Christopher N. Angstmann

Continuous Time Random Walks with Reactions Forcing and Trapping

Put abstract here.

Transmembrane Extension and Oligomerization of the CLIC1 Chloride Intracellular Channel Protein upon Membrane Interaction

Chloride intracellular channel proteins (CLICs) differ from most ion channels as they can exist in both soluble and integral membrane forms. The CLICs are expressed as soluble proteins but can reversibly autoinsert into the membrane to form active ion channels. For CLIC1, the interaction with the lipid bilayer is enhanced under oxidative conditions. At present, little evidence is available characterizing the structure of the putative oligomeric CLIC integral membrane form. Previously, fluorescence resonance energy transfer (FRET) was used to monitor and model the conformational transition within CLIC1 as it interacts with the membrane bilayer. These results revealed a large-scale unfolding between the C- and N-domains of CLIC1 as it interacts with the membrane. In the present study, FRET was used to probe lipid-induced structural changes arising in the vicinity of the putative transmembrane region of CLIC1 (residues 24-46) under oxidative conditions. Intramolecular FRET distances are consistent with the model in which the N-terminal domain inserts into the bilayer as an extended α-helix. Further, intermolecular FRET was performed between fluorescently labeled CLIC1 monomers within membranes. The intermolecular FRET shows that CLIC1 forms oligomers upon oxidation in the presence of the membranes. Fitting the data to symmetric oligomer models of the CLIC1 transmembrane form indicates that the structure is large and most consistent with a model comprising approximately six to eight subunits.

Turing Patterns from Dynamics of Early HIV Infection

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV.

A Fractional Order Recovery SIR Model from a Stochastic Process

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack–McKendrick age-structured SIR model, and it reduces to the Hethcote–Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

Pattern formation on networks with reactions: A continuous-time random-walk approach

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The non-trivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, whilst the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non \emph{ad-hoc} manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases; one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability.

Precision Isotope Shift Measurements in Calcium Ions Using Quantum Logic Detection Schemes.

We demonstrate an efficient high-precision optical spectroscopy technique for single trapped ions with nonclosed transitions. In a double-shelving technique, the absorption of a single photon is first amplified to several phonons of a normal motional mode shared with a cotrapped cooling ion of a different species, before being further amplified to thousands of fluorescence photons emitted by the cooling ion using the standard electron shelving technique. We employ this extension of the photon recoil spectroscopy technique to perform the first high precision absolute frequency measurement of the 2D(3/2)→2P(1/2) transition in calcium, resulting in a transition frequency of f=346 000 234 867(96)  kHz. Furthermore, we determine the isotope shift of this transition and the 2S(1/2)→2P(1/2) transition for 42Ca+, 44Ca+, and 48Ca+ ions relative to 40Ca+ with an accuracy below 100 kHz. Improved field and mass shift constants of these transitions as well as changes in mean square nuclear charge radii are extracted from this high resolution data.

Serum Levels of Human MIC-1/GDF15 Vary in a Diurnal Pattern, Do Not Display a Profile Suggestive of a Satiety Factor and Are Related to BMI

The TGF-b superfamily cytokine MIC-1/GDF15 circulates in the blood of healthy humans. Its levels rise substantially in cancer and other diseases and this may sometimes lead to development of an anorexia/cachexia syndrome. This is mediated by a direct action of MIC-1/GDF15 on feeding centres in the hypothalamus and brainstem. More recent studies in germline gene deleted mice also suggest that this cytokine may play a role in physiological regulation of energy homeostasis. To further characterize the role of MIC-1/GDF15 in physiological regulation of energy homeostasis in man, we have examined diurnal and food associated variation in serum levels and whether variation in circulating levels relate to BMI in human monozygotic twin pairs. We found that the within twin pair differences in serum MIC-1/GDF15 levels were significantly correlated with within twin pair differences in BMI, suggesting a role for MIC-1/GDF15 in the regulation of energy balance in man. MIC-1/GDF15 serum levels altered slightly in response to a meal, but comparison with variation its serum levels over a 24hour period suggested that these changes are likely to be due to bimodal diurnal variation which can alter serum MIC-1/GDF15 levels by about plus or minus 10% from the mesor. The lack of a rapid and substantial postprandial increase in MIC-1/GDF15 serum levels suggests that MIC1/GDF15 is unlikely to act as a satiety factor. Taken together, our findings suggest that MIC-1/GDF15 may be a physiological regulator of energy homeostasis in man, most probably due to actions on long-term regulation of energy homeostasis.

A fractional-order infectivity SIR model

Fractional-order SIR models have become increasingly popular in the literature in recent years, however unlike the standard SIR model, they often lack a derivation from an underlying stochastic process. Here we derive a fractional-order infectivity SIR model from a stochastic process that incorporates a time-since-infection dependence on the infectivity of individuals. The fractional derivative appears in the generalised master equations of a continuous time random walk through SIR compartments, with a power-law function in the infectivity. We show that this model can also be formulated as an infection-age structured Kermack–McKendrick integro-differential SIR model. Under the appropriate limit the fractional infectivity model reduces to the standard ordinary differential equation SIR model.

Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations

Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space- and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker--Planck equations in the diffusi...

A discrete time random walk model for anomalous diffusion

The continuous time random walk, introduced in the physics literature by Montroll and Weiss, has been widely used to model anomalous diffusion in external force fields. One of the features of this model is that the governing equations for the evolution of the probability density function, in the diffusion limit, can generally be simplified using fractional calculus. This has in turn led to intensive research efforts over the past decade to develop robust numerical methods for the governing equations, represented as fractional partial differential equations.Here we introduce a discrete time random walk that can also be used to model anomalous diffusion in an external force field. The governing evolution equations for the probability density function share the continuous time random walk diffusion limit. Thus the discrete time random walk provides a novel numerical method for solving anomalous diffusion equations in the diffusion limit, including the fractional Fokker-Planck equation. This method has the clear advantage that the discretisation of the diffusion limit equation, which is necessary for numerical analysis, is itself a well defined physical process. Some examples using the discrete time random walk to provide numerical solutions of the probability density function for anomalous subdiffusion, including forcing, are provided.