Membrane Dynamics

Rotational Diffusion; Cell Membrane Dynamics

In our earlier discussion of diffusion we learned that the translational random motions of macromolecules and microscopic objects is due to constant thermal collisions with the background fluid. For a molecule, in addition to center of mass motion, there will also be rotational motion about the center of mass. Under the influence of numerous collisions with the fluid, there will be rotational motions occurring due to random (in both direction and magnitude) torques acting on the molecule (see Figure 1). Just as in the case of translational motion, where there is a frictional force acting that is proportional to the velocity, there will be a frictional torque acting that, to a good approximation, is proportional to the angular velocity of the molecule

Even if the molecule is spherical in shape, it may be asymmetric in other ways such as its electrical or optical properties, and these properties may allow one to distinguish different orientations. For an isolated spherical molecule of radius r, Perrin showed that the rotational frictional coefficient, which is the proportionality constant between the frictional torque and the angular velocity, is

where h is the fluid viscosity. In general, the rotational frictional coefficient for a few other simple shapes, such as ellipsoids or rods, has been calculated and the common result is a third order dependence on the largest spatial dimension. The rotational diffusion coefficient is related to the rotational frictional coefficient through the general relation

where kB is the Boltzmann constant and T is the absolute temperature. DR has units of 1/s and its reciprocal (1/2DR) is known as the rotational relaxation time. It represents the time for a molecule to lose its "memory" of its initial orientation due to rotational diffusion. Characteristic rotational relaxation times for small molecules are very fast, from ps to ns (10^-12 - 10^-9 s), while larger macromolecules may have time constants of 10^-3 s or longer.

Figure 1. Cartoon of a macromolecule undergoing rotational diffusion due to random collisions with solvent molecules.

Example A spherical virus with electrical properties that allow one to distinguish its orientation is in a water solution at 20^oC (293K). By studying the time dependence of its interaction with light, the rotational diffusion time is measured to be 0.2 ms. Calculate the hydrodynamic radius of the virus. Use a value of 0.001 (SI units) for the viscosity of water.

Solution: From our discussion we know that the rotational time constant is related to the rotational diffusion coefficient by

and is further related to the sphere radius from the above equations. Substituting for these, we find that

.

Solving for the sphere radius r, we have

Figure 2 A lipid, the structural unit of biological membranes, with polar head and nonpolar tail

One interesting area of biophysical research that involves rotational diffusion is the study of cellular membranes. Membranes are made up of a variety of lipid molecules that have electrically charged head groups and linear hydrocarbon tail portions (Figure 2) and serve as a boundary for cells and other organelles. The charged head group is highly attracted to polar water molecules (hydrophilic) while the tail groups are repelled by water molecules (hydrophobic). Biological membranes are bilayers, composed of two layers of lipid molecules arranged with the hydrophilic tails inside the membrane and with the hydophilic head groups on the outer surface in contact with the water-based fluid inside and outside of the cell. Synthetic bilayers can be made from purified lipid molecules, but natural biological membranes contain large numbers of proteins in addition to other smaller molecules. Membrane proteins are classified according to their association as either integral or peripheral. Integral proteins are those that are tightly bound to the membrane, some of them even spanning across the full width of the membrane. These latter proteins are important in allowing small molecules and proteins to cross the membrane barrier. Peripheral proteins are more loosely bound to one of the surfaces of the membrane and can be dissociated by changes in pH or ionic concentrations.

In the 1970's it was first discovered that the individual lipid molecules in a membrane, as well as the imbedded proteins, are quite fluid, diffusing about on the two dimensional surface of the membrane at rates of several micrometers per second. Up until that time membranes were viewed as static structures but measurements in the 70's showed that lipids actually can not only diffuse about in their own monolayer but even "translocate" from one monolayer to the other in rare events. A model of biological membranes known as the fluid-mosaic model was developed to describe this dynamic structure and modified versions of it are still useful. Proteins in the membrane are confined (to various degrees) in different domains or regions of the membrane. Many proteins and other macromolecules bind to specific cellular receptor proteins on the membrane. Often these will first bind to the membrane surface through non-specific binding and then diffuse on the two-dimensional membrane surface until a specific receptor is found. This greatly speeds the binding kinetics.