Hydrodynamic stability of helical growth at low Reynolds number

A cylindrical object growing at a low Reynolds number can spontaneously develop a helical shape. We have studied this phenomenon numerically, and our results may shed some light on the spontaneous formation of helical tails of a dense protein network observed in experiments on actin based motility. We also identify an unstable critical pitch angle which separates helices that straighten into rods from helices that flatten into planar curves as they grow. At the critical angle the pitch angle remains constant, whereas both helical diameter and pitch increase with the helical contour length.

Figure 1

Illustration of one turn of a helix “wrapped” and “unwrapped.” As $z$ increases along the vertical edge by one pitch, the distance along the helix increases by the hypotenuse, $s=p∕\alpha$.

Figure 2

Representative simulation results showing helices, which grow out of a range of seeds. (a) The result of a single simulation run for a helix growing out of the $x\text{−}y$ plane. The “older” units are shaded darker. This helix was grown for 200 additional units from a random seed three units long. (b) A helix grown at $\chi ={\chi }_{\mathit{crit}}$ for 10 000 steps from a seed helix 300 steps long. (c) A helix grown for 10 000 steps from a seed with a large pitch angle $(\chi =70°)$ 300 steps long. (d) A helix grown for 10 000 steps from a seed with a small pitch angle $(\chi =20°)$ 300 steps long.

Figure 3

Simulation results showing progression of helical shapes depending on the initial and early stage helical angles. (a) The inverse of the magnitude of the rotation vector, ${\mid \mathbf{k}\mid }^{-1}$, increases linearly with length as the helix grows. This result agrees with the theory. (b) The slope, $c$, of the line ${\mid \mathbf{k}\mid }^{-1}=cl$ for various values of $\chi$ shows a transition at ${\chi }_{\mathit{crit}}$, where $c_{\mathit{inv}\left(k\right)}=2∕\pi$. ${\chi }_{\mathit{\text{early}}}$ is the value of $\chi$ after it has stabilized in about 150 time steps.

Figure 4

Evolution of helical pitch and diameter above and below the critical pitch angle. (a) The pitch shows a transition from a regime of nonlinear growth at high pitch angle to linear growth at below ${\chi }_{\mathit{crit}}$. (b) In order to illustrate the transition at the critical pitch angle ${\chi }_{\mathit{crit}}$, deviation is plotted for the lines for $\chi$ near ${\chi }_{\mathit{crit}}$ from the line for ${\chi }_{\mathit{crit}}$, $\mathit{\text{pitch}}\left(\chi \right)-\mathit{\text{pitch}}\left({\chi }_{\mathit{crit}}\right)$. (c) $c_{\mathit{\text{diameter}}}$ for various values of $\chi$ shows a transition from $c<1$ to $c>1$ at ${\chi }_{\mathit{crit}}$. (d) To further illustrate that the slope is greatest at ${\chi }_{\mathit{crit}}$, the slopes of pitch and diameter are plotted as functions of $\chi$. The diameter exhibits a fairly linear increase with the contour length l. For the pitch, we calculate $c_{\mathit{\text{pitch}}}$ for only the slope of the last 10% of the curve.

Figure 5

Simulation results relating helical angles of an initial seed, the early progression, and the stabilized shape. (a) The only value of $\chi$ for which $\chi$ remains constant is ${\chi }_{\mathit{crit}}$. Either above or below this value, the pitch angle diverges from ${\chi }_{\mathit{crit}}$. (b) The ultimate value of $\chi$ during growth is a function of ${\chi }_{\mathit{\text{seed}}}$, the pitch angle of the seed helix. Notice that for $\chi <{\chi }_{\mathit{crit}}$ points lie below the dotted line showing ${\chi }_{\mathit{\text{final}}}={\chi }_{\mathit{\text{initial}}}$, and above the line for $\chi >{\chi }_{\mathit{crit}}$. Thus $\chi$ diverges away from ${\chi }_{\mathit{crit}}$ during growth. Note that the dotted line does not cross the graph at ${\chi }_{\mathit{crit}}$ as it should if ${\chi }_{\mathit{crit}}$ really represents the value for which $d\chi ∕ds=0$. The reason is that the seed helix is a regular helix, and therefore does not represent stable growth. That is the origin of the counterintuitive result that, for a regular helix seed to end up with $\chi ={\chi }_{\mathit{crit}}$, the original value of $\chi$ is not ${\chi }_{\mathit{crit}}$. To get around this, we look at the “stabilized” value of $\chi$ after the initial jump. We could avoid this by using a seed helix with the growth pattern that spontaneously emerges, but this is difficult to do in practice. (c) The initial jump in $\chi$ is most noticeable for addition of the first segment. (d) The value of $\chi$ stabilizes after about 200 time steps out of 6000.