Response to Kang et al.

The Comments by Minchul Kang, Emmanuele DiBenedetto, and Anne K. Kenworthy miss the objectives of our original article (1). Kang et al. focus on mathematical details of a particular molecular mobility model that are not important for the central issues that are represented in our research and that of our diverse cited references.

Here is more detail of the motivation of Feder et al. The essential relevant features of cell membranes, and sometimes of the labeled mobile molecules themselves, are spatial and temporal inhomogeneities in their varied properties, as shown in the many publications we cited that preceded our article. The effective mobility of a molecule on a living cell membrane varies with its identity, reactivity, and location in the inhomogeneous environment of the membrane. See, for example, the early molecular tracking detail by Ghosh and Webb (2). Molecular mobility on and in cells can also vary with the timing of the measurements during the life of the cells. If the molecule is spontaneously diffusing around, as the living cell membrane is changing its local composition and thus its effective local viscosity and active driving forces (a ubiquitous occurrence), these effects complicate diffusion dynamics. Furthermore, lipid phaselike separations associated with cell structures lead to inhomogeneous molecular diffusibilities, driving forces, and temporally variable binding sites (2,3).

The trajectories of individual molecules moving in a nonuniform, two-dimensional medium can be approximated by various power laws that describe erratic displacement distances versus time and position, and these do not usually result in the expected exponent that can be fitted to a photobleaching recovery equation for a uniform medium. It is virtually always a nonquadratic power law of transport distances versus time that is detected by microscopic tracking of fluorescible membrane molecules on cell surfaces. Because the effective viscous inhibitions of molecular mobility vary with time as well as vector direction and location on the cell surface, the apparent density varies with location and time. Any probabilistic diffusion formulas are approximations to the time, location, and direction of the effective mobility of each molecule, and these can only roughly represent the actual complex time and position mobility. Therefore, formulation of the cell-surface diffusion dynamic is always a crude approximation. The formulaic approximation we offered in 1996 is one appropriate approximation based on the many measurement publications we cited.

Many researchers have subsequently confirmed this surprising issue of diverse molecular mobilities in cell membranes where the environment is drastically inhomogeneous on the spatial scale of the experimental data. See Bouchaud and Georges (4) for the early basic theoretical concept. The next 30 or so relevant articles cited after the theory reference in our article report variations of these phenomena. Few membrane diffusion experiments are consistent with homogeneous diffusion! Subsequently, two of the most diverse but understandable transports occur on systematic perturbations of living cells (5), and in vesicles comprised of coexisting quasiequilibrated phases of lipid mixtures (6).

The purpose of our article (1) was to introduce a simplified connection between very specific and simple molecular tracking experiments and the simplified fluorescence photobleaching recovery experiments to judge more conveniently the mobility of molecules on cell surfaces despite their various complex dynamics. That strategy is reported in our article with an attempt to keep it simple. A canonical method for measuring a local value of effective diffusion coefficient on a uniform membrane was subsequently derived and reported by Thompson et al. (7). The mathematical manipulations suggested by Kang et al. are not incorrect, but they provide another specific, although necessarily realistic, new meaning in biophysics, because the underlying cell-membrane heterogeneities causing anomalous diffusion are so complex and ill-defined.

To us, the most delightful surprise about molecular transport in cells came when we perturbed cells by severe osmotic swelling, which disconnected the plasma membrane from the cytoplasm and pulled out connections with many plasma proteins by controlling attachments to the cytoskeleton or intracellular structures. Sometimes, the postswelling results approximated classical homogeneous diffusion of membrane molecules, as has been frequently reported. Interesting recent results of our more controlled examples of cell membranes are presented by Baumgart et al. (5), who reveal the elimination at physiological temperatures of lipid rafts comprising optically detectable membrane heterogeneities. Lipid rafts, under various elusive definitions, are an alleged major source of anomalous cell-surface diffusion. However, we found the most satisfying molecular physics of membrane diffusion in another membrane system (8).

The content of our response to the Comments by Kang et al. expresses what we feel is the essence of the complex situation. Specific formulation of the exponent changes derived from fitting of molecular trajectories or photobleaching recoveries depends on the selection of the spatial diversities of membrane structures or differing interacting properties of the mobile membrane components that are labeled. Frequently, the apparent exponent in a logarithmic plot of molecular displacements appears to have exponential dependence varying with time. In some cases, this is due to confined diffusion; in other cases, it is recognizable as due to membrane flows, which can be driven by the intracellular flows or cytoskeleton.

Thus, in considering the Comments by Kang et al., it should be made clear just what properties of the diffusion medium have been assumed or will be revealed by their revised mathematics. Simple properties are not generally representative of typical cell-membrane properties, as described in our 1996 publication.