mathematical modeling and analysis of chemotaxis-like behavior · 2012. 5. 18. · (chemotaxis is...

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Mathematical Modeling

and Analysis of

Chemotaxis-Like Behavior

Angela Stevens

University of Munster

Germany

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One biological model-system for development

is the life-cycle of the cellular slime mold,

Dictyostelium discoideum, Dd.

Under starvation conditions the amoebae produce cAMP.

The cells sense this signal and move towards higher concentrations.

(Chemotaxis is very common in many developmental processes,

where cells are spatially reorganized.)

After aggregation and mound formation in Dd,

cells start to pre-differentiate into pre-stalk and pre-spore cells.

In the mound these cells vary w.r.t. their chemotactic sensitivity.

Mathematical Modeling and Analysis of Chemotaxis-Like Behavior 1

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%Chemotaxis and self-organization of Dictyostelium discoideum, (Dd)


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Chemotactic Variability and Cell Sorting:

Cells with stronger chemotactic abilities

can be found at the top of the mound.

Cells which are weaker w.r.t. to chemotaxis

can be found at the bottom of the mound

Later, these differently sorted cells differentiate into two cell types.

The cells from the top of the mound form the stalk.

Cells from the bottom of the mound become the surviving spores.

The outcome of cell sorting is a pre-stage for cell differentiation.


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Jager/Luckhaus, ’92, showed, that chemotaxis can serve

as the main mechanism for self-organization of Dd.

The main mechanism for such a phenomenon in a 2-dim model

must exhibit blow-up of solutions.

The mathematical system derived from the Keller-Segel model:

u density of cells, v concentration of chemo-attractant

∂tu = ∆u− χ∇ · (u∇v)

0 = ∆v + u− β

with Neumann boundary conditions.

χ > 0 is the chemotactic sensitivity.


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For the radial symmetric situation JL, ’92, proved rigorously:

There exist parameter regimes for which global solutions exist.

There exist parameter regimes for which blow-up happens,

in finite time.

Among others, a strong chemotactic sensitivity χ

gives rise to self-organization.

Can chemotaxis also serve as the main mechanism

for cell sorting in the mound stage of Dd?


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A suitable model for chemotactic cell sorting in the mound

would be spatially 3-dimensional and show ordering of the cells

w.r.t. to their chemotactic sensitivity.

Test problem:

Model with two chemotactic species in 2 dimensions,

which exhibits blow-up for one species,

whereas the solution for the other species still exists.


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Let u1, u2 be the densities of the two cell types,

and v be the concentration of the chemo-attractant. Consider

∂tu1 = µ1∆u1 − χ1∇ · (u1∇v)

∂1u2 = µ2∆u2 − χ2∇ · (u2∇v)

∂tv = D∆v + α1u1 + α2u2 − βv

in BR(0), t > 0, with Neumann boundary conditions.

χ1, χ2 > 0 are the chemotactic sensitivities,

µ1, µ2, D are the diffusion coefficients.

For χ2 = 0, the system is decoupled.

Then u1 can - ’just’ - blow up, whereas u2 does not.

What about χ2 << 1?


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1 Dimensional Analysis

... results in

∂tu1 = ∆u1 − χ1∇ · (u1∇v)

∂1u2 = µ2∆u2 − χ2∇ · (u2∇v)

0 = ∆v + u1 + u2 − 1

in B1(0), t > 0 with Neumann boundary conditions.

Existence of classical solutions was shown in the PhD-thesis

of E.E. Espejo (2008).

Global existence for general multi-component chemotaxis systems

with Dirichlet boundary conditions was shown by Wolansky (2002).


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Consider radial symmetric solutions.

Rewrite the system in terms of the rescaled mass functions

Mi(t, r) =

∫ r

0

ui(t, ξ)ξ dξ

with r = |x|. Then

∂rv =M1 +M2

r−r

2and

∂tM1 = r∂r(

1r∂rM1

)

− χ1

(

r2 − M1+M2

r

)

∂rM1

∂tM2 = µ2r∂r(

1r∂rM2

)

− χ2

(

r2 − M1+M2

r

)

∂rM2

with boundary condition M1(t, 1) = m1 , M2(t, 1) = m2.

GOAL: Find sufficient conditions for blow-up in finite time.


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2 Result:

Global solutions exist, if

m1 < min

{

µ2

χ2,2(2 − χ1m2)

χ1

}

and

m2 < min

{

1

χ1,2(2µ2 − χ2m1)

χ2

}

.

Blow-up in finite time happens if

m1 >2

χ1and

∫

B1(0)

|x|2u1(t, x)dx << 1 .


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3 Theorem:

If u1 blows up in finite time,

than also u2 blows up at the same time.

Proof by methods of Herrero/Velazquez ’96, ’97.

Assume u1 blows up in finite time and u2 is bounded,

then we obtain a lower bound for M1 and an upper bound for M2.

Construction of a sub-solution and comparison arguments

lead to a contradiction.

Formal derivation of the blow-up profile:

Calculate asymptotics for parameters χ1, χ2, µ2

for which M2 << M1 near the blow-up point.

Then singularity formation for u2 is possible

without mass aggregation, whereas u1 shows mass aggregation.


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In developmental processes also local signaling plays a role.

Signals may be localized in the extracellular matrix (ECM)

or filamentous structures might give rise to directional cues.


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Haptotaxis and Trail Following:

We consider a Keller-Segel model with non-diffusive memory, namely

∂tu = ∆u−∇ · (u∇ log(v)) , ∂tv = uvλ .

Earlier results:

λ = 0 global solutions (Chen Hua et al),

λ = 1 blow-up for specific initial data (Levine and Sleeman).

By setting θ = 11−λ and z = 1

1−λv(1−λ) = θv

1θ we obtain

∂tu = ∆u− θ∇ · (u∇ log(z)) , ∂tz = u , θ ∈ (0,∞)


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[Kang - S.- Velazquez]:

Space Dimension 1:

For θ = 1 formally every space dependent function

is asymptotically a steady state for t→ ∞.

It could be shown, that the long time dynamics

are strongly dependent on the initial data.

For 1 < θ < 3 it was rigorously proved

that solutions converge to a Dirac mass for t→ ∞.


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Here we only give the heuristic argument:

Assume I = [−1, 1],∫

Iudx = m and consider

zt =zθ

∫

Izθdx

,

which results from setting ut = 0,

then solving the equation for u,

and plugging the solution into the z-equation.

We assume that this is a good approximation

for the original problem for t→ ∞.


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For z(0, 0) > z(0, x) we obtain

z1−θ(t, x) = z1−θ(0, x) − (θ − 1)∫ t

0ds

R

Izθ(s,x)dx

Assume the following expansion:

z1−θ(0, x) = z1−θ(0, 0) +Bx2 + h.o.t. for x→ 0.

Thus z1−θ(t, x) ≈ z1−θ(0, 0) +Bx2 − (θ − 1)∫ t

0ds

R

Izθ(s,x)dx

.

So z1−θ(t, x) :≈ Bx2 + ψ(t), and

therefore z(t, x) ≈ (Bx2 + ψ(t))1

1−θ .

Explicit calculations show that

1−θψ′(t) ≈

∫

Idx

(Bx2+ψ(t))θ

θ−1, and

ψ′(t) ≈ −Kψθ+1

2(θ−1) (t), so ψ(t) ≈ At2(1−θ)3−θ .


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With this we obtain

Theorem:

There exist initial data (u0, z0) ∈ C2,α such that the

corresponding solutions (u, z) of our system satisfy

u(t, x) → mδ(x) and

z(t, x) ≈t

23−θ

(

Bx2t2(θ−1)3−θ + A

)1

θ−1

for t→ ∞, and where A,B are constants,

which depend on the initial data.


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4 Intuitive Understanding of the Model

The exponent λ measures the strength of reinforcement of the trail,

thus the tendency for aggregation increases with larger values of λ.

The dynamics of the cells are described by random motility

and by chemotactic drift towards higher concentrations of v.

The number of times that a brownian particle approaches a given

point in space depends very strongly on the space dimension.

Thus the environment, where the cells moves, is modified stronger

in lower dimensions than it is in higher dimensions. So in this mo-

del the tendency to aggregate increases for smaller spatial dimension.

In contrast to this, in the original Keller-Segel model with diffusion

finite time blow-up is more likely in higher dimensions.


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Asymptotic Results [S.-Velazquez]

Summary for: z0 (x) → z0 (∞) > 0 for x→ ∞, z0 ∈ C2.

N = 1 N = 2

θ < 1

Nondiffusive self-

similar behavior with-

out mass aggregation

Nondiffusive self-similar

behavior without

mass aggregation

θ = 1

Mass aggregation

in regions of order

one in infinite time

Nondiffusive self-similar

behavior without

mass aggregation

θ ∈ (1, 1 + 2

N]

Mass aggregation

in infinite time

Mass aggregation

in infinite time

1 + 2

N< θ

Mass aggregation

in finite time

Mass aggregation

in finite time


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N ≥ 3:

θ ≤ 1+ 2N

- Diffusive self-similar behavior without mass aggregation

1 + 2N< θ < N

N−2 - Mass aggregation in finite time

- Diffusive self-similar behavior

without mass aggregation

NN−2 ≤ θ - Singularity formation in finite time


- Diffusive self-similar behavior without mass aggregation

Further, the size of the region w.r.t. time can be calculated,

where an amount of mass of order one is distributed

during the aggregation process.


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The classical Keller-Segel model.

N = 1 N = 2 N ≥ 3

No singularities

Mass aggregation

in finite time

for M > Mcrit

Mass aggregation

in finite time

with arbitrary mass

Non-diffusive self-

similar behavior

for M < Mcrit

Singularity formation


in finite time

Diffusive self-similar

behavior without

mass aggregation


mathematical modeling and analysis of chemotaxis-like behavior · 2012. 5. 18. · (chemotaxis is...

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