Non-linear elution effects in split-peak chromatography

Journal of Chromatography, 436 (1988) Ill-135 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands CHROM. 20 108 NON-LINEAR ELUTION EFFECTS IN SPLIT-PEAK CHROMATOGRAPHY I. COMPUTER SIMULATIONS FOR THE CASES OF IRREVERSIBLE FUSION- AND ADSORPTION-LIMITED KINETICS DIF- DAVID S. HAGE Department of Chemistry, Iowa State University. Ames, IA 50011 (U.S.A.) and RODNEY R. WALTERS* Drug Metabolkm Research, The Upjohn Company, Kalamazoo, MI 49007 (U.S.A.) (First received May 5th, 1987; revised manuscript received September 29th, 1987) SUMMARY The split-peak effect, in which even a small sample of pure analyte elutes in both non-retained and retained fractions, has been shown to be a useful chromatographic tool for such applications as the determination of protein adsorption rate constants and diffusion coefficients. To evaluate such parameters, it is necessary to obtain data independent of sample size, or to work under linear elution conditions. In this paper, computer simulations were used to determine how non-linear elution conditions affect such measurements. The two cases studied were those in which the rate of analyte retention was limited either by diffusion or adsorption of analyte on the column. The simulation data were then compared to results obtained with two experimental systems: the retention of hemoglobin on reversed-phase columns and the binding of immunoglobulin G to protein A affinity columns. From the simulations, guidelines were developed for minimizing or eliminating non-linear elution effects in both of the cases studied. INTRODUCTION The effect of non-linear elution conditions, or column overloading, has long been of interest in chromatography. This has been true for both analytical and preparative-scale work since such conditions may not only result in changes in column capacity, but can also affect solute retention’+, band-broadening’-j, and resolutions. A number of studies have been performed to better understand these effects and to develop methods by which they can be quantitated. Due to the complexity of the systems and calculations involved, computer simulations have often been employed in such studies1J.6. One area in which non-linear effects have been reported is split-peak chromatography. The split-peak effect occurs when a single solute elutes from a column 0021-9673/88/%03.50 @ 1988 Elsevier Science Publishers B.V. 112 D. S. HAGE, R. R. WALTERS in two fractions: a non-retained peak and a strongly retained peak7-lo. It has been shown theoretically that this can occur even when small amounts of solute are used, or under linear elution conditions’. Such behavior is believed to be a result of the kinetic nature of the chromatographic process, being caused by either slow diffusion and/or slow adsorption of analyte in the column7. First predicted by Giddings and Eyring in 195511, the split-peak effect has since been used in a number of applications. These include the comparison of the kinetic properties of affinity matrices 7, the design of affinity chromatographic systems’ 2, and the determination of protein diffusion coefficients’ j. Attempts have also been made to use this phenomenon in the determination of rate constants for macromolecular interactions7v*. In these applications, the equations used to describe the split-peak effect generally assume that the relative sizes of the non-retained and retained fractions are independent of sample size, or that linear elution conditions are present7J2J3. Experimentally, however, a sample size dependence of these fractions has been noted for a number of systems. Non-linear effects in split-peak studies are typically seen as an increase in the relative size of the non-retained fraction with sample load. This has been observed in affinity chromatography for the adsorption of immunoglobulin G (IgG) on protein A columns7 and the binding of IgG’O, insulinsJo, and interferon14 on immunoaffinity columns. This behavior, however, has not been noted for IgG retention on reversed-phase columns7. It is important to consider these effects in the use of split-peak chromatography, especially if physical constants such as diffusion coefficients or adsorption rate parameters are being measured. In this experiment, the effect of non-linear elution conditions was studied using computer simulations. The two chromatographic cases examined were those in which the rate of analyte retention was either diffusionor adsorption-limited, the two cases currently most useful in the evaluation of physical constants7J3. From the simulations, guidelines were developed for minimizing non-linear effects in both cases. The simulation results were also compared to data obtained for two experimental systems: the retention of IgG on protein A columns and the binding of hemoglobin on reversed-phase columns. THEORY zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Chromatographic and simulation models The chromatographic model used here is the same as that presented by Hethtote and DeLisi (see ref. 15 and refs. cited therein). In this model, the column is divided into three distinct phases: the stationary phase, which contains the immobilized ligand or adsorption sites, the stagnant mobile phase, and the flowing mobile phase. The volume of the stagnant mobile phaseis given by V,,, the pore volume of the support, and the volume of the flowing mobile phase is given by Ve, the elution volume of an excluded, non-retained solute. As solute E passes through the column in the flowing mobile phase, it is viewed in this model as undergoing the following reactions leading to its adsorption: kl E e k?, EP (1) NON-LINEAR ELLJTION IN SPLIT-PEAK CHROMATOGRAPHY. E, + L kg I. 113 E,-L 3 where E, and E, represent the solute in the flowing mobile phase and stagnant mobile phase, respectively, and L is the immobilized ligand. Mass transfer of E between the flowing and stagnant mobile phases is described by the first-order rate constants kr and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA k - 1, while the binding of E with L is described by the second-order adsorption and first-order desorption rate constants, k3 and k - 3. In this model, kI and k- 1 are related to the excluded volume and pore volume of the column, V, and VP, by where mEPmand mEc, are the numbers of moles of E, and E, at equilibrium, and is the mass transfer equilibrium constant 7.15. Also, k3 and km3 are related to K3, the equilibrium binding constant, by the expression K1 K3 _ k”_”- [E!-‘-L] I%1PI 3 (4) where [ ] represents the concentrations of the given species in the stagnant mobile phase7J5. In previous work this model was used to develop an equation to describe the split-peak effect in cases where mass transfer and/or adsorption are rate-limiting. The relationship that was derived is as follows: -1 cf -F - 1 - +k,V, 1 k3mL > wherefis the fraction of solute eluting in the non-retained peak (i.e. eluting without interacting with the stationary phase), F is the flow-rate, and mL is the number of moles of binding sites in the column7. Eqn. 5 predicts that a plot of - l/In f VS.F will give a straight line with an intercept of zero and a slope equal to ( I/k1 V, + l/k3mL). The slope of this plot is useful in studying retention processes since it is related to the kinetics of the chromatographic system. For example, the slopes of such plots have been used to compare the overall kinetic properties of affinity supports 7*1?. By obtaining more information about the system, such as the values of mL and V,, it is also possible to evaluate the rate constants kI and k3 (ref. 7). The derivation of eqn. 5 is based on the assumptions that linear elution conditions are present (i.e., [L] B [E,]) and that E adsorbs irreversibly on the time scale of the experiment (i.e., km3 x 0 or the capacity factor, k’, is large so that the nonretained and retained peaks are resolved from one another). The net result of these assumptions is that eqn. 2 reduces to the simple first-order reaction D. S. HAGE, R. R. WALTERS 114 E zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA k&L1 E -L P P where ka[L] is the apparent first-order adsorption rate constant7. One problem with this simplified model is that experimentally obtaining linear elution conditions can be difficult. For example, non-linear effects may occur even when small sample loads are applied to a column due to the presence of a local excess of sample vs. ligand, such as takes place at the head of the column immediately following injection. To study the effect of such non-ideal conditions on the results obtained using eqn. 5, computer simulations were used. The computer model used was the same as presented earlier16. In this model, the column is divided into a large number of slices, each of which is further divided into three distinct regions: the flowing mobile phase, the stagnant mobile phase, and the stationary phase. All material injected on the column begins in the flowing mobile phase of the first slice. The simulation is performed by repeatedly carrying out two alternating operations. In the first, the material in each slice is distributed between the three phases according to the given set of kinetic equations and rate constants describing the system of interest. This is done for one unit of time, or one iteration. Once this has been done throughout the column, the material in the flowing mobile phase of each slice is shifted down the column one unit in order to simulate flow or convective mass transfer. At the same time, the amount of material leaving the last slice of the column is monitored, corresponding to detection of the chromatographic peak. This is repeated until all but a given fraction of solute has eluted from the column, in this case all but 1 ppm of the remaining free analyte. Two effects not considered directly by this model are extracolumn band-broadening prior to the column and band-broadening within the excluded volume of the column. One cause of the first is the use of large sample loops. This can be important in split-peak chromatography since injection volumes approaching or exceeding the excluded volume of the column are often used 7+12,13.Although not a problem under linear elution conditions’, this factor can potentially affect split-peak measurements made under non-linear conditions. This effect can be studied in simulations by changing the number of iterations over which a given sample is applied. Band-broadening within the excluded volume of the column can occur by such processes as eddy diffusion or longitudinal diffusion. Although such an effect is not easily studied with this model, its role is probably not large compared to extracolumn band-broadening or other effects, especially when considering the relatively short column residence time of the non-retained peak and the small columns typically used in split-peak measurements7*’ 2*13. In order to compare the simulation results under non-linear elution conditions to those predicted under linear conditions, an alternate form of eqn. 5 was used’. -1 24, 1 K--=X- + ( ki k-1 WJ[Ll > In this equation, ue is the linear velocity of an excluded, non-retained solute, h is the column length, and [L] is the initial concentration of ligand in the column, mL/Vp. NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 115 As in the case of eqn. 5, the above expression predicts a linear relationship between - l/lnSand a term related to the column residence time of E, uJh. Also, the slope is again comprised of two terms: the first, Ilki, being related to the kinetics of mass transfer or diffusion, and the second, (k- i/klk3[L]), being a function of the system’s adsorption kinetics. Using eqn. 7, it is possible to calculate the theoretical value of - l/lnfunder linear elution conditions given uJh, [L], and the rate constants of the system. This equation is particularly useful in simulations of the type done here since the total residence time of E in the column can be varied by changing the column length h (i.e., the number of slices in the column) while keeping ue and the reaction time in each slice constant (i.e., an excluded linear ,velocity of one slice/iteration and a reaction time of one iteration). Equations for the case of d#usion-limited kinetics Two cases were considered in looking at non-linear elution effects: mass-transfer or diffusion-limited kinetics, and adsorption-limited kinetics. The first studied was that of simple diffusion-limited kinetics. In this case, adsorption is assumed to be much faster than mass transfer, or k3[L] B k - 1, such that any solute entering the stagnant mobile phase binds to free ligand rather than diffusing back to the flowing mobile phase. The effect on eqn. 7 as adsorption becomes infinitely fast (i.e., k3 approaches cc) is that the second slope term, (k- l/klk3[L]), becomes much smaller than the first. This reduces eqn. 7 to -1 -- Inf- 24, 1 h 0 kI Eqn. 8 predicts that for the diffusion-limited cast under linear elution conditions, a plot of - l/lnf vs. u,/hkI should yield a line with a slope of one and an intercept of zero. The results for this case under non-linear conditions can be obtained by representing the system by the following reactions: fast E, + L + E,-L Note that in these reactions two situations may occur. immobilized ligand is present. In this situation, any mobile phase immediately binds to ligand, preventing flowing mobile phase. The net reaction of eqns. 9 and be written as kl E, + E,-L The first takes place when free E diffusing into the stagnant E from diffusing back to the 10 under these conditions can (11) 116 D. S. HAGE, R. R. WALTERS This can be described by the integrated rate expression for a simple first-order reaction” -k,t = WS (12) mE,,,e where t is the time of reaction, and mE,, and mE,, are the moles of E, present at times t and 0. The subscript “s” is used here to refer to the fact that these parameters are those for the particular slice studied during a given iteration. The second situation which can occur in this case takes place when all free L in the slice has been depleted so that E is no longer able to adsorb onto the stationary phase in that region. The result is that E can only undergo diffusion into and out of the pores of the support, making the net reaction h E e k<, EP (13) This reaction can be described by the integrated rate expression for a reversible firstorder reaction’ 7, which is as follows: = mE,,m + (mE - mE,,m)e-(kl+ ‘-$ 510 (14) where mE,,mis the moles of E present in the flowing mobile phase of the slice studied at equilibrium. The point at which the system switches from the situation given in eqn. 11 to that in eqn. 13 occurs when enough solute has entered the stagnant mobile phase to totally bind any free ligand present. The reaction conditions required for this can be determined by using eqn. 12 and the mass balance expressions for the system. The result is the following relationship: mL, = mE,,(l - emkIt) (15) which gives the time and mole conditions at which all free L is depleted. If the reaction conditions are such that mL,, is not less than the right-hand side of eqn. 15, then the system can be described by eqn. 12. If m L, is less than this expression, eqn. 12 is used until all ligand in the slice has been depleted and then eqn. 14 is used to describe the system. Equations for the case of adsorption-limited kinetics The second case studied was that of simple adsorption-limited kinetics. In this case, diffusion is assumed to be much faster than adsorption (i.e., kI and k_ 1 g kJL]), giving rise to an equilibrium in mass transfer of solute between the stagnant and flowing mobile phases. The effect on eqn. 7 is that its diffusional slope term, l/kl, becomes small with respect to that for adsorption, reducing eqn. 7 to NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. 117 I. The expected result, then, for the adsorption-limited case under linear elution conditions is that a plot of - l/in f vs. (u,/h)(k- i/klk3[L]) will give a line with a slope of one and an intercept of zero. To calculate the results under non-linear conditions, the system is represented by the reactions Kl E e-‘E (17) P k (18) E, + L -? E,-L These reactions can be described by the integrated rate expressions given below (see derivations in the Appendix). For the situation in which mn,, + mapso# mL,, the rate expression is (19) and for mE,, + mnP= = mL,, the rate expression is 1 -_= L.+ mL, %o (+?I> (20) ms where I’,, is the total void volume per slice (i.e., vm/h) and mL, is the number of free ligand sites in the slice of interest. EXPERIMENTAL Reagents The protein A, rabbit IgG, and bovine hemoglobin were from Sigma (St, Louis, MO, U.S.A.) and were the purest grades available. The morpholine and l,l’-carbonyldiimidazole (CDI) were from Aldrich (Milwaukee, WI, U.S.A.). The n-octyldimethylchlorosilane was from Petrarch (Bristol, PA, U.S.A.). The LiChrospher Si-500 (lo-pm particle diameter, 500-A pore size) was obtained from Alltech (Deerfield, IL, U.S.A.). Apparatus The chromatographic and data acquisition systems used were the same as described earlier7. The detector used for the IgG studies was a Hitachi 100-10 (Tokyo, Japan) operated at 280 nm. For the hemoglobin studies, a Kratos 757 (Ramsey, NJ, U.S.A.) detector was used operated at 414 nm. Computer simulations were performed on a National Advanced Systems 9160 Computer (Mountain View, CA, U.S.A.). Methods Computer simulations. All simulations were performed in Fortran G using 118 D. S. HAGE, R. R. WALTERS double-precision logic. The simulations were initiated by placing the desired amount of material in the flowing mobile phase portion of the first slice and were ended when all but 1 ppm of the remaining non-adsorbed material had eluted off the column. Programs were tested for convergence by performing a series of equivalent simulations in which columns were divided into increasingly larger numbers of slices while proportionately decreasing the rate constants for the system. All values reported are within 20 ppm of the estimated value for a column divided into an infinite number of slices as determined in this manner. zyxwvutsrqponmlkjihgfedcbaZY Chromatography . The LiChrospher reversed-phase matrix was prepared according to previously published procedures’ *J 9 using 5.0 g of n-octyldimethylchlorosilane per gram of silica and 50 g of carbon tetrachloride per gram of silica. The diol-bonded LiChrospher was also prepared as described previously20. The diol coverage of the LiChrospher prior to activation was 200 pmol per gram of silica, as determined by the periodate oxidation method2’J2. Protein A was immobilized onto the diol-bonded LiChrospher using the CD1 methodZ3. Immobilization was performed at pH 4.0 using 10 mg of protein A per gram of silica. As determined earlier, the immobilization yield of protein A under these conditions is cu. 100°h7. The weak mobile phase for the reversed-phase matrix was 0.02 M ammonium phosphate, 0.01% (v/v) morpholine (pH 7.0). All hemoglobin solutions were prepared in this buffer. The strong mobile phase was 2-propanol containing 0.01% morpholine. Protein retained on the reversed-phase support was eluted by using a linear 20-min gradient from 0 to 100% 2-propanol at a flow-rate of 0.25 ml/min. The application buffer for the protein A matrix was 0.10 M potassium phosphate buffer (pH 7.0) and the elution buffer was 0.10 M potassium phosphate (pH 3.0). All IgG solutions were prepared in the pH 7.0 phosphate buffer. Elution of IgG adsorbed on the protein A was done by a step change in pH. Both the reversed-phase and protein A matrices were placed into their respective weak mobile phases and vacuum-slurry packedI into columns of a previouslypublished design24. Kinetic studies on these columns were performed at 25°C. All other chromatography was performed at room temperature. Prior to the kinetic studies, both the protein A and reversed-phase columns were pretreated several times with either excess IgG or hemoglobin to remove any residual active groups or irreversible adsorption sites. To test for the removal of such sites, the static capacity of each matrix was measured by integration of the resulting breakthrough curves25. The capacities were found to be 8.6 f 0.4 (1 S.D.) mg of IgG per gram of silica for the protein A matrix and 47.1 f 0.4 mg of hemoglobin per gram of silica for the reversed-phase matrix. The static capacities of the matrices were estimated to decrease by less than 5% over the course of the kinetic studies. The split-peak behavior of the matrices was studied using the method described earlier’. For the protein A, split peaks were obtained by injecting 10 ~1 of 0.14-0.55 mg/ml IgG on a 6.35 mm x 4.1 mm I.D. column at flow-rates of 0.02-0.5 ml/min. The areas of both the non-retained and the retained peaks were determined by computer integration, normalized vs. flow-rate and corrected for any sample impurities (0.8% of the total area) or background shifts as discussed previously’. The corrected area of the IgG peaks was found to vary linearly with sample size over the entire sample range used. The free fraction f was calculated from these corrected areas as described earlier’. NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 119 For the reversed-phase columns, split peaks were observed by injecting 3 ~1 of 0.5-2.0 mg/ml hemoglobin on a 6.35 mm x 1.0 mm I.D. column at flow-rates of 0.06-0.63 ml/min. The non-retained area was found as described above, normalized vs. flow-rate and corrected for the solvent background level. No corrections for sample impurities (less than 0.2% of the total area) were made. The corrected area was again found to vary linearly with sample size over the entire sample range studied. The free fraction was calculated by dividing the non-retained area by the total area of the sample when injected through an open tube, as described previously’. RESULTS AND DISCUSSION Simulation results for d@uion-limited kinetics The first case studied was that in which analyte retention is diffusion-limited and adsorption is irreversible on the time scale of the experiment. This case is of interest since split-peak measurements on such systems have been used in the determination of diffusion coefficients. This is possible since the diffusion coefficient of the analyte is directly related in this model to the value of its mass transfer rate constant, k17,13. This type of experiment is performed by measuring the non-retained fraction cf) of the analyte on a diffusion-limited system at a variety of flow-rates. From this data, a plot of - l/in ,f vs. solute velocity or flow-rate is then made. Assuming linear elution conditions are present, eqn. 8 and related expressions predict that a straight line should result with a slope related to k17J3. Simulations were used to study the effect of non-linear elution conditions on such measurements by applying a known load of analyte to a column with a fixed number of binding sites and measuring the relative size of the non-retained peak. In zyxwvu 1.5 0.0 00 a5 11) 5 "e"'kl Fig. 1. Normalized split-peak plots at various column loads for the case of diffusion-limited kinetics with irreversible adsorption. The plots given are for column loads of 16-128%, in intervals of 16%. The line shown is the response predicted under linear elution conditions by eqn. 8. 120 D. S. HAGE, R. R. WALTERS this and in all following studies, the column load was defined as the ratio of the total moles of solute injected to the moles of free ligand initially present on the column, or mEt,@Ln (F or a summary of this and other mathematical terms, see the Symbols section.) Based on the simulation data, plots were then prepared according to eqn. 8 and used to determine the effect of sample size on the resulting slope. The results for the diffusion-limited case are given in Fig. 1. Note in Fig. 1 that - l/lnfis plotted vs. a reduced velocity parameter, u,/hkl, rather than an absolute velocity term, such as u, or ue/h as would be used in an actual diffusion coefficient measurement. This was done to better illustrate the effect of non-linear elution conditions for this case by giving plots independent of the system rate constants. The normalized plots given in Fig. 1 for the diffusion-limited case are for loads of 16128% of the column capacity. Each plot in Fig. 1 is the best-fit curve through 27 data points distributed over the entire range of reduced velocities shown. By plotting the data in Fig. 1 as a function of the reduced velocity u,/hkl, the results were not only found to be independent of the absolute value of the system rate constant kr, but were also independent of the relative size of the excluded and pore volumes of the column, V, and VP (i.e., the porosity term VP/Vet or k&-i). This was observed for VP/I’, values of 0.1-2.0, those found with most common chromatographic matricePJ ‘. In testing for the effect of using different injection volumes or the presence of extracolumn band-broadening prior to the column, it was found that the data in Fig. 1 were independent of the volume of analyte applied as long as the total moles of analyte applied was constant. This was observed for the load range of l&128% and for reduced velocities of 0.06-1.28, where no change in - l/lnfwas noted in going from an application volume of one slice to twice the excluded volume of the column. As already mentioned, the plots in Fig. 1 would be expected under linear elution conditions to give the linear relationship predicted by eqn. 8. Under the nonlinear conditions used to generate the data, however, deviations from the ideal response were found to occur for all loads studied. In general, the simulation values of - l/lnf and the relative size of the non-retained peak were larger than or equal to those predicted, with the extent of the deviations increasing with column load. These deviations typically occurred when the value of reduced velocity u,/hkl was small, or the residence time large, but disappeared as the reduced velocity was increased. Also, the range of reduced velocities over which deviations occurred increased in proportion to the column load applied. For instance, a load of 64% caused deviations up to a reduced velocity of cu. 0.64 while a load of 128% gave deviations up to a reduced velocity of cu. 1.28. It was further observed in Fig. 1 that plots for loads of less than 100% appeared to have a zero intercept, while those for loads of greater than 100% had an intercept greater than zero. This is due to the fact that as the reduced velocity approaches zero, or the solute residence time becomes infinitely long, the amount of sample adsorbed reaches its maximum value. When the sample applied is less than or equal to the column capacity, all would be expected to adsorb causing - l/lnfto approach zero. If the load is larger then the column capacity, then the amount adsorbed approaches the column capacity, leaving some free solute behind and giving - l/in f a value greater than zero. NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 121 Experimental systems believed to exhibit diffusion-limited kinetics include the adsorption of some proteins on reversed-phase columns7~13. One such protein is hemoglobin13. To compare such a system to the simulation results presented, split-peak plots were made for injections of hemoglobin on a Cs reversed-phase column at known loading levels. The results are given in Fig. 2 for loads of 1.7 and 6.8%. In order to compare the experimental results in Fig. 2 to the simulation data in Fig. 1, it was necessary to determine what range of reduced velocities was represented by the data in Fig. 2. This was done by making use of the linear region of the 0.50 mg/ml data, which occurred at flow-rates of cu. 0.25 ml/min or greater. Since this region followed the relationship between residence time and - l/in f predicted by eqn. 8, it was used to represent the theoretical response of the system under linear elution conditions. A linear least-squares fit to the data in this range gave a slope of 1.48 f 0.06 min/ml and an intercept of 0.01 f 0.03. By multiplying each flow-rate in Fig. 2 by this slope, it was determined that the experimental data shown represented a range in the reduced velocity u,/hkl of 0.09-0.93. Several similarities can be seen in Figs. 1 and 2. For example, the data in Fig. 2 show the same deviation patterns as those in Fig. 1 in that the obtained values of - l/in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f are larger than predicted at long residence times, or slow flow-rates, but approach the expected response for linear elution conditions as the reduced velocity or flow-rate increases. Also, in both figures the size of the deviations and the flowrate range over which they occur increase with the load. Furthermore, this flow-rate range again appears to increase roughly in proportion to the load, with the 0.50 mg/ml data showing non-ideal effects up to a flow-rate slightly over 0.2 ml/mm and the 2.0 mg/ml data showing deviations up to a flow-rate of cu. 0.8 ml/min. zyxwvutsrqpo Fb wra to hl/ m in) Fig. 2. Split-peak plots for hemoglobin on a reversed-phase column. The plots shown are for 3 ~1 injections of 0.50 mg/ml (0) and 2.00 mg/ml (m) bovine hemoglobin. The chromatographic conditions were the same as described in the text. The line shown is the linear fit for the 0.50 mg/ml data over the flow-rate range of 0.25463 ml/min. 122 D. S. HAGE, R. R. WALTERS Figs. 1 and 2 differ in that the deviations in - lllnfseen experimentally were larger than those predicted from the simulations. This can be due to a number of secondary effects not taken into account by the kinetic model used. Examples include site heterogeneity’ and reversible binding 28. Site heterogeneity can be important for the diffusion-limited case if ligands with different mass transfer properties are present, such as those located at different depths within the matrix or on particles of different diameters. Under linear elution conditions this results in k1 becoming an apparent rate constant, or a function of the individual mass transfer rates present. This also occurs under non-linear conditions but with the additional possibility that some types of sites may saturate before others. The result is a change in k1 and - I/in f with sample load, giving greater deviations than would be expected for a simple homogeneous matrix. In previous work done with proteins on Ca LiChrospher Si-500 and essentially non-porous matrices of the same diameter, the apparent kl was noted to vary by 17-fold depending on whether the ligands sampled were only on the surface of the particle or on the surface plus in the pores ‘. This makes site heterogeneity a likely explanation for the differences noted between Figs. 1 and 2. Reversible binding may affect the results if weakly-retained solute elutes near the non-retained peak, increasing the apparent free fraction and the value of - l/lnf measured. In previous simulation studies, it was shown for linear elution conditions that this is a significant problem only for systems with k’ values of 10 or 1esP. Although deviations due to this effect may increase under non-linear conditions, the fact that hemoglobin was estimated to have a k’ over 1500 in this experiment suggests that this was not a major factor. Regardless of which secondary effects were present, the data in Figs. 1 and 2 clearly show that non-linear effects in split-peak chromatography under diffusionlimited conditions can be minimized or even eliminated by choosing the proper load and/or flow-rate. In this case, non-linear effects decrease as the load decreases or as the flow-rate and solute velocity increase and disappear beyond a given flow-rate and load combination. This behavior explains why such effects were not seen in previous work examining the retention of IgG on Ca reversed-phase columns’. In this earlier study two different matrices were used: Nucleosil Si-50, which acts as a pellicular or nonporous support for IgG, and LiChrospher Si-500, which acts as a porous matrix7. For the Ca Si-50, which should have behaved similarly to the homogeneous case in Fig. 1, no non-linear effects were seen for loads of 7.3-14.5% over a reduced velocity range of 0.13-0.36. A comparison of these values with those in Fig. 1 shows that no observable deviations would have been expected under these conditions. For the Ca Si-500, which should have given results resembling the heterogeneous case in Fig. 2, no deviations were noted for loads of 244.0% and reduced velocities of 0.24-0.94 under the same chromatographic conditions as used in Fig. 2. Assuming IgG and hemoglobin behaved similarly on this matrix, a comparison of these values to those in Fig. 2 indicates that no significant deviations would have been expected for the data at 2.4% load or for the majority of the 4.0% load data. Fig. 2 does predict some observable deviations at the lower reduced velocities used with the 4.0% load, but the fact that these were not seen may be due to differences in the response of IgG and hemoglobin to secondary effects. For example, IgG may have been less susceptible than hemoglobin to site heterogeneity due to its larger size, a Stoke’s diameter NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 123 of 104 8, for IgG vs. 62 A for hemoglobin29*30, preventing it from sampling as many diffusionally-distinct ligands within the pores. The absence of non-linear effects for the diffusion-limited case at small residence times or column loads was further examined by performing a second series of simulations in which the effect of increasing loads on - l/in zyxwvutsrqponmlkjihgfedc fwa s measured at constant values of the reduced velocity u,/hkl. This corresponds to an experiment in which - l/lnfis determined at a constant flow-rate while the sample load is varied. Fig. 3 shows the results obtained at several different reduced velocities. In Fig. 3, the load is normalized vs. the reduced velocity u,/hkl to illustrate its relationship to this parameter as deviations begin to occur. The y-axis is also normalized vs. u,/hkl, but this is done only for ease of presentation. For each reduced velocity monitored, the resulting plot showed no deviations from the expected response under linear conditions at small values of the ratio Load/ (u,/hkl) (Le., small sample loads). But when this ratio exceeded 1.0, deviations in - ljnfbegan to occur. It was also noted that the relative amount of these deviations increased with the size of the reduced velocity monitored. Based on the simulation and kinetic models used, an equation was derived to explain why the deviations in Fig. 3 began to occur at such a well-defined point, and why the reduced velocity at which deviations began to occur in Figs. 1 and 2 increased proportionally with the load applied. To do this, it was assumed that all deviations began in the first slice of the column during the first iteration, when the amount of free analyte was largest with respect to that of free ligand. Under these conditions, the initial amount of free ligand present in the slice, mL,, is equal to NsmL/h, where mL/h is the original amount of free ligand per slice and N, is the number of slices examined. Since only the first slice considered in this case, N. is equal to one. Also under these conditions, the initial amount of free analyte in the slice, mu,,, is equal to the total amount of analyte applied, or (Load) using the definition of load given earlier. Furthermore, it is possible from the simulation model to replace the slice reaction time t by the quantity N&. Using each of these expressions along with eqn. 15 gives the following relationship: Load I [h(l - e-klY”e)/N,I-l (21) This equation gives the range of column loads that can be used at a given value of u,/hkl before deviations begin to occur in a system composed of a finite number of slices. The equivalent expression for a real system, or one approaching an infinite number of slices, can be obtained by finding the limit of eqn. 21 at a constant value of u,/hkl by simultaneously increasing the number of slices in the column and decreasing the amount of time per slice or increasing ue to keep the total residence time constant. As this is done, the exponential term of eqn. 21, e-klNd”c, approaches zero and may be replaced by (1 - kl N,/u,), the zeroth- and first-order terms of its Taylor series expansion3*. Substitution of this into eqn. 21 gives Load/(u,/hkl) I 1 (22) which is the same relationship noted empirically in Fig. 3. This expression shows that 124 D. S. HAGE, R. R. WALTERS 0.0 1 .5 2 .0 1 .0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 45 {Loa dM u& k l) Fig. 3. The effect of increasing load on - l/lnfat a constant value of U./~/C,. The plots shown are for u,/hkl values of 0.16 (a), 0.32 (+), 0.64 (A), and 1.28 (m). The horizontal line is the response predicted under linear elution conditions by eqn. 8. either a small load or large value of the reduced velocity u,/hkI minimizes deviations because both cause the ratio in eqn. 22 to decrease. Eqn. 22 also explains why the range over which deviations occur appears to be directly proportional to the load applied. This occurs since as larger loads are used, a proportionately larger reduced velocity is needed to bring the ratio in eqn. 22 back to a value of 1.0 or less. By combining eqn. 22 with eqn. 8, an alternate relationship is obtained for predicting when deviations due to non-linear effects occur even when kinetic parameters of the system or the value of the reduced velocity are not known. Load/( - l/lnJ I 1 (23) Thus, by measuring - l/lnffor a known sample load at the flow-rate of interest and computing the above ratio, an estimate of whether or not non-linear effects are occurring can be made for diffusion-limited systems following the model used here. zyxwvutsr If the ratio is greater than 1.0, deviations would be expected to occur. As the ratio becomes less than or equal to 1.0, the chance of deviations occcurring would be expected to greatly decrease. The hemoglobin data suggests that this ratio might also be useful in the study of more complex diffusion-limited systems. This is indicated in Fig. 2 by the fact that the flow-rate at which deviations begin to occur appears to increase proportionately with the load. Based on the relationship between flow-rate and - lllnfgiven in eqn. 5, this is equivalent to saying that deviations start to appear at a constant value of Load/( - l/lnJ, in this case a ratio of 0.05. Though this value is much less than that predicted by eqn. 23, probably as a result of the presence of secondary effects in the NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. a5 1.0 O+lh)(k_llkl I. 125 1.5 kg IL11 Fig. 4. Normalized split-peak plots at various column loads for the case of irreversible adsorption-limited kinetics. The plots given are for column loads of 16128%, in intervals of 16%. The line shown is the response predicted under linear elution conditions by eqn. 16. Cs Si-500 studies, the fact that it appears to be constant still makes it useful in minimizing non-linear effects. For example, if the exact ratio at which deviations occur is known, as it is here, then the flow-rate conditions required to eliminate non-linear effects at any size load can be determined. Even if this particular value is not known, the ratio Load/( - l/lnfl is useful in indicating whether non-linear effects may be present, since non-linear effects are less likely to occur as the value of Load/( - l/lnfl decreases. In general, it is known from eqn. 23 that conditions giving a ratio of I 1.0 should always be used to avoid non-linear effects, while the Cs Si500 results further indicate that for complex diffusion-limited systems, such as those using porous supports, a ratio even as low as 0.05 or less is desirable. Simulation results for adsorption-limited kinetics The second set of simulation studies examined non-linear effects for systems with irreversible adsorption-limited kinetics. This case is potentially useful in splitpeak chromatography for the determination adsorption rate constants for macromolecular interactions7+2 *. As in diffusion coefficient measurements, this can be done by determining the non-retained fractions of analyte at various flow-rates on an adsorption-limited system containing the ligand of interest. Plots of - l/in f vs. flowrate or solute velocity are then made according to expressions such as eqn. 16. In this case, a linear relationship with a slope related to k3 should result under linear elution conditions’. The effect of non-linear elution conditions on such plots was studied through simulations in the same manner used for the diffusion-limited case. The plots obtained for the simulation data are shown in Fig. 4. Note that - l/in f is again plotted 126 D. S. HAGE, R. R. WALTERS against a reduced velocity, in this case (u,/h)(k-i/klk3[L]), rather than an absolute velocity for the same reason as stated earlier. The range of reduced velocities used in Fig. 4 for the adsorption-limited case is the same as that used for the diffusionlimited case in Fig. 1. Each plot given in Fig. 4 is the best-fit curve through 13-16 data points distributed throughout the entire reduced velocity range shown. In comparing the data in Fig. 4 to the results predicted for this case by eqn. 16 under linear elution conditions, deviations from the ideal response were again seen for all loads studied. As noted in Fig. 1 for the diffusion-limited case, the simulations for the adsorption-limited case gave values zyxwvutsrqponmlkjihgfedcbaZYXWVUTS off and - l/lnf larger than those predicted, with deviations increasing as the load increased. It was also again seen that plots for loads of less than 100% appeared to have zero intercepts while those for loads above 100% gave non-zero intercepts. This occurred for the same reasons as discussed previously. As in the diffusion-limited case, the plots in Fig. 4 were not only independent of the absolute value of the system rate constants, but were also independent of relative size of VP and V,. This was again noted over the VP/Ve range of 0.1-2.0. Unlike the diffusion-limited case, however, it was found that the sample concentration or degree of precolumn sample dispersion did affect the results slightly. When the sample application volume was increased and the total moles of analyte applied was held constant, - l/in f decreased at each given reduced velocity. This behavior was noted for each load studied, 1.6-128% of the column capacity, over injection volumes ranging from one slice to twice the column excluded volume and reduced velocities ranging from 0.061.28. Despite this decrease in - l/in f with injection volume, the plots in Fig. 4 were not significantly affected since the change in - l/lnf observed was negligible with respect to the overall deviations present in the curves. For example, the observed decrease in - l/lnfwas only 0.3 ppt or less of the total deviation from the ideal response seen at any given reduced velocity. Also, it was noted that the extent of this injection volume dependence became smaller as the simulated columns were divided into an increasing number of slices. A more significant difference between the data in Figs. 1 and 4 was that the values of - l/lnfin Fig. 4 for the adsorption-limited case were typically greater than those for the diffusion-limited case in Fig. 1 under the same load and reduced velocity conditions. The two cases also differed in that the results in Fig. 4 did not converge with the response predicted under linear elution conditions. Instead, no apparent decrease in absolute deviations were noted as the reduced velocity increased. This occurred for all loads studied and over a reduced velocity range of at least 0.03-20 (i.e., an expected free fraction range under linear elution conditions of 3 . 10-13% to 95%). It was further noted in Fig. 4 that small loads, such as 16 and 32%, gave an almost linear increase in - l/in f with the reduced velocity, behavior not noted in the diffusion-limited results. Differences in the adsorption- and diffusion-limited cases were also apparent when comparing their responses to increasing load at a constant residence time. The results for the adsorption-limited case, shown in Fig. 5, differ from those under equivalent conditions for the diffusion-limited case in Fig. 3 in that deviations were present at all loads and residence times studied while in the diffusion-limited case they occurred only once a certain load level had been reached. Also, it was again noted that the relative size of the deviations in the adsorption-limited case were larger than those for diffusion-limited systems under equivalent conditions. NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 127 Fig. 5. The effect of increasing load on - l/in fat a constant value of (u./h)(k- Jklkl[L]). The plots shown are for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (u./h)(k-,/k,k,[L]) values of 0.16 (a), 0.32 (a), 0.64 (A), and 1.28 (W). The horizontal line is the response predicted under linear elution conditions by eqn. 16. Such differences can be explained based on the two kinetic models used. In the case of diffusion-limited kinetics, the system is represented by a series of first-order reactions and, under linear elution conditions, the split-peak slope is independent of the number of ligand sites. The result, as shown earlier, is that deviations occur only after all ligand in the first segment of the column has been depleted. This produces the behavior seen in Figs. 1 and 3, where certain load and residence time conditions must be met to produce non-linear effects. In the adsorption-limited case, however, the model used is second-order in nature and the split-peak slope is a function of l/[L]. This means that any depletion of free ligand will cause the apparent rate constant kJ[L] to decrease and the split-peak slope to increase. As a result, the adsorption-limited case would be expected to be more susceptible to deviations than the diffusion-limited case at any given load, with deviations occurring as long as finite sample loads are used. An experimental system previously demonstrated to be adsorption-limited is the binding of IgG to certain CDI-immobilized protein A affinity columns’. Typical split-peak plots for this system are given in Fig. 6 for loads of 0.9 to 3.7%. These plots are similar to the simulation results in Fig. 4 in several ways. First, each data set in Fig. 6 gave an apparent linear relationship between - l/in zyxwvutsrqponmlkjihgfe f and flow-rate, as seen for loads of 32% or less in Fig. 4. Secondly, the IgG data at different loads did not converge to a single response at high flow-rates, as was seen with the simulation results, but continued to increase with flow-rate over the entire range studied. Because of this last factor, eliminating or even minimizing non-linear effects in split-peak measurements for such a system can prove difficult if done by only adjusting the flow-rate and/or load conditions. Instead, a technique is required to 128 D. S. HAGE, R. R. WALTERS I.! Flow -ra t e hl zyxwvutsrqponmlkjihgfedcbaZYXWVUT /mid Fig. 6. Split-peak plots for IgG on a protein A column. The plots shown are for IO-~1injections of 0.14 (a), 0.27 (A), and 0.55 mg/ml (B) rabbit IgG. The chromatographic conditions were the same as described in the text. The line given for each data set is its linear fit through origin. extrapolate the results under linear elution conditions from those obtained under non-linear conditions. A third set of simulation studies were performed to determine what extrapolation methods could be effectively used. The particular technique tested was one using linear extrapolation since previous work showed that the plots of - l/lnfvs. flow-rate for protein A columns appeared to increase linearly with sample load7. These studies were done by using the simulation model to generate split-peak plots according to eqn. 16 at various loads under adsorption-limited conditions. The slope of each plot was then measured using a linear least-squares fit through the origin, the same technique used in the previous study7. Each slope was measured using 14 points over the reduced velocity range of 0.063-I .24. This range corresponds to free fractions of 10e5-45% under linear elution conditions. The slopes obtained were then plotted against load, as shown in Fig. 7. The resulting curve showed an essentially linear increase in the measured slope for loads of 24% or less but with some curvature beginning to appear as the load increased further. The data in Fig. 7 was used to test the extrapolation procedure by performing linear least-square fits over various load regions of the plot. The intercept obtained (i.e., the extrapolated value of the slope at 0% load) was then compared to the value predicted under linear conditions, a true split-peak slope of 1BOO0according to eqn. 16. The results obtained are summarized in Table I. In general, the use of small loads in the extrapolation gave more accurate results than when larger loads were used, with an error of only 0.9 ppt being obtained with loads of 416%. However, even with the largest load range studied, l&64%, the error was still less than the best previously reported7 experimental precision of such measurements, a value of f 3.4%. NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. 129 I. 6 22 Load (Xl Fig. 7. The effect of increasing load on the measured split-peak slope for a system with irreversible adsorption-limited kinetics. The slope values were determined as described in the text. The line shown is the linear fit to the data over the load range of 416%. Fig. 8 shows plots obtained when this extrapolation method was applied to various protein A columns, data from ref. 7 taken under approximately the same free fraction conditions used to generate the simulation results. The plots given represent two matrices shown to be adsorption-limited, the CDI-500 and SB-50, and one in which both adsorption and diffusion may have contributed to the overall rate of IgG retention, the SB-500 (ref. 7). Also shown are the simulation results for the load range of 4-16%. In Fig. 8, each protein A matrix gave the same linear response between the measured slope and load seen with the simulation results. This demonstrates the apparent applicability of this extrapolation method to complex as well as simple adsorption-controlled systems and, in the case of the SB-500, even to some systems with some diffusional contribution to the kinetics. Although the reason for the linTABLE I ERROR OF LINEAR EXTRAPOLATION OVER VARIOUS LOAD RANGES FOR A SYSTEM WITH IRREVERSIBLE ADSORPTION-LIMITED KINETICS Load range* 416% S-32% 1664% Linear zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH least- squares jit parameters Error (%)** Slope ( f I a) . 10’ intercept (*I a) 3.62 f 0.03 3.87 f 0.06 4.43 f 0.15 0.9991 f o.ooo3 0.996 f 0.001 0.983 f 0.007 -0.09 -0.4 -1.7 l The results for each data set are for 4 points distributed evenly throughout the load range studied. ** Calculated using the intercepts and an expected value under linear elution conditions of 1.9000. D. S. HAGE, R. R. WALTERS 0 4 8 Load 12 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON 1 (Xl Fig. 8. The effect of varying load on the measured split-peak slopes for IgG on various protein A matrices. The protein A data are from ref. 7. The plots shown are for the SB-50 (A), SB-500 (+), and CD1400 (0) matrices described in the reference and for the simulation results over the load range of 416% (m) from Fig. 7. earity of SB-500 response is not totally clear, it may indicate that as the transition from simple diffusion-limited kinetics to simple adsorption-limited kinetics occurs, the increased sensitivity of intermediate systems with respect to the diffusion-limited case to column overloading may give them a response more closely resembling that of a simple adsorption-limited system. Another observation made in Fig. 8 was that the experimental data showed a much larger relative increase in slope with load than predicted from the computer modeling. For example, the slope of such a plot predicted from the simulations was 3.62 - 10e3 for the given loads while experimentally this value ranged from 0.043 to 1.18. This may, again, be a sign that secondary effects such as site heterogeneity or reversible binding were present. Of these, site heterogeneity may have been particularly important. For adsorption-limited kinetics this includes not only diffusional or mass transfer heterogeneity but also heterogeneity of the ligand. The differences in mass transfer heterogeneity of the two matrix materials used in Fig. 8 have already been discussed. The presence of ligand heterogeneity was suggested by earlier work where it was shown that the Schiff base and CD1 coupling methods used gave immobilized protein A with different apparent adsorption rate constants, with the Schiff base protein A having a k3 value ten times that produced by the CD1 method. One possible interpretation is that the CD1 method denatured the protein to a greater extent, producing a more heterogeneous population of ligand and lowering the apparent value of zyxwvutsrq k3’. Based on site heterogeneity, it is possible to predict the same order of deviations as seen in Fig. 8. For example, the SB-SO_support, with the largest adsorption rate constant and the smallest amount of mass transfer heterogeneity, would have NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 131 been expected to most closely resemble the simple adsorption-limited case. The SB500 data, obtained using the same type of protein A but on a more porous matrix, would have been predicted to give even larger deviations. Lastly, the CDI-500 results, acquired on the same matrix material as the SB-500 but with possibly more heterogeneous protein A, would have been expected to give the largest deviations. CONCLUSION The simulation results presented show that non-linear elution effects in splitpeak chromatography can be minimized by using the proper separation conditions or extrapolation techniques. For the diffusion-limited case, it was found that these effects could be reduced or even eliminated by using small sample loads and/or fast flow-rates. It was found that under these conditions the measured non-retained fraction for the simulation model approached the value predicted under linear elution conditions. An expression was then derived for this case to calculate the flow-rate and load conditions needed to eliminate non-linear effects for the simple kinetic system studied. Although such a calculation is accurate for an experimental system only if its kinetics follow the model used here, this expression should still be useful for more complex diffusion-limited systems as a guideline in determining the approximate range of load or flow-rate conditions that can be used without producing nonlinear effects. Such information is potentially useful in determining diffusion coefficients by split-peak chromatography, which are obtained assuming linear conditions are present. For the adsorption-limited case, simulations demonstrated that non-linear effects can not be totally eliminated in split-peak measurements by changing the flow-rate or load. However, the data did show that such effects can be minimized by using extrapolation techniques. This was done by making a plot of the measured split-peak slope vs. sample load and performing a linear fit to determine the slope at zero sample load. When using loads less than 16%, the extrapolated slope obtained for the case studied varied by less than 1 ppt from that expected under linear elution conditions. Although the error increased with the loads used, even over a load range of 16-64% its level was still acceptable, having a value of less than 2%. This type of information is important to consider in the use of adsorption-limited systems for the determination of rate constants for macromolecular interactions. The difference in non-linear effects seen with the split-peak plots obtained for these two cases suggests that such plots may be useful tools in determining the ratelimiting step for solute retention in chromatographic systems. For example, the split-peak plots for hemoglobin on a Cs reversed-phase column showed the same non-linear effects as the diffusion-limited case, while a similar study for IgG on a CD1 protein A column gave results resembling those obtained for the adsorptionlimited case. These results confirmed those of earlier experiments suggesting that the hemoglobin and IgG systems studied were diffusion- and adsorption-limited, respectively. Thus, by making split-peak plots at various loads and comparing the resulting curves to the simulation results presented here, it may be possible to determine the rate-limiting step in retention for a given matrix. This should not only be useful in obtaining kinetic data but also in the optimization of chromatographic separations. Although the hemoglobin and IgG studies showed the same general responses D. S. HAGE, R. R. WALTERS 132 predicted by the simulations, they also gave larger deviations than expected from the computer modeling. It was proposed that this was due to the presence of secondary effects such as site heterogeneity or reversible binding. Of these, site heterogeneity may be particularly important since the relative size of deviations seen with various protein A columns was noted to follow the order predicted based on only their ligand and mass transfer heterogeneities. Mass transfer heterogeneity was also implicated in the hemoglobin study. Further computer modeling needs to be done to better determine the influence of this and other such phenomenon on the nonlinear elution effects seen in split-peak chromatography. APPENDIX The derivation of the integrated rate expressions for the simple adsorptionlimited system given in eqns. 17 and 18 is similar to that for a one-phase, secondorder reaction as described in ref. 17. For the reaction given in eqn. 18, the rate law is as follows: - Wls ~ = ME,Is[Ll, (AlI dt where [E& and [L& are the concentrations of E, and L in the stagnant mobile phase of the slice studied and all other terms are as defined previously. Converting eqn. Al to an expression in terms of moles gives -dmL s= & k3 EmEPmL. 642) where I’,,, is the pore volume per slice, I/,/h. Using the reaction stoichiometry given in eqns. 17 and 18, the mass balance expression for this system can be written as (mEew - mE,,) + (mEpso- mE,,) = (ML_ - mL,) (A3) It is also given that diffusion is much faster than adsorption, or that mass transfer equal to V,,/V,, by eqn. 3. Substitution of is in equilibrium, making ma&a,, V,,/V,, into eqn. A3 and using the fact that I’,, + V,, = I’,,,,, the total slice void volume, gives the solution in terms ofma, as being mr,, = @E., + mE,,, - mL, Letting x = (ma,, + ma,, -mL,) following differential equation: - m& dmLS = (W J’msW + ML) + mL,) (VP,/ vms) (A4) and substituting eqn. A4 into eqn. A2 gives the (A5) NON-LINEAR ELUTION IN SPLIT-PEAK CHROMATOGRAPHY. I. 133 Integration of eqn. A5 between the limits of times 0 and t yields the expression given in eqn. 19 for the case where x # 0. For x = 0, eqn. A5 reduces to - dmi. d = (k3/I’,,,,)dt kkJ2 646) which, upon integration over the same limits, gives the expression shown in eqn. 20. SYMBOLS Analyte or molecule of interest Immobilized ligand or binding site Analyte located in the flowing mobile phase and stagnant mobile phase, respectively Analyte-ligand complex E,-L Column length; number of slices in the computer model h Void volume; elution volume of a small, non-retained solV, ute Excluded volume; elution volume of a large, non-retained V, solute; volume of flowing mobile phase in the column Pore volume; VP = V,,, - V,; volume of stagnant mobile VP zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA phase in the column Excluded linear velocity; linear velocity of an large, excluded solute through the column Volumetric flow-rate of solvent F Forward and reverse mass transfer rate constants for kl,k- 1 transfer of solute between the flowing and stagnant mobile phases Mass transfer equilibrium constant; K1 = kl/k-l = KI E L ES, VP/ Ve ks,k-3 Adsorption and desorption rate constants for the interaction of analyte and ligand in the stagnant mobile phase Equilibrium constant for the binding of analyte to ligand; K3 K3 = kJk-3 Moles of free, or unbound, ligand immobilized on the mL zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA stationary phase Moles of E, and E, in the column mE,,@E, Total amount of E applied to the column m%sl Concentration of E,, L, etc. in the stagnant mobile phase; [E,l,M etc. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA mn,/ VP, mt.1V,, etc. Moles of E,, L, etc. in the slice of interest during a given iteration Non-retained fraction of analyte eluting from the column; f free fraction Free fraction term as used in eqns. 5, 7, 8 and 16 -l/in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f Apparent first-order adsorption rate constant under linML1 ear elution conditions 134 D. S. HAGE, R. R. WALTERS u&r Reduced velocity for the diffusion-limited case (see eqn. 8) Reduced velocity for the adsorption-limited case (see eqn. 16) Relative amount of analyte applied to the column; Load = mEtots,/mL Diffusional slope term of eqn. 7 Adsorption slope term of eqn. 7 Capacity factor Number of slices examined in the simulation model in eqn. 21 Time One unit of time in the simulation model One unit of length in the simulation model (@I@ - dkML1) Load l/k1 (k- dW&I) k NS t Iteration Slice ACKNOWLEDGEMENTS This work was supported by the National Science Foundation under Grant CHE-8305057. D.S.H. was supported by an American Chemical Society Analytical Division Fellowship from the Proctor and Gamble Company. REFERENCES 1 J. L. Wade, A. F. Bergold and P. W. Carr, Annl. Chem., 59 (1987) 1286. 2 A. I. Kalinchev, A. Ya. Pronin, P. P. Zolotarev, N. A. Goryacheva, K. V. Chmutov and V. Ya. Fihmonov, J. Chromalogr., 120 (1976) 249. 3 K. Yamaoka and T. Nakagawa, J. 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R. Cantor and P. R. Schimmel, Biophysicul Chemistry, Vol. 2, Freeman, San Francisco, CA, 1980, pp. 583-586. 31 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961, 4th, ed., p. 132.