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Research Papers

A Force Based Model of Individual Cell Migration With Discrete Attachment Sites and Random Switching Terms

[+] Author and Article Information
J. C. Dallon

e-mail: dallon@math.byu.edu

W. V. Smith

Department of Mathematics,
Brigham Young University,
Provo, UT 84602

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received December 7, 2012; final manuscript received February 12, 2013; published online June 11, 2013. Assoc. Editor: Edward Sander.

J Biomech Eng 135(7), 071008 (Jun 11, 2013) (10 pages) Paper No: BIO-12-1599; doi: 10.1115/1.4023987 History: Received December 07, 2012; Revised February 12, 2013

A force based model of cell migration is presented which gives new insight into the importance of the dynamics of cell binding to the substrate. The main features of the model are the focus on discrete attachment dynamics, the treatment of the cellular forces as springs, and an incorporation of the stochastic nature of the attachment sites. One goal of the model is to capture the effect of the random binding and unbinding of cell attachments on global cell motion. Simulations reveal one of the most important factor influencing cell speed is the duration of the attachment to the substrate. The model captures the correct velocity and force relationships for several cell types.

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References

Figures

Grahic Jump Location
Fig. 1

(a) Depicts the way we model a cell mathematically. The cell is a center location (nucleus) with attached springs. The other ends of the springs are attached to sites which can interact with the extracellular matrix (membrane bound attachment sites) depicted by “x”. (b) An image of a fibroblast in a collagen lattice.

Grahic Jump Location
Fig. 2

Data for the duration time of actin foci from wild type Dd cells is plotted as bar graphs [30] and the gray line is a Poisson distribution with mean 20. The simulations use Poisson distributions with different means depending on the simulation.

Grahic Jump Location
Fig. 3

The cell configuration for a typical simulation is shown at time 0.0501 and 0.1663 hs. The large vector is the overall force on the cell center. The squares are the attachment points for the binding sites and the smaller vectors show the forces on the substrate due to the attachment sites. The cell is moving in the positive x direction. Compare with Figs. 1–5 in Ref. [31].

Grahic Jump Location
Fig. 4

(a) The speed of a single cell is plotted against the mean attach time and mean detach time. (b) The time average of the sum of the absolute value of the forces is shown. The contour lines are plotted over the shading. The plot shows the average of 50 runs for each data point. The speed was calculated by averaging the average speed of the cell per minute over the duration of the simulation. The spring constants were all αj=0.206 nN/μm, with a maximum of 10 adhesion sites.

Grahic Jump Location
Fig. 5

The force is plotted against the speed as blue dots for 40,300 simulations of a single cell on a 2 dimensional substrate where the mean attach time is varied from 10 to 70 s, the mean detach time is varied from 0 to 50 s, and for each set of values 50 different simulations were run. In (a) Red, green, and black indicate simulations with a fixed average detach time of 10 s, 30 s, and 50 s respectively. (b) Red, green, and black indicate simulations with a fixed average attach time of 14 s, 34 s, and 54 s, respectively. (c) Shows the simulations in black asterisks where the average detach time is 24 s and the average attach time is 20 s. The red squares are experimentally measured values for individual Dd cells taken from [31]. Fig. 4 was constructed from this data.

Grahic Jump Location
Fig. 6

(a) The speed of a single cell is plotted against the mean attach time and strength of the cell. The cell speed is remarkably constant with respect to the strength of the cell. (b) The time average of the sum of the absolute value of the forces is shown. The contour lines are plotted over the shading. The plot shows the average of 50 runs for each data point. The mean detach time is 25 seconds.

Grahic Jump Location
Fig. 7

(a) Red, green, and black indicate simulations with a fixed value for the spring constant where αj is 0.0347, 0.081018, and 0.231 nN per micron, respectively. (b) Red, green, and black indicate simulations with a fixed average attach time of 16 s, 30 s, and 60 s, respectively. The force is plotted against the speed as blue dots for 40,300 simulations of a single cell on a 2D substrate where the mean attach time is varied from 10 to 70 s, the mean detach time is varied from 0 to 50 seconds, and for each set of values 50 different simulations were run. Figure 6 was constructed from this data. The red squares are experimentally measured values for individual Dd cells taken from [31].

Grahic Jump Location
Fig. 8

The boxes roughly outline the regions where experimental data has been reported for different cells types. The scatter plots are simulation results with parameters used to mimic the behavior of the different cells. Red denotes fibroblasts, black denotes murine dendritic cells, cyan denotes neutrophils, green denotes endothelial cells, and blue denotes Dd cells.

Grahic Jump Location
Fig. 13

The total change in the x coordinate and the y coordinate is plotted for simulations using the first force rule (a) and simulations using the new force rule (b). The data for the old force rule is taken from the same simulations that are shown in Fig. 4 The only difference between the two sets of data is the rule for generating force. The polygon shown in (a) Is the convex hull of the points shown in (b). The average change in x and in y for each data set is plotted in (a) With the * indicating the new force rule and the square indicating the old force rule.

Grahic Jump Location
Fig. 12

(a) The speed of a single cell is plotted against the mean attach time and mean detach time for simulations with the new force rule which depends on the time the attachment site has been bound. (b) The time average of the sum of the absolute value of the forces is shown. The contour lines are plotted over the shading. The plot shows the average of 50 runs for each data point. The speed was calculated by averaging the average speed of the cell per minute over the duration of the simulation which was 1 h. The spring constant was αj=0. 206 nN/μm, with a maximum of ten attachment sites.

Grahic Jump Location
Fig. 11

A quadratic curve is shown in (a) Which represents the force of an attachment site plotted against the time the site has been bound when the force is modeled as a spring. (b) Shows a representative curve of the new more realistic force rule which includes a term that strengthens the attachment site force with time.

Grahic Jump Location
Fig. 10

The panel shows the simulations where the average detach time is 24 s and the total number of possible binding sites ranges from 5 to 30 and the average attach time ranges from 10 to 70. The data plotted was used to construct Fig. 9 The squares are experimentally measured values for individual Dd cells taken from [31].

Grahic Jump Location
Fig. 9

(a) Speed of a single cell is plotted against the mean attach time and total possible number of attachment sites. (b) The time average of the sum of the absolute value of the forces is shown. The contour lines are plotted over the shading. The plot shows the average of 50 runs for each data point. Mean detach time is 25 s.

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