MACROMOLECULAS BIOLOGICAS PATRICA ROJAS FIGUEROA BIOLOGÍA Y CIENCIAS.
1 Dinamica Molecular y el modelamiento de macromoleculas.
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Transcript of 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
5
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)
Molecular Dynamics: Introduction
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
Molecular Dynamics: Introduction
Potential energy includes terms for (contd.)
Electrostatic
Interactions
van der Waals
Interactions
18
Molecular Dynamics: Introduction
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
Molecular Dynamics: Introduction
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
Molecular Dynamics: Introduction
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
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
25
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
26
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
27
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
28
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
29
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
Mixed terms
Repulsion
30
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
Mixed terms
Repulsion
31
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
Mixed terms
- +- +
Repulsion
Attraction
32
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
Mixed terms
-+-+
Repulsion
Attraction
33
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
Mixed terms
-+-+
Repulsion
Attraction
-+
+- +
34
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
Mixed terms
-+-+
Repulsion
Attraction
+-+
+ -++-+
+-+
u(2)
+- +
u(2)
u(N)
35
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
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
Molecular Dynamics: Introduction
Equation for covalent terms in P.E.
)](cos1[)(
)()(
02
0
20
20
nAk
kllkRV
torsions
n
impropers
anglesbonds
lbonded
49
Molecular Dynamics: Introduction
Equation for non-bonded terms in P.E.
ijr
ji
ij
ij
ij
ij
ji
nonbonded r
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
60
61
Average number of hydrogen bonds within the first water shell around an ion
62
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
70
71
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