122
F. KRUGER
retical and experimental approach to the quantitative relations in the process
of growth. It may be possible that corrections will be necessary or that
better formulations will be found, but I think that simple models will be
more useful than complicated ones. The latter perhaps offer better approaches to special cases, but they do not allow further derivations.
It was an essential feature that the models which I used resulted in
solutions which are intellectually perspicuous and therefore offer the
possibility to control experimentally the mathematical conclusions. Here
seems to me given the heuristic value of the mathematical formulations
going beyond the reproduction of series of numbers.
References
1. BERTALANFFY, L. v.: Entw. Mech. Org. 131, 613 (1934).
2. - Hum. BioI. 10, 181 (1938).
3. - Theoretische Biologie, Bd. II, Stoffwechsel, Wachstum.
1. Auff. Berlin: Borntraeger 1942. 2. Auf I. Bern: Francke (1951).
4. - HelgoI. Wiss. Meeresunters. 9, 5 (1964).
5. KRUGER, F.: Naturwissenschaften 49, 54 (1962).
6. - ZooI. Anz. (SuppI. Bd.) 27, 249 (1964).
7. - HelgoI. Wiss. Meeresunters. 12, 78 (1965).
8. - HelgoI. Wiss. Meeresunters. 14, 302 (1966).
9. LOCKER, A.: Naturwissenschaften 48, 445 (1961).
10. PANDIAN, T. J.: Marine BioI. 1, 16: (1967).
11. PUTTER, A.: Pffiigers Arch. 180, 298 (1920).
12. SCHRODINGER, E.: Was ist Leben? Bern: Francke (1946).
13. ZUCKER, L. and ZUCKER, TH. F.: J. gen. PhysioI. 25,445: (1941).
Discussion
KLAMERTH:
If your formula is quite a general one, being applicable not only to growing
organisms but also to multiplying entities, e.g. a culture of E. coli, it seems to
me that the term X should be replaced by generation time.
KRUGER:
As I have shown the formula may be applied to the growth of yeast cells as
well as of Paramecium; however, I did not attempt to apply it also to the growth
of cells in culture. I suppose that in this instance X would represent the time
elapsed since the starting of the culture.
MEYER-DoRING:
What is the significance of the terms X and ~?
KRUGER:
In my formula X means age starting from birth, whereas ~ represents mathematically prenatal age. ZUCKER (1941) was the first to propose this formula,
but he was not able to calculate the value of ~ and therefore used arbitrary
values, e.g. age from conception onwards.
F. KRUGER
retical and experimental approach to the quantitative relations in the process
of growth. It may be possible that corrections will be necessary or that
better formulations will be found, but I think that simple models will be
more useful than complicated ones. The latter perhaps offer better approaches to special cases, but they do not allow further derivations.
It was an essential feature that the models which I used resulted in
solutions which are intellectually perspicuous and therefore offer the
possibility to control experimentally the mathematical conclusions. Here
seems to me given the heuristic value of the mathematical formulations
going beyond the reproduction of series of numbers.
References
1. BERTALANFFY, L. v.: Entw. Mech. Org. 131, 613 (1934).
2. - Hum. BioI. 10, 181 (1938).
3. - Theoretische Biologie, Bd. II, Stoffwechsel, Wachstum.
1. Auff. Berlin: Borntraeger 1942. 2. Auf I. Bern: Francke (1951).
4. - HelgoI. Wiss. Meeresunters. 9, 5 (1964).
5. KRUGER, F.: Naturwissenschaften 49, 54 (1962).
6. - ZooI. Anz. (SuppI. Bd.) 27, 249 (1964).
7. - HelgoI. Wiss. Meeresunters. 12, 78 (1965).
8. - HelgoI. Wiss. Meeresunters. 14, 302 (1966).
9. LOCKER, A.: Naturwissenschaften 48, 445 (1961).
10. PANDIAN, T. J.: Marine BioI. 1, 16: (1967).
11. PUTTER, A.: Pffiigers Arch. 180, 298 (1920).
12. SCHRODINGER, E.: Was ist Leben? Bern: Francke (1946).
13. ZUCKER, L. and ZUCKER, TH. F.: J. gen. PhysioI. 25,445: (1941).
Discussion
KLAMERTH:
If your formula is quite a general one, being applicable not only to growing
organisms but also to multiplying entities, e.g. a culture of E. coli, it seems to
me that the term X should be replaced by generation time.
KRUGER:
As I have shown the formula may be applied to the growth of yeast cells as
well as of Paramecium; however, I did not attempt to apply it also to the growth
of cells in culture. I suppose that in this instance X would represent the time
elapsed since the starting of the culture.
MEYER-DoRING:
What is the significance of the terms X and ~?
KRUGER:
In my formula X means age starting from birth, whereas ~ represents mathematically prenatal age. ZUCKER (1941) was the first to propose this formula,
but he was not able to calculate the value of ~ and therefore used arbitrary
values, e.g. age from conception onwards.