# Explicit differential equation based model of protein folding

I am an applied mathematician interested in the dynamics of potential systems - i.e., systems with multiple unique energy minima. One of the best examples of such systems are protein folding potentials. From sources like This one I know that the potential, $$V(x)$$ exists and is a function of the positions of each residue in space and their interactions with each other. Since such a function exists, there should (mathematically) be a set of ordinary differential equations that capture the dynamics described by the potential.

However, although I have seen definitions of $V(x)$ in the literature, for the life of me I can't figure out exactly what the set of differential equations associated with this system is - i.e., the system is usually written out in a generalized form that applies to all proteins, but the exact information required to build such a potential for one such protein is unclear.

Does anyone have an example of of a particular $V(x)$ (for a particular, short, set of amino acids, say) where all the parameters are known and the system can be solved numerically as a set of ordinary differential equations?

What you are looking for is a force field: http://en.wikipedia.org/wiki/Force_field_(chemistry)

Standard force fields for proteins include CHARRM and AMBER. These have relatively simple, well-defined expressions for defining the various types of interactions between bonded and non-bonded atoms. Parameters are available for all standard atom types, and are fit to experimental and/or electronic structure calculations. These force fields are considered "classical" in that they do not explicitly treat electronic interactions, but rather try to capture these interactions in a coarse-grained way. Typically, there are expressions for describing the following interactions:

• bonded interactions (bond stretching)
• angle interactions (bond bending)
• torsion interactions (dihedral bending)
• 12-6 Lennard-Jones interactions (non-bonded, dispersion forces)
• Short-range charge-charge interactions (Coulombic interactions between charged or partially charged atoms)
• Long-range charge-charge interactions (see Ewald summation method for more details)

Note that in these classical models, partial charges are generally statically assigned to each atom. The functional form of these different terms varies depending on the force field and sometimes implementation. These interactions are typically pairwise additive, so to get the total potential energy of the system you simply add together all the various interactions between the appropriate pairs of atoms.

There are certainly more rigorous models that attempt to capture other electronic structure affects, however for large molecules such as proteins, relatively simple force fields are generally applied.

Please note that finding the global minimum in potential energy of a protein is not a trivial problem. People have been working on this for years, it's extremely difficult to develop algorithms that do not get trapped in local minima.

• I have read a bit about the really bumpy potentials associated with protein folding. Are there any molecules for which the energy potential smoother, so that maybe there are fewer than 20 local minima? On average, do you know how many variables these models have? How many minima? – Danny W. Oct 2 '14 at 20:24
• This list also doesn't give me a great idea of which potential to choose - there are dozens! Even if I choose one, how do I get the (apparently many) parameters for it? Thank you for your help, by the way. It is really appreciated. – Danny W. Oct 2 '14 at 20:26
• Even a relatively small protein will have a huge number of local minima (thousands, millions, billions, depending on the size). You could start with something like a single amino acid or even an alkane chain to begin testing. – commanderdata Oct 2 '14 at 22:46
• Check out the AMBER page for specific functional forms: en.wikipedia.org/wiki/AMBER And you don't have to actually select which potential to use, that's determined by the force field you select. The hard work for you will be the programming effort and looking up the exact parameters to use for different atom types. That's why most people use free packages such as LAMMPS, Gromacs, NAMD, DL_POLY, etc. – commanderdata Oct 2 '14 at 22:49
• Ultimately the potential is a function of the positions (actually the relative positions) of all atoms in the molecule. – commanderdata Oct 2 '14 at 22:50