Biological and Biochemical Oscillators by B. Chance, etc. PDF

By B. Chance, etc.

ISBN-10: 0121678725

ISBN-13: 9780121678722

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Sample text

A third type of modification was suggested by Lotka himself and is known as his second mechanism (12). The reaction scheme is shown in Figure 7. There are two stationary states, a saddlepoint at the origin and a center within the first quadrant. The global phase portrait is as shown in Figure 8. 37 D I E T E R H. 2 By - 1 - Ί C0 Products » k υ + 4 . L o h x ( a - y ) y ( x - b ) Figure 7. Reaction mechanism and differential equation for Lotka's second mechanism· Figure 8. Global phase portrait for Lotka1s second mechanism.

This construction has a straight line with the maximum slope of fixjj), which passes through its inflection point with abcissa x£ n * . Now with the help of the new 54 BIOCHEMICAL OSCILLATORS *f n f l variable ξ = x*/x it is possible to evaluate N as shown in (13 and 14)? Ax. can be characterized by its corresponding transfer function T(s). This function can be found by solving for the ratio of the Laplace transformation of the last variable A X ^ s ) over the Laplace transformation of the first Ax^(s), assuming that all the initial conditions are identically zero [x^(0) ■ 0 ] .

11. , Z. Phys. , _72, 508 (1910). 12. , J. Am. Chem. , 1595 (1920). 40 STABILITY PROPERTIES OF METABOLIC PATHWAYS WITH FEEDBACK INTERACTIONS Gustavo Viniegra-Gonzalez Cardiovascular Research Institute University of California San Francisco Medical Center In the search for a basis of biochemical oscillations, various chemical schemes have already been considered. For example, Spangler and Snell (1961), Higgins (1967) and Pye and Chance (1966) have thought of actually occurring special reactions which might be ultimately describable by Volterra's equations and which therefore may very well lead to oscillations.

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Biological and Biochemical Oscillators by B. Chance, etc.


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