Nonmonotonic transients and some mathematical models of circadian rhythms

J. theor. BioL (1976) 56, 435-441

Nonmonotonic Transients and Some Mathematical Models of Circadian Rhythms CHARLES

BEROF.~

School of Medicine, Stanford University, Stanford, California 94305, U.S.A. (Received 2 May 1974, and in revisedform 18 March 1975) A variety of mathematical models have been proposed to describe circadian rhythms; two important models are those of Pavlidis (1967a,b, 1968, 1971) and Wever (1963, 1964, 1965). Tests of these models should help in specifying qualitative features of circadian oscillators and should suggest directions for the study of the underlying mechanisms. Under natural conditions, endogenous circadian rhythms are synchronized to environmental periodicities such as light and temperature. In such an entrained state, a complicated multi-oscillator system might behave in a manner indistinguishable from that of a single oscillator. If such a system were tightly coupled, then it might also have an uncomplicated transient response following a perturbation or change in the driving regime. Alternatively, the coupling might be sufficiently loose that the transients would show evidence of complexity. It will be argued here that experiments conducted by Eskin (1969, 1971) provide this sort of evidence for clock complexity and provide evidence against some features of the models proposed by Pavlidis and Wever. Eskin studied the long-term effects on daily activity rhythms in the house sparrow (Passer domesticus) which result from placing the birds in constant conditions following an entrainment regime. Birds from the field were entrained by various light-dark cycles and then were kept with a low level of constant illumination. The birds' behavior would thus show the effects of adjustment from entrained oscillation to free oscillation. These transients lasted three months or longer, and they were characterized by long-term nonmonotonic variation. That is, a bird's activity-rest cycle period would t The results presented are taken from a thesis submitted by the author in partial fulfillment of the requirements for a B.A. degree in the Honors Program at Swarthmore College. Reprint requests should be directed to: Charles Berde, School of Medicine, Stanford University, Stanford, Calif. 94305, U.S.A. 435

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grow longer and shorter for several months before arriving at a period length that was typically just under 25 hr. (Eskin's approximate measure of a late stage value of the free-running periods of 49 birds, FRP-PK, was 24.87 ___0.54 hr.) The duration of a particular bird's cycle often varied by more than 2 hr. In many cases these changes seemed to be grouped in 4- or 6-day periods. Often a bird's period would increase for the first 30 cycles, remain relatively constant for 20 cycles, and then increase in an irregular fashion for another 30 cycles before decreasing slightly to a value which remained constant for several months. Because of the constancy of the final period length and because of some related statistical calculations (Eskin, 1969), it seems appropriate to regard these variations not as statistical fluctuations in steadystate behavior, but rather as transient effects in a fairly precise system. For a more complete discussion of statistical problems in the estimation of pacemaker period values, see Pittendrigh & Daan (1976). Plate I shows activity records taken from Eskin's (1969) thesis. Figure l(a) and (b) show transients as calculated by Eskin (1969). Wever (1963, 1964, 1965) has described daily activity rhythms with the following differential equation.

y+O'5(y2 q-y-2--3)p+ y(l q-O'6y) = 5~q-~ q-x

(1)

In this model, the organism is active when the value of the oscillating variable y is above a certain threshold. The variable x corresponds to the intensity of illumination; in natural conditions this would be approximately a periodic function of time. The non-linear terms on the left-hand side make the system self-sustaining and make the oscillations obey the circadian rule (Aschoff, 1960); the derivati~/es of x on the right-hand side express the working of differential as well as proportional Zeitgeber. For details, see Wever (1963, 1964, 1965). This system was simulated by adapting the function x(t) and its derivatives to the conditions of Eskin's experiments. That is, x was made to oscillate with appropriate period and wave-form and was then set equal to a positive constant. The period, amplitude, and constant conditions value of x were varied for the different simulations, and the system was released into free run at various times in the circadian cycle. In all of these simulations, the period of y adjusted rapidly and monotonically to the steady-state value, and remained there for 30 or more cycles without deviation. In no case was there any evidence of the long-term nonmonotonic transients observed in Passer domesticus. Pavlidis' (1967a,b, 1968, 1971) models embody some of the generalizations developed by Pittendrigh (1960, 1965) and his colleagues (Pittendrigh & Bruce, 1959, Pittendrigh, Bruce & Kaus, 1958, Pittendrigh & Minis, 1964)

24 h---'-] } ~::i!

180

:

/-"L: " . . . .

- 61 ! ~

-~-= - -

165 i

~

_.

~ - ~

159 i

----------~-i ~ . ~

~ - ~

64 i

'-~

"-

PLATE I. Activity records of 5 birds, courtesy of Dr Arnold Eskin. (D :D denotes constant darkness.)

[facing p. 436

437

NONMONOTONIC TRANSIENTS 26 (o)

25

24

I00

I 300

200

25.5

24.5

I I0O

o

I 200

Doys in D:D

FIG. l(a) and (b). Transients for two birds as calculated by Eskin (1969).

from studies of D r o s o p h i l a eclosion rhythms. Pavlidis assumed that the clock system consists of a primary self-sustaining oscillator which is light sensitive, and a secondary oscillator, driven by the primary, which is directly responsible for the rhythms. On the basis of qualitative-topological study and experimental data on phase shifts and entrainment by pulses and by long periods of illumination, Pavlidis proposed the following pair of first order differential equations to describe the primary system. i" = r + d - O . 6 s - s 2 - K L

r >>. 0

(2) The secondary system, represented by the state variable q, is driven by the = r-O.5s.

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primary according to the following equation: (1 + cq = as

(3)

The variable L corresponds to the light intensity. K was either made constant or made to satisfy a differential equation which represents saturation of the effect of light at high intensities. The variables r, s, and q could be interpreted either as the concentration of an enzyme and two substances, respectively, or as abstract state functions whose magnitudes, periods, and phases are related to observable clock phenomena. The constant d adjusts the form of the phase response curve, the width of the entrainment range, and the dependence of the free running period on the level of illumination. Following Pavlidis (1967b), its value is set at 1.0 for this simulation and for subsequent simulations involving modifications of this model. This system was simulated as in the previous case by adjusting the function L(t) to the experimental conditions. Different cases used different entrainment schedules and different values of the various coefficients. All cases studied showed rapid and monotonic relaxation to a constant period value, and thus these systems do not accurately represent the sparrow data. Pavlidis' model involves forcing of a secondary system by a primary, with no feedback. The earlier Pittendrigh-Bruce (1959) model assumed that there was some feedback. Pavlidis, Zimmerman & Osborn (1968) have argued that certain temperature effects can be incorporated into a model without using feedback of the secondary on the primary, but the assumption of feedback has not been proved invalid in any general way. For this reason, the model equations (2) and (3) were modified to include a feedback term, and simulations were conducted as in the two previous cases. This system also failed to show long-term nonmonotonic transients. Since these systems could not represent the experimental findings, other equation systems were studied in order to determine what sort of system could simulate these transients. It was found that when non-linear oscillators of the type used in Pavlidis' phenomenological model were used to model the secondary system as well as the primary, then under certain circumstances transients similar to the sparrow data were observed. This system is represented by equations (4). r. = P l(r° + d - 0.6s°- s ~ - K L - F I s b - F 2 s c ) s. = Pt(ra-O'5sa) r b = P2(rb +

d - 0"6S b - - S2b-- D i s ° - Fast)

s b = P2(rb-- 0"5•b) f'c = P~(rc + d - 0.6s~- s ~ - D 2 s , - D3Sb) s~ = e 3 ( r ¢ - O ' 5 s c ) .

(4)

NONMONOTONIC

TRANSIENTS

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The subscripts a, b and c denote the state variables corresponding to the primary oscillator and the two secondary oscillators, respectively. The D~ and F~ are driving and feedback coefficients, respectively, and the P~ are coefficients used to adjust the natural periods of the three oscillators with respect to one another. Again, the driving of the primary oscillator by a light cycle is represented by the KL term in the first equation. By appropriate adjustment of the coefficients, this model can describe either a two oscillator or a three oscillator case, and it can represent varying degrees of series and parallel character (Pavlidis, 1971) in the coupling. A series of simulations was conducted with L(t) again adjusted to the conditions of Eskin's experiments and the coefficients were altered in the various cases. Several of these systems reproduce some features of the sparrow data. For example, series coupling of two or three oscillators with no feedback and parallel coupling of three oscillators produce nonmonotonic transients when the natural periods of the oscillators are slightly different and when the forcing coefficients are moderately weak. The best results were obtained with series coupling with stagewise feedback. (C feeds back on B and B feeds back on A; C does not explicitly interact with A.) Moderately weak feedback (F i ~ 0.2) works best; strong feedback brings the system rapidly and monotonically to a steady state value, while very weak feedback does not allow sufficient interaction between the component oscillators for some of the effects described below. These systems do not require that the natural periods of the oscillators be different, and they correspond to the experimental data in the following respects: (1) the transients extend over a long duration, in several cases for 90 cycles or longer; (2) the transients were highly nonmonotonic in character; (3) the transients generally do not follow a simple pattern of beats or ever-decreasing overshoot with convergence to a final value; (4) the period value can be relatively constant for several cycles and then begin to vary greatly; (5) the period length in some cases varied in cycles of four to six periods over part, but not all of the transient range. Figure 2 shows representative transients from equations (4) and from Wever's and Pavlidis' models. A preliminary study of Pavlidis' (1971) model involving parallel-coupled biochemical oscillators suggests that this system produces nonmonotonic transients only when the natural periods of the component oscillators are slightly different; this is consistent with the results obtained for the parallel coupled cases of the system described by equations (4). For the cases studied, the transients seem to extend for less than ten cycles, but further work is needed to determine whether other parameter values will give longer transients.

440

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FI<~. 2. Simulation of transients for Wever's model (O--O), Pavlidis' model ( I - - l ) , Pavlidis' model with linear feedback (I-q--D), and equations (4) (O--O). The last one shows nonmonotonic transients. Equations (4) are not intended as a new phenomenological model o f circadian rhythms. These simulations are used merely to demonstrate that a particular feature of circadian rhythmicity cannot be described by some existing simpler models, and t h a t insofar as the systems underlying circadian rhythms can be described by oscillators, probably at least several complicated coupled oscillators are involved. Since nonlinear oscillators differ so greatly f r o m case to case, it is difficult to say whether m a n y other oscillator systems can produce these transients. I would like to thank Professor Cyrus Cantrell, Professor Andrew De Rocco, Professor Arnold Eskin, Professor Theodosios Pavlidis, Professor Colin Pittendrigh and Ms Helen Plotkin for assistance in this work. The computer simulations presented here were made possible through the generosity of the Swarthmore College Computer Center and Professor Theodosios Pavlidis. REFERENCES ASCHOFF,J. (1960). Cold Spring Harb. Symp. quant. Biol. 25, 11. ESKIN,A. (1969). Ph.D. Thesis. University of Texas. EsKn~, A. (1971). In Bioehronometry (M. Menaker, ed.). Washington, D.C.: National Academy of Sciences, 55.

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PAVLIDIS,T. (1967a). Bull. math. Biophys. 29, 291. PAVLIDIS,T. (1967b). Bull. math. Biophys. 29, 791. PAVLIDIS,T. (1968). In Lectures on Mathematics in the Life Sciences (M. Gerstcnhaber, ed.), p. 88. Providence: American Mathematical Society.

PAVLIDIS,T. (1971). J. theor. Biol. 33, 319. PAVLIDIS,Z., ZIMMERMAN,W. & OSBORN,J. (1968). Jr. theor. Biol. 18, 210. PrrrENDRXGrI, C. S. (1960). Cold Spring Harb. Symp. quant. Biol. 25, 159. Prrr~NDR~GH, C. S. (1965). In Circadian Clocks (J. Aschoff, ed.), p. 277. Amsterdam: North-Holland Publ. Co. PIT'tEI,~DRXOH,C. S. & BRUCE, V. G. (1959). In Photoperiodism and Related Phenomena in Plants and Animals (R. Withrow, cd.), p. 475. Washington, D.C.: A.A.A.S. Pn'rENDPaGH, C. S., BRUCE, V. G. & KAUS, P. (1958). Proc. natn. Acad. Sci. U.S.A. 44, 965. PITTENDRIGH, C. S. & DAAN, S. (1976). J. Comp. Physiol. In press. PI~rENDRIOH, C. S. & MIr4Xs, D. H. (1964). Am. Nat. 98, 261. WEVER, R. (1963). Kybernetik 1, 213. WEVER, R. (1964). Kybernetic 2, 127. WEVER, R. (1965). In Circadian Clocks (J. Aschoff, ed.), p. 47. Amsterdam: North-Holland Publ. Co.