Abstract
The estimation of parameters in even moderately large biological systems is a significant challenge. This challenge is greatly exacerbated if the mathematical formats of appropriate process descriptions are unknown. To address this challenge, the method of dynamic flux estimation (DFE) was proposed for the analysis of metabolic time series data. Under ideal conditions, the first phase of DFE yields numerical representations of all fluxes within a metabolic pathway system, either as values at each time point or as plots against their substrates and modulators. However, this numerical result does not reveal the mathematical format of each flux. Thus, the second phase of DFE selects functional formats that are consistent with the numerical trends obtained from the first phase. While greatly facilitating metabolic data analysis, DFE is only directly applicable if the pathway system contains as many dependent variables as fluxes. Because most actual systems contain more fluxes than metabolite pools, this requirement is seldom satisfied. Auxiliary methods have been proposed to alleviate this issue, but they are not general. Here we propose strategies that extend DFE toward general, slightly underdetermined pathway systems.
Highlights
AND BACKGROUNDA Google Scholar search for the keyword “parameter estimation” yields over 3 million hits, which renders it abundantly evident that the topic is everything but trivial, especially for applications in biology
(3), the case of under-determined systems (m > n) is the most common situation in metabolic modeling, because most pathway systems contain more reaction steps than metabolites. This common occurrence makes the under-determined case important for the model-free phase of dynamic flux estimation (DFE) and suggests that we investigate if the pseudo-inverse solution v(ti) = A+ b(ti) constitutes a biologically feasible, or even optimal, solution
The goal of this article was to extend the utility of DFE to the relatively common scenario where the algebraic system of fluxes is underdetermined or some time series data are missing or incomplete
Summary
AND BACKGROUNDA Google Scholar search for the keyword “parameter estimation” yields over 3 million hits, which renders it abundantly evident that the topic is everything but trivial, especially for applications in biology. The challenges of finding optimal parameter values for biological systems are multifold and include mathematical, statistical, computational, and even biological aspects. Computational challenges are driven by the sheer size of the often high-dimensional parameter space, the need to solve systems of differential equations thousands of times, and an error structure between model results and biological data that can be incredibly rough and contain uncounted local minima where search algorithms can get trapped. Biological issues include the size and complexity of a system, noisy or missing data, ill-characterized processes, and unrealistic parameter values. All these challenges are tightly interwoven and often create situations where no (good) solutions are obtained, where too many possible solutions can be identified, or where the exclusive criterion of the quality of the fit is misleading
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