Mathematical basis of stability analysis),(),(yxgyyxfx==&&system of two coupled differential equationsfind nullclines and fixed point(s)step 10),(00),(0=→==→=ooooyxgyyxfx&&ooyyyxxx−≡−≡~~step 2consider small deviation from fixed pointooyyyxxx−≡−≡~~consider small deviation from fixed pointlinearize around fixed point(s)step 3ydxcygyxgxyybxayfyxfxxooooooooyxyxyxyx~~~~~~~~),(),(),(),(+≡∂∂+∂∂≈+≡∂∂+∂∂≈&&step 4determine matrix A⎥⎦⎤⎢⎣⎡=dcbaA⎥⎦⎤⎢⎣⎡=dcbaAdetermine matrix Adetermine trace and determinant of A:step 5bcadAdaAtrace−==∆+==)det()(τdetermine stability of fixed pointstep 6only if τ < 0 and ∆ > 0, (xo,yo) is a stable fixed point!!! be careful: only valid for 2 dimensional systems !!!Last lectures: Genetic SwitchesL3-4: Naturally occurring: lysis-lysogeny decisionL5-6: Engineered: genetic toggle switchSwitches are necessary for making ‘decisions’:- development & differentiation (e.g. stem cells)what to be ?- metabolismwhat to eat ?- molecule synthesis (e.g amino acids)what to produce ?time scales for genetic regulation ~ 10 min - hoursImages removed to due copyright considerations.What if faster response is needed ?- finding food- chasing bait- signal transductionImage 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 - minutesIntroducing the H atom for signal transduction:chemotaxis of Escherichia coliImage 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 µmImages removed due to copyright considerations.The FlagellumImage removed due to copyright considerations.Absence of chemical attractantImage by MIT OCW.TumbleRunPresence of chemical attractantImage by MIT OCW.TumbleChemical Gradient Sensed in a Temporal MannerRunAttractantImage 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 byreceptors, kinases,and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512.Chemotactic pathway in E. coliTowards more complex system networks.Image removed due to copyright considerations.Proteins in the chemotactic network can bemodified 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 tophosphorylate one of its histidines.ADPpCheAATPCheA +↔+CheA (CheAp)is bound to the Tar receptorthrough an adapter protein CheW. CheW isnot known to have any enzymatic activity.(these proteins are sometimes called‘scaffolding protein’)CheApis unstable and transfers itsphosphoryl group to CheY (highly soluble,diffuses through the cytoplasmCheAHis48Courtesy 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 byreceptors, kinases,and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512.I Phosphorylation (CheA, CheY, CheB)ADPpCheAATPCheA+↔+autophosphorylation:pCheYCheACheYpCheA+↔+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 CheYp(opposite function as CheA)pCheZCheYCheZpCheY+↔+logic:high levels of CheZ -> low levels of CheYp(straight swimming)II Methylation (tar receptor)Image removed due to copyright considerations.CheR adds methyl groupCheBpremoves methyl groupphosphorylation state of CheBis controlled by CheAMethylation - Phosphorylation couplingImage removed due to copyright considerations.phosphorylation state of CheBis controlled by CheACourtesy 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 byreceptors, kinases,and adaptation enzymes." Annu Rev Cell Dev Biol 13 (1997): 457-512.Role of ligand bindingImage removed due to copyright considerations.The rate of CheA phosphorylation is stimulatedby unoccupiedreceptorsImage by MIT OCW. After figure 4 in Falke, J. J., R. B. Bass, S. L. Butler, S. A. Chervitz, and M. A. Danielson. "Thetwo-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 mMand sense changes as small as 0.1% 12 3 4tumblingmethylationadd attractantCorrelation of Receptor Methylation with Behavioral Responseremove attractantadd moreattractantBefore 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 onE. coli chemotaxis for about 100 years nowSingle cell chemotactic analysisImage 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 expressioncells have plasmids with CheY-GFP under inducible promoterassumption: all CheY is phosphorylatedstrain: CheY-, CheZ-, CheB-Hill #: ~10Image 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