Control de Potencia en AC con TRIACs disparados por un microcontrolador

approaches

Selection of a modelling strategy is dependent upon the nature of the associated biological system and the desired output, and there exists numerous mathemat- ical frameworks that facilitate a wide variety of model analysis techniques [Voit, 2013]. Arguably, the more important choice relates to the degree of accuracy re- quired from the model. The trade-off between simplicity and accuracy is at the core

off all modelling investigations and useful results are unlikely to be obtained with- out establishing an effective compromise. Generally, accounting for minimal detail decreases mathematical investment whereas increased complexity requires greater investment. The former case is often referred to as black box modelling; black box models are typically over-simplified in order to provide generic, qualitative outputs and are so named due to their lack of mechanistic detail which boils down to a case of input-output with little knowledge or clarity regarding the relevant system structure. The latter case is often referred to mechanistic modelling; mechanistic models account for specific biological mechanisms and other structural details with a view to providing quantitative, physically valid outputs. Note that a spectrum of mechanistic complexity exists which places black box models at one extreme and white box models (white box models will be referred to as mechanistic models for the purposes of this work) at the other extreme with grey box models referring to any intermediate modelling approach that contains a mixture of elements.

The distinction between black box and mechanistic modelling can be illus- trated by considering the enzyme-substrate interaction model that forms the basis of Michaelis-Menten enzyme kinetics:

E+S−−₎k−−1* k−1

C −→k2 E+P, (2.1)

whereEis the enzyme,Sis the substrate,Cis the enzyme-substrate complex formed in the reaction andPis the product formed by the reaction;k1andk−1represent the

forward and backward reaction rates of the reversible complex formation respectively and k2 represents the reaction rate of the irreversible product formation. In a

black box modelling approach, we over-simplify these interactions by neglecting the formation of the enzyme-substrate complex and, instead, consider a single reaction that results in product formation via enzyme-substrate binding:

E+S−→k3 P, (2.2)

wherek3 represents the reaction rate of the irreversible product formation and en-

capsulates all three reaction rates from (2.1). The interactions denoted by (2.2) and (2.1) can therefore inform the derivation of black box and mechanistic models of enzyme kinetics respectively. The black box model exhibits the same dynamics for both the enzyme and the substrate as their concentrations decrease towards zero over time; the product is shown to increase in concentration as expected (Figure 2.4A). In contrast, the mechanistic model includes the added dynamics of the enzyme- substrate complex and exhibits subtly different outputs overall (Figure 2.4B). The

A

B

Figure 2.4: Comparison of black box and mechanistic enzyme kinetics model out- puts. A) The black box model accounts for product, P, formation by virtue of enzyme, E, and substrate, S, binding interactions. B) The mechanistic model ac- counts for additional detail regarding the formation of the intermediate complex,

C. All parameters and initial concentrations of enzyme and substrate used in sim- ulations set equal to 1; initial concentrations of complex and product set equal to 0.

enzyme dynamics in particular are significantly different since the black box model does not account for enzyme dissociation from the complex. Although the substrate and product dynamics are qualitatively similar for both models, there are clear distinctions that highlight the consequences of over-simplification.

Comparisons of the black box and mechanistic models demonstrate the un- derlying system dynamics that can be overlooked through generalisation. This ex- ample is, however, particularly simplistic even in the case of our mechanistic model considering the intricacy that can potentially arise in biological systems. The mech- anistic model consists of one more biological entity and two more reaction rates than the black box model which presents a negligible increase in mathematical investment. Numerous mathematical modelling techniques are required to produce the simula- tions in Figure 2.4, the details of which are given in the following sections. Note that these simulations require information regarding the parameter values corresponding to the associated reaction rates as well as the initial concentrations of the associated biological entities. The parameter values, k1, k−1, k2 and k3 in this example are

all arbitrarily set equal to 1 for comparable simulations. The initial conditions are selected given the context of the problem; only the enzyme and substrate are present at time zero and are therefore given arbitrary initial concentrations of 1 whereas the

complex and product are formed over time and are not present initially, resulting in initial concentrations of 0.

It is mechanistic modelling approaches that are adopted throughout this the- sis in order to construct the most structurally detailed biological models possible that can be validated quantitatively, and thus provide physically valid outputs. The systems-level understanding and experimental data available to us is particularly well founded by virtue of multiple collaborative efforts and extensive literature min- ing. Despite the substantial increase in mathematical investment presented by the large, complex models that we aim to produce, we are confident that rigorous math- ematical analysis poses a manageable obstacle, the cost of which is outweighed by the insight gained from our results.