Diffusion within the Small Volume of a Cell
Within several minutes of adding a specific inducer to bacteria or eukaryotic cells, newly synthesized active enzymes can be detected. These are the result of the synthesis of the appropriate messenger RNA, its translation into protein, and the folding of the protein to an active conformation. Quite obviously, processes are happening very rapidly within a cell for this entire sequence to be completed in several minutes. We will see that our image of synthetic processes in the cellular interior should be that of an assembly line running hundreds of times faster than normal, and our image for the random motion of molecules from one point to another can be that of a washing machine similarly running very rapidly.
The random motion of molecules within cells can be estimated from basic physical chemical principles. We will develop such an analysis since similar reasoning often arises in the design or analysis of experi - ments in molecular biology. The mean squared distance Bar R2 that a molecule with diffusion constant D will diffuse in time t is Bar R2= 6Dt (Fig. 1.10). The diffusion constants of many molecules have been measured and are available in tables. For our purposes, we can estimate a value for a diffusion constant. The diffusion constant is D=KT⁄f , where K is the Boltzmann constant, 1.38 × 10-16 ergs/degree, T is temperature in degrees Kelvin, and f is the frictional force. For spherical bodies, f =6πηr, where r is the radius in centimeters andηis the viscosity ofthe medium in units of poise.
The viscosity of water is 10-2 poise. Although the macroviscosity of the cell’s interior could be much greater, as suggested by the extremely high viscosity of gently lysed cells, the viscosity of the cell’s interior with
Figure 1.10 Random motion ofa particle in three dimensions be-ginning at the origin and the
respect to motion of molecules the size of proteins or smaller is more likely to be similar to that of water. This is reasonable, for small molecules can go around obstacles such as long strands of DNA, but large molecules would have to displace a huge tangle of DNA strands. A demonstration of this effect is the finding that small molecules such as amino acids readily diffuse through the agar used for growing bacterial colonies, but objects as large as viruses are immobile in the agar, yet diffuse normally in solution.
Since D=KT⁄6πηr, then D = 4.4 × 10-7 for a large spherical protein of radius 50 Å diffusing in water, and the diffusion constant for such a protein__ within a cell is not greatly different. Therefore R2=6×4.4×10−7t, and the average time required for such proteinmolecules to diffuse the length of a 1 µ bacterial cell is 1/250 second and to diffuse the length of a 20 µ eukaryotic cell is about 2 seconds. Analogous reasoning with respect to rotation shows that a protein rotates about 1/8 radian (about 7°) in the time it diffuses a distance equal to its radius.
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