Advanced Chemical Computing Using Discrete Turing Patterns in Arrays of Coupled Cells

Abstract

We examine dynamical switching among discrete Turing patterns that enable chemical computing performed by mass-coupled reaction cells arranged as arrays with various topological configurations: three coupled cells in a cyclic array, four coupled cells in a linear array, four coupled cells in a cyclic array, and four coupled cells in a branched array. Each cell is operating as a continuous stirred tank reactor, within which the glycolytic reaction takes place, represented by a skeleton inhibitor-activator model where ADP plays the role of activator and ATP is the inhibitor. The mass coupling between cells is assumed to be operating in three possible transport regimes: (i) equal transport coefficients of the inhibitor and activator (ii) slightly faster transport of the activator, and (iii) strongly faster transport of the inhibitor. Each cellular array is characterized by two pairs of tunable parameters, the rate coefficients of the autocatalytic and inhibitory steps, and the transport coefficients of the coupling. Using stability and bifurcation analysis we identified conditions for occurrence of discrete Turing patterns associated with non-uniform stationary states. We found stable symmetric and/or asymmetric discrete Turing patterns coexisting with stable uniform periodic oscillations. To switch from one of the coexisting stable regimes to another we use carefully targeted perturbations, which allows us to build systems of logic gates specific to each topological type of the array, which in turn enables to perform advanced modes of chemical computing. By combining chemical computing techniques in the arrays with glycolytic excitable channels, we propose a cellular assemblage design for advanced chemical computing.

Chemical Memory with Discrete Turing Patterns Appearing in the Glycolytic Reaction

“…Early experiments with BZ reaction systems showed oscillation death regimes (discrete Turing patterns) in reactors coupled by peristaltic pumps [ 44 , 45 , 46 ] or different types of valves [ 47 ] between adjacent reactors instead of cellular membranes. The experimental findings in terms of transitions between oscillations and discrete non-uniform patterns were supported by models of two, four, and twenty coupled cells [ 31 , 32 , 48 ]. These transitions represented the basics of our chemical computing techniques and were therefore used also for three and four-coupled cells with various layouts to work as multiargument logic operations [ 33 , 48 ].…”

“…The experimental findings in terms of transitions between oscillations and discrete non-uniform patterns were supported by models of two, four, and twenty coupled cells [ 31 , 32 , 48 ]. These transitions represented the basics of our chemical computing techniques and were therefore used also for three and four-coupled cells with various layouts to work as multiargument logic operations [ 33 , 48 ]. Here we are concerned with testing these patterns as potential memory states.…”

“…In order to minimize the numerical complexity of the problem and to neglect reaction steps not responsible for dynamic behavior, we selected the model with concentrations of just two reactants, namely ATP (the variable denotes its time-dependent concentration) and ADP (described by ). The same simple model was used in a few papers on binary logic gates operating on the information coded in discrete Turing patterns [ 32 , 33 , 48 ]. The progress of glycolytic reaction proceeding in a continuously stirred reactor (CSTR) is described with the following equations: Here describes the ATP inflow rate, is the inhibition rate coefficient, M is the Michaelis constant, n gives the Hill coefficient, L is the allosteric constant representing affinity of the PFK to its reactive conformation R rather than to its non-reactive conformation L [ 63 ], denotes the autocatalysis rate coefficient, is the ratio between dissociation constants of ATP to ADP, and, finally, represents the rate coefficient of ADP degradation.…”

“…The colored arrows directed toward the nodes mark time-dependent inflows of ATP (the first term on the right side of Equation ( 1 )). Following the previous studies on interconnected glycolytic reactors [ 32 , 33 , 48 ], we assume that the nodes interact due to flows of reagents between them. We consider a fully symmetric network with identical flows between nodes with the rate .…”

“…The equations describing network evolution have the form: where symbols i and j index the nodes. In our simulations, we followed previous studies on discrete Turing patterns in systems with glycolytic reactions and used [ 32 , 33 , 48 ].…”