<p>The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model.</p> / Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology

Riesenhuber \& Poggio recently proposed a model of object recognitionin cortex which, beyond integrating general beliefs about the visualsystem in a quantitative framework, made testable predictions aboutvisual processing. In particular, they showed that invariant objectrepresentation could be obtained with a selective pooling mechanismover properly chosen afferents through a {\sc max} operation: Forinstance, at the complex cells level, pooling over a group of simplecells at the same preferred orientation and position in space but atslightly different spatial frequency would provide scale tolerance,while pooling over a group of simple cells at the same preferredorientation and spatial frequency but at slightly different positionin space would provide position tolerance. Indirect support for suchmechanisms in the visual system come from the ability of thearchitecture at the top level to replicate shape tuning as well asshift and size invariance properties of ``view-tuned cells'' (VTUs)found in inferotemporal cortex (IT), the highest area in the ventralvisual stream, thought to be crucial in mediating object recognitionin cortex. There is also now good physiological evidence that a {\scmax} operation is performed at various levels along the ventralstream. However, in the original paper by Riesenhuber \& Poggio,tuning and pooling parameters of model units in early and intermediateareas were only qualitatively inspired by physiological data. Inparticular, many studies have investigated the tuning properties ofsimple and complex cells in primary visual cortex, V1. We show thatunits in the early levels of HMAX can be tuned to produce realisticsimple and complex cell-like tuning, and that the earlier findings onthe invariance properties of model VTUs still hold in this morerealistic version of the model.

The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model. / Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology