Archive for August, 2012

The multiple futile cycle

August 27, 2012

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 n 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 3n+3 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 n=1, 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 n=1 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 \omega- limit points on the boundary of the positive region. The case n=2 (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
 

Organizing posts by categories

August 25, 2012

I have a tendency to use the minimal amount of technology I have to in order to achieve a particular goal. So for instance, having been posting things on this blog for several years, I have made use of hardly any of the technical possibilities available.  Among other things I did not assign my posts to categories, just putting them in one long list. I can well understand that not everyone who wants to read about immunology wants to read about general relativity and vice versa. Hence it is useful to have a sorting mechanism which can help to direct people to what they are interested in. Now I have invested the effort to add information on categories to most of the posts. It was easy (though time-consuming) to do and I find that the results are useful. It is helpful for me myself to navigate through the material and it is interesting for me to see at a glance how many posts on which subjects there are. For now on I will systematically assign (most) new posts to a category and the effort to do so should be negligible. This post is an exception since it does not really fit into any category I have.

Stable big bang singularities

August 2, 2012

This week I am at a conference on mathematical relativity in Oberwolfach. Today Jared Speck gave a talk on work of his with Igor Rodnianski which answers a question I have been interested in for a long time. The background is that it is expected that the presence of a stiff fluid (a fluid where the speed of sound is equal to the speed of light) leads to a simplification of the dynamics of the solutions of the Einstein equations near spacetime singularities. This leads to the hope that the qualitative properties of homogeneous and isotropic solutions of the Einstein equations coupled to a stiff fluid near their big bang singularities could be stable under small changes of the initial data. This should be independent of making any symmetry assumptions. As discussed in a previous post it has been proved by Fuchsian techniques that there is a large class of solutions consistent with this expectation. The size of the class is judged by the crude method of function counting. In other words there are solutions depending on as many free functions as the general solution. This is a weak indication of the stability of the singularity but it was clear that a much better statement would be that there is an open neighbourhood of the special data in the space of all initial data which has the conjectured singularity structure. This is what has now been proved by Rodnianski and Speck.

This result fits into a more general conceptual framework. Suppose we have an explicit solution u_0(t) of an evolution equation and we would like to investigate the stability of its behaviour in a certain limit t\to t_*. If we expect that solutions with data close to the data for u_0 have the same qualitative behaviour then we may try to prove this directly. Call this the forward method. If there is no evidence that this idea is false but it seems difficult to prove it then we can try another method as an interim approach to gain some insight. This is to attempt to construct solutions with the expected type of asymptotics which are as general as possible. I call this the backward method, since it means evolving away from the asymptotic regime of interest. The forward method is preferable to the backward if it can be done. In the case of singularities in Gowdy spacetimes Satyanad Kichenassamy and I applied the backward method and Hans Ringström later used the forward method. It is perhaps worth pointing out that while the forward method is more satisfactory than the backward one both together can sometimes be used to give a better total result than the forward method alone. There are also examples of this in the context of expanding cosmological methods with positive cosmological constant. I applied the backward method while Ringström, Rodnianski and Speck later used the forward method. The result for the stiff fluid with which I started this post also fits into this framework using the forward method. The corresponding result for the backward method was done by Lars Andersson and myself more than ten years ago.

There were two other talks at this conference which can be looked at from the point of view just introduced. One was a talk by Gustav Holzegel on his work with Dafermos and Rodnianski on the existence of asymptotically Schwarzschild solutions. The second was my talk on an apsect of Bianchi models I have discussed in a previous post. Both of these used the backward method.


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