Computational Cell Biology Summer Course

Cold Spring Harbor Laboratory – 2012

Using of XPP, AUTO and VCell for qualitative analysis of building blocks of hemostatic system

by

Alexey Tokarev

Hemostasis = platelet + plasma subsystems

Blood flow

Plug/ thrombus growth

Falati et al (2002)

plateletsplatelets+fibrin

fibrin

• Plasma clotting

Fibrin mesh. SEM made by Jean-Claude Bordet (Lyon, France).

• Platelet activation, secretion, aggregation

Ohlmann (2000)

Project goal: analyze the behavior of different components (building blocks) of hemostatic system using VCell, XPP and AUTO

Tasks: 1)Run the best of published models of platelet signaling in VCell and reproduce Ca2+ oscillations2)Investigate oscillating and steady regimes of functioning of IP3 receptor using XPP/AUTO3)Study travelling wave solutions of the reduced model of blood coagulation using XPP/AUTO

Part I. Quantitative model of platelet activation

Model summary:•5 compartments•70 species•77 reactions•132 kinetic parameters•xml file obtained from the author

Purvis et al, Blood, 2008

• Two-week efforts to run this model in VCell (thanks to Ion and Olena for help!)• Failure to reproduce published results.

Part II. IP3-receptor model for Ca2+ oscillationsDeYoung and Keizer, PNAS, 1992; Li and Rinzel, J.Theor.Biol., 1994

Full model:1+8-1=8 equations

Reduced model:2 equations

h=x000+x100+x010+x110

τ(IP3) << τ(Ca,act) << τ(Ca,inact)

[IP3]=0.3 uM

[IP3]=0.5 uM

h – Ca2+ phase plane, 2-variables model of IP3R

h

[Ca2+]I, uM

h

[Ca2+]I, uM

Bifurcation diagrams of the full and reduced models

Full (dashed) and reduced (solid) models.Li and Rinzel, JTB, 1994

Full (9 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

Reduced (2 variables)

Marco is acknowledged for very helpful discussion

[IP3]

[IP3 ]

Properties of IP3 receptor module in the platelet activation model of Lenoci et al. (Mol. BioSyst., 2011)

Modification of IP3R parameters vs. DeYoung and Keizer:forward binding rates are 10 times faster, dissociation constants are 2 times smaller

2-variables IP3R model,[IP3]=0.1, 1, 10 uM

Conclusion:no IP3R-dependent Ca2+ oscillations possible at all [IP3]

h

[Ca2+]I, uM

[Ca2+]ss

[IP3]

Investigation of cell volume effect on Ca2+ oscillations

ODE, Vcyt=10 um3

Stochastic……under construction…

Part III. Travelling wave solution in the mathematical model of blood clotting

0,001 2 3 40

50

100 Experiment Theory

Thr

ombi

n ac

tivity

, nM

Distance from the activator, mm

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

Reduced model:(full model)

Test case 1: moving front solution of the Fisher-KPP (Kolmogorov-Petrovskii-

Piskunov) equation

Solutions exist for every c>2

Phase diagram for c=3:

Fisher, Ann. Eugenics, 1937; Kolmogorov, Petrovskii, Piscounov. Bull.Mocsow Univ., Math.Mech., 1937

u’t=u’’xx + u(1-u)u(+∞)=0, u(-∞)=1

c – velocity of the moving front

ξ=x-ctU(ξ)=u(x,t)

-cU’=U’’+U(1-U)

U’=VV’=-cV-U(1-U)(U,V)(+∞)=(0,0), (U,V)(-∞)=(1,0)

heteroclinic orbit

U

V

Stable manifolds of f.p.(1,0)Unstable manifolds of f.p.(1,0)

U

V

U

ξ

solution

Test case 2: moving front solution of the FitzHugh-Nagumo (FHN) equation

c=0.3535

ut = uxx + f(u), f(u)=u(1-u)(u-a)u(+∞)=0, u(-∞)=1

The solution exits at single unique c

c = 0

Stable manifolds of f.p.(1,0)Unstable manifolds of f.p.(1,0)

heteroclinic orbit

solution

c = 1, 0.5,0.3535,0

U

V

U

V

U

ξ

Exact heteroclinic trajectory

Exact solution

AUTO

Test case 3: finding the exact homoclinic trajectory in AUTO

u

v

u

ξ

u'=vv'=-cv+u(1-u) +auv

Autowave solution u(x,t)

Conclusions / advices / hopes

1. Importing/using of the non-proved sbml models may appear to be the waste of your time

2. Always think how to reduce your complex model

3. 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

Thank you for your attention!