Turing pattern formation on the sphere is robust to the removal of a hole.

The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace-Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability.

Occam's razor gets a new edge: the use of symmetries in model selection.

We demonstrate the power of using symmetries for model selection in the context of mechanistic modelling. We analyse two different models called the power law model (PLM) and the immunological model (IM) describing the increase in cancer risk with age, due to mutation accumulation or immunosenescence, respectively. The IM fits several cancer types better than the PLM implying that it would be selected based on minimizing residuals. However, recently a symmetry-based method for model selection has been developed, which has been successfully used in an in silico setting to find the correct model when traditional model fitting has failed. Here, we apply this method in a real-world setting to investigate the mechanisms of carcinogenesis. First, we derive distinct symmetry transformations of the two models and then we select the model which not only fits the original data but is also invariant under transformations by its symmetry. Contrary to the initial conclusion, we conclude that the PLM realistically describes the mechanism underlying the colon cancer dataset. These conclusions agree with experimental knowledge, and this work demonstrates how a model selection criterion based on biological properties can be implemented using symmetries.