Selkov Oscillation Lecturer

(YUGI, Katsuyuki)

Kuroda Lab., The University of Tokyo

Underlying mechanisms of biochemical oscillations

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Electrocardiograph

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k1.2x102sec1

BorisukandTyson(1998)

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Hes1(Notch signalling system)

mRNA2

Hirata et al. (2002) Science

MAPK2

, 2006

Nakayama et al. (2008) Curr. Biol.

In depth

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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

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Kennedy et al. (2007)

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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

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: 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

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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 = v2 = v3 = dsdt(1,:) = dsdt(2,:) =end

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: 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

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1David Baltimore

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RockefellerCaltech

Baltimore

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Baltimore

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Hoffmann et al. (2002) Science

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NF-B

IBNF-B IKKIB

IB NF-B

NF-BIB

NF-B

NF-B

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()

()

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

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Selkov (MATLAB)

([ADP],[F6P])2

MATLAB

2:

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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( , ); % 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) ()

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:

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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

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[ADP]

[F6P]

(1,2)

5

2

( x , y ) ( , )

{x = f(x, y)y = g(x, y)

{x = x + 2y + x2yy = 8 2y x2y

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x

( , ) = ( 0 , p )

: =0

: =0

[ADP]

[F6P]

= 0

= 0

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()

x()

y

3a: Selkov

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[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

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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( ); DX = DY = quiver( ); %(X,Y)(DX,DY)end

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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

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Lev Bar-Or et al. (2000) PNAS

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p53

()

p53Mdm2

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2

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[ADP]

[F6P]

= 0

= 0

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2

[ADP]

[F6P]

[ADP]

[F6P]

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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)

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Selkov (MATLAB)

(MATLAB)

(MATLAB)

(MATLAB)

4: Selkov

(xx

xy

yx

yy

)

xx =

x(x+ ay + x2y)

yx =

y(x+ ay + x2y)

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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

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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

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>> selkov_jacobian

fix =

0.6000 1.4286

J =

0.7143 0.4200-1.7143 -0.4200

ans =

0.1471 + 0.6311i0.1471 - 0.6311i>>

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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

)

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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

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(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

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(Spiral)

(Focus)

(Center)

StableSpiral

Center

UnstableSpiral

(Eulers formula)

x

y

x

y

x

y

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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

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> 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

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3

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2

8130

260

http://bsw3.naist.jp/courses/courses106.html

http://www.nig.ac.jp/hot/2006/saga0606-j.html

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NIH

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Q. 2hr?

A. Hes1, Smad, Stat

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()

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3

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(limit cycle)

(trajectory)

Center ()

Center

Poincar-Bendixson

Center

Limitcycle

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Re

Stable Spiral Unstable Spiral

Unstable Spiral

Hopf

Hopf

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Selkov 2Hopf ()

1: a=0.14, b=0.6

2: a=0.06, b=0.6

MATLAB

6: SelkovHopf

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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

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

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Relaxation oscillatorand

Negative feedback oscillator

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Relaxation oscillator ()

2

Negative feedback oscillator ()

3

? Bendixson

2

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D

D

Bendixson

f1x1

+f2x2

([F6P])[F6P]

+([ADP])[ADP]

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1(positive)

1negative feedback loop

()

Relaxation oscillator

J =(+ ??

)

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2negative feedback oscillators (Bendixson)

()

3

Bendixson

Negative feedback oscillator

J =( ?

? )

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7: SelkovBendixson

Selkov

> 0

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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

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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

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(bifurcation diagram)(phase diagram)

1 ()

2(a, b)

LacY

-GFP b

a2012724

ab

8: Selkov ()

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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

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for a , b

if

if ( isreal(v(1)) )

OK ()

if

if( real(v(1)) < 0 )

2

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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)

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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)

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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)

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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)

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ab

: a, b

Hopf

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Supercritical Hopf

Selkov

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!

Poincar-Bendixson

Hopf

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Negative feedback oscillator

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Elowitz and Leibler (2000) Nature

Repressilator

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(NOT)

1 / ( 2n )

Repressilator

3

hUp://133.6.66.95/mizutanilab3/ROC_ROCirc.html

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: Ring Oscillator

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3()

Repressilator

{d[mRNA]

dt = 0 +

1+[Repressor]n [mRNA]d[Protein]

dt = ([mRNA] [Protein])

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A n=2.1, 0=0

B n=2, 0=0

C n=2, 0/=103

X

Phase diagram

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inset

X

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40

0.2-0.5 h-1

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(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])

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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)

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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):

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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

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Plan A:

2

Plan B:

Repressilator6Plan B

Hopf

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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

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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

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=0.13 =0.14 (+1)2(2X+4)-3X2 = 0.0231 > 0 (+1)2(2X+4)-3X2 = -0.0355 < 0

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Hill

(0)m

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Toggle switchPitchfork

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Toggle switchPitchfork

: xy

Gardner et al. (2000)

x =a

1 + y2 x

y =a

1 + x2 y

LacI (x) CI (y)x y

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Pitchfork

a=2 a>2

a

xx

yy

pitchfork

Stable

Stable

Unstable

y

= 0

= 0

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:

1. Toggle switch

a 1 3

2. 11

Stable Node 3 Stable Node 2 Unstable

Node 1

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:

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

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1(2/2)

= 0 y = x 2

x

y

a

= 0

= 0

a

y = x

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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

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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

)

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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

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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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11

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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