Review Lecture 2 Michaelis-Menten kinetics E + S ES E + P k 1 k -1 k 2 d[S] dt =?k 1 [E][S]+k ?1 [ES] d[E] dt =?k 1 [E][S]+(k ?1 +k 2 )[ES] d[ES] dt =k 1 [E][S]?(k ?1 +k 2 )[ES] dP dt =k 2 [ES]≡v E o =[E]+[ES] d[S] dt =?k 1 E o [S]+(k 1 [S]+k ?1 )[ES] d[ES] dt =k 1 E o [S]?(k 1 [S]+k ?1 +k 2 )[ES] Initial conditions: [S] t=0 = S o [E] t=0 = E o [ES] t=0 = 0 [P] t=0 = 0 om omax 0 SK Sv v + = Good approximation if S o >> E o in this case S 0 ~ [S] at the start of quasi-steady state Review Lecture 2 Equilibrium binding and cooperativity j P 1j PS ? ? + n n n n [S]...K 2 K 1 K... 2 [S] 2 K 1 K[S] 1 K1 [S]...K 2 K 1 nK... 3 [S] 3 K 2 K 1 3K 2 [S] 2 K 1 2K[S] 1 K r ++++ ++++ = Adair’s Equation: ][S] 1j [P ] j [P j K ? = macroscopic association constant for transitions between state j-1 and j Note #1 Detailed balance P o P 1 P 2 ... P n-1 P n k +1 k -1 k +2 k -2 k +n k -n ] 2 [P 2 k][S] 1 [P 2 k ] 1 [P 1 k][S] o [P 1 k] 2 [P 2 k][S] 1 [P 2 k dt ] 1 d[P 0 ] 1 [P 1 k][S] o [P 1 k dt ] o d[P 0 ? + + ?= = ? ? + + ? + + ?== ? + + ?== ][S] 1j [P ] j [P j k j k j K ? = ? + ≡ etc. I Identical and independent binding sites S k + k + k - k - S S k + k + k - S k - K 1 =2K K 2 =K/2K=k + /k - K[S]1 2K[S] 2 [S] 2 K2K[S]1 2 [S] 2 2K2K[S] r + = ++ + = use Adair: II Non-identical and independent binding sites S S S k + k - k + * k - * Independent binding: [S] * K1 [S] * K K[S]1 K[S] r + + + = K=k + /k - K * =k + * /k - * S k + * k - * k + k - III Identical and interacting binding sites S k + * k - * k + k - K 1 =2K K 2 =K * /2K=k + /k - K * =k + * /k - * S S S k + k - k + * k - * 2 [S] * KK2K[S]1 2 [S] * 2KK2K[S] r ++ + = use Adair: Cooperativity 2 βx2x1 βx)x(1 Y 2 [S] * KK2K[S]1 2 [S] * 2KK2K[S] r ++ + = ++ + = β = K*/K x = K[S] β > 1: positive cooperativity β > 2: sigmoidal curve β < 1: negative cooperativity (always: d 2 Y/dx 2 < 0) Hill number for ‘real’ dimer Introduction phage biology Phage genome: 48512 base pairs ~ 12 kB ‘phage.jpg’ ~ 10 kB Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. The lysis-lysogeny decision: As the phage genome is injected phage genes are transcribed and translated by using the host’s machinery. Which set of phage proteins are expressed determines the fate of the phage: lysis or lysogeny Image by MIT OCW. Image removed due to copyright considerations. A lysogen is immune to invasion of another phage. Repressor dimers turn off genes in the injected phage chromosome. High concentration of repressor keeps cell in lysogenic state. The lysis-lysogeny decision is a genetic switch only ‘space’ for one RNA polymerase (mutual exclusion) Image by MIT OCW. After Ptashne, Mark. A genetic switch : phage lambda. 3rd ed. Cold Spring Harbor, N.Y. : Cold Spring Harbor Laboratory Press, 2004. Single repressor dimer bound - three cases: I Negative control, dimer binding to OR2 inhibits RNAp binding to right P R promoter. Positive control, dimer binding to OR2 enhances RNAp binding to left P RM promoter. Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. II Negative control, dimer binding to OR1 inhibits RNAp binding to right P R promoter. Negative control, dimer binding to OR1 inhibits RNAp binding to left P RM promoter (too distant). Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. III Negative control, dimer binding to OR3 inhibits RNAp binding to left P RM promoter. Positive control, dimer binding to OR3 allows RNAp binding to right P R promoter. Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. Repressor-DNA binding is highly cooperative intrinsic association constants: K OR1 ~ 10 K OR2 ~ 10 K OR3 However K OR2 * >> K OR2 (positive cooperativity) Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. Flipping the switch by UV: Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. In lysogenic state, [repressor] is maintained at constant level by negative feedback Image by MIT OCW. Repressor-DNA binding is highly cooperative intrinsic association constants: K OR1 ~ 10 K OR2 ~ 10 K OR3 However K OR2 * >> K OR2 (positive cooperativity) Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. Cro dimers bind non-cooperatively to OR sites K OR3 ~ 10 K OR2 ~ 10 K OR1 Note for repressor: K OR1 ~ 10 K OR2 ~ 10 K OR3 Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. Image removed due to copyright considerations. See Ptashne, Mark. A genetic switch: phage lambda. 3rd ed. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory Press, 2004. Cooperative effects make sharp switch (‘well defined’ decision) Note: several layers of cooperativity: dimerization, cooperative repressor binding Images by MIT OCW. 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 (mM)[S] Y n H =1, non cooperative n H =3, positively cooperative 100 50 Repressor concentration % R e p r e s s i o n promoter controlled by a single repressor-operator system 99.7% repression lysogen lP D How to create a mathematical model that captures the essence of the switch ? Images removed due to copyright considerations. See Arkin, A., J. Ross, and H. H. McAdams. "Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells." Genetics 149, no. 4 (Aug, 1998): 1633-48.