An integrated mathematical model of cellular cholesterol biosynthesis and lipoprotein metabolism

Cholesterol regulation is an important aspect of human health. In this work we bring together and extend two recent mathematical models describing cholesterol biosynthesis and lipoprotein endocytosis to create an integrated model of lipoprotein metabolism in the context of a single hepatocyte. The integrated model includes a description of low density lipoprotein (LDL) receptor and cholesterol synthesis, delipidation of very low density lipoproteins (VLDLs) to LDLs and subsequent lipoprotein endocytosis. Model analysis shows that cholesterol biosynthesis produces the majority of intracellular cholesterol. The availability of free receptors does not greatly effect the concentration of intracellular cholesterol, but has a detrimental effect on extracellular VLDL and LDL levels. We test our model by considering its ability to reproduce the known biology of Familial Hypercholesterolaemia and statin therapy. In each case the model reproduces the known biological behaviour. Quantitative differences in response to statin therapy are discussed in the context of the need to extend the work to a more {\it in vivo} setting via the incorporation of more dietary lipoprotein related processes and the need for further testing and parameterisation of {\it in silico} models of lipoprotein metabolism

Clustering processes

The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for the case of a known number of clusters, consistency can be achieved under the only assumption that the joint distribution of the data is stationary ergodic (no parametric or Markovian assumptions, no assumptions of independence, neither between nor within the samples). If the number of clusters is unknown, consistency can be achieved under appropriate assumptions on the mixing rates of the processes. (again, no parametric or independence assumptions). In both cases we give examples of simple (at most quadratic in each argument) algorithms which are consistent.Comment: in proceedings of ICML 2010. arXiv-admin note: for version 2 of this article please see: arXiv:1005.0826v