RG Application -- Mathematics
Title of Proposed Project: Mathematical Modeling of the Mechanics of Cell Movements
Abstract:
The movement of individual cells plays an important role in many biological functions.
The processes involved in cellular motion are very complex, and mathematical modeling
is becoming increasingly vital in furthering the understanding on how they are integrated.
In this project I plan on modifying a previously developed mathematical model of the
mechanical processes involved in cell movement to include more detail on the
interactions that take place between a cell and a substrate on which it is crawling. The
project will involve investigating background literature, reformulating the governing
equations that I previously developed, and redesigning computer code that I have
previously written. Through this grant I am requesting one course release to complete the
work associated with this project and to write a manuscript on the mathematical model
and computational results.
NARRATIVE:
BACKGROUND, CONTEXT, AND GOALS : Cell crawling, which in literature is
more commonly referred to as cell motility, in multicellular organisms plays an essential
role in various processes including embryonic development, immune response, wound
healing, cancerous tumor growth, and the spread of cancer from a primary tumor to other
organs [1]. The mechanism of cell movement is very complex, and it involves tight
spatio-temporal regulation and integration of a number of sub-processes including
transduction of a chemical or mechanical signal from the environment into intracellular
biochemical signals and the transformation of these signals into a mechanical response.
Experiments designed to investigate various features of the phenomena typically focus on
single aspects of cell-motility. Mathematical modeling, on the other hand, affords a
significantly greater freedom (and speed) in exploring interactions between various subprocesses and can provide valuable guidance as to what kind of experiments should be
performed, and as a result, is becoming increasingly important in biological research.
While cell motility is instigated by biochemical signals, movement is inherently a
mechanical process, and my work on cell motility has thus far focused on modeling
mechanics involved in cell motility. The preliminary model that I have developed is
based on a modification to the system of partial differential equations that is frequently
used to compute stresses and displacements in solids. The basic premise of the model is
that a cell can deform itself, (unlike passive materials that only deform when an external
force is exerted upon them) and through these deformations is able to move. This
premise is incorporated into the equations describing the process of cell movement, and
the equations are solved numerically using the finite element method within a computer
code that I have written and by using Comsol Multiphysics [3], a finite element software
package. A paper describing this model and preliminary numerical results has been
submitted [12].
The forces that develop by the cell in order for it to move can, to some degree, be
determined experimentally by placing motile cells on a deformable substrate with known
mechanical properties. As the cell moves, experimentalists can measure the deformations
of the substrate and calculate the force that a cell exerts onto the substrate at the cellsubstrate interface [2, 6, 7, 8, 10, 11]. As an initial application and test of the
mathematical model I have developed, in [12], the model is used to computationally
investigate the forces that arise in cell-substrate interaction. Given a simplified criteria
for regions at which the cell is attached to the substrate, we show that the forces at the
cell-substrate interface that are computed by our model for a single set of attachments
correspond, both qualitatively and quantitatively, to the experimental results [5, 8].
Furthermore, we are able to compute the entire profile of forces that develops inside the
cell as it moves, which cannot be done experimentally.
While the mathematical model with the above attachment criteria produced results
resembling those seen in experiments, it is highly over-simplified. First, the preliminary
model does not incorporate the process of releasing old attachments and making new
attachments, which must take place in order for the cell to ‘crawl’ a significant distance
[6]. Second, the current model assumes that at the point of attachment the cell cannot
move relative to the substrate. In reality cells can make strong attachments with the
substrate as well as weaker attachments at which the cell can slightly move relative to the
substrate [6]. Finally, forces can develop in the regions where the cell and substrate are
in contact but not attached, and this is also not incorporated into the current model.
In order to gain a better understanding of how the cell-substrate attachments affect
cell motility, the above aspects must be incorporated into the mathematical model. While
the mechanical issues of cell-substrate contact have not been mathematically addressed,
various models of contact between two surfaces have been formulated in the context of
industrial processes. My goal for this project is to adopt the mathematical principles of
the mechanics of contacts [4], introduce them with specifications appropriate for cell
motility to the preliminary model that I have formulated, and to use the new model to
computationally investigate cell-substrate interaction. This goal is part of a long-term
goal to develop a computational model that will allow both biologists and applied
mathematicians to investigate how modifications to its mechanical and/or biochemical
aspects affect the entire process of cell motility.
PLAN OF WORK: The preliminary work that must be accomplished to reach the goals
of this project is an investigation of the mathematical models that have been used in
contact problems. Once this is complete, these contact models need to be reformulated to
account for unique aspects of cell motility. Specifically, the contact boundary conditions
of the moving cell must (i) account for friction between the bottom of the cell and the top
of the substrate (ii) be formulated so as to allow for the presence of stronger and weaker
contact zones, (iii) account for the forces that may develop when two surfaces in contact
attempt to penetrate one another, and (iv) allow for regions of contact to change with time
and space. Once the model is reformulated to account for these conditions the existing
computer code must also be redesigned so that computational experiments can take place.
These experiments include investigating how the strength, size, and location of adhesions
affect the speed of a motile cell [9]. By the end of Spring of 2010, when I am requesting
a course release through this Research Grant, I would like to have concluded running
numerical experiments and written a journal paper. The specific details of when each
component of this work will take place are included in the Timetable.
PROJECT'S VALUE: The results of this project will be valuable to multiple academic
communities. For biologists, the computational investigation of cell-substrate interaction
provides insight into the mechanical aspects of cell motility. It also gives guidance to
experimentalists as to the nature and design of the experiments aiming at verification of
specific research hypotheses. For applied mathematicians, applying the principles of
contact mechanics in combination with temporally and spatially varying contact
boundary conditions provides a novel modeling approach that can be applied to other
systems.
While the initial computational formulation of the proposed model will involve
writing my own computer code, I plan to also incorporate the model into Comsol
Multiphysics, a software package that can be used by undergraduate students. This will
provide an opportunity to engage undergraduate students in the research in which I am
involved. I have already worked with undergraduate researchers on some aspects of
modeling biochemical reactions through the CAM Summer Research Program, and the
use of Comsol will allow for more student involvement in the computational
implementation of a model of the biochemistry to the proposed model of cell mechanics,
which is another component of my long-term research plans.
EVALUATION AND DISSEMINATION: My goal is that a peer-reviewed publication
in the Bulletin of Mathematical Biology will result from this project. In addition, I would
like to present this work at the 2010 Annual Meeting of the Society of Mathematical
Biology. Finally, with this project I hope to have expanded the computational program
that I have been writing to include a more general treatment of cell-substrate interactions.