Review Lecture 2Michaelis-Menten kineticsE + S ES E + Pk1k-1k2 d[S]dt=−k1[E][S] + k−1[ES]d[E]dt=−k1[E][S] + (k−1+ k2)[ES]d[ES]dt= k1[E][S] − (k−1+ k2)[ES]dPdt= k2[ES] ≡ vEo=[E]+[ES] d[S]dt=−k1Eo[S] + (k1[S] + k−1)[ES]d[ES]dt= k1Eo[S] − (k1[S] + k−1+ k2)[ES]Initial conditions:[S]t=0= So[E]t=0= Eo[ES]t=0= 0[P]t=0= 0omomax0SKSvv+=Good approximation if So>> Eoin this case S0~ [S] at the start of quasi-steady stateReview Lecture 2Equilibrium binding and cooperativityjP1jPS↔−+nnnn[S]...K2K1K...2[S]2K1K[S]1K1[S]...K2K1nK...3[S]3K2K13K2[S]2K12K[S]1Kr++++++++=Adair’s Equation:][S]1j[P]j[PjK−=macroscopic association constantfor transitions between state j-1 and jNote #1 Detailed balancePoP1P2... Pn-1Pnk+1k-1k+2k-2k+nk-n]2[P2k][S]1[P2k]1[P1k][S]o[P1k]2[P2k][S]1[P2kdt]1d[P0]1[P1k][S]o[P1kdt]od[P0−++−==−−++−++−==−++−==][S]1j[P]j[PjkjkjK−=−+≡etc.I Identical and independent binding sitesSk+k+k- k-SSk+k+k-Sk-K1=2KK2=K/2K=k+/k-K[S]12K[S]2[S]2K2K[S]12[S]22K2K[S]r+=+++=use Adair:II Non-identical and independent binding sitesSSSk+k-k+*k-*Independent binding: [S]*K1[S]*KK[S]1K[S]r+++=K=k+/k-K*=k+*/k-*Sk+*k-*k+k-III Identical and interacting binding sitesSk+*k-*k+k-K1=2KK2=K*/2K=k+/k-K*=k+*/k-*SSSk+k-k+*k-*2[S]*KK2K[S]12[S]*2KK2K[S]r+++=use Adair:Cooperativity2βx2x1βx)x(1Y2[S]*KK2K[S]12[S]*2KK2K[S]r+++=+++=β = K*/Kx = K[S]β > 1: positive cooperativityβ > 2: sigmoidal curveβ < 1: negative cooperativity(always: d2Y/dx2< 0)Hill number for ‘real’ dimerIntroduction phage biologyPhage genome:48512 base pairs ~ 12 kB‘phage.jpg’ ~ 10 kBImage 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 injectedphage genes are transcribed andtranslated by using the host’smachinery.Which set of phage proteins areexpressed determines the fate of thephage: lysis or lysogenyImage by MIT OCW.Image removed due to copyright considerations.A lysogen is immune to invasion of another phage.Repressor dimers turn off genesin the injected phagechromosome. High concentrationof repressor keeps cell inlysogenic state.The lysis-lysogeny decision is a genetic switchonly ‘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 inhibitsRNAp binding to right PRpromoter.Positive control, dimer binding to OR2 enhancesRNAp binding to left PRMpromoter.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 inhibitsRNAp binding to right PRpromoter.Negative control, dimer binding to OR1 inhibitsRNAp binding to left PRMpromoter (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 inhibitsRNAp binding to left PRMpromoter.Positive control, dimer binding to OR3 allowsRNAp binding to right PRpromoter.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 cooperativeintrinsic association constants:KOR1~ 10 KOR2~ 10 KOR3However KOR2* >> KOR2(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 levelby negative feedbackImage by MIT OCW.Repressor-DNA binding is highly cooperativeintrinsic association constants:KOR1~ 10 KOR2~ 10 KOR3However KOR2* >> KOR2(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 sitesKOR3~ 10 KOR2~ 10 KOR1 Note for repressor:KOR1~ 10 KOR2~ 10 KOR3Image 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 bindingImages by MIT OCW.1.00.80.60.40.20.00 1 2 3 4 5(mM)[S] YnH=1, non cooperativenH=3, positively cooperativenH=3, positively cooperative10050Repressor concentration% Repression promoter controlled by asingle repressor-operator system 99.7% repressionlysogenλPDHow to create a mathematical model thatcaptures 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):