128 TEMPORAL ORGANIZATION IN CELLS 



The equation x]=~X2 gives us 



^22 ^12 



|//|1^2 



//ntan~i 



/f 



1/2 





/»i: 



or 



hn\H\^'Khn-h22) 



hii tan~i 

 (/?ii-/;22)tan-i 



H 1/2 



^12 

 //11/2 



One root of this equation is clearly hn = hn- The other roots are given by 



hi2\H\^'^ ^ J//|i/2 

 --- — ^ — = tan-i^ — ^ — 



(87) 



or 





tan 



Writing 



the equation becomes 



^11^22^12 



|i/|l/2 



a 



h\2 

 h\2 



2 

 12 



h'' 



hi2 ' '^ hnhi 

 tanax = X 



< 1 



(88) 



This will always have roots since a<\, which is the condition /i?2 < ^11^22 

 ensuring oscillatory motion in the coupled system. 



The equation x\ = Xi X2 gives 

 /?22 ^12 



1 



hv. 



or 



I//I1/2 



tan 



/7ntan~i 



// 



1/2 



/i 



tan-i 



//|l/2 |//[l/2 



12 



//i: 



.J//|l/2^ |//p/2 (/,^^+/,^^) 

 /?12 ^11 (^12 + ^22) 



I//|l/2/ //„/;i2 + //?2 



or, finally, 



tan 



hy2 \fJnhi2+hnh22/ 



'■IJh 



Uii 



/?12 \/lll/ll2 + /'ll/'22 



Again because //11/J22 > ^iL the expression 



^11^12 + ^12 



'12 



^11^12 + ^11^22 

 Writing x = l//|'''2///i2, the equation becomes 



i3.nbx = X 



= b < \ 



(89) 



