BFS/Model reduction

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

Modeling and Analysis

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.

Power-law formalisms

Savageau has developed an approach to modeling chemical reaction networks that approximates the mass action kinetics by a set of power laws. This formalism is referred to as biochemical systems theory (BST). This is also called the power-law formalism. A good description with a worked example is in the paper

• A. Sorribas and M. A. Savageau, A Comparison of Variant Theories of Intact Biochemical Systems. I. Enzyme-enzyme interactions and biochemical systems theory. Mathematical Biosciences, 94:161-193, 1989.

In the most general form, you approximate the chemical reactions using equations of the form

<amsmath>

\frac{dX_i}{dt} = \sum_j \mu_{ij} \left( \gamma_j \prod_k X_k^{f_{jk}} \right)

</amsmath>

where $X_{i}$ is the concentration of species $i$, $\mu _{ij}$ is the stochiometric coefficient, $\gamma _{j}$ is the rate constant and $f_{jk}$ is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).

A simplified form of these dynamics can be obtained by approximating the sum by just two terms: synthesis and degradation. In this case, the equations have the form

<amsmath>

\frac{dX_i}{dt} = \alpha_i \prod_k X_k^{f_{k}} - \beta_i prod_k X_k^{g_{k}}.

</amsmath>

The advantage of this form of the equations is that the equilibrium points can now be computed by setting the derivative to zero and taking the logarithm of the resulting balance equation between synthesis and degredation. Due to the power law form, this becomes a linear equation in terms of the various terms in the power law and this can be solved using linear algebra. A good description of this is given in

In addition, the variation of the equilibrium concentration as a function of the parameters can be computed and it has been demonstrated in a variety of systems that it gives a good approximation to full dynamics.

Steady state solutions and sensitivity

There is a fairly substantial body of work on looking at the steady state solutions for chemical reaction systems and studying the sensitivity of those solutions to changes in parameters (including things like enzyme levels). One common area of research is "metabolic control analysis" (MCA), also sometimes called "metabolic control theory" (MCT). The basic idea here is to look at how the steady state solutions vary as a function of parameters. A linearized analysis is used, which essentially works out to be the same as looking at sensitivity functions in control theory (with the parameters serving as the inputs). The analysis is of the zero frequency response (steady state), but has been extended to frequency domain by Ingalls:

• D. A. Fell, Metabolic Control Analysis: a survey of its theoretical and experimental development. Biochemical Journal, 328:313-330, 1992.
• B. P. Ingalls, A frequency domain approach to sensitivity analysis of biochemical networks. Journal of Physical Chemistry B-Condensed Phase, 2004.

There is quite a lot of debate in the literature about the relationship between MCA/MCT and the earlier biochemical systems theory (BST) described earlier. The short answer is that you can recover everything in MCA/MCT using the power-law formalism of BST. For an explicit comparison, see

• M. Savageau, E. O. Voit and D. H. Irvine, Biochemical systems theory and metabolic control theory: 1. Fundamental similarities and differences. Mathematical Biosciences, 86:127-145 (1987)

From my perspecitive, MCA/MCT is just a special case of classical concepts of sensitivity in control theory (as pointed out by Ingalls) and that seems like a good way to approach it. BST is an approximation that agrees with MCA/MCT, but really goes beyond that in terms of a (approximate) modeling methodology. In writing anything up, it would be good not to get pulled into the apparent battle between the two camps (though maybe this has finally died down).

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.