Gerold Baier and Sven Sahle- Spatio-Temporal Patterns with Hyperchaotic Dynamics in Diffusively Coupled Biochemical Oscillators

8/3/2019 Gerold Baier and Sven Sahle- Spatio-Temporal Patterns with Hyperchaotic Dynamics in Diffusively Coupled Biochem

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Discrete Dynamics in Nature and Society, Vol. 1, pp. 161-167Reprints available directly from the publisherPhotocopying permitted by license only

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Spatio-Temporal Patterns with Hyperchaotic Dynamics inDiffusively Coupled Biochemical Oscillators

GEROLD BAIER* and SVEN SAHLEInstitute fo r Plant Biochemistry, University of Tfibingen, D-72076 Tfibingen, Germany

(Received 9 October 1996)

We present three examples how complex spatio-temporal patterns can be l inked tohyperchaotic attractors in dynamical systems consisting of nonlinear biochemical oscilla-tors coupled linearly with diffusion terms. The systems involved are: (a) a two-variableoscillator with two consecutive autocatalytic reactions derived from the Lotka-Volterrhscheme; (b ) a minimal two-variable oscillator with one first-order autocatalytic reaction;(c) a three-variable oscillator with first-order feedback lacking autocatalysis. The dy-namics of a finite number of coupled biochemical oscillators may account for complexpatterns in compartmentalized living systems like cells or tissue, and may be testedexperimentally in coupled microreactors.

Keywords: Biochemical oscillations, Spatio-temporal hyperchaos, Complex patterns, Informationprocessing

1 INTRODUCTIONComplex biochemical spatio-temporal self-orga-nization with high-dimensional chaotic attractorscannot only occur in continuous reaction-diffu-sion systems with infinitely many degrees of free-dom in phase space. The discovery that lineardiffusive coupling of only two oscillatory reactionschemes can produce chaos [1 ] nurtures the ideathat finitely many coupled oscillators can be em-ployed to generate spatio-temporal patterns withunderlying 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 positiveLyapunov characteristic exponents (LCEs), i.e.hyperchaos, in three examples of coupled bio-chemical oscillators.

Following the introduction of the prototypicflow with a hyperchaotic attractor we first de-monstrate hyperchaos of coupled oscillators inone spatial dimension where the two coupled spe-cies have equal diffusion coefficients. The secondexample illustrates the transition from a stableTuring structure to spatio-temporal hyperchaos,both spatially trapped between two oscillatory

Corresponding author.161


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162 G. BAIER AND S. SAHLE

regimes of differing amplitude and frequency in a in two directions in phase space together withparameter gradient. The third example shows the the flow being folded back in two orthogonaltransition from low to high-dimensional hyper- directions. The numerical calculation of the spec-chaos in a two-dimensional grid of biochemical trum of Lyapunov exponents yields two positiveoscillators. LCEs. A Poincar6 cross section of a numericallysimulated hyperchaotic attractor of Eq. (1) is

shown in Fig.2 ABSTRACT HYPERCHAOS Equation (1) solves the mystery of R6sslersfirst equation for hyperchaos [4]. That pioneer-To introduce hyperchaos we present a four-vari- ing system possessed a hyperchaotic attractorable 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 twoequations it turns out that only the coupling ofJ2- J(l- Z, (1) expanding variable Y to the chaotic subsystem

I;" Z + ae Y, is different. Coupling the two linear subsystems2=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 prototypicFor bt 1,b2--0, the subsystem (X|,Xe, Z) for chaotic flows. Incidentally, the system ofis R6sslers equation for continuous chaos. Eq. (1) can be transformed into the simplestParameter al can be tuned to change the sys- hyperchaotic flow [5 ] (containing only one non-tems stable attractor from a single-loop limit linearity) by means of a mode transformation ofcycle via a period-doubling sequence to a chao- the linear variables [6]. Thus, the extensions totic attractor. The stretching and folding of the chaos with even more than two directions offlow is realized with the switching nonlinearity mean divergence work naturally as in [5].in the equation for Z [3]. For b 0, be 1, thesubsystem (Y,Z) forms an attracting limit cyclewhen a2 exceeds the value of the Hopf bifurca-tion. As this limit cycle works with the samenonlinearity as the first subsystem, one can viewsuch a structure as the oscillatory source of thechaotic 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 (XI,X2) plane, vBoth subsystems keep expanding for small valuesof variable Z du e to nonzero expansion para-meters a and a2, respectively. The two switchingnonlinearities in dZ/dt then increase the valuesof variables X2 and Y, and return the value of 0Z sharply close to zero whenever the threshold -0,3 x, 3,1is passed. We thus have an explicit implementa- FIGURE Poincar6 cross-section of hyperchaotic flow intion of the principle of an oscillator expanding Eq. (I) taken at x2 0 with al 0.0, a2 --0.3, DI b2 1.


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COUPLED BIOCHEMICAL OSCILLATORS 163

SPATIO-TEMPORAL HYPERCHAOSWITH TWO EQUAL DIFFUSIONCOEFFICIENTS

We now describe the occurrence of spatio-tempor-al hyperchaos in diffusively coupled (bio)chemicaloscillators. The first oscillator employed is a che-mical realization of the (Y , Z) limit cycle subsys-tem (b --0) of Eq. (1). The rate equations are:

" 0.01 + bX( Y- d),= Y- bXY+ aY 2 (2)with a,b,d > O.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 extendedby the instability-inducing second-order autocata-lysis governed by parameter a. Homogeneous os-cillating units with the kinetics of Eq. (2) cannow be coupled linearly by diffusive terms forboth variables as follows:

J( fx + Dx(X2 X),fy + Dy(Y2 Y1),

J(i fx + Dx(Xi+ 2Xi + Xi-1 ),i f + Dy Yi+ 2 Yi + Yi-kU --ix -- Dx(XN-1 Xu),"N --.fy -!- Dy( YN-1- YN) (3)with i= 2, 3,..., (N- 1).fx and fy denote kinetic

terms, e.g. as in Eq. (2); Dx and Dy are therespective diffusion coefficients of variables Xand Y; and N is the number of cells. The schemeis written with zero-flux boundary conditions, i.e.for an open chain of reaction cells. However, theresults presented can qualitatively be achievedwith periodic boundary conditions as well.

Previous results on hyperchaos with a finitenumber of diffusively coupled cells were obtainedwith a ratio of diffusion coefficients considerablysmaller or larger than one (see e.g. [2,7]). In con-

10 0

M= 80FIGURE 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 variablesXi were grey coded from -4 (white) to 2 (black).

trast, we now couple five cells of the oscillator inEq. (2) in the oscillatory parameter domain withequal diffusion coefficients, i.e. a ratio Dx/Dy 1.For a coupling strength of Dx D 0.05 weobserve a chaotic attractor with the four largestLCEs (0.2, 0.1,0,- 0.3). The attractor is hyperch-aotic with two positive LCEs. Under otherwiseidentical parameters the hyperchaotic attractoralso exists for a ratio of diffusion coefficientsslightly smaller or slightly larger than one. In -creasing the number of coupled cells leads tohyperchaotic 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 hypercha-otic regime. Again the hyperchaos exists for ratiosDx/Dy slightly smaller and slightly larger thanone. Thus, in addition to the cases Dx > D, andDx < D, spatio-temporal hyperchaos has beenfound to exist generically for the caseDx/Dy 1.

TURING PATTERN AND HYPERCHAOSTRAPPED BETWEEN TWO LIMITCYCLES

The simplest biochemical oscillator with two vari-ables contains only one first-order autocatalysis:

fi(-- a + XY- c1X/(c2 -] - X),--b-XY (4)


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164 G. BALER AND S. SAHLE

with a,b, cl,c2 > 0. This system has been foundto produce spatio-temporal hyperchaos forDx > D), [2 ] and for Dx < D), [8 ] in the arrange-ment of Eq. (3). In the latter case, i.e. underTuring conditions, the hyperchaos is a chaoticmixing of unstable Hopf and Turing modes. Wehave studied the dynamics of diffusively coupledoscillators Eq. (4) in a gradient of parameter a:

with i= 2, 3,..., (N- 1) .The gradient in the input to variable X was

chosen such that the first cell of the decoupledsystem oscillates in a small-amplitude sinusoidalfashion and the last oscillator possesses a large-amplitude relaxational limit cycle with lower fre-quency (parameters are given in the caption ofFig. 3). For strong coupling under Turing condi-tions (Dx < Dy) the system Eq. (5) with kineticterms taken from Eq. (4) organizes itself into astable asymmetric Turing pattern, e.g. for N 100.Decreasing the coupling strength leads to the ap-pearance of oscillating behavior at both spatialends of the system. Figure 3(a) is a space-timeplot of this situation. A stationary Turing patternis located in the central area. It is hemmed in byoscillatory cells. Interestingly, the oscillations arerelated to those of the uncoupled system andthus differ from each other in amplitude and fre-quency. The Turing structure dynamically sepa-rates the two oscillatory domains. This is aremarkable situation as it seems impossible toextract information about the distant oscillatorfrom a time series of one oscillator if only finiteprecision 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 concentrationsof 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 structurebreaks up and makes way for a ribbon of chaoticbehavior caught between the two oscillatoryhems (Fig. 3(b)). Again the situation is such thatif an attractor reconstruction would be tried froma single time series the astonished researchercould find three qualitatively different results (si-nusoidal periodic, chaotic, relaxational periodic)depending on which time series she or he woulduse, even though the system has settled to a sin-gl e well-defined attractor. Figure 4 shows twophase space projections of the trajectory. Vari-ables X1 and X98 (left side) flu a rectangularplane as if they were two independent periodicoscillators, whereas variables X5 1 and X53 (rightside) form a familiar chaotic attractor as in thecase of hyperchaos in diffusively coupled cellsunder Turing conditions [9]. Due to the closerelationship with the hyperchaos in [8 ] we suspectthat the chaotic region of Fig. 3(b) possesses


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X53

XFIGURE 4 Phase space projection of two variables for the attractor in Fig. 3(b). Left: One variable from the high and lowfrequency oscillation, respectively. Right: Two variables from the chaotic region.

more than one positive LCE but a reliable LCEestimation of the two-hundred dimensional sys-tem has yet to be performed.

HYPERCHAOTIC PATTERNS IN TWOSPATIAL DIMENSIONS

In our final example we employ Throns oscilla-tor [10] which lacks autocatalysis and instead cre-ates the oscillatory instability of a fixed point byfeedback inhibition in a three-variable scheme:

f( a/ (k m + Z) X,f =x-r,2 + Z)

(6)

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 threetimes 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 bound-aries. 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 wealso 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 du e to the hy-perchaos is reflected in the noncoherent irregularpattern of grey values.

In order to visualize the spatio-temporal patternassociated with a hyperchaotic attractor in moredetail we keep the length of the system in Fig. 5(a)constant but cover the area with a grid of100 100 oscillators with properly adjusted diffu-sion coefficient Dz. Figure 5(b) shows an aperiodicpatterns at one instant during the simulation. Thepattern is nonstationary and modulated by chao-tic oscillations in space and time. A full accountof its structural richness necessarily has to includethe temporal dimension. However, the strategy tofind the 2D spatio-temporally hyperchaotic pat-tern starting from the dynamics of only a fewcoupled cells worked reliably as in the 1D cases.


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166 G. BAIER AND S. SAHLE

a)

b)

FIGURE 5 Two-dimensional grid of oscillators Eq. (6) witha 0.1, k4 0.001, kl 0.48. Each cell is diffusively coupledto its four next neighbors and the grid has periodic boundaryconditions. Concentrations of variables Z! i were grey coded.(a) 10 x10 cells, Dx=D,=O, D=0.005, (b) 100 xl00cells; Dx Dy 0, Dz 0.5.

6 DISCUSSIONSpatio-temporally hyperchaotic patterns are ageneric feature of diffusively coupled biochemicaloscillators. In particular, the case where twocoupled oscillators possess a chaotic attractorcan be exploited to generate hyperchaos in a linearchain or in a closed ring of oscillators. Just ashyperchaos can be created out of ordinary chaos

by adding new variables with proper coupling asin Eq. (1), the dimension and the number ofpositive LCEs can be increased stepwise if thenumber of spatially coupled oscillatory cells isincreased. In addition to the case where the diffu-sion coefficients of two oscillator variables differby one magnitude, an explicit example could befound to demonstrate spatio.-temporal hyper-chaos in a system with two equal diffusion coeffi-cients. Thus no principal requirements areimposed beforehand on the ratios of diffusioncoefficients in experimental systems.

In systems with one spatial dimension hyper-chaotic patterns have now been observed in bio-chemical models of kinetic oscillations withsecond-order autocatalysis, as in Eq. (2), first-order autocatalysis, as in Eq. (4), or in a modelwith no autocatalysis at all, Eq. (6). The impor-tant feature of the oscillators is not necessarilytheir kinetic structure. For instance, equivalenthyperchaotic dynamics were described for twodifferent autocatalytic models, one an activator-inhibitor system, the other a substrate-depletionsystem [8]. The trick is to find a set of kineticparameters which lead to at least two positiveLCEs under diffusive coupling of a small numberof cells (Eq. (3)). These parameters can then beused for the higher-dimensional cases with thecoupling 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 biologicalor biochemical systems in general comprise a vari-ety of cells with a distribution of kinetic para-meters or coupling constants. A first step in thatdirection was the study of a one-dimensional chainof oscillators with a gradient of one parameter. Inaddition to the fact that hyperchaos can survivenonidentical parameters in coupled cells the possi-bility arises to combine different dynamic sub-structures in one composite pattern. In one case anonhomogeneous fixed point (Turing) structureseparated two oscillatory events of differing ampli-tude and frequency ("dynamic wall"), and in


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another case an aperiodic spatio-temporal patterncould be confined to a subset of the system byperiodic neighbors (Figs. 2 and 3) . Such compositestructures may for example occur in experimentalsystems where the presence of frequency gradientshas been established (see e.g. [12]).

Aperiodic patterns on two-dimensional gridsindicate that the connection between low-dimen-sional hyperchaotic attractors and high-dimen-sional hyperchaotic spatio-temporal patterns canbe expected in experimental systems with two spa-tial extensions, e.g. in thin liquid layers or ina single layer of biological cells. We expect thatspatio-temporal patterns with low spatial correla-tion and rich structure can naturally arise in tissueof living cells with oscillatory characteristics. Re-cently we have observed spatio-temporal hyper-chaos in diffusively coupled kinetic systemsproposed for the explanation of cytosolic calciumoscillations by Goldbeter, Dupont and Berridge[131.An important property of all systems presentedis the fact that the high-dimensional patternswere generated in finite-dimensional systems. Di-rectly interpreted the equations model a finitenumber of (bio)chemical reactors with continu-ous supply of at least one (bio)chemical species.They can thus easily be extended to describe aset of coupled continuous flow well-stirred tankreactors. To build and run such a set on amacroscopic scale appears tiresome. Yet in thecase of enzymatic reactions one could think of arealization as a set of microreactors. In that casethe assumption that all cells are identical andthat a component is pumped at equal ratethrough all reactors is just. The technology ofmicroreactors will allow the detailed investigationof high-dimensional chaotic attractors in finite-dimensional systems with an accuracy which cannever be expected in living systems.The idea of coupled microreactors with full

control of flows and coupling strengths leads us tospeculate whether such a system could undergostructure-formation on an even higher level. Sup-pose the concentrations in each cell could be mea-

sured by a sensor and the couplings between cellswere adjustable. Then the coupling strength be-tween cells could easily be made a function of theexchange of (bio)chemicals (i.e. information)between cells. The coupling strengths could thenbe made functions of the dynamics similar to theweights in artificial neural networks. The resultwould be a chemical network capable of learningto interpret external patterns (provided e.g. in theform of flow rates) in terms of a set of adjustedcoupling parameters. At present there seems to beno clue as to whether living systems use this prin-ciple of adjusted couplings of compartments tohandle information processing in enymatic reac-tion networks. However, Bray has elaborated howthe kinetics of metabolic reactions may relate to"learning" in neural networks [14]. A proper setof coupled microreactors would be an ideal modelexperiment to test both the complex self-organiza-tion and the information processing capabilitiesof diffusively coupled biochemical reactors.

AcknowledgementsWe thank Peter Plath for inspiration and discus-sion; Thomas Kirner, Peter Strasser, Otto R6sslerand Jack Hudson for discussion; and theDeutsche Forschungsgemeinschaft and the Fondsder Chemischen Industrie for financial support.

References

[10][lll[121[13][141

[1 ] O.E. R6ssler (1976), Z. Naturforsch. 31a, 1168.[2 1 G. Baier, S. Sahle, U. Kummer and R. Brock (1994),Z. Naturforsch. 49a, 835.[3] O.E. R6ssler (1976), Phys. Lett. A 57, 397.

[4] O.E. R6ssler (1979), Phys. Lett. A 71, 155.[5 ] G. Baier and S. Sahle (1995), Phys. Rev. E 51, R2712.[6] M. Btinner and T. Meyer, private communication.[7 ] G. Baier, P. Strasser and U. Kummer (1995), Z.

Naturjbrsch. 50a, 1147.[8 ] P. Strasser, O.E. R6ssler and G. Baier (1996), J. Chem.

Phys. 104, 9974.[9 ] P. Strasser (1995), diploma thesis, University of Ttibingen[in German].C.D. Thron (1991), Bull. Math. Biol. 53, 383.

G. Baler and S. Sahle (1997), J. Theor. Biol., submitted.E. Diamant and A. Bortoff (1969), Am. J. Physiol. 216,301.G. Baier, J.-P. Chen and A.A. Hoff [unpublished results].D. Bray (1990), J. Theor. Biol. 143, 215.


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Gerold Baier and Sven Sahle- Spatio-Temporal Patterns with Hyperchaotic Dynamics in Diffusively Coupled Biochemical Oscillators