Helices at Interfaces

Helically coiled filaments are a frequent motif in nature. In situations commonly encountered in experiments coiled helices are squeezed flat onto two dimensional surfaces. Under such 2-D confinement helices form ”squeelices” peculiar squeezed conformations often resembling looped waves, spirals or circles. Using theory and Monte-Carlo simulations we illuminate here the mechanics and the unusual statistical mechanics of confined helices and show that their fluctuations can be understood in terms of moving and interacting discrete particle-like entities the ”twist-kinks”. We show that confined filaments can thermally switch between discrete topological twist quantized states, with some of the states exhibiting dramatically enhanced circularization probability while others displaying surprising hyperflexibility. Introduction. – Helically coiled filaments are found everywhere in living nature. The list of examples is close to innumerable with the most prominent ones: FtsZ [1], Mrb [2], bacterial flagella [3, 4], tropomyosin [5] and intermediate filaments [6]. More recently microtubules were suggested to spontaneously form large scale superhelices [7, 8]. Even whole microorganisms exhibit helicity inherited from their constituent filaments [9]. The superhelicity of filaments is in some cases of strong evolutionary benefit as in the example of swimming bacteria utilizing the rotational motion of their helical flagellar filament for propulsion [10] and tropomyosin’s helical ”Gestalt-binding” around actin [5]. In other cases, like for microtubules the purpose of superhelicity remains so far unknown [8]. Paralleling biological evolution artificial, man made helically coiled structures have been created including coiled carbon nanotubes [11], DNA nanotubes [12], coiled helical organic micelles [13]. As happens often in experiments our justified desire to simplify observation conditions by confining filaments to a surface (coinciding with the focal plane) changes the physical properties of the underlying objects in initially unanticipated but physically rather interesting manner. With filament helices being such a profoundly ubiquitous structure it is the purpose of this work to investigate the rich physical effects of helical filaments confinement. As we will show, the confinement changes dramatically the shape as well as statistical mechanics of the confined helix generating several notable and surprising effects: a) Enhancement of cyclisation probability, b) Enhancement of end-to-end fluctuations and c) Generation of conformational multistability (despite apparent linearity of constitutive relations). We will see that the conformational dynamics of confined helices is most naturally described in terms of discrete particle like entities the ”twist kinks”, cf. Fig. 1. We show that these ”twist kinks” are completely analogous to overdamped Sine-Gordon-kinks from soliton physics [14] as well as loops in stretched elastic filaments [15]. These analogies will help us to rather intuitively develop a phenomenological understanding of the underlying physics. The Phenomenology of Squeezed Helices. – Confined biofilaments throughout literature exhibit often abnormal, wavy, spiral, and circular shapes that appear not be rationalized by the conventional Worm Like Chain model. This riddle of peculiar filament shapes is the starting point of our investigation. In this letter, we propose a new augmented model of confined intrinsically curved and twisted chains that leads to a variety of 2-D shapes matching experimental observations. The corresponding 2-D geometrical curves we call squeezed helices or more briefly squeelices. Filaments are modelled as 2-D confined Helical Worm Like Chains (cHWLC) with bending modulus B, twist modulus C and a preferred curvature ω1 and twist ω3.