  A statistical theory is presented of the magnesium ion interacting with
lysozyme under conditions where the latter is positively charged. Temporarily
assuming magnesium is not noncovalently bound to the protein, I solve the
nonlinear Poisson-Boltzmann equation accurately and uniformly in a perturbative
fashion. The resulting expression for the effective charge, which is larger
than nominal owing to overshooting, is subtle and cannot be asymptotically
expanded at high ionic strengths that are practical. An adhesive potential
taken from earlier work together with the assumption of possibly bound
magnesium is then fitted to be in accord with measurements of the second virial
coefficient by Tessier et al. The resulting numbers of bound magnesium ions as
a function of MgBr2 concentration are entirely reasonable compared with
densitometry measurements.
