10.
HP model for Protein Folding
Protein structures can be investigated using energy functions and trying to ﬁnd
the native structure by looking for the conﬁguration with lowest energy. Often
ignored in these computations are the facts that proteins obey laws of statistical
mechanics, and that temperature is an important factor.
Statistical mechanics tells us that not every protein in a large group of them
has the lowest energy. The energies are random but they obey certain statistical
laws based on the Boltzmann distribution. Also it is known that proteins become
disordered and unfolded at high temperature.
The Boltzmann distribution is a law of statistical mechanics related to random
distributions of energies of a large number of molecules. The following general
principle applies to many phenomena in statistical mechanics:
The most likely thing
to happen is what is observed.
This tendency towards the most likely conﬁguration
is related to
entropy
. Entropy is a measure of randomness, and physical systems
move towards higher entropy. The Boltzmann distribution is also important in the
computational technique of simulated annealing. In this techniques parameters are
varied randomly to search for the minimum of a function.
Probability is measured by counting the number of possibilities. A large number
of molecules called an ensemble. Each molecule has a diﬀerent energy. The prob
ability of a certain energy is given by the number of molecules having that energy
divided by the total number of molecules. Using combinatorial arguments we can
explain the Boltzmann distribution
(1)
P
(
E
)
∼
A
exp (

βE
)
where
P
(
E
) is the probability that a molecule has energy
E
. Here
β
is a constant
which is a function of temperature, and
A
is a constant chosen to assure that the
total probability is 1. This distribution occurs in the situation where the energy is
random, but the total energy is ﬁxed. We see that the lowest energy is the most
likely, but there is a probability that the molecule has higher energy.
10.1.
Random distribution of balls in boxes.
To understand the Boltzmann
distribution start with a simple combinatorial problem, balls tossed randomly into
boxes. The balls can be thought of as molecules and the boxes as states. At ﬁrst
we ignore energy.
Combinatorial problem I: Given
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 Fall '11
 staff
 Statistical Mechanics, Entropy, NJ

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