This work is focused on analysis of model systems of linear and cyclic arrays of coupled cells capable of parallel chemical computing based on reaction-diffusion patterns originally described by A. Turing.
In each cell glycolytic oscillatory reaction is assumed, which we model using a core model of glycolysis proposed by Moran and Golbeter (1984). This model reflects a major oscillatory component of glycolysis - the allosteric enzyme phophofructokinase (PFK) - and includes negative feedback loop provided by other enzymes in the glycolytic reaction chain. Mass exchange between cells is described by diffusion-like coupling. The two components of core model of glycolysis - ATP and ADP - have their own transport rate coefficients. While it is well known that Turing patterns (nonuniform stationary states) can emerge spontaneously when the inhibitor transport rate coefficient is higher than the activator transport rate coefficient, we found conditions such that Turing patterns occur even for equal transport rate coefficients and can be accessed by applying suitable perturbations.
Each type of array is analysed using numerical continuation methods to find parameters necessary for the occurrence of Turing patterns. Solution/bifurcation diagrams providing information about concentration levels of Turing patterns for chosen parameters are determined.
We show that precise sequences of perturbations induce specific transitions between Turing patterns. These input perturbations can be taken as input logic signals 1 and 0. The concentration level of ADP in each cell is taken either as logic 1 or logic 0 output signal. In case the input perturbation will cause the system to oscillate, additional "knockout" perturbations are applied so that the system makes transition to a specific Turing pattern. Thus each cell has its own steady state output signal (either 1 or 0) after a controlled transient settles on a Turing pattern. The system provides as many output signals as is the number of cells in the array, thus it is capable of parallel computing.
An array of two coupled cells is a core system displaying simple logic functions AND, OR, NAND, NOR. It can be used as the smallest piece of chemical computing processor.
A linear array of three coupled cells provides insight into more complex functions combining two simple logic functions including exclusive OR (XOR), whereas a cyclic 3-array provides tautology in addition to logic functions provided by the linear 3-array
A linear array of four coupled cells can be used to transmit parallel synchronous signals due to inherent symmetry, while a cyclic 4-array performs four different complex logic functions at a time.