We studied the response of the actin filament model network by examining the network over time as mechanical stimulation was imposed on the system. show initial, intermediate, and final configurations of an actin filament network subjected to cyclic stretching corresponding to 1, 5, 10, and 25 cycles of stretching. There are 16 nodes on the perimeter of a circle of radius 100 units and 30 interior nodes with 138 filaments. In dividing non-muscle vertebrate cells, the number of actin filaments has been estimated to be in the tens of thousands (Alberts, 2002
; Lee et al., 2010
; Mitsui & Schneider, 1976
) but due to the computational challenges with these number of filaments, we are assuming a model with fewer filaments. We have though examined the changes associated with our model response with respect to the number of filaments. Nodes located on the perimeter from 80° to 100° and 260° to 280° are not stretched. Nodes from −60° to 60° and 120° to 240° are stretched 10% for 25 cycles. After each cycle, the normalized filament stress determines the probability for breaking. “Breaking” is a term used to encompass all of the different mechanisms by which a filament might cease linking two nodes, whether this is through depolymerization, disassembly, or physical breaking (Janmey et al., 1991
; Krendel et al., 1998
; Mitchison & Cramer, 1996
; Ono, 2003
; Tsuda et al., 1996
). These are replaced by filaments which are randomly assigned new locations such that the total number of filaments is conserved.
Figure 2 A filament network of 138 filaments, 16 prescribed perimeter nodes, and 30 randomly placed internal nodes under 10% uniaxial horizontal stretch after (a) 1, (b) 5, (c) 10, and (d) 25 iterative cycles. Stretch was imposed in 1% increments on all peripheral (more ...)
In order to quantify filament orientation, we create histograms of all filament angles in the network, averaged over 10 uniquely generated networks (). A clear affinity for 90° vertical alignment appears as cycle number increases. This shift though can be detected in as few as 5 cycles. The error bars reveal that there is a small amount of variation in the distribution of the filament alignment, which is not unexpected in a stochastic model. Note that all filament angles have been transposed to the first quadrant of the Cartersian coordinate system to create these histograms so that we are able to similarly portray filaments from 90° to 360°.
Figure 3 Histograms of angular orientation of filaments in nodal equilibrium after (a) 1, (b) 5, (c) 10, and (d) 25 iterative cycles of 10% uniaxial horizontal stretch. Averaged over ten uniquely generated filament networks, each of which consisted of 138 filaments (more ...)
To quantify filament angle alignment over all cycles, we normalized all filament lengths to 1 and determined the dot products between each filament angle and the mean filament angle of the network. We calculated this normalized dot product for each cycle from 1 through 25 and averaged the results over ten uniquely generated 138-filament networks (). The normalized dot product increases immediately in the early cycles until a steady state is reached at approximately 14 cycles. The decision to normalize all filament lengths to 1 was made to allow equal representation among all filaments. If all of the 138 filaments aligned exactly with the filament network mean angle, the normalized dot product sum would be 1.0 for that cycle.
Figure 4 Normalized dot product between filament angles and the filament network mean angle at nodal equilibrium over cycles 1–25. For each cycle, filament lengths were all assumed to equal 1 and the average dot product between each filament angle and (more ...)
To determine the changes to our model under important parameters including filament number and internal node number, we increased the number of filaments to 414 and 1242 while varying the number of internal nodes between 15, 30, and 60. We tracked filament angle orientation and present the results at cycle 1 and 20 (). While there is no obvious alteration in filament orientation when increasing filament number, there is a slight shift towards vertical alignment when the number of internal nodes is decreased. We observed that with decreased number of internal nodes, more breakages occur while holding the filament number steady (data not shown). It is possible that fewer degrees of freedom to alleviate stress are causing higher filament strains when node number decreases, which subsequently leads to faster convergence to the perpendicularly aligned state.
Figure 5 Histograms of angular orientation of filaments in nodal equilibrium after 1 cycle and 20 cycles for networks with (A) 414 and (B) 1242 filaments, which are 3 fold and 9 fold multiples of the previous filament number, 138. The number of internal nodes (more ...)
The filament stresses for each of the 138 filaments in the network for 1, 5, 10, and 25 cycles of stretching are analyzed (). Positive stress (i.e., tension) denotes an increase in the length of the filament relative to the start of the stretch cycle and negative stress (i.e., compression) denotes a decrease in length. As the cycle number increases in , a clear trend towards decreasing stress is observed. At cycle 1, the range of filament stresses is [−150,250] MPa. By cycle 9, this has been reduced to [−100,100] MPa and by cycle 25, the range is [−50,50] MPa with one outlier at 100 MPa. These results are non-obvious as the model implies that stresses in not only tension but also compression are reducing in this uniaxial stretch based approach.
Figure 6 Individual filament stress values in nodal equilibrium for 138 filaments, 16 perimeter nodes, and 30 internal nodes after a network was stretched uniaxially by 10% for (a) 1, (b) 5, (c) 10, and (d) 25 cycles. Positive stress (i.e., tension) denotes an (more ...)
quantifies the filament stress through histograms comparing tensile (positive) and compressive (negative) stresses for 1, 5, 10, and 25 cycles of stretching. Each category bin covers a range of 10 MPa. The left-most and rightmost starred (*) bins contain filaments with stresses less than and greater than −190 and 190 MPa, respectively, as these contained much smaller numbers of filaments. Initially the filament stresses are relatively widely distributed but as the number of cycles increases, a shift to a much higher number of filaments in the bins of lower tension and compression occurs. This result is consistent with the results presented in . The peak in the right-most bin for the 1st cycle in is the result of a large number of high-stress filaments initially in the network, which decreases with greater number of cycles ().
Figure 7 Histogram of filament stresses in nodal equilibrium after (a) 1, (b) 5, (c) 10, (d) 25 iterative cycles of 10% uniaxial stretch averaged over 10 uniquely generated filament networks of 138 filaments, 16 perimeter nodes, and 30 internal nodes. The starred (more ...)
The orientation of cytoskeletal actin in NIH 3T3 fibroblasts exposed to 1 Hz cyclic uniaxial stretch after 3, 6, 12, and 24 hours was observed (). The cells were strained using the elastomeric membrane, fixed with paraformaldahyde, and then labeled using 6 µM Alexa Flour® 488 phalloidin stain for F-actin (green pseudo-colored). The filament outlines were enhanced in software using the convolution feature in ImageJ. For each time point, the angles of ten representative filaments relative to that of the cyclic stretch were measured and their average computed (). At 3 hours, we see that prominent actin filaments are arranged both parallel to and perpendicular to the axis of stretch. Over time, with an increasing number of stretch cycles, steadily increasing F-actin alignment in the direction normal to stretch is observed. To further support our observations, we measured the orientation of whole cells at the same time points as whole cell orientation generally follows intracellular cytoskeletal alignment (see ). This alignment is quantified showing a strong pattern moving toward vertical alignment (). These experimental results reflect the modeling results in and .
Figure 8 Images of NIH 3T3 fibroblasts exposed to 1 Hz vertical cyclic stretch uniaxial stretching after (a) 3, (b) 6, (c) 12, (d) 24 hours. Red lines indicate representative actin filaments measured for angle relative to stretch direction. (e) Average filament (more ...)
Figure 9 Angular orientation histogram of NIH 3T3 fibroblasts after (a) 3, (b) 6, (c) 12, and (d) 24 hours of uniaxial horizontal cyclic stretch. Following mechanical stimulation, the cells were stained for F-actin and fluorescent images were imported into ImageJ (more ...)