Although the WAVE complex stimulates actin nucleation via the Arp2/3 complex, sheet-like protrusions are still observed in ARP2-null, but not WAVE complex-null, cells. The WAVE complex recruits IRSp53 to sites of saddle curvature but does not depend on IRSp53 for its own localization. This pattern of enrichment could explain several emergent cell behaviors, such as expanding and self-straightening lamellipodia and the ability of endothelial cells to recognize and seal transcellular holes. Using super-resolution microscopy, we find that the WAVE complex forms actin-independent 230-nm-wide rings that localize to regions of saddle membrane curvature. Here we investigate how the WAVE complex organizes sheet-like lamellipodia. Here’s a little function to convert the fft() output to the animation output: # cs is the vector of complex points to convertĬonvert.fft <- function(cs, sample.How local interactions of actin regulators yield large-scale organization of cell shape and movement is not well understood. The fft() function returns a sequence complex numbers, while the animation returns pairs strength:delay (in degrees). The cycles shown here for the trajectory 1,2,3,4 is 2.5 0.71:135 0.5:180 0.71:-135 which is just another way to represent the output of the fft() R function. So :180 means that that cycle starts at the initial rotation of 180 degrees, or \(\pi\) radians. In the Cycles textbox the values after the colons mean the starting point of that cycle (in degrees), ie, the cycle’s delay. Here’s an animation for the same trajectory: Here’s an animation taken shamelessly from Better Explained describing a circular path:įiddle in the Cycles/Time textboxes to see what happens.Īnyway, remember this output? fft(1:4) / 4 # to normalize # 2.5+0.0i -0.5+0.5i -0.5+0.0i -0.5-0.5i The speed will be represented by the rate of change of \(d\) over time.The delay, or starting point, is given by an initial value of \(d\).The strength is represented by the circle size, which is controlled by \(z\).We mentioned that each cycle has a strength, a delay and a speed. So, \(si = \frac = -1\), can be understood: it means a 180 degrees ( \(\pi\) radians) rotation, placing the point in the x-axis, in the opposite side of the unit circle, ie, at point (-1,0) in the complex plane, which is the integer \(-1\). This time interval is called the sample interval, si, which is the fundamental period time divided by the number of samples \(N\). For simplicity we will make the time interval between samples equal. The sampling rate, sr, is the number of samples taken over a time period (aka acquisition frequency).The fundamental period, T, is the period of all the samples taken, the time between the first sample and the last. A non-periodic wave does not have a frequency or wavelength. A periodic wave has a frequency \(f\) and a wavelength \(\lambda\) (a wavelength is the distance in the medium between the beginning and end of a cycle, \(\lambda = v/f_0\), where \(v\) is the wave velocity) that are defined by the repeating pattern. Plot(xs,wave.4,type="l",ylim=c(-1.25,1.25)) title("overflowed, non-linear complex wave") abline(h=0,lty=3)Īlso, the Fourier Series only holds if the waves are periodic, ie, they have a repeating pattern (non periodic waves are dealt by the Fourier Transform, see below). If there is, eg, some overflow effect (a threshold where the output remains the same no matter how much input is given), a non-linear effect enters the picture, breaking the sinusoidal wave and the superposition principle. The Fourier Series only holds while the system is linear. Joseph Fourier showed that any periodic wave can be represented by a sum of simple sine waves.
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