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l7_notes - Mathematical basis of stability analysis ) , ( )...

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Unformatted text preview: Mathematical basis of stability analysis ) , ( ) , ( y x g y y x f x = = & & system of two coupled differential equations find nullclines and fixed point(s) step 1 ) , ( ) , ( = → = = → = o o o o y x g y y x f x & & o o y y y x x x − ≡ − ≡ ~ ~ step 2 consider small deviation from fixed point o o y y y x x x − ≡ − ≡ ~ ~ consider small deviation from fixed point linearize around fixed point(s) step 3 y d x c y g y x g x y y b x a y f y x f x x o o o o o o o o y x y x y x y x ~ ~ ~ ~ ~ ~ ~ ~ ) , ( ) , ( ) , ( ) , ( + ≡ ∂ ∂ + ∂ ∂ ≈ + ≡ ∂ ∂ + ∂ ∂ ≈ & & step 4 determine matrix A ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = d c b a A ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = d c b a A determine matrix A determine trace and determinant of A: step 5 bc ad A d a A trace − = = ∆ + = = ) 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 µ m, diameter ~ 0.5 µ m 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....
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This note was uploaded on 11/11/2011 for the course BIO 7.344 taught by Professor Bobsauer during the Spring '08 term at MIT.

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l7_notes - Mathematical basis of stability analysis ) , ( )...

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