bio molecular simulation

8/3/2019 Bio Molecular Simulation

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Simulation of Biomolecules

1. Force fields:Energy terms, Topology and parameter files


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2. Molecular Dynamics - Preceding Information:

Computation of nonbonded interactions, IntroductMechanics


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3. Molecular Dynamics I:The Idea, Integrating the equations of motion, MD

ensembles, Thermostats


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ensembles, Thermostats4. Molecular Dynamics II:

Langevin dynamics, Brownian dynamics


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5. Monte Carlo Simulations:

The Idea: Importance Sampling and the MetropolM M t C l i l ti i i


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5. Monte Carlo Simulations:

The Idea: Importance Sampling and the MetropolM M t C l i l ti i i


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Force fields:

Energy terms in all-atom force fields Non-covalent interactions

Parameterization of force fields Topology and parameter files Coarse-grained force fields

Knowledge-based force fields


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All-atom force fields

classical biomolecular simulations: approximation todynamics

classical force fields parameterized based on quant

simulations the potential ( force field) is a function of the cooparticles (N atoms)

energy expressed in terms of atom pairs, triples a

using the concept of chemical bonding:N 1 bond lengths, N 2 bond angles (involving torsion angles (between 3 bonds)

together with 6 degrees of freedom (DOF) for over

and rotation: 3N DOFs


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the potential can be expressed as

U =a U(1)(ra) +ab U

(2)(ra, rb)

+abc

U(3)(ra, rb, rc) +abcd

U(4)(ra, rb, r

Higher order terms arise from, e.g., polarisation winduce other multipoles. The latter interact with theon. Thus, the problem can only be solved iterative

An alternative expression to Eq. (1) is the division

and nonbonded contributions. Bonded contributions arsise from bond vibrations


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Nonbonded interactions are calculated between thdifferent molecules and for atoms of the same moseparated by more than thre bonds.

All nonbonded interactions are added up:


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Two-body potentials

Pairwise interactions are functions of the distance atoms a and b: rab = |ra rb|

Simple pairwise interactions:


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Two-body potentials: Lennard-Jones potentia

U(2)LJ = Um

rvdWij

r12

2rvdWij

r6

d

m

0.8 1 1.2

U

The attractive r6

term originates from quantum mto electron correlation, the so-called London or disinteractions.

The repulsive r12 term has a quantum origin in th

the electron clouds with each other (Pauli exclusiointernuclear repulsions.


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Two-body potentials: Lennard-Jones potentia

U(2)LJ = Um

rvdWij

r12

2rvdWij

r6

d

m

0.8 1 1.2

U

For r , U(2)

LJ 0: short range van der Waals ( At r = rvdWij the LJ potential has its minimum with

rvdWij = rvdWi + r

vdWj : sum of the vdW radii of atom

The potential depth is determined by the polarisabb f l t N i th t l t h


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Two-body potentials: Coulomb potential

U(2)coul = Kcoul qiqjr

distance (A

energy(kcal/mol)

2000

1000

0

1000

2000

0.1 0.2 0.0

For ionic interactions between fully or partially cha Repulsive if particles have the same charge, attrac

opposite charges.Th di l t i t t d ib th i


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Two-body potentials: Coulomb potential

U(2)coul = Kcoul

qiqjr

Unlike vdW interactions, the Coulomb interactions

with distance. The Coulomb potential constant is given by Kcoul

where the permittivity of a vacuum is 0 = 8.8542 C2 m1 J1.

With 1 m = 1010 , 1 C = 1/(1.6022 1019) esu (eelectrostatic charge unit for the elementary charge, e0,proton), and 1 J 6.0221 1023/4184 kcal mol1 f

Kcoul =1J m 332 kca


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Electrostatic interactions in molecules

Most interactions in molecules are electrostatic. An electric dipole consists of two charges q and

a distance l. The dipole is represented by a vecto

from the negative to the positive charge, q qmoment unit: 1 Debye = 1 D = 0.208 e0. In an external field the dominant molecular multipo All heteronuclear two-atomic molecules are polar d

difference of electronegativity of both atoms resultcharges (e.g., 1.08 D in HCl).

Depending on the symmetry, a multiatom molecul(e.g., 1.85 D in H2O).

Apolar molecules do not have a permanent electri


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Electrostatic interactions in molecules

Permanent and induced dipole moments are important in

TypicalType of Distance energy valuesinteraction dependence [kcal/mol] Emonopole-monopole 1/r 50 to 5 smonopole-dipole 1/r2 3.5 L

adipole-dipole 1/r3 0.5 b

London (vdW) 1/r6 0.1 a


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Hydrogen bonds

Hydrogen bonds derive from electrostatic interactithe positive partial charge of a H atom bound to anelectron-withdrawing donor (D) and the lone pair the acceptor atom (A): D H+ A whereand F.

The hydrogen bond energy depends on the geomA. The optimal DH A angle is 180.

The optimal H A AA angle (AA = anterior accon the D, A and AA elements and the hybridization


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Hydrogen bonds

Hydrogen bonds in proteins:

D AType distance [

amide-carbonyl >NH O=C< 2.90.1 hydroxyl-carbonyl OH O=C< 2.80.1 hydroxyl-hydroxyl OH OH 2.80.1

amide-hydroxyl >NH OH 2.90.1 amide-imidazole >NH N 3.10.2 ammonium-carboxyl NH+3

OOC 2.70.1 guanidinium-carboxyl

...NH+

2

OOC 2.70.1


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Two-body potentials: Bond length potentials

Bond length potentials model small-scale deviatioequilibrium bond length r0.

The harmonic potential using Hookes law is the s

molecular-mechanics formulation for bond vibratio According to Hookes law, the force F is proportiondisplacement, r r0, and the acceleration, d2r/dt

F(r) = kvib(r r0) = m d2

rdt2

, kvib =

From the angular frequency (number of radians which is connected to the wavelength via = 2c/mass m, the force constant kvib can be calculated


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Two-body potentials: Bond length potentials

From (2) the bond length potential follows:

U(2)HO =

kvib2

(r r0)2 (3)

r0

bondpotential

where the subscript HO is for harmonic oscillator. More realistic functions for bond vibration is, for ins

the Morse potential.

But since kvib usually large and and bond lengths (


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Three-body potentials: Bond angle potentials

Bond angle arrangement around each atom in a mgoverned by the hybridization stage of this atom, e sp hybridization linear geometry with 18

sp2

hybridization trigonal planar with 12 sp3 hybridization tetrahedral with 109.5

Exact orbits only exist if all bonded atoms are of the.g., as in methane. In propane, for instance, the C

angle is 112.5

, the HCH bond angles 107.5 109.5.

Electron lone pairs influence the geometry too. The bond angle is determined by


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Three-body potentials: Bond angle potentials

e1 =rb ra|rb ra|

, e2 =rc ra|rc ra|

The bond angle potential is often expressed as a h

function in terms of angle cosines:

U(3)HO() = kbend(cos cos 0)

2

Advantage of this trigonometric potential is its bouits ease of implementation and differentiation.

It avoids the calculation of inverse trigonometric fusingularity problems for linear bond angles.


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Four-body potentials: Bond torsion potentials

Origin of torsional potential: relief of steric congesstabilization and electronic repulsions (quantum meffects)

For example, in ethane the torsional strain is highemethyl groups are nearest (eclipsedor cis) state, athey are optimally separated (anti, transor stagge

The functional form is force fields is

U(4)() =n

Vn2

[1+cos (n 0)]


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Often 0 = 0, so that U(4)() =

nVn2 [1 cos n

The torsional angle is determined by the the posatoms involved, a, b, c, d and the three bonds i, j,

them:

(ra, rb, rc, rd) = arccos

(ei ej)(ej

sin ij sin

Combination of two- and threefold symmetry to repcis/transand trans/gaucheenergy differences:

U (4)()V2

[1 (2)] y[kcal/mol]

2 fold

3 fold

sum

O3

C3

C2

O2

3

4


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n and Vn are determined for low molecular weightexperimentally using NMR, IR, Raman and microwspectroscopy and/or quantum mechanical calclatiocompounds.

Examples for model compounds: hydrocarbons like ethane, propane or cyclohex

rotations about single CC bonds

ethylbenzene for rotations about CAC bonds(CA is an aromatic carbon here) alcohols like methanol and propanol for HC

CCOH, HCCO and CCCO seq

for serine and threonine) methylamine ethanal and N-methyl formamide


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Four-body potentials: Improper torsion

improper torsion is also known as out-of-plane b To enforce planarity or maintain chirality about cer

U(4)HO() =kimp

22 CA

where is the improper angle defined for the four for which the central atom a is bonded to b, c, d, asbetween the planes a b c and b c d.

Examples:

CCAN0 : for planarity around peptide bon C CA O 0 : for planarity of the carboxy grou


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Cross terms

Cross terms couple the as independent assumed the potential energy function arising from bonded

Schematic representation of various cross terms:


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Cross terms

For example, a stretch/bend cross term for a bondsequence abc allows bond lengths a b and b cincrease/decrease as abc decreases/increases.

A variation of the stretch/bend term is the Urey-Brawhich is commonly used in force fields (e.g., in CHUB potential is a simple harmonic function of the idistance between atoms i and k in the bonded seq

A torsion/torsion cross term is included in CHARMwhich accounts for the interdependence of the Raangles and .

Cross terms are corrections to the potential nergy

better agreement with the results from experimentchemical calculations


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Summary: Potential energy function

The total energy function for the CHARMM potential is:

U =

bondskb(b b0)

2 +

anglesk( 0)

2

+

dihedrals

n

Vn2

[1 + cos (n 0)] +

impropers

+

nonbondedpairs ij

UijrvdWij

r

12 2

r

vdW

ij

r

6++

UreyBradley

kUB(s s0)2 +

residues

UCMAP(, )


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Summary: Potential energy function

All force fields consider a molecule as a collectionheld together by some sort of elastic forces.

The atoms of a molecule may be thought of as joinmutually independent springs, restoring "natural" vlengths and angles.

All forces are defined in terms of potential energy internal coordinates of the molecules that constitu

force field. In these expressions, the sums extend over all bontorsions, and nonbonded interactions between all bound to each other or to a common atom.

More elaborate force fields may also include eithet (1 3 b d d i t ti ) i t


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Atom types in force fields

Transferability: The principle of transferability assumpotentials can be developed to incorporate all expfor model compounds and the applied successfullyof large biological molecules composed of the sam

subgroups. A reasonable assumption since bond lengths and

tend to adopt similar values in different molecular

To represent the different chemical environment (ecarbon in a phenyl ring) and hybridization, differenused for the same element.

In CHARMM22, for instance, there are around 57

carbons, 12 hydrogens, 11 nitrogens, 7 oxygens, 3heme iron one calcium cation one zinc cation


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Examples of atom types defined in CHARMM:


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Examples of atom typesas used in polypeptides inCHARMM:


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Force fields: Topology file

RESI ALA 0.00GROUPATOM N NH1 -0.47 ! |ATOM HN H 0.31 ! HN-NATOM CA CT1 0.07 ! | HB1

ATOM HA HB 0.09 ! | /

GROUP ! HA-CA--CB-HB2ATOM CB CT3 -0.27 ! | \

ATOM HB1 HA 0.09 ! | HB3ATOM HB2 HA 0.09 ! O=CATOM HB3 HA 0.09 ! |GROUP !ATOM C C 0.51ATOM O O -0.51BOND CB CA N HN N CABOND C CA C +N CA HA CB HB1 CB HB2 CB HB3DOUBLE O CIMPR N -C CA HN C CA +N OCMAP -C N CA C N CA C +N

DONOR HN NACCEPTOR O C


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Force fields: Parameter file

BONDS!!V(bond) = Kb(b - b0)**2!Kb: kcal/mole/A**2!b0: A!

!atom type Kb b0...CT3 CT1 222.500 1.5380...

ANGLES!

!V(angle) = Ktheta(Theta - Theta0)**2!Ktheta: kcal/mole/rad**2!Theta0: degrees!!V(Urey-Bradley) = Kub(S - S0)**2!Kub: kcal/mole/A**2 (Urey-Bradley)!S0: A!!atom types Ktheta Theta0 Kub S0


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DIHEDRALS!!V(dihedral) = Kchi(1 + cos(n(chi) - delta))!Kchi: kcal/mole!n: multiplicity!delta: degrees

!!atom types Kchi n delta...C CT1 NH1 C 0.2000 1 180.00NH1 C CT1 NH1 0.6000 1 0.00...

0

180

ene

rgy[kcal/mol]

0.4

0.8

1.2


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IMPROPER!!V(improper) = Kpsi(psi - psi0)**2!Kpsi: kcal/mole/rad**2!psi0: degrees!

!atom types Kpsi psi0...NH1 X X H 20.0000 0.00...

The X is a wildcard, i.e., N H 1 X X H includes the improper dihedral group, NH1 C CT1 H.

CMAP! 2D grid correction data. The following surfaces ! to the CHARMM22 phi, psi alanine, proline and gly! surfaces....

! alanine mapC NH1 CT1 C NH1 CT1 C NH1 24


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NONBONDED!!V(Lennard-Jones) = Eps,i,j[(Rmin,i,j/ri,j)**12 - 2!epsilon: kcal/mole, Eps,i,j = sqrt(eps,i * eps,j)!Rmin/2: A, Rmin,i,j = Rmin/2,i + Rmin/2,j!

!atom epsilon Rmin/2...C -0.110000 2.000000CA -0.070000 1.992400...


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Parameterization

Parameterization process for potential energy funcdifficult task.

The combinations of parameters that can be usedUnrealistic choices for one group of parameters cacompensated for by adjustment of another in ordeset of structural and energetic data.

The energy terms should have clear physical signparameters calibrated by empirical fitting of crystabarriers of analogous small molecules, vibrational and quantum-chemical data.

Problems arise from

approximations made in the extension of datalarge systems


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Parameterization

In summary, much freedom and manipulation are parameterizing a given energy function.

Only if constructed and parameterized correctly wmodel allow reliable structural predictions.

Energy parameters are not transferable from one force fi

Need of improvement: Determination of partial charges.

Improvement of electrostatic potential, e.g., usmultipoles instead of atomic point charges onlyforce fields).

Solvent representation and interpretation of re

absence of solvent.


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Overview of all-atom force fields

Program Force fields Molecules Comments

AMBER ff94, ff96, ff98 proteins all-atom ff, ff96 cellation

ff99 proteins, RNA,

DNA

good nonbonded

bilization of helicff99SB proteins, RNA,DNA

fixed the overstaff99

ff03 proteins, RNA,DNA

reparameterizedprovision for po

ased towards hebsc0 proteins, RNA,

DNAreparameterizatand torsional forbackbone

GAFF any molecule general Amber ffor the ligands in


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Overview of all-atom force fields

CHARMM22 proteins all-atom quantum-rameterizused with

CHARMM22/CMAP proteins the CHA

CMAP baCHARMM27 proteins, lipids,RNA, DNA,sugars

based on

CHARMM force fields implemented in CHARMM, N

GROMOS GROMOS96,GROMOS43a2,GROMOS53a6

alkanes, pro-teins, sugars

united-atoff availabmodel

GROMOS force fields implemented in GROMOS anOPLS OPLS-UA proteins united-ato

models


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Coarse-grained force fields

Several nuclei (and electrons) are lumped into pseso-called beads.

For proteins one typically has 2 beads per amino abackbone and, depending on the amino acid, betwbeads for the sidechain.

Bonded and nonbonded interactions between beathose between atoms in all-atom models.


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An alternative to physics-based force fields are knor statistical, energy functions, which derive from tknown protein structures.

The probabilities are calculated that residues appeconfigurations (e.g., rotamer conformations or bursurface environments), and that pairs of residues in a particular relative geometry.

These probabilities are converted into an effectiveenergy using the Boltzmann equation

G = kBT lnp(c1, c2)

p(c1)p(c2),


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Advantage: any behaviour seen in known protein sbe modeled, even if a good physical understandinbehaviour does not exist.

Disadvantage: these energy functions are phenomcannot predict new behaviour absent from the trai

Examples: Go models, Associated Memory HamilWolynes), Rosetta (David Baker)

References:S. Wodak and M. Rooman, Generating and testing protein folds, Cur3, 249-259 (1993)M.J. Sippl, Knowledge-based potentials for proteins, Curr. Opin. Struc(1995)

R.L. Jernigan and I. Bahar, Structure-derived potentials and protein si


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Solvation

ExplicitExplicit vs.vs. Implicit SolventImplicit Solvent

li i l


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Every water molecule is modeled explicitly.

Commonwater models are:

TIP3P TIP5P

Explicit solvent interactions are replaced by anenergy term based on solvent's mean fieldbehavior.

Most implicit solvent models start from a continuumelectrostatic description for the solvent.

solvent-inaccessible lowdielectric cavity containingthe solute

high dielectric mediumfor the solvent

Explicit solvent : Implicit solvent :

ExplicitExplicit vs.vs. Implicit SolventImplicit Solvent


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Approximation !

explicit solvent continuum electrostatics

lost information :

Efficient ?

Enormous decrease in degrees of freedom.

But not every implicit-solvent implementation isefficient !

Implicit solvent 1 : EEF1Implicit solvent 1 : EEF1


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EEF1 = Effective Energy Function W(R) for proteins with coordinates R in solution.

Assumption 1 : sum over groups i

Assumption 2 :

Assumption 3 :

Implicit solvent 2 : Generalized Born modelsImplicit solvent 2 : Generalized Born models


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The reaction field potential can be computed by solving the Poisson-Boltzmann equation:

dielectric constantof the solvent

Debye-Hckel

screening factor

solute charge density

Generalized Born modelsGeneralized Born models


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Solving the PB equation for a spherical solute with radius R, charge q, and dielectric constantgives the Born formula(Born, Z. Phys. 1 (1920), 45) :

Inspired by the Born formula, the generalized Born (GB) theory was developped. The mostreliable GB formula is Still's equation (Still et al., JACS 112 (1990), 6127) :

Needs the Born radii as input, which are conventionally computed within Coulomb-fieldapproximation (CFA) :

Born radii

Born radiiBorn radii


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Calculation of Born radiiCalculation of Born radii


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pairwise summation :

GenBornDominy and Brooks, JPCB 103 (1999), 3765

ACE

Schaefer and Karplus, JPC 100 (1996), 1578Schaefer et al., JCC 22 (2001), 1857

GBr6

Tjong and Zhou, JPCB 111 (2007), 3055

numerical volume integration :

GBMVLee et al., JCP 116 (2002), 10606

GBSW

Im et al., JCC 24 (2003), 1691

Beyond Coulomb-field approximation ?

in GBSW :

in Gbr6 :

Coulomb-fieldapproximation :

bio molecular simulation

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