Zum Mechanismus der biologischen 24-Stunden-Periodik

Summary

Using the physical and mathematical basis given in two foregoing papers, a differential equation is proposed for a model of the biological 24-hour-periodicity. This oscillation equation contains two characteristic non-linearities describing the self-sustaining property and the “circadian rule”. The right side of the equation (“external force”) represents the controlling environmental conditions, mainly the intensity of illumination. Solutions were obtained for different environmental conditions using a digital computer.

Under “constant conditions” the solution of the equation describes oscillations self-sustained within a certain range of environmental conditions. In this range the oscillations fulfil the circadian rule, e.g. for light-active organisms: The frequency and the mean value of the oscillation increase with increasing light intensity; with an additional (arbitrary) threshold separating activity time and rest time for describing an activity rhythm, the α∶ρ (activity time ∶ rest time) ratio and the total amount of activity also increase.

Under periodically changing environmental conditions five properties of the “Zeitgeber” used (two distinct intensities with twilight transitions) are variable and varied: The range of oscillation of the Zeitgeber, its frequency, its mean value, its L ∶ D ratio (time relation of light time and dark time), and the duration of the twilights. The most important of the examined properties was the phase angle difference between the (forced) oscillation and the (forcing) Zeitgeber. The general result for light-active organisms was: The phase of the oscillation advances relative to the Zeitgeber (in sofar as the oscillation is synchronized) if the period of the Zeitgeber, or its mean value, or its L∶D ratio, or the duration of the twilights increase. In dark-active organisms, the relation between phase angle difference and the mean value or the L∶D ratio is reversed. Exceptions to this general rule exist in the relation between phase angle difference and L ∶ D ratio if the “free running” period of the oscillation deviates too much from the period of a “weak” Zeitgeber (mainly in dark-active organisms) or if the duration of the twilights is too short (especially if the transitions are rectangular).

Single exposures to light (or darkness) during constant conditions result in phase shifts depending in direction and amount on the phase of the oscillation at which the disturbance occured. The resulting response curves depend in range and form on the one hand on the time of measuring the phase shifts (either immediately or after several periods — in the steady state — following the disturbance) and, on the other hand, on the intensity of the initial illumination, on the duration, and on the intensity of the exposures, each in a different manner. Moreover, response curves effective in LD conditions deviate from those measured under constant conditions; the reason being the difference in the energy state of the oscillations in the two conditions. Therefore, it is impossible to derive the phase angle difference between the oscillation and a Zeitgeber in self-sustained oscillations from the measurement of response curves alone.

The oscillation equation used contains only one free parameter, the frequency coefficient. If this coefficient is changed, the equation describes other biological rhythms. For instance, with a high value it describes the behaviour of single nerve cells, and that not only in cases of spontaneous rhythmicity (e.g. receptor cells) but also in cases of reactions to single or rhythmic stimuli. Moreover, the derived characteristics of the equation — especially the non-linearities — seem to be significant for other biological problems such as control mechanisms.