Chap6. Atomic-Level Simulation and Modeling of Biomacromolecules

6.1 Introduction

• In principle, all the problems in biology could be solved by solving the time-dependent Schroedinger equation (Quantum mechanics,QM)

• But, QM is impractical with big scale biological objects like organ. So, the solution is hierarchical strategy! (figure 6.1)

  • QM(electron scale) for the forces on atoms(FF)
  • MD(atom scale) for motions of large biomolecules
  • Molecular scale for membranes
  • etc

• It is possible
  • to predict 3-D structure and dynamics
  • to analyze interactions
• These atomic-level description is valuable for understanding
  • reaction chemistry at active sites in enzyme
  • binding of small molecule to DNA and protein, antibody to antigen, protein to DNA
  • conformational changes in proteins and how these cahnges function

6.2 Molecular Dynamics

1. QM methods

   • ab initio QM

     • Hartree-Fock(HF), density function theory(DFT), configuration interaction(CI)
     • most accurate
     • use no experimental data
   • semiempirical methods
     • MINDO, extended Huckel, AM1
     • based partly on comparisons with experiments

   • essential for describing systems in which the nature of the bonds changes(e.g., chemical reaction, excited states of molecules, electron transfer)

2. Molecular Dyanmics

   • bonds in biomolecules is relatively intact -> QM is less useful.

   • instead, the focus involves usually packing and conformation.

   • electrons -> spring, Schroedinger's eq -> Newton's eq

6.2.1 The Force Field

• Forces on atom
• Etot = Enonbond + Evalence
• Nonbond energy = electrostatic(Coulomb) + van der Waals(VDW) + hydrogen bond(HB)

  • required calculation for all pairs of atoms -> this is bottleneck for simulations of very large systems.

    • - ignore interactions longer than some cutoff radius - multipole techniques

{{| The multipole technique is a numerical method based on approximation of the potentials in/out of the head using multipoles. Multipoles are functions which fulfill the Laplace’s equation. For instance, one of the most frequently used multipoles is the elementary charge (a multipole of the first order). |}}

• Valence energy = bond + angle + torsion + inversion
• Standard Force Fields
  • - FF is defined by the particular choices for the parameters in the FF (force constants and equilibrium geometries) 1) FFs for a specific class of molecules
    • AMBER, CHARMM
      • - parameterized to describe the naturally occurring amino acids and cucleic acids - but, unusual ligand, amino acid, and bases are difficult to incorporate.
    • OPLS, MM3
      • - include unusual things.
    2) generic FFs
    • - rule-based FFs - Dreiding(B, C, N, O, F columns in periodic table), UFF(all elements of the periodic table) - useful for systems with unusual arrangements of atoms or for new molecule for which there are no experimental or QM data - but, may not sufficiently accurate
    3) spectroscopic quality FFs
    • - predict vibrational spectra, geometry and energy of molecule - the biased Hessian method : general procedure for developing such spectroscopic FFs from QM
• Effect of Solvents
  • - secondary and tertiary structures of proteins are determined by the nature of the solvent. - in earliest study, it is replaced by using a dielectric constant.

    - the most accurate MD include an explicit dscription of the water -> required very long time for calculation - dielectric continuum model : excellent compromise for attaining most of the accuracy of explicit water, while eliminating the atoms and time scale of the solvent.

6.2.2 Molecular Dynamics Methods

- These methods determine the atomic positions.

1. The Fundamental Equations

   • leads to 3N coordinats and 3N velocities that describe the trajectory of the system as a function of time

   • $$ -F_i = m_ix_i (6.13) $$ (F, dd(x) is vector with 3 dimension)

   • $$ x_i = f_i/m_i = -\bigtriangledown{E_tot}/m_i (6.14) $$

   • $$ x_n = \frac{x'_{n+\frac{1}{2}} - x'_{n-\frac{1}{2}}}{\delta} (6.15) $$ (delta is timestep)

   • $$ x'_{n+\frac{1}{2}} = x'_{n-\frac{1}{2}} - \frac{\delta}{m_i}\bigtriangledown{E_n} (6.16) $$
   • integrating (6.16) $$ x_{n+1} = x_n + \deltax_{n+\frac{1}{2}} $$
   • This integration is usually performed using the Verlet leapfrog algorithm.
   • Timestep $$ \delta $$ must be short enough to provide several points during the period of the fastest vibration.
2. NPT and NVT Dynamics
   • Methods to control the temperature of an MD simulation in fixed volume of pressure.
3. Constrained Internal Coordinates
4. MPSim
5. Periodic Boundary Conditions
6. Monte Carlo Methods

6.3 Application to Biological Problems