Hydrodynamic Forcing of Spontaneous Helical Growth

Inspired by the biological system of actin ``comet tails'' formed during bacterial motility, we have modelled the natural evolution of a cylindrical gel growing from a fixed object. We have found that a stable solution, is a kind of helical growth in which the helix pitch and diameter increase, and the pitch angle varies, but the axis remains constant. The growth can be defined by a rotation vector $\vec k$. The instantaneous magnitude of this vector is related to the intantaneous pitch and diameter, and is a simple inverse linear function of the contour length, $s$, along the helix: $\left| \vec k \left( s \right) \right| \approx c s$ for some constant c. The instantaneous pitch angle $\chi(s)$, is a function of the components of $\vec k$. Helical growth can spontaneously emerge from random initial conditions. However, by starting with "seed" helices of various shapes we found that a critical value of the pitch angle, $\chi_{crit} \approx \tan^{-1}\frac{4}{\pi}$, governs the growth of the helices. For example, when $\chi (s) = \chi_{crit}$ then $\frac{d\chi}{ds} = 0$. However, this critical value is an unstable fixed point, so growth does not converge to $\chi_{crit}$.