Slender elastic filaments with intrinsic helical geometry are encountered in a wide range of physical and biological settings, ranging from coil springs in engineering to bacteria flagellar filaments. The equilibrium configurations of helical filaments under a variety of loading types have been well studied in the framework of the Kirchhoff rod equations. These equations are geometrically nonlinear and so can account for large, global displacements of the rod. This geometric nonlinearity also makes a mathematical analysis of the rod equations extremely difficult, so that much is still unknown about the dynamic behaviour of helical rods under external loading.

An important class of simplified models consists of 'equivalent-column' theories. These model the helical filament as a naturally-straight beam (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Such theories have long been used in engineering to describe the free vibrations of helical coil springs, though their validity remains unclear, particularly when distributed forces and moments are present. In this talk, we show how such an effective theory can be derived systematically from the Kirchhoff rod equations using the method of multiple scales. Importantly, our analysis is asymptotically exact in the small-wavelength limit and can account for large, unsteady displacements. We then illustrate our theory with two loading scenarios: (i) a heavy helical rod deforming under its own weight; and (ii) axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, as well as yielding analytical insight into their tensile instabilities.

Advances in spatial profiling technologies are providing insights into how molecular programs are influenced by local signalling and environmental cues. However, cell fate specification and tissue patterning involve the interplay of biochemical and mechanical feedback. Here, we propose a new computational framework that enables the joint statistical analysis of transcriptional and mechanical signals in the context of spatial transcriptomics. To illustrate the application and utility of the approach, we use spatial transcriptomics data from the developing mouse embryo to infer the forces acting on individual cells, and use these results to identify mechanical, morphometric, and gene expression signatures that are predictive of tissue compartment boundaries. In addition, we use geoadditive structural equation modelling to identify gene modules that predict the mechanical behaviour of cells in an unbiased manner. This computational framework is easily generalized to other spatial profiling contexts, providing a generic scheme for exploring the interplay of biomolecular and mechanical cues in tissues.

Underlying many biological models are chemical reaction networks (CRNs), and identifying allowed and forbidden dynamics in reaction networks may 
give insight into biological mechanisms. Algebraic approaches have been important in several recent developments. For example, computational 
algebra has helped us characterise all small mass action CRNs admitting certain bifurcations; allowed us to find interesting and surprising 
examples and counterexamples; and suggested a number of conjectures.   Progress generally involves an interaction between analysis and 
computation: on the one hand, theorems which recast apparently difficult questions about dynamics as (relatively tractable) algebraic problems; 
and on the other, computations which provide insight into deeper theoretical questions. I'll present some results, examples, and open 
questions, focussing on two domains of CRN theory: the study of local bifurcations, and the study of multistationarity.