Statistical Physics of Run-and-Tumble bacteria Julien Tailleur M. - - PowerPoint PPT Presentation

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Statistical Physics of Run-and-Tumble bacteria Julien Tailleur M. - - PowerPoint PPT Presentation

Oct 26, 2023 331 likes •1.15k views

Statistical Physics of Run-and-Tumble bacteria Julien Tailleur M. Cates, D. Marenduzzo, I. Pagonabarraga, A. Thompson Laboratoire MSC CNRS - Universit Paris Diderot Large Fluctuations in Non-Equilibrium Systems J. Tailleur (CNRS-Univ Paris

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SLIDE 1

Statistical Physics of Run-and-Tumble bacteria Julien Tailleur

  • M. Cates, D. Marenduzzo, I. Pagonabarraga, A. Thompson

Laboratoire MSC CNRS - Université Paris Diderot

Large Fluctuations in Non-Equilibrium Systems

  • J. Tailleur (CNRS-Univ Paris Diderot)

LAFNES 2011 1 / 27

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SLIDE 2

The big picture

Micrometer scale Centimeter scale

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 2 / 27

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SLIDE 3

Outline

Intro Effective temperature and non-equilibrium dynamics Pattern formation in bacterial colonies Large deviations of bacteria on lattice

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 3 / 27

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SLIDE 4

Run-and-Tumble bacteria

Escherichia coli – Unicellular Organism (1µm × 3µm) Flagella (few µm long) Electron microscopy

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 4 / 27

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SLIDE 5

Schematic trajectory [Berg & Brown, Nature, 1972]

Run: straight line (velocity v ≃ 20 µm.s−1) • Tumble: change of direction (rate α ≃ 1 s−1, duration τ ≃ 0.1s) •

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 5 / 27

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SLIDE 6

Schematic trajectory [Berg & Brown, Nature, 1972]

Run: straight line (velocity v ≃ 20 µm.s−1) • Tumble: change of direction (rate α ≃ 1 s−1, duration τ ≃ 0.1s) •

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 5 / 27

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SLIDE 7

Schematic trajectory [Berg & Brown, Nature, 1972]

Run: straight line (velocity v ≃ 20 µm.s−1) • Tumble: change of direction (rate α ≃ 1 s−1, duration τ ≃ 0.1s) • Diffusion at large scale

Run-and-Tumble D =

v2 dα(1+ατ) ∼ 100 µm2.s−1

Brownian Motion Dcol =

kT 6πηr ∼ 0.2µm2.s−1

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 5 / 27

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SLIDE 8

In and out of “equilibrium”

10µm colloids in a bacterial bath [Wu Libchaber 2000]

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 6 / 27

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SLIDE 9

In and out of “equilibrium”

10µm colloids in a bacterial bath [Wu Libchaber 2000] t ≫ α−1 “Effective temperature” t ≪ α−1 Superdiffusive regime

  • J. Tailleur (CNRS-Univ Paris Diderot)

Introduction LAFNES 2011 6 / 27

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SLIDE 10

Master Equation (d > 1)

˙ P(x, u) =

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 7 / 27

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SLIDE 11

Master Equation (d > 1)

˙ P(x, u) = − ∇ · [P(x, u)v] − α(u)P(x, u) + 1 Ω

  • du′α(u′)P(x, u′)
  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 7 / 27

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SLIDE 12

Master Equation (d > 1)

˙ P(x, u) = − ∇ · [P(x, u)v] − α(u)P(x, u) + 1 Ω

  • du′α(u′)P(x, u′)

ρ(x) =

  • duP(x, u)
  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 7 / 27

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SLIDE 13

True equilibrium... perturbatively...

If v(u) = v u then ˙ ρ =

t≫ 1

α

D0∆ρ with D0 = v2

dα

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 8 / 27

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SLIDE 14

True equilibrium... perturbatively...

If v(u) = v u then ˙ ρ =

t≫ 1

α

D0∆ρ with D0 = v2

dα

Sedimentation v(u) = v u + vτ with vτ = −µ δm g uz

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 8 / 27

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SLIDE 15

True equilibrium... perturbatively...

If v(u) = v u then ˙ ρ =

t≫ 1

α

D0∆ρ with D0 = v2

dα

Sedimentation v(u) = v u + vτ with vτ = −µ δm g uz ρ(z ≫ v/α) ∝ e−λz; 2λv α = ln λ(vT + v) + α λ(vT − v) + α

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 8 / 27

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SLIDE 16

True equilibrium... perturbatively...

If v(u) = v u then ˙ ρ =

t≫ 1

α

D0∆ρ with D0 = v2

dα

Sedimentation v(u) = v u + vτ with vτ = −µ δm g uz ρ(z ≫ v/α) ∝ e−λz; 2λv α = ln λ(vT + v) + α λ(vT − v) + α

  • If vτ/v ≪ 1

λ = vτ

D0

ρ ∝ e−µ δm g z/D0

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 8 / 27

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SLIDE 17

True equilibrium... perturbatively...

If v(u) = v u then ˙ ρ =

t≫ 1

α

D0∆ρ with D0 = v2

dα

Sedimentation v(u) = v u + vτ with vτ = −µ δm g uz ρ(z ≫ v/α) ∝ e−λz; 2λv α = ln λ(vT + v) + α λ(vT − v) + α

  • If vτ/v ≪ 1

λ = vτ

D0

ρ ∝ e−µ δm g z/D0 = e−βeffVext(z) Effective kTeff = D0

µ

bacteria ≡ hot colloids

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 8 / 27

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SLIDE 18

Generic Potential

v(u) = vu + vτ where vτ = −µ∇Vext

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 9 / 27

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SLIDE 19

Generic Potential

v(u) = vu + vτ where vτ = −µ∇Vext If µ∇Vext ≪ v and µ∇2Vext ≪ α ρ( r) ∝ exp

  • −µVext(

r) D0

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 9 / 27

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SLIDE 20

Generic Potential

v(u) = vu + vτ where vτ = −µ∇Vext If µ∇Vext ≪ v and µ∇2Vext ≪ α ρ( r) ∝ exp

  • −µVext(

r) D0

  • Effective Temp. kTeff = D0

µ

bacteria ≡ Hot colloids Non-perturbatively? Only quantitative difference?

  • J. Tailleur (CNRS-Univ Paris Diderot)

External Potentials LAFNES 2011 9 / 27

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SLIDE 21

Fishing lobsters at the micrometer scale

[P

. Galajda, J. Keymer, P . Chaikin, R. Austin, J. Bacteriol. 189, 8704 (2007)]

  • J. Tailleur (CNRS-Univ Paris Diderot)

A bacterial ratchet LAFNES 2011 10 / 27

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SLIDE 22

Fishing lobsters at the micrometer scale

Colloids

  • J. Tailleur (CNRS-Univ Paris Diderot)

A bacterial ratchet LAFNES 2011 10 / 27

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SLIDE 23

Fishing lobsters at the micrometer scale

Bacteria

  • J. Tailleur (CNRS-Univ Paris Diderot)

A bacterial ratchet LAFNES 2011 10 / 27

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SLIDE 24

Where does the ratchet effect come from ?

Bacteria align with walls upon collisions •• Asymmetric walls no left-right symmetry Interactions with walls no time-reversal symmetry

10 20 30 40 50 60 v 1 2 3 4 5 6 7 Ratio

  • J. Tailleur (CNRS-Univ Paris Diderot)

A bacterial ratchet LAFNES 2011 11 / 27

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SLIDE 25

Where does the ratchet effect come from ?

Bacteria align with walls upon collisions •• Asymmetric walls no left-right symmetry Interactions with walls no time-reversal symmetry Elastic collisions No ratchet effect

10 20 30 40 50 60 v 1 2 3 4 5 6 7 Ratio ρ2 ρ1

v

  • J. Tailleur (CNRS-Univ Paris Diderot)

A bacterial ratchet LAFNES 2011 11 / 27

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SLIDE 26

Patterning: Math. Biol. vs Stat. Mech.

Specific vs generic (universal?) Quantitative vs qualitative Accurate description vs general mechanisms

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 12 / 27

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SLIDE 27

Pattern formation: extra ingredients

[Woodward et al., 1995]

Cells division & ‘death’ Interactions (chemical, steric...) clusters are non-motile Is this enough ?

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 13 / 27

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SLIDE 28

Interactions: fluctuating hydrodynamics

Microscopic: α, v, τ Mesoscopic: density field ρ(x, t)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 14 / 27

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SLIDE 29

Interactions: fluctuating hydrodynamics

Microscopic: α, v, τ Mesoscopic: density field ρ(x, t) ˙ ρ(x, t) = −∇J(x, t); J(x, t) = −D∇ρ + V ρ D = v2 dα(1 + ατ) V = − v dα∇ v 1 + ατ

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 14 / 27

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SLIDE 30

Interactions: fluctuating hydrodynamics

Microscopic: α, v, τ Mesoscopic: density field ρ(x, t) ˙ ρ(x, t) = −∇J(x, t); J(x, t) = −D∇ρ + V ρ D = v2 dα(1 + ατ) V = − v dα∇ v 1 + ατ = −D′(ρ)∇ρ/2 Interactions: tumble rate α constant, tumble duration τ = 0 swimming speed v′[ρ(x)] < 0 J(x, t) = −Deff(ρ)∇ρ Deff(ρ) = D(ρ) + ρD′(ρ)/2

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 14 / 27

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SLIDE 31

Interactions and phase separation

D(ρ) = v2(ρ)/dα Deff(ρ) = D(ρ) + ρD′(ρ)/2 v′(ρ) < 0 D′(ρ) < 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 15 / 27

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SLIDE 32

Interactions and phase separation

D(ρ) = v2(ρ)/dα Deff(ρ) = D(ρ) + ρD′(ρ)/2 v′(ρ) < 0 D′(ρ) < 0 Flat profile ρ(x) = ρ0 unstable if Deff(ρ0) < 0 Example: v(ρ) = v0 exp(−ρ/¯ ρ) Instability for ρ > ¯ ρ

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 15 / 27

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SLIDE 33

Interactions and phase separation

D(ρ) = v2(ρ)/dα Deff(ρ) = D(ρ) + ρD′(ρ)/2 v′(ρ) < 0 D′(ρ) < 0 Flat profile ρ(x) = ρ0 unstable if Deff(ρ0) < 0 Example: v(ρ) = v0 exp(−ρ/¯ ρ) Instability for ρ > ¯ ρ Phase separation of the bacterial colony

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 15 / 27

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SLIDE 34

Surface tension

Gradient expansion of ρ(x) Phase separation large gradients at the interfaces

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 16 / 27

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SLIDE 35

Surface tension

Gradient expansion of ρ(x) Phase separation large gradients at the interfaces Expand to higher orders: surface tension −κ(ρ)∆2ρ ˙ ρ = ∇[Deff(ρ)∇ρ] − κ∆2ρ

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 16 / 27

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SLIDE 36

Surface tension

Gradient expansion of ρ(x) Phase separation large gradients at the interfaces Expand to higher orders: surface tension −κ(ρ)∆2ρ ˙ ρ = ∇[Deff(ρ)∇ρ] − κ∆2ρ Phase separation with smooth interfaces

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 16 / 27

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SLIDE 37

Cells division and death

Logistic growth ˙ ρ = ∇[Deff(ρ)∇ρ] − κ∆2ρ + µρ(1 − ρ/ρ0) ρ ≤ ρ0 division dominates ( ˙ ρ > 0) ρ ≥ ρ0 death dominates ( ˙ ρ < 0)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 17 / 27

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SLIDE 38

Birth-death vs phase separation

D′(ρ0) < 0 Deff[ρ0] < 0 phase separation

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 18 / 27

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SLIDE 39

Birth-death vs phase separation

D′(ρ0) < 0 Deff[ρ0] < 0 phase separation ρlow ≤ ρ0 division; ρhigh ≥ ρ0 death

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 18 / 27

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SLIDE 40

Birth-death vs phase separation

D′(ρ0) < 0 Deff[ρ0] < 0 micro-phase separation • ρlow ≤ ρ0 division; ρhigh ≥ ρ0 death

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 18 / 27

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SLIDE 41

Birth-death vs phase separation

D′(ρ0) < 0 Deff[ρ0] < 0 micro-phase separation • ρlow ≤ ρ0 division; ρhigh ≥ ρ0 death Regular (E. coli) or amorphous patterns (S. typhimurium)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 18 / 27

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SLIDE 42

Salmonella typhimurium

[Woodward et al, Biophys. J. 68 2181-2189 (1995)] Numerics •

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 19 / 27

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SLIDE 43

A simple mechanism

Mechanism simple and generic based on microscopic models Interactions D′(ρ) < 0 Non-equilibrium instability Division/death stabilize large scales Competition patterns of fixed size

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 20 / 27

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SLIDE 44

Summary of Biophysics part

Effective temperature perturbatively Strong nonequilibrium effect otherwise (ratchet etc.) Interactions give rise to interesting phenomena (patterning etc.) Many more interesting questions! (experiments...)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 21 / 27

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SLIDE 45

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 46

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 47

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 48

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 49

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 50

Be wise—discretize!

1 2 3 4 5 6 7 8 9 10 d2 d8 d6 d6 d4 + i αi αi Simulations really fast (107 particles in few hours) Analytics simpler (no horrible CE expansion) Interesting cases (ZRP , exclusion process...) Biologists don’t want to talk to you anymore Fluctuating hydrodynamics and large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 22 / 27

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SLIDE 51

Trajectory: n±

i (tj) particles hoping to the right/left at site i

P[n+

i (tj), n− i (tj)] ∝

L

  • i=1

N

  • j=1

δ

  • n+

i (tj+1) − n+ i (tj) − J+ i (tj)

  • × δ
  • n−

i (tj+1) − n− i (tj) − J− i (tj)

  • J,
  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 23 / 27

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SLIDE 52

Trajectory: n±

i (tj) particles hoping to the right/left at site i

P[n+

i (tj), n− i (tj)] ∝

  • L
  • i=1

N

  • j=1

dˆ n±

i (tj)

  • eˆ

n+

i (tj)

  • n+

i (tj+1)−n+ i (tj)−J+ i (tj)

  • +eˆ

n−

i (tj)

  • n−

i (tj+1)−n− i (tj)−J− i (tj)

  • J
  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 23 / 27

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SLIDE 53

Trajectory: n±

i (tj) particles hoping to the right/left at site i

P[n+

i (tj), n− i (tj)] ∝

  • L
  • i=1

N

  • j=1

dˆ n±

i (tj)

  • eˆ

n+

i (tj)

  • n+

i (tj+1)−n+ i (tj)−J+ i (tj)

  • +eˆ

n−

i (tj)

  • n−

i (tj+1)−n− i (tj)−J− i (tj)

  • J

The probability of a trajectory can then be written P[n+

i (t), n+ i (t)] = i

D[ˆ n+

i , ˆ

n−

i ]e−S[n+,n−,ˆ n+,ˆ n−]

where the action S is given by S = − T dt

  • i
  • ˆ

n+

i ˙

n+

i + ˆ

n−

i ˙

n−

i − H[ˆ

n±

i , n± i ]

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 23 / 27

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SLIDE 54

density and current ρi= n+

i + n− i ;

Ji = d(n+

i − n− i );

ˆ ρi= 1 2(ˆ n+

i + ˆ

n−

i );

ˆ Ji = 1 2(ˆ n+

i − ˆ

n−

i )

S[ˆ n±

i , n± i ]

S[ˆ ρi, ˆ Ji, ρi, Ji]

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 24 / 27

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SLIDE 55

density and current ρi= n+

i + n− i ;

Ji = d(n+

i − n− i );

ˆ ρi= 1 2(ˆ n+

i + ˆ

n−

i );

ˆ Ji = 1 2(ˆ n+

i − ˆ

n−

i )

S[ˆ n±

i , n± i ]

S[ˆ ρi, ˆ Ji, ρi, Ji] Continuum limit (lattice spacing a = 1/L) – Diffusive scaling ρi ρ(x); ˆ ρi ˆ ρ(x); da v;

  • i

L 1 dx; Ji J(x)/L; ˆ Ji ˆ J(x)/L; ∇i a∇ + a2 2 ∆; t L2t

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 24 / 27

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SLIDE 56

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v ρ, ˆ ρ, J, ˆ J ∼ O(1) P[ρ, J] ∼ O(e−L) Large deviations

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 25 / 27

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SLIDE 57

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v ρ, ˆ ρ, J, ˆ J ∼ O(1) P[ρ, J] ∼ O(e−L) Large deviations Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

η(x, t)η(x′, t′) = 1

Lδ(x − x′, t − t′)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 25 / 27

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SLIDE 58

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v ρ, ˆ ρ, J, ˆ J ∼ O(1) P[ρ, J] ∼ O(e−L) Large deviations Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

η(x, t)η(x′, t′) = 1

Lδ(x − x′, t − t′)

η(x, t) ∼ O(1/ √ L) typical trajectories (P ∼ O(1))

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 25 / 27

slide-59
SLIDE 59

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v ρ, ˆ ρ, J, ˆ J ∼ O(1) P[ρ, J] ∼ O(e−L) Large deviations Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

η(x, t)η(x′, t′) = 1

Lδ(x − x′, t − t′)

η(x, t) ∼ O(1/ √ L) typical trajectories (P ∼ O(1)) η(x, t) ∼ O(1) Large deviations (P ∼ O(e−L)) This is why we can compute LDF using macroscopic approaches

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 25 / 27

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SLIDE 60

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

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SLIDE 61

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

Transition probability P[ρ0(x) → ρ1(x), t] δS0 = 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

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SLIDE 62

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

Transition probability P[ρ0(x) → ρ1(x), t] δS0 = 0 δS0 δˆ ρ = 0 ˙ ρ = −∇J

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

slide-63
SLIDE 63

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

Transition probability P[ρ0(x) → ρ1(x), t] δS0 = 0 δS0 δˆ ρ = 0 ˙ ρ = −∇J δS0 δ ˆ J = 0 J = − v α∇[vρ] + 2αρ ˆ J

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

slide-64
SLIDE 64

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

Transition probability P[ρ0(x) → ρ1(x), t] δS0 = 0 δS0 δˆ ρ = 0 ˙ ρ = −∇J δS0 δ ˆ J = 0 J = − v α∇[vρ] + 2αρ ˆ J ˆ J noise of the most probable trajectory.

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

slide-65
SLIDE 65

Intuitive feeling for ˆ J fields

S0 = −L τ dt 1 dx ˆ ρ ˙ ρ − vρ∇ ˆ J − J∇ˆ ρ + αρˆ J2 + αJ ˆ J v Same action as ˙ ρ = −∇J; J = − v

α∇[vρ] + √2Dρ η

Transition probability P[ρ0(x) → ρ1(x), t] δS0 = 0 δS0 δˆ ρ = 0 ˙ ρ = −∇J δS0 δ ˆ J = 0 J = − v α∇[vρ] + 2αρ ˆ J ˆ J noise of the most probable trajectory. Solutions of δS0 δρ = − ˙ ˆ ρ − v∇ ˆ J + α ˆ J2 = 0; δS0 δJ = −∇ˆ ρ + α ˆ J v = 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 26 / 27

slide-66
SLIDE 66

Acknowledgments

  • M. Cates, D. Marrenduzzo, R. Nash, I. Pagonabarraga,
  • A. Thompson

Non-interacting bacteria [M.J. Schnitzer, PRE 48, 2553 (1993)] From micro to macro [J.T., M.E. Cates, PRL 100, 218103 (2008)]

  • Ext. Pot./ratchet [J.T., M.E. Cates, EPL 86, 60002 (2009)]

Pattern formation [M.E. Cates, D. Marenduzzo, I. Pagonabarraga,

J.T., PNAS 107 11715 (2010)]

Lattice models of R&T particles [A. G. Thompson, J. Tailleur, M. E.

Cates, R. A. Blythe, JSTAT P02029 (2011)]

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 27 / 27

slide-67
SLIDE 67

Many ways of doing nothing: v(ρ) vs ν(ρ) vs τ(ρ)

The motility can decrease for many reasons v′(ρ) < 0 or ν′(ρ) > 0 or τ ′(ρ) > 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 28 / 27

slide-68
SLIDE 68

Many ways of doing nothing: v(ρ) vs ν(ρ) vs τ(ρ)

The motility can decrease for many reasons v′(ρ) < 0 or ν′(ρ) > 0 or τ ′(ρ) > 0 e.g. v, ν constant and τ = τ0 eρ/¯

ρ

D = v2 ν(1 + ντ); V = −D ντ 1 + ντ ∇ρ ¯ ρ Deff(ρ0) = D

  • 1 −

ρ0ντ ¯ ρ(1+ντ)

  • can become negative
  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 28 / 27

slide-69
SLIDE 69

Many ways of doing nothing: v(ρ) vs ν(ρ) vs τ(ρ)

The motility can decrease for many reasons v′(ρ) < 0 or ν′(ρ) > 0 or τ ′(ρ) > 0 e.g. v, ν constant and τ = τ0 eρ/¯

ρ

D = v2 ν(1 + ντ); V = −D ντ 1 + ντ ∇ρ ¯ ρ Deff(ρ0) = D

  • 1 −

ρ0ντ ¯ ρ(1+ντ)

  • can become negative

Same instability

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 28 / 27

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SLIDE 70

The meaning of death

Bacteria don’t die quickly: they become non-motile or simply stop dividing e.g. division for ρ < ρ0, no death for ρ > ρ0 Same qualitative behavior

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 29 / 27

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SLIDE 71

Why bother with the microscopic equations ?

D1

>

D2

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

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SLIDE 72

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

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SLIDE 73

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 i > 0 : j−i−1,−i= D1(n−i−1 − ni); ji,i+1 = D2(ni − ni+1); j−1,1 = D1n−1 − D2n1

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

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SLIDE 74

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

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SLIDE 75

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

slide-76
SLIDE 76

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0) D1 D1 D2 D2 −2 −1 1 2

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

slide-77
SLIDE 77

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0) D1 D1 D2 D2 −2 −1 1 2 i ≥ 0 : j−i−1,−i= D1(n−i−1 − ni); ji,i+1 = D2(ni − ni+1);

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

slide-78
SLIDE 78

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0) D1 D1 D2 D2 −2 −1 1 2 j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

slide-79
SLIDE 79

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 D1n−(0) = D2n+(0) D1 D1 D2 D2 −2 −1 1 2 x n j(x < 0)= −D1∇n−(x) = 0; j(x > 0) = −D2∇n+(x) = 0 n−(0) = n+(0)

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27

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SLIDE 80

Why bother with the microscopic equations ?

D1 D1 D1 D2 D2 D2 −3 −2 −1 1 2 3 x n γ = Cst; T (x); ∇(D(x)n(x)) = 0 D1 D1 D2 D2 −2 −1 1 2 x n γ(x); T = Cst; D(x)∇n(x)= 0

  • J. Tailleur (CNRS-Univ Paris Diderot)

Pattern formation LAFNES 2011 30 / 27