mpri - bio-informatique formelle - lc part 1 : theory

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Part 1 : Theory

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Standard laws of biochemical kinetics applied to molecular

networks

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Xi Xaka

Rate of Mass Action: forward reaction

Biocham model:

present(Xi).absent(Xa).

ka*[Xi] for Xi=>Xa.

parameter(ka,0.2).

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0 0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

Xa

d[Xa]0

dt

d[Xa]0

dt

Xi Xaka

a

a

d[Xa]k [Xi]

dt k (Xtot [Xa])

d[Xa]0, Xa* Xtot 1

dtsince Xtot Xi Xa 1

Steady State solution

Rate of Mass Action: forward reaction

d[Xa]

dt

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Xi Xaka

ki

Rate of Mass Action: reversible reaction

Biocham model:

present(Xi).absent(Xa).

ka*[Xi] for Xi=>Xa.ki*[Xa] for Xa=>Xi.

parameter(ka,0.2).parameter(ki,0.1).

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Xa0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

d[Xa]0

dt

d[Xa]0

dt

Xa*

d[Xa]0

dt

a

a i

d[Xa] k Xtot0, Xa* 0.67

dt k ksince Xtot=Xi+Xa=1

a i

a a i

d[Xa]k [Xi] k [Xa]

dt k Xtot (k +k ) [Xa]

Steady State solution

Xi Xaka

ki

Rate of Mass Action: reversible reaction

d[Xa]

dt

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production+

elimination-

a id[Xa]

k [Xi] k [Xa]dt

Xa0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

d[Xa]0

dt

d[Xa]0

dt

Xa*

d[Xa]0

dt

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Xa

d[Xa]0

dt

d[Xa]0

dt

Xa*

d[Xa]0

dt

Rate of Mass Action: reversible reaction

rated[Xa]

dt

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0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

1

2

34

5

B

Xa12345

Rate of Mass Action: catalyzed reversible reaction

Xi Xaka

ki

production+

elimination-

a id[Xa]

k [Xi] k [B] [Xa]dt

rate

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0 2.5 50

0.5

1

Xa*

B1 2 3 4

d[Xa]0

dt

d[Xa]0

dt

d[Xa]0

dt

Nullcline

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

1

2

34

5

B

Xa12345

Xi Xaka

ki production+

elimination-

a id[Xa]

k [Xi] k [B] [Xa]dt

rate

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Xi Xaka

Michaelis-Menten: forward reaction

Biocham model:

present(Xi).absent(Xa).

ka*[Xi]/(Ja+[Xi]) for Xi=>Xa.

parameter(ka,0.2).parameter(Ja,0.05).

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0 0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

Xi Xa

a a

a a

d[Xa] k [Xi] k (Xtot-[Xa])

dt J [Xi] J Xtot-[Xa]

where Xtot=Xi+Xa=1

ka

d[Xa]0, Xa* Xtot

dt

Steady State solution

Xa

d[Xa]0

dt

d[Xa]0

dt

Michaelis-Menten: forward reaction

d[Xa]

dt

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Xi Xaka

Michaelis-Menten: reverse reaction

Biocham model:

present(Xi).absent(Xa).

ka*[Xi]/(Ja+[Xi]) for Xi=>Xa.ki*[Xa]/(Ji+[Xa]) for Xa=>Xi.

parameter(ka,0.2).parameter(ki,0.1).parameter(Ja,0.05).parameter(Ji,0.05).

ki

Goldbeter-Koshland switch

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

Xa*

a i

a i

d[Xa] k (Xtot Xa) k [Xa]

dt J Xtot-Xa J [Xa]

production+

elimination-

d[Xa]0

dt

d[Xa]0

dt

d[Xa]0

dt

Michaelis-Menten: reversible reaction

rate

Xi Xaka

ki

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

0.1

Xa*

rate

Michaelis-Menten: catalyzed reversible reaction

B

Xi Xaka

ki

a i

a i

d[Xa] k (Xtot Xa) k [B] [Xa]

dt J Xtot-Xa J [Xa]

production+

elimination-

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

.1

Xa*

rate

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Xa*

[B]

d[Xa]0

dt

d[Xa]0

dt

d[Xa]0

dt

Nullcline

B

Xi Xaka

ki

a i

a i

d[Xa] k (Xtot Xa) k [B] [Xa]

dt J Xtot-Xa J [Xa]

production+

elimination-

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0

CycB*

APC

0

0.2

0.4

0.6

0.8

1

APC*

CycB

d[APC]0

dt

d[CycB]0

dt

' "syn deg deg

a i

[CycB]( [APC]) [CycB]

[APC] ( 20) (1 [APC]) ( 2) [CycB] [APC]1 [APC] [APC]

20 concentration of proteins that activates APC at Finish

concentration of proteins

dk k k

dtd B Cdc A Clndt J J

Cdc

Cln2

that inactivates APC at Start

CycB

APC

APC

Assume Cdc28 always present and in excess

Cln2Cdc20

Positive feedback

0.5

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0

0.2

0.4

0.6

0.8

1

CycB

Saddle Node bifurcationChange of parameter R (function of Cln2 and Cdc20)

APC

0

0.2

0.4

0.6

0.8

1

CycB0

0.2

0.4

0.6

0.8

1APCAPC

CycB

Saddle Node bifurcation point

Saddle Node bifurcation point

X Y

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X Y

CANNOT OSCIL

LATE

Negative feedback

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Y

X

ZYtot-Y Y

kci

kca

Xtot-X X

kai

kaa

Ztot-Z Z

kba

kbi

aa ai

ba bi

ca ci

d[X]k (Xtot [X]) k [X] [Y]

dtd[Y] k (Ytot [Y]) k [Y] [Z]

dt J Ytot [Y] J [Y]

d[Z]k (Ztot [Z]) k [Z] [X]

dt

Negative feedback

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X

Y

Z

Y

X

Z

The third element introduces a delay that allows the system to oscillate.

Negative feedback can create an oscillatory regime

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The importance of choosing the right parameters

Choose different values for the parameter kaa (activation of X)

• if kaa=0.015

• if kaa=0.1

• if kaa=0.2

Z

X

Y

XY

Z

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kaa : activation of X

Act

ivity

of

X

region of oscillations stable steady state

Hopf bifurcation points

HB HB

1. Choose a parameter: kaa

2. Vary its value. different solutions can be observedaccording to its value

3. The system oscillates between kaa=0.022 and kaa=0.114

4. At the point of bifurcation HB, the stable steady state changed into an unstable steady state and oscillationswere created.

5. The points surrounding the unstablesteady states show the amplitude of theoscillations.

Hopf bifurcationChange of parameter kaa (activation of X)

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Introduction to bifurcation theory

1. Saddle Node (SN) bifurcation2. Hopf (H) bifurcation3. SNIC bifurcation : when SN meets H4. Numerical Bifurcation theory5. Signature of bifurcations

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Bifurcation : Qualitative change in dynamics of the solutions of a system

Bifurcation point : Border line between two behaviours of solutions

Basic Definitions

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1. Saddle Node bifurcation

2 at equilibrium x=d[x]

([x]) [x] dt

rf r

r < 0, 2 solutionsone stable, one unstable

r = 0, 1 solutionsemi-stable

r > 0, 0 solution

x’

x x x

Bifurcation diagram

x’ x’

x

r

=> Vary the parameter, r

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2. Hopf bifurcation

dx dy(x,y) , (x,y)

dt dtf g

center

x

y

Bifurcation diagram x

p

g(x,y)=0

f(x,y)=0

stable focus (solutions converge to the steady state in a spiral)

x

y

g(x,y)=0

f(x,y)=0

unstable focus (solutions diverge from the steady state) + stable limit cycle (solutions convergeto the cycle)

x

y

g(x,y)=0

f(x,y)=0

Let p be a parameter of g(x,y) => vary p.

Supercritical Hopf bifurcation

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3. Saddle Node on Invariant Circles (SNIC)or when a saddle node meets oscillations

Combine cases 1 (Saddle Node) and 2 (Hopf)

parameter p

Positive feedback Negative feedback

When decreasing p, oscillations die at a saddle node bifurcationWhen increasing p, oscillations are created from a saddle node bifurcation

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4. Numerical bifurcation theory

How to solve numerically a system of n ODEs : the case of n=2

1 2 1 2where and fx =f(x) x=(x , x ) =(f , f )

1. Consider the following system of ODEs:

2. Solve at the equilibrium and determine the fixed points:

x =f(x)=0

*x

3. Determine the stability of the fixed points by computing the Jacobian A at thesevalues (Jacobian is the matrix of the partial derivatives of the functionswith respect to the components computed at the fixed points)

1 2f=(f , f )

1 2x=(x , x )

* *1 2 1 2

1 1

1 2

2 2

1 2 (x ,x ) (x ,x )

f f

x xA

f f

x x

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5. The eigenvalues can inform on the stability of the fixed points

4. Compute the characteristic equation in terms of the eigenvalues λ and where theequation is determined as follows:

2 4= where =trace of A and =determinant of A

2

* *1 2 1 2

1 1

1 2

2 2

1 2 (x ,x ) (x ,x )

f f

x xA- I 0

f f

x x

The solution of the equation is the following:

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2 4=

2where A is the jacobian

=trace of A => tr(A)

=determinant of A => A

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1

1,2 1,2

1 2

Let A be the jacobian

0, => saddle node bifurcation

and Re( ) 0 => Hopf bifurcation

For higher dimension bifurcations :

0 and 0

i

1 1,2

=> CUSP

0, and Re( ) 0 => Takens-Bogdanov

...

5. Signature of bifurcations

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Continuation of a saddle node in one-parameter –

One parameter bifurcation graph

1 1 2

2 1 2 1 2 1 2

f (x ,x ;p) 0

f (x ,x ;p) 0 where f , f , x , x and p are scalars

and where A 0 (det(A) 0)

Example of a system of 2 ODEs

2 equations, 3 unknowns.

Fix p=p* and solve for the steady state (x1, x2).

We seek an equation of x (either 1 or 2) in terms of p. That way, we can followa steady state as a parameter changes.

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For the case of the saddle node bifurcation, the following graph is obtained :

p

x1

*1 1 11 1

1 2 *2 2

2 2 2 *

1 2

1 1 1* *

1 2 *1 1 1 1* *

2 2 22 2 2 2

1 2

f f fx x 0

x x px x 0

f f fp p 0

x x p

or also

f f fx x px x x x

(p p ) or f f fx x x xx x p

*

1

* -1ss

2

1* * nss

f

p(p p ) A

f

p

and generalized to any n as long as A 0 :

f(x x ) (p p ) A

p

p1

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Part 2 : application to biology

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Quelques faits- 13 cycles rapides et synchronisés juste après fécondation

- Alternance entre les phases S et M (sans G1 ni G2)

- 6000 noyaux partagent le même cytoplasme

- Le niveau total des cyclines n’oscille qu’après le cycle 8 ou 9

- En interphase du cycle 14, arrêt en G2Quelques questions

- Pourquoi ne voit-on pas le niveau des cyclines osciller plus tôtpuisqu’il y a division nucléaire ?

- Pourquoi les cycles s’arrêtent-ils au 14e cycle ?

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Données expérimentales et simulation

CycBT

Stg/Cdc25

MPFb

Edgar et al. (1994) Genes and Development

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Pourquoi ne voit-on pas le niveau des cyclines osciller plus tôt puisqu’il y a

division nucléaire ?

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Un modèle simple du Xenope

CycB/Cdk1 = MPFCycB/Cdk1-P = preMPF

Wee1

Cdc25P

Fzy/APC

IEP

Cdk1

CycBP

Fzy/APC

IE

Cdc25

Wee1P

Cdk1

CycB

MPF

Wee1

Cdc25P

IEP

Fzy

P P

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D’un modèle de Xenopus …

' "s,mpf d,cb d,cb

' " ' "wee wee stg stg

' "s,mpf d,cb d,cb

[MPF] ( [FZY]) [MPF]

t

( [Wee1]) [MPF]+( [Stg]) [preMPF]

[preMPF]( [FZY]) [preMPF]

t

+(

dk k k

d

k k k k

dk k k

d

k

' " ' "wee wee stg stg

a,ie i,ie

a,ie i,ie

a,fzy i,fzy

a,fzy i,fzy

a,

[Wee1]) [MPF] ( [Stg]) [preMPF]

[MPF] (1 [IE]) [IE][IE]

t 1 [IE] [IE]

[IE] (1 [FZY]) [FZY][FZY]

t 1 [FZY] [FZY]

([Stg]

t

k k k

k kd

d J J

k kd

d J J

kd

d

' "

stg a,stg i,stg

a,stg i,stg

' "a,wee i,wee i,wee

a,wee i,stg

[MPF]) ([StgT] [Stg]) [Stg]

[StgT] [Stg] [Stg]

[Wee1] ( [MPF]) ([Wee1T] [Wee1])[Wee1]

t [Wee1] [Wee1T] [Wee1]

k k

J J

k k kd

d J J

Wee1

Cdk1/CycB

FZY

Cdc25

CycB/Cdk1 = MPFCycB/Cdk1-P = preMPF

Wee1

Cdc25P

Fzy/APC

IEP

Cdk1

CycBP

Fzy/APCIE

Cdc25

Wee1P

Cdk1

CycB

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… à un modèle de Drosophila

CycB/Cdk1 = MPFCycB/Cdk1-P = preMPF

Wee1

Cdc25P

Fzy/APC

IEP

Cdk1

CycBP

Fzy/APCIE

Cdc25

Wee1P

Cdk1

CycBLe noyau

CycB/Cdk1 = MPFCycB/Cdk1-P = preMPF

Wee1

Cdc25PCdk1

CycBP

Cdc25

Wee1P

Cdk1

CycBLe cytoplasme

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Des compartiments différents

Cdk1/CycBFZY

234

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Des compartiments différents

Wee1c

Stgc

CycB/Cdk1 = MPFCycB/Cdk1-P = preMPF

Wee1n

StgnCdk1

CycBn

IEP

Fzy

Cytoplasm

Nucleus

Cdk1

CycBn

P

Cdk1

CycBc

Cdk1

CycBn

P

Fzy

Wee1n

MPFn

Stgn/Cdc25

Stgc/Cdc25

MPFc

Wee1c

CycBT

Cytoplasm

Nucleus

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Pourquoi les cycles s’arrêtent-ils au 14e cycle ?

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String/Cdc25, facteur limitant (1)

Son ARN : -Stable pendant 13 cycles-Dégradation abrupte

Le niveau total de laprotéine :- est faible au début- augmente pendant les 8 premiers cycles- est dégradé graduellement jusqu’au 14eme cycle

Son degré dePhosphorylation :oscille a partir du 5eme cycle

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String/Cdc25, facteur limitant (2)

Traitement alpha-amanitin : 14 cycles

MPFT

MPFb

Xm

Stgm

Xp

Treatment at t=55 min Treatment at t=70 min

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Diagramme de bifurcation: MPFn et CycBT en fonction du nombre de cycles

0 2 4 6 8 10 12 14 16 18

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 18

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 18

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14 16 18

0.0

0.2

0.4

0.6

0.8

1.0

StgT=1 StgT=0

Cycles

MPFn

CycBT

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Ce que la théorie de la bifurcation nous permet de conclure :

=> String est responsable de l’endroit où se trouve le saddle node (feedback positif) Si on réduit la valeur de String, le saddle node va bouger.

=> Si on élimine le feedback négatif, on perd les oscillations (dans le cytoplasme, il n’y a pas de feedback negatif).