# Non-Conservative Potential for Charged Residues in Proteins

I am struggling with this question. I want to determine whether the potential of a protein is conservative or not if its dielectric constant depends on the local density of the protein. My model only considers a Lennard-Jones potential and Coulomb potential.

Considering two fixed charge residues separated by a distance $$r$$ and a neutral chain approaching them, density increases, and therefore the dielectric constant decreases. The force between the two charged residues is stronger, decreasing the potential energy of the system. In this model, no work should be done to approach the neutral molecule, as it does not feel Coulomb repulsion. Does this imply the potential is non-conservative?

Another argument I can think is that, as the dielectric constant depends on the position of all particles, a gradient of this potential and therefore a force, will exist when a neutral particle approaches the charged residues. This would make the potential conservative. Can someone help me with this issue?

Physics Asked on November 21, 2021

Dielectric constant is a phenomenological way of accounting for the rearrangement of charges in a medium, without diving into their microscopic description. In other words, one can do either one or the other: either using a dielectric constants OR describing the charges microscopically. If the charges are the protein residues, then they should be described as charges, by their respective potentialss, not as a dielectric constants.

However, it is worth noting that there is a lot more happening here than charge-charge interaction: protein residues are complex molecules that have their dipole moment (i.e. polarizability) and potentially higher order moments. This may justify inclusion of a dielectric constant. Moreover, protein is not in vacuum, but in a solvent, i.e. one should account for the dielectric constant/charges of the solvent, which may also vary.

There is likely no generic answer to this question - it depends on a specific model used for describing protein and its environment. As it is a rather mature field, there are books written about it...

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