Calculations of the binding affinities of protein-protein complexes with the fast multipole method

Abstract

In this paper, we used a coarse-grained model at the residue level to calculate the binding free energies of three protein-protein complexes. General formulations to calculate the electrostatic binding free energy and the van der Waals free energy are presented by solving linearized Poisson-Boltzmann equations using the boundary element method in combination with the fast multipole method. The residue level model with the fast multipole method allows us to efficiently investigate how the mutations on the active site of the proteinprotein interface affect the changes in binding affinities of protein complexes. Good correlations between the calculated results and the experimental ones indicate that our model can capture the dominant contributions to the protein-protein interactions. At the same time, additional effects on protein binding due to atomic details are also discussed in the context of the limitations of such a coarse-grained model.

Calculations of the second virial coefficients of protein solutions with an extended fast multipole method

“…In combination with our previous work on the binding affinity calculations [29] it is reasonable to expect that our residue level model can be used as a reliable model to describe protein-protein interactions in solutions. Naturally there are several immediate ways to improve the model, such as bettering the nuclear polarizability model of amino acids and improving treatment of the electrolyte solution modeling beyond Debye-Hückel theory.…”

“…the protein may have a shifted pK a as the case of P 1 Glu and P 1 Asp mutations in the BPTI-trypsin complexes [29] and the PROPKA 2.0 is an accurate program for the pK a prediction [40]. The dependence of B 2 of BPTI molecules on the concentration of the sodium chloride solution and the comparison with the experimental B 2 data will be described in Sec.…”

“…This iteration is called the "outer" iteration to separate the term with the "inner" iteration which is used to solve the one-body linear system with an iterative solver, such as GMRES. The "outer" iteration can reduce the size of system from O (2N × 2N ) to O(N × N ) and the "inner" iteration can be accelerated by introducing the Fast Multipole Method (FMM) [29]. Figure 3 shows how the double tree structures are defined to cover one body in one tree and the interactions between two separated bodies are allowed in the FMM algorithm to calculate matrix-vector products in Eqs.…”

“…[22]. Other properties listed in Table I from Kim et al [29] are based on the calculated results from Millefiori et al [23] An accurate parametrization of the dielectric function ε(iω) of water based on the experimental data is taken from Parsegian's work [35].…”

“…The Boundary Element Method (BEM) in combination with the Fast Multipole Method [26,27] is implemented to circumvent the extensive memory problem similar to the recent work by Lu et al [28]. The validity of our model was already tested by binding affinity calculations of several protein complexes [29]. Direct comparisons between our calculations of B 2 and experimental measurements under various solution conditions were made and reasonable agreements from these comparisons provide further concrete evidence that our model can be used as a universal model for studies of nonspecific protein-protein interactions in aqueous solutions.…”