Mini-Project I: Intercellular Calcium Wave Propagation
Supervisor: Yulia Timofeeva
Abstract: Calcium is crucial component in a plethora of cellular processes from differentiation to even cell death. Calcium oscillations within the cytosol represent the most widespread oscillatory behaviour at the cellular level, acting in cases as a communication mechanism within and between cells. Here we study intercellular propagation of these calcium oscillations using the well-established Fire-Diuse-Fire (FDF) model, as a means of mimicking the biological processes within the cell that induce these calcium waves, namely, Calcium Induced Calcium Release (CICR). Transforming the model into Laplace space, we uncover an elegant solution to the FDF model, within the framework of a ’sum over trips’ approach. This exemplifies the rules governing propagation of the solution between cells, leading to high-fidelity approximation to the global solution, backed up with numerical simulations. Furthermore, this approximation can be used within a high degree of accuracy to establish the ’speed’ of the calcium wave and parameter dependent conditions on wave propagation or failure.
Mini-Project II: Cluster-Cluster Aggregation with Lévy Diffusion
Supervisor: Colm Connaughton
Abstract: Here we focus specifically on cluster-cluster aggregation with a monomer source and binary aggregation. The standard approach to deriving scaling properties of non-equilibrium problems of this type is to make explicit assumptions about whether the system inhabits a diffusion or reaction limited regime. While the reaction-limited case has been extensively explored using the mean-field description that classical Smoluchowski equation provides, the diffusion-limited case remains relatively unstudied by comparison. We show that the methods and insights an effective statistical field theory provides transcends this problem. In particular, we demonstrate that the flux-carrying correlation function exhibits scaling that is independent of both the spatial transport mechanism and the particular regime the system inhabits. Since the scaling of the mass spectrum does not share in this independence, this property is highly counter-intuitive. We also provide the machinery which allows us to relate the theoretical properties that this field theory offers with the physical quantities we can measure in numerical simulation. Using Levy diffusion to manually control the spatial transport mechanism, we then managed to numerically test these scaling predictions via Monte Carlo simulation across the transition between diffusion and reaction limited regimes.
PhD Project Outline: Waves and Turbulence in Models of Geophysical Flows
Supervisor: Colm Connaughton
Co-supervisor: Miguel Bustamante
- To understand the role played by resonant wave triads in geophysical turbulence, particularly atmospheric waves.
- To identify the signature of resonant wave interactions in noisy atmospheric data.
- To obtain a theoretical description of atmospheric turbulent cascades that includes strongly nonlinear waves.
Work in progress: Integrability of the Forced Triad
Co-supervisor: Miguel Bustamante We know that the resonant triad is completely integrable: an explicit solution can be found in terms of Jacobi elliptic functions. However, what happens when we add a forcing term to the unstable mode, one which corresponds to the one of the waves in the cluster being forced by topology. Does the resulting motion remain integrable, and if so, what can we say about boundedness and periodicity.
Work in progress: A Fast and Scaleable Algorithm for Neuronal Cable Theory Problems
This is a collaborative effort with Quentin Caudron, also a member of the Complexity DTC. Using the current 'sum-over-trips' framework, solving the dynamics of spatially-extended neurons for physically-relevant geometries proves to be both impractical to implement and computationally intensive. This is based on the vast quantity of paths that must be sampled in order to guarantee convergence. We are developing an algorithm that bypasses this problem, making simulation of physically realistic spatially-extended neurons viable.
Scaling properties of one-dimensional cluster-cluster agregation with Lévy diffusion
Journal: Journal of Statistical Mechanics: Theory and Experiment, Volume 2010, May 2010.
Authors: C Connaughton and J Harris
Intercellular calcium waves in the fire-diffuse-fire framework: Green's function for gap-junctional coupling
Journal: Physical Review E 82, 051910 (2010)
Authors: J Harris and Y Timofeeva