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

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

2 MPRI - Bio-informatique formelle - LC Standard laws of biochemical kinetics applied to molecular networks

3 MPRI - Bio-informatique formelle - LC XiXa kaka Rate of Mass Action: forward reaction Biocham model: present(Xi). absent(Xa). ka*[Xi] for Xi=>Xa. parameter(ka,0.2).

4 MPRI - Bio-informatique formelle - LC 00.20.40.60.81 -0.1 0 0.1 0.2 Xa XiXa kaka Steady State solution Rate of Mass Action: forward reaction

5 MPRI - Bio-informatique formelle - LC Xi Xa kaka kiki 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).

6 MPRI - Bio-informatique formelle - LC Xa 00.20.40.60.81 -0.1 0 0.1 0.2 Xa* Steady State solution Xi Xa kaka kiki Rate of Mass Action: reversible reaction

7 MPRI - Bio-informatique formelle - LC production + elimination - Xa 00.20.40.60.81 -0.1 0 0.1 0.2 Xa* 00.20.40.60.8 1 0 0.05 0.1 0.15 0.2 0.25 Xa Xa* Rate of Mass Action: reversible reaction rate

8 MPRI - Bio-informatique formelle - LC 00.20.40.60.8 1 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 B Xa 1 2 3 4 5 Rate of Mass Action: catalyzed reversible reaction Xi Xa kaka kiki production + elimination - rate

9 MPRI - Bio-informatique formelle - LC 02.55 0 0.5 1 Xa* B 1 2 3 4 Nullcline 00.20.40.60.8 1 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 B Xa 1 2 3 4 5 Xi Xa kaka kiki production + elimination - rate

10 MPRI - Bio-informatique formelle - LC Xi Xa kaka 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).

11 MPRI - Bio-informatique formelle - LC 00.20.40.60.81 -0.1 0 0.1 0.2 Xi Xa kaka Steady State solution Xa Michaelis-Menten: forward reaction

12 MPRI - Bio-informatique formelle - LC Xi Xa kaka 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). kiki Goldbeter-Koshland switch

13 MPRI - Bio-informatique formelle - LC 0.00.20.40.60.81.0 0.0 0.2 Xa* production + elimination - Michaelis-Menten: reversible reaction rate Xi Xa kaka kiki

14 MPRI - Bio-informatique formelle - LC 0.00.20.40.60.81.0 0.0 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 Xa kaka kiki production + elimination -

15 MPRI - Bio-informatique formelle - LC 0.00.20.40.60.81.0 0.0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5.1 Xa* rate 012345 0 0.2 0.4 0.6 0.8 1 Xa* [B] Nullcline B Xi Xa kaka kiki production + elimination -

16 MPRI - Bio-informatique formelle - LC 0 CycB* APC 0 0.2 0.4 0.6 0.8 1 APC* CycB APC Assume Cdc28 always present and in excess Cln2 Cdc20 Positive feedback 0.5

17 MPRI - Bio-informatique formelle - LC 0 0.2 0.4 0.6 0.8 1 CycB Saddle Node bifurcation Change of parameter R (function of Cln2 and Cdc20) APC 0 0.2 0.4 0.6 0.8 1 CycB 0 0.2 0.4 0.6 0.8 1 APC CycB Saddle Node bifurcation point Saddle Node bifurcation point X Y

18 MPRI - Bio-informatique formelle - LC X Y C A N N O T O S C I L L A T E Negative feedback

19 MPRI - Bio-informatique formelle - LC Y X Z Ytot-Y Y k ci k ca Xtot-XX k ai k aa Ztot-Z Z k ba k bi Negative feedback

20 MPRI - Bio-informatique formelle - LC 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

21 MPRI - Bio-informatique formelle - LC 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 X Y Z

22 MPRI - Bio-informatique formelle - LC kaa : activation of X Activity of X region of oscillations stable steady state Hopf bifurcation points HB 1. Choose a parameter: kaa 2. Vary its value. different solutions can be observed according 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 oscillations were created. 5. The points surrounding the unstable steady states show the amplitude of the oscillations. Hopf bifurcation Change of parameter kaa (activation of X)

23 MPRI - Bio-informatique formelle - LC Introduction to bifurcation theory 1. Saddle Node (SN) bifurcation 2. Hopf (H) bifurcation 3. SNIC bifurcation : when SN meets H 4. Numerical Bifurcation theory 5. Signature of bifurcations

24 MPRI - Bio-informatique formelle - LC Bifurcation : Qualitative change in dynamics of the solutions of a system Bifurcation point : Border line between two behaviours of solutions Basic Definitions

25 MPRI - Bio-informatique formelle - LC 1. Saddle Node bifurcation r < 0, 2 solutions one stable, one unstable r = 0, 1 solution semi-stable r > 0, 0 solution x xxx Bifurcation diagram xx x r => Vary the parameter, r

26 MPRI - Bio-informatique formelle - LC 2. Hopf bifurcation 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 converge to 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

27 MPRI - Bio-informatique formelle - LC 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 feedbackNegative feedback When decreasing p, oscillations die at a saddle node bifurcation When increasing p, oscillations are created from a saddle node bifurcation

28 MPRI - Bio-informatique formelle - LC 4. Numerical bifurcation theory How to solve numerically a system of n ODEs : the case of n=2 1. Consider the following system of ODEs: 2. Solve at the equilibrium and determine the fixed points: 3. Determine the stability of the fixed points by computing the Jacobian A at these values (Jacobian is the matrix of the partial derivatives of the functions with respect to the components computed at the fixed points)

29 MPRI - Bio-informatique formelle - LC 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 the equation is determined as follows: The solution of the equation is the following:

30 MPRI - Bio-informatique formelle - LC

31 5. Signature of bifurcations

32 MPRI - Bio-informatique formelle - LC Continuation of a saddle node in one-parameter – One parameter bifurcation graph Example of a system of 2 ODEs 2 equations, 3 unknowns. Fix p=p* and solve for the steady state (x 1, x 2 ). We seek an equation of x (either 1 or 2) in terms of p. That way, we can follow a steady state as a parameter changes.

33 MPRI - Bio-informatique formelle - LC For the case of the saddle node bifurcation, the following graph is obtained : p x1x1 p1p1

34 MPRI - Bio-informatique formelle - LC Part 2 : application to biology

35 MPRI - Bio-informatique formelle - LC 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 noscille quaprès le cycle 8 ou 9 - En interphase du cycle 14, arrêt en G2 Quelques questions - Pourquoi ne voit-on pas le niveau des cyclines osciller plus tôt puisquil y a division nucléaire ? - Pourquoi les cycles sarrêtent-ils au 14e cycle ?

36 MPRI - Bio-informatique formelle - LC Données expérimentales et simulation CycBT Stg/Cdc25 MPFb Edgar et al. (1994) Genes and Development

37 MPRI - Bio-informatique formelle - LC Pourquoi ne voit-on pas le niveau des cyclines osciller plus tôt puisquil y a division nucléaire ?

38 MPRI - Bio-informatique formelle - LC Un modèle simple du Xenope CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Wee1 Cdc25P Fzy/APC IEP Cdk1 CycB P Fzy/APC IE Cdc25 Wee1P Cdk1 CycB MPF Wee1 Cdc25P IEP Fzy P P

39 MPRI - Bio-informatique formelle - LC Dun modèle de Xenopus … Wee1 Cdk1/CycB FZY Cdc25 CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Wee1 Cdc25P Fzy/APC IEP Cdk1 CycB P Fzy/APC IE Cdc25 Wee1P Cdk1 CycB

40 MPRI - Bio-informatique formelle - LC … à un modèle de Drosophila CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Wee1 Cdc25P Fzy/APC IEP Cdk1 CycB P Fzy/APC IE Cdc25 Wee1P Cdk1 CycB Le noyau CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Wee1 Cdc25P Cdk1 CycB P Cdc25 Wee1P Cdk1 CycB Le cytoplasme

41 MPRI - Bio-informatique formelle - LC 1 Des compartiments différents Cdk1/CycB FZY 234

42 MPRI - Bio-informatique formelle - LC Des compartiments différents Wee1 c Stg c CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Wee1 n Stg n Cdk1 CycB n IEP Fzy Cytoplasm Nucleus Cdk1 CycB n P Cdk1 CycB c Cdk1 CycB n P Fzy Wee1 n MPF n Stg n /Cdc25 Stg c /Cdc25 MPF c Wee1 c CycB T Cytoplasm Nucleus

43 MPRI - Bio-informatique formelle - LC Pourquoi les cycles sarrêtent-ils au 14e cycle ?

44 MPRI - Bio-informatique formelle - LC String/Cdc25, facteur limitant (1) Son ARN : -Stable pendant 13 cycles -Dégradation abrupte Le niveau total de la protéine : - est faible au début - augmente pendant les 8 premiers cycles - est dégradé graduellement jusquau 14eme cycle Son degré de Phosphorylation : oscille a partir du 5eme cycle

45 MPRI - Bio-informatique formelle - LC 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

46 MPRI - Bio-informatique formelle - LC Diagramme de bifurcation: MPF n et CycB T en fonction du nombre de cycles StgT=1StgT=0 Cycles MPF n CycB T

47 MPRI - Bio-informatique formelle - LC Ce que la théorie de la bifurcation nous permet de conclure : => String est responsable de lendroit 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 ny a pas de feedback negatif).