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Part 1 : Theory

Part 1 : Theory. Standard laws of biochemical kinetics applied to molecular networks. Rate of Mass Action: forward reaction. k a. Xi. Xa. Biocham model: present(Xi). absent(Xa). ka*[Xi] for Xi=>Xa. parameter(ka,0.2). Rate of Mass Action: forward reaction. k a. Xi. Xa. 0.2. 0.1.

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Part 1 : Theory

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

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

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

  4. Rate of Mass Action: forward reaction ka Xi Xa 0.2 0.1 Steady State solution 0 Xa -0.1 0 0.2 0.4 0.6 0.8 1 MPRI - Bio-informatique formelle - LC

  5. Rate of Mass Action: reversible reaction ka Xa Xi ki Biocham model: present(Xi). absent(Xa). ka*[Xi] for Xi=>Xa. ki*[Xa] for Xa=>Xi. parameter(ka,0.2). parameter(ki,0.1). MPRI - Bio-informatique formelle - LC

  6. Rate of Mass Action: reversible reaction ka Xa Xi ki 0.2 0.1 Steady State solution 0 Xa -0.1 0 0.2 0.4 0.6 0.8 1 Xa* MPRI - Bio-informatique formelle - LC

  7. Rate of Mass Action: reversible reaction production + elimination - rate 0.25 0.2 0.2 0.1 0.15 0.1 0 0.05 Xa Xa 0 -0.1 1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 Xa* Xa* MPRI - Bio-informatique formelle - LC

  8. Rate of Mass Action: catalyzed reversible reaction ka Xa Xi ki B rate 3 4 0.25 5 2 0.2 production + elimination - 0.15 1 0.1 0.05 Xa 0 1 0 0.2 0.4 0.6 0.8 1 4 5 3 2 MPRI - Bio-informatique formelle - LC

  9. 1 0.5 0 0 2.5 5 Nullcline ka Xa Xi ki production + elimination - B rate Xa* 3 4 0.25 5 2 0.2 0.15 1 0.1 0.05 Xa 0 1 0 0.2 0.4 0.6 0.8 2 1 4 3 1 4 5 3 2 B MPRI - Bio-informatique formelle - LC

  10. Michaelis-Menten: forward reaction ka Xi Xa Biocham model: present(Xi). absent(Xa). ka*[Xi]/(Ja+[Xi]) for Xi=>Xa. parameter(ka,0.2). parameter(Ja,0.05). MPRI - Bio-informatique formelle - LC

  11. Michaelis-Menten: forward reaction ka Xi Xa 0.2 0.1 Steady State solution 0 Xa -0.1 0 0.2 0.4 0.6 0.8 1 MPRI - Bio-informatique formelle - LC

  12. Michaelis-Menten: reverse reaction ka Xi Xa ki 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). • Goldbeter-Koshland switch MPRI - Bio-informatique formelle - LC

  13. Michaelis-Menten: reversible reaction ka Xa Xi ki rate 0.2 production + elimination - 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Xa* MPRI - Bio-informatique formelle - LC

  14. 5 0.5 4 0.4 3 0.3 2 0.2 1 0.1 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Michaelis-Menten: catalyzed reversible reaction ka Xa Xi ki rate B production + elimination - Xa* MPRI - Bio-informatique formelle - LC

  15. 1 0.8 0.6 0.4 Xa* 0.2 0 5 0.5 0 1 2 3 4 5 4 0.4 3 0.3 2 0.2 1 0.1 .1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 [B] Nullcline ka Xa Xi ki production + elimination - B rate Xa* MPRI - Bio-informatique formelle - LC

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

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

  18. Negative feedback X Y CANNOT OSCILLATE MPRI - Bio-informatique formelle - LC

  19. Negative feedback kai Xtot-X X kaa X kci Ztot-Z Y Z kca Z kbi Ytot-Y Y kba MPRI - Bio-informatique formelle - LC

  20. Negative feedback can create an oscillatory regime Z X X Y Z Y The third element introduces a delay that allows the system to oscillate. MPRI - Bio-informatique formelle - LC

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

  22. Hopf bifurcation Change of parameter kaa (activation of X) • 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. region of oscillations stable steady state HB HB Activity of X Hopf bifurcation points kaa : activation of X MPRI - Bio-informatique formelle - LC

  23. Introduction to bifurcation theory 1. Saddle Node (SN) bifurcation2. Hopf (H) bifurcation3. SNIC bifurcation : when SN meets H 4. Numerical Bifurcation theory 5. Signature of bifurcations MPRI - Bio-informatique formelle - LC

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

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

  26. 2. Hopf bifurcation Let p be a parameter of g(x,y) => vary p. g(x,y)=0 g(x,y)=0 g(x,y)=0 x x x f(x,y)=0 f(x,y)=0 f(x,y)=0 y y y stable focus (solutions converge to the steady state in a spiral) unstable focus (solutions diverge from the steady state) + stable limit cycle (solutions converge to the cycle) center x Bifurcation diagram p Supercritical Hopf bifurcation MPRI - Bio-informatique formelle - LC

  27. 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 bifurcation When increasing p, oscillations are created from a saddle node bifurcation MPRI - Bio-informatique formelle - LC

  28. 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) 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: MPRI - Bio-informatique formelle - LC

  29. 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: 5. The eigenvalues can inform on the stability of the fixed points MPRI - Bio-informatique formelle - LC

  30. MPRI - Bio-informatique formelle - LC

  31. 5. Signature of bifurcations MPRI - Bio-informatique formelle - LC

  32. 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 (x1, x2). 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. MPRI - Bio-informatique formelle - LC

  33. x1 For the case of the saddle node bifurcation, the following graph is obtained : p p1 MPRI - Bio-informatique formelle - LC

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

  35. 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 G2 Quelques questions • Pourquoi ne voit-on pas le niveau des cyclines osciller plus tôt • puisqu’il y a division nucléaire ? • Pourquoi les cycles s’arrêtent-ils au 14e cycle ? MPRI - Bio-informatique formelle - LC

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

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

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

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

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

  41. Des compartiments différents Cdk1/CycB FZY 1 2 3 4 MPRI - Bio-informatique formelle - LC

  42. Cytoplasm CycBn CycBc Wee1c P Cdk1 Cdk1 Stgc Nucleus CycBn CycBn Wee1n P Cdk1 Cdk1 IEP Stgn Fzy Fzy CycB/Cdk1 = MPF CycB/Cdk1-P = preMPF Des compartiments différents Cytoplasm CycBT Stgc/Cdc25 MPFc Wee1c Nucleus Wee1n Stgn/Cdc25 MPFn MPRI - Bio-informatique formelle - LC

  43. Pourquoi les cycles s’arrêtent-ils au 14e cycle ? MPRI - Bio-informatique formelle - LC

  44. 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 jusqu’au 14eme cycle Son degré de Phosphorylation : oscille a partir du 5eme cycle MPRI - Bio-informatique formelle - LC

  45. Stgm MPFT MPFb Xm Xp Treatment at t=55 min String/Cdc25, facteur limitant (2) Traitement alpha-amanitin : 14 cycles Treatment at t=70 min MPRI - Bio-informatique formelle - LC

  46. Diagramme de bifurcation: MPFn et CycBT en fonction du nombre de cycles MPRI - Bio-informatique formelle - LC Cycles

  47. 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). MPRI - Bio-informatique formelle - LC

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