Fig 1.
Schematic representation of existing data-driven methods for full system identification and Jacobian matrix estimation as a function of the number of variables and data points. Each method occupies a distinct region based on its data and dimensionality requirements. No methods can provide accurate estimations in the low variable-low datapoints region.
Fig 2.
The methodology proposed to identify regions for valid models when system identification methodologies fail. The methodology starts with the availability of a biological dataset, which is directed either to a direct system identification methodology, when it is possible, or to the framework proposed here, when the first is not possible.
Fig 3.
Regions of valid reduced models in the μ- plane.
The regions of validity of sQSSA (pink: ), rQSSA (green :
) and PEA (shaded blue:
) in the μ-
plane, along with the trajectories that are analyzed (see Table 2 for the related parameters and ICs). Circles and squares denote the starting and ending point of each trajectory, respectively. The thick solid and dotted lines denote
and
, respectively. The thin dashed curve denotes the points at which
and encapsulates the shaded region where
. Reduced models in this region are of low accuracy [60].
Table 1.
The algebraic relation that defines the SIM and the differential equation(s) that governs the flow on the SIM, according to sQSSA, rQSSA and PEA.
Table 2.
Parameters and initial conditions for simulation of the cases considered.
Fig 4.
Solution comparison between full (solid) and reduced (dotted) Michaelis-Menten simulations on the SIM, for Case 1 - sQSSA (left), Case 2 - rQSSA (middle) and Case 3 - PEA (right). Parameters and initial conditions on Table 2.
Fig 5.
Validity comparison between the sQSSA (left) and rQSSA (right) and PEA solutions of Case 3 (see Table 2 for parameters and ICs).
Table 3.
Comparison of the right-hand side expressions between the ground truth and the identified models, using data obtained from the deterministic full (top) and reduced (bottom) models.
Fig 6.
Evolution of (solid) and
(dashed) for Case 3. In the first part where sQSSA is valid,
and
, while in the second part where rQSSA is valid,
and
.
Table 4.
Comparison of the right-hand side expressions between the PEA and the identified models by the Weak SINDy, using the CSP-based split of the dataset of Case 3.
Fig 7.
System evolution under noisy conditions.
Temporal profiles of the variables reconstructed from NODE for the PEA case: deterministic data (left), data with 2% additive noise (middle) and data with 1% multiplicative noise (right).
Table 5.
Performance of Weak SINDy on identifying the corresponding models of the data produced from the full and reduced stochastic models.
Table 6.
Comparison of the right-hand side expressions between the PEA and the identified models, using the CSP-based split of the dataset of Case 3.
Fig 8.
Solution comparison between the actual deterministic (solid) and identified from Weak SINDy (dotted) models of Michaelis-Menten simulations of Case 3 on the SIM, after spliting the data into regions where sQSSA (top) and rQSSA (bottom) models are valid. The data for the cases of no noise (left), additive (middle) and multiplicative noise (right) have been smoothened by the NODE. Parameters and initial conditions on Table 2.
Table 7.
Performance of Weak SINDy on identifying the corresponding models of the data produced from the full and reduced deterministic and stochastic models.