Wrapping up E. coli Chemotaxis (L7 & L8) Main points of last 2 lectures: L7: Biological background what is the function of the individual molecules ? L8: modeling of all possible chemotactic reactions why doesn’t this model reproduce experimentally observed perfect adaptation ? L8-9: strip down full model to essentials based on assumptions that are experimentally justified (or sometimes not) reduction 1 100, no. 23 (Nov 11, 2003): 13259-63. Copyright (2003) National Academy of Sciences, U. S. A. Figure 1A in Mittal, N., E. O. Budrene, M. P. Brenner, and A. Van Oudenaarden. "Motility of Escherichia coli cells in clusters formed by chemotactic aggregation." Proc Natl Acad Sci U S A. Images removed due to copyright considerations. 3 Absence of chemical attractant 4 Image by MIT OCW. Tumble Run Presence of chemical attractant 5 Image by MIT OCW. Tumble Chemical Gradient Sensed in a Temporal Manner Run Attractant 6 Figure 1 of Spiro, P. A., J. S. Parkinson, and H. G. Othmer. Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. "A model of excitation and adaptation in bacterial chemotaxis." key player: Tar-CheA-CheW complex Copyright (1997) National Academy of Sciences, U. S. A. fast slow i n t e r m e d i a t e Copyright (1997) National Academy of Sciences, U. S. A. Figure 2 of Spiro, P. A., J. S. Parkinson, and H. G. Othmer. Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. "A model of excitation and adaptation in bacterial chemotaxis." T 2 LT 2 T 3 LT 3 T 2p LT 2p T 3p LT 3p k eff1 (L) k eff2 (L) k eff3 (L) k eff4 (L) k eff3 (L) k eff4 (L) k pt k pt First reduction α1-α [3][2] [3] α + ≡ in steady state: 8 (L) eff2 αk(L) eff1 α)k(1 phos k +?= safe zone net phosphorylated rate perfect adaptation for large L perfect adaptation for small L 9 fine-tune: net phoshorylation rate and k eff1 and k eff2 so that α falls in safe zone net phosphorylated rate no safe zone never perfect adaptation 10 T 2 LT 2 T 3 LT 3 T 2p LT 2p T 3p LT 3p k eff1 (L) k eff2 (L) k eff3 (L) k eff4 (L) k eff3 (L) k pt k pt Second reduction additional assumption: - CheB only demetylates phosphorylated receptors experimental backup: - not possible to directly measure if CheB demethylates only active receptors - rate of methylation drops immediately after addition of ligand indicates that CheB works on active receptors 11 T 3 LT 3 T 3p LT 3p k eff2 (L) r in k eff4 r in k pt Third reduction additional assumption: - [CheR]<< [receptors], methylation operates at saturation (r in is independent of receptor concentrations) experimental backup: - Michaelis constant of CheR binding << [receptors] so R tot ~ R bnd ok 12 T 3 LT 3 T 3p LT 3p k eff2 (L) r in k eff4 r in k pt Fourth reduction additional assumption: - demethylation is identical for bound and unbound receptors, so k eff4 is independent of L. experimental backup: - kinetics of demethylation almost independent of level of methylation and ligand binding. ok 13 T 3 LT 3 T 3p LT 3p k eff2 (L) r in k eff4 r in k pt This final module obeys perfect adaptation for any value of L. eff4 k in 2r ] p [3 = 14 Coming lectures: Biological oscillators Biological relevance: Cell division cycle Circadian Rhythms etc. What do you need to make an oscillator ? How is an oscillator different from the systems we already discussed (switches, chemotactic network) ? ? Graphical way to represent differential equations. 15 Refresh: Autocatalysis (Problem set #1). XXA kk 2 1,1 ??→←+ +? x=[X] a=[A] = constant (e.g. enormous surplus) 2 11 xkaxkx ? ?= & )(xfx = & This equation is of the type: first order differential equation Analysis recipe: 1. determine fixed points 2. stability analysis 16 1 1 * 2 * 1 0 ? = = k ak x x Fixed points: f(x*) = 0: 0>x & 0<x & 2 fixed points: one stable and one unstable 17 Guestimate of dynamics for different initial conditions [X] time 18 more quantitative stability analysis: small perturbation from fixed point: ))('exp(~)( )(' )(')()(')()( )()( * * *2*** * txft xf xfOxfxfxf xtxt η ηη ηηηη η ≈ ≈++=+ ?= & 0)(' 0)(' * * < > xf xf unstable fixed point stable fixed point 19 0)(' * <xf 0)(' * >xf ‘stable’ ‘unstable’ 20 Other example: T 3 LT 3 T 3p LT 3p k eff2 (L) r in k eff4 r in k pt C* C ineffpt ineffeffpt rCkCkC rCkCkkC +?= ++??= 2 * 2 * 4 * )( & & in in rdycxy rbyaxx ++= ++= & & bd kc kb kka Cy Cx pt eff effpt ?≡ ≡ ≡ ??≡ ≡ ≡ 2 4 * )( 21 Nullclines: x k k k r x d c d r y x k kk k r x b a b r y eff pt eff inin eff effpt eff inin 22 2 4 2 +=? ? = + + ? =? ? = 0 0 = = y x & & y fixed point (stable or unstable ?) 0=y & x 0=x & 22 23 y x ineffpt ineffeffpt rykxky rykxkkx +?= +++?= 2 24 )( & & 0=y & 0=x & ),( ** yx 0,0 >> yx && 0,0 >< yx && 0,0 <> yx && 0,0 << yx && 24 y x ineffpt ineffeffpt rykxky rykxkkx +?= +++?= 2 24 )( & & 0=y & 0=x & ? ? ? ? ? ? ? ? + = )( 2 , 2 ),( 24 4 4 ** Lkk krkr k r yx effeff ptineffin eff in increased ligand concentration 25 y x 0=y & 0=x & ? ? ? ? ? ? ? ? + = )( 2 , 2 ),( 24 4 4 ** Lkk krkr k r yx effeff ptineffin eff in TOT Cyx =+ C TOT C TOT Guestimated response C* 26 ‘activity’ (phosphorylation level) time C TOT (methylation level) time Oscillators ? yxayby yxayxx 2 2 ??= ++?= & & model for glycolysis 2 2 xa b y xa x y + = + = nullclines: 27 0 0 > > y x & & 0 0 > < y x & & 0 0 < > y x & & 0 0 < < y x & & 0=x & 0=y & y x 28 0 0 > > y x & & 0 0 > < y x & & 0 0 < > y x & & 0 0 < < y x & & 0=x & 0=y & y x 29 limitcycle x x time y 30 Dynamical response of switches, chemotactic network and oscillators ‘switch’ adaptation (differentiator, at least for small frequencies) oscillator 31 Dynamical response of switches, chemotactic network and oscillators two stable fixed points one stable fixed point unstable fixed point 32 nullclines: γ u1 2 α v β v1 1 α u + = + = Image removed due to copyright considerations. v γ u1 2 α dt dv u β v1 1 α dt du ? + = ? + = 33 y x ineffpt ineffeffpt rykxky rykxkkx +?= +++?= 2 24 )( & & 0=y & 0=x & ? ? ? ? ? ? ? ? + = )( 2 , 2 ),( 24 4 4 ** Lkk krkr k r yx effeff ptineffin eff in Adaptation (one stable fixed point) sfp 34 increased ligand concentration 35 y x 0=y & 0=x & ? ? ? ? ? ? ? ? + = )( 2 , 2 ),( 24 4 4 ** Lkk krkr k r yx effeff ptineffin eff in TOT Cyx =+ C TOT C TOT Oscillator (unstable fixed point) 0 0 > > y x & & 0 0 > < y x & & 0 0 < > y x & & 0 0 < < y x & & 0=x & 0=y & ufp y x 36 0 0 > > y x & & 0 0 > < y x & & 0 0 < > y x & & 0 0 < < y x & & 0=x & 0=y & y x 37 38 Image removed due to copyright considerations. See Figures 1, 2, 3 in Elowitz, M. B., and S. Leibler. "A synthetic oscillatory network of transcriptional regulators." Nature 403, no. 6767 (Jan 20, 2000): 335-8. 39