ENHANCEMENT AND RESTRICTION OF SYSTEM COORDINATION BY INTERACTION OF PATHWAYS

System coordination for a system with interaction of two pathways is investigated. Sufficient conditions for guaranteeing system coordination subject to any type, amplitude or period of external forcing are analytically derived. Effects of a pathway operating in the opposite direction on system coordination are examined in detail, and it is revealed that the pathway may enhance or restrict system coordination, depending on the values of kinetic parameters. It is concluded that increasing the complexity of enzymatic network by introducing interaction of pathways does not necessarily enhance system coordination. Numerical results using rectangular and sinusoidal forcing support the analytical results, and they quantitatively show how the system with interaction of two pathways maintain its coordination. When the sufficient conditions are satisfied, the system always develops to a finite stable (time-dependent) state under the forcing of any type, amplitude or period. When not, system coordination depends quantitatively on parameter values and the type, amplitude or period of external forcing. When two parameters are forced simultaneously, an additional factor, namely phase shifts between two forced parameters, also affects system coordination. It is shown that, unless the sufficient coordination conditions are maintained, external forcing may induce 'loss of coordination', resulting in the system having no biological functioning. The implications of the results for understanding biochemical coordination and for assessing possible consequences of modulating biochemical systems are discussed.

SUFFICIENT CONDITIONS FOR COORDINATION OF A FORCED BIOCHEMICAL SYSTEM WITH THE INTERPLAY OF ACTIVATION AND INHIBITION

A system of ordinary differential equations representing a biochemical system with the interplay of enzyme activation and inhibition is studied. Each term of the equations describes an enzymatic reaction and has an upper limit — the maximum activity of the enzyme. The sufficient conditions for guaranteeing coordination of the system subject to any type, period and amplitude of external forcing are analytically derived. Numerical analysis using three types of external signals, namely rectangular, sinusoidal and noisy signals, supports the analytical results. The sufficient conditions require kinetic parameter values (in particular, maximum enzyme activities) be related, implying system coordination requires enzymes in the system properly work together. It is shown that, when the sufficient conditions are satisfied, the system always develop to a stable (possibly time-dependent) state when it is subject to any type, period and amplitude of external forcing. When not, whether system coordination is destroyed depends quantitatively on the parameter values and the type, period and amplitude of external signals. Once system coordination is destroyed by external forcing, the system does not have any stable state on phase plane, and one of the species concentrations accumulates infinitely. Finally, I discuss the implications of the results for understanding the coordination of nonlinear enzyme-catalysed reaction systems with the interplay of activation and inhibition, and the possible consequences of modulating such systems.

Sufficient Conditions for Coordination of Coupled Nonlinear Biochemical Systems: Analysis of a Simple, Representative Example

Under conditions of stress and environmental fluctuations, how do coupled biochemical systems maintain the balance of metabolic fluxes? Using a model derived from a representative glycolysis example, this work derives sufficient coordination conditions for guaranteeing the balance of metabolic fluxes in the coupled systems subjected to any fluctuations. For diffusion-like coupling, it is explicitly shown that the sufficient conditions are the linear summation of individual uncoupled systems. For this case, coupling strength is not a key factor affecting the balance of metabolic fluxes, although it may affect the emergence of certain dynamic patterns. When the sufficient coordination conditions are satisfied, coupled systems always settle onto stable finite states, independently of the nature (type, period or amplitude) of external signals. When they are not, external signals may force the stable states of coupled systems to lose their coordination, resulting in the coupled systems having no stable finite states which is inconsistent with most normal biological functions. Numerical simulations support the analytical results. The generalization of this result for other coupled nonlinear biochemical systems is discussed. Finally, the maintenance of such biological coordination is discussed in the context of genetic engineering and environmental fluctuations.

Collapse of single stable states via a fractal attraction basin: analysis of a representative metabolic network

The impact of external forcing on an enzymatic reaction system with a single finite stable state is investigated. External forcing impacts on the system in two distinct ways: firstly, the reaction system undergoes a series of discontinuous changes in dynamical state. Secondly, a critical level of forcing exists, beyond which all finite states become unstable. It is shown that the results stem from the conditions for global stability of the system. Competition between the attractor for stable states and the unbounded states leads to a loss of integrity and the fractal fragmentation of the attraction basin for the finite state. The consequences of a fractal basin in this context are profound. Initial states which are infinitesimally close diverge to a finite and an unbounded state where only the finite state is consistent with biological functionality. Furthermore, above a critical forcing amplitude, the system does not converge to a finite state from any initial state, implying that there is no configuration of metabolite concentrations that is consistent with sustained evolution of the system. These results point to opportunities for constraining uncertainty in cell networks where nonlinear saturating kinetics form an important component.

Kinetic constraints for formation of steady states in biochemical networks

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. This work introduces the concept of kinetic constraints. It is shown that maximal reaction rates are appropriate constraints only for isolated enzymatic reactions. For biochemical networks, it is revealed that constraints for formation of a steady state require specific relationships between maximal reaction rates of all enzymes. The constraints for a branched network are significantly different from those for a cyclic network. Moreover, the constraints do not require Michaelis-Menten constants for most enzymes, and they only require the constants for the enzymes at the branching or cyclic point. Reversibility of reactions at system boundary or branching point may significantly impact on kinetic constraints. When enzymes are regulated, regulations may impose severe kinetic constraints for the formation of steady states. As the complexity of a network increases, kinetic constraints become more severe. In addition, it is demonstrated that kinetic constraints for networks with co-regulation can be analyzed using the approach. In general, co-regulation enhances the constraints and therefore larger fluctuations in fluxes can be accommodated in the networks with co-regulation. As a first example of the application, we derive the kinetic constraints for an actual network that describes sucrose accumulation in the sugar cane culm, and confirm their validity using numerical simulations.

State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.