ON THE NUMERICAL APPROXIMATION OF REACTION AND DIFFUSION MECHANISM IN A CELL: A NON-STANDARD COMPARTMENT MODEL.
ABSTRACT: Polycyclic aromatic hydrocarbons (PAHs) were potent carcinogenic and mutagenic environmental pollutants which were present in the environment in large quantity because of the process of incomplete combustion. These compounds were lipophilic and thereby diffused through the membranes and reached desoxyribonucleic acid (DNA) of the cell causing toxicity and cancer. Earlier we developed a spatially distributed model to study intracellular dynamics where a homogenization approach was used to overcome the difficulty caused by the complex and heterogeneous structure of cytoplasm. Since the spatially distributed model was a complex model and the inclusion of further processes and adding more species would make it even more complex a simpler model than the spatially distributed model was needed.
Thus a simplified model was developed here. This model was based on a compartment approach for some of the involved components while the spatially distributed modeling was retained where diffusion was expected to be an important factor. In this paper we showed that the numerical results obtained from the non-standard compartment model converge to the numerical results of the spatially distributed model where the numerical results of the latter model have earlier been validated against in-vitro experimental results.
Key words: Polycyclic aromatic hydrocarbons reaction and diffusion compartment model spatially distributed model. INTRODUCTION
The mathematical modeling of reaction and diffusion mechanisms of intracellular metabolisms is a challenging and active area of research. The biological cell is a fundamental unit of living organisms. A cell can be divided into four main compartments namely cell membrane cytoplasm nuclear membrane and nucleus. The modeling of reaction and diffusion mechanisms in a cell becomes a challenging task because of the complex and heterogeneous nature of a cell. The model becomes even more complicated when partitioning phenomena are taken into account (Dreij et al. 2011). We have developed a first model describing the fate of polycyclic aromatic hydrocarbons (PAHs) in complex cell geometry.
The presence of many thin membrane structures in cytoplasm really add to the difficulty level in the modeling because lipophilic compounds concentrate there.
The diffusion-reaction mechanism yields a PDE system the details of which can be found in the study of (Dreij et al. 2011) which is summarized below Here denotes the PAH diol epoxides (PAH DEs) or tetrols. The subscript water" stands for extra- cellular medium (Compartment I) and represents the reaction rate constant. denotes the diffusion constant.
Here denotes the PAH diol epoxides (PAH DEs) or tetrols. The subscript water" stands for extra- cellular medium (Compartment I) and represents the reaction rate constant. denotes the diffusion constant.
where denotes the DNA adducts. In the cellular and nuclear membranes no reaction process takes place. Only PAH DEs and tetrols diffuse through the membranes.
The model was further developed by (Dreij et al. 2012) to mimic cellular exposure to PAH DEs of benzo[a]pyrene (BPDE) and dibenzo[al]pyrene (DBPDE). To reduce the system a compartment modeling (CM) approach was used by (Dreij et al. 2011) which resulted in a system of ordinary differential equations (ODEs). A description of compartment modeling can be found in the work conducted by (Jacquez 1996; Godfrey 1983; Holz and Fahr 2001). It was also found that a standard compartment model cannot capture the essential features of metabolism for certain set of physical and chemical parameters (Dreij et al. 2011). Thus a non-standard compartment model (NSCM) will be developed here.
MATERIALS AND METHODS
In order to show the essential steps for the derivation of our model we only considered the subsystem by considering one species only namely PAH DEs. In the present model in three sub-domains namely water compartment cytoplasm and the nucleus diffusion were considered to be very fast. The other two sub- domains namely cell membrane and nuclear membrane would be considered as spatially distributed sub-domains where diffusion had an important effect. The reason for considering these two sub-domains (membranes) as spatially distributed was that membranes play a vital role for the transmission of the substance. Hence the dynamics of the concentrations in the former sub- domains would be modeled using ODEs whereas the concentrations in the membrane compartments would be modeled using partial differential equations (PDEs). Thus this model dealt with the system of ODEs and PDEs together
whereas in a standard compartment model only ODEs were discussed and in spatially distributed model PDEs were solved. A simple model showing the diffusion mechanism of PAH DEs (denoted by DE here) from water (compartment I) to cytoplasm (compartment III) has been sketched in Figure-1.
RESULTS AND DISCUSSION
The set of chemical reactions and diffusion process shown in Figure-1 gave rise to a mix system of ordinary and partial differential equations. The ordinary differential equations (ODEs) were obtained in aqueous parts namely extracellular domain cytoplasm and nucleus whereas partial differential equations (PDEs) whereas ktetrol stood for rate constant. The mass balance in Compartments III and V was modeled analogously. were obtained in lipid compartments namely cellular and nuclear membranes. The complete system of equations obtained from the previous section was implemented in Matlab. The physical and chemical constants which appeared in the system of equations were taken from (Dreij et al. 2011) and have been summarized in Table-1. Numerical simulations were performed for a time span of 600 seconds.
Table-1. Showing chemical and physical constants
Diffusion coefficient in cell membrane###Dmemcell
Diffusion coefficient in nuclear membrane###Dmemnuc
Rate constant for forming Tetrols###ktetrol
Partition coefficient for DEs###K P DE
Partition coefficient for Tetrols###K P tetrol
Cell membrane area###Amemcell
Nuclear membrane area###Amemnuc
Cell/nuclear membrane thickness
Volume of compartment I###V1
Volume of compartment II###V3
Volume of compartment III###V5
Only PAH diol epoxides were available in extracellular compartment at the start of the process whereas all the other chemical species in all the compartments including extracellular compartment had zero initial values therefore the degradation of PAH DEs was observed in extracellular medium. PAH DEs and other chemical species in other compartments were formed during the process as it happened in vitro or in vivo. Since the objective of this study was to show that the simplified non-standard compartment model showed the similar behavior as of a spatially distributed model which was more complex and computationally expensive as compared to a non-standard compartment model therefore instead of showing the profiles of all the chemical species in the cellular compartments it was important to show some of the profiles in selected compartments. Thus some of the results showed the concentration of PAH DEs in cell membrane and cytoplasm are presented in Figure-2.
From Figure-2 we saw that the concentration in different sub-domains depended on the value of the constant M the idea of which was also given by (Chaudhry et al. 2009b).We obtained similar results for the concentration of different species in various sub- domains where M played a vital role in the model. For obtaining reliable results we needed to find a reasonable value for M. For this purpose simulations were performed for different values of M. From these experiments it turned out that the results obtained from the non-standard compartment model converged to the results obtained by spatially distributed model as shown in Figure-3. In Figure-3 (a) the molar mass of PAH DEs in cytoplasm had been calculated for different values of M. Similarly the molar mass of PAH DEs in nucleus had been calculated for different values of M as shown in Figure-3 (b). From both figures it was quite clear that as the value of M increased the numerical results of non- standard compartment model provided results which were close of the results for the spatially distributed model. It turned out that with the value of M = the non- standard compartment model converged to spatially distributed model. Earlier (Dreij et al. 2011) validated our results of spatially distributed model when compared with in-vitro cell experimental results. Thus the convergence of non-standard compartment model to spatially distributed model showed the validity of the results of the model discussed in this work. For the estimation of more accurate size of M one of the optimization techniques would be used as is found in the work presented by (Chaudhry et al. 2009a).
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|Publication:||Pakistan Journal of Science|
|Date:||Dec 31, 2014|
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