6.4. MOLECULAR DIFFUSION IN BIOLOGICAL SOLUTIONS AND GELS

6.4A. Diffusion of Biological Solutes in Liquids

1. Introduction

The diffusion of small solute molecules and especially macromolecules (e.g., proteins) in aqueous solutions are important in the processing and storing of biological systems and in the life processes of microorganisms, animals, and plants. Food processing is an important area where diffusion plays an important role. In the drying of liquid solutions of fruit juice, coffee, and tea, water and frequently volatile flavor or aroma constituents are removed. These constituents diffuse through the liquid during evaporation.

In fermentation processes, nutrients, sugars, oxygen, and so on diffuse to the microorganisms, and waste products and at times enzymes diffuse away. In the artificial kidney machine various waste products diffuse through the blood solution to a membrane and then through the membrane to an aqueous solution.

Macromolecules in solution having molecular weights of tens of thousands or more were often called colloids, but now we know they generally form true solutions. The diffusion behavior of protein macromolecules in solution is affected by their large sizes and shapes, which can be random coils, rodlike, or globular (spheres or ellipsoids). Also, interactions of the large molecules with the small solvent and/or solute molecules affect the diffusion of the macromolecules as well as the small solute molecules.

Besides the Fickian diffusion to be discussed here, mediated transport often occurs in biological systems where chemical interactions occur. This latter type of transport will not be discussed here.

2. Interaction and “binding” in diffusion

Protein macromolecules are very large compared to small solute molecules such as urea, KCl, and sodium caprylate, and often have a number of sites for interaction or “binding” of the solute or ligand molecules. An example is the binding of oxygen to hemoglobin in the blood. Human serum albumin protein binds most of the free fatty acids in the blood and increases their apparent solubility. Bovine serum albumin, which is in milk, binds 23 mol sodium caprylate/mol albumin when the albumin concentration is 30 kg/m³ solution and the sodium caprylate is about 0.05 molar (G6). Hence, Fickian-type diffusion of macromolecules and small solute molecules can be greatly affected by the presence together of both types of molecules even in dilute solutions.

3. Experimental methods for determining diffusivity

Methods for determining the diffusivity of biological solutes are similar to those discussed previously in Section 6.3 with some modifications. In the diaphragm diffusion cell shown in Fig. 6.3-1, the chamber is made of Lucite^® or Teflon^® instead of glass, since protein molecules bind to glass. Also, the porous membrane through which the molecular diffusion occurs is composed of cellulose acetate or other polymers (G5, G6, K1).

4. Experimental data for biological solutes

Most of the experimental data in the literature on protein diffusivities have been extrapolated to zero concentration since the diffusivity is often a function of concentration. A tabulation of diffusivities of a few proteins and also of small solutes often present in biological systems is given in Table 6.4-1.

Table 6.4-1. Diffusion Coefficients for Dilute Biological Solutes in Aqueous Solution

Solute	Temperature		Diffusivity (m2/s)	Molecular Weight	Ref.
	°C	K
Urea	20	293	1.20 × 10−9	60.1	(N2)
	25	298	1.378 × 10−9		(G5)
Glycerol	20	293	0.825 × 10−9	92.1	(G3)
Glycine	25	298	1.055 × 10−9	75.1	(L3)
Sodium caprylate	25	298	8.78 × 10−10	166.2	(G6)
Bovine serum albumin	25	298	6.81 × 10−11	67 500	(C6)
Urease	25	298	4.01 × 10−11	482 700	(C7)
	20	293	3.46 × 10−11		(S6)
Soybean protein	20	293	2.91 × 10−11	361 800	(S6)
Lipoxidase	20	293	5.59 × 10−11	97 440	(S6)
Fibrinogen, human	20	293	1.98 × 10−11	339 700	(S6)
Human serum albumin	20	293	5.93 × 10−11	72 300	(S6)
γ-Globulin, human	20	293	4.00 × 10−11	153 100	(S6)
Creatinine	37	310	1.08 × 10−9	113.1	(C8)
Sucrose	37	310	0.697 × 10−9	342.3	(C8)
	20	293	0.460 × 10−9		(P3)

The diffusion coefficients for the large protein molecules are on the order of magnitude of 5 × 10⁻¹¹ m²/s compared to the values of about 1 × 10⁻⁹ m²/s for the small solutes in Table 6.4-1. This means macromolecules diffuse at a rate about 20 times as slow as small solute molecules for the same concentration differences.

When the concentration of macromolecules such as proteins increases, the diffusion coefficient would be expected to decrease, since the diffusivity of small solute molecules decreases with increasing concentration. However, experimental data (G4, C7) show that the diffusivity of macromolecules such as proteins decreases in some cases and increases in others as protein concentration increases. Surface charges on the molecules appear to play a role in these phenomena.

When small solutes such as urea, KCl, and sodium caprylate, which are often present with protein macromolecules in solution, diffuse through these protein solutions, the diffusivity decreases with increasing polymer concentration (C7, G5, G6, N3). Experimental data for the diffusivity of the solute sodium caprylate (A) diffusing through bovine serum albumin (P) solution show that the diffusivity D_AP of A through P is markedly reduced as the protein (P) concentration is increased (G5, G6). A large part of the reduction is due to the binding of A to P so that there is less free A to diffuse. The rest is due to blockage by the large molecules.

5. Prediction of diffusivities for biological solutes

For predicting the diffusivity of small solutes alone in aqueous solution with molecular weights less than about 1000 or solute molar volumes less than about 0.500 m³/kg mol, Eq. (6.3-9) should be used. For larger solutes the equations to be used are not as accurate. As an approximation the Stokes–Einstein equation (6.3-8) can be used:

Equation 6.3-8

Probably a better approximate equation to use is the semiempirical equation of Polson (P3), which is recommended for a molecular weight above 1000. A modification of his equation to take into account different temperatures is as follows for dilute aqueous solutions:

Equation 6.4-1

where M_A is the molecular weight of the large molecule A. When the shape of the molecule deviates greatly from a sphere, this equation should be used with caution.

EXAMPLE 6.4-1. Prediction of Diffusivity of Albumin Predict the diffusivity of bovine serum albumin at 298 K in water as a dilute solution using the modified Polson equation (6.4-1) and compare with the experimental value in Table 6.4-1. Solution: The molecular weight of bovine serum albumin (A) from Table 6.4-1 is MA = 67 500 kg/kg mol. The viscosity of water at 25°C is 0.8937 × 10−3 Pa · s and T = 298 K. Substituting into Eq. (6.4-1), This value is 11% higher than the experimental value of 6.81 × 10−11 m2/s.

6. Prediction of diffusivity of small solutes in protein solution

When a small solute (A) diffuses through a macromolecule (P) protein solution, Eq. (6.3-9) cannot be used for prediction for the small solute because of blockage to diffusion by the large molecules. The data needed to predict these effects are the diffusivity D_AB of solute A in water alone, the water of hydration on the protein, and an obstruction factor. A semitheoretical equation that can be used to approximate the diffusivity D_AP of A in globular-type protein P solutions is as follows, where only the blockage effect is considered (C8, G5, G6) and no binding is present:

Equation 6.4-2

where c_p = kg P/m³. Then the diffusion equation is

Equation 6.4-3

where c_A1 is concentration of A in kg mol A/m³.

7. Prediction of diffusivity with binding present

When A is in a protein solution P and binds to P, the diffusion flux of A is equal to the flux of unbound solute A in the solution plus the flux of the protein–solute complex. Methods for predicting this flux are available (G5, G6) when binding data have been experimentally obtained. The equation used is

Equation 6.4-4

where D_P is the diffusivity of the protein alone in the solution, m²/s, and free A is that A not bound to the protein, which is determined from the experimental binding coefficient. Then Eq. (6.4-3) is used to calculate the flux, where c_A is the total concentration of A in the solution.

6.4B. Diffusion in Biological Gels

Gels can be looked upon as semisolid materials which are “porous.” They are composed of macromolecules which are usually in dilute aqueous solution with the gel comprising a few wt % of the water solution. The “pores” or open spaces in the gel structure are filled with water. The rates of diffusion of small solutes in the gels are somewhat less than in aqueous solution. The main effect of the gel structure is to increase the path length for diffusion, assuming no electrical-type effects (S7).

Recent studies by electron microscopy (L4) have shown that the macromolecules of the gel agarose (a major constituent of agar) exist as long and relatively straight threads. This suggests a gel structure of loosely interwoven, extensively hydrogen-bonded polysaccharide macromolecules.

Some typical gels are agarose, agar, and gelatin. A number of organic polymers exist as gels in various types of solutions. To measure the diffusivity of solutes in gels, unsteady-state methods are used. In one method the gel is melted and poured into a narrow tube open at one end. After solidification, the tube is placed in an agitated bath containing the solute for diffusion. The solute leaves the solution at the gel boundary and diffuses through the gel itself. After a period of time the amount diffusing in the gel is determined to give the diffusion coefficient of the solute in the gel.

A few typical values of diffusivities of some solutes in various gels are given in Table 6.4-2. In some cases the diffusivity of the solute in pure water is given so that the decrease in diffusivity due to the gel can be seen. For example, from Table 6.4-2, at 278 K urea in water has a diffusivity of 0.880 × 10⁻⁹ m²/s and in 2.9 wt % gelatin, has a value of 0.640 × 10⁻⁹ m²/s, a decrease of 27%.

Table 6.4-2. Typical Diffusivities of Solutes in Dilute Biological Gels in Aqueous Solution

Solute	Gel	Wt % Gel in Solution	Temperature		Diffusivity (m2/s)	Ref.
			K	°C
Sucrose	Gelatin	0	278	5	0.285 × 10−9	(F2)
		3.8	278	5	0.209 × 10−9	(F2)
		10.35	278	5	0.107 × 10−9	(F2)
		5.1	293	20	0.252 × 10−9	(F3)
Urea	Gelatin	0	278	5	0.880 × 10−9	(F2)
		2.9	278	5	0.644 × 10−9	(F2)
		5.1	278	5	0.609 × 10−9	(F3)
		10.0	278	5	0.542 × 10−9	(F2)
		5.1	293	20	0.859 × 10−9	(F3)
Methanol	Gelatin	3.8	278	5	0.626 × 10−9	(F3)
Urea	Agar	1.05	278	5	0.727 × 10−9	(F3)
		3.16	278	5	0.591 × 10−9	(F3)
		5.15	278	5	0.472 × 10−9	(F3)
Glycerin	Agar	2.06	278	5	0.297 × 10−9	(F3)
		6.02	278	5	0.199 × 10−9	(F3)
Dextrose	Agar	0.79	278	5	0.327 × 10−9	(F3)
Sucrose	Agar	0.79	278	5	0.247 × 10−9	(F3)
Ethanol	Agar	5.15	278	5	0.393 × 10−9	(F3)
NaCl (0.05 M)	Agarose	0	298	25	1.511 × 10−9	(S7)
		2	298	25	1.398 × 10−9	(S7)

In both agar and gelatin the diffusivity of a given solute decreases more or less linearly with an increase in wt % gel. However, extrapolation to 0% gel gives a value smaller than that shown for pure water. It should be noted that in different preparations or batches of the same type of gel, the diffusivities can vary by as much as 10 to 20%.

EXAMPLE 6.4-2. Diffusion of Urea in Agar A tube or bridge of a gel solution of 1.05 wt % agar in water at 278 K is 0.04 m long and connects two agitated solutions of urea in water. The urea concentration in the first solution is 0.2 g mol urea per liter solution and is 0 in the other. Calculate the flux of urea in kg mol/s · m2 at steady state. Solution: From Table 6.4-2 for the solute urea at 278 K, DAB = 0.727 × 10−9 m2/s. For urea diffusing through stagnant water in the gel, Eq. (6.3-3) can be used. However, since the value of xA1 is less than about 0.01, the solution is quite dilute and xBM ≅ 1.00. Hence, Eq. (6.3-5) can be used. The concentrations are cA1 = 0.20/1000 = 0.0002 g mol/cm3 = 0.20 kg mol/m3 and cA2 = 0. Substituting into Eq. (6.3-5),