In this second remark, we present a revised correction to Algorithm 1010 [A. Orellana and C. De Michele 2020] with respect to the one already proposed in the remark on Algorithm 1010 [C. De Michele 2022].

We present a correction and an improvement to Algorithm 1010 [A. Orellana and C. De Michele 2020].

Aiming to provide a very accurate, efficient, and robust quartic equation solver for physical applications, we have proposed an algorithm that builds on the previous works of P. Strobach and S. L. Shmakov. It is based on the decomposition of the quartic polynomial into two quadratics, whose coefficients are first accurately estimated by handling carefully numerical errors and afterward refined through the use of the Newton-Raphson method. Our algorithm is very accurate in comparison with other state-of-the-art solvers that can be found in the literature, but (most importantly) it turns out to be very efficient according to our timing tests. A crucial issue for us is the robustness of the algorithm, i.e., its ability to cope with the detrimental effect of round-off errors, no matter what set of quartic coefficients is provided in a practical application. In this respect, we extensively tested our algorithm in comparison to other quartic equation solvers both by considering specific extreme cases and by carrying out a statistical analysis over a very large set of quartics. Our algorithm has also been heavily tested in a physical application, i.e., simulations of hard cylinders, where it proved its absolute reliability as well as its efficiency.

In this paper we introduce a fully flexible coarse-grained model of immunoglobulin G (IgG) antibodies parametrized directly on cryo-EM data and simulate the binding dynamics of many IgGs to antigens adsorbed on a surface at increasing densities. Moreover, we work out a theoretical model that allows to explain all the features observed in the simulations. Our combined computational and theoretical framework is in excellent agreement with surface-plasmon resonance data and allows us to establish a number of important results. (i) Internal flexibility is key to maximize bivalent binding, flexible IgGs being able to explore the surface with their second arm in search for an available hapten. This is made clear by the strongly reduced ability to bind with both arms displayed by artificial IgGs designed to rigidly keep a prescribed shape. (ii) The large size of IgGs is instrumental to keep neighboring molecules at a certain distance (surface repulsion), which essentially makes antigens within reach of the second Fab always unoccupied on average. (iii) One needs to account independently for the thermodynamic and geometric factors that regulate the binding equilibrium. The key geometrical parameters, besides excluded-volume repulsion, describe the screening of free haptens by neighboring bound antibodies. We prove that the thermodynamic parameters govern the low-antigen-concentration regime, while the surface screening and repulsion only affect the binding at high hapten densities. Importantly, we prove that screening effects are concealed in relative measures, such as the fraction of bivalently bound antibodies. Overall, our model provides a valuable, accurate theoretical paradigm beyond existing frameworks to interpret experimental profiles of antibodies binding to multi-valent surfaces of different sorts in many contexts.