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### Coordinates and Pathways in MM and QM/MM modeling

Haiyan Liu

School of Life Sciences,

University of Science and Technology of China

In MM and QM/MM modeling of biomolecules,we often aim at understanding mechanisms of processes, many of which too slow to be investigated by direct simulations.

Examples

To study protein functions:

Possible chemical/conformational (sub)states ? Mechanism of transitions between them?

To study protein/peptide folding:

any preferred “pathways” or “order of events”? Roles of topologies and sequences?

Two (?) basic causes for macroscopic slowness

- Need to overcome major enthalpic barriers (e.g., chemical reactions…)
- Need to “zoom” into a very limited region in the conformational space

(e.g., protein folding, binding…)

B state

A state

time

Among major obstacles in simulations- Sampling (in)efficiency

Transition time

Waiting time

Two basic types of approaches

A. Connecting known terminal states

A1 “forced” barrier crossing

Umbrella sampling, Targeting or Steered MD,

Drawbacks: projecting a many-dimensional system onto a few pre-assumed reaction coordinates

A projected representation of the many-dimensional problem

Problem associated with Improper projection

Environ.

Degrees of Freedom

Reaction coordinates (Rc)

Restrained optimization: discontinuous environment

Potential of mean forces along Rc:

sampling minima but not transition states

A2 Chain of states or path optimization methods

Discrete representation of pathways (a pathway is represented by a chain of replicas)

“enforced” continuity of the pathway

A parametric representation of the many-dimensional problem

B. Introducing more frequent transitions between states

Accelerate minimum-escaping (elevated temperature simulations, conformational flooding or local elevation, parallel replica simulations, potential energy function deformation)

The key is to avoid over-expanding the accessible conformational space.

Accelerated sampling approaches

- Potential energy-based v.s. kinetic energy-based
- Equilibrium v.s. non-equilibrium sampling
- Degree of freedom (DOF)-specific and degree of freedom-nonspecific
- delocalized (collective) DOF or local DOF

coordinates (or order parameters) are essential,provided that we have good enough energy model…

- “forced” transitions and free energy surfaces: which coordinates to project onto?
- Chain of states method: enforcing continuity on which coordinates?
- Accelerated sampling: which coordinates to apply the bias?

Examples

- Local elevation
- Potential energy-based, non-equilibrium,DOF-specific, local DOFs,
- Conformational flooding
- Potential energy-based, non-equilibrium, DOF-specific, delocalized DOFs
- Temperature REMD
- Kinetic energy-based, equilibrium, DOF-non specific
- Amplified collective motion (ACM) model
- Kinetic energy-based, non-equilibrium, DOF-specific,

delocalized DOFs

…

Our works in recent years

Amplified collective motion MD simulation (B)

Obtaining minimum energy paths in QM/MM modeling of enzymatic reactions with a modified nudged elastic band method (A2)

coarsely-guided sampling of folding trajectories of a small protein domain in implicit solvent (A1)

Hamiltonian replica change simulation with free energy-surface-derived umbrella potentials (B)

Accelerate conformation search by

Amplifying collective motions

Collective coordinates have been used in the analysis

of protein dynamics for a long time:

Normal mode analysis

Principal component (or essential dynamics) analysis

of conformational sets

Coarse grained elastic net work models.

Several important observations from such studies:

Protein motions (e.g. atomic positional fluctuations) are

dominated by a very small number of slow modes.

These slow modes often correspond to functional motions.

The low frequency space is insensitive to details of models

Zhang et al Biophys. J., 2003, 84, 3583

He，et al J. Chem. Phys. 2003, 119, 4005.

Derive low frequency collective modes using the coarse-grained

elastic network model

no need for exact minimum but use only a single conformation; low frequency modes can be updated on the fly in a simulation; correctly captures the low frequency modes along the “valley” on the energy surface (for compact structures)

Advantages

Sampling in conformational space extended along “valleys” of the energy landscape. No “melting” of local structures.

Lower frequency subspace updated on the fly.

No deformation of potential energy surface.

No pre-definition of “path” or “reaction coordinates”.

Drawbacks

- Functionally important motions may not correspond to the slowest few modes
- Does not correspond to any equilibrium ensemble. Difficult to be quantitative

Test systems

Inter-domain motions of T4 lysozyme in explicit solvent.

Folding of a S-peptide analog (in implicit solvent described by a Generalized-Born model)

Bacteriophage T4 lysozyme

X-ray structures

env 2 (0.13 nm)

env 1 (0.40 nm)

First three modes of the coarse grained model: 80% of the variations

ACM-MD produces larger fluctuations

Atom position RMS fluctuations in MD (300 K dashed line) and ACM-MD (Three slowest modes: 800 K, other modes: 300 K)

Zhang et al Biophys. J., 2003, 84, 3583

ACM-MD sampled larger variations in the two PCA direction.

Projection on the two largest principal components

of the crystal structures(dots), MD trajectory (red), and

ACM-MD trajectory(blue).

Zhang et al Biophys. J., 2003, 84, 3583

ACM-MD and normal MD are similar in intra-domain motions

Number of residues

In secondary structures

RMSD from native structure

N-term domain

C-term domain

Solid: MD

Dotted: ACM-MD

Zhang et al Biophys. J., 2003, 84, 3583

ACM-MD refolds the peptide while normal MD cannot

MD，start from native

ACM-md，start from native

MD，start from unfolded

ACM-md，start from unfolded

Solid： RMS deviation from unfolded as functions of time

Dotted： RMSD from native as functions of time

Zhang et al Biophys. J., 2003, 84, 3583

The ACM method:

Collective DOF; kinetic energy based;

improves sampling;

non-equilibrium ensemble thus difficult to go quantitative

Application by another group: Biochemistry , 2006, 45 (51) : 15269-15278

Chain of states method in path optimization

The nudged elastic band method

Each replica moves to minimize

the force perpendicular to the path.

and to maintain even distribution

of the replicas along the path

- Force:

Reaction coordinate driven

Advantages:

No pre-assumed reaction coordinate.

Suits for parallel computations

Problems for enzymatic reactions

Enzyme systems contain many floppy degrees of freedom.

Impractically small radius of convergence.

Soft spectator degree of freedom Y spoils the

NEB calculation

Xie et al J.Chem. Phys., 2004, 120,8039.

Heuristic solution: Exclude spectator degrees of freedom

Use a set of inter-atomic distances (chemical subspace)

f(d)

Multiple step reactions

d

Xie et al J.Chem. Phys., 2004, 120,8039.

The acylation step of type A

beta-lactamase

An application

Metal-preferences of metallo-proteases

E-coli peptide deformylase: prefers Fe++ over Zn++

Thermolysin: prefers Zn++

Dong et al, J.Phys.Chem. B, 2008（112）

,10280-10290.

comparative modeling of Zn-TLN and Zn-PDF using NEB

Dong et al, J.Phys.Chem. B, 2008（112）

,10280-10290.

Dong et al, J.Phys.Chem. B,

2008（112）,10280-10290.

Summary

- Some general discussions on “coordinates”-based or DOF specific approaches to accelerate the modeling of slow processes
- Two particular types of approaches
- Amplified collective motions
- NEB adapted for the simulations of enzyme reactions
- An example showing comparative modeling provides biochemical insights

Acknowledgements

Zhiyong Zhang, Jianbin He (ACM)

Li Xie (adapted NEB)

Minghui Dong (PDF and TLN)

All former and current group members

Adapted NEB: Weitao Yang and group

Funding: CAS, NSCFC

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