In this paper, we have presented a strategy for model simplification and reduction based on the CSP method. For this purpose and in order to make the method publicly available we implemented the original CSP algorithm in the COPASI software.
The CSP method is restricted to ODE models. Previously described simplification routines based on CSP mainly focus on the conversion of ODEs into DAE systems. In contrast, we use the CSP method to simplify models by lumping those reactions together that could be identified as being in QE. In addition, algebraic equations are used for species that are identified by Radical Pointers. Accordingly, we redefine chemical equations and kinetic rate laws of affected reactions. We demonstrated the usability of this approach using the COPASI implementation of the CSP method for a simple one-enzyme reaction and for a rather complex model of yeast glycolysis [16
The time scale separation analysis of the glycolysis model revealed five reactions (vPGI, vALD, vTIM, vlpPEP, and vPK) for which the simplification strategy can be applied. We demonstrated that the resulting reduced model is capable of maintaining characteristics of the full model within an acceptable error range:
(i) same dynamic regimes, e.g. Hopf bifurcation point at [Glcx]0 = 18.5 mM; (ii) similar steady state levels of metabolite concentrations; (iii) similar periods for both and amplitudes for the second oscillatory regimes.
Studying different dynamics underlines again (as in [11
]) the importance of time-resolved analyses since the contribution of the players in the system may vary over time and in between different dynamical regimes. This is ignored if either steady state data (or single time point data in general) or single dynamic regimes are studied.
Compared to our previous work on the ILDM method [10
] - or the ILDM method in general - the CSP allows a more straightforward interpretation of its results with respect to the identification of QSS species and especially QE reactions. In addition, the Importance Index of CSP allows to analyze the impact of individual reactions on the dynamics of the species in the system.
An interesting outcome of our analysis is that it is possible to follow the general inherent temporal organization of the entire system when analyzing the distinctive time scales. Thus, we could observe that for the second oscillatory regime, all time scales oscillate in phase, partially overlapping each other which indicates that the whole system shows slower or faster dynamics in the course of a period.
Moreover, the number of fast modes changes over time and is also different for different dynamic regimes. Both factors prohibit the use of a fixed number of modes for time scale decomposition.
Furthermore, we suggest that the results of the CSP analysis can also be used for studying the relative importance of different reactions for the dynamics of the system. As an example, we observed that the overall participation of PFK in the slow modes increases with increasing glucose levels. In a simple way, this may be explained by the increasing energy charge (ATP concentration) which inhibits the PFK. Therefore, the relative importance of the PFK to the slower modes of the system increases.
Another beneficial result of the simplification process is of course that the number of system parameters is considerably reduced, especially concerning parameters which are involved in processes on a faster time scale than the time scale of interest which are then usually hard to identify. Therefore, using this process less system parameters will be unidentifiable.
Our study is not the first trying to reduce the original glycolysis model by [16
] analyzed exclusively the limit cycle of the second oscillatory regime ([Glcx
= 24 mM) employing CSP without taking into account transient behavior. In contrast, we analyzed the model with original initial values taking into consideration also the initial transient time period. In addition, there are major methodological differences. First, our approach focuses on simplifying the underlying biochemical reaction network rather than on approximating the ODE system with a DAE system. Second, we do not fix the number of fast modes. Third, we compute the normed sum of PIs over the entire fast subspace in order to justify QEA.
A completely different approach was taken by [22
]. Their sole criterium for the reduction was the fulfillment of a Stuart-Landau equation which is in principle only valid in the vicinity of a Hopf bifurcation and therefore does not offer a general strategy for system reduction.
Obviously, there are some relations between CSP output data and sensitivity analyses like metabolic control analysis (MCA). Learning e.g. about the impact of individual reactions on systems properties like dynamics could in principle also result from sensitivity analyses. We did a preliminary comparison of the results of our CSP analysis and a conventional MCA for the steady state. This resulted in a similar global picture, but the CSP gave a more fine-grained picture w.r.t. the relative importance of reactions on species. In addition, the time-resolved analysis for oscillations is not possible with MCA.
With all the mentioned benefits of using CSP for systems analysis, there are also problems and limitations arising from this approach. We employed several heuristic thresholds for the discrimination of the reactions and species mainly contributing to the fast subspace of the system. These were based on our experience and obviously, this might not be optimal for arbitrary systems. Thus, other systems might demand slightly altered thresholds. This is underlined by the fact that we observed one reaction - AK - that in principle fulfilled all of our criteria for elimination, but in the end, it turned out to be impossible to eliminate from the system without introducing a large error. Therefore, it is always important to carefully check the behavior of the reduced system. The CSP can only support this process in a rational way, but does not allow for a fully automated analysis.
Even though, accordingly, scientists will always have to be on top of this method, it would be useful to support the reduction of the system in a stronger way than just providing the CSP. A semi-automated reduction which then quickly allows to be checked for error compared to the original model would reduce workload considerably and is currently planned to be included in the software. An additional planned extension of the software is the support of different compartment sizes (if multi-compartment models are analyzed) which is currently not the case.
All in all, we were surprised that taking into account different dynamic regimes only allowed the elimination of 5 reactions and 5 species of the glycolysis model which is considerably less than previous attempts that focused on particular regimes. This once again supports the view that it is crucial to define which systems behaviors should be reproduced by the simplified model before entering reduction strategies and these initial decisions might result in different models in the end.