Mathematical tools for cell chemotaxis · Theorem (HLS inequality with logarithmic kernel, Carlen & Loss, Beckner) In any dimension we have: ZZ Rd dR f (x)log jx yjf (y)dxdy M d Z

Analysis of KS Refined analysis Polarization Kinetic models E. coli

Mathematical tools for cell chemotaxis

Vincent CalvezCNRS, ENS Lyon, France

CIMPA, Hammamet, March 2012


Chemotaxis = biased motion in response to a chemical cue.


From individual to collective behaviour


Modeling issue: clusters and waves


Contents

Mathematical analysis of the Keller-Segel model – basics

Refined analysis of KS in dimension d = 1

(Spontaneous cell polarization)

Kinetic models for chemotaxis – analysis

The journey of E. coli


Contents







The Keller-Segel model

The Keller & Segel model involves two species:

• the cell density ρ(t, x),

• the chemoattractant concentration S(t, x).{∂tρ = ∆ρ− χ∇ · (ρ∇S) , t > 0, x ∈ Rd , d = 1, 2, 3

−∆S = ρ

In dimension d = 2, we have the following representation

S(t, x) = − 1

2π

∫R2

log |x − y |ρ(t, y) dy

Parameters: the chemosensitivity χ and the total number ofcells M


Modeling features

• No cell division, no death: only motion.

• Competition between dispersion of cells (diffusion) andaggregation.

• Rich model from the point of view of mathematical analysis.

• Poor model from the point of view of pattern formation.

• Unbounded solutions are an idealization of patterns (clusters).


Stationary statesWe seek stable clusters, i.e. densities µ(x), solutions of thestationary equation,{

0 = ∆µ− χ∇ · (µ∇S) , x ∈ Rd , d = 1, 2, 3

−∆S = µ

N. Mittal, E.O. Budrene, M.P. Brenner, A. van Oudenaarden, PNAS


Stationary states in 1D

0 = ∂xxµ− χ∂x (µ∂xS) , x ∈ R ,

−∂xxS = µ ←→ S(x) = −1

2|x | ∗ µ(x)

We set η = −∂xS . We end up with

η(x) =M

2Φ(χMx) , Φ(y) =

ey/2 − 1

ey/2 + 1,

µ(x) =M

2·1Lφ(x

L

), φ(y) = Φ′(y) =

ey/2(1 + ey/2

)2,

∫Rφ(y) dy = 2 .

Conclusion: there exist stationary states for any mass M > 0.The space scale is L = (χM)−1.


Stationary states in 2DWe seek a stationary state with radial symmetry µ(x) = µ(r)

0 =1

r∂r (r∂rµ)− χ1

r∂r (rµ∂r S) , x ∈ R ,

−1

r∂r (r∂r S) = µ

Again, we set η = −∂r S . We end up with

rη(r) =4

χ· 1

1 + λr 2,

The limit limr→+∞ rη(r) = M2π forces χM = 8π. We have

henceforth

µ(r) =4

χλφ(√

λr), φ(y) =

2

(1 + |y |2)2,

∫φ(y) dy = 2π .

Conclusion: there exist stationary states only for the critical mass.Any space scale L = λ−1/2 is admissible.


Critical mass in R2

In dimension d = 2 there is a nice and simple dichotomy:

Theorem (Blanchet, Dolbeault & Perthame)

Assume the initial data ρ0

(| log ρ0|+ (1 + |x |2)

)∈ L1.

• If χM < 8π solution are global in time (dispersion).

• It blows up in finite time if χM > 8π (aggregation).

• In the subcritical regime χM < 8π, the density converges to aself-similar profile.

There is a vast literature on this subject (prior to this theorem):Nanjundiah; Childress & Percus; Jager & Luckhaus; Nagai; Biler;Herrero & Velazquez; Gajewski & Zacharias; Horstmann; Senba &Suzuki. . .


Critical mass: numerical evidence


Critical mass, ctd.


Possible behaviours in other dimensions

The various criteria for global existence/blow-up strongly dependon the space dimension.

Theorem (Biler; Nagai; Corrias, Perthame & Zaag)

• In dimension d = 1 solutions are global in time (no blow-up).

• In dimension d = 3 solution is global in time if

‖ρ0‖L3/2 < C

Solution blows-up if

3M

2log

(∫R3

|x |2ρ0(x) dx

)+ F [ρ0] < −C .


Proof of global existence (d = 1)

In the one-dimensional case we have

S(t, x) = −1

2|x | ∗ ρ(t, x) .

In particular

∂xS(t, x) = −1

2(sign x) ∗ ρ(t, x) , ‖∂xS(t, x)‖∞ ≤

M

2.

Therefore the transport speed is bounded and blow-up neveroccurs.


Proof of global existence (d = 2)[Jager & Luckhaus ’92]: compute the Lp norm of the density.

d

dt

(1

p − 1

∫R2

ρp dx

)= −4

p

∫R2

∣∣∣∇ρp/2∣∣∣2 dx + χ

∫R2

ρp+1 dx

Need for functional analysis in order to compare the terms.Recall the classical Sobolev embedding in dimension 2:

‖u‖Lq? . ‖∇u‖Lq , q? =2q

2− q

‖u‖L4 . ‖u‖L2‖∇u‖L2

In the case p = 1:

d

dt

(∫R2

ρ log ρ dx

)= −4

∫R2

∣∣∣∇ρ1/2∣∣∣2 dx + χ

∫R2

ρ2 dx

≤ (−4 + CχM)

∫R2

∣∣∣∇ρ1/2∣∣∣2 dx


Proof of global existence (d = 3)

Same story, but different Sobolev inequality.

d

dt

(2

∫R3

ρ3/2 dx

)≤(−8

3+ Cχ‖ρ‖L3/2

)∫Rd

∣∣∣∇ρ3/4∣∣∣2 dx .

Thus, if ‖ρ0‖L3/2 is small enough, the L3/2 norm of ρ is decreasing.


Proof of blow-up (d = 2)The variance (second momentum) of the cell density can becomputed explicitly:

d

dt

(1

2

∫R2

|x |2ρ(t, x) dx

)= 2M

(1− χM

8π

)In the super critical regime χM > 8π a singularity must appear infinite time.

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Blow up with radial symetry

"macro.20" u 1:3"macro.30" u 1:3"macro.40" u 1:3"macro.50" u 1:3


Global existence (d = 2) – close the gap

Up to now, we have proven the following statement: solutionsblow-up when χM > 8π, whereas they are global when χM < C(Sobolev embedding).

It is possible to close the gap.

The argument uses the energy structure of the Keller-Segel system.

F [ρ] =

∫R2

ρ log ρ dx +χ

4π

∫∫R2×R2

ρ(x) log |x − y |ρ(y) dxdy

d

dtF [ρ(t)] = −

∫R2

ρ |∇ (log ρ− χS)|2 dx ≤ 0

Need for refined functional analysis in order to compare theterms. . .


Hardy-Littlewood-Sobolev inequality

Theorem (HLS inequality with logarithmic kernel, Carlen &Loss, Beckner)

In any dimension we have:

−∫∫

Rd×Rd

f (x) log |x − y |f (y) dxdy ≤ M

d

∫Rd

f log f dx + C

Consequence: the energy F is ”coercive” in the sub-critical case:

F [ρ0] ≥ F [ρ(t)] ≥(

1− χM

8π

)∫R2

ρ log ρ dx − C


Asymptotics for the heat equationRecall the long-time asymptotics of the heat equation:

∂tρ = ∆ρ

Rescale space and time (diffusive scaling): ρ(t, x) −→ u(τ, y),

ρ(t, x) =1

tu

(τ,

x√t

), τ =

1

2log t

∂τu = ∆u +1

2∇ · (uy)

The stationary state is the Gaussian kernel:

∇u +1

2uy = 0 ↔ u = λ exp

(−|y |

2

4

)More precisely, limτ→+∞ u(τ, y) = u(y) (exponentially fast inrelative entropy).


Asymptotics for the Keller-Segel equation

ρ(t, x) =1

tu

(τ,

x√t

), S(t, x) = v

(τ,

x√t

)We get the Keller-Segel equation with an additional drift:{

∂τu = ∆u +∇ ·(

12 uy − χu∇v

)−∆v = u

The rescaled energy has an additional confinement potential:

Frescaled [u] =

∫R2

u log u dy +1

4

∫R2

|y |2u(y) dy

+χ

4π

∫∫R2×R2

u(x) log |x − y |u(y) dxdy

We have limτ→+∞ u(τ, y) = u(y) in L1 (Blanchet, Dolbeault &Perthame).


Partial conclusion

• The Keller-Segel equation is equipped with an energy. Thishelps the analysis in dimension d = 2.

• The virial argument is very convenient, but gives fewinformations about how it blows-up.

• When diffusion dominates, it is important to rescalespace/time in order to visualize something.

• In dimension d = 3 it is not easy to make the distinctionbetween global existence and blow-up.


Contents







Generalized Keller-Segel equation in 1D

No blow-up in 1D, so we make the equation more flexible.

We consider nonlinear diffusion (porous-medium type), andnonlocal interaction:

∂tρ =∂2ρα

∂x2− χ ∂

∂x(ρ∂xS) ,

∫Rρ(x) dx = 1

S = −W ∗ ρ , W (x) =|x |γ

γ, α ≥ 1 , γ ∈ (−1, 1) .

The free energy writes:

F [ρ] =1

α− 1

∫Rρ(x)α dx +

χ

2γ

∫∫R×R

ρ(x)|x − y |γρ(y) dxdy


Cumulative distribution function

M(x) =

∫ x

−∞ρ(y) dy , X (m) = M−1(m) , X : (0, 1)→ R , X ↗

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Alternative formulation of the energy F [ρ] = G[X ]:

G[X ] =1

α− 1

∫(0,1)

(X ′(m))1−α dm+χ

2γ

∫∫(0,1)2

|X (m)−X (m′)|γ dmdm′


Gradient flow interpretation

Claim [Jordan, Kinderlehrer & Otto]: the Keller-Segel system isthe gradient flow of the energy G[X ]

∂tX = −∇G[X ]

Key observations:

• The functional G[X ] is not convex

• Each contribution is homogeneous.

If α− 1 + γ = 0 the two contributions have the samehomogeneity: the competition is fair.

G[λX ] = λ1−αG[X ] ∀λ > 0


The logarithmic case α = 1, γ = 0

The functional is almost zero-homogeneous.

G[X ] = −∫

(0,1)log(X ′(m)) dm +

χ

2

∫∫(0,1)2

log |X (m)− X (m′)| dmdm′

G[λX ] = G[X ] +(−1 +

χ

2

)log λ

Consequence:

∇G[X ] · X =(−1 +

χ

2

)−∂tX · X =

(−1 +

χ

2

)d

dt

(1

2|X (t)|2

)=(

1− χ

2

)Singularity if χ > 2: blow-up!


The logarithmic case, ctd.(Given a gradient flow of a convex energy G , and a critical point

∇G [A] = 0. Then limt→+∞ X (t) = A,

d

dt

(1

2|X (t)− A|2

)≤ 0

If G is uniformly convex: D2G ≥ νId,

d

dt

(1

2|X (t)− A|2

)≤ −ν|X (t)− A|2

)

Surprisingly, the same holds true here. If A is a critical point of theenergy,

d

dt

(1

2|X (t)− A|2

)≤ 0

Problem: there exists a critical point only when χ = 2. . .


The logarithmic case, ctd.. . . Rescale space/time!

Grescaled [X ] = G[X ] +1

4|X |2

There exists a critical point if χ < 2:

∇G[A] +1

2A = 0(

−1 +χ

2

)+

1

2|A|2 = 0

Theorem (C, Carrillo)

In the sub-critical case χ < 2

d

dt

(1

2|X (t)− A|2

)≤ −|X (t)− A|2

Explanation: the interaction part (concave) is ”digested” by thediffusion contribution.


Fair competition – blow-up

Assume α− 1 + γ = 0.

The functional is (1− α)-homogeneous.

G[λX ] = λ1−αG[X ]

Consequence:

∇G[X ] · X = (1− α)G[X ]

−∂tX · X = (1− α)G[X ]

d

dt

(1

2|X (t)|2

)= (α− 1)G[X ] ≤ (α− 1)G[X0]

Singularity if G[X0] < 0: blow-up!


Fair competition – critical parameter

In the case of fair competition there is a dichotomy which is similarto (KS2D).

Theorem (C, Carrillo)

Assume 1 < α < 2, γ = 1− α. There exists χc(α) > 0 such that:

• if χ < χc the energy F is everywhere positive. The densityconverges to a self-similar profile. The profile is unique.

• si χ > χc there exists a cone of negative energy. The densityblows-up in finite time if F [ρ0] < 0.

• The case F [ρ0] ≥ 0 and χ > χc is open.

• In higher dimension, the fair competition regime readsd(α− 1) + γ = 0.


Diffusion-dominating caseAssume α− 1 + γ > 0.

Typically, standard Keller-Segel equation in 1D.

The solution converges towards a unique stationary state µ.µ is compactly supported if (α > 1, γ ≥ 0).

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Attraction-dominating case

Assume α− 1 + γ < 0.

Typically, standard Keller-Segel equation in 3D.

Several criteria for blow-up are available. For instance,

C

(1

1− α− 1

γ

)(∫R|x |2ρ0(x) dx

)(1−α)/2

+ F [ρ0] < 0 .

The solution is global in time if

‖ρ0‖Lp < C

where p =2− α1 + γ

> 1.


Time discretization

Jordan, Kinderlehrer & Otto have proposed the following scheme(time-implicit Euler’s scheme):

Xn+1 − Xn

∆t= −∇G[Xn+1]

Xn+1 minimizes G[Y ] +1

2∆t‖Y − Xn‖2

Theorem (Blanchet, C, Carrillo)

In the subcritical case χ < 2, the JKO scheme converges towards aweak solution of the Keller-Segel equation as ∆t → 0.

• The JKO scheme is well adapted to the energy structure.

• But it is difficult to handle practically.


Space discretizationThe continuous functional G can be discretized using finitedifferences for X (m).

G [X ] = −∆mN−1∑i=1

log(X i+1 − X i

)+χ

2(∆m)2

∑i 6=j

log |X j − X i |

+∆m

4

N∑i=1

|X i |2 .

(see the lecture of Lucilla Corrias next Sunday)

TheoremThe critical parameter is χc = 2(1−∆m)−1.

• If χ > χc the solution of the discrete gradient flow blows-upin finite time (meaning that ∃i0 : X i0+1 − X i0 = 0 after afinite number of steps).

• If χ < χc the solution of the rescaled gradient flow convergestowards a unique stationary state at exponential rate.


Numerical illustrations

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Convergence of the solution towardsthe unique stationary state in self-similar variables when χ < 2.

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Convergence towards the unique sta-tionary state with porous-medium dif-fusion α > 1 (without rescaling).

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Blow-up of the discrete gradient flowwhen χ > 2.

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Blow-up of the discrete gradient flowwhen χ > 4

.


Partial conclusion

• Behind the dissipation of energy there is a nice gradient flowstructure (after appropriate change of viewpoint).

• The energy is homogeneous, and ”convex+concave”.

• Some results can be extended to higher dimension, but not all.

• The numerical scheme is a sort of ”particle method”, wherediffusion is deterministic (particles keep ordered).

• Using a very coarse space grid one gets an interestingfinite-dimensional reduction of the Keller-Segel PDE.


Collaborations

• Adrien Blanchet (Univ. Toulouse 1)

• Jose A. Carrillo (Univ. Autonoma Barcelona)

• Lucilla Corrias (Univ. Evry - Val d’Essonne)

• Abderrahman Ebde (Univ. Paris 13)

• Benoıt Perthame (Univ. Paris 6)


Contents







Cell polarization

• Cell polarization is a key process which yields symmetrybreaking of a cell.

• It is preliminary to cell division, cell migration (eukaryoticchemotaxis), mating (yeast), etc.

• It happens generally in reponse to spatial cues = bud scars(internal), pheromone signaling (external stimulus).


Spontaneous polarization

Mutant yeast is able to polarize spontaneously

• Wedlich-Soldner et al. (Science 2003)

• Irazoqui et al. (Nat Cell Biol 2003 )


Possible scenarios for spontaneous polarisation

• Diffusion-driven instability (ex: Turing): molecules diffusefaster on the cytoplasm than on the boundary.

• Flux-driven instability (ex: Keller-Segel): fluxes are updateddepending on the distribution of molecules.

Previous modeling efforts: Wedlich-Soldner et al. (Science 2003, JCell Biol 2004); Marco et al. (Cell 2007); Altschuler et al. (Nature2008); Goryachev et al. (FEBS Lett 2008); Howell et al. (Cell2009);

Hawkins, Benichou, Piel and Voituriez (Phys. Rev. E 2009)


The model of Hawkins et al.

It describes the transport of molecular markers (Cdc42) in thecytoplasm. Transport may be:

• passive (diffusion in the cytoplasm)

• active (transport along the cytoskeleton).

∂tρ(t, x , z) = ∆ρ(t, x , z)−∇ · (ρ(t, x , z)u(t, x , z))

t > 0 , (x , z) ∈ RN−1 × (0,+∞) .

Cell domain = half-space.

• ρ(t, x , z) = density of molecular markers in the cytoplasm.

• µ(t, x) = density of markers attached to the boundary.

• u(t, x , z) = advection field due to the cytoskeleton(microtubules or actin filaments).


Focus on the advection field

Two options for the advective field u:

1. The case of actin (diffuse cable network)

u(t, x , z) = ∇φ(t, x , z) , where

{−∆φ(t, x , z) = 0 ,

−∂zφ(t, x , 0) = µ(t, x) .

2. The case of microtubules (stiff cable network)

u(t, x , z) = −µ(t, x , 0)ez ,


Dynamics of markers on the boundary

Dynamics of markers on the boundary = attachment/detachmentto the cell membrane:

∂tµ(t, x) = ρ(t, x , 0)− L−1µ(t, x) , x ∈ RN−1 .

The flux of markers on the boundary is:

∂zρ(t, x , 0) + µ(t)ρ(t, x , 0) = ∂tµ(t, x) .

This guarantees mass conservation:∫∫RN−1×(0,+∞)

ρ(t, x , z) dxdz +

∫RN−1

µ(t, x) dx = M .


The one-dimensional case

In the one-dimensional case, both options reduce to the sameequation:

∂tρ(t, z) = ∂zzρ(t, z) + µ(t)∂zρ(t, z) , t > 0 , z ∈ (0,+∞) .

Goal of the 1D analysis: understand instability in the normaldirection = markers are transported on the boundary . . . or not.

Questions: dispersion, blow-up, non-trivial steady states?


First step: simple but irrealistic model

We set for simplicity: ∀t > 0 µ(t) = ρ(t, 0).

∂tρ(t, z) = ∂zzρ(t, z) + ρ(t, 0)∂zρ(t, z) , t > 0 , z ∈ (0,+∞) .

(Keller-Segel system with coupling at the boundary {z = 0}).

We seek stationary states:

0 = ∂zzν(z) + ν(0)∂zν(z) , z ∈ (0,+∞)

The solutions are ν(z) = ν(0) exp(−ν(0)z). In particular we have∫z>0 ν(z) dz = 1.

Conclusion: there exist stationary states only for the critical massM = 1 (as for KS 2D).


Critical mass

Theorem (C, Meunier & Voituriez)

There is a nice and simple dichotomy:

• If M < 1, the solution converges towards a (unique)self-similar profile:

limt→+∞

∥∥∥∥ρ(t, z)− 1√t

GM

(z√t

)∥∥∥∥L1

= 0 .

• If M = 1 there is a one-parameter family of steady states:να(z) = α exp(−αz).Convergence holds + the first moment is conserved:α−1 =

∫z>0 ρ

0(z) dz.

• If M > 1 and ∀z > 0 ∂zρ0(z) ≤ 0, the solution blows up in

finite time.


Sketch of proof: M = 1

Surprisingly, entropy is dissipated in the critical case M = 1:

H(t) =

∫z>0

ρ(t, z)

να(z)log

(ρ(t, z)

να(z)

)να(z) dz

d

dtH(t) = −

∫z>0

ρ(t, z) (∂z log ρ(t, z))2 dz + ρ(t, 0)2

= −∫

z>0ρ(t, z) (∂z log ρ(t, z) + ρ(t, 0))2 dz

≤ 0 .


Sketch of proof: M < 1We again introduce a Lyapunov functional in the subcritical case:

H(τ) =

∫y>0

u(τ, y)

GM(y)log

(u(τ, y)

GM(y)

)GM(y) dy

J(τ) =

∫y>0

yu(τ, y)dy

L(τ) = H(τ) +1

2(1−M)(J(τ)− α(1−M))2 .

The corrected entropy is nonincreasing:

d

dtL(τ) = −D(τ) ≤ 0 ,

where

D(τ) =

∫y>0

u(τ, y) (∂y log u(τ, y) + y + u(τ, 0))2 dy

+1

(1−M)

(d

dτJ(τ)

)2

.


Sketch of proof: M > 1

The blow-up result relies on some virial calculation involving thefirst moment J(t) + some interpolation inequality:

d

dtJ(t) = (1−M)ρ(t, 0)

≤ (1−M)M2

2

1

J(t).

We recover the critical threshold M = 1!


Back to the coupled system (1D)

Recall conservation of mass:∫z>0

ρ(t, z)dz + µ(t) = M .

Intuitively, blow-up is prevented in this case (the transport speed isbounded!).

TheoremAssume M > 1. The partial mass m(t) :=

∫z>0 ρ(t, z) dz

converges to 1 and the density ρ(t, z) converges towards theexponential profile

ν(z) = (M − 1)e−(M−1)z .


Back to entropy dissipationRe-define the relative entropy:

H(t) =

∫z>0

ρ(t, z)

m(t)ν(z)log

(ρ(t, z)

m(t)ν(z)

)ν(z) dz .

The following Lyapunov functional is again nonincreasing:

L(t) = m(t)H(t) +1

2(µ(t)− ν(0))2 + µ(t) log

(µ(t)

ν(0)

)+ m(t) log m(t) .

d

dtL(t) = −D(t) ≤ 0 .

The dissipation reads as follows:

D(t) =

∫z>0

ρ(t, z)

(∂z log ρ(t, z) +

ρ(t, 0)

m(t)

)2

dz

+ m(t)

(ρ(t, 0)

m(t)− µ(t)

)2

+ (ρ(t, 0)− µ(t)) log

(ρ(t, 0)

µ(t)

)+ µ(t) (µ(t)− ν(0))2 .


Partial conclusions

• We understand quite well the dynamics thanks to unexpectedentropy dissipation.

• Similar analysis can be performed in higher dimension, in theradially symmetric case.

• However this does not help to investigate symmetry breaking!


Collaborations

• Nicolas Meunier (Univ. Rene Descartes, Paris 5)

• Nicolas Muller (Univ. Rene Descartes, Paris 5)

• Raphael Voituriez (Univ. Pierre & Marie Curie, Paris 6)


Contents







Motion of swimming bacteria

Alternatively:

• Straight swimmingtrajectories (∼ 1sec.)= run

• Reorientation events(∼ 0.1sec.)= tumble

Howard Berg’s lab


J. Saragosti, A. Buguin, P. Silberzan,

Institut Curie


Traveling pulses

see also Adler, Science (1966)


Kinetic modeling

• Bacterial density f (t, x , v) is described at time (t), position(x) and velocity (v).

• Velocity space V is bounded (speed of bacteria is almostconstant ≈ 20µms−1).

The Othmer-Dunbar-Alt model (’88) :

∂t f + v · ∇x f︸︷︷︸run

=

∫v ′∈V

T[S ](v , v ′)f (t, x , v ′) dv ′ − λ[S ]f (t, x , v)︸︷︷︸tumble

• The tumbling kernel T[S ](v , v ′) describes the frequency ofreorientation v ′ → v .

• λ[S ] =∫v ′∈V T[S ](v ′, v) dv ′ is the intensity of the Poisson

process governing reorientation.


Chemoattractant release

• Bacteria can sense multiple chemical substances along theirtrajectories.

• Bacteria are able to produce some of these chemicals (e.g.amino-acids).

• Positive feedback: accumulation of bacteria in opposition tothe natural dispersion.

The chemical signal is secreted by the cells, following areaction-diffusion equation.

∂tS = DS∆S − αS + βρ

ρ(t, x) =

∫v∈V

f (t, x , v) dv


A critical mass phenomenon in 2D

We proposeT [S ](v , v ′) = χ(v · ∇S(x))+

The kinetic equation writes:

∂t f + v · ∇x f = χ(v · ∇S)+ρ− ω|∇S |f

The constant ω ensures conservation of mass. The intensity of thePoisson process is proportional to |∇S |.

Assume V = B(0,R) or S(0,R), and f0 has spherical symmetry:f0(θ · x , θ · v) = f0(x , v).

Theorem (Bournaveas, C)

• if χM|V | > C then solution blows-up in finite time.

• if χM|V | < C (and f0 is not too singular) then solution isglobal in time.


Blow-up: virial argumentDifferentiate twice the second moment:

I (t) =1

2

∫∫R2×V

|x |2f (t, x , v) dvdx

d

dtI (t) =

1

2

∫∫R2×V

(x · v)f (t, x , v) dvdx

d2

dt2I (t) ≤ c1M − c2χM2 − ω

∫R2

(x · j)|∇S | dx

Fortunately, under spherical symmetry,

ω

∫R2

(x · j)|∇S | dx =d

dtJ(t) , J(t) ≥ 0

Heuristics:{∂tρ = −∇ · jρ = −∇ · ∇S

→ ” j∇S ≈ ∂t

(∇−1ρ

)2”


Global existence: comparison argument

If χM|V | is small, we exhibit a supersolution for 0 < γ < 1:

k(x , v) =

{|x |−γ if (v · x) < 0

|Πv⊥x |−γ if (v · x) > 0

More precisely,

v · ∇xk ≥ χ (v · ∇S)+

∫v ′

k(x , v ′) dv ′

We can prove: ∀t > 0 f (t, x , v) ≤ k(x , v).


Partial Conclusion

• The kinetic models lack energy structure. Analysis is moredelicate to perform.

• Some results for the Keller-Segel system have been extendedat the kinetic level.

• General principles are moreless the same (dispersion vs.aggregation).


Collaborations

• Nikolaos Bournaveas (Univ. Edinburgh)

• Susana Gutierrez (Univ. Birmingham)



Contents







Description of E. coli movements

Alternatively:

• Straight swimmingtrajectories (∼ 1sec.): run

• Reorientation events(∼ 0.1sec.): tumble

Howard Berg’s lab


Chemical signaling

• Bacteria can sense multiple chemical substances along theirtrajectories.Chemoattractants: amino-acids (e.g. aspartate), glucose. . .

• Bacteria are able to produce some of these chemicals.Positive feedback: accumulation of bacteria in opposition tothe natural dispersion.

N. Mittal, E.O. Budrene, M.P. Brenner, A. van Oudenaarden, PNAS


Response to the chemical signal

E. coli reacts to the time variations of the signal: tumbling eventsdecrease when the signal concentration increases.

Complex signal integration insideeach individual:

• ”Memory effects”

• High sensibility to signalchanges (excitation)

• AdaptationJ.E. Segall, S.M. Block, H.C. Berg, PNAS


Recap kinetic modeling

• Bacterial density f (t, x , v) is described at time (t), position(x) and velocity (v).

• Velocity space V = S(0, c) (speed of bacteria is almostconstant ≈ 20µms−1).

The Othmer-Dunbar-Alt model (’88) :

∂t f + v · ∇x f︸︷︷︸run

=

∫v ′∈V

T[S ](v , v ′)f (t, x , v ′) dv ′ − λ[S ]f (t, x , v)︸︷︷︸tumble


What about the tumbling frequency?

.

Following Erban & Othmer, Dolak & Schmeiser,

T[S ](v , v ′) = ψ

(DS

Dt

∣∣∣∣v ′

)= ψ

(∂tS + v ′ · ∇xS

)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

rate

of t

umbli

ng

!


Relevance of the diffusive limitWhen the motion is unbiased at zeroth order:

T[S ](v , v ′) = T0(v , v ′) + εT1[S ](v , v ′) , T0(v , v ′) symmetric ,

then rescale time/space:

t → t

ε2, x → x

ε.

−→ diffusive limit

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n ru

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)

Position relative to the band (µm)

Mutant (ε ≈ 0.1) vs. Wild Type (ε ≈ 0.5).


Diffusive limitWe assume chemoattractant has only a slight influence:

T[S ] = ψ0 + εφ

(DS

Dt

∣∣∣∣v ′

), ε� 1

Write the diffusive scaling

ε∂t f + v · ∇x f =ψ0

ε(ρ− |V |f )

+

∫v ′φ(ε∂tS + v ′ · ∇S

)f ′ dv ′ − |V |φ (ε∂tS + v · ∇S) f

Taking formally the limit ε→ 0 we obtain at the leading order:

∂tρ+∇ · (J) = 0 ,

J = −∇((

1dψ0|V |2

∫V |v |

2 dv)ρ)

︸︷︷︸Fick’s law

+ ρ(

1ψ0|V |

∫v∈V vφ(v · ∇S) dv

)︸︷︷︸

chemotactic drift

.


Macroscopic equations

Simplified macroscopic model:

{∂tρ = ∆ρ−∇ · (ρu[S ])

−∆S = ρ

u[S ] = −∫

v∈Vvφ (v · ∇S) dv

The macroscopic flux is quite different from the classicalKeller-Segel model for chemotaxis (u[S ] = χ(S)∇S):

• It is not the gradient of a chemical potential.

• It is bounded.

• It is more nonlinear (in some sense).


Link with scalar conservation laws

Starting from the 1D equation. . .{∂tρ = ∂2

xxρ− ∂x (ρu[S ])

−∂2xxS = ρ

. . . we can write a scalar conservation law for the gradient η = ∂xS :∂tη + ∂x (F (η)) = ∂2

xxη , η(±∞) = ∓M

2

F (η) = −∫

v∈VΦ(vη) dv , Φ′ = φ .

• F is convex: F ′′(η) = −∫

v 2φ′(vη) dv ≥ 0.

• η is nonincreasing: ∂xη = −ρ ≤ 0.

Conclusion. There exist stationary states for any (M, χ, φ). Theyare globally stable.


Example of the step response function• We restrict to dimension d = 1.

• We assume a stiff response function φ(Y ) = −sign (Y ). Thenu[S ] = χsign (∂xS).

• We can compute easily the stationary state:

M−1µ(x) = χµ0(χx) , µ0(x) =1

2e−|x | .

−1000 −500 0 500 10000

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bact

eria

l den

sity


Cluster formation• Mittal et al. have observed stationary patterns of motile

bacteria.• The typical size does not depend on the number of cells.

Mittal et al., PNAS 2003.−1000 −500 0 500 1000

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sity


Influence of the stiffness

In the case φ(Y ) = − tanh(τY ), the stationary state getssmoother as τ decreases.

−1000 −500 0 500 10000

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Traveling pulses

Experiments by Adler (1966).


Plausible scenario

• Bacteria initially lie on the left side of a channel,

• They secrete a chemoattractant (presumably glycine),

• A fraction travels to the right with constant speed andconstant profile (asymmetric).


Mathematical model

• Bacteria gather due to secretion of a chemottractant (as forcluster formation),

• They consume another chemical (the nutrient N). Thistriggers the motion of a pulse.

Kinetic description:

∂t f + v · ∇x f = Q[S ,N]f

And reaction-diffusion equations:

∂tS = DS∆S − αS + βρ

∂tN = DN∆N − γρN .


Derivation of a simpler model

In the case ψ = ψ0 + εφ we can perform a diffusive limit:

ε∂t f + v · ∇x f =1

εQ[S ,N]f

Taking the limit when ε→ 0 leads to a parabolic equation for thedensity ρ(t, x):

∂tρ = Dρ∆ρ︸︷︷︸diffusion

−∇ · (ρu[S ] + ρu[N])︸︷︷︸chemotactic flux

u[S ] = −∫

v∈Vvφ (v · ∇S) dv

In the case of a stiff response function φ(Y ) = −sign (Y ):

uS = χS∇S

|∇S |


Numerical evidence for traveling pulses

.


Numerical evidence, ctd.

0 0.5 1 1.5 2

x 104

0

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40

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.



Limited nutrient: coexistence of a stationary state and a travelingpulse


Analytic computationsIn the case of a stiff response function φ(Y ) = −sign (Y ), weobtain an implicit formula for the speed of the pulse σ:

χN − σ = χSσ√

4DSα + σ2

The profile is a combination of two exponential tails.

ρ(z) =

{ρ0 exp (λ−z) , z < 0

ρ0 exp (λ+z) , z > 0

Asymmetry of the profile is given by:

λ−

|λ+|=

√4DSα + σ2 + σ√4DSα + σ2 − σ

> 1

It is strongly asymmetric when σ � 2√

DSα (speed of chemicaldiffusion).


Is the diffusive scaling relevant?

Recall ψ = ψ0 + εφ.

The diffusive scaling is not valid for all experimental settings.

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n ru

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n (s

)

Position relative to the band (µm)−800 −400 0 4000.1

0.2

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0.6

Mea

n ru

n an

d tu

mbl

e du

ratio

n (s

)Position relative to the band (µm)

Mutant (ε ≈ 0.1) vs. Wild Type (ε ≈ 0.5).


Traveling pulses in the kinetic model

2500 3000 3500 4000 4500 5000 55000

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space (mm)

cell

dens

ity

Comparison Experiments vs. Numerical simulation

τ = 20sec

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space (mm)

cell

dens

ityComparison numerical profiles with/without dispersion

τ = 80sec


Angular distribution of the runsAssume that the tumbling frequency T[S ](v , v ′) does not dependon the outgoing velocity v .

The distribution of runs should be homogeneous...

... but it is not!


Directional persistence

We have included this effect into the model:

T[S ](v , v ′) = ψ(v ′)K (v , v ′) ,

∫K (v , v ′)dv = 1 .

As soon as bacteria tumble, they pick a new velocity v withdensity probability K (v , v ′).

For instance

K (v , v ′) = exp

(cosα− 1

σ2

), α = angle (v ′, v)

Standard deviation σ is a function of v ′:

σ = σ0 + σ1ψ(v ′)


Macroscopic effect

The effect on the speed of the wave is significant: +30%


Conclusions

• There is a hierarchy of mathematical models for collective cellmotion (micro-, meso-, macroscopic).

• The appropriate choice relies on a compromise betweenaccuracy of description and simplicity of formulation.

• The ODA kinetic model is suitable for bacterial motion. It ispossible to derive simplified model (of parabolic types) whichare better adapted than the usual ones.

• Existence of stable traveling pulses is linked to the stationarychemotaxis problem (without nutrient).


Collaborations

• Nikolaos Bournaveas (Univ. Edinburgh)


• Jonathan Saragosti, Axel Buguin and Pascal Silberzan(Institut Curie, Paris)


Thank you for your attention!

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