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Data-driven identification of biological systems using multi-scale analysis [1]

['Ismaila Muhammed', 'Department Of Mathematics', 'Khalifa University Of Science', 'Technology', 'Abu Dhabi', 'United Arab Emirates', 'Dimitris M. Manias', 'Dimitris A. Goussis', 'Department Of Mechanical Engineering', 'Haralampos Hatzikirou']

Date: 2025-11 

Equation-based model reduction.

The MM reaction mechanism describes the interaction between a substrate S and an enzyme E, forming a reversible complex C. The complex C subsequently undergoes an irreversible reaction, yielding a product P and releasing the enzyme E, which can then participate in another cycle of substrate binding [26]. This process can be represented as follows:

(1)

where k f (L ) and k b (s−1) denote the forward and reverse rate constants of the enzyme-substrate complex formation, respectively, while k 2 (s−1) represents the catalytic rate constant, also referred to as the turnover number. According to the law of mass action, the MM mechanism is modeled by the following set of ordinary differential equations (ODEs):

(2)

Here, the square brackets denote the concentrations of the respective chemical species. Assuming the system is closed, where initially no product or complex is present, i.e., , the conservation relations and hold ([E](0) and [S](O) are the initial enzyme and substrate concentrations), the system in Eq (2) simplifies to:

(3)

where the simplified symbols c = [C], s = [S] and e 0 = [E](0) have been used. Assuming R1f and R1b are the rates of the forward and backward directions of the enzyme-substrate complex formation reaction, we set the bidirectional reaction rate and is the rate of the catalytic reaction. and are the stoichiometric vectors of the two reactions. It should be noted that the origin, , is the only equilibrium point of the system.

The multi-scale nature of the two-dimensional system in Eq (3) is manifested through the significant difference in magnitude between the two time scales, and ( ), which govern the system’s dynamics. These time scales can be approximated by the inverse modulo of the eigenvalues and of the Jacobian matrix J of the two-dimensional ODE system described in Eq (3) [35,42,57]:

(4)

where

(5)

and is the dissociation constant, is the Van Slyke-Culen constant and is the Michaelis-Menten constant [31]. The value of ε is indicative to the gap that develops between the two eigenvalues (and corresponding time scales) and defines the stiffness of the system; i.e., , so that when and when [58,59]. Both μ and are non-dimensional and non-negative and quantify the relative availability and initial distribution of enzyme and substrate in dimensionless form, determining the model’s dynamic behavior and the separation between fast and slow reaction processes. The analytical expression of the eigenvalues in terms of μ and allows us identify regions in the μ- plane where different reduced models can be constructed [60].

Fig 3 displays the regions of validity of different reduced models. In particular, the region of a valid sQSSA model is highlighted with pink, the region of a valid rQSSA model is highlighted with green and the region of valid Partial Equilibrium Approximation (PEA) model is highlighted with shaded blue lines. The regions are separated by the lines and . Along the narrow neighborhood of the line (solid), indicated by fading to white color, no valid QSSA model can be obtained and only the PEA model is valid. Along the part of the line (solid) , so that no reduced model can be constructed there since . Moving away from this part, ε progressively decreases, as it is indicated by the dashed curve [31,60]. Table 1 displays the reduced models that are valid in the specified portions of the plane; i.e., the sQSSA model (c is considered fast), the rQSSA model (s is considered fast) and the PEA model (the bi-directional reaction is in equilibrium, ). It is shown in Table 1 that the PEA model simplifies to the sQSSA model when and to the rQSSA model when .

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TIFF original image Download: Fig 3. Regions of valid reduced models in the μ- 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]. https://doi.org/10.1371/journal.pcbi.1013193.g003

The fact that the full model in Eq (3) is valid throughout the plane and the three reduced models in Table 1 are valid in portions of this plane, will be the basis for the assessment of the proposed framework for system identification. In particular, datasets originating from the three trajectories shown in Fig 3 will form the starting point of the identification process. As shown in the figure, one trajectory is located in the region where the sQSSA and the PEA models are valid (Case 1), another is located in the region where the rQSSA and the PEA models are valid (Case 2) and a third trajectory is located in a region that includes the domains of validity of the sQSSA/PEA models (first part of the trajectory), the PEA model (middle part) and the rQSSA/PEA models (last part) (Case 3). The parameters and initial conditions for the three trajectories are displayed in Table 2.

The validity of the reduced models in Table 1 is demonstrated in Fig 4, where profiles of the three trajectories considered here obtained with the full and appropriate reduced model are compared. It is shown that an excellent agreement is obtained.

The simplification of the PEA model to the two QSSA models in Case 3 is demonstrated in Fig 5 that compares the solution of the PEA model with the solutions provided by the two QSSA models of Table 1. The sQSSA-based profiles of c and s in the left panel and the rQSSA profiles in the right panel are denoted by dashed lines, while the PEA profiles in both panels are denoted by crosses. It is evident that the sQSSA solution closely follows the original trajectory in the first region (pink, as in Fig 3), where the sQSSA is valid, but begins to diverge as the system transitions into the second region (green, as in Fig 3), where only rQSSA is valid. Conversely, the rQSSA solution initially deviates from the original trajectory in the first region, where rQSSA is not valid, but progressively aligns with it upon entering the second region, where rQSSA is valid.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1013193

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