BFS/Model reduction

This page has some notes on modeling and modeling reduction for biomolecular feedback systems.

Modeling

There's a quite large literature on both stochastic and deterministic modeling of biomolecular systems. This is something that I will write about in detail in BFS, but here are some of the basic techniques.

Model reduction

There has been considerable work done on the general problem of model reduction for chemical systems. A fairly good overview of the main techniques is available in the paper by Okino and Mavrovouniotis:

• M. S. Okino and M. L. Mavrovouniotis, Simplification of mathematical models of chemical reaction systems. Chemical Reviews, 98(2), 1998.

While the work in this area is often motivated by problems in combustion and other complex, chemically reacting flows, the basic ideas seems applicable to biomolecular systems as well. The three main techniques that Okino and Mavrovouniotis call out are:

• Lumping - gather together species that interact with each other in collective ways
• Sensitivity analysis - eliminate reactions or species that are insignificant in the question the model is posed to answer
• Time-scale analysis - replace fast reactions with steady state solution, assume slow reactants are at constant concentration

Although it doesn't quite fit into this taxonomy, a fairly common approach to eliminating reactions is simply to solve an integer programming problem to try eliminating reactions that don't affect the evolution of species concentrations. One such approach is that of Petzold and Zhu:

• L. Petzold and W. Zhu, Model reduction for chemical kinetics: An optimization approach. AIChE Journal, 45(4):869-886, 1999.

This approach does not take into account inputs and outputs explicitly, nor does it account for uncertainty. It is somewhat tuned for situations where you have lots of reactions and a reasonable number of species (eg, combustion), though there is also a discussion of reducing the number of species (using a similar integer programming approach).

A more input/output based approach has been proposed by Liebermeister, Baur and Klipp, using the ideas of balanced truncation:

• W. Liebermeister, U. Baur and E. Klipp, Biochemical network models simplified by balanced trunction. FEBS Journal, 272:2034-4043, 2005.

This looks at removing the "external" metabolites by replacing them with a low-order, linear model obtained by balanced truncation from the full system dynamics (linearized about an operating point). It uses a metabolic control analysis (MCA) style terminology. It might be possible to extend this technique to use nonlinear balanced truncation (ala Lall, Glavaski and Marsden work) or some sort of simple gain scheduled set of models of the external environment. These techniques are somewhat related to work that Henrik Sandberg did as a postdoc at Caltech:

• H. Sandberg and R. M. Murray, Model reduction of interconnected linear systems, Optimal Control Applications and Methods, 30(3):225-245, 2008.

and there is also some work by Vandendorpe and Van Dooren that seems similar (I'm checking with Henrik about this):

• A. Vandendorpe and P. V. Dooren, Model Reduction of Interconnected Systems. In Model Order Reduction: Theory, Research Aspects and Applications, pp 305-321, 2008.

Stochastic approaches

An area that I haven't looked into yet is model reduction of Markov processes, along the lines of things that Martha Grover has done, and also Mustafa Khammash and Brian Munsky.