Pattern Formation in Biological Systems

Markus Bär

Dept. Mathematical Modelling & Data Analysis

Physikalisch-Technische Bundesanstalt, Germany

TU Berlin, Lectures GRK 1558, May, 27th, 2013.

1. Introduction

2. Basic Concepts

3. Examples: Reaction-Diffusion

Systems vs. Active Fluids

- Calcium Waves - Protein Oscillations in E. Coli

vs.

- Intracellular Mechanochemical Patterns - Turbulence in Bacterial Suspensions

Nonequilibrium Systems

• continuous flow of matter and energy • entropy export –> selforganisation, patterns

Open chemical reactor cell

Chemical vs. Biological Patterns

CIMA reaction

BZ reaction

Angel fish

Dictyostelium aggregation

Turing Instability and Patterns

A. M. Turing (1952):

,, On the Chemical Basis of Morphogenesis´´:

...Interplay of chemical reactions and diffusion

may lead to spontaneous formation of spatial

patterns......

I. Prigogine et al. (1960s):

Generalization – dissipative structures,

structures in space and time,

nonequilibrium thermodynamics

Balance Equation

Apply to concentrations in chemical reaction

-> reaction-diffusion equations

density current source/sink

i

Limitations of RD Models

• low concentration of reactands, no physical

interactions -> random walk, diffusion

• fluctuations, stochastic effects are neglected ->

pattern scale >> particle size

• heterogeneities, temporal fluctuations neglected

• no boundary effects, infinite systems resp.

periodic or no-flux boundary conditions

Activator-Inhibitor Systems

Schematic view Spatiotemporal dynamics

(Meinhardt & Gierer, 1972)

Patterns from long-range

inhibitor diffusion (Turing 1952)

Example: Brusselator Model • model for dissipative structures (Prigogine et al.)

• mechanism: autocatalytic step, rates = 1

• reaction-diffusion equations = set of coupled

nonlinear partial differential equations:

• variables u, v and control parameters a, b

Belousov-Zhabotinsky Reaction

,,Drosophila´´ BZ waves & patterns

BZ mechanism (FKN)

BZ model (Oregonator)

Petrov et al., Nature 1997

2. Basic Concepts

Linear Stability Analysis

1. Set up reaction-diffusion model

2. Compute uniform steady states u0

And consider small pertubations du(x,t)

3. u0 is stable, if all Re wj (q) < 0

Pattern Formation Issues

• Linear stability of spatially uniform states

• Linear stability of patterns and waves, more difficult: eigenfunctions often

unknown

• Boundary conditions may be important ! infinitely extended/ periodic systems vs.

separated systems (Neumann/Dirichlet)

Bifurcation Theory

• Re wj(qC,mC) = 0 -> instability, bifurcation point

• Patterns and waves with wavenumber qC and frequency WC = Im wj(qC,mC) emerges

• Supercritical (forward) bifurcation produce stable patterns -> amplitude equations

Turing Instability

Eigenvalues w(q) Example: Brusselator

a. uniform steady state

b. Turing unstable, if

Chemical Turing Patterns First experiment – CIMA reaction (de Kepper et al., PRL 1990)

(Ouyang, Swinney, Nature 1991)

Hopf Instability –

Extended Oscillatory Media

Hopf bifurcation: Example: Brusselator

Eigenvalues w(q)

a. uniform steady state

b. Hopf unstable, if

competes with Turing inst. !

Hopf Amplitude Equations

Ansatz for slowly varying A (x,t)

Complex Ginzburg-Landau equations

Apply to Brusselator:

Nonlinear Waves and Chaos

• Consider

• Nonlinear phase equation (,,a(x,t) is slaved´´)

• Benjamin-Feir-Newell instability criterion

-> waves

unstable, spatiotemporal chaos

Wave Instability

nonzero WC, qC

left and right traveling

waves

nonlinear competition:

traveling vs. standing

waves

Minimum: 3 variables !

Excitable Media (EM) Definition: Excitable media have a stable uniform

rest state, large finite perturbations can cause

nontrivial dynamics and patterns

Function: signal propagation, e. g. action potential

propagation, calcium waves

Analogies: Excitable media have three typical states –

rest, excited and refractory state, compare

a. Forest fire – green tree, burning tree and treeless

b. Epidemics – healthy, infected and immune state

Cellular Automaton for EM 1947: Wiener-Rosenblueth 3-state automaton for EM

With rest state, excited state and refractory state

(J. Weimar, TU Braunschweig)

Result: excitation spreads, open arms curl into spirals

EM-Reaction-Diffusion Models

1952: Hodgkin and Huxley model for squid giant

axon -> electrophysiology, neurophysiology

1962: Simplification by FitzHugh and Nagumo (Dv = 0)

FitzHugh-Nagumo Model

phase plane – isoclines pulse and wave train

space

u,v

Rotating Spiral Waves

rigidly rotating spiral meandering spiral (outward)

(Software: Ezspiral –D. Barkley / Warwick)

Phase Diagram for Spiral Waves

• Diagram for Barkley-model

• Spiral core dynamics described by 5 ODE model

-> 3 symmetry modes, 2 meander Hopf modes

(D. Barkley, Phys. Rev. Lett. 1992, 1994)

Far-field Breakup of Spirals

- oscillatory conditions: spirals break far away from ,,core´´ region

- inner part survives near instability (M. Bär & M. Or-Guil, PRL´99)

Core Breakup of Spirals

Modified Barkley model, excitable condition,

influence of meandering M. Bär, M. Eiswirth, Phys. Rev. E (1993)

Arrhythmias in the Heart ?

From: http://thevirtualheart.org (E. Cherry, F. Fenton)

Summary Part 2

• Different linear instabilities (Turing, Hopf) of uniform states give rise to different patterns

(stripes & hexagons, oscillations, waves & chaos)

• Excitable media have a stable uniform rest state,

But large perturbations create pulses, spirals or

chaotic patterns

• Excitable dynamics important in physiological systems:

Neurons, cardiac tissue,…..

Alternative Summary

1. Stationary (Turing) patterns: Slow activator diffusion,

fast inhibitor diffusion

2. Oscillations (Hopf bifurcation): Fast activation, slow

inhibition (feedback)

3. Waves: Activator diffusion faster or equal to inhibitor

diffusion

3. Examples for Biological

Pattern Formation

a. Intracellular Reaction-Diffusion Patterns

Difficulty: Typical diffusion length (D/k)1/2 of proteins

is > 1-10 m m -> no patterns in most cells, one or few wavelength in some cases, extended patterns only

in very big cells

History: Since 1980s - calcium waves in many cells

1992 – Ca spirals in oocytes (Lechleiter, Clapham, Science)

1999 - protein oscillations in E. Coli (Raskin, de Boer, PNAS)

2000 – NADH waves in neutrophils (Petty et al., PRL, PNAS)

………..

Calcium Waves

heart myocytes frog oocytes

(Wussling, Biophys. J.,1999) (Lechleiter, Clapham, Science 1992)

IP3 Receptor Dynamics • endoplasmatic reticulum (ER) is main Ca storage

in cells; ER channels for Ca are regulated by IP3R

• calcium binding to receptor provides fast activation

and slow inhibition (M. Berridge et al.)

Calcium Waves and Mitochondria

mitochondria can be activated as additional calcium stores

Experiment

Model

L. Jouaville et al., Nature 96;

M. Falcke et al., Biophys. J. 99, PRL 00.

Protein Oscillations in E. Coli

• MinC, MinD, MinE proteins regulate cell division • rapid pole to pole oscillations in E. Coli observed

cell division suppressed (Raskin, de Boer, PNAS 1999)

Model for MinD, MinE Waves • Cytosolic (free) and membrane bound concentrations

• MinD, MinE oscillations result from wave instability

(Turing-type II)

Models: Howard et al. PRL 01, Kruse, Biophys. J. 02, Meinhardt & de Boer, PNAS 01,

Huang et al. PNAS 03, Meacci & Kruse, Phys. Biol. 05.....

reaction-diffusion

equations

In-vitro experiment: MinD,E-waves

Loose et al.,

Science 2008.

b. Active Biological Fluids

Active fluids:

Complex fluids wherein energy is injected by active internal

units (molecular motors, self-propelled bacteria etc)

Examples:

a. Intracellular Mechanochemical Patterns:

Cytoskeleton, Molecular Motors, Cytosolic Flows

b. Turbulence in Dense Suspensions of Swimming Bacteria

Alternative to reaction-diffusion mechanism (Turing)

Cytoskeleton

F. Wottawah et al. PRL (2005).

• Cytoskeleton is a network of filaments

• Response to perturbations

solid-like at short time

fluid-like at large times

• Characteristics of cytoskeleton:

active (motors)

B. Alberts et al. Molecular biology of the cell (2007).

Mechanics and Biochemistry

J. Howard, S. W. Grill and J. S. Bois. Nature Rev. Mol. Cell Biol. (2011).

Interaction of biochemical and mechanical processes

Motor or cytoskeletal regulation,

viscosity, elasticity

Stress, transport (fluid motion)

Biochemistry Mechanics

Intracellular active fluid

Idea: Cytoskeleton as active fluid phase

Motors are distributed in the fluid

Local increase of motor concentration ->

active transport of fluid

Positive feedback on

motor concentration

Cytoskeleton

Motors

J.S. Bois, F. Jülicher and S.W. Grill. Phys. Rev. Lett. (2011)

Digression: Mechanochemical waves

in Physarum Polycephalum (C4)

M. Radszuweit (C.4): „A physarum droplet is an active poroelastic medium

coupled to a calcium oscillator“

Experiments by Takagi, Ueda, Physica D 2008, Physica D 2010.

Simulations

Spiral Antiphase oscillation

Turbulence in a living fluid

Cisneros et al (2007)

Bacillus subtilis, dense PIV

Wensink, Dunkel, Heidenreich, Drescher et al., PNAS (2012).

Experiments vs. continuum modelling

quasi-2D experiment 2D-simulation

PIV

...)(

0

uu

u

t

Vorticity maps

Equations for turbulence in active fluids

uuuuu2

20

2)( CAt

pisot uuu )((Navier-Stokes equation: )

2

1

2

20

2

0 )()(

u

uuuuuu

p

CAt

add advection

minimal model:

( Swift-Hohenberg type equation, 0 < 0 )

How good is the theory ?

Experiments vs. continuum modelling

quasi-2D experiment 2D-simulation

PIV

...)(

0

uu

u

t

Vorticity maps

Turing instability in velocity field + nonlinear hydrodynamic

coupling (J. Dunkel, S. Heidenreich)

Summary: Patterns in active fluids

• Active units (motors & cytoskeletal filaments, self-propelled particles/ bacteria …)

• Intracellular dynamics: Mechanochemical patterns, cell

polarization, cell motility, motility assays, …..

• Multicellular dynamics: Biofilms, bacterial suspensions,

aggregation phenomena, tissue dynamics, ……

•

Summary Part 3

• Various experiments on intracellular biological pattern formation in recent years

• Pattern formation organizes social behavior of amoebae and bacteria

• Reaction-diffusion type models reproduce many observations qualitatively

• Active processes enable and enhance pattern formation

• Quantitative comparisons experiment – theory needed !

• Perspectives: Developmental biology, ,,virtual human´´, neural systems ...

References 1. A. S. Mikhailov, ,,Foundation of Synergetics I´´, 2nd Ed. (1994).

2. G. Nicolis, ,,Introduction into Nonlinear Science´´, Cambridge (1996).

3. J. Keener & J. Sneyd, ,,Mathematical Physiology´´, Springer, 2nd Ed. (2008).

4. J. Murray, ,, Mathematical Biology´´, Vols. 1 + 2, Springer, 3rd Ed. (2001).

5. M. Cross & P. Hohenberg, Rev. Mod. Phys. Vol. 65 (1993).

6. R. C. Desai & R. Kapral, ,,Dynamics of Self-Organized and Self-Assembled

Structures´´, Cambridge (2009).

7. M. Cross & H. Greenside, ,,Pattern Formation and Dynamics in Non-

equilibrium Systems´´, Cambridge (2009).

8. L. M. Pismen, ,,Patterns and Interfaces in Dissipative Dynamics´´, Springer (2006).