Introduction to Molecular Modeling Techniques
Molecular Modeling Techniques are a Critical Component of Determining a Protein
Structure by NMR:
• Protein structures are calculated by augmenting traditional modeling functions with
experimental NMR data
Molecular Modeling/Molecular Mechanics is a method to calculate the structure and energy of
molecules based on nuclear motions.
• electrons are not considered explicitly
• will find optimum distribution once position of nuclei are known
• Born-Oppenheimer approximation of Shrödinger equation
 nuclei are heavier and move slower than electrons
 nuclear motions (vibrations, rotations) can be studied separately
 electrons move fast enough to adjust to any nuclei movement
molecular modeling treats a molecule as a collection of weights
connected with springs, where the weights represent the nuclei
and the springs represent the bonds.
Introduction to Molecular Modeling Techniques
Force Field used to calculate the energy and geometry of a molecule.
• Collection of atom types (to define the atoms in a molecule), parameters (for bond lengths,
bond angles, etc.) and equations (to calculate the energy of a molecule)
• In a force field, a given element may have several atom types.
 For example, phenylalanine contains both sp3-hybridized carbons and aromatic carbons.
 sp3-Hybridized carbons have a tetrahedral bonding geometry
 aromatic carbons have a trigonal bonding geometry.
 C-C bond in the ethyl group differs from a C-C bond in the phenyl ring
 C-C bond between the phenyl ring and the ethyl group differs from all other C-C
bonds in ethylbenzene. The force field contains parameters for these different
types of bonds.
Introduction to Molecular Modeling Techniques
Force Field used to calculate the energy and geometry of a molecule.
• Total energy of a molecule is divided into several parts called force potentials, or potential
energy equations.
• Force potentials are calculated independently, and summed to give the total energy of the
molecule.
 Examples of force potentials are the equations for the energies associated with bond
stretching, bond bending, torsional strain and van der Waals interactions.
 These equations define the potential energy surface of a molecule.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Bonds Length)
• Whenever a bond is compressed or stretched the energy goes up.
• The energy potential for bond stretching and compressing is described by an equation
similar to Hooke's law for a spring.
• Sum over two atoms
Sum over all bonds in the structure
lo – expected/natural bond length
kl – force constant
l – actual/observed bond length
From what we know about protein structures 
what we have been discussing up to this point
From the structure
Plot of Potential
Energy Function
for Bond Length
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Angles)
• As the bond angle is bent from the norm, the energy goes up.
• Sum over three atoms
Sum over all bond angles in the structure
fo – expected/natural bond angle
kf – force constant
f – actual/observed bond angle
From what we know about protein structures 
what we have been discussing up to this point
From the structure
Plot of Potential
Energy Function
for Bond Angle
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Improper Dihedrals)
• As the improper dihedral is bent from the norm, the energy goes up.
• Sum over four atoms
Sum over all improper
dihedrals in the structure
wo – expected improper dihedral (usually set to 0o)
kw – force constant
w – actual/observed improper dihedral
Plot of Potential
Energy Function
for Improper
Dihedrals (wo = 0)
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Dihedral Angles)
• As the dihedral angle is bent from the norm, the energy goes up.
• The torsion potential is a Fourier series that accounts for all 1-4 through-bond
relationships
• Sum over four atoms
Sum over all
dihedrals in the structure
… Fourier Series
fo – expected improper dihedral
An – force constant for each Fourier term
f – actual/observed improper dihedral
n – multiplicity (same parameter seen in the XPLOR constraint file)
Plot of Potential
Energy Function
for Dihedrals
Multiple minima
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Dihedral Angles)
• Need to include higher terms non-symmetric bonds
 Distinguish trans, gauche conformations
Different multiplicities identify which
torsion angles are energetically equivalent
For c1, 60, -60 & 180 are all equivalent
and should yield 0 torsion energy
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Nonbonded interactions)
• van der waals interaction
 Act only at very short distances
 Attractive interaction by induced dipoles between uncharged
atoms ~r6
 When atoms get too close, valence shell start to overlap and
repel ~r12
Van der Waals potential energy function
Than becomes
repulsive
Interaction first
attractive
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Nonbonded interactions)
• electrostatic interaction
 Electrostatic interaction of charged atoms
 Long-range forces
 Coulomb’s Law
Coulomb’s Law
Negative interaction if of
the same charge
Positive interaction that
inversely increases
distance
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Nonbonded interactions)
• electrostatic interaction
 Problem  defining dielectric constant (e)
 dielectric constant differs in solvent and protein interior
 eprotein ~ 2-4
 esolvent ~80
 For protein calculations using NMR constraints, typically
turn electrostatics off
 How to properly define solvent, buffers, salts, etc?
 Can explicitly define solvent  increases complexity
of calculations.
 With electrostatics off during the structure
calculations, can use the potential energy calculation
after the fact to determine the quality of the NMR
structure
Introduction to Molecular Modeling Techniques
Potential Energy Equation (Nonbonded interactions)
•
electrostatic interaction

Problem  defining dielectric constant (e)
1) Don’t use electrostatic potential energy during structure
calculation
2) Use a single dielectric constant

eprotein ~ 2-4; esolvent ~80
3) Use explicit solvent in structure calculation

Improved structure quality

Increased computational time

Properly defining solvent

Properly defining force fields
behavior in solvent
PROTEINS: Structure, Function, and Genetics 50:496–506 (2003)
Introduction to Molecular Modeling Techniques
The Potential Energy Function Not Sufficient to Fold A Protein
• it is not even sufficient to keep a folded structure folded.
Dynamic simulations, even
with the “best” force fields,
ALWAYS results in the
structure drifting away from
the original NMR, X-ray, or
homology model
GDT-TS – measure of percent
similarity to original structure
Proteins 2012; 80:2071–2079
Introduction to Molecular Modeling Techniques
Why Is the Potential Energy Function Not Sufficient to Fold A Protein?
• It is Not A complete function
primarily short-range geometry with many equal solutions
 VDW and electrostatics only contribute over short distances
How do you bring distant regions of the primary chain into contact?
• Too many possible conformations
N
 3 where N is the number of amino acids
• Other factors that drive the protein folding process
 hydrophobic interactions, hydrogen-bond formation, secondary structure interactions (helix
dipole), effects of solvent, compactness of structure, etc
 How do you define a mathematical equation defining these contributions?

Improving the Potential
Energy Function,
improving the parameters
and defining alternative
ab inito methods of
folding a protein are
major areas of Molecular
Modeling research.
Introduction to Molecular Modeling Techniques
One Way to View an NMR Protein Structure Determination Is as a Hybrid Modeled
Structure
ETOTAL = Echem + wexpEexp
Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara
Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr
• NMR structure calculations modify the standard potential energy function to include NMR
experimental constraints
 distance constraints (NOEs)
 dihedral constraints (NOEs, coupling constants, chemical shifts)
1
13C)
 chemical shifts ( H,
 residual dipolar coupling constants (RDCs)
• Recently, additional potential functions have been added that are not NMR experimental
constraints but are developed by analyzing databases and structural trends.
• Controversial
 not true experimental data
 but similar to other parameterized geometric functions (bond length, angles etc)
 bias structures to structures in PDB
 but this is the criteria used to determine the quality of a protein structure
Introduction to Molecular Modeling Techniques
Ramachandran Database
 similar in concept to bond length, bond angle, etc. parameters but directed to f, y, c1
& c2.
 based on observed values in the PDB.
 Radius of Gyration
 based on observed trends in the PDB related to sequence length
 tries to improve the compactness of NMR structures
 general tendency of a structure in the absence of explicit solvent to move towards an
open/expanded chain conformation
 Empirical Backbone-Backbone Bonding Potential
 based on observed trends in the PDB related to hydrogen bonds in secondary
structures and long range isolated H-bonds.
 optimizes h-bond distance and angle parameters

Convergence of NMR structure
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 target distance with upper and lower bounds
ENOE 

class
E
 NOE
classes constraint s
assign ( resid 14 and name HD* ) ( resid 97 and name HD* ) 4.0 2.2 3.0
No contribution to the
overall potential energy if
the distance between the
atoms is between the upper
and lower bounds of the
NOE constraint
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 Sample XPLOR Script
Sets-up the NOE target function and
clears any existing constraints
Defines the number of constraints
and sets the maximal violation
energy for a single constraint
Assigns a class name to constraints
and reads in file containing distance
constraints. Can read in multiple
files with different class labels.
Allows flexibility to treat different
classes of NOEs differently.
.
.
.
noe
reset
nrestraints = 20000
ceiling 100
class
all
@noe.tbl
averaging all cent
potential all square
scale
all $knoe
sqconstant all 1.0
sqexponent all 2
end
assign ( resid 2 and name HA ) ( resid 2 and name HG2# ) 4.0 2.2 1.6
assign ( resid 2 and name HA ) ( resid 3 and name HN ) 2.5 0.7 1.0
assign ( resid 2 and name HA ) ( resid 3 and name HD1# ) 4.0 2.2 2.0
assign ( resid 2 and name HA ) ( resid 3 and name HD2# ) 4.0 2.2 2.0
assign ( resid 2 and name HA ) ( resid 100 and name HB# ) 4.0 2.2 2.0
assign ( resid 2 and name HG2# ) ( resid 3 and name HN ) 4.0 2.2 2.0
assign ( resid 2 and name HG2# ) ( resid 3 and name HD* ) 4.0 2.2 4.4
assign ( resid 2 and name HG2# ) ( resid 100 and name HB# ) 4.0 2.2 3.0
assign ( resid 2 and name HG2# ) ( resid 103 and name HN ) 4.0 2.2 2.0
.
.
.
Defines how distances and
energies are calculated
.
.
.
Which NOE class
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 Sample XPLOR Script
 Averaging – determines how the distance between the
selected sets of atoms (pseudoatoms) is calculated.
Scaling factor for ambiguous
peaks in symmetric multimers
R-6 dependence of NOE
AVERAGE = R-6
R  ( Rij6 )1/ 6
AVERAGE = R-3
R  ( Rij3 )1/ 3
AVERAGE = SUM
R  ( Rij6 / nmo no) 1/ 6
ij
Average of all possible distance combinations
Eq.
if two distances 3 and 10 Å
((3-6 + 10-6)/2)-1/6 = 3.37 Å
Sum of all possible distance combinations
Eq.
if two distances 3 and 10 Å
(3-6 + 10-6)-1/6 = 2.99 Å
SUM is preferred over R-6 for ambiguous NOESY crosspeaks
AVERAGE = CENTER
1
2
R  (rcenter
 rcenter
)
Difference between geometric centers of atoms
(distance constraints have to be corrected for center averaging)
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 Sample XPLOR Script
 Averaging
Which is Best (R-6, SUM,CENTER)?
Point of Discussion in NMR Community
Lower limit
Upper limit
1.2
5.5
Averaging: none
Val
H
HG11
Averaging: Sum
(smaller than not averaging because of summing effect)
Val
H
HG11 - HG13
1.0
4.6
Averaging: R-6
(same as not averaging because limits are the same for different pairs).
Val
H
HG11 - HG13
1.2
5.5
Averaging: Center
(distance from H to the center of atoms HG11 through HG13)
Val
H
HG11 - HG13
2.1
4.9
Different upper/lower
limits from Val H’s to Hg1
depending on various
distance averaging
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 Sample XPLOR Script
 Potential – determines how the energies are calculated for violations of the distance
constraints
 Square-Well Function is commonly used
 scale – determines magnitude of function (force constant) typically 20-50 kcal/mole
 ceiling – maximum violation energy per constraint
Biharmonic Function
Square-Well Function
Soft-Square Function
Switch
point
Energy violation for any
distance outside target distance
Flat region around target
distance where violation
energy is zero. Equal energy
for upper/lower violations
Same flat region around target
distance but violation energy
for upper violations are
reduced at a “switch” point.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• distance constraint (NOE)
 An even “softer” approach suggests refinement directly against intensities based on a lognormal distribution
 IMPORTANT - measurements are not weighted equally but are penalized depending on the
degree of disagreement with the structure
 No difference for refinements against intensities or distance
J. Am. Chem. Soc. (2005) 127, 16026
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Dihedral Angle Restraints
 target dihedral angle with upper and lower bounds
 similar to NOE square-well function
 difference in dihedral angle (f) has to account for circular
nature (0-360o) of angles
assign (resid 1 and name c ) (resid 2 and name n )
(resid 2 and name ca ) (resid 2 and name c )
1.0
-93.57 30.0 2
Force constant (C) – determines the magnitude of the violation energies
 Scale (S) – overall weight factor (allows for changes in contribution to overall
violation energy during structure calculation
 Exponent (ed) – increases violation energies for larger differences in dihedral angles,
usually 2 for harmonics.

Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Dihedral constraints
• Sample Xplor script
Sets-up the dihedral target function
and clears any existing constraints
Defines the number of constraints
and sets the maximal violation
energy for a single constraint
Reads in file containing dihedral
constraints. Turns off message so
Xplor output file doesn’t contain the
reading of all the constraints
.
.
.
!! g2
assign (resid 1 and name c ) (resid 2 and name n )
(resid 2 and name ca) (resid 2 and name c ) 1.0 70.0 20.0 2
!! k3
assign (resid 2 and name c ) (resid 3 and name n )
(resid 3 and name ca) (resid 3 and name c ) 1.0 60.0 30.0 2
!! f4
assign (resid 3 and name c ) (resid 4 and name n )
(resid 4 and name ca) (resid 4 and name c ) 1.0 -55.0 20.0 2
!! s5
assign (resid 4 and name c ) (resid 5 and name n )
(resid 5 and name ca) (resid 5 and name c ) 1.0 -65.0 30.0 2
!! q6
assign (resid 5 and name c ) (resid 6 and name n )
(resid 6 and name ca) (resid 6 and name c ) 1.0 -70.0 20.0 2
!! t7
assign (resid 6 and name c ) (resid 7 and name n )
(resid 7 and name ca) (resid 7 and name c ) 1.0 -105.0 30.0 2
.
.
.
restraints dihed
reset
scale $kcdi
nass = 10000
set message off echo off end
@dihed.tbl
set message on echo on end
end
.
.
.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Other NMR empirical constraints follow the same pattern as the NOE and dihedral angles
 a difference between the target and observed value determines a violation
 a force constant to scale the energy associated with the violation
Coupling Constants:
EJ  kJ
Nrestrain t s
 (J
obs
 Jcalc) 2
m 1
Residual Dipolar Coupling Constants:
ERDC  kRDC
Nrestrain t s
2
(
D
obs

D
calc
)

m 1
Radius of Gyration:
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Coupling constant constraints
• Sample Xplor script
Sets-up the coupling target function
defines maximum number of constraints
and energy function shape (like NOE)
.
.
.
! T7
assign (resid 6 and name c ) (resid 7 and name n )
(resid 7 and name ca) (resid 7 and name c )
9.7 0.5
! C8
assign (resid 7 and name c ) (resid 8 and name n )
(resid 8 and name ca) (resid 8 and name c )
9.6 0.5
! Y9
assign (resid 8 and name c ) (resid 9 and name n )
(resid 9 and name ca) (resid 9 and name c )
8.0 0.5
! N10
assign (resid 9 and name c ) (resid 10 and name n )
(resid 10 and name ca) (resid 10 and name c )
7.5 0.5
! S11
assign (resid 10 and name c ) (resid 11 and name n )
(resid 11 and name ca) (resid 11 and name c )
5.4 0.5
! A12
assign (resid 11 and name c ) (resid 12 and name n )
(resid 12 and name ca) (resid 12 and name c )
8.0 0.5
couplings
nres 400
potential harmonic
class phi
.
degen 1
.
force 1.0
set echo off message off end
coefficients 6.98 -1.38 1.72 -60.0
@coup.tbl
class gly !for gly, don't know stereoassignement
degen 2
force 1.0 0.2
coefficients 6.98 -1.38 1.72 -60.0
@gly_coup.tbl
set echo on message on end
end
.
Defines the force constant ,
degeneracy and coefficients and reads
in the experimental constraint files
Same for Gly residues, two HA
protons, don’t know which is HA1
or HA2. Degeneracy accounts for
this
.
.
.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Radius of Gyration
• Sample Xplor script
.
.
.
Turns on the
radius of target
function
Force constant
collapse
assign (resid 1:111) 100.0 12.67
scale 1.0
end
.
.
.
Function will be applied
to this residue range
Radius of gyration
([2.2*(number residues)0.38]-0.5)
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Empirical Backbone-Backbone Hydrogen Bond Constraints
 based on an empirical formula derived from high quality X-ray structures in the PDB
 violation energy is based on deviations from expected h-bond length (R) and angle ( f)
Violation occurs if this term is not zero
(relationship between R and f)
EHB  kHB
Nrestraint s
3
3 2
(
1
/
R

A

[
B
/{
2
.
07

cos
f
NHO} ])

m 1
where A  0.019 and B  0.21Å 3
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Empirical Backbone-Backbone Hydrogen Bond Constraints
 based on an empirical formula derived from high quality X-ray structures in the PDB
 violation energy is based on deviations from expected h-bond length (R) and angle ( f)
.
Excerpt of an XPLOR script
illustrating how to implement
empirical h-bond constraints
!hb database must be read in after psf file
hbdb
kdir = 0.20
!force constant for directional term
klin = 0.08
!force constant for linear term (ca. Nico's hbda)
nseg = 1
! number of segments that hbdb term is active on
nmin = 11
!range of residues (number of 1st residue in protein sequence)
nmax = 110
!range of residues (number of last residue in protein sequence )
ohcut = 2.60
!cut-off for detection of h-bonds
coh1cut = 100.0
!cut-off for c-o-h angle in 3-10 helix
coh2cut = 100.0
!cut-off for c-o-h angle for everything else
ohncut = 100.0
!cut-off for o-h-n angle
updfrq = 10
!update frequency usually 1000
prnfrq = 10
!print frequency usually 1000
freemode = 1
!mode= 1 free search
fixedmode = 0
!if you want a fixed list, set fixedmode=1, and freemode=0
mfdir = 0
! flag that drives HB's to the minimum of the directional potential
mflin = 0
! flag that drives HB's to the minimum of the linearity potential
kmfd = 10.0
! corresp force const
kmfl = 10.0
! corresp force const
renf = 2.30
! forces all found HB's below 2.3 A
kenf = 30.0
! corresponding force const
@HBDB:hbdb.inp
end
.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Carbon chemical shift constraint
 differences in expected and observed like NOE and dihedral
 not a continuous function
o
“look-up” table with bins (10 ) correlating f,y angles with C and Cb chemical shifts
ECshift(f ,y )  kCshift
Nrestraint s
[(C  (f ,y ))
2
 C b (f ,y )) 2 ]
m 1
Bins of expected chemical shift differences
(relative to random coil chemical shifts) for Cb
chemical shifts plotted as a function of (f,y)
Expected Cb secondary shifts
n
n
where C n (f ,y )  Cexp
(
f
,
y
)

C
(n   or b )
ected
observed (f ,y )
3 ppm
2.5
2
1.5
1
0.5
f-11
f
2
f+13
0
S1
y-1
S2
y
S3
y+1
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Carbon chemical shift constraints
–
proton chemical shifts setup similarly
• Sample Xplor script
Sets-up the dihedral target function
and number of steps in expectation
array
Defines the number of constraints,
sets the class name (like NOE), the
force constant and shape (like NOE)
Reads in standard tables for random
coil carbon chemical shifts and
expected secondary structure
chemical shifts
Reads in experimental carbon
chemical shifts constraint file
.
.
.
!! Q6
assign (resid 5 and name c ) (resid 6 and name n )
(resid 6 and name ca) (resid 6 and name c )
(resid 7 and name n )
57.81 29.28
!! T7
assign (resid 6 and name c ) (resid 7 and name n )
(resid 7 and name ca) (resid 7 and name c )
(resid 8 and name n )
60.70 69.29
!! C8
assign (resid 7 and name c ) (resid 8 and name n )
(resid 8 and name ca) (resid 8 and name c )
(resid 9 and name n )
56.57 49.96
!! Y9
assign (resid 8 and name c ) (resid 9 and name n )
(resid 9 and name ca) (resid 9 and name c )
(resid 10 and name n )
56.53 40.37
!! N10
assign (resid 9 and name c ) (resid 10 and name n )
(resid 10 and name ca) (resid 10 and name c )
(resid 11 and name n )
53.56 36.29
carbon
phistep=180
psistep=180
nres=300
class all
.
.
.
force 0.5
potential harmonic
@rcoil.tbl
!rcoil shifts
@expected_edited.tbl !13C shift database
set echo off message off end
@carbon.tbl
!carbon shifts
set echo on message on end
end
.
.
.
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Proton chemical shift constraints
similar to carbon chemical shifts
– Different class (file) for each type of proton
–

–
Degeneracy is dependent on steroassignment

–
Lists the observed chemical shifts
1 or 2 chemical shifts
Error is conservative uncertainty in chemical shifts

Square-well potential
• Sample Xplor script
OBSE (resid 10 and (name HA)) 4.414
OBSE (resid 11 and (name HA)) 5.362
OBSE (resid 12 and (name HA)) 4.480
OBSE (resid 13 and (name HA)) 5.022
OBSE (resid 14 and (name HA)) 4.472
OBSE (resid 16 and (name HA)) 4.460
OBSE (resid 17 and (name HA)) 4.480
OBSE (resid 18 and (name HA)) 5.083
OBSE (resid 19 and (name HA)) 5.261
OBSE (resid 20 and (name HA)) 4.320
OBSE (resid 21 and (name HA)) 4.803
.
.
.
Kuszewski et al. (1995) Protein Sci. 107:293
proton
class alpha
degeneracy 1
force $kprot
error 0.3
@alpha.tbl
class gly
degeneracy 2
force $kprot $kprotd
error 0.3
@alpha_degen.tbl
class md
degeneracy 2
force $kprot $kprotd
error 0.3
@methyl_degen.tbl
class ms
degeneracy 1
force $kprot
error 0.3
@methyl_single.tbl
class os
degeneracy 1
force $kprot
error 0.3
@other_single.tbl
class od
degeneracy 2
force $kprot $kprotd
error 0.3
@other_degen.tbl
end
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Ramachandran database constraint
 similar to carbon chemical shift constraints
 not a continuous function
o
“look-up” table with bins (8 ) correlating f,y, c1 &c2 angles with the probability of
occurrence based on PROCHECK analysis of PDB
EDB(i )  kDB
Nrestrain t s
 (log P )
i
m 1
Violation energy is related to the probability of the
observed f,y or c1,c2 pair occurring and comparison
to neighboring bins.
Probability or number of observed occurrences
for given pairs of dihedral angles
Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)
• Ramachandran database
• Sample Xplor script
Turns on the
Ramachandran
database
function
Automatically
sets-up all the
expected
torsion angles
for the protein
sequence
.
.
.
set message off echo off end
rama
nres=10000
!intraresidue protein torsion angles
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/shortrange_gaussians.tbl
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/new_shortrange_force.tbl
!interresidue protein torsion angle correlations i with i+/-1
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/longrange_gaussians.tbl
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/longrange_4D_hstgp_force.tbl
end
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/newshortrange_setup.tbl
@/home/PROGRAMS/xplor-nih-2.9.1/databases/torsions_gaussians/setup_4D_hstgp.tbl
.
.
.
Introduction to Molecular Modeling Techniques
For a Given Set of Atomic Coordinates, An Energy for the Structure Can Be
Calculated Based on the Set of Potential Energy Functions
ETOTAL = Echem + wexpEexp
Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara
Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr
What Relationship Does This Energy Value Have to Any Experimental Observation?
NOTHING!
The energy value simply indicates how well the structure conforms to the expected parameters.
It does not indicate the relative stability of one protein to another.
It does not indicate the stability of the protein (G).
Calculating a E between the protein with/without a ligand does not indicate the binding
affinity of the ligand or the induced stability of the complex
Do Not Over Interpret the Meaning of this Energy Function!
Introduction to Molecular Modeling Techniques
Consider These Facts About Correctly Folded Proteins:
• Typical free energies of protein denaturation (Gd) are ~ 10 kcal/mol
implies the relative stability of the folded protein compared to the denatured (unfolded) protein
• In a native structure of a globular protein of 100 amino acids residues there might be:
 ~ 100 intramolecular hydrogen bonds
 ~ 10 salt links
 ~ 10 buried hydrophobic residues
• This apparently imparts a stability ca. -1000 kcal/mol to the folded state!
• The strength of interactions in the unfolded state must be very similar ( ca. -990 kcal/mol).

This suggests that an exceptionally high level of accuracy is needed to correctly analyze
energies of different conformers and correctly predict the most stable structure.
ETOTAL = Echem + wexpEexp
Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara
Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr
We are currently far from this laudable goal.
Introduction to Molecular Modeling Techniques
What Relationship Is There Between the Force Constants and Experimental
Observations?
For geometric parameters (bonds, angles), force constants
come from IR, raman spectroscopy and ab inito calculations.
For experimental parameters (NOE, dihedral), There is No Relationship!
Experimental force constants have been
determined by “trial & error” or empirically to
obtain a proper balance and weighted
contribution of each experimental parameter to
the calculated structure.
Introduction to Molecular Modeling Techniques
What Do We Mean By a Proper Balance?
As example:
It is more desirable to have experimental values like NOEs to have more influence on the
resulting structure than an empirical function like the Ramachandran database.
Thus, the force constant for distance constraints (kNOE) should be higher than the
corresponding force constant for the Ramachandran database potential (K DB).
Typical values of experimental/empirical
force constants in XPLOR scripts
.
.
.
evaluate ($knoe = 25.0) ! noes
evaluate ($kcdi = 10.0) ! torsion angles
evaluate ($kcoup = 1.0) ! coupling constant
evaluate ($kcshift = 0.5) ! carbon chemical shifts
evaluate ($krgyr = 1.0)
! radius of gyration
evaluate ($krama = 1.0) ! intraresidue protein
evaluate ($kramalr = 0.15) ! long range protein
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Introduction to Molecular Modeling Techniques
What Do We Mean By a Proper Balance?
These Force Constants Are Not Absolute and Are Routinely Changed
During a Structure Calculation
.
Excerpt of an XPLOR script illustrating
how force constants are modified during a
structure calculation
.
evaluate ($final_t = 100) { K }
evaluate ($tempstep = 50) { K }
evaluate ($ncycle = ($init_t-$final_t)/$tempstep)
evaluate ($nstep = int($cool_steps*2.5/$ncycle))
evaluate ($bath = $init_t)
evaluate ($k_vdw = $ini_con)
evaluate ($k_vdwfact = ($fin_con/$ini_con)^(1/$ncycle))
evaluate ($radius= $ini_rad)
evaluate ($radfact = ($fin_rad/$ini_rad)^(1/$ncycle))
evaluate ($k_ang = $ini_ang)
evaluate ($ang_fac = ($fin_ang/$ini_ang)^(1/$ncycle))
evaluate ($k_imp = $ini_imp)
evaluate ($imp_fac = ($fin_imp/$ini_imp)^(1/$ncycle))
evaluate ($noe_fac = ($fin_noe/$ini_noe)^(1/$ncycle))
evaluate ($knoe = $ini_noe)
evaluate ($kprot = $ini_prot)
evaluate ($prot_fac = ($fin_prot/$ini_prot)^(1/$ncycle))
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Introduction to Molecular Modeling Techniques
What Do We Mean By a Proper Balance?
Not Only can the Magnitude of the Force Constant be Modified During a
Structure Calculation, but Specific Target Functions can be Either Turned
Off or On During a Structure Calculation.
Turn off all target function
Excerpt of an XPLOR
script illustrating how
Target Functions are
turned off and on.
Turn on the selected
target function
.
.
flags
exclude *
include bonds angles impropers vdw
end
.
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flags
exclude *
include bond angle dihed impr vdw noe cdih carb rama coup coll
end
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Introduction to Molecular Modeling Techniques
What Do We Mean By a Proper Balance?
How can the force constant impact the structure calculation?
A Simple Illustration: incorrect distance constraint
C-H bond length of 1.1Å with 410 kcal/mol force constant
H-H distance constraint of 3.0 Å with 25 kcal/mol force
(ceiling of 100 kcal/mol)
10 Å
H
H
C
C
Distance constraint is violated (properly)
with no distortion in bond lengths
C-H bond length of 1.1Å with 10 kcal/mol force constant
H-H distance constraint of 3.0 Å with 500 kcal/mol force
H
3Å
H
C
Want to Keep All Known Geometric Values
Within Proper Ranges
C
Distance constraint is satisfied (improperly)
with large distortion in bond lengths
Introduction to Molecular Modeling Techniques
How Do We Use the Energy Function To Calculate a Protein Structure?
ETOTAL = Echem + wexpEexp
Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara
Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr
Molecular Minimization starting from some structure (R), find its potential energy using the
potential energy function given above. The coordinate vector R is then varied using an optimization
procedure so as to minimize the potential energy ETOTAL(R).
Molecular Dynamics the motion of a molecule is simulated as a function of time. Newton's
second law of motion is solved to find how the position for each atom of the system varies with
time. To find the forces on each atom, the derivative vector (or gradient) of the potential energy
function given above is calculated. Factors such as the temperature and pressure of the system
can be included in the treatment.
Introduction to Molecular Modeling Techniques
Anfinsen's Thermodynamic Hypothesis
The native conformation of a protein is the conformation with the lowest free energy (G)
 global minimum of the free energy surface.
 Rather difficult (and expensive) to calculate free energies
 by definition these involve averaging over a large number of conformations
 Primary sequence determines the protein fold.
In 1957, Anfinsen showed that
denatured ribonuclease A (124
amino acids, 4 disulphides)
produced in 8 M urea and reducing
agent (b -mercaptoethanol) could be
re-activated by dialysing out the
denaturant in oxidizing conditions
Introduction to Molecular Modeling Techniques
Levinthal Paradox
If the entire folding process was a random search, it would require too much time
 Initial stages of folding must be nearly random.
-13 seconds.
 Conformational changes occur on a time scale of 10
 Proteins are known to fold on a timescale of seconds to minutes.
 Consider a 100 residue protein:
 if each residue has only 3 possible conformations (far less than reality)
3100 conformation x 10-13 seconds = 1027 years
 Even if a significant number of these conformations are sterically disallowed, the folding
time would still be astronomical
 Energy barriers probably cause the protein to fold along a definite pathway
Introduction to Molecular Modeling Techniques
Molecular Minimization
• moves the Cartesian coordinates (X,Y,Z position) for each atom to obtain minimal energy
ETOTAL = Echem + wexpEexp
Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara
Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr
• result is dependent on the starting structure
• finds local not global minima
• typically, only small movements in atom position are made
 starting structure looks similar to ending structure
 large changes may occur for significantly distorted structures (stretch bonds)
Large bond change
could invert chirality
Introduction to Molecular Modeling Techniques
Molecular Minimization
• minimization will fail for severely distorted structures
 a poorly docked ligand onto a protein where bonds or atoms are overlapped
Highly unlikely that this structure
would minimize since the Cd of the
Leu side-chain penetrates the center
of the benzene ring
In order to properly refine this poor structure, the minimization protocol would need to pull
the Cd back through the ring which would require first going to a higher energy structure.
 This will not occur since the trend for minimization is to move towards a lower
energy.
 The “minimized” structure will probably result with a stretched and distorted Cd-Cg
bond as it moves the Cd away from the ring from the other direction
 the benzene ring and the remainder of the Leu side-chain will also be distorted in an
effort to accommodate the overlapped structure

Introduction to Molecular Modeling Techniques
Local versus Global Minimum problem
Structural landscape is filled with
peaks and valleys.
Minimization protocol always
moves “down hill”.
No means to “see” the overall
structural landscape
No means to pass through higher
intermediate structures to get to a
lower minima.
The initial structure determines the results of the minimization!
Introduction to Molecular Modeling Techniques
Local versus Global Minimum problem
Another perspective of the Structural Landscape is a 3D funnel view that
leads to the global minimum at the base of the funnel.
Introduction to Molecular Modeling Techniques
Molecular Minimization
• Process Overview
The molecular potential U depends on two types of variables:
Potential energy gradient g(Q), a vector with 3N components:
The necessary condition for a minimum is that the function gradient
is zero:
Where xi denote atomic Cartesian coordinates and N is the number
of atoms
The sufficient condition for a minimum is that the second derivative
matrix is positive definite, i.e. for any 3N-dimensional vector u:
A simpler operational definition of this property is that all
eigenvalues of F are positive at a minimum. The second derivative
matrix is denoted by F in molecular mechanics and H in mathematics,
and is defined as:
One measure of the distance from a stationary point is the rms gradient:
or
Introduction to Molecular Modeling Techniques
Molecular Minimization
• Process Overview
 minima occurs when the first derivative is zero and when the second derivative is positive.
• U(Q) is a complicated function varying quickly with atomic coordinates Q
 molecular energy minimization is often performed in a series of steps
 the coordinates at step n+1 are determined from coordinates at previous step n
where dn is called a step.
the initial step is a guess
 a systematic or random search is not practical (Levinthal Paradox)
• Steepest Descent Method
 search step (dn) is performed in the direction of fastest decrease of U,
opposite of the gradient g
Always makes right angle

where  is a factor determining the
length of the step.
not efficient, but good for initial distorted
structures
 may be very slow near a solution

Introduction to Molecular Modeling Techniques
Molecular Minimization
• Process Overview
• Conjugate Gradient (Powell)
 modify steepest descent to increase efficiency
 Initial steps are steepest descent
 current step vector is not similar to previous step vectors
 accumulates information about the energy function from one iteration to the next
One of two factors determines when a
minimization calculations is completed:
a) Number of defined steps (dn) have
been calculated.
b) a predefined value of the gradient (g)
has been reached. (gradient very
rarely actually reaches exactly zero)
Dependent on all previous steps
Introduction to Molecular Modeling Techniques
Molecular Dynamics (MD)
• moves the Cartesian coordinates (X,Y,Z position) for each atom by integrating their equations of
motion
 change in position with time gives velocity
 change in velocity with time (acceleration) gives force
 follow the laws of classical mechanics, most notably Newton's Second law:
The force on atom i can be computed directly from the derivative of the potential energy
function (U) with respect to the coordinates ri, Fi = -dU/dri.
 to initiate MD we need to assign initial velocities
 This is done using a random number generator using the constraint of the MaxwellBoltzmann distribution.

where:
Hamiltonian H(G) where G represents the set of positions and momenta
Target Temperature (T)
Boltzman constant (kB)
Introduction to Molecular Modeling Techniques
Molecular Dynamics (MD)
The temperature is defined by the average kinetic energy of the system according to the
kinetic theory of gases.
– internal energy of the system is U = 3/2 NkT
– kinetic energy is U = 1/2 Nmv2
where :
N is the number of atoms
v is the velocity
m is the mass
T is the temperature
k is the Boltzman constant
 By averaging over the velocities of all of the atoms in the system the temperature can
be estimated.
 Maxwell-Boltzmann velocity distribution will be maintained throughout the
simulation.
• If system has been energy minimized  potential energy is zero and temperature is zero
• Need to “heat” system up to desired temperature
scale velocities: v = (3kT/m)1/2
• Calculate a trajectory in a 6N-dimensional phase space (3N positions and 3N momenta)
 measure trajectories in small time steps, usually 1 femto-second (fs)
 typical duration of dynamics run is 10-100 pico-seconds (ps)

Introduction to Molecular Modeling Techniques
Molecular Dynamics (MD)
• Force Fi exerted on atom i by the other atoms in the system is given by the negative gradient of
the potential energy function (V) which in turn depends on the coordinates of all N atoms in the
system
For small time steps (δt), the following approximated hold:
• Typically a time step of 1 to 10 fs is used for molecular systems.
• Thus a 100 ps (10-10 seconds) molecular dynamics simulation involves 105 to 104 integration
steps.
• Even using the fastest computers only very rapid molecular processes can be simulated at an
atomic level.
Introduction to Molecular Modeling Techniques
Typical Time Regions For Molecular Motion
Time scale
Amplitude
Description
short femto, pico
10-15 - 10-12s
0.001 - 0.1 Å
- bond stretching, angle bending
- constraint dihedral motion
medium pico, nano
10-12 - 10-9s
0.1 - 10 Å
- unhindered surface side chain motion
- loop motion, collective motion
long nano, micro
10-9 - 10-6s
1 - 100 Å
- folding in small peptides
- helix coil transition
very long micro, second
10-6 - 10-1s
10 - 100 Å
- protein folding
Introduction to Molecular Modeling Techniques
XPLOR Script For Calculating An Extended Structure From PSF File
Read PSF and Parameter files
structure @PROTEIN.psf end
{*Read structure file.*}
parameter @/PROGRAMS/xplor-nih-2.9.1/toppar/parallhdg_new.pro end
Mathematical manipulation
of atom properties
vector ident (X) ( all )
vector do (x=x/10.) ( all )
vector do (y=random(0.5) ) ( all )
vector do (z=random(0.5) ) ( all )
{*Friction coefficient, in 1/ps.*}
{*Heavy masses, in amus.*}
Set parameters for non-bonded
potential energy function
Define which potential
energy functions to use
Execute set of Minimization
and Dynamics
vector do (fbeta=50) (all)
vector do (mass=100) (all)
parameter
nbonds
cutnb=5.5 rcon=20. nbxmod=-2 repel=0.9 wmin=0.1 tolerance=1.
rexp=2 irexp=2 inhibit=0.25
end
end
flags exclude * include bond angle vdw end
minimize powell nstep=50 nprint=10 end
flags include impr end
minimize powell nstep 50 nprint=10 end
dynamics verlet
nstep=50 timestep=0.001 iasvel=maxwell firsttemp= 300.
tcoupling = true tbath = 300. nprint=50 iprfrq=0
end
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Introduction to Molecular Modeling Techniques
XPLOR Script For Calculating An Extended Structure From PSF File
Change non-bonding parameters
Another round of Minimization
and Dynamics where hydrogens
are added and to the structure
and refined
Write out the structure and stop
.
.
parameter
nbonds
rcon=2. nbxmod=-3 repel=0.75
end
end
minimize powell nstep=100 nprint=25 end
dynamics verlet
nsteps=500 timestep=0.005 iasvel=maxwell
firsttemp = 300.
tcoupling = true tbath = 300. nprint=100 iprfrq=0
end
flags exclude vdw elec end
vector do (mass=1.) ( name h* )
hbuild selection=( name h* ) phistep=360 end
flags include vdw elec end
minimize powell nstep=200 nprint=50 end
{*Write coordinates.*}
remarks extended strand (PROTEIN)
write coordinates output=PROTEIN.ext end
stop