zero eigenvalue in the reduced Jacobian

[Irene]

Hello everyone! I'm trying to understand the possible reasons behind a zero eigenvalue in the reduced Jacobian of the ODE system. My understanding is that it should not be the case after Copasi removed mass conservations. Also, surprisingly I can still perform MCA on the model even though the reduced Jacobian is degenerate. I'm attaching the SBML file just in case. Any insight is much appreciated!

[Hoops, Stefan (sh9cq)]

Hello Irene,

On Sat, 2021-12-04 at 09:09 -0800, Irene wrote:
> Hello everyone! I'm trying to understand the possible reasons behind
> a zero eigenvalue in the reduced Jacobian of the ODE system. My
> understanding is that it should not be the case after Copasi removed
> mass conservations.

There are other situation than mass conservation which may cause zero
Eigenvalues in the reduced Jacobian. Mass conservation is a structural
property which causes the full Jacobian to be degenerated. However,
numerical situations may have the same effect at least at some points
in time or in the steady state.

> Also, surprisingly I can still perform MCA on the model even though
> the reduced Jacobian is degenerate. I'm attaching the SBML file just
> in case. Any insight is much appreciated!

Traditionally MCA (Reeder algorithm) could not be applied for
degenerated cases. However COPASI uses in those cases the Smallbone
algorithm: https://arxiv.org/pdf/1305.6449.pdf. Thus, MCA is still
possible.

Thanks,
Stefan


--
Stefan Hoops, Ph.D.
Research Associate Professor
Biocomplexity Institute & Initiative
University of Virginia
995 Research Park Boulevard
Charlottesville, VA 22911

Phone: +1 540 570 1301
Email: sho...@virginia.edu

[Mendes,Pedro]

Irene,

A zero eigenvalue appears when the system has so-called "marginal stability". In this situation a perturbation of the steady state causes the system to deviate from the original steady state and reach a new one in the vicinity of the first. This is different from a negative eigenvalue, where after perturbation it goes back to the original steady state, or a positive eigenvalue, where the perturbation moves the system away from the original steady state (at an exponential rate).

The most ewll-know system that has this property is the Lotka-Volterra system. When you find a steady state in LV, it  almost always has a zero eigenvalue.

Pedro

On 12/4/21 12:09 PM, Irene wrote:

Hello everyone! I'm trying to understand the possible reasons behind a zero eigenvalue in the reduced Jacobian of the ODE system. My understanding is that it should not be the case after Copasi removed mass conservations. Also, surprisingly I can still perform MCA on the model even though the reduced Jacobian is degenerate. I'm attaching the SBML file just in case. Any insight is much appreciated! --
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-- 
Pedro Mendes, PhD
Professor and Director,
Richard D. Berlin Center for Cell Analysis and Modeling
University of Connecticut School of Medicine
group website: http://www.comp-sys-bio.org

[sven....@gmail.com]

Hello Irene,

in addition to Stefan's and Pedro's I noticed one point in your model that may be related to the problem. I used the time scale analysis feature to find out which part of the model is related to the 0 eigenvalue. It seems it is the pair of reactions OXPHOS/MAS. These two reactions have two things in common: They have non integer stoichiometries, that should be fine, but could potentially make it more difficult for COPASI to identify mass conservation relations. But in addition they have negative stoichiometries. While in theory a product with negative stoichiometry should behave like a substrate with positive stoichiometry in a reaction, I have actually never tested how COPASI handles this. Did you enter the model this way in COPASI, or was it imported from an SBML file? 

Sven

[Irene]

Dear Stefan, Pedro, and Sven,

thank you for the detailed replies and helpful clarifications!

@Sven: the model I shared is parsed from the SBML file. Copasi seems to deal fine with the negative stoichiometry in it. I still reworked this SBML to avoid both negative and non-integer coefficients (file attached) and it does not affect anything.

Another common quality between MAS and OXPHOS reactions is the shared rate law which probably explains why they both are implicated.

Could you please explain briefly (or just refer me to the respective docs) how do you determine which part of the model is related to the zero EV? I get an error while trying to run Time Scale Separation analysis saying that it does not work for the model with multiple compartments.

[sven....@gmail.com]

Dear Irene,

good to know that the negative stoichiometries were not the problem!

What I did with the time scale analysis is actually a kind of off-label use of the feature. Usually the method is used to identify parts of the model that participate in fast time scales that can be used to simplify the model. In this case I did the opposite, assuming that 0 eigenvalue corresponds to a infinitely slow time scale mode. I used the CSP method with default settings. COPASI then identifies one  time scale mode that is 10 orders of magnitude slower than the rest of the model, and I just assumed that this time scale corresponds to the zero eigenvalue. the "participation index" then shows that the two reactions  MAS and OXPHOS are the only ones that participate in this extremely slow mode. (roughly that means that these reactions are the only ones for which the projection of their direction in phase space on the eigenvalue corresponding to the zero eigenvalue is not zero).

The CSP method as implemented in COPASI is described in:

Surovtsova, I., Simus, N., Hübner, K. et al. Simplification of biochemical models: a general approach based on the analysis of the impact of individual species and reactions on the systems dynamics. BMC Syst Biol 6, 14 (2012). https://doi.org/10.1186/1752-0509-6-14

And you are right, I think the fact that OXPHOS and MAS have identical rates is the reason for the extra zero eigenvalue. The way the model is written, it behaves exactly as if the two reactions were one which implies a different model structure. That means when COPASI tries to figure out the mass conservation it misses one. The reason is that COPASI calculates the the mass conservation relations from the structure of the model as defined by the stoichiometries, but in this model the rate laws of the two reactions change the structure. (This is sometimes called a structural instability, where the structure and behavior of a model depends on one specific value of a parameter. A tiny change to make the two rate laws a little bit different changes the model behavior completely).

Sven

[Irene]

Dear Sven,

thanks so much for your help and this detailed explanation! As a student, I should definitely invest more time into learning about Time Scale Separation analysis since it is such a powerful and interesting tool!

All the best,

Irene