Mathematics and Pattern Formation in Chemistry and Biology Peter - - PowerPoint PPT Presentation

Mathematics and Pattern Formation in Chemistry and Biology Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA

Tagung über Schulmathematik TU Wien, 28.02.2006

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

1. Equilibrium structures and dissipative patterns 2. Spatio-temporal patterns in chemical reactions 3. Patterns in development 4. Genetic and metabolic networks 5. Patterns in neurobiology

• 1. Equilibrium structures and dissipative patterns

2. Spatio-temporal patterns in chemical reactions 3. Patterns in development 4. Genetic and metabolic networks 5. Patterns in neurobiology

Equilibrium thermodynamics is based on two major statements: 1. The energy of the universe is a constant (first law). 2. The entropy of the universe never decreases (second law). Carnot, Mayer, Joule, Helmholtz, Clausius, ……

D.Jou, J.Casas-Vázquez, G.Lebon, Extended Irreversible Thermodynamics, 1996

Time Fluctuations around equilibrium Approach towards equilibrium Spontaneous processes S > 0

• Smax

E n t r

• p

y E n l a r g e d s c a l e ( ) < 0

U,V,equil

d S

2

Entropy and fluctuations at equilibrium

dSi > 0 dS = + < 0 dSi dSe

Selforganization

dSe < 0

Environment

dS = + dS > 0

tot env

dS dS > 0

env

Entropy is equivalent to disorder. Hence there is no spontaneous creation of order at equilibrium. Self-organization is spontaneous creation of

• rder.

Self-organization requires export of entropy to an environment which is almost always tantamount to an energy flux or transport of matter in an open system.

Entropy production and self-organization in open systems

Snowflakes as examples of equilibrium structures that do not require energy for their maintenance

Five examples of self-organization and spontaneous creation of order

• Hydrodynamic pattern formation in the atmosphere of Jupiter
• Pattern formation in heated fluids
• Pattern formation in chemical reactions
• Morphogenesis in the development of embryos
• Patterns in neurobiology

Examples of self-organization and pattern formation

Red spot South pole View from south pole

Jupiter: Observation of the gigantic vortex

Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988

Computer simulation

• f the gigantic vortex
• n Jupiter

Particles turning counterclockwise Particles turning clockwise View from south pole Jupiter: Computer simulation of the giant vortex

Philip Marcus, 1980. Picture taken from James Gleick, Chaos. Penguin Books, New York, 1988

Rayleigh-Benard convention cells in heated fluids

Alberto Petracci. Particle image velocimetry (PIV) measurement of convective flow in a Raleigh-Benard convection cell of dimension 606020 mm

Spatio-temporal pattern in the Belousov-Zhabotinskii reaction

Pattern formation in the Belousov-Zhabotinskii reaction

Anna L. Lin, Matthias Bertram, Karl Martinez, and Harry L. Swinney, Phys.Rev.Letters 84, 4240 (2000)

blastulation gastrulation

Pattern formation in animal development

Specific pattern formation in the brain correlates with certain activities.

Pictures from the web-page of the Neurobiology Research Unit, Rigshospitalet, Copenhagen, Denmark

reading aloud silent generation of words

Brain activity relative to the resting state

Pictures from the web-page of the Neurobiology Research Unit, Rigshospitalet, Copenhagen, Denmark

1. Equilibrium structures and dissipative patterns

• 2. Spatio-temporal patterns in chemical reactions

3. Patterns in development 4. Genetic and metabolic networks 5. Patterns in neurobiology

Isolated system dS U = const., V = const.,

• dS 0
• dS 0
• dS 0
• Closed system

dG dU pdV TdS T = const., p = const., =

• Open system

dS dS d S d S d S

i e i

dS = + = +

• dSenv

p T T

Stock Solution

Reaction Mixture

d S

i

deS dSenv

Entropy changes in different thermodynamic systems

Stock Solution [a] = a0 Reaction Mixture [a],[b]

A A A A A A A A A A A A A A A A A A A B B B B B B B B B B B B

Flow rate r =

1

R

• * A

A B A Ø B A B Ø

Reactions in the continuously stirred tank reactor (CSTR)

2.0 4.0 6.0 8.0 10.0 Flow rate r [t ]

• 1

1.0 0.8 0.6 1.2 Concentration a [a ]

A B

k = 1

• k = 1
• Reversible first order reaction in the flow reactor

Stock Solution [A] = a Reaction Mixture [A],[X]

A A A A A A A A A A A A A A A A A A A X X X X X X X X X X X X

Flow rate =

r

1

R- A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

Flow rate r

Stationary concentration x

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.1 0.2 0.3 0.4 0.5

rcr,1 rcr,2

Bistability Thermodynamic branch

r

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x Steady states: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

2 4 8 ) 5 12 ( ) 8 6 ( D 216

4 2 2 3 2 2 2 2 3

= + + + − + − + = a a a r a r r α α α α α α

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Polynomial discriminant of the cubic equation: Kinetic differential equations:

A

*

A X X A A X

+2 3

X k3 k4 k1 k2

r r r

x x k k a x k k a a r a ) ( ) ( ) ( t d d t d ] A [ d

2 4 2 2 3 1

+ + + − − = =

x x k k a x k k x r x

) ( ) ( t d d t d ] X [ d

2 4 2 2 3 1

+ − + + − = =

2 4 8 ) 5 12 ( ) 8 6 ( D 216

4 2 2 3 2 2 2 2 3

= + + + − + − + = a a a r a r r α α α α α α

) ( ) (

1 2 1 3 2 4 3 3

= − + + + − + a k r k k x a k x k k x

) 2 ( 2 : 1 ,

2 3 4 3 2 1

= − + + − = = = = a r x a x x k k k k α α α

Steady states: Polynomial discriminant of the cubic equation: Kinetic differential equations: D < 0 r : 3 roots , 2 are positive =

• r , r , and r

r r

1 2 3 1 2

0.4 0.6 0.2 0.0 r 0.00 0.01 0.02 0.03

• 0.5

1.0 1.5 2.0 2.5 a0

Range of hysteresis as a function of the parameters a0 and

diffusion reaction , , 2 , 1 ; ) , , , ( diffusion reaction chemical , , 2 , 1 ; ) , , , (

2 1 2 2 2 2 2 2 2 1

− = + ∆ = ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∆ ∆ = ∂ ∂ = = n i c c c F c D t c z c y c x c c c D t c n i c c c F dt dc

n i i i i i i i i i i i n i i

L L L L

Autocatalytic third order reactions A + 2 X 3 X

• Direct,

, or hidden in the reaction mechanism (Belousow-Zhabotinskii reaction). Multiple steady states Oscillations in homogeneous solution Deterministic chaos Turing patterns Spatiotemporal patterns (spirals) Deterministic chaos in space and time

Pattern formation in autocatalytic third order reactions

G.Nicolis, I.Prigogine. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through

• Fluctuations. John Wiley, New York 1977

1. Equilibrium structures and dissipative patterns 2. Spatio-temporal patterns in chemical reactions

• 3. Patterns in development

4. Genetic and metabolic networks 5. Patterns in neurobiology

a h h h a a

D D h a h D t h a h a k a a D t a > + − + ∆ = ∂ ∂ + − + + ∆ = ∂ ∂ − ; ) 1 ( model Meinhardt Gierer

2 2 2

ρ ν ρ ρ µ

Cascades, A B C ... , and networks of genetic control Turing pattern resulting from reaction- diffusion equation ? Intercelluar communication creating positional information

Development of the fruit fly drosophila melanogaster: Genetics, experiment, and imago

1. Equilibrium structures and dissipative patterns 2. Spatio-temporal patterns in chemical reactions 3. Patterns in development

• 4. Genetic and metabolic networks

5. Patterns in neurobiology

1 2 3 4 5 6 7 8 9 10 11 12 Regulatory protein or RNA Enzyme Metabolite Regulatory gene Structural gene

A model genome with 12 genes

Sketch of a genetic and metabolic network

Active states of gene regulation

Promotor

Repressor

RNA polymerase State : inactive state

III

Promotor

Activator Repressor

RNA polymerase State : inactive state

III

Activator binding site

Inactive states of gene regulation

Cross-regulation of two genes

2 , 1 , ) ( : Repression ) ( : Activation

n n n

= + = + = j i p K K p F p K p p F

j j i j j j i

Gene regulatory binding functions

2 P 2 2 P 2 2 1 P 2 1 P 1 1 2 Q 2 1 2 Q 2 2 1 Q 1 2 1 Q 1 1

) ( ) ( p d q k dt dp p d q k dt dp q d p F k dt dq q d p F k dt dq − = − = − = − =

2 2 1 1 2 2 1 1 2 1

] P [ , ] P [ , ] Q [ , ] Q [ . const ] G [ ] G [ p p q q g = = = = = = = 2 , 1 , ) ( : Repression ) ( : Activation

n n n

= + = + = j i p K K p F p K p p F

j j i j j j i

P 2 Q 2 P 2 Q 2 2 P 1 Q 1 P 1 Q 1 1 1 2 2 2 1 2 2 1 1 1

, ) ( , )) ( ( : points Stationary d d k k d d k k p F p p F F p = = = = − ϑ ϑ ϑ ϑ ϑ

Qualitative analysis of cross-regulation of two genes: Stationary points

( )

1 2 2 1 1 2 2 1 2 1

, p F p F p F p F p p ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ∂ ∂ − = Γ

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ = =

P P P P Q Q Q Q Q Q K D K D j i ij

d d k k p F k p F k p F k p F k d d P P Q Q x x a

2 1 2 1 2 2 2 1 2 2 2 1 1 1 1 1 2 1

A &

K K D D D K K D D K K D

P Q P Q P P Q Q Q P P Q ⋅ − ⋅ = ⋅ = ⋅ hence and

Qualitative analysis of cross-regulation of two genes: Jacobian matrix

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

( )

1 2 2 1 1 2 2 1 2 1

, p F p F p F p F p p ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ∂ ∂ − = Γ

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

Eigenvalues of the Jacobian of the cross-regulatory two gene system

) , ( ) ε ( ) ε ( ) ε ( ) ε (

2 1 P 2 P 1 Q 2 Q 1 P 2 P 1 Q 2 Q 1

p p k k k k D D d d d d Γ − = = + + + + +

Eigenvalues of the Jacobian of the cross-regulatory two gene system

2 P 2 P 1 Q 2 Q 1 P 2 P 1 P 2 Q 2 P 1 Q 2 P 2 Q 1 P 1 Q 1 Q 2 Q 1 Hopf P 2 P 1 Q 2 Q 1 OneD

) ( ) )( )( )( )( )( ( d d d d d d d d d d d d d d d d D d d d d D + + + + + + + + + = − =

Regulatory dynamics at D 0 , act.-act., n=2

Regulatory dynamics at D 0 , act.-rep., n=3

Regulatory dynamics at D < DHopf , act.-repr., n=3

Regulatory dynamics at D > DHopf , act.-repr., n=3

Regulatory dynamics at D 0 , rep.-rep., n=2

Hill coefficient: n Act.-Act. Act.-Rep. Rep.-Rep. 1 S , E S S 2 E , B(E,P) S S , B(P1,P2) 3 E , B(E,P) S , O S , B(P1,P2) 4 E , B(E,P) S , O S , B(P1,P2)

An example analyzed and simulated by MiniCellSim

The repressilator: M.B. Ellowitz, S. Leibler. A synthetic oscillatory network of transcriptional

• regulators. Nature 403:335-338, 2002

Stable stationary state Limit cycle oscillations Fading oscillations caused by a stable heteroclinic orbit Hopf bifurcation Bifurcation to May-Leonhard system Increasing inhibitor strength

1e+07 2e+07 3e+07 4e+07 5e+07 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Proteins

1e+07 2e+07 3e+07 4e+07 5e+07 0.02 0.04 0.06 0.08 1e+07 2e+07 3e+07 4e+07 5e+07 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mRNAs

1e+07 2e+07 3e+07 4e+07 5e+07 0.05 0.1 0.15 0.2 0.25 0.3

The repressilator limit cycle

P1 P2 P3

start start

The repressilator limit cycle

A B C D E F G H I J K L 1

Biochemical Pathways

2 3 4 5 6 7 8 9 10

The reaction network of cellular metabolism published by Boehringer-Ingelheim.

The citric acid

• r Krebs cycle

(enlarged from previous slide).

The bacterial cell as an example for the simplest form of autonomous life The human body: 1014 cells = 1013 eukaryotic cells + 9 1013 bacterial (prokaryotic) cells; 200 eukaryotic cell types

1. Equilibrium structures and dissipative patterns 2. Spatio-temporal patterns in chemical reactions 3. Patterns in development 4. Genetic and metabolic networks

• 5. Patterns in neurobiology

A single neuron signaling to a muscle fiber

B A

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

Hogdkin-Huxley OD equations ) ( ) ( ) ( 1

4 3 l l K K Na Na M

V V g V V n g V V h m g I C t d V d − − − − − − =

m m dt dm

m m

β α − − = ) 1 ( h h dt dh

h h

β α − − = ) 1 ( n n dt dn

n n

β α − − = ) 1 (

A single neuron signaling to a muscle fiber

Gating functions of the Hodgkin-Huxley equations

Temperature dependence of the Hodgkin-Huxley equations

) ( ) ( ) ( 1

4 3 l l K K Na Na M

V V g V V n g V V h m g I C t d V d − − − − − − =

m m dt dm

m m

β α − − = ) 1 ( h h dt dh

h h

β α − − = ) 1 ( n n dt dn

n n

β α − − = ) 1 (

Hogdkin-Huxley OD equations

Hhsim (2).lnk

Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current

L r V V g V V n g V V h m g t V C x V R

l l K K Na Na

π 2 ) ( ) ( ) ( 1

4 3 2 2

− + − + − + ∂ ∂ = ∂ ∂ m m t m

m m

β α − − = ∂ ∂ ) 1 ( h h t h

h h

β α − − = ∂ ∂ ) 1 ( n n t n

n n

β α − − = ∂ ∂ ) 1 (

Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = V() with = x + t

[ ]

L r V V g V V n g V V h m g d V d C d V d R

l l K K Na Na M

π ξ θ ξ 2 ) ( ) ( ) ( 1

4 3 2 2

− + − + − + =

m m d m d

m m

β α ξ θ − − = ) 1 ( h h d h d

h h

β α ξ θ − − = ) 1 ( n n d n d

n n

β α ξ θ − − = ) 1 (

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

50

• 50

100 1 2 3 4 5 6 [cm] V [ m V ]

T = 18.5 C; θ = 1873.33 cm / sec

T = 18.5 C; θ = 1873.3324514717698 cm / sec

T = 18.5 C; θ = 1873.3324514717697 cm / sec

• 10

10 20 30 40 V [ m V ] 6 8 10 12 14 16 18 [cm]

T = 18.5 C; θ = 544.070 cm / sec

T = 18.5 C; θ = 554.070286919319 cm/sec

T = 18.5 C; θ = 554.070286919320 cm/sec

Propagating wave solutions of the Hodgkin-Huxley equations

The human brain 1011 neurons connected by 1013 to 1014 synapses

Acknowledgement of Support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 14898 Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Österreichischen Akademie der Wissenschaften Universität Wien and the Santa Fe Institute

Universität Wien

Universität Wien

Contributors

Christoph Flamm, Gottfried Köhler, Andreas Svrček-Seiler, Stephanie Widder, Universität Wien, AT Erwin Gaubitzer, Lukas Endler and Rainer Machne, Universität Wien, AT Heinz Engl, Philipp Kuegler, James Lu, Stephan Müller, RICAM Linz Paul Phillipson, University of Colorado, Boulder

Web-Page for further information: http://www.tbi.univie.ac.at/~pks