A theoretical treatment of damped oscillations in the transient state kinetics of single-enzyme reactions

An extension of the available kinetic theory for reactions in the transient state is presented which establishes that single-enzyme reactions may exhibit damped oscillations under the conditions of standard kinetic experiments performed by stopped-flow techniques. Such oscillations may occur for reasonable magnitudes of rate constants in the enzymic reaction mechanism and at physiological concentrations of enzyme and substrate. In the simplest reaction systems, the oscillations will be strongly damped and lead to progress curves resembling those of a reaction governed by standard exponential transients; statistical regression methods may then have to be applied for their detection and characterization. The observation that single-enzyme reactions may exhibit oscillatory behaviour points to a previously unrecognized possible source of the damped oscillations observed in metabolic systems such as the pathways of glycolysis or photosynthesis.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

Identification of possible two-reactant sources of oscillations in the Calvin photosynthesis cycle and ancillary pathways

A systematic search for possible sources of experimentally observed oscillations in the photosynthetic reaction system has been performed by application of recent theoretical results characterizing the transient-state rate behaviour of metabolic reactions involving two independent concentration variables. All subsystems involving two independent reactants in metabolically fundamental parts of the Calvin cycle and the ancillary pathways of starch and sucrose synthesis have been examined in order to decide on basis of their kinetic and stoichiometric structure whether or not they may trigger oscillations. The results show that no less than 20 possible oscillators can be identified in the examined reaction system, only three of which have been previously considered as potential sources of experimentally observed oscillations. This illustrates the superiority of the method now applied over those previously used to identify possible two-reactant sources of metabolic oscillations and indicates that there should be no difficulty in complex metabolic pathways to point to a multitude of interactions that may trigger an oscillatory rate behaviour of the system.     

A Theoretical Graph Method for Search and Analysis of Critical Phenomena in Biochemical Systems. II. Kinetic Models of Biochemical Oscillators Including Two and Three Substances

Four kinetic models of hypothetical complex reactions containing minimal two-substance or three-substance oscillators were constructed on the basis of the graphical rules suggested in the preceding work. The kinetic models are thought to be a part of one of four general biochemical systems: 1) system of mutual protein phosphorylation/dephosphorylation; 2) autophosphorylation of multisubunit protein; 3) association/dissociation of proteins or protein-containing structures during protein–protein or protein–ligand interaction; and 4) two-substrate enzymatic reaction with substrate inhibition by one substrate. Graphical rules of oscillator association with surrounding medium were considered. The graphical criteria of the oscillation generator elimination and criteria of oscillation damping were obtained. Both damped and undamped oscillations of reaction components were obtained by numerical integration of the mathematical models of these reactions. The areas of changes of model parameters and variables, within which the oscillations exist, were found.     

On the dynamics of controlled metabolic network and cellular behavior

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negetive and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.     

Chemical oscillations arise solely from kinetic nonlinearity and hence can occur near equilibrium.

A minimal kinetic scheme for a system displaying sustained chemical oscillations is presented. The system is isothermal, and all steps in the scheme are kinetically reversible. The oscillations are analyzed and the crucial points elucidated. Both positive and negative feedback, if properly introduced, support oscillations, provided the state responsible for feedback is optimally buffered. It is shown that the requisite nonlinearity is introduced at the kinetic level because of feedback regulation and not, as is usually assumed, by large affinities that introduce nonlinearity at the thermodynamic level. Hence, sustained oscillations may occur near equilibrium.     

Structure and kinetic behavior of linear multicompartment systems

The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system. This work was supported by U.S. Public Health Service Research Grant NS 08710 from the National Institute of Neurological Diseases and Stroke.     

Spatial eigenmodes and synchronous oscillation: co-incidence detection in simulated cerebral cortex

 Zero-lag synchronisation arises between points on the cerebral cortex receiving concurrent independent inputs; an observation generally ascribed to nonlinear mechanisms. Using simulations of cerebral cortex and Principal Component Analysis (PCA) we show patterns of zero-lag synchronisation (associated with empirically realistic spectral content) can arise from both linear and nonlinear mechanisms. For low levels of activation, we show the synchronous field is described by the eigenmodes of the resultant damped wave activity. The first and second spatial eigenmodes (which capture most of the signal variance) arise from the even and odd components of the independent input signals. The pattern of zero-lag synchronisation can be accounted for by the relative dominance of the first mode over the second, in the near-field of the inputs. The simulated cortical surface can act as a few millisecond response coincidence detector for concurrent, but uncorrelated, inputs. As cortical activation levels are increased, local damped oscillations in the gamma band undergo a transition to highly nonlinear undamped activity with 40 Hz dominant frequency. This is associated with ``locking' between active sites and spatially segregated phase patterns. The damped wave synchronisation and the locked nonlinear oscillations may combine to permit fast representation of multiple patterns of activity within the same field of neurons. Received: 20 January 1999 / Revised version: 22 September 2000 / Second revised version: 20 December 2001 / Published online: 26 June 2002     

Selection advantage of metabolic over non-metabolic replicators: A kinetic analysis

A kinetic analysis and simulation of the replication reactions of two competing replicators—one non-metabolic (thermodynamic), the other metabolic, are presented. Our analysis indicates that in a rich resource environment the non-metabolic replicator is likely to be kinetically selected for over the metabolic replicator. However, in the more typical resource-poor environment it will be the metabolic replicator that is the kinetically more stable entity, and the one that will be kinetically selected for. Accordingly, a causal relationship between the emergence of a simple replicator and the emergence of a metabolic system is indicated. The results lend further support for the “replication first” school of thought in the origin of life problem by providing a mechanistic basis for the emergence of a metabolism, once a simple non-metabolic replicating system has itself been established. The study reaffirms our view that the roots of Darwinian theory may be found within standard chemical kinetic theory.     

Mitochondrial oscillation and activation of H+/cation exchange

Mitochondria incubated aerobically in the presence of tetrapropylammonium and weak acids and in the presence of trace amounts of tetraphenylboron undergo a series of damped oscillations reflecting cycles of osmotic swelling and shrinkage. The matrix volume changes are consequent to transport of tetrapropylammonium catalytically stimulated by tetraphenylboron. The amplitude and frequency of the oscillations increase with the concentration of tetrapropylammonium, as required for critical rates and extents of ion influx. Addition of bovine serum albumin abolishes both the uptake of tetrapropylammonium and the oscillations. Volume oscillations are paralleled by cyclic activation and depression of the respiratory rate. Two lines of evidence suggest that the train of damped oscillations depends on the cyclic activation of an electroneutral exchange of H⁺ with organic cations rather than on cyclic uncoupling. First, further increase of cation permeability due to a pulse of tetraphenylboron, after initiation of cation efflux, restores cation influx. Second, addition of Mg²⁺, which abolishes the oscillations, has a much more marked inhibitory effect on the process of cation efflux than on cation influx. Conversely, addition of A23187, which removes membrane-bound Mg²⁺, promotes cation efflux and thus the oscillations. It is suggested that, in the present system, stretching of the inner membrane and Mg²⁺ depletion result in activation of an electroneutral H⁺/organic cation exchange, and that cyclic activation of this reaction results in damped oscillations.     

Dynamics of a biological fixed film for phenol degradation in a fluidized-bed bioreactor

Experimental and modeling studies were conducted to analyze the dynamic response behavior of a phenol-oxidizing fixed film using a differential, fluidized-bed bioreactor in a recycle loop with a well-mixed reservoir. With the bioreactor at steady state, a pulse of phenol was added to perturb the system, and the phenol concentration was monitored continuously until steady state was again achieved.The experimental dynamics were compared with a dynamic mathematical model based on diffusion and reaction within the biofilm, liquid mixing, and biofilm growth. Constant-pH experiments could be adequately described using an unstructured, double-Monod kinetic expression with substrate inhibition by phenol.However, in dynamic experiments without pH control, the pH of the liquid phase dropped, and damped oscillations were observed in the phenol concentration and reaction rate trajectories. Oscillatory solutions could not be induced in the model, even when product inhibition was included, and a linear stability analysis did not reveal tendencies toward instability. The cause of the experimental oscillations remains unknown.     

Self-organization and dynamics of an open futile cycle

The dynamic behaviour of an open futile cycle composed of two enzymes has been investigated in the vicinity of a steady-state. A necessary condition required for damped or sustained oscillations of the system is that enzyme E₂, which controls recycling of the substrate S₂, be inhibited by an excess of this substrate. In order for the system to be neutrally stable and therefore to exhibit sustained oscillations, it is not necessary for antagonist enzyme E₁ to be activated by its product S₂. If it is enzyme E₁ which is inhibited by an excess of its substrate S₁, the system has a saddle point. Other conditions for stability or instability of the system have been determined. If the enzyme E₁, which is not inhibited by the substrate, exhibits a slow conformational transition of the mnemonical type, this transition dramatically alters the stability behavior of the system. If the mnemonical enzyme E₁ were exhibiting a positive kinetic co-operativity, decreasing the rate of the conformational transition of the mnemonical enzyme will increase the stability of the whole system and will tend to damp the oscillations in the vicinity of the steady-state. If conversely the mnemonical enzyme E₁ were exhibiting a negative kinetic co-operativity, decreasing the rate of the enzyme conformational transition will decrease the stability of the system and will tend to create or amplify oscillations of the system taken as a whole. If these results may be extended to more complex metabolic cycles, involving more than two enzymes, it may be tentatively considered that positive co-operativity associated with slow transition has emerged in the course of evolution in order to limit temporal instabilities of metabolic cycles. Alternatively one may speculate that the “biological function” of negative co-operativity is to create or amplify these temporal instabilities.     

A new kinetic model for biochemical oscillations: Graph-theoretical analysis

A graphical analysis demonstrates the ability of slow substrate activation and certain types of cooperativity between the two enzyme active sites to generate sustained oscillations. The analysis allows us to estimate kinetic parameter values for which oscillations exist. The scheme analyzed can explain the cyclical changes in functioning of various motor enzymes. Moreover, this scheme does not generate bistability for any parameter values. The graphical analysis presented is simple and visually clarifies the regulatory role of the details in the kinetic schemes.     

Comparison of the enzymatic behavior of high molecular weight and free lysyl-tRNA synthetase from rat liver: kinetic analysis of lysylation of tRNA

Lysyl-tRNA synthetase occurs in the high molecular weight form in rat liver. The high molecular weight lysyl-tRNA synthetase has been previously demonstrated to exist as multienzyme complexes of aminoacyl-tRNA synthetases. The multienzyme complexes can be dissociated by hydrophobic interaction chromatography and yield fully active, free lysyl-tRNA synthetase. The free form is found to be twice as active as the complexed form in lysylation. Bisubstrate and product inhibition kinetics of lysylation are systematically carried out for highly purified free lysyl-tRNA synthetase and the 18 S synthetase complex. Surprisingly, the two enzyme forms exhibit distinctly different kinetic patterns in bisubstrate and product inhibition kinetics under identical conditions. The 18 S synthetase complex shows kinetic patterns consistent with an ordered bi uni uni bi ping pong mechanism, while the results of free lysyl-tRNA synthetase do not. We conclude that structural organization of lysyl-tRNA synthetase beyond quaternary structure of proteins may alter the enzyme behavior.     

Linear nonequilibrium thermodynamics describes the dynamics of an autocatalytic system.

A model simulating oscillations in glycolysis was formulated in terms of nonequilibrium thermodynamics. In the kinetic rate equations every metabolite concentration was replaced with an exponential function of its chemical potential. This led to nonlinear relations between rates and chemical potentials. Each chemical potential was then expanded around its steady-state value as a Taylor series. The linear (first order) term of the Taylor series sufficed to simulate the dynamic behavior of the system, including the damped and even sustained oscillations at low substrate input or high free-energy load. The glycolytic system is autocatalytic in the first half. Because oscillations were obtained only in the presence of that autocatalytic feed-back loop we conclude that this type of kinetic nonlinearity was sufficient to account for the oscillatory behavior. The matrix of phenomenological coefficients of the system is nonsymmetric. Our results indicate that this is the symmetry property and not the linearity of the flow-force relations in the near equilibrium domain that precludes oscillations. Given autocatalytic properties, a system exhibiting liner flow-force relations and being outside the near equilibrium domain may show bifurcations, leading to self-organized behavior.     

Computer simulation of damped oscillations during peroxidase-catalyzed oxidation of indole-3-acetic acid

Oscillation patterns in horseradish peroxidase (HRP)-catalyzed oxidation of indole-3-acetic acid (IAA) at neutral pH were studied using computer simulation. Under certain conditions, such as the presence of a reaction promoter and continuous intake of oxygen from the gaseous phase, the simulated system exhibits damped oscillations of the concentrations of oxygen in the aqueous phase, [O(2)](aq), and of all the reaction intermediates. The critical concentration of oxygen in aqueous phase, [O(2)](cr)(aq), was used to describe the nature of the oscillations. The critical concentration is the concentration at which the system abruptly changes its properties. If [O(2)](aq) is higher than [O(2)](cr)(aq) then the reaction develops as an avalanche, otherwise, the reaction stops. The nature of oscillations is accounted for by the interaction of two processes: the consumption/accumulation of oxygen and the accumulation/consumption of reaction intermediates. Oscillations are always damped. Neither HRP or umbelliferone (Umb) deactivation nor IAA consumption can account for the damping. The nature of the damping is determined by the termination reactions of free radical intermediates and ROOH. The three major parameters of oscillations: period of oscillations, initial amplitude of oscillations and the rate of damping were studied as functions of: (i) oxygen concentration in the gaseous phase, (ii) initial oxygen concentration in aqueous phase, (iii) the concentration of IAA and (iv) the initial concentration of HRP.     

Kinetic behaviour of proenzymes activation in the presence of different inhibitors for both activating and activated enzymes

In the present paper, a kinetic analysis of a general model for proenzyme activation, where the activating enzyme and also the activated one are reversibly inhibited in two steps by two different inhibitors, has been performed. The cases in which both inhibitors are the same, or in which the inhibition is irreversible (only one or the two inhibition routes) are treated as particular cases of the general model. In addition, the kinetic behaviour of many other proenzyme activation systems involving inhibition, particular cases of the reaction scheme under study, can be obtained. The total number of particular cases for the general model under study is 370, so this approach offers to the scientific community working in limited proteolysis regulation for the first time a method based on general solutions which only needs to be specified to their concrete problem of zymogen activation. Finally, new adimensional parameters are introduced, allowing the knowledgement, in the case that any of the inhibition routes is irreversible, the relative weight of both activation and irreversible inhibition routes.     

Interaction of D-fructose and fructose 1-phosphate with yeast phosphofructokinase and its influence on glycolytic oscillations.

Fermentation of D-fructose- and D-glucose induced glycolytic oscillations of different period lengths in Saccharomyces carlsbergensis. Recent studies suggested, that D-fructose or one of its metabolites interacted with phosphofructokinase (ATP:D-fructo-6-phosphate 1-phosphofructokinase, EC 2.7.1.11), the core of the glycolytic 'oscillator'. In order to explore the kinetics of interaction, the influence of D-fructose and fructose 1-phosphate on purified yeast phosphofructokinase was studied. D-fructose concentrations up to 0.3 mM stimulated the enzyme, while a further increase led to competitive inhibition. The Hill coefficient for fructose 6-phosphate decreased from 2.8 to 1.0. Fructose 1-phosphate acted in a similar way, up to 1 mM activation and inhibition competitive to fructose 6-phosphate at higher concentration (2.0--3.5 mM) with the same effect on the Hill coefficient. The inhibition patterns obtained with D-fructose or fructose 1-phosphate suggest a sequential random reaction mechanism of yeast phosphofructokinase with fructose 6-phosphate and MgATP2-. The mode of interaction of phosphofructokinase with D-fructose and fructose 1-phosphate is discussed. The influence of both effectors resulted in altered enzyme kinetics, which may cause the different period lengths of glycolytic oscillations.     

Inhibition enhances memory capacity: optimal feedback, transient replay and oscillations

Recurring sequences of neuronal activation in the hippocampus are a candidate for a neurophysiological correlate of episodic memory. Here, we discuss a mean-field theory for such spike sequences in phase space and show how they become unstable when the neuronal network operates at maximum memory capacity. We find that inhibitory feedback rescues replay of the sequences, giving rise to oscillations and thereby enhancing the network’s capacity. We further argue that transient sequences in an overloaded network with feedback inhibition may provide a mechanistic picture of memory-related neuronal activity during hippocampal sharp-wave ripple complexes.