Stability of all 3- and 4-node interaction networks

We invite you to browse the data here: https://oceanchaos.shinyapps.io/motif/
Protein-protein interaction networks (PPIs) are large and complex, yet self-organize to sustain all known life. To help understand the building blocks of these systems, we analyze small subgraphs (called "motifs" in this study) where all proteins can interact positively, negatively or not at all. There are 132 such 3-protein networks that are topologically unique (A->B->C is the same as B->C->A and C->B->A and B->A->C etc, so only such network is simulated), and 22,662 4-protein networks. Using the model d[A]/dt = (synthesis rate) - (degradation rate)*[A] + kBA*[B]*[A]^(Hill coefficient) + kCA*[C]*[A]^(Hill coefficient) + ..., we choose 1,000 biologically plausible values for the parameters, find all fixed points of the system, and determine the stability of those points. Previous research had suggested that a switch from a stable fixed-point equilibrium, to a limit cycle, to divergence to infinity was to be expected as the three possible behaviors. Instead, we determined that the behavior space of the typical motif included many different numbers of fixed points in complex combinations of stable and unstable (a median of 6 and 12 distinct fixed point combinations). We also discover that divergence to infinity is not at all uncommon at known biological values, despite such a result likely being fatal to the cell, suggesting a substantial degree of selection and regulation.

The zip file includes:
- Lists of which motifs are isomorphic to each other (3-motifs: isomorph_list.txt; 4-motifs: isomorph_list4.txt)
- Julia code for generating our results for 3- and 4-motifs (farm3_log.jl and farm4_log.jl respectively). These were run on a supercomputer cluster for ~25k hours of CPU time to get 1,000 results for one member of each isomorph group. The results are stored in:
- stab[n]_[motif id]_pars.txt files, storing the parameter values of all simulations for n-motifs
- stab[n]_[motif id]_spec.txt files, which provide the spectral radius of the system's Jacobian at each fixed point. (-999 means no fixed points were found at the parameter values)
- stab[n]_[motif id]_aux.txt files, in the format [time to solve the system] [n numbers representing log-protein concentrations at the first fixed point] [the second fixed point] ...
Note that while these results are informed by biochemical networks, they are very general and would apply to any network.