Continuum Modeling: Finite Elements & Meshing

1.4 Continuum Modeling: Finite Elements & Meshing

Despite tremendous research in cell structural and computational biophysics, most existing approaches are phenomenological rather than micro-structurally motivated [16, 17], and the driving forces for non-equilibrium processes such as growth and remodeling remain poorly understood. Although current simulations might be able to reproduce a single specific experimental result, they fail to reliably predict the biophysical response in general. To address these fundamental shortcomings, we will develop physics based simulation tools that allow us to create a close iterative loop between experiment, modeling, and simulation and will yield fundamentally new insights into the dynamic events occurring on the bacterial cell surface associated with bacterial infection and antibiotic resistance. The Liszt DSL described in Section 1.1 is our start at a Mesh-based DSL, and we will extend it with the computational algorithms created in this section.

1.4.1 Create a computational modeling paradigm for cell surfaces

The cell surface of Gram-negative bacterial cells is composed of three thin layers, the inner membrane, the cell wall, and the outer membrane, interconnected by macromolecular protein complexes. From a computational point of view, the cell surface can be viewed as a set of two-dimensional topological manifolds embedded in the three-dimensional space. We will create methods for smooth reconstruction of free form surfaces from live-cell image data, which will define the finite element mesh. The surface meshing will be linked to edge detection algorithms that will be developed as a quantitative analysis tool for live-cell imaging as part of the mesoscale DBP. From a mechanics point of view, the cell surface can be modeled as a combination of a thin membrane with a resistance to normal forces and as a thin-walled shell with a resistance to normal, shear, and bending forces. We will model the extremely thin nature of the three surface layers that are crosslinked with one another at discrete points of singularity. We will appropriately characterize membrane fluidity to incorporate the different mobility of proteins in each of the layers, calibrating as described below.

1.4.2 Design a finite-element-based toolbox for cell surface modeling

The major challenge of the finite-element-based model is the up-scaling of microscale protein properties to mesoscale cell level parameters to bridge the scales from discrete to continuous. We will include in our computational toolbox multiscale finite element algorithms that allow us to systematically probe the impact of microscale sub-cellular parameters such as mobility, stiffness, or compliance on the overall mesoscale response of the cell. For example, we will explore the influence of different lateral protein mobilities on surface protein organization. We will model the constrained movement of proteins on restricted manifolds such as shells or membranes, whereby different proteins may exhibit different mobility. As another example, we will use the theory of finite kinematic growth, originally developed for entire tissues and organs, to develop rate equations for cell growth and cell remodeling. A crucial part of this theory that has not been addressed satisfactorily is the choice of appropriate evolution equations for the underlying growth tensor, i.e., the kinematic metric that characterizes the deposition of biomaterial and the related growth. Unlike existing approaches, we will closely tie these mesoscale evolution equations to the microscale model of de novo polarization of key proteins for viral infection that are probed experimentally in DBP 2. This will allow us to precisely predict heterogeneous growth of the bacterial cell wall in terms of both location and growth rate.

1.4.3 Calibrate the cell surface model using live cell imaging data.

We will calibrate our finite element based computational toolbox by means of large-scale quantitative video microscopy-based live-cell image data. For example, we will use high resolution, real-time tracking of outer membrane protein movements. We will design novel high-throughput parameter identification tools capable of using large scale data sets, for example from high resolution, real-time tracking of outer membrane protein movements. This will allow us to calibrate and validate our mesoscale cell model parameters. A typical example is membrane diffusivity which will be calibrated using the apparent diffusion coefficients of tagged single proteins on the bacterial surface. The identification of material parameters is a non-convex optimization problem. We will identify an efficient and robust protocol able to handle multiple local minima. After appropriate calibration and validation, the physics-based simulations of cell wall diffusion and growth will complement the experimental findings and allow us to study and understand surface protein motion and its biophysical origins.

Figure 1.5. Physics-based simulation of human embryonic stem cell-derived cardiomyocytes. In vitro grown cells (left) and in silico simulated cells (right). The cytoskeletal elements are modeled as discrete structural elements with nuclei stained in blue, actin filaments in green, and integrins in red.

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