J. Chem. SOC., Faraday Trans. I, 1982, 78, 3357-3367 Experimental Measurements of Polymer Unidirectional Fluxes in Polymer + Solvent Systems with Non-zero Chemical-potential Gradients BY MARIE-PAULE I. VAN DAMME, WAYNE D. COMPER AND BARRY N. PRESTON* Biochemistry Department, Monash University, Clayton, Victoria 3 168, Australia Received 1st March, 1982 The diffusion properties of two polymers in saline, namely dextran and albumin, have been followed by measurements of the unidirectional fluxes of their tracer (labelled) counterparts. The relationships between the diffusion coefficients obtained by measurement of the forward and back unidirectional fluxes to the polymer mutual-diffusion and intradiffusion coefficients have been analysed for semi-dilute polymer concentrations. The dynamic behaviour of macromolecules in a concentrated phase is of interest for many diverse aspects in biology and chemistry.Our studies, in particular, have been aimed at obtaining a better understanding of the dynamic transport behaviour of macromolecules in the extracellular matrices of connective tissues. We have previously devoted attention to the diffusional properties of various p o l y m e r ~ . ~ - ~ The standard form of the mutual-diffusion coefficient, which corresponds to the net flux of material resulting from relaxation of a concentration gradient, and the intradiffusion coefficient, which refers to the flux of trace amounts of labelled molecules through a solution of uniform concentration, have been extensively s t ~ d i e d . ~ - ~ The two other forms of diffusion coefficient that can be measured in a polymer + solvent system are those associated with the unidirectional fluxes of trace amounts of labelled material in systems with non-zero chemical-potential gradients as brought about by macroscopic concentration gradients.These types of diffusion coefficients have hitherto received only minor attention. It is the purpose of this paper to present an experimental analysis of the unidirectional-diffusion coefficients of binary systems containing either dextran + water or albumin + water (buffered with phosphate, pH 7.2). This study extends our earlier analysis4 of the measurements of the concentration dependence of the mutual-diffusion and intradiffusion coefficients for these polymers. EXPERIMENTAL MATERIALS - _ FDR7783 dextran (M, = 158 000; M,/M,= 1.32) and fluoresceinylthiocarbamoyl-dextran (FITC-Dx-150, Lot F-108) (z, = 153700; M,/M, = 1.51) were kindly donated by Dr K.Granath, AB Pharmacia (Uppsala, Sweden). These materials have been subject to extensive investigations in terms of their various physicochemical properties.3* 4 3 The two dextrans have similar properties. Bovine serum albumin (BSA, no. A-4503 fraction V, lot 109C-0081) was obtained from Sigma Chemical Company (St Louis, U.S.A.). [14C]Sorbitol (333 Ci rnol-')t was obtained from the Radiochemical Centre, Amersham. 7 1 Ci = 3.7 x 1 O ' O Bq. 33573358 MEASUREMENT OF POLYMER U N I D I R E C T I O N A L FLUXES PREPARATION OF POLYMER SOLUTIONS Stock solutions of polymers of known moisture content were made up by weight in phosphate-buffered saline (PBS) which was 0.14 mol dm-3 NaCl, 2.68 x lop3 mol dmP3 KCl, 1.5 x mol dm-3 KH2P04 and 8.1 x lop3 mol dm-3 Na2HP04 (pH 7.2).PREPARATION OF RADIOACTIVE-LABELLED MATERIALS [1251]BSA was prepared using the lactoperoxidase technique of Mar~halonis.~ [3H]BSA was prepared using the technique of Tack et aL8 The radioactive-labelled BSA was separated from the excess free 12512 or 3H on Sephadex G-25 PD-10 Column (AB Pharmacia, Uppsala, Sweden). The labelled monomeric BSA used in the diffusion studies was freed from dimer and any residual free 1251 or 3H by gel chromatography on Ultrogel AcA 54 (LKB, Bromma, Sweden) or G-150 (AB Pharmacia) at 4OC, using PBS buffer as eluate. The specific activity of the [1251]BSA was ca.2 Ci kg-' and of the [3H]BSA 260 Ci kg-l. Immediately before each experiment the [1251]BSA was further purified from free iodide on a Sephadex G-25 PD-10 column (AB Pharmacia). There was no release of free 12512 during the course of the diffusion experiments. [3H]FDR7783 dextran was prepared using the technique of Preston et al.4 with slight modifications. Dextran (100 mg) was dissolved in 0.1 mol dmP3 NaOH (1 cm3) to which Phenol Red (2 drops) was added; NaB3H4 solution (Radiochemical Centre, Amersham) (0.1 cm3 of 100 mCi in 0.1 mol dmp3 NaOH) was added and the reaction was carried out at 4 "C overnight. At the end of the reaction, acetic acid was added in order to acidify the medium. After 1 h, the medium was neutralized with NaOH. The polymer was separated from tritiated water on a Sephadex G-25 PD-10 column (AB Pharmacia).The preparation was then dialysed extensively against water and then purified on a Sepharose CL6B (AB Pharmacia) column (1.5 x 65 cm) in PBS. It has been shown previously4 that this labelling procedure resulted in minimal change in the molecular size distribution of the dextran sample. The specific activity of the [3H]FDR7783 was in the range 20-30 Ci kg-l. METHODS DETERMINATION OF DIFFUSION COEFFICIENTS Unidirectional-diffusion coefficients of labelled polymers, which were present in trace quantities, were measured in Sundelof diffusion cells. This technique has been described in detail el~ewhere.~ The cells consist of two cylindrical chambers which, by a shearing mechanism, can be moved in two different positions, as shown schematically in fig.1; a filling or emptying position, where the two chambers are separated, and a measuring position, where both chambers are brought over each other, forming thus a horizontal boundary. We define the concentrations in the bottom and top chambers as c b and C,, respectively. The mean concentration across the initial boundary is C[ = ( c b + C,)/2] and the difference in concentration is given as AC( = c b - C,). The total amount (mass) of diffusing solute Q transported across the boundary during the time t is given by Q2 = A2Ci D t / n where C, corresponds to the initial concentration of the labelled material in one of the chambers, A is the cross-sectional area of the diffusion compartments and D is the diffusion coefficient.The forward unidirecti%nal-diffusion coefficient representing the flux from the bottom to the top chamber is given by D while the back unidirectional-diffusion coefficient is given by z. For a system where C, = 0 the forward unidirectional-diffusion coefficient B is identical to the mutual-diffusion coefficient D. In the case where c b = C,, the back and forward unidirectional diffusion coefficients should be identical and are known as the intradiffusion coefficient D+. However, it is very difficult in practice to carry out these experiments owing to the absence of any stabilising gradient. Values of D+ were thus obtained at c b - C, N 5 kg mp3. For non-zero values of C, we shall examine the relationship of the mutual-diffusion coefficient and unidirectional-diffusion coefficients through the use of an additivity equation where D = (DC,,-BCJ/AC.FIG.1.- C, and M-P. I. V A N DAMME, W. D . COMPER A N D B. f i I l i -Schematic diagram of the formation C, represent the concentrations of N. P R ng positi ESTON on m ea sur i n g posit ion emptying posit ion 3359 of the free-liquid boundary in the Sundelof diffusion material in the bottom and top chambers of the respectively. cell. cell, Mutual-diffusion coefficients of BSA were also measured by free diffusion in a Beckman model E analytical ultracentrifuge using Schlieren optics. The initial concentration gradient was ca. 5 kg mP3 for routine measurements. RADIOACTIVE COUNTING PROCEDURES [3H]dextran samples were prepared for liquid scintillation counting with Aquasol (NEF-934 New England Nuclear, Boston, U.S.A.). Studies have shown that the degree of quenching of 3H counts is dependent on the total quantity of dextran present in the vial, as shown in fig.2. It is evident that there is a marked increase in quenching with increasing concentration of dextran, and we have chosen for routine measurements to work in the dextran concentration range 30-50 mg per vial, since although the counting efficiency is decreased by 40%, it remains relatively constant over this range. Furthermore, under these conditions the quenching does not change over a period of 3 days (fig. 2). In addition, the counting efficiency for these [3H]dextran samples is sensitive to the water/Aquasol ratio, as shown in fig. 3. A 22.5% (v/v) aqueous sample in Aquasol was used, conditions in which a stiff gel was formed.Note that preparation of counting samples with < ca. 19% H,O resulted in incomplete gel formation and precipitation of the sample with time. [3H]BSA and [14C]sorbitol were counted under similar conditions to [3H]dextran. 1251 was determined in an Autogamma analyser PW5420 (Philips, Holland).3360 MEASUREMENT OF POLYMER UNIDIRECTIONAL FLUXES 0 10 20 30 40 50 60 FIG. 2.-Effect of the addition of unlabelled dextran on the counting of [3H]dextran. The [3H]dextran samples (volume 1.3 ~ m - ~ ) were prepared for liquid scintillation counting with 4.5 cmP3 Aquasol. The experimental points represent the mean value of five determinations (0) together with their standard deviation (+). Some vials were left at 4OC for 3 days and then recounted, as indicated by the symbol dextran mass/10-6 kg 20 21 22 23 24 25 26 percentage of water in sample FIG.3.-Counting efficiency for [3H]dextran as a function of the percentage of the volume of aqueous sample with respect to the total volume of the solution (which contains Aquasol and aqueous sample) that is counted. The total volume was kept constant at 5.8 ~ m - ~ . The experimental points represent the mean value of three determinations (0) together with their standard derivation (+). Symbols represented by (0) correspond to vials which had been maintained at 4OC and recounted 3 days later. The dashed area corresponds to incomplete gel formation and precipitation of sample.M-P. I. V A N DAMME, W. D.COMPER A N D B. N. PRESTON 336 I FLUORESCENCE MEASUREMENTS Fluorescence measurements were performed on an Hirachi model 101 spectrofluorimeter (Hitachi Ltd, Japan) at 492 nm (excitation) and 515 nm (emission). RESULTS The use of the Sundelof cells allows studies to be made of the transport kinetics of labelled material across the initial boundary. The diffusion studies involved measurements associated with the transfer of < 15% of the total labelled material across the boundary. As shown in fig. 4, the plot of Q2 as a function of time is linear, as predicted by eqn (1); this is in agreement with a diffusional mechanism. In all diffusion studies reported here the regression coefficient of results plotted according to eqn (1) I s > 0.975. 0 1 2 3 4 time/ 1 O5 s FIG. 4.-A plot of Q2/C,2 (where Q is the total amount of material crossing the boundary after time t and C,, corresponds to the initial concentration of the labelled material in one of the chambers) as a function of time for the measurement of the unidirectional-diffusion coefficient of [3H]dextran from the bottom chamber to the top chamber where initially C, = 69 kg m-3 and C,= 64 kg mP3.The concentration dependence of the forward unidirectional-diffusion coefficient (n) of [1251]BSA and E3H]FDR7783 were studied when Ct = 0. The data are presented in fig. 5 and 6, respectively. For [lZ5I]BSA, the values of the a (= D) obtained do not show any concentration dependence and are in good agreement with the coefficients obtained from the analytical ultracentrifuge using Schlieren optics (fig.5), although some differences are apparent at high albumin concentrations. These values are also in good agreement to those obtained by the open-ended capillary technique2 and laser light scattering . lo In contrast to the diffusion of albumin, the diffusion coefficient of dextran shows - significant concentration dependence. When the diffusion values are plotted against C [ =(C,+ Ct)/2] we find that they are in good agreement with the mutual-diffusion coefficients (evaluated from ultracentrifuge and proton correlation spectroscopy) published previ~usly.~ As with the albumin transport, even at high C, values the Q2 value for dextran was linear with time, in accord with eqn (1). Therefore, while transfer of material across the boundary will produce a time-dependent change in AC, we do not detect any significant time-dependent variation in the diffusion coefficient.In general, transport across a free liquid boundary where high polymer concentration3362 MEASUREMENT OF POLYMER UNIDIRECTIONAL FLUXES 0 11' ri [ iI t ii i I 1 I 0 100 200 300 mean albumin concentration/kg m-3 FIG. 5.-Plot of the unidirectional-diffusion coefficient of [1251]albumin obtained from the Sundelof diffusion cell as a function of the mean concentration of albumin across the boundary (C) when C, = 0 (0). The value of the mutual-diffusion coefficient of albumin obtained by the boundary relaxation method in the analytical ultracentrifuge is also shown (0). The variation of the intradiffusion coefficient of [1251]albumin is shown as (m).The error bar corresponds to 95% confidence limits. gradients exist yields a mutual-diffusion coefficient at C which is essentially independent of AC. The values of both the forward-flux and the back-flux unidirectional-diffusion coefficients, a and D, respectively, have been measured when Cb was maintained constant while varying Ct (table 1 for [1251]BSA and table 2 for [3H]FDR7783). When minimal values of c b - Ct are studied the values of 25 and converge to the same value: that of the intradiffusion coefficient D+ whose values are presented in fig. 5 and 6 . Values of Td at intermediate values of Ct appear to correspond to some average of the mutual-diffusion and intradiffusion coefficient measured at the corresponding value of Cb. Values of D for [1251]BSA measured at constant Cb were found to be slightly dependent on Ct, suggesting that is directly related to C, for this particular polymer.We also find thatB is ca. 15% lower than the value of D+ at the concentration c b . In the case of [3H]FDR7783 (table 2), however, t h e n values tend to increase with increasing C, ; again, the values of the are significantly lower than the intradiffusion coefficients at corresponding concentrations. The calculated values of D, using eqn (2), are presented in the last columns of table 1 (for [1251]BSA) and table 2 (for [3H]FDR7783). These calculated values are in good agreement with mutal-diffusion coefficients evaluated experimentally. Estimates of D from eqn (2) when ( c b - C,) is small are subject to large error, which may explain the anomalous values obtained with albumin.Note that when the magnitude of D is low, then the difference between values of D a n d z is apparent even at small values of ACM-P. I. VAN DAMME, W. D . COMPER A N D B. N . PRESTON 3363 i i I t I 0 20 40 60 80 100 mean dextran concentration/kg m-3 FIG. 6.-Plot of the unidirectional-diffusion coefficient of [3H]dextran obtained from the Sundelof diffusion cell as a function of the mean concentration of dextran across the boundary when C, = 0 (0). The corresponding intradiffusion coefficient of [3H]dextran is also shown (m). The error bar represents 95 % confidence limits. TABLE 1 .-UNIDIRECTIONAL-DIFFUSION COEFFICIENTS OF ['251]ALBUMIN WITH c, MAINTAINED AT A CONSTANT VALUE OF (a) 157 kg m-3 AND (b) 327 kg mP3 WHILE VARYING C, IN EACH CASE diffusion coefficienta/ lo-" m2 s-' C,/kg m-3 + D (a) Ob 25.3 51.0 103.3 151.7 50.6 146.3 235.0 321.9 (4 Ob 157.0 131.8 106.2 53.8 5.5 327.0 276.6 181.0 92.3 5.5 6.13 (5.27-7.00) 5.86 (5.24-6.47) 5.38 (4.68-6.08) 4.58 (4.18-4.98) 3.29 (2.91-3.67) 6.01 (5.36-6.66) 5.96 (5.26-6.65) 3.31 (2.81-3.82) 1.63 (1.41-1.85) 1.47 (1.07-1.87) 2.82 (2.40-3.24) 2.80 (2.42-3.18) 2.78 (2.52-3.04) 1.07 (0.95-1.19) 0.83 (0.7 1-0.95) 0.91 (0.72-1.10) 0.63 (0.53-0.73) 0.70 (0.55-0.85) - - 6.2 6.4 6.6 7.9 16.1 6.0 6.8 5.2 4.2 35.1 a The numbers in parentheses indicate 95% confidence limits.For back-flux diffusion coefficient C, was initially 5 kg m-3. (i.e. ca. 5 kg m-3). This points to the necessity to extrapolate to zero AC in order to obtain an accurate value of the intradiffusion coefficient under these conditions.However, we have routinely employed the convenient practice of evaluating D+ values at AC % 5 kg m-3 in spite of the inaccuracy introduced, especially when dealing with low-magnitude diffusion coefficients. A more extensive study was undertaken in the comparison between t h e z and D+ values over a wide range of Cb values (tables 3 and 4). The values of t h e z for [1251]BSA were found to be consistently ca. 20% lower than the values of D+. For the [3H]FDR7783, the magnitude of theD is significantly less than the D+, at corresponding concentrations. These low values of D were also obtained when FITC-dextran was3364 MEASUREMENT OF POLYMER UNIDIRECTIONAL FLUXES TABLE 2.-UNIDIRECTIONAL-DIFFUSlON COEFFICIENTS OF [3H]DEXTRAN WITH c b MAINTAINED AT A CONSTANT VALUE OF (a) 45 kg rnP3 AND (b) 92.5 kg m-3 WHILE VARYING C, IN EACH CASE diffusion coefficienta / 1 O-ll m2 s-' CJkg m-3 D (a> O6 4.5 13.5 26.9 40.5 22.4 45.0 68.5 87.8 (4 O6 45.0 40.5 31.5 18.1 4.5 92.5 70.0 47.5 23.9 4.6 3.25 (2.71-3.75) 3.20 (2.81-3.59) 2.66 (2.52-2.80) 1.93 (1.63-2.23) 1.07 (0.87-1.27) 3.8 7 (3.33-4.40) 3.6 1 (3.57-3.65) 1.57 (1.32-1.81) 0.74 (0.67-0.81) 0.66 (0.49-0.83) - 0.45 (0.40-0.50) 3.51 0.64 (0.55-0.73) 3.51 0.69 (0.62-0.65) 3.65 0.70 (0.66-0.74) 4.41 a The numbers in parentheses indicate 95% confidence limits.For back-flux diffusion Diffusion coefficient the dextran concentration in the upper solution was initially 5 kg m-3. coefficients too low to measure with any accuracy.TABLE 3 .-COMPARISON OF THE BACK-FLUX UNIDIRECTIONAL-DIFFUSION COEFFICIENJ (B) OF ['251]ALBUMIN WITH ITS INTRADIFFUSION COEFFICIENT (D'). FOR THE DETERMINATION OF D, c, WAS 5 kg mP3, WHEREAS FOR D+ MEASUREMENTS C, - C, WAS ca. 5 kg M - ~ . ~~ ~~~ - diffusion coefficientsa/ 1 0-l1 m2 s-' 45.8 4.68 (4.28-5.08) 5.38 (4.70-6.10) 103.3 3.48 (3.08-3.88) 3.72 (3.38-4.06) 164.7 2.61 (1.97-3.24) 3.22 (2.91-3.43) 2 12.4 2.03 ( 1 37-2.19) 2.57 (1.97-3.17) 263.7 1.44 (1.36-1.52) 2.20 (1.88-2.52) 33 1.4 1.07 (0.95-1.19) 1.47 (1.07-1.87) a The numbers in parentheses indicate 95% confidence limits. used as an alternative tracer (table 4). A similarity between the values o f z and D+ has been noted previously for diffusion measurements with hyaluronate for low values for [3H]FDR7783 may be due to a boundary disturbance occurring in this system (for details see Discussion).We have used trace quantities of labelled solvent markers of different sizes, namely [14C]sorbitol and [3H]albumin, to monitor any such disturbance. The results of the unidirectional- diffusion coefficient of these trace materials under different polymer concentration conditions are shown in table 5. For convenience, we describe the direction of unidirectional transport of the solvent marker the same way as its dextran counterpart. It is clear from the studies of transport of [14C]sorbitol in dextran that the D for [14C]sorbitol into a dextran solution of either 50 or 100 kg mP3 is not significantly different from the diffusion coefficient when Cb = C,.Therefore, in using this particular solvent marker no substantial anomalies could be detected in the system. of Cbe5 This marked retardation of theM-P. I. V A N DAMME, W. D . COMPER A N D B. N. PRESTON 3365 In the case of using [3H]albumin as a solvent marker, using essentially the same procedure as for studies with [14C]sorbitol, we find now that t h e 3 is significantly lower as compared with the diffusion coefficient of [3H]albumin when C, z Ct. These results would substantiate the depressed back-flux diffusion coefficients of the [3H]dextran in similar systems (see table 4). TABLE 4.-cOMPARISON OF THE BACK-FLUX UNIDIRECTIONAL-DIFFUSION COEFFICIENTS (B) OF [3H]FDR7783 DEXTRAN AND FITC-DEXTRAN _WITH THEIR CORRESPONDING INTRADIFFUSION COEFFICIENTS (D+). FOR THE DETERMINATION OF D, C, WAS 4.45 kg mP3, WQEREAS FOR INTRADIF- FUSION MEASUREMENTS C, - C, WAS ca.5 kg m-3. diffusion coefficientsa/ lo-" m2 s-' D D+ c ~~ fluorescein- fluorescein- C,/kg m-3 [3H]FDR7783 dextran I3 HI FD R77 8 3 dextran 45.0 0.45 0.59 1.05 1 .08b 68.9 0.21 0.20 1.10 0.90b 92.5 0.10 0.10 0.66 0.67 (0.40-0.50) (0.55-0.63) (0.85-1 -25) (1.05-1.1 1) (0.18-0.24) (0.17-0.23) (0.94-1.26) (0.87-0.93) (0.07-0.13) (0.09-0.12) (0.49-0.8 3) (0.57-0.76) a The numbers in parentheses indicate 95% confidence limits. From ref. (3). TABLE TH THE TRANSPORT OF SOLVENT MARKERS, ['4C]SORBITOL AND [3H]ALBUMIN, IN SYSTEMS WITH ZERO AND NON-ZERO CONCENTRATION GRADIENTS OF FDR7783 DEXTRAN~ dextran FDR7783 concentration/kg mP3 diffusion coefficient of solvent marker (solvent marker, [14C]sorbitol = S*) 50+S* 4 50, 50 +- 5+S* lOO+S* -+ loob 100 +- 5+s* (solvent marker, [3H]albumin = A*) lOO+A* -+ 95 100 t 5+A* 47.3 (42.8-52.0) 5 I .9 (45.6-58.2) 37.7 (33.1-41.9) 41.8 (36.5-47.2) 1.67 (1.33-2.02) 1.02 (0.81-1.21) a Arrow indicates direction in which the flux of solvent marker was measured.Unlabelled sorbitol at a concentration of 5 kg m-3 was added to the bottom compartment solution in order to stabilize the boundary. Further insights into the relationship of and Z7 have been made in studying an 'ideal' system of the diffusion of [14C]sorbitol in sorbitol. We have performed unidirectional diffusion analysis by a similar procedure to that described above for polymer diffusion, with the results shown in table 6.We find that (i) there is a moderate concentration dependence of D for sorbitol_up to a concentration of 200 kg mP3, (ii) there is no major difference in the value of D when C, = 5 kg mP3 as compared with3366 MEASUREMENT OF POLYMER UNIDIRECTIONAL FLUXES TABLE 6.-uNIDIRECTIONAL-DIFFUSION COEFFICIENTS OF [ ''C]SORBITOL IN SORBITOL diffusion coefficient/ 10-l' m2 s-l C,/kg m-3 CJkg rnP3 e D 5 0 52.7 (48.9-56.5) - 104.5 0 48.4 (46.0-50.8) 47.5 (43.7-51.3)" 104.5 99.1 41 .O (37.6-44.4) 43.1 (41.2-45.0) 217.8 0 44.5 (39.9-49.1) 3 8.0 (3 6.6- 39. 4)a 217.8 210.1 30.0 (28.7-3 1.3) 32.0 (29.2-34.8) a For back-flux diffusion coefficient C, was initially 5 kg m-3. its value when AC is low and (iii) D a n d D are approximately the same for low AC values. These results would emphasize the fact that the anomalously low values for dextran are the result of the polymeric nature of this material.DISCUSSION Analysis of the flux of labelled polymers in the Sundelof diffusion cell allows the examination of polymer transport in systems with large polymer concentration gradients across a free liquid boundary. This has only been previously possible with the use of a diaphragm cell1' in which a millipore filter or glass filter is used to stabilise the initial boundary. The use of such a diffusion apparatus to measure polymer diffusion may appear to be unsatisfactory. The measurements of Keller et aZ.12 of the mutual diffusion of albumin or haemoglobin using the diaphragm cell has subsequently come under criticism due to the very low values of the mutual-diffusion coefficient obtained by this technique as compared with values obtained by laser light scattering13 and the open-ended capillary technique.2 Re-analysis of the polymer flux results of Keller et d.12 for albumin and haemoglobin on the basis of diffusion in a column of cross-section A l4 [i.e.eqn (l)] can yield mutual-diffusion coefficients comparable to those obtained by others. This analysis involves arbitrarily choosing one set of their experimental flux data as an internal standard for the evaluation of all diffusion-cell constants as embodied in the term A of eqn (1). Therefore, the anomalous nature of the mutual-diffusion coefficients measured by Keller et a l l 2 may stem either from the mathematical treatment of the results or the effect of interposing a membrane between the two interdiffusing solutions.With regard to dextran diffusion, the agreement between the unidirectional-diffusion coefficient of the dextran from Sundelof diffusion cells with the mutual-diffusion coefficient from the centrifuge (where low values of AC are employed) may indicate that the mutual-diffusion coefficient is not significantly dependent on AC. Caution is introduced here, however, as we have previously shown4 that the mutual-diffusion coefficient of dextran measured in the ultracentrifuge can be dependent on AC when studied over a limited range of AC. With the presence of large concentration gradients of dextran we have observed anomalous behaviour nssociated with the unidirectional back fluxes of this polymer.This behaviour may be regarded as some form of boundary disturbance, being equivalent to an overall displacement in the position of the initial boundary. An approximate calculation of the distance the boundary would have to be dis- placed in order to account for the difference between the back flux and intradiffusion flux indicates that this distance is extremely small and would be difficult to detect byM-P. I. V A N DAMME, W. D. COMPER AND B. N. PRESTON 3367 conventional measurements. It is the relatively lower-magnitude diffusion coefficients which would be more greatly affected by this boundary movement, such as those obtained with E3H]dextran transport. We have previously shown that boundary disturbances may result from certain hydrodynamic instabilities that yield countercurrent volume flows of high ordered and intricate nature in multicomponent polymer Obviously while no macroscopic changes in the volume of the system did occur, microscopic volume fluxes were visualized.Although this type of behaviour has not been visualized in the binary polymer +water systems studied here, it is conceivable that a form of microscopic volume flow may occur. This would be consistent with the results obtained by using [3H]albumin as a trace solvent marker (see table 5). It is informative to consider the case of macroscopic dimensions where a membrane of finite thickness is interposed between the two solutions. Kedem and Katchalsky16 have derived the commonly used expression for the total volume flow Jv = - L, oAII (3) where L, is the membrane filtration coefficient or hydraulic permeability, o is the reflection coefficient and A l l the difference in osmotic pressure across the membrane.It is often claimed that o is a measure of the selectivity of the system to solute and solvent (see Ogston and Michell' for more accurate expressions). A relationship could then be considered, in a binary system with a free-liquid boundary, of the interplay of dynamic polymer-polymer interactions, yielding a pseudo-membrane which is selective to the movement of the polymer itself and an osmotic gradient generated by the polymer concentration gradient. Note, however, that eqn (3) has been derived for discontinuous membrane systems in which absolute differences in thermodynamic parameters across the membrane, such as pi and n, are taken as the driving forces.The situation in a free boundary system is more complex, particularly with respect to an understanding of the effective osmotic pressure at any particular point in the boundary and the microscopic volume flows that may occur. This project was supported by the Australian Research Grants Committee (grant no. D68/16898, D2 73/14137 and DS 79/15252). We thank Gregory Checkley and Geoffrey Wilson for their expert technical assistance. B. N. Preston, T. C. Laurent and W. D. Comper, in Glycosaminoglycan Assemblies in the Extracellular Marrix, ed. D. A. Rees and S. Arnott (to be published by Humana Press). R. G. Kitchen, B. N. Preston and J. D. Wells, J. Polymer Sci., Polym. Symp., 1976, 55, 39. T. C. Laurent, L-0. Sundelof, K-0. Wik and B. Warmegird, Eur. J. Biochem., 1976, 68, 95. B. 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