"Sensing some resistance: A mathematical model for the contractile mechanosensory mechanism within cells"
It is becoming increasingly clear that physical force and the mechanical properties of their microenvironment play a crucial role in determining cellular behaviour and coordination. Un- derstanding these differences has significant implications for tissue engineering applications and to determine how the mechanical microenvironment may affect, for example, cancer growth and invasion. We use a continuum elasticity-based model with an active contractile component to describe the mechanosensory mechanism of a cell or cell layer adhered to a substrate. The model context focuses on the most common biophysical experimental set-ups investigating cel- lular contractility, which infer the cell-generated force from observed deformations of substrates with experimentally known mechanical properties. The mathematical model is analysed and solved using both analytical approaches (exploiting approximations and symmetry arguments) and Finite Element Methods. We use the model to explain observed cellular adaptations to changes in the mechanical properties of the underlying gel. In particular, we consider the dis- tribution of adhesion throughout a cell. For experimentally realistic distributions of adhesion points, the model is capable of recreating cell shapes and deformations that are consistent with those experimentally observed. Furthermore, energy considerations are shown to have significant implications for the optimisation of cell adhesion. Thus, we demonstrate the neces- sity of considering the whole cell-substrate system, including the patterning of adhesion, when investigating cell stiffness sensing.
Ashlee N. Ford Versypt
"Investigation of Short Chain Fatty Acids on the Gut-Bone Axis: From Mechanism to a Computational Systems Approach"
Short chain fatty acids produced by the microbial community in the gut play a role in regulating the immune system. Butyrate is a specific short chain fatty acid that can be modulated through diet. In this collaborative project, we seek to formulate a series of ordinary differential equations to provide a mechanistic description of the dynamic effects of butyrate on cells of the immune system locally in the intestine and on bone health through systemic consequences. The team involves three mathematical biologists with chemical engineering backgrounds and an experimental collaborator with expertise in the physiological interactions between the digestive, immune, and musculoskeletal systems.
"Mathematical Modeling of Human Erythropoiesis"
As red blood cells (RBCs), also called erythrocytes, are known for their vital function of transporting oxygen to tissues, their production rate unsurprisingly depends on the tissues need for oxygen (hypoxia). The hormone erythropoietin (Epo) is the main factor regulating erythropoiesis, the process by which erythrocytes are produced. In adult humans, Epo is produced principally in the kidney and released in response to hypoxia in order to stimulate the production of more erythrocytes. Patients suffering from severe kidney impairment therefore under-produce RBCs and subsequently suffer from severe anemia. Exogenous administration of synthetic Epo is required, but what is the optimal dose, and how frequently should it be given? To even begin to address such questions, and to understand the underlying process requires a mathematical model. We develop a model consisting of two delay differential equations that reflects the effect of Epo on increasing the production rate of erythrocytes by preventing the apoptosis of its precursor cells. The explicit modelling of the Epo concentration will allow for later use of the model in the case of exogenous dosing of Epo. We analyse our proposed model, and as expected, are able to prove the non-negativity of solutions and the existence of a unique steady-state solution for the system which represents the normal number of mature erythrocytes and the concentration of Epo for a healthy human. We find a simple condition on the other parameters under which the steady state is locally asymptotically stable for arbitrary large delays. When this parameter condition is violated we show that the steady state is stable for small delays and loses its stability through a Hopf bifurcation.
"Möbius invariances in biology"
We will discuss several examples where using the extension of our rigid transformations to the class of Möbius mappings will gain insights into both the morphogenesis and dynamic use of anatomical features in fungi, mammals and especially primates. The examples will range from 'Growth and Form' to data augmentation in AI.
"Examining the mechanical forces driving vascular regression using a fluid-structure-growth model"
Vascular regression is a critical process concluding the maturation of developing capillary networks, in which redundant blood vessels are removed. Recent research suggests that forces from the local blood flow (haemodynamic forces) trigger polarized endothelial cell migration against the flow, resulting in capillary collapse and regression. However, vascular regression is also driven by several additional pathways including local adhesion forces and cellular signalling factors. Due to the delicate nature of these microvessels, it is difficult to experimentally untangle the roles of each pathway during vascular development. As such, the development of computational models to analyse the relationship between the local haemodynamic forces and the surrounding vasculature during regression are invaluable. In this talk, we will present a novel computational framework to mathematically study and isolate the role of haemodynamic and adhesive forces in vessel deformation and endothelial cell migration during vascular regression. To model regression, we describe the capillary wall as a discretised hyperelastic membrane, hosting a multicellular model of endothelial cells. The capillary wall and the endothelial cells interact with and respond to the local blood flow in an iterative manner, creating a coupled fluid-structure growth simulation. This discrete approach provides a natural framework to consider the relationship between the capillary wall and the local blood flow, and allows for the easy inclusion of structural heterogeneities across the capillary wall. Using this model we are able to examine the relative roles of the haemodynamic forces and the local adhesion forces in vascular regression, and the network level ramifications of local regression.