Strong disorder leads to scale invariance in complex biological systems

Despite the innate complexity of the cell, emergent scale-invariant behavior is observed in many biological systems. We investigate one example of this phenomenon: the dynamics of large complexes in the bacterial cytoplasm. The observed dynamics of these complexes is scale invariant in three measures of dynamics: mean-squared displacement (MSD), velocity autocorrelation function, and the step-size distribution. To investigate the physical mechanism for this emergent scale invariance, we explore minimal models in which mobility is modeled as diffusion on a rough free-energy landscape in one dimension. We discover that all three scale-invariant characteristics emerge generically in the strong disorder limit. (Strong disorder is defined by the divergence of the ensemble-averaged hop time between lattice sites.) In particular, we demonstrate how the scale invariance of the relative step-size distribution can be understood from the perspective of extreme-value theory in statistics (EVT). We show that the Gumbel scale parameter is simply related to the MSD scaling parameter. The EVT mechanism of scale invariance is expected to be generic to strongly disordered systems and therefore a powerful tool for the analysis of other systems in biology and beyond.