Round 2, Reviewer 5 comments

[labarba]

While the manuscript reports an interesting development, it lacks two important components, listed below:

• [x] 1) Acknowledging previous works of electrostatic modeling of electrostatic potential and electric field of viruses. See for example Fig. 3 in J. Comput. Chem. 2019, 40, 2502–2508.

• [x] ) Better assessment of the accuracy of the delivered electrostatic energy by comparing with analytical solution. For example, see several cases Fig. 1,2,3 in BMC Biophysics 2012, 5:9.

• [x] Furthermore, the abstract and the entire paper should make clear that number of atoms and the dimensions of the molecule/virus are not the crucial factor resulting in computational complexity, rather the mesh size determines the computational efforts. In addition, it should be clearly stated that the report is for linearized PB, but not for non-linear PB.

[labarba]

The Reviewer suggested to use a better analytical solution, as proposed in Gilson1985. Looking through the paper, it's unclear its difference with Kirkwood’s method.

[cdcooper84]

1) Acknowledging previous works of electrostatic modeling of electrostatic potential and electric field of viruses. See for example Fig. 3 in J. Comput. Chem. 2019, 40, 2502–2508.

The reviewer is pointing to an important reference that performs similar calculations to us, further supporting the notion that Poisson-Boltzmann calculations are useful, even at large scales. We have added this reference in the Discussion section of the revised manuscript.

[cdcooper84]

2) Better assessment of the accuracy of the delivered electrostatic energy by comparing with analytical solution. For example, see several cases Fig. 1,2,3 in BMC Biophysics 2012, 5:9.

The test cases in the reference mentioned by the reviewer consist of one or two spherical cavities interacting with a planar surface. Even though it is true that they are harder tests for accuracy, we believe it does not add to the manuscript. In the context of BEM, these systems are modeled by larger matrices that consist of combinations of the same boundary operators that we are already using. Our results already show convergence down to 1e-4 accuracy for a spherical cavity with an off-centered charge, which we believe is enough evidence of correctness.

[cdcooper84]

Furthermore, the abstract and the entire paper should make clear that number of atoms and the dimensions of the molecule/virus are not the crucial factor resulting in computational complexity, rather the mesh size determines the computational efforts. In addition, it should be clearly stated that the report is for linearized PB, but not for non-linear PB

The reviewer is correct to point out that the crucial factor is the number of elements, rather than the number of atoms, however, these two factors are related: a larger number of atoms usually yields a larger number of boundary elements. Perhaps this confusion arises from Table 8, where in the comparison with APBS we are not reporting the mesh size. We believe including this information does not add to the main point of the table, which is to compare with a different code. This because the mesh sizes for Bempp and APBS are not comparable, as one is a surface mesh and the other a volumetric one. Moreover, comparing mesh sizes between Bempp runs of that Table is also irrelevant.

As the reviewer says, a BEM approach can only solve the linear Poisson Boltzmann equation. We state that in the first paragraph of the original manuscript. To make it more evident, we added the word "linear" to the abstract.