Question

One of the simplest known circadian clocks is found in the photosynthetic cyanobacterium Synechococus Elongatus and involves a set of three interacting proteins: KaiA, KaiB, and KaiC. In a purified mixture of these three proteins plus ATP, the ratio of phosphorylated KaiC to total KaiC (P- KaiC/KaiC) oscillates periodically between 0.3 and 0.6. The ratio starts at its maximum value of 0.6 at time t = 0 and does not bottom out at 0.3 until time t = 43,200 s. a) What is the period of this oscillation, in seconds? How about in hours? b) What is the frequency of this oscillation? c) Write down a formula that specifies the value of the ratio P-KaiC/KaiC at any given time t, assuming that the oscillation is harmonic. (Hint: You'll have to introduce an additive constant to the formula we've used in class, because the "equilibrium" value is not zero here.)

Answer 1

Since the ratio stars at its maximum value: 0.6 and then reaches 0.3 after 43200 s A period corresponds to one oscillation (go back to 0.6) Then, the period is: T = 2(43200) = 86400 s In hours: 1h T = 86400 s (2010-) = 24 h b) The frequency is given by: 1 1 f=i=86400 = 1.157 x 10-5Hz c) since the oscillation is harmonic: r = B + Acos(wt) Where: A is the amplitude and B is a constant The amplitude is: A _ 0.6 – 0.3 A = = 2 -= 0.15 The constant B = 0.45 The angular frequency: 21 21 W = T 86400 s Then: r = 0.45 + 0.115 cos(7.27 x 10-5 t) 10-5 rad