Versuch einer mathematischen Analyse der „Normalkurve“ von Krogh

Summary

The mathematical expression of biological temperature functions on the basis ofVan't Hoffs rule is unsatisfactory. This fact ledKrogh (1914) to present a standard curve, which was meant to demonstrate the basic uniformity of temperature relationships of numerous biological processes. This standard curve has served as a reference point in many papers up to the present time. Simple comparison of curve shapes is difficult for carious reasons and does not allow exact statements.Krogh himself has not attempted to define the shape of his standard curve mathematically. A pertinent proposal byJørgensen (1916) remained unnoticed by most biologists, although it was suitable mathematically.

Last year I have proposed a modification of theArrhenius formula for expressing biological processes mathematically, in which, instead of the absolute zero (−273° C), a smaller value appeared as biological zero temperature:

$$\begin{gathered} Y_t = \frac{m}{{l/t - z}} \hfill \\ n \hfill \\ \end{gathered} $$

where Y_t=speed of the process at temperature t; m=maximum speed of the process at t=; z=biological zero; n=a very great number. The three parameters can be calculated from observed data. Only the determination of z is somewhat more difficult.

The application of the new formula to several examples from literature led to very good results. In the present paper it is applied to the observed values on whichKrogh based his standard curve. This procedure demonstrates that the new formula represents the values in many cases with a high degree of exactness. In addition, the new curve allows the representation of the numbers given byKrogh on the relation between temperature and respiration in Tenebrio-pupae, which could not be brought into coincidence with the standard curve.

A number of further examples have thus demonstrated that the new formula is suited for an exact representation of biological temperature functions. Single numbers — like the Q₁₀ or μ values — cannot express biological temperature functions. The three parameters contained in the new formulare are the minimum requirements for a quantitative reproduction of a curve pattern.

The new formula has also been applied to express the relation between temperature and the speed of development ofAedes taeniorhynchus. In this case too it proves to be superior to the method ofJørgensen.

Zusammenfassung

• 1. Auf die Zahlenbeispiele, dieKrogh (1914) dem Entwurf seiner Normalkurve zugrunde legte, die den Zusammenhang zwischen der Geschwindigkeit biologischer Vorgänge und der Temperatur wiederzugeben versuchte. wird die im vergangenen Jahr vorgeschlagene Funktion:

  $$\begin{gathered} y_t = \frac{m}{{ \frac{l}{{ t - z}}}} \hfill \\ n \hfill \\ \end{gathered} $$

  angewandt.

• 2. Es kann gezeigt werden, daß die neue Funktion die Zahlenbeispiele vonKrogh mit hinreichender Genauigkeit wiederzugeben gestattet, so daß man in ihr den formelmäßigen Ausdruck für dieKroghsche Normalkurve sehen darf.

• 3. Gegenüber einem einfachen Vergleich des Kurvenverlaufes gestattet die mathematische Formel

  • a. einen exakten zahlenmäßigen Vergleich der gefundenen und berechneten Werte,

  • b. die Kennzeichnung des Kurvenverlaufes durch Zahlenwerte,

  • c. die Darstellung von Kurvenverläufen, die von der Normalkurve abweichen.

• 4. Am Beispiel vonAedes taeniorhynchus läßt sich zeigen, daß die neue Temperaturfunktion bis zu einem gewissen Umfang auch auf Entwicklungsgeschwindigkeiten anwendbar ist.