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Implications of network topology on stability.

Kinkhabwala A - PLoS ONE (2015)

Bottom Line: Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution).Any further explicit algebraic specification of the network will project onto this stability phase space.Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

View Article: PubMed Central - PubMed

Affiliation: Department of Systemic Cell Biology, Max Planck Institute of Molecular Physiology, Dortmund, Germany.

ABSTRACT
In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network's "influence topology", a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network's steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

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Related in: MedlinePlus

Sel’kov network.(A) Influence topology. Temporal scaling of all Jacobian edges to /r0/ gives the parameters ρ1 = r1//r0/, ρ2 = r2//r0/, and ρ3 = r3//r0/. The positive stoichiometric terms σ1 and σ2 must also be specified independently. The stability phase space is shown for σ1 = σ2 = 1 and (B) ρ3 = 1/10; (C) ρ3 = 1/4; and (D) ρ3 = 1. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
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pone.0122150.g010: Sel’kov network.(A) Influence topology. Temporal scaling of all Jacobian edges to /r0/ gives the parameters ρ1 = r1//r0/, ρ2 = r2//r0/, and ρ3 = r3//r0/. The positive stoichiometric terms σ1 and σ2 must also be specified independently. The stability phase space is shown for σ1 = σ2 = 1 and (B) ρ3 = 1/10; (C) ρ3 = 1/4; and (D) ρ3 = 1. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.

Mentions: Sel’kov in 1968 [30] proposed the following simple model to account for glycolytic oscillations:x˙1=−k1x1+σ1k2x12x2+σ2k3x2(122)x˙2=k0−k2x12x2−k3x2,(123)with σ1 and σ2 equal to 1. The generalized version is:x˙1=−V11+σ1V122+σ2V23(124)x˙2=V0−V122−V23.(125)The influence topology is shown in Fig. 10A, with principal minors:b1=−1+σ1ρ1−ρ2−ρ3(126)b2=ρ2+ρ3−σ1ρ1ρ3−σ2ρ1ρ3,(127)and Hurwitz determinants:Δ1=1+ρ2+ρ3−σ1ρ1(128)Δ2=(1+ρ2+ρ3−σ1ρ1)(ρ2+ρ3−σ1ρ1ρ3−σ2ρ1ρ3).(129)This result can also be obtained directly from the cycle-based definitions (where I have already removed terms that are zero):Δ1=−c1(130)Δ2=−c1·c1c1¯+c1·c2.(131)The corresponding stability phase space is displayed in Figs. 10B-D for σ1 = σ2 = 1 and different values of ρ3.


Implications of network topology on stability.

Kinkhabwala A - PLoS ONE (2015)

Sel’kov network.(A) Influence topology. Temporal scaling of all Jacobian edges to /r0/ gives the parameters ρ1 = r1//r0/, ρ2 = r2//r0/, and ρ3 = r3//r0/. The positive stoichiometric terms σ1 and σ2 must also be specified independently. The stability phase space is shown for σ1 = σ2 = 1 and (B) ρ3 = 1/10; (C) ρ3 = 1/4; and (D) ρ3 = 1. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
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getmorefigures.php?uid=PMC4380337&req=5

pone.0122150.g010: Sel’kov network.(A) Influence topology. Temporal scaling of all Jacobian edges to /r0/ gives the parameters ρ1 = r1//r0/, ρ2 = r2//r0/, and ρ3 = r3//r0/. The positive stoichiometric terms σ1 and σ2 must also be specified independently. The stability phase space is shown for σ1 = σ2 = 1 and (B) ρ3 = 1/10; (C) ρ3 = 1/4; and (D) ρ3 = 1. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
Mentions: Sel’kov in 1968 [30] proposed the following simple model to account for glycolytic oscillations:x˙1=−k1x1+σ1k2x12x2+σ2k3x2(122)x˙2=k0−k2x12x2−k3x2,(123)with σ1 and σ2 equal to 1. The generalized version is:x˙1=−V11+σ1V122+σ2V23(124)x˙2=V0−V122−V23.(125)The influence topology is shown in Fig. 10A, with principal minors:b1=−1+σ1ρ1−ρ2−ρ3(126)b2=ρ2+ρ3−σ1ρ1ρ3−σ2ρ1ρ3,(127)and Hurwitz determinants:Δ1=1+ρ2+ρ3−σ1ρ1(128)Δ2=(1+ρ2+ρ3−σ1ρ1)(ρ2+ρ3−σ1ρ1ρ3−σ2ρ1ρ3).(129)This result can also be obtained directly from the cycle-based definitions (where I have already removed terms that are zero):Δ1=−c1(130)Δ2=−c1·c1c1¯+c1·c2.(131)The corresponding stability phase space is displayed in Figs. 10B-D for σ1 = σ2 = 1 and different values of ρ3.

Bottom Line: Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution).Any further explicit algebraic specification of the network will project onto this stability phase space.Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

View Article: PubMed Central - PubMed

Affiliation: Department of Systemic Cell Biology, Max Planck Institute of Molecular Physiology, Dortmund, Germany.

ABSTRACT
In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network's "influence topology", a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network's steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

No MeSH data available.


Related in: MedlinePlus