Ammm:Coarse grain modeling: Difference between revisions

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Created page with "= Introduction = *The time and length scales accessible by all-atom molecular models are limited. *Common bead-springs coarse-grain models cannot represent anisotropy of proteins such as differentiation of <span class="texhtml">φ</span> and <span class="texhtml">ψ</span> angles. *Brownian dynamics used to study diffusion limited reactions. = A Coarse-Grain Model with Anisotropic vdW and Electrostatic Multipoles = *Energy functions has been developed for rigid bodi..."
 
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Latest revision as of 16:13, 10 May 2022

Introduction

  • The time and length scales accessible by all-atom molecular models are limited.
  • Common bead-springs coarse-grain models cannot represent anisotropy of proteins such as differentiation of φ and ψ angles.
  • Brownian dynamics used to study diffusion limited reactions.

A Coarse-Grain Model with Anisotropic vdW and Electrostatic Multipoles

  • Energy functions has been developed for rigid bodies composed of anisotropic Gay-Berne and multipole interaction sites.
  • Generalized Kirkwood method is applied to account for the solvation effect.

Anisotropic vdW (Gay-Berne Potentials)

  • Gay-Berne particles are defined by its length (l), breadth (d)
    • well-depth of the cross configuration(e0), well-depth of end-end/face-face configuration (eE), well-depth of side-by-side configuration (eS)
    • dw is the "softness" of the potential
    • Has been applied to methanol-water mixtures[1][2]
  • The Gay-Berne (GB) potential is a function of the principal axes of the two interacting particles ûi, ûj, and ri'j, the vector representing the difference between the positions of particle i and particle j.[3]
  • Relative positions and orientations of GB particles
  • When l=d and e0=eE=eS, the Gay-Berne potential reduces to the Lennard-Jones 6-12 form.

Electrostatic Multipole and Implicit Solvation

Dialanine Model

  • The dialanine model is composed of 5 rigid bodies
  • Bonded interactions between particles utilize energy functions similar to those of all-atom models.
  • Bond lengths, and angles are parameterized by fitting to all-atom MD samples via Boltzmann inversion.
  • Gay-Berne parameters were fit to all-atom homodimer binding energy for cross, end-end, face-face, and side-by-side configurations.
  • EMP parameters were obtained from all-atom model (AMOEBA) via multipole expansion.

Results

Dialanine Conformational Energy

Coarse-grain (GK) All-atom (GK)
Gas-phase energy Gas-phase energy
Solvation-phase energy Solvation-phase energy
Solution energy only Solution energy only

Deca-alanine Simulated Annealing

  • Simulated annealing simulations has been performed to fold structures. Structures were heated up to 1000K and then quenched to 0K over 40ns. The resulting structures (mapped back to all-atom) exhibit alpha-helices as shown below.
  • Structures may also contain a mixture of beta-sheets and alpha-helices
Coarse-grain (GK) All-atom (GK)
Alpha-helices Alpha-helices
Beta-sheets Beta-sheets

Summary

  • A preliminary coarse-grained model based on anisotropic vdW and multipole electrostatics has been developed.
  • Conformational energy map resembles the all-atom energy surface.
  • Folding of polyalanine using simulated annealing resulted in structures containing alpha-helices and beta-sheets.

Brownian Dynamics

  • Used to model diffusion limited systems
  • The Ermak-McCammon algorithm to generate trajectories:
  • Ri(t) is position of particle i
    • δt is time step
    • D0 = k'T / 6πηa (Stokes-Einstein equation)
      • η is the viscosity of solvent
      • a is the radius of the particle
    • Si is the random displacement

Effects of crowding on diffusion

  • A monodisperse hard sphere system is studied using BD[4]
  • Molecule A is observed until it reacts with its target upon collision surrounded by other crowding molecules.

Crowding reduces diffusion

  • D obtained from
  • where is the survival probability
    • ρB is the concentration of the molecule that reacts with molecule A
    • NA(t) is the number of simulations at time t in which molecule A has not reacted yet.
    • N0A is the total number of simulations.
  • Theoretical value for
    • Smoluchowski's analytical expression
    • Dx is diffusion constant of particle x and ax is radius

Time-step independence

Crowding slows down association

  • Association is determined by the survival probability

Larger crowding molecules produce lesser crowding effects

  • BD simulations were performed for crowding molecules of r=2, 4, and 6nm.

Importance of hydrodynamic interactions (HI)

  • Hydrodynamic interactions (HI) are often neglected when using Brownian Dynamics
  • Hard sphere interaction only models overestimate mobility of particles
  • HI is simulated by modifying diffusion coefficient in
    • where
    • According to Heyes[5]
  • Fig 6a) The diffusion constant calculated from BD simulation with HI correction for monodisperse system with sphere radius of 2 nm is shown in comparison with results from other theoretical methods: Solid line, Tokuyama and Oppenheim; dashed line, Medina-Noyola; squares, our BD simulation with HI correction.

  • Found that HI is more important when there is less crowding.

Electrostatic interaction dominate in a charged environment

  • Electrostatic interaction is essentially a Debye-Huckle type potential
    • Uij is the max interaction between molecules i and j
      • -1kT, -2kT, and -5kT
    • aij is sum of radii of i and j
    • κ is Debye solution parameter
      • salt concentration of 0.1M is used

Other Applications

Extension of Crowding

  • Association rate of hen egg lysozym (HEL) and the HyHEL-5 antibody in presence of crowding[6]
  • Crowding particles represented with 18 A radius spheres
  • No electrostatic nor hydrodynamic interactions.
  • Atomic detail for HEL and HyHEL-5 antibody
  • Binding criteria

  • Result
    • Increase of association rate with increased crowding is consistent with experiment and theoretical predictions
    • Explanation is that crowding reduces translational diffusion and rotational diffusion
    • Larger crowding particles still allow particles to rotate and find binding sites.

Application of Poisson-Boltzmann equation to BD

  • Same hen egg lysozyme/HyHEL-5 system[7]
    • No crowding
  • Electric field around the antibody is solved once with the Poisson-Boltzmann equation
    • Resulting force is used to generate trajectories.
  • HEL represented with two spheres
  • Binding criteria is when small HEL sphere and delta-carbon of Glu-H50 of HyHEL-5 are within 6.5 A of each other.

  • Rate constant

  • Result
    • Also predicted reaction rates with antibody mutants, but did not compare with experimental data.
  • Rojnuckarin applied similar method[8]
    • Extended method by implementing enhanced sampling method
    • Binding of superoxide (O2-) with copper-zinc superoxide dismutase
    • Binding of antibody NC6.8 with N-(p-cyanophenyl)-N'-(diphenylmethyl)-guanidinium acetic acid

Review of protein-protein interactions/docking[9]

  • Nonpolar solvation using solvent accessible surface area

References

  1. ↑ Golubkov, P. A. & Ren, P. Y. (2006). Generalized Coarse-Grained Model Based on Point Multipole and Gay-Berne Potentials. Journal of Chemical Physics 125, -.
  2. ↑ Golubkov, P. A., Wu, J. C. & Ren, P. Y. (2008). A Transferable Coarse-Grained Model for Hydrogen-Bonding Liquids. Physical Chemistry Chemical Physics 10, 2050-2057.
  3. ↑ D. J. Cleaver, C. M. Care, M. P. Allen, and M. P. Neal, Phys. Rev. E 54, 559 (1996).
  4. ↑ Sun, J. & Weinstein, H. (2007). Toward Realistic Modeling of Dynamic Processes in Cell Signaling: Quantification of Macromolecular Crowding Effects. Journal of Chemical Physics 127, -.
  5. ↑ D. M. Heyes, Mol. Phys. 87, 287 (1996).
  6. ↑ Wieczorek, G. & Zielenkiewicz, P. (2008). Influence of Macromolecular Crowding on Protein-Protein Association Rates-a Brownian Dynamics Study. Biophysical Journal 95, 5030-5036.
  7. ↑ Kozack, R. E., Dmello, M. J. &amp;amp; Subramaniam, S. (1995). Computer Modeling of Electrostatic Steering and Orientational Effects in Antibody-Antigen Association. Biophysical Journal 68, 807-814.
  8. ↑ Rojnuckarin, A., Livesay, D. R. & Subramaniam, S. (2000). Bimolecular Reaction Simulation Using Weighted Ensemble Brownian Dynamics and the University of Houston Brownian Dynamics Program. Biophysical Journal 79, 686-693.
  9. ↑ Elcock, A. H., Sept, D. & McCammon, J. A. (2001). Computer Simulation of Protein-Protein Interactions. Journal of Physical Chemistry B 105, 1504-1518.
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