Organizational remarks: PS#2,1b: Correction plot k and k for l=0.2/K (NOT: Plot ka and kb for L=0.2KL) Tomorrow's recitation topic PS #2 support
Organizational Remarks: PS #2, 1b: Correction: Plot k a and k b for L = 0...2/K L (NOT: Plot k a and k b for L = 0...2KL ) Tomorrow’s recitation topic: ‘PS #2 support’ 1
Dynamical response of switches, chemotactic network and oscillators switch adaptation ( differentiator at least for small frequencies oscillator
Dynamical response of switches, chemotactic network and oscillators ‘switch’ adaptation (differentiator, at least for small frequencies) oscillator 2
Dynamical response of switches chemotactic network and oscillators two stable fixed points one stable fixed point unstable fixed point
Dynamical response of switches, chemotactic network and oscillators two stable fixed points one stable fixed point unstable fixed point 3
nullclines 1+ Image removed due to copyright considerations u dt 1+V dt 1+u
nullclines: γ 1 u 2 α v β 1 v 1 α u + = + = Image removed due to copyright considerations. v γ 1 u 2 α dt dv u β 1 v 1 α dt du − + = − + = 4
Adaptation (one stable fixed point) y sf1 x,y ,"的+2rnk, 4 eff eff 2 X x=0 x k,+kru)x+k eff e2少 x ff 2
y x pt eff in pt eff eff in y k x k y r x k k x k y r = − + = − + + + 2 4 2 ( ) & & y & = 0 x & = 0 ⎟⎟⎠⎞ ⎜⎜⎝⎛ + = ( ) 2 , 2 ( , ) 4 2 4 4 * * k k L r k r k k r x y eff eff in eff in pt effin Adaptation (one stable fixed point) sfp 5
Oscillator (unstable fixed point y x>0 x>0 少0 x<0 0 <0 X
Oscillator (unstable fixed point) 0 0 > > y x & & 0 0 > y x & & 0 0 < < y x & & x& = 0 y& = 0 ufp y x 6
Oscillators continued 2 x=-xtaytx y model for glycolysis y-x y x nullclines a+x a+x x=b fixed point stable or unstable atbi
Oscillators continued .... y b ay x y x x ay x y 2 2 = − − = − + + & & model for glycolysis 2 2 a x b y a x x y + = + = nullclines: 2 * * a b b y x b + = = fixed point: stable or unstable ? 7
j=0 y x>0 x>0 少0 x<0 0 <0 X
0 0 > > y x & & 0 0 > y x & & 0 0 < < y x & & x & = 0 y & = 0 y x 8
1.0 0.8 oscillations 0.6 (stable limit cycle) 0.4 no oscillations 0.2 (stable fixed point 0.0 0.000.020.040.060.080.100.120.14
9
limitcycle X time
limitcycle x x time y 10