Particle-surface interactions govern a myriad of interface phenomena, that span from technological applications to naturally occurring biological processes. In the present work, particle-collector DLVO interactions are computed with the grid-surface integration (GSI) technique, previously applied to the computation of particle colloidal interactions with anionic surfaces patterned with O(10 nm) cationic patches. The applicability of the GSI technique is extended to account for interactions with collectors covered with topographical and chemical nanoscale heterogeneity. Surface roughness is shown to have a significant role in the decrease of the energy barriers, in accordance with experimental deposition rates that are higher than those predicted by the DLVO theory for smooth surfaces. An energy- and force-averaging technique is presented as a reformulation of the GSI technique, to compute the mean particle interactions with random heterogeneous collectors. A statistical model based on the averaging technique is also developed, to predict the variance of the interactions and the particle adhesion thresholds. An excellent agreement is shown between the models' predictions and results obtained from GSI calculations for large number of random heterogeneous collectors. Brownian motion effects for particle-collector systems governed by nanoscale heterogeneity are analyzed by introducing stochastic Brownian displacements in particle trajectory equations. It is shown that for the systems under consideration and particle sizes usually used in experiments, it is reasonable to neglect the effects of Brownian motion entirely. Computation of appropriately defined P ́eclet numbers that quantify the relative importance of shear, colloidal and Brownian forces validate that conclusion. An algorithm for the discretization of spherical surfaces into small equal-area elements is implemented in conjunction with the GSI technique and mobility matrix calculations of particle velocities, to obtain interactions and dynamic behaviors of patchy particles in the vicinity of uniform flat collectors. The patchy particle and patchy collector systems are compared in detail, through the computation of statistical measures that include adhesion probabilities and maximum residence times per patch. The lessened tendency of the patchy particle to adhere on the uniform collector is attributed to a larger maximum residence time per patch, which precludes interactions with multiple surface nano-features at a given simulated time. Also briefly described are directions for future work, that involve the modeling of two heterogeneous surfaces, and of surfaces covered with many types of heterogeneity, such as patches, pillars and spring-like structures that resemble polymer brushes or cellular receptors.