Load response of shape-changing microswimmers scales with their swimming efficiency Benjamin M. Friedrich 1, ∗ 1 cfaed, TU Dresden, Dresden, Germany (Dated: September 25, 2018) External forces acting on a microswimmer can feed back on its self-propulsion mechanism. We discuss this load response for a generic microswimmer that swims by cyclic shape changes. We show that the change in cycle frequency is proportional to the Lighthill efficiency of self-propulsion. As a specific example, we consider Najafi’s three-sphere swimmer. The force-velocity relation of a microswimmer implies a correction for a formal superposition principle for active and passive motion. PACS numbers: 47.63.Gd, 87.16.Qp, 47.63.-b Keywords: microswimmer, low Reynolds number, fluid-structure interaction, force-velocity relation Microswimmers that periodically change their shape can swim actively in a fluid. For example, biological cells such as sperm cells or motile green alga are propelled in a fluid by long, slender cell appendages known as cilia and flagella, which perform regular bending waves [1]. At the relevant length and time scales of microswimming, inertia is negligible and propulsion relies solely on viscous forces, corresponding to a regime of low Reynolds numbers [2, 3]. The fluid-structure interaction between a shape- changing microswimmer and the viscous fluid is bidi- rectional: active shape-changes set the surrounding fluid in motion; conversely, hydrodynamic friction forces feed back on the active propulsion mechanism of the mi- croswimmer and can change speed and shape of its swim- ming stroke. This load response becomes important when the swimmer is subject to an external force, and has implications for cargo transport and interactions be- tween several microswimmers, as well as for swimming in fluids of different viscosities. Furthermore, the load response may provide insight into the active propulsion mechanism itself. Previous theoretical work considered shape-changing microswimmers towing a load [4–6]. In these studies, shape and timing of the swimming stroke was prescribed. Other authors have formulated dynamic equations for the swimming stroke of microswimmers that employ active driving forces [7–11]. In this case, the speed of the swim- ming stroke depends on the external load. Experiments showed that the instantaneous phase speed of beating flagella indeed changes as a function of fluid viscosity [12–14] or external flow velocity [15–17]. The feedback between hydrodynamic friction forces and the speed of the flagellar beat is a prerequisite for the striking phenomenon of flagellar synchronization by hydrodynamic coupling [18]. Collections of beating cilia and flagella can phase-lock their oscillatory bending waves [19–22]. Theory explains this phenomenon by hy- drodynamic coupling between the cilia, where the hydro- dynamic load acting on each ‘flagellar oscillator’ depends on the phases of the other oscillators [23–26]. The load response of shape-changing microswimmers has also implications for a formal superposition principle for active and passive motion used in the literature [27– 30]. This superposition principle states that active self- propulsion can be characterized by a fictitious propulsion force, such that the motion of a multi-component swim- mer is characterized by the sum of the fictitious propul- sion forces of its individual components [29]. This ap- parently simple superposition principle provides formu- las that formally resemble a force balance. The use of fictitious propulsion forces was critically commented on by Felderhof [31]. In fact, the use of fictitious propulsion forces seemingly contradicts the fact that self-propelled microswimmers do not exert any net force on the fluid as a consequence of Newton’s third law. Furthermore, it is not clear if the formal superposition principle also holds in the presence of a load response of the active compo- nents. Here, we highlight generic aspects of the load response of active swimmers, and introduce a minimal model of a shape-changing microswimmer subject to external load. This model abstracts from the intricate force-generation mechanisms of cilia and flagella, and other biological mi- croswimmers. We explicitly account for the conversion of energy during active motion, from an energy reser- voir into work performed on the surrounding fluid, and possibly dissipation inside the microswimmer itself. We make the idealizing assumption that the energy expendi- ture per shape-change cycle is independent of load. This case corresponds to a maximal load response. We briefly sketch a case of a load-dependent driving force. Our contribution is two-fold: First, we report a direct relationship between the response of a shape-changing microswimmer to external load, and its swimming ef- ficiency. The swimming efficiency generalizes the hy- drodynamic propulsion efficiency of Lighthill [32], and measures the ratio between the power required to tow a passive swimmer of constant shape through the fluid, and the average rate of energy expenditure of the active, shape-changing swimmer [33]. Second, we propose that arXiv:1711.03597v2 [physics.bio-ph] 30 Apr 2018