Anal. Chem. 1991, 63, 2007-2011
fringing field source sensitivity results from an inability to trap a large proportion of the ions that do reach the analyzer cell. In order to avoid the problem of ion dispersion during gated transfer between cells, kinetic energies may be made more uniform by cooling the ions or by reducing the depth of the potential well. Clearly, however, unless the fringing field source is essential for other reasons, the better solution is simply to reduce the path length. One additional advantage derived by placing the source trap in the strong homogeneous region of the magnetic field is the ability to execute the FTICR experiment directly in the source. Although used only as an ion trap in this work, the high field source was actually configured for full FTICR operation. An example of increased flexibility that is specifically related to high duty cycle experiments is the opportunity to eliminate matrix and other high abundance ions during GC/FTICR measuremente; the dynamic range of the experiment is greatly enhanced by allowing low abundance ions to accumulate during a continuous beam experiment while a SWIFT (27) wave form is applied to the excitation plates of the source cell to eject the unwanted ions. This approach is superior to the combination of a crude time-of-flightmass filter and selective trapping, which is possible with longer path length dual cells.
LITERATURE CITED (1) Comlsarow, M. 8.; Marshall, A. G. Chem. Fhys. Len. 1974. 25. 282-283. (2) Comhrow, M. 8.; Marshall, A. G. Chem. Fhys. Len. 1974, 26, 489-491. (3) Kofel, P.; Alleman, M.; Kellerhals, H. P.; Wanczek, K. P. Int. J . Mess Spectrom. Ion R O C ~ S S 1989, . 87, 237-247. (4) Kofel, P.; Alleman. M.; Wanczek, K. P. Int. J . Mess Spectrum. Ion Process. 1965, 65, 97-103. (5) Kofel, P.; Alleman, M.; Kellerhals. H. P.; Wanczek, K. P. Int. J . Mess Spec!”. Ion Processes 1986, 72, 53-837. (8) Marayuma, S.; Anderson, L. R.; Smalley, R. E. Rev. Scl. Insbwn. 1990, 67, 3886-3693.
2007
(7) Lebrwle, C. B.; Amter, I. J.; Mclver, R. T. Int. J . Mesa &ecm. Ion Recesses 1969, 87, R7413. (8) Henry, K. D.; McLafferty, F. W. ap. Mess spscbwn. 1990, 25, 490-492. (9) Hunt, D. F.; Shabanowltz, J.; Yates, J. R.; Zhu, N.-2.; Rusrre#, D. H.; Castro, M. E. Roc. Nan. Aced. Scl. U.S.A. 1987, 84, 620-823. (IO) Beu, S. C.; Laude, D. A., Jr. Int. J . Mass spect”.Ion Pmwsses 1981, 704, 109-127. (11) (a) Cody, R. 8.; Kbrskrger,J. A.; Ghaderl, S.: Amster, I. J.; McLafferty, F. W.; Brown, R. S. Anal. chem.Acta 1985. 778, 43-86. (b) U S . Patent 4581533, 1988. (12) Gkncaspro, C.; Verdun, F. R.; Muller, J. H. Int. J . Mass Spsc!”. IOn Processes 1966, 72, 83-71. (13) Qiencaspro, C.; Vefdun, F. R. Anal. Chem. 1966, 58, 2097-2099. (14) Kerley, E.; R W U , D. AM/. chem.1969, 67, 53-57. (15) Hanson, C. D., Kerley, E. L.; Russell, D. H. Anal. Chem. 1989, 67, 83-85. (18) Hofstadler, S. A.; Laude, D. A., Jr. Int. J . Mess Spechn. Ion Pro~ ~ 8 8 1990, 88 701. 85-76. (17) De Konig, L. J.; Fokkens, R. H.; Plnske, F. A.; Nlbberlng, N. M. M. Int. J . Me= Spscirom. Ion Prooesses 1987, 77, 95-105. (18) Whlte, R. L.; Ledford, E. 8.; Qhaderl, S.; Spencer, R. 8.; Kulkaml, P. S.; Wklns, C. L.; Gross, M. L. Anal. Chem. 1980, 52, 463-468. (19) Comlsarow, M. 8.; M a , J. Anal. Chem. 1979, 57, 2198-2202. (20) Comlsarow, M. 8.; Marshall, A. 0. Can. J . Chem. 1974, 52, 1997-2000. (21) Riegner, D. E.; Hofstadler, S. A.; Laude, D. A., Jr. AM/. Chem. 1991, 63, 261-268. (22) Honovlch, J. P.; Markey, S. P. Int. J . Maw Spscfrom. Ion Rocmses 1990, 98, 51-68. (23) Hofstadler, S. A,; Laude, D. A., Jr. J . Am. Soc. Mess Specfrom. 1990, 7 , 351-360. (24) Van De ouchte, W. J.; Van Der Hart, W. J. Int. J . Mess Spectfom. Ion Procsaseo 1990, 95. 317-326. (25) Laude, D. A., Jr.; Beu, S. C. Anal. Chem. 1989, 67, 2422-2427. (28) Hofstadler, S. A.; Leude, D. A.. Jr. Int. J . Mess Spectfom. Ion ProC88888 1990, 07, 151-164. (27) Chen, L.; Marshall, A. 0. Int. J . Mess S p ” . Ion Fnm9ss.98 1987, 79, 115-125.
RECEIV~,for review March 28,1991. Accepted June 14,1991. Support from the Welch Foundation (F-1138), the Texas Advanced Research and Technology Program, and the National Science Foundation (CHEW13384 and CHE9057097) is gratefully acknowledged.
Liposome Flow Injection Immunoassay: Model Calculations of Competitive Immunoreactions Involving Univalent and MuItivalent Ligands William T. Yap,* Laurie Locascio-Brown, Anne L. Plant, Steven J. Choquette, Viola Horvath,’ and Richard A. Durst Center for Analytical Chemistry, NIST,Gaithersburg, Maryland 20899 The use of Ilpomnes as detectable reagents In wlld-phase Immunoassays has been explored in a How InJection bnmunoanalysts (FIIA) system. Model calculations are presented for FIIA basedon the ampdtlve binding ofunlvaknt analyte and mul(lvaientHposomes to hnmoblllzed antlbodles. Parameters ouch as Mndm constants, concentratbns of llporromes and antibody, and steric hindrance are conddered for their relative effects on detectable llposome signal response to analyte ancenlratiom. Quaillathre comparisons of the model with the experimental data are made.
INTRODUCTION The specificity of immunological recognition coupled to chemical and/or biochemical amplification strategies can provide highly selective and sensitive analytical measurements. Current address: Technical University of Budapest, Hungary. 0003-2700/91/0363-2007$02.50/0
Our immunoassay system (1) for determining the concentration of clinically relevant analytes in serum is based on the use of flow injection analysis for transport of small volumes of samples and reagents in a continuously flowing stream. Antigen-derivatized liposomes (henceforth referred to as liposome for simplicity) are used as the label for detection of an antibody-antigen binding event. Competitive binding of liposomes and analyte molecules occurs in a packed bead column containing immobilized antibodies. The use of liposomes provides significant signal amplification since ca. 106 marker molecules can be encapsulated in their aqueoua centers. In this format, the system can be used to quantitate picomole amounts of the drug theophylline. In addition to signal enhancement, the use of derivatized liposomes has the practical advantage of the ease of regeneration of an immunoreactor column because the bound multivalent liposomes can subsequently be broken up and washed away (1-3). The liposome immunoreagent is significantly different from the typical competitive immunoassay reagent. In competitive 0 1991 American Chemlcal Society
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 18, SEPTEMBER 15, 1991
immunoassays such as radioimmunoassays, a detectable derivative of the analyte with similar binding characteristics for the antibody competes with unlabeled analyte for binding to an antibody. In contrast, the liposome can be a multivalent reagent with binding constants and dissociation kinetics that differ by orders of magnitude from the binding characteristics of the analyte, since the outer leaflet of the liposome bilayer can contain thousands of analyte molecules for binding to immobilized antibody. In this paper, we report a model addressing the case of competitive binding between a monovalent hapten and a multivalent ligand (liposome). The model assumes equilibrium condition. In our FIIA system, equilibrium in the immunoreactor can easily be achieved by making the residence time of the sample in the immunoreactor sufficiently long. The model aids in the optimization of the sensitivity and dynamic range of our flow injection immunoanalysis system by evaluating the effecta of parameters such as concentrations of liposomes and immobilized antibodies, binding constants of liposomes and haptens, and steric considerationsassociated with the relatively bulky liposome reagent. The observed succe88 of multiple sequential injections (1)on increasing assay signal is also explained by the model. EXPERIMENTAL SECTION Immunoreactor Column. The flow injection immunoanalysis (FIIA) system, described previously in detail (I), consists of small-diameter tubing and computrer-controlledvalves that allow for programmed delivery of reagents and samples in a continuously flowing buffer stream. The reagents and samples are carried by this stream to a flow-through immunoreactor column containing solid silica particles (150-190 wm in diameter) to which monoclonal anti-theophyllineIgG is covalently immobilized as described in ref 4. The total number of antibody binding sites present in the column was quantitated by equilibration with saturating concentrations of [3H]theophylline. This column is the site of a competitive binding reaction between derivatized liposomes and the analyte, theophylline. The dimensions of the column were 2 mm i.d. by 10 cm in length. The void volume of the packed column was 0.15 mL. The flow rate was 0.2 mm min-’, which corresponded to a linear flow rate through the packed column of approximately0.4 cm 6’.The sample volume injected was 0.05 mL/analysis, and the total sample time on the column was approximately 107 s. Antigen-Derivatized Liposome. Liposomes, containing a solution of 100 mM carboxyfluorescein in their aqueous centers, were rendered immunoreactive by the inclusion of 5 mol % of theophylline-derivatized phosphatidylethanolamine(theePE) in their bilayer membranes, which included dimyristoylphosphatidylcholine, dicetyl phosphate, and cholesterol. Preparation and characterizationof theo-PE liposomes is described in detail in ref 1. The mean diameter of the liposomes was determined by quasielastic light scattering to be 135 f 65 nm. Estimation of the concentration of liposomes in reaction mixtures was made on the basis of the concentration of totallipid, the size of the liposome, and estimates for area of the lipid head group and by assuming unilamellarity. Assay Protocol. The assay protocol for the single-injection mode wm follows: (1)liposomes plus theophylline were injected into the flowing stream and carried onto the column; (2) unbound liposomes were allowed to flow to waste; (3) a detergent solution waa injected to disrupt the liposomes and release the fluorophores, which flowed downstream to be quantitated at the detector; (4) flow was continued to wash monomer theo-PE from the column, thereby regenerating the binding sites; and finally, ( 5 ) a subsequent theophylline sample plus liposomes was injected. In the double-injection mode, step 1 was repeated before step 3 was performed. The procedure used for antibody immobilization, coupled with the protocol used for regeneration of antibody active sites, resulted in a stable immunoreactor column, with consistent activity over hundreds of sequential assays. The analysis time is ca. 15 min/sample, including ca. 9 min for eluting off bound theophylline and theo-PE. This assay is therefore suitable for clinical and environmental analyses.
5
>
4
t fn z w I-
z w3
u 2
w
s W
22 U w
E
4UI R
1 I
0 -11
-10
-9 -8 -7 log [THEOPHYLLINE]
-6 (M)
-5
-4
Flgure 1. Results of FIIA: relathre Ruonwcence htenslty of liposomes as a function of varying concentrations of analyte, theophylline. Concentrations refer to the concentratlons of theophylline in the vold volumeofthecokmn. Cuvesland2aretheresultsofsingle-inand double-injection protocols, respecthrely.
FIIA System Result. Figure 1 shows competitive immunoassay results for theophylline quantitation collected with the FIIA system. The graph indicates the decrease in relative fluorescence signal from liposomes that were bound to immobilized antibody in the presence of increasing concentrationsof theophylline. Curve 1 is the result of single injections of theophylline sample plus liposomes onto the immunoreactor column. Curve 2 is the response from the double-injection protocol. The use of double injections increased signal response by about a factor of 2.
THEORETICAL MODEL Competitive Binding of Theophylline and Liposome to Anti-Theophylline Immobilized on a Glass Bead. Let us consider the general case of binding of two types of ligands at an immobilized IgG antibody molecule with two antigen binding sites, and we will assume that the two sites are equivalent. The antibody interacts with either type of the ligands, theophylline and liposome, and may exist in any of the following states: no bound ligands, one bound theophylline, two bound theophyllines, one bound liposome and one bound theophylline, one bound liposome, and two bound liposomes. These states are schematically shown as: Y, Yo, O Y O , O Y O , Yo,and OYO,respectively. If the binding of the ligands to the two sites were equivalent and independent, the probabilities of occurrence of each of the respective possible states would assume the following proportions: 1:2kxl: (krl)2:2kr1K’xz:2K’x2:(K’n2)2; where k denotes the binding constant of theophylline and K’the apparent binding constant of the liposome to the immobilized anti-theophylline,x denotes the concentration of free ligands, and the subscripts 1 and 2 indicate theophylline and liposome, respectively. Since there is a size difference of approximately 10-foldbetween the antibody and liposome (the distance between the two binding sites of an antibody is ca. 12 nm (5) and the liposome is ca. 135 nm in diameter (I)), significant steric effects are expected on the second antibody binding site once the first site is occupied by a liposome. Therefore, the relative probabilities for’ xone ~ theophylline and one were modified to 2 ~ ‘ ~ k x ~ K
ANALYTICAL CHEMISTRY, VOL. 63,NO. 18, SEPTEMBER 15, 1991
liposome bound and a'2(K'x2I2for two liposomes bound; here a1' and a 5 are parameters relating to the interaction energies between a site occupied by a liposome and a neighboring site occupied by theophylline or another liposome, respectively. Also, there is the probability of a multivalent liposome occupying both of the sites of an antibody, giving P, with a relative probability of a$ compared to that of singly attached species (cf. refs 6 and 7). Assuming that the bindings at the n antibodies on the bead are independent, then we can write the partition function Z as
w 1.0
0
\ LN
d C 0
a
E" 0)
+-
where for convenience we have defined: as = ( 2 + a j ) / 2 ,K = K'a3, al = ai/a3,and a2 = a$/a2%Then the average number of bound theophylline molecules per bead is
.-+
In Z / d In x1 = 2n(kxl + (kx1I2+ alkzlKx2)/z (1')
and the average number of bound liposomes per bead is u i = d In Z/d In x 2 = 2n(Kx2 + alkx,Kx2 + a2(Kx2)2)/z(2') These values can be converted into concentrations of bound theophylline and liposome by multiplying both sides of eqs 1' and 2' by the concentration of beads in the reactor column (number of beadslliquid void volume) and dividing by Avgadro's number to obtain
0.6
0
a2(Kx2)2)n = zn
u< = d
0.8
3
.--
z = (1+ 2kXl + ( k X J 2 + 2 K x 2 + 2UlkX1KX2+
2009
0.4
0
0'
0.2
0
E?
Lc
0.0 - 2 - 1
0
1
2
3
4
log (kc,) Figure 2. Uposome binding as a function of log (kc 1) for various values of a,. For all curves, Kc, = 5, Kc3 = 10, kc, = 0.056, and a 1 = 2. For the solM curves top to bottom, a , = 2, 1, and 0.5. For the dashed curve, a 2 = 0.
They are solved numerically for y 1and y2,given P, Q, R, and r, with the obvious physical constraints that 0 < y 1 < 1 and 0 < y 2 < 1. The fraction of liposome bound, u2/c2 = 1- y2, which is proportional to the experimentally measured fluorescence signal, is determined as a function of P, which 2ca(kx1 (kx1I2+ alkxlKx2) is proportional to the total analyte concentrate cl. u, = 1+ 2kx1 + (kx1I2+ 2Kx2 + 2alkxlKx2 + a2(Kx2I2 Let us consider first the steric effect of binding two liposomes to the same antibody. Figure 2 shows representative (1) plots of the fraction of bound liposomes as functions of total concentrations of theophylline for various values of a2,for the "2 1 2kxl ( k ~ 1+) 2~ K ~ 2 2alkxlK~z u ~ ( K x ~ ) same ~ fixed values of the other parameters and variables as indicated in the f i i e caption. As the value of a2 decreases, (2) the curves asymptotically approach the case for a2 = 0, where where c3 is the moles of immobilized antibody per liter and no two liposomes could simultaneously occupy the two binding u1 and u2 are moles per liter of bound theophylline molecules sites of the same antibody. The values for the fixed paramand liposomes, respectively. eters and variables used in generating the curves are chosen We also have the following conservation equations for to magnify the differences between the curves for illustrative theophylline and liposomes: purposes, and they are different from those we used in practice, which generate curves with very small differences c1 = x1 + u1 (3) in u2/c2 for various a2 such that the curves are hardly disc2 = x2 + u2 (4) tinguishable one from the others. From considerations of the geometry of immobilized antibody and the large relative diwhere c denotes total concentration, Le., free plus bound. mension of our liposome compared to the dimension between These four equations, eqs 1-4, determine the values of the the two binding sites on the same antibody, we expect that four unknowns: x l , x2, u,, and u2. The theoretical curves of our system will have a large steric hindrance effect and will the fraction of bound liposomes as functions of total analyte be described by the asymptotic case of having a2 = 0. In our concentration are determined by the thermodynamic parampresent immunoassay system, the effect of a2is relatively small eters k,K , a,, and a2and by the experimentally fiied variables and is unlikely to be detected experimentally under the c2 and cg. For the purpose of model calculations, we define conditions we currently employed. the following dimensionless parameters and variables: P = We shall now briefly consider the effect of the value of a, kxl, Q = Kx2,R = k / K , r = 2c3/c2,yl = xl/cl, and y 2 = x2/c2. on the characteristics of the binding curves. Figure 3 shows Substituting eqs 3 and 4 for u1 and 0 2 into the left-hand side u2/c2, the fraction of liposomes bound, as functions of log (kcl) of eqs 1 and 2, respectively; then dividing the resulting for several values of al, with the other parameters assigned equations through by c1 and c2, respectively, we obtain the fued values consistent with experimentalconditions. Curves following two equations for the two unknowns y1 and y2: a and b illustrate cases where the binding of a theophylline molecule on one site and a liposome on the other site resulted 1- rQNyi + Pyi2 + a i Q ~ l ~ 2 l in a stabilization energy of R T In a, = 1.37 (a, = 10) and 0.95 y 1 = 1 + 2Py1 Py12+ 2Qy2 + 2alPQyly2 + a2Q2y22 kcal/mol (al = 5 ) , respectively, compared to the case where (5) the binding of theophylline on one site and liposome on the other are independent of each other, i.e., the dashed curve c. Curves d and e illustrate the opposite cases where binding of 1 2Py, P2y12+ 2Qy2 + 2alPQy~y2+ a2Q2yz2 theophylline and liposome to the same antibody resulted in (6) a destabilization of 4.95 (a, = 0.2) and -1.37 kcal/mol (a,
+
+
+
+
~
+
+
+
+
2010
ANALYTICAL CHEMISTRY, VOL. 63, NO. 18, SEPTEMBER 15, 1991 1 .o 1.5
\ 0
1.0
\
2 0.5
0.0 -3 -2
0
-1
2
1
3
Flew 9. Uposome bhdhgas a furctbn of log (kc,) forverkusvakres of a , . For all curves, Kc, = 0.16, Kc, = 2, kc, = 0.0112, and a 2 = 0.001. For curves 8-8: a , = 10, 5, 1, 0.2, and 0.1, respectively.
,
-4
1
-3
-2
-1
0
log (kc,/Kc,a,
2
1
-1
0
1
2
3
log (kc 1)
W ( k c 1)
1.0
-2
3
1
Figure 4. Pbts of v 2 / c 2versus log ( k c l / K c @ , ) for various combinations of parameters. For all solid curves, k = 0.005 n M 1 . For all dashed curves, k = 0.5 n M ' . For all curves, c2 = 0.016 nM, c, = 0.25 nM, and a , = 0. Values of ( K , 8 ,) for the four paks of sdid and dashed curves from top to bottom are (20, 5), (20, l ) , (2, 5), and (2, 1).
= O.l), respectively. As expected, the curve shifts to the right as a1 increases, reducing the concentration sensitivity of theophylline binding response. For a1 > 1, i.e., positive stabilization energy, the curve shows a maximum in the neighborhood of kc, = 1, which we observed experimentally, as in Figure 1. As c1 approaches 0, Le., as In (kc,) becomes very negative, all of the curves asymptoticallyapproach a constant value of u2/cz, which is equal to (1+ 2K(cz + c3) - ((1+ 2K(cs+ ~ K C ~ ) ~ / ~for ) / cases ~ K Cwhere ~ , a2 is negligible. As c1 becomes very large, uz/cz asymptotically approaches 1/(1+ kcl/2Kc3al). Therefore plotting u2/c2versus log (kcl/Kc3al) will bring the various curves into coincidence at large cl, as illustrated in Figure 4. Comparing the solid curves in this
Figure 5. Plots of v , / c 2 as a function of log (kc,)for multiple Injections. Bottom curve is for the single injection; top curve Is for the doubblnin)ectlonprotocol. K = 10 n M ' , k = 0.056 n M ' , c2 = 0.016 nM, c3 = 0.25 nM, a , = 1, and a , = 0.0001. figure with their corresponding dashed curves where the value of k was increased 100-foldover that of the solid curves, shows that k has a relatively minor effect on the characteristics of the curves compared to the effects exerted by the other parameters. These provide convenient working curves for the system and also suggest a graphical means of estimating from experiments some of the thermodynamic parameters of the system as follows. Superimpose an experimentally measured u2/c2 vs log c1 plot on a set of these theoretical curves until a best fit is obtained. The horizontal shift of the origin of the experimental plot from that of the best fitted theoretical curve is simply log (k/Kc3al),and the asymptotic value of uz/c2 at small c1 is approximately (1+ 2K(c2+ c3) - ((1+ 2Kk3 - cJI2 + ~ K C ~ ) ' / ~ ) /these ~ K Ctwo ~ ; quantities, plus a rough estimate of a1 from the shape of the best fit curve, give an estimate of k and K. Multiple Injections in FIIA. The double-injection mode of operation was examined as a means of enhancing the response signal from the FIIA system. In this mode, theophylline sample plus liposomes are injected onto the column, unbound materials are removed by the flowing stream, and before the regeneration step, another injection of theophylline plus liposomes is performed. Since the off-rate of the multivalent liposomes is small (I), we assume that the total liposome concentration in the reactor column after the second injection is essentially equal to the sum of the liposome concentration in the sample plus the concentration of bound liposomes retained in the reactor column from the first injection. After the second injection, u2 is calculated as above, using this larger total liposome concentration. Figure 5 shows curves calculated for double injections of samples of varying concentrations of theophylline. In Figure 5, which was calculated with values of the parameters that are similar to the experimental values, the maximum u2 for double injection is almost twice that for single injection; these results are qualitatively observed in our experiments, aa seen by comparing curves 1 and 2 of Figure 1.
DISCUSSION AND CONCLUSIONS The overall goal in our developing this model was to predict conditions that would allow us to optimize the concentrationdependent response of FIIA. This model provides a useful guide in selecting suitable assay conditions for improving limit of detection and sensitivity, although improvements of pa-
ANALYTICAL CHEMISTRY, VOL. 63, NO. 18, SEPTEMBER 15, 1991
rameters are frequently accompanied by tradeoffs as well. For example, to measure at a lower range of theophylline concentrations by this system, one would like to work with a large k / K ratio and small c3, as described previously (Figure 4). While k is more or less constant, the apparent constant K can be varied. An important advantage of liposomes is the ability to alter their binding constant by altering the concentration of analyte in their membranes. Liposomes with large binding constants shift the range of analyte concentration where the greatest change of signal (proportional to u2) lies to higher analyte concentrations, which can be disadvantageous where analytical sensitivity at a lower limit of detection is required. On the other hand, large binding constants are often accompanied by small dissociation rate constants, which are an advantage for signal enhancement using the double-injection technique. Sensitivity at low analyte concentrations can also be achieved by keeping c3, the concentration of antibody, small. On the other hand, while decreasing c3 moves the response curve to a lower concentration of analyte, i.e. lower detection limit, it also tends to decrease the range of uz/cz attainable, thus decreasing the slope of the curve thereby diminishing the sensitivity. In our present FIIA system, the value of c3 used is such that the average separation between two immobilized antibodies is ca. twice the diameter of the liposomes. As c3 increases, the antibody density on the glass bead surface increases and the separation between antibodies decreases. And at high enough antibody surface density, binding of a liposome at an antibody would sterically interfere with bindings at the neighboring antibodies; then interactions between neighboring antibodies cannot be neglected in writing the partition function for the system and the binding equations, given by eqs l and 2 for the independent case, become more complex (6. refs 8-10). Altering the size of the liposomes obviously will also affect the steric effects at the binding sites. Liposome-based homogeneous immunoassays have been developed previously (e.g. see ref ll) and model calculations for these have been made on the basis of the direct binding between the antigen-derivatized liposomes and the antibody in
2011
solution (12). The heterogeneous FIIA system we developed involved the binding of both liposome and antigen to antibody immobilized on glass beads, and the model analyses for this system presented here focus on the competitive binding of antigen and relatively bulky multivalent liposome to immobilized bivalent antibody with steric effects. In applying this equilibrium model to the flow injection system, the residence time of the samples should be long enough so that equilibrium is approached. Although this has yet to be systematically investigated for our system, the fair agreement between the preliminary values of apparent dissociation constants determined from batch experiments and those estimated from the comparison of FIIA data with the model seems to suggest the adequacy of this model in describing our FIIA system. More careful experiments and further refinements of the model will allow a more rigorous treatment of data from this system. The simple model presented herein, notwithstanding all the simplifying assumptions, makes it possible for us to estimate assay sensitivity for given experimental conditions and, conversely, to choose conditions that result in better sensitivity.
LITERATURE CITED (1) Locescio-Brown. L.; Plant, A. L.; Horvath, V.; Durst, R. A. Anal.
chem.1990, 62. 2587-2593. (2) Wu, T. K.; Durst, R. A. "&h.Acta 1990, 7 , 187-195. (3) Plant, A. L.: Brkgys, M. V.; LocasbBrown, L.; Durst, R. A. Anal. Bochem. 1969, 176, 420-426. (4) Plant, A. L.: Locascio-Brown, L.; Durst, R. A,: Haller, W. Appl. Bk&em. Bbtechnd. 1991, 30, 83-98. (5) Cathiu, R. E. In Immunogkbws; Qood, R. A., Day, S. E.. E&.; Plenum Medical Book Co.: New York, 1978; p 43. (6) Crothers. D. M.: Metzger, H. I m " . 1972, Q, 341-357. (7) Kanmh, F. In Immuncgbbulhs; Good, R. A,, Day, S. E., Eds.; Plenum Msdlcel Book, Co.: New York, 1978: Chapter 3. (8) Stankowski. S. Biochlm. Bkphvs. Acta 1964, 777. 167-182. (9) Tamm, L. K.: Bartoidus, I.Bkxhmkby 1968, 27, 7453-7458. (10) Hago, P. J.: Wlmor, D. J. Arch. Bkchem. skphvs. 1867, 254, 92-101. (11) Lltchfieid, W. J.; Freytag, J. W.; Adamich. M. CHn. Chem. 1964, 30, 144 1- 1445. (12) Chang. E. L.: Wake, E. A. J . Immunol. Me1987. 702. 33-43.
RIEE~VED for review February 21,1991. Accepted July 1,1991,
