34 



TEMPORAL ORGANIZATION IN CELLS 



When Xi<Pi, dYJdKO. Thus y, must decrease. At some moment 

 y, < q,, and so long as X,- < /?, this inequality will continue since dYj/dt < 0. But 

 for Yi < qi, dXJdt > 0, so that Xj must now increase. We thus have a contra- 

 diction : if we keep Xi < Pi indefinitely, then dXJdt becomes and remains a 

 positive quantity so that Xi increases without bound. Thus Xi cannot remain 

 always less thanj?,-. A similar argument holds for 7,- and ^,, and we have the 

 result that the variables must oscillate about their steady state values, /;, and qi. 



Xiit) 



Y^t) 



Figure 3. 



providing only that they do not start at these values or reach these values 

 simultaneously. In this case the derivatives dXJdt and dYJdt are both zero, 

 and there is no motion in the system: it is completely stationary. 



The shape of the closed trajectories has been studied briefly on an analogue 

 computer, and it has the form shown in Fig. 3. As functions of time the varia- 

 ables Xi and F, have the periodic wave-forms shown in Fig. 4. Clearly these 

 oscillations are quite non-linear, and they are unlike any of the non-linear 

 oscillations which have been studied in mechanics. It is, however, hardly 

 surprising that functions deriving from biological considerations should be 



