Changes in net charge of glucocorticoid receptors by activation, and evidence for a biphasic activation kinetics

563

Molecular and Cellular Endocrinologv, 28 (1982) 563-586 Elsevier Scientific Publishers Ireland, Ltd.

CHANGES IN NET CHARGE OF GLUCOCORTICOID RECEPTORS BY ACTIVATION, AND EVIDENCE FOR A BIPHASIC ACTIVATION KINETICS

Peter A. ANDREASEN

*

Institute DK-2100

of Experimental Hormone Copenhagen 0 (Denmark)

Received

18 March

1982; revision

Research,

received

University

of Copenhagen,

11 July 1982; accepted

N&re All& 71,

23 July 1982

The kinetics of glucocorticoid receptor activation and the changes in molecular properties of the receptors by the activation were studied, employing aqueous two-phase partitioning of rat thymocyte, rat liver and mouse S49.1 lymphoma cell cytosol labelled with tritiated glucocorticoid. By a mathematical analysis of the time-course of the receptor partition coefficient during activation, we demonstrate that at least two different receptor conversions take place during this process. Partitionings at conditions excluding receptor aggregation allowed an evaluation of differences in net charge between activated and non-activated forms of native and chymotrypsin-treated receptors. The net charge of the chymotrypsinized receptor changes little by the activation, being between 0 and - 10 at pH 8 both in the non-activated and the activated state. In contrast, the activation changes the net charge of the native receptor from around -50 to around - 10. Keywords:

glucocorticoid

receptors;

two-phase

partitioning.

The accumulation of receptor-glucocorticoid complexes in nuclei in vitro and in vivo requires the preceding activation of the complexes (Higgins et al., 1979; Munck and Foley, 1979; Markovic and Litwack, 1980). However, the details of the activation reaction are not understood. The activation of glucocorticoid receptors in vitro is associated with a change in their partition coefficient in aqueous two-phase systems (Andreasen, 1976; Andreasen and Mainwaring, 1980). Phase partitioning was also used for demonstrating that the so-called nt’ receptors of certain glucocorticoid-resistant variants of the mouse lymphoma cell line S49.1 are very similar to the receptor forms produced by chymotrypsination of wild-type receptors (Andreasen and Gehring, 1981). The method offers * Present address: Laboratory of Tumor Biology, Institute of Pathology, Copenhagen, Frederik V’s Vej 11, DK-2100 Copenhagen 0 (Denmark).

0303-7207/82/OOOO-O/$02.75

0 Elsevier Scientific

Publishers

Ireland,

University

Ltd.

of

564

P.A. Andreasen

the opportunity of assessing net charge and surface properties of proteins (Albertsson, 1978). However, a couple of factors have formed an obstacle to the utilization of this opportunity in receptor studies. These factors include unexplained variations in the receptor partition coefficients with the salt composition of the phase systems (Andreasen and Gehring, 198 1) and between cytosols from different tissues, for instance between S49.1 cytosol (Andreasen and Gehring, 1981) and rat liver cytosol (Andreasen and Mainwaring, 1980; Andreasen, 198 1). The work presented here was aimed at overcoming these obstacles. We assess the contribution from receptor aggregations and from partial proteolysis by endogenous proteases to differences in the receptor partition coefficients between cytosols from different tissues, and use the method for determining changes in the net charge of the receptors in connection with activation and partial proteolysis by chymotrypsin. In addition, a mathematical analysis of the time-course of the partition coefficient during activation has shown that the activation does not proceed as a single, first-order reaction, but that at least two different receptor conversions determine the course of the approach to steady state.

MATERIALS

AND

All procedures

METHODS

were performed

at 0-4°C.

Chemicals Blue dextran 2000 (Pharmacia), bovine catalase (Sigma), chicken conalbumin (type I, Sigma), equine cytochrome c (type VI, Sigma), dextran T500 (lot no. FA 13748, Pharmacia), equine ferritin (type I, Sigma), bovine /%lactoglobulin (Sigma), whale myoglobin (type II, Sigma), chicken ovalbumin (grade V, Sigma), bovine serum albumin (Boehringer-Mannheim), Sephadex G- 150 Superfine (Pharmacia), bovine thyroglobulin (type I, Sigma). All other chemicals were as described previously (Andreasen and Gehring, 198 1). Buffers R, buffer, which was used if nothing else is indicated, contains 10 mM K,HPO, (pH 7.4) 1 mM dithioerythritol, 1 mM EDTA and 10% glycerol. R, buffer was identical to R, buffer except that it did not contain glycerol. Cells and tissues Rat livers and

thymi

were from

adrenalectomized

female

Sprague-

Activation

of glucocorticoid

receptors

565

Dawley rats, weighing approx. 125 g. The livers were used immediately after their removal from the animals. Thymocytes were prepared from the thymi as described by Schaumburg and Bojesen (1968) and stored as a pellet at -2O’C. The S49.1 mouse lymphoma sublines S49.1 TB.4 (containing a normal receptor) and S49.1 TB.4 55R (containing an nt’ receptor) were grown, harvested and stored as previously described (Andreasen and Gehring, 1981). Cytosol preparations S49.1 mouse lymphoma cell cytosol was prepared and labelled with [ 3H]triamcinolone acetonide as described previously (Andreasen and Gehring, 1981). Rat thymocyte cytosol was prepared in the same manner, except that the [3H]triamcinolone acetonide was added to the homogenate before the centrifugation. 1.7 ml of R, buffer were used per 1 g wet weight of thymus. Rat liver cytosol was prepared, labelled with [ 3H]triamcinolone acetonide and stored as described previously (Andreasen and Mainwaring, 1980) using R, buffer for the homogenization. The protein concentrations of the S49.1 cytosols, the thymocyte cytosol and the liver cytosol were 6.5 * 1.5 mg/ml (mean 2 SD, 17 determinations), 2.05 2 0.27 mg/ml (26 preparations) and 26.2 2 5.0 mg/ml (42 preparations), respectively. Before the experiments, the labelled cytosols were treated with dextran-coated charcoal as described previously (Andreasen, 198 1; Andreasen and Gehring, 1981). Chymotrypsin treatment of [ 3H]triamcinolone acetonide-labelled, charcoal-treated rat thymocyte cytosol was performed by making the cytosol 10 pg/ml in cu-chymotrypsin and leaving it for 10 min a 0°C. Previously published experiments (Andreasen and Gehring, 198 1) showed that a maximal effect of the protease on the non-activated receptor is achieved by that treatment. Phase partitioning Phase partitionings were performed as described previously (Andreasen and Gehring, 1981). Each 1 ml dextran T500-poly(ethylene glycol) 6000 two-phase system contained 6.25% of each of the polymers in buffer R, with 6 mM Tris and various concentrations of KCl, NaCl, LiCl, Na,SO, and Li,SO,. The pH of the system was approx. 8.2. The partition coefficients of the glucocorticoid-receptor complexes were calculated as the ratio between the tritium concentrations in the top and bottom phase, after a correction for the non-specifically bound steroid (Andreasen, 1981; Andreasen and Gehring, 1981). The partition coefficients of pure, commercially available proteins were calculated as the

566

P.A. Andreasen

ratios between the absorbances at 280 nm of diluted aliquots of the phases of systems with 1 mg protein per ml. DNA-Sepharose

chromatography

The separation of receptor forms able to bind to DNA-Sepharose in 0.1 M KC1 and those not able to do so was performed by DNA-Sepharose chromatography as described previously (Andreasen, 1981). The 0.5 M KC1 buffer used for eluting bound receptor from the column contained 2 mg bovine serum albumin per ml (Andreasen and Mainwaring, 1980). Gel filtration 200 ~1 cytosol aliquots were applied to an approx. 40 ml column of

Sephadex G-150 Superfine, equilibrated in R, buffer with 0.4 M KCl. The column was eluted with the same buffer, using a flow rate of about 3 ml/h. The column was calibrated with blue dextran, ferritin, catalase, bovine serum albumin, myoglobin and cytochrome c. Stoke’s radii of receptors were calculated as described by Siegel and Monty (1966).

RESULTS Factors contributing different cytosols

to differences

in receptor partition

coefficients

between

In order to identify sources of variation in the receptor partition coefficients between cytosols from different tissues, we partitioned rat liver and thymocyte cytosol samples in phase systems with different cytosol concentrations and different salt compositions (Tables 1 and 2). The cytosol samples had or had not been incubated with 0.4 M KC1 at 0°C i.e. a receptor-activating condition. There are three important points to be noted in Tables 1 and 2: (1) Increments in the cytosol protein concentration of 0.1 M KC1 phase systems above approx. 125 pg/ml lead to decreases in the partition coefficient of the activated receptors. Protein partition coefficients do not in general depend on the protein concentration of the phase systems (Albertsson, 1971). Therefore, the decreases are indicative of variations in the state of the receptor protein. Since the 15 set used for each partitioning is too short a time to allow any significant activation, the decreases are not due to variations in the fraction of activated receptors. The phenomenon is, however, readily explainable by an effect on the partition coefficient of receptor aggregations equilibrating during the partitionings, the aggregated forms having the lowest partition coefficient. But

Activation

561

of glucocorricoid receptors

Table 1 Partition coefficients of glucocorticoid systems with different salt compositions

salt

Cytosol

protein

receptors of rat thymocyte cytosol and different cytosol concentrations

in two-phase

concentration

composition Untreated

cytosol

Chymotrypsinized

cytosol

40 &ml

10 &ml

80 Wml

40 Wml

10 jWm1

0.0819 * 0.0276

0.0870 eo.0168

0.0913 kO.0143

0.0973 eo.0453

0.109 * 0.027

0.1 M KCI+ 0.9 M NaCl

(18) 0.363 AO.046

(4) 0.445 ‘0.103

(4) 0.343 * 0.035

(36) 0.350 * 0.066

(4) 0.391 * 0.086

0.22 M KCl+ 0.26 M Li,SO,

(18) 0.499 zk 0.073

(4) nd.

(4) n.d.

(38) 0.237 -co.024

(6) n.d.

Non-activated: 0.1 M KC1

(18) Activated: 0.1 M KC1

0.1 M KClt 0.9 M NaCl

(8)

1.11 20.07

1.20 LO.16

1.03 fO.05

1.16 *0.17

1.29 20.13

(4) 2.41 kO.34

(4) 2.17 20.26

(4) 3.01 f0.62

(18) 2.60 20.32

(4) 2.43 eO.85

(6)

(4)

(4)

(34)

(10)

Aliquots of [ 3H]triamcinolone acetonide-labelled, charcoal-treated rat thymocyte cytosol were partitioned in phase systems with the indicated salt compositions and cytosol protein concentrations, before (non-activated) or after (activated) a 3 h incubation of the IO-fold diluted cytosol with 0.4 M KC1 at O’C. The indicated cytosol protein concentrations of the phase systems are approximate, calculated from the determinations of the protein concentrations of the cytosol given in Materials and Methods. The table shows mean and standard deviations of the receptor partition coefficients. The numbers of determinations given in parentheses are twice the number of independent experiments. n.d.=not determined.

the effect of the protein concentration disappears with protein concentrations below a certain limit, and is also counteracted by a high salt concentration in the phase systems. In addition, at these conditions, the values for rat thymocyte and liver cytosol are very similar or indistinguishable, and very similar to the previously published values for S49.1 cytosol (Andreasen and Gehring, 198 1). Thus, with sufficiently low cytosol concentrations and sufficiently high salt concentrations in the phase systems, receptor aggregations do not appear to influence the receptor partition coefficients.

P.A. Andreasen

568 Table 2 Partition coefficients of glucocorticoid receptors of rat liver cytosol with different salt compositions and different cytosol concentrations Salt composition

Non-activated: 0.1 M KC1

0.1 M KCl+ 0.9 M NaCl 0.22 M KCl+ 0.26 M Li,SO,

Cytosol

protein

in two-phase

systems

concentration

1000 pg/ml

250 pg/ml

125 pg/ml

50 pg/ml

0.0220 -c 0.0097

0.0259 * 0.0092

0.027 1 -0.0103

0.0312 f0.0091

(18) 0.273 * 0.022

(10) 0.327 kO.032

(8) 0.362 * 0.084

(10) 0.342 20.048

(6) n.d.

(6) n.d.

(6) n.d.

(6) 0.557 ‘0.036 (2)

Activated: 0.1 M KC1

0.1 M KCl+ 0.9 M NaCl

0.255 kO.086

0.522 -co.113

0.806 kO.167

0.934 -0.103

(24) 1.46 50.25

(12) 2.05 20.33

(10) 2.20 20.35

(6) 2.07 20.13

(16)

(10)

(10)

(6)

Rat liver cytosol with 50% glycerol was diluted 5-fold with R, buffer and incubated for 3 h at 0°C with 0.4 M KCl. Cytosol aliquots were then taken for partitioning. For further details, see the legend to Table 1.

(2) The partition coefficients of non-activated receptors of rat liver and thymocyte cytosol in systems with 0.22 M KC1 + 0.26 M Li,SO, are about 0.5 (Tables 1 and 2), while that of non-activated receptors of S49.1 wild-type cytosol is about 2.4 (Fig. 5). In contrast to S49.1 cell homogenates, in which the receptors are not degraded by endogenous proteases (Yamamoto et al., 1976) endogenous proteases in homogenates from other tissues may cause partial proteolysis of receptors identical to that produced by chymotrypsin and trypsin (Carlstedt-Duke et al., 1977, 1979; Wrange and Gustafsson, 1978). Since chymotrypsin treatment of S49.1 wild-type cytosol changes the receptor partition coefficient in 0.22 M KC1 + 0.26 M Li,SO, from 2.4 to 0.2, and trypsin treatment has a similar effect (Andreasen and Gehring, 1981), the lower receptor partition coefficients in rat liver and thymocyte cytosol could be due to a partial proteolysis of a fraction of the receptors in these cytosols. We therefore estimated the fractions of partly proteolysed receptors by gel

Activation of ~~~c~~rtic~idreceptors

569

filtration in 0.4 M KC1 buffers, i.e. the activated receptors were investigated. In rat liver cytosol, 41.4 2 1.1% were in a 59 A Stoke’s radius form and 58.6 zt: 1.1% in a 40 A form (mean ‘- SD, 2 determinations). In the thymocyte cytosol, 16.0 C 10.7% were in the 59 A form, 60.2 -C7.1% in the 40 A form, and 23.8 k 6.4% in a 28 A form (mean * SD, 4 determinations). Chymotrypsin treatment of thymocyte cytosol converted all 59 A receptors to 40.” receptors, while trypsin treatment converted all receptors to the 28 A form (not shown). Thus, these forms correspond to the (native) 60 A form, the 36 A form and the 19 A form of Carlstedt-Duke et al. (1977). Chymotrypsin treatment of thymocyte cytosol caused a reduction of the partition coefficient in 0.22 M KC1 + 0.26 M Li,SO, to a value identical to that of chymotrypsinized receptors of 549.1 cytosol (Table 1). Therefore, the difference in partition coefficients in this system between S49.1 cytosol on one side and rat liver and thymocyte cytosol on the other is readily explained by the high fraction of partly proteolysed receptors in the thymocyte and liver cytosols. (3) In systems with 0.1 M KCl, the partition coefficient of receptors of thymocyte cytosol not exposed to activating conditions (Table 1) is much higher than those of the receptors of rat liver cytosol (Table2) and of S49.1 cytosol (Andreasen and Gehring, 1981). Thus, the relatively high partition coefficient is correlated with the occurrence of ‘tryptic’ receptor fragments. Activated ‘tryptic’ receptor fragments were previously found to have partition coefficients indist~g~shable from those of activated ‘chymotryptic’ fragments (Andreasen and Gehring, 1981). However, non-activated ‘tryptic’ fragments, initially having partition coefficients identical to those of non-activated ‘chymotryptic’ fragments, ‘are converted to the activated form with a rate constant of approx. 0.004 rnin- ’ in 0.05 M KC1 at O”C, at which condition there is no measurable activation of ‘chymotryptic’ fragments; this activation does not require the continued presence of trypsin activity (Andreasen, unpublished). Therefore, a possible explanation of the high partition coefficient is the submission of the receptors to trypsin-like proteases during the preparation of the thymocyte cytosol, resulting in the conversion of a fraction of the receptors to ‘tryptic’ fragments, part of which is becoming activated ‘spontaneously’ and thereby increasing the partition coefficient of the receptor population as a whole (the partition coefficients of the entire receptor population in systems with higher partition coefficients of the non-activated receptors are far less sensitive to small variations in the amount of activated receptors, explaining the absence of any difference for instance in 0.1 M KC1 + 0.9 M NaCl). This idea is supported by measurements of the partition coefficient of rapidly activated receptors in the presence of dextran sulphate. In

570

P.A. Andreasen

systems with 0.1 M salt, dextran sulphate partitions exclusively to the bottom phase and, when present in the systems together with the receptors, lowers the partition coefficients of receptors binding to it. Thus, whereas the partition coefficients of activated ‘tryptic’ receptor fragments are not lowered by dextran sulphate, the presence of dextran sulphate in systems with 0.1 M salt reduces the partition coefficient of ‘chymotryptic’ receptor fragments of S49.1 cytosol from about 1 to approx. 0.05 (Andreasen and Gehring, 1981). In addition, activated ‘chymotryptic’ receptor fragments of thymocyte cytosol, partly purified by adsorption on a DNA-Sepharose column and eluted with 0.5 M KCl, have a partition coefficient with dextran sulphate of a similar magnitude (not shown). But the receptors of freshly prepared, chymotrypsinized thymocyte cytosol, fully activated by incubation with 1 M NaCl for 15 min, have a partition coefficient in the presence of dextran sulphate of 0.209 * 0.015 (mean * SD, 4 determinations). This difference is calculated to be explainable by approx. 28% of the receptors in the cytosol being ‘tryptic’ fragments, unable to bind to dextran sulphate (using eqn. (2) of Andreasen and Mainwaring, 1980, and phase volumes of 0.69 and 0.41 ml for the top and bottom phases, respectively). This figure agrees well with the fraction of ‘tryptic’ fragments determined by gel filtration. On the other hand, the receptor partition coefficient in the presence of dextran sulphate remains lower than or indistinguishable from the value given above for up to 6 h of incubation of the IO-fold diluted thymocyte cytosol with 0.05, 0.1, 0.15 or 0.4 M KCI. Only after 24 and 48 h is a slight increase in the partition coefficient with dextran sulphate seen (not shown). The observed increase is explainable by an increase in the percentage of activated tryptic fragments to approx. 40% and 55% at 24 and 48 h, respectively. Alternatively, since free triamcinolone acetonide also does not bind to dextran sulphate, but has a partition coefficient of 1.63, the increase in the partition coefficient with dextran sulphate between 6 and 24 h of incubation may be due to some dissociation of steroid from the complex, the actually observed figures corresponding to approx. 23% of the complexes being dissociated during this time-period. Thus, the 25% ‘tryptic’ receptor fragments observed by gel filtration appear to be the result of proteolytic activity during the preparation of the cytosol, while the production of such fragments during incubations of the cytosol is at best very slow. Conclusively, the differences in receptor partition coefficients between the different cytosol preparations can be accounted for by aggregations of receptors at high protein concentrations in the phase systems and by partial proteolysis of the receptors during the preparation of some cytosols.

Activation

571

of glucocorticoid receptors

The kinetics of activation The time-course of the partition

coefficient of the glucocorticoid-receptor complex during activation by incubation with high salt concentrations at 0°C was studied, using phase systems with 0.1 M KC1 plus 0.9 M NaCl. The results of typical experiments with rat thymocyte cytosol are shown in Fig. 1. In order to characterize the receptor conversions underlying the timecourse of the partition coefficient, we developed formulae describing the time-courses to be expected from reaction schemes with two, three or four receptor forms, interconvertible through one, two or three monomolecular reactions (Fig. 2). Bimolecular reactions were not considered; previous findings of the absence of any effect of the total glucocorticoid-receptor complex concentration on the time-course of activation argue strongly against this possibility (Andreasen, 1978; Atger and Milgrom, 1976; Bailly et al., 1977). The expressions giving the time-course of the concentration of each receptor form in the schemes of Fig. 2 can be

0

1

2

3 INc”BAT~

TIME

;)

I

6

T”

48

Fig. I. Time-course of the partition coefficient of the glucocorticoid receptor of rat thymocyte cytosol during incubations at 0°C. [ 3H]Triamcinolone acetonide-labelled, charcoal-treated and chymotrypsinized rat thymocyte cytosol was diluted IO-fold and incubated with 0.05 M (m), 0.10 M (0) 0.15 M (0) or 0.40 M KC1 (0) at O’C. At the indicated time-points, aliquots were taken for partitioning in phase systems with 0.1 M KCl+0.9 M NaCl, and approx. 40 cg cytosol protein per ml. The symbols show means and standard deviations of the receptor partition coefficient for 2 determinations at each time-point in one typical experiment at each KC1 concentration. The fully drawn curves were constructed from the 01, h,, X,, K, and C values fitted on the basis of eqn. (2) as described in the text. The dashed curves were constructed from o, K, and C values fitted on the basis of eqn. (1).

P.A. Andreasen

7.

a-b X

Fig. 2. Reaction schemes with two, three or four different receptor forms, interconvertible through one, two or three monomolecular reactions. The different receptor forms are denoted by a, b, c, x and y.

found by conventional methods for solving differential equations. These expressions can then be used to derive the corresponding expressions for the time-course of the partition coefficient K, of the receptor population as a whole; this is the measured parameter. The following expression is obtained in the cases of reaction schemes 1, 4 and 7: K =C.K,-Be-“’

I

(1)

CS Te-”

and in the cases of reaction schemes 2, 3, 5 and 6: K = C. K, - B. (a eFhlr + (1 - (Y)e-‘z’) f

C+T*(ae

-A,t + (1 - a)e-bt)

(2)

where T and B are the volumes of the top and bottom phase, respectively (equal to 0.6 and 0.4 ml, respectively, in 1 M ionic strength phase systems), K,, and K, are the partition coefficients of the receptor population as a whole at the start and the end of the reaction, respectively, C equals (K,T + B)/( K, - K,,), and X, X,, A, and (Yare positive constants. (The development of these formulae is described in the Appendix.) Thus, the receptor partition coefficient may follow either a monophasic (eqn. (1)) or a biphasic approach (eqn. (2)) to the steady-state value.

Activation

513

of glucocorticoid receptors

In order to decide which of these equations provides the best description of the observed time-course of the receptor partition coefficient during incubations at O”C, a statistical analysis was performed. Experiments with thymocyte cytosol incubated with 0.4M KC1 were first considered. Inspecting the data of 3 independent experiments, the measured partition coefficients were found to vary slightly but significantly between experiments; therefore, each experiment was analysed separately. First, experimental data pairs (t, In K,) were submitted to non-linear regression analyses on the basis of eqn. (1) and eqn. (2), respectively, estimating the parameter sets (h, K,, C) and (a, A,, X,, K,, C), respectively, by the method of least squares. Since the double determination variances of the measured In K, values were found not to vary with the incubation time, non-weighted analyses were performed. The NLIN procedure published by the Statistical Analysis System Institute (1979) was used. The curves corresponding to the parameter sets determined in a typical experiment with 0.4 M KC1 are shown in Fig. 1.

Table 3 Analysis of variance of the logarithm to measured and calculated receptor coefficients during incubation of rat thymocyte cytosol with 0.4 M KC1 at 0°C Source of variation

Sum of squares

d.f.

Mean square

Between

0.0541

20

0.0027 1

0.0695

15

0.00463

In K,,j and lnK,

F

P

1.71 Between m,

and In &,

Between

In K,,j and ln t2,,

0.1236

35

0.00353

Between

In I,,, and In Is,,

0.1724

2

0.08620

Between

In K,,j and In I,,,

0.2960

37

0.00800

partition

24.45

13%

aO.l%

Test of the fit of the time-course of the receptor partition coefficient expected from eqns. (1) and (2), respectively, to the experimentally observed time-course during incubation of rat thymocyte cytosol with 0.4 M KC1 at O’C. The experiment shown in Fig. 1 was analysed. In K,,, denotes the natural logarithm to the jth value of the receptor partition coefficient measured at the incubation time t (j = 1 or 2); In K, denotes the mean of the natural logarithm to the values measured at the time t; I,,, and &., denote the values expected to be assumed by the receptor partition coefficient at the time t on the basis of eqn. (1) and eqn. (2), respectivekd.f. denotes degrees of freedom. The comparison of the variation between In K,,j and In K, and the variation between m, and In Is,, tests for divergence between the experimental data and a biphasic kinetics. The comparison of the variation between ln K,,j and In&, and the variation between In&, and lnf,,, tests for differences between the quality of the fits provided by eqn. (1) and eqn. (2).

574

P.A. Andreasen

Next, analyses of variance like that exemplified in Table 3 were performed in order to assess the quality of the fits of the two models to the data. Approximate F-tests showed that the observed time-courses of the receptor partition coefficient were not significantly different from those expected from eqn. (2) (in the 3 experiments, P = 13%, 2% and 44% respectively). It was then tested whether the observed time-courses were equally well described by eqn. (1). For all 3 experiments with rat thymocyte cytosol incubated with 0.4M KC1 at 0°C it could be concluded that eqn. (2) provides a significantly better description of the observed time-course of the receptor partition coefficient than eqn. (1) (P c O.l%,, < 0.1% and < 0.2%, respectively). The same conclusion was reached with respect to the time-course of the receptor partition coefficient during incubations of rat thymocyte cytosol with 0.15 M KC1 at 0°C (Fig. 1). Similarly, with S49.1 wild-type and S49.1 nt i cytosol incubated with 0.4 M KC1 at 0°C eqn. (2) provided a significantly better description of the data than eqn. (1) although the non-systematic divergence of the data from the curve predicted by eqn. (2) was somewhat bigger than with thymocyte cytosol (data not shown). Table4 shows the (Y,X, and A, values determined on the basis of eqn. (2) in the 3 experiments with thymocyte cytosol incubated with 0.4 M KCl, and in typical experiments with thymocyte cytosol incubated with 0.15 M KC1 and with S49.1 cytosol incubated with 0.4 M KCl. The parameters determined in the experiments with S49.1 cytosol are seen to be very similar to those determined in experiments with thymocyte cytosol. Thus, in all the cases investigated, the receptor conversions underlying the change of the partition coefficients are highly unlikely to be those described by reaction schemes 1, 4 and 7, while no significant divergence between the observed time-course of the receptor partition coefficient and that expected from reaction schemes 2, 3, 5 and 6 has been found. Obviously, however, it cannot be excluded that more complicated reaction schemes may underlie the observed kinetics. The divergence from simple monophasic kinetics is not due to the production and subsequent activation of ‘tryptic’ receptor fragments, since it is observed with S49.1 cytosol, in which no ‘tryptic’ receptor fragments are present. In addition, the presence of a constant fraction of ‘tryptic’ fragments in thymocyte cytosol does not affect the conclusion concerning the biphasic course of the partition coefficient, since the time-course of the partition coefficient of trypsinized receptors in 0.15 and 0.4M KC1 are indistinguishable from or very similar to that of chymotrypsinized receptors (not shown). Furthermore, the biphasic kinetics is not accountable for by even the maximal dissociation of steroid

0.4 M KCl, 0°C

S49.1 nt’

0.546 = 0.069

0.418rtO.023

receptors

0.0245 2 0.0042

0.0150”0.0028

0.00498”0.00142

0.0330 * 0.0055 0.0464 -c 0.0949 0.0514*0.0048

h, a (min-‘)

of glucocorticoid

during

activation

0.154co.033

0.234r2rO.025

0.0332 -c 0.00875

0.221~0.051 0.559-cO.123 0.614=kO.337

X, a (mm-‘)

a The parameters shown were fitted as described in the text. Estimates* standard errors are given. b The parameters estimated in 3 independent experiments with IO-fold diluted rat thymocyte cytosol incubated with 0.4 M KC1 at O*C are shown; the upper set of paremeters corresponds to the experiment shown in Fig. 1. The set of parameters shown for 0.15 M KC1 corresponds to the experiment of Fig. 1. ’ The parameters for S49.1 cytosols were obtained by analysing previously published data (Andreasen and Gehring, 1981).

0.4 M KCl, 0V.J

s49.1 wt=

0.518‘0.113

b

Rat thymocyte

0.15 M KC.& O°C

0.547~0.071 0.586~0.042 0.804*0.053

0.4 M KCI, WC

b

coefficient

Rat thymocyte

of the partition aa

the time-course

Incubation condition

characterizing

Cytosol

Table 4 Parameters

n

b

P. A. Andreasen

516

from the complex that is compatible with the control experiments with dextran sulphate (previous section), since a correction of the 24 h values of the partition coefficients in 0.1 M KC1 + 0.9 M NaCl for this amount of free steroid would make the deviation from simple monophasic kinetics even bigger. We next investigated the relationship between the changes in the receptor partition coefficient and changes in the amount of complex able to bind to DNA-Sepharose during incubations of thymocyte cytosol (Fig. 3). The maximally attainable binding, produced by 3 h of incubation with 0.4M KC1 at 0°C corresponds to 66% of the tritium in the

“.

0

i

i

j INCUBATION WE

5

i

(hfi

Fig. 3. Time-course of the amount of receptors able to bind to DNA-Sepharose during incubations with 0.15 and 0.4 M KC1 at O’C. [3H]Triamcinolone acetonide-labelled, charcoal-treated and chymotrypsinized rat thymocyte cytosol was diluted IO-fold and incubated with 0.15 (0) or 0.4 M KC1 (0) at 0°C for the indicated time-periods. The cytosols were then analysed for the fraction of receptors able to bind to DNA-Sepharose in 0.1 M KCl. Mean*SD of at least 2 determinations at each time-point is indicated. The fully drawn lines were constructed under the assumption that the amount of DNA-binding receptors follows the same approach to steady state as the partition coefficient, that is, B)/(K, - Ka)).((K, - K,)/(K,T+ B)), where a de(a, - e,)/(GJ - a,)=((KeT+ notes the fraction of binding receptors, and K,, K,,, K,, T and B have the meaning described in the text. The K,, K,, and K, values used were the means of those obtained in 3 independent experiments with 0.4 M KC1 and 3 independent experiments with 0.15 M KCl. The n, value used for the 0.15 M KC1 curves (equal to 60%) was calculated from the experimental 0, in 0.4 M KCl, assuming that the difference between the partition coefficients at steady state at the two KC1 concentrations represents a difference in the amount of DNA-binding receptors (using eqn. (2) of Andreasen and Mainwaring, 1980). The dashed and the dotted lines were constructed under the assumption that (a, - a,)/( a, - a,,) = e-x’ with X equal to 0.00498 min-’ and 0.221 mm-‘, respectively. These correspond to the X, value in 0.4 M KC1 and the h, value in 0.15 M KCl, respectively (Table 3).

Aciivation

of glucocorticoid receptors

517

cytosol. The main part of the 34% not binding is accountable for by the ‘tryptic’ receptor fragments in the thymocyte cytosol. This notion is supported by the finding that partitionings with dextran sulphate after an additional 3 h incubation of the tritium not binding to DNA-Sepharose after the first incubation demonstrated the absence of dextran sulphate affinity of that tritium fraction. In addition, the partition coefficients in systems with 0.1 M KC1 + 0.9 M NaCl and 0.22 M KC1 + 0.26 M Li,SO, of the activated receptors eluted from DNA-Sepharose are indistinguishable from those of the total tritium population of the cytosol incubated with 0.4M KC1 at 0°C showing that more than 90% of the receptors in the cytosol are in the activated state. The time-course of the amount of receptors able to bind to DNA-Sepharose will not be affected by the presence of a constant fraction of ‘tryptic’ receptor fragments, since here only a relative measure of the amount of activated complexes is required. Furthermore, as described in the previous section, the production of new ‘tryptic’ fragments is insignificant, at least during the first 6 h of incubation at 0°C. The measurements of the amount of receptors able to bind to DNA-Sepharose are not by themselves accurate enough to reveal whether a biphasic approach towards steady state is being followed. However, the data do allow the conclusion that the approach of the amount of DNA-binding receptors towards steady state in 0.4 M KC1 is slower than that expected from the rapid phase of the increase of the partition coefficient alone, that the approach towards steady state in 0.15 M KC1 is faster than that expected from the slow phase of the increase of the partition coefficient alone, but that at both KC1 concentrations the approach towards equilibrium is compatible with that expected if the entire change of the partition coefficient is due to activation (Fig. 3). It appears, therefore, that the activation is the only process changing the receptor partition coefficient. Conclusively, the existence of the biphasic kinetics may be due either to the occurrence of receptor conversions, different from the activation, which do not by themselves change the receptor partition coefficient (reaction schemes 3 and 6), to the existence of non-interconvertible receptor forms, which are not differing in their partition coefficients but are activated at different rates (schemes 2 and 5), or to an activation kinetics more complicated than those considered here. Changes in receptor net charge by activation and partial proteolysis The relationship between the partition coefficient K of a substance and its net charge Z is given by the following equation (Albertsson, 1971):

578

P.A. Andreasen

(3) K, denotes the contribution to the partition coefficient of the non-electrostatical properties of the substance. A, is independent of the partitioned substance, but varies with the nature of the salts in the phase system. We partitioned ovalbumin and bovine serum albumin in systems with varying pH and varying salt composition, but a constant total ionic strength of either 0.4 or 1.0 M. The net charges of the proteins at the different pH values were obtained from the literature (Cannan et al., 1941; Tanford et al., 1955). A linear relationship between In K and Z was found with ‘all the tested salt compositions of the systems. The slopes of the lines give the A, value characteristic of each salt composition. As expected, identical A, values were found in experiments with ovalbumin and bovine serum albumin. The values found with ovalbumin are listed in Table 5. If K, and Z of glucocorticoid receptors are independent of the salt composition of the phase systems, Z will be given by the slope of the straight line obtained in semilogarithmic plots of the receptor partition Table 5 A, values of phase systems Salt composition 0.4 0.1 0.1 0.1 0.1 0.1

M M M M M M

with 0.4 and 1.0 M ionic strength AS&SD

of phase systems

ionic strength: KC1+0.3 M NaCl KCl+0.12 M NaCl+0.06 KCl+0.3 M LiCl KCI+O.I M Na,SO, KCl+O.l M Li,SO,

1.O M ionic strength: 0.1 M KC1+0.9 M NaCl 0.1 M KCl-tO.43 M NaC1+0.16 0.1 M KC1 + 0.9 M LiCl 0.22 M KCl+0.26 M Na,SO, 0.1 M KC1+0.3 M Na,SO., 0.22 M KCl+0.26 M Li,SO, 0.1 M KC1+0.3 M Li,SO,

M Li,SO,

M Li,SO.,

0.0730-t0.0015 0.0387 * 0.0039 0.0458*0.0012 0.0243r+O.O019 0.0133~0.0008

0.0559~0.0014 0.0405~0.00010 0.0321~0.0019 0.0282’0.0017 0.0246 *0.0023 0.0147~0.0012 0.0163 -to.0022

Ovalbumin was partitioned in phase systems with the indicated salt compositions. For each salt composition, the partition coefficient K was determined at 7 different pH values in the range 4-7, with at least 2 determinations at each pH. The net charge Z at the different pH values was read from the titration curves for ovalbumin given by Cannan et al. (1941). The A, values are the slopes of the lines defined by the (Z, In K) data pairs, and were determined by simple linear regression analysis.

Actioation

of glucocorticoid receptors

519

coefficient vs. the A, values of the systems. However, plotting the partition coefficients of activated and non-activated S49.1 wild-type and nt’ receptors in phase systems with 1 M ionic strength in that way, the partition coefficients are scattered without any obvious pattern (Fig. 4A). Thus, Z or K, of the receptors must depend on the salt composition of these phase systems. Among a number of other proteins partitioned in 1 M ionic strength systems, similar deviations from linearity were found with conalbumin, cytochrome c, P-lactoglobulin and thyroglobulin. Since the deviations are observed with so many proteins, it seems likely that the structure of the phase system changes with the salt composition, influencing the K, of the proteins. However, a linear relationship does exist

0

I

IT Y 0I

I 0.01

.5

n T A

0.02

0.03

0.b

oh5

I

1, AsO

0 NONACTIVATED

,

,

0.01

0.02

0 ACTIVATED

0.03

0.04

0.05

Fig. 4. Relationship between the partition coefficients of S49.1 mouse lymphoma cell wild-type and nti receptors in 1 M ionic strength phase systems and the A, values of the systems. Aliquots of [ ‘Hltriamcinolone acetonide-labelled, charcoal-treated cytosol from. S49.1 wild-type or S49.1 nt’ cells were partitioned before or after a 3 h incubation of the IO-fold diluted cytosols with 0.4 M KC1 at O’C. Each 1 ml two-phase system contained approx. 130 cg cytosolic protein. The two-phase systems used are listed in Table 4 togethti with the corresponding A, values. A. Partition coefficients of non-activated wild type (D) and nt’ (V) receptors, plotted semilogarithmically vs. the A, values. The symbols incf&ste mean and standard deviation for at least 6 determinations in at least 3 independent experiments. B. Ratios between the partition coefficients of non-activated (0) and activated (0) wild-type and nt’ receptors, plotted semilogarithmically vs. the A, values. At kas~6 determinations of the partition coefficients of each receptor in at least 3 independent experiments were undertaken. The symbols indicate the ratios between the means. and the corresponding standard deviations. The parameters of the lines drawn were deterinined by simple linear regression analysis. Slopes of -5l.Sk6.2 and -9.7k4.9 are found for non-activated and activated receptors, respectively.

580

P.A. Andreasen

between the logarithm of the ratio between the partition coefficients of the 529.1 wild-type and nt’ receptors and the A, values (Fig. 4B). According to eqn. (3), the following relationship should exist between this ratio and the difference in net charge between the two receptor forms:

It is conceivable that concomitant non-systematic variations in the K, ratio and the Z difference may outbalance each other, resulting in the linear relationship between ln(K,,/K,,,) and A,. However, by far the most likely interpretation of the linear relationship is that the K, ratio does not vary with the salt composition of the phase system, and that the slopes of the lines therefore reflect the charge differences between the two receptor forms. According to this interpretation, the difference in net

5-

x

6

lt

w 0

k 0.5-

B 8 E z %

z

0.1-

0.05;

Obl Ob2 Ob3 oh

ohs

0106 0:07 obr

AS

Fig. 5. Relationship between the partition coefficients of chymotrypsinized rat thymocyte glucocorticoid receptors in 0.4 M ionic strength phase systems and the A, values of the systems. Aliquots of [ 3H]triamcinolone acetonide-labelled, charcoal-treated and chymotrypsin&d rat thymocyte cytosol were partitioned before (a) or after (0) a 3 h incubation of the IO-fold diluted cytosol with 0.4 M KC1 at 0% Each 1 ml phase system contained approx. 40 pg cytosolic protein. The phase systems used are listed in Table 4 together with the corresponding A, values. The receptor partition coefficients are plotted semilogarithmitally vs. the A, values. The symbols indicate means and standard deviations for 4 determinations in 2 independent experiments. The parameters of .the lines drawn were determined by simple linear regression analysis. The slopes for non-activated and activated receptors are - 6.8 * 4.5 and - 3.1 -C 1.4, respectively.

Activation

of glucocorticoid receptors

581

charge of the receptors in the non-activated state is about 50 at the pH of the phase systems (approx. 8), the wild-type receptor being most negative. Also, the K, value of the wild-type receptor is much larger than that of the nt i receptor, indicating a difference in non-electrostatical properties, for instance surface hydrophobicity. In the activated state the differences in charge and non-electrostatical properties of the two receptors are only small. Since previous studies have shown that nt’ receptors are indistinguishable from chymotrypsinized wild-type receptors (Andreasen and Gehring, 1981) the same differences exist between native and chymotrypsinized receptors. Using phase systems with a total ionic strength of 0.4 M, a reasonable approximation to linearity in In K/A, plots is obtained (Fig. 5). The plots show that the net charge of chymotrypsinized receptors is low and not changed measurably by activation. Therefore, the increase in the partition coefficient indicates that the non-electrostatical properties of chymotrypsinized receptors change during activation. The absence of any charge change by the activation of chymotrypsinized receptors and the large difference in charge between the native and the chymotrypsinized receptors in the non-activated but not in the activated state, show that the activation of native receptors is associated with a large change in the net charge towards less negative values.

DISCUSSION The investigations presented here show that partial proteolysis by endogenous proteases and rapidly equilibrating aggregations of the receptor during the partitioning are able to explain differences in the measured receptor partition coefficients between rat thymocyte, rat liver and mouse S49.1 lymphoma cell cytosol. For instance, aggregations are able to account for the relatively low partition coefficients of activated receptors in previous studies with rat liver cytosol, where a high cytosol protein concentration in the phase systems was used (Andreasen and Mainwaring, 1980; Andreasen, 1981). Chymotrypsin treatment has previously been shown to result in a reduction of the molecular weight of activated receptors from 102000 to 46000 (Wrange and Gustafsson, 1978; Carlstedt-Duke et al., 1979). This paper shows that chymotrypsin treatment results in a large change in the net charge of the non-activated receptor, but that the difference in net charge between the activated native and chymotrypsinized receptors is small. It is not known whether the chymotrypsin treatment is associated

582

P.A. Andreasen

with the release from the non-activated receptor of the 56000 mw peptide, which is lost from the activated receptor. If this is not the case, the chymotrypsin treatment only resulting in a cleavage of the receptor polypeptide chain, and the release of the 56000 mw peptide taking place during activation, the charge difference must be due to a conformational change of the non-activated receptor and/or possible associated proteins, which does not persist after the activation. If the 56000 mw peptide is released from the non-activated receptor, the additional possibility exists that the charge change following activation of the native receptor may be confined to this peptide. In a previous paper (Andreasen, 1978) observations were reported which suggested that the receptor activation is not adequately described as a single, first-order reaction. In that study, the time-course of the receptor partition coefficient during incubations of rat thymocyte cytosol with 0.115 M and 0.230 M KC1 at 4 and - 5°C was followed. Apparently, the steady-state value of the partition coefficient was much higher in 0.230 M KC1 than in 0.115 M KCl, and was approached relatively slowly, being reached after 2-3 h at both ionic strengths. This slow change of the partition coefficient was ascribed to the activation, since this process, as assayed by accumulation of receptor in chromatin, approached an apparent steady-state value with the same rate during incubations with 0.125 M salt (Andreasen, 1976). Indications of a different, rapid change of the receptor partition coefficient were also found but this was thought to reflect a change different from the activation (Andreasen, 1978). However, the relatively high receptor partition coefficient reached during incubation with 0.230 M KC1 could not be lowered by changing the KC1 concentration back to 0.115 M. This suggested that the apparent steady-state value reached at the low ionic strength did not represent a true equilibrium but a metastable state, at which activation was still going on, but at a rate much lower than during the initial phase. In the present study, by using longer incubation times, by taking samples for partitioning more frequently, and by undertaking a mathematical analysis of the data, we demonstrate that the receptor partition coefficient approaches approximately the same steady-state value in 0.15 M and 0.40 M KCl, but that the approach to steady state occurs through two steps with different rates, both of which appears to be associated with an increase in the amount of receptor able to bind to DNA. Thus, contrary to what was believed previously (Andreasen, 1978, 1981), the very rapid change of the partition coefficient in 0.23 and 0.40 M KC1 also appears to be associated with receptor activation. In addition, the value reached after 2-3 h at the lower ionic strengths only corresponds to completion of the rapid phase of activation; thus, it does

Activation

of glucocorticoid receptors

583

indeed not represent a true equilibrium, since the second slower phase of activation follows. Thus the data presented here confirm the idea suggested by the previous work. However, the mechanism behind the biphasic kinetics remains unknown, since it is compatible with a number of different reaction schemes. The most attractive interpretation of the data is that the rapid phase represents the approach to equilibrium between the non-activated and activated states of one form of the receptors, while the slow phase depends on a conversion of that receptor form to a different one, resulting in a shift in the equilibrium in favour of the activated state. Such a conversion could play a regulatory role in intracellular glucocorticoid receptor function.

ACKNOWLEDGEMENTS The statistical analysis of the time-course experiments was performed by Dr. Anders M&up Jensen, of the Statistical Research Unit of the Danish Medical and Social Science Research Councils. Mrs. Aase M. Frederiksen and Mrs. Kirsten Olsen are thanked for technical assistance. This work was supported financially by the Danish Cancer Society and the Danish Medical Research Council.

REFERENCES Albertsson, P.-A. (1971) Partition of Cell Particles and Macromolecules (Almquist and Wiksell, Stockholm). Albertsson, P.-A. (1978) J. Chromatogr. 159, 111-122. Andreasen, P.A. (1976) B&him. Biophys. Acta 428, 792-807. Andreasen, P.A. (1978) B&him. Biophys. Acta 540, 484-499. Andreasen, P.A. (1981) Biochim. Biophys. Acta 676, 205-212. Andreasen, P.A. and Gehring, U. (1981) Eur. J. B&hem. 120,443-449. Andreasen, P.A. and Mainwaring, W.I.P. (1980) B&him. Biophys. Acta 631, 334-349. Atger, M. and Milgrom, E. (1976) J. Biol. Chem. 251,4758-4762. Bailly, A., Sallas, N. and Milgrom, E. (1977) J. Biol. Chem. 252, 858-863. Cannan, R.K., Kibrick, A. and Palmer, A.H. (1941) Ann. N.Y. Acad. Sci. 41, 243-266. Carlstedt-Duke, J., Gustafsson, J.-I\. and Wrange, G (1977) Biochim. Biophys. Acta 497, 507-524. Carlstedt-Duke, J., Wrange, G., Dahlberg, E., Gustafsson, J.-A. and Hogberg, B. (1979) J. Biol. Chem. 254, 1537-1539. Higgins, S.J., Baxter, J.D. and Rousseau, G.G. (1979) In: Glucocorticoid Hormone Action, Eds.: J.D. Baxter and G.G. Rousseau (Springer-Verlag, Heidelberg) pp. 135-160. Markovic, R.D. and Litwack, G. (1980) Arch. B&hem. Biophys. 202, 374-379. Munck. A. and Foley, R. (1979) Nature (London) 278, 752-754.

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Schaumburg, B.P. and Bojesen, E. (1968) B&him. Biophys. Acta 170, 172-188. Siegel, L.M. and Monty, K.J. (1966) Biochim. Biophys. Acta 112, 346-362. Statistical Analysis System, User’s Guide (1979) Statistical Analysis System Institute

Inc.

(Gary, NC). Tanford, C., Swanson, S.A. and Shore, W.S. (1955) J. Am. Chem. Sot. 77, 6414-6421. Wrange, 6. and Gustafsson, J.-A. (1978) J. Biol. Chem. 253, 856-865. Yamamoto, K.R., Gehring, U., Stampfer, M.R. and Sibley, C.H. (1976) Recent Progr. Hormone Res. 32, 3-32.

APPENDIX Two-phase partitioning of glucocorticoid receptors: derivation of the timecourse of the receptor partition coefficient to be expected from receptor conversions following given reaction schemes The purpose of this appendix is to show how the time-course of the partition coefficient in aqueous two-phase systems of the receptor population as a whole can be predicted, when one assumes that the receptors are converted according to a given reaction scheme. Such predictions are useful when considering the reaction mechanisms underlying receptor activation on the basis of partitioning experiments (see the accompanying paper), and may also be useful when studying the kinetics of other receptor conversions resulting in changes of the receptor partition coefficient, for instance partial proteolysis. The following model reaction scheme for receptor conversion is considered as an example:

xk_$yk_gz k-,

k-,

X, Y and Z denote the different interconvertible receptor forms, and k + ,, k_ ,, k+2 and k_, the rate constants of the individual reactions. The following differential equations describe the changes with time of the concentrations of the three receptor forms: dX dt=k_,Y-k+,X

(1)

dY --k+,X+k_2Z-k_,Y-k+zY dt

(2)

g=k,,Y-k_,Z

(3)

Activaiion

585

of glucocorticoid receptors

The solution

of this set of equations

is

C, + C, eP1’+ C, eP2’

X=

(4)

(5)

y=C,+~,~~k+,C,e"l'+~'+k+,C,e"l'

k-,

I

+P,+k+,+k_lC,eP~,+P2+k+,+k-,C

z=c

c

2 eP21

k-,

k-,

(6)

where C,, C, and C, are constants depending on k, ,, k_ ,, k,2 and k_,, C, and C, are constants determined by the limit conditions, and p, and p2 the roots of the polynomium

,8’+(k+, +k, + ((k,, Since k, These inserted partition partition forms:

+k+,+k-,)P

+k-,).(k+,+k-,)

-k-,k+z)

(7)

,, k_ ,, k+* and k_, are all positive, p, solutions for X, Y and Z, expressed as into the following equation, giving the coefficient of the receptor population coefficients K, and the molar fractions

and & are both negative. molar fractions, are then relationship between the as a whole, K,, and the r, of n different receptor

SI(Kirl/(K,T+B)) K,=

i,,

(8)

2 (c/(K,T+

B))

Here, T and B are respectively. Rearranging is obtained:

the volumes of the top and bottom phase, the resulting expression, the following equation

K = Pi - Be ( a;ePI’ + a;ep2’) *

Pi+

(9)

T.(a;eP1’+a;eP2’)

where (K,-KK,).(KyT+B)+

pll_:+,

.(K,-KY).

(KJ+B))

SC,

586

P.A. Andreasen

and

(11) K,, and KY and K, being the partition coefficients of the receptor forms X, Y and Z, respectively, and P,’ and Pi are also constants not changing with time. Dividing numerator and denominator of the right-hand side of eqn. (9) by a; + cy;, one obtains K = P, - Be (cq eDI’ + a2 ep2’) ’

P,-tT.(a,

ea1’-ta2ePZ’)

(14

where (Y,+ (Ye= 1, P, = P;/(a; + a;), and P2 = P;/(a; + a;). Since /I, and & are both negative, K, + PI/P2 for t --+co. Therefore K = K,P, - Be (a, ePlf + a2 ePz’) I P,+T.( a1 eBt’ + a2 ePzr)

(13)

With t = 0, one finds K,=

K,P,-B P2 + T

(14)

or K,T+ p2= Km-K,

B

(15)

Thus, eqn. (13) corresponds to eqn. (2) in the accompanying paper. The time-courses expected from other reaction schemes may be derived by the same principles.