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The Multiensemble Sampling Method

The Multiensemble Sampling Method. Kyu-Kwang Han Department of Physics, Research Center for Bioiformatics, Pai Chai University Daejeon, 302-735, Korea (South). Contents of Talk. Brief theoretical review Usage of the method.

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The Multiensemble Sampling Method

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  1. The Multiensemble Sampling Method Kyu-Kwang Han Department of Physics, Research Center for Bioiformatics, Pai Chai University Daejeon, 302-735, Korea (South)

  2. Contents of Talk • Brief theoretical review • Usage of the method

  3. Major roadblocks in simulation studies of complex molecular systems like proteins • Accuracy of the potential energy function • Ability of the simulation method to sample enough the relevant configurations (same problem in the free energy calculation)  interest of our research

  4. Local-minima problem Conventional MC and MD simulations at low temperatures tend to get trapped into a few of a huge number of local-minimum-energy states so that the relevant configurations of the system are not sampled properly and the results strongly depend on the initial configuration. It is natural to consider the use of non-Boltzmann sampling

  5. Suppose that one has to investigate n similar systems in a single simulation, whose potential energy functions and temperatures are Ul and Tl, l = 1,…, n. • when a conventional simulation for system m is performed, The parts of configuration space relevant to system m are not broad enough, in general.  Impossible to obtain data of other n-1 systems!!

  6. when sampled upon a general (non-Boltzmann) weighting function W, Powerful when W is chosen appropriately to cover all the parts of configuration space relevant to the investigated systems

  7. Non-Boltzmann sampling methods: - umbrella sampling method (USM) [Valleau, J. P. J Comput Phys 1977, 23, 187] - multicanonical method(MCM) [Berg, B. A.; Neuhaus, T, Phys Lett 1991, B267, 249] - multiensemble sampling method (MESM) [Han, K.-K. Phys Lett A 1992, 165, 28] - entropic sampling method (ESM) [Lee, J Phys Rev Lett 1993, 211] - replica exchange method (REM) [Hukushima, K. et al. J Phys Soc Jpn 1996, 65 1604] and so on. In USM, MCM and ESM, the forms of W are not a priori known and have to be determined by iterations of short preliminary simulations.  Nontrivial process In REM, the required number of replicas increases, as the number of degrees of freedom increases, whereas only a single replica is simulated in the other methods  REM demands a lot of computer power for complex systems MESM, which was developed originally for accurate estimation of the free energy as like as USM and has not been used yet in applications to complex systems such as proteins, has no such difficulties of USM, MCM, ESM and REM.

  8. Multiensemble Sampling (MES) A universal form of W for investigating several systems is given where Cl are adjustable parameters and p is an arbitrary constant Originally, W with p=2 was derived by the functional minimization of the sum of the squares of expected relative errors in so that can be calculated with equal accuracy for every system (K.-K. Han, Phys Rev E 1996, 54, 6906) Optimum when Cl is taken to be the free energy of system l

  9. Dependence onCl in a two-ensemble sampling simulation (F1 - F0 = -12) system 0 : water with an uncharged sodium like particle system 1 : water with a sodium like particle chargedby 0.3e

  10. Dependence onCl in a two-ensemble sampling simulation (F1 - F0 = -12) system 0 : water with an uncharged sodium like particle system 1 : water with a sodium like particle chargedby 0.3e

  11. Optimum condition can be achieved very quickly by iterations of replacing the value of C1 – C0 by the estimated value of F1-F0

  12. Free energy of charging a sodium ion in water (Han, K.-K. et al J Comput Chem, 2001, 22, 1004) 11 systems (q = 0,0.1e, ...,0.9e, 1e) were included Two-ensemble sampling runs for the pairs of nearest systems first Eleven-ensemble sampling simulations in final

  13. Simulation of proteins Start with two highest temperatures Increase the number of investigated states one by one in order of high temperature betanova (20 residues, three-stranded β-sheet)

  14. 1fsd (28 residues, one β-hairpin and one a-helix)

  15. Summary • Ability of the MESM to explore all the parts of configuration space relevant to the systems of interest  W : a superposition of the Boltzmann factors of the systems • Simple and easy to obtain an optimal set of parameters of W  Variety of its application • Research on the protein folding using the method is being done, hoping to find out a way of resolving the local-minima problem.

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