Question

2.) Activator and helper. Figure 19.29B gives the fold change for a genetic network with an activator and helper. However, it contains several mistakes that we will fix here.

a.) Derive the formula for the fold change under the assumptions of a weak promoter and �!"# $% ≫ �. Show that the fold change is given by the formula below rather than that in the book. Hint: The schematic in Figure 19.29B should be helpful in determining the states and weights. �� = 1 + ��$&'!" 1 + ℎ�$&'#! 1 + ℎ 1 + � 1 + ℎ�$&'#! 1 + ℎ

b.) How would you interpret the effective number of activators in the system? How would you interpret the effective number of polymerase in the system? How does this support the idea of hierarchical regulation?

c.) Discuss the limits when � → 0, ℎ → 0, and �() → 0.

d.) Plot the fold change for �)* = −3.9�+�, �() = −5�+�, and ℎ = 1, 10, 10,. Show that, rather than the shape of the curves in Figure 19.29B, the curves converge to a given value of ��. What is this value given by? Show that continuing to increase the number of helper molecules does not benefit the activation of the gene, i.e. the curves converge as ℎ → ∞. On the same graph, plot the curve corresponding to �)* = −3.9�+�, �() = −8�+� and ℎ = 10. Does it benefit to continue to increase the helper-activator interaction energy?

2.) Activator and helper. Figure 19.29B gives the fold change for a genetic network with an activator and helper. However, it contains several mistakes that we will fix here. a.) Derive the formula for the fold change under the assumptions of a weak promoter and Freg »p. Show that the fold change is given by the formula below rather than that in the book. Hint: The schematic in Figure 19.29B should be helpful in determining the states and weights. | 1+ ae-Rap - 1.-Be 1 + he-Beha 1+h FC = - 1+a- 1.1+ he -Bena Tu 1 +h b.) How would you interpret the effective number of activators in the system? How would you interpret the effective number of polymerase in the system? How does this support the idea of hierarchical regulation? c.) Discuss the limits when a → 0, 0, and Ena + 0. d.) Plot the fold change for eap = -3.9kgT, Ena = -5kgT, and h = 1, 10, 102. Show that, rather than the shape of the curves in Figure 19.29B, the curves converge to a given value of FC. What is this value given by? Show that continuing to increase the number of helper molecules does not benefit the activation of the gene, i.e. the curves converge as h→0. On the same graph, plot the curve corresponding to Eap = -3.9kgT, Eha = -8kBT and h = 10. Does it benefit to continue to increase the helper-activator interaction energy?

Answer 1

Answer (a)- Formula for the fold change will be as below.

repression (R) = tfold-change (R)]=[fregler Site efort

Answer (b)- The probability of binding when the number of polymerase molecule is p=500

and activator is shown belowin the figure.

CM Eaps-5257 UN 301 Exp: -4KBT Cap=-3KBT 20 40 60 80 100 These are the number of activator o molecules, The probability of binding when the number of polymerase molecule is P = 500 and the binding paralniteres are ac bd = 5,3k ot, and stad=-13.12 pa siSKBT.