Objectives
- 1.
Define diffusion as the migration of molecules down a concentration gradient.
- 2.
Recognize that diffusion is the result of the purely random movement of molecules.
- 3.
Define the concepts of flux and membrane permeability and the relationship between them.
Diffusion is the migration of molecules down a concentration gradient
Experience tells us that molecules always move spontaneously from a region where they are more concentrated to a region where they are less concentrated. As a result, concentration differences between regions gradually diminish as the movement proceeds. Diffusion always transports molecules from a region of high concentration to a region of low concentration, because the underlying molecular movements are completely random . That is, any given molecule has no preference for moving in any particular direction. The effect is easy to illustrate. Imagine two adjacent regions of comparable volume in a solution ( Fig. 2.1 ). There are 5200 molecules in the left-hand region and 5000 molecules in the right-hand region. For simplicity, assume that the molecules may move only to the left or to the right. Because the movements are random, at any given moment approximately half of all molecules would move to the right and approximately half would move to the left. This means that, on average, roughly 2600 would leave the left side and enter the right side, whereas 2500 would leave the right and enter the left. Therefore a net movement of approximately 100 molecules would occur across the boundary going from left to right. This net transfer of molecules caused by random movements is indeed from a region of higher concentration into a region of lower concentration.
Fick’s first law of diffusion summarizes our intuitive understanding of diffusion
The preceding discussion indicates that the larger the difference in the number of molecules between adjacent compartments, the greater the net movement of molecules from one compartment into the next. In other words, the rate at which molecules move from one region to the next depends on the concentration difference between the two regions. The following definitions can be used to obtain a more explicit and quantitative representation of this observation:
- 1.
Concentration gradient is the change of concentration, Δ C , with a change in distance, Δ x (i.e., Δ C/ Δ x ).
- 2.
Flux (symbol J ) is the amount of material passing through a certain cross-sectional area in a certain amount of time.
With these definitions, the earlier observation can be simply restated as “flux is proportional to concentration gradient,” or
J∝ΔCΔx
By inserting a proportionality constant, D , we can write the foregoing expression as an equation:
J= -DΔCΔx
The proportionality constant, D , is referred to as the diffusion coefficient or diffusion constant . The minus sign accounts for the fact that the diffusional flux, or movement of molecules, is always down the concentration gradient (i.e., flux is from a region of high concentration to a region of low concentration). The graphs in Fig. 2.2 illustrate this sign convention.
Equation 2.2 applies to the case in which the concentration gradient is linear, that is, a change in concentration, Δ C, for a given change in distance, Δ x. For cases in which the concentration gradient may not be linear, the equation can be generalized by replacing the linear concentration gradient, Δ C /Δ x, with the more general expression for concentration gradient, dC / dx (a derivative). The diffusion equation now takes the form
J= -DdCdx
This equation is also known as Fick’s First Law of Diffusion . It is named after Adolf Fick, a physician who first analyzed this problem in 1855.
To complete the discussion of Fick’s First Law, we should examine the dimensions (or units) associated with each parameter appearing in Equation 2.3 . Because flux, J , is the quantity of molecules passing through unit area per unit time, it has the dimensions of “moles per square centimeter per second” (= [mol/cm 2 ]/sec = mol·cm −2 ·sec −1 ). Similarly, the concentration gradient, dC / dx , being the rate of change of concentration with distance, has dimensions of “moles per cubic centimeter per centimeter” (= [mol/cm 3 ]/cm = mol·cm −4 ). For all the units to work out correctly in Equation 2.3 , the diffusion coefficient, D , must have dimensions of cm 2 /sec (= cm 2 ·sec −1 ).
Essential aspects of diffusion are revealed by examining random, microscopic movements of molecules
Random movements result in meandering
The most important characteristics of diffusion can be appreciated just by considering the simplest case of random molecular motion—that of a single molecule moving randomly along a single dimension. The situation is presented graphically in Fig. 2.3 .
The molecule is initially (at Time = 0) at some location that for convenience we simply refer to as 0 on the distance scale. During every time increment, Δ t , the molecule can take a step of size δ either to the left or to the right. A typical series of 20 random steps is shown in Fig. 2.3 . Two features are immediately apparent from the figure. First, when a molecule is moving randomly, it does not make very good progress in any particular direction; it tends to wander back and forth aimlessly—it meanders. Second, because the molecule meanders, its net movement away from its starting location is not rapid. These two features manifest themselves in important ways when we consider the aggregate behavior of a large number of molecules. Fig. 2.4 presents the results of a numerical simulation of diffusive spreading of 2000 molecules initially confined at x = 0 ( Fig. 2.4 A). At each time point, each molecule takes a random step, with forward and backward steps being equally probable. After each molecule has taken 10 random steps ( Fig. 2.4 B), some molecules are seen to have moved away from the initial position, and the number of molecules remaining at precisely x = 0 has dropped to approximately 250. After 100 steps have been taken ( Fig. 2.4 C), many molecules have moved farther afield, with a corresponding drop in the number remaining at x = 0 to approximately 100. The trend continues in Fig. 2.4 D (after 1000 steps). Clearly, over time, the spatial distribution of molecules is gradually broadened by diffusion.
One may ask what the average position of all the molecules is after diffusion has caused the spatial distribution to broaden. Fig. 2.4 shows that as the molecules move randomly, they spread out progressively, but symmetrically , so that their average position is always centered on x = 0. This is reasonable: because moves to the right and left are equally probable, at any time, there should always be roughly equal numbers of molecules to the right and to the left of 0. The average position of such a distribution must be x = 0 at all times. This observation indicates that the average position is not an informative measure of the progress of diffusion.
The root-mean-squared displacement is a good measure of the progress of diffusion
We seek a quantitative description of the fact that, with time, the molecules will cluster less and will progressively spread out in space. The desired measure is the root-mean-squared (RMS) displacement, d RMS ( Appendix C ). For diffusion in one dimension,
dRMS1-D=2Dt
where D is the diffusion coefficient (as in Fick’s First Law) and t is time. For diffusion in two and three dimensions, the RMS displacements are given by, respectively,
dRMS2-D=4Dt
and
dRMS3-D=6Dt
An example of one-dimensional diffusion could be a repair enzyme randomly scanning DNA for single-strand breaks. A phospholipid molecule moving within a lipid bilayer undergoes two-dimensional diffusion. A glucose molecule moving in a volume of solution exemplifies three-dimensional diffusion.
Square-root-of-time dependence makes diffusion ineffective for transporting molecules over large distances
The most important aspect of the RMS displacement is that it does not increase linearly with time. Rather, random molecular movement involves meandering and thus causes spreading that increases only with the square root of time. Fig. 2.5 A shows the mathematical difference between displacement that varies directly with time and displacement that varies with the square root of time. The feature to notice is that over long distances the square root function seems to “flatten out.” This means that to diffuse just a little farther takes a lot more time. In fact, because of the square-root dependence of the RMS displacement on time, to go 2 times farther takes 4 times as long, 10 times farther takes 100 times as long, and so on. A more intuitive illustration of the qualitative difference between random and rectilinear movement is shown in Fig. 2.5 B. The conclusion is that, over long distances, diffusion is an ineffective way to move molecules around.
Diffusion constrains cell biology and physiology
The practical significance of the fact that diffusion has a square-root dependence on time ( Equations 2.4–2.6 ) can be shown by a simple calculation. Diffusion constants for biologically relevant small molecules (e.g., glucose, amino acids) in water are typically approximately 5 × 10 −6 cm 2 /sec. For such molecules to diffuse a distance of 100 μm (0.01 cm, approximate thickness of a human hair) would take (0.01) 2 /6 D = 3.3 seconds (use Equation 2.6 and solve for t ). For the same molecules to diffuse a distance of 1 cm (slightly less than the width of a fingernail), however, would take 1 2 /6 D = 33,000 seconds = 9.3 hours! These results show that diffusion is sufficiently fast for transporting molecules over microscopic distances but is extremely slow and ineffective over even moderate distances. Not surprisingly, therefore, most cells in the body are within 100 μm of a capillary and thus only seconds away from both a source of nutrient molecules and a sink for metabolic waste ( Box 2.1 ). These calculations also demonstrate why even small insects (e.g., a mosquito or a flea) must have a circulatory system to transport nutrients into, and waste out of, the body.
Oxygen (O 2 ) diffuses passively from tissue capillaries to cells in the tissue. To provide adequate O 2 to meet cellular metabolic needs, capillaries must be spaced closely enough in tissue to ensure that that O 2 concentration does not fall below the level required for mitochondrial function. We would expect capillary density in a particular tissue to depend on the metabolic rate of that tissue. Thus in slowly metabolizing tissue (e.g., subcutaneous), cells are typically separated by larger average distances from tissue capillaries. In contrast, in metabolically active tissues, cells are much closer to capillaries. In the cerebral cortex or the heart, for example, cells are typically only 10–20 μm from a capillary. In skeletal muscle the density of active capillaries depends strongly on the level of physical activity. At rest, skeletal muscle fibers are, on average, 40 μm from a functioning capillary. During strenuous exercise, many more capillaries are “recruited,” and the average separation between muscle fibers and capillaries falls to less than 20 μm.
The necessity of capillaries in delivering O 2 to cells can be exploited clinically. Solid tumors require an adequate supply of O 2 for growth. Angiogenesis (growth of new blood vessels) is therefore essential for tumor growth. As a result of the pioneering research of Judah Folkman, new therapeutic regimens, involving drugs that inhibit angiogenesis, are being developed to promote the destruction of solid tumors.
Fick’s first law can be used to describe diffusion across a membrane barrier
A membrane typically separates two compartments in which the concentrations of some solutes can be different. We may designate the two compartments as i (inside) and o (outside), corresponding, for example, to the cytosol and extracellular fluid, respectively. The concentration difference between the two compartments, Δ C = C i – C o , gives rise to a concentration gradient across the membrane, which has a certain thickness, say Δ x . The concentration gradient, Δ C /Δ x , drives the diffusion of the solute across the membrane, thus leading to a flux of material, J , through the membrane. This description suggests that Fick’s First Law in the form of Equation 2.2 would be well suited for analyzing such a situation:
J=-DΔCΔx=-DCi-CoΔx
In this form the equation applies to a solute diffusing across a membrane of thickness Δ x, provided that the solute dissolves as well in the membrane as it does in water (i.e., the concentration of the solute just inside the membrane is the same as the solute concentration in the adjacent aqueous solution; Fig. 2.6 A).
In Fig. 2.6 A, C o mem is the concentration of solute in the part of the membrane in immediate contact with the outside aqueous solution; C i mem is the concentration of solute in the part of the membrane in immediate contact with the inside aqueous solution. Realistically, because biological membranes are hydrophobic and nonpolar, whereas the aqueous solution is highly polar, solutes typically show different solubilities in the membrane relative to aqueous solution. To take such differential solubilities into account, we can define a quantity, β, the partition coefficient :
β =CmemCaq
where C aq is the solute concentration in aqueous solution and C mem is the solute concentration just inside the membrane. With the use of the partition coefficient, the solute concentrations just inside either face of the membrane can be written:
Cimem=β×CiandComem=β×Co
The diffusion equation can now be cast in the following form:
J=-D[ΔCΔx]across membrane= -DCimem-ComemΔx=-DβCi-βCoΔx= -DβCi-CoΔx
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