A power law for cells

Darwin observed that multiple, lowly organized, rudimentary, or exaggerated structures show increased relative variability. However, the cellular basis for these laws has never been investigated. Some animals, such as the nematode Caenorhabditis elegans, are famous for having organs that possess the same number of cells in all individuals, a property known as eutely. But for most multicellular creatures, the extent of cell number variability is unknown. Here we estimate variability in organ cell number for a variety of animals, plants, slime moulds, and volvocine algae. We find that the mean and variance in cell number obey a power law with an exponent of 2, comparable to Taylor's law in ecological processes. Relative cell number variability, as measured by the coefficient of variation, differs widely across taxa and tissues, but is generally independent of mean cell number among homologous tissues of closely related species. We show that the power law for cell number variability can be explained by stochastic branching process models based on the properties of cell lineages. We also identify taxa in which the precision of developmental control appears to have evolved. We propose that the scale independence of relative cell number variability is maintained by natural selection.

Nonuniformity of the constitutive law parameters for shear rupture and quasistatic nucleation to dynamic rupture: a physical model of earthquake generation processes.

Based on the recent high-resolution laboratory experiments on propagating shear rupture, the constitutive law that governs shear rupture processes is discussed in view of the physical principles and constraints, and a specific constitutive law is proposed for shear rupture. It is demonstrated that nonuniform distributions of the constitutive law parameters on the fault are necessary for creating the nucleation process, which consists of two phases: (i) a stable, quasistatic phase, and (ii) the subsequent accelerating phase. Physical models of the breakdown zone and the nucleation zone are presented for shear rupture in the brittle regime. The constitutive law for shear rupture explicitly includes a scaling parameter Dc that enables one to give a common interpretation to both small scale rupture in the laboratory and large scale rupture as earthquake source in the Earth. Both the breakdown zone size Xc and the nucleation zone size L are prescribed and scaled by Dc, which in turn is prescribed by a characteristic length lambda c representing geometrical irregularities of the fault. The models presented here make it possible to understand the earthquake generation process from nucleation to unstable, dynamic rupture propagation in terms of physics. Since the nucleation process itself is an immediate earthquake precursor, deep understanding of the nucleation process in terms of physics is crucial for the short-term (or immediate) earthquake prediction.