Research Papers

Complex Matrix Remodeling and Durotaxis Can Emerge From Simple Rules for Cell-Matrix Interaction in Agent-Based Models

[+] Author and Article Information
Daniel A. Krakauer

Department of Biomedical Engineering,
The Ohio State University,
270 Bevis Hall, 1080 Carmack Rd.,
Columbus, OH 43210

Keith J. Gooch

Department of Biomedical Engineering,
The Ohio State University,
270 Bevis Hall, 1080 Carmack Rd.,
Columbus, OH 43210;
Dorothy M. Davis Heart & Lung Research Institute,
The Ohio State University,
473 W. 12th Ave.,
Columbus, OH 43210
e-mail: gooch.20@osu.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received December 22, 2012; final manuscript received April 23, 2013; accepted manuscript posted May 8, 2013; published online June 11, 2013. Assoc. Editor: Edward Sander.

J Biomech Eng 135(7), 071003 (Jun 11, 2013) (10 pages) Paper No: BIO-12-1632; doi: 10.1115/1.4024463 History: Received December 22, 2012; Revised April 23, 2013; Accepted May 08, 2013

Using a top-down approach, an agent-based model was developed within NetLogo where cells and extracellular matrix (ECM) fibers were composed of multiple agents to create deformable structures capable of exerting, reacting to, and transmitting mechanical force. At the beginning of the simulation, long fibers were randomly distributed and cross linked. Throughout the simulation, imposed rules allowed cells to exert traction forces by extending pseudopodia, binding to fibers and pulling them towards the cell. Simulated cells remodeled the fibrous matrix to change both the density and alignment of fibers and migrated within the matrix in ways that are consistent with previous experimental work. For example, cells compacted the matrix in their pericellular regions much more than the average compaction experienced for the entire matrix (696% versus 21%). Between pairs of cells, the matrix density increased (by 92%) and the fibers became more aligned (anisotropy index increased from 0.45 to 0.68) in the direction parallel to a line connecting the two cells, consistent with the “lines of tension” observed in experiments by others. Cells migrated towards one another at an average rate of ∼0.5 cell diameters per 10,000 arbitrary units (AU); faster migration occurred in simulations where the fiber density in the intercellular area was greater. To explore the potential contribution of matrix stiffness gradients in the observed migration (i.e., durotaxis), the model was altered to contain a regular lattice of fibers possessing a stiffness gradient and just a single cell. In these simulations cells migrated preferentially in the direction of increasing stiffness at a rate of ∼2 cell diameter per 10,000 AU. This work demonstrates that matrix remodeling and durotaxis, both complex phenomena, might be emergent behaviors based on just a few rules that control how a cell can interact with a fibrous ECM.

Copyright © 2013 by ASME
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Fig. 1

Graphical representations of model details. (a) Large view of a cell composed of 30 “membranes,” 1 “nucleus” (center), 30 membrane-nucleus links, and intermembrane links (not visible). (b) A graphical illustration of how the Fruchterman-Reingold algorithm is used to straighten fibers. Every “binding site” becomes an anchor (blue) once per iteration in random order and its neighboring “binding sites” (red) behave like charged particles. (c) As the neighbors repel each other, they pivot about the anchor and this section of the bent fiber straightens, reducing its potential energy. As each section of a fiber successively undergoes this process, the entire fiber straightens.

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Fig. 2

(a) A simulation of a single fiber straightening under the rules described in Fig. 1(b). The fiber is composed of 22 “binding sites” and is initially s-shaped (left, 0 AU). From left to right the following four frames are the same fiber at 250, 500, 1000, and 2500 AU. (b) Legend for the strain experienced by fiber segments. (c) At 13 AU a single proximal “binding site” on the unstrained fiber (blue) is just beyond the proximity within which a “membrane” on cell can interact with it. (d) At 14 AU the proximal “binding site” on the fiber falls just within the proximity of a single “membrane” that extends a pseudopod (white) to bind to the fiber. (e) After binding, the pseudopod retracts, pulling on the fiber inducing strain (represented by light blue and green) and causing the “membrane” to move in the direction of the fiber.

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Fig. 3

(a) The two-cell free-floating matrix model at initial conditions. The intercellular ROI (yellow box) is defined as being five patches in height (approximately 1 cell diameter) and having borders five patches from the “nuclei” of the two cells. (b) At 4600 AU aligned fibers between the cells show more strain compared to other regions and at (c) 22,500 AU fibers have pulled away from the edge allowing for macroscopic compaction. (d)–(f) Time course of collagen remodeling from McLeod et al. [15]. HUVEC cultured in 2 mg/ml collagen that were fixed and stained with phalloidin and DAPI, and collagen were imaged after (d) 0, (e) 4, and (f) 16 h of culture. Panels show overlay of collagen (green) with cells (red/blue). (d) Scale bar is 25 μm for images (d), (e), and (f) [15]. (g) A compressed z stack of confocal reflectance microscopy images of 3T3 cells (outlined in yellow) initially on collagen featuring prominent fiber alignment and increased matrix density between two cells. Scale bar is 20 μm.

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Fig. 4

(a) Average matrix density and (b) Pericellular matrix density. Free-floating matrices (edge weight = 0.995) have a perimeter of “binding sites” that move 0.5% of what they would were a weighting factor not to be included in the model. Constrained matrices (edge weight = 1.000) possess a perimeter of “binding sites” around the edge of the world that cannot move. The pericellular matrix density at all time points has increased significantly from 0 AU (p < 0.05) (significance not labeled on this graph). Confidence intervals signify the SEM. *p < 0.05 compared to 0 AU. p < 0.05 between the free-floating and constrained matrices at a specific time point.

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Fig. 5

(a) The anisotropy index, or extent of alignment, of the fibers in the intercellular region. (b) The angle of alignment represents the difference in angle between the principal angle of the fiber bundle and the intercellular axis. (c) The percent change in matrix density in the intercellular region. N = 10 for all time points except for the free-floating matrix at 15,000 AU (n = 9) and 20,000 AU (n = 7). Confidence intervals signify the SEM. *p < 0.05 compared to 0 AU. p < 0.05 between the free-floating and constrained matrices at a specific time point.

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Fig. 6

Overlays of all cell tracks (n = 10) for (a) the free-floating matrix and (b) constrained matrix. Cell 1 is on the left and cell 2 is on the right. (c) Average location + / − SEM for the free-floating and constrained matrices. For (a)-(c), cells start with an X coordinate of + and – 2.5 cell diameters, marked with a ◊. (d) The distance between the two cells does not change significantly throughout the simulation. Cells on free-floating matrices migrated at roughly a constant rate towards each other of approximately 1 cell diameter per 20,000 AU (p = 0.003). Cells on constrained matrices migrated towards each other, but at a more modest rate of 0.45 cell diameters per 20,000 AU not achieving the criteria for statistical significance (p = 0.068).

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Fig. 7

(a) The anisotropy index of the fibers in the intercellular ROI. (b) The angle of alignment in the intercellular ROI. (c) The matrix density in the intercellular ROI. For all graphs, group 1 includes all simulations where cells do not migrate close together (n = 7) and group 2 includes all simulations where cells do migrate close together [n = 3, except at 15,000 AU (n = 2)]. Values for anisotropy index, angle of alignment, and intercellular matrix density for group 2 at 20,000 AU could not be calculated as the intercellular ROI had disappeared. (d) The distance between cells is significantly different between groups 1 and 2 from 2500 AU on. Confidence intervals signify the SEM. *p < 0.05 compared to AU. p < 0.05 between group 1 and group 2 at a specific time point.

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Fig. 8

(a) Single-cell model featuring a uniform square lattice arrangement of fibers. (b) Cells on this uniform matrix without a stiffness gradient do not migrate significantly in the –X direction regardless of whether the matrix is free-floating or constrained. When a stiffness gradient is present, decreasing from −X to +X, the cells move in the direction of greater stiffness in the −X direction with and without force strengthening of cell-matrix bonds. Cells on a matrix with a stiffness gradient migrate significantly farther when that matrix is free-floating than when the matrix is constrained. Cells never migrate more than 0.1 cell diameters in the Y direction. Data is presented as the mean +/− SEM. *p < 0.05 as compared to the same matrix type without a stiffness gradient. p < 0.05 between the different matrices under the same conditions. **p < 0.05 between free-floating matrices with and without force strengthening.




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