Thermodynamic precision of a chain of motors: the difference between phase and noise correlation

Inspired by recent experiments on fluctuations of flagellar beating in sperm and C. reinhardtii, we investigate the precision of phase fluctuations in a system of nearest-neighbor-coupled molecular motors. We model the system as a Kuramoto chain of oscillators with a coupling constant k and noisy driving. The precision p is a Fano-factor-like observable, which obeys the thermodynamic uncertainty relation (TUR), which is an upper bound related to dissipation. We first consider independent motor noises with diffusivity D: in this case, the precision goes as k/D , coherently with the behavior of spatial order. The minimum observed precision is that of the uncoupled oscillator punc ; the maximum observed precision is Npunc , saturating the TUR bound. Then we consider driving noises which are spatially correlated, as may happen in the presence of some direct coupling between adjacent motors. Such a spatial correlation in the noise does not evidently reduce the degree of spatial correlation in the chain, but sensibly reduces the maximum attainable precision p, coherently with experimental observations. The limiting behavior of the precision, in the two opposite cases of negligible interaction and strong interaction, is well reproduced by the precision of the single chain site punc and the precision of the center of mass of the chain Neffpunc with Neff<N : both do not depend on the degree of interaction in the chain, but Neff decreases with the correlation length of the motor noises.