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# MC - Monte Carlo BIOL 7110 CHEM 8901 BIOL 4105 CHEM 4804...

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2/22/11 1 Monte Carlo BIOL 7110 / CHEM 8901 / BIOL 4105 / CHEM 4804 February 22, 2011 Determination of Quantities of Interest Macromolecular thermodynamic properties Configurational properties (radius of gyration, etc.) Hydrodynamic properties (sedimentation coefficient, etc.) Many others … Experimentally , thermodynamic properties are macroscopic averages, i.e., ensemble averages Computationally , thermodynamic properties must be calculated from ensemble averages Computationally , we can use molecular dynamics or Monte Carlo to generate ensembles of structures How to Calculate Macroscopic Properties from Molecular Mechanics Simulations? Generate a very large number of conformations, N , using either molecular dynamics or Monte Carlo This collection approximates the thermodynamic ensemble Calculate the quantity of interest α i and the energy E i for each conformation i = 1, N. The expected (macroscopic) value of α is This emphasizes the importance of the partition function < α >= α i e E i / RT i e E i / RT i Z = e E i / RT i

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2/22/11 2 How to Calculate Macroscopic Properties from Molecular Mechanics Simulations? How accurate is our estimate of α ? < α >= α i e E i / RT i e E i / RT i How accurate is the energy function? How accurately does the collection of N conformations represent the true thermodynamic ensemble? How well have we covered conformational space? (How well converged is our estimate of α ?) LATER: How to improve the rate of convergence using importance sampling Monte Carlo is a Sampling Method Used to Estimate Quantities of Interest Example 1: The game of craps Throw a pair of dice 7 or 11 = instant winner 2, 3 or 12 = instant loser Any other throw –> your number (4, 5, 6, 8, 9, 10) Keep throwing the dice until e ither: You throw your number again (winner) or You throw 7 (loser) What are the odds at craps?
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