A comparison is made between conventional Michaelis-Menten kinetics and two models that take into account the duration of the conformational changes that take place at the molecular level during the catalytic cycle of a monomer. The models consider the time that elapses from the moment an enzyme-substrate complex forms until the moment a product molecule is released, as well as the recovery time needed to reset the conformational change that took place. In the first model the dynamics is described by a set of delayed differential equations, instead of the ordinary differential equations associated to Michaelis-Menten kinetics. In the second model the delay, the discretization inherent to enzyme reactions and the stochastic binding of substrates to enzimes at the molecular level is considered. All three models agree at equilibrium, as expected; however, out-of-equilibrium dynamics can differ substantially. In particular, both delayed models show oscillations at low values of the Michaelis constant which are not reproduced by the Michaelis-Menten model. Additionally, in certain cases, the dynamics shown by the continuous delayed model differs from the dynamics of the discrete delayed model when some reactant become scarce.
1 Introduction.
For over a century, the conventional way to represent many enzyme reactions has been based on the Michaelis-Menten kinetics (Michaelis and Menten, 1913) (hereafter MM model), in which substrate S binds reversibly with enzyme E to form an enzyme-substrate complex ES, which is later transformed into product P and free enzyme E. Derivation of the Michaelis-Menten equation is based on the following scheme (Segel, 1975):
However, there is strong evidence suggesting that enzime activity depends on conformational changes in the enzyme structure (Bennett et al., 1978;Herschlag, 1988;Hakansson et al., 1997;Agmon, 2000;Eisenmesser et al., 2002;Benkovic and Hammes-Schiffer, 2003). It is therefore reasonable to assume that a conformational change takes place from the moment an ES complex forms to the moment a product P is released; such change requires a certain time for its completion. This conformational change must then be reset before the enzyme is ready to bind substrate again, and so a recovery time must elapse once P is released. Single-enzyme studies (Lu et al., 1998;Xie and Lu, 1999;Eisenmesser et al., 2005;Walter, 2006) and nuclear magnetic resonance spectroscopy studies (Eisenmesser et al., 2005;Huang and Montelione, 2005) support these notions. A scheme that considers enzyme regeneration is presented by English et al. (2006),
where E 0 represents recovering enzyme. Conventional MM kinetics do not account in an explicit way for the times required by these conformational changes.
The aim of this paper is to present two models of enzyme catalysis, one continuous and one discrete, that take into account the times needed by enzymes to process ES complexes and to reset to its original state once products has been released; these models can be considered as an extension of conventional MM kinetics which includes delays. The main motivation behind the continuous delayed model is to establish the limitations of the MM description in non-equilibrium conditions. In order to examine the limitations associated to the continuous nature of the first model, we consider a discrete delayed model in which the catalytic cicle of each enzyme in the system is represented by a recursive function, hereafter referred to as the enzyme map. Previous work by Stange et al. (1998Stange et al. ( , 1999) ) present an analogous description of enzyme action in which each protein molecule is represented by a clock-like automaton that binds substrate S and transforms it into product P. The time needed by a single automaton to complete this transformation and return to its original waiting state represents the enzyme turnover time τ . The conformational changes of the enzyme (and the different stages of the catalytic cycle) are represented by motion along an internal ‘phase’ coordinate. Under the aproppriate conditions this model exhibits complex behavior such as oscillations and cluster formation; thus another motivation is to explore the emergence of such behavior in our model. Unlike other microscopic models based on stochastic simulation (Haseltine and Rawlings, 2002;Kierzek, 2002;Rao and Arkin, 2003;Puchalka and Kierzek, 2004), our discrete model is completely deterministic, and each enzyme is represented by a map instead of a clock-like unit.
In Section 2 the continuous delayed model is presented by means of a simple reaction, and a comparison is made between numerical integration provided by MM kinetics and those produced by the delayed model. Section 3 presents the equivalent discrete model; the results of the discrete simulation are compared with results obtained through the continuous delayed model.
In the conclusions presented in Section 4 we highlight the respective merits and limitations of each model.
To illustrate the effect of delays on the dynamics of enzyme catalysis we consider the simple reaction A
, where conversion between two substrates A and B is catalyzed by two unidirectional monomeric enzymes α and β.
As mentioned in the introduction, each conformational change taking place along the catalytic cycle will require a certain time for its completion. We then distinguish three stages in the cycle: 1) the enzyme is free, waiting to bind substrate; 2) the enzyme has bound substrate forming an ES complex; 3) product P has been released and the enzyme is resetting to its initial conformation. The processing time τ p required by stage 2 is a fraction c of the turnover time τ , while the recovery time τ r associeted to stage 3 is (1 -c)τ . The turnover time is the time that elapses from the moment an ES complex forms until the moment the enzyme is ready to bind substrate again; such time is given by
where V max is the maximum velocity of the enzyme in µmol/(min-mg of protein) and µ e its molecular weight in Daltons.
When processing and recovery times are considered the reaction is described by a set of four coupled delay differential equations (DDEs):
where [A], [B] are substrate concentrations in µM
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