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Pattern Formation in Biological Systems
Markus Bär
Dept. Mathematical Modelling & Data Analysis
Physikalisch-Technische Bundesanstalt, Germany
TU Berlin, Lectures GRK 1558, May, 27th, 2013.
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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
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Nonequilibrium Systems
• continuous flow of matter and energy • entropy export –> selforganisation, patterns
Open chemical reactor cell
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Chemical vs. Biological Patterns
CIMA reaction
BZ reaction
Angel fish
Dictyostelium aggregation
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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
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Balance Equation
Apply to concentrations in chemical reaction
-> reaction-diffusion equations
density current source/sink
i
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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
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Activator-Inhibitor Systems
Schematic view Spatiotemporal dynamics
(Meinhardt & Gierer, 1972)
Patterns from long-range
inhibitor diffusion (Turing 1952)
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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
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Belousov-Zhabotinsky Reaction
,,Drosophila´´ BZ waves & patterns
BZ mechanism (FKN)
BZ model (Oregonator)
Petrov et al., Nature 1997
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2. Basic Concepts
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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
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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)
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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
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Turing Instability
Eigenvalues w(q) Example: Brusselator
a. uniform steady state
b. Turing unstable, if
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Chemical Turing Patterns First experiment – CIMA reaction (de Kepper et al., PRL 1990)
(Ouyang, Swinney, Nature 1991)
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Hopf Instability –
Extended Oscillatory Media
Hopf bifurcation: Example: Brusselator
Eigenvalues w(q)
a. uniform steady state
b. Hopf unstable, if
competes with Turing inst. !
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Hopf Amplitude Equations
Ansatz for slowly varying A (x,t)
Complex Ginzburg-Landau equations
Apply to Brusselator:
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Nonlinear Waves and Chaos
• Consider
• Nonlinear phase equation (,,a(x,t) is slaved´´)
• Benjamin-Feir-Newell instability criterion
-> waves
unstable, spatiotemporal chaos
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Wave Instability
nonzero WC, qC
left and right traveling
waves
nonlinear competition:
traveling vs. standing
waves
Minimum: 3 variables !
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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
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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
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EM-Reaction-Diffusion Models
1952: Hodgkin and Huxley model for squid giant
axon -> electrophysiology, neurophysiology
1962: Simplification by FitzHugh and Nagumo (Dv = 0)
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FitzHugh-Nagumo Model
phase plane – isoclines pulse and wave train
space
u,v
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Rotating Spiral Waves
rigidly rotating spiral meandering spiral (outward)
(Software: Ezspiral –D. Barkley / Warwick)
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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)
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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)
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Core Breakup of Spirals
Modified Barkley model, excitable condition,
influence of meandering M. Bär, M. Eiswirth, Phys. Rev. E (1993)
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Arrhythmias in the Heart ?
From: http://thevirtualheart.org (E. Cherry, F. Fenton)
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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,…..
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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
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3. Examples for Biological
Pattern Formation
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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)
………..
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Calcium Waves
heart myocytes frog oocytes
(Wussling, Biophys. J.,1999) (Lechleiter, Clapham, Science 1992)
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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.)
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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.
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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)
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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
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In-vitro experiment: MinD,E-waves
Loose et al.,
Science 2008.
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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)
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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).
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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
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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)
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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.
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Simulations
Spiral Antiphase oscillation
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Turbulence in a living fluid
Cisneros et al (2007)
Bacillus subtilis, dense PIV
Wensink, Dunkel, Heidenreich, Drescher et al., PNAS (2012).
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Experiments vs. continuum modelling
quasi-2D experiment 2D-simulation
PIV
...)(
0
uu
u
t
Vorticity maps
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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 )
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How good is the theory ?
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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)
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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, ……
•
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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 ...
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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).