Energy Minimization

1
Energy Minimization


Local minima
Global minima

• At a maximum or minimum i.e., the gradient is
  zero.
• At a maximum the curvature is negative, i.e., lt
  0 (2nd derivative less than zero)
• At a minimum the curvature is positive, i.e., gt
  0 (2nd derivative more than zero)

2
Energy Minimization
For a multivariate system
the gradient is the first derivative vector
curvature is determined from the 2nd partial
derivative matrix - the Hessian matrix
3
The Line Search Method

• Efficiency of minimizer judged by number of
  iterations required to converge and number of
  function evaluations per iteration

4
Steepest Descent

• The line search direction is the direction of the
  gradient
• The direction of each step is perpendicular to
  the previous step
• Leads to oscillations inefficient behavior

5
Conjugate Gradient
hi1 new direction vector gi1 new gradient
vector

• Progresses along mutually conjugate directions
• Each successive step refines the direction
  towards the minimum

6
Newton-Raphson Method
Expensive but rapidly converging
where A0 is the Hessian Matrix of second
derivatives
Derived from the expansion of Taylor Series
At the minimum xx, E'(x)0
7
Rules for Selecting Minimizer
8
Conformational Sampling Potential Energy Scan
9
Conformational Sampling Temperature Dependence
Local minima
Global minima
10
Steps in a Molecular Dynamics Simulation
11
Setting Up The Simulation

• Obtaining the structure (PDB) file
• Amber parameter and topology library files
• Massage the input file to read in Leap
• Workout the missing parameters if any
• Add solvent box
• Neutralize charge of the system with counterions
• Write the coordinate and parameter file
• Run minimization
• Heating and Equilibration dynamics
• Production dynamics

12
Temperature Control

• To generate NVT, NPT ensembles
• To study the system behavior as temperature
  changes
• Protein unfolding
• Perform simulated annealing
• Searching conformational space

13
Constant Temperature Dynamics

• Kinetic Energy
• Velocity Reassignment Maxwell-Boltzmann
  Distribution
• Velocity Scaling

14
Constant Temperature Dynamics

• Berendsen Temperature Coupling Scheme

Rate of change of temperature is proportional to
the difference in temperature between bath and
the system ? coupling parameter whose
magnitude determines how tightly the bath and
system are coupled
Amber simulation ? 0.5- 5.0 ps
15
Constant Temperature Dynamics

• Stochastic collisions method Anderson et al.
• Randomly choose a particle and reassign its
  velocities from a Maxwell-Boltzmann distribution
• Trajectory Collection of mini microcanonical
  simulations
• Extended system method Nosé et al., Hoover et
  al.
• Thermal reservoir is part of the system,
  represented by additional degrees of freedom
• A fictitious mass parameter of the extra degree
  of freedom controls energy flow between the
  reservoir and system, larger the value of this
  parameter, slower the energy flow

16
Constant Pressure Dynamics

• Isothermal-isobaric ensemble
• Pressure is related to virial product of
  position and the derivative of PE
• Pressure is maintained by volume fluctuations of
  the simulation cell
• Volume fluctuations is related to isothermal
  compressibility, ?

17
Constant Pressure Dynamics

• Weak Coupling method Berendsen
• Extended Pressure-Coupling method Anderson et
  al.
• Introduce an extra degree of freedom (piston)
  corresponding to the volume of the box

18
Constant Pressure Dynamics

• Box side 20 Å (volume 8000 Å3) at 300 K
• For an ideal gas ? 1 atm-1, fluctuation 18100
  Å3
• for water ? 44.75x10-6 atm-1, fluctuation 121
  Å3