Analytical Gel Chromatography of Proteins

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

.

By GARY K ACKERS Deportment of Biochemistry. University of Virginia. Chorlottesville. Virginia

. . . . . . . . . . . . . I . Introduct.ion . I1 . General Aspects of Solute Part.itioriirig in Porous Setworks . . . . A . The Partitioning System . . . . . . . . . . . . . . . . . . B . Partition Coefficients . . . . C . Elution Chromatograms . . . . . . . . . . . D . Porous Materials . . . . . . . . . . . . . . . . . . . 111. Theory of Column 0perat.iori . . . . . . . . A. General Equation of Continuit'y . . . . . . . . B . Solutions for Single-Solute Systems . . . . C . Interpretation of the Axial-Dispersion Coefficient . . . . . . . . IV . Determination of Molecular Size and Weight. . . . . . . . . A . Treatment of Chromatographic Data . . . . . B . Interpretation of the Partition Coefficient-Calibration Functions . C . Multiple Porosity Columns . . . . . . . . . . D . Combination of Chromatographic R.esults wit.h Other Kinds of Data . E . Molecular Weight Determinations with Deriat.ured Proteins . . . F . Evaluation of Conformational Changes in Chemically Modified Proteins V . Nonelution Methods . . . . . . . . . . . . . . . . . . A . Direct Opt.ica1Scanning of Gel Columns . B . Thin-Layer Gel Chromatography . . . . . . . . . . . . . . . . . C Equilibrium Solute Part.itioriing . . . . . . . . . VI . Studies of Multicomponent Systems . A. Partition Coefficient for Total Solute . . . . . . . . B . Experimental Determination of Weight-Average PartiCion Coefficients . . . C . Het.erogeneous Systems of Noninteracting Components . D . Studies of Protein Subunit Int.eractions . . . . . . . E . Studies of Small-Molecule Binding Equilibria . . . . . . VII . Concluding Remarks . . . . . . . . . . . . References . . . . . . . . . . . . . .

.

343 346 346 349 3512 357 363 364 366 373 378 378 379 392 395 396 398 398 398 411 413 41.5 416 417 421 423 441 442 443

I. INTRODUCTION The widespread use of porous gels for separation and characterization of protein components has developed alniost entirely within the last ten years . During this time a diverse group of techniques has been generated and applied to a broad range of problems in the physical chemistry of biopolymers . Whereas the earliest applications of gel chromatography were aimed a t preparative procedures, the rapid dcvelopnierit of its analytical aspect has occurred largely within the last half.decade . As a result of this development the protein chemist now has a t his disposal a variety of powerful analytical techniques for the chromatographic study and characterization 343

344

GARY K. ACKERS

of macromolecular systems. These include procedures for determination of molecular size and weight, studies of heterogeneous distributions of biopolymers, conformational transitions, subunit interactions, ligand binding, and reaction rates. The field of gel chromatography is substantially a n outgrowth of the introduction by Porath and Flodin (1959) of cross-linked dextrans for chromatographic separations of proteins. Although early observations (Wheaten and Baumann, 1953; Lathe and Ruthven, 1956) had previously suggested the principle involved in these separations, the major impetus to development and wide utilization of the technique was the commercial production of dextrans cross-linked with epichlorohydrin ( I ‘Sephadexes”), following the basic discovery of Porath and Flodin. Their successful application to protein separations also provided the stimulus for development of other gel-forming materials presently in wide use. For a given experimental problem it is now possible to select from a wide variety of gelforming substances depending on the solvent conditions and porosities desired. This diversity of chromatographic systems has been increasingly exploited in recent years, both for preparative and analytical purposes. The extremely rapid growth of analytical gel chromatography has resulted from two niajor lines of research. First was the early realization by biochemical workers using preparative gel chromatography that the technique could also be used for analytical determinations. Since the basis of the separations attained was the fact that molecular species of differing size exhibited different characteristic elution volumes, it was apparent that the method could be used to provide a fundamental characterization of the solute. A calibrated column can be used for determination of the molecular radius and to provide an estimate of molecular weight. This realization quickly led to the development of a variety of new techniques and theories. Second has been the rapid exploitation by polymer chemists of cross-linked polystyrene gels for molecular weight characterization of synthetic polymers in nonpolar solvents. This technique, frequently called “gel permeation chromatography” (Moore, 1964) appears to have been first used by Vaughan (1960) (who called it “gel filtration”), and has been expanded by many workers during the past few years. As often happens in the rapid development of convergent lines of research by workers with different scientific frames of reference, the literttture produced by these two groups reveals only an occasional mutual awareness of parallel developments. A part of the lack of coherence in the literature arises from the confusing array of names that have been applied to the subject. The first name to appear was “gel filtration” (Porath and Flodin, 1859), a term that has come to be identified with the comniercial dextran products called “Sephadexes.” This was followed by L ‘ m o l e ~ ~ lsieve a r filtration’’ (Fasold et al., 1961). The next year three new names were proposed : “Molecular sieve chromatog-

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

345

raphy” (Hjerten and Mosbach, 1962), “exclusion chromatography” (Pedersen, 1962), and “restricted diffusion chromatography” (Steere and Ackers, 196213). I n 1964 “gel permeation chromatography” was proposed (Moore, 1964) with the claim that it was a new technique. I n the same year Determann suggested the simple term “gel chromatography.” More recently Haller (1968) has suggested “steric chromatography” in view of the fact that porous glass column media appear to behave in a manner similar to the gels. Each of these names has arguments in its favor, some based on mechanistic interpretations, some on historical precedent. However, in order to decrease the confusion generated by continued proliferation and use of many different terms to describe the same general methodology, it would appear highly desirable for workers t o adopt a common name. I n this review the terminology suggested by Determann (1964) has been adopted in view of its simplicity and adequacy. It is realized that “gel chromatography,” like “gas chromatography,” encompasses a somewhat diverse group of techniques and processes. The analysis developed for chromatography based on partitioning within gels can be extended to other systems, such as porous-glass networks. The objective of this article is to describe the fundamental aspects of gel chromatography as they comprise a set of analytical techniques that the protein chemist may use. The unifying principles that underlie these techniques have been the major focus of interest. Application to individual molecular systems have been included as they serve to illustrate the chromatographic principles or their utility, or as they represent important landmarks in the field. However, no comprehensive review of the literature has been attempted. An earlier review in this series (Porath, 1962) has dealt with basic aspects and preparative applications of cross-linked dextran gels for chromatography of proteins. A great deal of useful information is contained in the pioneering monograph by Flodin (1962). An admirable laboratory handbook has recently been written (Determann, 1967a) which contains descriptions of many practical applications and techniques. Other reviews and general references on the subject of gel chromatography that the reader may wish to consult include Flodin and Porath (1961), Tiselius et al. (1963), Gelotte (1964), Determann (1964), Andrew (1966), Pecsok and Saunders (1966), Altgelt (1967), Ackers and Steere (1967), Kellett (1967), and W inzor (1969). Articles by a number of workers are included in symposia edited by Peeters (1967), Lederer (1967), and Johnson and Porter (1968). Special emphasis has been placed throughout this discussion on the two fundamental principles that underlie the use of gel chromatographic systems as analytical tools in protein chemistry: (1) partition coefficients depend on molecular size and shape, in contrast to surface or charge properties; (2) gel columns are, to a first approximation, linear chromatographic

346

GARY K. ACKERS

systems, and can therefore be used for analysis of nonlinear molecular phenomena such as occur in chemically reacting multicomponent systems. Although deviations from these two principles of ideal behavior are almost always present, their magnitude is almost always small. With appropriate experimental arrangement the “second-order” effects can either be ignored or evaluated, and their contribution subtracted from the phenomena of interest. For a minority of protein systems the two principles stated above fail completely, and it is important to be able to detect these deviations. I n spite of the nearly ideal behavior of gel partitioning systems, the amount of information obtainable from an individual Chromatographic experiment has been severely limited by the experimental necessity of measuring solute zone profiles only after elution from a column. The monitoring and collection of samples as they are eluted from a column is clearly the desired procedure for preparative separations. However, this is not the ideal mode of information retrieval for analytical determinations. A great deal of useful information contained in the history of the sample during the experiment is lost by this procedure. The recent introduction of direct ultraviolet column scanning has greatly improved this situation by permitting direct analysis of solute profiles a t many stages during a single column experiment. This has placed analyticaI gel chromatography on the same footing as other transport methods (e.g., analytical ultracentrifugation) in terms of data acquisition during an experiment. The possibilities of scanning nonelutiori chromatography appear sufficiently promising that it may be expected largely to replace the traditional elution method for many analytical determinations. Therefore the basic aspects of this approach have been reviewed in some detail. Throughout the discussion attention has been focused on the distinction between those properties of a gel chromatographic system that are invariant and those that must be experimentally controlled. In particular, emphasis is placed on the partition coefficient as a precisely determinable quantity and on its interpretation for both single component and multicomponent solute systems. The analysis of elution behavior has been developed as a consequence of the fundamental partitioning process and the processes that produce axial dispersion of solute within a chromatographic column. This approach facilitates understanding of the inherent limitations to the technique as well as its possibilities in analytical applications.

11. GENERALASPECTS OF SOLUTEPARTITIONING IN POROUS NETWORKS A . The Partitioning System The primary process in gel chromatography is the cliff usional partitioning of solute niolecules between a mobile solvent phase and the interior solveut

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

347

FIG.1. Diagrammatic representation of partitioning system. The stippled area (a) represents solvent of the mobile phase. A porous gel particle enclosed by the dashed line makes up the stationary phase. It consists of two regions-the internal solvent (b) and gel matrix (solid lines). When solute molecules are introduced into the system, partitioning occurs between regions (a) and (b).

spaces within porous particles that make up the stationary phase. The stationary phase usually consists of a gel-forming material that has been allowed to imbide solvent until a swelling equilibrium has been achieved. In porous glass chromatography, it consists of a rigid porous glass structure containing solvent within its pores. The mobile phase consists of solvent that occupies the spaces exterior to the porous particles (Fig. 1). For any gel partitioning process it is useful to consider three distinct regions within the experimental system: . . . (1) First is the solvent region exterior to the gel particles, which has a volume Vo, termed the void volume. The void volume is thus the volume of the mobile solvent phase in a chromatographic column. (2) The second region consists of solvent within the interior of the gel particles. It is this region, of internal volume Vi, that is involved in diffusional exchange of solute with the void spaces. (3) The solid, gelforming material (the “gel matrix”) comprises the third region of the system and occupies a volume V,. The total system is the sum of these three regions and occupies a volume:

Vt

=

+ vi +

VO

V g

(1)

For any gel-forming material there is a fixed relationship between Vi and V , determined by the swelling equilibrium under given environmental parameters of state (temperature, pressure, solvent, ionic strength, etc.) . The relationship is conveniently expressed in terms of the solvent regain volume, X,. This quantity is the volume of solvent imbibed per unit weight W , of anhydrous gel-forming material.

348

GARY K. ACKERS

Since Sr is constant for a given gel material and environmental conditions, Vi is seen to be an extensive property of the system. The vohime, V,, is related to the anhydrous partial specific volume of the gel matrix material, and the weight, W,, by

vg,

vg = v,wg

(3) From the Eqs. (2) and (3) the fixed relationship between Vi and V , can be expressed as :

This invariant property of the gel phase is a feature upon which reproducibility of partitioning experiments depends from one sample to another of the same gel. The theoretical basis of swelling equilibria in polymer gels has been developed by Flory (1953). In his analysis the equilibrium is formulated as a balance between the entropy of swelling and the internal osmotic pressure of the gel. In principle, an externally imposed osmotic pressure produced by a solution of nonpenetrating solute molecules could change the swelling equilibrium and hence the relationship between Vi and V,. Such osmotic effects have been observed with Sephadex gels by Edmond et al. (1968), using high Concentrations of excluded dextran. Appreciable changes were observed in the intcrnal volumes of Sephadex beads a t dextran concentrations as high as 20 gm/dl. For protein solutions partitioned on most gels at low concentration, the osmotic effects of solute are negligible. However, in some cases it may be desirable to take these effects into account. Recent studies of “bound” water in cross-linked dextrans (Sephadexes) have been carried out using nuclear magnetic resonance spectroscopy (Kuntz et aE., 1969). It was found that, for swollen gels of greater porosity than G-15 (lower degree of cross-linking), approximately 0.3 gm of water per gram of anhydrous dextran did not freeze normally and could be detected by assay a t -35°C. In contrast, most of the water inside the gel (2-20 gm per gram of Sephadex) was found to freeze normally. The amount of this “nonfreezing” water per gram of dextran does not depend on porosity of the gel network. This water appears to be intimately associated with the individual polymer chains and does not depend on the geometric relationship between chains within the matrix. In the case of small-pore Sephadex (G-IO), however, approximately twice as much “nonfreezing” water appeared to be associated with the dextran chains. It was suggested that the “extra” water might be held within cavities th a t

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

349

are too small for a rigid ice lattice to form. At present it is not known to what extent this “water of hydration” is accessible to solute molecules. On operational grounds Vi may be defined as the volume of all soIvent penetrable by a given small solute molecule and an “effective” V , taken as V,-Vo-Vi. The amount of nonpenetrable water may then be estimated from this number and the V , of Eq. (3).

B. Partition Coeficients When solute is introduced into a gel-solvent system it is distributed by diffusion between the solvent regions inside and outside the gel. A t equilibrium the distribution is described by a partition isotherm which defines the relationship between weight of solute Qi inside the gel and solute concentration C in the void space exterior to the gel. &i

=

f(C)

(5)

It is often convenient to formulate the isotherm (5) in terms of equilibrium partition coefficients that provide a thermodynamic description of the system. There are several ways in which partition coefficients can be defined for systems of this kind. However, there are two particularly convenient formulations widely used in gel chromatography. These are described below. 1. Partition CoeBcient Rejerred to Internal Solvent Volume

A partition coefficient u can be defined as the amount of solute distributed into the gel per unit internal volume Vi and external concentration C.l In terms of this coefficient the isotherm (5) can be written Q i = UViC (6) The dimensionless quantity u is a measure of the extent of solute penetration within the gel’s interior solvent region. This isotherm Eq. (6) is applicable to thermodynamically nonideal systems as well as ideal systems. However, a particularly simple interpretation of u can be made for the ideal case of simple diffusional distribution in which all parts of the system are thermodynamically ideal. The penetrable volume T’, within the gel which is occupied by solute molecules a t equilibrium is

VP

=

c, Qi

(7)

1 The symbol K D has been most, commonly used for this partition coefficient. However u has frequently been used instead for studies of chemically reacting multicomponeiit systems in which equilibriiim constants are denoted by various subscripted K’s. This convention has been adopted here. K O is used to denote an apparent dimerization constant.

350

GARY I<. ACKERS

where C, is the solute concentration within the region of distribution V,. For an ideal system C, = C and Eqs. (6) and (7) imply the relationship: f J

V Vi

= p-

Under these ideal conditions, then, the partition coefficient can be interpreted simply as the fraction of the internal volume that is penetrable by the solute molecule under consideration. This relationship is a good approximation for many systems of interest, especially a t low solute concentration where the partition isotherm is found to be very linear, and u is a constant. However, under more general conditions (e.g., higher solute concentration) the partition isotherm is found to have the form

Qi

=

KiC

+ KzC2

(9)

The second term on the right represents nonideality of the system and is almost always small relative to the first term. Although the source of this nonideality is not presently understood in molecular terms, its existence can be inferred on thermodynamic grounds. At equilibrium the equality of thermodynamic activities within the gel phase and void phase leads to the relation

roc = YrCp

(10)

where yo and yg are respectively the activity coefficients of solute in the void phase and gel solvent phase. From Eqs. (6), (7), and (10):

I n general then a correction term equal to the ratio of activity coefficients must be applied to the volume ratio V,/Vi of Eq. (8) in order to satisfy the criterion of thermodynamic equilibrium, and u becomes the volume ratio only in the limit of infinite dilution where yo = 7%. Nonideality of the isotherm which takes the form of a concentration dependence of u is to be expected since yo and yg will generally have different concentration dependencies. At finite concentrations C we can write ^lo

=

1

+ K’IC +

*

..

Yg

Taking the first two terms on the right side of (12) and substituting into (11) VP u =(1

Vi

+ K’lC)

=

UO(1

+ K’lC)

(13)

where u0 = V,/Vi, the limiting value of u at infinite dilution. Then substituting the right side of Eq. (13) for u into Eq. (6), we have the form

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

351

of the isotherm given by Eq. (9) in which Kz = Viu"K'1. Experimentally the values of KI1are always found to be positive, indicating a positive value for KB. Numerical values of K'l are generally in the range 0.001-0.01 depending on the protein and particular gel. Experiments with ovalbumin on Sephadex G-100 a t p H 6.8 have yielded values of K', as high as 0.014 as caIculated from the experimentally determined partition isotherm: Q = 0.324 Cr 0.0046 C2 (Winzor and Xichol, 1965). It has been proposed that a measured concentration dependence of CJ might be combined with an independent deterniinatiori of yo as a function of concentration to provide an estimate of the activity coefficient for solute inside the gel (Brumbaugh and Ackers, 1968).

+

2. Partition Coeficient Referred to Total Volume The partition coefficient u has been defined in Eq. (6) relative to the internal volume of the gel, Vi. It is possible also to define a partition coefficient with respect to total volume of the gel phase, Vt - Vo (Laurent and Killander, 1964). I n this case the partition isotherm can be written: Qi

=

KAv(Vt - V O ) ~

Then from Eqs. (7), (lo), and (15)

KAV=

vt - vo

."/o -lip yg

Vt -

vo

(1

(15)

+ K'1C)

Thus it is seen that this formulation of the partition coefficient can be characterized by the sa.me coefficient of concentration dependence (KI1) as U . I n the limiting case of infinite dilution, KAVOis the volume fraction of the total stationary phase occupied by the solute molecule

The relationship between the coefficients KAY and from Eqs. (13) and (16) as

Substituting from Eq. (1) for Vt - Vn

and from Eq. (4), Eq. (19) becomes

u

can be expressed

352

GARY K . ACKERS

It is thus see11 that, for a given gel system, a constant ratio exists between the two partition coefficients CJ and KAv. This ratio is independent of the particular values of these coefficients (i.e., i t is the same for all molecular species) and is independent of solute concentration. If the partial volume and solvent regain are known, a measured value of one partition coefficient can always be converted to the other. I n this article equations will be formulated in terms of u except in cases where KAv is substantially more convenient. All relationships for both single and multiple component systems can, of course, be formulated in terms of either coefficient. It should also be noted that partition coefficients have been formulated here in terms of volume ratios in keeping with the primary volume partitioning mechanism involved in gel chromatography. An alternative formulation can be made in terms of concentration ratios: CJ=-

C, C

where C, is the concentration of solute per unit internal volume of the gel and C is concentration outside as usual. Since & i = C,Vi, this expression is equivalent to Eq. (6). Similarly

where Cav is concentration per unit volume of the total gel phase. C . Elution Chromatograms I n a chromatographic transport experiment a pulse of solute is allowed t o flow with the niobiIe phase between stationary porous particles, and diffusional distribution occurs between the solvent regions of the two phases. The limiting condition for diffusional exchange is determined by the equilibrium partition coefficients defined above. However, in transport experinients a norieyuilibrium perturbation is always present so that a steady state is usually achieved with respect to the partitioning process, and true equilibrium does not exist. I n addition, axial diffusion within the column occurs as well as effects of nonuniform flow around the stationary phase particles (Section 111). The resulting macroscopic behavior of the system depends on the initial configuration of the solute sample and the history of all interactions between solute and column. I n liquid Chromatography the transport behavior of a solute of interest is most frequently studied by means of the conventional elution experiment

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

353

in which a concentration-volume profile is measured for the sample emerging at the bottom of the column. The usual arrangement is a cylindrical tube into which the swollen gel particles or beads have been packed. The vertical tube is fitted with a suitable arrangement of inlet and outlet devices, which retain the gel bed while permitting the flow of solvent through the column. Sometimes provision is made for pumping of solvent in either direction (upward or downward) through the vertical column and for controlled introduction of a solute sample. As the sample is eluted from the bottom of the column, solute concentration is monitored as a function of the volume of solvent that has passed through the column after introduction of the sample at the top. For chromatography in aqueous solvents the detector is usually a spectrophotometric device. For organic solvents it is usually a refractometer. The output signal from the detector is commonly fed to a strip chart recorder with time as abscissa. If the rate of flow is known, the correlation between concentration and volume is established. For precise analytical determinations and subsequent automated computational procedures the output may be fed t o a digital data acquisition device that provides for interfacing of the experiment with a computer. A diagrammatic representation of this general concept for analytical elution chromatography is shown in Fig. 2. Many systems of this general scheme have been employed experimentally with various specific components, some of which are manufactured commercially. Although the packed cylindrical chromatographic column is used most widely SAMPLE INJECTOR

I C I

F 1 I

PRINTOUT

1 ,,DETECTOR U

I

1 1

I

FRACTION COLLECTOR

FIG.2. General scheme of automated system for elution chromatography.

354

GARY K. ACKERS

Volume -+

FIG.3. Schematic elution diagram to illustrate three types of solute behavior; (a) total exclusion; (b) a penetrant molecule; (c) a small totally nonexcluded molecule.

in gel chromatography other arrangements of stationary and mobile phase have been used as well, in particular for thin-layer techniques (Section V,B). The essential results observed in the column experiment are the solute’s characteristic volume and the spreading of solute zones. These operational features are described briefly in this section for several important types of experiment. The theoretical basis of the observed behavior of solute zones is discussed in Section 111. 1. Characteristic Elution Volumes There are two extreme types of solute load that can be applied to the column : a. Small Zone Experiments. If a solution containing a macromolecular component is introduced a t the top of the column in a sample of very small volume compared to the bed volume, Vt, followed by a flow of solvent, it is found that the effluent volume, V,, of solvent which passes through the column between introduction of the sample and subsequent emergence of its maximum concentration can be described by the equation (Wheaten and Baumann, 1953).

+

Ve = V O

UVi

(21)

The constant of proportionality, U, is found to be identical to the partition coefficient described previously (Section I1,B) and satisfies the conditions: =

0 for a molecular species totally excluded2 from the gel phase.

2 Conceptually the term “total exclusion” refers to an imaginary srirface around the gel particles that separates the internal voliime and void volrime of the system. However, since exclrision is always a matter of degree, the “totally excluded” moleciile must be defined on operational grounds: (1) It belongs to B class of all molecules that cannot be distinguished on the basis of molecular size differences to within limits of the experimental rneasnremerits. ( 2 ) This class of molecules has the smallest attainatde volume for distribution within the gel-solvent system (or elution from a column).

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

355

1 for a molecule that diffuses freely with no restrictions within the gel network. 0 < u < 1 for a penetrant macromolecule in the absence of specific interaction (e.g., adsorption, ion exchange). u > 1 when specific interaction effects sufficiently retard the elution velocity within the column u =

From Eq. (21) and the above conditions on u, it is seen that the larger molecular species are eluted Jirst in chromatography of this type (Fig. 3). An equivalent relationship to Eq. (21) is obtained if one measures “retention times” (for constant flow rate) (Carmichael, 1968). t, = t o

+ uti

(22)

The retention time represents the time interval between introduction of the sample pulse and emergence of its peak concentration. As usual, to and ti refer to totally excluded and “included” molecules, respectively. A third expression for the elution position (Laurent and Killander, 1964) is written in terms of the partition coefficient K A V .

ve = VO + K A V ( V t - VO)

(23)

b. Large Zone Experiments. If the sample is introduced in a volume S which is large in relation to the bed volume of the column and if the effluent concentration of the solute species is measured as a function of the volume, V , that has flowed through the column, it is found that a “plateau region” of constant initial solute concentration is bounded by leading and trailing boundaries which are diffuse, i.e., they do not possess sharp edges identifiable as the elution positions. However, an equivalent sharp boundary may be defined on the basis of conservation of mass. The procedure is shown diagrammatically in Fig. 4 for a trailing boundary. Consider a n arbitrary

V,

V-

-

V’

FIG.4. Equivalent sharp boiiridary for the trailing elution bouiidary of a large zone.

356

GARY K . ACKERS

reference position V , chosen within the plateau region of the elution diagram (Fig. 4). Then the equivalent boundary elution position P’ is chosen in such a way that the mass of solute Co(P’ - V,) represented by the area under the resulting “idealized” diagram equals the true mass represented by the area under the experimentally determined elution curve (solid line, Fig. 4). This equality can be expressed as

cop - V,) =

(V’ - V,) dC

(24)

By evaluating terms in Eq. (24) it is seen that

Thus the equivalent boundary position (centroid volume) P’is independent of the particular reference position V , chosen. Its value is obtained by graphical or numerical integration of the elution diagram. When this satisfies the relationship: procedure is carried out it is found that 7’

P’ = P + s = vo + u v i + s

(26)

A similar calculation for the leading boundary yields a centroid volume 7 that is equal to the right-hand side of Eq. (21). The parameter u is found to have the same numerical value for both the small zone peak-position determination and the large-zone “plateau experiment.” This coefficient characterizes the interaction between the column’s gel phase and a given molecular species and is to a first approximation independent of concentration. It can be related to the molecular radius or molecular weight as well as to the gel porosity. These two examples illustrate the simplest type of elution behavior that can be observed in gel chromatography. An objective of the analysis in Section I11 will be to provide a rational basis for this “linear” chromatographic behavior. The analysis will subsequently be extended to more complex phenomena involving multicomponent systems. 2. Spreading of Solute Zones

a. Redistribution of Sample. Addition of a solute sample to the column is immediately followed by a lengthwise spreading of the solute zone that takes place as a result of the redistribution of solute into the void spaces. If a sample element of volume Si is added to the top of a column of total cross-sectional area A , the length L, of the resulting zone is &‘/A. Upon entry into the column void spaces, the sample elenient may spread to a length as great as L’, = Si/, ( a = void volume per unit length). The factor by which the zone has been spread thus has an upper h i t :

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS-

I,,‘ _ --_A- _ Vt Ls

vo

a!

357

(27)

This axial dispersion is partially counteracted by diffusional distribution of solute into the gel phase and is therefore always somewhat less than the upper limit value given by Eq. (27). The value of Vt/Vo will depend on the geometry and packing of the stationary phase. For spherical gel particles it will have as its upper limit the numerical value 3.85, which is the reciprocal of the fractional void volume for close-packed spheres. This initial spreading generally represents a transient effect, as diffusional distribution into the gel will always quickly reduce this spreading factor. For a column operating a t conditions close to equilibrium, the final “volume redistribution” spreading factor will approach the value

_ -Vt LS’ L, Vo U V i

+

b. Continuous Axial Dispersion. As the sample moves down the column, additional lengthwise spreading (axial dispersion) occurs owing to several kinds of interaction processes superimposed on the basic solute partition effect: (1) Local nonuniform flow velocity of the mobile phase due t o effects of finite gel particle size, inhomogeneity of particle size, and nonuniform packing of the stationary bed. (2) Diffusion of solute along the axis of the column, both in the mobile and stationary phases. (3) Disequilibrium between the mobile and stationary phases of the column, with respect to diffusional exchange of solute. The net result of these processes is a dispersion of the solute zone or boundary along the axis of the column. This axial dispersion then appears in the eluted sample as zone spreading on the volume axis of the chromatogram (Figs. 3 and 4). Whereas exact formulation of equations to describe mechanistically the effects on zonespreading of each of the above factors is a formidable task, considerable progress has been made by the use of approximate theories to describe these three basic processes which lead to zone spreading. A discussion of this problem is presented in Section 111. For purposes of component separation it is obviously desirable t o minimize the axial dispersion in order t o minimize overlap of solute zones. However, this is not necessarily the case in analytical studies, as will be seen. The zone-spreading phenomenon can be characterized by an axial dispersion coefficient which is related t o molecular size and is therefore a useful parameter in its own right.

D . Porous Materials The term “porosity” has frequently been used to denote a critical molecular size limit for penetration into the gel network by a solute molecule. This concept has sometimes led to confusion because it suggests a n

358

GARY K . ACKERS

“all or none’’ type of penetration mechanism for the gel as a whole. Thus a number of commercially manufactured porous gels have been described as having “exclusion limits” which refer to a given molecular weight. However, the relevant property determined by the structure of the gel network is the partition coefficient (Section I1,B). This parameter represents the extent of internal solvent penetration by the solute molecules instead of an exclusion size-limit. The term “porosity” is also used to denote the range of molecular sizes that can be accommodated by a given porous material. This property depends on the nature of the curve relating molecular size to partition coefficient. As will be seen in Section IV, two parameters are generally required to define the state of the gel with regard to its partitioning properties. Adoption of a precise and universally useful means of “porosity” specification would be an aid toward minimizing confusion on this point. The specification of two molecular radius values and corresponding partition coefficients would suffice. The molecular radii corresponding to partition coefficients of 0.1 and 0.9, for example, would fully characterize the gel with respect to partitioning properties of all molecules. With column bed materials now available, it is possible to study molecules ranging in diameter from several angstroms to several hundred angstroms, or in molecular weight from several hundred to several hundred million daltons. The physical and chemical properties of these materials vary greatly. However, for gel chromatography they share a common set of basic requirements: (1) It is essential that the gel-forming materials have a strong affinity for the desired solvent in order to swell with a high solvent regain. For gels, the equilibrium structure consists of a three-dimensional network formed by the cross-linking of long polymeric chains. The structure is swollen by imbided solvent to an equilibrium limit determined by the nature of the matrix material and solvent. Hydrophilic gels usually contain polar groups such as hydroxyl groups, whereas aromatic and other less polar groups are incorporated into the hydrophobic materials. This solvent affinity is not a necessary requirement for porous glass column materials since the rigidity and porosity conferred upon them in their initial formation is maintained regardless of the presence of solvent. (2) The column material must have a low affinity for the solute molecules of interest in order that adsorptive or charge interactions be minimized in analytical applications (for separation purposes such side effects are not always undesirable and may actually be helpful). (3) The three-dimensional network formed by the swollen gel must have sufficient rigidity to withstand being packed into a column and subjected t o flow of solvent under (at least slight) pressure. In addition it should be sufficiently rigid that osmotic swelling and contraction is not sufficiently

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

359

large to produce errors. (4) For experiments that involve direct optical scanning of gel columns, it is desirable that the material have a minimum of chromophoric groups in the wavelength region used for detection of solute. A number of materials that satisfy these criteria have been employed for chromatographic use. Specific details of preparation and chemical structure have been described elsewhere and will not be reviewed here. However a brief cataloging will be made of the principal porous materials in order to indicate the ranges of molecular size and weight with which they have been effectively employed. For most protein systems it is possible to find a gel or porous glass material which, under a desired set of ionic and pH conditions, is free of detectable adsorptive interactions and has a desired porosity. 1 . Cross-Linked Dextrans

The most extensively used gel-forming materials have been the crosslinked dextrans, since their introduction by Porath and Flodin (1959). These materials are commercially available in bead form under the trade name Sephadex. The dextrari itself is a soluble polysaccharide of glucose and is produced during the growth of the microorganism Leuconostoc mesenteroides. The glucose residues are predominantly in cu-1,6-glycosidic linkage, and there are three hydroxyl groups per glucose unit, which can be TABLEI Properties of Commercial Dextran Gels (Sephadex)a

Type

Particle size* (dry; in p )

G-10 G-15 G-25, coarse G-26, medium G-25, fine G-50, coarse G-50, medium (3-50, fine G-75 G- 100 G-150 G-200

40-120 40-120 100-300 50-150 20-80 100-300 50-150 20-80 40-120 40-120 40-120 40-120

Water regain (ml/gm)

Approximate separation range Gel bed (ml/gm)

1.0 k 0.1 2-3 1.5 0.1 2.5-3.5

Peptides and glob. proteins Up to 700 Up to 1,500

Dextrans fractions Up t,o 700 Up to 1,500

+ 2.5 + 0.2

4-6

1,000-5,000

100-5,000

5.0 k 0.3

9-1 1

1,000-30,000

500-10,000

0.5 1.0 1.5 2.0

12-15 1520 20-30 30-40

3,000-70,000 4,000-150,000 5,000-400,000 t5,000-800,000

1,000-50,000 l,OOO-lOO,OOO 1,000-150,000 1,000-200,000

7.5 k 10.0 k l5,O & 20.0 k

Specificatioiis provided by the manufacturer: Pharmacia Fine Chemicals, Uppsala, Sweden. All porosities are also manufactured as beads in the particle size-Superfine (10-40 p diameter).

'

360

GARY K . ACKERS

cross-linked between chains with epichlorohydrin. The resulting crosslinkage is by two glycerin ether bonds. These gels are hydrophilic and their water regain is determined by the relative percentage of epichlorohydrin reacted and the molecular weight of the starting dextran material (Flodin, 1962). The Sephadex gels are relatively free of adsorptive interactions with large molecules in moderate to high salt concentration. However, a high affinity for small aromatic compounds has been noted by many workers (Janson, 1967; Eaker and Porath, 1967; Determann and Walter, 1968). Evidence has recently been presented which indicates that the sites of interaction are a t the ether cross-linkages (Determann and Walter, 1968). Table I summarizes the types and general specifications of commercial dextrans. 2. Polyacrylamide Polymerization of acrylamide by the use of a bifunctional agent such as N,N’-methylenebisacrylamide leads t o the formation of a cross-linked hydrophilic network. The polymerization is carried out in aqueous solutions, and the resulting hydrophilic gel can be granulated and used for chromatographic purposes (HjertBn and Mosbach, 1962). Polyacrylamide gels are similar in range of porosity to the cross-linked dextrans. The most widely used acrylamide gels for chromatographic purposes have been the commercially produced spherical beads Bio-Gel P. The types available and their corresponding specifications are listed in Table 11. Under TABLEI1 Bio-Gel P Types and Specifications

TYPO

Particle size (wet mesh) ~

Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel Bio-Gel

P-2 P-2 P-4 P-6 P-10 P-20 P-30 P-60 P-100 P-1,50 P-200 P-300

50-100 100-200 50-1.50 50-150 50-150 50-150 50- 150 50- 150 50-150 50-150 50-150 50-150

Approximate “exclusion limit” (molecular weight)

Hydrated bed volume (ml/gm dry gel)

Water regain (gm wat,er/ gm dry gel)

1,600 1,600 3,600 4,600 10,000 20,000 30,000 60,000 100,000 150,000 200,000 300,000

3.8 3.8 6.1 7.4 12.0 13.0 14.0 18.0 22.0 27.0 47.0 70.0

1.6 1.6 2.6 3.2 5.1 5.4 6.2 6.8 7.5 9.0 13.5 22.0

~~

361

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

TABLE111 Properties of Commercial Agarose Gels According to the Manufacturer ~~

Type Sag 2 Sag 4 Sag 6 Sag 8 Sag 10 Sepharose 2B Sepharose 4B Bio-Gel A-150 m Bio-Gel A-50 m Bio-Gel A-15 m Bio-Gel A-5 m Bio-Gel A-1.5 m Bio-Gel A-0.5 m

Supplier Seravaca Mann

-

Pharmacia Bio-Rad

-

Average particle size (ca. P )

Approximate fractionation range for proteins

70-140 (crushed) -

50 x 104-1.5 x 108 20 X lo4-15 X lo6 5 x 104-2 x 106 2 . 5 X lo4-70 X lo4 1 X lo4-25 X lo4 8 x 104-20 x i o 6 b 1 X lo4-3 X 1 X lO"1.5 X 10'

60-300 30-200 50-100 100-200

(beads) (beads) (beads) (beads)

10 X 4 X 1X 1x I x

lo4-50 X lo4 lo4-15 X lo6 10e5 X lo6 104-1.5 x 106 104-50 x 105

a Seravac Laboratories (PTY). Lts., Holyport Maidenhead, Berkshire, England. Also Mann Research Laboratories, Inc., New York, 10006 under the name "Ago-gel." b Refer to dextran fractions.

some experimental conditions polyacrylamide gels have been found to be more inert than dextran gels with respect to adsorptive interactions, particularly a t low ionic strength. 3. Agar and Agarose

Agar is a mixture of two polysaccharide components (Araki, 1956). The main component is a linear polymer of D-galactose and 3,6-anhydro-~galactose, termed agarose. The second component is also a galactose polymer, agaropectin, which contains a substantia1 number of carboxyl and sulfate groups. These can produce ion exchange effects within the gel, and the agarose is therefore much more desirable for analytical chromatography purposes. Early applications of granulated agar for chromatographic separations (Polson, 1961; Steere and Ackers, 1962a) indicated a wide range of use for separation of components and for determination of molecular size (Steere and Ackers, 1962b) and weight (Andrew, 1962). Procedures for separation of the agarose component have been developed by Hjerten (1962) and by Russell et al. (1964). The formation of agarose beads can be effectively carried out by procedures described by Hjerten (1964) or by Bengtsson and Philipson (1964). Commercial agarose products are listed in Table 111. The size and shape properties of agarose molecules have been investigated by Polson and Katz (1968).

362

GARY K. ACKERS

4. Porous Glass A procedure has been developed (Haller, 1965a,b) for making rigid, highsilica glass with a network of interconnected pores. The pores within these materials appear to consist of tortuous channels with roughly circular cross section but varying diameter. A very narrow distribution of pore diameters can be achieved. Structures have been made in which the mean pore diameters range from 170 to 1700 A. These materials have been used for studies on the mechanism of separation (Haller, 1968), Commercially produced versions of the porous-glass media are listed in Table IV. Some TABLEIV Fractionation Range of Commercially Available Porous Glass. Approximate

Bio-Glass-200 Bio-Glass-500 Bio-Class-1000 Bio-Glass-1500 Bio-Glass-2500

200 500 1000 1500 2500

0 Specifications provided by the manufactiirer: Bio-Rad Laboratories, R.ichmond, California.

structural studies of porous-glass materials have been carried out by Barral and Cain (1968). The special advantages of porous-glass columns is that they can be cleaned with acid or sterilized by autoclavirig without disturbing the packing. Also, they form an extremely rigid column, which does not change in volume when pressure is applied. This permits considerable variability in the useful flow rates that can be attained. 5. Polystyrene Gels

The cross-linking of polystyrene with divinylbenzene can be carried out t o produce gels of varying porosity, depending on the solvent used in the polymerization of styrene with divinylbenzene (Moore, 1964). These gels are swollen by nonpolar solvents, such as toluene, methylene chloride, and dimethyl formamide. The commercially available polystyrene beads (Styragel) are listed in Table V along with some of their properties. 6. Other Materials

A great variety of gel materials have been studied in relation to their use for gel chromatography. These include cross-linked products of locust

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

363

TABLEV Fractionation Range o j Diflerent Styragel Typesa Tyope (A) 60 100 400 1 x 103 5 x 103 10 x 103 30 x 103 1 x 105 3 x 106 5 x 106 10 x 106

Approximate fractioriatiori range for vinyl polymers

Approximate “exclusion limit”

(mw)

(Mlim)

800 2,000 8,000 20,000 100,000 200,000 600,000 2,000,000 6,000,000 10,000,000 20,000,000

1,600 4,000 16,000 40,000 200,000 400,000 1,200,000 4,000,000 12,000,000 20,000,000 40,000,000

a Specifications provided by the mariufacturer : Waters Associates, Inc., Framingham, Massachusetts.

bean gum (Deuel and Neukom, 1954), polyvinyl alcohol, sorbitol, cellulose, starch, gelatin treated with tannic acid (Polson and Katz, 1968), silica gel beads (Le Page and De Vries, 1966), and elastin fibers (Partridge, 1967). Modified dextrans have found wide use for separation in nonaqueous systems (Nystrom and Sjovall, 1965). Commercial preparations of methylated Sephadex are found to have lipophilic properties and can be used with a variety of nonaqueous solvents.

111. THEORY OF COLUMN OPERATION Theoretical analyses of chromatographic transport processes provide a rational basis for the observable behavior of gel partitioning systems described in the preceding section. An understanding of the mechanics of gel chromatography for single solute systems is also prerequisite to an extension of the technique to multicomponent systems and those in which chemical reactions are superimposed on the transport behavior. A number of approaches to the theory of chromatographic behavior have been taken in the past and more than one of these can be shown to be applicable to gel chromatographic situations. A useful quantitative theory of liquid chromatography that takes into account all the essential effects present in the gel chromatographic case can be formulated using a n approach developed by Glueckauf and co-workers (Glueckauf et al., 1949; Gluecltauf, 1955a,b). More extensive descriptions of chromatographic theory can be found, for example, in books by Yforris and Morris (1964) and Giddings (1965).

364

GARY K . ACKERS

In the early history of gel chromatography a number of processes were proposed as possible contributing factors to the separation process in addition to the equilibrium partitioning of solute. One proposal was that separations occurred as a result of differential flow velocity within the packed gel bed (Pedersen, 1962). This idea was based on the greater tendency of a larger particle to move into the more rapidly moving regions of liquid within the column. Thus the larger molecule would be eluted first. It was demonstrated that limited separations of macromolecules could be obtained on columns packed with impenetrable glass beads. Another suggestion was that elution volumes were determined in part by the diffusion rates of molecules within the gel phase (Steere and Ackers, 1962b; Ackcrs, 1964). This proposal was based on the severe restriction to diffusion within gels that had been experimentally observed for protein molecules and viruses (Ackers and Steere, 1961, 1962). Owing to their restricted diffusion rate into the stationary phase, larger molecules might be eluted from the column earlier than the smaller ones. A third proposal was that separations occurred according to the Bronsted principle of phase equilibria (Determann, 1967a),similar to the observed partitioning of macromolecules between phases of high polymeric materials (Albertsson, 1960). Although all these processes may play some limited role in determining elution positions of macromolecules on gel columns, it now appears that their main contribution under most conditions is to the axial dispersion of solute zones within the column. The essential features of column behavior can most simply be developed theoretically as a consequence of equilibrium solute partitioning onto which a steady-state nonequilibrium perturbation is superimposed. This perturbation which produces axial dispersion is the result of the factors listed in Section II,B. The contribution of these processes is developed in some detail in Section III,B. First, we consider the basic equations of continuity for transport within chromatographic columns.

A. General Equation of Continuity For the exchange of solute between mobile and stationary phases in a chromatographic process, it is useful to have continuity equations th a t express the conservation of mass for each of the components undergoing the exchange process. A number of such equations of continuity have been employed in studies of liquid chromatography. Some of these are based on analogies between chromatography and other transport processes. For example, partition chromatography is similar to countercurrent distribution or fractional distillation in which almost exact equilibrium partitioning takes place a t a number of stages. It was this likeness which led to the concept of “theoretical plates’’ in chromatography (Martin and

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

365

Synge, 1941). On the other hand, the role of nonequilibrium between mobile and stationary phases is always an important one, and it may sonietimes be the controlling factor in determining shapes of chromatographic zones and boundaries. I n this respect the analogy is closest between chromatography and sedimentation or electrophoretic transport in which diffusion plays a substantial role. In this discussion, attention will be focused on the role of the two fundamental processes: solute partitioning and axial dispersion. The general equation of continuity for exchange of solute between stationary and mobile phases with simultaneous axial dispersion has been derived by many workers (Glueckauf et al., 1949; Glueckauf, 1955a,b; Lapidus and Amundson, 1952; Thomas, 1948). For gel chromatography the equation can be written (Ackers, 1967a)

I n this equation L is an axial dispersion coeflcient and [ is the cross-sectional area into which solute partitioning occurs. The parameter [ is the solute distribution volume per unit length and is related to the partition coefficient u by the expression: (30)

t=CY+Pu

where cx is the void volume per unit length and fi is the internal volume per unit length. Equation (29) is conveniently written in terms of volume coordinates. Assuming constant flow rate F, the volume is V = Ft and

F -ac =-

ac

av

at

Substituting for aC/at into Eq. (29), we have

-ac + + - =ac A’ax av

azc

ax*

where

L’

Lt = F

(33)

These continuity equations (29) and (32) are based on considerations of solute flux within the column. The total solute flux, for unit cross section is J = TdQ -

dt

(34)

where Q T is the weight of solute within the column. The total flux J , can be considered to be the sum of two fluxes:

366

GARY K . ACKERS

(1) a flux, J,, due to solute partitioning and volume flow, and (2) an axial dispersion flux, J,, due to the iioriequilibriuni perturbation of the system. The solute partitioning flux J , can be written as the product C(dx/dt) of concentration and velocity, where dx/dt is the average solute velocity within the column. Then, since

The partitioning flux term is

J,

FC

= -

4

The second flux term J , due to nonequilibrium perturbation within any infinitesimal section of the column is

in which L is the coefficient of axial dispersion. The total flux, ( J , then is written

J = -FC -L-

6

+ J,)

ac ax

This is the fundamental flow equation for solute transport on the column. It has the form of Fick’s law of diffusion for a moving frame of reference (Gosting, 1956) and is analogous, for example, to the ultracentrifuge flow equation J = Csw2x - D(dC/dx). In this case, the moving frame of reference has a velocity F / ( and is described by the partitioning term J,. The chromatographic transport equation (29) results from the substitution of J from Eq. (38) into the general equation of continuity, for uniform cross section. aC - -_ - - a J (39) at ax applicable to transport processes of all kinds (cf. Gosting, 1956).

B. Solutions for Single-Solute Systems I n order to arrive a t macroscopic descriptions of the observable behavior of solute within the chromatographic column, it is necessary to solve one or more of the equations of continuity subject to the appropriate initial and boundary conditions of the experiment. There are a number of permutations of conditions and special cases that can be chosen for this purpose. In this section the general solutions to Eq. (32) will be described for the

367

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

case in which both partition transport and axial dispersion transport occur, and the partition coefficient u is constant (linear isotherm). Subsequently, the corresponding solutions are described for partition transport without axial dispersion. The comparison between these two cases illustrates the relative contributions of the two kinds of transport. In Section III,B,3, the more complicated and general case of transport will be described for a single solute with linearly concentration-dependent partition coefficient. 1. Transport with Axial-Dispersion and Constant-Partition Coeficient

As noted before, Eq. (38) has the form of Fick's first law of diffusion for a moving frame of reference. Now consider the flux, J ' , relative to this moving frame of reference F / i . From Eq. (38)

Then, since Eq. (39) is independent of the frame of reference, Eq. (29) can be written

CONFIGURATION OF INITIAL COLUMN LOAD

1

EFFLUENT CONFIGURATION

(a)

S

v=0 x=o

V-

i-v, =vo+uv,

leading edge

,f--

s-& a'8

I '

I

/

*>.I

v; =v,+uvi+s trailing edge

FIG.5. Two t,ypes of chromatographic experiment; (a) small zone experiment; (b) large zone (plateau) experiment. Vertical dashed lines are equivalent boundary positions. Curved dashed lines represent the concentration gradient across the zone.

368

GARY K. ACKERS

where the new position coordinate is $I bgcomes, with LV = L/F

=

x

-

( F t / [ ) . Likewise Eq. (32)

These last two equations have the form of the well-known parabolic equations of heat flow and diffusion. Solutions are well-known for a great variety of initial and boundary conditions (cf. Carslaw and Jeager, 1947; Crank, 1956; Jost, 1960). Here we will consider the two experiments described in Section II,C. For the small zone experiment the initial condition is that a quantity M [ of solute is placed on the top of the column (x = 0, V = 0) in a sample of infinitesimal volume, i.e., a n instantaneous solute pulse is applied and its elution position V , is measured (Fig. 5). The formulation of the initial conditions represented by this experiment is best accomplished by means of the Dirac delta function3 (unit impulse function) 6(x). I n terms of this function the conditions imposed on Eq. (41) by the experiment under consideration can be written:

V = 0, C = 6 ( ~ .)M V>O, c=o, x = o

(43)

The solution can be obtained by application of Laplace transforms (Crank, 1956) yielding

The concentration is thus a Gaussian function of the moving frame of reference coordinate 9. It is also distributed in a Gaussian form along the distance coordinate x within the column. However, the zone is not strictly Gaussian in the volume coordinate V , an observation first made by 3

The delta function may be defined by: 6(x)

=

lim xo+o

where: (l/xo;

Ax)=

10;

0

f(5)

< 5 < xo > xo

x < o

The function 6(x) is thus defined to be zero for all x except for x becomes infinite such that /-+*m

=

0, a t which point it

6(s)dx = 1

The delta function 6(x) thus denotes a unit impulse at 5 = 0. In general a unit impulse distributed a t some point x' is written 6(s - x').

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

369

Glueckauf for “continuous exchange” analysis of linear chromatographic systems (Glueckauf, 1952). The maximum concentration of the elution zone will occur very near the coordinates:

4=o=x-- V 5 Therefore, when the peak elution volume V , is measured, corresponding t o the boundary condition x = I, the approximate result is Eq. (21)

vo = vo +

UVi

This result provides the theoretical basis for the empirical use of peak positions to determine partition coefficients described in Section 11. For a solute zone of volume S = L A , the initial conditions are

v = 0,

c = co,

c = 0,

-L, < X < - - L ” 2

2

LS (xj > 2

(45)

and the corresponding solution is4

The maximum concentration, Cm, will occur again a t 4 = 0, and its measured value upon elution from the column is given by:

+

+

where VE = Vo uVi as. When S is sufficiently large that a plateau is maintained of concentration CO,the solution for the leading edge, corresponding to the initial conditions:

v=o,

is :

c=o, c=co,

x > o x
The error function of a number z is

2/01

erf(z) = __

ePq2dq

and the error function complement, erfc(z) is given by erfc(z) = 1 - erf(z)

(48)

370

GARY K. ACKERS

The corresponding solution for the trailing edge, with V' = V 4' = z - V ' / [

V'=O,

c=o,

C=Cor

+ S,

xo

is :

The coordinate position 4' = 0 (C = C0/2) coincides with the centroid volume 7' of the trailing boundary (Eq. 25) and satisfies the condition of mass conservation for the equivalent boundary defined in Eqs. (24) and (25). The conservation of mass condition is

6"

4' dC = 0

Substituting 1 - V ' / [ in Eq. (52) for +', we find that 61 = (Eq. 25) and thus Eq. (52) reduces to Eq. (26). This result provides the theoretical basis for Eq. (26) and the operational procedure for determination of from P'. The same type of analysis can be applied to the leading boundary.

(r

2. Ideal Solute Transport without Axial Dispersion I n order to illustrate the relative effects of solute partitioning and axial dispersion, i t is useful to consider solutions to the continuity Eq. (3) for the limiting case in which instantaneous equilibrium occurs a t each stage of the column, a n assumption embodied in the early chromatographic theories of Wilson (1940), Weiss (1943), and De Vault (1943). It will be seen that the most fundamental relationships, Eqs. (21) and (26), for determination of partition coefficients can be obtained even in this restricted case, although the zone shapes are not predicted realistically. This limited correspondence with reality also provides a useful background for consideration of the idealized solutions for transport behavior of interacting systems described in Section VI. Equation (32) for this case (I; = 0) becomes

The solution to this equation, for the conditions, Eq. (43), of a unit impulse solute load 6(x) is

c = 6 (.

-

);

(54)

371

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

Thus, it is seen that for any given volume V that has flowed through the column, the solute is distributed as a delta function at the position:

x=-

V

+

a

(55)

Pa

Substituting Vo = a1, V i = pl, Eq. (55) becomes X _

2

--

V Vo+c~Vi

As usual, the elution volume V eis defined by the condition x 1 =

V,

+

VO

UVi

=

I , and (57)

which is identical with the basic operational equation (21). The second type of experiment (large zone experiment) requires solution of Eq. (53) for an initial step function solute load. For the leading edge, the solution is

C=H(x-'f) where H is the Heaviside unit step function. The position of the leading edge step function is then Eq. (21).

x = -V 4 For the trailing edge the corresponding solution is Eq. (26)

V',

=

Vo

+

OVi

+s

The operational Eqs. (21) and (26), which describe empirically the most basic features of these two types of chromatographic experiment, have been derived therefore without consideration of axial dispersion. It is evident, however, that these solutions provide only a partial (highly idealized) representation of the experiment because the effects of axial dispersion and zone spreading have been ignored. It is seen in Section VI that idealized analyses of this type are useful in predicting qualitative features of reaction boundaries, even though analytical solutions cannot be obtained. 3 . Xflect of Concentration Dependence of Partition CoefJicient

The greatest degree of complexity exhibited by a single solute system in gel chromatography is described by the solutions of continuity equations for linear concentration dependence of the partition coefficient (Eq. 13). The general solution for isotherms of the type described by Eq. (9) has been

372

GARY K. ACRERS

-15

-10

-5

0

+5

+I0

+I5

FIG.6. Effects of concentration dependence of partition coefficient for ovalbumin solutions on Sephadex G-100. Plateau concentrations are, from top to bottom, 12.0 mg/ ml, 7.0 mg/ml, and 1.6 mg/ml. Taken from Winzor and Nichol (1965). obtained by Houghton (1963). For the concentration dependent partition isotherm, the complete continuity equation is a complicated nonlinear equation for which analytic solutions have not been obtained in the general case. However, it can be written in simple form by limiting considerations to the small concentration dependencies of c which are present in gel chromatography with most single-component solutes. The approximation is

p2a0K'1C

5

<< 1

and the corresponding equation of continuity is

ac- fl2K'luOC -ac= L azc v@ av t2 at

(59)

The solution of this equation for the solute band of finite width L, is (Houghton, 1963)

_ C --

CO

1 - erf(p)

exp(g)[erf(p

+ h ) - e r f h + h)l - erf(q + h)l + exp(m)[l + erf(q)l

+ exp(q)[erf(p + h )

(60)

373

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

where =

g= *

+

4 3L. 2dFVJ

4 - 3L,

q=21/Lvv

[(#) + 2" + B'"'lct2eV1

B

The effect of positive K1' on the shapes of solute bands as they aredeveloped within the column is to produce asymmetric leading and trailing boundaries. Thus, for a single solute component there is a boundary sharpening effect on the trailing edge and a boundary spreading effect on the leading edge of the zone. Experimental observations of this effect in gel chromatography were first described by Winzor and Scheraga (1963). For the case of ovalbumin a t high concentration on Sephadex G-100 columns, the effect is particularly pronounced. Experimental results of Winzor and Nichol (1965) for this system are shown in Figs. 6 and 7. Another consequence of Eq. (59) is seen in Fig. 6, namely that the (normalized) boundary shapes depend on the initial (plateau) concentration of solute applied.

C. Interpretation of the Axial-Dispersion Coeficient The general phenomenological treatment of the previous sections has led to the solutions of continuity equations for various cases of solute transport. These provide useful descriptions of the behavior of real experimental sys-

I

3

1

6

I 9

12

CONCENTRATION (mg/ml)

FIG.7. Concentration dependence in centroid elution volume for ovalbumin solutions chromatographed on Sephadex G-100 in phosphate buffer (pH 6.8, ionic strength 0.1). Data are obtained from centroid boundary positions of large zone experiments. These positions correspond to partition coefficients for the plateau concentrations of the respective zones. Taken from Winzor and Scheraga (1964).

374

GARY K . ACKERS

tems. However, in this approach the axial-dispersion coefficient has been described only in terms of a general nonequilibrium perturbation. The purpose of this section is to present a quantitative description of the factors that contribute t o the axial-dispersion coefficient and determine its nurnerical value. The importance of understanding the parameters that make up this coefficient is based on two considerations from the viewpoint of analytical applications. First, the axial-dispersion Coefficient, like the partition coefficient, is a function of molecular size. A column can in principle be used, with appropriate calibrations, to determine molecular size from zone spreading. Second, the axial dispersion effects play a n important role in the interpretation of solute zone profiles of chemically reacting systems of macromolecules. A third incentive to the analysis of axial dispersion lies in its use in improving the “efficiency” of chromatographic columns for separation purposes, i.e., reducing the overlap between zones. The basic contributing factors will be considered in turn, using the approach first developed by Glueckauf et al. (1949).

I . Finite Gel Particle Size and Nonuniform Velocity of Flow The internal statistical summation of nonuniformities within the column can lead to a Gaussian zone shape. For example, in the case of nonuniform velocity of flow, the column packing is visualized to consist of laycrs of particles, the passage through each of which produces a certain distribution of times of passage. The mixing of flowstreams which effectively occurs between layers, ensures that the successive distributions are uncorrelated and consequently the distribution of times of passage for the aggregate of many layers will approach a Gaussian probability curve. This effect has been observed in columns packed with impenetrable glass beads (Pedersen, 1962). For low values of the flow rate F , this effect should be independent of F, but is directly proportional to gel particle size d. Thus the contribution of this term to L is simply a constant A,. 2. Axial-Difusion Flux

There are two diffusion fluxes along the axis of the column that must be taken into account as contributing to axial dispersion. The first of these is diffusion within the mobile phase. This flux will be given by Fick’s law, which for a column of unit total cross section is

where a,the void fraction, is the cross-sectional area of the mobile phase, and D is the diffusion coefficient. The flux is again taken relative to the moving frame of reference 4. Fick’s second law, written for volume flow, then becomes

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

375

In order to evaluate the corresponding flow within the stationary phase we have

Then

PuD a2C F a p

dC

av

and the total Contribution due to diffusional flux

+ + b)I) a”c

J D = J,D

J,D

- - (a

F

w2

Thus the contribution LD of diffusion to L is

It is seen that the diffusional dispersion should be greater a t low flow rates. 3 . Nonequilibrium between Phases

As the solute moves with the mobile phase past the gel particles, diffusional distribution does not occur with instantaneous equilibrium between phases. This results in a nonequilibrium perturbation in which the concentration-distance profile within the stationary phase will lag the corresponding solute distribution within the mobile phase for a positive gradient and trail for a negative gradient. The conservation of mass condition is:

The total solute Q*T within the section per unit column length equals Q* aC where Q* is the (nonequilibrium) solute per unit length in the stationary gel phase. Then Eq. (67) can be written

+

ac

ac ag* dv av

-+a-+-=o

ax

It is assumed that in the time dt = dV/F in which an infinitesimal volume passes through the colum~i,the rate of change in amount of solute in the internal space is proportional to the disequilibrium perturbation Q - Q*

y$)z

=

const.(& - Q’)

376

. GARY K. ACKERS

This first-order approximation will be valid as long as the nonequilibrium perturbation is not very great (Glueckauf, 1955a,b). The constant in Eq. (69) is the reciprocal of the mean time dtD required for solute to equilibrate by diffusion with the stationary phase, so that in this time dtD, the change in solute is dQ* = aQ' - dtr,

at

Q - Q*

=

(70)

The mean equilibration time is related to the diffusion coefficient D. within the gel by

q is a geometrical (packing) factor and d is the bead size.

Noting that

F -aQ* = -aQ* av at and combining Eqs. (69), (70), and (71)

Taking derivatives with respect to V and rearranging (72)

Substituting for aQ*/aV from (73) into (68)

ac

- + (y

ax

ac + aQ -

av av

=

I

qd2F __ a2Q* D, av2 ~

(74)

Defining C' as the (nonequilibrium) concentration of solute within the , noting that Q = PuC, Eq. (74) penetrable regions such that Q' = ~ u C *and becomes

ac -+ ax

k-

ac =

aZc*

av

a v2

where

L'

=- 9d Z F @

DS Going back to Eq. (67)

aq- pu- ac* = - ac -- a

av

a2c*

a 7

av

ax

dc

av - 9

ac*ZV-

(75)

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

Then (75) becomes

-ac+ { - +ac- - +qd2F - - - a2c

av

ax

D,

axav

To a first approximation

-=-[&I(%)=-+($)

ac ax

aqd2F a2c

D,

av2

-O

377

(77)

ac

dv

so that Eq. (77) becomes

TILe coefficient o a2C/ax2 here is a coefficient of the type L’ [Eq. ( 3 2 ) ] . Then combining expressions for LV ( = L’/[) we have from Eqs. (66) and (78):

This expression is seen to have the form of the well-known Van Deempter equation (Van Deempter and Klinkenberg, 1953). In this equation th e “height equivalent to a theoretical plate” is given by

BD CF H = A + F + x

(80)

in which A , B, and C are constants. The relationship between the axial dispersion coefficient LV and height equivalent to a theoretical plate is

H

=

2+Lv

(81)

Equation (79) shows that the significant molecular parameters that determine the axial dispersion Coefficient are the partition coefficient (r (through the term +) and free diffusion coefficient D. A calibrated column can in principle be used to determine these constants. Although this approach has not been utilized to date, it appears sufficiently promising that its future development may be anticipated. The use of direct optical scanning (Section IV,A) would appear most suited to the analysis of axial dispersion since the effect of variations in column packing within the column can be taken into account. To date there have been very few critical studies of axial dispersion. An expression relating plate height to flow rate in chromatographic columns has been proposed by Giddings and Mallik (1966). However they failed to take into account the very substantial

378

GARY K. ACKERS

variation of D,with molecular size, and assumed D,/D = 35 for all molecular species. This might in part account for the almost total lack of correspondence found between theory and experimental values taken from a variety of literature sources. IV. DETERMINATION OF MOLECULAR SIZEAND WEIGHT Calibrated porous gel columns have been used extensively in recent years for determination of molecular size and weight. These determinations can be carried out rapidly with a minimum of equipment a t very small expense. The procedures employed encompass a considerable range of accuracy and reliability. I n general the best of them compare quite favorably with other methods of molecular size determination (e.g., diffusion, viscosity, lightscattering, electron microscopy). The determination of molecular weight, however, usually depends on a somewhat less reliable correlation between the molecular radius and molecular weight. Under special circumstances, when the solute molecules exist as random coils, the correlation is found to to be extremely good and the molecular weights are determined with correspondingly greater accuracy. There are, in all cases, certain limitations and precautions that must be taken in the intelligent application of these techniques to a given experimental problem. In this section we will consider the progress that has been made in understanding the utility of these procedures as well as their limitations for determination of molecular size and weight.

A . Treatment oj Chromatographic Data The experimental procedure most commonly employed for molecular size and weight determination is the small zone elution experiment, in which the primary quantity measured is the peak elution volume, Ifp. Data from such experiments have been represented in many different ways by various workers. The adoption of a standard convention for data representation would greatly facilitate comparison of results from different studies. A measured elution volume on a particular column cannot be compared directly with any other quantity except another elution volume from the same column, since this quantity reflects contributions of gel particle packing as well as partition coefficient and volume of stationary phase. Some workers have normalized the elution volumes with respect to total column volume or void volume and expressed their results in terms of V J V , or Ve/Vo,respectively. However, there is very little to be gained by this procedure since gel particle packing always varies significantly from one column to another. On the other hand, the calculation of partition coefficients from elution volumes provides an expression of the fundamental property of the system which is invariant for a given gel from one column to

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

379

the next. Clearly all published data should be expressed in terms of partition coefficients. A specification of the gel preparative procedure (or lot number of a commercial preparation) and conditions of the experiment will then enable other workers to reproduce a partition coefficient as precisely as possible. Representation of data in terms of partition coefficients also facilitates estimation of the significant errors involved in determination of molecular size or weight. A given relative error in the determination of elution volume may have a devastating effect on the accuracy of a molecular size or weight determination if the molecular species is eluted close to the void volume, whereas the same error may have a relatively minor effect for a molecule eluted toward the end of the column. From a theoretical point of view the partition coefficient is the most basic phenomenological parameter that can be used to characterize the interaction between a solute molecule and a porous gel network. However, from an experimental standpoint, the partition coefficient is always a derived quantity, subject to errors resulting from the reduction of primary data such as elution volumes [Eqs. (21) and (27)], volume-distance plots, migration distances on thinlayer plates, etc. In relating experimental data to molecular parameters, it is clearly desirable in some cases to perform numerical calculations directly on the primary data (such as elution volumes all obtained from a given column) in order to eliminate errors of data reduction. I n these cases the specification of partition coefficients should still be made as well as the other results computed from the data.

B. Interpretation of the Partition Coeficient-Calibration

Functions

The accuracy and reliability of a molecular size or weight determination depends on (1) the accuracy with which the partition Coefficient or other relevant parameter is determined, and (2) the degree f reliability with which an interpretation can be placed on this measured parameter in terms of molecular properties. With methods presently available, the partition coefficients can be determined reproducibly to an accuracy of three places. Since their values range between zero and unity, the accuracy is greatest for the larger partition coefficients. With regard to the second point there are two questions that must be answered: (a) Which properties of a molecule determine its partition coefficient with respect to a given porous material? (b) What relationships exist between molecular properties of the solute and the measured partition coefficient? At the present time these questions cannot be answered as rigorously as one would like on the basis of experimental evidence available although many theories exist. Nevertheless, a considerable amount of progress has been made toward a I:

380

GARY K . ACKERS

practical understanding of the molecular properties measured in a gel chromatographic experiment. Under most experimental conditions the measured parameter u (or K A Y ) corresponds to the distribution of solute within the gel matrix a t thermodynamic equilibrium (Section 111). This coefficient then represents the steric constraints that lead to exclusion of the solute molecule from a fraction of the gel’s internal solvent region. Although charge interactions and surface adsorption, when present, undoubtedly play some role, the partition coefficient may be viewed primarily as a result of steric exclusion effects. The major factors which determine the steric constraints are molecular size and shape and the structure of the gel matrix, including nonaccessible water of hydration. I n many respects the ideal porous material would be one with the “all or none” molecular sieving effect illustrated in Fig. 8a. For a gel of given II porosity,” then, all molecules larger than a critical size would have partition coefficients equal to zero, while all smaller molecules would have partition coefficients of unity. This would provide for maximum efficiency in the resolution of molecular species, both for preparative and analytical purposes. Indeed, some manufacturers of commercial gel materials have indicated molecular weight “exclusion limits” suggestive of this type of mechanism. However, the reatistic curves experimentally obtainable for partitioning of molecules on porous materials always exhibit a more gradual exclusion property, and are characterized by a nonlinear decreasing curve, such as that shown in Fig. 8b.

MOLECULAR RADIUS CI

FIG.8. Variation of partition coefficient with molecular size for “all or none” type partitioning (a) and for conical pores (b). The curves represent the same limit ( r = 10) for total exclusion.

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

381

There are a number of reasons for the curve having this general shape. For structural models involving any conceivable pore shapes or arrangement of constraining surfaces, the curve must decrease continuously with increasing molecular size. For example, consider equilibrium partitioning within uniform cylindrical pores oriented in such a way that all pores are totally accessible to penetration by molecules in the surrounding solvent regions. Even such a structure as this would not exhibit an “all or none” type of sieving, and the partition coefficient would be (1 - a/r ).2 Similarly, for conical-shaped pores, the function becomes u = (1 -

;>”

A plot of this function is shown in Fig. 8b and is compared with the corresponding curve (Fig. 8a) for the LLall-or-none”type of exclusion. A more useful conical pore model (Porath, 1963) is described below. For a given column, the most straightforward way of establishing the relationship between partition coefficient and a molecular size parameter is to calibrate the column with known standard particles in a completely empirical fashion. The resulting calibration curve then can be used t o determine the molecular size parameter of an unknown molecular species of interest (Steere and Ackers, 196213). For molecules of similar shape and density a direct correlation of partition coefficient with molecular weight becomes meaningful (Andrews, 1962, 1964; Ackers, 1964; Laurent and Killander, 1964; Siege1and Monty, 1966). In order to minimize the number of standard molecules that must be used to establish a reliable calibration curve, it is useful to have a general equation expressing the relation between molecular size or weight and partition coefficient. Equations have been proposed for column calibration based on empirical curve-fitting, on various geometric models of the gel, or on statistical assumptions. These calibration functions are summarized below.

1. Logarithmic Plots Nonlinear curves can frequently be straightened out over some region by plotting the variables as various powers or logarithms of each other. This device was first employed by Granath and Flodin (1961) for the treatment of gel chromatographic data, and subsequently by many other workers (Andrews, 1962, 1964, 1965; Whitaker, 1963; Leach and O’Shea, 1965). A logarithmic relationship found empirically to obtain for many gel systems can be written in terms of the partition coefficient: u =

-AlogM+

B

where A and B are empirical constants and M is molecular weight.

(83)

For

382

GARY K . ACKERS 250 -

7

- 3.0

--

- 2.5 170

- 2.0

I30

-

110

-

90

-

70

105

104

9

\

k*

10‘

Molecular weight

FIG.9. Logarithmic calibration plot for proteins of different molecular weight on Sephadex G-200. Taken from Andrews (1965).

spherical molecules or random coils the molecular weight can be expressed in terms of some power, p , of molecular radius a

M

=

Kap

and Eq. (83) becomes u =

where A’

=

-A’ log a

(84)

+ B’

(55)

p A and B’ = ( B - A log K ) .

A very thorough investigation of these relationships has been carried out by Andrews in a series of studies (Andrews, 1962, 1964, 1965) aimed a t evaluation of the general validity of direct molecular weight determinations using gel chromatography. A representative set of experimental data is shown in Fig. 9 for a Sephadex (2-200 gel. Here the elution volume V, has been correlated with molecular weight M . This correlation can be derived from Eqs. (21) and (83), giving

v, = - A

u

log M

+

+ B”

(86)

where A” = ViA and B” = (BVi Vo). It can be seen (Fig. 9) that this relationship holds only over the central portion of the curve. It was found that highly asymmetric molecules and those containing carbohydrate deviate significantly from the curve generated from elution volumes of the “normal” proteins. Furthermore, the behavior of all molecular species could be better correlated by a curve

AXALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

383

relating molecular weight to diffusion coefficient. Thus Eq. (85) is a more general representation of the data than Eq. (83), provided a is taken as the equivalent hydrodynamic radius (Stokes radius). The conclusion that elution volumes are best correlated with niolecular size rather than weight has been made by a number of workers (Steere and Ackers, 1962a,b; Laurent and Killander, 1964; Ackers, 1964; Siege1and Monty, 1965; Giddings et al., 1968). A careful investigation by Ward and Arnott (1965) of four glycoproteins indicated large deviations from the molecular weight calibration curves obtained with unconjugated globular proteins. Since the effect was observed both for gels of dextran (Sephadex) and polyacrylamide (Bio-gel) , it was attributed to the different partial specific volumes of the two groups of proteins rather than surface adsorption or charge effects. These studies indicate that the logarithmic calibration procedure is subject to uncertainties arising from the necessity to establish identical shape and density for calibrating standards and unknown molecule of interest. For proteins this difficulty can largely be circumvented if chromatography is carried out in solutions of denaturing agents so that all molecules are denatured “random coils” (Section IV,E). 2 . Conical Pore Model

The earliest geometrical model for calibration of gels (Porath, 1963) visualized the penetrable voids within the gel matrix as a collection of conical pores. I n this model the average pore has a diameter 2 T and depth H so that its total volume Vi is given by:

A molecule of radius a can penetrate to a maximum depth h, resulting in a n effective conical pore for distribution, whose volume is

v, = sh(r 3- a ) 2 The partition coefficient u is then proportional to the penetrable volume fraction V,/Vi. Since h / H = (1 - a / r ) :

k is a constant of proportionality. Assuniing that a is proportional to M”2 (for random coils) and r is proportional to the cube root of “accessible” solvent, his final equation is

a=k

I

l-lc’

I

( S ,h1“2 - u)l‘3

384

GARY K . ACKERS

where S , is the solvent regain and w is the part of S, from which all solute molecules are excluded. A plot of u1/3against A/ll2 should result in a straight line, according to this model. Such a correlation was indeed found for niolecules which exist as random coils (Porath, 1963). Andrew (1964) has found that such plots are linear even for compact globular proteins, for which the theory should not hold. 3. Mixtures of Cones, Cylinders, and Crevices

A siniilar analysis to that described above can be carried out using a broader collection of constraining shapes (Squire, 1964). For a mixture of pores comprised of cones, cylinders, and crevices, the equation derived by Squire takes the form

v, = vo[l + g(l

-9

1 3

where the constant g denotes the relative proportion of the different kinds of pores. By a slight extension of Squire’s treatment, this equation can be cast in the form of a partition coefficient U . Noting that a/r approaches zero for solvent molecules when V , = Vo Vi, we have

+

(T=-

v e

- v o

Vi

The adjustable calibration constants are g and r . Equation (91)has been cast in terms of molecular weight (Squire, 1964). If C is the molecular weight of the smallest protein molecule that cannot enter the gel

v.= v,

[1+g(l-z)]3 (94)

This relationship has been used by Sorof et al. (1966)to characterize the molecular weights of soluble liver protein fractions. An excellent correlation was obtained between V J V 0and both molecular weight and sedimentation coefficient.

4. Cylindrical Rod Model By assuming the gel network to be made up of rigid cylindrical rods, Laurent and Killander (1964) were able to obtain an expression for the partition coefficient referred to total volume of gel.

KAV = exp[--?rd:(a

+

T ) ~ ]

(95)

385

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

This equation is a n application of the expression derived by Ogston (1958) for the fractional volume available to spheres within a random collection of cylinders. The same relationship has been derived by Giddings et al. (1968) on the basis of a statistical mechanical approach and by Chun et al. (1969a) using a moment generating function. I n Eq. (95) T is the radius of the obstructing rod and d: is the concentration of rods within the system. Since the polymer chains of a gel network are not really cylindrical rods, these parameters can be viewed as adjustable constants, and provide a good correlation between molecular radius and partition coefficient (Fig. 10). An extensive investigation of this relationship for partitioning within polyacrylamide gels has been carried out by Fawcett and Morris (1966) as a function of gel composition. In this study the apparent rod concentration, S, was found to be a linear function of monomer concentration for constant degree of cross-linking while the apparent rod diameter T was constant. Furthermore, it was found that 2 increased t o a maximum with increasing degree of cross-linking a t constant monomer concentration, and then dropped sharply. The apparent rod diameter increased to a plateau under these conditions. This increase of r was interpreted as a lateral aggregation of the gel strands by these workers. The rod model was taken literally by Laurent (1967) in a determination of the structure of agarose

7

RADIUS OF EQUIVALENT SPHERE

(cm)

FIG.10. Correlation of partition coefficient ( K A V ) values with molecular radius by means of the cylindrical rod model (Eq. 95). Data are for niolecriles partitioned into Sephadex G-100 (upper curve, I) and G-75 (lower curve, 11) gels. Taken from Laurent and Killander (1964).

386

GARY K . ACKERS

gels by partition coefficient measurements. IJsing homogeneous Ficoll fractions with known Stokes radii, the gel was found to be made up of randomly oriented rodlike fibers 50 A in diameter. The structural model inferred from these studies differs somewhat from results of a physical chemical characterization of the agarose molecule by Hickson and PoIson (1968). By hydrodynamic methods these authors found the molecule to have a diameter of 10-16 A, and electron micrographs revealed helical coiling of these flexible polymer chains. It was suggested that such coiling might be responsible for the apparently greater fiber diameter inferred from the chromatographic studies. It is also possible that the fibers exist within the gel as laterally aggregated agarose molecules. In general the validity of inferences regarding the structure of gels drawn from Chromatographic studies must always be highly suspect, since equally good correlation between molecular size and partition coefficient can usually be obtained with a variety of calibrating functions based on different models (Siege1 and Monty, 1966; Ackers, 1967b). Although none of these equations is accurate in a structural sense, they all contain adjustable constants that enable them to be fitted to a wide range of data. The point clearly illustrated is that the gel can be represented by a variety of equivalent models just as molecules in solution can be represented as spheres, rods, ellipsoids, coils, etc. 5. Cylindrical Pores with Steric and Frictional Efects

The equilibrium partitioning concept of gel chromatography was extended (Ackers, 1964) to include nonequilibrium restricted diffusion effects using a model of cylindrical pores. An equation developed by Renkin (1955) was used for calculation of partition coefficients. c =

(1 -

:y [

1 - 2.104

(:)

+ 2.09 (1T)s

I")!(

- 0.95

(96)

This equation takes into account both steric and frictional interaction between a particle of radius a and a system of uniform cylindrical pores of radius r. Calibration of a column with standard molecular species provide an estimate of the apparent pore radius r. Subsequently, an unknown molecular radius, a, can be calculated for a molecule of interest from the experimentally determined partition coefficient. Although Eq. (96) provides an accurate means of correlating molecular radius with partition Coefficient for largc-pore gels, thc concept of nonequilibrium mechanism upon which it is based does not appear valid under most operating conditions. Most experiments with gels in which noriequilibriuin effects can clearly be shown to influence elution position also show a high sensitivity to flow rate and trailing of zones. Furthermore, the initial

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

387

observation of unequal partition coeficients for proteins on Sephadex G-200 obtained under equilibrium and nonequilibrium conditions does not appear to be a general phenomenon (Ackers, 1964; Edmond et al., 1968). Studies using direct optical scanning of columns (Brunibaugh and Ackers, 1968) provide strong evidence of identity between partition coefficients obtained in static equilibrium and nonequilibrium experiments. Under some conditions a t least the nonequilibriuni effects may play a dominant role in determining elution positions on porous-glass columns. Haller (1968) has found a large discrepancy between the equilibrium partition coefficients and those obtained by elution experiments. A nonequilibrium theory has also been proposed by Yau and Malone (1967). 6. BrBnsted-Type Partitioning Model

For partitioning of solute between two phases, a thermodynamic relatioriship of Bronsted has often been found applicable (Albertsson, 1960)

KAV = e--X/RT (97) where R is the gas constant, T the absolute temperature, arid X a constant. This approach has been extended by Fischer (1967) using a cell model in which he calculated thermodynamic parameters for solute transfer between bulk liquid and gel phase. The resulting equation can be written

The solvent regain is denoted by S,, and K1 and Kz are constants. This relationship has been successfully fitted to the data of Granath and Flodin (1961) for partitioning of dextrans on Sephadex gels (Gelotte and Porath, 1967). Although the exponential function Eq. (98) containing two adjustable constants can be fitted to experimental data, the mechanism upon which it is predicated does not appear to be of general validity. I t is based on the assumption that solute molecules and gel network interpenetrate each other in such a way that no solvent niicroregioris exist within the gel from which solute is excluded. 7 . Spheres, Cylinders, and Slabs

The distribution probabilities of flexible niacromolecules within voids represented as spherical cavities, cylindrical pores, and slab-shaped cavities has been calculated by Casassa using random-flight statistics (Casassa, 1967). The resulting equation for partitioning of a flexible linear polymer chain within a spherical cavity of radius r is

rn = 1

388

GARY K. ACKERS

FIG.11. Theoretical partition coefficients KAVof homogeneous linear polymer chains. R is the root mean square molecular radius. Curve (A) is for spherical cavities of radius a, (B) for long cylinders with radius a, (C) for slabs wit,h thickness za. The experimerltal

points show data for porous glass with 121 d pore radius ( 0 )and 900 d radii. from Casassa (1967).

Taken

where a is the radius of gyration of the polymer chain. The corresponding expression for cylindrical cavities of radius r and infinite length is

where the Pm’sare the roots of a zero-order Bessel function of the first kind J ( P ) = 0. For a “slab-shaped” cavity between two infinite planes separated by a distance 2 r m

1 m =O

These expressions have been compared with experimental data of Moore and Arrington (1966) for polystyrenes partitioned into porous glass column media. The slabornodel was found to give the best fit to data for glass with pores of 121 A and 900 (Fig. 11). 8. Random Plane Model

For an isotropic network of random planes in which all plane orientations are equally represented, the partition coefficient for rigid molecules has been calculated (Giddings et al., 1968) to be: u = exp[-

SL/2] (102) In this expression pore size is represented by S, the pore surface per unit

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

389

internal volume within the system. The molecular parameter is the "mean external length" defined as the mean length of projection of the molecule along the various axes. These authors have proposed the mean external length as a superior correlate of partition coefficient t o either the hydrodynamic radius or radius of gyration on the basis of model pore calculations involving a variety of geometric pore shapes. These contentions have not yet been tested against experimental observation and the calculations all depend on idealized geometric models of gel structure. The mean external length parameter does a t least have the important property of reflecting both the size and shape of molecules. 9. Random Distribution of Penetrable Volume Elements

I n order to circumvent the necessity of postulating specific geometric shapes for the pores within gel partitioning systems, a statistical calibrating function was proposed (Ackers, 196713). This function is based on three assumptions: (a) the microregions within the porous network are heterogeneous in size; (b) the individual region may be characterized by the radius, a, of the largest molecule that it can accommodate; and (c) the frequency distribution of sizes follows a normal curve. The differential fraction of the internal volume penetrable by a molecule of radius, a,then can be represented by a Gaussian probability curve. The total fractional volume within the gel that can be occupied by molecular species of radius a is the partition coefficient, given by the error function complement of the Gaussian distribution: n -nn

The constant a. is the position of the maximum value of the distribution and b, is a measure of the standard deviation. These parameters, a. and bo, are calibration constants for a given gel. Solving Eq. (103) for the molecular radius, a linear relation is predicted to exist between a and the inverse error function complement (erfc-l) of u. a = a.

+ bo erfc-'

u

(104)

Representative plots showing this correlation are shown in Fig. (12) for a variety of column materials and molecular species. The two calibration constants a. and bo appear, respectively, as the ordinate intercept and slope of each plot. 10. Generalizations

Each porous gel of given composition and history exhibits a characteristic relationship between partition coefficient and molecular size. As has been

390

GARY K . ACKERS

erfc-1 r FIG.12. Correlation of partition coefficient u with molecular radius according to Eq. (103), assuming random distribution of penetrable volume. elements. Taken from Ackers (1967b).

seen, this relationship can be represented by equations based on a variety of different structural models. The models are equivalent in the sense that they can be fitted to the experimental curve through adjustable constants. This is reminiscent of the fact that molecules can be represented by a variety of geometric structures (spheres, rods, coils, etc.) that are equivalent in terms of their measurable transport properties. A consequence of both the geometric pore model calculations and the statistical models is that, generally, two adjustable constants must be used to characterize the state of a porous network with respect to its solute partitioning properties? A second generalization can be drawn from the theoretical analyses of random constraints within porous systems : The penetrable volume fractions of such systems are always linearly independent functions of molecular size and are usually linearly independent functions of the concentration Equation (96), which was originally proposed for large-pore gels arid contains only one adjustable constant, can be applied to small-pore gels as well by incorporating a second adjustable constant, K , and replacing ( n / r ) by ( K - a / r ) in the expression for u. A single calibration constant, or partition coefficient is not adequate in the general case to define the partitioning system. Generally the specification of two calibration constants (and the calibrating function used) or partition coefficients (with corresponding molecular radii) is required.

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

391

within the system of the obstructing network. In general the column calibration function can be written for a given gel x, 0- =

d a , Pzi)

(105)

where u is the partition coefficient, a is molecular size, and the psi are calibration constants. A gel of different porosity is characterized by a different set of calibration constants psi. The linear independence property of partition coefficients means that if, for two given gels, partition coefficients are related to molecular size by two functions, ul(a, p l i ) and up(a,p p i ) which belong in common to a family of curves, a column made from a mixture of these gels (a “hybrid” column) in an arbitrary proportion would have partition coefficients related to molecular size by some function ~ ( apli, , pzi) that is not a member of the same family of functions to which u1 and up belong. This linear independence is a characteristic feature of all column calibration functions that have been shown to provide good correlation between molecular radii and partition coefficients. The property can be demonstrated experimentally utilizing a linear transformation of any of these calibration functions. With the error function complement representation of u, the linear transformation is Eq. (104) in which a0 and bo are the calibration constants pzl and pzp for a given gel x. If a “hybrid” column is now constructed with a mixed bed comprised of each of two gels which conform to Eq. (104), the partition coefficient uh will be ’

where w1 and 1 - w1 are, respectively, the fractions of the mixed bed’s internal volume Vi associated with each of the two gels. Then since the sum of two error function complements is not itself an error function complement, the plot of molecular radius versus erfc-’uh should not be linear. A precise way of stating this property is to say that there exists no set of nonzero constants el and c2 such that for all values of a, erfc-1 Uh

= €1 erfc-I u1

+ e2 erfc-I u2

(107)

Experimental data illustrating this fact have been obtained for hybrid columns made of Sephadex G-75, G-100, and G-200 (Ackers, 1968). These results are shown in Table VI and Fig. 13; Thus the question of whether a mixture of two gels of different porosity is equivalent to a single gel of intermediate porosity can be answered in the following way. The mixed bed gel is equivalent to a single gel of intermediate porosity in the sense that Eq. (106) is satisfied. However, it is not equivalent in terms of the property defined by Eq. (107). It is clear that the gels retain a n element of “individuality” when mixed together inasmuch as the only kind of

392

GARY K . ACKERS

average molecular sieve behavior that results is that described by Eq. (106), but not the kind implied by Eq. (107). The calibrating function is analogous to an equation of state for the gel phase where the “state” of the gel with respect to molecular sieve properties is completely defined by the parameters pZi. Calibration plots of three columns according to Eq. (106) are shown in Fig. (13). Here it is seen that whereas linear plots are obtained for the G-75 and G-200 columns, the data points for the hybrid column show a pronounced deviation from linearity. The dashed line is the expected curve for the hybrid coIumn calculated by means of Eq. (106) and the respective calibration constants of columns I and I1 (for G-75: a. = 5.7, bo = 22.5; for G-200: ao = 7.3, bo = 44.8). For comparison, the solid line is a hypothetical calibration plot for a G-100 column having approximately the same overall density of dextran as the hybrid column 111.

C. Multiple Porosity

Columns

Sometimes columns are used for separation or analysis in which the beds contain a composite of porous materials of different porosity (Steere and Ackers, 1962a; Spragg et al., 1969). The different gels are either stacked in layers within the column or thoroughly mixed together before the column is packed. I n either case the same elution volume is obtained for a given molecular species. Consider a column stacked with, say, n gels of different porosity. The TABLEVI Molecular Sieve Coeficients for Sephadex G-76 and G-,900 and “Hybrid” (Mixed Red) Column Molecule Dextran” Cytochrome c M yoglobin Ribonuclease Trypsin Ovalbumin Serum albumin Aldolase ?-Globulin Mean

10.6 16.4 18.8 19.2 19.4 27.3 36.1 46.0 52.0

m76

c2W

0.758 0.513 0.427 0.423 0.386 0.170 0.062 0.012

0.917 0.778 0.715 0.725 0.689 0.504 0.382 0.208 0.172

-

ch

0.811 0.589 0.528 0.526 0.495 0.280 0.179 0.080 0.060

WP6

0.661 0.690 0.650 0.660 0,641 0.670 0.634 0,653 0.650 0.657 f 0.012

a A sample of dextran 40 was fractionated on the Sephadex G-200 column, and a sample of narrow molecular size distribution was obtained. The value u = 10.6 A was calculated from the value of 5 2 0 0 for this fraction.

393

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

I

0

I

02

I

0.4

I

I

I

0.8 1.0 erfc-’ (r

0.6

I

1.2

I

1.4

I

1.6

i

1.8

FIG.13. Calibration plots for Sephadex G-75 and G-200 and a “hybrid” column ( 0 ) of mixed Sephadex G-75 and G-200. The ratio of the two gels was mixed so as to obtain an overall density of dextran comparable to that of Sephadex G-100. The calculated water regain obtained for the hybrid column was approximately 9.6 gm of water per gram of anhydrous dextran. The solid line is a hypothetical calibration plot for a Sephadex G-100 column having this overall density of dextran. Linear independence of the partition coefficients for these gels is indicated by the nonlinearity of the calibration plot for the hybrid column (dashed line). Taken from Ackers (1968).

elution volume of a given molecular species on such a column is the sum of elution volumes from each of the segments

1 n

VeT

=

VeK

(108)

k=l

where V e K is the volume of solvent which passes through the column between entry and exit of the molecular species from the K t h segment. Similarly the total void volume and internal volume are given by

2 n

VOT

=

VOK

k=l n

4 k =1

Then the partition coeffcient rTfor a molecule on the total column is

394

GARY K . ACKERS

I n order to relate UT to the partition coefficients U K pertaining to the individual segments of the column

we define W K as the fraction of internal volume associated with the K t h gel

2

ViK

k =l

Then by forming the product over K we have

WKUK

from Eqs. (112) and (113) and summing

Then from Eqs. (111) and (114) uT =

WKuK

(115)

k=l

and the elution volume of a species on the stacked gel column is

That this is the same elution volume as obtained on a mixed gel column is seen by considering the partition isotherm for the mixed gel. At any level within the column the total amount Q of solute per unit length distributed within the stationary phase is given by the sum of amounts of solute QK distributed within the various gels. n

n

where PK = w K P is the internal volume fraction per unit length associated with gel K and U K is the solute’s partition coefficient on gel K .

If we form the product

WKUK

n

k=l

and sum over K

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

395

Thus the partition coefficient UT for the solute molecule on a mixed bed column is the same as the total partition coefficient for a column stacked with the same gels (Eq. 115), and the elution volume is given by Eq. (116).

D. Combination of Chromatographic Results with Other Kinds of Data Conibination of chromatographic data with those obtained by a second method, such as sedimentation, viscosity, or light scattering, can be used to provide an accurate molecular weight, or to estimate the degree of molecular asymmetry, provided molecular weight is already known. For most purposes the diffusion coefficient is calculated from the molecular Stokes radius, a, by means of the Einstein relation (cf. Gosting, 1956).

yielding the standard value D20,wcorrected to water when T is 20°C. R is the gas constant, N is Avogadro’s number, and 7 is the viscosity. Such a value can then be combined with the sedimentation coefficient s20,w and partial specific volume to yield the anhydrous molecular weight by means of the Svedberg equation

where p is the density of water a t 20°C. Similar calculations can be performed for combination viscosity data with the diffusion coefficient. If the molecular weight is known, the frictional ratio f/fo may be calculated from the molecular radius, a. The friction coefficient f is related to a by Stokes’s law

f = 6aqa (122) while the frictional coefficient for an equivalent anhydrous spherical molecule is given by

The frictional ratio then is

This ratio provides a measure of both molecular asymmetry and hydration. If the latter can be estimated, then the axial ratio for an equivalent ellipsoid of revolution can be calculated (Siege1 and Monty, 1966). A protein of desired molecular weight may, in some cases, not be suffi-

396

GARY K. ACKERS

ciently pure to be studied successfully except by methods in which assay can be carried out by activity measurement. I n such cases the partition coefficient, measured by activity assay on a calibrated column can be used to calculate an upper limit value for the hydrated molecular weight (Ackers, 1964). By combining the molecular radius, a, with Avogadro’s number, N , the molecular weight, M,,,, is

The partial specific volume must be known or reasonably estimated. The true anhydrous molecular weight will generally be less than this calculated value M,,, by a large percentage. Based on an assumed hydration of 0.35 gm of water per gram of anhydrous protein, an approximate correction of 1.46 can be applied.

E. Molecular Weight Determinations with Denatured Proteins The inherent difficulties of direct molecular weight determination of compact globular proteins by gel chromatography stem largely from differences in molecular shape and density of the various molecular species employed, as has been described above. These difficulties can be largely overcome by carrying out the determinations under conditions where the proteins are denatured. The most effective applications of this approach to date are those in which a high concentration of guanidine hydrochloride (5 M to 8 M ) is present in solution, and was first used by Small and coworkers (Small et al., 1963; Cebra and Small, 1967). Studies of immunoglobulin derivatives were carried out on Sephadex G-200 columns in the presence of 5 M guanidine hydrochloride. They found a linear correlation between the square root of molecular weight and the square root of the partition coefficient. The method seems to have been rediscovered by Davison (1968), who used a 6% agarose equilibrated with 6 M guanidine hydrochloride containing mercaptoethanol and a denaturing agent. I n this study the log-molecular weight plot (Eq. 86) was found to hold approximately for a series of polypeptide chains ranging in molecular weight from 2000 to 100,000. A much more extensive investigation into the accuracy and reliability of this technique for molecular weight determinations has recently been carried out by Fish et al. (1969). These workers carried out a very careful investigation of polypeptide behavior on 6% agarose columns in G M guanidine hydrochloride, 0.1 M mercaptoethanol. In addition, these workers attempted to use the relation between molecular weight and radius of gyration for randomly coiled polypeptide chains in order to test the data against column calibrating equations such as Eqs. (95) and (103). They

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

397

derived a relationship between M0.656and radius of gyration based on the Flory-Fox parameter as obtained by Tanford et al. (1967). It was found that the calibrating functions proposed by Porath (Eq. 90), Laurent and Killander (Eq. 95), or Ackers (Eq. 103) could not be fit to the data. This result would appear to impIy a structural difference between the gel matrix in the presence of the denaturing solvents and under the more common nondenaturing conditions. Both Eqs. (95) and (103) have previously been shown to apply t o 6% agarose gels over the same range of partition coefficients as those employed in this study (Ackers, 1967b). In spite of this failure of the column calibrating functions a very high degree of correlation was obtained between the molecular weights and partition coefficients. Some representative data are shown in Fig. (14). It may be expected that this method will become widely used in the future. It may be especially useful in determining the number and identity of subunit polypeptide chains in a protein. It appears that an accuracy of a t least 10% can be achieved routinely in the determination of molecular weights by this method. A related technique to that described above has also undergone critical evaluation by Weber and Osborn (1969). It is the dodecyl sulfate polyTRANSFERRIN TAKA-AMYLASE

20

11

MG-HEAVY

f

0 X

OVALBUMIN

I

ALDOLASE

I

MDH

a

a

J

I>

0

w

$

4

2

0

0. I

0.2

0.3

U

FIG. 14. Correlation between molecular weights and partition coefficients for proteins chromatographed on 6% agarose colamns in 6 M guanidine HC1, 0.1 M mercaptoethanol. BSA is bovine serum albumin, MDH is malate dehydrogenase. Taken from Fish et al. (1969).

398

GARY K . ACKERS

acrylamidc gel electrophoresis technique introduced by Shapiro et al. (1967). Weber and Osborn have studied the behavior of a large series of polypeptides in order to assess the accuracy with Iyhich the mobilities niay be correlated with molecular weight. Since the mobilities, under these conditions, appear to be determined essentially by partitioning of the random coils within the gel matrix, a very high degree of correlation niay be expected, and was found. Again, an accuracy of a t least 10% appears to be routinely obtainable by this technique. I n addition to studies carried out under completely denaturing conditions, the chromatographic behavior of partially denatured proteins may be studied as a guide to their molecular size and shape changes when perturbed by a denaturing solvent.

F . Evaluation of Conjormational Changes in Chemically Modified Proteins Because of the sensitivity of partition coefficients to molecular size and shape i t is possible to detect conformational changes brought about by chemical modification of proteins. Such studies have been carried out by Habeeb (1966). I t was found that succinylated bovine serum albumin gave two components of Stokes radii 7.75 and 11.2 mp, acetylated albumin showed two components also of radii 5.5 mp and 8.4 mp. Nitroguanylated, guanylated, and amidinated albumin had radii of 4.5, 3.78, and 3.88 mp, respectively, whereas the untreated bovine serum albumin had a measured Stokes radius of 3.7 mp. The use of gel chromatography to study conformational changes may be expanded in the future to include studies of changes associated with allosteric transitions of proteins.

V. NONELUTION METHODS A . Direct Optical Scanning of Gel Columns It is evident from a comparison of the conventional elution experiment (Section 11) with solutions of the continuity equations (Section 111) that only a small fraction of the quantitative information obtainable from a given experiment is utilized by conventional procedures of solute zone analysis. The measurement of solute zones a t only a single point of the distance coordinate (the point corresponding to the bottom end of the chromatographic column) leads to a two-dimensional elution profile. However, the solutions of continuity equations which describe solute zone configurations are three-dimensional surfaces in the variables : concentration, distance within the column, and volume or time. By restricting nieasuremerits to a single value of the distance coordinate, only a single curve within the surface is sampled, corresponding to the intersection of the surface with a fixcd plane a t the distance coordinate chosen (Ackers, 1967a). This proce-

ANALYTICAL GEL CHROMATOGRAPHY O F P R O T E I N S

399

dure might be compared to a sedimentation velocity experiment in which only a single picture or scan is taken after the solute boundary has been allowed to sediment toward the bottom of the cell. It was proposed in 1967 that the direct ultraviolet optical scanning of gel columns could be utilized to provide a feasible solution to this problem of information retrieval (Ackers, 1 9 6 7 ~ ) . The development of column scanning procedures (Brumbaugh and Ackers, 1968) has now made it possible for precise determination of solute profile shapes to be made a t many intervals during a single experiment of short duration. These studies have made clear the fact that, for many analytical applications of gel chromatography to protein systems, the desired experimental information can best be achieved by direct optical scanning of solute within the column. Therefore the basic principles of this new approach will be reviewed here. For specific details of technique and instrumentation, the reader should consult the article by Brumbaugh and Ackers (1968). 1. Column Scanning Systems

MONOCHROMATOR COLUMN GUIDE \

X

Y PHOTOMULTIPLIER

MTMFTFR

I I

COLUMN--+IU

POTENTIOMETER

FIG. 15. Schemat.ic diagram of chromatographic column scanning system. Taken from Brumbaugh and Ackers (1968).

400

GARY X. ACKERS

,

220

2kO

300

350

400

440

WAVELENGTH, mp

FIG. 16. Baseline absorbance values for a series of Sephadex gels as a function of wavelength (pathlength 0.95 em). The baselines reflect primarily the internal light scattering of the column bed materials. From Ackers and Brumbaugh, unpublished results .

instrumental details see Brumbaugh and Ackers, 1968). I n this system the column is passed through a beam of horizontally collimated monochromatic light by means of a drive system. The drive system provides a signal denoting the position of the column with respect to the light beam. This signal is fed to the x-axis of an z-y plotter. The photomultiplier output is amplified and the logarithm of this signal is taken, giving linear absorbance on the Y-axis of the recorder. The rate of scanning can be varied over wide limits. A convenient gear setting allows a 15-cm length of column to be scanned a t a rate of 1 mm per second. The essential feature of design in the spectrophotometric system is the close positioning of the column to the end-on photomultiplier and the horizontal collimation of the light beam. It is also essential that the monochromator system have very low stray light characteristics in the wavelength regions of interest since the internal light scattering of the gel

401

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

produces a high baseline absorbance and the protein must be monitored accurately above this baseline. When the gel itself contains no chromophoric groups with significant absorption bands in the wavelength region employed, these high apparent absorbances are attributable to internal scattering of light within the gel. A series of such “baseline” absorbances for Sephadex gels of different cross-linking are shown in Fig. 16.

2. Determination of Solute Concentration within the Column The validity of results obtained from column scanning procedures of course depends on the reliable and accurate determination of protein concentration within the column. Because of the high internal light scattering by the column bed material, it is of primary importance to determine whether a Beer’s law relationship obtains for solute distributed within the gel. The determination of solute concentration depends on such a known relationship. If deviations are found to occur it may still be possible to calibrate the system empirically for concentration against apparent absorbance. However, when appropriate quality is achieved in the spectrophotometric system it is possible to demonstrate strict adherence of most protein gel partitioning systems to a linear Beer’s law plot (Brumbaugh and Ackers, 1968). Absorbance measurements are made a t the desired I

I

I

I

I

SOLUTION IN GEL 220 m p

3.25.

3.00w

0

2.75

z U

2 m a

2.50 2.25. 2.00

I J

I1.75 .75 0.00

1

1

I

I

I

0.0464

0.0696

0.0928

0116 0.116

0139 0.139

CONC.( m l ) FIG. 17. Absorbance of sperm whale myoglobin solutions in a saturated column of Sephadex G-100 (Superfine grade). Baseline absorbance a t 220 mp is approximately 1.85. Linearity of plot indicates adhrence to Beer’s 1:~wfor the system. Taken from Brumbaugh and Ackers (1968).

402

GARY K. ACKERS

wavelength for a column saturated with a fixed concentration of a given protein. The column is saturated by flowing a solution of protein into the column until a solution of equal concentration is eluted a t the bottom. A resulting Beer’s law plot of absorbance versus concentration for niyoglobin solutions is shown in Fig. 17 at 220 niH. Linearity of the plots obtained in this way reflects both conforniity to Beer’s law and linearity of the partition isotherni for the protein over the concentration range used. Results of this type have established the validity of protein concentration determinations by absorbancc measurements within gel columns. 3. Optical Eflects within the Gel

I n studies carried out to date it has been found that to a very high degree of approximation, the internal scattering of the gel has no effect on the measured absorbance of solute molecules partitioned into the gel. In general, it may not be expected that all types of column materials will exhibit this “ideal” behavior. Butler, for example, has studied the effect of densely packed scattering materials, such as polystyrene latex particles, on absorbance spectra of chromaphoric materials (Butler, 1962). In these studies he found a change in apparent path length for absorbance attributable to internal light scattering. I n the case of a molecule being partitioned into a densely scattering gel, this effect, if present, would be superimposed on the apparent optical pathlength change due to partitioning itself. I n addition, the effect of internal scattering could be different for differing molecular species, which penetrate different fractions of the gel’s internal solvent spaces. In penetrating the gel to different degrees, the various molecules may equilibrate into microregions having different densities of light-scattering elements (or, for some gels, different environments of chemical structure and hence diff eririg reflection-absorbance characteristics). I n addition t o light scattering effects on absorbance, spectral changes could result from interactions between gel and solute. The possibility of wavelength shifts and band broadening have been tested with proteins in a number of gels. The spectrum for the protein within the column can be compared with the corresponding spectrum obtained for the same path length in the free solution above the gel. Representative spectra of this type are shown in Fig. 18 for myoglobin solutions. The absorbance values taken within the column are seen to be lower than the free solution values, reflecting the fact that only part of the column cross-sectional area is occupied by solute. A rigorous test for identity of the two spectra is provided by the linearity of the plot shown in Fig. 19. Here absorbance values above and within the gel are plotted against each other for each wavelength. The slope of this plot reflects solute partitioning a t the point of measurement within the column.

403

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

4. Determination of Partition Coejicients by Column Saturation For a column saturated with solution of concentration C , absorbance measurements can be used to determine the partition coefficient, U, from a comparison of the measured absorbance, Ab, of the solute within the column bed and the corresponding absorbance, A,, of the free solution above the column bed. Since the pathlength is the same for both measurements, the ratio of these values can be written:

where tb and ta are, respectively, the extinction coefficients for solute within the column and in solution above the column. The solute concentration with respect to the total column bed is denoted by c b . It represents the total amount, Q T , of solute within a unit length of column divided by the total column cross-sectional area A . For a small ‘Lslice”of the column of height Ax, &TAX is the sum of amounts of solute within the void spaces and

I

220

I

240

I 260

I

280

295 300

400

420

WAVELENGTH ( m p )

FIG. 18. Effect of gel on myoglobin spectrum. Spectra designated ABOVE GEL were measured with respect to buffer (0.1 M sodium phosphate, pH 7.4) formyoglobin solutions (0.1 mg/ml) in the column above the gel bed. Those designated I N GEL were taken with respect to buffer plus gel a t one point in the myoglobin-saturated gel bed. Taken from Brumbaugh and Ackers (1968).

404

GARY K . ACKERS

IE I .4

1.2 1

w

c3

y

1.0

a

g

0.8

2LL 0

3 a

0.6

0.40.2 -

0.00

I

0.2

I

I

I

0.6 0.8 ABSORBANCE IN GEL 0.4

T

1.0

Fro. 19. Test for coincidence of spectra (shown in Fig. 18) for rnyoglobin solutions in column (abscissa) and in free solution above the gel l e d (ordinate). Taken from Brumhaugh and Ackers (1968).

within the gel. The void volume term is Axa C A , and the second term can be written from the partition isotherm as pa C A A x. Then QT = aCA PuCA. Since the total column volume per unit length is A , we can write, for CI,:

+

c*= (a + PU>C

(127)

Substituting for CI, from Eq. (127), Eq. (126) becomes (128)

Then defining a' = C Y ' E ~ / E ~ p', = &,/E, arid noting that when the column is scanned, these parameters are functions of x, we can write

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

+

P ( x ) = a’(x) P’(x)u(x)

405 (129)

The ratio, P ( x ) , of solute absorbance in the gel (corrected for baseline absorbance) to absorbance above the gel is thus a measure of solute partitioning and can be used to determine u(x). I n order to do this it is necessary to evaluate the functions a’(x)and p’(x). The first of these can be evaluated by using a molecular species sufficiently large to be totally excluded from the gel particles. Then, for all x,u = 0 and the experimentally determined P ( x ) = a’(x). Subsequently, the function p’(z) is determined using a small species such that u = 1for all x. Then p’(x) = P ( x ) - a’(x) for this case. Equation (129) has a particularly simple interpretation if it can be assumed that e, = t b and that concentrations C, in every region of penetration by the solute molecules are equal to the concentration, C, with which the column has been saturated. I n that case a’ p‘u becomes the fraction of the column’s cross-sectional area into which solute is distributed. Then the difference between measured absorbance A b within the column and A , in free solution represents simply the difference in optical pathlength for light absorption by the solution of concentration C. This set of circumstances represents an approximation t o the general case demanded by conditions of thermodynamic equilibrium (Section 11). For measurements a t sufficiently high protein concentration that a second-order term must be applied to the isotherm, the partition coefficient u becomes linearly concentration dependent and can be written u = uO(l P I C ) . For this case QT = aC pu°C puoK’,Cz and Eq. (129) becomes

+

+

+

+

P

= a’

+

P’U0(

1

+ K’IC)

A typical set of column scans before and after saturation is shown in Fig. 20. The lower traces are baseline scans of a Sephadex G-100 Superfine column,

while the upper scans were taken after saturation with a solution of ovalbumin. The offscale peak in the left-hand portion of the plot is an opaque porous polyethylene disk a t the top of the gel b,ed. The scale for the baseabove the gel and 1.5-2.5 A inside the gels, while the line scan is 0-1 scans of the saturated column are on scales of 1-2 8 above the gel and 1.52 . 5 A inside the gel. Subtracting the absorbance of the baseline gives values for absorbance of sample a t each point in the column. Table VII shows typical results of saturation measurements for ovalbumin, myoglobin, and cytochrome c . Values of a’ and p’ vary with different columns packed with the same gel and sometimes vary within the same column. If the gel has been thoroughly washed before the column is carefully packed, [(fi’(x))/(l - LY’(s))] will be constant throughout the length of the column and slight variations in a‘(x) will occur as a result of variations in gel

406

GARY K. ACKERS

particle packing. This effect is shown in Fig. 20 by the variation in distance between the (bottom) baseline trace and (upper) saturation trace for a G-100 column saturated with ovalbumin. This variation in particle packing is corrected for completely by measurements of d ( x ) using a ) obtained for totally excluded molecule. Then, a constant value of ~ ( z is a given molecular species. However, if care is not taken in packing the column, a variation with distance can be observed in all three functions a ( z ) ,/3(x),and u(x) as a result of small dextran particles that wash out of the Sephadex upon swelling and may still be present in the slurry with which the column is packed. These particles appear to establish a gradient within the column as they are washed down through the bed and trapped. They then contribute to the exclusion properties of the column resulting in vnriations of [(@'(z))/(l- a'(z)>]and u(r) with z as well as cr'(x>. It is also possible to observe variations in these quantities with x if the gel bed has been subjected to high pressures or packing forces producing distortion of the gel particles. With careful packing it is possible to eliminate almost all such variations over a suitable length of column. Since it appears likely that a t least some of the effects described above may have been present in a vast majority of columns used to date, it may TABLEVII Pmtition Coeficients for Proteins in Sephadex G-100 SuperJine Gels Partition coefficients Species

I u centroid

r-Globulin" Ovalbumin

0 0.162

M yoglobin

Cytochrome c

0.448 0.538

Gly cylglycine* Chromate

0,998 -

I1 u saturation 0.160 0.155 0.441 0.542

0

k 0.003" f 0.006 i: 0.008

4 0.004

0.998 -

I11 u small zorie 0.158 0.156 0.447 0.539 0.540

0

k 0.004d

& 0.004e 0.005 rt 0.003d k 0.003e 1

a The value of zero used for the partition coefficient of ?-globulin was verified by measurements with two other excluded moleciiles, thyroglobulin and Blue Dextrari (Pharmacin). The value of unity for the partition coefficient of glycylglyeine was used in the calculation of partition coefficients of the three protein molecules. The value (0.998) was calculated from independently determined values of the three parameters of Eq. (6). CDuplicate sets of determinations were made on two columns using ovalbumin solutioris of differing concentration. Ovalbumin and cytochrome c run individually on the same column. Ovalbuniin and cytochrome c run in mixture. Peak concentrations in both cases were 0.037 mg/ml.

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

~

407

~ _ _ _

MSTANE

FIG.20. Representative scan of a saturated column. Recorder plots show the column baseline (low traces) and the absorbance of the column after saturation with ovalbumin. Scans were taken a t 220 mp. The plots are divided into two sections by the offscale peak, which represents the opaque porous disk a t the top of the gel bed. The section to the left of this peak is the solution above the column bed. Recorder span was 1 absorbance unit. For both scans of the gel bed (rzght of disk) the scale was 1.5-2.5 A. Above the gel bed (left of disk) the lower trace (buffer) had a scale of 0 to 1 A, and the upper trace (ovalbumin) is on a scale of 1 to 2 A. Taken from Brumbaugh and Ackers ( 1968).

be useful to consider the consequences for elution experiments of a variation of the parameters a,P, and u with distance. It is evident that a nonconstant solute velocity will result through the column since

dx dt

-

F

a

+

Pu

The quantity measured as void volume from elution volume of an excluded molecule is: vo =

/01

a ( . )

dx

which is, of course, the total void volume of the column bed. Likewise for a molecule which is totally nonexcluded, the elution volume measurement yields

and the internal volume

408

GARY K . ACKERS

For the molecular species of intermediate size

V, =

[a(x) dz + ,( p(z)u(x) dx

(133)

The value of u determined from Eq. (21) then is seen to be a weighted average of u(x) with respect to p(z) over the length I of the column. u =

/01 P(x)u(x) dx [P(x> dx

(134)

A significant feature of the elution experiment is that, whereas different

molecular species may be partitioned differently a t various points within the column, the integrated average value of the partition coefficient, represented by elution volume, is taken over the same path for all molecular species. This is not the case for thin-layer chromatography, a fact which presents inherent limitations to the accuracy of this latter method for analytical determinations (Section V , B ) . 5 . Small Zone Experiments

A second procedure for determination of partition coefficients involves the scanning of a zone within the column in order to measure peak position as a function of volume flow. In this procedure a small sample (0.1 ml or less) of protein solution is applied to a small column (e.g., 5 nil bed volume), allowed to enter the colunin and followed by a fresh buffer. The colunin is allowed to flow a t a constant rate (e.g., 3 4 ml/hour) for a desired period

X

FIG.21. Multiple scans taken daring passage of a small zone (originally 0.1 ml) of potassium chromate through a Sephadex G-100 column. Taken from Brumbaugh arid Ackers (1968).

449

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

r

I

I

I

I

I

X

FIG.22. Peak position plotted against volume flow for small zone experiments. The abscissa represents peak position and the ordinate gives corresponding values of volume passed through the column. The slopes of these plots can be used to calculate the partition coefficient for myoglobin. Taken from Brumbaugh and Ackers (1968).

and then scanned. This procedure is repeated for a series of volume increments, and measurements are made of volume flow corresponding to each interval. The information obtained in this type of experiment, is the distance coordinate, x, of the solute zone’s peak position as a function of the volume, V , of liquid passed through the column. Some multiple scans of a small solute zone are shown in Fig. 21 and typical plots of elution volume versus peak position are shown in Fig. 22. The slopes of these plots can be used to calculate the partition coefficient, a, using the relationship:

C dx= a + p .

(135)

The small-zone experiment is equivalent to a set of conventional small zone elution experiments equal in number to the number of scans taken. The multiple determinations of u from a single experiment may be analyzed statistically. Representative values obtained from the data for several proteins are listed in Table VII. The three experiments shown in Fig. 22 are equivalent t o twenty-seven conventional elution experiments, but required only about 3 hours’ time. The efficiency of data acquisition can be improved considerably beyond this by the use of smaller columns and faster scanning systems.

410

GARY K . ACKERS

3.0-

W 0

z a m cc

w

m Q

2.0-

I

I

1

I

I

6.6 6.4 6.2 6.0 5.8

I

5.6

VOLUME

FIG.23. Leading edge monitored a t a single point within the column. Observation of the leading edge in a saturation experiment may be used to determine partition coefficients by a procedure analogous to that with flow cells. This plot was obtained by monitoring absorbance against time a t a fixed distance, 5, within the column during saturation by R solrltion of cyt,ochrome c. The absorbance scale is 2-3 A. Taken from Brumbaugh and Ackers (1968).

6. Large Zone Experiments

Useful information can also be obtained by means of integral boundary experiments in which the solute zone is sufficiently large to establish a plateau in concentration. This may be done by measurement of the rate of movement of the boundary centroid with respect to volume flow and application of Eq. (26). I n addition, the column scanner may be used as a single-point monitor by observing absorbance a t some level in the column. A leading boundary is shown in Fig. 23. This trace was obtained by holding the column at a fixed position and recording absorbance versus time (volume). Here the absorbance scale span is 2-3 A. Experiments of this type were performed simultaneously with the saturation experimerits for the molecules listed in Table I by driving the column to the lower limit of the scan and monitoring a t that point during the process of saturation. The centroid volume is then related to the void volume, VO,internal volume, Vi, and partition coefficient, cr, by Ey. (26). Adsorption of protein to some gels is known to occur under conditions of low ionic strength. Column scanning provides a convenient means of

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

411

detecting such interactions. When adsorption is present or contamination (e.g., from tubing) occurs, these effects are immediately “seen” as baseline elevations. I n addition the scanning approach appears to be ideally suited to studies of interacting components. The direct measurement of solute zone profiles within the column enables the investigator to obtain a record of the actual development of a reaction boundary. I n saturation experiments a weight-average partition coefficient can be obtained for the system at various concentrations under conditions of strict thermodynamic equilibrium (Section VI). Moreover, these parameters can be determined for a series of different gels in a single experiment and the results used to calculate the reaction parameters. This saturation technique can be applied to systems with slowly interacting components for which transport experiments are not feasible. By scanning the column at several wavelengths, concentration profiles of individual components can be obtained within a reaction boundary or zone. Preliminary results indicate that this approach will be particularly useful for ligand binding studies of the Hummel-Dreyer type (Section VI). Binding ratios of ligand to macromolecule can be immediately obtained as a function of macromolecule concentration across the zones.

B. Thin-Layer Gel Chromatography Migration of proteins on thin layers of porous gels was first investigated by Johansson and Rymo (1962). Using a modification of standard thinlayer chromatography technique they were able to demonstrate separation of serum proteins on a variety of Sephadexes (G-25, G-50, G-75, G-200). The method consists of preparing a thin (0.5 mm) gel solvent layer on a glass plate from a slurry of swollen gel particles. The particles must be small for good resolution (e.g., Superfine grades of Sephadex) and must be applied very evenly. A variety of spreaders are manufactured commercially for this purpose. With the plate either in a vertical or tilted position solvent is fed through the bed, usually by means of a filter paper wick. The flow rate can be controlled by the angle of inclination. Determann and Michel (1965) have used a “sandwich” arrangement of gel layer between double glass plates. After formation arid buff er-equilibration of the thinlayer plate a series of protein solutions may be applied in spots a t a starting line as in paper chromatography. After development with solvent, the distances of migration are measured. Usually detection is by some staining procedure with a dye such as Amido Black (Johansson and Rymo, 1964). The migration distances of proteins on such thin layers have been usually correlatcd with their molecular weights on empirical grounds (Andrews, 1964; Morris and Morris, 1964; Determarin and Michel, 1965). A smooth curve of the type obtained by Andrew is shown in Fig. 24. Usually some

412

GARY K. ACKERS

Thyroglobulin

100

t-

/

Ovalburnin

Chymotrypsinogen

104

105 Moleculm weight

106

FIG.24. Calibration plot of migration distance against logarithm of molecular weight, for thin-layer gel chromatography of proteins on Sephadex G-100. Taken from Andrews (1964).

marker molecule is employed and ratios of migration distance of various molecules to that for the marker are used as correlates of molecular weight. Determann has applied the technique to the study of size differences in autolyzates of pepsin and has demonstrated a concentration-dependent apparent molecular weight for isozymes of lactate dehydrogenase (Determann, 1967b). Although the principle for determining partition coefficients from thinlayer chromatograms is simple, it seems never to have been applied. I n the thin-layer experiment the solvent volume V passed through the bed is constant (assuming constant F ) for all molecular species, and their different distances of migration are measured. By contrast, in the column elution experiment, each molecular species travels the same distance through the chromatographic bed and the corresponding volumes for elution are measured. In the case of nonelution thin-layer experiments the total volume of the chroniatographic system is the product of the bed thickness, h, its length xi,and width yt. The distance coordinate in the direction of solvent flow is denoted by x, and the lateral distance coordinate by y. If the thickness of the gel bed is constant as well as the packing density of gel particles, the void fraction per unit distance x of the bed is a and the corresponding internal fraction is 0, as usual. The mean migration rate (corresponding to peak concentration) is given by

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

413

dx F -=dt a Pn

+

If F is not a function of t and and P do not vary with x or y, then the mean distance x, migrated by a molecular species in time t = V / F is related to the partition coefficient u by 2 ,

=

(5) + (5)

u

In analogy with elution chromatography, the excluded molecule (u = 0) migrates a distance xo = V / a , and the included molecule ( n = 1) migrates a distance xi = V / ( a p ) . Therefore the partition coefficient can be calculated in general -1 - -1

+

o=- x m

20

1-1 xi 20

( 137)

The main advantage of the technique appears to be the convenience with which large numbers of samples can be run simultaneously. However, the reproducibility of relative migration distances for different proteins leaves much to be desired (Determann, 1967a,b). This is probably due to variations in the packing of gel particles which result in variations of flow rate in the different parts of the gel bed, and dependence of a on both x and y. The migration distances measured after time t are integrated averages over different paths for the different molecules. This problem is not encountered in single-column elution experiments.

C. Equilibrium Solute Partitioning A number of investigations have involved the determination of partition coefficients by static equilibrium experiments (Acliers, 1964; Fasella et al., 1965; Stone and Metzger, 1968; Brumbaugh and Ackers, 1968). I n the simplest procedure a known weight of gel-forming material is swollen in excess solvent in a volumetric container. Then a measured volume of solution containing a quantity QT of the molecular species of interest is added, making up a total volume, Vt, for the system. The contents are then stirred or shaken, and equilibration is allowed to take place. After a period of time, the gel is allowed to settle; an aliquot of the supernatant liquid is removed, and its concentration Co is assayed. The apparent total volume V’, occupied by the solute then is calculated as the ratio QT/CO. The experiment is first carried out with a large molecule ( n = 0) for determination of Vo,the volume of liquid exterior to the gel. Then the internal volume Vi is determined either with a small molecule (u = 1) or from the

414

GARY K . ACKERS

known partial volume and water regain of the gel. Then the partition coefficient of a molecule of interest is calculated from Q T and Co

In the use of this static method, particular care must be taken to ensure thorough washing of the swollen gel particles. The gel-forming material exhibits large exclusion properties while constituting only a few percent by weight of the gel phase. Consequently release of soluble material upon swelling into the exterior spaces can result in apparently anomalous partition coefficients. I n early static partitioning experiments with Sephadex G-200 the amount of soluble dextran released from the dry gel-forming material was not taken into account, resulting in errorieously high values for V Oas determined tvith particles of tobacco mosaic virus (Ackers, 1964). Consequently the partition coefficierits calculated from Eq. (138) were substantially higher than those calculated from elution peak positions by Eq. (21). This effect was considerably less pronounced in the more tightly cross-linked G-75 and G-100 gels. A high degree of batch variability exists with regard to the amount of soluble material released. In addition to removal of materials which might have substantial exclusion properties it is desirable to wash out small molecules that might react chemically with the protein or absorb light and thus interfere with spectrophotometric assay. For washing purposes it is desirable to pack the gel particle into a glass tube forming a small column. After washing with several volumes of the desired buffer the gel bed can be transferred to a volumetric flask and the partitioning experiment carried out as described above. An alternative arid convenient procedure is to carry out the experiment right in the column. A solution of the molecule to be partitioned is passed into the column arid the effluent is maintained until solute concentration emerging a t the bottom equals that of the solution applied a t the top. Then fresh buffer is applied to the top and the effluent is collected in a volumetric container (e.g., a burette) until the concentration of solute emerging from the column becomes zero. Subsequently the volume and concentration of the collected elutant are determined. The product of these quantities is QT, and division by the concentration of initially applied solution yields the apparent volume within the total column occupied by solute. This volume V , = V o uV,. The partition Coefficient u is determined for a molecule of interest after first performing the experiment with an excluded molecule ( u = 0) yielding V , = VO,arid then \+ith a totally included molecule ( u = 1) yielding 8,= Vo V , . Knowledge of the two

+

+

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

415

luantities V o and Vi then provides a basis for determination of u for a nolecule of interest. Corresponding procedures can be applied to the ietermination of KAY, the total stationary phase volume partition coeffi:ient. Fasella et al. (1965) have used equilibrium partitioning experiments ;o determine ligand binding by proteins (Section VI). Stone and Metzger :1968) have developed a procedure using marker molecules for determina;ion of partition volumes within a gel system and have utilized the procelure for determination of binding of antibody fragment to its antigen [Section VI). Rrumbaugh and Ackers (1968) have measured equilibrium Dartition coefficients by direct optical scanning of saturated gel columns.

VI. STUDIESOF MULTICOMPONENT SYSTEMS The behavior of macromolecules undergoing transport processes can usefully be described as either “cooperative” or ‘inoncooperative.” The term zooperative is applied to a process in which the individual components of the system do riot behave independently and the behavior of one molecular species is linked in some fashion to that of others. The linkage may be through direct chemical reaction or through indirect means such as the binding of intermediates. The simplest systems to analyze are, of course, the noncooperative ones, because their behavior can most easily be referred to that which the individual components would exhibit by themselves. On operational grounds the transport behavior of multicomponent systems can be divided into two categories. The first category includes those systems for which separation of the individual species by the transport technique a t hand is possible. These include all systems of noninteracting components (i.e., all noncooperative systems) and those for which interaction is sufficiently slow that a t least partial separation occurs during the transport experiment. The second category includes those systems of components that cannot in principle be separated because interactions between species are rapid in comparison with the rates of their separation. Under these circumstances only average behavior is observed. Appropriately there are two fundamental approaches that can be taken in the analysis of multicomponent systems by transport methods. The first includes methods of analysis dependent on the macroscopic separation characteristics of species. This approach generally depends on the shapes and positions of solute zones. The second approach involves analyses based on appropriate average equilibrium properties and the experiments are designed in such a way that separation properties of the components may be ignored. As will be seen, both approaches can be effectively applied to both kinds of systems. Transport methods in which single molecular species exhibit complicated “nonlinear” behavior are poorly suited to the study of multicomponent systems. Prior to the advent of gel chromatography almost all liquid

416

GARY K. ACKERS

chromatographic methods fell into this category. However the porous gel materials presently available are ideally suited to the analysis of multicomponent systems because of (a) the essentially “linear” behavior of single macromolecular species (Sections I1 and 111)and (b) the dependence of partition coefficients on molecular size. In this section we will consider the analysis of both noninteracting multicomponent systems and systemf in which the various species undergo chemical reactions. But first we will consider the formulation of partition coefficients for multicomponent systems in general and some procedures for their experimental determination.

A . Partition Coeflcient for Total Solute For a multicomponent system containing various molecular species, the equilibrium partition coefficient for total solute is a weight average of the partition coefficients of the individual components (Ackers and Thompson, 1965). The weight average partition coefficient a, is defined

3

where Ciis the concentration of species j within the mobile phase and uj is the corresponding partition coefficient. This relationship can be seen in the following way. For each component the partition isotherm can be written

Qj = pujcj

(140)

The total solute QT partitioned into the stationary phase per unit length is 3

and the partition coefficient for total solute is equal to the ratio: QT

pc, Thus this coefficient is a weight average, a,, of the partition coefficients of components. I n some systems it is convenient to consider classes of components. For example, if aggregating subunits exist in a variety of isomeric forms representing different geometric arrangements of subunits, then each j-meric class (containing j subunits) may be considered to consist of mi isomers, each with partition coefficient uj; and concentration Cj;. The

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

417

partition coefficient for total solute can be expressed then as a weight average (Ackers, 1967a).

-=

(143)

rJw

where

cj =

c cj; mi

i=l

is the total Concentration of species belonging to class j . It can be seen from Eq. (139) that the weight average partition coefficient will be independent of total solute concentration (except for the slight linear concentration dependence of individual partition coefficients) since 5 change in total concentration will change all the (7,’s proportionately. However if interactions are present between species the various equilibria will be shifted by a change in total concentration and will exhibit a marked concentration dependence as a result of the interactions (Section

V1,D).

B. Experimental Determination of Weight-Average Partition Coeficients There are essentially three different methods whereby the characteristic phenomenological parameter % can be determined experimentally. 1. Direct Optical Scanning of Saturated Columns

The column saturation scanning method described in Section IV,A,4 can

be effectively employed to measure solute partitioning in multicomporient systems. For these systems the weight average partition coefficient a t any

distance coordinate, x, within the column is given by

in which P ( x ) is the ratio of solute absorbance a t point x within the column (corrected for baseline absorbance) to absorbance above the gel. The ) p’(z) are, respectively, the corresponding ratios measparameters ( ~ ’ ( xand ured for totally excluded and totally nonexcluded molecules. This method permits a very high degree of accuracy to be achieved in the determination

418

GARS K . ACKERS

of a,since several hundred data points can be taken conveniently in a single scan of the column. Although a’(z)and /3‘(z) may vary with distance (due to variations in gel particle packing) the weight-average partition coeficient is a n invariant property of the system. 2. Integral Boundary Method

If a large zone experiment (Section II,C,l) is carried out on a multicomporient system, the weight average partition coefficient may be determinec by measurement of the centroid positions of the boundaries on either sidc of the plateau (Ackers and Thompson, 1965; Ackers, 1967a). Within the plateau region of the zone, where C = Co, has a value a t each poinl within the column that is independent of concentration. The fraction o the total column into which partitioning occurs a t each point is (145:

I n order to determine a,experimentally from an elution diagram it is usefu to identify a particular value of the volume coordinate V’ for which V’ = and hence V’ = Vo ZVi. Then the desired parameter 8, can be determined from this particular volume coordinate V’ (knowing Vo Vi) in thc same way that a partition coefficient is determined for a single component The volume coordinate which satisfies this requirement is the centroic volume defined by Eq. (25). This follows from the conservation o mass condition for the trailing boundary:

+

+

v’

jo”2dC

=

0

(146:

in which J is the distance coordinate of the frame of reference moving with the velocity of solute in the plateau region. (147:

Then a t x

=

1

(148:

Since

is independent of C we have (148;

ANALYTICAL GEL CHROMATOGRAPHY OF P R O T E I N S

Therefore the weight average partition coefficient elution diagram as

419

is determined from the

The validity of the conservation of mass condition (Eq. 146) may be illustrated in the following way. Consider a column which has been saturated with solution a t concentration Co and the solute subsequently eluted with solvent. The resulting elution diagram is represented in Fig. 4 (V’, = 0). The equivalent boundary position is evaluated by a similar procedure t o t h a t described in Section II,C,l. The volume V’ is the volume of solvent with which the solute is eluted. The total amount of solute which was contained within the saturated column is

locnV’ dC

=

COP’

This is also equaI to the sum of solute in the column mobile phase (VoCo) and stationary phase. This latter quantity is

Then

which is equivalent to Eq. (149) and implies Eq. (146). A similar analysis can be applied to the leading boundary, leading to the relationship u, =

v-vo Vi

~

(154)

where

p = l co

0

V dC

(155)

and V represents the volume flow through the column since introduction of the sample’s leading edge. The centroid volumes of the leading and trailing boundaries of a multicomponent system move a t the same rate, a t a given position, 2 , within the column since these values are, respectively px = and P’, = $2. This identity of migration velocity was verified (Brumbaugh and Ackers, 1968) for plateau zones of tobacco mosaic virus protein under conditions of subunit association-dissociation equilibria (Fig. 25).

420

GARY K. ACKERS I

I

1

I

I

3.0 -

=E

2.0-

c

W

r

DISTANCE (mm)

FIG.25. Volume-distance plot for equivalent boundary positions of tobacco mosaic virus protein (0.2 mg/ml, 0.1 M sodium phosphat,e buffer, pH 7.4). Centroid position plotted against volume for leading and trailing boundaries. B, centroid points for trailing boundaries shown in Fig. 9. 0 , corresponding centroid of leading boundaries from the same experiment. Ident,ity of rates of centroid migration is indicated by cotlinearity of the two sets of points. Taken from Brumbaugh and Ackers (1968).

It is obvious from the foregoing that the centroid elution volume is a weight average of the elution volumes Vj which the components would have if present individually. If vj

=

vo +

UjVi

(156)

Then solving for uj and substituting into Eq. (139) leads directly to the weight average relationship.

3. Equilibrium Distribution Method The equilibrium distribution method described in Section V,C can be directly applied to the determination of weight average partition coefficients. From Eqs. (138) and (141) it is immediately seen that the coefficient measured by this technique is in fact the weight average Z. Of the

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

421

three methods for determination of G this one is the least satisfactory owing to the difficulties mentioned previously (Section V,C).

C. Heterogeneous Systems of Noninteracting Components I n order to characterize a heterogeneous system of the noncooperative (noninteracting) type, the only desired information is the weight fraction of all species present and their respective partition coefficients (which are then transformed into molecular sizes or weights). The simplest type of polydisperse system to analyze is one that contains only a few discrete species. I n principle each species will appear as a separate peak in a smallzone experiment or as two separate boundaries in a plateau experiment. However for practical purposes it may not be possible to achieve complete separation of components th at have closely similar partition coefficients, and more indirect procedures must be employed. The principles of multicomponent system analysis are the same for both discrete and continuous distributions, and consequently these will be discussed together. 1. Detection of Polydispersity

I n order to ascertain the existence of a polydisperse system, it is necessary to look for deviations from the expected behavior of single-component systems described in Section 111. The presence of zone asymmetry has been the most commonly used indicator of the presence of more than one species. However, this is not in all cases a reliable indicator since asymmetry can also be produced by the normal concentration dependence of partition coefficients (Winzor and Scheraga, 1963) or by high column flow rates which tend to produce “tailing” of zones and boundaries. SeveraI sensitive tests can be applied for the detection of polydispersity. If the sample is chromatographed and the eluted zone is collected sequentially into two equal fractions, then these fractions will behave differently when rechromatographed on the same column if polydispersity is present and they will both behave differently from a sample of the original material diluted by a factor of two and run on the same column. Quantitative differences will appear in the determined values of peak position or centroid elution volumes or in the axial dispersion coefficients obtained under identical flow rate. The differences between the rechromatographed fractions indicate the presence of noninteracting or slowly interacting species. These two cases can frequently be distinguished by variations in flow rate or column length. If slowly interacting species are present the differences in behavior of the rechromatographed zones should increase with flow rate. The third case is that in which rapidly equilibrating species are present. In this case the rechromatographed halves of the original sample will behave identically. However, their peak positions or centroid elution volumes will differ

422

GARY K. ACKERS

significantly from that of the original sample (at twice the concentration) and will be the same as that of a one-half dilution of the original sample, applied in the same sample volume. For self-associating systems these quantities increase with decreasing concentration. Systems of this type are discussed in Section V1,D. 2. Component Distribution Analysis

In order to determine the weight fractions of all species present in a heterogeneous mixture of macromolecules there are two fundamental difficulties that must be taken into account. First is the normal linear concentration dependence of partition coefficient for each species. This concentration dependence, although small, leads to a false evaluation of the relative amounts of components from the observed shapes of elution diagrams. At protein concentrations below about l mg/ml this effect becomes negligibly small for most purposes (Winzor and Nichol, 1965) and the use of low concentration samples is desirable for this reason. The second difficulty to the analysis lies in the fact that the various components generally have different axial dispersion coefficients, and zone spreading effects must be taken into account. For systems containing broad distributions of molecular size and under conditions where components have similar axial dispersion coefficients, the axial dispersion effects can be ignored and a n idealized approach used. Consider a sample containing a distribution f(a) of molecular radius values were f(a) da represents the weight fraction of the sample characterized by molecular size between a and a da. Such a sample when chromatographed will be dispersed on an elution diagram (ignoring axial dispersion) according to the distribution f (a) and the relationship .(a) between partition coefficients and molecular size. The experimentally determined weight fraction g ( u ) of the elution diagram corresponding to partition coefficient u is

+

g(u) = f ( a ) d a >

(158)

Therefore if the column has been calibrated so that the relationship between u and a is known, the component distribution f(a) is calculated a t each point across the elution diagram as the ratio of g(u) to .(a). I n practice, the elution diagram is divided into a series of intervals in u and the calculations are performed corresponding to the midpoints of the intervals. The size parameter used in the application of this method, of course, need not be molecular radius but can be any parameter for which an adequate calibration of the column can be made. Both molecular length (for linear polymer chains) and molecular weight have been used as well as molecular radius. An example of this type of analysis is shown in Figs. 26 and 27 for a series of polystyrene fractions (Hazel1 et al., 1968).

423

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

I

100 -

90

-

80 -

#7#6

8 3 #2

I

14

1

15 16

1

1

17 18

1

1

1

1

1

1

1

1

19 20 21 22 23 24 25 26 27 28 29 30 31

:

ELUTION VOLUME ( A S 5 M L COUNTS)

FIG.26. Chromatogram of a mixture of 10 samples of polystyrene, each having a narrow molecular size (chain length) distribution. Taken from Hazel1 et al. (1968).

I n the more general approaches to the problem of component distribution analysis, the axial dispersion effects are either taken into account mathematically or the experiment is carried out in such a way that a rigorously exact analysis is possible without taking them into account. The first approach has been taken by Tung (1966), who assumed a contiriuuin of Gaussian type dispersions representing all species present. The original integral equation can be written

424

GARY K . ACKERS I

I

50

lo2

I

I

I

I

1

!

I

IOC

5

W

$

2 2 E3

9c

8C 7c

6C 5c

4c

30 20 10 10

lo3 lo4 MOLECULAR LENGTH

,8

I o6

FIG.27. Molecular length distribution curves calculated from data of Fig. 26. Solid line represents the actual distribution from the known composition. The other curves are calculated from the gel chromatogram, calculated by two different procedures. Taken from Hazel1 et al. (1968).

c(V)=

Jb

W(y>

(:y’2

exp [-h(v - y)7 dy

(159)

Here the observed concentration-volume profile, C( V ) , of the elution diagram is related to the distribution W(y) of elution volumes that would be obtained in the absence of axial dispersion and h is an axial-dispersion parameter. Initially the method suffered from the assumed constancy of h for all molecular species. More recent refinements have included calibration of the column for variations in h. A method of solving Tung’s equation for W(y) has recently been developed by Pierce and Arnionas (1968) using Fourier transform techniques. Other procedures for component distribution analysis in which axial dispersion is taken into account have been developed by Hess and Kratz (1966), Smith (1967), and Pickett et uZ. (1967). An entirely different approach to the problem involves the determination only of weight average partition coefficients and is therefore independent of the axial dispersion effects (Ackers, 1968). If a sample of the hetero-

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

425

geneous mixture is chromatographed on a series of columns of different porosity, then the weight fractions f j of all species (for a discrete distribution) present can be determined by the set of linearly independent equations : k

j=l

where is the weight average partition coefficient of the mixture on a column of porosity x and a(a,,pzi) is the column calibrating function for which the calibration constants have been determined independently. For a system containing components there must be at least (2 Ic - 1) different porosities to provide a completely determined system. In practice it is desirable t o have a t least a severalfold overdetermined system in order that a least-squares analysis can be applied.

D. Studies of Protein Subunit Interactions Porous-gel partitioning experiments have proved to be one of the most powerful and promising new approaches to the study of protein subunit interactions. Many recent studies have shown these interactions to play an important role in the structural organization of biologically functional protein complexes and in the regulation of their biochemical activities. A number of such interacting systems have been cataloged in an article in this series by Reithel (1963) and their properties reviewed. I n many instances protein subunits are found to undergo association reactions for which no easily observed spectroscopic changes can be used to measure the degree of reaction. I n these cases an average property related to molecular size or weight can be employed to measure the degree of reaction a t various well-defined equilibrium states of the system. The average property may be a weight average partition coefficient determined in a gel chromatographic transport experiment. Transport methods6 generally have been among the most useful means of study for chemically reacting systems of macromolecules (Nichol et aZ., 1964). They are of especially advantageous use in these situations where the only conveniently measured differences between reactants and products are differences of molecular size or weight. In addition to systems of this type, however, transport behavior has been used effectively to study “mixed association” reactions between two or more different molecular species. In principle they can also be used to study kinetic properties of protein subunit interactions 6 The term “transport method” here is used for both the dynamic and equilibrium experiments that can he carried out by such approaches as sedimentation, electrophoresis, chromatography, and countercu:rerit distribution.

426

GARY K . ACKERS

(Bethune, 1967). This application is presently in a somewhat primitive stage of development. In this section we will consider the use of partitioning experiinents in porous media arid related transport experiments as a means of study for subunit association equilibria. The equations will be presented for reactions of the self-association type as these have been most widely studied. The same general considerations are applicable to reactions of the mixed AB). association type ( A B

+

1. Partition Isotherm

Consider a self-associating system comprised of molecular species j in equilibrium, where j represents a quaternary structure comprised of j subunits. At constant temperature and pressure the formation of each j-mer can be characterized by an equilibrium constant :

where Cj and C1 represent constituent concentratioris of j-mer arid monomer, respectively. The total concentration of solute is, with association up to n-mer: 71

When such a system is subjected to a partitioning experiment, the total amount of solute per unit column length distributed into the gel phase at equilibrium is [Eq. (141)] 11

QT

= B C ~iK$ij i=1

(163)

The parametric equations (162) arid (163) define the partition isotherm for total solute under equilibrium conditions. The corresponding partition coefficient for total solute is the weight average defined for all multicomporierit systems by Eq. (142). From Eys. (142) and (161) the coefficient can be expressed - ZUjKjClj 0, = (164) ZCj When the total concentration CT is changed, the various equilibria between species are shifted according to the law of mass action so that the average degree of aggregation increases with increasing solute concentration. Consequently the value of Z decreases with increasing concentration, since ui decreases with increasing molecular size.

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

427

2. Small-Zone Experiments

A number of studies have been carried out in which small samples of a n interacting protein system have been chromatographed on a gel column and an apparent molecular weight calculated from the peak elution position (cf., Andrews, 1964; Sullivan and Riggs, 1967). Whereas this type of experimerlt is a useful qualitative means of detecting interaction, there is a t present no theoretical basis for the quantitative interpretation of such experiments. As a result of continuous dilution caused by axial dispersion the peak concentration of the zone moves with a decreasing velocity down the column. The apparent partition coefficicnt is then a function of column length as well as the equilibrium constants and initial concentration of applied sample. It may be anticipated that a relatively complete quantitative description of these effects will be achieved through computer simulation techniques such as has been made possible in electrophoresis (Cann and Goad, 1965) and sedimentation (Cox, 1969). Small-zone experiments are useful for the determination of minimum subunit size and niolecular weight. I n the limit of infinite dilution the measured apparent partition coefficient must correspond to the completely dissociated species. This approach has been used for estimation of L-glutamate dehydrogenase subunit molecular weight (Rogers et al., 1965; Andrews, 1965) and similarly for D-amino acid oxidase (Henn and Ackers, 1969a,b), and for P-lactoglobulin A (Andrews, 1964) (Fig. 28). 3. Large-Zone Experiments

If the column is loaded with a sample of sufficient volume that a plateau region is present throughout the duration of the experiment, then the con40 r

16L

0

'

I

I

I

I

1

I

'

1

J

6 12 18 24 30 Amount of P-lactoglobulin A used (mg )

1 k . 28. .Apparelit moleciilar weights of p-lactoglobdin A from elution positiorls of small-zone experiments. The abscissa is the anlourit of p-lactoglobulin initially applied. Taken from A4ridrews(1964).

428

GARY K . ACKERS

siderations of Section VI,A are applicable. The centroids of leading and trailing boundaries move at velocities that depend only on the parameter $. The centroid elution position can be used to determine the weight average partition coefficient G pertaining to each plateau concentration Coaccording to Eqs. (150) and (154). The theory of integral boundary experiments has been developed for interacting systems on the basis of partition coefficients (Ackers and Thompson, 1965; Ackers, 1967a) and equivalently, on the basis of elution volumes (Gilbert, 1966a; Nichol et al., 1967). I n the theoretical treatment initially presented (Ackers and Thompson, 1965), it was shown that the chromatographic transport equations could be cast in the same form as those for sedimentation and electrophoresis if elution volumes are substituted for velocity terms. This predicted behavior was confirmed on experimental grounds by Gilbert (1966a), who showed that the dissociation constant for reversible reaction between trypsin and soybean trypsin inhibitor could be correctly calculated (from the data of Winzor and Nichol, 1965) when elution volumes were used to replace velocities in the appropriate moving boundary equation. The information that can be obtained from large-zone experiments with interacting systems depends on two different kinds of analysis. These will be discussed in turn. a. Analysis of Characteristic Boundary Shapes. When a chemically reacting system is subjected to a transport experiment, there are two kinds of processes that operate to produce dispersion of the boundaries between solution and solvent. The first of these is the free diffusion or axial dispersion present in the transport behavior of noninteracting systems as well. Second is the dispersion (or contraction) of boundaries which arises from the chemical reactions. This effect is the result of competition between the normal tendency of the various molecular species to move a t different velocities and the tendency on the other hand for them to move with the same velocity because of molecular association. The shapes of boundaries therefore depend on the rates of equilibration between species (reactants and products) and on the differences in their respective transport coefficients. The two processes of boundary dispersion do not, of course, operate independently of each other and therefore a complete analysis of the system behavior can be obtained only by soIution of the continuity equation. Unfortunately analytical solutions are not possible for the transport behavior of interacting systems (which are formally analogous to singlecomponent systems exhibiting nonlinear concentration dependence of transport coefficients). In spite of these difficulties, considerable progress has been achieved toward understanding of the way in which boundary shapes are influenced by the chemical reaction in systems where the equilibrium

429

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

can be considered to be established instantaneously (i.e., rapid in comparison to the length of time required for the transport experiment under consideration). This problem was investigated by Gilbert (1955) and by Gilbert and Jenkins in an elegant series of papers (Gilbert and Jenkins, 1959; Gilbert, 1959, 1963). The behavior of free boundaries (i.e., those in which no stationary phase is present) was described theoretically for the idealized situation in which diffusion is ignored. In spite of the inherent limitations imposed by this idealization, the Gilbert theory has been notably successful in predicting the correct qualitative features of reaction boundaries. The corresponding theory for transport behavior in the presence of a stationary phase has been developed more recently (Ackers and Thompson, 1965; Ackers, 1967a) and has been applied to a number of experimental systems. Before describing the quantitative aspects of this theory, it is useful to consider some of the qualitative aspects of boundary shapes to be expected in the behavior of chromatographic systems. The general qualitative features that are found in the boundary shapes of associating systems were demonstrated by Winzor and Scheraga (1963), who showed that a boundary-sharpening was present on the leading edge of the solute zone and a corresponding boundary spreading occurred on the trailing side. This effect, shown in Fig. 29, arises from the tendency of the 0

i

VOLUME ,ml

FIG.29. Elution profile of a-chymotrypsin (3.8 mg/ml) chromatographed on Sephndex G-100. (a) Concentration-volume profile showing boundary sharpening effect on leading edge (left) and boundary spreading of trailing edge (right). Arrows indicate inflection points. (b) First derivate curves of absorbance versus volume for the elution profile shown in (a). Taken from Winzor and Scheraga (1963).

430

GARY K. ACKERS

-

0.5 0 0.5

0

-

(a) I

1

I

I

-

1

n

(b)

-

0.5 0

I

2 Volume

FIG.30. Derivative curves of the trailing edge of the protein zone in the chromat,ography of a-chymotrypsin on Sephadex G-100. Concentrations (mg/ml) are (a) 0.6, (b) 1.2, (c) 1.4, (d) 2.8, (e ) 3.4, (f) 5.0. Taken from Wineor and Scheraga (1963).

larger molecules to move faster within the column than the smaller ones. However, on the leading boundary those molecules that move ahead of these in the plateau find themselves in a region of lower concentration which promotes their dissociation, leading to decreased velocity. Thus the boundary sharpening effect is in continuous operation a t the leading edge. These effects produce a continuous spreading of the trailing boundary. Winzor and Scheraga (1963) also demonstrated that the qualitative features predicted by the Gilbert theory for behavior of reaction boundaries were analogous to those of chromatography. The gradient of concentration across the trailing boundary of a-chymotrypsin in low ionic strength thus exhibited two maxima and a single minimum above a certain critical plateau concentration (Fig. 30). This is analogous to the behavior of a monomer-n-mer associating system in sedimentation experiments when n is greater than 2. On the other hand, the gradient for chymotrypsin under conditions of dimerization (n = 2) exhibited only a single maximum and no minimum. Furthermore, the elution volume of the leading boundary centroid could be correlated linearly with independently determined weight average molecular weight (Winzor and Scheraga, 1964). This empirical correlation procedure was used to determine the minimum subunit molecular weight of bovine thrombin.

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

431

The quantitative theory for these effects was proposed by Ackers and Thonipson (1965). Since chromatographic transport is based on solute partitioning with the nonmobile phase, the theory of this process is appropriately formulated in ternis of the partition isotherm (Eqs. 162 and 163). The idealized equation of continuity is (Ackers, 1967a) :

In this equation, axial dispersion has been ignored except as it arises from the chemical reaction between molecular species transported on the column. The solution of Eq. (165) for the trailing boundary of a large-zone experiment subject to the conditions of Eq. (50) is

where

Equation (167) describes the monomer solute zone profile as a threedimensional surface in the coordinates V , 2, and C1. A corresponding surface for total solute concentration CT is defined by the parametric Eqs. (167) and (162). In elution experiments where solute profiles are measured at a fixed value of the distance coordinate x = I, the geometrical description of the solute zone is the curve determined by intersection of the surface (167) with the plane x = 1. The curve obtained by making this substitution is

V

=

Vo

+ (P(C1)Vi + S

(169)

For conditions that obtain across the boundary, this equation expresses the concentration of monomer as a continuous function of the effluent volume. It has the same form as Eqs. (26) and (150), in which the continuous function @(Cl) has replaced the constants u and a,. All these equations have been written in such a way that elution profiles depend on a particular column size and fractional void space (Le., on V ;and V,). It is useful, however, to express them in a more general form in which all elution profiles can be compared directly. For this purpose the system of description can be normalized in terms of reduced volume coordinates 'V and V. For the trailing boundary we have

(170)

432

GARY K . ACKERS

and for the leading boundary: v=-

v - vo Vi

1

v>vo

(171)

All elution diagrams can be compared on a common scale when expressed in terms of these coordinates. Equation (169) becomes simply vl = @(C,) for all columns. This equation was used to predict the characteristic trailing boundary shapes expected for a two-component monomer d n-mer system (Ackers and Thompson, 1965). For this case, the equation v’ = O(Cl)can be solved explicitly for C1 and the result substituted into Eq. (162). The resulting equation for the elution profile is

in which ul and an are partition coefficients for monomer and n-mer, respectively, and K is the equilibrium constant. Equation (172) is the analog of Gilbert’s equation for the velocity ultracentrifugation of a polymerizing system (Gilbert, 1955). The reduced volume coordinate v’ bears the same relation to the partition coefficients here as does velocity, x / t , to sedimentation coefficients or mobilities in the corresponding theory for free boundary transport. The concentration gradient across the boundary can be expressed

where 6 = (a1 - u’)/(al - a,). n > 2 a t the position dmin

=

This gradient has a single minimum for

2n - 1 3(n - 1)

n-2 + 3(n - 1) an

(174)

As in the case of sedimentation and electrophoresis the reduced coordinate position of the minimum is independent of K and C T and depends only on the stoichiometry n of the reaction. From the minimum in the gradient and values of a1 and an, the value of n may be calculated.

n=

3u’min- a1 - 2an 3V’,in - 2a1 - an

(175)

I n practical terms the calculation of n depends on evaluation of a1 and u,, which is described later. The other qualitative case for monomer-n-mer association is that of dimerization, n = 2 . For this case Eq. (173) predicts only a single maximum and no minimum in the gradient. These qualitative features of the

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

433

ac, a V'

(a)

FIG.31. Ideal concentration gradient patterns (in reduced volume coordinates v') for the trailing edge of a solute zone where monomer is in reversible equilibrium with a single polymeric species. (a) When n > 2, the gradient exhibits two maxima and a single minimum. (b) For the dimerization reaction n = 2, a single maximum only is present. From Ackers and Thompson (1965).

reaction boundaries are shown in Fig. 31 for two cases in terms of the reduced volume coordinates. For the leading boundary, solution of the continuity Eq. (165) leads to a physically impossible solution, leading to the conclusion that the boundary should be the self-sharpening type (Ackers, 1967a). The idealized theory of boundary shapes described in this section has been successful in the prediction of qualitative features observed with real systems. I t can be used, therefore, as a guide to the explanation of certain qualitative results. However, it does not generally provide an adequate basis for the quantitative characterization of experimental systems. This limitation results from the nature of the approximations that must be made to obtain analytical solutions to the equation of continuity. It may be expected in the future that computer simulation techniques (Cann and Goad, 1965; Cox, 1969) will greatly improve this situation by providing realistic boundary shapes that take into account all sources of axial dispersion. b. Analysis of Partition Coeficient Averages. I n order to determine reliable stoichiometries and equilibrium constants from integral boundary experiments, use can be made of the weight average partition coefficient.

434

GARY K . ACKERS

The determination of this parameter is rigorously exact (Section V1,B) and its value depends on plateau concentration in a way that is determined by the equilibrium constants for the reactions as well as the partition coefficients of individual participating species. These latter quantities are functions of gel porosity as well as molecular size in a linearly independent fashion (Section IV,B). There are, therefore, two kinds of experimental variations that can be introduced in order to make inferences regarding the nature of a given reaction system. As the plateau concentration Co is varied, the equilibria are shifted so that concentration terms, C,, pertaining to the individual species are altered in Eq. (164), while the corresponding partition coefficient terms, c j , remain constant. Alternatively the plateau concentration can be maintained constant while the porosity of the gel is varied (i.e., measured for a series of gels saturated a t the same plateau concentration (Co). I n this case the partition coefficient terms change while the C, terms remain constant. I n elution chromatography, the simplest variation that can be introduced is t ha t of the plateau concentration, Co. The dissociation curve obtained for a given column (i-e., a,as a function of CO) can then be compared with various models for reaction stoichiometries and equilibrium constants until a best fit is obtained. For most studies (except at very low protein concentration) it is necessary to make corrections for the linear concentration dependency of individual partition coefficients. If it is assumed that K’, is the same for all species, substitution of Eq. (13) into Eq. (139) leads to a on C T (Ackers, 1967a). This procedure has been linear dependence of carried out for a number of systems including human hemoglobin (Ackers and Thompson, 1965; Chiancone et al., 1968), a-chymotrypsin (Ackers, 1967a), L-glutamate dehydrogenase (Chun et al., 1969b) and D-amino acid oxidase (Henn and Ackers, 1969a,b). A critical evaluation of this approach has been carried out by Gilbert (1967) and by Chiancone et al. (1968) (Fig. 32). Usually a large amount of accurate experimental data is required in order virtually to eliminate all ambiguities in the po-sible models that can be fit to the data. In a very careful study of human oxyhemoglobin (Chiancone et al., 1968), it was not found possible to distinguish unequivocally between various dissociation models. It was possible, however, to ascertain that predominantly dimer-tetramer association was present over the concentration range studied and to determine an accurate value for the dimer-tetramer equilibrium constant. The use of nonlinear least-squares parameter-fitting procedures made it possible for data to be critically tested against a variety of models. Each model entails a different special case of Eq. (164) for the interpretation of measured data. With the development of direct column scanning, it is now possible to determine accurate values conveniently and rapidly on column beds

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

435

stacked with different porosity gels. By use of the saturation technique (Section VI,B) the Z values can be determined in a single experiment for all the porosities in the column bed. It has been proposed (Ackers, 1968) that variations in porosity as well as concentration can be used to determine the number of components present in the reaction mixture (corresponding to C,) as well as the weight fractions of all species present. If M different reaction mixtures are each chromatographed on N different gels of different porosity where M and N are at least as great as k, the number of components, then the M N experimental parameters of the type

e=uur~o

(176)

define a k-dimensional linear vector space, and the calculated rank of the matrix of experimental observations 812 ... ............. 811

OM1

OM2

'

'

'

OMN

I

(177)

is equal to the number k of components present. I n addition, the relationships, Eq. (176), constitute a completely determined system of equaN ) (Ackers, 1968), and can in principle be tions whenever M N 2 k ( M solved for all the Cj and a,. The component distribution analysis becomes much simpler if calibration functions u(aj, pzi) are determined independently for the various gels (Eq. 176). Then the weight average partition coefficient on gel x can be written (for any plateau concentration)

+

where f j is the weight fraction of species j , and aj its molecular size. Since the calibration constants pzi are known independently there are 2 k unknowns ( f j and aj) for each species. With the additional relationship

i

j=1

f3

= 1

(179)

there remain 2k - 1 unknowns. In practice it is, of course, desirable to have a manyfold overdetermined set of data in order that a least squares approach can be applied. For two-component systems, this approach provides an effective means of determining the partition coefficient un of n-mer (ul can usually be obtained reasonably well by extrapolation of to infinite dilution). An alternative method of determining u , ~has been proposed by Winzor el al. (1967) based on a conibination of the z-average

436

GARY K. ACKERS

elution volume and centroid elution volume. The method involves evaluation of a series of iterated integrals and is based on the idealized expression for volume within the reaction boundary (Eq. 169) in which axial dispersion has been neglected. Nevertheless the method has been successfully applied to the association of a-chymotrypsin to obtain reasonable values of stoichiometry and equilibrium constant.

4. Differential Methods An alternative approach to the use of integral methods described above for the analysis of interacting systems is the use of methods which make use of small differences in plateau concentration Co(finite difference boundaries) or in slight differences in partition coefficient of samples run in series in a column (layering technique). Both these approaches have been developed by Gilbert and co-workers (Gilbert, 1966b; Chiancone et al., 1968; Gilbert and Gilbert, 1965, 1968). a. Difference Boundaries. If a solution at some concentration Co is applied to a column until a plateau is established, and subsequently followed by additional solution of the same solute a t slightly different concentration Co ACo until a second plateau is established, then a finite difference boundary will be established between the two plateau concentrations. The elution volume VA of the equivalent boundary position for such a boundary has been shown by Gilbert to be:

+

where is the centroid elution volume of the boundary corresponding to the plateau concentration Co. I n the limit as ACo approaches zero, the finite difference boundary approaches a true differential boundary, so that

From Eqs. (153)-(157) it can be seen that thisvolume V coincides withthe V of Eq. (169) and thus reflects the “diffusion free” profile of the reaction boundary (Chiancone et al., 1968), which is defined in terms of partition coefficients by Eqs. (168) and (169). Using these expressions

(Ackers, 1967a). The differential boundary pertaining to a particular concentration Co can be determined by extrapolating measurements on finite difference boundaries to zero concentration difference. I n their study of hemoglobin dissociation, Chiancone rt al. (1968) carried out finite

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

437

CONCENTRATION g /d I

FIG.32. Elution volume as a function of human oxyhemoglobin concentration for a column, 50 X 0.8 cm, of Bio-Gel P-100 a t 2.5"-3.03 equilibrated with buffer, 0.1 iM Na+ (0.09 as chloride, plus phosphate, pH 7.00 at 20"). 0 , integral boundary formed with finite difference boundary; -, theoretical curves. Taken from Chiancone solvent; 0, et al. (1968).

difference boundary measurements using differences ACO equal to 10% of Co and compared models of the reactions t o the resulting data (Fig. 32). It was found that the accuracy of finite difference boundary determination was less than that of the corresponding integral boundary measurements. Nevertheless this appears to be a highly promising approach which provides a different kind of average property determination than the weight average. Jenkins (1965) has shown on theoretical grounds that a differential boundary will split, under appropriate conditions, into a number of boundaries that can be related to the individual partition coefficients aj and concentration terms Cj of a reaction mixture. b. Layering Technique with Constant Plateau Concentration. A second differential technique closely related to that described above is based on the differential rate of migration of two boundaries for solute zones of identical plateau concentration (Gilbert, 1966b). This method provides a very sensitive means of detecting differences in degree of association; for example, in closely related molecular species. If a solution containing one species is layered over another in the column, a discontinuity will arise a t the interface between the solutions whenever solute moves a t different rate

438

GARY K . ACKERS

within the two plateau regions. If the second sample applied to the column is more highly aggregated than the first it will tend to move faster and will move into the first zone producing a ‘(hump” in the concentration profile of the region of interface between the two. If the opposite condition obtains, a “trough” will result between the two zones instead. Gilbert has shown (1966b) that the area of this dip TC, is related to the difference in degree of dissociationf‘ of the two solutes. For two closely related solutes which both undergo a monomer-n-mer association:

r2

AT

= (f)2

- f’i)(Vi

- V,)

(153)

where and are the respective degrees of dissociation of total solute in first and second samples applied to the column. The volume terms V , and V z are elution volumes of the monomer and n-mer, respectively. This technique is especially useful for the investigation of dissociation behavior in closely related proteins such as hemoglobins with slight differences in amino acid composition. The application of column scanning affords a n additional measurement of the differences in degree of association since the concentrations ‘(seen” by the scanning system in the plateau region (of constant mobile phase concentration C,) will differ if molecular species occupy different fractions of the column cross-sectional area. 5. Temperature Dependence Studies Procedures of study for interacting protein systems can be carried out a t different temperatures in order to obtain more complete thermodynamic characterization of the reactions involved. From the temperature dependence of the equilibrium constant the enthalpy and entropy changes for the reaction can be calculated. A study of D-amino acid oxidase apoenzyme (Henn and Ackers, 1969a,b) revealed a sharp transition in the dimerizatiori constant K o over a narrow temperature range (12-14OC) and a molar enthalpy change of 72 kcal (Fig. 33). The corresponding changes in molar entropy over the transition region was found to be 230 entropy units, whereas the value at temperatures above and below this region were approximately 30 entropy units. These findings are consistent with the discovery by Massey and co-workers (1966) of a sharp thermal transition in catalytic and physical properties of the holoenzyme, presumably due t o a large conformational transition (“melting”). 6. Combination wiih Other Methods

Analytical molecular sieve chromatography can be effectively combined with other techniques in the study of multicomponent interacting systems. Since partition coefficients are sensitive t o molecular shape as well as size, the combination of weight average partition coefficient data with weight-

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

4.80

t

I

3.40

I

3.50

I

3.60

439

2 '0

V T x lo3

FIG.33. Van't Hoff plot for the dimeriaation of D-amino acid oxidase apoenzyme a t temperature of 4"-20". The protein undergoes a sharp thermal transition in the region 12"-14". Taken from Herin and Ackers (1969b).

average molecular weight can be used to provide inferences regarding the mode of subunit aggregation. In a study of association by L-glutamate dehydrogenase subunits (Chun et al., 1969b) theoretical values of a,were calculated as a function of concentration for both coinpact aggregation (in which all species are assumed to be spherical) and linear aggregation models. These coefficientswere calculated on the basis of weight fractions of species. and equilibriuni constants obtained from measurements of weight-average molecular weight which is independent of shape factors. The resulting plots are shown in Fig. 34. The experimental data points were found to fall on the theoretical curve for linear aggregation (lower curve, Fig. 34). The middIe curve is the theoretical plot for compact aggregation and the upper curve is the predicted dissociation curve based on direct molecular weight calibration of the (agarose) column and assumed logarithmic relation between molecular weight and partition coefficient. The extreme failurc of this model results in part from the fact that the glutanlate dehydrogensse "monomeric subunit" is a hexameric polypeptide chain structure which contains voids so that the relation between molecular weight

440

GhRY K. ACKERS

OB 0.8

I

04 -

I

I

I 2.0

1.0

3.0

I 4.0

I

5.0

CO

FIG. 34. The weight-average partition coefficients of bovine liver L-glutamate dehydrogenase as a function of concentration generated from various calibration models. (A) Association model based on the relationship between ui and log Mi (log model).

G

=

(I,

-

AC

jjln j

j

(A)Association based on a spherical model (sphere model) where al = 54.5 d, a. = 9.1 d, bo = 157.5 d. ( 0 )Indefinite linear association model from the weight fractions G=

2 j

Ujfj

where UJ' = erfc[(f/jo)ja$ - ao/bol a1 = 54.5 A, a0 = 9.1 d, bo = 157.5 A. ( 0 ) Experimentally obtained curve at 0.2 M sodium phosphate buffer-10-3 M EDTA (pH 7.0) a t 2.5". Column gel was composed of Sepharose4B. Taken from Chun et al. (1969a).

441

ANALYTICAL GEL CHROMATOGRAPHY O F PROTEINS

and size is quite different from that of the compact globular proteins used to obtain the logarithmic calibration of the column.

E . Studies of Small-Molecule Binding Equilibria The determination of binding ratios for small molecule (ligand) to a protein (moles of small molecule bound per mole of protein) can be efficiently carried out using a procedure first reported by Hummel and Dreyer (1962). A gel column is equilibrated with solution containing the ligand a t a desired concentration. A small sample of protein solution in which the total ligand concentration equals that of the column saturating solution is then added to the column. If the protein binds ligand, the solvent of the sample in which the protein is equilibrated will be depleted with respect to ligand. When the sample is added to the column the protein will be separated from the ligand depleted solvent, moving ahead with its bound ligand. The resulting elution profile will exhibit a peak in ligand concentration above the ligand-saturation baseline, which represents the excess ligand bound. A corresponding trough will follow, representing the depletion that 1.2

o.2

I-

I

t

OO ~

20

40

Volume ( ml)

FIG.35. Binding of 2’-cytidylic acid and ribonuclease as detected by the Hiimmel and Dreyer procedure on a Sephadex C-25 colurnri equilibrated with 0.1 M acetate buffer, pII 5.3. Absorhancy measurements a t 28.5 mp (ordinate) indicate a positive peak (left) and negative trough (right) with respect to the (horizontal) baseline absorbancy. Taken from Hummel and Dreyer (1962).

442

GARY K . ACKERS

resulted from the ligand bound. These effects are illustrated in Fig. 35. The area of the trough and peak are equal. However, measurement of the trough area is usually simpler due to possible protein absorbance and light-scattering effects (depending on wavelength used) which may produce artifacts. From the area of the trough and known total amount of protein applied, the binding ratio is determined corresponding to the ligand saturation level used. Subsequent treatment of binding ratio data can be used to calculate binding constants and number of sites. This technique has been used by a number of investigators. The first thorough and critical evaluation was provided by Fairclough and Fruton (1966), who carried out a study of binding by serum albumin of L-tryptophan or tryptophan derivatives. They concluded that the method is a t least as precise as equilibrium dialysis and offers many advantages in speed, convenience, and flexibility. A more complicated type of experiment has been studied by simulation (Gilbert and Gilbert, 1968) in which protein plus ligand is applied to a column equilibrated with buffer. The resulting behavior was calculated using idealized approximations to simulate the way in which the zone separates. VII. CONCLUDING REMARKS

It is evident on both theoretical and experimental grounds that the group of techniques based on partitioning of molecules into porous gel networks constitutes an extremely versatile and useful array of analytical tools for studies of protein systems. It may be expected therefore that the application of these techniques will be extended to a much wider group of experimental systems. Clearly, the useful limits of technical precision obtainable in gel chromatography have not yet been achieved, and a development of more effective instrumentation for this purpose may be expected. I n addition, there are several fundamental processes, such as axial dispersion within chromatographic columns, for which a much more adequate understanding will be attained experimentally in the near future. It may also be expected that new porous materials will continue to be developed for use as chromatographic media. In this review attention has been focused primarily on the general principles that form the basis of analytical usage of these porous materials in protein chemistry. ACKNOWLEDGMENTS The author wishes to express his indebtedness to Drs. W. W. Fish, K. G. Mann, C. Tanford, and I(.Weber for providing him with results of investigations in advance of publication. He also wishes to thank Dr. G. A. Gilbert for many valuable suggestions and criticisms of this manuscript. This work has been supported by a Grant from the United States Public Health Service.

ANALYTICAL GEL CHROMATOGRAPHY OF PROTEINS

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