Modeling of cellular mechanics

Mechanical interactions between biological cells are increasingly understood to be important in normal tissue function, and have been shown to play a critical role in diseases such as cancer. This has led to a renewed effort to understand the mechanics of cells and their interactions with their local environment. Since 2010, I have been involved in the Physical Sciences in Oncology Center (PS-OC) program at UC Berkeley, LBL, and UC San Francisco, which is part of a pilot initiative from the National Cancer Institute to encourage more interaction between the physical and mathematical sciences and cancer research. This has provided an ideal environment to carry out interdisciplinary science, and I have worked on several collaborative projects with biologists, bioengineers, and physicists.

References

  1. B. D. Hoffman, J. C. Crocker, Cell mechanics: dissecting the physical responses of cells to force, Annu. Rev. Biomed. Eng. 11 (2009) 259–288. [Link].
  2. M. J. Paszek, N. Zahir, K. R. Johnson, J. N. Lakins, G. I. Rozenberg, A. Gefen, C. A. Reinhart-King, S. S. Margulies, M. Dembo, D. Boettiger, D. A. Hammer, and V. M. Weaver, Tensional homeostasis and the malignant phenotype, Cancer Cell 8 (2005) 241–254. [Link]

Experimental collaborations

I am currently working on several collaborations with experimental groups in the Bay Area PS-OC, where computational modeling can be used to understand cells and their environment. The studies have focused on mammary acini, which are groups of around one hundred cells that form hollow spherical structures in the human breast, and are responsible for milk production. They are implicated in a significant fraction of breast cancers and are therefore of great interest to oncologists.

In one study, I have worked with researchers in the labs of Dan Fletcher, Hana El-Samad, and Valerie Weaver to examine changes in stiffness in acini as they progress towards malignancy; computations can be used to separate out the effects of changes in mechanical properties and changes in acinus geometry. In a separate collaboration with the group of Jan Liphardt, I have been using reference map simulations to understand the interactions of acini and the extracellular environment.

References

  1. G. Venugopalan, D. Camarillo, K. D. Webster, C. D. Reber, H. El-Samad, J. A. Sethian, V. M. Weaver, D. A. Fletcher, and C. H. Rycroft, Multicellular Architecture of Malignant Breast Epithelia Influences Mechanics PLOS ONE 9, e101955 (2014). [Link]
  2. Q. Shi, R. P. Ghosh, H. Engelke, C. H. Rycroft, L. Cassereau, J. Sethian, V. M. Weaver, J. Liphardt, Rapid disorganization of mechanically interacting systems of mammary acini, Proc. Natl. Acad. Sci. 111, 658–663 (2014). [Link]

Mechanical interactions between cells and their environment

As a parallel effort to the experimental collaborations, I have also designed new numerical algorithms for studying how cells interact mechanically, working with James Sethian. On a large scale, it is possible to model cells using a continuum, as moving spheres or ellipsoids, or by using a cellular automata simulation. However, in any problem on a smaller scale, a simulation must necessarily have a more detailed description of the the shapes of the cell boundaries and how they deform, but currently the techniques available are limited. Some two-dimensional models have represented the boundaries as points and lines, and used these to answer questions in cell morphology and acinus formation. However, generalizing these models to three dimensions is difficult, since it would require representing the cell boundary using a mesh, which would have to be reconfigured if the cell undergoes significant deformation, such as during division.

To address this, I have developed an Eulerian framework for tracking multiple deforming interfaces in three dimensions. An example that uses this framework is shown below. Initially a single cell is introduced. The cell is embedded within fluid evolving under the Navier–Stokes equations, and a surface tension force is applied at the cell membrane. The volume of each cell increases over time, and the cell undergoes division above a certain threshold. By proceeding in this manner, it is possible to construct a representation of a multicellular cluster as shown in the rightmost image.

While these pictures may bear some resemblance to existing cell modeling techniques (such as the ellipsoid representation) they are distinctly different from what is currently available, in that the computational method individually resolves the three-dimensional shapes of each cell within the cluster.

References

  1. K. A. Rejniak and A. R. A. Anderson, A computational study of the development of epithelial acini: I. Sufficient conditions for the formation of a hollow structure, Bull. Math. Biol. 70 (2007) 677–712. [Link]
  2. E. Palsson, A three-dimensional model of cell movement in multicellular systems, Future Generation Computer Systems 17, 835–852 (2001). [Link]

Basement membrane interactions

The above framework can be used to answer questions about how cells interact with each other and their environment. Typically, mammary acini are surrounded by a layer of extracellular matrix referred to as the basement membrane, which provides mechanical stability. One of the hallmarks of the progression of breast cancer occurs when malignant cells are able to break through the membrane, due to a combination of chemical degradation and mechanical breakage, and begin invading the surrounding tissue. To examine this, a pre-malignant mammary acinus is initialized as a cluster of cells, and surrounded by a basement membrane that is modeled as a discrete elastic mesh. Here, a scenario is considered in which a single malignant cell within the cluster begins to grow and divide. In this simulation, the membrane is assigned a mechanical toughness, and eventually the stress exerted on it by the cells causes it to rupture. While this simulation involves a highly simplified picture of the mechanics, the underlying mathematical tools could be extended to incorporate a wide variety of other mechanical effects, such as cell–cell adhesion and cell motility. Given advances in experimental techniques for making measurements of three-dimensional biological systems, I believe that modeling techniques of this form will become increasingly effective.