Eur. Phys. J. B 3, 237–245 (1998) T HE EUROPEAN P HYSICAL JOURNAL B c  EDP Sciences Springer-Verlag 1998 Field-induced winding of chiral polymers P.I.C. Teixeira and E.M. Terentjev a Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK Received: 15 November 1997 / Accepted: 16 February 1998 Abstract. We propose a microscopic model of a chiral polymer chain with permanent transverse dipoles interacting with an external electric field. Its behaviour has been investigated by computer simulation in the limit of weak chirality. Large-scale (tertiary) helical winding induced along the field direction has been found above a threshold field Ec, and the helix parameters have been calculated as functions of the field strength. Below Ec there is no coherent helical structure of the chain conformation. We find a characteristic scaling of the threshold and the winding radius a with the chain bending modulus ε,(Ec/kBT ) ∼ ε 1/4 and a ∼ (kBT /E)ε. PACS. 61.20.Ja Computer simulation of liquid structure – 36.20.-r Macromolecules and polymer molecules – 87.15.By Structure, bonding, conformation, configuration, and isomerism of biomolecules 1 Introduction Notwithstanding the very large configuration space avail- able to them, many biopolymers, such as polypeptides and proteins, often adopt unique conformations [1]. Of partic- ular interest amongst these are secondary helical struc- tures, of which DNA is perhaps the best-known example. Helical geometry is a consequence of the interplay of hy- drogen bonds forming between different groups within the molecule, and the sterical constraints which the same must satisfy. Upon variation of some external parameter such as temperature, solvent composition, or pH, a conforma- tional transformation can be induced where an ordered, rod-like helix transforms into a random coil; this is the so- called helix-coil ‘transition’, which can be detected, e.g., by monitoring the optical rotation or the viscosity of the sample [1]. The classical theories thereof date from the 1960s (see [2] for a review); more recent efforts, includ- ing computer simulations, have concentrated on detailed calculations for particular molecules (see, e.g., [3,4] and references therein). The natural question to ask at this stage is, what is the minimal model of a helix? Unsurprisingly, most theoretical work on helical polymers has been concerned with DNA. In ‘supercoiled’ DNA, occurring in bacteria, circular (i.e., closed-loop) molecules arrange themselves into helices on a length scale (typically 50 nm) which is much larger than the double-helix repeat distance (3.4 nm) (for a review of supercoiling see [5]). Such ‘tertiary structures’ can be accounted for by treating DNA as a circular elastic rod [6]. Also in the semi-microscopic models [7,8] supercoil- ing is a consequence of twist rigidity, which presupposes a non-trivial (effectively ribbon-shaped) backbone. By con- a e-mail: emt1000@cus.cam.ac.uk trast, in Marko and Siggia’s more microscopic model [9, 10], chirality is introduced by imposing conservation of the linking number, i.e., the number of times the two threads wind around each other. Elegant though this formulation might be, it is none the less restricted to ring molecules. Here we present what is, to our knowledge, the first microscopic model of the tertiary structure of a chiral, helix-forming polymer in an external (electric) field. This paper is organised as follows: we start by writing down the Hamiltonian of a semi-flexible, chiral polymer chain with transverse dipoles along its backbone. We then proceed to eliminate the transverse variables and obtain an effective Hamiltonian which is a function of the unit tangent vector only, but contains an effective chiral coupling to the ex- ternal field. Simulation results are then presented for the effect of the external field upon chain conformations. We find a non-monotonic behaviour in the chain correlations, indicating a coherent large-scale helical winding around the direction of the field. This ordering occurs above a threshold value of the field, below which no consistent di- rection of the helix has been observed; the threshold elec- tric field exhibits a clear scaling dependence on the chain bending modulus. We conclude with a summary and a discussion of future directions. 2 Theory Several theoretical models and approaches have been de- veloped to treat chiral, and especially helically twisted, polymers; these originate mainly in the physical chemistry of biopolymers. Here we shall use the basic and explic- itly tractable formalism of a semi-flexible polymer chain [11], in which chirality is introduced by means of a spe- cific interaction between neighbouring segments. Strictly