Research Areas
Our research interests may be roughly divided into following areas:
Mechanistic and design studies of chemical oscillators
Pattern formation:
autonomous homogeneous chemical systems
reaction-diffusion systems under external forcing
nonhomogeneous reaction-diffusion media
Systems of biological interest
Chemomechanical transduction
Coupled oscillators
Within each of these areas, there may be several sub-areas as well as projects that are more experimentally or more theoretically oriented. On the right, for each area we give a more detailed list of sub-areas, with a few relevant projects/publications described below.
List of Main Research Areas
Mechanistic and Design Studies of Chemical Oscillators
Design of New Reactions
Pattern Formation - autonomous homogeneous chemical systems
Effects of Global Feedback on Pattern Formation
Pattern Formation - reaction-diffusion systems under external forcing
Pattern Formation - nonhomogeneous reaction-diffusion media
Systems of Biological Interest
Mathematical Modeling
Chemomechanical Transduction
Coupled Oscillators
Mechanistic and Design Studies of Chemical Oscillators
Our group pioneered the systematic design and mechanistic study of oscillating chemical reactions. We continue to be interested in developing new reactions that have particularly desirable features, e.g., producing specific types of patterns or being photosensitive. We also carry out studies to elucidate the mechanisms of reactions that display complex dynamical behavior.
F. Sagués and I. R. Epstein, “Nonlinear Chemical Dynamics,” Dalton Trans. 1201-1217 (2003) (cover article).
I. Szalai, K. Kurin-Csörgei, I. R. Epstein and M. Orbán, “Dynamics and Mechanism of Bromate Oscillators with 1,4-Cyclohexanedione,” J. Phys. Chem. A 107, 10074-10081 (2003).
A. K. Horváth, I. Nagypál, G. Peintler and I. R. Epstein, “Autocatalysis and Selfinhibition: Coupled Kinetic Phenomena in the Chlorite-Tetrathionate Reaction,” J. Am. Chem. Soc. 126, 6246-6247 (2004).
I. Berenstein, L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, “Dynamic mechanism of photochemical induction of Turing superlattices in the chlorine dioxide-iodine-malonic acid reaction-diffusion system,” J. Phys. Chem. A 109, 5382-5387 (2005).
V. K. Vanag and I. R. Epstein, “Resonance-Induced Oscillons in a Reaction-Diffusion System,” Phys. Rev. E 73, 016201-1-7 (2006).
K. Kovács, M. Leda, V. K. Vanag and I. R. Epstein, “Small amplitude and mixed mode pH oscillations in the bromate-sulfite-ferrocyanide-aluminum(III) system,” J. Phys. Chem. A 113, 146–156 (2009).
V. Horvath, I.R. Epstein and K. Kustin, “Mechanism of the ferrocyanide–iodate–sulfite oscillatory chemical reaction,” J. Phys. Chem. A 120, 1951-1960 (2016). DOI: 10.1021/acs.jpca.5b11152
Design of New Reactions
Pattern formation in uncatalyzed bromate oscillatory system visualized by various indicators
M. Orbán, K. Kurin-Csörgei, A. M. Zhabotinsky, and I. R. Epstein, “A New Chemical System for Studying Pattern Formation: Bromate - Hypophosphite - Acetone - Dual Catalyst,” Faraday Disc. 120, 11-19 (2001).
A. K. Horváth, I. Nagypál and I. R. Epstein, “Oscillatory Photochemical Decomposition of Tetrathionate Ion,” J. Am. Chem. Soc. 124, 10956-10957 (2002).
K. Kurin-Csörgei, M. Orbán and I. R. Epstein, “Systematic Design of Chemical Oscillators Using Complexation and Precipitation Equilibria,” Nature 433, 139-142 (2005).
K. Kurin-Csörgei, I. R. Epstein and M. Orbán, “Periodic Pulses of Calcium Ions in a Chemical System,” J. Phys. Chem. A. 110, 7588-7592 (2006).
V. Horváth, K. Kurin-Csörgei, I. R. Epstein and M. Orbán, “Oscillations in the Concentration of Fluoride Ions Induced by a pH-Oscillator,” J. Phys. Chem. A, 112, 4271-4276 (2008).
Pattern Formation - autonomous homogeneous chemical systems
One of the most visually striking occurrences in chemical systems is the formation of spatial patterns and waves. These phenomena are thought to be of importance in a variety of pattern formation phenomena in living systems. We are studying the dynamical origin of pattern formation, attempting to design new kinds of patterns, and investigating how introducing feedback can induce or alter pattern formation.
Pattern Formation Associated with the Wave Instability: Standing wave patterns on a disc - pattern with C6 symmetry (top), pattern with C9 symmetry (bottom)
M. Dolnik, A. B. Rovinsky, A. M. Zhabotinsky and I. R. Epstein, "Standing Waves in a Two-Dimensional Reaction-Diffusion Model with the Short-Wave Instability," J. Phys. Chem. A 103, 38-45 (1999).
M. Dolnik, A. M. Zhabotinsky, A. B. Rovinsky, I. R. Epstein, "Spatio-temporal patterns in a reaction-diffusion system with wave instability", Chem. Eng. Sci. 55, 223-231 (2000).
V. K. Vanag and I. R. Epstein, “Subcritical Wave Instability in Reaction-Diffusion Systems,” J. Chem. Phys. 121, 890-894 (2004).
Effects of Global Feedback on Pattern Formation
Pattern Formation Associated with the Wave Instability: Standing wave patterns on a disc - pattern with C6 symmetry (top), pattern with C9 symmetry (bottom)
V. K. Vanag, L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, " Oscillatory cluster patterns in a homogeneous chemical system with global feedback", Nature 406,389-391 (2000).
V. K. Vanag, A. M. Zhabotinsky and I. R. Epstein " Pattern formation in the Belousov-Zhabotinsky reaction with photochemical global feedback", J. Phys. Chem 104A, 11566-11577 (2000).
H. G. Rotstein, N. Kopell, A. Zhabotinsky and I. R. Epstein, “A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators,” SIAM J. Appl. Math. 63, 1998-2019 (2003).
H. G. Rotstein, N. Kopell, A. M. Zhabotinsky and I. R. Epstein, “Canard Phenomenon and Localization of Oscillations in the Belousov-Zhabotinsky Reaction with Global Feedback,” J. Chem. Phys. 119, 8824-8832 (2003).
Pattern Formation - reaction-diffusion systems under external forcing
When an oscillator is driven by an external force, a variety of phenomena, including resonance, may arise. When two or more chemical oscillators are coupled to one another, they can exhibit a much richer range of behavior than the individual component oscillators. We are seeking to understand what happens when systems that oscillate not only in time but also in space (pattern formation) are forced and/or coupled. Biological examples include organisms under the periodic forcing resulting from circadian oscillations in light intensity.
Temporally Periodic Forcing
V. K. Vanag, A. M. Zhabotinsky and I. R. Epstein,"Oscillatory Clusters in the Periodically Illuminated, Spatially Extended Belousov-Zhabotinsky Reaction", Phys. Rev. Lett. 86, 552-555 (2001).
M. Dolnik, I. Berenstein, A. M. Zhabotinsky and I. R. Epstein, “Spatial Periodic Forcing of Turing Structures,” Phys. Rev. Lett. 87, 238301-1-4 (2001).
Spatiotemporal Forcing of Turing Structures:
Modulation of Turing structures in CDIMA reaction with spatial periodic forcing (PRL 87,238301,2001)
M. Dolnik, A. M. Zhabotinsky and I. R. Epstein,"Resonant Suppression of Turing Patterns by Periodic Illumination", Phys. Rev. E 63, 026101-1-10 (2001).
A. Sanz-Anchelergues, A. M. Zhabotinsky, I. R. Epstein and A. P. Muñuzuri, "Turing Pattern Formation Induced by Spatially Correlated Noise", Phys. Rev. E 63, 056124-1-4 (2001).
M. Dolnik, I. Berenstein, A. M. Zhabotinsky and I. R. Epstein, "Spatial Periodic Forcing of Turing Structures", Phys. Rev. Lett. 87, 238301-1-4(2001).
I. Berenstein, L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, “Superlattice Turing Structures in a Photosensitive Reaction-Diffusion System,” Phys. Rev. Lett. 91, 058302-1-4 (2003).
I. R. Epstein, I. B. Berenstein, M. Dolnik, V. K. Vanag, L. Yang and A. M. Zhabotinsky, “Coupled and Forced Patterns in Reaction-Diffusion Systems,” Phil. Trans. Roy. Soc. A 366, 397–408 (2008).
L. Haim, A. Hagberg, R. Nagao, A.P. Steinberg, M. Dolnik, I.R. Epstein and E. Meron, “Fronts and patterns in a spatially forced CDIMA reaction,” Phys. Chem. Chem. Phys. 16, 26137-26143 (2014).
Pattern Formation - nonhomogeneous reaction-diffusion media
Most studies of patterns and waves in chemical systems assume or attempt to create uniform, homogeneous media. Real reaction-diffusion systems, particularly those of biological or industrial importance, involve media that are not homogeneous.We are seeking to understand how the structure of the medium affects the behavior of such systems. Recent work has focused on a microemulsion, a mixture of oil, water and a surfactant, where we are able to “tune” the structure of the medium, and hence the behavior of the system, by varying the composition.
Fully developed inwardly moving spirals and target patterns in BZ-AOT microemulsion.
Microemulsion Systems
V. K. Vanag and I. R. Epstein, “Inwardly Rotating Spiral Waves in a Reaction-Diffusion System,” Science 294, 835-837 (2001).
V. K. Vanag and I. R. Epstein, “Dash-waves in a Reaction Diffusion System,” Phys. Rev. Lett. 90, 098301-1-4 (2003).
V. K. Vanag and I. R. Epstein, “Segmented Spiral Waves in a Reaction-Diffusion System,” Proc. Nat. Acad. Sci. USA 100, 14635-14638 (2003) (cover article).
A. Kaminaga, V. K. Vanag, and I. R. Epstein “`Black spots’ in a surfactant-rich Belousov-Zhabotinsky reaction dispersed in a water-in-oil microemulsion system,” J. Chem. Phys. 122, 174706-1-11 (2005).
A. Kaminaga, V. K. Vanag, and I. R. Epstein, “A reaction-diffusion memory device,” Angew. Chem. Int. Ed. 45, 3087-3089 (2006).
T. Bánsági Jr., V. K. Vanag, and I. R. Epstein, “Two- and Three-Dimensional Standing Waves in a Reaction-Diffusion System,” Phys. Rev. E 86, 045202-1-4 (2012) doi: 10.1103/PhysRevE.86.045202.
Coupled Patterns
L. Yang and I. R. Epstein, “Oscillatory Turing Patterns in Reaction-Diffusion Systems with Two Coupled Layers,” Phys. Rev. Lett. 90, 178303-1-4 (2003).
L. Yang and I. R. Epstein, “Symmetric, Asymmetric and Antiphase Turing Patterns in a Model System with Two Identical Coupled Layers,” Phys. Rev. E 69, 026211-1-6 (2004).
I. Berenstein, M. Dolnik, L. Yang, A. M. Zhabotinsky and I. R. Epstein, “Turing Pattern Formation in a Two-Layer System: Superposition and Superlattice Patterns,” Phys. Rev. E 70, 046219-1-5 (2004).
D.G. Míguez, M. Dolnik, I.R. Epstein and A.P Muñuzuri, “Interaction of Chemical Patterns in Coupled Layers,” Phys Rev. E 84, 046210-1-6 (2011).
Systems of Biological Interest
We are always interested in applying insights and techniques from our studies of chemical systems to systems of significance in other areas, particularly biology. Although theseprojects primarily involve mathematical modeling, we also collaborate with experimentalists working on the relevant biological systems.
Synaptic Memory: Schematic of the first two steps of autophosporylation of CaMKII holoenzyme and hysteresis loop in the system with the Ca2+ independent protein phosphatase in vitro
A. M. Zhabotinsky, "Bistability in the Ca2+/calmodulin-dependent protein kinase-phosphatase system" Biophys. J. 79, 2211-2221 (2000).
J. E. Lisman and A. M. Zhabotinsky,"A Model of Synaptic Memory: A CaMKII/PP1 Switch that Potentiates Transmission by Organizing an AMPA Receptor Anchoring Assembly", Neuron 31, 191-201 (2001).
A. M. Zhabotinsky, R. N. Camp, I. R. Epstein and J. E. Lisman, “Role of the Neurogranin Concentrated in Spines in the Induction of Long-term Potentiation,” J. Neurosci. 26, 7337-7447 (2006).
I. R. Epstein “Predicting complex biology with simple chemistry,” Proc. Nat. Acad. Sci. USA 103, 15727-15728 (2006).
Gene Expression Networks: Synthetic gene network for entraining and amplifying cellular oscillations
J.Hasty, J. Pradines, M. Dolnik, J. J. Collins,"Noise-based switches and amplifiers for gene expression" Proc. Nat. Acad. Sci. 97, 2075-2080 (2000).
J.Hasty, F. Isaacs, M. Dolnik, D. McMillen, J.J. Collins, “Designer gene networks: Towards fundamental cellular control” Chaos 11, 207-220 (2001).
C. Konow, Z. Li, S. Shepherd, D. Bullara and I. R. Epstein, “Influence of survival, promotion, and growth on pattern formation in zebrafish skin,” Sci. Rep. 11, 9864-1-14 (2021) DOI:10.1038/s41598-021-89116-4
M. Liu, L. Yuan, C. Zhu, C. Pan, Q. Gao, H. Wang, Z. Cheng, and I. R, Epstein, “Peptide-modulated pH rhythms,” ChemPhysChem 23, e202200103 (2022) DOI: 10.1002/cphc.202200103.
Mathematical Modeling
A significant portion of our work involves the development and analysis of mathematical models for a variety of systems in chemistry, biology and other areas that exhibit interesting behavior in time and/or space. Wherever possible, this theoretical work is closely coupled to our experimental studies.
Synaptic Memory: Schematic of the first two steps of autophosporylation of CaMKII holoenzyme and hysteresis loop in the system with the Ca2+ independent protein phosphatase in vitro
Reaction-Diffusion Systems
L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, “Spatial Resonances and Superposition Patterns in a Reaction-Diffusion Model with Interacting Turing Modes,” Phys. Rev. Lett. 88, 208303-1-4 (2002). (more ... )
V. K. Vanag and I. R. Epstein, “Stationary and Oscillatory Localized Patterns, and Subcritical Bifurcations,” Phys. Rev. Lett. 92, 128301-1-4 (2004).
L. Yang, A. M. Zhabotinsky and I. R. Epstein, “Stable Squares and other Oscillatory Turing Patterns in a Reaction-Diffusion Model,” Phys. Rev. Lett. 92, 198303-1-4 (2004).
L. Yang, A. M. Zhabotinsky and I. R. Epstein, “Jumping solitary waves in an autonomous reaction-diffusion system with subcritical wave instability,” Phys Chem Chem Phys. 8, 4647 - 4651 (2006) (cover article).
Networks
B. Shargel, H. Sayama, I. R. Epstein and Y. Bar-Yam, “Optimization of Robustness and Connectivity in Complex Networks,” Phys. Rev. Lett. 90, 068701-1-4 (2003).
Y. Bar-Yam and I. R. Epstein, “Response of Complex Networks to Stimuli,” Proc. Natl. Acad. Sci. 101, 4341-4345 (2004).
M.A. M. de Aguiar, I. R. Epstein and Y. Bar-Yam, “Analytically Solvable Model of Probabilistic Network Dynamics,” Phys. Rev. E 72, 067102-1-4 (2005).
Chemomechanical Transduction
Top: Center of mass trajectories (red lines) of an array of nine interacting BZ gels under different illumination intensities - low intensity (left): irregular motion; medium intensity (center): swing-forward motion; high intensity (right): circular motion. Bottom: Bird flocks with center of mass trajectories indicated by dashed red lines.
In a process analogous to what occurs in the muscles in our bodies, the BZ reaction, when run in an appropriate gel medium, can convert the energy stored in the chemical bonds of its reactants into mechanical force. Such gels, which typically employ a photosensitive ruthenium-based BZ catalyst, can generate motion that can be controlled with an external light source. In collaboration with colleagues at the China University of Mining and Technology, we have carried out a number of experimental and modeling studies of such systems.
X. Lu, L. Ren, Q. Gao, Y. Zhao, S. Wang, J. Yang and I.R. Epstein, “Photophobic and phototropic movement of a self-oscillating gel,” Chem. Commun. 49, 7690-7692 (2013). doi: 10.1039/C3CC44480E
L. Ren, W. She, Q. Gao, C. Pan, C. Ji and I. R Epstein, “Retrograde and direct wave locomotion in a photosensitive self-oscillating gel,” Angew. Chem. Int. Ed., 55, 14301-14305 (2016). DOI: 10.1002/anie.201608367
L. Ren, M. Wang, C. Pan, Q. Gao, Y. Liu and I. R. Epstein, “Autonomous reciprocating migration of an active material,” Proc. Nat. Acad. Sci. 114, 8704-8709 (2017). DOI:10.1073/pnas.1704094114
L. Ren, L. Wang, Q. Gao, R. Teng, Z. Xu, J. Wang, C. Pan, and I. R Epstein, “Programmed Locomotion of an Active Gel Driven by Spiral Waves,” Angew. Chem. Int. Ed. 59, 7106-7112 (2020). DOI: 10.1002/anie.202000110
L. Ren, L. Yuan, Q. Gao, R. Teng, J. Wang and I.R. Epstein, “Chemomechanical Origin of Directed Locomotion Driven by Internal Chemical Signals,” Sci. Adv. 6, eaaz9125 (2020). DOI: 10.1126/sciadv.aaz9125
R. Teng, Y. Li, L. Ren, I.R. Epstein and Q. Gao, “Emergence of collective motion driven by phase reset,” ChemPhysChem e202300054 (2023). doi.org/10.1002/cphc.202300054 (cover article)
Coupled Oscillators
A complex symmetry-breaking rhythm in a pair of identical coupled Van der Pol oscillators. Top panel shows time series with oscillator 1 (blue) undergoing small-amplitude oscillations and oscillator 2 (red) showing large-amplitude relaxation oscillations. Bottom panel shows phase plane analysis of trajectory shown at top using our Cartesian product quilt method that emphasizes the role of folded singularities in guiding trajectories.
The behavior of oscillators when interacting with one another has fascinated scientists at least since Huygens' observations in the seventeenth century that pendulum clocks on a wall tended to synchronize with one another. We have investigated systems of coupled chemical and physical oscillators both experimentally using microfluidic techniques and theoretically using mathematical and computational analyses of model systems.
Experimental
M. Toiya, V.K. Vanag and I.R. Epstein, “Diffusively Coupled Chemical Oscillators in a Microfluidic Assembly,” Angew. Chem. Int. Ed. 47, 7753-7755 (2008).
M. Toiya, H.O. González-Ochoa, V. K. Vanag, S. Fraden, and I.R. Epstein, “Synchronization of Chemical Micro-oscillators,” J. Phys. Chem. Lett. 1, 1241-1246 (2010).
J. Delgado, N. Li, M. Leda, H.O. González-Ochoa, S. Fraden and I.R. Epstein, “Coupled Oscillators in a 1D Emulsion of Belousov-Zhabotinsky Droplets,” Soft Matter 7, 3155-3167 (2011).
V. Horvath, P. L. Gentili, V. K. Vanag and I. R. Epstein, “Pulse-Coupled Chemical Oscillators with Time Delay,” Angew. Chem. Int. Ed. 51, 6878-6881 (2012).
N. Tompkins, N. Li, C. Girabawe, M. Heymann, G. B. Ermentrout, I. R. Epstein and S. Fraden, “Testing Turing's theory of morphogenesis in chemical cells,” Proc. Nat. Acad. Sci. USA 111, 4397-4402 (2014). DOI: 10.1073/pnas.1322005111
Theoretical
V.K. Vanag and I.R. Epstein, “Periodic perturbation of one of two identical chemical oscillators coupled via inhibition,” Phys. Rev. E 81, 066213 (2010)
N.M. Awal, D. Bullara and I.R. Epstein, “The Smallest Chimera: Periodicity and Chaos in a Pair of Coupled Chemical Oscillators,” Chaos 29, 013131-1-12 (2019). doi.org/10.1063/1.5060959.
V. Horváth, D. J. Kutner, M.D. Zeng and I.R. Epstein, “Phase-Frequency Model of Strongly Pulse-Coupled Belousov-Zhabotinsky Oscillators,” Chaos 29, 023128-1-7 (2019). .doi.org/10.1063/1.5082161.
N.M. Awal, I.R. Epstein, T.J. Kaper and T. Vo, “Symmetry-breaking rhythms in coupled, identical oscillators,” Chaos 33, 011102 (2023) doi.org/10.1063/5.0131305