A Method for Efficient Calculation of Diffusion and Reactions of Lipophilic Compounds in Complex Cell Geometry

Abstract

A general description of effects of toxic compounds in mammalian cells is facing several problems. Firstly, most toxic compounds are hydrophobic and partition phenomena strongly influence their behaviour. Secondly, cells display considerable heterogeneity regarding the presence, activity and distribution of enzymes participating in the metabolism of foreign compounds i.e. bioactivation/biotransformation. Thirdly, cellular architecture varies greatly. Taken together, complexity at several levels has to be addressed to arrive at efficient in silico modelling based on physicochemical properties, metabolic preferences and cell characteristics. In order to understand the cellular behaviour of toxic foreign compounds we have developed a mathematical model that addresses these issues. In order to make the system numerically treatable, methods motivated by homogenization techniques have been applied. These tools reduce the complexity of mathematical models of cell dynamics considerably thus allowing to solve efficiently the partial differential equations in the model numerically on a personal computer. Compared to a compartment model with well-stirred compartments, our model affords a more realistic representation. Numerical results concerning metabolism and chemical solvolysis of a polycyclic aromatic hydrocarbon carcinogen show good agreement with results from measurements in V79 cell culture. The model can easily be extended and refined to include more reactants, and/or more complex reaction chains, enzyme distribution etc, and is therefore suitable for modelling cellular metabolism involving membrane partitioning also at higher levels of complexity.

Diffusion-influenced reactions in a hollow nano-reactor with a circular hole

“…Methods motivated by homogenization techniques have been applied to make such problem treatable, yielding good agreement with experiments 29 . Along similar lines, the theory of irreversible diffusion-controlled reactions has been applied to describe reactions between substrates and enzymes in a whole-cell model 30 .…”

On the numerical approximation of drug diffusion in complex cell geometry

“…This model was earlier defined in [1]. In order to distinguish between the concentrations within the different sub-domains, we add an index.…”

“…For this, we use homogenization techniques. The process of the derivation of an averaged model of cytoplasm has been very extensively described in [1], which uses the following steps:…”

“…For the estimation of the effective diffusion coefficient, 3,eff for the homogenized cytoplasm, the numerical experiments with real membrane structures have already been performed [1,7] and the resulting effective diffusion coefficient constants using Monte Carlo procedure are 3 ,eff = 2.77 × 10 −10 m 2 s −1 , 3 ,eff = 2.44 × 10 −10 m 2 s −1 .…”

“…In order to validate the specific model and mathematical approach, we have collected sets of data from in vitro experiments and cells in culture describing the partitioning, intracellular metabolism, and reactivity of polycyclic aromatic hydrocarbons (PAHs) [2][3][4][5][6]. For the numerical treatment of the model without changing the essential features of metabolism, the approach for homogenizing the cytoplasm has already been used in [8,9] using periodic homogenization, but in this paper we will use iterative homogenization including periodic and stochastic homogenization which has been done in [1]. For our modeling strategy, we will use the assumptions of [7], which are summarized here:…”