Ammm:Coarse grain modeling
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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)
- 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
- Electrostatic multipole interactions are implemented as described in a previous lecture.
- Solvation energy is computed with the Generalized Kirkwood method.
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
- Uij is the max interaction between molecules i and j
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
- ↑ Golubkov, P. A. & Ren, P. Y. (2006). Generalized Coarse-Grained Model Based on Point Multipole and Gay-Berne Potentials. Journal of Chemical Physics 125, -.
- ↑ 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.
- ↑ D. J. Cleaver, C. M. Care, M. P. Allen, and M. P. Neal, Phys. Rev. E 54, 559 (1996).
- ↑ Sun, J. & Weinstein, H. (2007). Toward Realistic Modeling of Dynamic Processes in Cell Signaling: Quantification of Macromolecular Crowding Effects. Journal of Chemical Physics 127, -.
- ↑ D. M. Heyes, Mol. Phys. 87, 287 (1996).
- ↑ Wieczorek, G. & Zielenkiewicz, P. (2008). Influence of Macromolecular Crowding on Protein-Protein Association Rates-a Brownian Dynamics Study. Biophysical Journal 95, 5030-5036.
- ↑ Kozack, R. E., Dmello, M. J. & Subramaniam, S. (1995). Computer Modeling of Electrostatic Steering and Orientational Effects in Antibody-Antigen Association. Biophysical Journal 68, 807-814.
- ↑ 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.
- ↑ Elcock, A. H., Sept, D. & McCammon, J. A. (2001). Computer Simulation of Protein-Protein Interactions. Journal of Physical Chemistry B 105, 1504-1518.
