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 faster 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.
Transcript
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 faster 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.
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
lo – expected/natural bond lengthkl – force constantl – actual/observed bond length
Introduction to Molecular Modeling Techniques
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
Introduction to Molecular Modeling Techniques
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
…
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 1, 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
Interaction first attractive
Than becomes repulsive
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
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
Introduction to Molecular Modeling Techniques
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 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.
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Introduction to Molecular Modeling Techniques
Potential Energy Equation (NMR Constraints)• hydrogen bond constraint
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 )
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!
Introduction to Molecular Modeling Techniques
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.
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
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.
Introduction to Molecular Modeling Techniques
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
Introduction to Molecular Modeling Techniques
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)
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
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
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:
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 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
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)
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:
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 = -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 peco-seconds (ps)
Energy Only (Univariate) Method
• Simplest to implement• Proceeds one
direction until energy increases, then turns 90º, etc.
• Least efficient – many steps– steps are not guided
• Not used very much.
Steepest Descent Method
• Simplest method in use• Follows most negative
gradient (max. force)• Fastest method from a
poor starting geometry• Converges slowly near
the energy minimum• Can skip back and forth
across a minimum.
Conjugate Gradient Method
• Adds ‘history’ to simplicity of steepest descent method to implicitly gather 2nd derivative information to guide the search.
• Variations on this procedure are the Fletcher-Reeves, the Davidon- Fletcher-Powell and the Polak-Ribiere methods.
Second Derivative Methods
• The 2nd derivative of the energy with respect to X,Y,Z [the Hessian] determines the pathway.
• Computationally more involved, but generally fast and reliable, esp. near the minimum.
Structure• Dihedral driving (systematic)• Randomization-minimization (Monte Carlo)• Molecular dynamics (Newton’s laws of motion)• Simulated Annealing (reduce T during MD run)• Genetic Algorithms (start with a population of
conformations; modify slightly; retain lowest energy ones, repeat)
• Trial & error (poor)
Methods are tedious, but absolutely necessary if the result is to be meaningful!
Caveats about Minimum Energy Structures
• What does the global minimum energy structure really mean?
• Does reaction/interaction of interest necessarily occur via the lowest energy conformation?
• What other low energy conformations are available? (Boltzmann distribution/ensemble of conformations and probability/entropy considerations may be important).
