk1

(∂c∂t

+ ξ.∇c)dξ

= ∇c K(∂c∂t

, |∇c|)

Flux-limited Keller-Segel because U is bounded


ε∂

∂tfε + ξ ·∇xfε = M(ξ)

∫V

k0 + εk1(x, ξ′)

εfε(t, x, ξ

′)dξ′ −k0 + εk1(x, ξ)

εfε

fε −→ε→0

M(ξ)n(x, t)

ξ

εfε = ξ[1 +

k1(x, ξ)

k0]fε(t, x, ξ)−O(ε)−

ξ

k0 + εk1(x, ξ)ξ · ∇xfε

Jε =∫V

ξ

εfε =

∫Vξk1(x, ξ)

k0fε(t, x, ξ)dξ +O(ε)−

∫V

ξ ⊗ ξk0· ∇xfε

Jε −→ε→0

∫Vξk1

(∂c∂t

+ξ

k0.∇c

)M(ξ)dξ n(x, t)−

A

k0.∇n

Conclusion

ε∂

∂tf(t, x, ξ) + ξ · ∇xf =

1

εK[c, f ],

K[c, f ] =∫VK(c; ξ, ξ′)f(ξ′)dξ′ −

∫VK(c; ξ′, ξ)dξ′ f,

Rule No 1. K(c; ξ, ξ′) = k−(c(x− εξ′)

)Gives the standard Fokker-Planck (Keller-Segel) equation

Rule No 2. K(c; ξ, ξ′) = k(∂c∂t + ξ′.∇c

)= k0 + εk1

(∂c∂t + ξ′.∇c

)Gives the Flux Limited Fokker-Planck (Keller-Segel) equation

So far

ε∂

∂tf(t, x, ξ) + ξ · ∇xf =

1

εK[c, f ],

K[c, f ] =∫VK(c; ξ, ξ′)f(ξ′)dξ′ −

∫VK(c; ξ′, ξ)dξ′ f,

−∆c(t, x) = n(t, x) :=∫Vf(t, x, ξ)dξ,

Rule No 1. K(c; ξ, ξ′) = k−(c(x− εξ′)

)Rule No 2. K(c; ξ, ξ′) = k

(∂c∂t + ξ′.∇c

)= k0 + εk1

(∂c∂t + ξ′.∇c

)

So far

ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε = M(ξ)

∫V

k(x− εξ′)ε


k(x− εξ)ε

fε

Theorem (Rule No 1) The limit is

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)

∂

∂tn(t, x)− div

[ A

k(x)∇n(t, x)

]+ div

[nA∇k(x)

k(x)] = 0

(Keller-Segel, Drift-diffusion)

So far

K(c; ξ, ξ′) = k(∂c∂t

+ ξ′.∇c)

= k0 + εk1

(∂c∂t

+ ξ′.∇c)

Theorem (Rule No 2) The limit is

∂

∂tn(t, x)− div

[ ak0∇n(t, x)

]+ div[nU ] = 0

U(∂c∂t

,∇c)

=1

k0

∫ξk1

(∂c∂t

+ ξ.∇c)dξ

= ∇c K(∂c∂t

, |∇c|)

Flux-limited Keller-Segel because U is bounded

Bacterial movement

We have seen the movement by run and tumble

There is a macroscopic consequence

Traveling bands

• Adler’s famous experiment for E. Coli (1966) is much simpler

• Medium contains nutrients

• Explain this pattern ; its asymmetry (Buguin, Saragosti, Silberzan,Curie institute)

• Is it a phenomena explanable at the macroscopic scale ?

Traveling bands

• Medium contains nutrients

∂∂tn(t, x)−∆n(t, x) + div

[n(χc∇c+ χS∇S)

]= 0,

−dc∆c(t, x) + c = n(t, x), ∂tS = −n S.

Traveling bandsIn dimension 2, for Keller and Segel model :

{∂∂tn(t, x)−∆n(t, x) + div(nχ∇c) = 0, x ∈ R2,−∆c(t, x) = n(t, x),

Theorem (d=2) (Method of energy) (Blanchet, Dolbeault, BP)

(i) for ‖n0‖L1(R2) <8πχ , there are smooth solutions,

(ii) for ‖n0‖L1(R2) >8πχ , there is creation of a singular measure

(blow-up) in finite time.

(iii) For radially symmetric solutions, blow-up means

n(t) ≈ 8 πχ δ(x = 0) +Rem (M. Herrero, J.-L. Velazquez)

Traveling bands

. From A. Marrocco (INRIA, BANG)

Traveling bands

Traveling waves of speed σ are 1-D solutions n(x− σt), c(x− σt)

Traveling pules satisfy also n(±∞ = 0) > 0

Theorem There are not such solutions for the Fokker-Planck

equation

∂n

∂t= ∆n−∇ · (n∇c)

Except for particular singular caes (Z.-A. Wang), one cannot observ traveling

pulses (bands)

Traveling bands

Traveling waves of speed σ are 1-D solutions n(x− σt), c(x− σt)−σn′ = n′′ − χ(nc′)′

−c′′ = nf − rc...etc,

−σn = n′ − χnc′

ln(n)′ = −σ + χc′

ln(n) = −σx+ χc+ µ

This is incompatible with any rule for production/regulation of c

Traveling bands

Traveling waves of speed σ are 1-D solutions n(x− σt), c(x− σt)−σn′ = n′′ − χ(nc′)′

−c′′ = nf − rc...etc,

−σn = n′ − χnc′

ln(n)′ = −σ + χc′

ln(n) = −σx+ χc+ µ −→|x|→∞

−∞

This is incompatible with any rule for production/regulation of c

Traveling bands

Conclusion (rule No1)

The Keller-Segel system, and therefore the rule No 1, is not

compatible with traveling pulses.

Consider rule number 2 and FLKS

Traveling bands∂∂tn(t, x)−∆n(t, x) + div[n(Uc + US)] = 0,

Uc = χ(ct, cx) ∇c|∇c|, US = χ(St, Sx) ∇S|∇S|

Theorem Asymmetric traveling pulses to the FLKS model existwith stiff response

20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

1.4

space

concentrations

Numerical solution with stiff response

Traveling bands−σn′ − n′′+ [n(Uc + US)]′ = 0,

Uc = χc sign(c′), US = χS sign(S′)

−dCc′′+ c = n − σS′ = −n

−σn− n′+ n(Uc + US) = 0,

20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

1.4

space

concentrations

Traveling bands −σn− n′+ n(Uc + US) = 0,

Uc = χc sign(c′), US = χS sign(S′) n = n(0) exp[x (−σ + χc + χS)] = 0, for x ≤ 0,

n = n(0) exp[x (−σ − χc + χS)] = 0, for x ≥ 0,

20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

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1.2

1.4

space

concen

tration

s

Traveling bands

Left : angular distribution of runs Right : duration of runs depending on

blblablablablblablabla the position

K might depend on ξ and ξ′

(number of flagella involved in the run ?)

Traveling bands

Superimposition of the calculated (pink) and the experimental (blue) concentration profiles

Traveling bands

Conclusion (rule No2)

the rule No 2 and FLKS give a quantitavely correct answer.

References : Construction of traveling pulse/waves solutions at the

kinetic level

V. Calvez, G. Raoul, Ch. Schmeiser

E. Bouin, V. Calvez, G. Nadin

Traveling bands

Next step : Can one explain the reason of a tumbling rate

K(c; ξ, ξ′) = K(∂c∂t

+ ξ′.∇c)

?

Biochemical pathwaysCan one explain the tumbling rate

K(c; ξ, ξ′) = K(∂c∂t

+ ξ′.∇c)?

Extracellular medium

——membrane——

Cytosol

Biochemical pathwaysCan one explain the tumbling rate

K(c; ξ, ξ′) = K(∂c∂t

+ ξ′.∇c)?

Use the internal biochemical pathway controling tumbling,

f(t, x, ξ,m) m= receptor methylation level (internal state)

c = external concentration

See Erban-Othmer, Dolak-Schmeiser, Zhu et al, Jiang et al

Biochemical pathwaysThis is used for droplets models in kinetic theory

f(t, x, ξ, r) r= radius of the droplets

∂

∂tf(x, ξ, t) + ξ.∇xf︸︷︷︸

Transport

+F∂rf︸︷︷︸change of radii

= Q(f, f)︸︷︷︸Binary collisions

Biochemical pathways

f(t, x, ξ,m) m= receptor methylation level (internal state)

c = external concentration

Extracellular medium c(x, t)

——– membrane ——–

Cytosol m


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

∂

∂m[R(m, c)f ] =︸︷︷︸

Change in methylation level

K[m, c][f ]

K[m, c][f ]=∫ [K(m, c, ξ, ξ′)f(t, x, ξ′,m)−K(m, c, ξ′, ξ)f(t, x, ξ,m)

]dξ′

Question : Can it be related to the tumbling kernel

K(c; ξ, ξ′) = K(∂c∂t

+ ξ′.∇c)


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

∂

∂m[R(m, c)f ] =︸︷︷︸

Change in methylation level

K[m, c][f ]

R(m, c) is an adaptation rate

Equilibrium is : m = M(c).

Simple models

R(m, c) = m−M(c)

R(m, c) = R(m−M(c)) = (m−M(c) G((m−M(c)

)G > 0


Introduce new scales

∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m−M(c))f ] = Kε[m, c][f ]

(fast adaptation)

Kε[m, c][f ]=∫[K(

m−M(c)

ε, ξ, ξ′)f(t, x, ξ′,m)−K(..., ξ′, ξ)f(t, x, ξ,m)

]dξ′

(stiff response)


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m−M(c))f ] = Kε[m, c][f ]

(fast adaptation)

Kε[m, c][f ]=∫[K(

m−M(c)


]dξ′

(stiff response)

Theorem : As ε→ 0,

fε(t, x, ξ,m)→ f(t, x, ξ) δ(m−M(c)

)and the tumbling kernel for f(t, x, ξ) is K

(∂c∂t + ξ′.∇c

).

Biochemical pathwaysProof. Step 1

∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m−M(c))f ] = Kε[m, c][f ]

In the limit∂

∂m[R(m−M(c))f ] = 0

R(m−M(c)) f(m) = Cst = 0

Therefore f(m) = 0 except when R(m−M(c)) = 0 i.e. whenm = M(c)

f(m) = f(t, x, ξ) δ(m−M(c)

)


Proof. Step 2

f(t, x, ξ) =∫f(t, x, ξ,m)dm

∂

∂tf(t, x, ξ,m) + ξ · ∇xf =

∫mKε[m, c][f ]dm

This implies

‖f(t)‖∞ ≤ C(t)


Proof. Step 3

∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m, c)f ] =

∫ [K(

m−M(c)


]dξ′

One cannot pass to the limit

f(m) = f(t, x, ξ) δ(m−M(c)

)∫ [

K(m−M(c) ≈ 0

ε ≈ 0, ξ, ξ′)f(t, x, ξ′,m)−K(..., ξ′, ξ)f(t, x, ξ,m)

]dξ′


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m, c)f ] =

∫ [K(

m−M(c)


]dξ′


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m, c)f ] =

∫ [K(

m−M(c)


]dξ′

f(t, x, ξ,m) = εq(t, x, ξ,m−M(c)

ε), y =

m−M(c)

ε


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m, c)f ] =

∫ [K(

m−M(c)


]dξ′


ε), y =

m−M(c)

ε

∂

∂tq(t, x, ξ, y) + ξ · ∇xf +

1

ε

∂

∂y[yG(εy)q] =

1

εDtM∂yq

+∫ [

K(y, , ξ, ξ′)q(t, x, ξ′, y)−K(y, ξ′, ξ)q(t, x, ξ, y)]dξ′


∂

∂tf(t, x, ξ,m) + ξ · ∇xf +

1

ε

∂

∂m[R(m, c)f ] =

∫ [K(

m−M(c)


]dξ′


ε), y =

m−M(c)

ε


∂

∂tq(t, x, ξ, y) + ξ · ∇xf +

1

ε

∂

∂y[yG(εy)q] =

1

εDtM∂yq

+∫ [

K(y, , ξ, ξ′)q(t, x, ξ′, y)−K(y, ξ′, ξ)q(t, x, ξ, y)]dξ′

Forces q −→ δ(y − DtMG(0))


0 100 200 300 400 500 600 700 8000

1

2

3

4

5x 10−3 (a) ρ

0 100 200 300 400 500 600 700 800−0.5

0

0.5

1

1.5

2

2.5x 10−3 (b) j

bacterial population inside the observation channel sepa-rates into two groups centered near the two control points.The migration of cells between the two groups is signifi-cant when the period is longer than 200 s [Figs. 2(b)–2(d)],and it is much reduced for higher driving frequencies[Fig. 2(a)]. Our experiments clearly show that for fast-varying gradients, the cell population cannot follow theattractant and exhibits oscillatory behaviors out of phasewith the stimulus waveform, in contrast to its responses toslowly varying gradients.

The average motion of the bacterial population can becharacterized by the center of mass (c.m.) of the bacteriawithin the observation window ½"300 !m; 300 !m#. InFig. 3(a), we show the c.m. position versus time fordifferent driving period T. Both the amplitude (A) of thec.m. and the phase difference (!") between the c.m.oscillation and the attractant oscillation change signifi-cantly with T. A increases with T and saturates whenT $ 600 s. Conversely, !" increases with decreasing T,from ð0:12& 0:02Þ# at T ¼ 1200 s to ð0:74& 0:22Þ# atT ¼ 80 s, as shown in Figs. 3(b) and 3(c).

Bacterial chemotaxis motion follows a run and tumblepattern with the tumbling frequency determined by thechemoeffector concentration and the chemoreceptor

methylation level, which is controlled by the adaptationprocess with a finite adaptation time $. The cell’s driftvelocity v, determined by the bias of the tumbling fre-quency, should therefore also follow a relaxation dynamics(see SM [15] for a detailed derivation). To capture thisrelaxation dynamics of v in the simplest way possible,we use a Langevin equation where v follows a linearrelaxation dynamics with a constant time $:

dx

dt¼ v;

dv

dt¼ "ðv" vdÞ=$þ %; (1)

where vd is the chemotaxis velocity. The Langevin equa-tion is coarse grained in time beyond the average run time$r of individual cells; therefore, the fluctuation in velocitychange %ðtÞ can be treated as a white noise: h%ðtÞ%ðt0Þi ¼2!&ðt" t0Þ, with strength! ¼ !0v

20$r=$

2, where!0 is anorder unity constant and v0 is the average run speed. Forsimulations of chemotaxis motions of individual cells, weuse the signaling pathway-based E. coli chemotaxis simu-lator (SPECS) model [16], where the internal signalingpathway dynamics is described by the interaction betweenthe average receptor methylation level and the kinaseactivity which determines the switch probability of theflagellar motor and eventually the cell motion. Accordingto the SPECS model simulations [16], the chemotaxis

velocity can be approximately expressed as vd *C @Gð½L#Þ

@x ½1þG"1c

@Gð½L#Þ@x #"1, where [L] is the ligand con-

centration andGð½L#Þ¼ lnð1þ½L#=KiÞ" lnð1þ½L#=KaÞ isthe free-energy difference between active and inactive

100µm

A T=100s B T=200s

C T=400s D T=800sTim

e/T

0

0.5

1

x(µm)200( )-200( ) 0 RL 200( )-200( ) 0 RL

0

0.5

1

Nor

mal

ized

Cel

l Den

sity 1

0.5

0

0.25

0.75

1

0.5

0

0.25

0.75

Nor

mal

ized

Lig

and

conc

entr

atio

n

RL L Rx(µm)

FIG. 2 (color online). The spatiotemporal profiles of the celldensity for different driving periods. (a) T ¼ 100 s,(b) T ¼ 200 s, (c) T ¼ 400 s, (d) T ¼ 800 s. The normalizedcell density at a given position x and time t (scaled by T) isrepresented by the color (see color bar). L-aspartate concentra-tions at the two source channels oscillate between 0.1 and0.9 mM with opposite phases. The gray scale stripes at thetwo sides of each panel show the normalized ligand concen-trations at the left (L) and right (R) control point as a function oftime, t ¼ 0 is when ligand concentration is maximum at the rightcontrol point. The cell density is measured 20 frames per period.

5mm

Modulating part

Source channel

BA

Observation partAgarose

Source channel

Buffer area

Observation channelAttractant

inputBufferinput

Outlet

Inlet for bacteria

100µm

Control point

x

x

C D

Time/T

Nor

mal

ized

Cel

l Den

sity

Con

cent

ratio

n

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0

T=800s

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0

T=100s

FIG. 1 (color online). Experimental setup. (a) A panoramicpicture of the two-layer PDMS (polydimethylsiloxane) chip.(b) A zoomed-in view of the observation channel. The timelapse of the normalized E. coli density and attractant concentra-tion at a control point is shown for periods T ¼ 800 s (c) and100 s (d). The ligand concentration is measured by adding asmall amount of fluorescein into attractant stocks. Bacterialdensities are determined by averaging the fluorescence intensityfrom the fluorescence-labeled cells in a region of &25 !maround the control points.

PRL 108, 128101 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

23 MARCH 2012

128101-2


Is there a reason to think that classical diffusion does not apply ?

what would be abnormal diffusion in this context ?

Abnormal diffusion

Is there a reason to think that classical diffusion does not apply ?

what would be abnormal diffusion in this context ?

ARTICLEReceived 28 Mar 2015 | Accepted 19 Aug 2015 | Published 25 Sep 2015

Swarming bacteria migrate by Levy WalkGil Ariel1, Amit Rabani2, Sivan Benisty2, Jonathan D. Partridge3, Rasika M. Harshey3 & Avraham Be’er2

Individual swimming bacteria are known to bias their random trajectories in search of food

and to optimize survival. The motion of bacteria within a swarm, wherein they migrate as a

collective group over a solid surface, is fundamentally different as typical bacterial swarms

show large-scale swirling and streaming motions involving millions to billions of cells. Here by

tracking trajectories of fluorescently labelled individuals within such dense swarms, we find

that the bacteria are performing super-diffusion, consistent with Levy walks. Levy walks are

characterized by trajectories that have straight stretches for extended lengths whose variance

is infinite. The evidence of super-diffusion consistent with Levy walks in bacteria suggests

that this strategy may have evolved considerably earlier than previously thought.

DOI: 10.1038/ncomms9396 OPEN

1 Department of Mathematics, Bar-Ilan University, Ramat Gan 52000, Israel. 2 Zuckerberg Institute for Water Research, The Jacob Blaustein Institutesfor Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Midreshet Ben-Gurion, Israel. 3 Department of MolecularBiosciences, University of Texas at Austin, Austin, Texas 78712, USA. Correspondence and requests for materials should be addressed to G.A.(email: [email protected]).

NATURE COMMUNICATIONS | 6:8396 | DOI: 10.1038/ncomms9396 | www.nature.com/naturecommunications 1

& 2015 Macmillan Publishers Limited. All rights reserved.

Abnormal diffusiontAssume that

M(ξ) ≥ 0,∫RdM(ξ)dξ = 1,

∫Rd|ξ|2M(ξ)dξ <∞,∫

RdξM(ξ)dξ = 0, 0 < k− ≤ k(x) ≤ k+,

ε∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

k(x)

ε


Theorem (usual limit) The limit is

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)

∂

∂tn(t, x)− div

[ A

k(x)∇n(t, x)

]= 0

Abnormal diffusion

The ‘standard’ derivation

Mellet, Mischler, MouhotBen Abdallah, Mellet, M. PuelAceves-Sanchez, MelletCrouseilles, Hivert, Lemou

What happens for fat tail∫Rd|ξ|2M(ξ)dξ =∞,

|ξ|2M(ξ) ≈ |ξ|−α for |ξ| � 1,

α > 1 is the Diffusion case

Abnormal diffusion

Assume that

M(ξ) ≥ 0,∫RdM = 1

M(ξ) = M(|ξ|) (radial symmetry)

|ξ|2M(ξ) ≈ |ξ|−α for |ξ| � 1, −1 < α < 1

εα∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

1

ε

[nε(t, x)M(ξ)− fε

]

nε(t, x) =∫Rdfε(t, x, ξ)dξ

Abnormal diffusionM(ξ) ≥ 0,

∫RdM = 1

M(ξ) = M(|ξ|) (radial symmetry)

|ξ|2M(ξ) ≈ |ξ|−α for |ξ| � 1, −1 < α ≤ 1

εα∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

1

ε


nε(t, x) =∫Rdfε(t, x, ξ)dξ

Theorem (Fractional Laplacian) The limit is

fε(t, x, ξ) −→ε→0

n(t, x)M(ξ)

∂

∂tn(t, x)− div

[Dαn(t, x)

]= 0

Abnormal diffusion

What is Dαn(t, x)?

Define the Fourier transform

n(k) := Fn(k) =∫Rdn(x)eix.kdx

Then

FDαn(t, k) =k

|k||k|αn(t, k)

Abnormal diffusion

εα∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

1

ε


Proof. (i) Because∫nε(t, x)M(ξ)dξ = nε(t, x) =

∫fεdξ

we conclude that∂

∂tnε + divJε = 0

Jε(t, x) =∫

ξ

εαfε(t, x, ξ)dξ

Abnormal diffusionProof. (ii) Prove that∫

Rd×Rdfε(t, x, ξ)2

M(ξ)dxdξ ≤ C

∫ ∞0

∫Rd×Rd

|fε(t, x, ξ)− nεM(ξ)|2

M(ξ)dxdξdt ≤ Cε1+α

But one cannot conclude∫ ∞0

∫Rd×Rd

|Jε(t, x)|2dxdt ≤ C

Abnormal diffusionProof. (iii) Fourier transform the equation

fε(t, k, ξ)[1 + iεk.ξ] = nε(t, k)M(ξ)− ε1+α ∂

∂tfε(t, k, ξ)



∂tfε(t, k, ξ)

ξ

εαfε = ε−α

ξM(ξ)

1 + iεk.ξnε(t, k) +O(ε)



∂tfε(t, k, ξ)

ξ

εαfε = ε−α

ξM(ξ)


Jε = ε−α∫

ξM(ξ)

1 + iεk.ξdξ nε(t, k) +O(ε)



∂tfε(t, k, ξ)

ξ

εαfε = ε−α

ξM(ξ)


Jε = ε−α∫

ξM(ξ)


Jε = ε−α∫

ξM(ξ)




∂tfε(t, k, ξ)

ξ

εαfε = ε−α

ξM(ξ)


Jε = ε−α∫

ξM(|ξ|)1 + iεk.ξ

dξ nε(t, k) +O(ε)

Jε = ε−α∫

ξM(|ξ|)1 + iεk.ξ

dξ nε(t, k) +O(ε)

Jε =k

|k|ε−α

∫ξ1M(|ξ1|)1 + iε|k|ξ1

dξ1 nε(t, k) +O(ε)

Abnormal diffusion

Jε =k

|k|ε−α

∫ξ1M(|ξ1|)1 + iε|k|ξ1


It remains to identify

ε−α∫ξ1M(|ξ1|)1 + iε|k|ξ1

dξ1

Abnormal diffusion

Jε =k

|k|ε−α

∫ξ1M(|ξ1|)1 + iε|k|ξ1



ε−α∫ξ1M(|ξ1|)1 + iε|k|ξ1

dξ1

= ε−α∫ [ ηM(|η|)

1 + iε|k|η− ηM(|η|)

]dη

Abnormal diffusion

Jε =k

|k|ε−α

∫ξ1M(|ξ1|)1 + iε|k|ξ1



ε−α∫ξ1M(|ξ1|)1 + iε|k|ξ1

dξ1

= ε−α∫ [ ηM(|η|)

1 + iε|k|η− ηM(|η|)

]dη

= −ε1−α|k|∫η2M(|η|)1 + iε|k|η

dη

Abnormal diffusion

= −ε1−α|k|∫η2M(|η|)1 + iε|k|η

dη

= −ε1−α|k|∫ |η|−α

1 + iε|k|ηdη

And notice that only large values of η count.

η2M(|η|) = |η|−α1|η|>A + integrable function

Abnormal diffusion

= −ε1−α|k|∫η2M(|η|)1 + iε|k|η

dη

= −ε1−α|k|∫ |η|−α

1 + iε|k|ηdη

= |k|α∫ |u|−α

1 + iudu

Abnormal diffusion

= −ε1−α|k|∫η2M(|η|)1 + iε|k|η

dη

= −ε1−α|k|∫ |η|−α

1 + iε|k|ηdη

= |k|α∫ |u|−α

1 + iudu

= |k|α∫ |u|−α

1 + u2du

Abnormal diffusionTherefore

Jε ≈k

|k||k|α

∫ |u|−α1 + u2

du nε(t, k)

And the result is proved

Abnormal diffusion

For bacterial movement this does not apply because the velocity are

on the sphere.

Degeneracy should be on the turning kernel.

Free boundary

For the hyperbolic regime, another non-standard asymptotics in

kinetic equations is due to

E. Bouin and V. Calvez

∂

∂tfε(t, x, ξ) + ξ · ∇xfε =

1

ε


Assuming that the initial data has the form

f0ε = Q0(ξ)e

S0(x)ε

Free boundary

Theorem (Free boundary) The limit is

fε(t, x, ξ) ≈ Q[S](ξ)eS(x,t)ε

∂S(x, t)

∂t= H

(∇xS

)

For a convex Hamiltonian H(p) defined later.

Free boundary

Proof. Set

fε(t, x, ξ) = qε(t, x, ξ)eS(x,t)ε

nε(t, x) = Rε(t, x) eS(x,t)ε

the equation

ε[ ∂∂tfε(t, x, ξ) + ξ · ∇xfε

]= nε(t, x)M(ξ)− fε

becomes

qε(t, x, ξ)[ ∂∂tS(x, t) + ξ · ∇xS(x, t) + 1

]= Rε(t, x)M(ξ) +O(ε)

Free boundary

qε(t, x, ξ)[ ∂∂tS(x, t) + ξ · ∇xS(x, t) + 1

]= Rε(t, x)M(ξ) +O(ε)

Consider for α ∈ R, p ∈ Rd, the first eigenvalue problem in ξ

Q(α,p)(ξ) [α+ ξ.p+ 1] =∫Q(α,p)(ξ)dξ M(ξ)−H(α, p) Q(α,p)(ξ)

Q(α,p)(ξ) > 0

Free boundary

qε(t, x, ξ)[ ∂∂tS(x, t) + ξ · ∇xS(x, t) + 1

]= Rε(t, x)M(ξ) +O(ε)

Consider for α ∈ R, p ∈ Rd, the first eigenvalue problem in ξ


Q(α,p)(ξ) > 0

Q(α,p)(ξ) =∫Q(α,p)(ξ)dξ

M(ξ)

α+ ξ.p+ 1 +H(α, p)

Notice that

H(α, p) = −α+ H(p), H(0) = 0

Free boundary


Q(α,p)(ξ) =∫Q(α,p)(ξ)dξ

M(ξ)

α+ ξ.p+ 1 +H(α, p)

Integrating, we find H using

1 =∫

M(ξ)

α+ ξ.p+ 1 +H(α, p)dξ

Free boundaryCome back to the equation

qε(t, x, ξ)[∂S(x, t)

∂t+ ξ · ∇xS(x, t) + 1

]= Rε(t, x)M(ξ) +O(ε)

At first order

qε(t, x, ξ) = Rε(t, x)M(ξ)

∂S(x,t)∂t + ξ · ∇xS(x, t) + 1 +H(α, p)

In other words

qε(t, x, ξ) ≈ Q(α,p)(ξ), α =∂S(x, t)

∂t, p = ∇xS(x, t)

and

H(α, p) = 0.

Conclusions

Problems of biology open many interesting questions in terms of

modeling, analysis, numerics.

I hope some day you’ll join us

Thanks to my collaborators

. F. Chalub, P. Markowich, C. Schmeiser

. N. Bournaveas, V. Calvez

. A. Buguin, J. Saragosti, P. Silberzan (Curie Intitute)

. M. Tang, N. Vauchelet, S. Yashuda






THANK YOU Roberto






THANK YOU Roberto

THANK YOU ALL

Kinetic equations and cell motion Beno^ t Perthame

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