Multiscale Mechanochemical Interactions Between Cell Membrane and Actin Filaments - Innovative Approaches to Cell Biomechanics - page 88

Details for the method to extract the cell outline curve are described in Sect. 6.3.3
in Chap. 6 . Briefly, first, 3 × 3 neighbourhood-averaging is applied twice to reduce
noise. A Sobel filter with a kernel size of 3 × 3 is applied on the image to detect the
cell boundaries. Then, the cell outline curve r = [ x (s), y ( s )] are acquired by extracting
8-connected contour pixels from the binarized images.
The curvature ʺ ( s ) of a parameterized curve r = [ x ( s ), y ( s )] can be calculated as
k s

( )

( ) =− ( )

2 32

/

. Here, we approximate the spatial derivatives with
the second-order accurate central-difference scheme, and obtained the curvature
ʺ ʔs ( ʸ , t ) at time t and at angle ʸ along the cell outline, where the subscript ʔs is
the spatial interval (pixels) used for the approximation. In the analysis, an outward
curvature is defined as positive.

x    yxy / xy  

2

+

7.4 Relationship Between Cell Peripheral Activity

and Shape: A Spatiotemporal Pattern

In this section, we will explain a methodology to explore the correlation between the
cell peripheral activity and that of the cell peripheral shape. In the analysis, the curva-
tures are estimated at various spatial intervals, ʔs , ranging from sub-micrometers to
tens of micrometers. The spatiotemporal changes in the curvatures are compared with
that of the cell peripheral activity shown in Fig. 6.6 in Chap. 6 , and the relationship
between cell peripheral activity and shape is qualitatively demonstrated.

7.4.1 Detection of Multiscale Concavities and Convexities

of Cell Periphery

The approximated curvature, ʺ ʔs ( ʸ , t ), changes depending on the level of coarse
graining with different spatial intervals, ʔs . In Fig. 7.2a , the approximated curvatures
are plotted against angle ʸ and the spatial interval ʔs for the cell outline curve at a fixed
time (for the data shown in Fig. 6.4 ) . To clearly visualize the characteristic change

Fig.7.2 (continued) approximation, and then are mapped against angle ʸ and spatial interval ʔ s.
The curvatures are normalized for each ʔ s by dividing each data set with the maximum value.
The boxed area indicates the spatial change in the curvature, ʺ ʔ s , estimated with ʔ s = 125 pixels.
( b ) Enlarged figure of the region of the ordinate from ʔ s = 1.25 pixels (0.1 ʼm) to ʔ s = 24 pixels
(2.3 ʼm) of ( a ). The boxed area indicates the spatial change in the curvature estimated with
ʔ s = 12.5 pixels (1 ʼm). ( c ) The map of the curvature, ʺ ʔ s , estimated with ʔ s = 125 pixels (10 ʼm).
( d ) The map of the curvature, ʺ ʔ s , estimated with ʔ s = 12.5 pixels (1 ʼm). The scale at the bottom
of each map is the approximated arc length corresponding to the angle ʸ indicated by the abscissa.
( e , f ) Enlarged sections of the map indicated by the boxes in ( d ) (Adapted with permission from
The Royal Society of Chemistry: [Integrative Biology], copyright (2012))