Analysis of KS
Refined analysis
Kinetic models
Mathematical tools for chemotaxis
Vincent Calvez
CNRS, ENS Lyon, France
Instructional conference, ICMS, Edinburgh, April 2010
E. coli
Analysis of KS
Refined analysis
Kinetic models
Chemotaxis = biased motion of cell in response to a chemical
cue.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Contents
Mathematical analysis of the Keller-Segel model – basics
Refined analysis in dimension d = 1
Kinetic models for chemotaxis – analysis
The journey of E. coli
E. coli
Analysis of KS
Refined analysis
Kinetic models
Contents
Mathematical analysis of the Keller-Segel model – basics
Refined analysis in dimension d = 1
Kinetic models for chemotaxis – analysis
The journey of E. coli
E. coli
Analysis of KS
Refined analysis
Kinetic models
The Keller-Segel model
The Keller & Segel model involves two species:
• the cell density ρ(t, x),
• the chemoattractant concentration S(t, x).
(
∂t ρ = ∆ρ − χ∇ · (ρ∇S) t > 0, x ∈ Rd
−∆S = ρ
Parameters of the model are the chemosensitivity coefficient χ and
the total number of cells M.
In dimension d = 2, we have the following representation
Z
1
S(t, x) = −
log |x − y |ρ(t, y ) dy
2π R2
E. coli
Analysis of KS
Refined analysis
Kinetic models
Modeling features
• No cell division, no death: only motion.
• Competition between dispersion of cells (diffusion) and
aggregation.
• 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.
E. coli
Analysis of KS
Refined analysis
Kinetic models
A simple dichotomy in R2
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 a
self-similar profile.
There is a rich literature for this subject (prior to this theorem):
Nanjundiah; Childress & Percus; Jäger & Luckhaus; Nagai; Biler;
Herrero & Velázquez; Gajewski & Zacharias; Horstmann; Senba &
Suzuki. . .
E. coli
Analysis of KS
Refined analysis
Kinetic models
Other dimensions
The behaviour of solutions strongly depends on the space
dimension.
Theorem (Biler; Nagai; Corrias, Perthame & Zaag)
• In dimension d = 1 blow-up never occurs (more details will be
given later)
• In dimension d ≥ 3 solution is global in time if
kρ0 kLd/2 < C
Solution blows-up if
R
M d/(d−2)
>C
2
Rd |x| ρ0 (x) dx
E. coli
Analysis of KS
Refined analysis
Kinetic models
E. coli
Proof of blow-up – the case d = 2
The second momentum of the cell density can be computed
explicitely:
Z
d 1
χM
2
|x| ρ(t, x) dx = 2M 1 −
dt 2 R2
8π
In the super critical regime χM > 8π a singularity must appear in
finite time.
Blow up with radial symetry
1e+08
"macro.20" u 1:3
"macro.30" u 1:3
"macro.40" u 1:3
"macro.50" u 1:3
2
1.8
1e+06
1.6
1.4
10000
cell density
1.2
1
100
0.8
0.6
1
0.4
0.2
0.01
0
−3
−2
−1
0
1
2
3
-10
-8
-6
-4
log radius
-2
0
2
Analysis of KS
Refined analysis
Kinetic models
E. coli
Proof of blow-up – the case d ≥ 3
d
dt
Z
ZZ
1
1
|x|2 ρ(t, x) dx = dM−ωd
ρ(x)
ρ(y ) dxdy
2 R2
|x
−
y |d−2
d
d
R ×R
Adapt Cauchy-Schwarz inequality:
M
d/2+1
Z
2
1−d/2
|x| ρ(x) dx
R2
ZZ
≤C
ρ(x)
Rd ×Rd
Conclude:
Z
d 1
|x|2 ρ(t, x) dx ≤ dM
dt 2 R2
1−C R
1
ρ(y ) dxdy
|x − y |d−2
!
M d/2
R2
|x|2 ρ(t, x) dx
d/2−1
Analysis of KS
Refined analysis
Kinetic models
E. coli
First attempt to prove global existence – the case d = 2
[Jäger & Luckhaus ’92]
Z
Z
Z d
1
4
p/2 2
p
ρp+1 dx
ρ dx = −
∇ρ dx + χ
dt p − 1 R2
p R2
2
R
Need for functional analysis in order to compare things.
Recall the Sobolev embedding in dimension 2:
kukLq? . k∇ukLq ,
q? =
2q
2−q
kukL4 . kukL2 k∇ukL2
In the case p = 1:
Z
Z
Z
d
√
ρ log ρ dx = −4
|∇ ρ|2 dx + χ
ρ2 dx
dt
2
2
2
R
R
R
Z
√ 2
≤ (−4 + C χM)
|∇ ρ| dx
R2
Analysis of KS
Refined analysis
Kinetic models
E. coli
Global existence – the case d ≥ 3
Same story, but different Sobolev inequality.
d
dt
2
d −2
Z
Rd
ρd/2 dx
≤
Z
8
− + C χkρkLd/2
d
Rd
d/4 2
∇ρ
dx
Analysis of KS
Refined analysis
Kinetic models
Global existence – the case d = 2
The Keller-Segel system has an energy structure:
Z
ZZ
χ
F[ρ] =
ρ log ρ dx +
ρ(x) log |x − y |ρ(y ) dxdy
4π
R2
R2 ×R2
Z
d
ρ |∇ (log ρ − χS)|2 dx ≤ 0
F[ρ(t)] = −
dt
2
R
Need for refined functional analysis in order to compare things. . .
E. coli
Analysis of KS
Refined analysis
Kinetic models
Hardy-Littlewood-Sobolev inequality
Theorem (HLS inequality with logarithmic kernel)
In any dimension we have:
ZZ
Z
M
f (x) log |x − y |f (y ) dxdy ≤
−
f log f dx + C
d Rd
Rd ×Rd
Consequence: the energy F is ”coercive” in the sub-critical case:
Z
χM
F[ρ0 ] ≥ F[ρ(t)] ≥ 1 −
ρ log ρ dx − C
8π
R2
E. coli
Analysis of KS
Refined analysis
Kinetic models
Asymptotics for the heat equation
Recall the long-time asymptotics of the heat equation:
∂t ρ = ∆ρ
Rescale space and time (diffusive scaling): ρ(t, x) −→ u(τ, y ),
1
x
1
ρ(t, x) = u τ, √
, τ = log t
t
2
t
∂τ u = ∆u + ∇ · (uy )
The stationary state is the Gaussian kernel:
|y |2
∇u + uy = 0 ↔ u = λ exp −
2
More precisely, u(τ, y ) → u(y ) (exponentially fast in relative
entropy).
E. coli
Analysis of KS
Refined analysis
Kinetic models
E. coli
Asymptotics for the Keller-Segel equation
1
x
ρ(t, x) = u τ, √
,
t
t
S(t, x) = v
x
τ, √
t
We get the Keller-Segel with an additional drift:
(
∂τ u = ∆u + ∇ · (uy − χu∇v )
−∆v = u
The rescaled energy has an additional confinement potential:
Z
Z
1
Frescaled [u] =
u log u dy +
|y |2 u(y ) dy
2
2
2
R
R
ZZ
χ
+
u(x) log |x − y |u(y ) dxdy
4π
R2 ×R2
We have u(τ, y ) → u(y ) in a weak sense (Blanchet, Dolbeault &
Perthame).
Analysis of KS
Refined analysis
Kinetic models
Conclusion
• The Keller-Segel equation is equipped with an energy. This
helps the analysis in dimension d = 2.
• The virial argument is very convenient, but gives few
informations about how it blows-up.
• When diffusion dominates, it is important to rescale
space/time in order to visualize something.
• In dimension d ≥ 3 it is not easy to make the distinction
between global existence and blow-up.
• See the talk of N. Meunier next week for related issues
(concerning cell polarization).
E. coli
Analysis of KS
Refined analysis
Kinetic models
Contents
Mathematical analysis of the Keller-Segel model – basics
Refined analysis in dimension d = 1
Kinetic models for chemotaxis – analysis
The journey of E. coli
E. coli
Analysis of KS
Refined analysis
Kinetic models
Generalized Keller-Segel equation in 1D
No blow-up in 1D, so we generalize the equation to make it more
flexible.
We consider nonlinear diffusion (porous-medium type), and
nonlocal interaction:

Z
∂ 2 ρα
∂


−χ
(ρ∂x S) ,
ρ(x) dx = 1
 ∂t ρ =
∂x 2
∂x
R

|x|γ

 S = −W ∗ ρ , W (x) =
, α ≥ 1 , γ ∈ (−1, 1) .
γ
The free energy writes:
Z
ZZ
1
χ
F[ρ] =
ρ(x)α dx +
ρ(x)|x − y |γ ρ(y ) dxdy
α−1 R
2γ
R×R
E. coli
Analysis of KS
Refined analysis
Kinetic models
E. coli
Cumulative distribution function
Z
x
M(x) =
ρ(y ) dy ,
X (m) = M −1 (m) ,
X : (0, 1) → R ,
X %
−∞
2
4
1.8
3
1.6
2
1.4
1
1.2
−→
1
0.8
0
−1
0.6
−2
0.4
−3
0.2
0
−3
−2
−1
0
1
2
3
−4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Alternative formulation of the energy F[ρ] = G[X ]:
Z
ZZ
χ
1
0
1−α
G[X ] =
(X (m))
dm+
|X (m)−X (m0 )|γ dmdm0
α − 1 (0,1)
2γ
(0,1)2
Analysis of KS
Refined analysis
Kinetic models
Gradient flow interpretation
Claim:
The Keller-Segel system is the gradient flow of the energy G[X ]
(in the classical L2 (0, 1) sense):
∂t X = −∇G[X ]
[Jordan, Kinderlehrer & Otto, Otto]
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Analysis of KS
Refined analysis
Kinetic models
Key observations
• The energy G is not convex, because γ ∈ (−1, 1).
• Each contribution is homogeneous. If α − 1 + γ = 0 the two
contributions have the same homogeneity: the competition is
fair.
G[λX ] = λ1−α G[X ] ∀λ > 0
E. coli
Analysis of KS
Refined analysis
Kinetic models
E. coli
The logarithmic case α = 1, γ = 0
The functional is almost zero-homogeneous.
Z
ZZ
χ
0
G[X ] = −
log(X (m)) dm +
log |X (m) − X (m0 )| dmdm0
2
2
(0,1)
(0,1)
χ
log λ
G[λX ] = G[X ] + −1 +
2
Consequence:
χ
∇G[X ] · X = −1 +
2
χ
−∂t X · X = −1 +
2
d 1
χ
2
|X (t)| = 1 −
dt 2
2
Singularity if χ > 2: blow-up!
Analysis of KS
Refined analysis
Kinetic models
E. coli
The logarithmic case, ctd.
Given a gradient flow of a convex energy G , and a critical point
∇G [A] = 0. Then X (t) → A,
d 1
2
|X (t) − A| ≤ 0
dt 2
If G is uniformly convex: D 2 G ≥ νId,
d
dt
1
|X (t) − A|2
2
!
≤ −ν|X (t) − A|2
Surprisingly, the same holds true here. If A is a critical point of the
energy,
d 1
2
|X (t) − A| ≤ 0
dt 2
Problem: there exists a critical point only when χ = 2. . .
Analysis of KS
Refined analysis
Kinetic models
The logarithmic case, ctd.
. . . Rescale space/time!
1
Grescaled [X ] = G[X ] + |X |2
2
There exists a critical point if χ < 2:
∇G[A] + A = 0
χ
−1 +
+ |A|2 = 0
2
Theorem (C, Carrillo)
In the sub-critical case χ < 2
d 1
2
|X (t) − A| ≤ −|X (t) − A|2
dt 2
Explanation: the interaction part (concave) is ”digested” by the
diffusion contribution.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Fair competition – blow-up
Assume α − 1 + γ = 0.
The functional is (1 − α)-homogeneous.
G[λX ] = λ1−α G[X ]
Consequence:
∇G[X ] · X = (1 − α)G[X ]
d
dt
−∂t X · X = (1 − α)G[X ]
1
2
|X (t)| = (α − 1)G[X ] ≤ (α − 1)G[X0 ]
2
Singularity if G[X0 ] < 0: blow-up!
E. coli
Analysis of KS
Refined analysis
Kinetic models
Fair competition – critical parameter
In the case of fair competition there is a dichotomy which is similar
to (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 density
converges to a self-similar profile. The profile is unique.
• si χ > χc there exists a cone of negative energy. The density
blows-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 reads
d(α − 1) + γ = 0.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Diffusion-dominating case
Assume 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).
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−3
−2
−1
0
1
2
3
E. coli
Analysis of KS
Refined analysis
Kinetic models
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
−
1−α γ
Z
2
(1−α)/2
|x| ρ0 (x) dx
R
The solution is global in time if
kρ0 kLp < C
where p =
2−α
> 1.
1+γ
+ F [ρ0 ] < 0 .
E. coli
Analysis of KS
Refined analysis
Kinetic models
Time discretization
Jordan, Kinderlehrer & Otto have proposed the following scheme
(time-implicit Euler’s scheme):
Xn+1 − Xn
= −∇G[Xn+1 ]
∆t
Xn+1
minimizes
G[Y ] +
1
kY − Xn k2
2∆t
Theorem (Blanchet, C, Carrillo)
In the subcritical case χ < 2, the JKO scheme converges towards a
weak solution of the Keller-Segel equation as ∆t → 0.
• The JKO scheme is well adapted to the energy structure.
• But it is more difficult to handle practically.
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Analysis of KS
Refined analysis
Kinetic models
E. coli
Space discretization
The continuous functional G can be discretized using finite
differences for X (m).
G̃ [X ] = −∆m
N−1
X
i=1
X
χ
log X i+1 − X i + (∆m)2
log |X j − X i |
2
i6=j
N
∆m X i 2
+
|X | .
2
i=1
Theorem
The critical parameter is χ̃c = 2(1 − ∆m)−1 .
• If χ > χ̃c the solution of the discrete gradient flow blows-up
in finite time (meaning that ∃i0 : X i0 +1 − X i0 = 0 after a
finite number of steps).
• If χ < χ̃c the solution of the rescaled gradient flow converges
towards a unique stationary state at exponential rate.
Analysis of KS
Refined analysis
Kinetic models
E. coli
Numerical illustrations
2
4
1.8
2
4
1.8
3
1.6
3
1.6
2
2
1.4
1.4
1
1.2
1
0
0.8
1
1.2
1
0
0.8
−1
0.6
−1
0.6
−2
−2
0.4
0.4
−3
0.2
0
−3
−2
−1
0
1
2
3
−4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Convergence of the solution towards
the unique stationary state in selfsimilar variables when χ < 2.
2
0
−3
−2
−1
0
1
2
3
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
4
1.8
3
1.6
3
1.6
2
2
1.4
1.4
1
1.2
1
1
1.2
0
0.8
1
0
0.8
−1
0.6
−1
0.6
−2
−2
0.4
0.4
−3
0.2
0
−3
−4
Convergence towards the unique stationary state with porous-medium diffusion α > 1 (without rescaling).
4
1.8
−3
0.2
0
−2
−1
0
1
2
3
−4
−3
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Blow-up of the discrete gradient flow
when χ > 2.
1
0
−3
−2
−1
0
1
2
3
−4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Blow-up of the discrete gradient flow
.
when χ > 4
Analysis of KS
Refined analysis
Kinetic models
Conclusion
• Behind the dissipation of energy there is a nice gradient flow
structure (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”, where
diffusion is deterministic (particles keep ordered).
• Using a very coarse space grid one gets an interesting
finite-dimensional reduction of the Keller-Segel PDE.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Collaborations
• Benoı̂t Perthame (Univ. Paris 6)
• José A. Carrillo (Univ. Autònoma Barcelona)
• Adrien Blanchet (Univ. Toulouse 1)
• Lucilla Corrias (Univ. Evry - Val d’Essonne)
• Abderrahman Ebde (Wolfgang Pauli Institute, Vienna)
E. coli
Analysis of KS
Refined analysis
Kinetic models
Contents
Mathematical analysis of the Keller-Segel model – basics
Refined analysis in dimension d = 1
Kinetic models for chemotaxis – analysis
The journey of E. coli
E. coli
Analysis of KS
Refined analysis
Kinetic models
Kinetic modeling
• Bacterial density f (t, x, v ) is described at time (t), position
(x) and velocity (v ).
• Velocity space V is bounded.
The Othmer-Dunbar-Alt model (’88) :
Z
∂t f + v · ∇ x f =
T[S](v , v 0 )f (t, x, v 0 ) dv 0 − λ[S]f (t, x, v )
v 0 ∈V
|
{z
} |
{z
}
run
tumble
• The tumbling kernel T[S](v , v 0 ) denotes the frequency of
reorientation v 0 → v .
R
• λ[S] = v 0 ∈V T[S](v 0 , v ) dv 0 is the intensity of the Poisson
process governing reorientation.
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Analysis of KS
Refined analysis
Kinetic models
Chemoattractant release
Again, the chemical signal is secreted by the cells, following a
reaction-diffusion equation:
Z
∂t S = DS ∆S − αS + ρ , ρ(t, x) =
f (t, x, v ) dv
v ∈V
Or, simply:
−∆S + αS = ρ
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Analysis of KS
Refined analysis
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Tumbling kernel
Assume that T[S](v , v 0 ) depends only on the posterior velocity v .
The kinetic equation reads:
∂t f + v · ∇x f = T[S]ρ(t, x) − λ[S]f (t, x, v )
Example:
Tε [S](v , v 0 ) = Ψ(S(x + εv )) ,
Ψ%
E. coli
Analysis of KS
Refined analysis
Kinetic models
Diffusive limit
It is required that T = Tε [S] is a small perturbation of an unbiased
process: T[S] = T0 + εT1 [S].
Diffusive scaling :
ε∂t fε + v · ∇x fε =
1
{Q0 fε + εQ1 [S]fε }
ε
Theorem (Chalub, Markowich, Perthame & Schmeiser)
Under reasonable hypotheses, fε (t, x, v ) → ρ(t, x)F (v ), avec
Z
Z
Q0 F = 0 ,
vF (v ) dv = 0 ,
F (v ) dv = 1
v ∈V
v ∈V
∂t ρ = ∇ · (D[S]∇ρ − Γ[S]ρ) , Γ[S] = χ(S, |∇S|)∇S
Z
Z
1
1
D[S] =
|v 2 |F (v ) dv , Γ[S] = −
v Q1 [S]F (v ) dv .
2λ[S0 ] v ∈V
λ[S0 ] v ∈V
Other scales/limit (e.g. hyperbolic) are possible (Dolak & Schmeiser,
Filbet, Laurençot & Perthame)
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Analysis of KS
Refined analysis
Kinetic models
Diffusive limit, ctd.
In the case
Tε [S](v , v 0 ) = Ψ(S(x + εv ))
= Ψ(S(x)) + εΨ0 (S(x)) (v · ∇S(x)) + O(ε2 )
we get:
Z
1
D[S] =
|v 2 | dv
2Ψ(S)|V | v ∈V
Z
Ψ0 (S)
χ[S] =
|v 2 | dv .
2Ψ(S)|V | v ∈V
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Analysis of KS
Refined analysis
Kinetic models
The free transport operator
First step in analysis: only runs, no tumble.
∂t f + v · ∇ x f = 0
Explicit solution: f (t, x, v ) = f0 (x − tv , v ).
Recall the dispersion structure of the heat equation ∂t ρ = ∆ρ:
1
∀t > 0 kρ(t)kLp ≤ √ d(1−1/p) kρ0 kL1
( t)
The free transport operator satisfies:
kρ(t)kLp ≤
1
t d(1−1/p)
kf0 kL1x Lpv
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Analysis of KS
Refined analysis
Kinetic models
Duhamel’s formula
Second step: include a right-hand-side (velocity jump process):
∂t f + v · ∇x f = g (t, x, v )
The Duhamel’s formula follows the characteristics:
Z t
f (t, x, v ) = f0 (x − tv + v ) +
g (t − s, x − sv , v ) ds
s=0
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Analysis of KS
Refined analysis
Kinetic models
Global existence – first example
Here T [S](v , v 0 ) . S(x + εv ) (or even ∇S(x + εv )).
Z
t
f (t, x, v ) . f0 (x − tv + v ) +
S(t, x + εv − sv )ρ(t, x − sv ) ds
s=0
Z t
kρ(t)kLp . C0 (t) +
kS(t − s, x + εv − sv )ρ(t − s, x − sv )kLpx L1 ds
v
s=0
Z t
1
kρ(t)kLp . C0 (t) +
kS(t − s, x + εv )ρ(t − s, x)kL1x Lpv ds
d(1−1/p)
s=0 s
Z t
1
kS(t − s)kLp kρ(t − s)kL1 ds
kρ(t)kLp . C0 (t) + Cε
d(1−1/p)
s=0 s
Main message: the transport competes with the aggregation.
The dispersion estimate helps to measure this competition.
Conclusion: delocalization effects ensure global existence (see also
[Hillen, Painter & Schmeiser]).
E. coli
Analysis of KS
Refined analysis
Kinetic models
Global existence – second example
What is the critical nonlinearity?
We assume a global bound in dimension d = 2.
Assume either
T [S] . k∇SkLp ,
p<∞
T [S] . k∇SkθL∞ ,
θ<1
or
Then the solution is global in time.
Same strategy of proof: dispersion overcomes aggregation.
Strong suspicion: T [S] ≈ ∇S may get into trouble.
E. coli
Analysis of KS
Refined analysis
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A critical mass phenomenon in 2D
We propose
T [S](v , v 0 ) = 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 the
Poisson 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 is
global in time.
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Blow-up: virial argument
Differentiate twice the second moment:
ZZ
1
I (t) =
|x|2 f (t, x, v ) dvdx
2 R2 ×V
ZZ
d
1
I (t) =
(x · v )f (t, x, v ) dvdx
dt
2 R2 ×V
Z
d2
2
I (t) ≤ c1 M − c2 χM − ω
(x · j)|∇S| dx
dt 2
R2
Fortunately, under spherical symmetry,
Z
d
ω
(x · j)|∇S| dx = J(t) ,
dt
R2
Heuristics:
∂t ρ = −∇ · j
ρ = −∇ · ∇S
→
J(t) ≥ 0
2
” j∇S ≈ ∂t ∇−1 ρ ”
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Global existence: comparison argument
If χM|V | is small, we exhibit a supersolution:
|x|−γ
if (v · x) < 0
k(x, v ) =
|Πv ⊥ x|−γ
if (v · x) > 0
More precisely,
Z
v · ∇x k ≥ χ (v · ∇S)+
We can prove: ∀t > 0
k(x, v 0 ) dv 0
v0
f (t, x, v ) ≤ k(x, v ).
E. coli
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Compatibility
• The diffusion limit yields the standard Keller-Segel model.
• The blow-up threshold is compatible.
E. coli
Analysis of KS
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Conclusion
• The kinetic models lack energy structure. Analysis is more
delicate to perform.
• Some results for the Keller-Segel system have been extended
at the kinetic level.
• General principles are moreless the same (dispersion vs.
aggregation).
E. coli
Analysis of KS
Refined analysis
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Collaborations
• Benoı̂t Perthame (Univ. Paris 6)
• Nikolaos Bournaveas (Univ. Edinburgh)
• Susana Gutiérrez (Univ. Birmingham)
E. coli
Analysis of KS
Refined analysis
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Contents
Mathematical analysis of the Keller-Segel model – basics
Refined analysis in dimension d = 1
Kinetic models for chemotaxis – analysis
The journey of E. coli
E. coli
Analysis of KS
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Kinetic models
Description of E. coli movements
Alternatively:
• Straight swimming
trajectories (∼ 1sec.): run
• Reorientation events
(∼ 0.1sec.): tumble
Howard Berg’s lab
E. coli
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Chemical signaling
• Bacteria can sense multiple chemical substances along their
trajectories.
Chemoattractants: amino-acids (e.g. aspartate), glucose. . .
• Bacteria are able to produce some of these chemicals.
Positive feedback: accumulation of bacteria in opposition to
the natural dispersion.
N. Mittal, E.O. Budrene, M.P. Brenner, A. van Oudenaarden, PNAS
E. coli
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Response to the chemical signal
E. coli reacts to the time variations of the signal: tumbling events
decrease when the signal concentration increases.
Complex signal integration inside
each individual:
• ”Memory effects”
• High sensibility to signal
changes (excitation)
• Adaptation
J.E. Segall, S.M. Block, H.C. Berg, PNAS
E. coli
Analysis of KS
Refined analysis
Kinetic models
J. Saragosti, A. Buguin, P. Silberzan,
Institut Curie
E. coli
Analysis of KS
Refined analysis
Kinetic models
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 almost
constant ≈ 20µms −1 ).
The Othmer-Dunbar-Alt model (’88) :
Z
∂t f + v · ∇ x f =
T[S](v , v 0 )f (t, x, v 0 ) dv 0 − λ[S]f (t, x, v )
0
|
{z
} | v ∈V
{z
}
run
tumble
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E. coli
What about the tumbling frequency?
• First approach, not including memory effects
(Erban & Othmer,
& Schmeiser):
Dolak
DS = ψ ∂t S + v 0 · ∇ x S
T[S](v , v 0 ) = ψ
Dt v 0
• Phenomenological approach by Segall et al.:
memory effect = time convolution along the
past trajectory:
Z 4sec.
0
0
K (s)S(t − s, x − sv ) ds
T[S](v , v ) = ψ
0sec.
Analysis of KS
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E. coli
Diffusive limit
Assume that chemoattractant has only a slight influence:
DS T[S] = ψ0 + εφ
Dt v 0
Diffusive scale:
Z
ψ0
ε∂t f +v ·∇x f =
(ρ − |V |f )+ φ v 0 · ∇S f 0 dv 0 −|V |φ (v · ∇S) f
ε
v0
1
ρ(t, x). What about ρ(t, x)?
At first order, f (t, x, v ) ≈
|V |
j
∂t ρ + ∇ ·
=0
ε
Z
Z
j
v φ(v · ∇S)f dv
ε∂t j + ∇ · (v ⊗ v )f dv = −ψ0 |V | − |V |
ε
v ∈V
V
R
R
j
−→ = −∇ dψ01|V |2 V |v |2 dv ρ + ρ ψ01|V | v ∈V v φ(v · ∇S) dv
ε |
{z
} |
{z
}
Fick’s law
chemotactic speed
Analysis of KS
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Kinetic models
Macroscopic model
(
∂t ρ = Dρ ∆ρ − ∇ · (ρ u[S])
∂t S = DS ∆S + βρ
Z
u[S] = −
v φ (v · ∇S) dv
v ∈V
The macroscopic flux is quite different from Keller-Segel model
(recall u[S] = χ(S)∇S):
• It is not a gradient.
• It is bounded (no possible singularity).
• It is more nonlinear (in some sense).
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Stationary state – general φ
We restrict to d = 1.
The equation for stationary states writes:

Z

 Dρ ∂x µ + µ
v φ(v ∂x S0 ) dv = 0
v ∈V

 −∂ S = µ
xx 0
Z
Dρ ∂x µ − ∂x
Φ(v ∂x S0 ) dv = 0
v ∈V
Z
Dρ ∂x (∂x S0 ) +
Φ(v ∂x S0 ) dv = Cste
v ∈V
Conclusion: solve an ODE for ∂x S0 .
E. coli
Analysis of KS
Refined analysis
Kinetic models
Cluster formation
E. coli
Analysis of KS
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E. coli
Step response function
We assume a step response function φ(Y ) = −sign (Y ).
Then u[S] = χsign (∂x S).
Stationary state:
χ
µ(x) = exp − |x|
Dρ
160
140
bacterial density
120
100
80
60
40
20
0
−1000
−500
0
space (µ m)
500
1000
Analysis of KS
Refined analysis
Kinetic models
E. coli
Influence of the internal response function
160
160
140
140
120
120
bacterial density
bacterial density
φ(Y ) = − tanh(Y /δ). δ −1 is the slope at the origin.
100
80
60
40
80
60
40
20
0
100
20
−1000
−500
0
500
0
1000
−1000
−500
160
160
140
140
120
120
100
80
60
40
500
1000
500
1000
100
80
60
40
20
0
0
space (µ m)
bacterial density
bacterial density
space (µ m)
20
−1000
−500
0
space (µ m)
500
1000
0
−1000
−500
0
space (µ m)
Analysis of KS
Refined analysis
Traveling pulses
Experiments by Adler (1966).
Kinetic models
E. coli
Analysis of KS
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Kinetic models
Possible 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 and
constant profile (asymmetric).
E. coli
Analysis of KS
Refined analysis
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Mathematical model
• Bacteria gather due to secretion of a chemottractant (as for
cluster formation),
• They consume another chemical (the nutrient N). This
triggers the motion of a pulse.
Kinetic description:
∂t f + v · ∇x f = Q[S, N]f
And reaction-diffusion equations:
∂t S = DS ∆S − αS + βρ
∂t N = DN ∆N − γρN .
E. coli
Analysis of KS
Refined analysis
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Derivation of a simpler model
In adimensional form it writes:
1
ε∂t f + v · ∇x f = Q[S, N]f
ε
Taking the limit when ε → 0 leads to a parabolic equation for the
density ρ(t, x):
∂t ρ = Dρ ∆ρ − ∇ · (ρu[S] + ρu[N])
{z
}
| {z } |
chemotactic flux
diffusion
Z
u[S] = −
v φ (v · ∇S) dv
v ∈V
In the case of a stiff response function φ = step:
uS = χS
∇S
|∇S|
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Analysis of KS
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E. coli
Numerical evidence for traveling pulses
60
bacterial density
50
40
30
20
10
0
0
0.5
1
space (µ m)
1.5
2
4
x 10
Analysis of KS
Refined analysis
Kinetic models
Numerical evidence, ctd.
.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Numerical evidence, ctd.
.
E. coli
Analysis of KS
Refined analysis
Kinetic models
Numerical evidence, ctd.
Limited nutrient: coexistence of a stationary state and a traveling pulse
E. coli
Analysis of KS
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Some quantitative features
In the case of a stiff response function φ = step, we obtain a
formula for the speed of the pulse σ:
σ
χN − σ = χS p
4DS α + σ 2
The profile is a combination of two exponential tails.
(
ρ0 exp (λ− z) , z < 0
ρ(z) =
ρ0 exp (λ+ z) , z > 0
Asymmetry of the profile is given by:
p
λ−
4DS α + σ 2 + σ
p
=
|λ+ |
4DS α + σ 2 − σ
√
It is strongly asymmetric when σ 2 DS α (speed of chemical
diffusion).
E. coli
Analysis of KS
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E. coli
Traveling kinetic pulses
In fact...
The diffusion scaling is not valid for all experimental settings.
Traveling waves are likely to exist at the kinetic level too.
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
Analysis of KS
Refined analysis
Kinetic models
Conclusions
• There is a hierarchy of mathematical models for collective cell
motion (micro-, meso-, macroscopic).
• The appropriate choice relies on a compromise between
accuracy of description and simplicity of formulation.
• The ODA kinetic model is suitable for bacterial motion. It is
possible to derive simplified model (of parabolic types) which
are better adapted than the usual ones.
• Existence of stable traveling pulses is linked to the stationary
chemotaxis problem (without nutrient).
E. coli
Analysis of KS
Refined analysis
Kinetic models
Collaborations
• Benoı̂t Perthame (Univ. Paris 6)
• Nikolaos Bournaveas (Univ. Edinburgh)
• Jonathan Saragosti, Axel Buguin and Pascal Silberzan
(Institut Curie, Paris)
E. coli
Analysis of KS
Refined analysis
Kinetic models
Thank you for your attention!
E. coli