In probability theory, a stochastic process, sometimes called a random process, is the counterpart to a deterministic process. Instead of dealing with only one possible reality of how the process might evolve over time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possible paths the process might take, but some paths may be more probable and others less. In the simplest possible case (discrete time), a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values.
Published in Chapter:
Mechanical Models of Cell Adhesion Incorporating Nonlinear Behavior and Stochastic Rupture of the Bonds: Concepts and Preliminary Results
Jean-François Ganghoffer (LEMTA – ENSEM, France)
Copyright: © 2011
|Pages: 29
DOI: 10.4018/978-1-60960-491-2.ch027
Abstract
The rolling of a single biological cell is analysed using modelling of the local kinetics of successive attachment and detachment of bonds occurring at the interface between a single cell and the wall of an ECM (extracellular matrix). Those kinetics correspond to a succession of creations and ruptures of ligand-receptor molecular connections under the combined effects of mechanical, physical (both specific and non-specific), and chemical external interactions. A three-dimensional model of the interfacial molecular rupture and adhesion kinetic events is developed in the present contribution. From a mechanical point of view, this chapter works under the assumption that the cell-wall interface is composed of two elastic shells, namely the wall and the cell membrane, linked by rheological elements representing the molecular bonds. Both the time and space fluctuations of several parameters related to the mutual affinity of ligands and receptors are described by stochastic field theory; especially, the individual rupture limits of the bonds are modelled in Fourier space from the spectral distribution of power. The bonds are modelled as macromolecular chains undergoing a nonlinear elastic deformation according to the commonly used freely joined chains model, while the cell membrane facing the ECM wall is modelled as a linear elastic plate. The cell itself is represented by an equivalent constant rigidity. Numerical simulations predict the sequence of broken bonds, as well as the newly established connections on the ‘adhesive part’ of the interface. The interplay between adhesion and rupture entails a rolling phenomenon. In the last part of this chapter, a model of the deformation induced by the random fluctuation of the protrusion force resulting from the variation of affinity with chemiotactic sources is calculated, using stochastic finite element methods in combination with the theory of Gaussian random variables.