The multiple futile cycle is a simple type of network of chemical reactions which is often found in biological systems. In a previous post I mentioned it as a component of a slightly more complicated network found in many cells, the MAP kinase cascade. One concrete realization of the multiple futile cycle is a protein which can be phosphorylated at up to sites. All the phosphorylation steps are carried out by one kinase while all dephosphorylation steps are carried out by one phosphatase. Each step is modelled in the Michaelis-Menten way, including an enzyme-substrate complex as one of the species and using mass action kinetics. There results a system of ordinary differential equations with three conservation laws. These represent the conservation of the total amount of the two enzymes and of the substrate protein. In the case , which might be called the simple futile cycle, using the conservation laws to eliminate some of the variables leads to a three-dimensional dynamical system. A basic question is what can be said about the dynamics of solutions of this system.
It has been shown by Angeli and Sontag (Nonlinear Analysis RWA, 9, 128) that in the case every solution converges to a stationary solution and that this stationary solution is unique for given values of the conserved quantities. The proof uses the theory of monotone dynamical systems. The original dynamical system is not monotone and so the first step in their proof is to replace it by another system which is monotone and show that convergence properties of solutions of the second imply convergence properties of solutions of the first. The second step is to prove the convergence of solutions of monotone systems under the additional condition of the existence of a translational symmetry. The paper mentions that this result is dual to a previously known result due to Mierczyński about monotone systems with a conserved quantity. Up to now I thought that the only benefit of knowing that a dynamical system is monotone is the possibility of reducing it to an effective system of one dimension less. This is only interesting if the initial system is of dimension no more than three. What this work has shown me is that knowing that a system is monotone can sometimes be the key to concluding much more. One aspect of the paper of Angeli and Sontag which was a source of confusion for me was a difference in conventions to what I am familiar with from chemical reaction network theory. This seems to be essential for the monotonicity argument and not just a matter of taste. The stoichiometric matrix (or stoichiometry matrix) is defined differently because a reversible reaction is treated as a single reaction rather than as a pair.. I feel a spontaneous preference for the CRNT convention but here it seems that a different one can be a real advantage. In the case of the simple futile cycle an important effect is that the dimension of the kernel of the stoichiometric matrix is three with the CRNT convention and one with the Angeli-Sontag convention.
In another paper (J. Math. Biol. 61, 581) Angeli, De Leenheer and Sontag present a more general theory related to this. Here the hypotheses needed to obtain a suitable monotone system involve the properties of certain graphs constructed from the reaction network. In this theory the notion of persistence of the dynamical system plays an important role. This is the property that a positive solution can never have any limit points on the boundary of the positive region. The case (dual futile cycle) has been considered in a paper of Wang and Sontag (J. Nonlin. Sci. 18, 527). There they are able to show that for certain ranges of the parameters generic solutions converge to stationary solutions. To emphasize the power of the techniques developed in these papers it should be pointed out that they can be applied to systems with arbitarily large numbers of unkowns and parameters and that when they apply they give strong conclusions.
D. Angeli and E. D. Sontag (2008). Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles Nonlinear Analysis: Real World Applications, 9 DOI: 10.1016/j.nonrwa.2006.09.006
D. Angeli, P. De Leenheer and E. D. Sontag (2010). Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. Journal of mathematical biology, 61 (4) PMID: 19949950