Spatiotemporal patterns enhanced by intra- and inter-molecular fluctuations in arrays of allosterically regulated enzymes

Abstract

Enzymatic reactions often involve slow conformational changes, with reaction cycles that sometimes require milliseconds or seconds to complete. The dynamics are strongly affected by fluctuations at the nanoscale. However, such enzymes often occur in small numbers in a cell; hence, the fluctuations caused by finite numbers of molecules should also be substantial. Because of these factors, the behavior of the system is likely to deviate from that of classical reaction-diffusion equations, in which immediate reaction events are assumed to occur without memory and between a huge number of molecules. In this work, we model each enzyme as a unit represented by a phase variable to investigate the effects of fluctuations in arrays of enzymes. Using an analysis based on partial differential equations and stochastic simulations, we show that fluctuations arising from internal states of enzymes (intramolecular fluctuations) and those arising from the stochastic nature of interactions between molecules (intermolecular fluctuations) have distinctive effects on spatiotemporal patterns; the former generally disturb synchronization at high frequencies, whereas the latter can enhance synchronization. The balance of the two types of fluctuations may determine the spatiotemporal behavior of biochemical processes.

An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations

“…Enzymes are complex macromolecules and their reaction cycles may depend on their conformational states. Therefore, the prediction of biological phenomena caused by small-number effects in real biochemical reactions, would entail further analytical challenges for catalytic reaction networks including arbitrary higher-order mixed reactions (rather than first- and second-order reactions only) or internal dynamics of the enzymes (as modeled and analyzed in Togashi and Casagrande, 2015 ) as important issues.…”

Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings

“…In the same way as our previous works on enzymes, , we consider a stochastic diffusional drift process (biased Brownian motion) for state ϕ i , such that where v is the mean velocity, and η i ′ ( t ) is white noise with ⟨η i ′ ( t )⟩ = 0, ⟨η i ′ ( t )­η j ′ ( t ′)⟩ = 2μ ϕ σ ϕ δ ij δ­( t – t ′). Here, two parameters are introduced: μ ϕ is the “mobility”, or susceptibility, of the phase, and σ ϕ specifies the intensity of internal noise affecting the operation.…”

“…Here, two parameters are introduced: μ ϕ is the “mobility”, or susceptibility, of the phase, and σ ϕ specifies the intensity of internal noise affecting the operation. (In our previous model, a single parameter σ = μ ϕ σ ϕ was used for the intensity of intramolecular fluctuations. )…”

“…Note that the definition of α and β differs from Model 1 and. 2 The regulatory product dissociates at rate κ, and also immediately upon the initiation of the cycle. Likewise, at φ i = φ p in the cycle, a new product is released from the enzyme.…”

“…Moreover, the number of each species of protein molecules is not always large, in which case we must consider discrete molecules. Considering these aspects, in our previous works on reaction-diffusion systems with enzymes, we used a phase model for each enzyme to study the effects of the delay in the reaction cycle, which is related to conformational changes of the enzyme , (the reaction scheme was originally proposed in ref ). A variety of spatiotemporal patterns, such as traveling waves and spirals, were observed, and the effects of fluctuations on pattern formation were also investigated.…”