Constant-pH MD is a simulation methodology specially formulated for the
treatment of variable protonation states.
This is to be contrasted with conventional force-field based MD simulations,
which generally treat protonation states by assuming they are fixed.
Consider, for example, a protein with two titratable residues which may both be
either protonated or deprotonated (Figure 12);
the system has four possible protonation states.
In the conventional route, the user must enumerate these possibilities,
construct distinct topologies, and then simulate the cases individually.
The simulations for each state must then be connected by either asserting
knowledge about the system (*e.g.*, by assuming that only certain
states are of biological importance) or by performing additional simulations
to probe transitions between states directly (*e.g.*, by performing
free energy calculations).
In a constant-pH MD simulation, knowledge of the transformations is not
assumed and is instead actively explored by interconverting between the
various protonation states.
This is especially useful when the number of protonation states is extremely
large and/or prior information on the importance of particular states is
not available.

In formal terms, conventional MD samples from a canonical ensemble, whereas constant-pH MD samples from a semi-grand canonical ensemble. The new partition function,

is essentially a weighted summation of canonical partition functions, , each of which are defined by an occupancy vector, . The elements of are either one or zero depending on whether a given protonation site is or is not occupied, respectively. For a vector of length , the set of all protonation states, , has at most members. In order to sample from the corresponding semi-grand canonical distribution function, a simulation must explore

Although a constant-pH MD system may contain any number of titratable protons,
the base transformation is always the movement of *one* proton from a
molecule into a bath of non-interacting protons ``in solution.''
For a generic chemical species A, this corresponds to the usual deprotonation
reaction definition, except with fixed pH:

In the language of statistical mechanics the species HA and A refer to all terms in Eq. (80) which do and do not, respectively, contain the specific proton in question (

pH |

and then recast as a statistical mechanical analog of the Henderson-Hasselbalch equation by recognizing that is just the ratio of deprotonated / protonated fractions of species A. The

In practice, can be calculated from a simulation by simply counting the fraction of time spent in state HA (

In most experimental contexts, a different form of Eq. (81) is used which is often referred to as a ``generalized'' Hill equation. This corresponds to a specific choice of pH dependence such that

p |

The constant is then known as the Hill coefficient and the so-called apparent