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Chapter 5:

Simulating Protein Self-Assembly

Sic parvis componere magna solebam

(In this manner I was accustomed to compare great things to small.)

________________________________- Vergil


There are many dynamic systems within the cell, and in other small-scale systems, that are difficult to fully understand, and hard to study using conventional experimental analysis. The exact method by which proteins of the cytoskeleton combine, the way that viruses assemble out of their component proteins, and the general interaction of proteins and other macromolecules are all difficult to determine by observation alone.

Direct inspection of living tissue is usually unable to resolve these interactions, except in aggregate form, while higher magnification methods generally work only on specially prepared tissue. The processing needed to prepare specimens for most high magnification techniques, such as electron microscopy, usually kills and immobilises the tissue, halting any dynamic processes. While some preparatory techniques such as rapid freezing may preserve a snapshot of a dynamic process, it is still often difficult to determine the full process, although such frozen specimens can provide useful data about such dynamic processes. (1)

Indirect methods can, however, be very useful. Techniques such as protein sequencing and X-ray crystallography can reveal the shapes of proteins, and these and other techniques can be used to determine what areas of the protein are active and can bind to other molecules. They and may even suggest how a protein may change its conformation, and under what conditions. Other methods, such as analysing reaction rates and deducing early states, can also be very useful, as will be shown below.

Despite all these methods, however, it can still be difficult to establish the combinatorial nature of macromolecular interactions at the nanometre scale. Even if the initial conditions and the end result are known, the manner by which the reaction proceeded may still be uncertain. Establishing this method can reveal many interesting things about how structures at this scale function.

Mathematical Models

The most successful method for studying dynamic processes so far has been mathematical analysis of experimental results. Using observations of bulk properties such as gross polymerisation rates, a surprising amount of detail can be inferred.

The Oosawa Model

Fumio Oosawa was a pioneer of this approach. Using a simple mathematical model of polymerisation similar to that used in polymer chemistry, he and his co-workers were able to accurately model the assembly of a number of self-assembling proteins. Much of their work concentrated on actin and bacterial flagella, but they also reproduced a variety of other, less easily modelled, self-assembling protein structures. (2)

The Oosawa model, in essence, treats the growth rate of a polymer as being independent of its length, once the polymer is larger than a certain critical size, known as the 'stable nucleus' size. For a polymer larger than this size, the growth of the polymer can be expressed as a combination of a growth rate, multiplied by the concentration of 'available' free monomers that can be bound, minus the disassociation rate (or loss rate) of bound monomers breaking away from the polymer.

For a given polymer of sufficient size, the growth rate G is

G = (5.1)


k + = the binding rate of free monomers to the polymer

k -- = the loss rate of bound monomers from the polymer

c1 = the concentration of free monomers to be bound

Applied to a single polymer, the above equation must be interpreted in terms of probabilities, since the binding and loss rates are only accurate in aggregate. However, this equation can be used to provide a description of the change in the population of all polymers with sub-units of size i, and hence a description of the population of all polymers; (3)



ci = the concentration of polymers of size 'i' sub units

t = time

k+ = the binding rate of free monomers to the polymer

k -- = the loss rate of bound monomers from the polymer

c1 = the concentration of free monomers to be bound

This equation tells us that the number of polymers of a particular size i subunits