Biomedical Engineering Reference
In-Depth Information
outcomes in well-defined situations, and (4) current theoretical work aims at find-
ing more detailed links between structure and function. However, it is well known
that “... In the real world, exact solutions are the notable exception” [164]. Further,
analytic solutions may be too complicated to be of real use.
During the past decades, the increasing availability and power of computers trig-
gered the development of simulation as a general tool for trying to bring insights into
models that are too complex to allow simple mathematical description. The basic
idea consists of observing the behavior of a model system containing enough ingre-
dients to mimic a realistic situation. We shall now give a few examples to illustrate
the kind of information that can be obtained with this approach. However, a word of
caution as found in the introduction of a recent treatise of molecular dynamics simu-
lation may be useful: “... Simulations must be kept honest, because seing is believing,
and animated displays can be very convincing irrespective of their veracity” [164].
We shall sequentially consider two approaches of growing complexity.
Simulations as numerical solutions of simple equations.
Even fairly simple mod-
els cannot be described with plain mathematical formula. As an example, when
flow chambers are used to study the motion of receptor-coated microspheres near
ligand-bearing surfaces, a quantitative determination of the rate of bond formation
requires a knowledge of the frequency and duration of molecular encounters. The
distance between spheres and surfaces must therefore be known with millisecond
and nanometer accuracy, which is difficult to achieve with present day techniques.
This was an incentive to mimic the motion of microspheres near surfaces in the pres-
ence of hydrodynamic shear and account for Brownian motion and hydrodynamic
interactions between the sphere and the wall [149] [152]. This approach was neces-
sary to obtain quantitative information on the kinetic rate of bond formation [149].
The validity of simulations was subjected to theoretical [152] and experimental [168]
check. In a later study, numerical simulations were found useful to demonstrate that
the bond formation between a ligand and a receptor might be viewed as the progres-
sion of the complex along a linear segment of a rugged energy landscape rather than
the passage across an energy barrier, as was usually considered [169].
Simulating complex systems: molecular dynamics and related approaches.
The basic
idea of molecular dynamics was reported by Alder and Wainwright [2] in the late
50s. They modeled a system made of hard spheres in a box, starting with a ran-
dom configuration defined as positions and velocities. Then, they applied the law of
mechanics to follow the displacements of all spheres through a series of ultrashort
steps, and the new positions and velocities of spheres were recorded at each step.
Fourteen years later, the motion of several tens of model water molecules interacting
with more realistic potentials was similarly simulated [163]. The continual growth
of computer power allowed investigators to study models of increasing complexity,
until they were able to simulate interactions between proteins molecules, including
tens of thousands of atoms, and surrounded by hundreds of water molecules. The
most detailed approach consisted of modeling interactions between pairs of atoms
or chemical groups with semiempirical potential functions derived from quantum
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