The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
Tikhonov Regularization and Total Least Squares
SIAM Journal on Matrix Analysis and Applications
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In this article, we present a very fast and easy to implement method for reconstruction of metabolic pathways based on time series data. To model the metabolic reactions, we use the well-established setting of ordinary differential equations. In the present article we consider a network leading to the accumulation of quercetin-glycosides in tomato (Solanum lycopersicum). Quercetin belongs to a group of plant secondary metabolites, generally referred to as flavonoids, which are extensively being studied for their variety of important functions in plants as well as for their potentially health-promoting effects on human. We use time series measurements of metabolite concentrations of quercetin derivatives. In the present setting, the observed concentrations are the variables and the reaction rates are the unknown parameters. A standard method is to solve the parameters by reverse engineering, where the ordinary differential equations (ODE) are solved repeatedly, resulting in impractical computation times. We use an alternative method that estimates the parameters by least squares minimization, and which is, in the order of hundred times faster than the iterative method. Our reconstruction method can incorporate an arbitrary a priori known network structure as well as positivity constraints on the reaction rates. In this way we can avoid over-fitting, which is another often encountered problem in network reconstruction, and thus obtain better estimates for the parameters. We test the presented method by reconstructing artificial networks and compare it with the more conventional method in terms of residuals between the observed and fitted concentrations, computing times and the proportion of correctly identified edges in the network. Finally we exploit this fast method to statistically infer the kinetic constants in the flavonoid pathway. We remark that the method as such is not limited to metabolic network reconstructions, but can be used with any type of time-series data that is modeled in terms of linear ODE's.