10 HP model and Boltzmann distribution

10 HP model and Boltzmann distribution - 10. HP model for...

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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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10 HP model and Boltzmann distribution - 10. HP model for...

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