Identifying network topologies that can generate turing pattern

Turing pattern provides a paradigm of non-equilibrium self-organization in reaction-diffusion systems. On the basis of many mathematical studies, it has been proposed that various biological development processes use Turing instability to achieve periodic patterns. In this paper, we introduce a framework to systematic identify network topologies that are capable for Turing pattern formation. All possible 2, 3-node genetic regulatory networks are enumerated and linear stability analysis is applied to access their ability to generate Turing instability. We find that all 3-node networks that can achieve Turing pattern can be mapped to either pure or cross activator-inhibitor mechanisms, and the pure activator-inhibitor system is more robust for Turing pattern formation than the other one. Additional linkages can further increase the performance of the circuit by either introducing another core topology or complementing existing regulations. Moreover, we find that addition of a fixed node enables the formation of Turing pattern even when the diffusion coefficients of two morphogens are fairly close to each other. Our results provide the design principle of robust circuits for Turing pattern generation and can be further applied for systematically exploring other bifurcation phenomena.