## When is a dynamical system of mass action type?

At the moment I am at the European Conference on Mathematical and Theoretical Biology in Krakow. This is a joint conference with the Society for Mathematical Biology and it is very large, with more than 900 participants. This leads to a huge number of parallel sessions and the need to choose very carefully in order to get the most profit from the conference. On one day, for instance, there were two cases with two sessions on immunology occurring simultaneously.

On Tuesday I went to a session on biochemical reaction networks. This included a talk by Gheorghe Craciun with a large expository component which I found enlightening. He raised the question of when a system of ODE with polynomial coefficients can be interpreted as coming from a system of chemical reactions with mass action kinetics. He mentioned a theorem about this and after asking him for details I was able to find a corresponding paper by Hars and Toth. This is in the Colloquia Mathematica Societatis Janos Bolyai, which is a priori not easily accessible. The paper is, however, available as a PDF file on the web page of Janos Toth. A chemical reaction network gives rise to a system of equations of the form $\dot x_i=p_i-x_iq_i$ where the $p_i$ and $q_i$ are polynomials with positive coefficients. They represent the contributions from reactions where the species with concentration $x_i$ is on the right and left side respectively. The result of Hars and Toth is that any system of this algebraic form can be obtained from a reaction network. It was pointed out by Craciun in his talk that this means that arbitrarily complicated dynamics can be incorporated into systems coming from reaction networks. If we have a system of the form $\dot x_i=f_i-g_i$ we can replace it by $\dot x_i=(x_1\ldots x_n)(f_i-g_i)$. This changes the system but does not change the orbits of solutions. If, for instance, we start with the Lorenz system with unknowns $x$, $y$ and $z$ we can simply translate the coordinates so as to move the interesting dynamics into the region where all coordinates are positive and then multiply the result by $xyz$. This preserves the strange attractor structure. This result may be compared with the Perelson-Wallwork theorem discussed in a previous post.

The construction of the reaction network reproducing the given equations is not very complicated. The main problem is keeping track of the notation. Suppose we start with a system of $n$ equations in $k$ variables $x_i$ which is polynomial and satisfies the necessary condition already mentioned. The reaction network can be constructed in the following way. (It is not at all unique.) Introduce one species $X_i$ for each $x_i$. The right hand side of each equation is a sum of terms of the form $Ax_1^{m_i}\ldots x_k^{m_k}$ and one reaction is introduced for each of these terms. To explain what it is suppose without generality that it belongs to the first equation. If $A>0$ hen the reaction transforms the complex $m_1X_1+m_2X_2+\ldots m_kX_k$ to the complex $(m_1+1)X_1+m_2X_2+\ldots m_kX_k$ with rate constant $A$. The only species where there is a net production is $X_1$ and so this reaction only contributes to the first equation. Moreover it does so with the desired term. On the other hand if $A<0$ the reaction transforms the complex $m_1X_1+m_2X_2+\ldots m_kX_k$ to the complex $(m_1-1)X_1+m_2X_2+\ldots m_kX_k$ with rate constant $-A$. The assumption on the system assures that $m_1-1\ge 0$.

### 3 Responses to “When is a dynamical system of mass action type?”

1. János Tóth Says:

I liked Gheoge’s idea first, then tried to do some calculations on the Lorenz model, and I think with this premultiplication we often get an equation the solutions of which blow up. Could you find appropriate constants to produce chaotic behaviour in the frst orthant? E. g.

NDSolve[{
x'[t] == x[t] y[t] z[t] (-3 (x[t] – y[t])),
y'[t] == x[t] y[t] z[t] (-x[t] z[t] + 26.5 x[t] – y[t]),
z'[t] == x[t] y[t] z[t] (x[t] y[t] – z[t]),
x[0] == 1, z[0] == 2, y[0] == 5}, {x, y, z}, {t, 0, 200},
MaxSteps -> \[Infinity]];

does not work.

Thanks for the post, János

• hydrobates Says:

Thanks for the comment. I see now that things are not as easy as I thought. It would be nice if this idea for treating the Lorenz system could be made watertight.

Alan

2. hydrobates Says:

After once again hearing a talk by Craciun on this subject and talking a little about it to Eduardo Sontag I decided that I had finally understood the essential point. I checked with Janos that he agreed with my conclusions. The essential point is that the behaviour which is to be reproduced in a reaction system should be contained in an invariant compact set. Then the translation can be chosen to bring this region into the positive orthant and no solutions can escape to infinity. Since in the Lorenz system the strange attractor is contained in a compact invariant set this argument works in that case.