The stochastic dynamics of flagellar beating for micro-swimmers, such as flagellated cells, sperms and microalgae, is dominated by a feedback mechanism between flagellar shape and the rate of activation/de-activation of the $N \gg 1$ driving molecular motors. In the context of the so-called rigid filament models, where the axoneme is described by a single degree of freedom $X(t)$, we investigate the effect of direct coupling between the activity dynamics of adjacent motors, parametrized by $K \ge 0$. A functional Fokker-Planck equation for $X$ and the state of the $N$ motors is obtained. In the limit of small coupling $K \ll 1$, we derive a system of equations governing the dynamics of the Fourier modes of the active motor density, obtaining estimates for several observables and the fluctuations' quality factor $Q$. For larger $K$ we resort to numerical simulations. The effect of introducing the coupling $K>0$ is to increase characteristic times and the beating period. Moreover at large $K$s the limit cycle becomes bi-stable, with abrupt avalanches of the motor dynamics. Increasing $K$ is similar to what observed in the case $K=0$ when the confining elastic force is strongly reduced. The quality factor of fluctuations has a non-monotonic behavior: it first increases with $K$, then decreases. This is accompanied by the reduction and eventual disappearance of regions where the fraction of activated motor is nor $0$ neither $1$.
The dynamics of active biological structures, such as motorized organelles, cilia and flagella, results from a complex interplay of elasticity, hydrodynamics and active driving [1]. The last ingredient, usually in the form of several molecular motors converting chemical energy into mechanical displacement, deserves investigation from many different perspectives: biochemistry, thermodynamics, non-equilibrium statistical physics [2]. The energy injection rate of molecular motors is a stochastic process, only partially understood, and its impact on the mechano-hydrodynamics of mesoscopic objects is still far from a clear picture [3][4][5]. Fluctuations of the observed oscillations in these systems can provide a powerful magnifying lens, disclosing internal details that are not yet directly accessible in vivo [6][7][8]. However this magnifying lens needs theory to be fully exploited, exactly as in the primordial example of this kind, Brownian motion: Einstein provided an essential model, Perrin exploited it in experiments, and the properties of atoms were disclosed to the eyes of scientists, previously blind [9]. Even more than a century later, now in the biological context, good models and their analysis are still crucial to fill the gap between microscopic mechanisms, occurring at the nano-scale, and mesoscopic observation, at the micro-scale [6,10,11]. The presence of energy conversion from a chemical storage into directed motion, and finally dissipation, imposes a time-arrow in these systems, breaks time-reversibility and implies the impossibility to apply the principles of equilibrium statistical physics [10,12,13].
A neat example of this problem is provided by flagellar beating, such as that occurring in cilia and flagella as appendices of eukaryotic cells, e.g. in sperms, certain micro-algae, or in several epithelia [1]. In all such examples, the internal structure of a flagellum is the same and takes the name of axoneme. An extreme simplification of this structure reduces it to 9 doublets (excluding the central one which is not fundamental for the rest of the discussion) of microtubules (MTs), running parallel to each other all along the length of the flagellum. Each doublet is decorated by thousands of dyneins, which are proteins (ATP-ases) acting as molecular motors by keeping an arm anchored to the first MT and attaching/detaching from the second one. This attaching/detaching process follows a stochastic dynamics that consumes ATP and induces local sliding between the MTs. Several mechanical constraints in the axoneme convert such a relative sliding into local bending. How hundreds of thousands of stochastic bending strokes coordinate to generate a smooth travelling wave is a fascinating and debated matter [14]. Consensus exists on the fact that a feedback mechanism is needed: active motors induce local bending, bending makes (locally) the motors de-activate, but then the bending relaxes and the motors activate again [15,16]. These loops, which take place in space and time, can be realised according to several scenarios. For instance the feedback response can be sensitive to different local observables, such as relative parallel displacement between the MTs [17], curvature [18,19] or its time-derivative [20], or normal/tangential forces/deformations [21]. Experiments have provided some discrimination, but not unique across different biological objects [14]. Much less studied is how fluctuations are affected by these feedback mechanisms. Theory and experiments, according to our view, are still in their infancy for this particular problem [6,10]. A fundamental model taking into account the finiteness of the number of motors, and therefore their fluctuations, has been used only a few times to analyse experimental data: in the following we call it Jülicher-Prost (JP) model [16].
A modification of the JP model has been recently proposed to understand new experimental observations: the modification consists in adding a direct coupling between adjacent motors, inducing stronger correlations and consequently more complex fluctuations in the spatio-temporal beating pattern [10]. The impact of noise correlation (analogous to motor coordination) for fluctuations of the beating limit cycle has been studied also in [7,11,22]. In the rest of the paper we call this model JPK i.e. JP with coupling constant K (when K → 0 the original JP model is reproduced).
We mention that another class of models exists, the so-called crossbridge or “power-stroke” models [15,23,24], where the motors are described as flexible springs, with their heads binding to specific sites of the filament and remaining stuck until unbinding. The motors switch between conformational states. The dynamic instability is associated to a variation of the unbinding rate with spring tension, in particular the rate should be increasing with the spring stretching. An interesting connection between the crossbridge model and the JP model -involving also n
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