Kinetic entrapment: a mechanism for periodic one-dimensional growth

Globular proteins and other irregularly-shaped but identical molecular objects often self-assemble into fibers such as sickle-cell hemoglobin fibers. Such fibers are typically one-dimensional aggregates of fixed width, indefinitely long length and strong periodic order. A recent numerical study [Lenz, Witten 2017] gave strong evidence that such one dimensional fibers arise generically when a) the constituents are identical, b) their shape is asymmetric, so that they do not tile space, c) they aggregate irreversibly under the influence of a short-range attraction, and d) the energetic cost of distorting the constituents into bonding configurations is comparable to the attachment energy gained by this bonding. Here we propose a common kinetic mechanism in which the next growth site is a generic, deterministic function of the current aggregate configuration. We take the growth function to depend only on a local neighborhood of the previous growth site. A small bias favoring convex regions causes the growth site to revisit a given neighborhood of the deposit sufficiently often that the sequence of neighborhoods and growth sites reliably falls into a fixed, repeating cycle. This self entrapment does not occur when the bias parameter is halved.