MPRI - Bio-informatique formelle - LC Part 1 : Theory
MPRI - Bio-informatique formelle - LC
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).