Spontaneous Polaron Transport in Biopolymers

Polarons, introduced by Davydov to explain energy transport in α-helices, correspond to electrons localised on a few lattice sites because of their interaction with phonons. While the static polaron field configurations have been extensively studied, their displacement is more difficult to explain. In this paper we show that, when the next to nearest neighbour interactions are included, for physical values of the parameters, polarons can spontaneously move, at T = 0, on bent chains that exhibit a positive gradient in their curvature. At room temperature polarons perform a random walk but a curvature gradient can induce a non-zero average speed similar to the one observed at zero temperature. We also show that, at zero temperature, a polaron bounces on sharply kinked junctions. We interpret these results in light of the energy transport by transmembrane proteins. Introduction. – Proteins, essential components of all biological cells are central to their proper functioning. As “form determines function”, a study of protein structure and dynamics is of utmost importance in elucidating their role in cellular behaviour. One of the key problems in biology is to understand energy transport from one part of the cell to another and to study the role of protein conformations and conformational transitions in this process. The mechanism of charge and energy transport in proteins and other bio-macromolecules at the atomic scale was proposed Davydov and co-workers [1]. In this approach the transport properties are considered in terms of the emergence of a ‘polaron’ whose properties and dynamics are used to describe the resultant transport. The polaron describes a localised excitation which carries energy corresponding to some vibrational modes of a group of molecules and a distortion of the chain containing these molecules. The system acts as a particle and its dynamical properties can be studied in terms of solutions of particular differential equations that describe its behaviour. The Davydov theory hinges on the assumption that an extra electron or energy quanta released in the hydrolysis of ATP (adenosine triphosphate) can be stored by the protein molecule in its vibrational mode. The non(a) E-mail: [email protected] (b) E-mail: [email protected] (c) E-mail: [email protected] linear coupling between the chain vibrational modes and the electron leads to the formation of a polaron which as it propagates along the polypeptide backbone leads to energy/charge transport. The soliton mediated transport mechanism has been applied to helical proteins [2] and is reviewed in [3,4]. Most polaron studies are based on simple one-dimensional model but recent studies also considered two and and three-dimensional models [5–8]. Studies of polaron transport on α helical polymers have recently been reported by Henning [9–12]. However, in most of these studies the polaron is “kicked” from its rest state via a perturbation [13]. In this paper we look at the spontaneous polaron transport via charge-conformational coupling. In particular, we study polaron transport on a flexible chain with an imposed initial bend at both zero (T = 0) and non-zero (T 6= 0) temperatures. We now summarise our main results. At T = 0 a polaron can undergo spontaneous motion via the coupling of charge with the conformational degrees of freedom. Thus an imposed bend on the chain causes the polaron to accelerate. However, we have found that when such an accelerating polaron encounters a kink, a slope discontinuity along the chain backbone, it gets reflected instead of continuing along its original direction of motion guided by inertia. The reason for this due to the longrange nature of the exchange coupling term. At finite temperature T 6= 0 thermal fluctuations wash away directed