Molecular simulations of proteins: how good an approximation is electrostatics?

Question

This question has to do with molecular dynamics simulation of molecules, especially proteins, using software such as GROMACS.

This type of software uses empirical force fields (eg AMBER) and some constraints on covalent bond length and geometry; the simulation proceeds in steps of typically a few femtoseconds and can cover tens to hundreds of nanoseconds at a time.

The forces considered here are pure electrostatics; there is no magnetic or coupled electric-magnetic component. Of course since there are electric charges in the simulation (eg polar groups, molecular dipoles, ...), and they move, and moving charges produce magnetic fields which exert force on other moving charges, the simulation is going to miss some things.

The question is: how good of an approximation is electrostatics in this context?

This is about developing some intuition and good order of magnitude estimates, not about any kind of precision. I think the pieces of this are: How large are the charges? and How fast do they move? (due to thermal motion, or when transitioning between stable conformations) and consequently: How strong is the resulting magnetic field, and what is the ratio of electrostatic force between charges to the magnetic force between moving charges?

As a biologist, the kinds of proteins I would be interested in especially as model systems are voltage-sensing domains (VSDs) of transmembrane proteins, and tubulin/microtubules. Both of these have large conformational changes between different states, and have mobile subunits with large dipoles. Looking up VSDs, it seems the distance of movement between states can be 10-40 A (1 and 2), the macrodipoles of subunits of the protein can be 20-25 Debye (3), and the motion time scale is nanoseconds to milliseconds (4 and 5).

Answer 1

The magnetic field of a moving charge can be obtained from the expression of the Liénard–Wiechert potential and it can be expressed as a function of the corresponding electric field as $$ {\bf B}({\bf r},t) = \frac{{\bf n}(t_r)}{c} \times {\bf E}({\bf r},t) $$ where $t_r = t- \frac{\left|{\bf r}-{\bf r^{\prime}}\right|}{c}$ is the retarded time, ${\bf n}$ is the unit vector pointing in the direction from the source and $c$ the speed of light.

It is clear from the above formula, that the Lorentz force on a charge moving with speed $v$ cannot be larger than $\frac{v}{c}E$, therefore the ratio between magnetic and electric force cannot be larger that $\frac{v}{c}$.

In a thermalized system $v\sim \sqrt{k_BT/m}$, therefore $$ \frac{\text{magnetic force}}{\text{electric force}}\sim\frac{v}{c}\sim \sqrt{\frac{k_BT}{mc^2}}\ll 1 $$ Even in the case of a proton, the ratio is of the order $10^{-5}$, at room temperature. Taking into account that effective force fields used in computer simulations have uncertainties orders of magnitude larger, and that numerical integration algorithms introduce in turn additional, larger inaccuracies, I do not see any reason to be worried about neglecting magnetic forces.