Lecture 9: Chemotaxis and Directed Cell Movements Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2016/17

Computational Biology Group (CoBi), D-BSSE, ETHZ

Contents

1 Continuous Models of Chemotaxis

2 Example

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Chemotaxis

Chemotaxis

http://2008.igem.org/Team:Heidelberg/Project/General information

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The presence of a gradient in a chemoattractant a(x , t) gives rise to movement of a species, density n(x , t), up the concentration gradient.

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Continuous Models of Chemotaxis

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Keller-Segel Model (1971) Fluxes in Keller-Segel Model

Chemotactic Flux:

JC = nχ(a)∇a

Diffusional Flux:

JD = −D∇n

Total Flux:

J = JC + JD

Keller-Segel Model (1971) Cells: Chemoatttractant: Computational Biology Group (CoBi), D-BSSE, ETHZ 23. November 2016 5 / 27

∂n = ∇(Dn (a)∇n − χ(a)n∇a) ∂t ∂a = Da ∇2 a − nδ(a) ∂t Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Diffusion Dn (a)

Diffusion 

Enhancement of Motility: Constant Motility:

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Dn (a) = D 1 + α

aK (a + K )2



Dn (a) = D

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Chemoattractant Degradation δ(a)

Chemoattractant Degradation

Typically neglected: Nonlinear: Linear:

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δ(a) = 0 a (a + K ) δ(a) ∝ a δ(a) ∝

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Chemotactic Sensitivity χ(a)

Experiments: The chemotactic effect increases as the chemoattractant concentration a(x , t) decreases.

Chemotactic Sensitivity

Log Law: Receptor Law:

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χ(a) =

χ0 a

χ(a) = χ0

k2 (k + a)2

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Interesting qualitative Behaviours

Interesting qualitative Behaviours

Travelling Waves: Aggregation:

1 a χ(a) = χ0 n χ(a) = χ0

.

Work this out as part of your homework!

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Non-dimensional Aggregation Model

Cells: Atttractant:

∂u = Du ∇2 u − α∇(uχ(v )∇v ) + f (u, v ) ∂t ∂v = Dv ∇2 v + g(u, v ) ∂t

Parameters: α, χ, Du , Dv > 0

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Linearization around the steady state The spatially homogenous problem u=

u v

!

u˙ =

f (u, v ) g(u, v )

!

has steady states (u ∗ , v ∗ ).

Linearization at the steady state We write u = u ∗ + u1 , v = v ∗ + v1 , where 0 <   1, such that

Cells: Atttractant:

∂u1 = Du ∇2 u1 − α∇(u ∗ χ(v ∗ )∇v1 ) + fu∗ u1 + fv∗ v1 ∂t ∂v1 = Dv ∇2 v1 + gu∗ u1 + gv∗ v1 ∂t

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Separable Solution

Time-dependent Solution ˙ = JΦ Φ

⇒

Φ(t) ∝ exp (λt)

where λ represents the eigenvalues of J.

SpatialSolution 0 = JW + D∆W

⇒

W (x ) ∝ exp (ikx )

where Dk 2 are the eigenvalues of J. k is referred to as wavenumber.

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Ansatz: Separable Solution

w(x , t) = Φ(t)W (x ) ∝ exp (λt + ikx ) λw = Jw − k 2 Dw

∀x

We can rewrite this as Hw = λw H=

H = J − k 2 D.

fu∗ − Du k 2 fv∗ + αχu ∗ k 2 gu∗ gv∗ − Dv k 2

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Prof Dagmar Iber, PhD DPhil

!

Lecture 9 MSc 2016/17

Eigenvalues of H

Hw = λw

H = J − k 2 D.

We then have tr (H) = −(Du + Dv )k 2 + (fu∗ + gv∗ ) < 0 det (H) = h(k 2 ) = Du Dv k 4 − (Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ )k 2 +fu∗ gv∗ − fv∗ gu∗

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Stability to temporal perturbations, k = 0

To obtain Re(λ(k 2 = 0)) < 0 we require tr (J) = (fu∗ + gv∗ ) < 0 det (J) = h(k 2 = 0) = fu∗ gv∗ − fv∗ gu∗ > 0

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Dispersion Relation For patterns to emerge we require det (H) = h(k 2 ) = Du Dv k 4 − (Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ )k 2 +fu∗ gv∗ − fv∗ gu∗ < 0 We thus want hmin < 0. The critical case occurs for hc (k 2 ) = Du Dv k 4 − (Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ )k 2 +fu∗ gv∗ − fv∗ gu∗ = 0 and thus kc2 =

Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ 2Du Dv

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Dispersion Relation Replacing kc2 =

Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ 2Du Dv

in hc (k 2 ) = Du Dv k 4 − (Du gv∗ + Dv fu∗ + αu ∗ χ∗ gu∗ )k 2 +fu∗ gv∗ − fv∗ gu∗ = 0 yields −(Du gv∗ + Dv fu∗ + 2 Du Dv (fu∗ gv∗ − fv∗ gu∗ )) u ∗ χ∗ gu∗ p

α =

s

kc2 =

fu∗ gv∗ − fv∗ gu∗ Du Dv

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Example: Chemotactic Aggregation

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Aggregation of Dictyostelium discoideum Model for the aggregation of the amoebae state of the slime mold Dictyostelium discoideum. http://www.youtube.com/watch?v=bkVhLJLG7ug The population n(x , t) secretes a chemical attractant, cyclic-AMP, a(x , t), that attracts the amoebae.

Cells: Chemoatttractant:

∂n = Dn ∇2 n − ξ∇(n∇a) ∂t ∂a = Da ∇2 a + hn − δa ∂t

Parameters: h, δ, ξ, Dn , Da > 0 Computational Biology Group (CoBi), D-BSSE, ETHZ 23. November 2016 19 / 27

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Linear stability of the steady state The stability of the steady states can be determined by studying the long-term behaviour of perturbations of the steady state w=

n − n∗ a − a∗

!

˙ = Jw + D∆w w

J=

fn f a gn ga

!∗

=

0 0 h −δ

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!

;

D=

Prof Dagmar Iber, PhD DPhil

Dn − ζ n ∗ 0 Da

!

Lecture 9 MSc 2016/17

Dispersion Relation For patterns to emerge we require det (H) = h(k 2 ) = Dn k 2 (δ + Da k 2 ) − hξn∗ k 2 < 0 We thus want hmin < 0. The critical case occurs for hc (k 2 ) = Dn k 2 (δ + Da k 2 ) − hξn∗ k 2 = 0 and thus kc2 =

hξn∗ − δDn Dn Da

In the infinite domain we thus only require kc2 > 0, i.e. ξn∗ > Dn for pattern to emerge. Computational Biology Group (CoBi), D-BSSE, ETHZ 23. November 2016 21 / 27

Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Finite Domain

In the finite domain [0, 1] solutions are with zero flux boundary conditions w ∝ exp (λt) cos (kx ), k = nπ We thus require kc2 =

hχn∗ − δDn > π2 Dn Da

for n = n∗ to be unstable. The critical wavelength is the first to go unstable, namely k1 = π.

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Dimensional Conditions

In the finite domain [0, L] solutions are with zero flux boundary conditions nπ w ∝ exp (λt) cos (kx ), k= L We thus require π2 χhδ 2 n∗ − δDn > 2 kc2 = Dn Da L for n = n∗ to be unstable. The critical wavelength is the first to go unstable, namely k1 = π/L.

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Minimal Domain Size

In the finite domain [0, L] solutions are with zero flux boundary conditions, domain size L must meet the following condition L2 >

π 2 Dn Da χhδ 2 n∗ − δDn

Note that higher expression rate h facilitates the emergence of pattern.

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Conditions for Chemotaxis

χ measures aggregation, Da , Dn dispersion. For pattern to emerge aggregation has to defeat dispersion. Minimal domain size Higher chemoattractant production facilitates pattern formation.

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Slime mold solving a maze in the lab

http://www.youtube.com/watch?v=F3z mdaQ5ac&feature=related

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17

Thanks!! Thanks for your attention!

Slides for this talk will be available at: http://www.bsse.ethz.ch/cobi/education

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Prof Dagmar Iber, PhD DPhil

Lecture 9 MSc 2016/17