Archive for January, 2012

Modelling Dictyostelium aggregation, yet again

January 29, 2012

In the last post I discussed the second chapter of the book ‘La Vie Oscillatoire’ which is concerned with glycolytic oscillations. The third chapter is on calcium oscillations, a theme which I have written about more than once in recent posts. Here I will say something about the subject of the fourth chapter, signalling by the cellular slime mould Dictyostelium. I have written some things about Dictyostelium in previous posts. It has fascinated many scientists by its ability to go from a state where the cells are independent to a state which looks like a multicellular organism as a reaction to a scarcity of food. The cells gather by means of chemotaxis. It is usual for mathematical talks about chemotaxis to start with nice pictures portraying the life cycle of Dictyostelium discoideum, the most famous organism of this type. It gave rise to the formulation of the Keller-Segel model which is rather popular among mathematicians. As I have mentioned in previous posts it is not so clear to what extent the Keller-Segel model, as attractive as it is, is really relevant to the life of D. discoideum. In his book Goldbeter seems to share this sceptical point of view while leaving open the possibility that the Keller-Segel model might be a reasonable model for aggregation in a less famous relative of D. discoideum, Dictyostelium minutum.

Returning to D. discoideum, it is a known fact that the cells of this organism signal to their neighbours by producing cAMP (cyclic adenosine monophosphate). This process can be modelled by a system of ODE (or a similar system with diffusion) called the Martiel-Goldbeter model. Experimentally it is seen that cultures of starving D. discoideum develop circular or spiral waves centered on certain pacemaker cells. In the book a decription is given of how this process can be understood on the basis of the Martiel-Goldbeter model. It is useful to draw a diagram of the dynamical properties of this model as a function of the activities of two key enzymes. There is a region where solutions of the MG model converge to a stationary solution, a region where they show excitable behaviour and a region where there is a limit cycle. As a cell develops following the beginning of a period of hunger it moves around in this parameter space. It starts from constant production (1), becomes excitable (2), produces pulsations (3) and then becomes excitable again (4). This is a statement which relates to the ODE system but when diffusion is added the idea is that the pacemaker cells are in stage (3) of their development while cells in stage (4) can then lead to waves which propagate away from the pacemakers. Because of the variability of the cells (of their intrinsic properties or their life histories) the different stages of development can be present at the same time.

I now want to say some more about the MG model itself. The variables in the system are the fraction \rho_T of receptor in the active state and the intracellular (\beta) and extracellular (\gamma) concentrations of cAMP. The evolution equation for \rho_T is of the form \frac{d\rho_T}{dt}=-f_1(\gamma)\rho_T+f_2(\gamma)(1-\rho_T) where f_1 and f_2 are ratios of linear functions. The evolution equation for \gamma is \frac{d\gamma}{dt}=\frac{k_t\beta}{h}-k_e\gamma where the quantities other then the unknowns are constants. The most complicated evolution equation is that for \beta which is \frac{d\beta}{dt}=q\sigma\Phi(\rho_T,\gamma,\alpha)-(k_i+k_t)\beta. Here \alpha is the concentration of ATP, taken to be constant in this model, and all other quantities except the unknowns are constants. The function \Phi is complicated and will not be written down here. It is obtained from a system with more equations by a quasi-steady state assumption and in its dependence on \gamma it is a ratio of two quadratic functions. A further quasi-steady state assumption leads to a simplified system for \rho_T and \gamma alone which is more tractable for analytical considerations.

Albert Goldbeter and glycolytic oscillations

January 21, 2012

This Christmas, at my own suggestion, I was given the book ‘La Vie Oscillatoire’ by Albert Goldbeter as a present. This book is concerned with oscillatory phenomena in biological systems and how they can be explained and modelled mathematically. After the introduction the second chapter is concerned with glycolytic oscillations. I had a vague acquaintance with this subject but the book has given me a much better picture. The chapter treats both the theoretical and experimental aspects of this subject.

If yeast cells are fed with glucose they convert it into alcohol. Those of us who appreciate alcoholic beverages can be grateful to them for that. In the presence of a supply of glucose with a small constant rate alcohol is produced at a constant rate. When the supply rate is increased something more interesting happens. The output starts to undergo periodic oscillations although the input is constant. It is not that the yeast cells are using some kind of complicated machine to produce these. If the cells are broken down to make yeast extract the effect persists. In fact for yeast extract the oscillations go away again for very high concentrations of glucose, an effect not seen for intact cells. This difference is not important for the basic mechanism of production of oscillations. The breakdown of sugar in living organisms takes place via a process called glycolysis consisting of a sequence of chemical reactions. By replacing the input of glucose by an input of each of the intermediate products it was possible to track down the place where the oscillations are generated. The enzyme responsible is phosphofructokinase (PFK), which converts fructose-6-phosphate into fructose-1,6-bisphosphate while converting ATP to ADP to obtain energy. Now ADP itself increases the activity of PFK, thus giving a positive feedback loop. This is what leads to the oscillations. The process can be modelled by a two-dimensional dynamical system called the Higgins-Selkov oscillator. Let S and P denote the concentrations of substrate and product respectively. The substrate concentration satisfies an equation of the form \dot S=k_0-k_1SP^2. The substrate is supplied at a constant rate and used up at a rate which increases with the concentration of the product. (Here we are thinking of ADP as the product and ignoring other possible effects.) The product concentration correspondingly satisfies \dot P=k_1 SP^2-k_2 P.

The Higgins-Selkov oscillator gives rise to a limit cycle by means of a Hopf bifurcation. The ODE system is similar to the Brusselator. There are two clear differences. The substance which is being supplied from ouside occurs linearly in the nonlinear term in the Higgins-Selkov system and quadratically in the Brusselator. In the Higgins-Selkov system the nonlinear term occurs with a negative sign in the evolution equation for the substance being supplied from outside while in the Brusselator it occurs with a positive sign. In the book of Goldbeter the Higgins-Selkov oscillator seems to play the role of a basic example to illustrate the nature of biological oscillations.

The NFAT signalling pathway

January 6, 2012

The role of T cells in the immune system is to recognize foreign substances and then take appropriate action. In order for this to happen information must be propagated from the surface of the cell, where the T cell receptor is, to the nucleus in order to initiate DNA transcription. The last step in this process is the binding of a suitable combination of transcription factors to the DNA. NFAT (nuclear factor of activated T cells) is one of these transcription factors. The fact that the associated signalling pathway plays an important role in the activation of T cells explains the name. In fact this substance (or class of substances – there are actually five different ones) are important for signalling in many cells of the immune system. I already mentioned the NFAT signalling pathway, its connection to calcium and a paper on the subject by Salazar and Höfer in a previous post. Now I have written a paper where I look into mathematical aspects of the activation of NFAT by means of dephosphorylation and the role of calcium in this process. Salazar and Höfer introduced a high-dimensional dynamical system and computed stationary solutions in a slightly simplified version of that system. I now proved, using chemical reaction network theory, that for each choice of the many parameters in the system there exists exactly one stationary solution of the full system for each value of the total amount of NFAT in the cell. Every solution with that total amount of NFAT converges to the stationary solution at late times. Furthermore, this solution is well approximated by the explicit solution of the simplified system under a biologically motivated assumption that certain parameters are small enough. The main tool in the proof is the Deficiency Zero Theorem.

The result just mentioned concerns the model for the dephosphorylation process with the stimulation of the cell expressed through fixed choices of the parameters. In reality the stimulation is communicated through the calcium concentration in the cytosol. This means that the parameters in the model for desphosphorylation should be replaced by time-dependent functions which themselves are the result of a dynamical process. The situation is described by Salazar and Höfer with the help of a two-dimensional dynamical system closely related to one introduced by Somogyi and Stucki to describe calcium oscillations in liver cells. In the paper I did some analysis of the model, giving criteria for the stability of the unique stationary solution for given parameter values and the existence of periodic solutions. Hopf bifurcations play a role. The model is closely related to the Brusselator and techniques of proof can be imported from that case. In particular it is important to identify explicit invariant regions for the flow. When a solution of the model for the calcium concentration is such that it tends to a constant at late times then it can be shown that the resulting configuration of the phosphorylation states of NFAT also converges to the situation with constant coefficients previously analysed. When a solution converges to a periodic solution at late times it is not clear what can be said.


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