Computational method for analysis of polyethylene biodegradation

In a previous study concerning the biodegradation of polyethylene, we proposed a mathematical model based on two primary factors: the direct consumption or absorption of small molecules and the successive weight loss of large molecules due to β-oxidation. Our model is an initial value problem consisting of a differential equation whose independent variable is time. Its unknown variable represents the total weight of all the polyethylene molecules that belong to a molecular-weight class specified by a parameter. In this paper, we describe a numerical technique to introduce experimental results into analysis of our model. We first establish its mathematical foundation in order to guarantee its validity, by showing that the initial value problem associated with the differential equation has a unique solution. Our computational technique is based on a linear system of differential equations derived from the original problem. We introduce some numerical results to illustrate our technique as a practical application of the linear approximation. In particular, we show how to solve the inverse problem to determine the consumption rate and the β-oxidation rate numerically, and illustrate our numerical technique by analyzing the GPC patterns of polyethylene wax obtained before and after 5 weeks cultivation of a fungus, Aspergillus sp. AK-3. A numerical simulation based on these degradation rates confirms that the primary factors of the polyethylene biodegradation posed in modeling are indeed appropriate.