  The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial
differential equation used in applications such as protein interactions and
biophysical chemistry (among many others). It describes the nonlinear
electrostatic potential of charged bodies submerged in an ionic solution. The
kinetic presence of the solvent molecules introduces randomness to the shape of
a protein, and thus a more accurate model that incorporates these random
perturbations of the domain is analyzed to compute the statistics of quantities
of interest of the solution. When the parameterization of the random
perturbations is high-dimensional, this calculation is intractable as it is
subject to the curse of dimensionality. However, if the solution of the NPBE
varies analytically with respect to the random parameters, the problem becomes
amenable to techniques such as sparse grids and deep neural networks. In this
paper, we show analyticity of the solution of the NPBE with respect to analytic
perturbations of the domain by using the analytic implicit function theorem and
the domain mapping method. Previous works have shown analyticity of solutions
to linear elliptic equations but not for nonlinear problems. We further show
how to derive \emph{a priori} bounds on the size of the region of analyticity.
This method is applied to the trypsin molecule to demonstrate that the
convergence rates of the quantity of interest are consistent with the
analyticity result. Furthermore, the approach developed here is sufficiently
general enough to be applied to other nonlinear problems in uncertainty
quantification.
