Treating the ligand explicitly in addition to the protein clearly is more demanding of computational resources, but offers the advantage of greater realism. In particular, the interactions between these species can be treated in some detail. This may be important for the separated species, e.g., to allow for higher-order multipole components of electrostatic coupling, possibly varying with conformational fluctuations of the ligand. But, it is likely to be particularly important in the encounter complex, where steric interactions between the flexible species may strongly perturb the gating dynamics and the entry of the ligand to the reactive site. These might be colorfully described as "foot in door" effects. This level of treatment has been attempted in a few Brownian dynamics simulations, but is likely to become more common.
4.1. Triosephosphate Isomerase
Wade et al. conducted Brownian dynamics simulations of a model substrate interacting with a flexible, gated enzyme that is known to exhibit diffusion-controlled kinetics [5]. The enzyme is chicken muscle triosephosphate isomerase, a homodimeric structure with flexible polypeptide loops gating the two active sites. In the computer model, the enzyme subunits were taken to be rigid except for the flexible loops, which were represented as chains of 17 spherical residues, of which the central 11 residues were allowed to move. The substrate, glyceraldehyde 3-phosphate (GAP) was represented as a pair of touching spheres of radius 0.2 nm with central charges of -0.3 e and -1.7 e to represent the glyceraldehyde and phosphate moieties, respectively. The charges account for the monopole and dipole moments of the doubly-charged form of GAP, and hydrodynamic interactions between the two spheres were included to give the correct diffusion constant for the substrate. Electrostatic and steric interactions between GAP and the rigid and flexible portions of the enzyme were included by use of a coarse-grained model, and an ionic strength of 0.1 M was assumed.
The Brownian dynamics simulations were performed with the Ermak-McCammon algorithm [22]. Analysis of the trajectories showed that both translational and rotational electrostatic steering contributed significantly to the kinetics of successful docking of the substrate to the active site. More importantly from the perspective of gating, the opening and closing of the flexible loops of the unliganded enzyme is seen to be on a timescale of about 1 ns, and the diffusional relaxation time of the substrate is on a timescale of about 16 ns. Thus, the system is in the fast gating regime, and the estimated rate constant is close to that observed experimentally. The conclusions therefore support those of an earlier "protein centric" study [23], but the explicit simulation shows clearly how a substrate may approach a closed active site and remain nearby long enough for the site to open so that reaction can ensue. It should be noted that some more recent studies (reviewed in [24]) suggest slower gating motions and that simulations of other peptide loop motions are somewhat slower (see below). It is possible that approximations in this pioneering study yielded somewhat rapid loop motions, though additional study is clearly warranted. In any case, since all studies point to the gate being open at least half of the time in the unliganded system, there would still only be a small departure from the always-open rate, however, even in the limit of slow gating.
4.2. HIV-1 Protease
HIV-1 Protease, an essential enzyme for HIV replication and an important target of antiviral drugs, is a small homodimeric protein with two polypeptide "flaps" covering the active site at the dimer interface. Chang et al. used coarse-grained models of the enzyme and a nonapeptide substrate to study the gating of substrate binding by the opening and closing of the flaps [25]. As in the study of triosephosphate isomerase by Wade et al. described above [5], Brownian dynamics was used to simulate the internal and overall diffusion the flexible molecules [22]. Again, the coarse-grained model comprised spherical interaction centers for the amino acid residues, but the parameters for this model were developed by reference to a large number of experimental structures of the protease [26, 27]. With these methods, it was possible to run many encounter trajectories on the microsecond timescale.
Analysis of the results showed that the gating timescale was of the order of 60 ns, while the timescale for diffusional escape of the peptide substrate was of the order of 40 ns [25]. Thus, the rate of binding is somewhat reduced by the gating. Interestingly, the fractions of gate-open time were found to be about 14% and 2% for the wild-type enzyme and a drug resistant mutant (G48V/V82A/I84V/L90M), raising the possibility that slowed binding of large antiviral drugs could contribute to resistance in the mutant [25].
It is worth noting that "protein centric" studies of gating in HIV-1 protease have also been reported. These have shown that the experimental order of binding rate constants for different antiviral drugs can be rationalized in the slow-gating picture [28] and that the effects of crowding by other macromolecules alter the gating dynamics and, presumably, the kinetics of ligand binding [29, 30].