Archive for November, 2011

Calcium oscillations

November 19, 2011

There is evidence to suggest that oscillations in levels of calcium inside and outside cells are used as a signalling mechanism. A variety of mathematical models have been introduced to study this phenomenon. Here I will discuss some aspects of the subject. A more general review can be found in this Scholarpedia article. In the plasma membrane and the endoplasmic reticulum there are pumps which transport calcium ions out of the cytosol. The result is a huge concentration difference between the cytosol on the one hand and the extracellular space and the lumen of the endoplasmic reticulum on the other hand. This can be several orders of magnitude. There are also ion channels in these membranes which, when open, allow the calcium to flow down its gradient. This provides a way to change the calcium concentration in the cytosol very fast and this can cause rapid changes in the behaviour of a cell. In this context it is important that the endoplasmic reticulum has such a high surface area and is so widely distributed in the cell. One type of calcium channels in the ER reacts to the binding of the substance IP{}_3 (inositol 1,4,5-trisphosphate) to the channel by opening. This effect is also modulated by the calcium concentration in the cytosol. There are calcium channels in the plasma membrane and there is also a certain amount of leakage through both membranes. Transport of calcium in and out of mitochondria can be an important effect. Some combination of these features can lead to oscillations in the calcium concentration in the cytosol. This presents a challenge for mathematical modelling. Ideally a dynamical system consisting of ODEs for the concentrations of various substances would exhibit periodic solutions. Of course a system of this kind must have dimension at least two and several two-dimensional models have been proposed. It could be that several of these models are useful since calcium signalling in different cell types may use different mechanisms. The difficult thing is not to find a model exhibiting oscillations but to find the right model for a particular type of cell. In what follows I consider one type of model. I have chosen this type for two reasons. The first is its simplicity. The second is that it may be relevant to explaining the role of calcium in the activation of T cells.

I consider first a model due to Somogyi and Stucki (J. Biol. Chem. 266, 11068). It is a two-dimensional dynamical system. The two variables are the calcium concentrations in the lumen of the ER and the cytosol, call them x and y. The concentration of IP{}_3 is taken to be constant. The rates of change of x and y are given by k'y-kx-\alpha f(y)x and kx-k'y+\alpha f(y)x+\gamma-\beta y. The quantities k,k',\alpha,\beta,\gamma are positive constants while f is a positive function which describes the behaviour of the IP{}_3 receptor and must be further specified to get a definite model. The inventors of the model remark that setting k=0 and f(y)=\frac{y^2}{a^2+y^2} causes this system to reduce to the famous Brusselator, which I have commented on elsewhere. Thus the model can be thought of as a kind of generalized Brusselator and indeed it exhibits similar qualitative behaviour. The choices which are suggested to be appropriate for the cells being studied (in this case hepatocytes) is that k>0 and f is given by a Hill function, f(y)=\frac{y^n}{a^n+y^n}. Nice features of this system is that it has a unique stationary solution which can be written down explicitly and that it is also possible to get an explicit formula for the characteristic equation of the linearization at that stationary solution. In this way the stability of the stationary solution can be determined, with instability corresponding to the existence of a limit cycle. It is stated that a Hopf bifurcation occurs but there is no discussion of proving this. The general picture seems to be that oscillatory behaviour occurs at intermediate levels of IP{}_3 stimulation and disappears at levels which are too low or too high. In this paper an alternative version of the model is introduced where in some places x is replaced by x-y. This happens when modelling effects driven by the difference of concentrations in the two compartments. Given that x is normally much larger than y it is plausible to replace the difference of concentrations by the concentration in the ER.

The dephosphorylation of the transcription factor NFAT during the activation of T cells has been studied in a paper of Salazar and Höfer (J. Mol. Biol. 327, 31). An important step in the activation process is an influx of calcium caused by release of IP{}_3. The calcium binds to calmodulin. It also binds to the phosphatase calcineurin which can then be activated by calmodulin. Finally calcineurin removes phosphate groups from NFAT. In this paper a model for calcium dynamics is used which is closely related to the (alternative model) of Somogyi and Stucki. There are three equations but two of them form a closed system which is more or less the Somogyi-Stucki model with a specific choice of receptor activity as a function of the concentration of IP{}_3. The last equation essentially means that the calcium level is integrated in time to give the concentration of active calcineurin.


The global attractor conjecture in chemical reaction network theory

November 2, 2011

In a previous post I wrote about chemical reaction network theory and, in particular, about a result belonging to this theory called the deficiency zero theorem. Now I have realized that in that post I claimed more than was justified. I will correct this point here. The assumptions of the deficiency zero theorem are that we have a chemical reaction network which is weakly reversible and of deficiency zero. (For the terminology I refer to the previous post on CRNT.) One conclusion for the associated dynamical system is that there is a unique stationary solution c_* (in each stoichiometric compatibility class) where all concentrations are positive. A second conclusion is that c_* is asymptotically stable (a local statement). The Lyapunov function L used to prove the second statement also allows some further conclusions to be drawn. For solutions with positive concentrations L is strictly decreasing along solutions away from c_*. This means that a positive solution can have no positive \omega-limit points other than c_*. In addition L tends to infinity at infinity, thus showing that each solution stays in a compact set. It can be concluded that the \omega-limit set is compact and that unless it is c_* it must consist of points where at least one concentration is zero. In the post just quoted I claimed that every solution converges to c_*. Reading the original three basic papers on this subject by Horn, Jackson and Feinberg might easily give the impression that this is the case. Looking at the proofs in detail, as I have done in the meantime, reveals that this statement is not proved in those papers. There are also many later papers on CRNT where this issue is not raised. This does not mean that the problem escaped attention completely, even in the early days. In a paper by Horn from 1974 he explicitly states that the proof of the result on global stability in his paper with Jackson was not correct. He expresses the opinion that the statement is nevertheless probably true. In a paper from 2001 on the kinetic proofreading model Eduardo Sontag proved a result of this kind under some extra conditions.

Recently this issue has received renewed attention and has been given the name ‘global attractor conjecture’ by Craciun et. al. (J. Symbolic. Comput. 44, 1551). In a 2011 paper of Anderson (SIAM J. Appl. Math. 71, 1487) he writes that it ‘is considered to be one of the most important open problems in the field of chemical reaction network theory’. In that paper he proves the conjecture in the case of systems with a single linkage class and so perhaps the question is close to being resolved.