week 2 basics of md simulations

Week 2Basics of MD SimulationsLecture 3: Force fields (empirical potential functions among atoms) Boundaries and computation of long-range interactionsLecture 4: Molecular mechanics (T=0, energy minimization) MD simulations (integration algorithms, constrained dynamics, ensembles)

Empirical force fields (potential functions)Problem: Reduce the ab initio potential energy surface among the atomsTo a classical many-body interaction in the formSuch a program has been tried for water but so far it has failed to produce a classical potential that works. In strongly interacting systems, it is difficult to describe the collective effects by summing up the many-body terms.Practical solution: Truncate the series at the two-body level and assume (hope!) that the effects of higher order terms can be absorbed in U2 by reparametrizing it.

Interaction of two atoms at a distance R can be decomposed into 4 piecesCoulomb potential (1/R)Induced polarization (1/R2)Dispersion (van der Waals) (1/R6)Short range repulsion (e-R/a)The first two can be described using classical electromagnetism1. Coulomb potential:

2. Induced polarization:

Dipole field

Total polarization int.(Initial and final E fieldsin iteration)Non-bonded interactions

Here e corresponds to the depth of the potential at the minimum; 21/6 Combination rules for different atoms:The last two interactions are quantum mechanical in origin. Dispersion is a dipole-dipole interaction that arises from quantum fluctuations (electronic excitations to non-spherical states)Short range repulsion arises from Pauli exclusion principle (electron clouds of two atoms avoid each other), so it is proportional to the electron densityThe two interactions are combined using a 12-6 Lennard-Jones (LJ) potential

12-6 Lennard-Jones potential (U is in kT, r in )

Because the polarization interaction is many-body and requires iterations,it has been neglected in most force fields.The non-bonded interactions are thus represented by the Coulomb and 12-6 LJ potentials.ModelRO-H () qHOHqH (e)e (kT)s ()m (D)k (T=298 C)

SPC 1.0 109.50.4100.2623.1662.27655

TIP3P 0.957 104.50.4170.2573.1512.35977

Exp.Gas1.8680Ab initioWater3.00SPC: simple point chargeTIP3P: transferable intermolecular potential with 3 pointsPopular water models (rigid)

There are hundreds of water models in the market, some explicitly including polarization interaction. But as yet there is no model that can describe all the properties of water successfully.SPC and TIP3P have been quite successful in simulation of biomolecules,and have become industry standards.However, the mean field description of the polarization interaction is boundto break in certain situations:Ion channels, cavitiesInterfacesDivalent ionsTo deal with these, we need polarizable force fields.Grafting point polarizabilities on atoms and fitting the model to data has notworked well. Ab initio methods are needed to make further progress.

Covalent bondsIn molecules, atoms are bonded together with covalent bonds, whicharise from sharing of electrons in partially occupied orbitals. In order to account for the dipole moment of a molecule in a classical representation, partial charges are assigned to the atoms. If the bonds are very strong, the molecule can be treated as rigid as in water molecule. In most large molecules, however, the structure is quite flexible and this must be taken into account for a proper description of the molecules. This is literally done by replacing the bonds by springs. The nearest neighbour interactions involving 2, 3 and 4 atoms are described by harmonic and periodic potentials bond stretchingbendingtorsion (not very good)

Interaction of two H atoms in the ground (1s) state can be describedusing Linear Combinations of Atomic Orbitals (LCAO)From symmetry, two solutions with lowest and highest energies are:+ Symmetric- Anti-symmetric

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The R dependence of the potential energy is approximately given by the Morse potential Where De: dissociation energyRe: equilibrium bond distance

controls the width of the potentialClassical representation:k

Electronic wave functions for the 1s, 2s and 2p states in H atomH2 molecule:



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In carbon, 4 electrons (in 2s and 2p) occupy 4 hybridized sp3 orbitalsThus carbon needs 4 bonds to have a stable configuration.Nitrogen has an extra electron, so needs 3 bonds.Oxygen has 2 extra electrons, so needs 2 bonds.Tetrahedral structure of sp3

Bond stretching and bending interactions are quite strong and well determined from spectroscopy. Torsion is much weaker compared to the other two and is not as well determined. Following QM calculations, they have been revamped recently (e.g., CMAP corrections in the CHARMM force field), which resulted in a better description of proteins.For a complete description of flexibility, we need, besides bond stretching,bending and torsion. The former can be described with a harmonic form:While the latter is represented with a periodic function

An in-depth look at the force fields Three force fields, constructed in the eighties, have come to dominate the MD simulationsCHARMM (Karplus @ Harvard)Optimized for proteins, works well also for lipids and nucleic acidsAMBER (Kollman & Case @ UCSF)Optimized for nucleic acids, otherwise quite similar to CHARMM3. GROMOS (Berendsen & van Gunsteren @ Groningen)Initially optimized for lipids and did not work very well for proteins (smaller partial charges in the carbonyl and amide groups) but it has been corrected in the more recent versions.The first two use the TIP3P water model and the last one, SPC model.They all ignore the polarization interaction. Polarizable versions have been under construction for over a decade but no working code yet.

Bond lengths : kr (kcal/mol/2) r0 () NCA 320. 1.430(1) CAC 250. 1.490(2) CN 370. 1.345(2)(peptide bond)OC 620. 1.230 (2)(double bond) NH 440. 0.997(2) HACA 330. 1.080 (2) CBCA 222. 1.538 (3) HBCB 322. 1.111(3)NMA (N methyl acetamide) vibrational spectra Alanine dipeptide ab initio calculationsFrom alkanes

Bond angles : kq (kcal/mol/rad2)q0 (deg) C N CA 50. 120. (1) C N H 34. 123. (1) H N CA 35. 117. (1) N CA C 50. 107. (2) N CA CB 70. 113.5 (2) N CA HA 48. 108. (2) HA CA CB 35. 111. (2) HA CA C 50. 109.5 (2) CB CA C 52. 108. (2) N C CA 80. 116.5 (1) O C CA 80. 121. (2) O C N 80. 122.5(1)Total 360 deg.Total 360 deg.

Basic dihedral configurations transcisDefinition of the dihedral angle for 4 atoms A-B-C-DDihedrals:

Dihedral parameters: Vn (kcal/mol) nf0 (deg)NameC N CA C 0.2 1 180. f N CA C N 0.6 1 0. yCA C N CA 1.6 1 0. wH N CA CB 0.0 1 0. H N CA HA 0.0 1 0. C N CA CB 1.8 1 0. CA C N H 2.5 2 180. O C N H 2.5 2 180. O C N CA 2.5 2 180.

a helix structure

a-helix ~ -57o

~ -47o(Sasisekharan)

When one of the 4 atoms is not in a chain (e.g. O bond with C in CA-C-N), then an improper dihedral interaction is used to constrain that atom.Most of the time, off-the-chain atom is constrained to a plane using:CA

C O

NWhere q is the angle C=O bond makeswith the CA-C-N plane, and q0 = 180 which enforces a planar configuration.

BoundariesIn macroscopic systems, the effect of boundaries on the dynamics of biomolecules is minimal. In MD simulations, however, the system sizeis much smaller and one has to worry about the boundary effects. Using nothing (vacuum) is not realistic for bulk simulations.Minimum requirement: water beyond the simulation box must be treated using a continuum representation (reaction field). An intermediate zone is treated using stochastic boundary conditions.Most common solution: periodic boundary conditions. The simulation box is replicated in all directions just like in a crystal.The cube and rectangular prism are the obvious choices for a box shape, though there are other shapes that can fill the space (e.g. hexagonal prism and rhombic dodecahedron).

Periodic boundary conditions in two dimensions: 8 nearest neighboursParticles in the box freely move to the next box, which means they appear from the opposite side of the same box. In 3-D, there are 26 nearest neighbours.

Treatment of long-range interactionsProblem: the number of non-bonded interactions grows as N2 where N is the number of particles. This is the main computational bottle neck that limits the system size. A simple way to deal with this problem is to introduce a cutoff radius for pairwise interactions (together with a switching function), and calculate the potential only for those within the cutoff sphere. This is further facilitated by creating non-bonded pair lists, which are updated every 10-20 steps. For Lennard-Jones (6-12) interaction, which is weak and falls rapidly, this works fine and is adapted in all MD codes. Coulomb interaction, however, is very strong and falls very slowly.[Recall the Coulomb potential between two unit charges, U=560 kT/r ()]Hence use of any cutoff is problematic and should be avoided.

This problem has been solved by introducing the Ewald sum method.All the modern MD codes use the Ewald sum method to treat the long-range interactions without cutoffs. Here one replaces the point charges with Gaussian distributions, which leads to a much faster convergence of the sum in the reciprocal (Fourier) space. The remaining part (point particle Gaussian) falls exponentially in real space, hence can be computed easily using a cutoff radius (~10 ).The mathematical details of the Ewald sum method is given in the following appendix. Please read it to understand how it works in practice.

Appendix: Ewald summation methodThe coulomb energy of N charged particles in a box of volume V=LxLyLzHerewith integer values for n.The prime on the sum signifies that i=j is not included for n=0.Ewalds observation (1921): in calculating the Coulomb energy of ionic crystals, replacing the point charges with Gaussian distributions leads to a much faster convergence of the sum in the reciprocal (Fourier) space.The remaining part (point particle Gaussian) falls exponentially in real space, hence can be computed easily.

For a point charge q at the origin:The Poisson equation and its solution:When the charge q has a Gaussian distribution:

This result is obtained most easily by integrating the Poisson equationIntegrate 0 to r :

For the second part, the potential due to a charge qj at rj is given by:Writing the charge density asWhere erfc(x) is the complementary error function which falls as Thus choosing 1/ about an Angstrom, this potential converges quickly.Typically, it is evaluated using a cutoff radius of ~10 , so the original boxwith N particles is sufficient for this purpose (with the nearest image convention). The direct (short-range) part of the energy of the system is:

The Gaussian part converges faster in the reciprocal (Fourier) space hence best evaluated as a Fourier seriesThe Poisson equation in the Fourier spaceFor a point charge q at the origin:

When the charge q has a Gaussian distribution:Multiply each integral byto compete the squareEach Gaussian integral then gives The corresponding potential in the Fourier space

The Gaussian charge density for the periodic boxWhich yields for the potential

Transforming back to the real spaceThe reciprocal space (long-range) part of the systems energy:

This energy includes the self-energy of the Gaussian density which needsto be removedSo the total energy is:

Molecular mechanicsMolecular mechanics deals with the static features of biomolecular systems at T=0 K, that is, particles do not have any kinetic energy. [Note that in molecular dynamics simulations, particles have an average kinetic energy of (3/2)kT, which is substantial at room temp., T=300 K] Thus the energy is given solely by the potential energy of the system.Two important applications of molecular mechanics are:Energy minimization (geometry optimization):Find the coordinates for the configuration that minimizes the potential energy (local or absolute minimum).Normal mode analysis:Find the vibrational excitations of the system built on the absolute minimum using the harmonic approximation.

Energy minimization The initial configuration of a biomolecule whether experimentally determined or otherwise does not usually correspond to the minimum of the potential energy function. It may have large strains in the backbone that would take a long time to equilibrate in MD simulations, or worse, it may have bad contacts (i.e. overlapping van der Waals radii), which would crash the MD simulation in the first step!To avoid such mishaps, it is a standard practice to minimize the energy before starting MD simulations.Three popular methods for analytical forms: Steepest descent (first derivative) Conjugate gradient (first derivative) Newton-Raphson (second derivative)

Steepest descent:Follows the gradient of the potential energy function U(r1, ,rN) at each step of the iteration

where i is the step size. The step size can be adjusted until the minimumof the energy along the line is found. If this is expensive, a single step isused in each iteration, whose size is adjusted for faster convergence.Works best when the gradient is large (far from a minimum), but tends to have poor convergence as a minimum is approached because the gradient becomes smaller.Successive steps are always mutually perpendicular, which can lead to oscillations and backtracking.

A simple illustration of steepest descent with a fixed step size in a 2D energy surface (contour plots)

Conjugate gradient:Similar to steepest descent but the gradients calculated from previous steps are used to help reduce oscillations and backtracking

(For the first step, d is set to zero)Generally one finds a minimum in fewer steps than steepest descent,e.g. it takes 2 steps for the 2D quadratic function, and ~n steps for nD. But conjugate gradient may have problems when the initial conformation is far from a minimum in a complex energy surface. Thus a better strategy is to start with steepest descent and switch to conjugate gradient near the minimum.

Newton-Raphson:Requires the second derivatives (Hessian) besides the first.Predicts the location of a minimum, and heads in that direction.To see how it works in a simple situation, consider the quadratic 1D case

In general

For a quadratic energy surface, this method will find the minimum in onestep starting from any configuration.

Construction and inversion of the 3Nx3N Hessian matrix is computationally demanding for large systems (N>100). It will find a minimum in fewer steps than the gradient-only methods in the vicinity of the minimum. But it may encounter serious problems when the initial conformation is far from a minimum. A good strategy is to start with steepest descent and then switch to alternate methods as the calculations progress, so that each algorithm operates in the regime for which it was designed. Using the above methods, one can only find a local minimum. To search for an absolute minimum, Monte Carlo methods are moresuitable. Alternatively, one can heat the system in MD simulations, which will facilitate transitions to other minima.

Normal mode analysisAssume that the minimum energy of the system is given by the 3N coordinates, {r0i}. Expanding the potential energy around the equilibrium configuration gives

Ignoring the constant energy, the leading term is that of a system of coupled harmonic oscillators. In a normal mode, all the particles in the system oscillate with the same frequency w. To find the normal modes,first express the 3N coordinates as {xi, i=1,,3N}.

The potential energy becomes

where the spring constants are given by the Hessian as

Introducing the 3Nx3N diagonal mass matrix M

The secular equation for the normal modes is given by

For a 3Nx3N matrix, solution of the secular equation will yield 3N eigenvalues, wi and the corresponding eigenvectors, aiOf these, 3 correspond to translations and 3 to rotations of the system.Thus there are 3N-6 intrinsic vibrational modes of the system. At a given temperature T, the motion of the ith coordinate is given by

The mean square displacement of the coordinates and atoms:

A simple example: normal modes of water moleculeWater molecule has 9-6=3 intrinsic vibrations, which correspond to symmetric and anti-symmetric stretching of H atoms and bending.Because of H-bonding, water molecules in water cannot freely rotatebut rather librate (wag or twist).

Wave numbers: 3652 cm-1 3756 cm-1 1595 cm-1 (200 cm-1~1 kT)Excitation energies >> kT, which justifies the use of a rigid model for water.

Normal modes of a small protein BPTI (Bovine pancreatic tyripsin inhibitor)Some characteristic frequencies (in cm-1)StretchingBendingTorsionH-N: 3400-3500H-C-H: 1500C=C: 1000H-C: 2900-3000H-N-H: 1500C-O: 300-600C=C, C=O: 1700-1800C-C=O: 500C-C: 300C-C, C-N: 1000-1250S-S-C: 300C-S: 200

Applications to domain motions of proteins Many functional proteins have two main configurations called, for example, resting & excited states, or open & closed states. Proteins can be crystallized in one of these states the other configuration need to be found by other means. The configurational changes that connect these two states usually involve large domain motions that could take milli to micro seconds, which is outside the reach of current MD simulations. Normal mode analysis can be used to identify such collective motions in proteins and predict the missing state that is crucial for description of the protein function. Examples: 1. Gating of ion channels (open & closed states)2. Opening and closing of gates in transporters

Molecular dynamicsIn MD simulations, one follows the trajectories of N particles accordingto Newtons equation of motion:

where U(r1,, rN) is the potential energy function consisting of bonded and non-bonded interactions (Coulomb and LJ 6-12). We have already discussed force fields and boundary conditions in some detail. Here, we will consider:Integration algorithms, e.g., Verlet, Leap-frogInitial conditions and choice of the time stepConstrained dynamics for rigid molecules (SHAKE and RATTLE)MD simulations at constant temperature and pressure

Integration algorithmsGiven the position and velocities of N particles at time t, a straightforward integration of Newtons equation of motion yields at t+DtIn practice, variations of these equations are implemented in MD codes In the popular Verlet algorithm, one eliminates velocities using the positions at t-Dt,Adding with eq. (2), yields:(1)

(2)

This is especially useful in situations where one is interested only in the positions of the atoms. Velocities can be calculated from Some drawbacks of the Verlet algorithm: Positions are obtained by adding a small quantity (order Dt2) to large ones, which may lead to a loss of precision. Velocity at time t is available only at the next time step t+Dt It is not self starting. At t=0, there is no position at t-Dt. It is usually estimated using or

In the Leap-frog algorithm, the positions and velocities are calculated at different times separated by Dt/2To show its equivalence to the Verlet algorithm, considerSubtracting the two equations yields the Verlet result.If required, velocity at t is obtained from:

To iterate these equations, we need to specify the initial conditions.The initial configuration of biomolecules can be taken from the Protein Data Bank (PDB) (if available).In addition, membrane proteins need to be embedded in a lipid bilayer. VMD has a facility that will perform this step with minimal effort.All the MD codes have facilities to hydrate a biomolecule, i.e., fill the void in the simulation box with water molecules at the correct density.Ions can be added at random positions. Alternatively, VMD solves the Poisson-Boltzmann equation and places the ions where the potential energy are at minimum.After energy minimization, these coordinates provide the positions at t=0. Initial velocities are sampled from a Maxwell-Boltzmann distribution:

Choosing the time step:In choosing a time step, one has to compromise between two conflicting demands: In order to obtain statistically significant results and access biologically relevant time scales, one needs long simulation times, which requires long time stepsTo avoid instabilities, conserve energy, etc., one needs to integrate the equations as accurately as possible, which requires using short time steps.SystemTypes of motionRecom. time step (fs)Atoms translation10 fsRigid molecules + rotation5 fsFlex. mols, rigid bonds + torsion2 fsCompletely flex. mols .+ vibrations1 - 0.5 fs

Constrained dynamicsMD simulations are carried out most efficiently using the 3N Cartesian coordinates of the N atoms in the system. This is fine if there are no rigid bonds among the atoms in the system. For rigid molecules (e.g. water), one needs to impose constraints on their coordinates so that their rigid geometry is preserved during the integration. This is achieved using Lagrange multipliers. Consider the case of a rigid diatomic molecule, where the bond length between atoms i and j is fixed at d, which imposes the following constraint on their coordinatesThe force on atom i due to this constraint can be written as

Note that Thus the total constraint force on the molecule is zero, and the constraint forces do not do any work. Incorporating the constraint forces in the Verlet algorithm, the positions of the atoms are given byTo simplify, combine the known first three term on the r.h.s. as (1)

(2)

A third equation is given by the constraint conditionSubstituting eqs. (1) and (2) above yieldsThis quadratic equation can be easily solved to obtain the Lagrange multiplier l. Substituting l back in eqs. (1) and (2), one finds the positions of the atoms at t+Dt.For a rigid many-atomic molecule, one needs to employ a constraint for every fixed bond length and angle. (Bond and angle constraints among three atoms (i, j, k) can be written as three bond constraints.)

Assuming k constraints in total, this will lead to k coupled quadratic equations, which are not easy to solve.A common approximation is to exploit the fact that l is small, hence thequadratic terms in l can be neglected. This leads to a system of k linearequations, which can be solved by matrix inversion.In the SHAKE algorithm, a further simplification is introduced by solving the constraint equations sequentially one by one. This process is then iterated to correct any constraint violations that arise from the neglect ofthe coupling in the constraint equations. SHAKE is commonly used in fixing the bond lengths of H atoms.Another well-known constraint algorithm is RATTLE, which is designed for the velocity-Verlet integrator, where velocities as well as positions arecorrected for constraints.

MD simulations at constant temperature and pressureMD simulations are typically performed in the NVE ensemble, where all 3 quantities are constant. Due to truncation errors, keeping the energy constant in long simulations can be problematic. To avoid this problem,the alternative NVT and NPT ensembles can be employed. The temperature of the system can be obtained from the average K. E.Thus an obvious way to keep the temperature constant at T0 is to scale the velocities as:Because K. E. has considerable fluctuations, this is a rather crude method.

A better method which achieves the same result more smoothly is the Berendsen method, where the atoms are weakly coupled an external heat bath with the desired temperature T0If T(t) > T0 , the coefficient of the coupling term is negative, which invokes a viscous force slowing the velocity, and vice-versa for T(t) < T0 Similarly in the NPT ensemble, the pressure can be kept constant by simply scaling the volume. Again a better method (Langevin piston), is to weakly couple the pressure difference to atoms using a force as in above, which will maintain the pressure at the desired value (~1 atm).

Monte Carlo (MC)Metropolis algorithm:Given N particles interacting with a potential energy function U(r1,, rN)Probability:Assume some initial configuration with energy U0Move the particles by small increments to new positions with energy U1If U1 < U0, accept the new configurationIf U1 > U0 , select a random number r between 0 and 1, and accept the new configuration if Keep iterating until the minimum energy is found

week 2 basics of md simulations

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