BDflex: A method for efficient treatment of molecular flexibility in calculating protein-ligand binding rate constants from Brownian dynamics simulations

Abstract

A method developed by Northrup et al. [J. Chem. Phys. 80, 1517] for calculating proteinligand binding rate constants (k a ) from Brownian dynamics (BD) simulations has been widely used for rigid molecules. Application to flexible molecules is limited by the formidable computational cost to treat conformational fluctuations during the long BD simulations necessary for k a calculation. Here, we propose a new method called BDflex for k a calculation that circumvents this problem. The basic idea is to separate the whole space into an outer region and an inner region, and formulate k a as the product of k E andη d , which are obtained by separately solving exterior and interior problems. k E is the diffusion-controlled rate constant for the ligand in the outer region to reach the dividing surface between the outer and inner regions; in this exterior problem conformational fluctuations can be neglected.η d is the probability that the ligand, starting from the dividing surface, will react at the binding site rather than escape to infinity. The crucial step in reducing the determination ofη d to a problem confined to the inner region is a radiation boundary condition imposed on the dividing surface; the reactivity on this boundary is proportional to k E . By confining the ligand to the inner region and imposing the radiation boundary condition, we avoid multiple-crossing of the dividing surface before reaction at the binding site and hence dramatically cut down the total simulation time, making the treatment of conformational fluctuations affordable. BDflex is expected to have wide applications in problems where conformational fluctuations of the molecules are crucial for productive ligand binding, such as in cases where transient widening of a bottleneck allows the ligand to access the binding pocket, or the binding site is properly formed only after ligand entrance induces the closure of a lid.

Rate Constants and Mechanisms of Protein–Ligand Binding

“…A different method for treating protein flexibility was developed by Greives & Zhou (29) (Figure 4 d ). The aim is to break the full problem of k b calculation into two separate problems: an exterior problem, where the ligand is confined outside a dividing surface S * (to be signified by a superscript asterisk) and the protein is kept rigid; and an interior problem, where the ligand is confined within the dividing surface and the protein is fully flexible.…”

“…Transition probabilities in green and red are obtained from simulations where the protein is treated as rigid and flexible, respectively. ( d ) The BDflex algorithm of Greives & Zhou (29). The dividing surface S * is absorbing for the exterior problem and partially absorbing for the interior problem.…”

Modeling protein association mechanisms and kinetics

“…Recently a new algorithm was developed [29], based on breaking the association process into two problems confined to either the outer region, where the molecules can be modeled as rigid, or the inner region, which covers the binding site but is small enough such that modeling of internal motions can be affordable. The algorithm has been tested on the binding of small molecules to proteins [Qin, Greives, & Zhou, to be published].…”

Theoretical frameworks for multiscale modeling and simulation

“…From a technical standpoint, artificial coupling to a low-resolution model can facilitate conformational sampling of a high-resolution model [55,57,58]. In other cases, separation of the simulations in different regions [59] or of different components [61] can be designed for efficient calculation of biophysical properties. While the partitioning into high- and low-resolution parts is fixed in most current studies, switchable or adaptive partitioning is being developed [84–86].…”

“…4b upper panels). Interestingly, for calculating the binding rate constant one can even completely separate the simulations in the outer and inner regions, according to a method called BDflex [59]. Through simulations in which the ligand is confined to the outer region while the whole protein is treated as rigid and the boundary between the outer and inner regions as absorbing, one obtains the rate constant for absorption on the boundary (Fig.…”