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Bow-tie signaling in c-di-GMP: Machine learning in a simple biochemical network

Fig 5

Mathematical model reveals how the c-di-GMP network fitness depends on the strength of selection and on the number of sensors.

A: The environment history was modeled as a succession of n binary environments E, 0-black or 1-white, corresponding respectively to motility- or biofilm-favoring environments. Stimuli (X) were generated from each environment by introducing noise to the original signal; the expressed phenotypes Y were calculated from the β’s and the matrix X; the fitness of the network in each environment is the agreement between the expressed phenotype and the favored phenotype in that environment; the fitness across the n environments is the geometric mean fitness. B: The fittest network was calculated using logistic regression algorithm. and are the fitting parameters of the unbiased network for infinite history. C: The fittest network was presented to a new environment and a new set of stimuli and we calculated the expressed phenotype, as well as the fitness in that new environment. D: The fitness in changing environments depended strongly on n, the number of environments that tuned the c-di-GMP network during strain evolution. Strong selection selected for networks adapted to recent environment (small n) favoring specialists; weak selection provided the opportunity to learn from a long history of fluctuating environments (large n), favoring generalists. E: The fitness achieved by a c-di-GMP network depends on the number of sensory modules (m) and the length of evolutionary history (n, where small n corresponds to strong selection and large n corresponds to weak selection). Networks with too many sensors (m > n/2) performed well in the past but poorly in the future. The curves presented in D-E were obtained from numerical simulations of the scheme described in A-B-C (Logistic regression over a m × n matrix followed by the estimation of the fitness of the network on one new environment; 1000 independent simulations per conditions m, n; η = 0.6). Arithmetic mean was used to average these simulation results.

doi: https://doi.org/10.1371/journal.pcbi.1005677.g005