Generalizedensemble Algorithms for Molecular Simulations of Proteins

1
Generalized-ensemble Algorithms for Molecular
Simulations of Proteins

• Hongjun Yue, 4/19/2003

2
Outline

• Introduction
• Generalized Ensemble
• Generalized-ensemble Algorithms
• Generalized-ensemble Monte Carlo simulations of
  peptides as examples

3
Introduction

• The function of proteins and peptides are solely
  determined by their 3D shape, such as a-helix and
  ß-sheets
• One tries to understand the folding of a protein
  or peptide solely from the underlying physical
  laws, taking the interaction between all atoms of
  the protein in account.
• There are two types of simulations, Monte Carlo
  simulation and Molecular Dynamics simulation.

Reference 1,2,4,5
4
Introduction

• The energy of protein (can be described by ECEPP
  energy function) contains a huge number of local
  minima.
• At temperature of experimental interest(300K),
  traditional methods like Monte Carlo or molecular
  dynamics tend to get trapped in one of the local
  minima, because of the biased sampling.
• A simulation in generalized ensemble performs a
  free random walk in potential energy space and
  can overcome this difficulty.

Reference 1,2,4,5
5
Generalized Ensemble

• Generalized Ensemble (complete open system)
• The system can exchange energy, volume and
  particles with the surroundings
• Keeps the corresponding average values U, V and N
  constant

• ltEgt
• ltVgt
• ltNgt

Reference 2
6
Generalized Ensemble

• Partition function of Generalized Ensemble
• The probabilities of the microstates (Pi) in the
  Generalized Ensemble are

Reference 3
7
Generalized-ensemble Algorithms (GEA)

• Characteristic of GEA
• Each state is weighted by a non-Boltzmann
  probability weight factor
• Uniform distribution of pre-chosen physical
  quantity
• Weights are not priori known
• Estimators for these weights have to be
  determined often by iterative procedure
• In a single simulation run, one can obtain
• the global-minimum-energy state
• canonical ensemble averages as functions of
  temperature through some reweighting tech.

Reference 1,2,4,5
8
GEA--Multicanonical Ensemble

• Also Entropic sampling or adaptive umbrella
  sampling of the potential energy, perform a 1D
  random walk in potential energy space
• The probability distribution of energy is defined
  as Pmun(E)Wmu(E)const.
• The multicanonical weight factor then satisfies
• Wmu(E) n-1(E), n(E) is the density of states
  with energy E
• The weight factor is not a priori know, one has
  to determine it for each system by a few
  iterations of trial MC or MD simulation

Reference 2,5
9
GEA--Multicanonical Ensemble

• Once the weight factor is obtained, one performs
  a long production simulation run.
• The global-minimum-energy state thus obtained
• The canonical distribution PB(T,E) can be
  calculated
• The expectation value of a physical quantity
  A,ltAgt

Reference 2,5
10
GEASimulated tempering (ST)

• Also the method of expanded ensemble, performs a
  free random walk in temperature space.
• Both the configuration and the temperature are
  updated during the simulation with a weight
  WST(E,T).
• The function a(T) is chosen so that the
  probability distribution of temperature PST(T) is
  flat

Reference 2
11
GEASimulated tempering (ST)

• Function a(T) is not priori known and have to be
  determined by iterations of short simulations.
• The canonical distribution PB(E,T) and thus
  expectation of a physical quantity ltAgt and be
  calculation as following.

• Gm12tm, tm is the integrated autocorrelation
  time at Tm

Reference 2
12
GEMreplica-exchange method

• Also multiple Markov chain method and parallel
  tempering
• The system of REM consist of M non-interacting
  replicas of original system in the canonical
  ensemble at M different temperatures Tm.
• Each replica in canonical ensemble of the fixed
  temperature is simulated simultaneously and
  independently for a certain MC or MD steps.
• A pair of replicas at neighboring
  temperatures,say xmi and xm1 j with the
  probability w(xmi xm1 j)

Reference 2
13
Monte Carlo simulation of C-peptide

• (a) X-ray structure of C-peptide
• (b) the lowest conformations of C-peptide
  obtained from a multicanonical Monte Carlo run of
  1,000,000 MC sweeps in gas phase
• (C) and in aqueous solution represented by the
  distance-dependent dielectric funtion.

Reference 5
14
Monte Carlo simulation of C-eptide

• Some comments
• C-peptide, residues 1-13 of ribonuclease A
• XRD data of the whole enzyme exhibits a nearly
  3-turn a-helix
• Simulations predicted a similar a-helix in the
  lowest-energy conformation in aqueous solution
• The positions of a-helix in both situations are
  of the same

15
Monte Carlo simulation of BPTI(16-36)

• (a) X-ray structure (structure X) of BPTI(16-36)
• (b) the lowest conformations of BPTI(16-36)
  obtained from a multicanonical Monte Carlo run of
  1,000,000 MC sweeps in gas phase
• (C) and in aqueous solution represented by the
  term proportional to the solvent-accessibel
  surface area(structure S).

Reference 5
16
Monte Carlo simulation of BPTI(16-36)

• Some comments
• a-helix and ß-sheets from residues 16-36 of
  bovine pancreatic trypsin inhibitor (BPTI)
  studied
• Structure deduced for simulation is rather
  different from on deduced from the XRD
  experiments of the entire BPTI
• Obvious agreement between simulation and NMR
  experiment on fragment corresponding to residues
  16-36

17
References

• 1 Ulrich H.E. Hansmann, Physica A 254(1998)
  15-23.
• 2 Ayori Mitsutake, Yuji Sugita, Yuko Okamoto
  Biopolymers (Peptide Science) 60, 96-123(2001)
• 3 Material on General ensemblecan be found at
  http//www.ecm.ub.es/condensed/eduard/papers/massi
  eu/node3.html
• 4 Ulrich H.E. Hansmann, Physica A 321(2003)
  152-163
• 5 Material from internet, Yuko Okamoto, Ab
  Initio Predictions of Three-Dimensional
  Structures of Proteins by Monte Carlo
  Simulations.