Mathematical Modelling of Pattern Formation in Yeast Biofilms

We use mathematical modelling and experiments to investigate yeast biofilm growth and pattern formation. Biofilms are sticky communities of cells and fluid residing on surfaces. As yeast biofilms are a leading cause of hospital-acquired infections, researchers have developed methods of growing them on semi-solid agar. These biofilms initially form a thin circular shape, before transitioning to a non-uniform floral morphology. To quantify biofilm growth, we use a radial statistic the measure expansion speed, and an angular pair correlation function to quantify petal formation. These spatial statistics enable comparison between experiments and mathematical model predictions. Our motivation is to improve understanding of the physical mechanisms governing biofilm formation. One hypothesised mechanism is nutrient-limited growth, in which movement and consumption of nutrients drives growth and generates patterns. Another hypothesis is that yeast biofilms expand by sliding motility, where cell proliferation and weak biofilm–substratum adhesion drive growth. Mathematical modelling enables us to investigate the contribution of each hypothesised mechanism to biofilm growth and pattern formation. We use a reaction–diffusion system with non-linear, degenerate cell diffusion to model nutrient-limited biofilm growth. This model admits sharp-fronted travelling wave solutions that advance with constant speed, an assumption consistent with experimental data. To investigate whether the reaction–diffusion model can explain petal formation, we consider the linear stability of planar travelling wave solutions to transverse perturbations. There is good agreement between the theory and experimental data, suggesting that nutrient-limited growth can explain floral pattern formation. Next, we introduce biofilm mechanics by deriving a two-phase fluid model. We treat the biofilm as a mixture of cells and an extracellular matrix, and obtain governing equations from mass and momentum conservation. Since yeast biofilm height is small compared to their radius, we use the thin-film approximation in two scaling regimes to simplify the model. The extensional flow regime involves weak biofilm–substratum adhesion, and models expansion by sliding motility. In contrast, the lubrication regime involves strong biofilm– substratum adhesion, and large pressure and surface tension. We compute axisymmetric numerical solutions to both thin-film models to investigate how mechanics affects biofilm growth. There is good agreement between the extensional flow model and experimental data, suggesting that sliding motility can explain expansion speed. Parameter sensitivity analyses show that increased nutrient supply and biomass production rates generate faster expansion. The effect of surface tension, which represents the strength of cell–cell adhesion, is the key difference between the two regimes. In the extensional flow model, surface tension inhibits ridge formation close to the leading edge, but does not affect expansion speed. In contrast, surface tension generates radial expansion in the lubrication regime. Since the thin-film models enable us to predict biofilm height and nutrient uptake explicitly, they provide a more detailed description of biofilm growth than the reaction–diffusion model. However, their complexity makes it more difficult to use linear stability analysis to investigate two-dimensional patterns. This problem, and alternative expansion mechanisms such as osmotic swelling and agar deformation, provide avenues for future work.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 201

Microchamber Cultures of Bladder Cancer: A Platform for Characterizing Drug Responsiveness and Resistance in PDX and Primary Cancer Cells.

Precision cancer medicine seeks to target the underlying genetic alterations of cancer; however, it has been challenging to use genetic profiles of individual patients in identifying the most appropriate anti-cancer drugs. This spurred the development of patient avatars; for example, patient-derived xenografts (PDXs) established in mice and used for drug exposure studies. However, PDXs are associated with high cost, long development time and low efficiency of engraftment. Herein we explored the use of microfluidic devices or microchambers as simple and low-cost means of maintaining bladder cancer cells over extended periods of times in order to study patterns of drug responsiveness and resistance. When placed into 75 µm tall microfluidic chambers, cancer cells grew as ellipsoids reaching millimeter-scale dimeters over the course of 30 days in culture. We cultured three PDX and three clinical patient specimens with 100% success rate. The turn-around time for a typical efficacy study using microchambers was less than 10 days. Importantly, PDX-derived ellipsoids in microchambers retained patterns of drug responsiveness and resistance observed in PDX mice and also exhibited in vivo-like heterogeneity of tumor responses. Overall, this study establishes microfluidic cultures of difficult-to-maintain primary cancer cells as a useful tool for precision cancer medicine

Survival, extinction, and interface stability in a two--phase moving boundary model of biological invasion

We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two populations: each population undergoes linear diffusion and logistic growth, and the boundary between the two populations evolves according to a two--phase Stefan condition. This mathematical model describes situations where one population invades into regions occupied by the other population, such as the spreading of a malignant tumour into surrounding tissues. Full time--dependent numerical solutions are obtained using a level--set numerical method. We use these numerical solutions to explore several properties of the model including: (i) survival and extinction of one population initially surrounded by the other; and (ii) linear stability of the moving front boundary in the context of a travelling wave solution subjected to transverse perturbations. Overall, we show that many features of the well--studied one--phase single population analogue of this model can be very different in the more realistic two--phase setting. These results are important because realistic examples of biological invasion involve interactions between multiple populations and so great care should be taken when extrapolating predictions from a one--phase single population model to cases for which multiple populations are present. Open source Julia--based software is available on GitHub to replicate all results in this study.Comment: 31 pages. 9 figure

Dissecting the regulatory strategies of NF-kB RelA target genes in the inflammatory response reveals differential transactivation logics

Nuclear factor κB (NF-κB) RelA is the potent transcriptional activator of inflammatory response genes. We stringently defined a list of direct RelA target genes by integrating physical (chromatin immunoprecipitation sequencing [ChIP-seq]) and functional (RNA sequencing [RNA-seq] in knockouts) datasets. We then dissected each gene’s regulatory strategy by testing RelA variants in a primary-cell genetic-complementation assay. All endogenous target genes require RelA to make DNA-base-specific contacts, and none are activatable by the DNA binding domain alone. However, endogenous target genes differ widely in how they employ the two transactivation domains. Through model-aided analysis of the dynamic time-course data, we reveal the gene-specific synergy and redundancy of TA1 and TA2. Given that post-translational modifications control TA1 activity and intrinsic affinity for coactivators determines TA2 activity, the differential TA logics suggests context-dependent versus context-independent control of endogenous RelA-target genes. Although some inflammatory initiators appear to require co-stimulatory TA1 activation, inflammatory resolvers are a part of the NF-κB RelA core response

Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity

Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u$, which causes the density to disperse. However, RDEs with negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a non-classical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are initially multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u=1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce an new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position.Comment: 35 pages, 10 figure