J. theor. Bioi. (1978) 73, 657-677

Cross Inhibition Models for the Transmission of Hormonal Signals E. VAN CAUTER AND J. E. DUMONT Institut de Recherche Interdisciplinaire, School of Medicine, University of Brussels and Biology Department, Euratom, 115 Bd de Waterloo, B-1000 Brussels, Belgium (Received 16 January 1976, and in revisedform

9 February

1978)

Implications of a model where the formation of two mediators interacting through cross inhibition is stimulated by the same hormone or by two specific hormones are analyzed at steady ‘state, first in the case where no co-operative processes are involved, secondIy in the case where one of the inhibitory branches presents positive co-operativity characteristics. The possible occurrence of agonist concentration-mediators response curves with extrema of opposite type, multiple steady states, hysteresis and discontinuous transitions from one functional program to another is demonstrated. When steady-state hormonal levels fluctuate, it is shown that the cross inhibitory mediators may induce a strict temporal organization of the intracelhtlar processes and result in both amplification and frequency multiplication of the fluctuations. 1. InmdIIction

In 1965, Sutherland and coworkers proposed the first model for the mechanism

of hormone action, postulating cyclic adenosine monophosphate as the intracellular mediator of the extracellular messenger (Sutherland et al., 1965). Since then, other mediators of agonist action such as cyclic guanosine monophosphate (cGMP) and calcium have been demon-

(CAMP)

strated. Moreover, it was shown that, in many cells, the same response may be controlled by more than one agonist and that the agonist-receptor interaction could result in decreases as well as increases of intracellular mediators’ concentrations. Finally, cyclic nucleotides and calcium were found to interact and control their respective levels in various ways. Because of this complexity, the need for a theoretical framework to analyze data and formulate hypotheses became evident. The Ying-Yang model introduced by Goldberg et al. (1973, 1975) and related models of Ca”’ -cyclic AMP interactions proposed by Berridge (1975) and Rasmussen (1975) have been fist attempts of schematic, but qualitative, description. 651 0022~5193/78/0821-0657$02.00/O

0 1978Academic Press Inc. (London) Ltd.

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Recently, Rapp & Berridge (1977) have suggested that, in a number of experimental systems, the interactions between intracellular cyclic AMP and calcium could be described by a closed loop model with negative feedback. We proposed a general representation of the roles played by the mediators and their interactions in the mechanism of action of hormones and neurotransmitters (Van Cauter et al., 1977). From this representation, all possible models and sub-models which could theoretically apply were derived and a methodology for their systematic theoretical investigation was defined. Because of its relevance in different experimental systems such as smooth muscle (Rasmussen, 1976; Berridge, 1975), the rat parotid gland (Butcher, 1976), one of these models, where the formation of two mediators interacting through cross inhibitions is stimulated by one and the same agonist, was analyzed theoretically under general assumptions excluding any co-operative process (Van Cauter et al., 1976). The possible occurrence of extrema of opposite type in the mediators’ concentrations as functions of the agonist’s concentration was demonstrated. However, the recent literature suggests that co-operativity rather than non-co-operativity may be the rule in physiological systems. In particular, positive co-operativity in the negative control of CAMP level by calcium has been demonstrated in several hormones (Lin et al., 1974; Kakiuchi et al., 1973 ; Teo 62 Wang, 1973 ; Steer & Levitzki, 1975). In this work, we show that the introduction of positive co-operativity in one of the branches of the cross inhibition model results in the occurrence of multiple steady states of mediators’ concentrations for a given, constant extracellular hormone level. The hysteresis phenomena and the all or none transitions associated with these solutions may explain functional d&continuities observed in different hormone sensitive cells as well as prolonged effects of short term hormonal signals. The assumption of constancy of extracellular hormone levels is then abandoned in order to simulate the physiological oscillations, such as those resulting from the circadian rhythms and episodic secretions observed in man (Weitzman, 1976). We study the transmission of these oscillations through the intracellular system and demonstrate that cross inhibitory mediators’ interactions induce a strict temporal organization of the intracellular processes and may result in both amplification and frequency multiplication of the oscillations. 2. The Models and their Basic Assumptions Figure 1 illustrates the general cross inhibition models of the mechanism of hormone and neurotransmitter action considered here. In Fig. l(a), the agonists, Hi and H,, synthesized and degraded outside

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(a)

(b)

FIG. 1. (a) two agonists, HI and Hz, synthesized and degraded outside the cell, bind to two membrane receptors and stimulate the formation of two mediators, X and Y, interacting through cross inhibitions. (b) One agonist, H, binds to two different receptors.

their target cell, bind to two different receptors R, and R,. Agonists do not penetrate inside the cell. In Fig. l(b), one agonist, H, binds to thetwo receptor structures. The two agonist-receptor interactions stimulate the synthesis or release of mediator Xfrom precursor pool A and mediator Y from precursor pool C. X could correspond to CAMP, A representing the ATP pool, whereas Y could correspond either to cGMP or to free intracellular calcium, C representating the GTP pool or a sequestrated or extracellular calcium pool. The mediators X and Y are degraded or inactivated inside the cell or pumped out of the cell. In the case of free calcium, this inactivation could also consist of a pumping in a sequestrating structure such as sarcoplasmic reticulum. Mediator X inhibits, by various mechanisms, the accumulation of Y and vice versa. The mechanism of the agonist-receptor interaction has been definitely demonstrated in no tissues. Our previous study, in which different currently proposed hypotheses were considered, indicates that the type of agonistreceptor interaction does not play a role in determining the possible behaviours

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of the model (Van Cauter et al., 1976). Therefore, the simplest type of receptor model, so-called the non-dissociable receptor model (Boeynaems & Dumont, 1975), has been used here for both receptors. The agonist, H, binds reversibly to the receptor molecule R to form an active complex HR

H+R 2HR. k-1 The total concentration

of receptor molecules is conserved,

R+HR = Rtota, and the affinity of the agonist for the receptor is given by:

An activity of the effector or receptor-effector system in the absence of agonist has been detected in many tissues (Perkins, 1973). It was therefore assumed that, in the absence of agonist, X and Y were synthesized at a low rate, proportional to the concentration of the corresponding free receptor molecule, and that their basal concentrations were controlled by the crossinhibitions. The concentrations of A and C are assumed to be constant and saturating. The corresponding parameters were incorporated in the kinetic constants of the basal and stimulated synthesis of X and Y, respectively denoted by kA and lu,, kc and Kc. First order processes with kinetic constants II and 1, have been considered for the degradations of X and Y, respectively. The two types of inhibitory pathways which have been demonstrated in experimental systems (Lin et al., 1974; Kakiuchi et al., 1975), namely, inhibition by stimulating the disposal of the mediator and inhibition of the synthesis of the mediator, have been studied. They have the common property that the inhibiting species is not degraded through its action on the other species. They are detailed hereafter in the case where X represents the inhibited species and Y the inhibiting species. n, and n2 are, respectively, the number of X and Y molecules taking part in the cross inhibitory interaction. (A)

This inhibitory

STIMULATION

OF THE DEGRADATION

interaction

may be schematically represented as follows: x Ii ++BIY”l X’ , --, where X’ is the inactive form of X, the corresponding differential equation describing the evolution of the concentration of X is

dCx1 -dt = k,iChl +&C~iRJ--(Z~ +&Y"%-X] and B1 characterizes the strength of the inhibition.

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OF THE SYNTHBSIS

This mechanism may be represented as follows: &A’+ KJ A -4

x,

where

and the kinetic equation for X is:

dE = kA%ota~--CHAI) +

KA

C~AI - uxl.

dt 1 fB,[Yl”2 1 +B,[Y]“2 From the chemical scheme and the conservation equations of the receptors, the number of independant variables is defined and the set of non-linear differential equations representing the kinetic evolution is formulated. The steady-state solutions are obtained by setting the derivatives equal to zero and solving the corresponding algebraic system. It may be easily recognized that the steady-state solutions are of the same form for all four possible combinations of mechanisms I and II involved in the cross inhibition, provided adequate transformations of the coefficients which are given in Table 1. Models where each mediator pathway is stimulated by a specific agonist seem to apply in an increasing number of experimental systems. Therefore, we will first extend our previous analysis of the models with non-co-operative inhibitory interactions (Van Cauter et al., 1976) to the case where two agonists are present and obtain the steady-state solutions corresponding to the different types of relative evolution of their concentrations. This preliminary analysis allows the discussion of the results obtained in sections 4 and 5 where positive co-operativity is present in one branch of the cross inhibition in a more general framework.

3. The Models with Non-co-operative Inhibitory

Interactions

In this section, n, = n2 = 1. The steady-state concentration of X, denoted by X*, is given by a second degree equation of the form x*2+Bx*-c = 0, (1) where B and C are non-linear expressions depending on the parameters of the sub-model and on the agonists steady-state concentrations, HT and Hz. The same type of equation may be obtained for Y* Y*2+BY*-c = 0. (2)

Bz + &A

Bz -+ BJ,

Inhibition of X by stimulation of degradation, inhibition of synthesis of Y 4 -+ 4

of

Bi -+ 44

Both intqactions by inhibition synthesis

1

Transformations to apply to above expressions

TABLE

Bs --f Ba

& +- BA

Inhibition of synthesis of X, inhibition of X by stimulation of degradation

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ACTION

The expressions of B, B, C and C as functions of the parameters and of H: and H; are given in Table 1. C and Care positive for all values of H: and Hg and equations (1) and (2) have therefore one single positive real root. B and B may be positive or negative depending on the values of the parameters and the relative concentrations of H: and Hz. The following relations are verified :

=o, aH:

g>o, aH:

-E. o, aHT

!c>,

aH;

ac -o aT2’

aC ajzo’

’ (3)

Simulated concentration-response curves of such models should be obtained for three types of relative evolution of HT and Hz corresponding to three possible experimental situations. In the first situation, one of the agonists’ concentrations, say Hf, is progressively increased, whereas the other, say Hf, is maintained constant throughout the experiment. Simulated H: concentration-X, Y response curves arc obtained by solving equation (1) for increasing values of HT. Their pattern may be easily derived from relations (3). Indeed, we have ax* -=aH:

1 -a3 2 ( aH: I( laY* -=aHT

JB,B+&

i -aB 1-$& 2 ( aHf >(

ac/aH: + &jTc > >

’ 0,

0, /I IS 0. Then the possible patterns for the concentration-response easily deduced from relations (I), (2) and Table 1. Indeed, we have

(4) curves may be

(5) -=-

(6)

and, with (4), the sign of aB/aHT is still opposite to the sign of aB/aHf even though H, is now a dependent variable and not an independent parameter. It may be verified from equations (5) and (6) that X* and Y* may not undergo more than one extremum when HT and H3 are concomitantly and proportionally increased. Moreover, if one of the variables X* or Y* decreases, when H: increases, the other, Y* or X*, necessarily increases. Indeed, let X* decrease, ax*

-

aHy

< 0, then

zf. >o, aH:

since the factor

and the term

--ac/aHf J332+4c are always positive. ff!

aH;

-c 0 and

ay*

aF:

is necessarily positive, thus y* increases. This property, which results directly from the existence of the cross inhibitions, restricts the possible behaviours of X* and Y* when WY and H$ increase concomitantly and proportionally, to five cases: (1). both X* and Y* increase (2). one variable increases whereas the other decreases (3). one variable experiences a minimum, whereas the other steadily increases (4). one variable experiences a maximum, whereas the other steadily increases

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(5). one variable experiences a minimum whereas the other experiences a maximum. The minimum occurs necessarily for lower Hr concentrations than the maximum since otherwise the two variables would both decrease for a certain range of H, values which has been shown to be impossible. All the possible patterns of the cross inhibition models shown in Fig. 1 are summarized in these five cases. Behaviours where both X* and Y* decrease, or with more than two extrema, or two extrema of the same type are thus impossible. From equations (4), (S), and (6), it may be observed that behaviours with two extrema will only be obtained if the sign of aB/aHT and consequently of c%/aHf, changes when WY increases. The forms of B and B given in Table 1 imply that this condition will be met if asymmetrical characteristics of the pathways such as high receptor RI total concentrations vs. low receptor R2 total concentration, low afhnity of RI for the agonist vs. high affinity of RZ for the agonist, weak inhibiting potency of X vs. strong inhibiting potency of Y are present. It was previously shown that these asymmetrical conditions may be viewed as non-redundancy conditions (Van Cauter et al., 1976).

4. The Models with Co-operative Inhibitory

Interactions

In this section, we introduce positive co-operativity characteristics in one branch of the cross inhibition, let the inhibition exerted by X on the accumulation of Y. We thus have n, = 2, n2 = 1. For all sub-models, the steady-state concentration of X* is given by a third order equation of the form: P(X)=X3-pX’$qX-r=O,

(7) where p, 4 and r are positive non-linear functions of the parameters of the model (Rib Rzt, Ki, K,, kA, kc, Ki, Kc, Z1,12, B,, B2) and of the extracellular agonists concentrations HT and Hf. The solutions of the problem with one agonist are simply obtained by putting HI = Hz, The same type of third degree equation is obtained for the Y* concentration. The forms of p, 4 and r are given in Table 2 for the model with inhibitions by stimulation of the degradations. The Descartes rule of signs applied to P(X) and P(- X) indicates that for appropriate values of coefficients p, 4 and r, equation (7) admits at most three positive real roots corresponding to three theoretically possible different steady states for X*. Simulated agonists concentration-response curves have been obtained for large sets of parameter values for each sub-model. The solutions of the cubic equation (7) were computed by the trigonometric method.

Inhibitions by stimulation of the degradations 14 +KJf:)

&t&d-K.&H:)

P

la -+ &

4

B&l S&H;)

&R&c+K&H;)

TABLE 2

UMl +&H:)

hR&n+KJW:)

r

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Figures 2 and 3 illustrate for the three types of relative evolution of HI and H, previously considered possible behaviours of the models with two agonists. Similar S-shaped patterns were obtained for the three types of inhibition mechanisms and for the model with one agonist. These S-shaped behaviours seem thus to be typical of the general cross inhibition models presented in Fig. 1 when one of the inhibitory branches involves positive co-operativity. Their characteristics and possible implications will be discussed hereafter in the more general case of the model with two agonists. This discussion may be readily transposed to the case of the model with one agonist by considering HI identical to H,. In Fig. 3, the range of variation of the agonists’ concentrations has been separated into three parts, (a), (b) and (c), respectively corresponding to three different steady state behaviours. In part (a) of the patterns, equation (7) has only one real positive. solution and, for each pair of values HI and H,, (a)

He = 10 5

SF-----7

4

E5 .I? ‘-,4 L z3

3 2

s2 8 % I ‘: Oo

5 IO If, concentration

lr-Oo

15

5 IO HI concentration

15

(b)

5 6 4 s5 54

3

z3 8 =2 8 %I

2

~ Oo

5 IO 15 14, concentration 50 46 41 36 &oncentrotion

Ii-! O0-

5 IO 15 HI concentration 50 46 41 36 If2 Concentration

FIG. 2. Agonist concentration-response curves obtained for the model with stimulations of the degradation when HI is i&eased and Hz maintained equal to 10 (a) or decreased (b). Numerical values of the parameters are: ka = kc = 0.1, K.., = Kc = 1, Rl, = 4, Rzt = 2 Kl = 0.1, KS = 1, II = & = 0.4, Br = Bz = 2.

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2Hz.

4

hb(

If2 concentmtion

FIG. 3. Agonists’ concentration-response curves of the model with two agonists and inhibitions by stimulation of the degradation of the mediators. Values of the parameters are:

kA = kc = 0.1, KA = Kc = 1, RI+ = 4, Rzt = 2 & = O-1, Kz = 2, 11 = 12 = O-4, Bl = Ba = 2.

Xand Y reach, after a certain time of evolution of the system, the steady-state levels illustrated in this part of the plot. The time necessary for the system to approach this steady state depends on its initial conditions, i.e. the levels of X, Y and the other variables at time t = 0, but the steady-state levels obtained are the same whatever these initial conditions may be. This part of the agonist concentration-response patterns is of the same type as those reported for the models without positive co-operativity. Similarly, when HT is increased and Hg maintained constant or decreased, the X* concentration is monotonically increasing and the Y* concentration monotonically decreasing for the three sub-models. This situation is illustrated for the model with stimulations of the degradations in Fig. 2. When Hf and Hz are concomitantly and proportionally increased, extrema of opposite type are also observed for asymmetric pathway characteristics (Fig. 3). In part (b) of the patterns, equation (7) has three positive real roots for each pair of values of the agonists’ concentrations. Two of these roots are identical when HT = HI,, Hf = H,, and HT = Hlb, Hz = Hz,,. The study of the stability of the roots by normal mode analysis shows that the intermediate branch consists of unstable, physically unattainable, steady states. For a given pair of values of the agonists’ concentrations in part (b) of their range of variation, X and Y have two possible steady-state levels, one high and one low, the lower branch of X steady states corresponding to the higher branch of Y steady states and

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vice versa, and the system will generally evolve towards the steady state closer to its initial state. In part(c) of the patterns, equation (1) has again one positive real solution corresponding to a unique steady state. If the agonist concentrations are gradually increased from a pair of values in part (a) to a pair of values in part (c), the steady state of the system will evolve along the dotted path illustrated in Fig. 3 and for HF = HI,,, Hf = Hz,,, undergo a discontinuous transition from a steady-state with low X* and high Y* to a steady state with high X* and low Y*. If the agonists’ concentrations are decreased from a value in part (c) to a value in part (a), the steady state of the system will evolve along the dashed path and undergo a discontinuous transition from a steady state with high X* and low Y* to a steady state with low X* and high Y* for HT = HI,, Hz = Hza. The succession of steady states is thus different depending on whether the agonist concentration is increased or decreased, the system describing a hysteresis loop.

5. The Transmission of External Oscillations of Hormone Levels We now investigate how oscillations in the steady-state agonist concentration resulting from external fluctuations such as circadian rhythms or episodic secretion are transmitted to the intracellular mediators when a model with cross inhibitions applies. Our aim is to demonstrate that certain behaviours usually not looked for in biochemistry are likely to occur rather than to perform a step by step analysis of the processes involved and their possible implications. For the sake of simplicity, the model with one agonist and inhibitions by activation of the degradations will be used with H” = H; = 2 H:, &, = 0.1, Kzes = 1, and the values of the other parameters as given in the legend of Fig. 3. We will first consider sinusoidal oscillations of the hormone concentration and then simulate the transmission of hormone fluctuations as observed in human plasma. Let H* = H:+AH*.cos

2a

r t,

where Hlf; is the mean level, AH* the absolute amplitude, kH = AH*/H*, the relative amplitude and T the period. If f is the time necessary for the set of chemical reactions defining the model to approach steady state when no external fluctuations of hormone levels are applied, the validity of a steadystate treatment will be ensured for T % J. The steady-state assumption implies that the attainment of steady state is instantaneous when compared to the time scale of the external oscillations.

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Figure 4 illustrates the variation of the steady-state hormone level N* and the resulting variations of the steady-state mediators’ concentrations X* and Y* in three different cases. In Fig. 4(a), Hz = 5.0 and ka = O-90. The corresponding variation of H* is restricted to part (a) of the patterns illustrated in Fig. 3. Since the maximum Ii* level reached during one cycle is Hzax = 9.5 < H&, = 2H&,, no transition from one branch of the hysteresis loops to the other occurs. From a qualitative point of view, this case incorporates the behaviours obtained for the models without co-operative processes where extrema of opposite type were observed in the hormone concentration-mediators response curves. The existence of these extrema results in the multiplication of the frequency of the external oscillations at the level of the X* and Y* concentrations, each of the mediators undergoing two maxima during one cycle although for this particular set of values of the parameters, the double periodicity is likely to be detected in the Y* pattern only. In Fig. 4(b), H$ = 8 and kH = O-88. The variation of H* thus implies for Y* a transition from the lower branch to the upper branch of the hysteresis loop in the early phase of the cycle and a transition from the upper branch to the lower branch towards the end of the cycle. The opposite transitions occur simultaneously for X*. The resulting X* and Y* patterns are characterized

Time

FIG. 4. (a) The hormone fluctuations occur in the region of single steady-state including occurrence of extrema of opposite type. The resulting fluctuations of mediators’ concentrations exhibit a double periodicity. (b) The hormone fluctuations, include the multiple steady-state region and the range of occurrence of extrema of opposite type. A strict intracellular temporal organization is induced together with frequency multiplication. (c) The hormone fluctuations include the multiple steady-&®ion but exclude the range of occurrence of extrema. The fluctuations of the hormonal concentratjons are amplified at the level of the intracellular mediators.

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not only by the appearance of a double periodicity, consequence, as in the lkst case hereabove, of the existence of extrema but also by a strict temporal intracellular organization, the processes induced by mediator X being stimulated when the processes induced by Y are at then minimal activity and vice versa. The amplification potential of the hysteresis loop per se may be shown by considering a variation of H* not including the range of occurrence of the extrema. In Fig. 4(c), Hz = 8.0 and kH = 0.50. The term “ amplification ” is used here to characterize how the hormone fluctuations, considered as a signal arriving at the target cell, are transmitted to the mediators’ concentrations rather than to compare the respective levels of the stimulus H* to the responses X* and Y* as proposed for instance by Davies & Williams (1971). Our terminology thus involves the definition of the time scale over which the signal and its responses are observed. The mean level of X* over one cycle is obtained by numerical calculation of

whereas the absolute amplitude

of its fluctuations is defined as Hx&ix-

x&h

where X&,, and X$;i, are respectively the maximum and minimum level reached by X* during the cycle. Similar formulations are used for Y*. The lower right side diagrams of Fig. 4 illustrate the oscillations of X* and Y*. Their relative amplitudes are, respectively, kx = 0.92 and k, = 0.98, the hormone signal having thus been amplified by a factor of 1.84 (kx/k, = 0*92/ 0.50) at the level of mediator Xand by a factor of 1.96 at the level of mediator Y. Finally, we calculated the fluctuations of steady-state concentrations of mediators X and Y which would result from the stimulation of the target cell by hormone levels fluctuating following the pattern observed over 24 h in human plasma. These simulations are summarized in Fig. 5. The upper diagram shows the variation of plasma corticotrophin (ACTH) levels measured every 15 mm during 24 h in a healthy male volunteer (Copinschi et al., 1975). The now classical profile of ACTH secretion with basal circadian rhythmicity and superimposed secretory episodes is observed. The length of the sampling interval as compared to the characteristic times of the chemical reactions, usually all below one minute, allows the steady-state approximation for the calculation of the resulting fluctuations of X* and Y* illustrated by the lower diagrams of Fig. 5. The simulations have been performed using for the hormone concentration the ACTH levels expressed in pg ml -I, the range of variation of H* thus 23

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time

FIG. 5. (a) Plasma corticotrophin (ACTH) levels measured every 15 min during 24 h in a healthy male volunteer. Hormone levels are expressed in pg/ml of plasma. (b) Simulated resulting fluctuations of mediator’s X steady-state concentration if cross inhibition model with hormonal concentration-mediators’ response curves as illustrated by Fig. 3 applies. (c) Simulated resulting fluctuations of mediator’s Y steady-state concentration under same assumption as above.

comprising the multiple steady-state zone as may be observed from Fig. 3. The comparison of the X* and Y* fluctuations illustrated in Fig. 5 clearly shows the temporal organization induced inside the cell by the cross inhibitions. Intracellular processes mediated by X will be stimulated at times when intracellular processes mediated by Y will occur at their basal rate and vice versa. The phenomenon of frequency multiplication is evident on the Y* diagram where the circadian quasi-sinusoidal component has completely disappeared whereas the ultradian fluctuations are both more frequent and of larger magnitude. These spectral modifications of the hormonal signal are better quantified by the Fourier line spectra of ACTH, X and Y steady state concentrations represented in Fig. 6 as periodograms (Jenkins & Watts, 1968). The 24-h line is enhanced in the X* periodogram where the variance is distributed over higher frequencies, mainly 12 h, 6 h and l-5 h. Moreover, &radian variations with periods between 4 and 2 h apparent in the ACTH pattern are practically nonexistent in the X* pattern. Gallagher and coworkers (1973) have demonstrated that there is no strict correspondence between the number and the magnitude of the secretory episodes of ACTH and cortisol,

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50 40 30 20 IO 0

Periods

(h)

6. Periodograms of ACTH, X* and Y* fluctuations iIIustrated in Fig. 5. The percentage of total variance of each series, saySxl, XZ, . . . , XT,where T = 24h, associated with periodT/j, j = 1, . . . , T/2, is 4 + ba, FIG.

;,il

(x5- w ’

whereal and b, arethe Fouriercoefficients corresponding to frequencyj/Tand T 2=-1.x5. T5=1

respectively. Moreover, pulsatile secretion of cortisol seem to occur independently of ACTH stimulation (Holaday et al., 1977). Non-linear transmission properties of the adrenal systems such as those examplified hereabove are a possible explanation for these puzzling results. 6. Discussion and Conclusions The model studied here constitutes a mere mathematical translation of conceptual models suggested by experimental results difhcult to integrate and interpret. ‘Its assumptions aim to combine simplicity and realism and the number of parameters and variables are kept to the minimum compatible with their direct physiological interpretation. Yet, its theoretical analysis demonstrates the possible occurrence ,of behaviours much more complex than those commonly foreseen and looked for in the course of investigations

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on hormone and neurotransmitter action. The methodology used can be readily transposed to the study of other possible models. The implications on the transmission of external hormonal fluctuations of different properties of the hormone concentration-mediator response curves such as the existence of extrema and of multiple steady states are developed separately in order to allow their generalization to other systems where one or several of these properties could be observed. Most previous models of hormone controlled systems were specmc of certain hormones or target tissues and did not aim at their integration in a general theory of hormones and mediators interactions. They often implied such a number of variables, of parameters and of specihc assumptions that neither their methodology nor their results could be extended to other systems (Heinmets, 1971; Yates & Brennan, 1968; Distefano & Stear, 1968). The closed loop negative feedback model recently discussed by Rapp & Berridge (1977) stands as an exception in the field. These authors suggest that such a model applies when cyclic AMP enhances the level of cytosol calcium which, in turn, inhibits adenyl cyclase or, inversely, when cyclic AMP decreases the level of calcium which, in turn, activates the synthesis of cyclic AMP. The most general representation of these interactions would be a model similar to the one illustrated on Fig. l(a) where one of the cross inhibitions is replaced by a cross activation (Van Cauter et al., 1977). A number of theoretical works (Goodwin, 1963; Walter, 1970; Walter, 1974) have shown that sustained steady state oscillations of the variables may appear in closed loop reaction circuits with negative feedback. Therefore, Rapp & Berridge (1977) suggest that these oscillations form the basis of several experimentally observed biological rhythms with periods ranging from fractions of seconds to a few minutes. However, the interactions between extracellular signals, receptors and mediators are not mathematically formulated and the conditions under which they could indeed be described as a closed loop negative feedback circuit remain therefore to be investigated. In particular, the fact that two variables of the loop (cyclic AMP and cytosol calcium) may be controlled by different external signals results in the possible occurrence of a number of alternative schemes and behaviours. In what concerns the non-co-operative cross inhibitory interactions, the present work widely extends our previous conclusions obtained when both mediator systems are stimulated by one and the same agonist. The possible occurrence of extrema of opposite type in the agonists’ concentrationresponse curves seems to be a general property of cross inhibition models. This result has important consequences both for the interpretation of experimental data and the development of theoretical concepts regarding the mechanism of hormone and neurotransmitter action. Apparently

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contradictory effects of agonists on intracellular concentrations of mediators, e.g. CAMP, cGMP and Ca’+, in different experimental systems find here a possible explanation and the importance of investigating the largest range of agonists’ concentrations possible is strongly emphasized. On the other hand, the present study demonstrates that the interactions between mediators’ systems play a major role in determining the shape of the observed agonist concentration-response curves and that consistent theoretical formulations should take these interactions into account. In particular, it has been shown that, contrary to a generally accepted concept (Goldberg et al., 1973), a cross inhibition model may apply even when concentration-response curves where one of the mediators increases whereas the other decreases are not observed. The introduction of positive co-operativity in one of the inhibitory pathways results in the possible existence of multiple steady states with all or non transitions and hysteresis. Previous theoretical reports on cross inhibitory interactions in biology include the cross catalytic enzyme model studied by Spangler and Snell(1961) and works discussing the possible behaviours of Jacob and Monod type models for genetic induction and repression (Heinmets, 1964; Simon, 1965; Cherniavskii et al., 1967; Babloyantz & Nicolis, 1972). Multiple steady states were found by Spangler & Snell (1961) and Babloyantz & Nicolis (1972) when positive co-operativity was introduced in both branches of the cross inhibition. Simon (1965) obtained alternative quasi steady states by postulating time lags which is, from an analytical point of view, analogous to the introduction of an inmute number of differential equations. The introduction of a trigger device allowed Cherniavskii et al. (1967) to obtain two stable steady states. In our model, discontinuous transitions and hysteresis appear under less drastic conditions, positive co-operativity being introduced only in one branch of the cross inhibitions since no presently available experimental data support the concept of symmetrical non-linearities. Discontinuities in hormone concentrations-response curves suggest that, when the hormonal concentrations increase, the individual cell switches from one functional program to another. Several cellular functions indeed occur without gradual transition at the level of the single cell. The triggering of mitosis in all cells (Ryan & Hendrick, 1974) and of phagocytosis in the thyroid cell (Friedman, 1976) are examples of sudden switches to another functional program which are not yet understood. The possible existence of such d&continuities in the hormone concentration-response curves brings new insight to the problem of detining the effects controlled by each of the hormones and each of the mediators. Both agonists and both mediators could control one series of effects for low agonist concentrations and another

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series of effects for higher agonist concentrations. Referring to Fig. 3, in part (a) of the plots, Y could stimulate one metabolic pathway whereas the corresponding levels of X would be too low to exert any action and, in part (c) of the plots, the inverse situation could prevail. Such a behaviour could explain the apparent paradox that hormones which at one level elicit functional activation and differentiation, could at another level trigger the opposite processes of mitosis and cell multiplication (Teo & Wang, 1973; Friedman, 1976). The translation of relatively short term signals in longer term biochemical effects, e.g. in phenomena such as the prolonged thyroid secretory effect of a single injection of the rapidly disappearing thyrotropin, could be related in part to the existence of a hysteresis loop. If constant sources of A and C rather than the assumption of their constant concentration would be postulated into the model, sustained oscillatory behaviours typical of “ dissipative structures ” (Prigogine et al., 1969) would be expected. Recent theoretical works have challenged the historical dominance of the concept following which biological systems should act as pure transmitters of the chemical fluctuations to which they are submitted. Hyver (1973a,b) has shown that the frequency response of a chemical network could exhibit low-pass or band-pass tllter characteristics. In their analysis of a deprotonation reaction, Hahn et al. (1974) demonstrate that, because of the existence of multiple steady states, oscillations of the end product likely to be considered as quasi-periodic could result ftom purely random fluctuations of the input variable. In our cross inhibition model, external hormonal steady-state fluctuations may be amplitied and their frequency multiplied at the level of the intracellular mediators’ concentrations and their effects. Moreover, because of the nature of the mediators’interaction, the hysteresis phenomenon results in a strict temporal organization of the intracellular events when extracellular hormonal levels oscillate. High frequency non-periodic varitions referred to as “ u&radian rhythms “, have been observed for many biological parameters of higher vertebrates, including electroencephalogram and hormone levels (Halberg, 1969; Weitzman, 1976). Non-linear transmission properties of biochemical networks such as filtering, frequency multiplication and generation of quasi-periodicities, could play a role in their origin and characteristics.

We thank J. G. Hardman and M. Dhaenefor stimulating discussion.This work was realized under contract of the Mini&e Belge de la Politique Scientifique within the framework of the Association Euratom, Universite Libre de Bruxelles, University of Pisaand waspartially supportedby A.P.M.O. Foundation (Brussels).

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