• Computational Modelling Folate Metabolism and DNA Methylation: Implications for Understanding Health and Ageing

      Mc Auley, Mark T.; Mooney, Kathleen M.; Salcedo-Sora, J. Enrique; University of Chester; Edge Hill University; Liverpool Hope University (Oxford University Press, 2016-12-21)
      Dietary folates have a key role to play in health as deficiencies in the intake of these B vitamins have been implicated in a wide variety of clinical conditions. The reason for this is folates function as single carbon donors in the synthesis of methionine and nucleotides. Moreover, folates have a vital role to play in the epigenetics of mammalian cells by supplying methyl groups for DNA methylation reactions. Intriguingly, a growing body of experimental evidence suggests DNA methylation status could be a central modulator of the ageing process. This has important health implications because the methylation status of the human genome could be used to infer age-related disease risk. Thus, it is imperative we further our understanding of the processes which underpin DNA methylation and how these intersect with folate metabolism and ageing. The biochemical and molecular mechanisms which underpin these processes are complex. However, computational modelling offers an ideal framework for handling this complexity. A number of computational models have been assembled over the years, but to date no model has represented the full scope of the interaction between the folate cycle and the reactions which govern the DNA methylation cycle. In this review we will discuss several of the models which have been developed to represent these systems. In addition we will present a rationale for developing a combined model of folate metabolism and the DNA methylation cycle.
    • A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations

      Karakatsani, Fotini; University of Chester (Oxford University Press, 2015-07-22)
      We derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractional-step #-scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time.
    • Space-Time Discontinuous Galerkin Methods for the '\eps'-dependent Stochastic Allen-Cahn Equation with mild noise

      Antonopoulou, Dimitra; Department of Mathematics, University of Chester, UK (Oxford University Press, 2019-04-08)
      We consider the $\eps$-dependent stochastic Allen-Cahn equation with mild space- time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with space-time finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piece-wise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piece-wise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in 2+1-dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any Runge-Kutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.