Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

Spontaneous emergence of cell polarity intrinsically lies at the localization of signaling molecules on a particular region of cell membrane. Such a process necessarily contains a positive feedback loop to amplify the localized cluster. To describe the polarizing process and explore different feedback functions involved, deterministic and stochastic models with non-local kinetics are discussed in this paper. Stochastic Simulation Algorithm (SSA) is used to numerically simulate the polarizing behavior and analytical analysis by the power spectrum is applied to approximate the parameter regime for the spontaneous emergence of cell polarity. Compared to the results from the deterministic model, we can understand how the stochastic effect extends the parameter regime for achieving cell polarization under different types of feedback, including the forms of quadratic function, linear function, and Hill function. Both deterministic and stochastic methods fail to yield the polarity at a low number of molecules. Our results suggest that the parameter region for cell polarization under the Hill function feedback is smaller than that with the quadratic function feedback. ©2021

Budding yeast, which undergoes polarized growth during budding and mating, has been a useful model system to study cell polarization. Bud sites are selected differently in haploid and diploid yeast cells: haploid cells bud in an axial manner, while diploid cells bud in a bipolar manner. While previous studies have been focused on the molecular details of the bud site selection and polarity establishment, not much is known about how different budding patterns give rise to different functions at the population level. In this paper, we develop a two-dimensional agent-based model to study budding yeast colonies with cell-type specific biological processes, such as budding, mating, mating type switch, consumption of nutrients, and cell death. The model demonstrates that the axial budding pattern enhances mating probability at an early stage and the bipolar budding pattern improves colony development under nutrient limitation. Our results suggest that the frequency of mating type switch might control the trade-off between diploidization and inbreeding. The effect of cellular aging is also studied through our model. Based on the simulations, colonies initiated by an aged haploid cell show declined mating probability at an early stage and recover as the rejuvenated offsprings become the majority. Colonies initiated with aged diploid cells do not show disadvantage in colony expansion possibly due to the fact that young cells contribute the most to colony expansion.

Randomness often plays an important role in the spatial and temporal dynamics of biological systems. General stochastic simulation methods may lead to excessive computational cost for a system in which a large number of molecules involved. Therefore, multi-scale hybrid simulation methods become important for stochastic simulations. Here we build a spatially hybrid method which couples two approaches: discrete stochastic simulation and continuous stochastic differential equations. In our method, the locations of the interfaces between the two approaches are changing according to the distribution of molecules in a one-dimensional domain. To balance the accuracy and efficiency, the time step of the numerical method for the continuous stochastic differential equations is adapted to the dynamics of the molecules near the adaptive interfaces. The simulation results for a linear system and two nonlinear biological systems in different one-dimensional domains demonstrate the effectiveness and advantage of our new hybrid method with the adaptive time step control.