Spatio-Temporal Patterns with Hyperchaotic Dynamics in Diffusively Coupled Biochemical Oscillators

We present three examples how complex spatio-temporal patterns can be linked to hyperchaotic attractors in dynamical systems consisting of nonlinear biochemical oscillators coupled linearly with diffusion terms. The systems involved are: (a) a two-variable oscillator with two consecutive autocatalytic reactions derived from the Lotka-Volterrh scheme; (b) a minimal two-variable oscillator with one first-order autocatalytic reaction; (c) a three-variable oscillator with first-order feedback lacking autocatalysis. The dynamics of a finite number of coupled biochemical oscillators may account for complex patterns in compartmentalized living systems like cells or tissue, and may be tested experimentally in coupled microreactors.


INTRODUCTION
Complex biochemical spatio-temporal self-organization with high-dimensional chaotic attractors cannot only occur in continuous reaction-diffu- sion systems with infinitely many degrees of free- dom in phase space.The discovery that linear diffusive coupling of only two oscillatory reaction schemes can produce chaos [1] nurtures the idea that finitely many coupled oscillators can be em- ployed to generate spatio-temporal patterns with underlying attractors of arbitrarily high (finite) dimension in phase space [2].Here we investigate the concept of attractor dimension increase asso- ciated with an increasing number of positive Lyapunov characteristic exponents (LCEs), i.e. hyperchaos, in three examples of coupled bio- chemical oscillators.
Following the introduction of the prototypic flow with a hyperchaotic attractor we first de- monstrate hyperchaos of coupled oscillators in one spatial dimension where the two coupled spe- cies have equal diffusion coefficients.The second example illustrates the transition from a stable Turing structure to spatio-temporal hyperchaos, both spatially trapped between two oscillatory regimes of differing amplitude and frequency in a in two directions in phase space together with parameter gradient.The third example shows the the flow being folded back in two orthogonal transition from low to high-dimensional hyperdirections.The numerical calculation of the specchaos in a two-dimensional grid of biochemical trum of Lyapunov exponents yields two positive oscillators.
LCEs.A Poincar6 cross section of a numerically simulated hyperchaotic attractor of Eq. (1) is shown in Fig.

ABSTRACT HYPERCHAOS
Equation (1) solves the mystery of R6ssler's first equation for hyperchaos [4].That pioneer- To introduce hyperchaos we present a four-vari- ing system possessed a hyperchaotic attractor able ordinary differential equation composed of for special choices of parameters and initial con- two well-known subsystems.The equation is: ditions, most.choices in the neighborhood, how- ever, lead to an escape of the trajectory to ('1 =--X2--aX, either fixed point or infinity.Comparing the two equations it turns out that only the coupling of J2-J(l-Z, (1) expanding variable Y to the chaotic subsystem I;" Z + ae Y, is different.Coupling the two linear subsystems 2=0.1 + 3Z(blX2 +bzY)-SZ equivalently to Z (as in Eq. ( 1)) yields a hy- perchaotic flow that is prototypic in the same with al,az, bl,b2 > 0.
sense as the subsystem (X|,Xe, Z) is prototypic For bt 1,b2--0, the subsystem (X|,Xe, Z) for chaotic flows.Incidentally, the system of is R6ssler's equation for continuous chaos.
Eq. (1) can be transformed into the simplest Parameter al can be tuned to change the sys- hyperchaotic flow [5] (containing only one non- tem's stable attractor from a single-loop limit linearity) by means of a mode transformation of cycle via a period-doubling sequence to a chao- the linear variables [6].Thus, the extensions to tic attractor.The stretching and folding of the chaos with even more than two directions of flow is realized with the switching nonlinearity mean divergence work naturally as in [5].
in the equation for Z [3].For b 0, be 1, the subsystem (Y,Z) forms an attracting limit cycle when a2 exceeds the value of the Hopf bifurca- tion.As this limit cycle works with the same nonlinearity as the first subsystem, one can view such a structure as the oscillatory source of the chaotic attractor.Combining the two subsystems (b--1,b2 1), as in Eq. (1), yields a hyper- chaotic flow: the harmonic oscillator of the chao- tic subsystem oscillates in the (X'I,X'2) plane, v Both subsystems keep expanding for small values of variable Z due to nonzero expansion para- meters a and a2, respectively.The two switching nonlinearities in dZ/dt then increase the values of variables X2 and Y, and return the value of 0 Z sharply close to zero whenever the threshold -0,3 x, 3,1 is passed.We thus have an explicit implementa-

FIGURE
Poincar6 cross-section of hyperchaotic flow in tion of the principle of an oscillator expanding Eq. (I) taken at x2 0 with al 0.0, a2 --0.3, DI b2 1.
The mechanism consists of two consecutive autocatalytic steps in variables X and Y, a zero- order input to X and a first-order decay of vari- able X.This Lotka-Volterra scheme is extended by the instability-inducing second-order autocata- lysis governed by parameter a. Homogeneous os- cillating units with the kinetics of Eq. ( 2) can now be coupled linearly by diffusive terms for both variables as follows: J( fx + Dx(X2 X), fy + Dy(Y2 Y1), J(i fx + Dx(Xi+ 2Xi + Xi-1 ), i f + D y Yi+ 2 Yi + Yi- kU --ix --Dx(XN-1 Xu), "N --.fy -!-Dy( YN-1-YN) (3) with i= 2, 3,..., (N-1).fxand fy denote kinetic terms, e.g. as in Eq. (2); Dx and Dy are the respective diffusion coefficients of variables X and Y; and N is the number of cells.The scheme is written with zero-flux boundary conditions, i.e. for an open chain of reaction cells.However, the results presented can qualitatively be achieved with periodic boundary conditions as well.
Previous results on hyperchaos with a finite number of diffusively coupled cells were obtained with a ratio of diffusion coefficients considerably smaller or larger than one (see e.g. [2,7]).In con- 100 M= 80 FIGURE 2 Spatio-temporal evolution of 100 oscillators Eq.
(2) coupled in the form of Eq. ( 3) with a 1, b--3, d= 1, Dx Dy 5. Logarithms of the concentrations of variables Xi were grey coded from -4 (white) to 2 (black).trast, we now couple five cells of the oscillator in Eq. (2) in the oscillatory parameter domain with equal diffusion coefficients, i.e. a ratio Dx/Dy 1.
For a coupling strength of Dx D 0.05 we observe a chaotic attractor with the four largest LCEs (0.2, 0.1,0,-0.3).The attractor is hyperch- aotic with two positive LCEs.Under otherwise identical parameters the hyperchaotic attractor also exists for a ratio of diffusion coefficients slightly smaller or slightly larger than one.Increasing the number of coupled cells leads to hyperchaotic attractors with an increasing num- ber of positive LCEs for a finite window of cou- pling parameters Dx D. Figure 2 is a space- time plot for 100 coupled cells in the hyperchaotic regime.Again the hyperchaos exists for ratios Dx/Dy slightly smaller and slightly larger than one.Thus, in addition to the cases Dx > D, and Dx < D, spatio-temporal hyperchaos has been found to exist generically for the case Dx/Dy 1.
The gradient in the input to variable X was chosen such that the first cell of the decoupled system oscillates in a small-amplitude sinusoidal fashion and the last oscillator possesses a largeamplitude relaxational limit cycle with lower fre- quency (parameters are given in the caption of Fig. 3).For strong coupling under Turing condi- tions (Dx < Dy) the system Eq. (5)with kinetic terms taken from Eq. ( 4) organizes itself into a stable asymmetric Turing pattern, e.g. for N 100.Decreasing the coupling strength leads to the ap- pearance of oscillating behavior at both spatial ends of the system.Figure 3(a) is a space-time plot of this situation.A stationary Turing pattern is located in the central area.It is hemmed in by oscillatory cells.Interestingly, the oscillations are related to those of the uncoupled system and thus differ from each other in amplitude and fre- quency.The Turing structure dynamically separates the two oscillatory domains.This is a remarkable situation as it seems impossible to extract information about the distant oscillator from a time series of one oscillator if only finite precision is available, e.g. as provided by experi- mental data.Thus, with a moderate parameter lOO lOO 500 FIGURE 3 Spatio-temporal evolution of 100 oscillators Eq. ( 4) coupled in the form of Eq. ( 5) with a--0.057,b---0.12,c 0.5, c2 0.1, 5 0.01.Logarithms of the concentrations of variables Y) were grey coded.(a) D.,, 0.07, D), 0.4, (b) Dx 0.055, D., 0.2.gradient, diffusive coupling results in dynamic de- coupling of otherwise identical oscillators.
For weaker coupling the Turing structure breaks up and makes way for a ribbon of chaotic behavior caught between the two oscillatory hems (Fig. 3(b)).Again the situation is such that if an attractor reconstruction would be tried from a single time series the astonished researcher could find three qualitatively different results (sinusoidal periodic, chaotic, relaxational periodic) depending on which time series she or he would use, even though the system has settled to a sin- gle well-defined attractor.Figure 4 shows two phase space projections of the trajectory.Vari- ables X1 and X98 (left side) flu a rectangular plane as if they were two independent periodic oscillators, whereas variables X51 and X53 (right side) form a familiar chaotic attractor as in the case of hyperchaos in diffusively coupled cells under Turing conditions [9].Due to the close relationship with the hyperchaos in [8] we suspect that the chaotic region of Fig. 3(b) possesses X53 X FIGURE 4 Phase space projection of two variables for the attractor in Fig. 3(b).Left: One variable from the high and low frequency oscillation, respectively.Right: Two variables from the chaotic region.more than one positive LCE but a reliable LCE estimation of the two-hundred dimensional system has yet to be performed.

HYPERCHAOTIC PATTERNS IN TWO SPATIAL DIMENSIONS
In our final example we employ Thron's oscilla- tor [10] which lacks autocatalysis and instead cre- ates the oscillatory instability of a fixed point by feedback inhibition in a three-variable scheme: with a, k, km > O.
We have recently found that diffusive coupling of three such cells creates a hyperchaotic attrac- tor [11].Now we go further and couple three times three cells on a square grid with next- neighbor couplings of each cell.For simplicity, periodic boundary conditions (toroidal topology) is chosen but, as in Section 3, the qualitative result can also be obtained with zero-flux boundaries.Calculating the spectrum of Lyapunov ex- ponents for the set of parameters: a--0.1, k 0.48, km --0.001 with coupling in variable Z (Dz 0.005, and Dx Dy 0) we find four posi- tive LCEs.For the same set of parameters we also observe hyperchaos on a toroidal grid com- posed of 10 10 oscillators Eq. ( 6). Figure 5(a) is a simulation of this system.The decay of cor- relation between neighboring cells due to the hyperchaos is reflected in the noncoherent irregular pattern of grey values.
In order to visualize the spatio-temporal pattern associated with a hyperchaotic attractor in more detail we keep the length of the system in Fig. 5(a) constant but cover the area with a grid of 100 100 oscillators with properly adjusted diffu- sion coefficient Dz. Figure 5(b) shows an aperiodic patterns at one instant during the simulation.The pattern is nonstationary and modulated by chao- tic oscillations in space and time.A full account of its structural richness necessarily has to include the temporal dimension.However, the strategy to find the 2D spatio-temporally hyperchaotic pat- tern starting from the dynamics of only a few coupled cells worked reliably as in the 1D cases.FIGURE 5 Two-dimensional grid of oscillators Eq. ( 6) with a 0.1, k4 0.001, kl 0.48.Each cell is diffusively coupled to its four next neighbors and the grid has periodic boundary conditions.Concentrations of variables Z!i were grey coded.(a) 10 x10 cells, Dx=D,=O, D=0.005, (b) 100 xl00 cells; Dx Dy 0, Dz 0.5.

DISCUSSION
Spatio-temporally hyperchaotic patterns are a generic feature of diffusively coupled biochemical oscillators.In particular, the case where two coupled oscillators possess a chaotic attractor can be exploited to generate hyperchaos in a linear chain or in a closed ring of oscillators.Just as hyperchaos can be created out of ordinary chaos by adding new variables with proper coupling as in Eq. (1), the dimension and the number of positive LCEs can be increased stepwise if the number of spatially coupled oscillatory cells is increased.In addition to the case where the diffu- sion coefficients of two oscillator variables differ by one magnitude, an explicit example could be found to demonstrate spatio.-temporalhyperchaos in a system with two equal diffusion coeffi- cients.Thus no principal requirements are imposed beforehand on the ratios of diffusion coefficients in experimental systems.
In systems with one spatial dimension hyper- chaotic patterns have now been observed in bio- chemical models of kinetic oscillations with second-order autocatalysis, as in Eq. (2), first- order autocatalysis, as in Eq. ( 4), or in a model with no autocatalysis at all, Eq. ( 6).The impor- tant feature of the oscillators is not necessarily their kinetic structure.For instance, equivalent hyperchaotic dynamics were described for two different autocatalytic models, one an activator- inhibitor system, the other a substrate-depletion system [8].The trick is to find a set of kinetic parameters which lead to at least two positive LCEs under diffusive coupling of a small number of cells (Eq. ( 3)).These parameters can then be used for the higher-dimensional cases with the coupling strength (diffusion coefficients) as bifur- cation parameter.
A new direction of investigation will now be to study the dynamics of nonidentical coupled cells.The obvious motivation for this is that biological or biochemical systems in general comprise a vari- ety of cells with a distribution of kinetic para- meters or coupling constants.A first step in that direction was the study of a one-dimensional chain of oscillators with a gradient of one parameter.In addition to the fact that hyperchaos can survive nonidentical parameters in coupled cells the possi- bility arises to combine different dynamic sub- structures in one composite pattern.In one case a nonhomogeneous fixed point (Turing) structure separated two oscillatory events of differing ampli- tude and frequency ("dynamic wall"), and in another case an aperiodic spatio-temporal pattern could be confined to a subset of the system by periodic neighbors (Figs. 2 and 3).Such composite structures may for example occur in experimental systems where the presence of frequency gradients has been established (see e.g.[12]).
Aperiodic patterns on two-dimensional grids indicate that the connection between low-dimen- sional hyperchaotic attractors and high-dimensional hyperchaotic spatio-temporal patterns can be expected in experimental systems with two spa- tial extensions, e.g. in thin liquid layers or in a single layer of biological cells.We expect that spatio-temporal patterns with low spatial correla- tion and rich structure can naturally arise in tissue of living cells with oscillatory characteristics.Recently we have observed spatio-temporal hyperchaos in diffusively coupled kinetic systems proposed for the explanation of cytosolic calcium oscillations by Goldbeter, Dupont and Berridge [131.
An important property of all systems presented is the fact that the high-dimensional patterns were generated in finite-dimensional systems.Di- rectly interpreted the equations model a finite number of (bio)chemical reactors with continu- ous supply of at least one (bio)chemical species.
They can thus easily be extended to describe a set of coupled continuous flow well-stirred tank reactors.To build and run such a set on a macroscopic scale appears tiresome.Yet in the case of enzymatic reactions one could think of a realization as a set of microreactors.In that case the assumption that all cells are identical and that a component is pumped at equal rate through all reactors is just.The technology of microreactors will allow the detailed investigation of high-dimensional chaotic attractors in finite- dimensional systems with an accuracy which can never be expected in living systems.The idea of coupled microreactors with full control of flows and coupling strengths leads us to speculate whether such a system could undergo structure-formation on an even higher level.Suppose the concentrations in each cell could be mea- sured by a sensor and the couplings between cells were adjustable.Then the coupling strength be- tween cells could easily be made a function of the exchange of (bio)chemicals (i.e.information) between cells.The coupling strengths could then be made functions of the dynamics similar to the weights in artificial neural networks.The result would be a chemical network capable of learning to interpret external patterns (provided e.g. in the form of flow rates) in terms of a set of adjusted coupling parameters.At present there seems to be no clue as to whether living systems use this prin- ciple of adjusted couplings of compartments to handle information processing in enymatic reac- tion networks.However, Bray has elaborated how the kinetics of metabolic reactions may relate to "learning" in neural networks [14].A proper set of coupled microreactors would be an ideal model experiment to test both the complex self-organiza- tion and the information processing capabilities of diffusively coupled biochemical reactors.