Biochemical oscillations in cell biology

1 Show that when 6 = in Equation (5.6) for the non-saturating phosphorylation- 

2 For Equation (5.12), show that if one wants to increase f from 0.1 to 0.9 by changing 0, one needs to increase 0 by a factor of 81. That is, it takes about a one-hundred-fold increase in the kinase activity to turn on the switch. 

3 Consider a phosphorylation-dephosphorylation cycle for a substrate protein (S) with saturated kinase (K) but unsaturated, first-order, phosphatase (P). For simplicity, we neglect the cofactors such as ATP, ADP, and PI  

Show that the steady state concentration of the phosphorylated [S ]55 = x satisfies the quadratic equation 

4 Consider the biochemical reaction system with four species  

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