Features inside the living cell are complex and crowded; in such complex environments diffusion processes can be said to exhibit three distinct behaviours: pure or Fickian diffusion, superdiffusion and subdiffusion. Furthermore, the behaviour of biochemical processes taking place in these environments does not follow classical theory. Because of these factors, the task of modelling dynamical proceses in complex environments becomes very challenging and demanding and has received considerable attention from other researchers seeking to construct a coherent model. Here, we are interested to study the phenomenon of subdiffusion, which occurs when there is molecular crowding. The Reaction Diffusion Partial Differential Equations (RDPDEs) approach has been used traditionally to study diffusion. However, these equations have limitations due to their unsuitability for a subdiffusive setting. However, I provide models based on Fractional Reaction Diffusion Partial Differential Equations (FRDPDEs), which are able to portray intracellular diffusion in crowded environments. In particular, we will consider a class of continuous spatial models to describe concentrations of molecular species in crowded environments. In order to investigate the variability of the crowdedness, we have used the anomalous diffusion parameter $\alpha$ to mimic immobile obstacles or barriers. We particularly use the notation $D_t^{1-\alpha} f(t)$ to represent a differential operator of noninteger order. When the power exponent is $\alpha=1$, this corresponds to pure diffusion and to subdiffusion when $0<\alpha<1$. This thesis presents results from the application of fractional derivatives to the solution of systems biology problems. These results are presented in Chapters 4, 5 and 6. An introduction to each of the problems is given at the beginning of the relevant chapter. The introduction chapter discusses intracellular environments and the motivation for this study. The first main result, given in Chapter 4, focuses on formulating a variable stepsize method appropriate for the fractional derivative model, using an embedded technique~\cite{landman07,simpson07,simpson06}. We have also proved some aspects of two fractional numerical methods, namely the Fractional Euler and Fractional Trapezoidal methods. In particular, we apply a Taylor series expansion to obtain a convergence order for each method. Based on these results, the Fractional Trapezoidal has a better convergence order than the Fractional Euler. Comparisons between variable and fixed stepsizes are also tested on biological problems; the results behave as we expected. In Chapter 5, analyses are presented related to two fractional numerical methods, Explicit Fractional Trapezoidal and Implicit Fractional Trapezoidal methods. Two results, based on Fourier series, related to the stability and convergence orders for both methods have been found. The third main result of this thesis, in Chapter 6, concerns the travelling waves phenomenon modeled on crowded environments. Here, we used the FRDPDEs developed in the earlier chapters to simulate FRDPDEs coupled with cubic or quadratic reactions. The results exhibit some interesting features related to molecular mobility. Later in this chapter, we have applied our methods to a biological problem known as Hirschsprung's disease. This model was introduced by Landman~\cite{landman07}. However, that model ignores the effects of spatial crowdedness in the system. Applying our model for modelling Hirschsprung's disease allows us to establish an interesting result for the mobility of the cellular processes under crowded environmental conditions.
| Identifer | oai:union.ndltd.org:ADTP/254249 |
| Creators | Farah Abdullah |
| Source Sets | Australiasian Digital Theses Program |
| Detected Language | English |
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