Swimming sperm appear to break Newtonian laws of physics with their tails

Human sperm are famously good swimmers, yet the physics of their motion has puzzled scientists for decades. Thick cervical mucus or lab-made gels should throttle any cell only fifty microns long, but sperm shoot through with surprising ease.

That puzzle finally cracked when a Kyoto University team revealed that the sperm tail’s internal mechanics side step Newton’s third law, the rule that demands an equal pushback for every push forward.

Their leader, Kenta Ishimoto of the Research Institute for Mathematical Sciences, worked with colleagues Clément Moreau and Kento Yasuda to pin down the trick: a property they call odd elasticity, measured by a new “odd elastic modulus.”

Sperm tails and physics

At our scale a swimmer throws water backward and glides forward in balance. Shrink the scene a thousand times and inertia vanishes, leaving syrup thick drag called low Reynolds number flow.

One flick of a tiny flagellum usually stalls before it starts, so the cell must wiggle in a non reciprocal pattern that never repeats in reverse.

Newton’s equal and opposite law assumes forces act in isolated pairs without added energy.

But sperm tails aren’t passive springs, they’re powered by molecular motors that constantly inject energy into the system. That disrupts the clean symmetry Newton envisioned.

What the team actually measured

Ishimoto’s group used high speed video of human sperm and the green alga Chlamydomonas, both of which swim with whip-like flagella.

They tracked the tail’s position over time and mapped those shapes into a two dimensional coordinate system known as “shape space.” These patterns form stable loops, called limit cycles, that repeat every beat  

To link movement to internal forces, the researchers created an elastic matrix for the flagellum. The off-diagonal parts of this matrix, normally ignored in passive systems, revealed long-range, non reciprocal forces within the tail.

These are captured in the odd elastic modulus, a measure of how the tail deforms without a mirrored pushback  

Sperm tail elasticity explained

In sperm tails, a single localized bend sends tension through the entire tail. But instead of balancing out, these forces add energy that moves the next bend forward. The result is a traveling wave, one that moves without causing an equal push in the opposite direction  

The study shows that the tail’s odd elasticity directly controls the wave’s speed and efficiency. This isn’t just theoretical.

As the odd modulus increases, so does the propulsion velocity. That finding aligns with observed beat frequencies in human sperm, which can cycle at about 20 times per second.

How the math changes the game

In classical mechanics, elastic materials deform and then recover, storing energy like springs. Odd elasticity breaks this loop. Work done in one stroke isn’t recouped in the next, it drives the system forward.

Near the tail’s steady-state motion, the team found that the standard, symmetric (even) part of the modulus essentially vanished. Only the odd part mattered  

This behavior stays consistent even when randomness is added. The researchers tested what happens when the tail’s beat fluctuates. Surprisingly, the sperm still swam efficiently. That suggests that the linear odd modulus governs swimming stability in noisy or sticky environments  

Algae, robots, and medicine

Chlamydomonas cells move with two flagella that beat in asymmetrical strokes. Even with these different mechanics, the model held.

That points to a shared strategy in many swimming cells: generate odd elasticity and let the non-reciprocal forces do the work  

This insight could be used in soft robotics. Imagine tiny bots navigating the bloodstream or crawling through mud, not with motors, but by waving internal fibers tuned with odd modulus dynamics. The model gives a playbook for how to build them.

Why sperm tails matter

Newton’s third law still works, just not in systems that continually absorb and expend energy. What sperm demonstrate is that when you’re far from equilibrium, you don’t have to obey force symmetry anymore. Cells that swim, flap, or twist with internal motors all bypass this constraint  

The paper also shows that you can blend fluid and solid mechanics into a single, unified theory.

The tail’s odd elasticity doesn’t just describe motion, it connects elasticity, internal energy, and hydrodynamics into one model that can apply to many other living systems  

How cells tune their elasticity on the fly is still unclear. Sperm likely adjust their stiffness in response to chemical signals during the journey to the egg. Mapping how molecular motors distribute themselves could help decode this control system.

Another big question is how viscosity affects swimming. Sperm can face cervical mucus thousands of times thicker than water.

This model hints that tuning the odd modulus helps sperm compensate for the extra drag, but measuring it in live tissue remains a challenge.

Nature doesn’t break physical laws, it rewrites the assumptions they rely on. In the micro world, sperm swim not by brute force, but by flexing internal springs in asymmetric ways that push without pushback. And that’s how they get ahead.

The study is published in PRX Life.

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