麻省理工大学：《Systems Biology》Mathematical basis of stability analysis

Mathematical basis of stability analysis 文=f(x,y) system of two coupled differential equations y=g(x,y) step 1 find nullclines and fixed point(s) 文=0→>f(x2y)=0 j=0→>g(x2y)=0 step 2 consider small deviation from fixed point x≡x-X y-y

Mathematical basis of stability analysis ( , ) ( , ) y g x y x f x y = = & & system of two coupled differential equations step 1 find nullclines and fixed point(s) 0 ( , ) 0 0 ( , ) 0 = → = = → = o o o o y g x y x f x y & & o o y y y x x x ≡ − ≡ − ~ ~ step 2 consider small deviation from fixed point

x≡x-x consider small deviation from fixed point step 3 linearize around fixed point(s) y a0a (x0,y) ≡cx+ (x0,y) step 4 determine matrix A a b

o o y y y x x x ≡ − ≡ − ~ ~ consider small deviation from fixed point step 3 linearize around fixed point(s) cx dy y g y x g y x ax by y f y x f x x o o o o o o o o x y x y x y x y ~ ~ ~ ~ ~ ~ ~ ~ ( , ) ( , ) ( , ) ( , ) ≡ + ∂ ∂ + ∂ ∂ ≈ ≡ + ∂ ∂ + ∂ ∂ ≈ & & step 4 determine matrix A ⎥⎦⎤ ⎢⎣⎡ = c d a b A

A determine matrix A step 5 determine trace and determinant of A I= trace(a)=a+d △=det(A)=ad-bc step determine stability of fixed point only if t0, (o, yo) is a stable fixed point !be careful: only valid for 2 dimensional systems ! !

⎥⎦⎤ ⎢⎣⎡ = c d a b A determine matrix A step 5 determine trace and determinant of A: A ad bc trace A a d ∆ = = − = = + det( ) τ ( ) step 6 determine stability of fixed point only if τ 0, (xo,yo) 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

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

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

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

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

Figure 1A in Mittal, N, E.O. Budrene, M. P. Brenner, and Avan Oudenaarden Motility of Escherichia coli cells in clusters formed by chemotactic aggregation. " Proc Nat/ Acad SciUSA 100,no.23(Nv11,2003):13259-63 Copyright (2003) National Academy of Sciences, U.S. A

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 x 1-2 um diameter 0.5 um Images removed due to copyright considerations

cell length ~ 1-2 µm, diameter ~ 0.5 µm Images removed due to copyright considerations

The Flagellum Image removed due to copyright considerations

The Flagellum Image removed due to copyright considerations