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  Scheper, T O 

Why metabolic systems are rarely chaotic.  Scheper, T O (2008) Why metabolic systems are rarely chaotic. BioSystems, 94, pp. 145‐152. doi: 10.1016/j.biosystems.2008.05.020   This version is available: http://radar.brookes.ac.uk/radar/items/42a6b7f6‐7dc9‐2dee‐5fc9‐d47f8ffe2cda/1/Available in the RADAR: November 2010 Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non‐commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.   This document is the preprint version of the journal article. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.  

Why Metabolic Systems Are Rarely Chaotic

Tjeerd olde ScheperDepartment of Computing, Oxford Brookes University, Wheatley Campus, Oxford OX33 1HX, UK

Abstract

One of the mysteries surrounding the phenomenon of chaos is that it can rarely be found in biological systems. This

has led to many discussions of the possible presence and interpretation of chaos in biological signals. It has caused

empirical biologists to be very sceptical of models that have chaotic properties or even employ chaos for problem

solving tasks. In this paper, it is demonstrated that there exists a possible mechanism that is part of the catalytical

reaction mechanisms which may be responsible for controlling enzymatic reactions such that they do not become

chaotic. It is proposed that where these mechanisms are not present or not effective, chaos may still occur in biological

systems.

Key words: Control of Chaos, Rate Control, Biological Complexity, Chemical Chaos

1. Introduction

In biology, it can be argued that virtually allchemical reaction steps are under some form of en-zymatic control. This can be analysed to determinethe rate of production of the different metabolitesand then modelled using traditional Michaelis-Menten type kinetics [1]. More recently, the totalflux of a chemical pathway can be analysed usingcontrol analysis which allows the relative efficienciesof each intermediate step to be quantified [2]. Eventhough these pathways are of tremendous complex-ity, they appear to be stable [3]. Chaos occurs inchemical (and physical) processes regularly and itseems that biochemical processes can either preventit from occurring or have evolved to avoid chaoticdomains.

Because there appears to be little indication forchaotic behaviour to be a particularly bad (or pos-sibly good) survival trait, it is assumed that thebiochemical catalytical process itself avoids chaotic

Email address: tvolde-scheper@brookes.ac.uk (Tjeerdolde Scheper).

domains by virtue of its specific organisation. En-zymatic processes control the rate of production oftheir substrates and it is suggested in this paper thatthis is a built-in safeguard that limits the system tostable dynamics.

It is assumed that the Michaelis-Menton model issufficiently accurate to describe the rate of reactionwhich depends non-linearly on the substrate con-centrations. The Michaelis-Menton model exhibitsrate saturation at higher concentrations of the sub-strates and it is this concept that will be used todescribe a rate control method for chaotic control.

v =S Vm

S + Km

(1)

The Michaelis-Menton equation (1) describes arate curve v where S is the substrate concentrationand Km the Michaelis constant with Vm the limiting

rate. For this equation, it is assumed that enzyme-substrate reactions, as part of the catalytical pro-cess, are so rapid that the process is at virtual equi-librium. This effectively means that the breakdownof the enzyme-substrate complex into the product is

Preprint submitted to Elsevier 23rd May 2008

the rate limiting step of the catalysis [2]. Althoughthis is only a conceptual model of enzyme activitybecause it assumes that the rates are close to equi-librium, it seems to be adequate for most reactionrate analysis problems. The steady state assumption

implied by this can only be true if indeed most ofthese problems are near steady-state or in a statewhich is indistinguishable from a steady state. Thismay seem to be an arbitrary distinction but its rele-vance can be more readily understood if it is consid-ered that a system in steady-state and a controlledchaotic system are dynamically similar. A chaoticsystem that is under control of some effective controlmethod, shows the same dynamical properties as astable system because the control method forces thesystem to revert to more classical dynamics. Onlycareful analysis, and interference with the controlmethod, will show the difference. This property ofchaotic control is one of the reasons why chaotic con-trol has been studied in great depth and has beenused to control different systems [4].

The aim of controlling chaotic dynamic systemsis, generally, to stabilise specific points or orbitswithin the phase space. The chaotic nature of a sys-tem can thus be reduced to stable states. Differentmethods have been developed that are either vari-ations on the OGY system of control [5,6] or delaycontrol [7,8,9]. The OGY control methods requireknowledge of the unstable periodic orbits (UPOs)contained in the attractor. Therefore, an analyticalunderstanding of the chaotic system is necessary tocontrol the system. The delay control method usesthe control function F (y) = K(y(t)−y(t−τ)) whichdoes not require any knowledge of the UPOs but itneeds appropriate choices for the control constantK and the delay τ . If the choices for K and τ arenot correctly chosen then the system will not sta-bilise into an orbit. Note that some chaotic systemscan not be stabilised using the single delay controlmethod such as the Lorenz system [10] due to thefact that these contain negative Floquet exponentsbut these systems can be controlled by an extendeddelayed feedback control [11,9].

To study the possibility that metabolic processescontain a mechanism to prevent chaotic states fromdeveloping, two different biochemical kinetic modelswill be shown to be readily controlled using the novelrate control method described in this paper. Addi-tionally, the principles of the rate control method ofchaotic control are outlined using two classic chaoticmodels.

2. Rate Control

In this section, a novel method of chaotic controlis presented that does not depend on a priori knowl-edge of the presence of unstable periodic orbits ina chaotic system. Additionally, it requires only thecurrent state of each of the variables by limiting therelative rate of expansion of each of those variables.An extensive analysis of this method (including ananalysis of the method in terms of the Lyapunov ex-ponents) is in preparation [12]. This will also demon-strate how the rate control method can overcomethe limitations of the single delay control method forspecific chaotic systems, such as the Lorenz system.

Consider the form of n linear rate equations thatgovern a particular dynamic system:

x1 = F1(x1) + G1(x2)

x2 = F2(x2) + G2(x2)

...

xn = Fn(xn) + Gn(xn)

In this case, the system depends on one of thefunctions F or G to grow or decrease at some rate.Additionally, either function may be non-linear andmay depend on more than one variable. If one con-siders the nature of a chaotic flow equation, in therange where stretching and folding occurs, i.e. wherethe local Lyapunov exponents tend to have at leastone positive value, the global behaviour is domi-nated by some of the n equations. These tend to bethe non-linear parts of the system that allow the sys-tem to expand at an exponential rate. The local rateof expansion is proportional to the local behaviourof each of the variables. If a chaotic system is nearan unstable periodic orbit, it can be kept near or onthis orbit by small proportional adjustments to oneor more of the variables. This is the principle of theOGY method of control. In rate control, the expo-nential expansion of the system away from the orbitis used to limit the rate of expansion, thereby pre-venting the chaotic system from leaving the orbit.Note that the rate control method does not neces-sarily target specific orbits.

To achieve a rate control of a chaotic system, therate of expansion can be estimated from the growthterms in each equation of the system. By determin-ing the current proportion of the variable of thespace it occupies, a measure for each variable can befound, e.g. for the Rossler system,

2

dx

dt= −(y + z) (2)

dy

dt= x +

y

α(3)

dz

dt=

β

α+ (z x) − (γ z) (4)

the growth term for equation (4) is (z x) (and anadditional constant term). The proportion of eachof these two variables to the growth rate is given by

qx =x

x + µx

and qx =z

z + µz

(5)

where µx and µz are constant. For this non-linearequation, the divergence of the two variable x andz determines the increase in rate of growth of the z

variable itself. Therefore, by limiting the (x z) termin proportion to the divergence rate, the systemshould be stabilised. To express this in more gen-eral terms, the following equation describes a genericrate control function for the divergence of x and z:

σ(x, z) = f eξ qx qz = f e

{

ξ(x z)

(x z + x + z + µ)

}

(6)

where µ is a constant and ξ and f are variable scalarsthat can be used to stabilise different orbits. Thisrate control function can be used in many differentchaotic systems of which two examples are presentedbelow before the rate control is applied to biochem-ical models. The rate control function is capable ofboth limiting the rate of expansion of a variable aswell as promoting the rate of expansion which canalso lead to the stabilisation of an unstable periodicorbit depending on the local dynamics.

3. Application of rate control to chaotic

systems

3.1. Rossler system

To demonstrate the rate control on a flow system,the Rossler system [13] was modified to include therate control function (6). The rate function may beapplied to all the three Rossler variables but it is suf-ficient to apply the rate control to the third z vari-able alone. The resulting modified rate controlledRossler system is then as follows:

σ(x, z) = f e

ξ(xz)

(x z + x + z + µ) (7)

dx

dt= −(y + z) (8)

dy

dt= x +

y

α(9)

dz

dt=

β

α+ (σ(x, z) z x) − (γ z) (10)

where the Rossler variables are α = 5, β = 1 andγ = 5.7 and where the rate control parameter ξ canbe variable but by default f = 1, ξ = −1 and µ =150. With any appropriately chosen value for µ, theparameter ξ may be varied resulting in different un-stable periodic orbits to become stabilised. In figure1 the evolution of the Rossler z variable is shownwhen ξ = −1.5 and with the rate control enabled attimestep 50. The system rapidly becomes stabilisedinto a stable one period. The rate control function σ

is shown in figure 2 and is constant at 1 until the z

variable increases rapidly, it then decreases in pro-portion to the change in z and thereby “slows” downthe z variable. This results in the system becomingstable periodic. By changing the rate parameter ξ,it is possible (whilst all other parameters and ini-tial conditions are the same) to stabilise differentunstable periodic orbits. In figure 3 are shown threeorbits that can be stabilised when the values for ξ

are −1.5,−1.0 and −0.4. From right to left can beseen the stable period 1 orbit for ξ = −1.5, two or-bits of the stable period 4 orbit for ξ = −0.4, theperiod 1 orbit when ξ = −1.0 and the other two or-bits of the period 4 for ξ = −0.4. It can be recog-nised from these two figures that the shape of theRossler attractor is stretched in the x and y planewhen ξ < 1.0. Otherwise, it has a shape similar tothe chaotic attractor, including stretching and fold-ing when x > 0 and y > 0.

3.2. Ikeda map

To demonstrate that the rate control method ap-plies to chaotic maps as well as flows, the rate con-trol is introduced in the Ikeda map. The Ikeda mapis a good example of a complex chaotic map thatcontains several points of nonlinearity [14]. By plot-ting the real and imaginary parts of the complexequation

z(n + 1) = a − b zneiκ−

iη

1 + |zn|2 (11)

3

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

Figure 1. Rossler z variable rate controlled into unstableperiodic orbit. Super-imposed, starting at value 1, is therate control function σ.

0 20 40 60 80 100 120 140 160 180 2000.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 2. Evolution of the control function σ for the Rosslermodel, here ξ = −1.5. The control is turned on at timet = 50.

with x = Re(z) and y = Im(z), the Ikeda map canbe described as

φ = κ −η

(1 + x2n + y2

n)(12)

xn+1 = a + b(xn cosφ − yn sin φ) (13)

yn+1 = b(xn sin φ + yn cosφ) (14)

with a = 1, b = 0.9, κ = 0.4 and η = 6.0. Therate control equation (6) is applied to both xn andyn by limiting the function φ (12). The function φ

therefore becomes

φ ′ = κ −η

(1 + σ(xn, yn)x2n + σ(xn, yn)y2

n)(15)

where the rate control parameters are chosen to beµ = 5 and ξ is variable.

−10 −5 0 5 10 150

2

4

6

8

10

12

14

−0.4

−1.5 −1.0

Figure 3. Phase space plot of x versus z for three differ-ent unstable periodic orbits within the Rossler attractor,ξ = −1.5 (period 1), ξ = −0.4 (period 4) and ξ = −1 (pe-riod 1) respectively from right to left.

Some resulting unstable periodic orbits are shownin figure 4 where three different orbits, two periodthree orbits and a period six orbit, are indicatedon top of the uncontrolled Ikeda map. Orbits arestabilised depending on the proximity of the systemto the orbit when control is enabled. The controlis initially disabled and becomes enabled after 5000iterative time steps. The resulting evolution of the x

variable of the Ikeda map for ξ = 2 is shown in figure5 where the map will very quickly stabilise into thethree orbit when control is enabled.

The rate control function σ that is applied to themap is shown in figure 6, where the control functionis equal to one when rate control is disabled andbecomes periodic when control is enabled and sta-bilised. Note that σ is close to one for two of the un-stable points, which implies that the local dynamicsis small (i.e. one of the local Lyapunov exponents issmall positive), not much rate control is needed toprevent the system from leaving the orbit. The thirdunstable point requires a relative large limitation ofthe expansion rate for σ = 0.4.

4. Control of a chemical oscillator

The relevance of rate control of chaos to the bio-chemical processes can be illustrated by introduc-ing rate control to the growth terms of a knownmulti-variable chemical model. No assumptions onthe properties of the control parameters are initiallymade but different values for the control parame-ters ξ and f , see equation (6), have been tested. Themodel described below is a three variable model that

4

−0.5 0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Figure 4. Ikeda attractor with three different unstable pe-riodic orbits, ξ = 2 (3 orbit, circles), ξ = 2.25 (3 orbit,squares) and ξ = 1.75 (six orbit, diamonds).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−0.5

0

0.5

1

1.5

2

Figure 5. Evolution of controlled x variable of the Ikedamap, control is turned on at 5000.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 6. Rate control variable σ of the Ikeda map, withcontrol enabled at time step 5000.

is derived from a two-variable autocatalator system[15]. It has six reaction steps with rates given bylaw-of-mass-action kinetics:

P → A Rate= k0 p0

P+C → A+C Rate= kc p0 γ

A → B Rate= ku α

A+2B → 3B Rate= k1 α β2

B → C Rate= k2 β

C → D Rate= k3 γ

This system is considered to be open with a con-stant precursor p0. In a dimensionless form the lastthree relevant rate equations can be written as:

d a

d t= u (k + c) − a − a b2 (16)

sd b

d t= a + a b2 − b (17)

dd c

d t= b − c (18)

where

a =

√

k1 ku

k22

α, b =

√

k1

ku

β, c =

√

k1 k23

ku k22

γ

(19)

are dimensionless concentrations and t = ku τ is di-mensionless time. The parameters d = 0.02, k = 65and s = 0.005 are dimensionless and derived fromthe original rate equations [15]. The parameter u

shows asymptotic behaviour when varied througha supercritical Hopf bifurcation at u = 0.016, fol-lowed by a period doubling cascade that begins atu = 0.143. For 0.1534 < u < 0.1551, the systemis chaotic [15]. For the subsequent simulations, thevalue of u = 0.154, well inside the chaotic range.

To introduce the rate control method, the propor-tional quantities qa, qb, qc are determined in equa-tions (20), (21) and (22). For variable a, the rate ofgrowth is given by the term u (k + c), for variableb the rate of growth is given by the term (a + a b2)and for variable c the rate of growth is given by theterm b. Apart from equation (17), the rate of growthis given by a single variable. It can therefore be ex-pected that the rate of growth of equation (17) maydiverged faster than the other two. The completerate control of these equations can now be formu-lated, including the rate equations σn, as follows.

5

qa =a

a + µa

(20)

qb =b

b + µb

(21)

qc =c

c + µc

(22)

σa(qc) = fa eξa qc (23)

σb(qa, qb) = fb eξb qa qb (24)

σc(qb) = fc eξc qb (25)

d a

d t= σa(u (k + c)) − a − a b2 (26)

sd b

d t= σb(a + a b2) − b (27)

dd c

d t= σc b − c (28)

Note that the rate control functions σn will beequal to 1 when control is disabled. When control isactive, it will vary around the value 1 depending onthe rate of expansion. For this model, µa = 1, µb =150, µc = 30 and the rate control parameters ξn andfn may be varied. In figure 7 is shown the chaoticattractor of the model when control is disabled (i.e.ξn = 0). In figure 8 is shown the effect of controlwhen ξn = −1 for the variable b of the chaotic chem-ical model. At (non-discrete) time 9 is the controlenabled and the system rapidly stabilises into a pe-riod one orbit. The corresponding control functionσb is shown in figure 9. Notice that the rate controlvaries between 0.92 and 1, i.e. the rate limiting func-tion only reduces the rate of expansion with a factorof < 0.08. This is nevertheless sufficient to stabilisethe chaotic system. The values of the rate parame-ters ξa, ξb and ξc can be varied and may result in thestabilisation of different unstable period orbits.

5. Control of an autocatalytic system

To demonstrate the dynamic effect of the rate con-trol mechanism on an different autocatalytic system,the control method is introduced into a model ofextracellular matrix degradation balance [16]. Thismodel, illustrated in figure 10, describes the cyclicalgeneration and degradation of extracellular matrixproteins including the autocatalytic degradation ofthe enzymes that drive this cycle. Protein-cleavingenzymes, proteases p, degradate the unsoluble ex-tracellular matrix m into soluble fragments f . Thesefragments f are converted back into unsoluble ma-trix by an enzyme transglutaminase g (and otherenzymes not included in this model). The proteasesp and transglutaminase g are therefore antagonistic

Figure 7. Chaotic attractor of the chemical oscillator model.

6 7 8 9 10 11 12

20

40

60

80

100

120

Figure 8. Control of variable b of the chemical oscillatormodel into unstable periodic one orbit, control is enabledat time step 9, ξn = −1.

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 100.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Figure 9. Rate control variable σb of the chemical oscillatormodel, with control enabled at time step 9, ξn = −1.

6

Cells

Proteinase p

Extracellular matrix(unsoluble proteins)

m

Proteolysis fragments

(soluble)f

Transglutaminase g

Cells

rim p

p

p

⊕

⊕

Figure 10. Scheme of the bienzymatic cyclic model of the extracellular matrix degradation balance (after Berry [16]).

in building and degenerating the matrix. The ma-trix m is created de novo at a constant rate rim;the presence of the fragments f will stimulate theproduction of both enzymes p and g by surroundingcells. These cells detect the presence of the intercel-lular fragments which will stimulate the productionof p and g. All proteins are subject to degradationby the proteases p, including p itself. The model canbe described by the dimensionless equations (29),(30), (31) and (32).

dm

d t= kg

f g

KG + f−

m p

1 + m+ rim (29)

d f

d t= −kg

f g

KG + f+

m p

1 + m−

f p

1 + f(30)

d p

d t= α

fn

KnR + fn

− ka p2 (31)

d g

d t= β

f l

K lS + f l

− kdeg

g p

Kdeg + g(32)

Here the parameters n and l in equations (31) and(32) are the Hill-numbers for the two enzymes wheren = l = 4. α and β are the proteases and trans-glutaminase synthesis rate respectively and are as-sumed to be much smaller than the turnover ratesfor the matrix m and fragments f . The model pa-rameters are derived from exerimental data and for

the following experiments the parameters were set asα = 0.026, β = 0.00075, KG = 0.1, Kdeg = 1.1, kg =

kdeg = 0.05 and ka =kdeg

Kdeg= 0.0455. For differ-

ent values of rim the model exhibits a wide range ofdynamic behaviour, including periodic cycles, bista-bility and chaos. For the rate control experiments,the values of rim used were rim = 0.0098 for chaoticdynamics, rim = 0.00990 for bistability and rim =0.00995 for a period-6 cycle [16].

The rate control equation (6) can be introducedseperately in each or all of the equations of the au-tocatalytic model. If it is introduced into equation(29) or (30), the rate control will successfully sta-bilise the system into a periodic orbit. More inter-esting consequences of the rate control mechanismcan be demonstrated by introducing the rate controlequation in either the synthesis of the proteases p

(31) or the synthesis of the transglutaminase g (32)or both. The modifications necessary to include ratecontrol in these equations are shown in equations(33) to (37).

Because both the synthesis of p and g depend onthe presence of the protein fragments f , the controlis proportional to the rate of expansion of the phasespace in the direction of f only. This is sufficientto control the system into a stable 4-orbit which isshown superimposed in figure 11 and in figure 12

7

0.5 0.6 0.7 0.8 0.9 1 1.12

3

4

5

6

7

8

9

10

11

12

Figure 11. Phase space f versus m of the chaotic attractor(dotted) of the autocatalytic model with a 4-period con-trolled orbit superimposed (continuous line).

where is plotted f versus time. Parameter values forthe control equations are fp = 0.5, fg = 0.5, ξp =0.01, ξg = −0.01, µf = 1. In figure 13 is shown thevalues of the control equations σp (34) and σg (35)that stabilise the autocatalytic model into the 4-orbit. Very small changes to the rate of expansionof the enzyme synthesis rates is sufficient to pre-vent the model from becoming chaotic. This demon-strates that the periodic stabilisation of the extracel-lular matrix degradation balance can be controlledeffectively and efficiently by the surrounding cellsby monitoring the presence of the protein fragmentsand adjusting the rate of synthesis of the two en-zymes. The rate control method ensures that thesurrounding cells need not supervise all the pro-cesses outside the cell but can control these by mon-itoring only one aspect of the degradation balance.Even if the model parameters change due to a changein the balance, the rate control method can preventthe recurence of chaotic dynamics.

qf =f

f + µf

(33)

σp(qf ) = fp eξp qf (34)

σg(qf ) = fg eξg qf (35)

d p

d t

′

= σp(qf )αfn

KnR + fn

− ka p2 (36)

d g

d t

′

= σg(qf )βf l

K lS + f l

− kdeg

g p

Kdeg + g(37)

For example, if other surrounding cells increasethe rate of de novo generation of unsoluble extracel-lular matrix proteins, which could lead to a bifur-

0 0.5 1 1.5 2 2.5 3

x 105

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Figure 12. Controlled orbit of f in time of the autocatalyticmodel with initial chaotic transient.

0 0.5 1 1.5 2 2.5 3

x 105

0.997

0.998

0.999

1

1.001

1.002

1.003

Figure 13. Control functions σp (top) and σg (bottom) forthe autocatalatic model.

cation from chaos to bistability (or peridicity), thesame control method can maintain the periodic dy-namics at the cost of a change in period. This alsorequires a very small increase in the control param-eters to maintain effective control ξp = 0.02, ξg =−0.02. By increasing the rate of synthesis of the ex-tracellular matrix rim from 0.0098 to 0.00995, themodel changes dynamics from chaos to a period 6-orbit. This is shown in figure 14 where the extra-cellular matrix production rate is increased at thedashed line, switching the dynamics of the modelfrom a stabilised unstable periodic orbit to a dif-ferent periodic orbit. If the model is uncontrolled,the system would cycle in a 6-period as is shown infigure 15. In this figure, the uncontrolled behaviour

8

3 3.5 4 4.5 5 5.5 6 6.5 7

x 105

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Figure 14. Controlled periodic orbits of the autocatalyticmodel, where the model switches from controlled chaotic toperiodic dynamics at the dashed line.

Figure 15. Phase space of the autocatalatic model showingthe uncontrolled 6-orbit (thin line) and the controlled orbit(thick line).

of the model is shown with rim = 0.00995 (thinline) and the controlled dynamics with the same pa-rameters is shown superimposed (thick line). Thisdemonstrates clearly that the control modifies thedynamic behaviour of the model to remain periodicwhich does not dependent on the uncontrolled sys-tem being either chaotic or periodic. This impliesthat extracellular processes can be readily controlledby the surrounding cells without the need for com-plex feedback or other control mechanisms.

6. Conclusion

The rate control of chaotic systems is a novel andeffective method to stabilise unstable periodic or-bits contained in chaotic systems. It does not requirea priori knowledge of the UPOs contained in thesystem, however, it requires access to some of thesystem variables and the rate of change of some, orall, of those variables. Finding and stabilising dif-ferent UPOs can easily be achieved experimentally.The mechanism is only active (in the sense thatthe control function σ is significantly different fromone) when the variables in the rate control equation(6) are changing rapidly. This means that when thechaotic system is not near a folding or stretchingmanifold the control is inactive. The proportionalchange of the variables and therefore the local shapeof the attractor are the only limited elements of ratecontrol. It may be possible to include a scalar to de-termine the relative contribution of the rate limitingfunction to the chaotic system. This may be used todirect a chaotic system towards any unstable orbitwhich can then be stabilised using other methods,e.g. external periodic input or delay control.

The described method of rate limiting control canbe argued to be similar to other rate controllingmechanisms such as enzymatic control of biochemi-cal processes. In such systems the flux of the productis determined by the relative control of each of theenzymatic controlled steps [2]. Such a mechanismwould prevent the occurrence of chaotic dynamicsby preventing the system to become chaotic. A bi-furcation doubling cascade, for example, would leadto a stable (controlled unstable periodic orbit) pe-riod until the parameter has been changed back intothe stable domain.

The control of the autocatalytic model shows thatthe rate control mechanism is very effective at con-trolling the dynamics of an extracellular reactionbalance without the need to control each variable. Itmonitors the rate of expansion of the protein frag-ments which then influences the rate of synthesisof the enzymes that drive the system. Only smallmodifications are required to achieve a high level ofcontrol and will remain in stable periodic dynam-ics even if external parameters are modified (albeitwith different periods).

Modelling the rate control of chaos on a chemicalchaotic oscillator does not yet prove that the mecha-nism for rate control of chaos and the enzymatic ratecontrol function perform a similar task of prevent-

9

ing the occurance of chaos. The two mechanisms ofthe reaction rate control are not directly related butboth show similar properties. More extensive mod-elling of rate control of different chemical oscilla-tors and better simulation of the enzymatic reactionsteps may provide some ideas.

7. Future work

Attempts to control some different chaotic sys-tems have been successful (the Duffing system,Henon map and Hindmarsh-Rose system) but itappears that some variables in a system are moreeffective at controlling the chaotic dynamics thanothers. This is more obvious in simple system, suchas Duffing’s model and the Henon map, but inmore complex dynamic systems this is not readilyevident. A systematic analysis of the relative contri-butions of rate control of different system variablesmay be required to determine if this effect is due tothe control method or to properties of the chaoticsystem itself. The strength of the control parame-ters provides a useful indication of the efficacy ofthe method for a given set of equations.

If the rate control of different system variables in-deed result in variable control of the chaotic systemthis may give some clue to identifying which vari-ables are more effective rate limiting than others ascan be seen in the rate control of the autocatalyticmodel. This can give an indication of the relevanceof a particular enzyme or metabolite for the overallprocess. It may also be the case that the total con-trol of the chaotic system is distributed over all ratevariables in different proportions. This mechanismcould have evolved readily in biological systems andwould appear to be a simple way to eliminate un-desirable chaos. In those pathways or mechanismswhere rate control is not applicable, chaotic statesmay still be found. For example, even though thereare rate control mechanisms limiting the ionic flowthrough membrane channels, these may not be ableto control the transmembrane potential as previ-ously has been assumed. It has already been shownthat in pathological cases, neural activity can showchaotic activity [17]. It has been suggested, by dif-ferent authors [18,19,20,21,22] that chaos is a possi-ble means of information processing. Understandingthe rate control methods of normal pathways mayhelp in determining if this hypothesis is viable.

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