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

2 ( d S ) < 0 U,V,equil S max Approach towards e l a equilibrium c s d e g r a Fluctuations around y l n p Spontaneous processes E equilibrium o r t � S > 0 n E Time Entropy and fluctuations at equilibrium

Entropy is equivalent to disorder. Hence Environment dS > 0 env there is no spontaneous creation of order at equilibrium. Selforganization Self-organization is spontaneous creation of dS = dS i + dS e < 0 order. dSi > 0 dS e < 0 Self-organization requires export of entropy to an environment which is almost always tantamount to an energy flux or transport of dS = dS + dS > 0 tot env 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

View from south pole South pole Red spot Jupiter: Observation of the gigantic vortex Picture taken from James Gleick, Chaos . Penguin Books, New York, 1988

Computer simulation of the gigantic vortex on Jupiter View from south pole Particles turning counterclockwise Particles turning clockwise 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 60 � 60 � 20 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

Stock Solution Reaction Mixture p d e S T dS env T d S i � � dS 0 � dS 0 dS 0 Isolated system Closed system Open system U = const., V = const., T = const., p = const., � dS = dS env + dS 0 0 � � dS dG dU pdV TdS = - - 0 dS d S d S = + i e � d S 0 i Entropy changes in different thermodynamic systems

Stock Solution [a] = a0 Reaction Mixture [a],[b] A B B A A A A A B A * � A B A A � B A B A � Ø A A � R A - 1 B Flow rate r = B � A A B � Ø A A B A B A B B B B A A Reactions in the continuously stirred tank reactor (CSTR)

1.2 -1 Flow rate r [t ] Concentration a [a ] 0 4.0 6.0 8.0 2.0 10.0 1.0 0.8 k = 1 � A B 0.6 k = 1 � Reversible first order reaction in the flow reactor

Stock Solution [A] = a Reaction Mixture [A],[X] 0 r * A k 1 A X X A A A X A A k 2 A X A X k 3 A A +2 3 X A X X k 4 A A � R- A r X 1 Flow rate = A r A A 0 A X A X A X r 0 X X X X A A

0.5 � r Stationary concentration x Thermodynamic 0.4 branch 0.3 0.2 Bistability r cr,1 r cr,2 0.1 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Flow rate r

r Kinetic differential equations: * A d [ A ] d a = = − − + 2 + + 2 ( ) ( ) ( ) r a a k k x a k k x x 0 1 3 2 4 d t d t k 1 d [ X ] d x A X = = − + + − + 2 2 ( ) ( ) r x k k x a k k x x 1 3 2 4 k 2 d t d t k 3 +2 3 A X X k 4 r 0 A r 0 X

r Kinetic differential equations: * A d [ A ] d a = = − − + 2 + + 2 ( ) ( ) ( ) r a a k k x a k k x x 0 1 3 2 4 d t d t k 1 d [ X ] d x A X = = − + + − + 2 2 ( ) ( ) r x k k x a k k x x 1 3 2 4 k 2 d t d t k 3 Steady states: +2 3 A X X + − + + + − = 3 2 k 4 ( ) ( ) 0 x k k x k a x k k r k a 3 4 3 0 1 2 1 0 r 0 A r 0 X

r Kinetic differential equations: * A d [ A ] d a = = − − + 2 + + 2 ( ) ( ) ( ) r a a k k x a k k x x 0 1 3 2 4 d t d t k 1 d [ X ] d x A X = = − + + − + 2 2 ( ) ( ) r x k k x a k k x x 1 3 2 4 k 2 d t d t k 3 Steady states: +2 3 A X X + − + + + − = 3 2 k 4 ( ) ( ) 0 x k k x k a x k k r k a 3 4 3 0 1 2 1 0 r = = α = = − + + α − α = 3 2 , 1 : 2 ( 2 ) 0 k k k k x x a x r a 0 A 1 2 3 4 0 0 r 0 X

r Kinetic differential equations: * A d [ A ] d a = = − − + 2 + + 2 ( ) ( ) ( ) r a a k k x a k k x x 0 1 3 2 4 d t d t k 1 d [ X ] d x A X = = − + + − + 2 2 ( ) ( ) r x k k x a k k x x 1 3 2 4 k 2 d t d t k 3 Steady states: +2 3 A X X + − + + + − = 3 2 k 4 ( ) ( ) 0 x k k x k a x k k r k a 3 4 3 0 1 2 1 0 r = = α = = − + + α − α = 3 2 , 1 : 2 ( 2 ) 0 k k k k x x a x r a 0 A 1 2 3 4 0 0 Polynomial discriminant of the cubic equation: r 2 α 4 a a 2 2 = + α − + α − α + α + α + = 3 2 2 3 2 216 D ( 6 0 ) ( 12 5 ) 8 4 0 0 0 r r r a a X 0 0 8 2

r Kinetic differential equations: * A d [ A ] d a = = − − + 2 + + 2 ( ) ( ) ( ) r a a k k x a k k x x 0 1 3 2 4 d t d t k 1 d [ X ] d x A X = = − + + − + 2 2 ( ) ( ) r x k k x a k k x x 1 3 2 4 k 2 d t d t k 3 Steady states: +2 3 A X X + − + + + − = 3 2 k 4 ( ) ( ) 0 x k k x k a x k k r k a 3 4 3 0 1 2 1 0 r = = α = = − + + α − α = 3 2 , 1 : 2 ( 2 ) 0 k k k k x x a x r a 0 A 1 2 3 4 0 0 Polynomial discriminant of the cubic equation: r 2 α 4 a a 2 2 = + α − + α − α + α + α + = 3 2 2 3 2 216 D ( 6 0 ) ( 12 5 ) 8 4 0 0 0 r r r a a X 0 0 8 2 � � D < 0 : 3 roots r , r , and r , 2 are positive r = r - r 1 2 3 1 2

0.6 0.00 0.4 � r 0.2 0.01 0.0 � 2.5 0.02 2.0 1.5 a 0 1.0 0.03 0.5 Range of hysteresis as a function of the parameters a 0 and �

dc = = ( , , , ) ; 1 , 2 , , chemical reaction L L i F c c c i n 1 2 i n dt ∂ c = ∆ diffusion i D c ∂ i i t ⎛ ⎞ ∂ ∂ ∂ 2 2 2 c c c ⎜ ⎟ ∆ = + + i i i c ⎜ ⎟ ∂ ∂ ∂ 2 2 2 i ⎝ ⎠ x y z ∂ c = ∆ + = − ( , , , ) ; 1 , 2 , , reaction diffusion i L L D c F c c c i n 1 2 ∂ i i i n t

Autocatalytic third order reactions Multiple steady states � Oscillations in homogeneous solution Direct, A + 2 X 3 X , or hidden in the reaction mechanism Deterministic chaos (Belousow-Zhabotinskii reaction). 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