This review utilizes an optimized Rouse-Zimm discrete hydrodynamic model and the preaveraged Oseen tensor, which accurately consider hydrodynamic interactions to study model dendrimers. We report the analytical theories that have been previously developed for the creation of generalized analytical models for dendrimers. These generalized theories were used to assess the conformational and dynamical behavior of the dendrimers. By including stiffness in the bonds, the neglect of excluded volume interactions may be somewhat offset. This is true at least in the case of short spacers. While the topological limitations on the directions and orientations of the individual bond vectors in dendrimers implement semiflexibility, the intensity of these contacts was determined by the potential geometric orientations of the bonds, and later on the excluded volume interactions in dendrimers, which were described in terms of the effective co-volume between nearest non-bonded monomers and modeled using the delta function pseudopotential. With the aid of the models developed, the authors condensed various conformational and dynamic properties of dendrimers that depend on their degree of semiflexibility and the strength of the excluded volume. These analyses came to the conclusion that the flexible dendrimer in one limit and the earlier described freely rotating model of dendrimers in the other constitute a highly generalized way of capturing a wide range of conformations in the developed mathematical model in dendrimers.

The effects of macromolecular crowding on the thermodynamic properties of test proteins are determined by the latter\'s transfer free energies from a dilute solution to a crowded solution. The transfer free energies in turn are determined by effective protein-crowder interactions. When these interactions are modeled at the all-atom level, the transfer free energies may defy simple predictions. Here we investigated the dependence of the transfer free energy (Δμ) on crowder concentration. We represented both the test protein and the crowder proteins atomistically, and used a general interaction potential consisting of hard-core repulsion, non-polar attraction, and solvent-screened electrostatic terms. The chemical potential was rigorously calculated by FMAP (Qin and Zhou, 2014), which entails expressing the protein-crowder interaction terms as correlation functions and evaluating them via fast Fourier transform (FFT). To high accuracy, the transfer free energy can be decomposed into an excluded-volume component (Δμe-v), arising from the hard-core repulsion, and a soft-attraction component (Δμs-a), arising from non-polar and electrostatic interactions. The decomposition provides physical insight into crowding effects, in particular why such effects are very modest on protein folding stability. Further decomposition of Δμs-a into non-polar and electrostatic components does not work, because these two types of interactions are highly correlated in contributing to Δμs-a. We found that Δμe-v fits well to the generalized fundamental measure theory (Qin and Zhou, 2010), which accounts for atomic details of the test protein but approximates the crowder proteins as spherical particles. Most interestingly, Δμs-a has a nearly linear dependence on crowder concentration. The latter result can be understood within a perturbed virial expansion of Δμ (in powers of crowder concentration), with Δμe-v as reference. Whereas the second virial coefficient deviates strongly from that of the reference system, higher virial coefficients are close to their reference counterparts, thus leaving the linear term to make the dominant contribution to Δμs-a.