The Martiel-Goldbeter model (Biophys. J. 52, 807) is a system of ODE which models the production of the signalling molecule cAMP by Dictyostelium discoideum. There are a number of other competing models on the market but I will concentrate on just this one which seems to have been quite popular. A review of what this and other models can and cannot describe has been given in a recent article by Goldbeter (Bull. Math. Biol. 68, 1095) which also sketches some of the history of the problem. The applicability of this ODE model is to well-stirred cultures where there is no opportunity for pattern formation. In a situation where pattern formation is possible it is appropriate to add terms representing diffusion. This has been done in a paper by Tyson et. al. (Physica D, 34, 193) where spiral waves are found. The reason that so much effort is spent on studying this organism is that it is a model case for cell-cell communication and the development of structures from a homogeneous population of cells.
The basic biological mechanisms which are modelled by the equations concern the following processes. Extracellular cAMP can bind to a membrane receptor, thus activating it. This receptor can be phosphorylated. The activated receptor influences the enzyme adenylate cyclase which converts ATP to cAMP. There is another enzyme, phosphodiesterase, which eliminates cAMP. Martiel and Goldbeter derive a system of nine ordinary differential equations which describe the dynamics of these processes. By a quasi-steady state hypothesis, based on the smallness of certain parameters, they reduce this to a system of four equations. One of the variables in the latter system is the intracellular concentration of ATP. It is found experimentally that this concentration is almost time-independent. Numerical solutions of the four-equation system give results consistent with this. As a consequence it seems reasonable to set the concentration of ATP to a constant value in the other three equations, thus producing a system of three ODE. This three-equation system is what is usually known as the Martiel-Goldbeter system. The remaining unknowns in this system are the fraction of the receptor in the active state and the intracellular and extracellular concentrations of cAMP. For certain parameter values it is reasonable to make a further quasi-steady state assumption and reduce this system to one with only two equations. These parameter values are not appropriate for the biological system being studied and so the two-equation system cannot give useful quantitative results. It turns out, however, that solutions of the two-equation system show many of the qualitative features of solutions of the three-equation system and hence the two-equation system can be useful for obtaining a better intuitive understanding of the dynamics using phase-plane analysis.
The Martiel-Goldbeter system can be used to reproduce qualitative and quantitative features of experiments. Some of the main qualitative features are as follows. For certain parameter values the system has a stable limit cycle corresponding to spontaneous periodic production of cAMP by a cell. In this case the system is said to be oscillatory. It may be noted that oscillatory and excitable systems are often linked by bifurcations when parameters are varied, and this system is no exception. This dynamical behaviour may be important for the ‘pacemaker’ role of certain cells in the initiation of the aggregation process in Dictyostelium discoideum. For other parameter values the system is excitable and this results in cells amplifying an applied variation in cAMP concentration. This is also believed to be important in coordinating aggregation. Finally, the cells adapt in such a way that they do not react in a lasting way to a change from one time-independent value of an externally imposed cAMP concentration to another.