introduction to amber the theory and practice of biomolecular simulations using the amber suite of...

Introduction to AmberIntroduction to Amber

The theory and practice of biomolecular The theory and practice of biomolecular simulations using the Amber suite ofsimulations using the Amber suite of

programsprograms

Dr. Vladislav VassilievDr. Vladislav Vassiliev

NCI National Facility, The Australian National University, NCI National Facility, The Australian National University,

ACT 0200, Canberra, AustraliaACT 0200, Canberra, Australia

February 2011

Presentation OutlinePresentation Outline

Introduction to Amber 12Introduction to Amber 12

Hands-onHands-on• Setting up a standard Amber MD RunSetting up a standard Amber MD Run• Building non-standard ResiduesBuilding non-standard Residues• QM/MM: Using Amber-Gaussian QM/MM: Using Amber-Gaussian

InterfaceInterface• QM/MM: Using Amber inbuilt QM QM/MM: Using Amber inbuilt QM

methodsmethods

What is AMBER?What is AMBER?

AAssisted ssisted

MModel odel

BBuilding with uilding with

EEnergy nergy

RRefinementefinement

AMBERAMBER

What is Amber?What is Amber?

“Amber” refers to two things: 1) a set of molecular mechanical force fields for the

simulation of biomolecules

2) a package of molecular simulation programs (about 50 ) which includes source code and demos

The current version of the code is Amber version 12, which is distributed by UCSF (University of California, San Francisco) subject to a licensing agreement

Amber Home Page: http://ambermd.org/

What is AmberWhat is Amber

Amber is distributed in two parts:

AmberTools12 and Amber 12:

• AmberTools12 could be used without Amber12, but not vice versa

• AmberTools12 currently consists of several independently developed packages that work well by themselves, and with Amber itself

• Amber 12 centered around the sander and pmemd simulation programs and continues to be licensed as before, under a more restrictive license (Academic/non-profit/government: $400. Industrial (for-profit): $20,000 for new licensees, $15,000 for licensees of Amber 10).

AmberToolsAmberTools

NAB build molecules; run MD or distance geometry, etc.

antechamber & MCPB Create force fields for general organic molecules

ptraj & cpptraj Analyze trajectories from Amber or CHARMM

tleap and xleap Basic preparation program for Amber simulations

3D-RISM Solves integral equation models for solvation

sqm semiempirical and DFTB quantum chemistry

pbsa Performs numerical solutions to Poisson-Boltzmann models

Mdgx Code for explicit solvent molecular dynamics simulations

MMPBSA.py & amberlite Energy-based analyses of MD trajectories

AmberToolsAmberTools

• AmberTools is released under the GNU General Public License (GPL)

• A few components are included that are in the public domain or which have other, open-source, licenses.

• AmberTools is distributed in source code format, and must be compiled in order to be used. One needs C, C++, and fortran compilers to compile the AmberTools programs.

• The source code of AmberTools could be obtained here: http://ambermd.org/AmberTools-get.html

Versions of AmberVersions of Amber

Version Released

12 2012

11 2010

10 2008

9 2006

8 2004

AMBER HomeAMBER Home• Have a look at the Amber Home Page: http://ambermd.org/

Amber Main ReferencesAmber Main ReferencesA general overview of the Amber codes:

D.A. Case, T.E. Cheatham, III, T. Darden, H. Gohlke, R. Luo, K.M. Merz, Jr., A. Onufriev, C. Simmerling, B. Wang and R. Woods. The Amber biomolecular simulation programs. J. Comput. Chem. 26, 1668-1688 (2005)

An overview of the Amber protein force fields, and how they were developed:

W. Ponder and D.A. Case. Force fields for protein simulations. Adv. Prot. Chem. 66, 27-85 (2003). E. Cheatham, III and M.A. Young. Molecular dynamics simulation of nucleic acids: Successes, limitations and promise. Biopolymers 56, 232-256 (2001).

What is Amber?What is Amber?

““Amber” Amber” is a software package for modelling of Large Molecular Systems

Why Do We Need A Special Why Do We Need A Special Treatment for Large Molecular Treatment for Large Molecular

Systems?Systems?

Or, Why Do We Need Amber?Or, Why Do We Need Amber?

Quantum Chemistry Methods Provide a Quantum Chemistry Methods Provide a Rigorous Description of Molecular SystemsRigorous Description of Molecular Systems

But… they are very time consuming…

They solve Schrödinger equation

And they are generally applicable:

To Treat Large Molecular Systems We Need To Treat Large Molecular Systems We Need to Reduce the Complexity of the Systemto Reduce the Complexity of the System

As a result Molecular Mechanics methods are thousands times faster than Quantum Chemistry methods

Molecular Mechanics is a non-quantum mechanical technique for treating Large Molecular Systems

Molecular Mechanics vs Molecular Mechanics vs Quantum MechanicsQuantum Mechanics

Quantum Mechanics Molecular Mechanics

Considers atoms as collections of electrons and nuclei

Considers atoms as soft or hard spheres. Covalent bonds are treated as springs

Solves quantum Schrödinger equation Uses classical potential energy equations

Force FieldsForce FieldsThe potential energy equations to calculate the energy in Molecular Mechanics methods and the parameters/constants used in the equations are known as a Force Field

AMBER

CHARMM

CHARMm

CVFF

COSMOS-NMR

GROMACS

GROMOS

OPLS

ENZYMIX

ECEPP/2

QCFF/PI

UFF

CFF

MMFF

MM2, MM3, MM4

QVBMM

X-Pol

DRF90PIPF

SIBFA

AMOEBA

There are many force fields designed for different purposes.

Amber Force FieldAmber Force FieldThe total Energy in Amber force field consists of

1. bonded terms relating to atoms linked by covalent bonds and

2. nonbonded terms describing the long-range electrostatic and van der Waals interactions:

Etotal = Ebonded + Enonbonded

Amber Force Field: Amber Force Field: Bonded TermsBonded TermsThe bonded Energy in Amber force field consists of

bond stretching, angle bending, and torsion terms:

Ebonded = Estretch + Ebend + Etorsion

Amber Force Field: Amber Force Field: Bond StretchingBond StretchingAmber force field treats covalent bonds between atoms as springs (Hooke's Law, F = -kx)

bonds

eqrstretchingbond rrKE 2)(

where Kr is the empirical stretching force constant, r is the actual bond length and req is the “natural” (empirical) bond length

Amber Force Field: Amber Force Field: Bond StretchingBond Stretching

Amber Force Field: Amber Force Field: Angle BendingAngle BendingAmber force field treats angles that are bonded to the same central atom as springs (Hooke's Law, F = -kx)

angles

eqbendingangle KE 2)(

where Kθ is the empirical bending force constant, θ is the actual bond angle and θeq is the “natural” (empirical) bond angle

Amber Force Field: Amber Force Field: Angle BendingAngle Bending

Amber Force Field: Amber Force Field: Torsion EnergyTorsion Energy

dihedrals

ndihedral n

VE )cos1(

2

where Vn is the barrier to free rotation for the “natural” bond, n is the periodicity of the rotation (number of cycles in 360°), φ is the torsion angle and γ is the angle where the potential passes through its minimum value

Torsion Energy: torsional (dihedral) angle rotation between atoms that are vicinal (bonded to adjacent atoms) to each other

Amber Force Field: Amber Force Field: Torsion EnergyTorsion Energy

Amber Force Field: Amber Force Field: Nonbonded TermsNonbonded Terms

The nonbonded Energy terms in Amber force field describe the long-range electrostatic and van der Waals interactions:

Enonbonded = Eelectrostatic + EvdW

Amber Force Field: Amber Force Field: Electrostatic EnergyElectrostatic Energy

The Electrostatic Energy in the Amber force field represents the pair-wise sum of the electrostatic energies of all possible interacting non-bonded atoms i and j:

atoms

ji ij

jiticelectrosta R

qqE

where qi and qj are the point charges on atoms, Rij is the interatomic distance and ε is the dielectric constant

Amber Force Field: Amber Force Field: van der Waals Energyvan der Waals Energy

The van der Waals Energy in the Amber force field represents the pair-wise sum of the van der Waals energies of all possible interacting non-bonded atoms i and j:

atoms

ji ij

ij

ij

ijvdW R

A

R

BE

612

where the Aij and Bij parameters control the depth and position (interatomic distance) of the potential energy well for a given pair of non-bonded interacting atoms and Rij is the interatomic distance

Amber Force FieldAmber Force Field

- Empirical Parameters

Where Do Empirical Where Do Empirical Parameters Come From?Parameters Come From?

Parameter Derivation:Parameter Derivation: Partial ChargesPartial Charges

In AMBER:

1)Partial atomic charges are static

2)Quantum chemical methods (B3LYP/ccpVTZ//HF/6-31G**) are used to generate an electrostatic potential (ESP) around a molecule on the spheric grid

3) RESP (Restrained Electrostatic Potential) Method is used to derive the partial charges

QM = ab initio, DFT, semi-empirical

ConnollyConnolly

Parameter Derivation: Parameter Derivation: Van der Waals ParametersVan der Waals Parameters

It is the most difficult part…

1) Optimizing van der Waals parameters to reproduce the experimental or high-level Quantum Chemical data

Could be computationally expensive

2) Optimizing van der Waals parameters through the Monte Carlo or MD simulations to reproduce the experimental properties of bulk solvent (density, etc.).

For example, OPLS van der Waals parameters Could be computationally expensive

3) Reusing existing van der Waals parameters for similar atom types from the same or other force field

The simplest approach

Parameter Derivation:Parameter Derivation: Bond and Angle InteractionsBond and Angle Interactions

req and θeq come either from experimental data (X-ray, neutron diffraction) or Quantum Chemical calculations (geometry optimization)

Kr and Kθ force constants are usually optimized to reproduce the vibration frequencies calculated using high-level Quantum Chemical methods.

Or (the simplest approach)

Kr and Kθ force constants could be derived from the existing bond/angle parameters for similar bond/angle types from the same or other force field

Parameter Derivation:Parameter Derivation: Dihedral Angle InteractionsDihedral Angle Interactions

Vn, n, and γ are derived to reproduce the rotational profile from the high-level Quantum Chemical calculations.

J.Wang et al., Development and testing of a general amber force field, Journal of Computational Chemistry, 25 (2004), 1157

Or (the simplest approach)

Vn, n, and γ could be derived from the existing dihedral angle parameters for similar dihedral angle types from the same or other force field

Force Fields in Amber 12Force Fields in Amber 12


ff99SB ff10 ff12SB

Proteins ff99 +backbone torsion

Modifications

no change fromff99SB

ff99SB + newbackbone and

sidechain torsions

DNA ff99 ff99 +“Barcelona”

backbone torsionmodifications

no change fromff10

RNA ff99 ff99 +“Barcelona”

backbonechanges + “OL3”

changes for c

no change fromff10

Force Fields in Amber 12Force Fields in Amber 12


Lipid11: A modular lipid force fieldA new modular force field for the simulation of phospholipids and cholesterol designed to be compatible with the other pairwise additive Amber force field

Non-additive" force fields based on atom-centered dipole polarizabilities can also be used.

These add a "polarization" term to what was given above

where μi is an induced atomic dipole.

In addition, charges that are not centered on atoms, but are off-center (as for lone-pairs or "extra points") can be included in the force field.

Other Force Fields in Amber: Other Force Fields in Amber: Inclusion of PolarizationInclusion of Polarization

Other Force Fields in Amber: AMOEBAOther Force Fields in Amber: AMOEBA((AAtomic tomic MMultipole ultipole OOptimized ptimized EEnergetics for nergetics for BBiomolecular iomolecular AApplications)pplications)

• Atomic Multipoles: The model uses a polarizable atomic multipole description of electrostatic interactions. Multipoles through the quadrupole are assigned to each atomic center based on a distributed multipole analysis (DMA) derived from large basis set molecular orbital calculations at the MP2/aug-cc-pVTZ level and the experimental geometry of the gas-phase monomer.

• Polarization is treated via self-consistent induced atomic dipoles. Atomic dipole polarizabilities can be derived from an empirical fit to experimentally known molecular polarizabilities.

The induced dipole at each atomic site is computed as

where αi is the atomic polarizability and Ei,α is the sum of the fields generated by both permanent multipoles and induced dipoles

Other Force Fields: AMOEBAOther Force Fields: AMOEBA((AAtomic tomic MMultipole ultipole OOptimized ptimized EEnergetics for nergetics for BBiomolecular iomolecular AApplications)pplications)

• The functional forms for bond stretching and angle bending were taken from the MM3 force field:

• A Urey-Bradley functional form was chosen for the stretch-bend term:

Other Force Fields: AMOEBAOther Force Fields: AMOEBA((AAtomic tomic MMultipole ultipole OOptimized ptimized EEnergetics for nergetics for BBiomolecular iomolecular AApplications)pplications)

• Repulsion-Dispersion. The buffered 14-7 potential has been applied to model pairwise additive vdW interactions

where εij is the potential well depth, ρij= Rij/R0ij with Rij as the i-j separation and R0

ij the minimum energy distance. n = 14, m = 7, δ = 0.07, γ=0.12. The combining rules are:

• The buffered 14-7 function yields a repulsive region softer than the Lennard-Jones 6-12 function but steeper than typical Buckingham exp-6 formulations. • The buffered 14-7 form was found to outperform Lennard-Jones and Buckingham potentials in simultaneously reproducing gas phase ab initio results and liquid thermodynamic properties of noble gases and a series of diatomic species.

What we can do with Amber?What we can do with Amber?

Molecular Dynamics SimulationsMolecular Dynamics Simulations

The Molecular Dynamics simulation method is based on Newton’s second law or the equation of motion,

F=maF=ma,,

where F is the force exerted on the particle, m is its mass and a is its acceleration

Integration of the equations of motion then yields a trajectory that describes the positions, velocities and accelerations of the particles as they vary with time. From this trajectory, the average values of properties can be determined.

MD: Melting of IceMD: Melting of Ice

Human carboxyl Human carboxyl esterasecomplexed with morphineesterasecomplexed with morphine

MD: Translocation of DNAMD: Translocation of DNA

This movie shows the electrophoretically-driven translocation of a 58-nucleotid DNA strand through the transmembrane pore of alpha-hemolysin

Molecular Dynamics: Molecular Dynamics: Amber MD WorkhorsesAmber MD Workhorses

SANDERSANDER - Simulated Annealing with NMR-Derived Energy RestraintsPMEMDPMEMD - Particle Mesh Ewald Molecular Dynamics

SANDERSANDER

PMEMDPMEMD

GPU

PMEMD is up to 55% fasterPMEMD is up to 55% fasterthan SANDERthan SANDER

Molecular Dynamics: Molecular Dynamics: What Are Current Simulation What Are Current Simulation

Capabilities?Capabilities?

Time scales of biological processes

Femtosecond (fs) = 10-15 secondPicosecond (ps) = 10-12 secondNanosecond (ns) = 10-9 secondMicrosecond (μs) = 10-6 second

Molecular Dynamics TrajectoryMolecular Dynamics Trajectory

Time

Molecular Dynamics Simulation

Snapshots of Representative

Structures

Molecular dynamics trajectory is a file containing snapshots of the simulated system

Snapshots

For each Snapshot Amber Saves For each Snapshot Amber Saves Structure and Energy DecompositionStructure and Energy Decomposition

Energy Decomposition for each snapshot is written in the form:

NSTEP = 100 TIME(PS) = 10.200 TEMP(K) = 297.23 PRESS = -1257.4 Etot = -73238.8859 EKtot = 18131.6434 EPtot = -91370.5294 BOND = 654.2822 ANGLE = 1929.4666 DIHED = 775.1417 1-4 NB = 817.8912 1-4 EEL = 4242.6763 VDWAALS = 9440.1805 EELEC = -109230.1680 EHBOND = 0.0000 RESTRAINT = 0.0000 EKCMT = 7732.1695 VIRIAL = 16916.4636 VOLUME = 338304.5955 Density = 0.9012 Ewald error estimate: 0.1550E-03 ------------------------------------------------------------------------------

Snapshots of Representative

Structures

Plotting Molecular Dynamics Plotting Molecular Dynamics PropertiesProperties

Equilibration step

Equilibration step allows atoms and molecules to find more natural positions with respect to one another

MD Phase

During the MD Phase molecular properties (structures, energies, etc.) are accumulated for future analysis

Not all system properties Not all system properties reach equilibrium at the reach equilibrium at the

same timesame time

Advanced MD Analysis in AmberAdvanced MD Analysis in AmberCConformational clustering tools is available inonformational clustering tools is available in ptraj ptraj

ptraj uses several different algorithms for clustering trajectory frames into groups based on pairwise similarity

Trajectory snapshots after MD

Clustering

Statistical EnsemblesStatistical Ensembles

• The microscopic state of a system is defined by the atomic positions, q, and momenta, p; these can also be considered as coordinates in a multidimensional space called phase space

• A single point in phase space, denoted by G, describes the state of the system

• An ensemble is a collection of points in phase space satisfying the conditions of a particular thermodynamic state.

• A Molecular Dynamics simulations generates a sequence of points in phase space as a function of time;

• These points belong to the same ensemble, and they correspond to the different conformations of the system and their respective momenta

Statistical Ensembles Supported in Statistical Ensembles Supported in Amber Amber

• Microcanonical ensemble (NVE) : The thermodynamic state characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E. This corresponds to an isolated system.

• Canonical Ensemble (NVT): This is a collection of all systems whose thermodynamic state is characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed temperature, T.

• Isobaric-Isothermal Ensemble (NPT): This ensemble is characterized by a fixed number of atoms, N, a fixed pressure, P, and a fixed temperature, T.

Advanced MD Techniques in Amber Advanced MD Techniques in Amber AAdaptively daptively BBiased iased MMolecular olecular DDynamics (ynamics (ABMDABMD) method) method ABMD is a method for the computation of the free energy surface of a reaction

coordinate using non-equilibrium dynamics.• Chemical reactions, conformational transitions, etc, occur when the system migrates from one local equilibrium minimum to another, overcoming the usually large energy barriers that separate reagents from products.

• The probability of such an event occurring spontaneously depends exponentially on the energy barrier and easily exceeds the computational time regime that present-day computer technology can afford.

Advanced Molecular Dynamics Advanced Molecular Dynamics Techniques in Amber: Techniques in Amber: Path integral molecular dynamicsPath integral molecular dynamics

Path integral molecular dynamics simulations can be used to sample equilibrium canonical distributions using quantum dynamics rather than Newton's equations for nuclear motion. Both equilibrium and kinetic isotope effects can be estimated via thermodynamic integration over mass.

• Centroid Molecular Dynamics (CMD) is an approximate method for calculating real-time quantum correlation functions.

• Ring Polymer Molecular Dynamics (RPMD).

• Both CMD and RPMD simulations provide an efficient route for the calculation of approximate correlation functions, which can then be related to the true quantum correlation functions.

How to Treat Bulk System?How to Treat Bulk System?

Bulk (“infinite”) solvent

We run computer simulation to predict and study the properties of a system in bulk (very big or “infinite” system)

How to Treat Bulk System?How to Treat Bulk System?Treatable system

But…We can simulate only a relatively small number of particles in order not to slow down the computation.

Artificial surface effectProblem: we are not interested in surface effects

Possible Solutions?Possible Solutions?1) The system size should be extremely large to ensure that the surface has only a small influence on the bulk properties

But…Such system is too big to simulate…

Surface effect has small influence on the bulk properties

Possible Solutions?Possible Solutions?2) Surface effects can be ignored for all system sizes if we use periodic boundary conditions.

Central BoxNo surface

The cubical simulation box (Central Box) is replicated throughout space to form an infinite latticeAll other boxes are identical to the Central Box (its copies)

Periodic Boundary ConditionsPeriodic Boundary Conditions

1) If a molecule leaves the Central Box

2) Then one of its images will enter through the opposite face

Periodic Boundary ConditionsPeriodic Boundary Conditions

Periodic Boundary Conditions in Periodic Boundary Conditions in AMBERAMBER

Truncated octahedron

Rectangular parallelepiped

1)

2) Truncated octahedron has the advantage of being more nearly spherical than most other MD cells. This can be very useful when simulating a large molecule in solution, where fewer solvent molecules are required for a given simulation cell width.

Estimation of Binding Energies in Estimation of Binding Energies in Non-Covalent complexesNon-Covalent complexes

In general, non-covalent bonding refers to attractive intermolecular forces that are not covalent in nature.

Non-covalent interactions may include ionic bonds, hydrophobic interactions, hydrogen bonds and van der Waals forces.

Estimation of Binding Energies in Estimation of Binding Energies in Non-Covalent complexesNon-Covalent complexes

Protein-ligand complex

Evaluating Free Energies ofEvaluating Free Energies ofBinding using AmberBinding using Amber

Evaluating Free Energies ofEvaluating Free Energies ofBinding: Binding: MM-PBSAMM-PBSA

• The acronym MM-PBSA stands for Molecular Mechanics- Poisson Bolzmann Surface Area

• The MM-PBSA approach represents the postprocessing method to evaluate free energies of binding or to calculate absolute free energies of molecules in solution.

Acc. Chem. Res. 2000, 33, 889-897


1. One carries out a molecular dynamics simulation, typically in a periodic box with water and counterions (“regular” MD simulation), and correct representation of long-range electrostatic effects such as PME, saving a set of representative structures.

2. After MD Simulation any solvent and counterion molecules are removed, and the free energy, G, is calculated according to the following equation:


any solvent and counterion molecules

are removed

Evaluating Free Energies ofEvaluating Free Energies ofBinding:Binding: MM_PBSA MM_PBSA

where G is the calculated average free energy, and EMM is the average molecular mechanical energy:

where these correspond to the bond, angle, torsion, van der Waals, and electrostatic terms in the molecular mechanical force field, evaluated with no nonbonded cutoff.

Evaluating Free Energies ofEvaluating Free Energies ofBinding: Binding: MM_PBSAMM_PBSA

GPBSA is the solvation free energy calculated with a numerical solution of the Poisson-Bolzmann equation and an estimate of the nonpolar free energy with a simple surface area term.

-TSMM is the solute entropy, which can be estimated by quasi harmonic analysis of the trajectory or, in selected cases, by using normal-mode analysis. This final term is likely to be much smaller than the other two in many applications of estimating relative free energies.

Evaluating Free Energies ofEvaluating Free Energies ofBinding: Binding: Thermodynamic integrationThermodynamic integration

The thermodynamic integration (TI) technique allows to calculate the free energy difference between two systems, A and B, by slowly interconverting the Hamiltonian HA (representing system A) into the Hamiltonian HB (representing system B), during the course of the simulation.

This process could involve the annihilation or creation of atoms (“Computational alchemy” ).

Examples:

Atom → nothingGroup of Atoms (or Molecule) → nothingCharge on Atom → No charge on AtomCharge on Group of Atoms (or Molecule) →

→ No charge on Group of Atoms (Molecule)

““Computational alchemy”Computational alchemy”

• One common application of this model is pKa calculations, where the charges are mutated from the protonated to the deprotonated form

CO

O HC

O

O

Disappears during simulation

Evaluating Free Energies ofEvaluating Free Energies ofBinding: Binding: Thermodynamic integrationThermodynamic integration

The free energy difference is then given by

1

0

dH

A

The subscript λ at the pointed angles indicates that the average should be taken over an ensemble with Hamiltonian Hλ .

In MD simulations the integral is often replaced by a sum over a discrete set of values of λ:

i i

HA

where Δλ is chosen such that the result is statistically accurate while using a minimum of computer time.

Inclusion of Solvation Effects in Inclusion of Solvation Effects in AmberAmber

Practically all important biological Practically all important biological processes take place in processes take place in solventsolvent

Solvation methods can be devided into two Solvation methods can be devided into two main categories: main categories: explicit explicit

(supermolecule) (supermolecule) and and implicitimplicit solvation solvation methodsmethods

Explicit solvent modelExplicit solvent modelMolecular solvent models employ hundreds or thousands of discrete solvent molecules

Pros: Many of the properties of solutions and solutes can be reproduced

Cons: Such calculations converge only slowly to precise answers because of the large number of particles and states involved; expensive computationally.

Implicit Solvation Methods in AmberImplicit Solvation Methods in Amber

Implicit solvation schemes speed up the Implicit solvation schemes speed up the calculations by orders of magnitude and calculations by orders of magnitude and

are assumed to compromise little on are assumed to compromise little on essential features of the solvation essential features of the solvation

phenomenon. phenomenon.

Continuum solvation modelsContinuum solvation modelsContinuum model treat the solvent as a continuous medium having the average properties of the real solvent and surrounding the solute beginning at or near its van der Waals or Solvent-accessible surface.

Pros: Faster than molecular solvation models

Cons: Obtaining accurate numerical solutions for a large system such as a protein still has a significant computational cost

Implicit Solvation Methods in Amber:Implicit Solvation Methods in Amber:The Generalized Born/Surface Area ModelThe Generalized Born/Surface Area Model

To estimate the total solvation free energy of a molecule, ΔGsolv , one typically assumes that it can be decomposed into the "electrostatic" and "non-electrostatic“ parts:

nonelelsolv GGG

where ΔGnonel is the free energy of solvating a molecule from which all charges have been removed (i.e. partial charges of every atom are set to zero), and ΔGel is the free energy of first removing all charges in the vacuum, and then adding them back in the presence of a continuum solvent environment.

ΔGnonel comes from the combined effect of two types of interaction: the favorable van der Waals attraction between the solute and solvent molecules, and the unfavorable cost of breaking the structure of the solvent (water) around the solute.

Implicit Solvation Methods in Amber:Implicit Solvation Methods in Amber:The Generalized Born/Surface Area The Generalized Born/Surface Area

ModelModelCalculating ΔGnonel:

In the Amber code ΔGnonel is taken to be proportional to the total solvent accessible surface area (SASA) of the molecule, with a proportionality constant derived from experimental solvation energies of small non-polar molecules, and uses a fast Linear Combinations of Pairwise Overlaps (LCPO) algorithm [J. Comput. Chem. 20, 217-230 (1999)] to compute an analytical approximation to the surface accessible area of the molecule.

bAreaGnonel

Implicit Solvation Methods in Amber:Implicit Solvation Methods in Amber:The Generalized Born/Surface Area The Generalized Born/Surface Area

ModelModelCalculating ΔGel:

Within Amber GB models, each atom in a molecule is represented as a sphere of radius ρi with a charge qi at its center; the interior of the atom is assumed to be filled uniformly with a material of dielectric constant of 1. The molecule is surrounded by a solvent of a high dielectric εw (80 for water at 300 K)

ij jiijgb

jigbel RRrf

qqGG

,,2

1

where rij is the distance between atoms i and j, the Ri are the so-called effective Born radii of atoms i and j, and fgb is a certain smooth function of its arguments.

A common choice of fgb is

2/12

2

4exp

ji

ijjiijgb RR

rRRrf

Implicit Solvation Methods in Amber:Implicit Solvation Methods in Amber:ALPB (Analytical Linearized Poisson-ALPB (Analytical Linearized Poisson-

Boltzmann)Boltzmann)Based on an approximate analytical solution of the linearized Poisson-Bolzmann equation for a sphere (Kirkwood, 1934).The basic ALPB equation that approximates the electrostatic part of the solvation free energy is:

where β = εin /εex is the ratio of the internal and external dielectrics, α = 0. 571 412, and A is the so-called effective electrostatic size of the molecule. fgb is the same smooth function as in the GB model. The GB approximation is then just the special case of ALPB when the solvent dielectric is infinite; however, for finite values of solvent dielectric the ALPB tends to be more accurate.

Grigori Sigalov, Andrew Fenley, and Alexey Onufriev, J. Chem. Phys. 124, 124902 (2006)Grigori Sigalov, Peter Scheffel, and Alexey Onufriev, J. Chem. Phys. 122, 094511 (2005)

ij gbji

exinalpbel Af

qqGG

1

1

111

2

1

Implicit Solvation Methods in Amber:Implicit Solvation Methods in Amber:Poisson-Boltzmann solverPoisson-Boltzmann solver

An efficient finite-difference numerical solver is implemented for various applications of the Poisson-Boltzmann (PB) method.The electrostatic potential φj at atomic charge site is computed by solving the PB equation:

where ε(r) is the dielectric constant, φ(r) is the electrostatic potential, ρ(r) is the solute charge, zi is the charge of ion type i, ci is the number density of ion type i far from the solute, kB is the Boltzmann constant, and T is temperature; the summation is over all different ion types.

• This is the most rigorous method for treatment of implicit solvent in Amber• It can be used for both static (single point) and dynamic applications.• However, it is much slower than GB and ALPB and memory intensive for macromolecules.

i

Bii Ykrzzrrr /exp44

Inclusion of Solvation Effects in Inclusion of Solvation Effects in Amber: Amber: RISMRISM

RISM - Reference Interaction Site ModelRISM - Reference Interaction Site ModelRISM is an approximate solution to the Ornstein-RISM is an approximate solution to the Ornstein-

Zernike (OZ) equation:Zernike (OZ) equation:

where r12 is the separation between particles 1 and 2 while Ω1 and Ω2 are their orientations relative to the vector r12. The two functions in this relation are h, the total correlation function, and c, the direct correlation function.

RISM: RISM: Practical ConsiderationsPractical Considerations

• Calculating a 3D-RISM solution for a single Calculating a 3D-RISM solution for a single solute conformation typically requires about solute conformation typically requires about 100 times more computer time than the same 100 times more computer time than the same calculation with explicit solvent or PB.calculation with explicit solvent or PB.

• Memory: anywhere from a few megabytes Memory: anywhere from a few megabytes for the smallest solutes to gigabytes for large for the smallest solutes to gigabytes for large complexescomplexes

Exploring Conformational SpaceExploring Conformational Spaceof Biomoleculesof Biomolecules

Conformational SpaceConformational Spaceof Biomolecules Can Be Very Complexof Biomolecules Can Be Very Complex

Exploring Conformational SpaceExploring Conformational Spaceof Biomoleculesof Biomolecules

• Due to this property of the free energy landscape, efficient computational approaches for searching for low-energy minima in these complex systems present a great challenge.

Exploring Conformational Space: Exploring Conformational Space: Simulating AnnealingSimulating Annealing

Time

Temperature

Energy Profile Local Minima

Hea

ting

phas

e

Cooling phase

Hea

ting

phas

e

Cooling phase

Exploring Conformational Space:Exploring Conformational Space: REMDREMD

REMD stands for the Replica Exchange Method DynamicsIn REMD several noninteracting copies (replicas) are independently and simultaneously simulated at different temperatures.

Replica 1, T1

Replica 2, T2

Replica N, TN

At intervals during the otherwise standard simulations, conformations of the system being sampled at different temperatures are exchanged based on a Metropolis-type criterion

Exploring Conformational Space: Exploring Conformational Space: REMDREMD

As a result, the low temperature simulations (replicas) have the potential to escape kinetic traps by jumping to minima that are being sampled by the higher-temperature replicas where kinetic trapping is less prevalent.

Replica 1, T1

Replica 2, T2

Replica N, TN

Treating Long-Range Treating Long-Range Electrostatic InteractionsElectrostatic Interactions

atoms

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Cut-off

• The particle-mesh Ewald (PME) procedure (or, optionally, a "true" Ewald sum) is used to handle long-range electrostatic interactions.

Treating Long-Range Treating Long-Range Electrostatic Interactions in Electrostatic Interactions in

AmberAmber

Doing Semi-Empirical Quantum Doing Semi-Empirical Quantum Chemistry with AmberChemistry with Amber

• Amber 12 is packaged with sqm - a linear scaling semi-empirical program for calculation of energies, charges and geometries of systems up to ˜20,000 atoms.


sqm’s Available features include:• Linear scaling Divide and Conquer (D&C) calculations.• Single point AM1, PM3, MNDO, MNDO/d or PDDG-PM3

calculations.• Geometry Optimization (steepest decent, conjugate

gradient, BFGS, and LBFGS available)• Mulliken, CM1 and CM2 charge analysis• Nuclear Magnetic Resonance prediction and simulation

• Mixed quantum mechanics/molecular mechanics (QM/MM) linear scaling Semi-Empirical calculations.


• Amber 12 is packaged with SQM semi-empirical program.

MNDO: H, Li, Be, B, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, Sn, I, Hg, PbAM1: H, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, I, HgPM3: H, Be, C, N, O, F, Mg, Al, Si, P, S, Cl, Zn, Ga, Ge, As, Se, Br, Cd, In, Sn, Sb, PDDG/PM3: H, C, N, O, F, Si, P, S, Cl, Br, IPDDG/MNDO: H, C, N, O, F, Cl, Br, IRM1: H, C, N, O, P, S, F, Cl, Br, IPM3CARB1: H, C, OPM6: H, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Ar, K, Ca, Zn, Ga, Ge,Kr, Rb, DFTB/SCC-DFTB: (Any atom set available from the www.dftb.org website)

A Hybrid Quantum A Hybrid Quantum Mechanical/Molecular Mechanical Mechanical/Molecular Mechanical

(QM/MM) Approach(QM/MM) Approach

Why Do We Need a Hybrid QM/MM Why Do We Need a Hybrid QM/MM Approach?Approach?

Quantum Mechanics Molecular Mechanics

generally applicable restricted to the classes of molecule it have been designed for

allow the calculation of ground and excited state properties: molecular

energies and structures, energies and structures of transition states, atomic

charges, reaction pathways etc.

allow the calculation of ground state properties: relative molecular energies

and structures

CPU and memory hungry. Computationally efficient

Suitable for small and medium size systems

Suitable for large molecular systems

Why Do We Need a Hybrid QM/MM Why Do We Need a Hybrid QM/MM Approach?Approach?

CPU Time Memory

Method Seconds Time units KB Memory units

Quantum chemical*

273.00 1820 4889 85

Molecular Mechanical

0.15 1 58 1

The main bottleneck of quantum chemical methods is that they are CPU and memory hungry.

For example, for small peptide of 126 atoms one energy evaluation requires:

*Semi-empirical PM3 method

In general, CPU and memory requirements (N – number of atoms):

Molecular Mechanical methods ~ N2

Semiempirical Quantum Chemical methods

~ N2

Ab initio Quantum Chemical methods ~ N4

A Hybrid QM/MM ApproachA Hybrid QM/MM ApproachThe general idea of a hybrid QM/MM approach is that large chemical systems may be partitioned into 1) an electronically important region (QM region) which requires a quantum chemical treatment and 2) a remainder which only acts in a perturbative fashion and thus admits a classical description (MM region).

The Simplest Hybrid QM/MM ModelThe Simplest Hybrid QM/MM Model Hamiltonian for molecular system in the Born-Oppenheimer approximation:

esChExternalofEffect

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esch

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esch

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electrons

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electrons

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The main drawbacks of this simple QM/MM model are: it is impossible to optimize the position of the QM part relative to the external charges because QM nuclei will collapse on the negatively charged external charges.

some MM atoms possess no charge and so would be invisible to the QM atoms

the van der Waals terms on the MM atoms often provide the only difference in the interactions of one atom type versus another, i.e. chloride and bromide ions both have unit negative charge and only differ in their van der Waals terms.

“Standard” QM hamiltonian

The MM region is viewed in the QM calculations as a set of point charges

A Hybrid QM/MM ModelA Hybrid QM/MM Model So, it is quite reasonable to attribute the van der Waals parameters (as it is in the MM method) to every QM atom and the Hamiltonian describing the interaction between the QM and MM atoms can have a form:

The van der Waals term models also electronic repulsion and dispersion interactions, which do not exist between QM and MM atoms because MM atoms possess no explicit electrons.

A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103(1976), 227-49

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The Hybrid QM/MM ModelThe Hybrid QM/MM Model Now we can construct a “real” hybrid QM/MM Hamiltonian:

A “standard” MM force field can be used to determine the MM energy. For example, AMBER-like force field has a form:

nuclei

i

atomsMM

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MMMMQMQM HHHH ˆˆˆˆ/

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Choice of QM methodChoice of QM method... is a compromise between computational efficiency and practicality and the desired chemical accuracy.

The main advantage of semi-empirical QM methods is that their computational efficiency is orders of magnitude greater than either the density functional or ab initio methods

OP

O O

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Calculation times (in time units)

1800

1

36228

1

RHF/6-31G*

PM3

Ab initio method

Semi-empirical method

Calibration of the QM/MM potentialCalibration of the QM/MM potential

Crucial aspect is how the interaction between QM and MM parts is determined.

In choosing the appropriate form, it is required that the balance between attractive and repulsive forces must be preserved and the QM/MM interactions must be of the correct magnitude with respect to the separate QM and MM contributions

Calibration of the QM/MM potential:Calibration of the QM/MM potential:ParameterizationsParameterizations

1) Modification of the one-electron terms arising from interaction of the electron cloud of the QM fragment with the point charge of an MM atom.

2) By varying the radii in the van der Waals terms.

3) By varying 1)+2)

nuclei

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1) 2)


1) By hand, to find the optimum values of the parameters by calculating interaction curves for charge/ion systems and comparing them with the MP2/6-311++G** ab initio results.M.J. Field, P.A. Bash, M. Karplus, J.Comp.Chem., 11(1990), 700-733.

2) Fitting calculated H-bond energies to experimental data on ion-molecular complexes in the gas phase.V.V. Vasilyev, A.A. Bliznyuk, A.A. Voityuk, Int.J.Quant.Chem. 44(1992), 897-930.

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3) Optimizing van der Waals parameters on QM atoms to reproduce the 6-31G(d) interaction energies for H-bonded complexes in the gas phase.P.A. Bash, L. Lawrence, A.D. MacKerell, Jr., D. Levine, P. Hallstrom, PNAS USA, 93(1996), 3698-703.

4) Optimizing van der Waals parameters on QM atoms to reproduce the MP2/6-31G(dp) interaction energies for H-bonded complexes in the gas phase.J. Gao // Toward a molecular Orbital Derived Empirical Potential for Liquid Simulations // J.Phys.Chem. B 101(1997), 657-63

5) By varying the radii in the van der Waals terms to reproduce experimental free energies of solvation using MD simulations.P.L. Cummins, J.E. Gready, J.Comp.Chem., 18(1997), 1496-512.

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Dividing Covalent Bonds across the Dividing Covalent Bonds across the QM and MM RegionsQM and MM Regions

In many simulations it is necessary to have the QM/MM boundary cut covalent bonds, and a number of additional approximations have to be made.


A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103 (1976), 227-249

V. Thery, D. Rinaldi, J.-L. Rivail, B. Maigret, G.G. Ferenczy, J.Comp.Chem. 15 (1995), 269

Using a hybrid orbital on the frontier MM atom

O

O

QM Region

MM Region

Frontier MM Atom

Frontier QM Atom

Frontier MM Atom


“Link” atoms are used to gracefully cap the electron density.

This approach is used in Amber

Using “link” atoms

Implementation of “link” Atom Implementation of “link” Atom Approach in Amber 9 & 10Approach in Amber 9 & 10

The link atom is placed along the bond vector joining the QM and MM atom

The default link atom type is hydrogen

It interacts with MM region only electrostatically (no VDW term).

WdV interaction between QM and MM atoms which form 1-2 and 1-3 “bonded” pairs is not calculated.

Bond stretching, angle bending, and torsion interactions between QM and MM regions are calculated as those in MM if 1-2, 1-2-3, or 1-2-3-4 terms contain at least one MM atom

Reviews on QM/MMReviews on QM/MM

•H. Hu and W. Yang, Free energies of chemical reactions in solution and in enzymes with ab initio quantum mechanics/molecular mechanics methods, Annu Rev Phys Chem. 2008;59:573-601 •C. Bo and F. Maseras, QM/MM methods in inorganic chemistry, Dalton Trans., 2008, 2911–2919

• H.M. Senn and W. Thiel, QM/MM studies of enzymes, Current Opinion in Chemical Biology, 2007(11), 182-187

• R.A. Friesner and V. Guallar, Ab initio Quantum Chemical and Mixed Quantum Mechanics/Molecular Mechanics (QM/MM) Methods for Studying Enzymatic Catalysis, Annual Review of Physical Chemistry, 2005 (56), 389-427

• G. Monard, X. Prat-Resina, A. González-Lafont, J.M. Lluch, Determination of enzymatic reaction pathways using QM/MM methods, Int. J Quant Chem, 2003, 93 Issue 3, Pages 229 - 244

Hints for running QM/MM calculationsHints for running QM/MM calculationsChoosing the QM regionChoosing the QM region

There are no good universal rules here

One might want to have as large a QM region as possible

However, having more than 80-100 atoms in the QM region will lead to simulations that are very expensive.


For many features of conformational analysis, a good MM force field may be better than a semi-empirical or DFTB quantum description.

QM Methods in Amber 12QM Methods in Amber 12

Available semi- empirical Hamiltonians are MNDO, AM1, PM3, RM1, PDDG/PM3, PDDG/MNDO, and PM3CARB1, PM3-MAIS, MNDO/d, AM1/d (Mg from AM1/d and H, O, and P from AM1/d-PhoT) and PM6

They can be used for gas phase, generalized Born and PME periodic simulations.

QM Methods in Amber 12QM Methods in Amber 12

Support is also available the DFT methods:

1. The Density Functional Theory-based-tight-binding (DFTB) Hamiltonian

2. The Self-Consistent-Charge version, SCC-DFTB

In Amber 9 the DFTB/SCC-DFTB implementation does not support generalized Born, PME or Ewald calculations,

The elements supported by QM The elements supported by QM methods in Amber 12methods in Amber 12

MNDO: H, Li, Be, B, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, Sn, I, Hg, Pb

AM1: H, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, I, Hg

PM3: H, Be, C, N, O, F, Mg, Al, Si, P, S, Cl, Zn, Ga, Ge, As, Se, Br, Cd, In, Sn, Sb, Te, I, Hg, Tl, Pb, Bi

PDDG/PM3: H, C, N, O, F, Si, P, S, Cl, Br, IPDDG/MNDO: H, C, N, O, F, Cl, Br, IPM3CARB1: H, C, ODFTB/SCC-DFTB: H, C, N, O, S, Zn

QM/MM calculations: QM/MM calculations: ab initio ab initio and DFT methodsand DFT methods

Amber can support QM/MM simulations via an interface to external QM software packages:

ADF (Amsterdam Density Functional)

Gaussian

GAMESS-US Orca

NWChem TeraChem

QM/MM calculations: QM/MM calculations: ab initio ab initio and DFT methodsand DFT methods

Mechanical and electrostatic embedding:• Gaussian• Orca• TeraChem

Mechanical embedding:• ADF• GAMESS-US• NWChem

Importance of VisualizationImportance of Visualization

One quick look at the structure can help to detect errors and save days or weeks of your time

Freeware Visualization Programs:Freeware Visualization Programs:RasMolRasMol

http://www.openrasmol.org/

Freeware Visualization Programs:Freeware Visualization Programs:VMD (VVMD (Visualisual M Molecularolecular D Dynamicsynamics))

http://www.ks.uiuc.edu/Research/vmd/VMD is a molecular visualization program for displaying, animating, and analyzing large biomolecular systems using 3-D graphics and built-in scripting.

Freeware Visualization Programs:Freeware Visualization Programs:gOpenMolgOpenMol

http://www.csc.fi/gopenmol/

http://sirius.sdsc.edu/

Freeware Visualization Programs:Freeware Visualization Programs:ChimeraChimera

http://www.cgl.ucsf.edu/chimera/

Freeware Visualization Programs:Freeware Visualization Programs: MD Display MD Display

http://www.cgl.ucsf.edu/chimera/A Multi-platform 3D Stereo Molecular Dynamics Trajectory Visualization Package

Commercial ProgramsCommercial Programs

… they represent an expert molecular modeling environment which provides construction, editing, and visualization tools for both large and small molecules

Tripos (www.tripos.com)

Accelrys (http://www.accelrys.com)

and others…

Learning AmberLearning Amber

Amber Basic TutorialsAmber Basic Tutorials• http://ambermd.org/tutorials/

Simulating a small fragment of DNA

Basic introduction to LEaP, sander, and ptraj, to build, solvate, run MD and analyze trajectories.

Using VMD with AMBER Brief introduction to using VMD for visualising AMBER inpcrd, restrt and trajectory files

Folding TRP Cage Vreating structures using XLeap followed by running heating and long MD simulations to conduct protein folding experiments. Advanced analysis: RMSd fitting, mdcrd to binpos conversion, average structure calculation, hydrogen bond analysis and dihedral angle tracking using ptraj

Demo of Ptraj Commands How to use AMBER's ptraj analysis program to analyse a peptide simulation and gather a range of statistics from the trajectory.

Visualizing Amber Trajectories with Sirius

how to use Sirius visualization software to display and analyze AMBER MD trajectory files

Amber Advanced TutorialsAmber Advanced Tutorials• http://ambermd.org/tutorials/

Setting up an Advanced System (Including Charge Derivation)

Preparing a system, for simulation with sander, that contains several non-standard residues

A simple coupled potential QM/MM/MD simulation.

How to set up a simple QM/MM/MD simulation of NMA in solution using AMBER 9

MM-PBSA Step by step explanation of using the mm_pbsa script in AMBER 9 to calculate the binding energy of the RAS-RAF protein complex

Nudged Elastic Band (NEB) method

How use the NEB method to predict a pathway for a conformational change in alanine dipeptide.

pKa Calculations using Thermodynamic Integration

How to calculate the pKa value of the ASP residue in the protein thioredoxin

… and other

ResumeResume

• Amber package represents an expert molecular modelling environment with a reach functionality and good computer performance.

Hands-onHands-on

Web: sf.anu.edu.au/~vvv900/monash

Tutorial files (AMBER_INTRO_COURSE/):

•standard-setup.tar – “Standard” setup (long)

•nonstandard.tar – Handling non-standard residues (long)

•amber-gaussian.tar – QM/MM using Amber-Gaussian interface (short)

•qm-mm.tar – QM/MM using Amber inbuilt semiempirical methods

introduction to amber the theory and practice of biomolecular simulations using the amber suite of...

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