BFS/Model reduction

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

Deterministic 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.

Metabolic reaction networks

The basic starting point for much of the work on modeling of reaction networks is the kinetic mass action model, which can be written in the form

<amsmath>

\frac{dX}{dt} = S v(X)

</amsmath>

where <amsmath>X \in {\mathbb R}^m</amsmath> is the vector of species concentrations, <amsmath>S \in {\mathbb R}^N</amsmath> is the stoichiometry matrix and <amsmath>v:{\mathbb R}^m \to {\mathbb R}^N</amsmath> is the nonlinear reaction rate vector.

Once you have the equation in this form, there are a couple of standard modeling techniques that are used:

Flux Balance Analysis: the main idea here is to analyze the set of reaction rates <amsmath>v</amsmath> that lie in the null space of <amsmath>S</amsmath> (and hence correspond to steady state conditions). We don't worried about the species concentrations themselves; just the reactions. In the case where there are many reactions and fewer species, we will get lots of possible reactions and we can explore what the space of all possible reactions tells us. An example of this sort of analysis is done by Schilling et al, who look at what you can say about pathways using this formalism:

  • C. H. Schilling, J. S. Edwards, D. Letscher and B. O. Pallson, Combining pathway analysis with flux balance analysis for comprehensive study of metabolic systems. Biotechnology and bioengineering, 71(4):286-306, 2001.

Chemical Reaction Network (CRN) theory: CRN theory focus on the graph structure that is encoded in the stoichiometry matrix as well as the specific type of nonlinearities that arise from mass action kinetics. The rough idea here is to "lift" the dynamics to a higher dimensional space of complexes by considering every product that occurs in the rate laws as a separate element. This then gives a set of linear dynamics in the lifted space (but with nonlinear constraints between the "states") and some analysis can be done. In particular, you can find conditions on the reaction graph under which you can have single or multiple equilibria. A nice overview of this approach that has an appealing mathematical flavor is given in the notes by Gunawardena:

  • J. Gunawardena, Chemical reaction theory for in-silico biologists. Unpublished notes, 2003.

There is also some recent MATLAB software for doing CRN analysis called ERNEST. This software can read SBML files and do network analysis, which seems pretty handy.

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: production and consumption. 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

  • 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)

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. This is all described in a bit more detail in the next section.

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:

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

In addition to all of the deterministic modeling work described above, it is also possible to formulate everything in terms of stochastic reactions, as detailed in Dan Gillespie's work. What's missing in this approach is much work on the analysis of the resulting stochastic dynamics (versus just simulations).

One paper that I found is work by Kim and Sauro extending some of the ideas from metabolic control analysis (MCA/MCT) to the stochastic setting:

  • K. H. Kim and H. M. Sauro, Stochastic control analysis for biochemical reaction systems. arXiv:0904.3124v3 [q-bio.QM], 21 Aug 2009.

Along the lines described in the previous section, the main idea here is to look at the sensitivity of the species concentrations to various (stochastic) parameters. In the context of biochemical systems, one motivation would be to look at how stochastic fluctuations in enzyme concentrations affect metabolic networks. This type of analysis should apply to a variety of biological circuits.

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