A recently published study shows that human sperm can continue to swim efficiently in a highly viscous fluid environment that should almost immediately prevent movement according to conventional physics. Their movement method seems to effectively "bypass" Newton's third law that we are familiar with. The research was led by Kenta Ishimoto, a mathematical scientist at Kyoto University. He and his colleagues proposed that microscopic "active substances" such as sperm exhibit an unconventional elastic property by continuously injecting energy internally, thereby maintaining fluctuations and advancement in a high-resistance environment.
On an everyday scale, water is a relatively "light" fluid for the human body, but at a microscopic scale, the fluid behaves more like a thick wall of resistance: inertia is almost negligible, viscosity dominates, and the object stops almost immediately once it stops pushing hard. For small-scale swimmers like sperm, there is no "gliding" phase between each tail swing. If the flagellum stops flapping, the progress will be terminated instantly. This leads to the so-called "scallop theorem": In a highly viscous fluid, simply repeating completely reversible back-and-forth motion cannot produce a net displacement. If a microscopic swimmer wants to move forward, it must rely on an irreversible, time-directed movement pattern.
Sperm use flagella to "solve" this physical puzzle. The flagellum is a slender, flexible tail-like structure with a large number of molecular motors distributed inside it, which can generate traveling waves along the length of the flagellum, making the entire tail like an "active whip" that continuously transmits waves. Similar structures also exist in microorganisms such as the green algae Chlamydomonas, which also rely on flagella to swim in a viscous environment. Because molecular motors continuously inject energy into the system, the flagellum behaves less like a passive spring and more like an internally driven "active material."
The research team focused on a property called "odd elasticity" in the active material. In ordinary elastic materials, force and response are reciprocal: any way you stretch or bend the material, it rebounds in a similar way, following an action-reaction symmetry. However, in active materials, internal energy sources can allow the material to produce a non-reciprocal response, that is, the reaction force generated when subjected to an external force no longer simply "mirrors" the external force. This asymmetric mechanical behavior helps to maintain the traveling wave, even if the viscous fluid continues to consume the mechanical energy of the system.
To describe this process, researchers proposed the "odd elastohydrodynamics" theoretical framework. This framework aims to systematically characterize the "non-local, non-reciprocal" interactions exhibited by elastic materials in viscous fluids, and distinguish which effects originate from the resistance of the surrounding fluid and which originate from the active driving mechanism within the material. The research team pointed out that if we only start from the macroscopic drag effect, it will often obscure the true mechanical nature of the fluctuations inside the flagellum, so it is necessary to theoretically separate the two. They also introduced a "singular elastic modulus" as a mathematical tool to distinguish ordinary elastic responses from active non-reciprocal mechanical behavior.
In terms of model verification, the researchers applied this theory to experimental data of human sperm and the swing data of Chlamydomonas flagella. The results show that the flagellar waves of human sperm are mainly generated by internal active activities, while passive elasticity plays a role in stabilizing the waveform and helping it relax in time. For Chlamydomonas, the nonreciprocal response derived in the model is highly consistent with the wave pattern produced by the actual beating of its flagellum, further supporting the key role of "strange elasticity" in driving microscopic swimming.
The research team believes that this framework can reveal the "non-local, non-reciprocal" intrinsic interaction mechanism within active materials. In layman's terms, the tail of a sperm is not a small whip whipped by an external force, but a complex structure that continuously consumes energy. Its internal dynamics allow it to successfully swim forward in a physical world where "ordinary reciprocating motion cannot move forward." The author emphasizes that the so-called "bypassing" Newton's third law does not really mean violating the basic laws of physics, but because sperm is regarded as an "open system": a large number of microscopic active units continue to inject energy into the system, thereby breaking the mechanical symmetry we are used to seeing in closed, passive systems.
The implications of this research extend beyond the sperm itself. The researchers pointed out that this theoretical perspective helps to gain a deeper understanding of the movement patterns of "collective swimmers" from single cells to coordinated swarms in complex fluid environments. From an application perspective, the analytical thinking of "strange bomb-fluid dynamics" is also expected to provide theoretical guidance for the design of microscopic self-assembly robots, artificial micro-swimmers, and flexible materials that imitate life movements. The relevant results were published in the journal PRX Life in October 2023. The paper is titled "Odd Elastohydrodynamics: Non-Reciprocal Living Material in a Viscous Fluid".