On the nature of screening in charge regulated macroion solutions

We present a derivation of the screening length for a solution containing a charge regulated macroion, e.g. protein, with its counterions. We show that it can be obtained directly from the second derivatives of the total free energy by taking recourse to the uncertainty relation" of the Legendre transform which connects the Hessians or the local curvatures of the free energy as a function of density and its Legendre transform, i.e., osmotic pressure, as a function of chemical potentials. Based on the Fowler-Guggenheim-Frumkin model of charge regulation we then analyze the screening resonance" and the ``overscreening" of the screening properties of the charge-regulated macroion solution.