The entropy can also be written using statistical arguments without the use
of terms such as randomness. As originally shown by Ludwig Boltzmann
in the late 1800s, the entropy Sof a system can be written in terms of the
parameter W:
S=kBlnW (8.20)
The parameter Wrepresents the number of different configurations of mole-
cules that result in the same energy. If there is only one configuration for
a molecule the entropy is zero since ln 1 =0. As the number of possible
configurations increases, the entropy will increase. With these expressions,
it is possible to derive relationships among different parameters that are
identical to those found based upon the laws of thermodynamics.
Research direction: protein folding and prions
Statistical approaches toward understanding biological systems are becom-
ing increasingly important in biology and biochemistry. In genomics, the
expression levels of thousands of genes are now routinely monitored using
so-called DNA chips. As the number of organisms with their entire genome
sequenced increases, the next step of using the availability of every gene
to understand the properties of organisms remains a challenge. The genes
can be individually placed on microarrays, providing the opportunity for
massive parallel monitoring of the degree of gene expression in response
to outside stimuli, such as temperature changes. For a number of organ-
isms, commercial products are available that allow the monitoring of most
or all genes of certain organisms ranging from Escherichia colito humans.
Meaningful interpretation of the output of these arrays requires careful
consideration of the best statistical analysis of the data.
Aside from data analysis, the question of how states with conforma-
tions of different energy find the lowest-energy state is critical for protein
folding. In cells, proteins are constantly produced and fold into unique
conformations, despite the need for transport across cell membranes and
assembly into large complexes with cofactors, which is often assisted by
proteins called chaperones. If proteins were free to fold without direc-
tion the number of possible conformations would be too large for a unique
shape to always be achieved, in what is termed the Levinthal paradoxafter
Cyrus Levinthal who originally posed the paradox in 1968. For example,
if a protein has 50 amino acid residues that each can have one of 10
different conformations, then the entire protein has 10^50 different possible
conformations. If each of these conformations had equal weight then the
protein would never be able to randomly sample a sufficient number to
determine the best conformation. Even a very rapid sampling time of 1 ps
per conformation would require 10^38 s, and the cell containing the protein
would be long dead before even a modest set of possibilities were sampled.
168 PARTI THERMODYNAMICS AND KINETICS
