Glycolysis, connected with the consumption of the sugar (glucose), is one the most important metabolic processes supplying energy for cellular processes. Additionally to its value in fundamental biophysics, studies of the glycolytic processes attract recently especial attention as an important mechanism for the biofuel (namely, bioethanol) production. It is known that oscillating behaviour can have benefits over steady states in the sense of efficiency of glucose utilization. Thus, these reasons induce the question about external regulation of oscillating glycolytic reaction.

Since the propagation dynamics and shape of traveling reaction-diffusion waves in living cells (for example, neutrophils and yeast) depend on both intra- and intercellular states, it is supposed that propagation of traveling glycolytic waves in the living cells plays an important role in signal transduction and energy control.

Usually, the mathematical models providing the traveling waves and other patterns do not take into account a reactor thickness. At the same time, the terms containing positive and negative feedback loops can physically belong to the different parts of a reactor in the case of glycolytic experiment.

Thus, we propose the following scheme, where reaction terms in the bulk consists of only the terms corresponding to the irreversible chemical reaction. And the boundary conditions collects the terms describing substrate influx as well as product degradation and outflow through the reactor’s bottom in the normal direction.

We assume that inward waves are induced by the inhomogeneous influx. In particular, we model this substrate inflow profile by an axially symmetric paraboloid, which has a minimum in the bottom center of oneside fed gel  reactor. The simulation has been conducted within a whole disk. In order to present the details of wave propagation, one sector is cut out, see the animation below, where the product concentration is expressed via pseudocolor scale.

[hana-flv-player video=’/videos/Selkov3D.flv’/]

Thus, we have found that self-sustained oscillations can emerge in the systems, where only autocatalytic reaction exists. In our case, the required negative feedback can be replaced with the proper boundary condition.  In our simulation, the inward waves are definitely phase waves formed by the inhomogeneous influx that  quite good corresponds to the resently known experimental finding [Bagyan et. al] .

Details can be found in the articles:

• E.B. Postnikov, A.Yu. Verisokin, D. V. Verveyko, A.I. Lavrova. Self-sustained biochemical oscillations and waves with a feedback determined only by boundary conditions. Physical Review E. 2010. V. 81. 052901.
• A.Yu. Verisokin, D.V. Verveyko, E.B. Postnikov, A.I. Lavrova. Model of Glycolytic Traveling Waves Control in 3D Spatial Reactor. IEEE Control Applications, (CCA) & Intelligent Control, (ISIC), 2009. pp. 194 — 198