. -
Theory of Affinity Chromatographv Separations 4.
Phillip C. Wankat School of Chemical Engineering, Purdue University, West Lafayette, Ind. 47907
A theory for affinity chromatography Is developed which includes steric blocking of ligand sites and interactions between enzyme and bound ligand, enzyme and washing agent, and between two enzymes. For small loadings, only the initial ligand concentration and the dissociation constant are important. As loadings become larger, steric exclusion effects become important. Elution of enzyme from the column by use of a washing agent depends upon the ratio of washing agent concentration divided by the dissociation constant of the enzyme-washing agent complex. High enzyme concentrations can be obtained when washing agent is used to elute enzyme after a step input. Wlth two adsorbed enzymes present, superposition of the single enzyme results is valid if small pulses are used. For step inputs and frontal development, superposition of single enzyme results is not valid since enzyme-enzyme Interactions become important. The first enzyme to break through from the column can temporarily be concentrated to a concentration greater than its feed concentration.
Since 1968, affinity chromatography has become established as a standard laboratory technique for separation of enzymes. Hundreds of references have appeared detailing specific applications of affinity chromatography [for example, see "Science Citation Index" under Cuatrecasas or the review by Weetall ( I ) ] but very little has been published on the theoretical aspects of this technique. This paper presents a theory of affinity chromatography and considers the interactions between enzymes, bound ligand, and elution agent. Affinity chromatography is based on the idea of utilizing the specific interaction between an enzyme and its inhibitor, substrate, or substrate analog to purify the enzyme. Apparently, the first work on affinity chromatography for enzyme purification was by Lerman (2) who used modified cellulose columns. Further work was done by Arsenis and McCormick (3) who also used modified cellulose columns. Affinity chromatography research and use accelerated greatly after the work of Cuatrecasas e t al. (4) and Cuatrecasas and Wilchek ( 5 ) utilizing modified Sepharose as the support material. The laboratory technique has been spelled out in detail by Cuatrecasas et al. ( 4 ) , Cuatrecasas (6) and Cuatrecasas and Anfinsen (7). Besides one-step purifications, two-step procedures have been studied by Jervis ( 8 ) , and specific elution from a single affinity column has been studied by Ohlsson et al. (9). Specific elution from non-affinity columns has also been extensively studied (10) (1) H. H. Weetall, Separ. Purific. Methods, 2, 199 (1973). (2) L. S. Lerman, Proc. Natl. Acad. Sci. US., 39, 232 (1953). (3) C. Arsenis and D. B. McCormick, J. Biol. Chem., 241, 330 (1966). (4) P. Cuatrecasas. M. Wilchek, and C. Anfinsen, Proc. Nati. Acad. Sci. U.S., 61, 636 (1968). (5) P. Cuatrecasas and M. Wilchek, Biochem. Biophys. Res. Commun., 33, 235 (1968). (6) P. Cuatrecasas, J. Biol. Chem., 245, 3059 (1970). (7) P. Cuatrecasas and C. Anfinsen, Methods Enzymol., 22, 345-378 (1971). (8)L. Jervis, Biochem. J., 127. 839 (1972). (9) R Ohlsson, P Brodelius, and K. Mosbach, Fed. Eur. Biochem Soc Lett., 25, 234 (1972)
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and could be analyzed by adaption of the theory presented here. In contrast to the extensive experimental work, very little theoretical work has been done on affinity chromatography. Hixson (11) and Hixson and Nishikawa (12) presented a preliminary theory of affinity chromatography which included interactions between enzyme, bound inhibitor, and elution agent, but did not include interactions between two enzymes or the effect of area coverage by the enzyme-ligand complex. They compared their results with experiment but did not consider the effect of all the variables in detail. Nishikawa et al. (13) studied the quantitative effect of various parameters in affinity chromatography utilizing an adsorption isotherm model. The effect of various parameters on equilibrium binding was studied experimentally, but a theory of column operation was not developed. Wu et al. (14) developed a theoretical model based on a single perfectly mixed stage and theoretically studied the effect of different variables on the separation. Results from the single stage model were extended to column operation despite the fact that there are certain dangers inherent in this extrapolation. Standard chromatographic theory is not directly applicable to affinity chromatography for several reasons. With relatively high enzyme loading, the isotherms are not linear but are of a Langmuir form. Interactions with the elution agent and with other enzymes may be important, especially when columns which attract a group of enzymes are being used. Finally, affinity chromatography is often used as a preparative method so that large loadings of enzyme are applied in step inputs. Several chromatographic theories are available. We picked the simplest model of the column so that we could concentrate on the interactions within the column. Thus we have sacrificed some realism in describing the column operation in order to treat the enzyme-ligand interactions more accurately. The model used is the staged, discrete transfer and discrete equilibrium step model that is applicable to countercurrent distribution (15). This model can also be applied to chromatography, and is essentially the first model employed by Martin and Synge (16). As the number of stages increases (or equivalently HETP decreases or column length increases), this model approaches the continuous flow staged model. We thus expect our results to be qualitatively correct and to become more accurate as the number of stages is increased. Other models can be employed with a corresponding increase in mathematical complexity. We will first consider the interactions between enzyme and bound ligand, and after this the interactions between (10) B. M. Pogell and M. G. Sarngadharan, Methods fnzymol., 22, 379-385 (1971). (11) H. F. Hixson, Paper 41e, 65th Annual Meeting A.I.Ch.E., New York, N.Y., Nov. 30, 1972. (12) H. F. Hixson and A. H. Nishikawa, Fed. Proc., 30, 1078 (1971). (13) A. H. Nishikawa, P. Bailon, and A. H. Ramel, Paper 106, 25th Southeastern Regional Meeting, ACS. Charleston, S.C., Nov. 8, 1973. (14) Y.-T. Wu, D. J. Graves, and J. J. Ferguson, Paper 104, 25th Southeastern Regional Meeting, ACS, Charleston, S.C., Nov. 8, 1973. (15) L. C. Craig and D. Craig, in "Technique of Organic Chemistry," Vol. 3, A. Weissberger, Ed., Interscience, New York, N.Y., 1956, pp 149-332. (16) A. J. P. Martin and R. L. M. Synge, Biochem. J., 35, 1358 (1941).
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
washing agent, enzyme, and ligand will be considered. Then the countercurrent distribution (CCD) model will be formulated and the solution technique will be developed. Once this has been done, the alterations necessary to include the interactions with a second adsorbed enzyme will be discussed. After the theory has been developed, the theoretical responses for pulse, step, and break-through curves will be presented for a single enzyme with and without washing agent. Then the unusual effects that can occur with step and break-through curves when enzyme-enzyme interactions are included will be discussed. Enzyme-Ligand Interactions. In affinity chromatography, a small molecule which has specific affinity for a specific enzyme (as an inhibitor, substrate, product, etc.) is attached to an insoluble solid. Since the attached ligand is usually relatively small and the solid has a large surface area per unit volume, high loadings of ligand can be achieved. The enzyme which attaches itself to the ligand is usually considerably larger than the ligand so that the enzyme-ligand complex covers a relatively large area and blocks or covers other ligand sites which it is not attached to. This steric exclusion effect must be considered. We will assume that there is no non-specific adsorption of enzyme on the solid surface, that a maximum of a monolayer of enzyme can cover the surface (Langmuir type adsorption), and that free enzyme, available ligand, and enzyme-ligand complex are in equilibrium on each stage in the column. Experimental evidence for the area effects postulated here has been obtained by Hixson and Nishikawa (17) who found that the maximum loading capacity of trypsin was independent of the t.ype of ligand used and was considerably less than theoretical loading that would be obtained if all ligand sites had been used. A somewhat more realistic modeling of steric effects would utilize excluded volume effects. This would niodel the standard use of 10 to 14-A leashes ( 4 , 6) and the importance of using highly porous matrices ( 4 , 6, 13).Area effects were used here because they are easier to model mathematically. Define [E] as the concentration of free enzyme in solution in Mmoles/ml, and [B] and [EB] as the surface concentrations of bound available ligand and enzyme-ligand complex in pmoles/cm2. 'The reversible reaction
(1)
E 4 B Z E B
is assumed to be in equilibrium on each stage and has an effective dissociation constant defined as =
[ E1 [B] pmoles ~-m l
(2)
-[EBl
Not e that the units of K E Bused here are millimolar. A mass balance on bound ligand can be used to remove [B] in Equation 2 and then the resulting equation can be used to remove [EB] from the mass balances for the column. The B mass balance must include the blocking of ligand sites by the enzyme-ligand complex. Define Bo as the original concentration of ligand, wmoles/cm2; AT as the total surface area per stage, cm2; and AE as the area covered by the E B complex, cm2/fimole. With these definitions: B,, x A T = Total p m o l e s l i g a n d / s t a g e
[ E B ] x A, = pmoles of E B c o m p l e x / s t a g e
stage is just the area per mole multiplied by the number of moles on each stage. A E X ([EB] X AT) = Area covered by E B complex, cm2/stage
The area remaining for ligand which is available for attachment with free enzyme is then AT - AE [EB] AT. From this, the available ligand concentration is
This development assumes that Bo > ~ / A Eso that a t complete saturation, there is some ligand remaining that is not in the E B complex because it is sterically blocked. If this is not true, then ligand concentration will determine the saturation concentration of enzyme instead of AF,. In addition, in Equation 3 AE [EB] must be 51. Substitution of Equation 3 into Equation 2 and solution for [EB] gives
Equation 4 is a specific form of the Langmuir adsorption isotherm. In the development presented here, we have assumed that no other species are adsorbed either by non-specific or by specific interactions. Also the possible presence of a washing agent was ignored. We will next consider elution and then later consider the effect a second adsorbed enzyme will have. Elution Effects. Elution can be obtained by several techniques. First, if K E B is not too small, a pulse of feed can be eluted by flowing the buffer used in the feed solution through the system. This type of operation is typical of elution chromatography and produces no additional interactions. The results for this case are presented later. A second method for eluting the enzyme is to drastically change the pH or ionic strength of the feed stream. The enzyme will be eluted either because changes in the E B complex alter the value of K E B or because the free enzyme changes conformation and can no longer interact with the ligand. The latter case can be treated by the methods considered below. The former case requires either experimental data which gives the variation of K E Bwith pH or ionic strength (169, or a kinetic model which explains the apparent variation in KEB(29). This method is not considered in this paper, but is currently being studied. The third elution technique is to use some elution or wash agent, W, which forms a soluble EW complex with free enzyme. The EW complex is not attracted by ligand B. W would commonly be substrate or inhibitor but in certain cases it could be H + or some other ions. In all cases we assume that W does not interact with B or the E B complex and thus is not adsorbed. Thus Equations 1 to 4 are unchanged, but Reaction 1 will be shifted to the left since free enzyme E is removed from solution by formation of the E W complex. The EW complex is formed by the reversible reaction E + W Z E W
(5)
The area covered by the enzyme-ligand complex on one
This reaction is assumed to be in equilibrium on each stage and has a dissociation constant defined as
(17) H. F. Hixson and A. H. Nishikawa, Arch. Biochern. Biophys., 154, 501 (1973).
( 1 8 ) J. Lebowitz and M. L. Laskowski, Biochemistry, 1, 1044 (1962). (19) K . J. Laidler, "The Chemical Kinetics of Enzyme Action," Oxford University Press, London, 1958, ChaDter 5.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
1401
(11)
[W] can be removed from this equation by writing a mass balance on W for each stage. Define M W , , as ~ the pmoles of W (as both free W and in EW complex) on stage j after transfer step S, and VL as the volume of liquid on each stage. Since W is not adsorbed, the W mass balance becomes [W]
+
[EW] =
.\IwJ,s/T’L
Solving Equation 7 for [W], substituting this into Equation 6, and solving for [EW] we obtain
A similar equation can be written for washing agent W. The equation is simplified since W is not adsorbed.
Equations 11 and 12 represent two recursion relations which can be solved stage-by-stage and transfer step-bytransfer step regardless of how complex the chemical interactions are as long as fE,,s can be calculated. From the definition of f E j l s it can be calculated as M o l e s of e n z y m e as E and EW - f E j , s - Total m o l e s of e n z y m e ( E EW EB)
+
or ([El
Mw,,, will be found from a recursion relation that will be developed shortly. Other models for elution such as reaction of W with the E B complex or reaction of W with B can be formulated. The same principles used here could be applied to these cases. Chromatographic Model. The staged model with discontinuous transfer and equilibrium steps (CCD model) was chosen since complex chemical interactions can be included (since only a set of algebraic equations needs to be solved). This model with constant distribution coefficients was first employed by Martin and Synge (16) for chromatography and is discussed in detail for countercurrent distribution by Craig and Craig (15). We assume that a column of length L can be divided into N equilibrium stages each of which has a length equal to the H E T P (height equivalent to a theoretical plate).
=
fEJ,S
([El
+
- [EWI)~’L
[EWI)L’L
+ [EBIAT
+
(13)
Equations 4 and 8 allow one to calculate [EB] and [EW] once [E] is known. Thus to calculate f E , , s from Equation 13, the concentration of free enzyme on stage j after transfer step S must be calculated. This can be done by writing a mass balance for this stage after equilibrium has been obtained. AZIE
1,
s
= [Eli’,
+
[EWIVL
+
[EBM,
(14)
Substitution of Equations 4 and 8 into Equation 14 gives a cubic equation for [E].
(9)
(15)
As in the usual chromatographic developments, the H E T P is a variable which should be determined experimentally for a variety of conditions. In this model transfer steps and equilibrium steps are done discretely. Some fraction, q, of the liquid on each stage is transferred to the next stage a t each transfer step. Liquid that is transferred from different stages never mixes. In the usual CCD theory, it is assumed that all of the mobile phase is transferred a t each transfer step and q = 1.0. The possibility of q < 1.0 is included here since it generalizes the resulting equations without increasing their complexity. We will first assume that only one enzyme is specifically attracted to the bound ligand. None of the other enzymes in the feed are adsorbed by either the bound ligand or by the solid support. Thus these non-adsorbed enzymes can be neglected in the following development. For the mass balance on the enzyme which is specifically attracted to the bound ligand, we define ME,^ as the pmoles of enzyme (in all forms) in stage J after transfer step S Also, define f E , as the fraction of enzyme in the liquid phase (as free enzyme and in EW complex) of stage J after transfer step s With this definition of f E , s , (1 - f E , s ) is the fraction of enzyme in the stationary phase in stage J after transfer step S Once f E j s has been calculated we can write a mass balance on enzyme
Equation 15 can be solved using Cardan’s formulas (20) or numerically. The Newton-Rapheson technique (21) was employed for the numerical results obtained in this study. Once the initial conditions have been specified, the iterative procedure for calculating the separation expected proceeds as follows: 1)Solve Equation 15 for [E]. 2) Use Equations 4 and 8 to find [EB] and [EW]. 3) Calculate fE,,s-l from Equation 13. 4) Calculate ? d E j , s and from Equations 11 and 12. 5 ) Increment either stage or transfer step number and return to step 1. This procedure was programmed for an IBM 7094 computer. It will rapidly calculate the expected separation for a variety of cases. This program can be obtained on request from the author. These results are presented later. The model presented here is flexible since additional chemical interactions can easily be added. These additional interactions will increase the complexity of steps 1 to 3, but the equations to be solved remain algebraic equations. T o illustrate this approach, the next section will consider the interactions between two adsorbed enzymes and a washing agent. Two-Enzyme System. If two enzymes are attracted to the bound ligand they will compete with each other for the ligand and for the available area. During elution the enzymes will compete for washing agent W. To include these interactions both enzymes must be included in the calculation. The following reactions occur:
N x HETP = L
II, I,
s
= y m o l e s of e n z y m e
transferred in
+
y m o l e s of e n z y m e left behind
(10) After some simple manipulations, Equation 10 becomes 1402
(20) J. B. Rosenbach, E. A. Whitman. B. E. Meserve, and P. M. Whitman, “College Algebra.” 4th ed., Ginn and Co., Boston, 1958, pp 361-363. (21) L. Lapidus, “Digital Computation for Chemical Engineers,” McGraw-Hill, New York, N.Y., 1962, pp 288-292.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1 1 , SEPTEMBER 1974
E2 + W S E2W
KE~W = [EzI[WJ/[EzWI (19)
Complexes E1B and EzB both cover some of the bound ligand. The available ligand concentration can be found by the same approach used to obtain Equation 3.
Equation 20 can be substituted into Equations 16 and 17 and these can be solved for [ElB] and [EzB].
For the wash step, total mass of washing agent on a stage is given as [WI + [EiWI
+ [EzWI
= .M,j,s/V,
(23)
Rapheson technique for [Ez]. Once [Ez] has been found, [ElW] and [ElB] can be found and then Equation 26 can be used to check the guess for [El]. This technique was programmed on an IBM 7094 computer and was checked by giving both enzymes the same properties and then comparing the results with the single enzyme results. Convergence was slow for small values of KE~B. This program can be obtained on request from the author. Other solution techniques for coupled, nonlinear, algebraic equations could also be employed. Addition of a second enzyme increases the complexity of the algebraic manipulations, but in principle any number of chemical interactions could be included. This would increase the number of coupled, nonlinear algebraic equations which have to be solved and would require an efficient solution technique. Before considering the results, methods of experimentally determining the variables will be discussed. Equilibrium data of enzyme adsorption on the adsorbent can be plotted to form a straight line by plotting [E]/[EB] us. [E]. The slope will be A E and the intercept KEBIBo.Although all three of these variables are real, one of them (such as Bo) can be picked arbitrarily in the theory. This is discussed later. The number of stages or HETP of the column can be obtained from a standard chromatography experiment. With very low enzyme concentrations, Equation 4 simplifies to a linear isotherm. N can then be found by standard chromatography techniques (22). The pulse chromatography experiment can also give an independent estimate of (KEB/Bo)(VL/AT). The peak maximum exits at time t E given as
Substitution of this equation into Equations 18 and 19 and simultaneous solution for [ElW] and [EzW] gives where U is the average mobile phase velocity and RE is the retention of the enzyme. For the linear isotherm approximation of Equation 4, the retention can be calculated as
The stage-by-stage mass balances for El, Ez, and W have the same forms as Equations 11 and 12. f~~ and f~~ have the same form as Equation 13 with the proper subscripts added, and f~~and f~~can be calculated if [El] and [Ez] are known. The free enzyme concentrations can be determined from mass balances on a stage written after the equilibrium step.
Direct measurement of L, t ~and , U allows calculation of RE. From this (KEB/Bo)(VL/AT) can be calculated. Since (KEBIBo)is already known from equilibrium measurement, ( VL/AT)can be calculated. If the void fraction. t , is known, V Lis easily found as l',
= E~'col/LY
and AT can be obtained. The dissociation constant for the washing step, KEW,can be found by equilibrium measurements in solution. Bo can be obtained by a titration of moles of ligand in the column and then dividing this by N X AT
Bo = ( p m o l e s ligand i n column)/("\' x A T ) Equations, 21, 22, and 24 to 27 are six coupled, nonlinear, Some checks on the above values can be obtained. AT algebraic equations in the six unknowns [ElB], [E*B], can be obtained by a direct surface measurement. The [ElW], [EzW], [El], and [Ez]. Solution of these six equavalue resulting from a direct measurement should be larger tions gives the variables required to calculate f~~ and f ~ ~ .than that obtained previously since some of the pores will Then the recursion relations for the next transfer step can be too small for the enzyme. The value of AE can be estibe solved. mated by calculating the projected area of a globular proThe procedure is similar to the procedure for a single entein of the desired molecular weight. The loading of enzyme except steps 1 and 2 are now done simultaneously. zyme a t complete saturation is The solution method used here to solve for the six unSaturation Loading, wmoles = ( A T / A E ) N(very high [E]) knowns was to guess [El], use Equation 2 1 to remove [E*B] form Equation 22, substitute this result and Equation 25 (22) C. J. King, "Separation Processes." McGraw Hill, New York, N.Y.. into Equation 27, and solve this equation by the Newton1971, pp 408-415. ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
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1
20 staqer
z u 0 0
c
0 3 0
r0 x I
20
I
1
1
I
I
30 40 50 60 TRANSFER STEP
I
1
70 80 NUMBER
90
I
I
Figure 1. Chromatogram for single enzymes showing the effect of the number of stages in the column for pulse inputs 20
KEB= 1.0, && = 10, VL = 0.04, 6.47 = 0.2, FO = 0.2. Washing agent added at step 5,k& = 5.0 and WO= 200
22
24 26 TRANSFER
28 STEP
30 32 NUMBER
34
36
Figure 3. &B
Chromatogram for single enzymes showing the effect of for pulse inputs when a washing agent is used
N = 20, &A, = IO, VL = 0.04, &AT = 0.2,FO = 0.2. Washing agent added at step 5, KEW= 5.0 and Wo = 200. K E is~ rnrnolar
RESULTS
20
40
1
I
60
80
TRANSFER
Figure 2.
I
I
1
I
100 120 140 160 S T E P NUMBER
I
I80
1 200
Chromatogram for single enzymes showing the effect of
KEBfor pulse inputs when no washing agent is used N = 20, &AE = 10, V, = 0.04, &AT = 0.2,
Fo = 0.2. GBis mrnolar
Finally, the important parameter VL/(ATB~)= l/(lmoles ligand/ml liquid). Once the above values have been obtained, the expected separations can be determined. The theory can be experimentally tested by predicting break-through curves, the effect of large pulses, and washing from a saturated column. Since these experiments are different from the small pulse experiments used to find N and AT, they should provide a good test of theory. Unfortunately, this theory does not provide for a complete a priori prediction before ligand is bound to the solid. The most important single variable, K E B ,must be obtained by studying bound ligand. Values obtained for free ligand may be considerably lower than that for bound ligand. The literature should be referred to for proper coupling conditions to obtain a good adsorbent ( 1 , 4 , 6, 7). 1404
The results will be shown as chromatograms plotted as product concentratiodfeed concentration us. transfer step number. The abscissa can be converted to elution volume by multiplying the transfer step number by V,, As will be shown later, in most cases the value of VL used is arbitrary as long as the value of VL/AT remains constant (VLIAT is pore volume/surface area in pores, and is constant for each solid support). Thus, the abcissa can be considered as the relative volume of material which has passed through the column. The parameter values used correspond roughly to an inhibitor attached to agarose purifying an enzyme with a molecular weight of approximately 30,000. The single enzyme results for pulse inputs, frontal development, elution from a saturated column, and step inputs will be presented first and then the two enzyme results will be presented. The effect of increasing the number of stages for a single pulse of enzyme followed by buffer and then washing agent at step 5 is shown in Figure 1. As expected, the peak concentration decreases, the peak width increases, and a larger elution volume is required as N increases. Also, as an artifact of the CCD model, the curves become smoother as the number of stages is increased, but the basic shape of the curves does not change significantly. None of these peaks are Gaussian because the isotherms are not linear. The effect of N was also studied for break-through curves. All the break-through curves could be superimposed by translating along the abscissa. Since the number of stages does not have a major qualitative effect on the product concentrations, a value of N = 20 was used for the remainder of the calculations. Twenty stages is a convenient number to work with, and it is large enough to show all major effects. Straight lines will be used to connect the theoretically predicted points. In practice, these curves would be smoothed out. The effect of variation of the dissociation constant K E B is shown in Figures 2 and 3. It should be emphasized that the K E Bare the dissociation constants for ligand attached to a solid support, and not for free ligand. In Figure 2, no washing agent is used and a considerable volume of buffer
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
07
Y w
I
20
22
24 26 TRGNSFER
28 STEP
1
I
30 32 NUMBER
I
34
36
Figure 4. Chromatogram for single enzymes showing the effect of VL and &AT for pulse inputs when a washing agent is used KEB= 1.0, N = 20, FO= 0.2. Washing agent added at step 5, wo = 200
kW = 5.0 and
is required to elute the enzyme. In Figure 3, washing agent W was added a t transfer step 5 . Note that with K E B= 100, the enzyme is not held up strongly in the column and the washing agent is not required. K E Bvalues of 0.1 and 0.01 were also studied. These enzymes require more time to elute when washing agent was used, but are not qualitatively different than the curve for K E B = 1.0. When a washing agent was not used, the time required to elute material with K E B = 0.1 was much greater than for K E B = 1.0 and, with KEB = 0.01, significant amounts of enzyme did not elute after 300 transfer steps. The effects of the variables Bo, V L ,AT, A E ,and enzyme feed concentration Fo were studied next. The three variables Bo, AT, and AE always appear as Bo X AE or Bo X AT (or BoAEIB~AT= AE/AT).The effect of the variables depends upon the size of the pulse. If the pulse was small so that a t equilibrium the first stage was not saturated and the liquid concentration was less than the feed concentration, the only variable which has a large effect is the For these small pulses B ~ A E and Fo ratio of VL/(ATBO). have very little effect, and V Land AT& have little effect as is constant. Since VLI(ATBO) long as the ratio of VL/(ATBO) is the reciprocal of (moles ligand/volume fluid), this result says that initial ligand concentration is very important. This result is shown in Figure 4 as is the effect of increasing VL/(ATBO). With increased values of VL/(ATBO), the enzyme is not held up in the column as strongly. These results for small pulses show that for analytical purposes Fo and A d o have little effect as long as the outlet curves are normalized. If the coordinates are not normalized, peak height is directly proportional to enzyme feed concentration. Changing supports will have an effect since the ratio VL/(ATBO) will be changed. Since decreasing Bo will inthis model predicts that enzyme is held crease VL/(ATBO), less strongly as the initial ligand concentration is decreased. If the feed pulse has a high enzyme concentration, the first stage in the column may become saturated, and the feed concentration then has a large effect on the outlet chromatogram. This is shown in Figure 5 where two concentrated feeds are compared. These curves can be com-
TRGNSFER
STEP
NUMBER
Figure 5. Chromatogram for single enzymes showing the effect of feed concentration when the column is partially saturated. Operation is with a pulse input and washing agent is used KEB = 1.0, N = 20, ~ A =E IO, VL = 0.04, &AT = 0.2. Washing agent added at step 5, KEW= 5.0,Wo = 200
TRANSFER
STEP
NUMBER
Figure 6. Chromatogram for single enzymes showing the effect of washing agent for pulse inputs KEB= 1.0, N = 20, &AE = 10, V, = 0.04,&A, = 0.2,Fo = 0.2. Washing agent added at steps 5. Curve 1: Same results obtained for KEW= 0.05, WO = 200. and for K& = 0.5, WO = 2000. Curve 2: KEW = 0.5, WO = 200. Curve 3: Results obtained for GW= 5.0, WO = 200 and &w = 0.5, WO= 20 are within 0.7% of each other
pared with the K E B= 1.0 curve in Figure 3 (where Fo = 0.2) to show the effect of saturation of the first stage. For Fo = 2.0, the product concentration increases more rapidly and reaches a higher final value than it did for the unsaturated curve shown in Figure 3. For the extreme case of FO = 20.0, the pulse input saturates the entire column and the first peak is enzyme not retained by the column. The second peak occurs when the adsorbed material is eluted by washing agent W. In the model used here, saturation does not depend on Bo but depends upon the ratio of AT/AEwhich is
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
1405
20
30
40
TRANSFER
50 STEP
60 70 NUMBER
G I3 0 20
80
Figure 7. Break-through curves for single enzymes for continual enzyme feed. Effect of variation of KEBand of ratio V L / A ~ Sis, shown N = 20, &AE = 10, VL = 0.04, &AT = 0.2, Fo = 0.2. No washing agent. Curve A: KEe = 1.0 Same results obtained for V, = 0.04, &AT = 0.02, and for VL = 0.4, 5.47 = 0.2
20
30
40 50 TRANSFER
60 STEP
70
80 hUklBER
90
I00
Figure 8. Results for washing single enzymes from saturated columns when no washing agent is used N = 20, 5.4 = 10, VL = 0.04, &AT = 0.2. Initial free enzyme concentration = 0.2
the maximum enzyme loading on the solid per stage based on available area. The effects of variation of washing agent and concentration of washing agent are shown in Figure 6. As expected, elution is faster when KEW is small or when the concentration of washing agent is high. As explained in the caption of Figure 6, curves 1 and 3 are valid for several values of KEW and Wo as long as the ratio WO/KEWis constant. Thus elution depends only upon the ratio of WO/KE\Vif K E L>>~ [E]. This also follows from Equation 8 since M w , , ~ / Vis~ ,equal to Wo when q = 1.0. For pulses which do not saturate the column, the free enzyme concentration is usually very low so the ratio WO/KEW is the important elution variable. Effective elution can be obtained for relatively poor washing 1406
1
,
40 50 TRANSFER
I
1
1
60 70 80 STEP NUMBER
1
90
1
I30
Figure 9. Chromatogram for large pulse input of single enzymes showing the effect of pulse size when no washing agent is used
.4= 10, VL = 0.04, &AT = 0 2, FO= 0.2 Parameter KEa = 1.0, N = 20, 6 number of transfer steps that enzyme is added
IS
agents (large K E W )by using high concentrations of washing agent. This can be done since the washing agent usually has a much lower molecular weight than the enzyme. Enzyme can be eluted from columns with lower values of K E B by using high concentrations of washing agent with low values of KEW. In Figure 7, break-through curves are shown for various values of K E B .With poor affinity columns (high K E B ) ,the break-through curves are very sharp since very little enzyme is held up on the column. For very good affinity col, break-through curves are again sharp umns (low K E R ) the because very little enzyme leaves the column until the column is completely saturated. Curve A shows that increasing VIIATBOcauses the enzyme to break through much faster since the column capacity is decreased. Changes in VI, and ATBOseparately do not affect the break-through curves if the ratio of VLIATBO is constant. Although not shown on Figure 7, decreasing AE has a strong effect on the break-through curves. Lower values of AE increase the column capacity, and the break-through curve starts later and is less steep than the curves shown in Figure 7 . Increasing enzyme feed concentration will cause the break-through curve to appear sooner. It is also interesting to consider elution from a completely saturated bed. These results are shown in Figure 8 for elution with buffer only and no washing agent. For large values of K E R ,the product concentration drops rapidly because very little enzyme was held by the ligand, but instead was in the liquid phase. For small values of K E B ,the product concentration drops rapidly because the enzyme is strongly held to the ligand. After the initial drop, the enzyme is eluted very slowly. These curves have been run for 300 transfer steps to show that the product concentrations asymptotically approach zero. Comparison of these results with Figure 7 shows that adsorption and desorption are not symmetrical for nonlinear isotherms. In experimental studies, affinity chromatography has often used large pulse inputs. In Figure 9, the effect of pulse size is shown for the case where no washing agent is used. As expected, the enzyme appears in the product sooner for larger pulses, higher enzyme concentrations appear in the product, and the column can become temporar-
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
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Flgure 11. Break-through curves for two enzymes for continual feed of both enzymes. No washing agent is used
Figure 10. Chromatogram for large pulse input of single enzymes showing the effect of pulse size when washing agent is used
K E ~= B 1.0, KE,B = 10.0, N = 20, 0.2,Fo, = FO, = 0.1
= 54, = 10, VL = 0.04, &AT =
KEB= 1.0, 20, 54= 10, V, = 0.04,5 A 7 = 0.2, Fo = 0.2, kW = 5, WO= 200. Curve A: Ten steps enzyme, ten steps buffer only, then start adding washing agent W. Curve B: Ten steps enzyme, twenty steps buffer only, then start adding washing agent W. Curve C: Thirty steps enzyme, twenty steps buffer only, then start adding washing agent W
ily saturated. The results for a pulse size of 30 was extended to 300 transfer steps. The enzyme is slowly removed from the column and the product concentration asymptotically approaches zero. Comparison of these results with the K E R = 1.0 curve of Figures 7 and 8 shows that for large pulses (curves 30 and 40 of Figure 9) the initial part of the pulse response follows the break-through curve and the final part follows elution from a saturated bed. In Figure 10, results are shown for pulses followed by buffer solution and then washing agent. The product is removed from the column faster, is much more concentrated, and does not demonstrate the extreme tailing that occurred in the results shown in Figure 9. The larger the pulse, the smaller the enzyme feed concentration, or the more effective the washing agent, the greater the concentration effect will be. Concentration of product can also be enhanced by increasing AT& or decreasing KEBand AE since this will increase the enzyme loading in the column. Concentrations much greater than 9.5 were predicted for some cases. Curve C in Figure 10 shows an enzyme which had started to elute from the column before the washing agent exited from the column. Up to transfer step 69, this curve is exactly the same as the curve marked 30 in Figure 9. At transfer step 70, the washing agent appears in the product stream and the enzyme is rapidly eluted. Up to this point, all of the results presented have been for the case where a single enzyme is adsorbed. The interactions between two adsorbed enzymes are also of considerable interest. First, small pulses which do not saturate the column a t any point were studied. In these cases superposition of the single enzyme results were valid since at low loadings enzyme-enzyme interactions are unimportant. This result was found for a number of values of KEB and KEW.Thus, for analytical studies with small pulses, the interactions between two adsorbed enzymes can be ignored and the simpler single enzyme theory is valid.
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Flgure 12. Chromatogram for large pulse input for two enzymes when no washing agent is used
= 10.0, N = 20, 54,= 54, = 10, VL = 0.04,&AT = KE,B= 1.0, 0.2, Fo, = Fop= 0.2. Feed was 20 steps with enzyme followed by buffer only
For continuous feed and large pulse inputs, enzyme concentrations become high and the enzyme-enzyme interactions cannot be ignored. In Figure 11, the break-through curves for two interacting enzymes are shown. As expected, enzyme 2 with the larger value of K E B comes out first. However, the result that enzyme concentration becomes greater than the feed concentration for a long period of time was somewhat unexpected although multicomponent gas adsorption curves show the same effects (23).This occurs because enzyme 2 will saturate a stage first and then is (23) J. W. Carter and H. Husain, Chem. Eng. Scb, 29, 267 (1974).
ANALYTiCAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
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pushed off when the more strongly held enzyme 1 appears. If the total mass of enzyme 2 on a given stage is plotted, it first increases until saturation is reached. Then when enzyme 1 reaches the stage, the amount of enzyme 2 decreases and then approaches another asymptotic value when both enzymes have reached an equilibrium situation. Thus, to some extent, enzyme 1 acts as an elution agent for enzyme 2. A similar effect is seen for a large pulse input as shown in Figure 12. Again enzyme 2 product concentration becomes greater than the feed concentration because enzyme 1 acts to push enzyme 2 out of the column. The break-through and large pulse input results can not be obtained by superposition of the results obtained with a single enzyme. Also the curves for enzyme 1 are different from those obtained with a single enzyme. With two enzymes present, enzyme 1 comes out somewhat sooner than expected because enzyme 2 blocks some of the sites that would normally be available for adsorption. These results show that enzyme-enzyme interactions cannot be ignored in preparative separations if more than one enzyme is held up on the column.
DISCUSSION The theoretical results obtained in this study agree,qualitatively with the experimental results obtained in numerous studies. Very sharp peaks with a trailing edge have been observed by numerous investigators for both pulse and step inputs (4-7, 17). An experimental study of the effect of washing agent (17)produced a series of curves similar to those shown in Figure 6. Maximum loadings of enzyme which are independent of ligand type have been reported ( 17 ) . The effective dissociation constants studied here are within the range used successfully for affinity chromatography, and the qualitative effect of variation of K E Bis similar to that obtained experimentally (4-7, 17). The results presented here were obtained for the discrete transfer countercurrent distribution model of chromatography which has one serious flaw. With this model non-adsorbed material such as washing agent W is propagated through the column as a shock front and is not smoothed out by dispersion as it should be. This will cause the peaks to be sharper than they would be in practice. One method of partially compensating for this problem is to assume that not all of the liquid phase is transferred at each transfer step ( i e . , q < 1.0). This simulates dispersion and prevents formation of a shock front and will broaden enzyme peaks when washing agent is used to elute material. When a value of q = 0.9 was used in the calculations, the expected effect was observed. Use of q = 0.9 did not appear to affect any of the qualitative observations made previously. The results presented here were obtained mainly for a system with only 20 stages. This limited number of stages was used since the purpose of this study was to show the effect of the variables and not to predict any specific separation. T o predict a separation, enough data to calculate the HETP would be required. Use of a larger number of stages
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*
requires more computation time, but does not lead to any other difficulties. The results with two enzymes present showed that enzyme interactions cannot he ignored for continuous feeds or step inputs. This is important for group separations or when separation is based on specific elution. Other interactions such as non-specific adsorption when the solid support is not inert can he included by techniques similar to those used here. Other techniques of eluting the desired enzyme have been used experimentally. The most common elution technique has been to change the ion strength or the pH of the buffer. This technique was not studied here mainly because the required data are not readily available. Elution may also proceed by adding a material which reacts with the ligand. The theory presented here can readily be modified for these elution techniques.
NOMENCLATURE AE, A E ~ AE* , = Area occupied by enzyme-ligand complex, cm'/ pmole AT = Total surface area of support cm2/stage [B] = Surface concentration of available bound ligand, pmoles/ cm2 Bo = Initial surface concentration of bound ligand, pmoles/cm? E], [El], [Ez] = Free enzyme concentration, pmoles/ml = mmolar L B I , [EIB], [EzB] = Surface concentration of enzyme-ligand complex pmoles/cm2 [EW], [ElW], [EzW] = Concentration of enzyme-elution agent complex, pmoleslml = mmolar Fo = Enzyme feed concentration, icmoles/ml /E,,~,/ E 1 ,) /E*,, = Fraction of enzyme in liquid in stage j after transter step HETP = Height equivalent to a theoretical plate, cm/stage j = Stagenumber KEB,K E ~ BKE*B , = Effective dissociation constant for enzymeligand complex pmoledml = mmolar K E W , K E ~ wK, E * ~=; Dissociation constant for enzyme-washing agent complex, pmoles/ml = mmolar L = Length of column, cm ME,,,, ME,,,^,, ME^^,^, = Moles of enzyme on stage j after transfer step S, pmoles Mu;,,, = Moles of washing agent W on stage j after transfer step S , pmoles N = Total number of stages q = Fraction of liquid which is transferred a t each transfer step RF, = Retention of enzyme = fraction of time enzyme is in mobile phase s = Transfer step number t~ = Time enzyme peak exits, sec c' = Mobile phase velocity, cm/sec V,,i = Volume of column, ml VL = Volume of liquid phase on a stage, ml [W] = Concentration of free washing agent, pmoleslml = mmolar Wo = Concentration of washing agent in elution feed, pmoleslml = mmolar e = Volume fraction voids
9
ACKNOWLEDGMENT Discussions with Alden Emery and Harry Hixson were very helpful.
RECEIVEDfor review August 8, 1973. Accepted June 13, 1974. This research was supported by NSF Grant No. GI 34919 Al.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
