Frontiers in Applied Mathematics and Statistics (Jan 2021)
Emergence of Log-Normal Type Distributions in Avalanche Processes in Living Systems: A Network Model
Abstract
Actin is the major cytoskeletal protein of mammal cells that forms microfilaments organized into higher-order structures by a dynamic assembly-disassembly mechanism with cross-linkers. These networks provide the cells with mechanical support, and allow cells to change their shape, migrate, divide and develop a mechanical communication with their environment. The quick adaptation of these networks upon stretch or compression is important for cell survival in real situations. Using atomic force microscopy to poke living cells with sharp tips, we revealed that they respond to a local and quick shear through a cascade of random and abrupt ruptures of their cytoskeleton, suggesting that they behave as a quasi-rigid random network of intertwined filaments. Surprisingly, the distribution of the strength and the size of these rupture events did not follow power-law statistics but log-normal statistics, suggesting that the mechanics of living cells would not fit into self-organized critical systems. We propose a random Gilbert network to model a cell cytoskeleton, identifying the network nodes as the actin filaments, and its links as the actin cross-linkers. We study mainly two versions of avalanches. First, we do not include the fractional visco-elasticity of living cells, assuming that the ruptures are instantaneous, and we observe three avalanche regimes, 1) a regime where avalanches are rapidly interrupted, and their size follows a distribution decaying faster than a power-law; 2) an explosive regime with avalanches of large size where the whole network is damaged and 3) an intermediate regime where the avalanche distribution goes from a power-law, at the critical point, to a distribution containing both 1) and (ii). Then, we introduce a time varying breaking probability, to include the fractional visco-elasticity of living cells, and recover an approximated log-normal distribution of avalanche sizes, similar to those observed in experiments. Our simulations show that the log-normal statistics requires two simple ingredients: a random network without characteristic length scale, and a breaking rule capturing the broadly observed visco-elasticity of living cells. This work paves the way for future applications to large populations of non-linear individual elements (brain, heart, epidemics, … ) where similar log-normal statistics have also been observed.
Keywords
