Biology Reference
In-Depth Information
FIGURE 3.1
Self-assembly of
floating magnetic tiles into arrays.
occupied by the structure to be assembled. Phase space is an abstraction that represents, as
a single location in multidimensional space, the location and momentum of every subunit
in the system at any one time.
)
Neglecting momentum for a moment and concentrating on
location, the state space for a single particle can be represented in three-dimensional space
simply by the
x
,
y
,
z
coordinates of that particle. The locations of two particles can be rep-
resented together as a unique location in six-dimensional space, specified by the three
coordinates of each particle, and the location of
n
particles can be represented simultan-
eously by a unique location in (3
n
)-dimensional space [(6
n
)-dimensional when momenta
are included]. A system will tend to 'explore' its multidimensional state space, the move-
ment of the unique point in state space being dictated by the interactions between the
particles it contains. Some locations in state space will represent states in which all par-
ticles are unbound, some will represent partially completed assemblies and some will
represent a complete assembly. If the volume of state space that represents an acceptable
assembly is very small in comparison to the volume of state space itself, it is highly
unlikely that the system will ever visit this volume and assembly is therefore unlikely in
a reasonable amount of time. This would be the case if, for example, many subunits had
to come together in just one precise order and orientation in space for a structure to be
formed. If, on the other hand, the volume of state space that represents an acceptable
assembly occupies a reasonable fraction of state space, the system is much more likely
to enter this volume. This would be the case, for example, if all subunits were identical
and could come together in any order to make a filament that would be acceptable what-
ever its orientation.
The foregoing abstraction boils down to a common-sense conclusion: self-assembly works
well for uncomplicated structures but is less useful for ones that are complex or have to be
orientated very precisely.
An intuitive feel for the process of self-assembly can be provided by macroscopic models
of the process, one of the best known of which consists of small, light, magnetic tiles that are
floated on to water in a Petri dish (
Figure 3.1
). The dish is subjected to gentle agitation, to
provide a macroscopic analogue of
the Brownian motion that keeps the molecular
)
A clear and accessible introduction to phase space appears in Chapter 5 of R. Penrose's
The Emperor's
New Mind
.
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