Stochastic Chemical Oscillations on a Spatially Structured Medium
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1 Stochastic Chemical Oscillations on a Spatiall Structured Medium Michael Giver, Bulbul Chakrabort Department of Phsics, Brandeis Universit Zahera Jabeen Department of Phsics, Universit of Michigan
2 Motivations: Theor and Eperiment
3 Theoretical Motivation?
4 Theoretical Motivation Result: In a chemical reaction sstem, diffusion can be a destabilizing influence leading to oscillations, waves and spatial patterns.
5 Theoretical Motivation Problem: Patterns are often onl observed when the parameters are fine tuned Problem: Diffusion coefficients of different chemical species must differ b a large amount Solution? We must consider fluctuations intrinsic to our sstem
6 Eperimental Motivation: Belousov-Zhabotinsk(BZ) Increased Coupling = 2 /3 Dead state = M. Toia, V. K. Vanag, and I. R. Epstein, Angew Chem Int Ed Engl 47, 7753 (2008).
7 Building the Model Well Mied Sstem (point oscillator) + Intrinsic Fluctuations Spatiall Etended Sstem (Reaction-Diffusion) + Intrinsic Fluctuations Spatiall Structured Medium + Intrinsic Fluctuations
8 A Model of Chemical Oscillations Brusselator Model 0! X i : R 0 X i! Y i : R 1 2X i + Y i! 3X i : R 2 X i! 0 : R N Ila Prigogine ( ) Simple two species activator/inhibitor sstem that ehibits oscillator behavior Does not attempt to model actual BZ reactions
9 The well mied Brusselator (Point Oscillator) Mean-Field Rate Equations dx/dt = R 0 R 1 X + R 2 X 2 Y R N X dy/dt = R 1 X R 2 X 2 Y Two Possible Behaviors: Attracting fied point Unstable fied point within an attracting limit ccle
10 The well mied Brusselator (Point Oscillator) Mean-field R 0 = 3000 X,Y Stochastic Brusselator time Time Series of X Power Spectrum of X 180 P X ( ) X 3000 P( ) time
11 The well mied Brusselator (Point Oscillator) Mean-field X,Y R 0 = Stochastic Brusselator 3500 Time Series of X time 600 Power Spectrum of X P X ( ) X 2000 P( ) time
12 The well mied Brusselator (Point Oscillator) 600 P X ( ) P( ) Power Spectrum: P (!) = A! 2 + B (B! 2 ) 2 +(2 A + B) 2! 2 Near bifurcation, phase space trajectories spiral into fied point with frequenc ω 0 Fluctuations ecite sstem at characteristic frequenc
13 Spatiall etended Brusselator (1d) We model the one dimensional sstem as a lattice of well mied volumes Reactants can hop between volumes with specified rates Reactions: 0! X i : R 0 X i! Y i : R 1 2X i + Y i! 3X i : R 2 X i! 0 : R N w/ hopping: X i! X i±1 : D Y i! Y i±1 : D
14 Space Spatiall etended Brusselator (1d) ẋ = R 0 ẏ = R 1 Mean-Field: (R 1 + R N ) + R D r 2 R D r 2 Stochastic *T. Biancalani, D. Fanelli, F. Di Patti, Phs. Rev. E 81, (2010) 1. Uniform Stationar State 1. Uniform Stationar State 2. Uniform Oscillations 3. Stationar Turing Patterns 2. Uniform Oscillations 3. Stationar Turing Patterns 4. Stochastic Turing Patterns X
15 An Inhomogenous Lattice of Brusselators Reactions: 0! X i : R 0 X i! Y i : R 1 2X i + Y i! 3X i : R 2 X i! 0 : R N w/ hopping: X i! X i±1 : D Y i! Y i±1 : D Oil Oil Oil
16 An Inhomogenous Lattice of Brusselators Stochastic Simulations Increasing D /D Anti-phase snchronization is never observed Turing patterns observed in predicted parameter regime Intermediate state near Turing bifurcation not seen in mean-field
17 An inhomogenous Lattice of Brusselators Coarse-grained Picture Oil Oil Oil Oil Adapted Brusselator Model D 0! X i : R 0 X i! Y i : R 1 2X i + Y i! 3X i : R 2 X i! 0 : R N X i $ U i : R U i! U i±1 : Y i $ V i : R V i! V i±1 : D
18 An inhomogenous Lattice of Brusselators Coarse-grained Picture U V U V V U U V V U V U Adapted Brusselator Model D 0! X i : R 0 X i! Y i : R 1 2X i + Y i! 3X i : R 2 X i! 0 : R N X i $ U i : R U i! U i±1 : Y i $ V i : R V i! V i±1 : D
19 An inhomogenous Lattice of Brusselators 4 Species Brusselator : (Mean-Field) i = f( i, i )+R (u i i ) i = g( i, i )+R (v i i ) u i = R ( i u i )+D (u i 1 2u i + u i+1 ) v i = R ( i v i )+D (v i 1 2v i + v i+1 ) In-phase Anti-phase
20 An inhomogenous Lattice of Brusselators Anti-Phase Order Parameter = 1 N NX ep[i( j (j 1) )] j=1
21 An inhomogenous Lattice of Brusselators Anti-Phase Order Parameter Not Anti-Phase!
22 Summar Intrinsic fluctuations can give rise to interesting behavior not accessible in the mean-field limit. Simple heterogeneities lead to a qualitativel different sstem Anti-phase state requires fine tuning of parameters and is destroed b noise Thanks to the Fraden Lab for images and movies This work has been funded b the NSF IGERT and MRSEC programs at Brandeis Universit
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