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J. Phys. Chem. B 2000, 104, 9898-9903
Effect of Charge on Protein Diffusion in Hydrogels Naoki Hirota, Yasuhiro Kumaki, Tetsuharu Narita, Jian Ping Gong, and Yoshihito Osada* DiVision of Biological Science, Graduate School of Science, Hokkaido UniVersity, Sapporo 060-0810, Japan ReceiVed: April 17, 2000; In Final Form: July 24, 2000
To study the effect of charge on protein diffusion in hydrogels, mutual diffusion of a globular protein, myoglobin, has been investigated at various pH and ionic strength levels in two kinds of polysaccharide gels, neutral agarose gel and anionic carrageenan gel, by the recently developed electronic speckle pattern interferometry method. In the uncharged agarose gel, diffusions of myoglobin are not effected by the change in pH and the ionic strength, indicating no electrostatic interaction between the gel and myoglobin. The experimental data in agarose gel agree with the combined model proposed by Clague and Philips for the diffusion of spheres in hydrogels. While in the negatively charged λ-carrageenan gel, the diffusion of myoglobin is accelerated by electrostatic attraction when the pH is lower than the isoelectric point (pI) of the protein, but it is extensively hindered by the electrostatic repulsion when pH > pI. The diffusion of myoglobin in λ-carrageenan gel agrees with the Tsai and Strieder model to give an apparent radius of the protein at various pH values.
Introduction The study of molecule diffusion in hydrogels is crucial both for gaining a better understanding of solute-gel interactions and for the designing of novel applications of such materials.1,2 The rate of diffusion of proteins through gels is especially important in developing various electrophoretic, chromatographic, and membrane separation processes. Diffusion coefficients of protein through gels have already been measured by many methods including dynamic light scattering,3-6 pulsed field-gradient nuclear magnetic resonance (PFGNMR),7-10 and holographic interferometry (HI).11-14 However, relatively little information is available on the partitioning or diffusion of charged macromolecules in charged gels.15 The effect of charge on the diffusion coefficient D of micelles16 and linear polyelectrolytes17 in membranes with straight, cylindrical pores has been explained fully by theoretical predictions regarding the effect of fiber volume fraction φ, suggesting that there is little or no effect of charge on D for macromolecules in such pores. Whether this might also be true for macromolecules in charged gels is unknown. Diffusion studies using charged probes in semidilute polymer solutions indicated that the effect of ionic strength on D is not negligible. In tracer diffusion studies of BSA in DNA solutions,18 it was found that, upon increase of the DNA concentration from 0 to 30 mg/mL, the diffusibility of BSA in a 0.01 M NaCl solution decreased more than that in a 0.1 M NaCl solution. At 30 mg/mL DNA solution, the tracer diffusion coefficient of BSA in 0.01 M NaCl was about 20% lower than in 0.1 M NaCl. Studies on the diffusion of polystyrene latex spheres in poly(acrylic acid) solutions19,20 also showed that a decrease in the ionic strength lowered the diffusion coefficient. Whether any of these results are applicable to gels is unclear. Conventionally, the diffusion coefficients in gels are measured by monitoring the concentration of the diffusing solute in the solution outside the gel.21,22 Holographic interferometry technique offers a powerful tool for noncontact measurement. It has been used to measure the mutual diffusion coefficients in binary liquid systems in the past.23-27 Recently, this technique was
employed for liquid-gel or gel-gel systems.12-14 This method has several clear advantages over other conventional techniques, since it is a direct method which avoids sampling and analysis of the liquid solution outside the gel. The diffusion coefficients in gels could be conveniently determined from the fringes in interferograms based on Fick’s law. In the previous paper,28 we introduced an electronic speckle pattern interferometry (ESPI) to represent the holographic methods, and this method was proved to be a powerful tool for the characterization of molecular diffusion in gels. The purpose of this study is, by using ESPI, to elucidate the effects of electrostatic interactions on the diffusion in charged gel by measuring the mutual diffusion coefficient of globular protein myoglobin in neutral gel agarose and in anionic gel carrageenan over a range of pH values from 2.0 to 9.0 and over a range of ionic strength from 0.01 to 0.5 M. The advantage to using these kinds of polysaccharide gels is that they do not swell or shrink when exposed to ionic solutions, so the volume fraction of fibers remains constant. We have found that the diffusion of myoglobin in charged gel is greatly influenced by the electrostatic interaction. A quantitative analysis of the diffusion coefficients has been made to elucidate the mechanism of the diffusion process in the charged gels. Experiments Materials. Agarose (Agarose S, no. 312-01193) and κ-carrageenan (no. 033-09292) were obtained from Wako Co. Ltd. and λ-carrageenan (no. 24203-1610) was obtained from Junsei Chemical Co., Ltd. and used without further purification. The gelation temperatures of agarose and κ-, λ-carrageenan were approximately 42 °C. To prepare the gel phases for diffusion measurements, the polysaccharide solution at a prescribed weight percentage (0.5-4 wt %) was prepared by dissolving the polysaccharide powder in deionized water or in 40 mM buffer solution (Tris-HCl at pH ) 2, Tris-malate at pH ) 6.8 and pH ) 9.0) and slowly heating to the solution boiling temperature. The KCl salt concentration was changed from 0.01 to 0.5 M. The solution was kept at this temperature until the
10.1021/jp001438o CCC: $19.00 © 2000 American Chemical Society Published on Web 09/29/2000
Effect of Charge on Protein Diffusion in Hydrogels
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9899
polysaccharide was completely dissolved. Then it was cooled to approximately 80 °C, being stirred until it appeared homogeneous. After that, it was transferred to a glass spectrophotometric cuvette using a syringe. The cuvette was then cooled to the room temperature for at least an hour to ensure a complete gelation. A piece of rectangular plastic was inserted into the cuvette to keep the surface of the gel flat. The gel length in the cuvette was about 1.8 cm. Myoglobin from horse muscle (no. M-0630) was obtained from SIGMA (St. Louis, MO) and used without further purification. The myoglobin aqueous solution was prepared by dissolving the myoglobin (5 wt %) in deionized water or 40 mM buffer solution (Tris-malate at pH ) 2.0 and pH ) 6.8, Tris-HCl at pH ) 9.0). The KCl concentration was changed from 0.01 to 0.5 M. The molecular weight of the myoglobin is M ) 17 600, and its molecular size is 4.5 nm × 3.5 nm × 2.5 nm.11 Its isoelectric point is pI ) 6.73. The self-diffusion coefficient D0 of myoglobin is 1.07 × 10-6 cm2/s.11 Measurements. Diffusion Coefficient of Protein in Solution. The self-diffusion coefficient (D) of myoglobin in various buffer solutions was measured at 20 °C by the stimulated echo 1H NMR method using a JEOL ALPHA600 (600 MHz) which was equipped with a pulsed field gradient unit and actively shielded triple-resonance probe (Nalorac). The relationship between the echo signal intensity and pulse field gradient parameters is given by29
ln[A(δ)/A(0)] ) -γ G Dδ (∆ - δ/3) 2
2
2
(1)
where A(δ) and A(0) are echo signal intensity at t ) ∆, with and without the field gradient pulse length δ, respectively, γ is the gyromagnetic ratio of the proton, G is the field gradient strength, D is the self-diffusion coefficient, and ∆ is the gradient pulse interval. The echo signal intensity was measured as a function of G. The plots of ln[A(δ)/A(0)] against γ2G2δ2(∆ δ/3) give a straight line with a slope of -D. The ∆ and G values employed in these experiments were 800 ms and 0.275-54.725 G/cm, respectively. The field gradient pulse length δ was 0.9 ms. Diffusion Coefficient of Protein in Gels. The mutual diffusion coefficients of the protein in the gel were measured by the newly developed ESPI method. The principle of the experiment has been reported in the previous paper.28 A continuous wave HeNe laser emitting coherent light at 632.8 nm is used as the light source. The laser beam is divided into a reference beam and an object beam by a beam splitter. Each beam is focused through a pinhole spatial filter by a 25× microscope objective, and then each beam passes through collimating lenses. The object beam traverses the diffusion cell, and the reference beam is reflected in the same way as the object beam. The two beams impinge on the CCD array. The interference fringes are recorded with a cooled CCD camera. The total image area contains 1280 × 1024 pixels. The diffusion cell is a 1.0 cm × 4.5 cm spectrophotometric cuvette with a 5 mm light path. The myoglobin solution was put into the diffusion cell, and the dual-illumination images were sequentially taken and stored on the hard disk at 15 min intervals for 15 h. The background noise was subtracted from all the images taken during the image acquisition step. The diffusion constant was evaluated using the relations described in the previous report.28 The interface between gel and solution was defined as x ) 0 and the region of gel as x > 0. In the diffusion processes of proteins, we could not observe the fringes at x < x′ where x′ was the position of ∆Cmax because of the low velocities of the protein diffusion. So, positions of
Figure 1. Diffusion coefficients of myoglobin in D2O buffer solution at 20 °C: (9) pH ) 2.0; (b) pH ) 6.8 (pI of myoglobin); (2) pH ) 9.0.
three next neighboring fringes, x1, x2, and x3, were determined to estimate the diffusion constant using
∆Ci ) A
∫xx /2/2xxDtDt i
1
i
2
exp(-η2) dη
i ) 1, 2, 3, ... (2)
and
∆C2 - ∆C1 ) ∆C3 - ∆C2
(3)
where ∆Ci ) C(xi,t1) - C(xi,t2) and A was a constant. Measurements were made in at least duplicate at 20 °C. The main source of error is the coarseness of the fringe peak in the digital image processing. The diffusion coefficient was the average estimated from 30 images. The maximum value and the minimum value are shown as the range of error bars in the figures. A detailed discussion on the errors of ESPI was given in one of our previous papers.28 Partitioning of Myoglobin. The partitioning of the myoglobin with the 3 wt % λ-carrageenan gel was measured by immersing a piece of the gel (10 × 8 × 5 mm3) in myoglobin solution of various concentration and measuring the changes in the UV absorption of the myoglobin solution at 510 nm (molar absorbance of myoglobin: ) 7760) after the equilibrium (36 h). The amount of myoglobin diffused into the gel was expressed in terms of β, which is the molar ratio of the myoglobin molecules in the gel to the sulfonate groups of the gel. The detailed procedure and the analysis were reported elsewhere.30 The partition coefficient was calculated from the concentration ratio of myoglobin in the gel and in the surrounding solution. Results and Discussions To obtain the self-diffusion coefficients of myoglobin in solution at the infinite concentration, the myoglobin concentration dependence of the diffusion coefficients was first studied at pH ) 2.0, 6.8, and 9.0 using the NMR method. These pH values were chosen since myoglobin is in positively, neutrally (pI ) 6.8), and negatively charged gel at pH ) 2.0, 6.8, and 9.0, respectively, and this serves to elucidate the effects of charges and size change accordingly. As shown in Figure 1, the diffusion coefficients of myoglobin decreased with increasing myoglobin concentration due to increased interaction with adjacent macromolecules. Since the experimental data showed good linear concentration dependence, the values of selfdiffusion coefficient D0 at infinite concentration were obtained by extrapolating to zero concentration, and the results are listed in Table 1. The Stokes-Einstein radii of myoglobin were also
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TABLE 1: Diffusion Coefficient of Myoglobin at Infinite Concentration Obtained by NMR Measurement and the Hydrodynamic Radius Obtained from the Stokes Equation D0 ) (kT)/(6πηRh) pH D0 (×10-7 cm2/s) Rh (nm)
2
6.8
9
8.21 2.32
9.06 2.09
11.1 1.72
Figure 3. Dependences of diffusion coefficient of myoglobin in polysaccharide gel on the fiber volume fraction of the gel with various pH values and 0.01 M KCl at 20 °C. In agarose gel: (0) pH ) 2; (O) pH ) 6.8; (4) pH ) 9. In λ-carrageenan gel: (9) pH ) 2; (b) pH ) 6.8; (2) pH ) 9. Numbers in the figure are pH values.
Figure 2. 2. Dependences of diffusion coefficient of myoglobin in agarose gel (O) and in λ-carrageenan gel (b) on the fiber volume fraction at pH ) 6.8 containing 0.01 M KCl at 20 °C.
calculated from these D0 using the Stokes-Einstein equation,31 and the values were summarized in Table 1. Before investigating the effect of charge on diffusion in gels, we need to discuss the effect of the fiber radius of the gel on the diffusion. Figure 2 shows the diffusion coefficient of myoglobin in agarose gel and λ-carrageenan gels at pH ) 6.8 containing 0.01 M KCl as a function of fiber volume fraction. As has been reported in our previous paper, the result in agarose gel shows a good agreement with the results reported by Kong et al., measured by holographic interferometry.1,28 One can easily see that the diffusion coefficient in the λ-carrageenan gel is smaller than that in agarose gel under the same gel fiber volume fraction φ. The isoelectric point of myoglobin is pH ) 6.8, so the net surface charge is zero at pH ) 6.8, and there should be no distinct electrostatic interaction between myoglobin and λ-carrageenan gel; a notable difference in observed D/D0 in the two gels is apparently attributed to the different fiber radii of the gels. Agarose fiber has an average radius of 1.9 nm, whereas λ-carrageenan fiber radii range from 0.2 to 0.5 nm, depending on the concentration.15 Thus, a difference in fiber size exhibits a pronounced effect on the diffusion process in the gel network. According to Ogston,32 the probability PR>Rh that “space size” R is greater than the solute radius in a medium of straight cylindrical fibers of radius Rf and volume fraction φ is expressed as follows:
[ ( )]
PR>Rf ) exp -φ 1 +
Rh Rf
2
(4)
where Rh is the hydrodynamic radius of the solute. From eq 4, supposing Rh ) 2 nm, we calculated that 91% of total myoglobin can penetrate a 2 wt % agarose gel whose fiber radius Rf is 1.9 nm, while only 32% can penetrate λ-carrageenan gel whose fiber radius is 0.29 nm, indicating a substantial effect of fiber radius on the diffusion process. Since D/D0 is proportional to the penetration probability, the diffusion coefficient in agarose gel should be 2.8 times that in the λ-carrageenan gel at 2 wt %
fiber concentration. However, the D/D0 is 0.79 in agarose gel and 0.55 in λ-carrageenan gel at 2 wt % from Figure 2, which gives a ratio of 1.4, much smaller than the value estimated from eq 4. One should also notice that D/D0 in agarose linearly decreases with increasing fiber volume fraction while that in λ-carrageenan does not. These deviations might be associated with the “cage” effects. Although a protein can penetrate through the free “available” space that is defined from eq 4, the “cage” of surrounding fibers can trap it. Agarose has rather a rigid chain compared to λ-carrageenan, which leads to an enhanced “cage” effect and results in a decreased diffusion rate of protein. On the other hand, the fiber networks of the λ-carrageenan gel are quite flexible, undergoing active thermal motions in the fluid and behaving as if the real “space size” is larger than that of the “frozen” fibers to bring about increased diffusibility. Johansson et al. has studied the effect of the stiffness of the polymer chains in a network on the particle diffusion by considering the ratio of fiber persistence length to the fiber radius and found, by simulation, that for a constant fiber volume fraction, there is a remarkable decrease in D/D0 with the increase in the persistence length.33 So the differences between the experimental results and the estimated value from eq 4 in two polysaccharide gels could be explained by this reasoning. Figure 3 shows D/D0 at pH ) 2.0, 6.8, and 9.0. We can see that by changing pH, the diffusion of myoglobin is more strongly influenced in λ-carrageenan gel than in agarose gel, obviously due to the ionized nature of λ-carrageenan gel. The diffusion of myoglobin is enhanced and suppressed at pH ) 2.0 and 9, respectively, in comparison with that at pH ) 6.8, the isoelectric point of the protein. Since the sulfonic acid groups on the λ-carrageenan gel are fully ionized regardless of the change in pH, the pH-dependence of the diffusion suggests an electrostatic interaction between the protein and the charged gel. According to our previous study on the diffusion of ionic surfactant into an oppositely charged polymer gel, the observed mutual diffusion coefficient increases with the enhancement of the attractive interaction between the surfactant and the gel.21,22 It has been clarified that, though the driving force of surfactant diffusion into the gel is the concentration gradient of the surfactant, the binding of the surfactant with the polymer network sustains a high concentration gradient of the mobile surfactant between the outside and inside of the gel, which facilitates the subsequent surfactant diffusion to give an apparent diffusion coefficient much larger than that without binding.21,22 The enhanced diffusion at pH ) 2 in the present case might
Effect of Charge on Protein Diffusion in Hydrogels
Figure 4. Dependences of diffusion coefficient of myoglobin in λ-carrageenan gel on the fiber volume fraction of the gel with various pH values and KCl concentrations at 20 °C: (9, 0) pH ) 2; (b, O) pH ) 6.8; (2, 4) pH ) 9; (closed marks) 0.01 M KCl; (open marks) 0.5 M KCl. Numbers in the figure are pH values.
also be attributed to the attractive interaction or binding between myoglobin, which has a net positive surface charge at pH ) 2, and the λ-carrageenan gel, which is negatively charged. This result is not contradictory to that reported by Johannson et al. that an attractive electrostatic interaction between the diffusant and the fiber decreases the observed diffusion coefficient.34,35 Johannson et al. measured the self-diffusion coefficients at all concentrations using a radiolabel technique, whereas the present method measures mutual diffusion coefficients at all concentrations in the gel. The suppression of the diffusion rate of myoglobin at pH ) 9.0 can be, therefore, associated with the electrostatic repulsion between the solute protein and the gel. When the polymer network is repulsive to the protein, the accessible space of the protein in the gel decreases, which is equivalent to an increase in the fiber radius or the protein radius by approximately a thickness of Debye length for a constant fiber-fiber distance. It would be interesting, accordingly, to see how the ionic strength affects the diffusion at these pH values. As shown in Figure 4, when the KCl concentration is increased from 0.01 M (closed marks) to 0.5 M (opened marks), no differences in the diffusion coefficients could be observed at any pH in agarose gel. The diffusion coefficient in λ-carrageenan gel at pH ) 6.8 also does not change with the change in the ionic strength. These results indicate that there is no electrostatic interaction and no change in the gel structure upon change in the salt concentration. On the other hand, the diffusion of myoglobin in λ-carrageenan gel decreases at pH ) 2.0, while it increases at pH ) 9.0, especially at a low fiber volume fraction, with the increase in the salt concentration. That is, the electrostatic interaction (attraction at pH ) 2 and repulsion at pH ) 9) is partly shielded by increasing the salt concentration. However, the salt effect seems to be quite weak at 0.5 M considering a change in the Debye length from 3 nm (0.01 M) to 0.4 nm (0.5 M). Since the diffusion of myoglobin at its ioselectric point does not change with the salt concentration, the weak effect of salt should not be associated with the change in fiber conformation. The influence of salt on the protein conformation is not known and might be accounted for. Therefore, we further investigated the salt effect in a wider concentration range at pH ) 8.1, and the results are shown in Figure 5. As shown in Figure 5, there is no salt effect in agarose gels. D/D0 of myoglobin in λ-carrageenan gel systematically increases when KCl concentration increases from 0-1.0 M and approaches the data at its isoelectric point. The effect is especially notable at a lower fiber concentra-
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9901
Figure 5. Diffusion coefficients of myoglobin in polysaccharide gel with various KCl salt concentrations at 20 °C. The pH of the solution is 8.1. In agarose gels: (O) 0 M KCl; (4) 0.5 M KCl; (3) 1 M KCl. In λ-carrageenan gel: (b) 0 M KCl; ()) 0.01 M KCl; (2) 0.5 M KCl; (1) 1 M KCl.
Figure 6. Dependences of diffusion coefficient of myoglobin in various polysaccharide gels on the fiber volume fraction of the gel at 20 °C. The pH of the solution is 8.1: (() agarose gel; (2) λ-carrageenan gel; (1) λ-carrageenan gel.
tion due to more effective electrostatic shielding in a low fiber concentration of λ-carrageenan gel. κ-Carrageenan gel has one sulfonic acid per repeating disaccharide unit, and λ-carrageenan gel has three. In other words, the charge density of λ-carrageenan gel is 3 times larger than of that of κ-carrageenan gel at the same fiber volume fraction. Therefore, the electrostatic interaction of λ-carrageenan gel with the protein should be lower than that of κ-carrageenan gel. As shown in Figure 6, the diffusion coefficients of myoglobin in λ-carrageenan gel at pH ) 8.1 were the smallest among the three polysaccharide gels. Thus, it was found that the electrostatic interaction strongly affects the diffusion process. Now an important problem arises: how strongly does the myoglobin undergo interaction with charged network in the gel? To evaluate this quantitatively, the partitioning of myoglobin into the gel has been established at various pH values and ionic concentrations. We define β as the molar ratio of myoglobin diffused into the gel to the total sulfonate groups in the λ-carrageenan gel and plot β versus the equilibrium concentration of the surrounding myoglobin solution, as shown in Figure 7. The partitioning behavior substantially changed with pH values in 0.01 M KCl. Compared to the β at pH ) 6.8, in which myoglobin was neutral, we can see β values at pH ) 2, where myoglobin has net positive charge, show the largest values and β at pH ) 9 show the smallest values at any myoglobin concentration, and this difference disappears in 0.5 M KCl. Such
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Figure 7. Partitioning of myoglobin in λ-carrageenan gel with (a) 0.01 M KCl and (b) 0.5 M KCl at 20 °C: (9) pH ) 2; (b) pH ) 6.8; (2) pH ) 9. The dashed line is the concentration of myoglobin solution used in the ESPI diffusion experiment.
TABLE 2: Partition Coefficients KA of Myoglobin in λ-Carrageenan Gel
Figure 8. Clague and Phillips simulation results for the diffusion of myoglobin in polysaccharide gels compared with the experimental data (a) in agarose gel (opened marks) and (b) in λ-carrageenan gel (closed marks): (0, 9) pH ) 2; (O, b) pH ) 6.8; (4, 2) pH ) 9. The lines represent simulation results using the fiber radius Rf ) 1.9 nm and the myoglobin radius Rh at various pH values shown in Table 1. Note that no free parameters are used in the simulation.
ionic strength 0.01 M KA
0.5 M
pH ) 2
pH ) 6.8
pH ) 9
pH ) 2
pH ) 6.8
pH ) 9
1.08
0.58
0.22
0.68
0.72
0.58
different partitioning behaviors with change in pH and ionic strength clearly demonstrate the important contribution of the electrostatic interaction. The partition coefficient KA of myoglobin in 3 wt % λ-carrageenan gel was calculated at various pH values and ionic strengths from Figure 7, and the results are summarized in Table 2. The KA at pH ) 2.0 and 0.01 M KCl is 5 times larger than that at pH ) 9.0, and this value decreased at pH ) 2.0 and increased at pH ) 9.0 in the presence of 0.5 M KCl, showing almost the same value in any pH. Protein transport in the hydrogels may occur primarily within the water-filled regions in the space made by the polymer network.36 Any factors reducing the size of these spaces may decrease the diffusion of the solute. Such factors include the size of the solute in relation to the size of the space between polymer chains, polymer chain mobility, and the existence of charges on the network that we have demonstrated. The diffusivity of a solute through a physically cross-linked hydrogel decreases as the size of the solute increases, as the cross-linking density increases, and as the volume fraction of water within the gel decreases. The polymer chains may retard protein diffusion by reducing the average free volume available to the solute and by the hydrodynamic motion of the network, which acts as a physical obstruction, thereby increasing the path length of the solute. In the past, various mathematical models have been derived to explain and predict solute diffusion in hydrogels, and the models were tested with experimental data.32,33,37-39 In these previous models, any contributions of the electrostatic interactions between hydrogels and solutes were neglected. Among the previous models, we found that the diffusion data of spherical myoglobin (aspect ratio: R ) 1.6) in agarose can
be fitted best by using the combined hydrodynamic and obstruction model proposed by Clague and Phillips for hindered diffusion of spherical solutes in porous media comprised of straight, cylindrical fibers.37
2 -1 D ) 1 + R exp[-πφ0.174 ln(59.6(Rh/Rf))] D0 3
(
)
(5)
where
R)φ
(
)
R f + Rh Rf
2
(6)
This model accounts rigorously both for the enhanced hydrodynamic drag on a solute that is caused by the presence of a surrounding fibrous medium and for the steric effect on the path of diffusion, and the model allows for the predicting of the rates of protein diffusion in agarose accurately with no adjustable parameters. Figure 8 shows the comparison between the experimental data obtained from Figure 3 and the Clague and Phillips simulation results for agarose gel (Figure 8a) and λ-carrageenan gel (Figure 8b) using the value of protein radii at various pH values, as given by Table 1. As shown in Figure 8a, though no free parameters are used in the calculation, this model shows a good agreement with the experimental data of agarose gel. However, the model does not fit with our data for λ-carrageenan gel at all (Figure 8b). On the other hand, we have found that the obstruction model by Tsai and Strieder,38 which describes the transport properties of molecules in random arrays of overlapping fibers, shows a good agreement with our experimental data in λ-carrageenan.40
2 -1 D ) 1+ R D0 3
(
)
(7)
Effect of Charge on Protein Diffusion in Hydrogels
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9903 Acknowledgment. This research was supported by Grantin-Aid for the Specially Promoted Research Project, “Construction of Biomimetic Moving System Using Polymer Gels”, from the Ministry of Education, Science, Sports and Culture of Japan. References and Notes
Figure 9. Tsai and Streider simulation results for the diffusion of myoglobin in λ-carrageenan gel compared with the experimental data: (9) pH ) 2; (b) pH ) 6.8; (2) pH ) 9. The solid line represents the simulation result using the fiber radius Rf ) 0.29 nm and myoglobin radius Rh ) 2.09 nm obtained by NMR analysis at pH ) 6.8. Note that no free parameters are used in the simulation of the solid curve and it agrees well with the experimental data at the isoelectric point of myoglobin. The broken line and the dashed line are the best fit of the experimental results at pH ) 2 and pH ) 9, respectively, which gives apparent radii Rah of the protein as shown in Table 3.
TABLE 3: Hydrodynamic Radius of Myoglobin Obtained from NMR Measurement and ESPI Diffusion
pH ) 2 pH ) 6.8 pH ) 9
Rh in buffer from NMR (nm)
apparent Rah in gel (nm)
Rah/Rh
2.32 2.09 1.72
1.73 2.08 2.2
0.75 1 1.28
The solid line in Figure 9 shows the Tsai and Streider simulation result using the myoglobin radius obtained from NMR analysis at the isoelectric point (Rh ) 2.09 nm). The simulation is in good agreement with the experimental data at pH ) 6.8. Here, we can estimate apparent radii of protein, Rah, at pH ) 2 and pH ) 9, from the best fitting curve of the Tsai and Streider model by keeping Rf nm constant. Table 3 shows the comparison between the myoglobin radius Rh obtained from NMR measurement and the apparent value Rah. The apparent radius at pH ) 9 is larger than the NMR results, and the apparent radius at pH ) 2 is smaller than the NMR results. These results indicate that at pH ) 9 the apparent radius of myoglobin becomes larger than the true hydrodynamic radius due to the electrostatic repulsion between gel and solute. On the other side, the apparent myoglobin radius becomes smaller than the true radius at pH ) 2 due to the electrostatic attraction between gel and solute. The ratio of Rah/Rh is listed in Table 3. However, as shown in Figure 9, a considerable deviation from the experimented data is still observed, especially at the low fiber concentration at pH ) 2 and pH ) 9. This should be associated with the decreased ionic strength due to decrease in the fiber concentration, and an extended theory capable of more accurately predicting the diffusion in charged gels is necessary.
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