Proteins are large organic molecules composed of tens, hundreds, or
thousands of amino acid residues bound together by peptide bonds into
a necklace like structure. Unlike other organic molecules, there are
a large number of different conformations any one protein molecule can
take. These structures are determined by a large number of conflicting
and largely canceling forces exerted on the protein residues by the
surrounding solvent and other residues in the protein chain. In one
type of protein, globular protein, the protein molecule can
spontaneously and reproducibly fold to a compact, well defined
structure.
There are many pertinent questions one can ask. For example, how can
we model the folding forces? Can we predict the native structure? Is
it even proper to talk about a single native structure? Is the native
structure the equilibrium structure? In addition to these questions
there are questions about the folding process itself. How does the
protein quickly and reliably find its native structure, an exponentially
small region of its total phase space, and does every protein fold via a
unique pathway or through an ensemble of pathways?
In the strictest sense we know that there cannot be a single pathway by
which a protein folds. Consider this: if an ensemble of denatured proteins
all must pass through a single narrow pathway in their phase space, then
there must be a large reduction in entropy upon entering this path. This
step would consequently be very unlikely and rate limiting. It is much
more likely that proteins fold via many different pathways. Such a
mechanism would allow analysis of protein folding dynamics through
general equilibrium and non-equilibrium statistical mechanics.
This picture of protein folding dynamics, while yielding results
mathematically similar to classical transition state theory, is somewhat
different in spirit. In this picture of two state systems, the barrier
is a free energy barrier: an energetic barrier does not exist. Thus the
transition state is composed of a broad ensemble of structures rather than
one particular structure. This does not mean of course that the transition
state is completely random. The transition state may be characterised by
partial structure in the form of stable pieces of secondary structure or
partially correct backbone shape.
So protein folding can be given a lower order description as a quasi-static
evolution of an ensemble of equilibrium structures from the denatured state
to the native state over a relatively modest free energy barrier. One
method of doing this is to use the Fokker-Planck equation, which is an
approximate equation derived by truncating the full evolution equation
(Kramers-Moyal Equation). By grouping protein states according to a
reaction coordinate, this equation can thus be re-expressed as a diffusion
equation.
So what constitues a well-designed protein (well designed as a folder,
not well designed functionally)? Well designed proteins are those
proteins which have a large diffusion constant at temperatures below which
the native state is stable and populated. That is, well designed proteins
can quickly find the native state. (Of course, for real proteins this
temperature must also be between zero and one hundred celsius.) Probably
one reason why a protein's native state is only marginally stable is that
greater stability of the native state would result in less specific
residue-residue interactions leading to a much lower diffusion constant
and difficulty folding. This problem is evident in proteins with
sulfur bridges: misformed sulfur bridges lead to a slowing down in the
folding process.
As mentioned already, statistical thermodynamics along with a diffusion
constant should be used to characterize the early stages of protein folding.
This is often encapsulated in the now famous funnel picture.
