Binding units (BU) and the area under binding isotherms (AUI) new indices of effector-target conjugation

JOURNALOF IMMUNOLOGICAL METHODS ELSEVIER

Journal of Immunological Methods 170 (1994) 197-210

Binding units (BU) and the area under binding isotherms (AUI) New indices of effector-target conjugation Jesus Galvez a, Lourdes Cabrera b, Francisco Lajarin b, Pilar Garcia-Pefiarrubia ,,b a Laboratory of Physical Chemistry, Faculty of Science, 30100 Espinardo, Murcia, Spain, b Department of Biochemistry and Molecular Biology B and Immunology, School of Medicine, 30100Espinardo, Murcia, Spain (Received 12 July 1993; revised received 4 November 1993; accepted 15 December 1993)

Abstract New methods for simplified quantitation of effector-target conjugation have been developed. The binding unit (BU) is defined as the number of target cells required to bind a specified percentage of effector cells. The number of binding units is determined from binding isotherms in which effector conjugate frequencies are measured by holding constant the number of effector cells and by varying the number of target cells. Alternately, a binding unit can be defined as the number of effector cells required to bind a specified percentage of target cells. In this case, BU is computed from binding isotherms in which target conjugate frequencies are measured at different values of effector cells by holding constant the number of target cells. Also, the area under the curve (AUI) of these isotherms is another index that can be used as an overall measure of the binding capacity in an effector-target system. The experimental values of BU and AUI determined from effector and target isotherms agree well with theoretical predictions based on our previously developed binding model (J. Immunol. Methods (1992) 155, 133-147). The relationship between BU and AUI, and procedures to determine these parameters are shown. The value of these indices to express effector-target conjugation quantitatively has been confirmed by determining the values of BU and AUI for the NK-K562 effector-target system. Key words: Effector-target interaction; Binding unit; Area under the isotherm

I. Introduction * Corresponding author. This work was supported by grants from the DGICYT, project number PM 90-0042. Abbreviations: NK, natural killer cells; a, frequency of effector conjugation; amax, maximum value for the effector conjugate frequency; fl, frequency of target conjugation; flmax, maximum value for the target conjugate frequency; K o, dissociation constant; BU, number of binding units; AUI, area under the binding isotherms.

The first step in cell-mediated cytotoxicity is effector-target interactions that lead to the formation of conjugates between effector and target cells (Martz, 1977; Berke et al., 1975; Zagury et al., 1975; Bonavida et al., 1983). However, despite extensive research over the past years, the precise mechanisms by which effector cells recognize,

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J. Galvez et al. /Journal of lmmunological Methods 170 (1994) 197-210

bind and eventually lyse specific target cells are only partially understood (Young et al., 1988; Trinchieri, 1989). One of the major problems for the understanding of the binding process is the lack of a standardized method of analyzing the data to obtain a measure of the binding capacity in an effector-target system that can be used to compare results from one experiment to another and from different laboratories around the world. This also explains why in the literature large differences in conjugate frequencies have been reported for the same effector-target system (Berke, 1985; Bonavida et al., 1983; GarciaPefiarrubia and Bankhurst, 1989a; GarciaPefiarrubia and Galvez, 1993; Garcia-Pefiarrubia et al., 1989d). Recently we have developed a binding model that considers most of the experimental observations of effector-target interactions at the population level (Garcia-Pefiarrubia et al., 1992; Garcia-Pefiarrubia and Galvez, 1993). According to this model, the binding process is characterized by three parameters, the maximum effector (o/max) and target (flmax) conjugate frequencies, and the dissociation constant (K o) of the conjugates formed. These three parameters provide a complete description of the conjugation process, and they are determined from binding isotherms by applying procedures recently described (Garcia-Pefiarrubia and Bankhurst, 1989a; Garcia-Pefiarrubia et al., 1989d, 1992; GarciaPefiarrubia and Galvez, 1993). However, for routine assays it is more appropriate to have a single index that permits estimations of the binding efficiency more easily than by the rigorous procedure. In this paper we introduce two single parameters, the binding unit (BU), and the area under the binding isotherms (AUI), for the evaluation of the binding capacity in an effector-target system. Theoretical expressions for BU and AUI derived from our binding model reveal that these parameters provide a suitable measure of overall binding efficiency. Furthermore, experiments designed to assess the conditions for the binding assays under which BU and AUI are most sensitive are also described. The value of these indices has been tested by determining the binding capacity of the NK-K562 effector-target system.

2. Theoretical considerations

2.1. Binding assays Effector-target conjugation can be successfully analyzed by considering that effector (L) and target (T) interactions occur through a reversible bimolecular reaction described by the equation kl

L+T.

k-i

"LT

where LT are the conjugates formed and k I and k_ 1 represent the forward and reverse rate constants (Garcia-Pefiarrubia and Bankhurst, 1989a; Garcia-Pefiarrubia et al., 1989d, 1992). The dissociation binding constant, Ko, is related to the effector-target affinity and is defined by K o = k_ i/k~. Hence, effector-target systems with large affinities are associated with small values of K D. This model assumes that all receptors in the system (i.e., those located on both the effector and target cells) are equivalent and that all receptors bind independently of one another. Furthermore, we must consider saturability of effectortarget interactions, i.e., the fact that for a given number of effector and target cells only a finite number of specific receptor sites exists. With these assumptions, and taking into account that [LT] << Nmax + Tmax + K o, where Nmax and Tmax are the maximum number of specific receptor sites for effector and target cells (see GarciaPefiarrubia et al. (1992) for a detailed discussion), the following expression (showing the dependence of the effector conjugate frequency, a, on the effector-to-target ratio, R, when the binding assays are perfomed by holding constant the number of effector cells) was found O/

O/max

(1)

1 +yR

where a

KD

y=-~+b;a=[3m-'---~;b

O/max

2/3max

(2)

In Eqs. (1) and (2) R = N / T , N and T are the numbers of effector and target cells, and O/max and /3ma x are the maximum effector and target conjugate frequencies that appear as a conse-

J. Galvez et al. /Journal of Immunological Methods 170 (1994) 197-210

quence of saturability (Garcia-Pefiarrubia et al., 1992; Garcia-Pefiarrubia and Galvez, 1993). By following analogous considerations to those described for effector populations we find for target populations that the corresponding equation for the dependence of the target conjugate frequency, /3, on R when binding assays are performed holding the number of target cells constant, is (Garcia-Pefiarrubia et al., 1992; Garcia-Pefiarrubia and Galvez, 1993): flmax

/3 ---

(3)

~

I+-R

199

/,00°7

4O

o

/30oo0o /,ooo '

3~ 1 0 0 0 0 0 R 5

where C

KD

T

O~max

= -- + d ; c = - - ;

/3max

d= - 2O~max

(4)

In practice, most data about effector-target conjugation are reported at a given value of the effector-to-target ratio. However, the R value, and the corresponding numbers of effector and target cells are not standardized. Thus, values of R ranging from 0.1 to 5, and numbers of effector cells from 5 × 104 to 107 are frequent in the binding assays described in the literature. Therefore, even for a fixed value of R, data for effector-target conjugation are not normally reported at the same value of N (discussed in GarciaPefiarrubia et al., 1989d). This complicates comparisons of binding data since in binding assays in which the value of R is fixed, large differences in conjugate frequencies are obtained as N changes (Garcia-Pefiarrubia et al., 1989d, 1992). This fact is illustrated in Fig. 1 in which Eq. (1) has been used to plot effector conjugate frequency vs. R at different values of N. This plot is a surface in three-dimensional space with two dose variables (ratio and effector cells) and one response variable (the a values), and shows that the effector conjugate frequency is a function of both the ratio R and the number of effector cells N. Analogous plots are obtained if we plot the target conjugate frequency vs. R at different values of T. This demonstrates clearly that conjugate frequencies are not independent of an arbitrarily chosen fixed ratio R and, therefore, conventional

Fig. 1. Theoretical dependence of the effector conjugate frequency (~) on the effector-to-target ratio (R) and the number of effector cells ( N ) computed from Eq. (1). a values were calculated by using the following parameters: Otmax = 60t~, flmax = 50%, K D = 0.5 >( 105 cells/tube.

procedures to quantify effector-target conjugation based on these kinds of measurements are not appropriate.

2.2. Quantitation of effector-target conjugation The problem of quantifying binding efficiency has two principles of solution that can be implemented by following two different approaches. (1) A rigorous approach based on determining the three binding parameters, i.e., the two maximum conjugate frequencies, O~max and /3max, and the dissociation constant, KD, of the conjugates formed. These three parameters characterize the binding properties of an effector-target system and can be determined from binding isotherms by following procedures recently described (GarciaPefiarrubia et al., 1992). (2) However, for routine assays it is more practical to have a single index that allows us to estimate the overall binding capacity by using faster and easier procedures than those required in the previous approach. In this work we describe two single indices that can be used as a measure of overall binding efficiency. These indices will be illustrated by considering the two binding isotherms shown in Fig. 2 which have been computed from Eq. (1) for two effector-target systems with the same values of

200

J. Galvez et al. /Journal of Immunological Methods 170 (1994) 197-210 50 8

4O

30

Or%

A

i

i

i

i

20

,o/ 1/R Fig. 2. Effector binding isotherms computed from Eq. (1) showing the dependence of the effector conjugate frequency on the reciprocal of the effector-to-target ratio. N = 105 cell/tube, K D = 0.5 × 105 cells/tube, flma~ = 50%. A: areax = 40%; B: atmax = 60%.

K D and flmax but with different values of O/max (40% curve A, and 60% curve B). These isotherms represent effector conjugate frequencies determined as a function of the number of target ceils, and plotted as O/ vs. the values of the T : N ( = l / R ) ratio, where the number of effector ceils, N, is constant.

and 0.8 for curve B; and (2) multiplying this value of T / N by the number of effector cells (105) which directly gives the number of target cells required to bind 20% of the effectors. This shows that for isotherm A 1.4 × 105 targets are required to bind 20% of the effectors and thus, one binding unit consists of 1.4 × 105 target cells. Proceeding analogously we find that for isotherm B one binding unit consists of 8 × 104 target cells. Binding data can be reported as the number of binding units contained in a specified number, for example 107, of target cells. Thus, for isotherm A we find 107/(1.4 x 105) = 71.4 binding units/107 target cells, and for isotherm B we have 125 binding units/107 target cells. As expected, there is an increase of (125-71.4)/71.4---75% in the number of binding units for curve B because this isotherm relates to the effector-target system with a higher value of o/max"These computations can be summarized as follows n u m b e r of binding units ( B U ) / 1 0 7 targets

10 7 X Rp N

(5)

where N is the constant number of effector cells in the binding assays (105 in Fig. 2), p is the reference binding level (20% in the above computations), and Rp is the effector-to-target ratio required to bind p% of the effectors. Eq. (5) shows that there is a simple relationship between the dose-response curve, the corresponding value of Rp, and the binding unit, so that BU can be easily determined from experimental isotherms.

2.4. Derivation of a theoretical expression for BU 2.3. Binding units The first index is the binding unit that, by analogy with the well known lytic unit, is defined as the number of target cells required to bind a specified percentage of effector cells. Thus, for example, if in the binding assays shown in Fig. 2 we assume that the reference binding level, O/p, is 20%, and that the number of effector cells is 105, the size of a binding unit associated with these isotherms is obtained by: (1) determining the 1/Rp value (i.e., the T : N ratio) that causes 20% binding. From Fig. 2, this ratio is 1.4 for curve A

From Eq. (1) the value of Rp at which O/= ap is given by gp =

O/max -- O/p

(6)

TO/p

By inserting this value in Eq. (5) we have BU/IO 7 targets - 1 0 7 ( a m ~ - o/p )

yN ~

at,

(7)

J. Galvez et al. /Journal of lmmunological Methods 170 (1994) 197-210

This expression allows us to determine analytically the number of binding units if amax and 3' are known. From Eq. (2) it follows that in order to compute 3" the three binding parameters (o/max, /3ma x and K D) are necessary. Hence, BU depends also on the three binding parameters (through 3'), and this justifies the use of BU as an overall measure of binding efficiency. The binding isotherms shown in Fig. 2 were obtained using the following parameters: curves A and B, K D = 0.5 × 105 cells/tube, flmax = 50%; curve A, O/max = 4 0 % ; c u r v e B , O/max = 6 0 % . Thus, from Eq. (2) the 3' values for these curves are, respectively, 1.4 and 1.6. By introducing the values of O/m~x, Y, N = 105 c e l l s / t u b e and O/p = 20%, in Eq. (7) we find that BU is 74.1 binding units/107 targets for curve A and 125 for curve B in agreement with the results previously obtained from Eq. (5). Because in this case the dose-response curves in Fig. 2 have been plotted using data points computed from Eq. (1), an exact agreement between Eqs. (5) and (6) exists. When BU is determined from experimental binding isotherms some deviations between these equations should be expected. 2.5. Area under the isotherms The second index to quantify effector-target interactions is the area under the dose-response curves. Thus, in Fig. 2 curve B shows higher values of effector-conjugate frequencies at any value of the T : N ratio than curve A and, therefore, the area under curve B is also larger than the area under curve A. This suggests that these areas can also be used as an overall measure of binding capacity. The AUI values for a given isotherm are calculated by integration of the effector dose-response curves over a range [0, x ] of the values of the T : N ratio. In order to use AUI as an overall parameter of binding capacity, measurements of effector-conjugate frequencies as a function of the T : N ratio in the interval [0, x] must provide values of effector conjugate frequencies ranging from 0 to the maximal binding values that can be normally reached in binding assays (for normal effector populations these values of a range from 0 to approximately 80% of amax). Under these conditions the AUI values can

201

be obtained using the trapezium method that is an approximate procedure of integration based on adding together the areas of successive trapezia formed by adjacent data points on the curve. In formulae AUI(O, x ) x =

_

,

n [al + a2 + "'" +o/n-1 + "~(o/o x n [o/l +o/2 + . . . + o / n _ l + ~o/n]

(S)

since a 0, the effector-conjugate frequency at 1 / R = 0, is zero. In Eq. (8) n represents the number of data points in the interval [0, x], and O/j ( j = 1, 2 . . . . . n) are the values of the effector conjugate frequencies that must be chosen so that [0, x] is subdivided into n regions of identical amplitude. To illustrate the use of Eq. (8) we shall consider isotherm A in Fig. 2. Set x = 5 and the number of data points n = 5. Hence, conjugate frequencies must be determined at 1 / R = 1, 2, 3, 4 and 5. From Fig. 2 we find that these values of O/ are 16.7, 23.5, 27.3, 29.7 and 31.3, respectively. By introducing these values of x, n, and a in Eq. (8) we obtain A U I ( 0 , 5 ) = 112.8. By proceeding analogously, it follows that the corresponding value for isotherm B is 161.2. If the number of data points is larger, for example n = 10, then AUI(0,5) values for isotherms A and B are 114.3 and 163.3. The exact values for these isotherms are 114.9 and 164 (see below). Note that five data points are normally enough to obtain values of AUI within 2% of the correct ones, so that a double increase in the number of data points does not improve significantly the corresponding deviations (typical deviations for experimental a values may be within + 5%). As previously for the number of binding units, AUI for curve A is also smaller than for curve B, and this is in agreement with the larger effector-target conjugation associated to isotherm B. 2.6. Derivation o f a theoretical expression for A U I From Eq. (1) the area under effector binding isotherms in the interval [0, x] where z = 1 / R is

202

Z Galvez et al. / J o u r n a l o f Immunological Methods 170 (1994) 1 9 7 - 2 1 0

given by x

Z

AUI(0, x) = £ am~' -3'- d+ zz

= am~[X - 3" In 3`+x ]3`

(9)

Using this equation, AUI can be determined analytically if amax and 3` arc known. This situation is identical to that described in the above section for BU, i.e., AUI depends also on the three binding parameters (through 3') associated to the binding isotherms, and this justifies the use of AUI values as an index for overall binding efficiency. By introducing in Eq. (9) the values of am~x and 3" given in previous sections for the isotherms shown in Fig. 2, we find that the AUI values are AUI(0,5) = 114.9 for curve B and 164 for curve A.

2. 7. BU and A U I for target cell populations Effector binding isotherms are obtained by measuring effector conjugate frequencies in binding assays in which the number of effector cells is constant and the number of target ceils is increased until values of a near to saturation of the effector population are reached (see Fig. 2). Analogously, binding isotherms for target populations are obtained by holding constant the number of target cells and by increasing the number of effector cells so that values of target conjugate frequencies near to saturation are reached. Target conjugate frequencies are then plotted vs. the effector-to-target ratios used in the binding assays (Garcia Pefiarrubia et al., 1992). For these kinds of isotherms overall binding indices similar to those described for effector isotherms can also be given. Thus, the binding unit for target populations is defined as the number of effector cells required to bind a specified percentage of target ceils. Furthermore, if BU is reported as the number of binding units contained in 10 7 effector cells, the expression corresponding to Eq. (5) is number of binding units (BU)/107 effectors 107 -

-

-

TX

gp

(10)

where T is the constant number of target cells in the binding assays, p is the reference binding level, and Rp is the effector-to-target ratio required to bind p% of the targets. As for effector populations, Eq. (10) shows that there is also a simple relationship between the dose-response curve, the corresponding value of Rp, and the binding unit for target populations. By combining Eqs. (3) and (10) an analytical expression for BU is obtained 107( ~max- tip ) BU/10 7 effectors = ~ /3p

(11)

This expression shows that BU depends on the three binding parameters through /3m,x and (see Eq. (4)), and this provides a justification for the use of this index as an overall measure of binding capacity for target populations. The area under these isotherms also provides a suitable index for binding efficiency. This index is obtained by proceeding as in previous sections and so, we find AUI(0, x) X ~--- h i / 3 1

'f-/32 "q- "-" q - / 3 n - 1 "q- 1 ( / 3 0 - ] - / 3 n ) ]

X n [ / 3 1 + /3 2 + . . . + /3 n _ l + -ff/3 . ]

(12)

since the value of /3 at R = 0, /3o = 0. In this expression x represents the maximum value of R in the isotherm, n is the number of data points, and /3j (j = 1, 2 . . . . . n) are the target conjugate frequencies that must be determined at intervals of equal amplitude in the range [0, x]. From Eq. (3) the corresponding analytical solution is tX

X

AUI(0, x) = )o/3ma×~-+--xxdx =/3m~x[X - 6 In

(13)

This equation demonstrates that AUI can also be used as a global index for capacity of conjugation because through 6 is a function of the three binding parameters that characterize the corresponding isotherms.

J. Galvez et al. /Journal of Immunological Methods 170 (1994) 197-210 3. Material and methods

3.1. Media and reagents

Effector cells were prepared in RPMI 1640, (Flow, Irvine, Scotland) supplemented with 10% FCS (Flow), and 2 mM L-glutamine (Flow) and penicillin-streptomycin (10 4 U / m l ) (Flow). Fluorescein isothiocyanate (FITC)-conjugated and unconjugated monoclonal antibodies Leu-2, Leu-3, Leu-4, Leu-ll, and Leu-12 (1 mg/ml) were purchased from Becton-Dickinson (Mountain View, CA), and 10 /~1 aliquots were used in a 106 cell/ml suspension. Lymphoprep was purchased from (Nicomed, Oslo, Norway) and magnetic beads coated with goat anti-mouse IgG were obtained from Dynabeads (Dynal, Oslo, Norway). 3.2. Preparation of effector cells

Effector cells were enriched by negative selection either by panning or by immune magnetic separation (Garcia-Pefiarrubia et al., 1992). Briefly, heparinized blood obtained from healthy donors aged 20-40 years, was used as a source of NK cells. To prepare monocyte-depleted cells (MDC), peripheral blood mononuclear cells were separated by Lymphoprep gradient centrifugation, followed by successive adherence for 60 min to a glass petri plate, and then passed through a nylon wool column. In several experiments, T cells were eliminated by panning on plastic dishes. Thus, polystyrene petri dishes were coated with goat anti-mouse IgG (Sigma Chemical Co., St Louis, MO). MDC pretreated with anti-CD3 (Leu-4), anti-CD4 (Leu-3), and anti-CD8 (Leu-2) mAbs, were added to the dishes and incubated at 4°C for 1 h. The nonadherent cells were then gently poured off. The resulting populations were higher than 75% CD16 as assessed by cytofluorometric analysis by FACScan (Becton Dickinson, Mountain View, CA). In other experiments, magnetic beads coated with goat anti-mouse IgG, were used for the purification of mAb-pretreated NK cells, after lysis by rabbit complement (37°C, 1 h). The cells were incubated with magnetic beads (1 cell:10 magnetic beads) for 45 min at room temperature. A magnet was used to sepa-

203

rate beads with attached CD3 +, CD4 +, and CD8 + T cells from antibody-negative NK cells. This technique provides a percentage of CD16 ÷ cells > 85% as assessed by FACScan. 3.3. Target cells

K562 cells were maintained in vitro in RPMI 1640, supplemented with 10% fetal calf serum, 1% glutamate and 1% penicillin-streptomycin. Prior to use these cells were washed twice in medium at room temperature. Viability was assessed by trypan blue dye exclusion (Gibco) and was higher than 95%. 3.4. Binding assay

The standard procedures used (Berke et al., 1975; Bonavida et al., 1983; Glasebrook, 1978; Bradley and Bonavida, 1984) have previously been shown t o provide optimum conditions (temperature, time and speed of centrifugation) to facilitate conjugate formation. Thus, equal volumes (100/zl) of effector cells and washed K562 cells in concentrations ranging from 105-107 cells/ml of media, were added to 10 x 75 round bottom plastic tubes. The suspensions were incubated for 20 min at 24°C and pelleted at 100 x g for 5 min. The pellets were gently resuspended 3-4 times with a micropipette, and kept on ice until counted. 50/zl of the cell suspension were then placed on a microscope slide, covered, and viewed under light microscopy. Since above room temperature the conjugate frequencies change with time due to lysis (Perez et al., 1985), it is important that the binding assays are not performed above 24°C. 3.5. Frequency of conjugation

The effector conjugate frequency (a) was determined manually by counting the total number of effector cells bound to K562 target cell under light microscopy. In each experiment more than 300 effector cells were counted (usually between 350 and 500 counted effector cells) and the resuits expressed as per cent of total number of effector cells. In parallel, the target conjugate frequency (/3) was determined by counting the

J. Galvezet al./Journal of ImmunologicalMethods170 (1994)197-210

204

total n u m b e r o f target cells b o u n d to effector cells. M e a s u r e m e n t s were p e r f o r m e d in three separate experiments and s t a n d a r d deviations were always smaller than 5%. Binding isotherms were obtained by determining effector and target conjugate frequencies at several values of R (normally the R values were 0.2, 0.5, 1, 2, and 5), and by plotting the corresponding values of a vs. 1/R (i.e., vs. the T : N ratio), or /3 vs. R. This interval of R values normally gave values of a and /3 over a range f r o m 0 to 80% o f O/max and /3max"

3.6. Data analysis B U were d e t e r m i n e d f r o m binding isotherms in which effector or target conjugate frequencies were m e a s u r e d at several T : N ( = l / R ) or N : T ( = R ) ratios ( d e p e n d i n g on w h e t h e r effector isotherms or target isotherms were obtained). Next the Rp value binding a specified p e r c e n t a g e p (e.g., 20%) of the effector or target cells was estimated. This was accomplished by fitting a dose-response curve (isotherm) to the observed data points (conjugate frequencies). T h e n a T : N ratio for effector binding isotherms or N : T ratio for target binding isotherms was d e t e r m i n e d such that the height of the curve at 1/Rp or a t Rp was just equal to the reference binding level p % (see Fig. 2). T h e c o r r e s p o n d i n g n u m b e r of binding

units for effector and target populations was obtained by introducing Rp in Eq. (5) or in Eq. (10). Theoretical B U values were also d e t e r m i n e d using the binding model summarized in Eqs. (1) or (3) to fit conjugate frequencies to the corresponding isotherms. Thus, for effector binding isotherms that can be adequately described using this model, Eq. (1) predicts that a plot of the reciprocal of the conjugate frequency vs. R is linear, so that O/max and y were obtained from the reciprocal of the Y intercept and slope of these plots using linear regression analysis (Garcia Pefiarrubia et al., 1992). With these values of O/max and y, B U were directly c o m p u t e d from Eq. (7). F o r target binding isotherms Eq. (3) shows that a plot of the reciprocal of the values o f / 3 vs. 1/R was linear so that/3m~x and 8 were obtained f r o m the reciprocal of the Y intercept and the slope of these plots (Garcia Pefiarrubia et al., 1992). B U for target populations were then computed by introducing these values o f /3max and in Eq. (11). A U I values were directly c o m p u t e d from Eqs. (8) or (12) after binding isotherms were fitted to the observed conjugate frequencies. To this end, values of a o r / 3 at intervals of equal amplitude were estimated from these isotherms. In our experiments five data points (n = 5) at 1/R or R = 1, 2, 3, 4, 5 in the interval [0,5] (x = 5) were used for computation.

Table 1 Values of BU and AUI obtained from effector binding isotherms in the NK-K562 tumor target cell Donor a N × 10 5 b 'Y c O/max(O/~) d KD × 10-5 B U / 1 0 7 targets e exp. f theor.

AUI(0,5) g exp.

h theor.

5

2.0

1.22

52.4

0.59

60.0

66.3

152.3

157.8

6

1.0

2.27

50.5

0.87

70.0

67.2

113.4

119.1

7 7 7 7

0.5 1.0 2.0 5.0

5.63 2.35 1.03 0.83

45.2 45.6 48.1 50.9

0.91 0.76 0.42 0.69

52.0 56.0 71.5 42.8

44.8 54.4 68.2 37.4

68.9 108.6 144.6 164.8

64.3 105.7 152.9 172.3

8 8

1.0 3.0

4.34 0.97

50.2 53.7

ND 0.41

31.0 69.0

34.8 58.1

79.4 158.1

84.0 174.1

a,d Values of N and K D are given in cells/tube. b,c Values of 3' and Otmax were obtained from the plots 1/a vs. R using linear regression analysis. e,g Experimental values obtained from the binding isotherms. f,h Theoretical values computed from Eqs. (7) and (9) respectively.

J. Galvez et al. /Journal of lmmunological Methods 170 (1994) 197-210

Theoretical AUI values for effector and target cells were also computed by introducing the values of O/max and 3' or /3r,~x and 8 (which were determined as described above) in Eq. (9) or in Eq. (13), respectively. In all cases, [0,5] was the interval chosen for computation.

B 40

4O

30

o<% 20

4. Results

/

4.1. NK-K562 effector-target system BU and AUI values were determined from binding isotherms that were obtained by perform-

310

f t

The NK-K562 effector-target system was used to test whether the values of BU and AUI provide a suitable measure for the overall capacity of effector-target conjugation. The fitness of our binding model to assess the behaviour of the binding isotherms was tested by comparing experimental BU and AUI values to those computed theoretically from Eqs. (7) and (9) for effector isotherms or from Eqs. (11) and (13) for target isotherms. These results are summarized in Tables 1 and 2.

205

1

i

2

20

~

;

i

s

S i

tl

i

i

i

1

2

$

4

5

R

Fig. 3. NK-K562 effector-target system. A: binding isotherms for donor No. 6 obtained by holding constant the n u m b e r of effector cells ( N = 105 cells/tube). Data are plotted as a vs. 1 / R . T h e T : N ratio producing 20% binding is 1.42. B: binding isotherms for donor No. 1 were obtained by maintaining a constant n u m b e r of target cells (T = 105 cells/tube). Data are plotted as /3 vs R. T h e N : T ratio producing 20% binding is 1.92.

ing binding assays at several values of R (Garcia Pefiarrubia et al., 1992). Eight different donors were used as a source of NK cells. Fig. 3A shows an example of these kinds of curves for donor No. 6. This isotherm was obtained by performing binding assays in which the number of effector

Table 2 Values of BU and A U I obtained from target binding isotherms in the NK-K562 tumor target cell Donor

a T X 10 - 5

b8

c/3max(%)

d KD X 10 -5

B U / 1 0 7 effectors

AUI(0,5)

e exp.

f theor.

g exp.

h theor.

1 1 1

0.5 1.0 2.0

2.99 2.50 1.61

46.3 45.7 46.7

0.60 0.97 1.18

88.9 52.1 41.2

88.0 51.4 41.5

96.5 106.5 126.0

95.4 103.0 127.3

2 2

0.5 1.0

3.12 1,92

42.8 45.4

0.61 0.81

80.0 58.4

73.1 66.1

88.3 116.9

86.3 115.2

3

1.0

2,69

46.3

1.04

45.5

48.9

101.1

100.7

4 4 4

1.0 2.0 4.0

0,68 0.66 0,60

40.9 45.0 43.0

0.17 0.27 0.45

146.3 88.2 42.9

153.7 94.7 47.9

138.2 158.7 161.2

145.8 161.2 157.4

5

1.0

1.93

39.2

0.97

42.2

49.7

91.5

99.3

6

1.0

1,48

54.6

0.54

107.5

117.2

158.2

153.8

a,d Values of T and K o are given in cells/tube. b,c Values of 8 and /3max were obtained from the plots 1//3 vs. 1 / R using linear regression analysis. e,g Experimental values obtained from the binding isotherms. f,h Theoretical values computed from Eqs. (11) and (13) respectively.

206

J. Galvez et al. /Journal of Immunological Methods 170 (1994) 197-210

cells was constant (105 cells/tube). By plotting these values of o/ as a function of 1/R the smoothed curve displayed in Fig. 3A was obtained. From this curve it follows that the 1/Rp value resulting in 20% binding was 1.42, i.e., Rp = 1/1.42 = 0.70. Finally, and taking into account that the n u m b e r of effector cells was 105 cells/tube, we find by substituting into Eq. (5): 107 × 0.70/105 = 70 B U / 1 0 7 targets. The value of A U I was determined by estimating from Fig. 3A the effector conjugate frequencies at 1/R = 1, 2, 3, 4, 5, and by substituting these values in Eq. (8) with x = n = 5. This gives AUI(0,5) -- 17.5 + 24.0 + 27.2 + 29.4 + 30.6/2 -113.4. In order to compute the theoretical values of BU and A U I the o/max and y values are necessary (see Eqs. (7) and (9)). These were obtained from the slope and intercept of the plots l/o~ vs. R and so, for donor No. 6 we found O/max = 5 0 . 5 % , y = 2.27. By inserting these values in Eq. (7) we have B U / 1 0 7 targets = 67.2, while Eq. (9) gave AUI(0,5) = 119.1, in good agreement with the BU and A U I values determined experimentally (70 and 113.4, respectively). BU and A U I values obtained from effector binding isotherms are summarized in Table 1 for all donors tested. This table also gives the values of o/max and y used for computation of the corresponding theoretical indices. The dissociation constant (K o) was determined from these isotherms using procedures already described (Garcia Pefiarrubia et al., 1992) and is also included in Table 1. The corresponding values of BU and A U I for target populations were determined from binding isotherms obtained by performing binding assays in which the n u m b e r of target cells was constant. An example is shown in Fig. 3B for donor No. 1. F r o m this plot the Rp value resulting in 20% binding was 1.92. Since the n u m b e r of target cells was 105 cells/tube, from Eq. (10) we have 107/(105 × 1.92) = 52.1 B U / 1 0 7 effectors. The corresponding value of A U I was obtained by estimating from Fig. 3B the target conjugate frequencies at R = 1, 2, 3, 4, 5, and by substituting these values in Eq. (12) with x = n = 5. This gave AUI(0,5) = 106.5. BU and A U I values for all donors tested, as well as the values of [3max and 6

300 A

250

200

B

150 m

lOO

50

1.0

2.0

3.0

KoxlO-Scells//tube Fig. 4. Values of BU for effector cells computed from Eqs. (7) and (2) as a function of K D by assuming ap = 20%. The parameters used for computation were: amax = /3max= 60%. The number of effector cells (cells/tube) in these curves are: (A) 5X104; (B) 105; (C) 5×105; (D) 106.

used for computation of the theoretical indices (see Eqs. (11) and (13)), are summarized in Table 2.

4.2. Relationship between BU and AUI, and the binding parameters o/max, [3. . . . and K o Binding efficiency in an effector-target system increases as the K D values become lower, a n d / o r the values of o/max and flma~ are larger (Garcia Pefiarrubia et al., 1992). Hence, in order to use BU and A U I as a valid overall measure of effector-target conjugation, these indices should also increase in effector-target systems that display large values of a m xa and flmax and small values of KD. Fig. 4 illustrates the dependence of BU on K D for effector populations. Isotherms in this figure have been computed from Eqs. (7) and (2) for four different values of N (5 × 10 4, 10 5, 5 × 105 and 1 0 6 cells/tube) by assuming o/max = f l m a x = 60%, and ap = 20%. The corresponding depen-

J. G alvez et al. ~Journal of Immunological Methods 170 (1994) 197-210

207

250 A 200

200

D

B

0

~ 150

150

100

11111

5O

5O

C

1.0 2.0 Kox 10-Scells / tube

3.0

Fig. 5. D e p e n d e n c e of AUI(0,5) for effector cells computed from Eq. (9) on K o. O t h e r conditions as in Fig. 4.

dence of the AUI(0,5) values on K D has been computed from Eqs. (9) and (2) under the same conditions, and is shown in Fig. 5. There is an increase of the BU and AUI values as the dissociation constant becomes smaller and, in both cases, these indices also display a strong dependence on the number of effector cells. The influence exerted by amax on BU and AUI is shown in Figs. 6 and 7. Both BU and AUI increase with O/max and, as previously, the values of both indices depend significantly on the number of effector cells in the binding assays. Finally, the dependence of BU and AUI on ~max is displayed in Figs. 8 and 9. Analogous trends to those observed in Figs. 4 - 7 are shown, i.e., BU and AUI increase as /3ma x is larger, and there is a strong influence of the number of effector cells on these curves. The dependence of BU and AUI indices of target populations on the three binding parameters is similar to that illustrated in Figs. 4 - 9 for effector populations. Thus, these indices increase with O/max and flmax, and become larger as K o is smaller. The number of target cells also influenced these parameters significantly (data not shown).

20

40

60

80

O
Fig. 6. Values of B U for effector ceils computed from Eqs. (7) and (2) as a function of am~x. T h e parameters used for computation were: K D = 0.5 x 105 cells/tube, ¢tma~ = 60%, ap = 20%. N u m b e r of effector cells in these curves as in Fig. 4.

300

250 B

200 A

~

150

11111

g0

20

40

60

80

Otmax% Fig. 7. D e p e n d e n c e of AUI(0,5) for effector cells computed from Eq. (9) on amax. O t h e r conditions as in Fig. 6.

J. Galvez et aL /Journal of Immunological Methods 170 (1994) 197-210

208 250

A

200

B

@

150

m

100

50

20

40

60

80

,/3max% Fig. 8. Values of B U for effector cells computed from Eqs. (7) and (2) as a function of /3max. The parameters used for computation were: K D = 0.5 × 105 cells/tube, areax = 60%, ap = 20%. N u m b e r of effector cells in these curves as in Fig. 4.

250

200

B

A 5 150 ,<

100

20

40

60

80

j3max~o Fig. 9. D e p e n d e n c e of AUI(0,5) for effector cells computed from Eq. (9) on/3m~x. O t h e r conditions as in Fig. 8.

5. Discussion

The problem of developing a standard method for quantifying effector-target conjugation can be approached by determining the three binding parameters am~x, /3max, and KD (Garcia Pefiarrubia et al., 1992). However, in this paper we have shown that for routine assays it is possible to estimate the overall binding capacity in an effector-target system from single indices that can be determined using faster and easier procedures than those required in the more rigorous methods. For this purpose, we have introduced two indices, the binding unit (BU), and the area under the isotherms (AUI), which we have shown to provide a useful measure of binding efficiency. Their characteristics can be conveniently explained by the following analogy: binding isotherms in effector-target conjugation have an equivalent role to the plots of percentage cytotoxicity vs. effector-to-target ratio in CMC assays. Thus, the binding unit can be considered as an index of binding efficiency equivalent to the lytic unit used for the expression of CMC (see Bryant et al. (1992) for an excellent discussion of the calculation of lytic units for the expression of CMC). Similarly, Dye et al. (1991) have shown recently that the area under the curve (AUC) in CMC assays can be used to quantify cytolytic efficiency. In this context, AUI is equivalent to AUC. However, the above analogy is only formal and has been adopted to facilitate comparison with more familiar concepts. Obviously, plots in CMC assays are different to binding isotherms. Thus, in the first case these curves are characterized by two parameters, the maximal rate of target cell destruction (Vmax) and the apparent Michaelis constant ( K ~ p) (Thorn and Henney, 1976; Zeijlemaker et al., 1977; Callawaert et al., 1978; Thoma et al., 1978; Merril, 1982; Merril and Sathananthan, 1986), and they are obtained by measuring 51Cr release. In contrast, binding isotherms depend on the three binding parameters mentioned above and they are obtained by measuring effector conjugate frequencies. Because BU and AUI are indices for an over-

J. G alvez et al. /Journal of Immunological Methods 170 (1994) 197-210

all measure of binding efficiency their behaviour must be in agreement with the fact that effectortarget conjugation increases with larger values of otm~, and /3max and smaller values of K D. Figs. 4 and 5 show that both BU and AUI increase as K D becomes smaller, and that there is also a strong dependence of these indices on the number of effector cells. This fact is of great practical relevance when designing the best conditions for the performance of binding assays. From these figures it seems that the largest sensitivity is attained by decreasing the number of effector cells in the binding assays (curves A in Figs. 4 and 5). However, under these conditions the values of the conjugate frequencies are small, variations in the a values when R changes are also very low, and the corresponding isotherms are more sensitive to erratic data (Garcia Pefiarrubia et al., 1992). This normally implies that the reference binding level (ap) is not well defined in the isotherms and so the determination of BU from Eq. (5) is less precise. Conversely, if the binding assays are performed at large values of N, isotherms are less influenced by these sources of errors because both the values of a and its dependence on R are larger (Garcia Pefiarrubia et al., 1992). However, curves C in Figs. 4 and 5 reveal that under these conditions BU and AUI vary little with K D. Hence, it is convenient to perform binding assays with a number of effector cells that allow us to minimize these opposing effects. Figs. 3, 4 and 5 show that values of N around ( 1 - 2 ) × 105 cells/tube fulfil these requirements for those effector-target systems in which the K D values range from ( 0 . 2 - 3 ) × 105 cells/tube. Figs. 6 - 7 and Figs. 8 - 9 illustrate the dependence of BU and AUI on O~max and/3m~ ,, respectively. Both BU and AUI increase as Otmax and ~max are larger, and the above considerations about the influence exerted by the number of effector cells in the binding assays are also valid in these cases. This determines the other important requirement when comparing effector-target conjugation in different experiments: the values of BU and AUI must be determined from binding isotherms obtained at the same values of N or T. This requirement must also be fulfilled

209

when one determines lytic units in CMC assays, although, in practice, an explicit statement about this fact is not well established (Taswell, 1987). Finally, it was also found for target populations that the dependence of BU and AUI on °tmax, flmax, and K D was analogous to that displayed in Figs. 4 - 9 (plots not shown). An experimental verification of these predictions has been carried out for the NK-K562 tumor target cell. The results obtained from eight different donors for effector and target populations are summarized in Tables 1 and 2. For target populations the following conclusions can be drawn: (a) for a given number of target cells the values of BU and AUI are comparable if the donors do not exhibit large differences in the /3max a n d / o r K D values (donors Nos. 1, 2, 3 with N = 105 cells/tube in Table 2); (b) however, if the value of K D for a given donor is low, the corresponding values of BU and AUI increase (donor No. 4 with N = 105 c e l l s / t u b e in Table 2), and this fact can be related to an enhanced NK binding capacity for this particular donor; (c) the same conclusion also applies if a donor exhibits a high value of ~max (donor No. 6 with N = 105 cells/tube in Table 2); (d) for a given donor, the values of BU decrease, while the AUI values increase, as the number of target ceils becomes larger (donors Nos. 1, 2, 4 in Table 2). Note that this behaviour is in agreement with that shown in Figs. 4-9, and confirms that for comparative purposes the values of BU and AUI should be reported at the same values of T or N. For effector populations Table 1 shows that, in general, analogous conclusions to that described for target cells are also valid. Note, however, that for donor No. 7 the three first values of BU do not decrease as N increases. This behaviour is an anomaly in the trends displayed in Figs. 4, 6 and 8, although it should be noted that the corresponding AUI values exhibit the correct sequence. This can be explained by considering that BU values are determined from a single data point in the binding isotherms (see Figs. 2 and 3) whereas, in contrast, values of AUI are obtained from a set of data points in these isotherms (a minimum of five data points is strongly recommended). Therefore, the calculation of AUI is

210

J. Galvez et al. /Journal of lmmunological Methods 170 (1994) 197-210

more resistant to the distortional effect of occasional erratic data. In conclusion, two indices, the number of binding units and the area under the isotherms, permit simplified quantitation of the binding efficiency in effector-target systems. These indices are easily determined and fulfil a need for standardized and reliable methods for the expression of effector-target conjugation.

6. References Berke, G. (1985) Enumeration of lymphocyte-target conjugates by cytofluorometry. Eur. J. Immunol. 15, 337. Berke, G., Gabison, D. and Feldman, M. (1975) The frequency of effector cells in populations containing cytotoxic T lymphocytes. Eur. J. Immunol. 5, 813. Bonavida, B., Bradley, T.P. and Grimm, E.A. (1983) The single-cell assay in cell-mediated cytotoxicity. Immunol. Today 4, 196. Bradley, T.P. and Bonavida, B. (1984) Mechanism of cellmediated cytotoxicity at the single cell level. VII. Trigger of the lethal hit event is distinct for N K / K and LDCC effector cells as measured by the two target conjugate assay. Cell. Immunol. 83, 199. Bryant, J., Roger, D., Whiteside, T.L. and Herberman, R.B. (1992) Calculation of lytic units for the expression of cell-mediated cytotoxicity. J. Immunol. Methods 146, 91. Callawaert, D.M., Johnson, D.F. and Kearney, J. (1978) Spontaneous cytotoxicity of cultured human cell lines mediated by normal peripheral blood lymphocytes. III. Kinetic parameters. J. Immunol. 121, 710. Dye, J.F., Somers, S.S. and Guillou, P.J. (1991) Simplified quantitation of cytotoxicity by integration of specific lysis against effector cell concentration at a constant target cell concentration and measuring the area under the curve. J. Immunol. Methods 138, 1. Garcia-Pefiarrubia, P. and Bankhurst, A.D. (1989a) Kinetic analysis of effector cell recycling and effector target binding capacity in a model of cell-mediated cytotoxicity. J. Immunol. 143, 2101. Garcia-Pefiarrubia, P. and Bankhurst, A.D. (1989b) Quantitation of effector-target affinity in the human NK and K562 tumor cell system. J. Immunol. Methods 122, 177. Garcia-Pefiarrubia, P. and Galvez, J. (1993) Kinetic analysis of cell-mediated cytotoxicity at the population level; target and effector binding capacity, affinity, recycling and other implications. Inmunologia 12, 35. Garcia-Pefiarrubia, P., Koster, F.T., Bankhurst, A.D. and Galvez, J. (1989a) Model for population distribution of lymphocyte-target cell conjugates. J. Theor. Biol. 138, 77. Garcia-Pefiarrubia, P., Koster, F.T. Bankhurst, A.D. and Galvez, J. (1989b) Effect of the conjugate size on the

kinetics of cell-mediated cytotoxicity at the population level. J. theor. Biol. 138, 93. Garcia-Pefiarrubia, P., Koster, F.T. and Bankhurst, A.D. (1989c) Population distribution of natural killer and K562 target cell conjugates. Nat. Immun. Cell Growth Regul. 8, 57. Garcia-Pefiarrubia, P., Koster, F.T. and Bankhurst, A.D. (1989d) The maximum conjugate frequency (am~x) characterizes killer cell populations. J. Immunol. Methods 118, 199. Garcia-Pefiarrubia, P., Cabrera, L., Alvarez, R. and Galvez, J. (1992). Effector-target interactions: saturability, affinity and binding isotherms. A study of such interactions in the human NK cell-K562 tumour cell system. J. Immunol. Methods 155, 133. Glasebrook, A.L. (1978) Conjugate formation by primary and secundary populations of murine immune T lymphocytes. J. Immunol. 121, 1870. Martz, E. (1977) Mechanisms of specific tumor-cell lysis by alloimmune T lymphocytes: resolution and characterization of discrete steps in the cellular interactions. Contemp. Top. Immunobiol. 7, 301. Merril, S.J. (1982) Foundations of the use of an enzyme-kinetic analogy in cell-mediated cytotoxicity. Math. Biosci. 62, 219. Merril, S.J. and Sathananthan, S. (1986) Approximate Michaelis-Menten kinetics displayed in a stochastic model of cell-mediated cytotoxicity. Math. Biosci. 80, 223. Perez, P., Bluestone, J.A., Stephany, D.A. and Segal, D.M. (1985) Quantitative measurements of the specifity and kinetics of conjugate formation between cloned cytotoxic T lymphocytes and splenic target cells by dual parameters flow cytometry. J. Immunoi. 134, 478. Taswell, C. (1987) A solution to the problems of cytolysis assays with additional applications to other immunological and biochemical assays. J. Immunol. 138, 333. Thoma, J, A., Touton, M.H. and Clark, W.R. (1978) Interpretation of 51Cr-release data: a kinetic analysis. J. Immunol. 120, 991. Thorn, R.M. and Henney, C.S. (1976) Kinetic analysis of target destruction by effector cells. I. Delineation of parameters related to the frequency and lytic efficiency of killer cells. J. Immunol. 117, 2213. Trinchieri, G. (1989) Biology of natural killer cells. Adv. Immunol. 47,187. Young, D. J.-E. and Liu, C.-C. (1988) Multiple mechanisms of lymphocyte-mediated killing. Immunol. Today 5, 140. Zagury, D., Bernard, J., Thiernesse, N., Feldman, M. and Berke, G. (1975) Isolation and characterization of individual functionally cytotoxic T lymphocytes. Conjugation, killing and recycling at the single level. Eur. J. Imrnunol 5, 818. Zeijlemaker, W.P., Van Oers, R.H.J., De Goede, R.E.Y. and Schellekens, P.T.A. (1977) Cytotoxic activity of human lymphocytes: quantitative analysis of T cell and K cell cytotoxicity revealing enzyme-like kinetics. J. Immunol. 119, 1507.