Molecular Modeling Techniques are a Critical Component of Determining a Protein Structure by NMR:

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

Introduction to Molecular Modeling Techniques

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.

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.


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.


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

lo – expected/natural bond lengthkl – force constantl – 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

Sum over all bonds in the structure

Potential Energy Equation (Angles)• As the bond angle is bent from the norm, the energy goes up.• Sum over three atoms

o – expected/natural bond anglek – force constant– 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

Sum over all bond angles in the structure

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

o – expected improper dihedral (usually set to 0o)k – force constant– actual/observed improper dihedral

Plot of Potential Energy Function for Improper Dihedrals (o = 0)

Sum over all improper dihedrals in the structure

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

– expected improper dihedral An – force constant for each Fourier term – actual/observed improper dihedraln – multiplicity (same parameter seen in the XPLOR constraint file)

Sum over all dihedrals in the structure

Fourier Series

Plot of Potential Energy Function for Dihedrals

Multiple minima

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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 1, 60, -60 & 180 are all equivalent and should yield 0 torsion energy


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

Interaction first attractive

Than becomes repulsive


Potential Energy Equation (Nonbonded interactions)• electrostatic interaction

Electrostatic interaction of charged atoms Long-range forces Coulomb’s Law

Coulomb’s Law

Positive interaction that inversely increases distance

Negative interaction if of the same charge

Introduction to Molecular Modeling Techniques Potential Energy Equation (Nonbonded interactions)

• electrostatic interaction Problem defining dielectric constant () dielectric constant differs in solvent and protein interior

protein ~ 2-4solvent ~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


PROTEINS: Structure, Function, and Genetics 50:496–506 (2003)

Potential Energy Equation (Nonbonded interactions)• electrostatic interaction

Problem defining dielectric constant ()1) Don’t use electrostatic potential energy during structure

calculation2) Use a single dielectric constant

protein ~ 2-4; solvent ~803) Use explicit solvent in structure calculation

Improved structure quality Increased computational time Properly defining solvent Properly defining force fields behavior in solvent

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

3N 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.


One Way to View an NMR Protein Structure Determination Is as a Hybrid Modeled Structure

• NMR structure calculations modify the standard potential energy function to include NMR experimental constraints

distance constraints (NOEs) dihedral constraints (NOEs, coupling constants, chemical shifts) chemical shifts (1H, 13C) 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

ETOTAL = Echem + wexpEexp

Eexp = ENOE + Etorsion + EH-bond + Egyr + Erama + ERDC + ECSA + Epara

Echem = Ebond + Eangle + Edihedral + Evdw + Eelectr


Ramachandran Database similar in concept to bond length, bond angle, etc. parameters but directed to , , 1 & 2.

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

Potential Energy Equation (NMR Constraints)• distance constraint (NOE)

target distance with upper and lower bounds


classes sconstra

classNOENOE EE

int

No contribution to the overall potential energy if the distance between the atoms is between the upper and lower bounds of the NOE constraint

assign ( resid 14 and name HD* ) ( resid 97 and name HD* ) 4.0 2.2 3.0


Potential Energy Equation (NMR Constraints)• distance constraint (NOE)

Sample XPLOR Script

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 2end...

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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.

Which NOE class

Defines how distances and energies are calculated

assign ( resid 2 and name HA ) ( resid 2 and name HG2# ) 4.0 2.2 1.6assign ( resid 2 and name HA ) ( resid 3 and name HN ) 2.5 0.7 1.0assign ( resid 2 and name HA ) ( resid 3 and name HD1# ) 4.0 2.2 2.0assign ( resid 2 and name HA ) ( resid 3 and name HD2# ) 4.0 2.2 2.0assign ( resid 2 and name HA ) ( resid 100 and name HB# ) 4.0 2.2 2.0assign ( resid 2 and name HG2# ) ( resid 3 and name HN ) 4.0 2.2 2.0assign ( resid 2 and name HG2# ) ( resid 3 and name HD* ) 4.0 2.2 4.4assign ( resid 2 and name HG2# ) ( resid 100 and name HB# ) 4.0 2.2 3.0assign ( resid 2 and name HG2# ) ( resid 103 and name HN ) 4.0 2.2 2.0...

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.

AVERAGE = R-66/16 )( ijRR

AVERAGE = R-33/13 )( ijRR

AVERAGE = SUM6/16 )/( mononRR

ijij

AVERAGE = CENTER

)( 21centercenter rrR

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

Scaling factor for ambiguous peaks in symmetric multimers

Difference between geometric centers of atoms(distance constraints have to be corrected for center averaging)

SUM is preferred over R-6 for ambiguous NOESY crosspeaks

R-6 dependence of NOE

Lower limit Upper limit

Averaging: none

Val H HG11 1.2 5.5

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



Averaging

Which is Best (R-6, SUM,CENTER)? Point of Discussion in NMR Community

Different upper/lower limits from Val H’s to H1 depending on various distance averaging



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

Square-Well Function Soft-Square FunctionBiharmonic Function

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.

Switch point


• distance constraint (NOE) An even “softer” approach suggests refinement directly against intensities based on a log-normal 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

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 () has to account for circular nature (0-360o) of angles

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.


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


Potential Energy Equation (NMR Constraints)• Dihedral constraints• Sample Xplor script

restraints dihed reset scale $kcdi nass = 10000 set message off echo off end @dihed.tbl set message on echo on endend ...

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!! 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...

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


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

srestraN

m

calcobsJJ JJkEint

1

2)(

srestraN

m

calcobsRDCRDC DDkEint

1

2)(

Coupling Constants:

Residual Dipolar Coupling Constants:

Radius of Gyration:


Potential Energy Equation (NMR Constraints)• Coupling constant constraints• Sample Xplor script

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 endend

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! T7assign (resid 6 and name c ) (resid 7 and name n ) (resid 7 and name ca) (resid 7 and name c ) 9.7 0.5! C8assign (resid 7 and name c ) (resid 8 and name n ) (resid 8 and name ca) (resid 8 and name c ) 9.6 0.5! Y9assign (resid 8 and name c ) (resid 9 and name n ) (resid 9 and name ca) (resid 9 and name c ) 8.0 0.5! N10assign (resid 9 and name c ) (resid 10 and name n ) (resid 10 and name ca) (resid 10 and name c ) 7.5 0.5! S11assign (resid 10 and name c ) (resid 11 and name n ) (resid 11 and name ca) (resid 11 and name c ) 5.4 0.5! A12assign (resid 11 and name c ) (resid 12 and name n ) (resid 12 and name ca) (resid 12 and name c ) 8.0 0.5...

Sets-up the coupling target function defines maximum number of constraints and energy function shape (like NOE)

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


Potential Energy Equation (NMR Constraints)• Radius of Gyration• Sample Xplor script

collapse assign (resid 1:111) 100.0 12.67 scale 1.0end

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Turns on the radius of target function

Function will be applied to this residue range

Radius of gyration ([2.2*(number residues)0.38]-0.5) Force constant

3

1

233

Å21.0019.0

])}cos07.2/{[/1(int

BandAwhere

BARkEsrestra

NHO

N

m

HBHB


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 ()

Violation occurs if this term is not zero(relationship between R and )


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 ()

.!hb database must be read in after psf filehbdb 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.inpend.

Excerpt of an XPLOR script illustrating how to implement empirical h-bond constraints

Exp

ecte

d C

seco

ndar

y sh

ifts

-1

+1

-1 +1

)(),(),(),(

])),()),([(),(

exp

1

22int

ornCCCwhere

CCkE

nobserved

nected

n

N

m

CshiftCshift

srestra


• Carbon chemical shift constraint differences in expected and observed like NOE and dihedral not a continuous function

“look-up” table with bins (10o) correlating , angles with C and C chemical shifts

Bins of expected chemical shift differences (relative to random coil chemical shifts) for C chemical shifts plotted as a function of (,)


Potential Energy Equation (NMR Constraints)• Carbon chemical shift constraints

– proton chemical shifts setup similarly• Sample Xplor script

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 endend...

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!! 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

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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

Introduction to Molecular Modeling Techniques 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.tblend

Potential Energy Equation (NMR Constraints)• Proton chemical shift constraints

– similar to carbon chemical shifts– Different class (file) for each type of proton

Lists the observed chemical shifts– Degeneracy is dependent on steroassignment

1 or 2 chemical shifts– Error is conservative uncertainty in chemical shifts

Square-well potential

• Sample Xplor script

OBSE (resid 10 and (name HA)) 4.414OBSE (resid 11 and (name HA)) 5.362OBSE (resid 12 and (name HA)) 4.480OBSE (resid 13 and (name HA)) 5.022OBSE (resid 14 and (name HA)) 4.472OBSE (resid 16 and (name HA)) 4.460 OBSE (resid 17 and (name HA)) 4.480OBSE (resid 18 and (name HA)) 5.083OBSE (resid 19 and (name HA)) 5.261OBSE (resid 20 and (name HA)) 4.320 OBSE (resid 21 and (name HA)) 4.803 ...

Kuszewski et al. (1995) Protein Sci. 107:293

srestraN

m

iDBDB PkiEint

1

)(log)(


• Ramachandran database constraint similar to carbon chemical shift constraints not a continuous function

“look-up” table with bins (8o) correlating , angles with the probability of occurrence based on PROCHECK analysis of PDB

Violation energy is related to the probability of the observed , or 1,2 pair occurring and comparison to neighboring bins.

Probability or number of observed occurrences for given pairs of dihedral angles


Potential Energy Equation (NMR Constraints)• Ramachandran database• Sample Xplor script

set message off echo off endramanres=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.tblend@/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

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Turns on the Ramachandran database function

Automatically sets-up all the expected torsion angles for the protein sequence

For a Given Set of Atomic Coordinates, An Energy for the Structure Can Be Calculated Based on the Set of Potential Energy Functions





What Relationship Does This Energy Value Have to Any Experimental Observation?

NOTHING!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!Do Not Over Interpret the Meaning of this Energy Function!

• 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).


Consider These Facts About Correctly Folded Proteins:

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.




We are currently far from this laudable goal.


What Relationship Is There Between the Force Constants and Experimental Observations?

For geometric parameters (bonds, angles), force constants For geometric parameters (bonds, angles), force constants come from IR, raman spectroscopy and ab inito calculations.come from IR, raman spectroscopy and ab inito calculations.

For experimental parameters (NOE, dihedral), There is No Relationship!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.


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 (KDB).

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.evaluate ($knoe = 25.0) ! noesevaluate ($kcdi = 10.0) ! torsion anglesevaluate ($kcoup = 1.0) ! coupling constantevaluate ($kcshift = 0.5) ! carbon chemical shiftsevaluate ($krgyr = 1.0) ! radius of gyrationevaluate ($krama = 1.0) ! intraresidue proteinevaluate ($kramalr = 0.15) ! long range protein...

Typical values of experimental/empirical force constants in XPLOR scripts


These Force Constants Are Not Absolute and Are Routinely Changed During a Structure Calculation


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.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))..

Excerpt of an XPLOR script illustrating how force constants are modified during a structure calculation



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.

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.flags exclude * include bonds angles impropers vdw end..flags exclude * include bond angle dihed impr vdw noe cdih carb rama coup collend..

Excerpt of an XPLOR script illustrating how Target Functions are turned off and on.

Turn off all target function

Turn on the selected target function



How can the force constant impact the structure calculation?

A Simple Illustration: incorrect distance constraint

H H

C C

10 Å

H H

C C

3 Å

C-H bond length of 1.1Å with 410 kcal/mol force constantH-H distance constraint of 3.0 Å with 25 kcal/mol force (ceiling of 100 kcal/mol)

C-H bond length of 1.1Å with 10 kcal/mol force constantH-H distance constraint of 3.0 Å with 500 kcal/mol force

Distance constraint is violated (properly) with no distortion in bond lengths

Distance constraint is satisfied (improperly) with large distortion in bond lengths

Want to Keep All Known Geometric Values Within Proper Ranges

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.


How Do We Use the Energy Function To Calculate a Protein Structure?





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.

Anfinsen's Thermodynamic Hypothesis

In 1957, Anfinsen showed that denatured ribonuclease A (124 amino acids, 4 disulphides) produced in 8 M urea and reducing agent ( -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. Conformational changes occur on a time scale of 10-13 seconds. 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


Molecular Minimization• moves the Cartesian coordinates (X,Y,Z position) for each atom to obtain minimal energy

• 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


Molecular Minimization• minimization will fail for severely distorted structures

a poorly docked ligand onto a protein where bonds or atoms are overlapped

In order to properly refine this poor structure, the minimization protocol would need to pull the C 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 C-C bond as it moves the C 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

Highly unlikely that this structure would minimize since the C of the Leu side-chain penetrates the center of the benzene ring


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!


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

or

One measure of the distance from a stationary point is the rms gradient:

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:


• 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 n 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 (n) is performed in the direction of fastest decrease of U, opposite of the gradient g

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

Always makes right angle

a) Number of defined steps (n) have been calculated.

b) a predefined value of the gradient (g) has been reached. (gradient very rarely actually reaches exactly zero)


• 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:

Dependent on all previous steps


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 = -U/ri. to initiate MD we need to assign initial velocities

This is done using a random number generator using the constraint of the Maxwell- Boltzmann distribution.

where:

Hamiltonian H() where 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 atomsv is the velocitym is the massT is the temperaturek 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)


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

• 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.

For small time steps (δt), the following approximated hold:


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


XPLOR Script For Calculating An Extended Structure From PSF File

structure @PROTEIN.psf end {*Read structure file.*}parameter @/PROGRAMS/xplor-nih-2.9.1/toppar/parallhdg_new.pro end

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.*}vector do (fbeta=50) (all) {*Heavy masses, in amus.*}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 endendflags exclude * include bond angle vdw endminimize powell nstep=50 nprint=10 endflags include impr endminimize powell nstep 50 nprint=10 enddynamics verlet nstep=50 timestep=0.001 iasvel=maxwell firsttemp= 300. tcoupling = true tbath = 300. nprint=50 iprfrq=0end..

Read PSF and Parameter files

Mathematical manipulation of atom properties

Set parameters for non-bonded potential energy function

Define which potential energy functions to use

Execute set of Minimization and Dynamics

.

.parameter nbonds rcon=2. nbxmod=-3 repel=0.75 endendminimize powell nstep=100 nprint=25 enddynamics verlet nsteps=500 timestep=0.005 iasvel=maxwell firsttemp = 300. tcoupling = true tbath = 300. nprint=100 iprfrq=0endflags exclude vdw elec endvector do (mass=1.) ( name h* )hbuild selection=( name h* ) phistep=360 endflags include vdw elec endminimize powell nstep=200 nprint=50 end {*Write coordinates.*}remarks extended strand (PROTEIN)write coordinates output=PROTEIN.ext endstop


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

Documents

Molecular Modeling Techniques are a Critical Component of Determining a Protein Structure by NMR: