Centre for Advanced Computational Solutions

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Now showing 1 - 5 of 54
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    Novel numerical methods for stochastic ordinary and partial differential equations in modelling complex systems : A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy at Lincoln University
    (Lincoln University, 2023) Tiwari, Parul
    Many natural and engineered systems are complex due to inherent uncertainty. Stochastic Differential Equations (SDEs) and Stochastic Partial Differential equations (SPDEs) provide a rigorous mathematical foundation for modelling these systems. Understanding the dynamics of complex systems under stochastic influences is crucial for predicting system behaviour. Numerical techniques often struggle to handle the complexity and stochastic nature of these equations. This research focuses on adapting and enhancing numerical methods to provide efficient and reliable solutions. The numerical accuracy and stability of these methods are assessed through simulations and examples. This study introduces the synthesis of stochastic spectral methods to solve complex systems by representing random variables as a sum of orthogonal polynomials. We applied Polynomial Chaos Expansion (PCE) methods to contaminant transport problem and to differential equations with random forcing term. We compute the Wick exponentials and show that Wick product coincides with the ordinary product for deterministic functions. We use Malliavin calculus to find the derivatives of a stochastic quantity and are visualised through graphs. We discuss numerical challenges associated with the PCE methods and their solution strategies. In all examples, our chosen method does better and allows us to lead the way in developing robust and efficient strategies to deal with randomness, ultimately enhancing the reliability and resilience of complex systems across various scientific and engineering domains.
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    Computational techniques in mathematical modelling of biological switches
    (MSSANZ, 2015-12) Chong, KH; Samarasinghe, Sandhya; Kulasiri, Don; Zheng, J; Weber, T; McPhee, MJ; Anderssen, RS
    Mathematical models of biological switches have been proposed as a means to study the mechanism of decision making in biological systems. These conceptual models are abstract representations of the key components involved in the crucial cell fate decision underlying the biological system. In this paper, the methods of phase plane analysis and bifurcation analysis are explored and demonstrated using an example from the literature, namely the synthetic genetic circuit proposed by Gardner et al. (2000) which involved two negative loops (from two mutually inhibiting genes). Figure 1 shows a schematic diagram of the synthetic genetic circuit constructed by Gardner et al. (2000). Particularly, a saddle-node bifurcation is used as a signal response curve to capture the bistability of the system. The notion of bistability is obscure to most novice researchers or biologists because it is difficult to understand the existence of two stable steady states and how to flip from one stable steady state to another and vice versa. Thus, the main purpose of this paper is to unlock the computational techniques (bifurcation analysis implemented in a software tool called XPPAUT) in mathematical modelling of bistability through a simple example from Gardner et al. (2000). In addition, time course simulations are provided to illustrate: 1) the notion of bistability where the existence of two stable steady states and we demonstrated that for two different initial conditions one of the genes is ‘ON’ and the other gene is ‘OFF’; 2) hysteresis behaviour where the saddle-node bifurcation points as two critical points in which to turn ‘ON’ one gene happens at a larger parameter value than to turn ‘OFF’ this gene (at a lower parameter value). The hysteresis behaviour is important for irreversible decision made by cell to commit to turn ‘ON’. In conclusion, the understanding of the computational techniques in modelling biological switch is important for elucidating genetic switch that has potential for gene therapy and can provide explanation for experimental findings of bistable systems.
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    Modelling calmodulin dependent calcium signalling involved with synaptic plasticity : A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Lincoln University
    (Lincoln University, 2018) Stevens-Bullmore, Hamish Edward
    Neurotransmission of synapses is plastic in that they get modulated to increase or decrease conductivity (this is known as synaptic plasticity). Synaptic plasticity consists of two opposing forces: long term potentiation (LTP) which strengthens synapses; and long term depression (LTD) which weakens synapses. LTP and LTD are associated with memory formation and loss respectively. Synaptic plasticity is controlled at a molecular by Ca²⁺-mediated protein signalling. Here, Ca²⁺ binds the protein, calmodulin (CaM) which modulates synaptic plasticity in both directions. This is because Ca²⁺- bound CaM activates both LTD- and LTP- inducing proteins. Understanding how CaM responds to Ca²⁺ signalling, and how this translates into synaptic plasticity is therefore important to understand synaptic plasticity induction. In this thesis, CaM activation by Ca²⁺ and calmodulin binding to downstream proteins was mathematically modelled using differential equations. CaM was first modelled in isolation to determine its kinetic binding properties with Ca²⁺. By performing local and global sensitivity analyses of Ca²⁺ binding/unbinding parameters, distinct binding properties between the two Ca²⁺ binding lobes were found. The difference between the binding lobes was exacerbated as intracellular Ca²⁺ stimulation rose. A full model of the two opposing pathways of synaptic plasticity was then employed. Simulations were monitored, and global sensitivity analyses were performed to determine how Ca²⁺/CaM signalling occured at various Ca²⁺ signals. At elevated stimulations, the total CaM pool rapidly bound to its protein binding targets which regulate both LTP and LTD. This was followed by CaM getting redistributed from low affinity to high affinity binding targets. Specifically, CaM was redistributed away from LTD- inducing proteins to bind the high affinity LTP-inducing protein, calmodulin dependent kinase II (CaMKII). In this way, CaMKII acted as a dominant affecter and repressed activation of opposing CaM binding protein targets. This model thereby showed a novel form of CaM signalling by which the two opposing pathways can crosstalk indirectly. The model also investigated how cAMP is regulated by CaM. It was found that CaMKII can repress cAMP production by repressing CaM-regulated proteins which catalyse cAMP production. The model also found that at low Ca²⁺ stimulation levels typical of LTD- induction, CaM-signalling was unstable and is therefore unlikely to alone be sufficient to induce synaptic depression. Overall, this thesis showed how limiting levels of CaM may be a fundamental aspect of Ca²⁺ regulated signalling which allows crosstalk among proteins without requiring to interact directly. Understanding synaptic plasticity can help understand neurodegenerative disease and although the current study is focused on synaptic plasticity, understanding CaM regulation has implications in numerous other cell types.
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    Contouring and earthwork estimation for bordered strip irrigation
    (Lincoln College, University of Canterbury, 1977) Harrington, G. J.
    Computer programmes were developed for processing data from grid, direct, and random stadia field contouring systems. The three systems were evaluated for their use in providing contour plans for bordered strip irrigation design. A computer method of calculating the earthwork volumes associated with bordered strip irrigation was developed which uses terrain data from the above surveying methods or any other convenient source. This method was compared with land grading to form plane or warped paddock surfaces onto which levees may be formed, thus creating bordered strips. With the aid of the bordered strip earthwork calculating programme, the effect of changes of bordered strip paddock layout and slope restraints was investigated. An attempt to correlate estimated earthworks with earthmoving machine times was made.
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    A nutrient dependant switch explains mutually exclusive existence of meiosis and mitosis initiation in budding yeast
    (Elsevier, 2014-01-21) Wannige, C; Kulasiri, Don; Samarasinghe, Sandhya
    Nutrients from living environment are vital for the survival and growth of any organism. Budding yeast diploid cells decide to grow by mitosis type cell division or decide to create unique, stress resistant spores by meiosis type cell division depending on the available nutrient conditions. To gain a molecular systems level understanding of the nutrient dependant switching between meiosis and mitosis initiation in diploid cells of budding yeast, we develop a theoretical model based on ordinary differential equations (ODEs) including the mitosis initiator and its relations to budding yeast meiosis initiation network. Our model accurately and qualitatively predicts the experimentally revealed temporal variations of related proteins under different nutrient conditions as well as the diverse mutant studies related to meiosis and mitosis initiation. Using this model, we show how the meiosis and mitosis initiators form an all-or-none type bistable switch in response to available nutrient level (mainly nitrogen). The transitions to and from meiosis or mitosis initiation states occur via saddle node bifurcation. This bidirectional switch helps the optimal usage of available nutrients and explains the mutually exclusive existence of meiosis and mitosis pathways.