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
Motivations: Theor and Eperiment
Theoretical Motivation?
Theoretical Motivation Result: In a chemical reaction sstem, diffusion can be a destabilizing influence leading to oscillations, waves and spatial patterns.
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
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).
Building the Model Well Mied Sstem (point oscillator) + Intrinsic Fluctuations Spatiall Etended Sstem (Reaction-Diffusion) + Intrinsic Fluctuations Spatiall Structured Medium + Intrinsic Fluctuations
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 (1917-2003) Simple two species activator/inhibitor sstem that ehibits oscillator behavior Does not attempt to model actual BZ reactions
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
The well mied Brusselator (Point Oscillator) 12000 Mean-field 10000 8000 R 0 = 3000 X,Y 6000 4000 Stochastic Brusselator 6000 5000 2000 0 980 985 990 995 1000 time Time Series of X Power Spectrum of X 180 P X ( ) 160 140 4000 120 X 3000 P( ) 100 80 2000 60 1000 40 20 0 18950 18960 18970 18980 18990 19000 time 0 0 0.1 0.2 0.3 0.4 0.5
The well mied Brusselator (Point Oscillator) 20000 18000 Mean-field X,Y 16000 14000 12000 10000 R 0 = 5000 8000 6000 4000 2000 Stochastic Brusselator 3500 Time Series of X 0 0 200 400 600 800 1000 time 600 Power Spectrum of X P X ( ) 3000 500 2500 400 X 2000 P( ) 300 1500 1000 200 500 100 0 18950 18960 18970 18980 18990 19000 time 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
The well mied Brusselator (Point Oscillator) 600 P X ( ) 500 400 P( ) 300 200 100 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 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
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
Space Spatiall etended Brusselator (1d) ẋ = R 0 ẏ = R 1 Mean-Field: (R 1 + R N ) + R 2 2 + D r 2 R 2 2 + D r 2 Stochastic *T. Biancalani, D. Fanelli, F. Di Patti, Phs. Rev. E 81, 046215 (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
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
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
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
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
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
An inhomogenous Lattice of Brusselators Anti-Phase Order Parameter = 1 N NX ep[i( j (j 1) )] j=1
An inhomogenous Lattice of Brusselators Anti-Phase Order Parameter Not Anti-Phase!
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