2 (bifurcation) Saddle-Node Hopf Pitchfork 2.1 Saddle-Node( ) 2.1.1 Griffith : (bistability) ON/OFF 2 (bistability) (Stable node Stable spiral) 2 Griffith X mrna mrna X Griffith ( x y mrna ) 2.1: Griffith 2
ẋ = ax + y ẏ = x 2 1 + x by 2 Griffith a b Saddle Node Saddle-Node 2.1.2 (phase plane) Griffith mrna(y) Protein(x) (nullcline) 0 (nullcline) 2.2 2 2 (fixed point) 1: Griffith Griffith 2.1.3 (vector field) { ẋ = f(x, y) ẏ = g(x, y) 2 (x, y) (ẋ, ẏ) { ẋ = x + 2y + x 2 y ẏ = 8 2y x 2 y (x, y) = (1, 2) (x, y) = (1, 2) (ẋ, ẏ) = (5, 2) x 5 y 2 (1,2) 3
x (ẋ = 0) ( 2.2) 2.2: 2: Griffith 1 2.1.4 (linearization) 1 2 (Jacobian matrix) 1 d dt x 1 = F 1 (x 1,, x n ). d dt x n = F n (x 1,, x n ) x k(fp) 4
d dt x 1(fp) + x 1. x n(fp) + x n = F 1(fp) (x 1,, x n ). F n(fp) (x 1,, x n ) + 1 2 ( x 1,, x n ) + 2 F 1 x 2 1 F 1 F x 1 1 x n x 1... F n F x 1 n x n x n 2 F 1 x 1 x n x 1..... x n. 2 F n x n x 1 2 F n x 2 n (3 ) F k (x 1(fp),, x n(fp) ) = 0 0 2 ( x k(fp) ) 2 2 2 0 d dt x 1. x n = F 1 x 1 F 1 x n.. F n x 1 F n x n x 1. x n d x = J x dt 1 J 3: Griffith Griffith 2.1.5 (matrix exponential) d x = J x dt x = exp(jt) x 0 exp(jt) = I + Jt 1! + J2 t 2 + J3 t 3 + 2! 3! x 0 ( ) 5
(eigenvalue) J λ v ( exp(jt)v = I + Jt 1! + J2 t 2 2! ( = 1 + λt 1! + λ2 t 2 2! = exp(λt)v + J3 t 3 3! + λ3 t 3 3! ) + v ) + v x 0 x 0 = c 1 v 1 + c n v n 2.3 2.3: ( x ss ) x x(t) = exp(jt) x 0 = exp(jt)(c 1 v 1 + c n v n ) = c 1 exp(λ 1 t)v 1 + c n exp(λ n t)v n x ( ) ( 2.4) 6
2.4: ( ) ( ) ( 2.5 2.6) : Unstable node Stable node Saddle Stable spiral ( ) Unstable spiral ( ) Center 2 2 J = ( a c ) b d λ 2 (a + d)λ + (ad bc) = 0 2 λ 1 λ 2 2 τ = λ 1 + λ 2 = a + d = tr(j) = λ 1 λ 2 = ad bc = det(j) 7
2.5: ( ) 1 ( ) 2 ( ) τ τ 2 4 4: Griffith Griffith 2.1.6 (bifurcation) Griffith Saddle- Node 5: Griffith ab < 1 2 ab = 1 2 ab > 1 3 2 3 2.2 Hopf( ) 2.2.1 Sel kov : Sel kov 1968 2.7 8
2.6: ( ) ( ) phosphofructokinase(pfk) Sel kov Sel kov 2.7: Sel kov ADP PFK 2.7 { ẋ = x + ay + x 2 y ẏ = b ay x 2 y x ADP y F6P (limit cycle) Center ( 2.6) ( 2.7 ) λ = a + bi 2π (3) b 9
Hopf (Hopf bifurcation) ( ) Stable spiral Unstable spiral Hopf Unstable node Hopf Hopf 2.8: ( ) Stable spiral Unstable spiral ( ) Unstable spiral ( ) 2.2.2 : Sel kov 1. Sel kov 2. 1 Spiral 3. Sel kov 4. 2 ( : ) 5. Sel kov Stable Spiral Unstable Spiral a b ( : τ=0?) 10
6. 5 a b Stable Unstable 2.3 Pitchfork( ) 2.3.1 Toggle switch Collins lab. Gardner ( 2.9) (toggle switch) Griffith Saddle-Node Toggle switch 1 3 Pitchfork 2.9: Gardner Toggle switch 2 2.9 a ẋ = 1 + y x 2 a ẏ = 1 + x y 2 x y Pitchfork ( 2.10) 2.3.2 : Toggle switch 1. Toggle switch a 1 3 2. 1 1 Stable Node 3 Stable Node 2 Unstable Node 1 11
2.10: Pitchfork y 1 3 (pitchfork) 3. y x x x 5 ax 4 + 2x 3 2ax 2 + (1 + a 2 )x a = 0 4. 3 x x 5 ax 4 + 2x 3 2ax 2 + (1 + a 2 )x a = 0 (x 3 + x a)(x 2 ax + 1) = 0 5. x 3 + ax + b = 0 3 D = 4a 3 27b 2 D > 0 3 D = 0 D < 0 1 2 x 3 + x a x 3 + x a = 0 1 2 6. 2 x 2 ax + 1 a = 2 1 3 7. a > 2 2 x 2 ax + 1 8. 2 ( : ) 2.4 Further reading Strogatz, S.H, Nonlinear dynamics and chaos, Perseus Books Publishing, 1994. (ISBN 0-7382-0453-6) ( Borisuk and Tyson (1998) ) 12
Fall, C.P., Marland, E.S., Wagner, J.M. and Tyson, J.J. Computational cell biology, Springer, 2002. (ISBN 0-387-95369-8) (Bendixson ) Borisuk, M.T. and Tyson, J.J., Bifurcation analysis of a model of mitotic control in frog eggs, J. Theor. Biol. 195:69-85, 1998. ( ) Griffith, J.S. Mathematics of cellular control processes. II. Positive feedback to one gene. J. Theor. Biol. 20, 209-16, 1668. (Griffith ) Sel kov, E.E., Self-oscillations in glycolysis. 1. A simple kinetic model. Eur J Biochem. 4(1):79-86, 1968. (Sel kov ) Gardner, T.S., Cantor, C.R. and Collins, J.J., Construction of a genetic toggle switch in Escherichia coli., Nature 403(6767):339-42, 2000. (Toggle switch ) 13