THEORETICAL METHODS TO STUDY PROTEIN FOLDING : EMPIRICAL FORCE FIELDS.

THEORETICAL METHODS TO STUDY PROTEIN FOLDING: EMPIRICAL FORCE FIELDS

Individual components

System level(Networks)

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PDEs to describe reaction/diffusion

Network graphs

Fully-detailed

Atomistically-detailed

Coarse-grained

QM

QM/MM

All-atom

United-atom

Residue level

Molecule/domain level

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Anfinsen’s thermodynamic hypothesis.“The studies on the renaturation of fully denaturated ribonuclease required many supporting investigations to establish, finally, the generality which we have occasionally called the ‘thermodynamic hypothesis’. This hypothesis states that the three-dimensional structure of a native protein in its normal physiological milieu (solvent, pH, ionic strength, presence of other components such as metal ions or prosthetic group, temperature and other) is the one in which the Gibbs free energy of the whole system is lowest; that is, the native conformation is determined by the totality of interatomic interactions and hence by the amino acid sequence in a given environment.”

C.B. Anfinsen, Science, 181, 223-230, 1973.

To facilitate the implementation of this hypothesis in protein-structure prediction, “free energy” was replaced with “potential energy”.

Nature (and a canonical simulation) finds the basin with the lowest free energy, at a given temperature which might happen to but does not have to contain the conformation with the lowest potential energy.

The global-optimization methods are desinged to find structures with the lowest potential energy, thus ignoring conformational entropy. Technically this corresponds to canonical simulations at 0 K.

“Potential energy” or “free energy”?

The stability of the structures of biological macromolecules results from special structure of their energy landscapes, which can be termed “minimal frustration” or “funnel-like structure”. A good example is the pit dug by antlion larva.

Theoretical studies of protein structure and protein folding

• Need to express energy of a system as function of coordinates

• Need an algorithm to explore the conformational space

From Schrödinger equation to analytical all-atom potentials

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What is a force field? A set of formulas (usually explicit) and parameters to express the conformational energy of a given class of molecules as a function of coordinates (Cartesian, internal, etc.) that define the geometry of a molecule or a molecular system.

Features:

• Cheap• Fast• Easy to program

• Restricted to conformational analysis

• Non-transferable• Results sometimes

unreliable

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All-atom empirical force fields: a very simplified representation of the potential energy surfaces

Class I force fields

Multiplication of atom types in empirical force fields

NamePotential

type References

AMBER/OPLSall-atom,

united-atom

Weiner et al., 1984; 1986; Cornell et al., 1995; Jorgensen et al., 1996

http://ambermd.org/

CHARMm all-atomBrooks et al., 1983; MacKerrel et al.,

1998; 2001http://www.charmm.org/

GROMOS all-atomvan Gunsteren & Berendsen, 1987;

Scott et al., 1999http://www.gromos.net/

ECEPP/3all-atom; rigid

valence geometry

Nemethy et al., 1995; Ripoll et al., 1995

http://cbsu.tc.cornell.edu/software/eceppak/

http://www.icm.edu.pl/kdm/ECEPPAK

DISCOVER (CVFF)

all-atomDauber-Osguthorpe, 1988; Maple et

al., 1998

Force fields commonly used for protein simulations

20

2

1ddkdE d

s

d

d0 d

Es(

d)

Bond distortion energy

Typical values of d0 and kd

Bond d0 [A] kd [kcal/(mol A2)]

Csp3-Csp3 1.523 317

Csp3-Csp2 1.497 317

Csp2=Csp2 1.337 690

Csp2=O 1.208 777

Csp2-Nsp3 1.438 367

C-N (amide) 1.345 719

Comparison of the actual bond-energy curve with that of the harmonic approximation

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eddbes

ds

eDdE

ddddkdE Anharmonic potential

Morse potential (CVFF force field)

Potentials that take into account the asymmetry of bond-energy curve

d [A]

E [

kcal

/mol

]

Harmonic potential

Anharmonic potential

Morse potential

20

2

1 kEb

0

Eb()

k

Energy of bond-angle distortion

Typical values of 0 and k

Angle 0 [degrees] k

[kcal/(mol degree2)]

Csp3-Csp3-Csp3 109.47 0.0099

Csp3-Csp3-H 109.47 0.0079

H-Csp3-H 109.47 0.0070

Csp3-Csp2-Csp3 117.2 0.0099

Csp3-Csp2=Csp2 121.4 0.0121

Csp3-Csp2=O 122.5 0.0101

Single bond between sp3 carbons or between sp3 carbon and nitrogen

Example: C-C-C-C quadruplet

dihedral angle [deg]

Eto

r [k

cal/m

ol] 60

50

40

30

20

10

0

3cos16.1 torE

Double or partially double bonds

Example: C-C(carboxyl)-C(amide)-C quadruplet

2cos120 torE

Single bond between electronegative atoms (oxygens, sulfurs, etc.).

Example: C-S-S-C quadruplet

cos16.02cos15.3 torE

Basic types of torsional potentials

Potentials imposed on improper torsional angles

A

B

X

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Nonbonded Lennard-Jones (6-12) potential

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r0

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Lorenz-Berthelot combining rules

Sample values of i and r0i

Atom type r0

C(carbonyl) 1.85 0.12

C(sp3) 1.80 0.06

N(sp3) 1.85 0.12

O(carbonyl) 1.60 0.20

H(bonded with C) 1.00 0.02

S 2.00 0.20

1012

6exp

r

D

r

CrE

r

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nb

Other nonbonded potentials

Buckingham potential

10-12 potential used in some force fields (e.g., ECEPP) for proton…proton donor pairs

Coulombic (electrostatic) potential

Charge determination

• Mullikan population charges (ECEPP/3, other early force fields).

• Fitting to molecular electrostatic potentials + subsequent adjustment to reproduce potential-energy surfaces or experimental association energies, etc.

• Based on atomic electronegativities with corrections to topology and geometry (No and coworkers, J. Phys. Chem. B, 105, 3624–3634, 2001; Koca and coworkers, J. Chem. Inf. Model., 53, 2548–2558, 2013).

Charge determination: fitting to molecular electrostatic potential (MEP) maps

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Charge determination: fitting to molecular electrostatic potential (MEP) maps

Ab initio calculations Fitted by using CHELP-SV

Francl et al., J. Comput. Chem., 17, 367-383 (1996)

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Polarizable force fields

Energy contribution Source of parameters

Bond and bond angle distortion

Crystal and neutronographic data, IR spectroscopy

Torsional NMR and FTIR spectroscopy

Nonbonded interactionsPolarizabilities, crystal and neutronographic data

Electrostatic energy Molecular electrostatic potentials

AllEnergy surfaces of model systems calculated with molecular quantum mechanics

Sources of parameters

Class II force fields (MM3, MMFF, UFF, CFF)

Maple et al., J. Comput. Chem., 15, 162-182 (1994)

Maple et al., J. Comput. Chem., 15, 162-182 (1994)

Parameterization of class II force fields

n i ij ji

QMn

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p

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Solvent in simulations

Explicit water

• TIP3P

• TIP4P

• TIP5P

• SPC

Implicit water

• Solvent accessible surface area (SASA) models

• Molecular surface area models

• Poisson-Boltzmann approach

• Generalized Born surface area (GBSA) model

• Polarizable continuum model (PCM)

O

H H0.417 e

-0.834 e

104.52o

0.9572 ÅO

H H0.520 e

0.00 e

-1.040 eM

0.15 Å

TIP3P model TIP4P model

O=3.1507 Å

O=0.1521 kcal/mol

O=3.1535 Å

O=0.1550 kcal/mol

Solvent accessible surface area (SASA) models

atoms

iisolw AF

i Free energy of solvation of

atomu i per unit area,

Ai solvent accessible surface of

atom i dostępna

Vila et al., Proteins: Structure, Function, and Genetics, 1991, 10, 199-218.

Comparison of the lowest-energy conformations of [Met5]enkefalin (H-Tyr-Gly-Gly-Phe-Met-OH) obtained with the ECEPP/3 force field in vacuo and with the SRFOPT model

vacuum SRFOPT

vacuum SRFOPT

Compariosn of the molecular sufraces of the lowest-energy conformation of [Met5]enkefaliny obtained without and with the SRFOPT model

Molecular surface are model

AFcav

Surface tension

A molecular surface area

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Generalized Born molecular surface (GBSA) model

Protein structure calculation/prediction and folding simulations

• Single energy minimization (wishful thinking at the early stage of force-field development).

• Global optimization of the PES (ignores conformational entropy).

• Molecular dynamics/Monte Carlo (take entropy into account but slow) and liable to non-convergence).

• Generalized ensemble sampling (MREMD).

Force field validation

Structure of gramicidiny S predicted by using the build-up procedure with energy minimzation with the ECEPP/3 force field (M. Dygert, N. Go, H.A. Scheraga, Macromolecules, 8, 750-761 (1975). The structure turned out to be effectively identical with the NMR structure determined later.

Superposition of the native fold (cyan) and the conformation (red) with the lowest C RMSD (2.85 Å) from the native fold

Energy-RMSD diagram

Global optimization of the energy surface of the N-terminal portion of the B-domain of staphylococcal protein A with all-atom ECEPP/3 force field + SRFOPT mean-field solvation model (Vila et al., PNAS, 2003, 100, 14812–14816)

First successful folding simulation of a globular protein by molecular dynamics

Duan and Kollman, Science, 282, 5389, 740-744 (1998)

Folding proteins at x-ray resolution using a specially designed ANTON machine (x-ray: blue, last frame of MD) simulation (red): villin headpiece (left), a 88 ns of simulations, WW domain (right), 58 s of simulations. Good symplectic algorithm; up to 20 fs time step.D.E. Shaw et al., Science, 2010, 330, 341-346