## Oscillations in the MAP kinase cascade

In a recent post I mentioned my work with Juliette Hell on the existence of oscillations in the Huang-Ferrell model for the MAP kinase cascade. We recently put our paper on the subject on ArXiv. The starting point of this project was the numerical and neuristic work of Qiao et. al., PLoS Comp. Biol. 3, 1819. Within their framework these authors did an extensive search of parameter space and found Hopf bifurcations and periodic solutions for many parameters. The size of the system is sufficiently large that it represents a significant obstacle to analytical investigations. One way of improving this situation is to pass to a limiting system (MM system) by a Michaelis-Menten reduction. In addition it turns out that the periodic solutions already occur in a truncated system consisting of the first two layers of the cascade. This leaves one layer with a single phosphorylation and one with a double phosphorylation. In a previous paper we had shown how to do Michaelis-Menten reduction for the truncated system. Now we have generalized this to the full cascade. In the truncated system the MM system is of dimension three, which is still quite convenient for doing bifurcation theory. Without truncation the MM system is of dimension five, which is already much more difficult. It is however possible to represent the system for the truncated cascade as a (singular) limit of that for the full cascade and thus transport information from the truncated to the full cascade.

Consider the MM system for the truncated cascade. The aim is then to find a Hopf bifurcation in a three-dimensional dynamical system with a lot of free parameters. Because of the many parameters is it not difficult to find a large class of stationary solutions. The strategy is then to linearize the right hand side of the equations about these stationary solutions and try show that there are parameter values where a suitable bifurcation takes place. To do this we would like to control the eigenvalues of the linearization, showing that it can happen that at some point one pair of complex conjugate eigenvalues passes through the imaginary axis with non-zero velocity as a parameter is varied, while the remaining eigenvalue has non-zero real part. The behaviour of the eigenvalues can largely be controlled by the trace, the determinant and an additional Hurwitz determinant. It suffices to arrange that there is a point where the trace is negative, the determinant is zero and the Hurwitz quantity passes through zero with non-zero velocity. This we did. A superficially similar situation is obtained by modelling an in vitro model for the MAPK cascade due to Prabakaran, Gunawardena and Sontag mentioned in a previous post in a way strictly analogous to that done in the Huang-Ferrell model. In that case the layers are in the opposite order and a crucial sign is changed. Up to now we have not been able to show the existence of a Hopf bifurcation in that system and our attempts up to now suggest that there may be a real obstruction to doing so. It should be mentioned that the known necessary condition for a stable hyperbolic periodic solution, the existence of a negative feedback loop, is satisfied by this system.

Now I will say some more about the model of Prabakaran et. al. Its purpose is to obtain insights on the issue of network reconstruction. Here is a summary of some things I understood. The in vitro biological system considered in the paper is a kind of simplification of the Raf-MEK-ERK MAPK cascade. By the use of certain mutations a situation is obtained where Raf is constitutively active and where ERK can only be phosphorylated once, instead of twice as in vivo. This comes down to a system containing only the second and third layers of the MAPK cascade with the length of the third layer reduced from three to two phosphorylation states. The second layer is modelled using simple mass action (MA) kinetics with the two phosphorylation steps being treated as one while in the third layer the enzyme concentrations are included explicitly in the dynamics in a standard Michaelis-Menten way (MM-MA). The resulting mathematical model is a system of seven ODE with three conservation laws. In the paper it is shown that for given values of the conserved quantities the system has a unique steady state. This is an application of a theorem of Angeli and Sontag. Note that this is not the same system of equations as the system analogous to that of Huang-Ferrell mentioned above.

The idea now is to vary one of the conserved quantities and monitor the behaviour of two functions $x$ and $y$ of the unknowns of the system at steady state. It is shown that for one choice of the conserved quantity $x$ and $y$ change in the same direction while for a different choice of the conserved quantity they change in opposite directions when the conserved quantity is varied. From a mathematical point of view this is not very surprising since there is no obvious reason forbidding behaviour of this kind. The significance of the result is that apparently biologists often use this type of variation in experiments to reach conclusions about causal relationships between the concentrations of different substances (activation and inhibition), which can be represented by certain signed oriented graphs. In this context ‘network reconstruction’ is the process of determining a graph of this type. The main conclusion of the paper, as I understand it, is that doing different experiments can lead to inconsistent results for this graph. Note that there is perfect agreement between the experimental results in the paper and the results obtained from the mathematical model. In a biological system if two experiments give conflicting results it is always possible to offer the explanation that some additional substance which was not included in the model is responsible for the difference. The advantage of the in vitro model is that there are no other substances which could play that role.