Neural network Poisson–Boltzmann electrostatics for biomolecular interactions

Abstract

<p>We develop a neural network approach to solving the dielectric-boundary Poisson–Boltzmann (PB) equation (PBE) and estimating the electrostatic free energy of charged molecules in aqueous solvent. Such equation is the Euler–Lagrange equation of the classical PB electrostatic free-energy functional with the presence of a dielectric boundary. We construct a penalized dielectric-boundary PB functional to remove the constraint imposed by the boundary condition for the boundary of the computational region and show that such penalized functionals converge to the classical PB functional. We represent electrostatic potentials by fully connected feed-forward neural network functions with sigmoidal activation, use the penalized functional and Monte Carlo integration method to construct a neural network loss function, and minimize it by a stochastic gradient-descent (SGD) method. Numerical results are presented to show the convergence of the neural network simulations with varying learning rates, batch size, and network architecture. Moreover, the relation between the boundedness of network weights and learning rates in the loss optimization is explored. The neural network PB method is applied to the calculation of electrostatic free energy of the solvation of single ions and protein BphC, demonstrating that the new approach can handle both simple and complex geometries and predict qualitatively well the electrostatic energy. In particular, we find that using the trained neural network weights from one simulation as the initial weights for simulations with different settings significantly increases the simulation efficiency. Such transferability of network weights provides an advantage of our neural network PB approach to complex biomolecular systems.<br></p>