A spin glass representation of hierarchical gene regulation and the partial resolution of frustration

Gene regulatory networks are directed and often contain loops. A loop can place competing demands on its genes, so that no expression state satisfies every regulatory interaction. In statistical physics such a conflict is called frustration. Existing spin-glass models of tissues assume symmetric regulation, and studies of frustration consider single cells. We develop a nonreciprocal multicellular Ising model that couples a directed regulatory network in each cell to random, symmetric signalling between neighbouring cells. We define intracellular and intercellular frustration densities, which count the unsatisfied interactions within a cell and with its neighbours. We study six exemplar nine-gene networks that share a downstream cascade and differ in an upstream three-gene motif. In isolated cells the motif sets the frustration. Negative feedback loops pass frustration around the loop, and their genes keep switching. The incoherent feedforward loop holds frustration at a single gene and freezes. When cells communicate, they form spatial clusters with shared expression profiles, and conflict between neighbours concentrates at cluster boundaries. Stronger coupling lowers intercellular frustration and raises intracellular frustration. Frustration is therefore resolved only in part: cells satisfy more of the couplings to their neighbours by satisfying fewer of their own regulatory interactions. Varying the coupling strength reveals a transition near the value at which the combined input of a cell's eight neighbours matches a single regulatory interaction. Below it, cells follow their own networks and the tissue fragments. Our framework links the structure of gene regulatory networks to the stability of gene expression in cells and tissues.