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. 2017:2017:5924270.
doi: 10.1155/2017/5924270. Epub 2017 May 30.

Finding Solvable Units of Variables in Nonlinear ODEs of ECM Degradation Pathway Network

Shuji Kawasaki  1 , Dhisa Minerva  2 , Keiko Itano  2 , Takashi Suzuki  2
Affiliations

Finding Solvable Units of Variables in Nonlinear ODEs of ECM Degradation Pathway Network

Shuji Kawasaki et al. Comput Math Methods Med. 2017.

Abstract

We consider ordinary differential equation (ODE) model for a pathway network that arises in extracellular matrix (ECM) degradation. For solving the ODEs, we propose applying the mass conservation law (MCL), together with a stoichiometry called doubling rule, to them. Then it leads to extracting new units of variables in the ODEs that can be solved explicitly, at least in principle. The simulation results for the ODE solutions show that the numerical solutions are indeed in good accord with theoretical solutions and satisfy the MALs.

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Figures

Figure 1
Figure 1
Mechanism of activation of a (MMP2) by b (TIMP2) and c (MT1-MMP).
Figure 2
Figure 2
Binding sites of a (MMP2), b (TIMP2), and c (MT1-MMP).
Figure 3
Figure 3
Pathway network of association/dissociation pairs of complexes of a (MMP2), b (TIMP2), and c (MT1-MMP). Here, arrows indicating only association are depicted.
Figure 4
Figure 4
Doubling rules in the association/dissociation of b- and -cC.
Figure 5
Figure 5
Looking at PWN with unit variables.
Figure 6
Figure 6
X 1: simulation result and theoretical solution.
Figure 7
Figure 7
ξ 2581011: simulation result and theoretical solution and MCL (26).
Figure 8
Figure 8
ξ 4791112: simulation result and theoretical solution and MCL (24).
Figure 9
Figure 9
ξ 24: simulation result and theoretical solution.
Figure 10
Figure 10
ξ 3689: simulation result and theoretical solution and MCL (32).
Figure 11
Figure 11
ξ 357: simulation result and theoretical solution.
Figure 12
Figure 12
ξ 3689 + ξ581011 + ξ791112: simulation result and theoretical solution and MCL (36).
Figure 13
Figure 13
Time course of X9(t): comparison of the models with or without the doubling rule.
Figure 14
Figure 14
X 9(∞) versus X2(0): comparison of the models with or without the doubling rule.
See this image and copyright information in PMC

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