Functional sensitivity and mutational robustness of proteins

Sensitivity and robustness appear to be contrasting concepts. However, natural proteins are robust enough to tolerate random mutations, meanwhile be susceptible enough to sense environmental signals, exhibiting both high functional sensitivity (i.e., plasticity) and mutational robustness. Uncovering how these two aspects are compatible is a fundamental question in the protein dynamics and genotype-phenotype relation. In this work, a general framework is established to analyze the dynamics of protein systems under both external and internal perturbations. We introduce fluctuation entropy for the functional sensitivity and the spectrum entropy for the mutational robustness. The compatibility of sensitivity and robustness is analyzed by the optimization of two entropies, which leads to the power-law vibration spectrum of proteins. These power-law behaviors are confirmed extensively by protein data, as a hallmark of criticality. Moreover, the dependence of functional sensitivity and mutational robustness on the protein size suggests a general evolutionary constraint for proteins with different chain lengths. This framework can also establish a general link of the criticality with robustness-plasticity compatibility, both of which are ubiquitous features in biological systems.