Hydrodynamic Clustering of Human Sperm in Viscoelastic Fluids

We have numerically investigated sperm clustering behaviours, modelling cells as superpositions of regularised flow singularities, coarse-grained from experimentally obtained digital microscopy of human sperm, both in watery medium and a highly viscous–weakly elastic, methylcellulose medium. We find that the cell yaw and cell pulling dynamics inhibit clustering in low viscosity media. In contrast clustering is readily visible in simulations modelling sperm within a methylcellulose medium, in line with previous observations that bovine sperm clustering is much more prominent in a rheological polyacrylamide medium. Furthermore, the fine-scale details of sperm flagellar movement substantially impact large-scale collective behaviours, further motivating the need for the digital microscopy and characterization of sperm to understand their dynamics.


I. DOMAIN-SIZE DEPENDENCE
In the main text, we have analyzed the dynamics for a domain-size of L = 5, where lengthscales have been nondimensionalised by the flagellum length. The sperm clustering dynamics is a relatively local phenomenon, and thus the domain-size does not affect the dynamical properties. In particular, we plot the alignment order parameter, C 0 := C(0), the clustering order parameter C * 0 := C * (0) and the angular diffusion time, τ , in SM Figure 1 for simulations with the same cell density of N/L 2 , but with different values of the domain-size, L. As illustrated in the figure, all results except for the smallest domain with L = 2 show similar values, demonstrating domain-size independence once the domain is of the size considered in the main text.

II. HEAD STERIC INTERACTIONS
In this supplementary section, the effects of steric interactions are briefly considered. As summarised in the main text, the mammalian flagellar diameter is ≈ 0.5-1µm, which is in turn smaller than the 15µm scale for the height variation of boundary accumulated sperm above a surface [6]. Hence, as mentioned in the main text, near a surface flagella are generally expected to pass over each other in virtually all cases without steric influences, as further indicated by the observations that flagella smoothly cross in high-resolution microscopy of multiple swimming sperm [2]. Thus steric interactions between sperm flagella are not considered and instead we focus on steric interactions involving the sperm head. Attractive head-head steric interactions would be relevant for rodent sperm train modelling, where sperm-sperm attachments form [5]. However, the fluctuating clustering of bovine sperm observed in Tung et al. [8], which does not involve attachment, instead motivates the consideration of repulsive steric forces.
Thus we proceed to consider the prospective impact of short-range repulsive interactions involving the sperm head in the current modelling framework of regularised Stokes flow singularities. We therefore introduce an additional velocity-field induced by the presence of steric influences associated with spherical sperm heads of radius a, in addition to the flow due to the other cells' hydrodynamic interaction with the surrounding media.
The additional velocity due to steric influences for the triplet (n, m, l) is modelled via Here the vector r n indicates the relative position of the point x (n,m,l) 0 from the centre of the sperm head associated with the n cell, with the parameter b representing the strength of the short-range repulsion. The above expression is of the same form used for steric interactions in previous studies [3] and, in particular, once b is sufficiently large, any given cell head is an exclusion region for all other cells.
In SM Figure 2, we show the alignment ordering function, C(r), and its associated order parameter C 0 := C(0), obtained from the simulations with the repulsive strengths b ∈ {0, 1, 10, 100} and the head-sizes a ∈ {0.03, 0.06}, noting the unit of length is a flagellum length. The human sperm head with dimensions 4.5µm×2.8µm×1.1µm possess the same volume as a sphere of radius a ≈ 0.032 on using a flagellar length of 56µm to non-dimensionalize [7]. Bovine sperm have relatively larger heads, with a ≈ 0.05 for Bos taurus [1]. The instantaneous non-dimensional velocity scale of the human sperm in low viscous medium is of order unity [4] and thus the range of steric interaction strength, which includes values with b = 0 and b 1, covers cases with the absence of steric forces on one hand and extreme short-range repulsion on the other.
For each case of these steric interaction parameters, the alignment ordering functions almost coincide, as shown in the left and the middle columns of SM Figure 2, which indicates the head steric interactions do not impact the collective behavior. The right column of the figure also shows the alignment order parameter, C 0 is not affected. These results strongly support and motivate the assumption of negligible head steric interactions used in the main text.  In the left and middle column, the alignment ordering function is plotted for different values of the repulsive strength parameter, b, for LVM (left) and HVM (middle). In the right column, the alignment order parameter C0 is shown. Note that in plots (a,b,d,e) the bars are plus/minus the standard error, while in plots (c,f) the bars give the range corresponding to the confidence intervals generated in calculating the order parameters, as detailed in Method Section.