Thomas E. Woolley

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress–strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam’s razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.

Mathematical Modeling of Cortical Neurogenesis Reveals that the Founder Population does not Necessarily Scale with Neurogenic Output

Abstract The mammalian cerebral neocortex has a unique structure, composed of layers of different neuron types, interconnected in a stereotyped fashion. While the overall developmental program seems to be conserved, there are divergent developmental factors generating cortical diversity amongst species. In terms of cortical neuronal numbers, some of the determining factors are the size of the founder population, the duration of cortical neurogenesis, the proportion of different progenitor types, and the fine-tuned balance between self-renewing and differentiative divisions. We develop a mathematical model of neurogenesis that, accounting for these factors, aims at explaining the high diversity in neuronal numbers found across species. By framing our hypotheses in rigorous mathematical terms, we are able to identify paths of neurogenesis that match experimentally observed patterns in mouse, macaque and human. Additionally, we use our model to identify key parameters that would particularly benefit from accurate experimental investigation. We find that the timing of a switch in favor of symmetric neurogenic divisions produces the highest variation in cortical neuronal numbers. Surprisingly, assuming similar cell cycle lengths in primate progenitors, the increase in cortical neuronal numbers does not reflect a larger size of founder population, a prediction that has identified a specific need for experimental quantifications.

Graph-facilitated resonant mode counting in stochastic interaction networks

Oscillations in dynamical systems are widely reported in multiple branches of applied mathematics. Critically, even a non-oscillatory deterministic system can produce cyclic trajectories when it is in a low copy number, stochastic regime. Common methods of finding parameter ranges for stochastically driven resonances, such as direct calculation, are cumbersome for any but the smallest networks. In this paper, we provide a systematic framework to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations relying on real root counting algorithms and graph theoretic methods. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix J2, unlike deterministic oscillations which are determined by J. By using graph theoretic tools, analysis of stochastic behaviour for larger interaction networks is simplified and stochastic dynamical systems with multiple resonant modes can be identified easily.

Time to change your mind? Modelling transient properties of cortex formation highlights the importance of evolving cell division strategies.

The successful development of the mammalian cerebral neocortex is linked to numerous cognitive functions such as language, voluntary movement, and episodic memory. Neocortex development occurs when neural progenitor cells divide and produce neurons. Critically, although the progenitor cells are able to self-renew they do not reproduce themselves endlessly. Hence, to fully understand the development of the neocortex we are faced with the challenge of understanding temporal changes in cell division strategy. Our approach to modelling neuronal production uses non-autonomous ordinary differential equations and allows us to use a ternary coordinate system in order to define a strategy space, through which we can visualise evolving cell division strategies. Using this strategy space, we fit the known data and use approximate Bayesian computation to predict the founding progenitor population sizes, currently unavailable in the experimental literature. Counter-intuitively, we show that humans can generate a larger number of neurons than a macaques even when starting with a smaller number of progenitor cells. Accompanying the article is a self-contained piece of software, which provides the reader with immediate simulated results that will aid their intuition. The software can be found at www.dpag.ox.ac.uk/team/noemi-picco.

Changes in the retreatment radiation tolerance of the spinal cord with time after the initial treatment

Abstract Purpose: To estimate, from experimental data, the retreatment radiation ‘tolerances’ of the spinal cord at different times after initial treatment. Materials and methods: A model was developed to show the relationship between the biological effective doses (BEDs) for two separate courses of treatment with the BED of each course being expressed as a percentage of the designated ‘retreatment tolerance’ BED value, denoted and . The primate data of Ang et al. (2001) were used to determine the fitted parameters. However, based on rodent data, recovery was assumed to commence 70 days after the first course was complete, and with a non-linear relationship to the magnitude of the initial BED (BEDinit). Results: The model, taking into account the above processes, provides estimates of the retreatment tolerance dose after different times. Extrapolations from the experimental data can provide conservative estimates for the clinic, with a lower acceptable myelopathy incidence. Care must be taken to convert the predicted value into a formal BED value and then a practical dose fractionation schedule. Conclusions: Used with caution, the proposed model allows estimations of retreatment doses with elapsed times ranging from 70 days up to three years after the initial course of treatment.

PLCζ Induced Ca2+ Oscillations in Mouse Eggs Involve a Positive Feedback Cycle of Ca2+ Induced InsP3 Formation From Cytoplasmic PIP2

Egg activation at fertilization in mammalian eggs is caused by a series of transient increases in the cytosolic free Ca2+ concentration, referred to as Ca2+ oscillations. It is widely accepted that these Ca2+ oscillations are initiated by a sperm derived phospholipase C isoform, PLCζ that hydrolyses its substrate PIP2 to produce the Ca2+ releasing messenger InsP3. However, it is not clear whether PLCζ induced InsP3 formation is periodic or monotonic, and whether the PIP2 source for generating InsP3 from PLCζ is in the plasma membrane or the cytoplasm. In this study we have uncaged InsP3 at different points of the Ca2+ oscillation cycle to show that PLCζ causes Ca2+ oscillations by a mechanism which requires Ca2+ induced InsP3 formation. In contrast, incubation in Sr2+ media, which also induces Ca2+ oscillations in mouse eggs, sensitizes InsP3-induced Ca2+ release. We also show that the cytosolic level Ca2+ is a key factor in setting the frequency of Ca2+ oscillations since low concentrations of the Ca2+ pump inhibitor, thapsigargin, accelerates the frequency of PLCζ induced Ca2+ oscillations in eggs, even in Ca2+ free media. Given that Ca2+ induced InsP3 formation causes a rapid wave during each Ca2+ rise, we use a mathematical model to show that InsP3 generation, and hence PLCζs substate PIP2, has to be finely distributed throughout the egg cytoplasm. Evidence for PIP2 distribution in vesicles throughout the egg cytoplasm is provided with a rhodamine-peptide probe, PBP10. The apparent level of PIP2 in such vesicles could be reduced by incubating eggs in the drug propranolol which also reversibly inhibited PLCζ induced, but not Sr2+ induced, Ca2+ oscillations. These data suggest that the cytosolic Ca2+ level, rather than Ca2+ store content, is a key variable in setting the pace of PLCζ induced Ca2+ oscillations in eggs, and they imply that InsP3 oscillates in synchrony with Ca2+ oscillations. Furthermore, they support the hypothesis that PLCζ and sperm induced Ca2+ oscillations in eggs requires the hydrolysis of PIP2 from finely spaced cytoplasmic vesicles.

Predicting neuroblastoma using developmental signals and a logic-based model

Genomic information from human patient samples of pediatric neuroblastoma cancers and known outcomes have led to specific gene lists put forward as high risk for disease progression. However, the reliance on gene expression correlations rather than mechanistic insight has shown limited potential and suggests a critical need for molecular network models that better predict neuroblastoma progression. In this study, we construct and simulate a molecular network of developmental genes and downstream signals in a 6-gene input logic model that predicts a favorable/unfavorable outcome based on the outcome of the four cell states including cell differentiation, proliferation, apoptosis, and angiogenesis. We simulate the mis-expression of the tyrosine receptor kinases, trkA and trkB, two prognostic indicators of neuroblastoma, and find differences in the number and probability distribution of steady state outcomes. We validate the mechanistic model assumptions using RNAseq of the SHSY5Y human neuroblastoma cell line to define the input states and confirm the predicted outcome with antibody staining. Lastly, we apply input gene signatures from 77 published human patient samples and show that our model makes more accurate disease outcome predictions for early stage disease than any current neuroblastoma gene list. These findings highlight the predictive strength of a logic-based model based on developmental genes and offer a better understanding of the molecular network interactions during neuroblastoma disease progression.