K i + [R i ] n = [Rtot i ] (2) ] = β i [M i ] δ i [Ri tot ] (5) + α m j (6) 1 + r n i. dr i dt = βm i δr i (7) m j = βm i = δr i (9) 1 + r n + α.

Transcription

1 The lamba phage will remain in the lysogenic state if ci proteins preominate, but will be transforme into the lytic cycle if cro proteins preominate. cro: (Control of Repressor's Operator) Transcription inhibitor, bins OR3, OR2 an OR1 (affinity OR3 > OR2 = OR1, i.e. preferentially bins OR3). At low concentrations blocks the prm promoter (preventing ci prouction). At high concentrations ownregulates its own prouction through OR2 an OR1 bining. No cooperative bining. ci: (Clear 1) Transcription inhibitor, bins OR1, OR2 an OR3 (affinity OR1 > OR2 = OR3, i.e. preferentially bins OR1). At low concentrations blocks the pr promoter (preventing cro prouction). At high concentrations ownregulates its own prouction through OR3 bining. Bining of ci at OR1 stimulates an almost simultaneous ci bining to OR2 via cooperative bining (via ci C terminal omain interactions) N terminal omain of ci on OR2 tightens the bining of RNA polymerase to prm an hence stimulate its own transcription. Repressor also inhibits transcription from the pl promoter. Susceptible to cleavage by RecA* in cells unergoing the SOS response.

2 Operator states: (i = {λ : ci, teto1 : tetr, laco1 : laci}) an mrna prouction an ecay: [Di tot ] [D i ] = 1 + [R i ] n, [Di ] = [Dtot i ][R i ] n /K i [R i ] n (1) + K i [R i ] + n [R i] n [D tot i ] K i + [R i ] n = [Rtot i ] (2) t [M j] = α i [D i ] + α i [D i ] δ Mj [M j ] (3) tight regulation: α α an [R i ] [R tot i ] Protein concentrations: t [M [Di tot ] j] α i 1 + [Ri tot ] n + αi δ Mj [M j ] (4) /K i t [Rtot i ] = i [M i ] δ i [Ri tot ] (5) Caricature: α i [D tot i ] = α, α i = α, δ Mi = 1 (choice of time unit), K i = 1 (choice of concentration unit), i =, δ i = δ. m j t = α 1 + r n i + α m j (6) Steay state: thus, r i t = m i δr i (7) m j = α 1 + ri n + α (8) r = δ Linear stability: m = m 0 + m, r = r 0 + r. m i = δr i (9) ( ) α 1 + r n + α. (10) where X = r t m j = r i m j (11) t r i = m i δr i (12) [ ] α 1 + r n + α m j = αnrn 1 (1 + r n ) 2 > 0 (13) 1

3 t m ci r ci m laci r laci m tetr r tetr = 1 δ 1 δ 1 δ m ci r ci m laci r laci m tetr r tetr (14) 1 λ Roots I: δ λ λ < 0 (always): Roots II: instability: Reλ > 0 1 λ δ λ 1 λ δ λ = (1+λ) 3 (δ+λ) X 3 = 0 (15) (1 + λ)(δ + λ) = (16) 2λ = (1 + δ) ± (1 δ) 2 4X (17) δ + 1 < (1 δ) 2 4X (18) (δ + 1) 2 > (1 δ) 2 4X (19) 4δ > 0 > (20) (1 + λ)(δ + λ) = Xe ±iπ/3 (21) 2λ = (δ + 1) ± (δ 1) 2 + 2X(1 ± 3i) (22) δ + 1 < Re (δ 1) 2 + 2X(1 ± 3i) (23) [(δ 1) 2 + 2X] tan φ = 2 3X (24) Re (δ 1) 2 + 2X(1 ± 3i) = [(δ 1) 2 + 2X] cos φ (25) thus, Reλ > 0 for large enough δ. δ + 1 < [(δ 1) 2 + 2X] cos φ (26) 2

4 Oct 13, 03 16:21 Page 1/1 #! /usr/bin/octave2.0 global alpha_0=5e 4; global alpha=10; global n=2.1; global beta=5; function ret = m(m,p) global alpha alpha_0 n; ret= m + alpha/(1+p^n) + alpha_0; enfunction function ret = p(m,p) global beta; ret= beta*(p m); enfunction function xot = f(x,t) xot=zeros(6,1); xot(1)=m(x(1),x(6)); xot(2)=p(x(1),x(2)); xot(3)=m(x(3),x(2)); xot(4)=p(x(3),x(4)); xot(5)=m(x(5),x(4)); xot(6)=p(x(5),x(6)); enfunction x0=[1;1;0;0;0;0]; t=linspace (0,20,100); y=lsoe("f",x0,t); plot(t,y) input("press Enter"); osc3.m Printe by Anras Czirok Saturay September 30, /1

5 >> syms X b l; >> A=[ 1,0,0,0,0,; b,, 0,0,0,0; 0,, 1,0,0,0; 0,0,b,,0,0; 0,0,0,, 1,0; 0,0,0,0,b, ] >> B=A l*eye(6) B = [ 1 l, 0, 0, 0, 0, ] [ b, l, 0, 0, 0, 0] [ 0,, 1 l, 0, 0, 0] [ 0, 0, b, l, 0, 0] [ 0, 0, 0,, 1 l, 0] [ 0, 0, 0, 0, b, l] >> simple(et(b) (1+l)^3*(+l)^3) ans= X^3*b^3 osc3_calc Printe by Anras Czirok Sep 17, 10 10:59 Page 1/1 >> eig(a) ans = 1/2* 1/2+1/2*(^2 2*+1 4*X*b)^(1/2) 1/2* 1/2 1/2*(^2 2*+1 4*X*b)^(1/2) 1/2* 1/2+1/2*(^2 2*+1+2*X*b 2*i*X*b*3^(1/2))^(1/2) 1/2* 1/2 1/2*(^2 2*+1+2*X*b 2*i*X*b*3^(1/2))^(1/2) 1/2* 1/2+1/2*(^2 2*+1+2*X*b+2*i*X*b*3^(1/2))^(1/2) 1/2* 1/2 1/2*(^2 2*+1+2*X*b+2*i*X*b*3^(1/2))^(1/2) >> 2*eig(A) ans = 1+(^2 2*+1 4*X*b)^(1/2) 1 (^2 2*+1 4*X*b)^(1/2) 1+(^2 2*+1+2*X*b 2*i*X*b*3^(1/2))^(1/2) 1 (^2 2*+1+2*X*b 2*i*X*b*3^(1/2))^(1/2) 1+(^2 2*+1+2*X*b+2*i*X*b*3^(1/2))^(1/2) 1 (^2 2*+1+2*X*b+2*i*X*b*3^(1/2))^(1/2) Friay September 17, /1

6 The synthetic microbial consortium oscillator. (A) Circuit iagrams of the activator (top) an repressor (bottom) strains. In the activator strain, transcription of rhli an cfp are regulate by separate copies of the hybri promoter, Prhl/lac, which is up-regulate by C4-HSL an own-regulate by LacI. In the repressor strain, cini is riven by the hybri promoter Prhl/lac an yfp is regulate by the hybri promoter Pcin/lac, which is up-regulate by 3-OHC14-HSL an own-regulate by LacI. Both strains contain constitutively expresse copies of cinr an rhlr, which encoe transcription factors that respon to the HSLs to regulate their respective promoters, an aiia an laci riven by 3-OHC14-HSL responsive promoters. (B) Global topology of the ual-feeback consortium oscillator. The activator strain up-regulates genes in both strains. The repressor strain own-regulates genes in both strains. AiiA own-regulates signaling (ashe lines, omitte in Figs. 2 an 3 an figs. S2 an S7 for simplicity). (C) Representative time series of activator (blue) an repressor fluorescence (yellow), an activator population fraction (black, ratio of the area of activator cells to the area of the entire population of cells, as measure in pixels) for the consortium epicte in (A). Relative fluorescence values are the population average relative to the maximum after backgroun subtraction.