# Hydrobates

A mathematician thinks aloud

## Metabolic networks

The processes of life involve systems of coupled chemical reactions. Often these are assumed to be homogeneous in space so that the concentrations of the substances involved are functions of time alone. The dynamics of these quantities are described by a system of ODE satisfied by the concentrations. I will call a system of this kind a metabolic system. What I want to do here is to say what is special about metabolic systems compared to general ODE systems. I also want to say something about features of these systems which are typically studied in theoretical work. A feature of metabolic systems which should be mentioned immediately is that they are often very large and moreover depend on a large number of parameters. This makes rigorous analysis difficult and can lead to a strong temptation to move to numerical approaches. On the basis of my personal mathematical preferences I would like to see analytical work taken as far as possible. My main source of information on this topic is the book ‘Systems Biology in Practice’ by E. Klipp et. al.

Consider a system of $n$ substances taking part in $r$ reactions. The system is taken to be of the form $\frac{dS}{dt}=N\nu$ where $S$ is the vector of concentrations taking values in ${\bf R}^n$, $\nu$ is a vector of reaction rates, taking values in ${\bf R}^r$ and $N$ is a constant matrix, the stoichiometric matrix. It is $n$ by $r$. It contains information about how many molecules of each type are consumed or produced in each reaction. The whole system depends on $m$ parameters which form a vector in ${\bf R}^m$. Notice that any system of ODE can be put into this form – simply take $n=r$ and $N$ to be equal to the identity. One obviously interesting question about the system is how many stationary solutions it admits. Actually what is of interest is steady state solutions where $\nu$ is not identically zero. (Call these non-trivial.) The concentrations of all chemicals should be independent of time but there should be non-zero reaction rates so that individual reactions are converting certain chemicals into others. Non-trivial steady states are only possible if the rank of $N$ is less than $r$. In general the possible reaction rates form a vector space of dimension $r$ minus the rank of $N$. Notice that the definition of ‘non-trivial’ here does not only use information about the ODE system – it also uses information about a particular splitting of the right hand side into two factors.

Metabolic control analysis is an attempt to understand which changes in a metabolic system result in which changes in particular features of the solutions. The quantity $\nu$ depends on the variables $S$ and $p$. For certain purposes it may be helpful to consider the derivatives of $\nu$ with respect to these variables. In terms of components this means considering the partial derivatives $\frac{\partial\nu_i}{\partial S_j}$ or $\frac{\partial\nu_i}{\partial p_j}$. In fact it is common to consider normalized quantities such as $\frac{S_j}{\nu_i}\frac{\partial\nu_i}{\partial S_j}$, which is possible as long as the denominators do not vanish. The resulting quantities are called elasticities ($\epsilon$-elasticty for $S$ and $\pi$-elasticity for $p$). These relative quantities seem to require giving up the picture of certain quantities as vectors. Maybe they should be thought of as bunches of scalars or as points of a manifold admitting certain preferred transformations. A different type of coefficients are known as control coefficients. They are defined in terms of steady state solutions of the system. Steady states can be changed by changing parameters in the system. It seems to me that the definition of the control coefficients requires being able to define the rates of change of some aspect of the steady state solution (e.g. one of the concentrations) with respect to another.This appears to include the implicit requirement that the steady states are locally isolated for given values of the parameters. This means that both quantities of interest are functions of a common quantity (the parameter being varied) so that we have a chance to define the derivative of one with respect to the other.

There are certain types of nonlinearities which are typically used in modelling metabolic processes. The simplest is the mass action form. Here if $p$ molecules of a species with concentration $S_1$ and $q$ molecules of a species with concentration $S_2$ are the inputs for a certain reaction then the reaction rate is taken to be proportional to $S_1^pS_2^q$. This has a simple intuitive interpretation in terms of the probability that the necessary molecules meet. The other common type of nonlinearity results from the mass action form by a Michaelis-Menten reduction, a procedure described in a previous post. This leads to a non-polynomial nonlinearity but has the important advantage of reducing the number of equations in the system.

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This entry was posted on July 24, 2009 at 8:07 am and is filed under dynamical systems, mathematical biology. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

### 3 Responses to “Metabolic networks”

1. Jonathan Vos Post Says:
July 25, 2009 at 5:53 am | Reply

The Evolution of Controllability in Enzyme System Dynamics

http://www.interjournal.org/manuscript_abstract.php?1152005238

Abstract:

A building block of all living organisms’ metabolism is the “enzyme chain.” A chemical “substrate” diffuses into the (open) system. A first enzyme transforms it into a first intermediate metabolite. A second enzyme transforms the first intermediate into a second intermediate metabolite. Eventually, an Nth intermediate, the “product” diffuses out of the open system. What we most often see in nature is that the behavior of the first enzyme is regulated by a feedback loop sensitive to the concentration of product. This is accomplished by the first enzyme in the chain being “allosteric”, with one active site for binding with the substrate, and a second active site for binding with the product. Normally, as the concentration of product increases, the catalytic efficiency of the first enzyme is decreased (inhibited). To anthropomorphize, when the enzyme chain is making too much product for the organism’s good, the first enzyme in the chain is told: “whoa, slow down there.” Such feedback can lead to oscillation, or, as this author first pointed out, “nonperiodic oscillation” (for which, at the time, the term “chaos” had not yet been introduced). But why that single feedback loop, known as “endproduct inhibition” [Umbarger, 1956], and not other possible control systems? What exactly is evolution doing, in adapting systems to do complex things with control of flux (flux meaning the mass of chemicals flowing through the open system in unit time)? This publication emphasizes the results of Kacser and the results of Savageau, in the context of this author’s theory. Other publications by this author [Post, 9 refs] explain the context and literature on the dynamic behavior of enzyme system kinetics in living metabolisms; the use of interactive computer simulations to analyze such behavior; the emergent behaviors “at the edge of chaos”; the mathematical solution in the neighborhood of steady state of previously unsolved systems of nonlinear Michaelis-Menton equations [Michaelis-Menten, 1913]; and a deep reason for those solutions in terms of Krohn-Rhodes Decomposition of the Semigroup of Differential Operators of the systems of nonlinear Michaelis-Menton equations. Living organisms are not test tubes in which are chemical reactions have reached equilibrium. They are made of cells, each cell of which is an “open system” in which energy, entropy, and certain molecules can pass through cell membranes. Due to conservation of mass, the rate of stuff going in (averaged over time) equals the rate of stuff going out. That rate is called “flux.” If what comes into the open system varies as a function of time, what is inside the system varies as a function of time, and what leaves the system varies as a function of time. Post’s related publications provide a general solution to the relationship between the input function of time and the output function of time, in the neighborhood of steady state. But the behavior of the open system, in its complexity, can also be analyzed in terms of mathematical Control Theory. This leads immediately to questions of “Control of Flux.”

2. Hopf bifurcations and Lyapunov numbers « Hydrobates Says:
January 10, 2010 at 1:15 pm | Reply

[...] There are very many applications where the Hopf bifurcation plays a role. A first example is the Brusselator mentioned above. This is a schematic two-dimensional model for a chemical reactor. When I hear the name I get a mental picture of Brussels sprouts. This is of course nonsense. The name comes from the fact that the model was developed in Brussels and is a simplification of a three-dimensional model called the Oregonator which was developed in Oregon. The latter name was influenced by the fact that it is a kind of oscillator. The Oregonator is nothing other then the Field-Noyes model discussed in a recent post. As mentioned there the Field-Noyes model also exhibits Hopf bifurcations. Hopf bifurcations occur in the FitzHugh-Nagumo and Hogdkin-Huxley systems. Thus they are potentially relevant for electrical signalling by neurons. They may also come up in another kind of biological signalling, namely that by calcium. For an extensive review of this subject I refer to a paper of Martin Falcke (Adv. Phys. 53, 255). In section 5 of that paper the author discusses experimental evidence indicating that certain calcium oscillations cannot be modelled using Hopf bifurcations and that it might be better to use other types of bifurcation. On the other hand he suggests that the evidence for this is not conclusive. Oscillations in glycolysis are modelled by the Higgins-Selkov oscillator, a two-dimensional system bearing a superficial resemblance to the Brusselator. The unknowns are the concentrations of ADP and the enzyme phosphofructokinase. This simple system describing a part of glycolysis exhibits a Hopf bifurcation. More information on this and related systems can be found in the book of Klipp et. al. on systems biology quoted in a previous post. [...]

3. Chemical reaction network theory « Hydrobates Says:
October 6, 2010 at 8:33 am | Reply

[...] often include complicated networks of chemical reactions. I have made some comments on this in a previous post. From a mathematical point of view this leads to large systems of ordinary differential equations [...]

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