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The driving force for chemical processes is described by thermodynamics. The mechanism by which chemical processes occur is described by kinetics. Thermodynamics dictates the energetic relationships between different chemical states, whereas the sequence or rate of events that occur as molecules transform between their various possible states is described by kinetics:
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Conformational transitions and local vibrations are the usual subjects of molecular dynamics studies. Molecular dynamics alters the intramolecular degrees of freedom in a step-wise fashion, analogous to energy minimization. The individual steps in energy minimization are merely directed at establishing a down-hill direction to a minimum. The steps in molecular dynamics, on the other hand, meaningfully represent the changes in atomic position, ri, over time (i.e. velocity).
For the "i" atoms of the system:
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Newton's equation is used in the molecular dynamics formalism to simulate atomic motion:
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The rate and direction of motion (velocity) are governed by the forces that the atoms of the system exert on each other as described by Newton's equation. In practice, the atoms are assigned initial velocities that conform to the total kinetic energy of the system, which in turn, is dictated by the desired simulation temperature. This is carried out by slowly "heating" the system (initially at absolute zero) and then allowing the energy to equilibrate among the constituent atoms. The basic ingredients of molecular dynamics are the calculation of the force on each atom, and from that information, the position of each atom throughout a specified period of time (typically on the order of picoseconds = 10^-12 seconds).
The force on an atom can be calculated from the change in energy between its current position and its position a small distance away. This can be recognized as the derivative of the energy with respect to the change in the atom's position:
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Energies can be calculated using either molecular mechanics or quantum mechanics methods. Molecular mechanics energies are limited to applications that do not involve drastic changes in electronic structure such as bond making/breaking. Quantum mechanical energies can be used to study dynamic processes involving chemical changes. The latter technique is extremely novel, and of limited availability (CHARMM/GAMESS is an example of such a program).
Knowledge of the atomic forces and masses can then be used to solve for the positions of each atom along a series of extremely small time steps (on the order of femtoseconds = 10^-15 seconds). The resulting series of snapshots of structural changes over time is called a trajectory. The use of this method to compute trajectories can be more easily seen when Newton's equation is expressed in the following form:
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In practice, trajectories are not directly obtained from Newton's equation due to lack of an analytical solution. First, the atomic accelerations are computed from the forces and masses. The velocities are next calculated from the accelerations based on the following relationship:
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Lastly, the positions are calculated from the velocities:
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A trajectory between two states can be subdivided into a series of sub-states separated by a small time step, "delta t" (e.g. 1 femtosecond):
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The initial atomic positions at time "t" are used to predict the atomic positions at time "t + delta t". The positions at "t + delta t" are used to predict the positions at "t + 2*delta t", and so on.
The "leapfrog" method is a common numerical approach to calculating trajectories based on Newton's equation. The steps can be summarized as follows:
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The method derives its name from the fact that the velocity and position information successively alternate at 1/2 time step intervals.
Molecular dynamics has no defined point of termination other than the amount of time that can be practically covered. Unfortunately, the current picosecond order of magnitude limit is often not long enough to follow many kinds of state to state transformations, such as large conformational transitions in proteins.
Molecular dynamics calculations can be performed using AMBER, CHARMM, CHARMM/GAMESS, Discover, QUANTA/CHARMm, and SYBYL.