(Received for publication, February 21, 1995; and in revised form, May 9, 1995)
From the
Previously, we demonstrated that aggregates of the high affinity
receptor for IgE (FcRI), formed by the binding of chemically
cross-linked oligomers of IgE, continue to signal early and late
cellular responses long after the formation of new aggregates is
blocked. In the present work, we explore quantitatively the
relationship between aggregation of the receptors and one of the
earliest biochemical changes this initiates. We compare the time course
of aggregate formation, inferred from studies of the binding of dimers
of IgE, and the time course of phosphorylation of tyrosines on receptor
subunits when the receptors are aggregated. A simple model does not fit
the data. It appears that aggregates formed late in the response are
less effective signaling units than those formed initially. We propose
new explanations for the persistence of the response and the unusual
kinetics.
Several types of receptors central to the functioning of the immune system stimulate cellular responses when they are aggregated by ligand, directly or indirectly. Among the earliest observable responses to such aggregation is enhanced phosphorylation of tyrosines on one or more of the subunits of the receptor and on a variety of other proteins. In this respect, these receptors resemble many receptor kinases(1) ; however, unlike the latter, the multi-subunit immune response receptors (2) have no known intrinsic kinase domains. Instead, there is increasing evidence that their action is mediated by constitutively bound and newly recruited kinases(3, 4) .
In basophils and mast cells,
aggregation of the high affinity receptor for IgE (FcRI)
stimulates morphological, secretory, and biosynthetic changes in the
cells(5, 6) . As with other multi-subunit immune
response receptors, increased phosphorylation of tyrosine residues on
subunits of the receptor (
and
in this instance) and on
several other cellular proteins is among the earliest molecularly
defined consequences of the aggregation of
Fc
RI(7, 8, 9) . Recent studies of the
kinase activity induced by Fc
RI aggregation on rat basophilic
leukemia (RBL) (
)cells indicate that the kinases Lyn and Syk
are among those whose activity is observable following aggregation of
Fc
RI(10, 11, 12) . A fraction of
Fc
RI on RBL cells is associated with Lyn prior to activation, and
additional Lyn is associated with the receptor after
activation(13, 14, 15) . Syk appears not to
be pre-associated with the receptor, but there is evidence that it
associates with both the
and the
subunits, primarily
,
following activation(13, 14) .
The relation between
the degree of receptor aggregation and subsequent cellular events is
often complex. For example, receptor aggregation is not always
stimulatory but, depending on the concentration of ligand and receptor,
can turn on or turn off cellular responses(16) . It is
therefore important to understand the quantitative relationships
between the aggregation of such receptors and the early biochemical
signals they generate. To pursue these quantitative relationships, we
chose an experimental system that is particularly tractable, both
theoretically and experimentally. We used chemically cross-linked
dimers of IgE to aggregate FcRI on RBL cells. The only receptor
aggregates that form in this case consist of two Fc
RI, bound to
the distinct IgE molecules in single IgE dimers. Because the
dissociation of IgE from Fc
RI is slow(17) , receptor
aggregates are long-lived. Excess monomeric IgE can be used to stop the
formation of new aggregates without breaking up existing ones.
Previously, we used such a protocol and found that receptors aggregated by oligomers continue to signal both early and late events for protracted periods(18) . The early signal we monitored was phosphorylation of tyrosines on the receptor itself and on other proteins. Those studies also suggested that the stable aggregates undergo a dynamic process of phosphorylation and dephosphorylation. The goal of the present study was 2-fold. First, we wanted to determine the time course of receptor aggregation and compare it with the time course of phosphorylation of tyrosines on the receptor. Second, we wanted to see whether carefully quantitated results from similar experiments could be accounted for by a simple model in which the extent of phosphorylation was related to the extent of aggregation by a series of internally consistent rate constants. Aggregation of receptors is not readily assessed directly and must be inferred from binding studies. The use of dimeric rather than trimeric aggregates makes such inferences much more reliable, and so we principally used this type of oligomer for the present studies.
Figure 1:
Binding to and aggregation of IgE
FcRI by bifunctional dimers of IgE. The forward and reverse rate
constants are k
and k
for the binding (or dissociation) of one IgE in a dimer to (from)
a single Fc
RI and k
and k
for the binding (or dissociation) of the
second IgE in a monovalently bound dimer to (from) a second
receptor.
The model used to analyze the binding data is presented in detail under ``Appendix.'' Least squares fits of the model to binding data from a variety of experiments, described below, yielded estimates of the parameters.
Figure 2:
The kinetics of binding of I-IgE dimer (1.6 nM) to Fc
RI (2.5 nM (a), 3.2 nM (b), and 2.7 nM (c)) in the presence and absence of unlabeled IgE monomer
(53 nM) for each of the three replicate experiments.
shows the data for dimer binding when no monomer was present;
shows when monomer was added after 2 min; and
demonstrates when
monomer was added initially, along with dimer. The theoretical curves (solidlines) plot the best simultaneous fit of the
binding model (``Appendix'') to the nine data sets (three
experiments, each with three conditions). The parameter values are: P
= 0.54, P
=
1, k
= 0.8
10
M
s
,
k
= 1.4
10
M
s
, k
= k
= k
= 1
10
s
. Essentially the same fit is
obtained for any k
value within 3 orders of
magnitude of the diffusion limit.
One observation that will be important in the interpretation of data on phosphorylation of protein tyrosines from analogous experiments is that monomer, at the concentration used, terminates new dimer binding within minutes (Fig. 2, lowercurves).
Figure 3:
Binding kinetics of dimers when receptors
are limiting. In two separate experiments (a, b), RBL
cells, all of whose FcRI (3-4 nM) were initially
unoccupied (uppercurves) or 70-80% of whose
Fc
RI were occupied by unlabeled monomeric IgE (lowercurves), were incubated with
I-IgE dimer
(21 nM) for the indicated times. The best simultaneous fit of
the binding model to the data (average cpm from triplicate samples at
each time point) was obtained with P
= 1, k
equal to the diffusion limit
(``Appendix''), and corrections of 15% (a) and 1% (b) in the number of receptors per cell determined
experimentally. The P
and k
values giving the best fits were the same, with or without a
correction in the number of receptors per cell. Other parameters were
as previously determined (Fig. 2). The results of two analogous
experiments on Chinese hamster ovary cells transfected with Fc
RI
are consistent with the same parameter set (data not
shown).
The value of P that gave the best fit of the model to the data
is P
= 1, corresponding to the case where
either both or neither of the IgEs in a dimer can bind to Fc
RI.
Under the alternative assumption of independent inactivation of IgEs in
a dimer, so that P
= 0.19, the predictions
made by the model deviate markedly from the data both qualitatively and
quantitatively.
Figure 4:
Time
course of tyrosine phosphorylation of the and
subunits of
Fc
RI and two other cellular proteins, p72 and p30, in RBL cells
stimulated with IgE dimer (1.6 nM), in the absence of IgE
monomer, and with monomer (53 nM) added after 2 min. The
results shown are from densitometric scans of autophotographs of the
gels on which the samples were analyzed. The intensity values are
averages from duplicate samples in representative single experiments.
The cellular protein p38 had a time course of tyrosine phosphorylation
similar to those of p72 and p30 (not
shown).
In Fig. 5we present the result
of calculations of how the rate of formation of aggregates (dimers) of
FcRI is expected to vary with time under the conditions used for
the experiments shown in Fig. 4. It is apparent that
dimer-induced phosphorylation of protein tyrosines does not follow the
time course of the rate of aggregation of receptors. In contrast with
the time course of phosphorylation observed experimentally when
monomeric IgE blocked further binding of dimers (Fig. 4), the
predicted rate of formation of aggregated receptors decreases rapidly
to zero under the same conditions (Fig. 5). Even in the absence
of monomer, phosphorylation of protein tyrosines continued to increase
and then remained elevated well after the predicted rate of aggregation
had peaked and begun to decrease.
Figure 5:
Predicted variation of the rate of
dimerization of receptors with time, plotted with k 1000-fold lower than the diffusion limit.
The other parameters are as in Fig. 2. The difference between
the observed phosphorylation levels and the predicted aggregation rate
becomes even more pronounced if the forward rate constant is closer to
the diffusion limit. The model presented under ``Appendix''
was used to calculated the rate of
dimerization.
How the concentration of receptor
aggregates changes in time can be seen from Fig. 2where, for
the range of binding parameters identified in the binding experiments,
plots of the predicted concentration of receptors that are aggregated
are indistinguishable from the plotted binding curves. The kinetics of
phosphorylation of protein tyrosines on and
(Fig. 4)
paralleled more closely the time course for the total number of
aggregated receptors than the predicted time course of the rate of
formation of aggregates (Fig. 5). In particular, phosphorylation
levels were maintained for at least an hour after the addition of
monomeric IgE had blocked further binding and aggregation of Fc
RI
(compare the lowercurves in Fig. 2and Fig. 4).
However, there were also significant differences
between the patterns of aggregation of FcRI (Fig. 2) and
phosphorylation of tyrosine residues on the subunits of the receptor (Fig. 4). First, when IgE dimer bound in the absence of IgE
monomer, the level of phosphorylation of tyrosines on the
and
subunits of the receptor stopped rising after 10-20 min,
although receptor dimerization continued to increase for over an hour.
Second, the levels of dimer-induced phosphotyrosine in the presence and
absence of IgE monomer were much closer to each other than were the
predicted concentrations of receptors in dimers in the corresponding
samples. For example, in the first panel of Fig. 4, the
ratio of the apparent steady state levels of phosphotyrosine associated
with the
chain when dimer bound in the absence of monomer and
when monomer was added after 2 min was approximately 4. The ratio of
the concentrations of receptors in dimers, under the same two
experimental conditions, estimated from
I-IgE binding
data for cells with approximately 3
10
receptors,
was 7 at the end of an hour and was still increasing (Fig. 2a).
To compare in a more rigorous way the relationship between aggregation and phosphorylation, we extended the binding model to include dimer-induced phosphorylation of tyrosines. The model, illustrated in Fig. 6and detailed under ``Appendix,'' allows for both reversible and irreversible dephosphorylation.
Figure 6:
Extension of the binding model (Fig. 1) to include dimer-induced tyrosine phosphorylation (rate
), reversible dephosphorylation (rate µ
), and
irreversible dephosphorylation (rate
µ
).
Fig. 7shows the best fit of the model to
the data for tyrosine phosphorylation of the subunit, obtained
under the two conditions where monomer was added to block further dimer
binding and receptor aggregation (lowercurves). The uppercurve shows the prediction of the model for the
case when IgE dimer binds in the absence of monomer. The theoretical
curve was generated using the phosphorylation and dephosphorylation
rates that provided the best fit of the model to the experiments in
which monomer was added. That fit was obtained by assuming that
irreversible dephosphorylation was negligible on the time scale of the
experiments (µ
= 0) and that phosphorylation was
rapid (
10 min
). The fit was not sensitive
to the reversible dephosphorylation rate µ
. In
generating the theoretical curves in Fig. 7, we used the value
µ
= 1 s
. Values in the range
1 min
to 1 s
are consistent with
the time course of dephosphorylation in our earlier experiments in
which EDTA was added to inhibit kinase activity(18) .
Figure 7:
Time course of tyrosine phosphorylation of
the subunit of Fc
RI. The twolowercurves represent the best fit of the model (Fig. 5) to the data from experiments with excess monomer added
after 2 min (
) or along with dimer (
). The binding
parameters used are as previously determined (Fig. 2, 3). The
data do not determine the phosphorylation rate
or the reversible
dephosphorylation rate µ
. Values used in the fit were
= 10 s
and µ
= 1
s
, consistent with the observation that
phosphorylation increases rapidly after cell activation and
dephosphorylation is rapid after kinase removal(18) . The
irreversible dephosphorylation rate determined from the fit is
µ
= 0 s
. The uppercurve shows the result of using these parameters in the
model to predict tyrosine phosphorylation of
when dimer binds in
the absence of monomer. The corresponding data are indicated by
.
We also fit the model to the data obtained in the absence of monomer and used the resulting parameters to predict phosphotyrosine levels in the experiments where monomer was added (results not shown). The irreversible dephosphorylation rate needed to fit data from experiments where dimer bound in the absence of monomer predicted rapid dephosphorylation under the other two conditions. As it stands, the model cannot give an adequate simultaneous fit of both types of data, i.e. dimer-induced phosphotyrosine levels observed in the presence and absence of monomer. In particular, it cannot account for the ``squeezing'' together of the tyrosine phosphorylation curves obtained with and without the addition of excess IgE monomer.
A possible explanation for the leveling off of phosphorylation while
aggregation continues to increase is that the kinases responsible for
phosphorylation are in limited supply so that the phosphorylation
reaction saturates. The model can be modified to allow for this
possibility by replacing the constant phosphorylation rate (per
aggregated receptor) by a rate that depends on the concentration of one
or more kinases and adding equations to keep track of the changing
kinase concentration(s). However, Fig. 8shows that when the
concentration of IgE dimers was increased, the height of the plateau
increased, demonstrating that the phosphorylation reaction had not been
saturated.
Figure 8:
Time
course of phosphorylation of tyrosines on the subunit of
Fc
RI, following stimulation of RBL cells by IgE dimer at 3 nM (
) and 6 nM (
) concentrations. Data are from
duplicate samples in one of two experiments. The solidcurves show the corresponding predictions of the model.
The irreversible dephosphorylation rate used, µ
= 2.6
10
s
,
was determined from the best simultaneous fit of the model to six data
sets (two experiments, three dimer concentrations each; data at 1.6
nM not shown). The other parameters were as previously
estimated (Fig. 7).
The fundamental quantities calculated from the model are
fractions of FcRI in the distinct states illustrated in Fig. 5. Therefore, we could calculate the fraction of receptor
subunits that were phosphorylated over time in the various experiments.
In the experiments where dimer bound in the absence of monomer,
separate fits of phosphotyrosine levels measured on the
and
subunits of Fc
RI indicated that phosphorylation of tyrosines
peaked when about 8-9% of the receptors were phosphorylated.
Previously, we showed that stable aggregates of Fc
receptors continue to signal RBL cell responses without the formation
of new aggregates(18) . The use of chemically cross-linked
oligomers of IgE to induce receptor aggregation made the observation
possible, since aggregation could be stopped (with monomeric IgE)
without breaking up previously formed aggregates. This class of ligands
has the further advantage that, since few states form, one can
characterize the binding in detail and draw inferences about the time
course of receptor aggregation. In this study, we used the simplest IgE
oligomer, a dimer. We found that the response (phosphorylation) leveled
off while the number of aggregates was still rising steeply. By
comparing the observed time courses with predictions of a simple model,
we could eliminate a number of possible explanations for the data. The
surprising conclusion we are left with is that although the aggregates
that form initially are effective and persistent signaling units,
aggregates that form later are relatively ineffective.
We first
determined the kinetics with which dimers bound to the cells in the
presence and absence of monomeric IgE. Analyzing the data within the
framework of a model for the binding of IgE dimers to RBL cells, we
determined binding and aggregation parameters and used them to predict
the time course of the formation of dimerized FcRI. The model ( Fig. 1and ``Appendix'') is straightforward and gave
close fits to the data (Fig. 2, 3). The results also indicate
clearly that the concentration of IgE monomer used to block the binding
of IgE dimers was effective.
To investigate the relationship between
the aggregation of FcRI and the phosphorylation of tyrosines on
the
and
subunits of the receptor, we compared the predicted
time course of receptor aggregation with the time course of
phosphorylation of Fc
RI (Fig. 4). The comparison showed
that the kinetics of phosphorylation does not correspond with the rate of formation of the aggregates (Fig. 5); in
particular, once formed, the stable aggregates show persistent
activity. Initially, the time course of phosphorylation parallels quite
closely the time course of the number of receptors in aggregates, but
the comparison reveals a difference at later times (compare Fig. 2and Fig. 4). What was striking was that
phosphorylation reached a plateau while aggregate formation continued.
The shape of the phosphorylation curve when no monomeric IgE was
present was similar to the shape of the phosphorylation curve when
monomer was added after 2 min and the formation of new aggregates was
blocked.
To make quantitative comparisons between the predicted kinetics of aggregation and the experimentally determined kinetics of phosphorylation and to estimate the fraction of aggregated receptors that are phosphorylated, a mathematical description of the time course of phosphorylation is an essential tool. We therefore introduced a phenomenological model that allows for tyrosine phosphorylation and both reversible and irreversible (on the time scale of the experiments) dephosphorylation (Fig. 6). The role of this minimal model is to help identify the types of interactions of receptors, kinases, and phosphatases that are consistent with the data and those that can be rejected.
The model depicted in Fig. 5is inconsistent with the experimental results, in that it cannot account simultaneously for apparent steady state levels of phosphotyrosine observed both in the absence and presence of new aggregate formation (i.e. in experiments with and without the addition of monomeric IgE). Parameters that account for the early plateau in levels of phosphotyrosine in the absence of monomer predict a rapid decay of phosphotyrosine after monomer is added, but no such decay is observed. Parameters consistent with the continued elevation of levels of phosphotyrosine after monomer is added predict that in the absence of monomer, phosphorylation of tyrosines should reach higher levels than those observed and should continue to rise over the hour period of our experiments. Again, this contrasts with our observations (Fig. 7).
In rejecting the
phenomenological model, we reject a wide array of potential
phosphorylation/dephosphorylation schemes. The model has a constant
rate of phosphorylation of non-phosphorylated receptor dimers and a
constant rate of dephosphorylation of phosphorylated receptor dimers. A
constant rate of phosphorylation is consistent with a pool of specific
kinases that is not substantially depleted, either because the pool is
large or because kinases remain active and interact with multiple
receptor aggregates. Also consistent with the constant phosphorylation
rate is a model where some or all of the receptors are stably
associated with kinases that trans-phosphorylate adjacent receptors
when dimers form(34) . Even if only a small fraction of
FcRI is associated with kinase, phosphorylation of receptor
tyrosines should increase in proportion with the formation of
additional aggregates. Constant recruitment of additional kinases to
complexes of aggregated receptors and associated kinases is also
consistent with the model and therefore cannot fully explain the data.
A constant rate of dephosphorylation is consistent with an undepleted
pool of phosphatases. If we invoke the activation or recruitment of
phosphatases to explain why phosphorylation remains constant in the
absence of monomer, despite continued aggregation, we would then
predict the decay of phosphorylation in the presence of monomer,
contrary to observation.
Simple alternatives to the model do not
correct the problems with fitting the data. In particular, we have
shown that phosphorylation of tyrosines on the subunits of FcRI
does not appear to saturate under the conditions of our experiments (Fig. 8).
One possibility we have not ruled out is that
because of the dynamics of recruitment of kinases, aggregates that form
``late'' in the response are essentially left out of the
response. This could occur, for example, if kinases that were
pre-associated with receptors were capable of dissociating and binding
to newly created phosphotyrosines on other proteins. Since Lyn kinase
is only weakly associated with unaggregated receptors but is recruited
to phosphorylated tyrosines on aggregated FcRI(15) , the
creation of new high affinity binding sites that could compete for Lyn
might reduce the number of pre-associated Lyn-receptor complexes
available late in the response. If this were so, phosphorylation of
and
would level off while aggregates continued to
accumulate because such newly formed aggregates, lacking associated
kinase, would be ineffective.
Additional evidence that aggregates
formed late in a response are less effective than aggregates formed
initially comes from degranulation studies. We have observed that when
RBL cell degranulation is induced by either dimeric ()or
trimeric (18) oligomers of IgE, the kinetics of the response
(secretion of hexosaminidase) is similar whether or not high
concentrations of monomeric IgE are added shortly (2-3 min) after
oligomer addition. For at least the first 30 min, the curves increase
and tend to parallel each other. The addition of monomer reduces
release but not to the extent that it reduces binding. That is, the
additional aggregates that form late after the addition of oligomer
appear to contribute less to the secretory response than those formed
earlier.
In closing, we wish to extend our previous discussion (18) concerning how our results on persistent activity after
cessation of aggregate formation can be reconciled with the apparently
conflicting results of experiments in which aggregation of FcRI
was induced by highly multivalent antigens. In the latter studies, the
cellular response (secretion) halted abruptly after the addition of
competing monovalent hapten, even though receptor aggregation, as
judged by persistence of cell-bound antigen, was not fully
reversed(35, 36, 37, 38) . The
degree of persistence of bound antigen depended on the time between the
addition of antigen and the addition of hapten. The longer the delay,
the greater the fraction of antigen that became resistant to
dissociation(37) .
The molecular basis for the
dissociation-resistant state is not known. The resistant state is not
due to an induced change in the affinity of IgE for an antigenic site
since the aggregation of IgE-FcRI does not alter the affinity of
IgE for monovalent ligand(39) . The aggregation of three or
more Fc receptors does lead to the immobilization of Fc
RI (40) and, at supraoptimal antigen concentrations, to
interaction of the receptor with the detergent-resistant cell skeleton (41, 42, 43, 44, 45) . An
immobilized IgE cannot diffuse away from an antigen when it
dissociates. If the antigen is bound to two or more immobile IgEs, its
motion is also restricted when one of the bound IgEs dissociates.
Consequently, immobilization of the receptors may enhance the
interaction of their bound IgE with the antigen, thus reducing the
hapten's ability to break up aggregates effectively.
That the dissociation-resistant aggregates no longer signal suggests that the receptors in such aggregates have been specifically desensitized(46, 47, 48) . In turn, this has been interpreted to indicate that signal transduction is short lived; that to maintain signaling, new aggregates must constantly be made, and that cellular responses are proportional to the rate of formation of aggregates rather than to the number of such clustered receptors. This interpretation is not consistent with our result or previous results (49) on persistent signaling by dimers and trimers of IgE.
One way to reconcile the results in the two systems is to postulate that the rate of desensitization differs markedly from one system to the other(18) . However, there is an interpretation of the experiments with multivalent antigen that is consistent with the oligomer results and that does not depend on postulating differences between the systems. The new interpretation we propose is that desensitization is a slow process, that desensitized receptors become immobile, and that because of the rebinding effects discussed above, only a small number of desensitized, immobilized receptors is needed to keep some of the bound antigen on the cell surface after hapten addition. Since only desensitized receptors remain aggregated, signal transduction ceases. In the oligomer experiments, signaling persists because both active and inactive receptors remain aggregated.
The single site forward and reverse rate constants for binding
of IgE in solution to an unoccupied FcRI are
k
and k
for monomeric IgE and k
and k
for a functional IgE in a dimer; k
and k
are the
corresponding rate constants for aggregation, i.e. for a
bifunctional dimer to bind to (or dissociate from) a second Fc
RI.
The concentrations of bound ligand satisfy the following differential
equations.
We have modeled the binding of IgE to its receptor as a simple bimolecular reaction. There is evidence that the binding reaction is more complicated than this, with a conformational change occurring after the bound complex is formed(50) . For the limited range of binding experiments we analyze however, the simple model suffices.
where P is the inverse of the 1/b intercept, i.e.P
is the limit of
the bound dimer fraction as t
or, equivalently,
as 1/t
0.
is obtained by equating two expressions for the
fraction q of non-binding IgE in dimers. The expressions for q come from the following two relations. First, with random
mixing, the fraction of non-binding dimers 1 - P = q
. Second, the fraction of doubly
binding dimers P
P
= (1
- q)
. Substituting P
= 0.54 into , we find P
= 0.19.
where A, the fraction of the cell surface occupied by
FcRI, is assumed to be small. Measurements by both post-field
relaxation (51) and fluorescence photobleaching
recovery(40, 53, 54) give values for the
lateral diffusion coefficient for mobile Fc
RI on RBL cells of
approximately 3
10
cm
s
at ambient temperatures (19-24
°C). Because k
depends on A logarithmically, it is relatively insensitive to the exact value
of A. Thus if A varies 100-fold, e.g. between 0.002 and 0.20, then with D = 3
10
cm
s
, k
varies only 5-fold, from 1.3 to 6.6
10
cm
s
.
The actual rate at which a singly bound dimer encounters a free
receptor depends not only on the forward rate constant k but also on the concentration of vacant
receptors, R. If most receptors are unoccupied, i.e.R
R
, the product k
R
is the
encounter rate, and 1/k
R
is the mean encounter time. For an RBL cell with 300,000
receptors/cell and a surface area of roughly 600 square microns (6
10
cm
/cell), k
R
is in the
range 65-330 encounters per second. Then, the diffusion limit of
the mean time for a singly bound dimer, capable of binding doubly, to
bind to a second Fc
RI is on the order of 0.01 seconds. Because of
its convoluted structure, the RBL cell surface area is the least
reliable parameter in this estimate. We have taken it to be
approximately twice that of a 5-µm sphere, but it is possible that
it is larger than this, which would increase the estimate of the
diffusion limit of the mean cross-linking time.
Therefore, there are two additional states to follow: aggregated
receptors phosphorylated on one or more tyrosine residues
(concentration Y) and aggregated receptors that
are irreversibly dephosphorylated (concentration Y
). No account is taken either of the sites of
phosphorylation or of the extent to which the pairs of receptors are
phosphorylated. Y
is now interpreted as the
concentration of bivalently bound dimers associated with
unphosphorylated receptors capable of becoming phosphorylated. The rate
of tyrosines on aggregated receptors becoming phosphorylated is
,
and the rates of reversible and irreversible dephosphorylation are
µ
and µ
, respectively. In the equations
that follow, phosphorylation and dephosphorylation are not limited by
the supply of kinases and phosphatases. for Y
becomes
The equations for Y and Y
are as follows.
Equations 6, 8, and 9 remain the same. The conservation laws become
Phosphotyrosine (measured as densitometric intensity) is then
taken to be a linear function of Y, the fraction
of receptors that are phosphorylated on tyrosines. The slope and
intercept differ from experiment to experiment.