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Laubenbacher (2005): A Discrete Approach to Top-Down Modeling of Biochemical Networks
This book chapter presents a modeling framework "of time-discrete dynamical systems over a finite state set" which is "assumed to have a finite set of possible states for each variable". The modeling method discussed in it uses "tools from computer algebra and the theory of Groebner bases". The focus is on "cellular biochemical networks".
Such a top-down approach will start with little knowledge about the system, capturing at first only a coarse-grained image of the system with only a few variables. Then, through iterations of simulation and experiment, the number of variables in the model is increased. At each iteration, novel experiments will be suggested by simulations of the model, which when carried out will provide data to improve the model further, leading to a higher resolution in terms of mechanisms.
The first subchapter deals with Boolean network models:
Boolean network models [...] differ from ODE models in that molecules are considered present or absent, rather than ranging over a continuum of values. There is increasing evidence that certain types of regulatory networks have key features that can indeed be represented well through Boolean models. [...] One of the disadvantages of the Boolean modeling framework is the need to discretize real-valued expression data into an ON/OFF scheme, which loses a large amount of information.
As an alternative, the authors propose finite-state polynomial models:
We now describe a multi-state discrete model approach that leverages existing algorithmic methods from symbolic computation and computational algebraic geometry (Laubenbacher and Stigler 2004). It models a regulatory network as a time-discrete multi-state dynamical system, synchronously updated. The method shares many features with a recently developed continuous top-down method (Yeung et al. 2002), which we first describe in some detail. According to the authors, the method is intended to generate a "first draft of the topology of the entire network, on which further, more local, analysis can be based."
Another subchapter deals with data discretization:
The first important choice to make is the number of discrete states to use. The second choice is the method by which to map realvalued data to discrete states. There are various ways of labeling real-valued data using finite-state sets. Thresholds with biological relevance are one type of labeling that can be used. This is typically referred to as binning. [...] Any discretization method suitable for our purposes must preserve information about the dynamic relationship between the different variables, and must accommodate several heterogeneous time series simultaneously (e.g., transcription data as well as protein and metabolite concentrations).
The authors also address the question how a mathematical theory for discrete models could be devised:
Discrete models are not well understood at a theoretical level. In particular, the relationship between the structure of a model and its dynamics has remained elusive. There are no general results about the number of components of the state space of Boolean or multi-state discrete models or about the existence of steady states. Especially the question of steady states is an important one for biological models. Having fairly general results about the relationship between structure and dynamics for sufficiently large classes of models is an important problem. [...] Our goal is to make a phenomenological (and, ultimately, mechanistic) mathematical model of a multivariate system we can observe as well as perturb, and about which we may have partial knowledge. The major challenges, compared to typical engineered systems, are that the system is very often high-dimensional, the number of observations is small in comparison, and the information we have about the systems is very limited. [...] No corresponding mathematical theory exists yet for the identification of biological systems.
In the section "Conclusions" the authors write:
Our method using polynomial dynamical systems over finite fields has the advantageous feature that its mathematical underpinning provides access to a variety of mathematical algorithms and symbolic computation software. [...] We believe that the field of system identification can serve as a blueprint for a mathematical top-down modeling program in systems biology.
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