1. SMA5233 Particle Methods and Molecular Dynamics Lecture 2: Physical Principles and Design Issues of MD A/P Chen Yu Zong Tel: Room 08-14, level 8, S16 National University of Singapore

2. When is classical mechanics a reasonable approximation?
In Newtonian physics, any particle may possess any one of a continuum of energy values. In quantum physics, the energy is quantized, not continuous. That is, the system can accomodate only certain discrete levels of energy, separated by gaps.
At very low temperatures these gaps are much larger than thermal energy, and the system is confined to one or just a few of the low-energy states. Here, we expect the `discreteness' of the quantum energy landscape to be evident in the system's behavior.
As the temperature is increased, more and more states become thermally accessible, the `discreteness' becomes less and less important, and the system approaches classical behavior.

3. When is classical mechanics a reasonable approximation?
For a harmonic oscillator, the quantized energies are separated by DE=hf, where h is Planck's constant and f is the frequency of harmonic vibration.
Classical behavior is approached at temperatures for which KBT>>hf, where KB is the Boltzmann constant and KBT = kcal/mol at 300 K. Setting hf = kcal/mol yields f = 6.25/ps, or 209cm-1. So a classical treatment will suffice for motions with characteristic times of a ps or longer at room temperature.

4. Classical Mechanics - in a Nutshell
Building on the work of Galileo and others, Newton unveiled his laws of motion in According to Newton:
A body remains at rest or in uniform motion (constant velocity - both speed and direction) unless acted on by a net external force.
In response to a net external force, F, a body of mass m accelerates with acceleration a=F/m.
If body i pushes on body j with a force Fij, then body j pushes on body i with a force Fji=-Fij.

5. Classical Mechanics - in a Nutshell
For energy-conserving forces, the net force Fi on particle i is the negative gradient (slope in three dimensions) of the potential energy with respect to particle i's position: Fi= - ▼iV(R) where V(R) represents the potential energy of the system as a function of the positions of all N particles, R.
In three dimensions, ri is the vector of length 3 specifying the position of the atom, and R is the vector of length specifying all coordinates. In the context of simulation, the forces are calculated for energy minimizations and molecular dynamics simulations but are not needed in Monte Carlo simulations.
Classical mechanics is completely deterministic: Given the exact positions and velocities of all particles at a given time, along with the function , one can calculate the future (and past) positions and velocities of all particles at any other time. The evolution of the system's positions and momenta through time is often referred to as a trajectory.

6. Statistical Mechanics - Calculating Equilibrium Averages
According to statistical mechanics, the probability that a given state with energy E is occupied in equilibrium at constant particle number N, volume V, and temperature T (constant , the `canonical' ensemble) is proportional to exp{-E/ KBT}, the `Boltzmann factor.'
Probability ~ exp{-E/ KBT}
The equilibrium value of any observable O is therefore obtained by averaging over all states accessible to the system, weighting each state by this factor.

7. Statistical Mechanics - Calculating Equilibrium Averages
Classically, a microstate is specified by the positions and velocities (momenta) of all particles, each of which can take on any value. The averaging over states in the classical limit is done by integrating over these continuous variables:
where the integrals are over all phase space (positions and momenta _) for the N particles in 3 dimensions.
When all forces (the potential energy V) and the observable O are velocity-independent, the momentum integrals can be factored and canceled:
where is the total kinetic energy,
and E=K+V. As a result, Monte Carlo simulations
compare V's, not E's.

8. What is molecular mechanics?
Molecular Mechanics (MM) finds the geometry that corresponds to a minimum energy for the system - a process known as energy minimization.
A molecular system will generally exhibit numerous minima, each corresponding to a feasible conformation. Each minimum will have a characteristic energy, which can be computed. The lowest energy, or global minimum, will correspond to the most likely conformation.

9. Energy Minimization
Torsion angles
Are 4-body
Angles
Are 3-body
Bonds
Are 2-body
Non-bonded
pair
U is a function of the conformation C of the protein. The problem of
“minimizing U” can be stated as finding C such that U(C) is minimum.

10. Units in force fields Quantity AKMA unit Equivalent SI Energy
Most force fields use the AKMA (Angstrom – Kcal – Mol – Atomic Mass Unit) unit
system:
Quantity
AKMA unit
Equivalent SI
Energy
1 Kcal/Mol
4184 Joules
Length
1 Angstrom
10-10 meter
Mass
1 amu (H=1amu) Kg
Charge
1 e C
Time
1 unit
second
Frequency
1 cm-1
rd/s

11. Energy Minimization Local minima:
A conformation X is a local minimum if there exists a
domain D in the neighborhood of X such that for all Y≠X in D:
U(X) <U(Y)
Global minimum:
A conformation X is a global minimum if
for all conformations Y ≠X

12. Molecular Modeling: Energy minimization: Force guided approach:
Initialize:
Change atom position: xi -> xi+dxi
Compute potential energy change:
V -> V +dV
Determine next movement:
Force: Fxi=-dV/dxi; Fyi=-dV/dyi; Fzi=-dV/dzi
Atom displacement: dxi=C*Fxi
New position: xi=xi+dxi

13. Notes on computing the energy
Bonded interactions are local, and therefore their computation has a linear
computational complexity (O(N), where N is the number of atoms in the
molecule considered.
The direct computation of the non bonded interactions involve all pairs of
atoms and has a quadratic complexity (O(N2)).
This is usually prohibitive for large molecules.
Reducing the computing time: use of cutoff in UNB

14. Cutoff schemes for faster energy computation
wij : weights (0< wij <1). Can be used to exclude bonded terms,
or to scale some interactions (usually 1-4)
S(r) : cutoff function.
Three types:
1) Truncation: b

15. Cutoff schemes for faster energy computation
2. Switching a b
with
3. Shifting or b

16. The minimizers Some definitions: Gradient:
The gradient of a smooth function f with continuous first and second
derivatives is defined as:
Hessian
The n x n symmetric matrix of second derivatives, H(x), is called
the Hessian:

17. The minimizers Steepest descent (SD):
Minimization of a multi-variable function is usually an iterative process, in which
updates of the state variable x are computed using the gradient, and in some
(favorable) cases the Hessian.
Iterations are stopped either when the maximum number of steps (user’s input)
is reached, or when the gradient norm is below a given threshold.
Steepest descent (SD):
The simplest iteration scheme consists of following the “steepest
descent” direction:
Usually, SD methods leads to improvement quickly, but then exhibit
slow progress toward a solution.
They are commonly recommended for initial minimization iterations,
when the starting function and gradient-norm values are very large.
(a sets the minimum
along the line defined
by the gradient)

18. The minimizers Conjugate gradients (CG):
In each step of conjugate gradient methods, a search vector pk is
defined by a recursive formula:
The corresponding new position is found by line minimization along
pk:
the CG methods differ in their definition of b:
- Fletcher-Reeves:
- Polak-Ribiere
- Hestenes-Stiefel

19. The minimizers Newton’s methods:
Newton’s method is a popular iterative method for finding the 0 of a
one-dimensional function: x3 x2 x1 x0
It can be adapted to the minimization of a one –dimensional function, in which
case the iteration formula is:
The equivalent iterative scheme for multivariate functions is based on:
Several implementations of Newton’s method exist, that avoid computing
the full Hessian matrix: quasi-Newton, truncated Newton, “adopted-basis
Newton-Raphson” (ABNR),…

20. What is molecular dynamics simulation?
Molecular Dynamics (MD) applies the laws of classical mechanics to compute the motion of the particles in a molecular system.
Simulation that shows how the atoms in the system move with time
Typically on the nanosecond timescale
Atoms are treated like hard balls, and their motions are described by Newton’s laws.

21. What is molecular dynamics simulation?
Molecular dynamics simulations generate information at the microscopic level, including atomic positions and velocities.
The conversion of this microscopic information to macroscopic observables such as pressure, energy, heat capacities, etc., requires statistical mechanics.
MD Simulation
Microscopic information
Statistical Mechanics
Macroscopic properties

22. Characteristic protein motions
Type of motion
Timescale
Amplitude
Local:
bond stretching
angle bending
methyl rotation
0.01 ps
0.1 ps
1 ps
< 1 Å
Medium scale
loop motions
SSE formation
ns – ms
1-5 Å
Global
protein tumbling
(water tumbling)
protein folding
20 ns
(20 ps)
ms – hrs
> 5 Å
Periodic (harmonic)
Random (stochastic)

23. Molecular Dynamics Simulation
Molecule: (classical) N-particle system
Newtonian equations of motion:
with
Integrate numerically via the „leapfrog“ scheme:
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with
Δt  1fs!
(equivalent to the Verlet algorithm)

24. How do you run a MD simulation?
Get the initial configuration
From x-ray crystallography or NMR spectroscopy (PDB)
Assign initial velocities
At thermal equilibrium, the expected value of the kinetic energy of the system at temperature T is:
This can be obtained by assigning the velocity components vi from a random Gaussian distribution
with mean 0 and standard deviation (kBT/mi):

25. How do you run a MD simulation?
For each time step:
Compute the force on each atom:
Solve Newton’s 2nd law of motion for each atom, to get new
coordinates and velocities
Store coordinates
Stop
X: cartesian vector
of the system
M diagonal mass matrix
.. means second order
differentiation with
respect to time
Newton’s equation cannot be solved analytically:
Use stepwise numerical integration

26. What the integration algorithm does
Advance the system by a small time step Dt during which forces are considered constant
Recalculate forces and velocities
Repeat the process
If Dt is small enough, solution is a reasonable approximation

27. Choosing a time step Dt
Not too short so that conformations are efficiently sampled
Not too long to prevent wild fluctuations or system ‘blow-up’
An order of magnitude less than the fastest motion is ideal
Usually bond stretching is the fastest motion:
C-H is ~10fs so use time step of 1fs
Not interested in these motions?
Constrain these bonds and double the time step

28. Molecular Dynamics Ensembles
The method discussed above is appropriate for the micro-canonical ensemble: constant N (# of particles) V (volume) and ET (total energy = E + Ekin)
In some cases, it might be more appropriate to simulate under constant Temperature (T) or constant Pressure (P):
Canonical ensemble: NVT
Isothermal-isobaric: NPT
Constant pressure and enthalpy: NPH
How do you simulate at constant temperature and pressure?

29. Common Ensembles

30. MD in different thermodynamic ensemble
MD naturally samples the NVE (microcanonical ensemble): this is not very useful for most applications.
More useful ensembles include:
Canonical (NVT): requires the particles to interact with a thermostat
Isobaric-Isothermal (NPT): requires the particles to interact with a thermostat and a baristat

31. Temperature
Thermodynamics definition
Measuring T

32. Canonical (NVT) Several possible approaches
Berendsen Method (velocities are rescaled deterministically after each step so that the system is forced towards the desired temperature)
Stochastic collisions (Anderson: a random particle’s velocity is reassigned by random selection from the Maxwell-Boltzmann distribution at set intervals)
Extended Lagrangians (Nose and Nose-Hoover: the thermal reservoir is considered an integral part of the system and it is represented by an additional degree of freedom: set of deterministic equations that give sampling in NVT)

33. Simulating at constant T: the Berendsen scheme
Bath supplies or removes heat from the system as appropriate
Exponentially scale the velocities at each time step by the factor :
where  determines how strong the bath influences the system
system
Heat bath
T: “kinetic” temperature
Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:3684 (1984)

34. Simulating at constant P: Berendsen scheme
Couple the system to a pressure bath
Exponentially scale the volume of the simulation box at each time step by a factor :
where  : isothermal compressibility
P : coupling constant
system
pressure bath
where
u : volume
xi: position of particle i
Fi : force on particle i
Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:3684 (1984)

35. Nose-Hoover Thermostat
Introduce a new coordinate, s, representing reservoir in the Lagrangian
Q=effective mass associated
with s
g is a parameter we will fix
later
With momenta:

36. Nose-Hoover Thermostat
Extended-system Hamiltonian:
it can be shown that the molecular positions and momenta follow a canonical (NVT) distribution if g = 3N+1
With p’=p/s

37. Nose-Hoover Thermostat
Real and virtual variables are related by:
s can be interpreted as a scaling factor of the time step
treal = tsim/s
since s varies during the simulation, each “true” time step is of varying length

38. Nose-Hoover Thermostat
From the Hamiltonian, can derive the equations of motion for the virtual variables p, r and t.
Scaled-variables equations of motion
constant simulation Dt
fluctuating real Dt
Real-variables equation of motion (here, I have
Dropped the prime)

39. IMPLEMENTATION
Simplify expression (Nose-Hoover) by introducing a thermodynamic
Friction coefficient:
So equations of motions (dropping the prime):
(g=3N)

40. An experiment is usually made on a macroscopic sample that contains an extremely large number of atoms or molecules sampling an enormous number of conformations. In statistical mechanics, averages corresponding to experimental observables are defined in terms of ensemble averages;

41. AVERAGES Thermodynamic (Ensemble) average: average over all points in
phase space at a Single time
Dynamic average: average over a single point in phase space at
All times
Ergodic hypothesis: for infinitely long trajectory, thermo average=
Dynamic average
A large number of observations made on a single system at N
arbitrary instants in time have the same statistical properties
as observing N arbitrarily chosen systems at the same time.

42. To get a thermo average, need to know the probability of finding the system at each point (state) in phase space.
Probability density
Partition function
Ensemble (thermodynamic) average of observable A(rN,pN):

43. In an MD simulation, we determine a time average of A, which is expressed as
                                                                                                                                                                                                                                                                                                                                                    
where t is the simulation time, M is the number of time steps in the simulation and A(pN,rN) is the instantaneous value of A.

44. THERMODYNAMIC PROPERTIES FROM MD SIMULATIONS
Thermodynamic (equilibrium) averages can be calculated
via time averaging:
Examples:
Average potential energy
Average kinetic energy

45. Other quantities obtained from MD: Heat Capacity
Expressed in terms of fluctuations in the energy

46. Other Quantities
Root Mean Square Deviation (for instance, over Ca atoms)
Radius of Gyration

47. Relieve local stresses, Relax bond length/angle distortions
From PDB orXtal
Relieve local stresses,
Relax bond length/angle distortions
velocities (vx, vy,vz) are randomly assigned according to a Maxwellian distribution,
P(vi)dv = (m/2pikBT)1/2 exp[ -mv2/2kBT ]dv
to overcome “hot spots” due to
Initial high velocities
T(t) = (1/kB(3N-n)) Si (mivi2)
Scale velocities by (T'/T(t))1/2 to get new
temperatureT

48. Types of MD Systems Empirical Potentials
Empirical potentials either ignore quantum mechanical effects, or attempt to capture quantum effects in a limited way through entirely empirical equations. Parameters in the potential are fitted against known physical properties of the atoms being simulated, such as elastic constants and lattice parameters.
Empirical potentials can further be subcategorized into pair potentials, and many-body potentials
Pair potentials: the total potential energy of a system can be calculated from the sum of energy contributions from pairs of atoms.
Many-body potentials: the potential energy cannot be found by a sum over pairs of atoms.
Tersoff Potential involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential.
Tight-Binding Second Moment Approximation (TBSMA), the electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms, and the potential energy contribution is then a function of this sum.

49. Types of MD Systems Semi-Empirical Potentials
Semi-empirical potentials make use of the matrix representation from quantum mechanics. However, the values of the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical formulae are used once again to determine the energy contributions of the orbitals.
There are a wide variety of semi-empirical potentials, known as tight-binding potentials, which vary according to the atoms being modeled.

50. Types of MD Systems Ab-initio (First Principles) Methods
Ab-initio (first principles) methods use full quantum-mechanical formula to calculate the potential energy of a system of atoms or molecules. This calculation is usually made "locally", i.e., for nuclei in the close neighborhood of the reaction coordinate. Although various approximations may be used, these are based on theoretical considerations, not on empirical fitting. Ab-Initio produce a large amount of information that is not available from the empirical methods, such as density of states information.
Perhaps the most famous ab-initio package is the Car-Parrinello Molecular Dynamics (CPMD) package based on the density functional theory..

51. Types of MD Systems Coarse-Graining: "Reduced Representations"
The earliest MD simulations on proteins, lipids and nucleic acids generally used "united atoms", which represents the lowest level of coarse graining. For example, instead of treating all four atoms of a methyl group explicitly (or all three atoms of a methylene group), one represents the whole group with a single pseudoatom. This pseudoatom must, of course, be properly parameterized so that its van der Waals interactions with other groups have the proper distance-dependence. Similar considerations apply to the bonds, angles, and torsions in which the pseudoatom participates. In this kind of united atom representation, one typically eliminates all explicit hydrogen atoms except those that have the capability to participate in hydrogen bonds ("polar hydrogens").

52. Types of MD Systems Coarse-Graining: "Reduced Representations"
When coarse-graining is done at higher levels, the accuracy of the dynamic description may be less reliable. But very coarse-grained models have been used successfully to examine a wide range of questions in structural biology. Following are just a few examples.
protein folding studies are often carried out using a single (or a few) pseudoatoms per amino acid;
DNA supercoiling has been investigated using 1-3 pseudoatoms per basepair, and at even lower resolution;
Packaging of double-helical DNA into bacteriophage has been investigated with models where one pseudoatom represents one turn (about 10 basepairs) of the double helix;
RNA structure in the ribosome and other large systems has been modeled with one pseudoatom per nucleotide.
The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data.

53. Electrostatics and the `Generalized Born' Solvent Model
All problems in electrostatics boil down to the solution of a single equation, Poisson's equation:
where is the Laplacian operator, f is the electrostatic potential, r is the charge density (total charge per unit volume including all `free' and `polarization' charges), and e0 is the permittivity of free space. In Cartesian (xyz) coordinates,
The electrostatic potential at a given point in space is the potential energy per unit charge that a test charge would have if positioned at that point in the electric field E specified by f :

54. Solvent Model: Structure of water molecule

55. Solvent Model: Estimate the average distance between water molecules

56. Solvent Model: Layout of water molecule
Divide the cubic evenly into meshes, according to the distances between water molecules
Place oxygen atom of each water molecule on the node of the mesh, generate random direction for hydrogen atoms.

57. Add-in a Molecule (Nano-tube)

58. MD Simulation Result

59. MD-Experiments with Argon Gas

60. Role of environment - solvent
explicit or
implicit?
box or
droplet?

61. Surface (tension) effects?
periodic boundary conditions
and the minimum image convention

62. „conformational substates“
Proteins jump between many, hierarchically ordered
„conformational substates“
00-09
H. Frauenfelder et al., Science 229 (1985) 337

63. Molecular Dynamics Simulations
External coupling: Temperature (potential truncation, integration errors) Pressure (density equilibration) System translation/rotation
Analysis
Energies (individual terms, pressure, temperature) Coordinates (numerical analysis, visual inspection!)  Mechanisms
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64. Molecular Dynamics Simulations
Design Constraints:
Design of a molecular dynamics simulation can often encounter limits of computational power. Simulation sizes and time duration must be selected so that the calculation can finish within a useful time period.
The simulation's time duration is dependent on the time length of each timestep, between which forces are recalculated. The timestep must be chosen small enough to avoid discretization errors, and the number of timesteps, and thus simulation time, must be chosen large enough to capture the effect being modeled without taking an extraordinary period of time.
The simulation size must be large enough to express the effect without the boundary conditions disrupting the behavior. Boundary constraints are often treated by choosing fixed conditions at the boundary, or by choosing periodic boundary conditions in which one side of the simulation loops back to the opposite side.
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65. Molecular Dynamics Simulations
Design Constraints:
The scalability of the simulation with respect to the number of molecules is usually a significant factor in the range of simulation sizes which can be simulated in a reasonable time period.
In Big O notation, common molecular dynamics simulations usually scale by either O(nlog(n)), or with good use of neighbour tables, O(n), with n as the number of molecules
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