We discuss the dynamics of a generalized three-sphere microswimmer in which the spheres are connected by two elastic springs. The natural length of each spring is assumed to undergo a prescribed cyclic change. We analytically obtain the average swimming velocity as a function of the frequency of the cyclic change in the natural length. In the low-frequency region, the swimming velocity increases with the frequency and its expression reduces to that of the original three-sphere model by Najafi and Golestanian. In the high-frequency region, conversely, the average velocity decreases with increasing the frequency. Such a behavior originates from the intrinsic spring relaxation dynamics of an elastic swimmer moving in a viscous fluid.

We discuss the directional motion of an elastic three-sphere micromachine in which the spheres are in equilibrium with independent heat baths having different temperatures. Even in the absence of prescribed motion of springs, such a micromachine can gain net motion purely because of thermal fluctuations. A relation connecting the average velocity and the temperatures of the spheres is analytically obtained. This velocity can also be expressed in terms of the average heat flows in the steady state. Our model suggests a new mechanism for the locomotion of micromachines in nonequilibrium biological systems.Comment: 5 pages, 2 figure

We discuss the non-equilibrium statistical mechanics of a thermally driven micromachine consisting of three spheres and two harmonic springs [Y. Hosaka et al., J. Phys. Soc. Jpn. 86 (2017)]. We obtain the non-equilibrium steady state probability distribution function of such a micromachine and calculate its probability flux in the corresponding mode space. The resulting probability flux can be expressed in terms of a frequency matrix that is used to distinguish between a non-equilibrium steady state and a thermal equilibrium state satisfying detailed balance. We analytically show that the probability flux of a micromachine is proportional to the temperature difference between the first and the third spheres when the friction coefficients are all identical. The scale of non-equilibrium of a micromachine can quantitatively be characterized by the flux rotor. We obtain a linear relation between the flux rotor and the average velocity of a thermally driven micromachine that can undergo a directed motion in a viscous fluid.

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We discuss the shear viscosity of a Newtonian solution of catalytic enzymes and substrate molecules. The enzyme is modeled as a two-state dimer consisting of two spherical domains connected with an elastic spring. We take into account the enzymatic conformational dynamics, which is induced by the binding of an additional elastic spring that represents a bond between the substrate and enzyme. Employing the Boltzmann distribution weighted by the waiting times of enzymatic species in each catalytic cycle, we obtain the shear viscosity of dilute enzyme solutions as a function of substrate concentration and its physical properties. The substrate affinity distinguishes between fast and slow enzymes, and the corresponding viscosity expressions are obtained. Furthermore, we connect the obtained viscosity with the diffusion coefficient of a tracer particle in enzyme solutions.