Figure 1. Common diffusion phenomena in biomacromolecular media. (a) Polystyrene microspheres endocytosed by multinucleated giant cells[9]. (b) Diffusion trajectory of RNA-protein particles in Vibrio cholerae[14]. (c) Diffusion process of RNA molecules in E. coli[12]. (d) Diffusion trajectory of small quantum dot aggregates in chromatin[17]. (e) Diffusion trajectory of adenovirus-associated virus-infected HeLa cells[10]

Figure 2. (a) Transmission electron microscope (TEM) micrographs of oleic acid-covered quantum dots dispersed in polystyrene. (b) The experimentally measured diffusion coefficient from a quantum dot in polystyrene as a function of the ratio of theoretical diffusion coefficient calculated from S-E relation: True viscosity, Rouse viscosity, Blend viscosity[36]. (c) Schematic representation of individual nanoparticles in polymer melts, indicating critical length scales and two extreme states. The left is a "small" (2R<dT) particle not constrained by entanglement; the right is a "large" (2R>dT) particle trapped by an entangled network, strongly coupled to the exclusion of the surrounding polymer. (d~f) S-E violation ratio as a function of the ratio of nanoparticle diameter versus entangled mesh size. Different color lines represent different levels of entanglement (from bottom to top): N/Ne=1 (black), 2 (red), 4 (blue), 8 (pink), and 16 (green). (d) The solid curves correspond to the uncorrected SCGLE method, and the dashed lines correspond to constraint-release model. (e) The dashed lines correspond to the SCGLE results considering the numerical prefactor memory function, and the solid lines correspond to the results without numerical prefactor. (f) Black squares represent experimental results and lines represent SCGLE results. The solid lines uses a numerical prefactor of 3.6, and the dashed lines ignore prefactor[37]

Figure 3. (a) Schematic representation of particles in a four-functional macromolecular network. (b) The grid model of macromolecular network, the parameters are: g, genus; n, degree; k, functionality. (c) Mesh models with different topologies. (d~f) Trajectories of particles in three diffusion modes in a macromolecular network: Brownian (d), hopping (e) and trapped (f). (g) Mean square displacement (MSD) of particle diffusion in a macromolecules network of different stiffness and their non-Gaussian parameters. (h) Typical diffusion trajectory of particles in networks at g=6, d/ax of 1.00,1.65 and 1.90, respectively. (i) MSD of particles in different grids[43-44]

Figure 4. (a) The change of the waiting time with the chain stiffness for two grid sizes, where the solid line is the theoretical prediction result. (b) The change of the free energy barrier of hopping process at different chain stiffness. (c) Free energy (ΔF) isosurface experienced by particles in different grid topologies. (d) Changes of the free energy variation ΔF(z) with position z for different particle diameters. (e) The change of the free energy barrier Ub experienced by particles between adjacent grid cells in a grid of different topologies with the constrained parameter d/ax[44,46]

Figure 5. (a) A Schematic representation of the graphene oxide (GO) within the cell membrane. (b) Janus particles that diffuse in the environment surrounded by E. coli. (c) Schematic representation of particles with different charges spreading on the cell membrane surface. (d) The segment MSD of the persistence part (red) and oscillation part (blue) in the super diffusion trajectory. (e) Schematic representation of the super diffusion trajectory of GO within the cell membrane. (f) Hopping length distribution under different interaction energies. (g) Statistical distribution of angles between adjacent persistent segments of the trajectory, top: normal diffusion, bottom: super diffusion[49?-51]

Figure 6. (a) Schematic diagram of pores induced by GO. (b) The pore and GO area ratio and diffusion index vary with the interaction energy Ka, the red and blue curves depict the most likely pore area and the mean pore area, respectively, and the four colors of the background correspond to the four pore states. (c) Schematic representation of the GO motion at different times, Ka=25.29 kBT/nm2, t=22160τ, 22240τ, 22480τ, 22800τ, 23040τ (top) and Ka=34.74 kBT/nm2, t=22320τ, 22360τ, 22400τ, 22440τ, 22480τ (bottom). (d) Distributions of bacterial density around the particle with R=6.97 μm and c0=0.05 mmol/L (left), c0=1.0 mmol/L (right). (e) Fluctuations of lipid bilayer membranes containing nanoparticles with surface charge β=0.8 (left), 0 (middle), and －0.8 (right). (f) The changes of the distance l between the contact points of uniformly charged particles to the cell membrane and the membrane plane with charge |β|[49?-51]

Figure 7. (a) In slow relaxation environment, displacement distribution of an object with a diffusion distance r at a certain time t can be described by the non-Gaussian probability distribution function Gs(r,t), which can be decomposed into a set of basic diffusion Gaussian processes (white curves)[64]. (b) The relationship between appearance time of Gaussian behavior τclt, Fick diffusion τFick, average relaxation time of environment <τenv> and its fluctuation range. (c) Schematic representation of colloidal beads spreading along linear phospholipid bilayers with close radius, and the corresponding Brownian MSD and non-Gaussian DPDF[53]. (d) Schematic representation of the diffusion of colloidal beads in an entangled fiber-type actin grid, and the corresponding Brownian MSD versus abnormal DPDF. (e) Schematic representation of coarse rod diffusion in polymer network, MSD of different rod length particles. When the rod length to grid size is an integer multiple, the particle presents Brownian behavior, and the non-Gaussian DPDF under the corresponding rod length[65]

Figure 8. (a) Log mean square displacement of liposome diffusion in a nematic solution of F-actin filaments. Direction parallel to the actin filament results (open circles and cyan lines), direction perpendicular to the actin filament results (filled circles and orange lines), results of liposomes in water without actin (dashed lines), and the hypothesis inferred from the tail of the displacement probability distribution shown in (b) (gray symbol). (b) Displacement probability distribution under the corresponding system, with the same symbol as in (a). (c) Probability distribution P(D) of the diffusion coefficient of liposome diffusion in the system, calculated from a non-Gaussian probability distribution of liposome displacement, for motions perpendicular to actin filaments (yellow) and parallel (cyan). The vertical line represents the average diffusion coefficient. (d) For (c) results plotted in logarithmic coordinates. Where the vertical lines distinguish the "slow" sharp and "fast" broad peaks, which indicates an independent diffusion process acting intermittently on the liposomes[64]