Kinetic model for dynein oscillatory activity.

A kinetic model for dynein, a molecular motor, is considered. This model explains the oscillatory behaviour, observed by Chikako Shingyoji et al. [Ch. Shingyoji, H. Higuchi, M. Yoshimura, E. Katayama, T. Yanagida, Dynein arms are oscillatory force generators, Nature 393 (1998) 711-714.] and by Susumu Aoyama and Ritsu Kamiya [S. Aoyama, R. Kamiya, Cyclical interactions between two outer doublet microtubules in split flagellar axonemes, Biophys. J. 89 (2005) 3261-3268.] in surprisingly simple axonemal fragments. The model shows that sustained oscillations can be generated due to the obligate cooperative interaction of the two dynein heads in the axonemal fragments. No other feedback control interactions are involved in the model to explain oscillations, similar to those observed experimentally, for realistic dynein rate constants. The modified model shows how the ATP hydrolytic exhaustion influences the amplitude and frequency of dynein oscillatory activity.

Unusual kinetic behavior predicted for alpha-keto acid dehydrogenase complexes.

A novel regulation type, which may be observed as an unusual kinetic 'cooperativity', is predicted for the alpha-keto acid dehydrogenase complexes. The inter-relationship of this regulation with the well-known regulatory effect of enzyme phosphorylation is discussed.

Hormonal regulation of 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase: kinetic models.

A graphical method to reveal the so-called 'critical fragments' in schemes of biochemical systems is considered. These fragments produce multiple steady states or self-oscillations in systems. As an example, the bifunctional enzyme, 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase, regulated by glucagon through enzyme phosphorylation, is discussed. It is shown that this enzyme may act as a metabolic switching mechanism in discontinuous or oscillatory regimes, depending on the specific structure of its kinetic scheme. The boundaries of concentrational and parameter domains for these critical phenomena are also predicted.

Principles of symmetrical organization for the pyruvate dehydrogenase complex.

The experimentally observed phenomenon of non-equimolarity for enzyme components, assembled into multienzyme complexes of the 2-oxo acid dehydrogenases family, is structurally interpreted to predict the only possible stable symmetrical distribution of peripheral components on the complex core. To obey the equivalent neighboring, that is necessary for unique self-assembled structures, we should deduce discrete conformational states for core subunits, those with different affinity for peripheral components. Two kinetically different types of substrate-intermediate pathways through the lipoyl network of the mammalian pyruvate dehydrogenase complex follow from this structural theory. The theory predicts unusual kinetic behavior for the multienzyme complex.

Activity oscillations predicted for pyruvate dehydrogenase complexes.

A kinetic model for the pyruvate dehydrogenase complex is analyzed. The model takes into account intermediate channeling through the lipoyl network attached to the complex core, as well as inter-related regulatory effects of protein X acetylation and enzyme phosphorylation. The model predicts undamped oscillations of enzyme activity.

A model of stepping kinetics for rotary enzymes. Application to the F1-ATPase.

Our simple kinetic model, based on the classic "binding change mechanism", describes the stepping kinetics for the rotary enzyme motors. The model shows that the cooperative interactions between active sites in the motor enzyme F1-ATPase induce the stepping product release. This phenomenon results from non-harmonic oscillations in the enzyme forms. The found rate constants, corresponding to the stepping phenomenon, are close to the rate constants known for the F1-ATPase. The duration of dwells during the product release is shown to depend on the ATP concentration in accordance with the known experimental data.

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.

Oscillatory activity of P-type membrane adenosine triphosphatases: a kinetic model.

A kinetic model for membrane P-type adenosine triphosphatases is considered, the main application being to the erythrocyte Ca2+-ATPase. It is shown that a simple modification of the known catalytic mechanism of the ATPase by addition of a self-inhibition step and the steady calcium influx leads to damped oscillations in the system discussed. In this way, the model can explain the kinetic experimental results obtained for the purified enzyme in solution as well as for the enzyme incorporated into liposome membranes. The estimated kinetic parameters are close to the experimental ones. Alternative changes in time, demonstrated by the kinetic model for the conformational enzyme states, E(1 )and E(2), confirm the model of two alternatively functioning gates in the ion pumping Ca2+-ATPase.

Influence of Ca2+ oscillatory influx on membrane Ca2+-ATPase activity: a kinetic model.

A kinetic model for the membrane Ca2+-ATPase is considered. The catalytic cycle in the model is extended by enzyme auto-inhibition and by oscillatory calcium influx. It is shown that the conductive enzyme activity can be registered as damped or sustained Ca2+ pulses similar to observed experimentally. It is shown that frequency variations in Ca2+ oscillatory influx induce changes of pulsating enzyme activity. Encoding is observed for the signal frequency into a number of fixed levels of sustained pulses in the enzyme activity. At certain calcium signal frequencies, the calculated Ca2+-ATPase conductivity demonstrates chaotic multi-level pulses, similar to those observed experimentally.

Analysis of cyclic enzyme reaction schemes by the graph-theoretic method.

The development of the graph-theoretic method is proposed particularly for the analysis of closed cycles of elementary stages in enzyme reaction schemes. Some simplifications of the graph structure may be based on the application of Kirchhoff's laws to enzyme reaction graphs in the steady-state. The importance of the cyclic processes for enzyme regulations and a principle non-equilibrium of this phenomenon are emphasized. As an example of the regulatory role of cycles "the liberation" from substrate inhibition by substrate analogues is considered. The modification of the graph-theoretic method in the pre-steady-state kinetics for arbitrary initial conditions (for pre-mixing procedures) is also discussed. The necessary and sufficient conditions for damped oscillations in the pre-steady state are formulated which are the equality conditions for some of the rate constants along the cycle (both for reversible and irreversible stages).

Simplest kinetic schemes for biochemical oscillators.

The topological structure of the simplest critical fragments in biochemical systems has been characterized. The conditions are considered where the critical fragments induce oscillations of the concentrations of the system participants. To illustrate, three biochemical systems (transport of ions through a membrane, protein phosphorylation, and two-substrate reaction) are discussed. The kinetic schemes of these systems contain one of the discovered critical fragments. Relaxation oscillations of the concentrations of the system participants were demonstrated using the numerical integration method.

Kinetic manifestations of structural self-organization for the components of the pyruvate dehydrogenase complex. A mathematical model.

The structure-function organization of mammalian pyruvate dehydrogenase complex (PDC) is considered. The linear size of the complex components in relation to the size of their fragments and mean distances between potentially active sites are estimated. A kinetic model for the complex with core subunits divided into conformational classes characterized by their different activities is discussed. The kinetic features of this model are compared with the features of the model for the core subunits not divided into classes. Curves are described for the activity of PDC versus time for various activities of the kinase and the phosphatase as dependent on their binding to more or less active core subunits. The results show that spatially distributed multiple active sites in the multi-enzyme complex can drastically change their full activity in time without any marked changes in the rate constants of the elementary reaction steps due only to spatially changed reaction pathways. A new type of regulation discussed for PDC reveals unusual regulatory abilities for complexes of this family.

A model for the spatial location of pyruvate dehydrogenase phosphatase in mammalian pyruvate dehydrogenase complex.

Recent experimental findings on the structural--functional features of pyruvate dehydrogenase phosphatase (PDP) isolated from various sources are compared. Two alternative mechanisms (a and b) of dephosphorylation of the E1 component in the pyruvate dehydrogenase complex (PDC) are discussed: a) the reaction occurs as a result of stochastic collisions of PDP and PDC, and the generation of an enzyme--substrate complex (PDP--E1--PDC) and dephosphorylation of the E1 component occur independently at different PDP binding sites on the PDC core; b) the dephosphorylation is performed simultaneously by a certain number of PDP molecules symmetrically bound on the PDC core. The second mechanism is suggested by the self-assembly theory of multicomponent enzyme systems and can be proved by kinetic experiments. Based on self-assembly principles and data on feasible binding sites of peripheral components of the PDC, the stoichiometry and mutual location of PDP, pyruvate dehydrogenase kinase, and the E1 component on the core of mammalian PDC are postulated to provide optimal functioning of the PDC. Structural mechanisms of stimulation of PDP activity by Ca2+ and polyamines are also discussed.

Stability of multienzyme systems with feedback regulation: a graph theoretical approach.

Stability of multienzyme systems with feedback regulation has been analyzed on the basis of the Lienard-Chipart criteria. The rules governing the topological graph construction for multienzyme systems have been developed. A theorem about correspondence of the graph constructed and coefficients of the characteristic polynomial of linearized kinetic equations is proved. The graph-theoretical stability analysis proposed is illustrated by a number of examples of multienzyme systems with feedback regulation.

Graph-theoretic approach to metabolic pathways.

A graph-theoretic approach is shown to be applicable within the framework of the metabolic control analysis. Kinetic differential equations linearized near a steady state are presented as kinetic graphs (schemes), their structure being correlated with kinetic properties of corresponding metabolic networks. The global properties may be expressed in terms of the local properties for steady states of metabolic systems. Instability, bistability, and concentrational oscillations are shown to be induced by specific graph fragments. The approach is illustrated by an example of systems showing the oscillatory kinetic behaviour.

Regulation of pyruvate dehydrogenase complex activity in Ehrlich ascites carcinoma cells by Ca2+ and pyruvate.

The dependence of pyruvate dehydrogenase complex (PDC) activity on [Ca2+] was determined in Ehrlich ascites carcinoma cells at different pyruvate concentrations. The resulting family of curves had the following characteristics: a) bell-shaped appearance of all curves with maximum activity at 600 nM Ca2+; b) unchanged position of maxima with changes in pyruvate concentration; c) nonmonotonous changes in PDC activity with increasing pyruvate concentration at fixed [Ca2+]. Feasible mechanisms involving Ca2+-dependent phosphatase and kinase which are consistent with the experimental findings are discussed. To determine the steps in the chain of PDC reactions which determine the observed phenomena, a mathematical model is suggested which is based on the known data on the structural--functional relationships between the complex components--pyruvate dehydrogenase (E1), dihydrolipoyl acetyl transferase (E2), dihydrolipoyl dehydrogenase (E3), protein X, kinase, and phosphatase. To adequately describe the non-trivial dependence of PDC activity on [Ca2+] at different pyruvate concentrations, it was also necessary to consider the interdependence of some steps in the general chain of PDC reactions. Phenomenon (a) is shown to be due only to the involvement of protein X in the PDC reactions, phenomenon (b) to be due to changes in the activity of kinase, and phenomenon (c) to be due to dependence of acetylation and transacetylation rates on pyruvate concentration.