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Statistical Mechanics and Models: Probability, Ensemble Averages, and Helix Formation

This article covers topics in statistical mechanics and models, including probability, ensemble averages, and helix formation. It discusses the canonical partition function, probability expression rearrangement, ensemble averages, and the utility of the partition function. It also explores helix formation models and their parameters, as well as the partition function for different states.

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Statistical Mechanics and Models: Probability, Ensemble Averages, and Helix Formation

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  1. Stat Mech and Models • Probability • II.Ensemble Averages • III. Helix Models

  2. Probability and the partition function Using the Boltzmann distribution, the probability at a given temperature of a particular state is: Is the degeneracy The bottom is the canonical partition function.

  3. Probability rewriten Let us take the probability expression, and rearrange How did I do the last step? What does the top look like?

  4. Ensemble averages To take an ensemble average: Average Energy:

  5. Ensemble averages Average Energy: Notice: What does this tell us about the utility of the partition function?

  6. Ensemble averages We can write macroscopic variables in terms of the parition function Energy, pressure, entropy, helmholz free energy, and more

  7. Utility of Other Ensembles: Microcanonical: Isothermal/Isobaric:

  8. Notes on Gibbs Free Energy Also know by some people as the free energy or by others as the thermodynamics potential What might be a serious problem with simulating DG?

  9. Helix Formation Models Physical simple models for helix formation: amenable to paper and pencil calculations {although the object of computational tests} Each amino acid {or base pair} can exist in either unfolded or folded conformations A unfolded amino acid is denoted by 0, and a folded amino acid is denoted by 1. For a fixed number of amino acids {an n-mer}, we can enumerate the number of states: N=2; 4 states N=3; 8 states N=m 2m states What are the states for N=2? What are the states for N=3?

  10. Helix Formation Models The most straightforward of these models are two parameter models: These parameters are the nucleation parameter, s, and the propagation parameter, s. Physically what do you think these parametbers mean? To be precise: s is the equilibrium constant for folding at the end of a sequence of folded AA What does this assume about the energy of folding a single residue? ss is the equilibrium constant for folding in an unfolded sequence

  11. Zip Model A particular example of the simple helix formation models The only allowed states are those with contigious stretches of 1’s: Physically what does this mean in terms of structures? Is this realistic? What are the states for N=2? What are the states for N=3? As N increases the number of allowed states compared to possible states decreases

  12. Zip Model Each allowed state contributes to the partition function. A state contributes either 1 {the entirely unfolded state}, or ssm , where m is the number of 1’s {the size of the folded stretch}. Why do we multiply? What is the partition function for N=3? What is the partition function for N=2? Can you derive a expression for wk? How would this differ if we allowed all states?

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