Mutual Synchronization of Molecular Turnover Cycles in Allosteric Enzymes II. Product Inhibition

Multiparticle diffusion limited [Formula: see text] reaction in small volumes

A multiparticle reaction model in which particles A annihilate in clusters of size k as [Formula: see text] is investigated analytically. The system is studied for arbitrary reaction order k > 2, dimension d, and size L. Particles diffuse with diffusion constant D, and annihilate with rate σ which depends on the positions of kA particles in the cluster prior to a reaction. The particles are assumed to be spatially extended objects with radius a. Exclusion effects are not taken into account since A particles are allowed to overlap. The master equation is rephrased in the language of a field theory which, in turn, is used to derive the equations of motion for many-point densities. An approximate form of the equations of motion was solved analytically in the diffusion-controlled limit (infinite reaction rate). An explicit expression for the effective reaction rate has been found in the form of the Laplace transform. It was shown that the number of particles saturates to a constant value for large times. The value is approached through an exponential decay. The exponential decay constant is the non-algebraic function of particle size a and system size L.

Molecular synchronization waves in arrays of allosterically regulated enzymes

Spatiotemporal pattern formation in a product-activated enzymic reaction at high enzyme concentrations is investigated. Stochastic simulations show that catalytic turnover cycles of individual enzymes can become coherent and that complex wave patterns of molecular synchronization can develop. The analysis based on the mean-field approximation indicates that the observed patterns result from the presence of Hopf and wave bifurcations in the considered system.

Functional conformational motions in the turnover cycle of cholesterol oxidase

Reexamining experimental data of single-molecule fluorescence correlation spectroscopy for cholesterol oxidase, we find that the existing Michaelis-Menten models with dynamical disorder cannot explain strong correlations between subsequent turnover cycles revealed in the diagonal feature in the joint statistical distribution of adjacent "on" times of this enzyme. We suggest that functional conformational motions representing ordered sequences of transitions between a set of conformational substates are involved, along with equilibrium conformational fluctuations in the turnover cycle of cholesterol oxidase. A two-channel model of single-enzyme dynamics, including a slow functional conformational motion in one of the channels, is proposed that allows us to reproduce such strong correlations.

Interplay between chemical reactions and transport in structured spaces

The main motivation behind this study is to understand the interplay between the reactions and transport in a geometries that are not compact. Typical examples of compact geometries are a box or a sphere. A network made of containers C(1) , C(2),..., C(N) and tubes is an example of the space that is structured and noncompact. In containers, particles react with the rate lambda. Tubes connecting containers allow for the exchange of chemicals with the transport rate D. A situation is considered where a number of reactants is small and kinetics is noise dominated. A method is presented that can be used to calculate the average and higher moments of the reaction time. A number of different chemical reactions are studied and their performance compared in various ways. Two schemes are discussed in general, the reaction on a fixed geometry ensemble (ROGE) and the geometry on a fixed reaction ensemble, examples are given in the ROGE case. The most important findings are as follows. (i) There is a large number of reactions that run faster in a networklike geometry. Such reactions contain antagonistic catalytic influences in the intermediate stages of a reaction scheme that are best dealt with in a networklike structure. (ii) Antagonistic catalytic influences are hard to identify since they are strongly connected to the pattern of injected molecules (inject pattern) and depend on the choice of molecules that have to be synthesized at the end (task pattern). (iii) The reaction time depends strongly on the details of the inject and task patterns.

Spatial heterogeneity and the stability of reaction states in autocatalysis

The impact of stochasticity and spatial heterogeneity on the quadratic autocatalytic system is studied. In a nonspatial setting the reactive state of the system is found to be unstable in small volumes where internal fluctuations drive the system to the unreactive state. This phenomena is of potential importance to the stability of reactions in biological cells. A simple spatial model is constructed by linking N nonspatial models via migration of reactants controlled by a mixing rate lambda. Simulation of this stochastic process demonstrates the importance of such mixing in controlling the impact of internal fluctuations on the stability of the autocatalytic reaction. For high mixing rate the mean reactant levels in equilibrium correspond to the well-mixed deterministic system, although a significant degree of spatial heterogeneity remains. For intermediate mixing rates, mean reactant levels vary continuously with lambda, where the interaction of internal fluctuations with limited spatial mixing modifies the reactive states of the deterministic system. However, there is a threshold below which mixing is unable to control internal fluctuations which drive the system into the unreactive state. Thus a critical minimum level of communication between the cells is required to stabilize the reaction across the entire system. Approximate analytic results, obtained using moment-closure techniques, support these findings and demonstrate the relationship between the spatial stochastic and nonspatial deterministic models.

Microscopic self-organization in networks

We report our numerical studies on microscopic self-organization of a reaction system in three types of differently connected networks: a regular network, a small-world network, and a random network. Our simulation results show that the topology of the network has an important effect on the communication among reaction molecules, and plays an important role in microscopic self-organization. The correlation length among reacting molecules in a random or a small-world network is much shorter compared with that in a regular one. As a result, it is much easier to obtain microscopic self-organization in a small-world or a random network. We also observed a phase transition from a stochastic state to a synchronized state when we increased the randomness of a small-world network.

Transitions induced by the discreteness of molecules in a small autocatalytic system

The autocatalytic reaction system with a small number of molecules is studied numerically by stochastic particle simulations. A novel state due to fluctuation and discreteness in molecular numbers is found, characterized as an extinction of molecule species alternately in the autocatalytic reaction loop. Phase transition to this state with changes of the system size and flow is studied, while a single-molecule switch of the molecule distributions is reported. The relevance of the results to intracellular processes is briefly discussed.