FACTURACIÓN DE LAS DISTRIBUIDORAS EN EL PERÍODO 2000-2009

2.2.2.1 Non-linearity

Gene regulatory networks are inherently nonlinear. This is due to the complex, often not fully understood, interactions between cellular components of GNRs (genes, proteins, transcription factors, small molecules, etc.) manifested through intricate layers of feedback loops. By possessing this nonlinearity, GRNs are able to give rise to a diverse array of dynamical behaviours, including stable focus, sustained oscillations, hysteresis, toggle-switch, and multistability. However, nonlinearity also imposes great hurdles for GRN modelling. To obtain insights into these systems, nonlinear modelling frameworks must accordingly be developed, yet they are often analytically intractable. The problem gets even worse for large- scale networks.

To counter this problem, there have been attempts to simplify nonlinear models for ease of investigation while retaining their fundamental, important features. For instance, approaches based on linearisation have been developed in which systems are approximated by linearising around a fixed point, usually the steady state. Notably, stability analysis [71] has its

foundation based on local linearisation theory. S-system approximation is another example where a nonlinear system is approximated with a power-law representation around a steady state point. Although the resulting S-system is still nonlinear, it possesses the features that are more amicable for analysis [72, 73]. In a stochastic context, Kampen [74] developed the Linear Noise Approximation method to approximate the Chemical Master Equation (CME) by meaningfully separating stochastic variables into a deterministic part and a random,

fluctuating part (see appendix A for thorough discussion). The analysis of the CME now reduces to analysing change in probability distribution of the fluctuations.

Linearisation, however, comes with costs. Approximated models are often limited to locality and may fail to address global characteristics of the systems. Therefore, demand for tractable, efficient nonlinear modelling approaches is currently an important aspect of the field.

2.2.2.2 Multi-scale

In modelling systems, including biochemical systems, simplifications are unavoidable.

Depending on the level of detail that a model intends to capture, certain assumptions should be made to ignore the effects of some unnecessarily detailed processes without a significant loss of higher level of knowledge that can be acquired in a system. In physics, there are some well- defined rules. For example, a macroscopic object obeys the laws of classical mechanics, whereas these laws no longer hold true in mesoscopic and microscopic physics, which obey the laws of quantum mechanics. Modelling a biochemical system is similar because modelling effort has to focus on a particular scale depending on the purpose of the exercise. We can define the scales involved in biochemical reactions in general as follows:

Macroscopic scale: In this scale, we assume that the system is a well-mixed solution or, equivalently, is homogeneous. The behaviour of every particle is assumed to be the average behaviour of its kind. Therefore, particles are treated as concentrations (the number of molecules per unit volume) and models in this level are normally expressed by differential equations. Because the chemical reaction is described by increasing or decreasing

concentration levels, the changes in state of the system are continuous.

Microscopic scale: This is the lowest level of reactions, where atom-atom, atom-molecule or molecule-molecule collisions take place. The Avogadro number is the number of formula units in a mole and it describes the fundamental quantitative relationship between macroscopic and microscopic levels: one mole of atoms or molecules = 6.022 × 1023 atoms or molecules. The system at the microscopic level is represented by single molecules, each with a position and a momentum. Hence, the dynamics are stochastic in contrast to macroscopic computation where the dynamics can be described through averaging theorems.

Mesoscopicscale: This intermediate description of chemical reactions incorporates the information between the microscopic and macroscopic scales in a suitable way. The

boundaries are not sharp but can be roughly indicated. In the mesoscopic level we eliminate some irrelevant or poorly understood variables. For example, we assume the solution is well- mixed, therefore, we only count the molecules in a system, rather than keeping track of their individual properties. Because every particle is treated as an individual in this level, the dynamics of the system is stochastic with states changing discretely.

Inspired by the need to counter the multi-scale nature of biochemical systems, great efforts have been put into multi-scale modelling allowing adequate levels of description for different species and different reactions. The resulting models are sometimes referred to as hybrid

models. Some researchers [75, 76], for example, have considered partitioning the system into subsets depending on the basis of propensity function values: fast and slow reaction subsets. They then constructed the Chemical Master Equation (see Appendix A1) which describes the joint probability density functions of the number and the occurrences of both subsets. Cao et al. [77] have proposed a virtual fast system which is Markovian to represent the real non- Markovian system. The authors used the stationary (asymptotic) properties of the virtual fast system to make a stochastic algorithm to construct the slow-scale reactions, which the authors argued, is more reliable and transparent. Takahashi [78], on the other hand, developed a modular, object-oriented meta-algorithm to integrate the multi-scaled system running on different types of algorithms and driven by different timescales. They applied the meta-

algorithm to heat shock response model that combined Gillespie’s and Gibson’s Next Reaction algorithms and deterministic differential equations. The results showed a dramatic

improvement in performance without the loss of significant accuracy. Overall, multi-scale stochastic modelling is certainly a worthwhile exercise in the future when the scale of modelled systems becomes large.