Underlying mechanisms of biochemical oscillations

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1 Underlying mechanisms of biochemical oscillations (YUGI, Katsuyuki) Kuroda Lab., The University of Tokyo

2 Electrocardiograph

4 Borisuk and Tyson (1998) k<1.2x10 2 sec 1 k>1.2x10 2 sec 1

5 Hes1 (Notch signalling system) mrna 2 Hirata et al. (2002) Science MAPK 2, 2006 Nakayama et al. (2008) Curr. Biol. In depth

6 Relaxation oscillator (e.g. Sel kov model) Hopf Negative feedback oscillator (e.g. Repressilator)

7 Sel kov

8 Sel kov { ẋ = v2 v 3 v1 v2 v3 F6P(y) PFK v1 = b v2 = ay + x 2 y ẏ = v 1 v 2 ADP(x) v3 =x Strogatz (1994) pp.205

9 Kennedy et al. (2007)

10 insulin Bertram et al. (2007) Corkey et al. (1988)

11 Insulin Luteinizing hormone Growth hormone ( Adrenocorticotropic hormone )

12 15 11 min 11 min insulin insulin Continuous insulin insulin glucose production µ mol/kg/min 13min 26min

13 1(MATLAB): Sel kov model { ẋ = v2 v 3 v1 v2 v3 F6P(y) PFK ADP(x) v1 = b v2 = ay + x 2 y ẏ = v 1 v 2 v3 = x Strogatz (1994) pp.205 : a = 0.06, b = 0.6 : ADP = 1.0, F6P = 1.0

14 function selkov( ) time = 0.001:1:100; s0 = [1.0, 1.0]; % Initial values param = [0.06, 0.6]; % Constants [t,time_course] = ode15s(@(t,s) ODE(t,s,param),time,s0); figure; plot(t,time_course); end function dsdt = ODE(t,s,param) ADP = s(1); F6P = s(2); a = param(1); b = param(2); v1 = v2 = v3 = dsdt(1,:) = dsdt(2,:) = end

15 :F6P : ADP

16 function selkov( ) time = 0.001:1:100; s0 = [1.0, 1.0]; % Initial values param = [0.06, 0.6]; % Constants [t,time_course] = ode15s(@(t,s) ODE(t,s,param),time,s0); figure; plot(t,time_course); end function dsdt = ODE(t,s,param) ADP = s(1); F6P = s(2); a = param(1); b = param(2); v1 = b; v2 = a * F6P + ADP^2 * F6P; v3 = ADP; dsdt(1,:) = v2 v3; dsdt(2,:) = v1 v2; end

17 1 David Baltimore

18 Baltimore Rockefeller Caltech

19 Baltimore

20 Hoffmann et al. (2002) Science

21 NF-κB NF-κB IκB NF-κB IKK IκB IκB NF-κB NF-κB IκB NF-κB

22 ( ) ( ) EMSA(Electrophoretic Mobility Shift Assay)

23 (feedback)

24 1

25 Phase plane ( ) [F6P] 2 x-y ẏ =0 ẋ =0 Nullcline ( ) = 0 [ADP]

26 2: Sel kov (MATLAB) ([ADP],[F6P]) 2 MATLAB

27 function selkov_phaseplane( ) time = 0.001:1:50; plot_phase_plane(time_course,param(1),param(2)); end function plot_phase_plane(time_course,a,b) figure; hold on; % plot(, ); % ADP = 0:0.1:3; F6P = ADP./ ( a + ADP.^2 ); % dadp / dt == 0 plot(adp,f6p, r ); % r red r ADP = 0:0.1:3; % d F6P / dt == 0 plot(,, ); hold off; end function dsdt = ODE(t,s,param) ( )

28 :

29 function selkov_phaseplane( ) time = 0.001:1:50; plot_phase_plane(time_course,param(1),param(2)); end function plot_phase_plane(time_course,a,b) figure; hold on; % plot(time_course(:,1),time_course(:,2)); ADP = 0:0.1:3; F6P = ADP./ ( a + ADP.^2 ); % dadp / dt == 0 plot(adp,f6p, r ); % r red r ADP = 0:0.1:3; F6P = b./ ( a + ADP.^2 ); % d F6P / dt == 0 plot(adp,f6p, r ); hold off; end

30 { ẋ = f(x, y) ẏ = g(x, y) ( x, y ) ( ẋ, ẏ ) { ẋ = x + 2y + x 2 y ẏ = 8 2y x 2 y [F6P] 2 5 (1,2) [ADP]

31 x ( ẋ, ẏ ) = ( 0, p ) [F6P] ẏ = 0 ẋ = 0 : ẏ =0 : ẏ =0 [ADP]

32 3a: Sel kov ( ) x ( ) y

33 [F6P] ẏ = 0 ẋ = 0 [ADP]

34 3b: Sel kov Sel kov figure (MATLAB) quiver meshgrid

35 : MATLAB [X,Y] = meshgrid(0.01:0.2:2, 0.01:0.2:2); DX = -X + a * Y + X.^2.* Y; DY = b - a * Y - X.^2.* Y; (X,Y) (DX,DY) quiver(x,y,dx,dy);

36 function selkov_vector_field () ( ) function ODE(t,s,param)( ) function plot_phase_plane(time_course, a, b)( ) function plot_vector_field(a,b) figure(1); [X,Y] = meshgrid( ); DX = DY = quiver( ); % (X,Y) (DX,DY) end

38 function selkov_vector_field () ( ) function ODE(t,s,param)( ) function plot_phase_plane(time_course, a, b)( ) function plot_vector_field(a,b) figure(1); [X,Y] = meshgrid(0.01:0.2:3, 0.01:0.2:10); DX = -X + a * Y + X.^2.* Y; DY = b - a * Y - X.^2.* Y; quiver(x,y,dx,dy); % (X,Y) (DX,DY) end

39 2 Arnold Levine

40 p53 Levine Rockefeller

41 Lev Bar-Or et al. (2000) PNAS

42 p53 Mdm2 p53 ( )

44 2

45 [F6P] ẏ = 0 ẋ = 0 [ADP]

46 [F6P] [F6P] [ADP] [ADP] 2

47 dx 1 dt = F 1 (x 1,,x n ) F 1 x 1 dx 1 dt = F 1 (x 1,,x n ) d dt x 1 = F 1 (x 1,,x n ). d dt x n = F n (x 1,,x n ) F 1 x 1 F 1 x n d dt x 1. x n = F 1 F 1 x n x F n x 1 F n x n x 1.. x n

48 4: Sel kov Sel kov ( MATLAB) xẋ = x ( x + ay + x2 y) y ẋ = y ( x + ay + x2 y) ( MATLAB) ( MATLAB) ( ẋ x y (MATLAB) ẏ x ẋ ẏ y )

49 MATLAB f diff(f, x); f f = -x + a * y + x^2 * y; J =[ diff(f,x), diff(f,y) ; diff(g,x), diff(g,y) ];

50 Symbolic Math ToolBox syms x y a b; % x, y, a, b x, y S = solve( -x + a * y + x^2 * y=0,'b - a * y - x^2 *y=0', 'x', 'y'); S x,y x S.x

51 J = subs(j, [a,b],[0.06,0.6]) % J a,b 0.06,0.6 A eig(a)

52 function selkov_jacobian( ) syms x y a b; S = solve(' ',' ',' ',' '); % x, y f = -x + a * y + x^2 * y; g = J = [ ; %f x,y ]; %g x,y fix = subs([s.x,s.y],[, ],[, ]) % a,b 0.06,0.6 J = subs(j, [,,,a,b],[,,0.06,0.6]) %J a,b 0.06,0.6 end %

53 fix = >> selkov_jacobian J = ans = i i >>

54 xẋ = x ( x + ay + x2 y)= 1+2xy y ẋ = y ( x + ay + x2 y)=a + x 2 xẏ = x (b ay x2 y)= 2xy ( b, b a + b 2 ) y ẏ = y (b ay x2 y)= a x 2 ( 1+2xy a + x 2 2xy a x 2 ) ( 1+ 2b2 a+b 2 a + b 2 2b2 a+b 2 a b 2 )

55 function selkov_jacobian( ) syms x y a b; S = solve('-x + a * y + x^2 * y=0','b - a * y - x^2 * y=0','x','y'); % x, y f = -x + a * y + x^2 * y; g = b - a * y - x^2 * y; J = [ diff(f,x), diff(f,y); %f x,y diff(g,x), diff(g,y) ]; %g x,y end fix = subs([s.x,s.y],[a,b],[0.06,0.6]) % a,b 0.06,0.6 J = subs(j, [x,y,a,b],[fix(1),fix(2),0.06,0.6]) %J a,b 0.06,0.6 eig(j) %

56 (Node) d x = J x dt : (attractor) : (repellor) x(t) = exp(jt) x 0 = c 1 exp(λ 1 t)v 1 + c n exp(λ n t)v n (Saddle) y y : : Stable Node Saddle x x

57 (Spiral) (Focus) y y Stable Spiral x Unstable Spiral x y (Center) (Euler s formula) Center x

58 5: Sel kov Sel kov ( ) a=0.06, b=0.6 1: Ax=λx A-Iλ =0 2x2 λ 2 -tr(a) λ+det(a)=0 2: λ τ = tr(a) = λ 1 + λ 2 Δ = det(a) =λ 1 λ 2 τ < 0 Δ > 0 τ 2 4Δ > 0

59 ( J = Δ > b2 a+b 2 a + b 2 2b2 a+b 2 a b 2 ) a,b a=0.06, b=0.6 = det(j) =a + b 2 > 0 τ = tr(j) = (a + b 2 ) a b2 a + b 2 τ 2 4Δ < 0 τ > 0

60 3

62 NIH

63 Q. 2hr? A. Hes1, Smad, Stat

66 ( )

67 3

68 (limit cycle) (trajectory) Center ( ) Limit cycle Center Poincaré-Bendixson Center

69 Supercritical Hopf Im λ Re λ<0 Re λ

70 Hopf Stable Spiral Unstable Spiral Unstable Spiral Hopf

71 6: Sel kov Hopf Sel kov 2 Hopf ( ) 1: a=0.14, b=0.6 2: a=0.06, b=0.6 MATLAB

72 τ = (a + b 2 ) a b2 a + b 2 a=0.06, b=0.6 τ>0 a=0.14, b=0.6 τ<0 Δ = a + b 2 > 0 τ 4 = 5(a + b 2 ) a b2 a + b 2 1,2 τ-4δ < 0

73 a=0.14, b=0.6 a=0.06, b=0.6

74 Relaxation oscillator and Negative feedback oscillator

75 2 Relaxation oscillator ( ) 2 Negative feedback oscillator ( ) 3? Bendixson

76 Bendixson D D f 1 + f 2 x 1 x 2 ( [F6P]) [F6P] + ( [ADP]) [ADP]

77 Relaxation oscillator 1 (positive) 1 negative feedback loop J = ( +?? ) ( )

78 Negative feedback oscillator 2 negative feedback oscillators ( Bendixson ) J = (?? ) ( ) 3 Bendixson

79 7: Sel kov Bendixson Sel kov > 0

80 function selkov_bendixson( ) syms x y a b; f = % ODE g = J = % ODE % f, g J J = subs(j, [a,b],[0.06, 0.6]); % a,b B = % figure(1) hold on; for i=0:0.5:3 % x for j=0:1:10 % y B_value = subs( if( B_value > 0 ) plot(i,j,'ko','markerfacecolor','w'); else plot(i,j,'ko','markerfacecolor','k'); end end end end ); % B x,y i,j

82 function selkov_bendixson( ) syms x y a b; f = -x + a * y + x^2 * y; g = b - a * y - x^2 * y; J = [ diff(f,x), diff(f,y); diff(g,x), diff(g,y)]; J = subs(j, [a,b],[0.06, 0.6]); % a,b B = J(1,1)+J(2,2); % figure(1) hold on; for i=0:0.5:3 % x for j=0:1:10 % y B_value = subs( B, [x,y], [i,j] ); % B x,y i,j if( B_value > 0 ) plot(i,j,'ko','markerfacecolor','w'); else plot(i,j,'ko','markerfacecolor','k'); end end end end

83 (bifurcation diagram) (phase diagram) 1 (ρ) 2 (a, b) LacY-GFP b ρ a

84 8: Sel kov ( ) b a

85 1 plot(x,y,'ko','markerfacecolor','k'); plot(x,y,'ko','markerfacecolor','w'); plot(x,y,'ko','markeredgecolor','r','markerfacecolor','r'); plot(x,y,'ko','markeredgecolor','r','markerfacecolor','w'); 1 isreal(n) real(c)

86 2 for a, b if if ( isreal(v(1)) ) OK ( ) if if( real(v(1)) < 0 )

87 function selkov_phase_diagram() figure; hold on; for a=0.01:0.01:0.15 for b=0.1:0.1:1.0 (1/2) end end end function v=jacobian( p, q ) syms x y a b; S = solve(' -x + a * y + x^2 * y=0','b - a * y - x^2 * y=0','x','y'); f = -x + a * y + x^2 * y; g = b - a * y - x^2 * y; J = [ diff(f,x), diff(f,y); diff(g,x), diff(g,y)]; fix = subs([s.x,s.y],[a,b],[p,q]); J = subs(j, [x,y,a,b],[fix(1),fix(2),p,q]); v = eig(j); end

88 (2/2) function phase_diagram( a, b ) v=jacobian(a,b); if ( isreal(v(1)) ) % v(1) v(2) v(1) if( v(1) < 0 && v(2) < 0 ) plot( ); else plot( ); end else if( < 0 ) % v(1) < 0 end end plot( ); else plot( ); end

89 function selkov_phase_diagram() figure; hold on; for a=0.01:0.01:0.15 for b=0.1:0.1:1.0 phase_diagram(a,b); end end end (1/2) function v=jacobian( p, q ) syms x y a b; S = solve(' -x + a * y + x^2 * y=0','b - a * y - x^2 * y=0','x','y'); f = -x + a * y + x^2 * y; g = b - a * y - x^2 * y; J = [ diff(f,x), diff(f,y); diff(g,x), diff(g,y)]; fix = subs([s.x,s.y],[a,b],[p,q]); J = subs(j, [x,y,a,b],[fix(1),fix(2),p,q]); v = eig(j); end

90 (2/2) function phase_diagram( a, b ) v=jacobian(a,b); if ( isreal(v(1)) ) % v(1) v(2) v(1) if( v(1) < 0 && v(2) < 0 ) plot(a,b,'ko','markerfacecolor','k'); else plot(a,b,'ko','markerfacecolor','w'); end else if( real(v(1)) < 0 ) % v(1) < 0 end end plot(a,b,'ko','markeredgecolor','r','markerfacecolor','r'); else plot(a,b,'ko','markeredgecolor','r','markerfacecolor','w'); end

91 : a, b Hopf b a

92 Sel kov Supercritical Hopf

93 Hopf! Poincaré-Bendixson

94 Negative feedback oscillator

95 Repressilator Elowitz and Leibler (2000) Nature

96 (NOT) 1 / ( 2n ) hup:// /mizutanilab3/roc_rocirc.html Repressilator 3

97 : Ring Oscillator

98 Repressilator 3 ( ) { d[mrna] α dt = α [Repressor] [mrna] n d[protein] dt = β([mrna] [Protein])

99 Phase diagram A n=2.1, α0=0 B n=2, α0=0 C n=2, α0/α=10 3 X

100 X inset

101 h -1

102 9: { d[mrna] α dt = α [Repressor] [mrna] n d[protein] dt = β([mrna] [Protein]) (laci, ci, tetr, LacI, CI, TetR) = (0.2, 0.3, 0.1, 0.1, 0.5, 0.4) α=20, α0=0, β=0.2, n=2

103 MATLAB (1/2) function repressilator( input_args ) time = 0.001:1:200; s0 = [0.2, 0.3, 0.1, 0.1, 0.5, 0.4]; % Initial values param = [20, 0, 0.2, 2]; % Constants [t,time_course] = ode15s(@(t,s) ODE(t,s,param),time,s0); plot_time_course(t,time_course); plot_phase_plane(time_course); end

104 MATLAB (2/2): function dsdt = ODE(t,s,param) laci = s(1); ci = s(2); tetr = s(3); LacI = s(4); CI = s(5); TetR = s(6); alpha = param(1); alpha_zero = param(2); beta = param(3); n = param(4); dsdt(1,:) = dsdt(2,:) = dsdt(3,:) = dsdt(4,:) = dsdt(5,:) = dsdt(6,:) = end

105

106 function dsdt = ODE(t,s,param) laci = s(1); ci = s(2); tetr = s(3); LacI = s(4); CI = s(5); TetR = s(6); alpha = param(1); alpha_zero = param(2); beta = param(3); n = param(4); dsdt(1,:) = alpha_zero + alpha / ( 1 + CI^n ) - laci; dsdt(2,:) = alpha_zero + alpha / ( 1 + TetR^n ) - ci; dsdt(3,:) = alpha_zero + alpha / ( 1 + LacI^n ) - tetr; dsdt(4,:) = - beta * ( LacI - laci ); dsdt(5,:) = - beta * ( CI - ci ); dsdt(6,:) = - beta * ( TetR - tetr ); end

107 Hopf Plan A: 2 Plan B: Repressilator 6 Plan B

108 Hopf 6 J = X X X 0 β 0 0 β β 0 0 β β 0 0 β p3 ṁ 1 p 3 = αnpn 1 3 (1 + p 2 = X 3 )2 p 3 = α 1+p n 3 + α 0 (β + 1) 2 (2X + 4) 3βX 2 < 0

109 10: Hopf β (β + 1) 2 (2X + 4) 3βX 2 < 0 β α=10, α0=0, n=2 X = αnpn 1 3 (1 + p 2 3 )2 p 3 = α 1+p n 3 ( ) α=10, α0=0, n=2 p3 = 2 p3 + α 0

110 β=0.13 β=0.14 (β+1) 2 (2X+4)-3X 2 = > 0 (β+1) 2 (2X+4)-3X 2 = < 0

111 Hill (α0) Κm

112 Toggle switch Pitchfork

113 Toggle switch Pitchfork x LacI (x) λci (y) y ẋ = ẏ = a 1+y 2 x a 1+x 2 y Gardner et al. (2000) : x y

114 y ẏ = 0 Pitchfork y y ẋ = 0 x a=2 a>2 x Stable Unstable pitchfork Stable a

115 : 1. Toggle switch a Stable Node 3 Stable Node 2 Unstable Node 1

116 : 3. y x x x 5 - ax 4 + 2x 3-2ax 2 + (1 + a 2 )x - a = x x 5 - ax 4 + 2x 3-2ax 2 + (1 + a 2 ) x - a = 0 (x 3 + x - a)(x 2 - ax + 1) = 0

117 : (1/2) 5. x 3 + ax + b = 0 3 D = - 4a 3-27b 2 D > 0 3 D = 0 D < x 3 + x - a x 3 + x - a = 0 1 a

118 : (2/2) 6. x 2 - ax + 1 x 2 - ax + 1 = 0 a > 2 2 a = 2 1 a < x (x 3 + x - a)(x 2 - ax + 1) = 0 (x 3 + x - a) 1 (x 2 - ax + 1) a = Toggle switch 0 < a < 2 a = 2 a > 2 a = 2

119 : (1/2) 8. Toggle-switch ( 1 2ay 2ax (1+x 2 ) 2 (1+y 2 ) a > 2 x 2 - ax + 1 = 0 ( a ± a 2 4 (i), a ) a ) (ii) 2 ( : Δ )

120 : (2/2) 10. x 3 + x - a = 0 x (a ) ( ) a 2 3 x = a ( 2 + a ) 2 ( ) 3 3 x (i) (ii) a = 2 x = 1 x a 0 < a < 2 0 < x < 1 a > 2 x > 1 0 < x < 1 0 < a < 2 Δ > 0 x > 1 a > 2 Δ < 0

121 : Pitchfork a x 0<a<2 a=2 a>2 x 3 + x - a = Node x 1 x 2 - ax + 1 = 0 0 x 3 + x - a = x 2 - ax + 1 = 0 1 x 3 + x - a = Node x 2 x 2 - ax + 1 = 0 2 Saddle x 1

122 1 (1/2) ẏ = 0 y = 0 (1 + x2 ) a 2x (1 + x 2 ) 2 = 2ax (1 + x 2 ) 2 x 1 3 y - y = a 1+x 2 y = 2a(1 + x2 ) 2 ( 2ax) 2(1 + x 2 ) 2x (1 + x 2 ) 4 y a = 2a(1 + x2 ){(1 + x 2 ) 2x 2x} (1 + x 2 ) 4 = 2a(1 + x2 )(1 3x)(1 + 3x) (1 + x 2 ) 4 y y 1 3 ẏ = 0 x

123 1 (2/2) ẋ = 0 y = x 2 y a ẋ = 0 y = x a ẏ = 0 x

124 2 y ẋ = 0 y a ẏ = 0 a x y ẋ = 0 x a ẏ = 0 a x

125 3 x = a y = a 1+y 2 1+x 2 x = = a ( 1+ ) 2 a 1+x 2 a(1 + x 2 ) 2 (1 + x 2 ) 2 + a 2 (1 + x 2 ) 2 x + a 2 x = a(1 + x 2 ) 2 x +2x 3 + x 5 + a 2 x = a +2ax 2 + ax 4 x 5 ax 4 +2x 3 2ax 2 + (1 + a 2 )x a =0

126 4 (x 3 + x a)(x 2 ax + 1) = 0 x 5 ax 4 + x 3 + x 3 ax 2 + x ax 2 + a 2 x a = 0 x 5 ax 4 +2x 3 2ax 2 + (1 + a 2 )x a = 0

127 x 3 + x - a = 0 D = a 2 < x 2 - ax + 1 = 0 D = a 2-4 a > 2 D > 0 2 a = 2 D = 0 1 a < 2 D < <a<2 a=2 a>2 x 3 + x - a = x 2 - ax + 1 = 0 0 x 3 + x - a = x 2 - ax + 1 = 0 1 x 3 + x - a = x 2 - ax + 1 = x=1

128 ẋ = ẏ = a 1+y 2 x 8 a 1+x 2 y xẋ = 1 y ẋ = 2ay (1 + y 2 ) 2 y ẋ = 2ax (1 + x 2 ) 2 ( 1 2ay 2ax (1+x 2 ) 2 y ẏ = 1 (1+y 2 ) 2 1 )

129 =1 = 1 4a 2 xy (1 + x 2 ) 2 (1 + y 2 ) 2 4a 2 xy (1 + x 2 + y 2 + x 2 y 2 ) 2 9-(ii) xy = (a ± a 2 4)(a a 2 4) 4 =1 =1 = 1 = 1 4a2 a 4 4a 2 (2 + x 2 + y 2 ) 2 4a 2 (2 + 4a2 8 4 ) 2 = 1 4 > 0 ( a>2) a2 τ<0, Δ>0, τ 2-4Δ>0 2

130 10 (i) 0 < a < 2 x x a = 0 x = 0 a = 2 x = 1 x a 0 < a < 2 x < x < 1 (ii) y=x =1 4a 2 xy (1 + x 2 ) 2 (1 + y 2 ) 2 = 1 4a2 x 2 (1 + x 2 ) 4 d dx =1 8a2 x(1 + x 2 ) 3 (x 1) 2 (1 + x 2 ) 4 Δ x a = 0, x = 0 Δ = 1 a = 2, x = 1 Δ = 0 0 < a < 2 1 > Δ > 0 a > 2 Δ < 0

131 11 x x 2 - ax + 1 = 0 1 x 3 + x - a = 0 x 2 - ax + 1 = 0 2 a Node Saddle

132 ? (Saddle- Node, Pitchfork) (Hopf)

133 Further readings Strogatz, S.H, Nonlinear dynamics and chaos, Perseus Books Publishing, (ISBN ) Borisuk and Tyson (1998) Fall, C.P., Marland, E.S., Wagner, J.M. and Tyson, J.J. Computational cell biology, Springer, (ISBN ) 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.

ẋ = ax + y ẏ = x x by 2 Griffith a b Saddle Node Saddle-Node (phase plane) Griffith mrna(y) Protein(x) (nullcline) 0 (nullcline) (

ẋ = ax + y ẏ = x x by 2 Griffith a b Saddle Node Saddle-Node (phase plane) Griffith mrna(y) Protein(x) (nullcline) 0 (nullcline) ( 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