A Model for Integrin Binding in Cells By: Kara Huyett Advisor: Doctor Stolarska 31 st of August, 2015 Cell movement, and cell crawling in particular, has implications in various biological phenomena. For example, cells from primary cancer tumors have the ability to crawl into the blood stream and flow into different parts of the body and begin growing as a secondary tumor. For example, a cancer tumor in the liver could crawl a very short distance in order to catch a ride in the bloodstream. It could then travel up into the brain where it could begin to grow in the brain tissue. Another important aspect of cell movement is wound healing. If there was a way to control the ligands for cell attachments to bind to, the rate at which cells moved to a site of injury could be increased. The project I have been working on with my advisor, Doctor Stolarska, deals with cell crawling and spreading. Our goal was to make a model that portrayed how integrins, the primary protein used in the attachments to the substrate, are involved in cell movement. Succeeding this, we wanted to show that by manipulating integrin and ligands, the substrate-bound chemical to which a cell attaches, the movement of a cell could be controlled. Our model is very important in portraying cell movement and the manipulation of ligands and integrins in order to control the movement and fate of cells. As a result of our project, one day people may be able to stop the metastasis of cancer cells and contain or even control cancer cells.
Cells crawl by extending their leading edge which is pushed out by the polymerization of actin filaments. These actin filaments grow at a faster rate on their positive end which causes them to push out the edge of a cell. (If the cell is moving the actin filaments will push out the leading edge, but if the cell is spreading, every edge of the cell will be pushed outwards.) After the leading edge is pushed out, the lamella, the front-most edge of the cell, will bind to the substrate. After the lamella is attached, the rear of the cell is contracted by a myosin and actin interaction. Myosin is a globular protein that walks along the actin filaments and causes the filaments to be pulled forward which causes the rear end of the cell to be pulled back inwards towards the middle of the cell. Image 1: This image is of a cell crawling. It shows the extension of the leading edge. This edge is pushed out by the polymerization of actin filaments. It then attaches to the substrate and
contracts it s lagging edge. The contraction is caused by an interaction between myosin and actin filaments. Focal adhesions are the physical binding site of the cell and the substrate. These adhesions are made up of myosin, actin filaments, integrins and various motor proteins. These adhesions are important, for they are what allow the cell to bind to the substrate and move forward. These adhesions act much like friction does with people and animals. These are the interactions that allow for the cell to have contact with its surroundings and move forward (or spread). Corning Glass Company is interested in our model because it portrays the relationship between a cell and the substrate it is binding to. Corning is working with stem cells. They want to be able to place a stem cell onto a substrate of certain stiffness and know exactly what that cell will become. In experimental work, Engler et al (2006) placed stem cells on various substrates of different stiffness. What they found was very extraordinary. The stem cells placed in a substrate of stiffness 1KPa, which is approximately the stiffness of brain tissue, grew into nerve cells, cells put into substrate of stiffness 11KPa, the stiffness of muscles, grew into muscle cells and the cells put in a substrate of 34KPa, the stiffness of bones, grew into bone cells. In order to better understand how cells interact mechanically with their environment, Doctor Stolarska created a model and computation tool for simulations of a cell spreading over the substrate to which it is attached. This model includes a mathematical representation of spreading and contraction due to actin polymerization and actin-myosin interactions.
Doctor Stolarska wanted me to make a model that showed how integrins attached and detached from the substrate in which they are located on. Once these models are combined, we can show how and where a cell interacts with a substrate. Thus, as the cell grows, the focal adhesions can grow and new binding sites can be created. We can then apply the model to various situations to find out how substrate stiffness affects cell shape and how we can control where cells bind to and how they move. Overview of my one dimensional project: To start my model, I created a one dimensional version that showed the binding and unbinding of integrins within the cell. In this model, there was the cell and the substrate. Imbedded in the substrate were two ligands patches. These ligand patches were the sites where the integrins within the cell could bind to. In the Matlab code I programmed all of this in, the ligand patches were represented by a matrix, L, and within the matrix there were ones and zeros. The ones represented the region where a ligand patch was present and zeros represented the absence of a ligand patch. Where there were ones, integrins could bind and where there were zeros integrins could not bind.
(Image 2) Image 2: My one dimensional model. It shows the cell (green), the substrate (blue) and the ligand patches (red). The ligand patches are the areas where the ligands can bind to. Overview of my two dimensional model: The two dimensional model looks much like the one dimensional model. There is the cell and the ligand patches to where integrins will bind. Image 3: My two dimensional model. It shows the cell (blue) and the ligands (green).
Model Equations: The binding and unbinding of integrins is modeled by a system of reaction-diffusion equations: These equations describe how unbound integrins, Cu, bind to the ligands, L, and create bound integrins, Cb. Db is the diffusion coefficient for bound integrins, and Du is the diffusion coefficient for the bound integrins. These are the coefficients that tell us how quickly the bound and unbound integrins are moving from higher to lower concentrations within the cell. The coefficients kf and kb are the rates at which integrins are binding and unbinding within the cell, respectively. When the two equations above are added together, the reaction terms cancel and only the diffusion component remains. Because we are using no flux boundaries (where nothing can leave the cell), we know that the total number of integrins in the cell must always remain fixed.
In order to solve this equation for Cu and Cb we used the finite element method. The finite element method was used because it allowed us to take our system of partial differentiable equations and turn them into a system of algebraic equations. Furthermore, the finite element method allowed us to solve the system of partial differential equations on a domain of irregular shape. The solution to partial differential equations is a function of space a time. Graphically, this solution can be represented as a continuous surface. When the cell domain is meshed into nodes and elements, we can find an approximate solution to the system of partial differential equations by solving for the concentrations of the bound and unbound integrins at a finite number of nodes. This is represented in image 4. Image 4: This image shows how we found the concentrations of the bound and unbound elements at the nodes. We created these nodes by putting a mesh over the circular surface. Each element has three nodes. To use the finite element method we must first put the governing equations into weak form. To do this, we multiplied each side by a test function. This arbitrary function allowed us to manipulate both sides of the equation. The weak form is given by:
Since we were able to do this, we could work with the first derivatives and thus work with linear interpolation functions instead of quadratic interpolation functions. After putting the equations into weak form, we meshed the surface and created many elements(approximately 60) and nodes(approximately 180). Image 5 shows the mesh we created over the cell. Each element is represented by a triangle and the nodes are the vertexes. Image 5: This is the mesh we created over the circular shape. The nodes and elements can be observed. The circles in the middle of the larger circle represent the ligand patches. After creating the mesh, we made a local matrix, which was a descrete representation of the integrands in weak form, for each element. After creating these local matrices, we assembled them into one large global matrix. This global matrix combined the contributions of all of the
elements into one system of equations. Using this global matrix we solved for the concentration of the unbound integrins and bound integrins at each nodepoint. The discrete weak form for each element is given by the two equations below. Note that ξ and η, are coordinates of a reference triangle to which each triangular element in the mesh is mapped. The individual matrices in the discrete weak form above are given by:
Interpolation Functions: Above are the three interpolation functions that were used in each element. They are all linear, and they allowed us to approximate the concentration of bound and unbound integrins between the nodes. Derivative of the Interpolation Functions with respect to x or y:
The discrete weak form can then be represented by a matrix called Alocal. This matrix is then assembled into a much larger matrix, Aglobal. Aglobal was the matrix A in the linear algebraic equation Ac=f that represents the system of algebraic equations that we solved for Cu and Cb. We solved for Aglobal at each node point by solving for Alocal at each node point and then assembling all the local matrices into one global matrix. One dimensional results: We ran this program in Matlab. The initial concentration of the bound integrins was zero and the initial concentration of the unbound integrins was one. As we ran this program, over the areas where the ligands were there was a dramatic increase in the concentration of bound integrins and a dramatic decrease in the concentration of the unbound integrins. This is exactly what we predicted would happen.
Image 6: One dimensional results. As you can see, there is an increase in bound integrins over the ligand patches and a decrease of unbound integrins over the same ligand patches. Ligand patches exist where 4 < x < 8 and 12 < x < 16. Ligand patches were placed from 4 < x < 8 and 12< x < 16. As can be observed from the graph, over the regions where a ligand patch existed, the concentration of bound integrins increased and the concentration of unbound integrins decreased. Also, as time passed, the difference between the new time step and the previous time step decreased. This shows that if the program ran for an infinite amount of time, it would eventually reach a steady state, the equilibrium. Two dimensional results: The two dimensional results behaved similarly to the one dimensional results. At the regions where the ligand patches existed, which can be seen from the mesh in image number 5, the number of bound integrins increased as time increased, and the number of unbound integrins decreased.
Image 7: Two dimensional results. Each circle is a cell and in the middle of the cell you can see the ligand patches where the integrins bind to. As you can see, the concentration of bound integrins goes up in the ligand regions as times passes as the concentration of the unbound integrins goes down. If this program was run for longer, eventually the system would hit steady state and the concentration of the bound and unbound integrins at the ligands would no longer change. Future Work: In upcoming research, Doctor Stolarska and I would like to incorporate our models in order to create a global model that allows us to simulate a cell spreading and binding to a substrate. With my model, we will be able to manipulate where the ligands are located and thus where the cell will bind to. This will allow us to determine the shape of the cell and what are the stresses in the cell and the substrate. Ultimately, this will help us understand the interaction of a cell with a substrate. This will have implications in stopping cancer. We could also cause cells
to crawl faster which could be beneficial in wound healing. Being able to manipulate cell crawling could be revolutionary in the future. One day, we would love to use our model in reallife scenarios to control tumors and other dangerous cells..
Works Cited Engler, Adam J., Shamik Sen, H. Lee Sweeney, and Dennis E. Discher. "Matrix Elasticity Directs Stem Cell Lineage Specification." Cell 126.4 (2006): 677-89. Web. Théry, Manuel, Anne Pépin, Emilie Dressaire, Yong Chen, and Michel Bornens. "Cell Distribution of Stress Fibres in Response to the Geometry of the Adhesive Environment." Cell Motility and the Cytoskeleton Cell Motil. Cytoskeleton 63.6 (2006): 341-55. Web. Ananthakrishnan R, Ehrlicher A. The Forces Behind Cell Movement. International Journal of Biological Sciences. 2007: 3(5):303-317. Web.