Why metabolic systems are rarely chaotic

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.

Introduction

In biology, it can be argued that virtually all chemical reaction steps are under some form of enzymatic control. This can be analysed to determine the rate of production of the different metabolites and then modelled using traditional Michaelis–Menten type kinetics (Murray, 2002). More recently, the total flux of a chemical pathway can be analysed using control analysis which allows the relative efficiencies of each intermediate step to be quantified (Fell, 1997). Even though these pathways are of tremendous complexity, they appear to be stable (Poolman et al., 2001). Chaos occurs in chemical (and physical) processes regularly and it seems that biochemical processes can either prevent it from occurring or have evolved to avoid chaotic domains.

Because there appears to be little indication for chaotic behaviour to be a particularly bad (or possibly good) survival trait, it is assumed that the biochemical catalytical process itself avoids chaotic domains by virtue of its specific organisation. Enzymatic processes control the rate of production of their substrates and it is suggested in this paper that this is a built-in safeguard that limits the system to stable dynamics.

It is assumed that the Michaelis–Menton model is sufficiently accurate to describe the rate of reaction which depends non-linearly on the substrate concentrations. The Michaelis–Menton model exhibits rate saturation at higher concentrations of the substrates and it is this concept that will be used to describe a rate control method for chaotic control.

$$v=\frac{S{V}_m}{S+K_m}$$

The Michaelis–Menton Eq. (1) describes a rate curve $v$ where S is the substrate concentration and $K_m$ the Michaelis constant with $V_m$ the limiting rate. For this equation, it is assumed that enzyme-substrate reactions, as part of the catalytical process, are so rapid that the process is at virtual equilibrium. This effectively means that the breakdown of the enzyme-substrate complex into the product is the rate limiting step of the catalysis (Fell, 1997). Although this is only a conceptual model of enzyme activity because it assumes that the rates are close to equilibrium, it seems to be adequate for most reaction rate analysis problems. The steady state assumption implied by this can only be true if indeed most of these problems are near steady-state or in a state which is indistinguishable from a steady state. This may seem to be an arbitrary distinction but its relevance can be more readily understood if it is considered that a system in steady-state and a controlled chaotic system are dynamically similar. A chaotic system that is under control of some effective control method, shows the same dynamical properties as a stable system because the control method forces the system to revert to more classical dynamics. Only careful analysis, and interference with the control method, will show the difference. This property of chaotic control is one of the reasons why chaotic control has been studied in great depth and has been used to control different systems (Garfinkel et al., 1992).

The aim of controlling chaotic dynamic systems is, generally, to stabilise specific points or orbits within the phase space. The chaotic nature of a system can thus be reduced to stable states. Different methods have been developed that are either variations on the OGY system of control Ott et al., 1990, Starrett, 2003 or delay control Pyragas, 1992, Pyragas, 1995, Pyragas, 2001. The OGY control methods require knowledge of the unstable periodic orbits (UPOs) contained in the attractor. Therefore, an analytical understanding of the chaotic system is necessary to control the system. The delay control method uses the control function $F(y)=K(y(t)-y(t-\tau ))$ which does not require any knowledge of the UPOs but it needs appropriate choices for the control constant K and the delay $\tau$. If the choices for K and $\tau$ are not correctly chosen then the system will not stabilise into an orbit. Note that some chaotic systems can not be stabilised using the single delay control method such as the Lorenz system (Nakajima, 1997) due to the fact that these contain negative Floquet exponents but these systems can be controlled by an extended delayed feedback control Socolar et al., 1994, Pyragas, 2001.

To study the possibility that metabolic processes contain a mechanism to prevent chaotic states from developing, two different biochemical kinetic models will be shown to be readily controlled using the novel rate control method described in this paper. Additionally, the principles of the rate control method of chaotic control are outlined using two classic chaotic models.