Improved parameter estimation for systems with an experimentally located Hopf bifurcation

Inverse problems from biomedicine: inference of putative disease mechanisms and robust therapeutic strategies

Many complex diseases that are difficult to treat cannot be mapped onto a single cause, but arise from the interplay of multiple contributing factors. In the study of such diseases, it is becoming apparent that therapeutic strategies targeting a single protein or metabolite are often not efficacious. Rather, a systems perspective describing the interaction of physiological components is needed. In this paper, we demonstrate via examples of disease models the kind of inverse problems that arise from the need to infer disease mechanisms and/or therapeutic strategies. We identify the challenges that arise, in particular the need to devise strategies that are robust against variable physiological states and parametric uncertainties.

Elimination of the initial value parameters when identifying a system close to a Hopf bifurcation

One of the biggest problems when performing system identification of biological systems is that it is seldom possible to measure more than a small fraction of the total number of variables. If that is the case, the initial state, from where the simulation should start, has to be estimated along with the kinetic parameters appearing in the rate expressions. This is often done by introducing extra parameters, describing the initial state, and one way to eliminate them is by starting in a steady state. We report a generalisation of this approach to all systems starting on the centre manifold, close to a Hopf bifurcation. There exist biochemical systems where such data have already been collected, for example, of glycolysis in yeast. The initial value parameters are solved for in an optimisation sub-problem, for each step in the estimation of the other parameters. For systems starting in stationary oscillations, the sub-problem is solved in a straight-forward manner, without integration of the differential equations, and without the problem of local minima. This is possible because of a combination of a centre manifold and normal form reduction, which reveals the special structure of the Hopf bifurcation. The advantage of the method is demonstrated on the Brusselator.