- Snapshots of modern mathematics
- Pierre de Fermat and Andrew Wiles in Czech Republic stamps
- Stefan Banach (March 30, 1892 – August 8, 1945)
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler

# Coarse-Graining the Fluid Flow around a Human Sperm

*It has been observed that human sperm cells trace regular but complex trajectories, shaped as spiralling helices. A new mathematical model try to give a better description the movement of these micorsopic swimmers. Here the English version of an article by Antonio DeSimone, from the Italian site MaddMaths!.*

Cell motility is at the root of many biological processes that are of fundamental importance for life. From metastatic tumour cells invading nearby tissues, to the response of the immune system, to sperm cells swimming their way (within the female reproductive tract) to fertilize an egg cell, cell motility spans the whole range from the alfa to the omega of the fate of human beings.

The recent article “Coarse-Graining the Fluid Flow around a Human Sperm” by K. Ishimoto et al. ^{[1]}K. Ishimoto, H. Gadelha, E. Gaffney, D. Smith, and J. Kirkman-Brown: Coarse-Graining the Fluid Flow around a Human Sperm. Physical Review Letters 118, 124501 (2017) (“Coarse-Graining” from now) focuses precisely of the fluid flows induced by human sperm cells, and has attracted considerable attention from the general media. In spite of being a very well studied subject, the intricate trajectories traced by swimming sperm cells are still full of misteries and surprises.

Swimming is based on the principle of action and reaction. When beating its tail, a sperm cell exterts forces on the surrounding fluid, in order to displace it. As a consequence, it is subject to the reaction of the fluid, which it can use to propel itself. It then follows that there are two equivalent but complementary aproaches to the study of swimming of microscopic unicellular organism. One can either focus on the swimmer, or on the signature the swimmer leaves on the flow of the surrounding fluid.

In the first approach, the swimming problem is formulated through the equations of motion of the swimmer. This is put in motion by the forces that the fluid exerts on it. Since inertia is negligible at the microscopic scales of interest for unicellular organisms, the swimmer must be able to advance and win the viscous resistance of the fluid under zero external force. This is the translational analogue of the “falling cat” problem: starting from an arbitrary orientation, a cat manages to right itself and always land on its feet, in the absence of external torques. This is the realm of Geometric Control Theory, with its rich arsenal of mathematical tools (see e.g. the Encyclopedia article ^{[2]}A. DeSimone, F. Alouges, A. Lefebvre: Biological fluid dynamics, nonlinear partial differential equations. Springer Encyclopedia on Complexity and Systems Science (2009) for a brief introduction).

In the second approach, the swimming problem is formulated by looking at the equations governing the motion of the surrounding fluid. This is put in motion by the forces that the swimmer exerts on it.

To discuss this point of view, let us consider a hypothetical olympic swimmer, performing a breast stroke while keeping his legs fixed, in a fully extended position (a new, hypothetical olympic specialty, not that efficient in all honesty!).

In the power part of the stroke, the swimmer moves his fully extended arms, say, downwards, and the arms are pushed upwards by the reaction of the fluid. The viscous drag of the surrounding fluid pushes then the legs and the body against the direction of motion, hence downwards. As a consequence of its motion, the swimmer is *subject to* a tensile force dipole that tends to stretch him, and (by action and reaction) it pulls on the water (it’s a puller).

In the shape recovery phase of the stroke, the arms are moved upwards to recover the initial shape, but they are kept flexed, close to the swimmer’s torso, to minimize the viscous resistance of the fluid. Assuming that inertia is negligible, as it is at the microscopic scales of interest in the motility of unicellular organsims, the swimmer backtracks a bit, moving downwards, and the picture described above reproduces itself, with all signs changed. The swimmer is compressed by the action of the fluid, and it *exerts on* the water an extensile force dipole (it’s a pusher).

When averaged over a full cycle, since viscous forces are higher in the power phase of the stroke, when the arms are fully extended, the characters of the power phase dominate over those of the shape recovery phase and the swimmer is, in average, a puller.

This simplified analysis describes rather well the fluid flow generated by *Chlamydomonas reinhardtii*, a biflagellated alga that swims along an approximately straight line thanks to the symmetric beating of two flagella. These are placed in the anterior part of a round body, and, by performing an almost perfect breast stroke, power the swimming of this eukarytic unicellular swimmer. By contrast, bacteria such as *Escherichia coli*, propel themselves thanks to the rotational motion of a helical flagellar bundle, placed in the posterion part of an elongated body. The axis of the helix, the axis of rotation, and the axis of the body are all alingned, and the bactrium moves again along an approximately straight line (at least in between two successive sharp turns, which are accompished thanks to a characteristic “tumble” maneuver).

In sharp contrast with the unicellar swimmers described above which, thanks to their symmetric beating, follow trajectories wich are approximately straight, human sperm cells trace regular but complex trajectories, shaped as spiralling helices. To rationalize the origin these trajectories, the authors of “Coarse-Graining” find that the fluid flow generated by swimming sperm cells can be assimilated to the one generated by two “blinking force triplets”. This means that the strength of each force triplet is modulated in time so that the two triplets execute a regular dance, periodically exchanging their role as the leading partner.

For each triplet, two of the forces generate a force dipole aligned with the direction of motion, as in the cases described above. The third element of the triplet contributes a force with a strong component transversal with respect to the direction of motion, hence explaining the lateral steering, the deviations from a straight trajectory and, ultimately, the observed spiralling helices. In other words, while the sperm head is, on average, pushed by its flagellum, it is also periodically pulled backwards and sideways: this explains the osberved spiralling trajectories.

In order to arrive at their synthetic (coarse-grained) view of the signature of the swimmer motion on the fluid flow, the authors of “Coarse-Graining” use a combination of classical methods and of modern tools. The fluid flow is described as the superposition of fundamental solutions (mulitpole expansions) of the Stokes equations (the equations describing fluid flow at microscopic sclaes). Among these, the dominat contributions (modes) are extracted using the technique of Principal Component Analysis. In principle, the extraction of the dominant modes with PCA can be made automatic: it is, in a sense, a technique of machine learning.

The results in “Coarse-Graining” open the way to the systematic analysis of the behavior of unicellular swimmers through the analysis of their induced flows.

The induced flows carry the signature of the individual behavior (the beating pattern) of an organism, which is often of considerable biological significance. Using machine learning techniques, the analysis can be extended form the behavior of individuals to that of populations, where collective phenomena such as collective schooling, emergence of alignment order, formation of bacterial vortices, and emergence of bactierial turbulence are commonly observed. The hope is to be able to develop models at the population level, which however retain the specific biological features coming form the individual cell dynamics.

**Antonio DeSimone**

mathLab Director, SISSA Trieste

References

1. | ↑ | K. Ishimoto, H. Gadelha, E. Gaffney, D. Smith, and J. Kirkman-Brown: Coarse-Graining the Fluid Flow around a Human Sperm. Physical Review Letters 118, 124501 (2017) |

2. | ↑ | A. DeSimone, F. Alouges, A. Lefebvre: Biological fluid dynamics, nonlinear partial differential equations. Springer Encyclopedia on Complexity and Systems Science (2009) |

0 comments