Mathematical basis of stability analysis ),( ),( yxgy yxfx = = & & system of two coupled differential equations find nullclines and fixed point(s) step 1 0),(0 0),(0 =→= =→= oo oo yxgy yxfx & & o o yyy xxx ?≡ ?≡ ~ ~ step 2 consider small deviation from fixed point o o yyy xxx ?≡ ?≡ ~ ~ consider small deviation from fixed point linearize around fixed point(s) step 3 ydxc y g y x g xy ybxa y f y x f xx oo oo oo oo yx yx yx yx ~~~~ ~~~~ ),( ),( ),( ),( +≡ ? ? + ? ? ≈ +≡ ? ? + ? ? ≈ & & step 4 determine matrix A ? ? ? ? ? ? = dc ba A ? ? ? ? ? ? = dc ba A determine matrix A determine trace and determinant of A: step 5 bcadA daAtrace ?==? +== )det( )(τ determine stability of fixed point step 6 only if τ < 0 and ? > 0, (x o ,y o ) is a stable fixed point !!! be careful: only valid for 2 dimensional systems !!! Last lectures: Genetic Switches L3-4: Naturally occurring: lysis-lysogeny decision L5-6: Engineered: genetic toggle switch Switches are necessary for making ‘decisions’: - development & differentiation (e.g. stem cells) what to be ? - metabolism what to eat ? - molecule synthesis (e.g amino acids) what to produce ? time scales for genetic regulation ~ 10 min - hours Images removed to due copyright considerations. What if faster response is needed ? - finding food - chasing bait - signal transduction Image removed due to copyright considerations. genetics is too slow ! Protein switches (active/inactive states) (total amount active + inactive is constant, ignore gene expression) timescales 1 ms - minutes Introducing the H atom for signal transduction: chemotaxis of Escherichia coli Image removed due to copyright considerations. See Alberts, Bruce, et al. Chapter 13 in Molecular biology of the cell. 4th ed. New York: Garland Science, 2002. 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. cell length ~ 1-2 μm, diameter ~ 0.5 μm Images removed due to copyright considerations. The Flagellum Image removed due to copyright considerations. Absence of chemical attractant Image by MIT OCW. Tumble Run Presence of chemical attractant Image by MIT OCW. Tumble Chemical Gradient Sensed in a Temporal Manner Run Attractant Image by MIT OCW. After figure 4 in Falke, J. J., R. B. Bass, S. L. Butler, S. A. Chervitz, and M. A. Danielson. "The two-component signaling pathway of bacterial chemotaxis: a molecular view of signal transduction by receptors, kinases, and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512. Chemotactic pathway in E. coli Towards more complex system networks. Image removed due to copyright considerations. Proteins in the chemotactic network can be modified in differents ways: I Phosphorylation (CheA, CheY, CheB) II Methylation (Tar receptor) Image removed due to copyright considerations. I Phosphorylation (CheA, CheY, CheB) CheA (protein kinase), uses ATP to phosphorylate one of its histidines. ADP p CheAATPCheA +?+ CheA (CheA p )is bound to the Tar receptor through an adapter protein CheW. CheW is not known to have any enzymatic activity. (these proteins are sometimes called ‘scaffolding protein’) CheA p is unstable and transfers its phosphoryl group to CheY (highly soluble, diffuses through the cytoplasm CheA His48 Courtesy of Annual Review of Cell and Developmental Biology. Used with permission. Falke, J. J., R. B. Bass, S. L. Butler, S. A. Chervitz, and M. A. Danielson. "The two-component signaling pathway of bacterial chemotaxis: a molecular view of signal transduction by receptors, kinases, and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512. I Phosphorylation (CheA, CheY, CheB) ADP p CheAATPCheA +?+ autophosphorylation: p CheYCheACheY p CheA +?+ phosphoryltransfer: CheYp binds to the motor (FliM), motor rotates CW (= tumbles) logic: high levels of CheA -> high levels of CheYp (lots of tumbles) low levels of CheA -> low levels of CheYp (straight swimming) Image removed due to copyright considerations. CheZ dephosphorylates CheY p (opposite function as CheA) p CheZCheYCheZ p CheY +?+ logic: high levels of CheZ -> low levels of CheYp (straight swimming) II Methylation (tar receptor) Image removed due to copyright considerations. CheR adds methyl group CheB p removes methyl group phosphorylation state of CheB is controlled by CheA Methylation - Phosphorylation coupling Image removed due to copyright considerations. phosphorylation state of CheB is controlled by CheA Courtesy of Annual Review of Cell and Developmental Biology. Used with permission. Falke, J. J., R. B. Bass, S. L. Butler, S. A. Chervitz, and M. A. Danielson. "The two-component signaling pathway of bacterial chemotaxis: a molecular view of signal transduction by receptors, kinases, and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512. Role of ligand binding Image removed due to copyright considerations. The rate of CheA phosphorylation is stimulated by unoccupied receptors Image by MIT OCW. After figure 4 in Falke, J. J., R. B. Bass, S. L. Butler, S. A. Chervitz, and M. A. Danielson. "The two-component signaling pathway of bacterial chemotaxis: a molecular view of signal transduction by receptors, kinases, and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512. why is this all so complex ? Image by MIT OCW. methylation is important for adaptation (~ background subtraction) E. coli can sense aspartate from 10 nM - 1 mM and sense changes as small as 0.1% 1 2 3 4 tumbling methylation add attractant Correlation of Receptor Methylation with Behavioral Response remove attractant add more attractant Before starting with the modeling, first let’s look at some recent experiments Alon et al. Nature 397,168 (1999) Cluzel et al. Science 287, 1652 (2000) Sourjik et al., PNAS 99, 123 (2002) PNAS 99, 12669 (2002) Nature 428, 439 (2004) Remember scientists have been working on E. coli chemotaxis for about 100 years now Single cell chemotactic analysis Image removed due to copyright considerations. See figure 1 in Cluzel, P., M. Surette, and S. Liebler. Science 287, no. 5458 (Mar 3, 2000): 1652-5. "An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells." correlation CW bias & CheY-P gene expression cells have plasmids with CheY-GFP under inducible promoter assumption: all CheY is phosphorylated strain: CheY-, CheZ-, CheB- Hill #: ~10 Image removed due to copyright considerations. See figure 1 in Cluzel, P., M. Surette, and S. Liebler. Science 287, no. 5458 (Mar 3, 2000): 1652-5. "An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells." low YFP/CFP: unbound high YFP/CFP: bound FRET (fluorescence resonant transfer) Figures 1A, 1B in Sourjik, V., and H. C. Berg. "Binding of the Escherichia coli response regulator CheY to its Proc Natl Acad Sci U S A 99, no. 20 Copyright (2002) National Academy of Sciences, U. S. A. target measured in vivo by fluorescence resonance energy transfer." (Oct 1, 2002): 12669-74. CheY-YFP (yellow) CheZ-CFP (blue CheZ binds only to CheYp !! adding attractant leads to immediate lower concentration of CheY p -CheZ complex, lower [CheYp], less tumbling Figures 1A and 1B in Proc Natl Acad Sci U S A 99, no. 1 (Jan 8, 2002): 123-7. "Receptor sensitivity in bacterial chemotaxis." Sourjik, V., and Berg HC. Copyright (2002) National Academy of Sciences, U. S. A. Hill # ~ 1 Copyright (2002) National Academy of Sciences, U. S. A. Figure 2 in Proc Natl Acad Sci U S A 99, no. 1 (Jan 8, 2002): 123-7. "Receptor sensitivity in bacterial chemotaxis." Sourjik, V., and Berg HC. amplification between receptors and CheYp: ~35 amplification between CheYp and motor: ~10 total amplification ~ 350 our models should reproduce this (hint: receptor clustering) Copyright (2002) National Academy of Sciences, U. S. A. Figure 2 in Proc Natl Acad Sci U S A 99, no. 1 (Jan 8, 2002): 123-7. "Receptor sensitivity in bacterial chemotaxis." Sourjik, V., and Berg HC. adaptation slow perfect adaptation Models should also reproduce qualitative properties such as perfect adaptation excitation(fast) Figures 1a and 1 b in Proc Natl Acad Sci U S A 99, no. 1 (Jan 8, 2002): 123-7. "Receptor sensitivity in bacterial chemotaxis." Sourjik, V., and Berg HC. Copyright (2002) National Academy of Sciences, U. S. A. Images removed due to copyright considerations. See Figure 1 in Alon, U., M. G. Surette, N. Barkai, "Robustness in bacterial chemotaxis." Nature 397, no. 6715 (Jan 14, 1999): 168-71.and S. Leibler. Perfect adaptation is robust against changes in Che-protein concentrations not all parameters are robust ! Images removed due to copyright considerations. See Figure 2 in Alon, U., M. G. Surette, N. Barkai, "Robustness in bacterial chemotaxis." Nature 397, no. 6715 (Jan 14, 1999): 168-71.and S. Leibler. Goal of next lecture is develop models that qualitatively and quantitative reproduce these phenomena, such as: huge gain sensitivity perfect adaptation All these effects are ubiquitous in signal transduction pathways in general. ‘Fine tuned model for perfect adaptation’ Spiro et al. PNAS 94, 7263-7268 (1997) A model of excitation and adaptation in bacterial chemotaxis key player: Tar-CheA-CheW complex Copyright (1997) National Academy of Sciences, U. S. A. Figure 1 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. assumptions: 1. Tar is only receptor type, CheW and CheA always bound to Tar 2. Methylation occurs in specific order 3. Consider only 3 highest methylation states 4. Only CheB p demethylates 5. Phoshorylation of CheA does not affect ligand (un)binding 6. Tar-CheR binding does not affect ligand un(binding) and phosphorylation of CheA 7. CheZ is not regulated 8. Phosphotransfer from complex to CheY or CheB is not affected by occupancy or methylation state. 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. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. Ligand bound states generally have lower autophosphoryalation rates Copyright (1997) National Academy of Sciences, U. S. A. Figure 2 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. CheR methylates ligand-bound states more rapidly Copyright (1997) National Academy of Sciences, U. S. A. Figure 2 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. Consider step in aspartate concentration time ~ 1 ms, increase in ligand bound complex Copyright (1997) National Academy of Sciences, U. S. A. Figure 2 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. time ~ 5 s, total # of phosphorylated complexes decreases gradually because ligand bound complexes do not autophoshorylate very well also: CheB p decreases low CheA p , low CheY p , tumble suppression Copyright (1997) National Academy of Sciences, U. S. A. Figure 2 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. time ~ 50 s, slowly the unbound complex methylate. Note that demethylation is switched because of low levels of CheA p (low CheB p ). also low CheA p , low CheY p , tumble suppression Copyright (1997) National Academy of Sciences, U. S. A. Figure 4 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. Higher methylation states autophosphorylate easier, so slowly CheA p adapts to its initial level high CheA p , high CheY p , tumbling Copyright (1997) National Academy of Sciences, U. S. A. Figure 4 of Spiro P. A., J. S. Parkinson, and H. G. Othmer. "A model of excitation and adaptation in bacterial chemotaxis." Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8. CheY p CheB p Spiro, P. A., J. S. Parkinson, and H. G. Othmer. Figures 1, 2, and 4 in " Proc Natl Acad Sci U S A 94, no. 14 (July 8, 1997): 7263-8. A model of excitation and adaptation in bacterial chemotaxis." Copyright (1997) National Academy of Sciences, U. S. A. Image removed due to copyright considerations. See Figures 2 and 3 in Barkai, N., S. Leibler. "Robustness in simple biochemical networks." Nature 387, no. 6636 (Jun 26, 1997): 913-7. Image removed due to copyright considerations. See Figures 2 and 3 in Barkai, N., S. Leibler. "Robustness in simple biochemical networks." Nature 387, no. 6636 (Jun 26, 1997): 913-7. Image removed due to copyright considerations. See Figures 2 and 3 in Barkai, N., S. Leibler. "Robustness in simple biochemical networks." Nature 387, no. 6636 (Jun 26, 1997): 913-7.