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In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time-and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single second-order hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

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In the early Drosophila embryo, BMP-type ligands act as morphogens to suppress neural induction and to specify the formation of dorsal ectoderm and amnioserosa. Likewise, during pupal wing development, BMPs help to specify vein versus intervein cell fate. Here, we review recent data suggesting that these two processes use a related set of extracellular factors, positive feedback, and BMP heterodimer formation to achieve peak levels of signaling in spatially restricted patterns. Because these signaling pathway components are all conserved, these observations should shed light on how BMP signaling is modulated in vertebrate development.