epl draft Propulsion hydrodynamics of a butterfly micro -

Propulsion motion of a simple mechanical model at low Reynolds numbers is considered. The model consists of two spheroids (wings) connected by a hinge. Its non-reciprocal operation cycles represent combinations of flapping motions of the wings and of their rotations, resembling conformational motions characteristic for real protein machines and similar to the propulsion pattern of a butterfly. The net generated velocity and the net stall force, exhibited by an immobilized machine on its support, are calculated and their dependence on the model parameters is discussed. Introduction. – It is well known that bacteria and other microorganisms can swim through fluids by periodically changing their shape. The only restriction, imposed by the laws of hydrodynamics at low Reynolds numbers, has been pointed out by Purcell [1] in the form of the “scallop theorem”: a purely reciprocal cyclic motion, such as opening and closing of a scallop’s shell, cannot generate net propulsion. General analysis of propulsion effects of an object cyclically changing its shape is available [2, 3]. Several theoretical models of propulsion have been considered to achieve non-reciprocal motion. The Purcell’s three-link swimmer, a simplest model swimmer, consists of three rigid rods connected at two hinges each of which has one degree of freedom: the relative angle between two rods [1, 4–6]. The model with three linked spheres, presented by Najafi and Golestanian [7], has three spheres connected via two deformable rods(see also [8]). Examples of propulsion motion, characteristic for bacterial motions and involving cilia waves or rotation of flagella, have been considered [9–17]. Not only microorganisms, but even individual macromolecules operating as protein machines can cyclically change their shapes while being immersed into a fluid. Typically, a protein machine receives energy in the chemical form, with an ATP or other molecule binding to it as a ligand. This leads to a gradual change of the protein shape, representing a process of conformational relaxation of the protein-ligand complex to its equilibrium state . At some stage, the ligand is converted into a product (such as the ADP molecule) which then leaves the protein. After product detachment, the free protein molecule undergoes conformational relaxation back to its equilibrium state. Thus, the cycle of a machine consists of two relaxational motions, the forward one induced by ligand binding and the backward