A Method for the Analysis of Biological Transduction Phenomena JONI A. TORSELLA* Department of Biostatistics, Epidemiology, and Systems Science, Medical University of South Carolina, Charleston, South Carolina 29425 KENNETH M. PRUITT Department of Biochemistq The University of Alabama at Birmingham, Birmingham, Alabama 35294 CHAN F. LAM Department of Biostatistics, Epidemiology, and Systems Science, Medical University of South Carolina, Charleston, South Carolina 29425 Received 3 January 1991; revised 4 November 1991
ABSTRACT Biological transduction can be defined as the triggering of a cellular response by the binding of molecules of effector substances to specific cellular sites. An example of biological transduction, analyzed in this report, is the triggering of T-cell proliferation by the binding of T-cell growth factor (TCGF) to specific TCGF-binding sites on responsive T-cells. Sigmoidal or S-shaped curves often result when measurements of biological response are plotted as a function of concentration of effector substance. Such curves suggest that effector molecules must bind a critical number of cellular sites, and this critical number of bound complexes must undergo secondary events (cross-linking, association, internalization, second messenger release, etc.) in order to initiate the biological response. The method described here estimates the critical number of cellular sites (R) and the probability of these secondary events (Ps,B) as follows: (1) The total number of cellular sites (N) is estimated from binding data, and the probabilities (P,) of effector molecules binding to a site are estimated from response data. (2) The response data are assumed to follow the summed binomial distribution function, which is equated to the incomplete beta function. (3) R and Ps,B are estimated by applying nonlinear regression to the incomplete beta function. The T-cell data to which the method was applied gave N = 15,000, R = 5, and Ps,B = 7.22~ 10e4. These results show that the binding of very few TCGF molecules is required for activation of T-cells and that the probability of the secondary events leading to cell proliferation is much smaller than the probability of TCGF binding to
*Current 19101.
address:
M4THEMATICAL
Wyeth-Ayerst
BIOSCIENCES
Research,
P.O. Box 8299, Philadelphia,
110:191-200 (1992)
OElsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
PA
191 OO25-5564/92/$5.00
192
JONI A. TORSELLA
ET AL.
T-cells. The method described can be used to analyze any biological transduction experiments where both binding and biological response data are available.
INTRODUCTION Biological transduction can be defined as the triggering of a cellular response by the binding of effector molecules to specific cellular sites. An example, which will be considered in detail in the present study, is the triggering of T-cell proliferation by the binding of T-cell growth factor (TCGF, also known as interleukin-2) to high-affinity TCGFspecific binding sites on TCGF-responsive T-cells. Sigmoidal or S-shaped curves are often observed when the extent of the biological response in transduction experiments is plotted as a function of the concentration of the effector substance [3-6, 8, 12-14, 20, 221. Such curves suggest that a critical number of specific receptors must interact with effector molecules in order to trigger the cellular response. We describe here a general quantitative method for determining that critical number from appropriate experimental data. In particular, we use the hypothesis proposed by Alberty and Baldwin [2] and Kabat and Mayer [lo]. They proposed that the fraction of cells lysed by complement could be described by the summed binomial distribution equation. The method also provides an estimate of the probability of secondary events subsequent to the binding of effector molecules to the critical number of sites. SUMMED
BINOMIAL
R (R G N) or more of the receptor-specific sites ejjkctiuely interact with an effector molecule. (We are assuming that all of the cells in the system have the same sensitivity to the effector substance. This type of cellular homogeneity implies that R is constant.) The phrase “effectively interact” is used here to mean that the effector molecule binds to the receptor site and the molecule and its site undergo any secondary events (cross-linking, association, internalization, second messenger release, etc.) necessary for the binding to be an observable cellular response. To determine the number of available sites and the critical number of efsectiuely interacting sites needed for a biological response to occur, a mathematical model relating the proportion of cells affected, Y, to the concentration of effector substance, [El, must be used. From six postu-
METHOD FOR ANALYSIS OF TRANSDUCTION
193
lates based on hit-theory considerations. DInzer developed 17, 211 a probabilistic model of a summed binomial function to describe cell death induced by radiation. For our biological system we propose a different postulate (P3, below). We also generalize Dlnzer’s other five postulates by replacing some of his original phrases: we use “concentration of effector substance” in place of “dose of radiation,” effective interaction” in place of “destruction of critical sites,” and “biological response” in place of “death of cells.” All six of the postulates that we use are stated here for purposes of clarity. (Pl) The cell has a total of N receptor-specific sites. (P2) The cell responds if R CR < N) or more of these sites effectively interact with effector molecules. (P3) A cellular event will occur only if both binding and the necessary secondary events have occurred. (P4) The probability, pi, that a given effector molecule will efictiuely interact with the ith receptor-specific site is constant. This is called the Bernoulli postulate. Even though there is evidence that the structure of some sites changes as more and more binding sites are bound with effector molecules (e.g., oxygenation of hemoglobin [19]), the structure of the site remains relatively unchanged for many systems. (P5) If n[E] is the total number of effector molecules (where IZ= VA, with V equal to volume and A equal to Avogadro’s number), then we let n[E] be large and pi be small (i = 1,2,. . . , iV) in such a way that n[E]p; + h,[E] (finite). This is called the Poisson postulate. (P6) A, = A, = ... = A,. This is called the similar target postulate. Let Y equal the fraction of cells that have responded and X equal the fraction of specific receptor sites that have interacted. If the receptors are randomly distributed over the cell surface, and if they react with the effector molecules independently and with equal probabilities, several authors 12, 7, 211 have shown that the system will follow the summed binomial distribution function (SBINOM),
(1) In order to apply Equation (1) to experimental data, it is necessary to relate Y and X to the experimental observations. For many experiments, Y is measured directly and the concentration of the effector substance, [El, is reported. The problem then is to find a way to relate X and [El.
194
JONI A. TORSELLA ET AL.
THE PROBABILITY OF EFFECI’OR BINDING AND THE CONCENTRATION OF EFFECTOR SUBSTANCE Since binding can be measured in vitro independently of the biological response, it is convenient to consider the probability that a receptor is bound, Ps. Assuming that receptors are identical and effector molecules are identical, the probability that any effector molecule will bind to any receptor is constant under a fixed set of experimental conditions. Therefore, P, is the ratio of the number of bound molecules per cell, B, to the total number of sites per cell: PB = B/N.
(2)
Let Ps,B equal the probability of occurrence of given that binding of the effector molecule to the has occurred. Since our model implies that there without binding, Equation (3) follows directly from ity:
x = P,P,,,.
the secondary events specific receptor site can be no response the laws of probabil(3)
It should be noted that Ps,B is assumed to be constant in this model. The number of bound molecules per cell is
([El-[Fl)A
B=
c
’
(4)
where [E] is the total effector concentration (moles/liter), [F] is the concentration of free effector (moles/liter), A is Avogadro’s number, and C is cell concentration (cells/liter). Thus Ps is p
=
B
w+W~
(5)
NC
and X is
x= (PI-PIPp NC
S/B’
If the effector substance binds randomly, independently, and with the same affinity to identical binding sites, then from the definition of the dissociation constant, K7
we
=
[FW-
B
B)
’
(7)
get
NCPI
[B1= [F]+K,’
(8)
METHOD FOR ANALYSIS OF TRANSDUCTION
195
where [Bl is the total concentration of bound effector molecules and KD is the dissociation constant for the receptor-effector complex. If measurements of [B] as a function of [F] for a fixed value of C are available, N and KD can be estimated from Equation (8) by nonlinear regression or by a Scatchard analysis. The parameters R, of Equation cl), and Ps,s, of Equation (6), can be estimated by the simplex algorithm (a nonlinear regression program [ll, 17, 181). Instead of using the summed binomial distribution function in the estimation process, Equation (1) is equated to the incomplete beta function [l, 211: Z,(R,N-R+l)=
Xj[l-
x1”-‘.
Machine algorithms are available [l] that are computationally for evaluating the incomplete beta function. A PRACTICAL
APPLICATION:
efficient
INTERLEUKIN-2
The immune system is regulated by hormones in much the same way as most organ systems. Interleukins are the hormones of the immune system [15, 161. Interleukin-2 (IL-2) regulates T-cell proliferation by the following mechanism. After a macrophage ingests antigen, it presents the antigen to a T-cell. The antigen stimulates T-cells to secrete IL-2 and to make IL-2 receptors. The binding of IL-2 to IL-2 receptors on the T-cell signals the T-cell to divide. The daughter cells are then in turn activated by antigen, secrete IL-2, make IL-2 receptors, and so on. The net result of the process is the generation of a clone of identical, antigen-specific T-cells [15]. Robb et al. [131 studied the proliferation of T-cells as a function of IL-2 (TCGF). Their data provide a good example for the practical application of the analytical tools outlined above. Robb et al. [13] measured values of free TCGF and the fraction of the biological response for the [3H]TdR proliferation assay. We took data from Figure 7 of that work. These values are listed in Table 1, columns 1 and 2, and are plotted in Figure 1. Robb et al. also carried out TCGF T-cell binding studies (data shown in their Figure 4A). Using the results of those studies together with free TCGF data, we calculated the bound TCGF listed in column 3 of Table 1. From their binding studies they estimated N, the number of TCGF binding sites per cell, to be 15,000. We estimated values of Ps,B and R by fitting the incomplete beta function to the data given in columns 2 and 4 of Table 1. The best fit was obtained with Ps,B = 7.22X 10m4 and R = 5 (the actual esti-
196
JONI A. TORSELLA
ET AL.
TABLE 1 Free TCGF, Fraction of Biological Response, and Bound TCGF Data from Robb [13] for Binding of TCGF to Murine CTLL-2 Subclone 15H Cells and the Calculated Values for Ps and X
Free TCGF (PM)
Fraction of biological response
250.0 130.0 60.0 25.0 10.2 6.5
1.007 1.015 0.896 0.674 0.407 0.237
Bound TCGF molecules/cell 1.39 x 1.30 x 1.12x 8.26 x 5.00 x 3.62x
104 104 104 10’ 103 lo3
N=15000
1.00
X
0.925 0.864 0.746 0.551 0.333 0.242
4.6x 1O-4 4.3 x 10-4 3.7x 10-4 2.7~10-~ 1.7x 10-4 1.2x 10-4
0
0
+
+
+
0.70
Y FRACTION
FE (bound/N)
0
0.60
+ o
1
10
100
PRFiDICIED OBSERVED
moo
FREE TCGF pM FIG. 1. Plot
of observed and predicted fraction of biological response as a function of the concentration of free TCGF. Observed data were obtained from the [3H]TdR proliferation assay, CTLL-2, subclone 15H conducted by Robb et al. [13]. The estimated R and Ps,B are 4.87 and 7.22 X 10m4, respectively, with a residual sum of squares equal to 8.90 X 10m3.
mated value is 4.87). Predicted biological response values based on these parameter estimates are plotted in Figure 1. Robb et al. were able to estimate N with an accuracy of k 2000 sites [13]. We studied the effect of variations in N on the estimated parame-
METHOD FOR ANALYSIS OF TRANSDUCTION
197
When N was changed to 17,000 sites per cell ter values of R and Ps,B. or 13,000 sites per cell, the estimated R and Ps,Bwere, after rounding off to three significant figures, 4.87 and 7.22 X 10P4, respectively, which are identical to those obtained when N is 15,000. Thus variations of this particular N to at least +2000 do not have a significant effect on the estimated values of R and Ps,B. DISCUSSION Our analysis shows that introduction of the concept of conditional probability, Ps ,B,results in good agreement between biological transduction data and our SBINOM model. Our estimate of Ps,B (7.22X 10m4) was several orders of magnitude below the Pe values. In other words, the probability of occupation of a receptor site is much greater than the probability of subsequent occurrence of the secondary events leading to the cellular response (in the present example, cell proliferation). Is this result consistent with what is known about this biological system? Robb et al. [13] found that the binding of TCGF to T-cell receptors was much more rapid than the subsequent cellular response. Their measurements of the time course of association of [35S]methionineTCGF with receptor-bearing T-cells showed a very rapid uptake of radiolabel, with maximum levels achieved within 15 min. However, the subsequent cell proliferation stimulated by the TCGF binding required several hours. What is presently known about IL-2 intracellular signal transduction indicates that it is a complex, multistep process initially involving interaction of cytoplasmic domains of the IL-2 receptor with catalytic domains of cytoplasmic kinases [9]. Few molecular details are known about subsequent processes that must occur to produce the final cellular response, but these processes are certain to be complex, multistep events. It is reasonable, however, to assume that an event with a greater probability of occurrence would have a faster reaction rate. Thus our estimate of the low value of P,,, compared to the Ps values is consistent with the observations made by Robb et al. concerning reaction rate. Our model has two parameters: R and Ps,B.Are the particular estimates of these parameters that gave the best fit in this example unique, or would another pair of estimates fit as well? We explored this question by fixing values of R and then finding the estimates of Ps,B that gave the best fit for each R.The fits for R below (R = 1) and above (R = 10) the best estimate of 5 were worse than the best fit, as shown in Figures 2 and 3. The residual sums of squares are 0.108 for R = 1 and 0.035 for R = 10, compared to 0.009 for R = 5. Although this exercise does not prove that our best fit estimates are unique, it does demon-
198
JONI A. TORSELLA
1.00
N=15000
0 0 +
ET AL.
0 +
+
Y FRACTION OF
BIOLOGICAL BESPONSE
0.70 0.60 +
o 5. _ ’ . 0.40 -
+
0
0.30 0.20 -
+
0
o
PREDICTED OBSERVED
0.10 0.00 : 1
I
I
10
100
-l
1000
FREE TCGF pM
FIG. 2. Plot of observed and predicted fraction of biological response as a function of the concentration of free TCGF with N and R set equal to 15,000 and 1, respectively. Ps,Bis estimated to be 1.74X10m4, with a residual sum of squares equal to 1.08X 10-l.
strate that the chance of finding a different set of parameters having a better fit than R = 5 and PS,B = 7.22~ lo-4 is small. Finally we offer some speculation about the biological significance of the numbers resulting from this analysis. What meaning can be attached to the conditional probability, Ps,B? It is a number composed of the individual probabilities of the complex sequence of events leading to the cellular response. It contains contributions from rate constants, equilibrium constants, component concentrations, and other characteristics of the system components. The sorting out of these individual contributions to Ps,B will depend upon future studies of specific cellular processes. Our model assumes a constant value for Ps,B. This was also speculated by Robb et al. when they stated [13], “The similarity between the biological dose-response curve and the [35S]methionine-TCGF binding curve indicates that the biological response is nearly proportional to the TCGF-binding site occupancy.” Biological transduction is a key mechanism for controlling and coordinating the functions not only of the immune system but of all living systems. The process involves the triggering of a complex set of cellular events by interaction of effector molecules with cellular receptors. In
METHOD
199
FOR ANALYSIS OF TRANSDUCTION
N=15000
1.00
0.90
Q
0
0.80
+
0.70
0
Y FRACTION OF BIOLOGICAL
$
+
0.60 0.50 -
RESPONSE
0
0.40 0.30
-
0.20
-
+
0
+
o
PREDICl-ED OBSERVED
+
0.10 0xX) 1
I
I
1
10
100
loo0
FREE TCGF
phi
FIG. 3. Plot of observed and predicted fraction of biological response as a function of the concentration of free TCGF with N and R fixed at 15,000 and 10, respectively. Ps, s is estimated to be 1.48x10m3, with a residual sum of squares equal to 3.54X 10m2.
order for the system to function well, the process should be efficient. The relatively high number of receptors (N = 15,000 in this example) and the small threshold number that must be occupied (R = 5) mean that the effector molecules do not have to be present in very high concentrations in order for the system to work. Perhaps this is the reason a concentration of IL-2 on the order of lO_” M is effective in triggering the division of T-cells. We appreciate the insightjid comments and suggestions made by the reviewers. The result is a much better manuscript. REFERENCES M. Abramowitz and I. A. Segun, Handbook of Mathematical Functions, Dover, 1965, New York. R. A. Alberty and R. L. Baldwin, A mathematical theory of immune hemolysis, J. Immunol. 66:725-735 (1951). D. Baran, M. Komer, and J. Theze, Characterization of the soluble murine IL2R and estimation of its affinity for IL-2, J. tmmunol. 141:539-546 (1988). J. Chin and R. Horuk, Interleukin 1 receptors on rabbit articular chondrocytes:
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ET AL.
relationship between biological activity and receptor binding kinetics, FASEB 4:1481-1487 (1990). 5 J. Chin, P. Cameron, E. Rupp, and J. Schmidt, Identification of a high-affinity receptor for native human interleukin lp and interleukin 1 LYnormal human lung fibroblasts, J. Exp. Med. 165:70-86 (1987). 6 J. Chin, E. Rupp, P. M. Cameron, K. L. MacNaul, P. A. Lotke, M. J. Tocci, J. A. Schmidt, and E. K. Bayne, Identification of a high-affinity receptor for interleukin 1 cx and interleukin 1 p on cultured human rheumatoid synovial cells, J. Clin. Inuest. 82:420-426 (1988). 7 H. Danzer, eber einige Wirkungen von Strahlen VII, Z. Physik. 89:421 (1934). 8 T. Delaunay, J. Louahed, and H. Bazin, Rat (and mouse) monoclonal antibodies VIII. ELISA measurements of Ig production in mouse hybridoma culture supernatants, J. Immunol. Methods 131:33-39 (1990). 9 M. Hatakeyama, T. Kono, N. Kobayashi, A. Kawahara, S. Levin, R. Perlmutter, and T. Taniguchi, Interaction of the IL-2 receptor with the src-family kinase ~56”~. Identification of novel intermolecular association, Science 2521523-1528 (1991). 10 E. A. Kabat and M. M. Mayer, Experimental Immunochemistry, C. C. Thomas, 1971 Springfield, Ill. 11 C. F. Lam, Simplex procedure, Techniques for the Analysis and Modelling of Enzyme Kinetic Mechanisms, Research Studies Press, Chichester, 1981, pp. 140-147. 12 D. Langlois, J. Saez, and M. Befeot, Effects of angiotensin-II on inositol phosphate accumulation and calcium influx in bovine adrenal and Y-l tumor adrenal cells, Endocrine Res. 16(1):31-49 (1990). 13 R. J. Robb, A. Munck, and K. A. Smith, T cell growth factor receptors: quantitation, specificity, and biological relevance, J. Exp. Med. 154:1455-1474 (1981). 14 R. Santen, P. Langecker, S. Santner, S. Sikka, J. Rajfer, and R. Swerdloff, Potency and specificity of CGS-16949A as an aromatase inhibitor, Endocrine Res. 16(1):77-91
(1990).
15
K. A. Smith, Interleukin 2: a lo-year perspective, in Interleukin 2, K. A. Smith, ed., Academic, San Diego, Calif., 1988, pp. l-35. 16 K. A. Smith, Interleukin-2, Sci. Am. March:50-57 (1990). 17 G. Strang, Network flows and combinatorics, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, Mass., 1986, Chap. 7, pp. 629-664. 18 G. Strang, Optimization, Introduction to Applied Mathematics, WellesleyCambridge Press, Wellesley, Mass., 1986, Chap. 8, pp. 665-734. 19 L. Stryer, Oxygen transporters Biochemistry, W. H. Freeman, New York, 1988, Chap. 7, pp. 143-174. 20 E. Thomas and T. Aune, Lactoperoxidase, peroxide, thiocyanate antimicrobial system: correlation of sulfhydryl oxidation with antimicrobial action, Infect. Immun. 21
20(2):456-463
M. E. Turner,
(1978).
Some classes of hit-theory
models,
Math.
Biosci.
23:219-235
(1975). 22
A. E. Wadsworth, E. Maltaner, and F. Maltaner, The quantitative determination of the fixation of complement by immune serum and antigen, J. Immunol. 21:313-340
(1931).
ABSTRACT Biological transduction can be defined as the triggering of a cellular response by the binding of molecules of effector substances to specific cellular sites. An example of biological transduction, analyzed in this report, is the triggering of T-cell proliferation by the binding of T-cell growth factor (TCGF) to specific TCGF-binding sites on responsive T-cells. Sigmoidal or S-shaped curves often result when measurements of biological response are plotted as a function of concentration of effector substance. Such curves suggest that effector molecules must bind a critical number of cellular sites, and this critical number of bound complexes must undergo secondary events (cross-linking, association, internalization, second messenger release, etc.) in order to initiate the biological response. The method described here estimates the critical number of cellular sites (R) and the probability of these secondary events (Ps,B) as follows: (1) The total number of cellular sites (N) is estimated from binding data, and the probabilities (P,) of effector molecules binding to a site are estimated from response data. (2) The response data are assumed to follow the summed binomial distribution function, which is equated to the incomplete beta function. (3) R and Ps,B are estimated by applying nonlinear regression to the incomplete beta function. The T-cell data to which the method was applied gave N = 15,000, R = 5, and Ps,B = 7.22~ 10e4. These results show that the binding of very few TCGF molecules is required for activation of T-cells and that the probability of the secondary events leading to cell proliferation is much smaller than the probability of TCGF binding to
*Current 19101.
address:
M4THEMATICAL
Wyeth-Ayerst
BIOSCIENCES
Research,
P.O. Box 8299, Philadelphia,
110:191-200 (1992)
OElsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
PA
191 OO25-5564/92/$5.00
192
JONI A. TORSELLA
ET AL.
T-cells. The method described can be used to analyze any biological transduction experiments where both binding and biological response data are available.
INTRODUCTION Biological transduction can be defined as the triggering of a cellular response by the binding of effector molecules to specific cellular sites. An example, which will be considered in detail in the present study, is the triggering of T-cell proliferation by the binding of T-cell growth factor (TCGF, also known as interleukin-2) to high-affinity TCGFspecific binding sites on TCGF-responsive T-cells. Sigmoidal or S-shaped curves are often observed when the extent of the biological response in transduction experiments is plotted as a function of the concentration of the effector substance [3-6, 8, 12-14, 20, 221. Such curves suggest that a critical number of specific receptors must interact with effector molecules in order to trigger the cellular response. We describe here a general quantitative method for determining that critical number from appropriate experimental data. In particular, we use the hypothesis proposed by Alberty and Baldwin [2] and Kabat and Mayer [lo]. They proposed that the fraction of cells lysed by complement could be described by the summed binomial distribution equation. The method also provides an estimate of the probability of secondary events subsequent to the binding of effector molecules to the critical number of sites. SUMMED
BINOMIAL
R (R G N) or more of the receptor-specific sites ejjkctiuely interact with an effector molecule. (We are assuming that all of the cells in the system have the same sensitivity to the effector substance. This type of cellular homogeneity implies that R is constant.) The phrase “effectively interact” is used here to mean that the effector molecule binds to the receptor site and the molecule and its site undergo any secondary events (cross-linking, association, internalization, second messenger release, etc.) necessary for the binding to be an observable cellular response. To determine the number of available sites and the critical number of efsectiuely interacting sites needed for a biological response to occur, a mathematical model relating the proportion of cells affected, Y, to the concentration of effector substance, [El, must be used. From six postu-
METHOD FOR ANALYSIS OF TRANSDUCTION
193
lates based on hit-theory considerations. DInzer developed 17, 211 a probabilistic model of a summed binomial function to describe cell death induced by radiation. For our biological system we propose a different postulate (P3, below). We also generalize Dlnzer’s other five postulates by replacing some of his original phrases: we use “concentration of effector substance” in place of “dose of radiation,” effective interaction” in place of “destruction of critical sites,” and “biological response” in place of “death of cells.” All six of the postulates that we use are stated here for purposes of clarity. (Pl) The cell has a total of N receptor-specific sites. (P2) The cell responds if R CR < N) or more of these sites effectively interact with effector molecules. (P3) A cellular event will occur only if both binding and the necessary secondary events have occurred. (P4) The probability, pi, that a given effector molecule will efictiuely interact with the ith receptor-specific site is constant. This is called the Bernoulli postulate. Even though there is evidence that the structure of some sites changes as more and more binding sites are bound with effector molecules (e.g., oxygenation of hemoglobin [19]), the structure of the site remains relatively unchanged for many systems. (P5) If n[E] is the total number of effector molecules (where IZ= VA, with V equal to volume and A equal to Avogadro’s number), then we let n[E] be large and pi be small (i = 1,2,. . . , iV) in such a way that n[E]p; + h,[E] (finite). This is called the Poisson postulate. (P6) A, = A, = ... = A,. This is called the similar target postulate. Let Y equal the fraction of cells that have responded and X equal the fraction of specific receptor sites that have interacted. If the receptors are randomly distributed over the cell surface, and if they react with the effector molecules independently and with equal probabilities, several authors 12, 7, 211 have shown that the system will follow the summed binomial distribution function (SBINOM),
(1) In order to apply Equation (1) to experimental data, it is necessary to relate Y and X to the experimental observations. For many experiments, Y is measured directly and the concentration of the effector substance, [El, is reported. The problem then is to find a way to relate X and [El.
194
JONI A. TORSELLA ET AL.
THE PROBABILITY OF EFFECI’OR BINDING AND THE CONCENTRATION OF EFFECTOR SUBSTANCE Since binding can be measured in vitro independently of the biological response, it is convenient to consider the probability that a receptor is bound, Ps. Assuming that receptors are identical and effector molecules are identical, the probability that any effector molecule will bind to any receptor is constant under a fixed set of experimental conditions. Therefore, P, is the ratio of the number of bound molecules per cell, B, to the total number of sites per cell: PB = B/N.
(2)
Let Ps,B equal the probability of occurrence of given that binding of the effector molecule to the has occurred. Since our model implies that there without binding, Equation (3) follows directly from ity:
x = P,P,,,.
the secondary events specific receptor site can be no response the laws of probabil(3)
It should be noted that Ps,B is assumed to be constant in this model. The number of bound molecules per cell is
([El-[Fl)A
B=
c
’
(4)
where [E] is the total effector concentration (moles/liter), [F] is the concentration of free effector (moles/liter), A is Avogadro’s number, and C is cell concentration (cells/liter). Thus Ps is p
=
B
w+W~
(5)
NC
and X is
x= (PI-PIPp NC
S/B’
If the effector substance binds randomly, independently, and with the same affinity to identical binding sites, then from the definition of the dissociation constant, K7
we
=
[FW-
B
B)
’
(7)
get
NCPI
[B1= [F]+K,’
(8)
METHOD FOR ANALYSIS OF TRANSDUCTION
195
where [Bl is the total concentration of bound effector molecules and KD is the dissociation constant for the receptor-effector complex. If measurements of [B] as a function of [F] for a fixed value of C are available, N and KD can be estimated from Equation (8) by nonlinear regression or by a Scatchard analysis. The parameters R, of Equation cl), and Ps,s, of Equation (6), can be estimated by the simplex algorithm (a nonlinear regression program [ll, 17, 181). Instead of using the summed binomial distribution function in the estimation process, Equation (1) is equated to the incomplete beta function [l, 211: Z,(R,N-R+l)=
Xj[l-
x1”-‘.
Machine algorithms are available [l] that are computationally for evaluating the incomplete beta function. A PRACTICAL
APPLICATION:
efficient
INTERLEUKIN-2
The immune system is regulated by hormones in much the same way as most organ systems. Interleukins are the hormones of the immune system [15, 161. Interleukin-2 (IL-2) regulates T-cell proliferation by the following mechanism. After a macrophage ingests antigen, it presents the antigen to a T-cell. The antigen stimulates T-cells to secrete IL-2 and to make IL-2 receptors. The binding of IL-2 to IL-2 receptors on the T-cell signals the T-cell to divide. The daughter cells are then in turn activated by antigen, secrete IL-2, make IL-2 receptors, and so on. The net result of the process is the generation of a clone of identical, antigen-specific T-cells [15]. Robb et al. [131 studied the proliferation of T-cells as a function of IL-2 (TCGF). Their data provide a good example for the practical application of the analytical tools outlined above. Robb et al. [13] measured values of free TCGF and the fraction of the biological response for the [3H]TdR proliferation assay. We took data from Figure 7 of that work. These values are listed in Table 1, columns 1 and 2, and are plotted in Figure 1. Robb et al. also carried out TCGF T-cell binding studies (data shown in their Figure 4A). Using the results of those studies together with free TCGF data, we calculated the bound TCGF listed in column 3 of Table 1. From their binding studies they estimated N, the number of TCGF binding sites per cell, to be 15,000. We estimated values of Ps,B and R by fitting the incomplete beta function to the data given in columns 2 and 4 of Table 1. The best fit was obtained with Ps,B = 7.22X 10m4 and R = 5 (the actual esti-
196
JONI A. TORSELLA
ET AL.
TABLE 1 Free TCGF, Fraction of Biological Response, and Bound TCGF Data from Robb [13] for Binding of TCGF to Murine CTLL-2 Subclone 15H Cells and the Calculated Values for Ps and X
Free TCGF (PM)
Fraction of biological response
250.0 130.0 60.0 25.0 10.2 6.5
1.007 1.015 0.896 0.674 0.407 0.237
Bound TCGF molecules/cell 1.39 x 1.30 x 1.12x 8.26 x 5.00 x 3.62x
104 104 104 10’ 103 lo3
N=15000
1.00
X
0.925 0.864 0.746 0.551 0.333 0.242
4.6x 1O-4 4.3 x 10-4 3.7x 10-4 2.7~10-~ 1.7x 10-4 1.2x 10-4
0
0
+
+
+
0.70
Y FRACTION
FE (bound/N)
0
0.60
+ o
1
10
100
PRFiDICIED OBSERVED
moo
FREE TCGF pM FIG. 1. Plot
of observed and predicted fraction of biological response as a function of the concentration of free TCGF. Observed data were obtained from the [3H]TdR proliferation assay, CTLL-2, subclone 15H conducted by Robb et al. [13]. The estimated R and Ps,B are 4.87 and 7.22 X 10m4, respectively, with a residual sum of squares equal to 8.90 X 10m3.
mated value is 4.87). Predicted biological response values based on these parameter estimates are plotted in Figure 1. Robb et al. were able to estimate N with an accuracy of k 2000 sites [13]. We studied the effect of variations in N on the estimated parame-
METHOD FOR ANALYSIS OF TRANSDUCTION
197
When N was changed to 17,000 sites per cell ter values of R and Ps,B. or 13,000 sites per cell, the estimated R and Ps,Bwere, after rounding off to three significant figures, 4.87 and 7.22 X 10P4, respectively, which are identical to those obtained when N is 15,000. Thus variations of this particular N to at least +2000 do not have a significant effect on the estimated values of R and Ps,B. DISCUSSION Our analysis shows that introduction of the concept of conditional probability, Ps ,B,results in good agreement between biological transduction data and our SBINOM model. Our estimate of Ps,B (7.22X 10m4) was several orders of magnitude below the Pe values. In other words, the probability of occupation of a receptor site is much greater than the probability of subsequent occurrence of the secondary events leading to the cellular response (in the present example, cell proliferation). Is this result consistent with what is known about this biological system? Robb et al. [13] found that the binding of TCGF to T-cell receptors was much more rapid than the subsequent cellular response. Their measurements of the time course of association of [35S]methionineTCGF with receptor-bearing T-cells showed a very rapid uptake of radiolabel, with maximum levels achieved within 15 min. However, the subsequent cell proliferation stimulated by the TCGF binding required several hours. What is presently known about IL-2 intracellular signal transduction indicates that it is a complex, multistep process initially involving interaction of cytoplasmic domains of the IL-2 receptor with catalytic domains of cytoplasmic kinases [9]. Few molecular details are known about subsequent processes that must occur to produce the final cellular response, but these processes are certain to be complex, multistep events. It is reasonable, however, to assume that an event with a greater probability of occurrence would have a faster reaction rate. Thus our estimate of the low value of P,,, compared to the Ps values is consistent with the observations made by Robb et al. concerning reaction rate. Our model has two parameters: R and Ps,B.Are the particular estimates of these parameters that gave the best fit in this example unique, or would another pair of estimates fit as well? We explored this question by fixing values of R and then finding the estimates of Ps,B that gave the best fit for each R.The fits for R below (R = 1) and above (R = 10) the best estimate of 5 were worse than the best fit, as shown in Figures 2 and 3. The residual sums of squares are 0.108 for R = 1 and 0.035 for R = 10, compared to 0.009 for R = 5. Although this exercise does not prove that our best fit estimates are unique, it does demon-
198
JONI A. TORSELLA
1.00
N=15000
0 0 +
ET AL.
0 +
+
Y FRACTION OF
BIOLOGICAL BESPONSE
0.70 0.60 +
o 5. _ ’ . 0.40 -
+
0
0.30 0.20 -
+
0
o
PREDICTED OBSERVED
0.10 0.00 : 1
I
I
10
100
-l
1000
FREE TCGF pM
FIG. 2. Plot of observed and predicted fraction of biological response as a function of the concentration of free TCGF with N and R set equal to 15,000 and 1, respectively. Ps,Bis estimated to be 1.74X10m4, with a residual sum of squares equal to 1.08X 10-l.
strate that the chance of finding a different set of parameters having a better fit than R = 5 and PS,B = 7.22~ lo-4 is small. Finally we offer some speculation about the biological significance of the numbers resulting from this analysis. What meaning can be attached to the conditional probability, Ps,B? It is a number composed of the individual probabilities of the complex sequence of events leading to the cellular response. It contains contributions from rate constants, equilibrium constants, component concentrations, and other characteristics of the system components. The sorting out of these individual contributions to Ps,B will depend upon future studies of specific cellular processes. Our model assumes a constant value for Ps,B. This was also speculated by Robb et al. when they stated [13], “The similarity between the biological dose-response curve and the [35S]methionine-TCGF binding curve indicates that the biological response is nearly proportional to the TCGF-binding site occupancy.” Biological transduction is a key mechanism for controlling and coordinating the functions not only of the immune system but of all living systems. The process involves the triggering of a complex set of cellular events by interaction of effector molecules with cellular receptors. In
METHOD
199
FOR ANALYSIS OF TRANSDUCTION
N=15000
1.00
0.90
Q
0
0.80
+
0.70
0
Y FRACTION OF BIOLOGICAL
$
+
0.60 0.50 -
RESPONSE
0
0.40 0.30
-
0.20
-
+
0
+
o
PREDICl-ED OBSERVED
+
0.10 0xX) 1
I
I
1
10
100
loo0
FREE TCGF
phi
FIG. 3. Plot of observed and predicted fraction of biological response as a function of the concentration of free TCGF with N and R fixed at 15,000 and 10, respectively. Ps, s is estimated to be 1.48x10m3, with a residual sum of squares equal to 3.54X 10m2.
order for the system to function well, the process should be efficient. The relatively high number of receptors (N = 15,000 in this example) and the small threshold number that must be occupied (R = 5) mean that the effector molecules do not have to be present in very high concentrations in order for the system to work. Perhaps this is the reason a concentration of IL-2 on the order of lO_” M is effective in triggering the division of T-cells. We appreciate the insightjid comments and suggestions made by the reviewers. The result is a much better manuscript. REFERENCES M. Abramowitz and I. A. Segun, Handbook of Mathematical Functions, Dover, 1965, New York. R. A. Alberty and R. L. Baldwin, A mathematical theory of immune hemolysis, J. Immunol. 66:725-735 (1951). D. Baran, M. Komer, and J. Theze, Characterization of the soluble murine IL2R and estimation of its affinity for IL-2, J. tmmunol. 141:539-546 (1988). J. Chin and R. Horuk, Interleukin 1 receptors on rabbit articular chondrocytes:
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ET AL.
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