Searching for Hydrodynamic Orienting Effects in the Association of Tri-N-acetylglucosamine with Hen Egg-White Lysozyme

Using stopped-flow fluorometry, we determined rate constants for the formation of diffusional encounter complexes of tri-N-acetylglucosamine (NAG3) with hen egg-white lysozyme (kaWT) and its double mutant Asp48Asn/Lys116Gln (kaMT). We defined binding anisotropy, κ ≡ (kaWT – kaMT)/(kaWT + kaMT), and determined its ionic strength dependence. Our goal was to check if this ionic strength dependence provides information about the orienting hydrodynamic effects in the ligand-binding process. We also computed ionic strength dependence of the binding anisotropy from Brownian dynamics simulations using simple models of the lysozyme–NAG3 system. The results of our experiments indicate that in the case of lysozyme and NAG3 such hydrodynamic orienting effects are rather negligible. On the other hand, the results of our Brownian dynamics simulations prove that there exist molecular systems for which such orienting effects are substantial. However, the ionic strength dependence of the rate constants for the wild-type and modified systems do not exhibit any qualitative features that would allow us to conclude the presence of hydrodynamic orienting effects from stopped-flow experiments alone. Nevertheless, the results of our simulations suggest the presence of hydrodynamic orienting effects in the receptor–ligand association when the anisotropy of binding depends on the solvent viscosity.

Comments to data presented in Tables S1 to S6 At first glance, the differences between rate constants computed with hydrodynamic interactions included (Tables S1-S3) and neglected (Tables S4-S6), respectively, may seem much too large. Most of the previous studies reported a substantially smaller decrease in the association rate constants caused by the inclusion of HI between two spherical particles S1,S2 or a spherical target and dumbbell dimer. S3 In these studies, spherical elements used to model molecules were considered uniformly reactive over their surfaces, and a single distance criterion was used to define association reactions. On the other hand, Shushin S4 analyzed the effect of hydrodynamic interaction on diffusion-controlled reaction rate of molecules with highly anisotropic reactivity, modeled by small reactive hemispheres around the reactive centers on the surfaces of spherical molecules. He showed that the hydrodynamic interaction effect can lead to about 3-5 times and larger reduction of the rate. In the case of our simulations, we obtain such reductions as 50 (ionic strength 0) and 108 (ionic strength 500 mM) for the most restrictive reaction criterion, and as 10 (ionic strength 0) and 18 (ionic strength 500 mM) for the least restrictive reaction criterion. The increase in the ratio with going from the least to the most restrictive reaction criterion is in qualitative agreement with the results of Shushin. We believe that more complex molecular shapes and requirement to satisfy simultaneously four distance criteria instead of just one are responsible for the higher values in our case.
Finally, in a recent publication on encounter rates between xanthone and 2-naphtoic acid, S5 we also obtained substantial decrease of the association rate with hydrodynamic interactions included in comparison to hydrodynamic interactione neglected, i.e 10250/723=14.
In this case we deal with molecular models composed of several spherical elements, but the reaction criteria used a single distance. It is worthy to note that the rate constant obtained It may be also noted that that the decrease of the encounter rates with increased solvent viscosity is somewhat larger than predicted by the analytical Smoluchowsky equation for spherical particles, as according to this equation the association rate constant is proportional to the inverse of viscosity of the solvent. However, it should be also noted that no-HI simulation means that there is no hydrodynamic interactions between beads of the receptor model with the beads of the ligand model. The receptor is at rest in the center of the coordinate system, with its hydrodynamic radius of equivalent spere computed assuming hydrodynamic interactions between constituting beads. On the other hand, the ligand diffusion is simulated with hydrodynamic interactions between its constituting beads included.
Moreover, it is probably also important that four distances are used in definition of the reaction criteria instead of just one. Thus simple recalculation referring to association of single spheres may be misleading. .

Examples of the UHBD program inputs
Simulations with receptor-ligand hydrodynamic interactions included read mol1 file "./enzym-10bead-model.pdb" pdb end ! read in receptor read mol2 file "./ligand-3bead-model.pdb" pdb end ! read in ligand set charge radii file "./qrdata.dat" ! set charges and ra para par end ! using data in file