1 dinamica molecular y el modelamiento de macromoleculas

1

Dinamica Molecular y el

modelamiento de macromoleculas

2

Historical Perspective

1946 MD calculation 1960 force fields 1969 Levinthal’s paradox on protein folding 1970 MD of biological molecules 1971 protein data bank 1998 ion channel protein crystal structure 1999 IBM announces blue gene project

3

Proteins

Polypeptide chains made up of amino acids or residues linked by peptide bonds

20 aminoacids 50-500 residues, 1000-10000 atoms Native structure believed to correspond to energy minimum,

since proteins unfold when temperature is increased

4

Proteins: Local Motions

0.01-5 AA, 1 fs -0.1s Atomic fluctuations

• Small displacements for substrate binding in enzymes• Energy “source” for barrier crossing and other activated processes

(e.g., ring flips) Sidechain motions

• Opening pathways for ligand (myoglobin)• Closing active site

Loop motions• Disorder-to-order transition as part of virus formation

6

Levinthal paradox

Proteins simply can not fold on a reasonable time scale (Levinthal paradox; J. Chem. Phys., 1968, 65: 44-45)• Each bond connecting amino acids can have several (e.g., three)

possible states (conformations). A protein of, say, 101 AA could exist in 3100 = 5 x 1047 conformations. If the protein can sample these conformations at a rate of 1013/sec, 3 x 1020/year, it will take 1027 years to try them all. Nevertheless, proteins fold in a time scale of seconds.

7

Proteins: Rigid-Body Motions

1-10 AA, 1 ns – 1 s Helix motions

• Transitions between substrates (myoglobin) Hinge-bending motions

• Gating of active-site region (liver alcohol dehydroginase)• Increasing binding range of antigens (antibodies)

8

Quantum Mechanical Origins

Fundamental to everything is the Schrödinger equation•

• wave function

• H = Hamiltonian operator

• time independent form

Born-Oppenheimer approximation• electrons relax very quickly compared to nuclear motions

• nuclei move in presence of potential energy obtained by solving electron distribution for fixed nuclear configuration

it is still very difficult to solve for this energy routinely

• usually nuclei are heavy enough to treat classically

H it

( , , )R r t

Nuclear coordinates

Electronic coordinates

H E

22 im

H K U U

9

Force Field Methods

Too expensive to solve QM electronic energy for every nuclear configuration

Instead define energy using simple empirical formulas• “force fields” or “molecular mechanics”

Decomposition of the total energy

Force fields usually written in terms of pairwise additive interatomic potentials• with some exceptions

(1) (2) (3)( ) ( ) ( , ) ( , , )Ni i j i j ki i j i i j i k j

U u u u r r r r r r r

Single-atom energy(external field)

Atom-pair contribution 3-atom contribution

Neglect 3- and higher-order terms

10

Conformation optimization for molecular interaction

Molecular Mechanics Approach:

11

Energy minimisation

Calculation of how atoms should move to minimise TOTAL potential energy At minimum, forces on every

atom are zero.

Optimising structure to remove strain & steric clashes However, in general finds local rather than global minimum. Energy barriers are

not overcome even if much lower energy state is possible ie structures may be locked in. Hence not useful as a search strategy.

12

Energy minimisation

Potential energy depends on many parameters Problem of finding minimum value of a function with >1 parameters. Know value of

function at several points. Grid search is computationally

not feasible Methods

• Steepest descents

• Conjugate gradients

13

Molecular Dynamics: Introduction

Newton’s second law of motion

14

Molecular dynamics

F=ma F is calculated from molecular mechanical potential. Model conformational changes. Calculate time-dependent properties (transport properties).

15

We need to know

The motion of the

atoms in a molecule, x(t) and therefore,

the potential energy, V(x)


16Molecular Dynamics: Introduction

How do we describe the potential energy V(x) for amolecule?Potential Energy includes terms for

Bond stretching

Angle Bending

Torsional rotation

Improper dihedrals

17


Potential energy includes terms for (contd.)

Electrostatic

Interactions

van der Waals

Interactions

18


To do this, we should knowat given time t,

initial position of the atom

x1 its velocity

v1 = dx1/dt and the acceleration

a1 = d2x1/dt2 = m-1F(x1)

19


The position x2 , of the atom after time interval t would be,

and the velocity v2 would be, tvxx 112

tdx

dVmvtxFmvtavv x

1

111

11112 )(

20


In general, given the values x1, v1 and the potential energy V(x), the molecular trajectory x(t) can be calculated, using,

tdx

xdVmvv

tvxx

ixii

iii

1

)(11

11

21

How a molecule changes during MD

22

The Necessary Ingredients

Description of the structure: atoms and connectivity Initial structure: geometry of the system Potential Energy Function: force field

AMBERCVFFCFF95Universal

23

Contributions to Potential Energy

Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

24




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

25




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

26




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

27




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

28




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

29




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

Repulsion

30




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

Repulsion

31




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

- +- +

Repulsion

Attraction

32




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

33




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

-+

+- +

34




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

+-+

+ -++-+

+-+

u(2)

+- +

u(2)

u(N)

35




Intramolecular only

Ustr stretch

Ubend bend

Utors torsion

Ucross cross

UvdW van der Waals

Uel electrostatic

Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

+-+

+ -+

+-

+-+

u(2)

+- +

u(2)

u(N)

36

Modeling Potential energy

U(r) U(req ) dUdr rreq

(r req ) 12

d2Udr2

rreq

(r req )2

1

3

d3U

drrreq

(r req )3 ....1

n!

dnU

drn

rreq

(r req )n

37

Modeling Potential energy

dU

dr rreq

(r req )

U(r) 1

2

d2U

dr2

rreq

(r req )2 1

2kAB (r req )2

U(req )

U(r) 1

2

d2U

dr2

rreq

(r req )2

0 at minimum0

38

Stretch Energy

Expand energy about equilibrium position

Model fails in strained geometries• better model is the Morse potential

22

12 12 12 12 12 122( ) ( ) ( ) ( )

o o

o o o

r r r r

dU d UU r U r r r r r

dr dr

minimumdefine

212 12 12( ) ( )oU r k r r

(neglect)

harmonic

122

12( ) 1 rU r D e

dissociation energy force constant

250

200

150

100

50

0

Ene

rgy

(kca

l/mol

e)

0.80.60.40.20.0-0.2-0.4

Stretch (Angstroms)

Morse

39

Bending Energy

Expand energy about equilibrium position

• improvements based on including higher-order terms

Out-of-plane bending

22

2( ) ( ) ( ) ( )

o o

o o odU d UU U

d d

minimumdefine

2( ) ( )oU k

(neglect)

harmonic

2( ) ( )oU k

u(4)

40

Torsional Energy

Two new features• periodic

• weak (Taylor expansion in not appropriate)

Fourier series

• terms are included to capture appropriate minima/maximadepends on substituent atoms

– e.g., ethane has three mimum-energy conformations• n = 3, 6, 9, etc.

depends on type of bond– e.g. ethane vs. ethylene

• usually at most n = 1, 2, and/or 3 terms are included

1( ) cos( )nn

U U n

41

Van der Waals Attraction

Correlation of electron fluctuations Stronger for larger, more polarizable molecules

• CCl4 > CH4 ; Kr > Ar > He

Theoretical formula for long-range behavior

Only attraction present between nonpolar molecules• reason that Ar, He, CH4, etc. form liquid phases

a.k.a. “London” or “dispersion” forces

-+-+ - +- +

86

( )attvdW

CU O r

r

42

Van der Waals Repulsion Overlap of electron clouds Theory provides little guidance on form of model Two popular treatments

inverse power exponential typically n ~ 9 - 12 two parameters

Combine with attraction term• Lennard-Jones model Exp-6

repvdW n

AU

r

rep BrvdWU Ae

12 6

A CU

r r 6

Br CU Ae

r

a.k.a. “Buckingham” or “Hill”

10

8

6

4

2

0

2.01.81.61.41.21.0

LJ Exp-6

Exp-6 repulsion is slightly softer

20

15

10

5

0

x103

8642

Beware of anomalous Exp-6 short-range attraction

43

Electrostatics 1.

Interaction between charge inhomogeneities Modeling approaches

• point charges

• point multipoles

Point charges• assign Coulombic charges to several points in

the molecule

• total charge sums to charge on molecule (usually zero)

• Coulomb potential

very long ranged

0( )

4i jq q

U rr

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

44

Electrostatics 2. At larger separations, details of charge distribution are less important Multipole statistics capture basic features

• Dipole

• Quadrupole

• Octopole, etc.

Point multipole models based on long-range behavior• dipole-dipole

• dipole-quadrupole

• quadrupole-quadrupole

i iiq r

i i iiqQ r r

Vector

Tensor

0, 0Q

0, 0Q

Q

Q

1 21 2 1 23

ˆ ˆˆ ˆ ˆ ˆ3( )( ) ( )ddur

r r

21 21 2 1 2 24

3 ˆ ˆˆ ˆ ˆˆ ˆ ˆ( ) 5( ) 1 2( )( )2dQ

Qu Q Q

r

r r r

2 2 2 2 21 21 2 12 1 2 1 2 125

31 5 5 2 35 20

4QQQ Q

u c c c c c c c cr

Axially symmetric quadrupole

45

Polarization

Charge redistribution due to influence of surrounding molecules• dipole moment in bulk different

from that in vacuum

Modeled with polarizable charges or multipoles Involves an iterative calculation

• evaluate electric field acting on each charge due to other charges

• adjust charges according to polarizability and electric field

• re-compute electric field and repeat to convergence

Re-iteration over all molecules required if even one is moved

+ -+

+-

+-+

+ -++-+

+-+

46

Polarization

ind E

ind ,i Ei

Ei q jrij

rij3

ji

ijrij

3ji

3rij

rij

rij

1

Approximation

Electrostatic field does not include contributions from atom i

47

Common Approximations in Molecular Models Rigid intramolecular degrees of freedom

• fast intramolecular motions slow down MD calculations

Ignore hydrogen atoms• united atom representation

Ignore polarization• expensive n-body effect

Ignore electrostatics Treat whole molecule as one big atom

• maybe anisotropic

Model vdW forces via discontinuous potentials Ignore all attraction Model space as a lattice

• especially useful for polymer molecules

Qualitative models

48


Equation for covalent terms in P.E.

)](cos1[)(

)()(

02

0

20

20

nAk

kllkRV

torsions

n

impropers

anglesbonds

lbonded

49


Equation for non-bonded terms in P.E.

ijr

ji

ij

ij

ij

ij

ji

nonbonded r

qq

r

r

r

rijRV

0

6min

12min

4])(2)[(()(

50

-14 to –130.5 to 1Torsional vibration of buried

groups

-11 to –100.5 to 1Rotation of side chains at

surface

-12 to –111 to 2Elastic vibration of globular

region

-14 to –130.2 to 0.5Relative vibration of bonded

atoms

Log10 of characteristic time

(s)Spatial extent (nm)Motion

An overview of various motions in proteins (1)

51

Log10 of characteristic time (s)

Spatial Extent (nm)

Motion

-5 to 2???Protein folding

-5 to 10.5 to 1Local denaturation

-5 to 00.5 to 4Allosteric transitions

-4 to 00.5Rotation of medium-sized side chains

in interior

-11 to –71 to 2Relative motion of different globular

regions (hinge bending)

An overview of various motions in proteins (2)

52

A typical MD simulation protocol

Initial random structure generation Initial energy minimization Equilibration Dynamics run – with capture of conformations at regular

intervals Energy minimization of each captured conformation

53

Essential Parameters for MD (to be set by user)

Temperature Pressure Time step Dielectric constant Force field Durations of equilibration and MD run pH effect (addition of ions)

54

STARTING DNA MODEL

55

DNA MODEL WITH IONS

56

DNA in a box of water

57

SNAPSHOTS

58

Protein dynamics study

Ion channel / water channel Mechanical properties

• Protein stretching

• DNA bending

Movie downloaded from theoreticla biophysics group, UIUC

59

),()(),( jiHBijLJjiww XXUrUXXU

612

4)(ij

LJ

ij

LJLJijLJ rr

rU

Molecular Interactions

)1()1()(),(3

1,

ijllk

ijkHBijHBjiHB ujGuiGrrGXXU

)2/exp()( 22 xxG

water-water interaction

van der Waals’ term

Hydrogen bonding term

ion-water interaction

,

charge ),()(),( jiijLJjiiw XXUrUXXU

ij

ijHBjiij r

rzzrU

)exp()(charge

61

Average number of hydrogen bonds within the first water shell around an ion

63

Solvent dielectric models

V QiQ j

rij

Effetive dielectric constant

eff r r r 1

2rS 2 2rS 2 e rS

S 0.15Å 1 ~ 0.3Å 1

64

Introduction to Force FieldsIntroduction to Force Fields

•Sophisticated (though imperfect!) mathematical function

•Returns energy as a function of conformation

It looks something like this …

U(conformation) = Ebond + Eangle + Etors + Evdw + Eelec+ …

65

Why do we need force field? Force field and potential energy surface.

• Changes in the energy of a system can be considered as movements on a multidimentional surface call the “energy surface”. Force is the first derivative of the energy.

In molecular mechanics approach, the dimension of potential surface is 3N, N is number of particles.

The probability of the molecular system stay in certain conformations can be calculated if the underlying potential is known.

66

Three types of force field Quantum mechanics (Schrodinger equation for electrons),

usually deal with systems with less than 100 atoms. Empirical force field: molecular mechanics (for atoms), can

be used for systems up to millions of atoms. Statistical potential (flexible), no restriction.

67

Source of FF componentsSource of FF components

•Geometrical terms: bond, angle, torsion & vdw parameters come from empirical data.

•Electrostatic charges: two problems arise no exp.data for charges basis underlying molec. model

68

Molecular mechanical force field

Potential is the summation of the following terms : Bond stretching, Angle bending, Torsion rotation, Non-bonded interactions

• Vdw interaction, • Electrostatic interaction.

(Hydrogen bonds). (Implicit solvent). …

69

Figures are taken from NIH guide of molecular modeling

72

non-bonded terms

73

Class I CHARMM CHARMm (Accelrys) AMBER OPLS/AMBER/Schrödinger ECEPP (free energy force field) GROMOS

Class II CFF95 (Biosym/Accelrys) MM3 MMFF94 (CHARMM, Macromodel, elsewhere) UFF, DREIDING

Common empirical force fields

74

Assumptions

Hydrogens often not explicitly included (intrinsic hydrogen methods)• “Methyl carbon” equated with 1 C and 3 Hs

System not far from equilibrium geometry (harmonic) Solvent is vacuum or simple dielectric

75

Assumptions:Harmonic Approximation

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

76

Assumptions:Harmonic Approximation

d2Udx 2

x0

6.45 102 kgs2

k

HO k m 2

k

6.45 102 kg

s2

1.67 10 27 kg6.215 1014 Hz

2

9.8911013 Hz 3.30 103cm 1(Exp : 4.395 103cm 1)

77

Brief History of FFBrief History of FF

78

Force Field classificationForce Field classification

1.- with rigid/partially rigid geometries

ECEPP, …

2.- without electrostatics

SYBYL, …

3.- simple diagonal FF

Weiner, GROMOS, CHARMm, OPLS/AMBER, …

4.- more complex FF

MM2, MM3, MMFF, …

79

Force Field Electrostatics van der Waals

CHARMm empirical fit to quantum mechanics dimers

empirical (x-ray, crystals)

GROMOS empirical empirical (x-ray, crystals)

OPLS/AMBER empirical (Monte Carlo on liquids)

empirical (liquids)

Weiner ESP fit (STO-3G) empirical (x-ray, crystals)

Cornell RESP fit (6-31G*) empirical (liquids)

Comparison of the simple diagonal FFComparison of the simple diagonal FF

80

Transferability

AMBER (Assisted Model Building Energy Refinement)• Specific to proteins and nucleic acids

CHARMM (Chemistry at Harvard Macromolecular Mechanics)• Specific to proteins and nucleic acids • Widely used to model solvent effects• Molecular dynamics integrator

81

Transferability

MM? – (Allinger et. al.) • Organic molecules

MMFF (Merck Molecular Force Field)• Organic molecules

• Molecular Dynamics

Tripos/SYBYL• Organic and bio-organic molecules

82

Transferability

UFF (Universal Force Field)• Parameters for all elements

• Inorganic systems

YETI • Parameterized to model non-bonded interactions

• Docking (AmberYETI)

83

MMFF Energy

Electrostatics (ionic compounds) • D – Dielectric Constant

• - electrostatic buffering constant

nij

jiticelectrosta

RD

qqE

84

MMFF Energy

Analogous to Lennard-Jones 6-12 potential• London Dispersion Forces

• Van der Waals Repulsions

2

07.0

07.1

07.0

07.17*7

7*7

*

*

ijij

ij

ijij

ijijVDW

RR

R

RR

RE

The form for the repulsive part has no physical basis and is for computational convenience when working with largemacromolecules. K. Gilbert: Force fields like MM2 which is used for smaller organic systems will use a Buckingham potential (or expontential) which accurately reflects the chemistry/physics.

85

Pros and Cons

N >> 1000 atoms Easily constructed

Accuracy Not robust enough to describe

subtle chemical effects• Hydrophobicity• Excited States• Radicals

Does not reproduce quantal nature

86

Simple Statistics on MD Simulation

Atoms in a typical protein and water simulation 32000 Approximate number of interactions in force calculation 109 Machine instructions per force calculation 1000 Total number of machine instructions 1023

Typical time-step size 10–15 s Number of MD time steps 1011 steps Physical time for simulation 10–4 s Total calculation time (CPU: P4-3.0G ) days 10,000

87

Hardware Strategies

Parallel computation • PC cluster• IBM (The blue gene), 106 CPU

Massive distributive computing• Grid computing (formal and in the future) • Server to individual client (now in inexpensive)

Examples: SETI, folding@home, genome@homeprotein@CBL

88

# publications/year mentioning FF used to model proteins

1 dinamica molecular y el modelamiento de macromoleculas

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