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Anal. Chem. 1969, 6 1 , 362-366
Computer Simulation and Experimental Validation of the Electrophoretic Behavior of Proteins Richard A. Mosher,* Douglas Dewey, Wolfgang Thormann,’ Dudley A. Saville,2 and Milan Bier Center for Separation Science, University of Arizona, Tucson. Arizona 85712
A mathematical model of the electrophoretic behavior of protelns Is presented. The Debye-Huckel-Henry theory is used for the description of proteln mobillty, which has the important result of maklng net mobillty a functlon of Ionic strength. A net charge vs pH relatlonshlp and a diffusion coefflclent are required to descrlbe a speclfic proteln. The model is employed for the computer slmulatlon of three distinct electrophoretic modes: Isoelectric focusing, lsotachophoresls, and zone electrophoreds. The valldlty of the model Is tested by comparlng slmulatlon with experlmental data. Excellent qualtatlve agreement was found.
INTRODUCTION The most significant recent advance in electrophoretic theory is the use of digital computers to solve transport equations that contain both electromigrational and diffusional flux terms. This allows prediction not only of the composition of steady-state zones, as did prior theories, but also of the shapes and evolution of the intervening boundaries. An additional benefit provided by numerical techniques is the ability to describe the interaction of weak buffers, and the associated dynamics in local pH and conductivity gradients. This eliminates the need for assumptions concerning those gradients (1-3). Such assumptions often derived from attempts to simplify the description of a specific electrophoretic mode, such as isotachophoresis (ITP), isoelectric focusing (IEF),zone electrophoresis (ZE), or moving boundary electrophoresis (MBE). Our laboratory developed the first description of the transient electrophoretic behavior of electrolyte systems that encompassed biprotic ampholytes and monovalent weak and strong acids and basis (4-6). That model emphasized the unity that underlies all electrophoretic methods, reproducing the salient features of each mode. The unified theory revealed how the initial conditions, i.e. the initial distribution of the buffers and samples, and boundary conditions, i.e. the permeability of the ends of the separation column, combine to produce a specific mode. All experimental methods can be viewed as a set of initial conditions joined to specific boundary conditions and often overlaid with a constraint such as molecular sieving. Simulation data have added substantially to our understanding of fundamental electrophoretic phenomena. For example, the establishment of stable pH gradients for IEF using just one weak acid and one weak base has been described theoretically and utilized experimentally (7). The focusing mechanism of idealized multicomponent ampholyte mixtures has been thoroughly characterized (€9,as has the ITP behavior of amphoteric samples (9). This work has involved extensive experimental verification in several different capillary configurations (8, 10, 11) as well as in continuous flow devices (12). Relevance to the practitioner has been demonstrated Present address: Department of Clinical Pharmacology, University of Bern, Bern, Switzerland. *Department of Chemical Engineering, Princeton University, Princeton, NJ 08544. 0003-2700/89/036 1-0362$01.50/0
by revealing the source of several experimental phenomena, including anomalous spikes in isotachopherograms (13),and pH gradient instabilities in IEF termed the plateau phenomenon (14) and cathodic drift (11). The flexibility of the model is underscored by the capacity to simulate related processes such as electrodialysis (4) and sample behavior in the presence of immobilized pH gradients (15). One of the most important applications of electrophoresis is protein analysis and purification. Here we report the addition of a description of protein behavior to the model. The appropriate modifications are presented along with examples of simulation results and comparison to corresponding experimental data. MODEL FOR T H E ELECTROPHORETIC BEHAVIOR O F PROTEINS The model is one dimensional and based upon the principles of electroneutrality and conservation of mass and charge. Relationships between the concentrations of the various species of a component are described by equilibrium constants. Isothermal conditions are assumed, as is the absence of bulk flow. First, the required modifications of the existing mathematical model (5)to include protein behavior are presented. For purposes of simplicity the relationships are presented for one protein although the computer programs can handle any number of proteins. Dissociation Model. To describe the association/dissociation reactions of a protein with water, the designation [PI is used to represent the total concentration of the protein, one molecule of which binds a maximum of j protons. [Pj] is the concentration of the species with j dissociable protons. The dissociation reactions are shown in eq 1. From these, the
Pj + Pj-1 + H+
P1 + Po + H+ familiar equilibrium expressions are obtained, as shown in eq 2. Here Pj is the species with all possible protons bound and
K1 = [Pi-11W+I/ [Pi1 K2 = [Pj-,I [H+I/ [Pj-ll
(2)
Kj = [POI[H+I/ [PlI Po is the species that has lost all dissociable hydrogen ions. The total concentration of the protein can be expressed in terms of these dissociation constants
where i
UKl = KlK2 ... Ki
1=1
0 1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989
Table I. pH Dependence of the Ionization of Albumin and Hemoglobin" PH
albumin
3.0 3.5 3.7 4.0 4.2 4.5 5.0 6.5 7.0 9.0 10.0 10.5 10.85 11.0 11.5 12.0
net charge hemoglobin
58 35.5
34
z, is equivalent to standard valence for small ions. An expression for z2 for a macroion appears within the square
20.05 0.0 -20.5
-32 -35 -37 -44 -63.5 -84
" Ionization data were adopted from ref 17 for hemoglobin and from ref 18 for albumin. The diffusion coefficient used for hemoglobin is 6.8 X lo-" m2/s (19) and for albumin is 5.94 x mz/s (20)*
Similarly, the average concentration of hydrogen ions removed is given by [ P .] i [H+]-'h Kl i=l
f:i [H+]-'h Kl
Thus, if the net charge of the molecules possessing all j protons is zo, then the average charge at a given pH is z = zo - 8. This is obtained from titration data which is used to generate a tabular relationship of net charge as a function of pH. Also needed is the mean square charge which will appear in the expression for the current. This is calculated by first deriving an expression for the "mean square number of protons removed per molecule" i2[ H+]-i
T o extract
fiK, 1=1
i=l
1
d[P]
(6)
+ ih=[l H + ] - i1=1 hKl
(7)from this tabular
i+l
d t = -VC(Fi i=l
+ Ri)
where Ri is the rate of production of the ith species. Fi is the flux of the ith species in mol/(m2 s) and results from electromigration and diffusion Fi
= -ziQ[Pi]V4 - DV[P;]
(13)
where V 4 is the potential gradient. Since ERi is nil, the statement for conservation of mass of the protein becomes
1=1
i=l
(11)
where D is the diffusion coefficient (m2/s). Conservation of Mass. The general equation expressing the conservation of mass of the protein is equivalent to that for low molecular weight compounds
(5)
1+ i[H+]-'hK1
=
r = kT/6?rqD
1=1
i=l
3=
-
brackets of expression 17. n, is the ionic concentration (molecules/m3),N is the number of species present, c is the dielectric constant of the fluid (unitless, 80 for H 2 0 at 25 "C), eo is the permittivity of free space (8.854 X C2/(J m)), k is Boltzmann's constant (1.38 X lowz3 J/(deg molecule)), and T i s the Kelvin temperature (298). This has the important result of making the protein mobility a function of the ionic strength. The parameter g(Kr) in eq 9 is a dimensionless function that varies in a sigmoidal fashion from 1to 1.5 (16). The radius of the protein is calculated from the diffusion coefficient using the Stokes equation,
(4)
/=1
Dividing eq 4 by the right-hand side of eq 3 yields the number of protons removed per molecule
(v2)
viscosity (8.95 X kg/(m s), H 2 0 at 25 "C), e is the unit C), r is the particle radius (m), and K is the charge (1.6 X Debye parameter (m-l). This parameter is defined by the following equation:
68.5
13 3 1.7 -1.5 -10.5 -14.0
' f:
383
d[P]/dt = V((z0 - ij)Q[P]V@+ DV[P]J
(14)
Conservation of Charge. For protein-free systems, the charge conservation relation is N
0 = V C fzmFm m=l
(15)
where N is the number of components present and f is Faraday's constant (96500 C/mol). The contribution of the protein is expressed by i -E((zo- i)2fQ[P,-l]V4 + fD(zo - i)V[Pj-,]] (16) i=O
data we note
This can be expanded to (7)
-[(z0)'
- 232O + (y2)]fQ[P]V@-fD(zo - e)V[P] (17)
which is the term included within the summation in eq 15 to account for the contribution of the protein. Electroneutrality. The expression of electroneutrality in the absence of protein is
and thus
N
The ionization of hemoglobin and albumin as a function of pH is presented in Table I. Protein Mobility. The Debye-Huckel-Henry (16) description of protein mobility a t unit charge is eg(d Q = 6.lrar(l Kr)
(9)
+
where Q is the electrophoretic mobility (m2/(V s)),
9
is the
C z,n,
m=l
=0
(18)
where N is the total number of species present with concentration n. The contribution of the protein (zo - P ) Pis added to the left-hand side of eq 18.
EXPERIMENTAL SECTION All experimental data reported here were obtained with the Elphor VaP 22 (Bender and Hobein, Munich, FRG),a continuous
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989
Table 11. Electrochemical Parameters of Small Molecules Used in Simulation
l l
mobility, m2/(V s) x
compound
PK1
PK2
108
Glu '4%
2.16 9.04 3.75
4.29 12.48
2.97 2.26 5.64 5.19 2.41 7.91 36.27 19.87
formic acid Na+
Tris
c1-
H+ OH-
8.3
flow instrument. Fluid enters the top of the vertical, rectangular separation chamber (0.5 mm X 10 cm X 50 cm) and is fractionated by 90 outlet tubes. Five inlets are provided for infusion of buffer(s) and four for sample($ Interelectrode distance is 10 cm with the field applied transverse to the flow. Membranes are used to isolate the electrode compartments from the separation chamber. Just prior to exit a computer-controlled reflectance scanner records the absorbance at 280 nm along the separation axis. Bovine serum albumin (BSA) was obtained from Sigma, St. Louis, MO., as a 30% solution. Human hemoglobin was prepared as follows: 50 mL of fresh whole blood was mixed with 57 mg of Na8EDTA and centrifuged for 10 min at 5000 rpm in a Sorvall SS-34 rotor. The packed cells were washed six times with an equal volume of 0.15 M NaCl (ca. 25 mL), using the same centrifugation conditions. After the final centrifugation an equal volume of distilled water was added to the packed cells and the mixture was vortexed to promote lysis. Carbon monoxide (CO)was then gently bubbled through the solution for 10 min and the solution was again centrifuged for 30 min at 10000 rpm in the Sorval SS-34 rotor. The supernatant was decanted and stored under CO.
RESULTS AND DISCUSSION Each of the common electrophoretic modes has a characteristic sample behavior and background buffer distribution. IEF is unique in that a stationary steady-state sample and buffer distribution is produced. This method is dedicated to the fractionation of ampholytes with separation based on differences in isoelectric points (PI). The background buffer system must create a pH gradient that spans the PISof the samples of interest. In the presence of an electric field of the proper polarity, samples migrate until their position corresponds to the pH a t which they have a zero net charge. In ITP, a discontinuous buffer system is used in which the net mobility of the leading ion is greater than that of every sample ion. Sample zones stack according to their net mobility, display a steady-state shape, and migrate at the same velocity. This stack is followed by a terminator whose net mobility is less than that of all samples. In ZE a uniform buffer system is used in which separation is based on differences in electrophoretic mobility. ZE sample zones change relative position and normally display an approximately Gaussian profile which broadens throughout the experiment. The following section presents a qualitative comparison of simulation and experimental data. It should be noted that experiments were performed at 4 "C, whereas simulation conditions assume 25 "C. Isoelectric Focusing. Two ampholytes were used to generate the pH gradient, the amino acids arginine (Arg) and glutamic acid (Glu). (Their electrochemical parameters along with those of the other small molecules used in simulations are presented in Table 11.) The sample was BSA. Figure 1 displays simulation (right) and corresponding experimental (left) data showing the focusing dynamics of this mixture. At t = 0 (not represented in the figure) both the simulation and the experimental separation space are occupied by a uniform mixture of 15 mM Arg and Glu and 0.68 mg/mL BSA. The distribution in panel A is an early stage of focusing, panel B
5.8
5 2 4 -...3 2 9. 6
P
'2L 1 2
0 0
FRACTION NUMBER
1 4
0.2 0.2
0.4 0.4
0.6 O B COLUMN LENGTH crn
1.0
0
Figure 1. Isoelectric focusing behavior of bovine serum albumin. The anode is to the right in all panels. Residence time within the separation chamber, voltage, and current for the experimental data in panels A, B,andC,are11.7min,511V,30mA;11.7min,950V,50mA;and 11.7 min, 1740 V, 90 mA, respectively. These panels present absorbance profiles obtained with the scanner. A residence time of 11.7 min corresponds to an infusion rate of 2.1 mL/min. Temperature in the chamber was 4 "C throughout the experiment. A cation exchange membrane was used to isolate the anolyte (0.1 M H,PO,) from the separation chamber, an anion exchange membrane served this purpose for the cathoiyte (0.l M NaOH). Simulation data were obtained at a constant current of 5 A/mZ. Distributions are presented after 20, 30 and 100 min of current flow in panels D, E, and F, respectively. The glutamic acid (dotted) and arginine (dashed) profiles are presented to show the relationship between the behavior of the background buffers and that of the sample. The concentration of the protein is given by the left side ordinate of panels D-F and of the amino acids by the right ordinate.
is an intermediate stage, and panel C presents a final focus. Experimental data are absorbance profiles recorded by the scanner. The degree of focusing achieved in the instrument was varied by adjusting the applied voltage. Corresponding simulation data are presented in panels D, E, and F. In both experimental and simulation data a peak of albumin is generated at each column end and migrates inward, meeting with its counterpart to form the final focus. That which arises from the anode, on the right, is somewhat stronger in both sets of data. This type of sample behavior in IEF has been termed the transient double peak approach to equilibrium (21). The final pH profiles in both sets of data have two plateaus corresponding to the PI values of the amino acids (3.2 and 11.7, Glu and Arg, respectively, data not shown). Isotachophoresis. A cationic system consisting of 10 mM sodium formate as leading electrolyte and 10 mM formic acid as terminating electrolyte was used for the experimental and the simulation work. Sodium acts as the leading ion, hydrogen ion as the terminator, and formate as the counterion in this system. The experimental data are presented in the left half of Figure 2, with the cathode to the right. Leading electrolyte was infused through four of the buffer inlets, and the terminating electrolyte through the fiih, anodic-most, buffer inlet of the Elphor VaP 22. The sample, 5 mg/mL BSA, was infused into the leader, close to the interface between the electrolytes. Panel A shows the albumin distribution in the outlet fractions, with the apparent heterogeneity being an artifact of the collection process. This is confirmed by the insert which displays a profile obtained with the scanner. Panel B presents the corresponding pH and conductivity profiles. These data clearly show the protein in stack. This system is unusual in that it is an example of enforced migration (22)where the conductivity of the sample zone is less than that of the terminator. Corresponding simulation data
ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989 2.0
A
1
'
'
365
z 1.6
c
3,2k
, , , , 1 . 5 3 . 6-.-1 , , ,:, 1 2 . 7 : ,.-.--.____..._.-.....~~~...~.~ ....____ _ __ _' _______
2.2
0
I
0,
16
32
48
64
1.8 80
0
1.4
2.8
4.2
5.5
7.0
COLUMN LENGTH cm
FRACTION NUMBER
Flgure 2. Isotachophoretic behavior of bovine serum albumin. The experimental data, presented In panels A and B, represent measurements made on collected fractions. The insert in panel A is an absorbance profile recorded by the scanner and indicates that the small peaks in the measured profile represent an artifact of collection and not sample heterogeneity. Residence time within the separation chamber was 11.5 min, which corresponds to an infusion rate of 2.1 mL/min of the background buffers. The sample was infused at 1.5 mL/h. Applied voltage was 406 V, with a current of 40 mA. 100 mM formic acid was used for both anolyte and catholyte. Dialysis membranes were used to isolate those solutions from the separation chamber, which was maintained at 4 'C. The cathode is to the right in all panels. Simulation data are presented in panels C and D. I n the initial configuration for the simulation, the interface between the leading and terminating electrolytes was 1.4 cm from the anodic column end. The sample zone was centered on this interface, with a width of 8% of the column len h and a concentration of 0.03 mM. The current density was 12 A/m . The profiles represent 110 min of current flow. I n panels B and D the pH scales are on the left and conductivity on the right.
F
are presented in panels C and D, showing the sample in stack in the electrolyte system in an enforced configuration. The amount of sample is insufficient to produce the plateau-shaped zone usually associated with ITP. The measured peak concentration is 2.0 mg/mL with the predicted plateau concentration value being 5.2 mg/mL. Zone Electrophoresis. The buffer employed for the ZE simulations and experiments consisted of tris(hydroxymethy1)aminomethane (Tris) titrated to p H 8.6 with HC1. Two systems are presented, one in which the buffer controls the background pH and conductivity throughout the separation space and one in which the sample has a significant impact on the local conductivity. The former will be referred to as a noninteractive system and the latter as an interactive system. Panel A of Figure 3 presents two profiles of hemoglobin obtained with the scanner. The first displays the sample profile in the absence of applied voltage, the second after migration in the noninteractive system. The sample concentration is 12 mg/mL in the background buffer of 200 mM Tris-C1. The migrating zone retains its Gaussian profile. The simulation data, presented in panel B, were generated by using the-same buffer and sample concentrations. The sample zone profiles are shown after 0, 10,20, and 30 min of current flow (300 A/m2). These data also reflect the maintenance of a Gaussian shape of the sample zone. In panel C, experimental data are presented for a system in which the background buffer concentration has been reduced by a factor of 10. The initial profile is unchanged from that in panel A. Electrophoretic migration in this interactive system produces a severe distortion of the sample zone. Under approximately the same voltage (430 V vs 385 V for panel A) the zone moves substantially farther in the same time. The distortion and increased migration rate appear in the simulation data presented in panel D, which displays hemoglobin concentration profiles
g
1.1
5c
0 5
g
5 .'
0.4
06
1.2
1.6
2.0
0.4
0.6
1.2
1.6
2 0
COLUMN LENGTH cm
Flgure 3. Zone electrophoretic behavior of human hemoglobin. Experimental data, representing absorbance profiles obtained with the scanner, are presented in panels A and C. 200 mM Tris-CI, pH 8.6, was used as the electrolyte in panel A,. with dialysis membranes isolating the electrode compartments from the separation chamber. I n this noninteractive system the applied voltage was 385 V producing 200 mA. The residence time was 11.5 min, corresponding to a buffer infusion rate of 2.1 mL/min. Sample (12 mg/mL) was infused at 1 mL/h. The same rates apply to the interactive experiment in panel C, for which the buffer was diluted 10-fold. The voltage was 430 V, with 26 mA. For the simulation data in panels B and D, a Gaussian sample profile with a peak concentration of 0.1875 mM was superimposed upon the uniform background buffer to produce the initial configuration. The current density was a constant 300 A/m2 in the noninteractive example (panel 8)and 30 A/m2 in panel D. The anode is to the right in each panel. The numbers represent minutes of current flow.
after 0, 10, and 20 min of current flow (30 A/m2). This asymmetric distortion; termed electrophoretic dispersion, is caused by the deviation in conductivity across the sample zone. In the first example the higher ratio of buffer to sample eliminates this effect, and diffusion is the primary dispersive force acting on the sample.
CONCLUSIONS The work presented confirms that the electrophoretic behavior of proteins, in configurations which reflect the classical categories of ZE, ITP, and IEF, can be accurately described by a single set of equations. The model used is different from that used by Roberts (23) for transient processes and by Shimao (24)for steady-state behavior. Thcee models neglected the corrections in mobility provided by the Debye-HuckelHenry description and in our hands did not faithfully reflect the experimental data. For example, albumin did not migrate in stack in the sodium formate ITP buffer system. The qualitative agreement between simulation and experimental data is excellent. Quantitative comparisons are much more difficult and depend on the availability and accuracy of the electrochemicaldata for both the small molecules and the proteins. Furthermore, experimental conditions in the Elphor VaP 22 never reflect the assumption of a completely quiescent solution. The crescent phenomenon is a well-known result of the interplay between the parabolic flow profile of the buffer, the electroosmotic flow (EEO), and electrophoresis (25), and can produce a sample distortion similar to that present in Figure 3C. That this phenomenon is not the cause of the observed behavior is indicated by the minimal spreading present in Figure 3A. A second potential source of the distortion is an electrically driven fluid motion termed electrohydrodynamic distortion (EHD) (26), which produces flow at conductivity or dielectric constant discontinuities. The impact of EHD was examined by repeating the experiment of Figure 3C using an ac field of the same strength. This eliminates the electrophoretic migration and dispersion and EEO, thus isolating any effects due to EHD (26). No
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989
sample broadening was observed in the presence of the ac field. In spite of these limitations, it is useful to make some quantitative comparisons between the experimental and simulation data. In the IEF mode the most pertinent is related to the efficiency of focusing, which is a function of the amount of charge per unit volume expended to coalesce the two transient peaks. In the simulation this convergence occurs after passage of 1.65 C/mL. To calculate this value for the experiment, we have used a chamber volume of 25 mL, a residence time of 11.7 min, and a current of 70 mA (Figure 1, panel C), to obtain 1.97 C/mL. In a similar comparison with albumin using identical buffer and sample concentration and performing the experiment in a ribbonlike capillary (IO), the efficiencies were 1.8 C/mL for the simulation and 2.2 C/mL for the experiment. The best quantitative comparison for the ZE mode is sample net mobility. This is simply calculated by dividing the observed velocity by the field strength, yielding 3.38 X lo4 m2/(V s) for the noninteractive experiment (panel A, Figure 3). The value computed from the simulation is 8.74 X low8m2/(V s). Correction of the simulation value for the difference in viscosity, due to the difference in temperature, at which the values are determined (25 "C for the simulation and 4 OC for the experiment) yields 4.99 X mz/(V s). The parameters of primary interest in ITP are net mobility and the plateau concentration value of a sample in a specific buffer system. In the example presented in Figure 2 the position of the sample zone is enforced not by the relative net mobilities of the leading, sample and terminating ions as is usual in ITP, but rather by the pH gradient which is present between the sample zone and the terminating electrolyte. This can be illustrated by examination of the computed net mobilities. In classical ITP, these values increase from terminator to sample to leader. In this example, however, the net mobility of the hydrogen ion, which is the terminator, is 3.43 X m2/(V s), that of the albumin is 2.05 x lo4 m2/(V s), and the sodium ion is 5.19 X lo* m2/(V s). These values suggest that the terminator should overtake the albumin zone. If this were to happen however, the protein would be in an environment with a pH of 2.77 and its net mobility would increase to 7.83 X lo4 m2/(V s), due to a substantial increase in charge. Thus, it is the pH gradient between the albumin zone and the terminating electrolyte that enforces this migration order. The example clearly demonstrates that proteins can migrate in an enforced configuration in ITP. The sample amount in this example was insufficient to establish a plateau. Simulation predicts a plateau concentration of 5.2 mg/mL for albumin in this system. Experimental attempts to duplicate this value produced a zone that displayed a ragged profile with an es-
timated average concentration 2- to 3-fold higher than the prediction. Possible sources of the distortion are electrohydrodynamic forces, enhanced viscosity, droplet sedimentation, and the flow reorientation that occurs at the collection ports. Effective continuous flow ITP of proteins will require the use of buffer systems which produce lower adjusted sample concentrations.
ACKNOWLEDGMENT The authors thank the Bender and Hobein Company, Munich, FRG, for the loan of the Elphor VaP 22 and Terry Long for capable experimental assistance. LITERATURE CITED Almgren, M. Chem. Scr. 1971, I , 69. Cann, J. R. Biophys. Chem. 1980, 1 7 , 249. Weiss, G. N.; Catsimpoolas, N.; Rodbard, D. Arch. Bimhem. Biophys. 1974, 763, 106. Bier, M.; Palusinski, 0. A,: Mosher, R. A,; Saville, D. A. Science 1983, 279, 1281. Saville, D. A.; Palusinski, 0. A. AIChEJ. 1988, 3 2 , 207. Palusinski, 0. A.; Graham, A.; Mosher. R. A,; Bier, M.; Saviile, D. A. AIChE J . 1986, 3 2 , 214. Mosher, R. A.; Thormann, W.; Graham, A,; Bier, M. Electrophoresis 1985. 6 , 545. Thormann, W.; Mosher, R. A.; Bier, M. J . Chromafcg. 1986, 357, 17. Mosher, R. A,; Thormann, W. Electrophoresis 1988, 7, 395. Thormann, W.; Tsai, A.; Michaud, J.-P.; Mosher, R. A,; Bier, M. J . Chromatogr. 1987, 389, 75. Mosher, R. A.; Thormann, W.; Bier, M. J . Chromatcgr. 1988, 436, 191. Kuhn, R.; Wagner, H.; Mosher, R. A,; Thormann, W. Nectrophoresis 1987, 8 , 503. Mosher, R. A.; Thormann, W.; Bier, M. J . Chromatogr. 1985, 320, 23. Mosher, R. A.; Thormann, W.; Bier, M. J . C h m t c g r . 1988,357,31. Mosher, R. A.: Bier, M.; Rlghetti, P. G. Electrophwesis 1986, 7, 59. Henry, D. C. Proc. R . SOC.London, A 1931. 133, 106. Cohn, E. J.; Green, A. A.; Blanchard, M. H. J . Am. Chem. SOC.1937, 5 9 , 509. Linderstrom-Lang, K.; Nielsen, S. 0. I n Electrophoresis;Bier, M., Ed.; Academic Press: New York, 1959; Voi. 1, p 85. Edsall. J. T. In The Proteins; Neurath, H., Bailey, K., Eds.: Academic Press: New York, 1953 Vol. 1, Pt. B, p 637. Mahler, H. R.; Cordes, E. H. Blobgkal Chemistry, 2nd ed.;Harper and Row: New York. 1971; p 87. Behnke, J. N.; Dahger, S. M.; Massey, T. H.; Deal, W. C., Jr. Anal. 8iochem. 1975, 6 9 , 1. Everaerts, F. M.; Beckers, J. L.; Verheggen. Th. P. E. M. Isotachophwesis--Theory, Instrumentation and Applications ; Elsevier: Amsterdam, 1976. Roberts, G. 0. NASA Contract Report, 1984; NASAGR-171034. Shlmao, K. Electrophoresis 1988, 7, 297. Strickler, A.; Sacks, T. In Isoelectric Focusing and Isotachophoresis; Catslmpoolas, N., Ed. Ann. N . Y . Acad. Scl. 1973, 209, 497. Rhodes, P. H.; Snyder, R . S.; Roberts, G. 0. J . Colloid Interface Sci., in press.
RECEIVED for review August 16, 1988. Accepted November 15,1988. This work was supported by NASA Grant NAGW693 and was presented in part a t Electrophoresis '88, Copenhagen, Denmark, 1988.
