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 t<0 and A>0, (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 and ∆ > 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