We have briefly reviewed the variety of interactions which are important in protein interactions and seen suitable simple mathematical forms for their representation. These are drawn together to form a potential energy function:

This function can be used to calculate a value for the potential energy (PEF(R)) for any conformation of a given protein - defined by the (normally Cartesian) coordinate vector R. A number of important points can be made:

• We called the function a "potential" energy function as it does not contain contributions made to the total energy made by the motions of the atoms involved. It is possible to calculate these using molecular dynamics methods.
• The function aims to give reasonable values for the difference in "microstate" energies between two different conformations. The absolute value for the energy given does not mean anything (certainly NOT the free energy of formation). Only differences have meaning. The above equation also does not allow the examination of any process which involves the change in chemical bonding, e.g., one cannot simulate chemical reactions in an enzyme active site with it.
• This kind of function is normally of little use in estimating whether a protein adopts a particular fold - much more useful is the approach set out by Sippl which uses an empirically based approach to identify mis-folds. (See Sippl, M.J. (1990) Calculation of conformational ensembles from potentials of mean force - an approach to the knowledge-based prediction of local structures in globular-proteins, J. Mol. Biol. 213:859-883).
• To be able to calculate the potential energy of a protein using the above equation involves a large number of parameters (equilibrium bond lengths b_eq, bond stretching constants K_b ...). The process of finding these is arduous and in there are only around four potential energy functions in common usage for proteins (CHARMm, AMBER, GROMOS and ECEPP).
• Although results obtained with current potential energy functions are only approximate they have one great advantage - they are computationally cheap. This allows the introduction of realistic representation of environment - such as having large numbers of explicitly modelled water molecules surrounding a protein. It also allows the calculation of the potential energy for many different conformations of the same molecule. This facilitates the use of techniques such as molecular dynamics which allows the thermal motions of a system to be explored. This can be contrasted with quantum chemical methods which even for small systems are so expensive that only a limited number of calculations can be made but produce very accurate energies.

This section provides a very brief introduction into what the uses of potential energy functions are in protein studies.

Energy minimization

Very often these methods are used if a distorted structure is produced - e.g. a homology based model. Energy minimization can then relieve short interatomic distances while maintaining important structural features.

Energy minimization can be used to help to solve experimental structures:

• In X-ray crystallography measure a set of intensities I(h,k,l) for a large number of reflections (h,k & l are Miller indices listing the reciprocal lattice points of the crystal). These are proportional to F_obs(h,k,l)² the observed structure functions. Once there is an approximate idea of the structure (an initial model) the model's electron density can be used to calculate expected structure factors F_calc(h,k,l). The structure can be refined by minimizing:




The program XPLOR is commonly used to do this.

• In NMR (nuclear magnetic resonance) experiments give approximate distances between hydrogen atoms (NOE's) and some dihedral angles. To obtain a 3D structure a similar process is performed in which the objective function is the sum of PEF(R) and restraint terms for the distances and dihedral angles. Molecular dynamics simulated annealing is used to optimize the function to obtain a set of 3D structures consistent with the experimental data.

Molecular Dynamics

is solved to find how the position for each atom of the system x_i varies with t. To find the forces on each atom (F_i) the derivative vector (or gradient) of equation 1 is calculated. Factors such as the temperature and pressure of the system can be included in the treatment.

Molecular dynamics simulation procedures are very popular in the protein field. They have the advantage that they can treat systems where motion is essentially diffusive in character - important because of the role of water in protein structure. The procedures can be used to calculate "ensemble average" properties - recent advances have included the ability to calculate free energy differences between (slightly) different ligands or conformations of a protein. A disadvantage of conventional molecular dynamics procedures is that they can only tackle motions with a relatively short time scale - one nanosecond is the approximate upper limit with current computers.

Other methods

Bibliography

S.J. Weiner, P.A. Kollman, D.A. Case, U.C. Singh, G. Alagona, S. Profeta Jr. and P. Weiner. "A new force field for the molecular mechanical simulation of nucleic acids and proteins". J. Am. Chem. Soc. , 106:765-784 (1984).

The original paper describing the AMBER potential energy function. An interesting if slightly heavy going paper describing the process of developing an energy function, source of parameters etc.. For further details of the AMBER program and energy function then why not look at the AMBER home page at the University of Calfornia, San Francisco.

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