1. Computational Cell Biology Summer Course
Using of XPP, AUTO and VCell for qualitative analysis of building blocks of hemostatic system by
Alexey Tokarev
Cold Spring Harbor Laboratory – 2012

2. Hemostasis = platelet + plasma subsystems
Blood flow
Plug/ thrombus growth
Falati et al (2002)
platelets
platelets+fibrin
fibrin
Platelet activation, secretion, aggregation
Plasma clotting
Fibrin mesh. SEM made by Jean-Claude Bordet (Lyon, France).
Ohlmann (2000)
Project goal:
analyze the behavior of different components (building blocks) of hemostatic system using VCell, XPP and AUTO
Tasks:
Run the best of published models of platelet signaling in VCell and reproduce Ca2+ oscillations
Investigate oscillating and steady regimes of functioning of IP3 receptor using XPP/AUTO
Study travelling wave solutions of the reduced model of blood coagulation using XPP/AUTO

3. Part I. Quantitative model of platelet activation
Purvis et al, Blood, 2008
Model summary:
5 compartments
70 species
77 reactions
132 kinetic parameters
xml file obtained from the author
Two-week efforts to run this model in VCell (thanks to Ion and Olena for help!)
Failure to reproduce published results.

4. Part II. IP3-receptor model for Ca2+ oscillations
DeYoung and Keizer, PNAS, 1992; Li and Rinzel, J.Theor.Biol., 1994
Full model:
1+8-1=8 equations
τ(IP3)
<< τ(Ca,act)
<< τ(Ca,inact)
Reduced model:
2 equations
h=x000+x100+x010+x110

5. h – Ca2+ phase plane, 2-variables model of IP3R
[IP3]=0.3 uM
[IP3]=0.5 uM h
[Ca2+]I, uM h
[Ca2+]I, uM

6. Bifurcation diagrams of the full and reduced models
Full (9 variables)
Reduced (2 variables)
Conclusions:
full IP3R model may be too redundant for modeling of signal transduction in a cell
(general): robust properties of cellular building blocks are governed by time hierarchy of processes and thus can be described by low-dimensional models
Full (dashed) and reduced (solid) models.
Li and Rinzel, JTB, 1994
Marco is acknowledged for very helpful discussion

7. Properties of IP3 receptor module in the platelet activation model of Lenoci et al. (Mol. BioSyst., 2011)
[IP3] h
2-variables IP3R model, [IP3]=0.1, 1, 10 uM
[Ca2+]I, uM
Modification of IP3R parameters vs. DeYoung and Keizer:
forward binding rates are 10 times faster, dissociation constants are 2 times smaller
[Ca2+]ss
Conclusion:
no IP3R-dependent Ca2+ oscillations possible at all [IP3]
[IP3]

8. Investigation of cell volume effect on Ca2+ oscillations
ODE, Vcyt=10 um3
DeYoung->Vcell:
J->J/S(IM)
v3->v3/c1
c1 -> 1
Stochastic…
…under construction…

9. Part III. Travelling wave solution in the mathematical model of blood clotting
(full model)
Reduced model:
N. Dashkevich, M. Ovanesov, A. Balandina, S. Karamzin, P. Shestakov, N. Soshitova, A. Tokarev, M. Panteleev, F. Ataullakhanov. Thrombin activity propagates in space during blood coagulation as an excitation wave. To appear in Biophys. J.
Zarnitsina et al., Chaos, 2001

10. Phase diagram for c=3:
Test case 1: moving front solution of the Fisher-KPP (Kolmogorov-Petrovskii-Piskunov) equation V
Fisher, Ann. Eugenics, 1937; Kolmogorov, Petrovskii, Piscounov. Bull.Mocsow Univ., Math.Mech., 1937
Stable manifolds of f.p.(1,0)
u’t=u’’xx + u(1-u)
u(+∞)=0, u(-∞)=1
c – velocity of the moving front
ξ=x-ct
U(ξ)=u(x,t)
-cU’=U’’+U(1-U)
U’=V
V’=-cV-U(1-U)
(U,V)(+∞)=(0,0), (U,V)(-∞)=(1,0)
Unstable manifolds of f.p.(1,0) U V
heteroclinic orbit U U
solution
Solutions exist for every c>2 ξ

11. Test case 2: moving front solution of the FitzHugh-Nagumo (FHN) equation
Stable manifolds of f.p.(1,0)
Unstable manifolds of f.p.(1,0) V
ut = uxx + f(u), f(u)=u(1-u)(u-a)
u(+∞)=0, u(-∞)=1
The solution exits at single unique c U V
c = 1,
0.5,
0.3535,
heteroclinic orbit
AUTO U
Exact heteroclinic trajectory
Exact solution U
c=0.3535
solution ξ

12. Test case 3: finding the exact homoclinic trajectory in AUTO
u'=v
v'=-cv+u(1-u) +auv v u u ξ
Autowave solution u(x,t)

13. Conclusions / advices / hopes
Importing/using of the non-proved sbml models may appear to be the waste of your time 
Always think how to reduce your complex model
Using AUTO one can find the steady autowave solution of coupled PDEs (if it exists). Hope this will help in studying the plasma coagulation system 

14. Thank you for your attention!