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excelente articulo
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(YUGI, Katsuyuki)
Kuroda Lab., The University of Tokyo
Underlying mechanisms of biochemical oscillations
2012724
Electrocardiograph
2012724
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k1.2x102sec1
BorisukandTyson(1998)
2012724
Hes1(Notch signalling system)
mRNA2
Hirata et al. (2002) Science
MAPK2
, 2006
Nakayama et al. (2008) Curr. Biol.
In depth
2012724
Relaxation oscillator (e.g. Selkov model)
Hopf
Negative feedback oscillator (e.g. Repressilator)
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Selkov
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Strogatz(1994)pp.205
Selkov
{x = v2 v3y = v1 v2
v2 = ay + x2yv1 = bv3 =x
v1 v2 v3ADP(x)F6P(y)PFK
2012724
Kennedy et al. (2007)
2012724
Bertram et al. (2007) Corkey et al. (1988)
insulin
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Luteinizing hormone
Growth hormone ()Adrenocorticotropic hormone
Insulin
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15
2 2 211 min 11 min
insulin
insulin
insulin
Continuous insulin gluc
ose
prod
uctio
n
mol
/kg/
min
13min
26min
2012724
: a = 0.06, b = 0.6
: ADP = 1.0, F6P = 1.0
Strogatz(1994)pp.205
1(MATLAB): Selkov model
{x = v2 v3y = v1 v2
v2 = ay + x2yv1 = bv3 = x
v1 v2 v3ADP(x)F6P(y)PFK
2012724
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
2012724
: ADP
:F6P
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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
2012724
1David Baltimore
2012724
RockefellerCaltech
Baltimore
2012724
Baltimore
2012724
Hoffmann et al. (2002) Science
2012724
NF-B
IBNF-B IKKIB
IB NF-B
NF-BIB
NF-B
NF-B
2012724
()
()
EMSA(Electrophoretic Mobility Shift Assay)
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(feedback)
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1
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Phase plane ()
2
x-y
Nullcline ()
= 0
[ADP]
[F6P]
x = 0
y = 0
2012724
Selkov (MATLAB)
([ADP],[F6P])2
MATLAB
2:
2012724
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); % rredr ADP = 0:0.1:3; % d F6P / dt == 0 plot( , , ); hold off;end
function dsdt = ODE(t,s,param) ()
2012724
:
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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); % rredr ADP = 0:0.1:3; F6P = b ./ ( a + ADP.^2 ); % d F6P / dt == 0 plot(ADP,F6P,r); hold off;end
2012724
[ADP]
[F6P]
(1,2)
5
2
( x , y ) ( , )
{x = f(x, y)y = g(x, y)
{x = x + 2y + x2yy = 8 2y x2y
2012724
x
( , ) = ( 0 , p )
: =0
: =0
[ADP]
[F6P]
= 0
= 0
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()
x()
y
3a: Selkov
2012724
[ADP]
[F6P]
= 0
= 0
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Selkovfigure(MATLAB)
quiver
meshgrid
3b: Selkov
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[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);
: MATLAB
2012724
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
2012724
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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
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2Arnold Levine
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Rockefeller
p53Levine
2012724
Lev Bar-Or et al. (2000) PNAS
2012724
p53
()
p53Mdm2
2012724
2012724
2
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[ADP]
[F6P]
= 0
= 0
2012724
2
[ADP]
[F6P]
[ADP]
[F6P]
2012724
d
dtx1 = F1(x1, , xn)
...d
dtxn = Fn(x1, , xn)
d
dt
x1...xn
=
F1x1
F1xn...
. . ....
Fnx1
Fnxn
x1...
xn
F1x1
F1x1
F1xn
dx1dt
= F1(x1, , xn)
dx1dt
= F1(x1, , xn)
2012724
Selkov (MATLAB)
(MATLAB)
(MATLAB)
(MATLAB)
4: Selkov
(xx
xy
yx
yy
)
xx =
x(x+ ay + x2y)
yx =
y(x+ ay + x2y)
2012724
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) ];
MATLAB
2012724
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');
Sx,y
xS.x
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J = subs(J, [a,b] ,[0.06,0.6])
% Ja,b0.06,0.6
A
eig(A)
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function selkov_jacobian( )syms x y a b;S = solve(' ',' ',' ',' '); % x, yf = -x + a * y + x^2 * y;g = J = [ ; %fx,y ]; %gx,y
fix = subs([S.x,S.y],[ , ],[ , ]) %a,b0.06,0.6J = subs(J, [ , ,,a,b] ,[ , ,0.06,0.6]) %Ja,b0.06,0.6 %end
2012724
>> selkov_jacobian
fix =
0.6000 1.4286
J =
0.7143 0.4200-1.7143 -0.4200
ans =
0.1471 + 0.6311i0.1471 - 0.6311i>>
2012724
yx =
y(x+ ay + x2y) = a+ x2
xx =
x(x+ ay + x2y) = 1 + 2xy
xy =
x(b ay x2y) = 2xy
yy =
y(b ay x2y) = a x2
(1 + 2b2a+b2 a+ b2
2b2a+b2 a b2)
( 1 + 2xy a+ x22xy a x2
)(b ,
b
a+ b2
)
2012724
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, yf = -x + a * y + x^2 * y;g = b - a * y - x^2 * y; J = [ diff(f,x) , diff(f,y); %fx,y diff(g,x) , diff(g,y) ]; %gx,y
fix = subs([S.x,S.y],[a,b],[0.06,0.6]) %a,b0.06,0.6J = subs(J, [x,y,a,b] ,[fix(1),fix(2),0.06,0.6]) %Ja,b0.06,0.6eig(J) %end
2012724
(Node)
: (attractor)
: (repellor)
(Saddle)
:
: StableNode Saddle
d
dtx = Jx
x(t) = exp(Jt)x0= c1 exp(1t)v1 + cn exp(nt)vn
x
y
x
y
2012724
(Spiral)
(Focus)
(Center)
StableSpiral
Center
UnstableSpiral
(Eulers formula)
x
y
x
y
x
y
2012724
Selkov()a=0.06, b=0.6
1:
Ax=x|A-I|=02x2 2-tr(A) +det(A)=0
2: = tr(A) = 1 + 2 = det(A) =12
< 0 > 0
24 > 0
5: Selkov
2012724
> 0 2 a,b
a=0.06, b=0.6
24 < 0 > 0
= det(J) = a+ b2 > 0
J =
(1 + 2b2a+b2 a+ b2
2b2a+b2 a b2)
= tr(J) = (a+ b2) a b2
a+ b2
2012724
3
2012724
2
8130
260
http://bsw3.naist.jp/courses/courses106.html
http://www.nig.ac.jp/hot/2006/saga0606-j.html
2012724
NIH
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Q. 2hr?
A. Hes1, Smad, Stat
2012724
2012724
2012724
()
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3
2012724
(limit cycle)
(trajectory)
Center ()
Center
Poincar-Bendixson
Center
Limitcycle
2012724
Re
Stable Spiral Unstable Spiral
Unstable Spiral
Hopf
Hopf
2012724
Selkov 2Hopf ()
1: a=0.14, b=0.6
2: a=0.06, b=0.6
MATLAB
6: SelkovHopf
2012724
a=0.06, b=0.6 >0
a=0.14, b=0.6 0
1,2 -4 < 0
= (a+ b2) a b2
a+ b2
4 = 5(a+ b2) a b2
a+ b2
2012724
a=0.14, b=0.6 a=0.06, b=0.6
2012724
Relaxation oscillatorand
Negative feedback oscillator
2012724
Relaxation oscillator ()
2
Negative feedback oscillator ()
3
? Bendixson
2
2012724
D
D
Bendixson
f1x1
+f2x2
([F6P])[F6P]
+([ADP])[ADP]
2012724
1(positive)
1negative feedback loop
()
Relaxation oscillator
J =(+ ??
)
2012724
2negative feedback oscillators (Bendixson)
()
3
Bendixson
Negative feedback oscillator
J =( ?
? )
2012724
7: SelkovBendixson
Selkov
> 0
2012724
function selkov_bendixson( ) syms x y a b; f = % ODE g = % ODE J = % f, gJ 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( ); % Bx,yi,j if( B_value > 0 ) plot(i,j,'ko','MarkerFaceColor','w'); else plot(i,j,'ko','MarkerFaceColor','k'); end end endend
2012724
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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] ); % Bx,yi,j if( B_value > 0 ) plot(i,j,'ko','MarkerFaceColor','w'); else plot(i,j,'ko','MarkerFaceColor','k'); end end endend
2012724
(bifurcation diagram)(phase diagram)
1 ()
2(a, b)
LacY
-GFP b
a2012724
ab
8: Selkov ()
2012724
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)
1
2012724
for a , b
if
if ( isreal(v(1)) )
OK ()
if
if( real(v(1)) < 0 )
2
2012724
function selkov_phase_diagram()figure;hold on;
for a=0.01:0.01:0.15for b=0.1:0.1:1.0endendend
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
(1/2)
2012724
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();elseplot();endelseif( < 0 ) % v(1) < 0plot( );elseplot( );endendend
(2/2)
2012724
function selkov_phase_diagram()figure;hold on;
for a=0.01:0.01:0.15for b=0.1:0.1:1.0phase_diagram(a,b);endendend
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
(1/2)
2012724
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');endelseif( real(v(1)) < 0 ) % v(1) < 0plot(a,b,'ko','MarkerEdgeColor','r','MarkerFaceColor','r'); else plot(a,b,'ko','MarkerEdgeColor','r','MarkerFaceColor','w');endendend
(2/2)
2012724
ab
: a, b
Hopf
2012724
Supercritical Hopf
Selkov
2012724
!
Poincar-Bendixson
Hopf
2012724
Negative feedback oscillator
2012724
Elowitz and Leibler (2000) Nature
Repressilator
2012724
(NOT)
1 / ( 2n )
Repressilator
3
hUp://133.6.66.95/mizutanilab3/ROC_ROCirc.html
2012724
: Ring Oscillator
2012724
3()
Repressilator
{d[mRNA]
dt = 0 +
1+[Repressor]n [mRNA]d[Protein]
dt = ([mRNA] [Protein])
2012724
A n=2.1, 0=0
B n=2, 0=0
C n=2, 0/=103
X
Phase diagram
2012724
inset
X
2012724
40
0.2-0.5 h-1
2012724
(lacI, cI, tetR, LacI, CI, TetR) = (0.2, 0.3, 0.1, 0.1, 0.5, 0.4)
=20, 0=0, =0.2, n=2
9:
{d[mRNA]
dt = 0 +
1+[Repressor]n [mRNA]d[Protein]
dt = ([mRNA] [Protein])
2012724
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
MATLAB(1/2)
2012724
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
MATLAB(2/2):
2012724
2012724
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
2012724
Plan A:
2
Plan B:
Repressilator6Plan B
Hopf
2012724
6
Hopf
m1p3
=npn13(1 + p23)2
= X
p3 =
1 + pn3+ 0
( + 1)2(2X + 4) 3X2 < 0
J =
1 0 0 0 0 X0 1 0 X 0 00 0 1 0 X 0 0 0 0 00 0 0 00 0 0 0
p3
2012724
10: Hopf
( + 1)2(2X + 4) 3X2 < 0
p3 =
1 + pn3+ 0
() =10, 0=0, n=2 p3 = 2
p3
=10, 0=0, n=2
X =npn13(1 + p23)2
2012724
=0.13 =0.14 (+1)2(2X+4)-3X2 = 0.0231 > 0 (+1)2(2X+4)-3X2 = -0.0355 < 0
2012724
Hill
(0)m
2012724
Toggle switchPitchfork
2012724
Toggle switchPitchfork
: xy
Gardner et al. (2000)
x =a
1 + y2 x
y =a
1 + x2 y
LacI (x) CI (y)x y
2012724
Pitchfork
a=2 a>2
a
xx
yy
pitchfork
Stable
Stable
Unstable
y
= 0
= 0
2012724
:
1. Toggle switch
a 1 3
2. 11
Stable Node 3 Stable Node 2 Unstable
Node 1
2012724
:
3. y x
x x5 - ax4 + 2x3 - 2ax2 + (1 + a2)x -
a = 0
4. 3 x x5 - ax4 + 2x3 - 2ax2
+ (1 + a2) x - a = 0 (x3 + x - a)(x2 - ax + 1) = 0
2012724
: (1/2)
5. x3 + ax + b = 0 3
D = - 4a3 - 27b2 D > 0
3 D = 0D < 0 1
2
x3 + x - a x3 + x - a = 0 1
a
2012724
: (2/2)
6. x2 - ax + 1
x2 - ax + 1 = 0 a > 2 2a = 2
1a < 2
7. 56 x (x3 + x -
a)(x2 - ax + 1) = 0 (x3 + x - a)1
(x2 - ax + 1) a = 202
Toggle switch 0 < a < 2 a = 2 a > 2
a = 2
2012724
: (1/2)
8. Toggle-switch
9. a > 2 x2 - ax + 1 = 0
(i)
(ii) 2
(: )
(1 2ay(1+y2)2
2ax(1+x2)2 1
)
(aa2 4
2,aa2 4
2
)
2012724
: (2/2)
10. x3 + x - a = 0 x
x
(i) a = 2 x = 1 x a 0 < a < 2 0 < x < 1 a > 2 x > 1
(ii) 0 < x < 1 0 < a < 2 > 0 x > 1 a > 2 < 0
x =3
a2+
(a2
)2+(13
)3 1
3 3
a2 +
(a2
)2 + ( 13)3
2012724
: Pitchfork
11. 7.10.
a
x
0
1(1/2)
y =a
1 + x2 = 0
13x
y ---y - 0 +
y
y =0 (1 + x2) a 2x
(1 + x2)2
= 2ax(1 + x2)2
y =2a(1 + x2)2 (2ax) 2(1 + x2) 2x
(1 + x2)4
=2a(1 + x2){(1 + x2) 2x 2x}
(1 + x2)4
=2a(1 + x2)(13x)(1 +3x)
(1 + x2)4
x
ya
13
= 0
2012724
1(2/2)
= 0 y = x 2
x
y
a
= 0
= 0
a
y = x
2012724
2
x
y = 0
ax
y
= 0
a
x
ya
= 0
= 0
a2012724
3
y =a
1 + x2x =
a
1 + y2
x =a
1 +(
a1+x2
)2=
a(1 + x2)2
(1 + x2)2 + a2
(1 + x2)2x+ a2x = a(1 + x2)2
x+ 2x3 + x5 + a2x = a+ 2ax2 + ax4
x5 ax4 + 2x3 2ax2 + (1 + a2)x a = 0
2012724
4
(x3 + x a)(x2 ax+ 1) = 0x5 ax4 + x3 + x3 ax2 + x ax2 + a2x a = 0
x5 ax4 + 2x3 2ax2 + (1 + a2)x a = 0
2012724
5-7
5. x3 + x - a = 0 D = - 4 - 27a2 < 0
12
6. x2 - ax + 1 = 0 D = a2 - 4 a > 2
D > 0 2a = 2 D = 0
1 a < 2 D < 0
7.
0
8 x =a
1 + y2 x
y =a
1 + x2 y
xx = 1
yx = 2ay
(1 + y2)2
yy = 1
yx = 2ax
(1 + x2)2(1 2ay(1+y2)2
2ax(1+x2)2 1
)
2012724
9-(ii) =1 4a
2xy
(1 + x2)2(1 + y2)2
= 1 4a2xy
(1 + x2 + y2 + x2y2)2
xy =(aa2 4)(aa2 4)
4= 1
=1 4a2
(2 + x2 + y2)2
= 1 4a2
(2 + 4a284 )2
= 1 4a2
a4
= 1 4a2
> 0 ( a > 2)
0, 2-4>02
2012724
10
(i) 0 < a < 2 x x
a = 0 x = 0 a = 2 x
= 1 x a 0 < a < 2
x 0 1
0 < x < 1
(ii) y=x
=1 4a2xy
(1 + x2)2(1 + y2)2
= 1 4a2x2
(1 + x2)4
ddx
= 1 8a2x(1 + x2)3(x 1)2
(1 + x2)4 x
a = 0, x = 0 = 1a = 2, x = 1 = 0
0 < a < 2 1 > > 0a > 2 < 0
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x
a2
1
NodeSaddle
x3 + x - a = 0
x2 - ax + 1 = 0
x2 - ax + 1 = 0
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(Saddle-
Node, Pitchfork)
(Hopf)
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Further readings
Strogatz, S.H, Nonlinear dynamics and chaos, Perseus Books Publishing, 1994. (ISBN 0-7382-0453-6)
Borisuk and Tyson (1998)
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.
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