Population dynamics and chemical reactions

The seminar which I mentioned in a recent post has caused me to go back and look carefully at a number of different models in biology and chemistry. It has happened repeatedly that I felt I could glimpse some mathematical relations between the models. Now I have spent some time pursuing these ideas. One aspect is that many of the systems of ODE coming from biological models can be thought of as arising from chemical reaction networks with mass action kinetics, even when the unknowns are not chemical concentrations. In this context it should be mentioned that if an ODE system arises in this way the chemical network which leads to it need not be unique.

The first example I want to mention is the Lotka-Volterra system. Today it is usually presented as a model of population dynamics. Often the example of lynx and hares is used and this is natural due to the intrinsic attractiveness of furry animals. The story of Volterra and his son in law also has a certain human interest. The fact that Lotka found the equations earlier is usually just a side comment. In any case, the population model is equivalent to an ODE system coming from a reaction network which was described by Lotka in a paper in 1920 (J. Amer. Chem. Soc. 42, 1595). The network is defined by the reactions A_1\to 2A_1, A_1+A_2\to 2A_2, A_2\to 0 and A_1+A_2\to 0. The last entry in the list can be thought of as an alternative reaction producing another substance which is not included explicitly in the model. A simpler version, also considered in Lotka’s paper, omits this last reaction. In his book ‘Mathematical aspects of reacting and diffusing systems’ Paul Fife looks at the second system from the point of view of chemical reaction network theory. He computes its deficiency \delta in the sense of CRNT to be one. It has three linkage classes. The second model also has deficiency one. All the linkage classes have deficiency zero and so the deficiency one theorem does not apply. The chemical system introduced by Lotka was not supposed to correspond to a system of real reactions. He was just looking for a hypothetical reaction which would exhibit sustained oscillations.

Next I consider the fundamental model of virus dynamics as given in the book of Nowak and May which has previously been mentioned in this blog. Something which I only noticed now is that in a sense there is a term missing from the model. This represents the fact that when a virion enters a cell to infect it that virion is removed from the virus population. This fact is apparently not mentioned in the book. In an alternative model discussed in a paper of Perelson and Nelson (SIAM Rev. 41, 3) they also omit this term and discuss possible justifications for doing so. The fundamental model as found in the book of Nowak and May can be interpreted as the equations coming from a network of chemical reactions. This is also true of the modified version where the missing term is replaced. Both systems (at least the ones I found) have deficiency two.

Several well-known models in epidemiology can also be obtained from chemical networks. For instance the SIR model can be obtained from the reactions S+I\to 2I and I\to 0. This network has deficiency zero and is not weakly reversible. The deficiency zero theorem applies and tells us that there is no equilibrium. Of course this fact is nothing new for this example. The SIS model is similar but in that case the system has deficiency one and a positive stationary solution exists for certain parameter values. You might complain that the games I am playing do not lead to useful insights and you may be right. Nevertheless, seeing analogies between apparently unrelated things is a notorious strength of mathematics. There is also one success story related to the things I have been talking about here, namely the work of Korobeinikov on the standard model of virus dynamics mentioned in a previous post. He imported a Lyapunov function of a type known for epidemiological models in order to prove the global asymptotic stability of stationary solutions of the fundamental model of virus dynamics.


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