Mechanical Models of Cell Adhesion Incorporating Nonlinear Behavior and Stochastic Rupture of the Bonds: Concepts and Preliminary Results

Mechanical Models of Cell Adhesion Incorporating Nonlinear Behavior and Stochastic Rupture of the Bonds: Concepts and Preliminary Results

Jean-François Ganghoffer (LEMTA – ENSEM, France)
DOI: 10.4018/978-1-60960-491-2.ch027
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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.
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Cell adhesion is an important phenomenon in biology, especially in immune defence, wound healing, and the growth of tissues. The ability of the cell to divide and give birth of daughter cells is certainly a fundamental feature of the living cell. It is well known that this property strongly depends on adhesion phenomena: most cells can indeed proliferate only if they adhere to a convenient surface. The development and functioning of multicellular organisms includes very often migration of cells on surfaces; this motion relies on the coordination of attachment and detachment processes of molecular bonds (Ndri, 2001). Adhesion is further a key element for the development of vectors for the targeted delivery of medicaments – such as liposomes – which are lipidic pockets transporting active elements (Marques, 2001). Adhesion is a multistep process (Bongrand & Benoliel, 1999; Bongrand, 1982, Limberg, 2002), involving the approach of the cell towards the wall, followed by the critical phase of creation of the first bond. This step is followed by a consolidation step, consisting of an adaptation of the membrane shape, the concentration of receptors in the contact zone, and eventually the reinforcement of the cell membrane in the vicinity of the adhesion zone (Bongrand & Benoliel, 1999).

Cell adhesion is a multiscale phenomenon involved in cell rolling and cell migration (Bongrand et al., 1982; Bongrand & Benoliel, 1999), due to the induced cell motility. Cell adhesion involves complex phenomena that intervene in various biological processes such as the growth of the tissues and the immune response, due to the motion of leukocyte cells (Bongrand et al., 1982). The motion of cells along a wall occurs by two different mechanisms: rolling (e.g., the movement of leukocytes along a blood vein), due to the action of the fluid flow around the cell on the contact interface (Figure 1), and protrusion and retraction, resulting from a modification of the cytoskeleton structure (Bongrand & Benoliel, 1999; Sagvolden et al., 1999). Protrusion is usually associated to the creation of lamellipods and focal contacts on the adherent part of the contact interface (Figure 1).

Figure 1.

Top view: Rolling of a cell connected to an ECM by ligand-receptor pairs. Bottom view: distinct zones of bond adhesion and rupture at the cell-ECM wall interface.

Rolling and active deformation of the cell occur, for instance, during immune defence due to the action of leukocytes, which are transported by plasma flow, captured by the wall by rolling and further move towards the infected zone by active deformations (Jones, 1996).

The creation of new connections is the result of the junction between free and specific molecules, the ligands and the receptors (Bongrand, 1982; Marques, 2001); the failure of existing connections occurs by pulling effects due to the cell motion, which result from the action of various forces. Several models have been developed in the literature in order to describe the adhesion kinetics or the deformation of the cell during the adhesion process: those models can generally be classified into probabilistic approaches (Roberts et al. 1990; Haussy and Ganghoffer 2001) or deterministic modelling strategies (Combs et al. 2004). Other modelling or experimental studies concerning the deformation mechanisms during cell adhesion have been developed by several authors, see (Sagvolden et al, 1999, Lenormand, 2001, Edelman, 1976; Evans & Needham, 1987).

Key Terms in this Chapter

Rolling: Like velcro, ligands on the circulating leukocytes bind to selectin molecules on the inner wall of the vessel. This causes the leukocytes to slow down and begin rolling along the inner surface of the vessel wall. During this rolling motion, transitory bonds are formed and broken between selectins and their ligands.

Rupture: The rupture phenomenon for a physical entity (presently one bond) is the separation in two (or more) pieces under the action of a force or a stress.

Cell Adhesion: Cellular adhesion is the binding of a cell to a surface, extracellular matrix or another cell using cell adhesion molecules such as selectins, integrins, and cadherins.

Stochastic Fields: 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.

Ligand-Receptor Connections: The ligand (Latin ligare = to bind) is a substance that is able to bind to and form a complex with a biomolecule to serve a biological purpose. In a narrower sense, it is a signal triggering molecule, binding to a site on a target protein. The binding occurs by intermolecular forces, such as ionic bonds, hydrogen bonds and Van der Waals forces. Biochemical receptors are large protein molecules that can be activated by the binding of a ligand (such as a hormone or drug). Receptors can be membrane-bound, occurring on the cell membrane of cells, or intracellular, such as on the nucleus or mitochondrion. Binding occurs as a result of non-covalent interaction between the receptor and its ligand, at a location called the binding site on the receptor. A receptor may contain one or more binding sites for different ligands.

Viscoelasticity: This refers to the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time dependent strain.

Statistical Nonlocality: From a general point of view, nonlocality is a direct influence of one object on another distant object. The spectral distribution of power used to model the spatial distribution of rupture thresholds of the ligand-receptor bonds exhibits some nonlocality, as the rupture limit of one bond may be influenced by that of the adjacent and non-adjacent bonds.

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