Biomedical Engineering Reference
In-Depth Information
These mechanical and thermodynamic models are highly complex and with
many parameters, yet they are still idealized and rely on a number of nontrivial
assumptions, that often can not be supported by sufficient quantitative experimental
observations. Furthermore, discrepancies appear when one tries to equalize the
adhesion energy expression of the thermodynamic models with the one of the me-
chanical models. The disagreement is due to the inadequacy of the two-dimensional
approximation of the thermodynamic analyses in treating the contribution of all the
degree of freedom of a flexible tape in a three-dimensional space [
32
].
Recently a reaction kinetics approach has been reconsidered and it has been
developed along with the experimental technologies. Kinetic models describe
adhesive interaction using chemical reaction kinetics and relating the kinetic rates
to the force applied to the rolling cell. They were conceptualized by Bell [
2
]
and then refined by several authors [
3
,
8
,
16
] (See Appendix A for more details).
Reaction kinetics is used in biology for describing the binding of ligands to cell
surface receptors. The rationale for applying this framework to adhesive interactions
comes from the physical characteristics of these non-covalent specific interactions:
the receptor-ligand binding can be modeled as a key-to-lock type of interaction
dynamically governed by reaction rates. The nature of the key-to-lock molecular
interactions presents three inter-dependent features: (a) concurrency of different
kinds of interactions that contribute to the expression of the same phenomenon, (b)
inherently random competition between the various species of ligands in forming
bonds with the same species of receptors, (c) discrete character of the quantity of
components and low number of bond-mediated adhesion (single-bond condition).
The inclusion of the kinetic equations into the mechanical and thermodynamical
model enables the solution of the bond density, which, when combined with the
force on a bond, provides the adhesive stress.
The three categories of models, mechanic, thermodynamic and kinetics, are
usually deterministic and based on ordinary differential equations and algebraic
equations. However, an ordinary differential equation model of the kinetics of
the lymphocyte recruitment does not capture the concurrency and the parallelism
that are inherent to the network of interactions. Moreover, usually deterministic
differential equations are used to simulate a dynamics that is intrinsically stochastic,
as the amount of the adhesion molecules responsible for the processes of tethering
and rolling of the lymphocyte on the endothelium is low.
An alternative to the common approaches to model the kinetics of the molecular
interactions governing the lymphocyte recruitment is a process calculus approach.
Process calculi can represent interactions between independent concurrent and
parallel processes as communication (message-passing), rather than as the mod-
ification of shared variables. They can describe processes and systems using a
small collection of primitives, and operators for combining those primitives. Finally,
process calculi define algebraic laws for the process operators, which allow process
expressions to be manipulated using equational reasoning.
We focused on a stochastic extension [
25
,
26
]ofthe-calculus [
21
]. The
biochemical stochastic -calculus is an efficient alternative to the ordinary differ-
ential equations since it provides a stochastic modelling framework and it allows to
