where V is the potential energy of the system and the vector, x represents the state variables, such as positions and orientations of the solute molecules (solvent molecules are not typically explicitly included). The matrix, D (generalized diffusivity), represents the dynamic effects of the solvent, i.e., the hydrodynamic damping caused by motion through a fluid and the addition of thermal energy. The vector, W of uncorrelated random numbers following a unit Gaussian distribution gives stochastic effects. It is worth noting that the presence of the temperature, along with Boltzmann's constant, affects the thermal fluctuations. Like the MD models, there exists a force term (the gradient of the potential energy) in the BD model with components that include intermolecular electrostatics and also the averaged effects of the solvent, such as hydrophobic and solvent dielectric effects. As an example of the latter, the Browndye2 engine uses the software package APBS (Adaptive Poisson-Boltzmann Solver) to compute the electric field around a macromolecule, given the bulk proper- Computation of association rate using the Luty-McCammon-Zhou method ties of the solvent and dissolved ions . Because BD simulations encompass several approximations from MD simulations, the number of variables is significantly reduced, and the time step dt can be in orders of magnitude larger than that used in MD simulations. In the case of the receptorligand complexes presented in this tutorial, both the receptor and the ligand are treated as rigid bodies at the BD level. Given the simplifications and speedup of using BD simulations, it is possible to generate elongated trajectories of the ligand in the space around the receptor to estimate the association rate. Browndye2 uses the Luty-McCammon-Zhou algorithm (Figure ) (a variation of the earlier Northrup-Allison-McCammon algorithm ), which generates multiple trajectories, each of which ends either in an encounter with the receptor, or an escape. The ligand is started on a sphere surrounding the receptor and moves until it either reaches a final encounter with the receptor or reaches another sphere, larger and concentric with the first. If it reaches this second sphere, it either escapes, ending the trajectory, or it is placed back on the first sphere to continue the trajectory. The probability of the escape versus continuing from the inner sphere depends on several factors, such as the diameters of the spheres, the total charge on each system (ligand and the receptor), and the ionic strength of the solvent. Ultimately, the second-order association rate constant is computed from the proportion of encounters to escapes, and therefore a large number of trajectories are required to estimate that probability precisely. When used with SEEKR2, the encounter complex is defined as the ligand reaching the outermost milestone that encompasses the MD region. The milestoning calculations, with their parameters computed from the MD simulations, convert this initial rate constant into the final rate constant estimate, which accounts for the atomic details in the binding site, as described in section 3.2 and 3.3.