Interface methods for biological and biomedical problems

This special issue contains eight papers on interface methods for biological and biomedical problems. Interface problems arise frequently in science and engineering because of the spatial separation of different material properties between regions or because of a force exerted on the materials of differential physical, chemical, and/or biological properties. Theoretical modeling of interface problems often involves partial differential equations (PDEs) with discontinuous coefficients and singular sources. Mathematical techniques, such as the immersed boundary method, the immersed interface method, and the match interface and boundary method, have emerged in the last few decades as suitable candidates for interface modeling and computations. Some of the recent technical development emphasizes high-order schemes, geometric complexity, stability analysis, geometric singularities, and level set approaches for interface tracking. Indeed, interface techniques have been instrumental to a wide range of advances in real-world biological and biomedical problems. The focus of this special issue is on recent advances in computational methods and theoretical models for interface problems and their applications. A broad range of topics are addressed, including the theoretical, analytical, and computational aspects of interface problems and related biomolecular surfaces. The work of Zheng, Yang, and Wei [1] offers a new framework for the biomolecular surface generation based on the PDE transform. Biomolecular surfaces serve as interfaces in elliptic interface problems, such as those in the Poisson–Boltzmann model for electrostatic analysis and the Poisson–Nernst–Planck model for charge permeation. The PDE transform has been shown to be efficient, stable, and robust for the surface modeling of large proteins and virus molecules. The article by Griffith [2] presents the application of an adaptive, staggered-grid version of the immersed boundary method to the three-dimensional simulation of the fluid dynamics of the aortic heart valve. The paper by Zhao [3] provides an improvement to the differential-geometry-based multiscale solvation model for biomolecular surface representation and solvation analysis originally introduced by Wei. A time-dependent Poisson–Boltzmann equation is proposed to make it easier to deal with the nonlinear term in the original Poisson–Boltzmann equation, which is coupled to the Laplace–Beltrami equation for the surface modeling. A filtering process is employed to stabilize the time integration. The new model is validated by biological properties of macromolecules. Sohn, Li, Li, and Lowengrub [4] present an energy variation approach to investigate the role of hydrodynamic forces on three-dimensional axisymmetric multicomponent vesicles. A set of governing equations for the system is derived by the energy variation. Numerical methods are developed to solve governing PDEs, including fourth-order nonlinear and nonlocal ones. Research described in this article may shed light on the role of vesicles in biological processes. The study by Bouzarta, Layton, and Young [5] simulates the dynamics of an approximately inextensible flexible fiber immersed in a two-dimensional cellular background flow. The authors model the system as an immersed boundary problem, in which the fluid dynamics is described using the Stokes equations, and they compute the motion of the immersed fiber using the method of regularized Stokeslets. Cogan [6] uses a mathematical model of bacterial biofilm to investigate the effect of relatively large-scale variations in the spatial domain on the susceptibility of the biofilm. The model is numerically solved using a hybrid boundary integral method where boundary conditions are