Axonemal regulation by curvature explains sperm flagellar waveform modulation

Abstract Flagellar motility is critical to natural and many forms of assisted reproduction. Rhythmic beating and wave propagation by the flagellum propels sperm through fluid and enables modulation between penetrative progressive motion, activated side-to-side yaw and hyperactivated motility associated with detachment from epithelial binding. These motility changes occur in response to the properties of the surrounding fluid environment, biochemical activation state, and physiological ligands, however, a parsimonious mechanistic explanation of flagellar beat generation that can explain motility modulation is lacking. In this paper, we present the Axonemal Regulation of Curvature, Hysteretic model, a curvature control-type theory based on switching of active moment by local curvature, embedded within a geometrically nonlinear elastic model of the flagellum exhibiting planar flagellar beats, together with nonlocal viscous fluid dynamics. The biophysical system is parameterized completely by four dimensionless parameter groupings. The effect of parameter variation is explored through computational simulation, revealing beat patterns that are qualitatively representative of penetrative (straight progressive), activated (highly yawing) and hyperactivated (nonprogressive) modes. Analysis of the flagellar limit cycles and associated swimming velocity reveals a cusp catastrophe between progressive and nonprogressive modes, and hysteresis in the response to changes in critical curvature parameter. Quantitative comparison to experimental data on human sperm exhibiting typical penetrative, activated and hyperactivated beats shows a good fit to the time-average absolute curvature profile along the flagellum, providing evidence that the model is capable of providing a framework for quantitative interpretation of imaging data.

. Overview of the ARCH model showing a sketch of the active moment density m against curvature κ, for critical curvature threshold κc and preferred moment mc. Arrows show the direction of the gradient of active moment density in the ARCH model.

S1 Derivation of the Axonemal Regulation of Curvature, Hysteretic model (ARCH)
To briefly recap the main manuscript, the underlying hypothesis for the ARCH model is the simple and elegant concept of curvature control. In simple terms, that the dynein activity within the axoneme will cause the sperm flagellum to bend, increasing curvature, until a threshold is reached whereupon the process will reverse. In the ARCH model, we adapt this hypothesis to one of active moment control, wherein we relate the rate of change of active moment density, m, to the curvature, κ (Fig. S1). We derive the model by considering, in turn, the behavior of the top m > 0 and bottom m < 0 branches of the model respectively.
m > 0 A positive active moment density will cause a section of the axoneme to bend, thus increasing local curvature. We hypothesize that there is a preferred moment, m c , owing to the activity of dynein and the arrangement of the axoneme. Therefore, while the curvature is less than some critical curvature threshold, κ < κ c , the active moment density will approach the preferred moment as shown in Fig. S1. When the curvature is greater than the critical curvature threshold the active moment density will decrease to the negative preferred moment −m c . These considerations allow us to write the following equation for the rate of change of active moment density when m > 0, where τ −1 d is a characteristic dynein switching rate, and H (·) is the Heaviside function. We note that the first term on the left-hand side provides the attraction to the preferred moment, while the curvature threshold switching behavior comes from the second term.
Similarly, but in reverse to the case m > 0, when m < 0 the active moment density will approach the negative preferred moment, −m c , while the curvature κ is greater than the negative critical curvature threshold, −κ c . When κ < −κ c , the active moment density will increase to the preferred moment density m c . We write, Combining for all m Combining Eqns (S1) and (S2), we obtain where sgn(·) is the signum function. Using the fact that H (x) = 1 − H (−x), we obtain (after simplification) the dimensional ARCH model, i.e.
Generalizations of the ARCH model In the present work we consider the preferred moment density and critical curvature thresholds at each switch to be of the same magnitude, differing only in sign. However, we have taken care to set up the model to allow (via the process outlined above), the choice of the thresholds to differ with only minor modifications to the model.

Flagellum discretisation
The flagellum is discretised into n T = 60 straight-line segments of equal length ∆s = 1/n T . Segment end points are denoted , t), n = 0, . . . , n M − 1 denoting the active moment at the actively bending segment joints. Here, n M = 57 < n T has been chosen to represent the approximate 95% of the flagellum that actively bends.

Discretised system
The spatially discretised equations can then be written, for the force-and moment-free conditions as for the elastic behaviour of the flagellum as for the hydrodynamic equations evaluated for points on the flagellum, with n = 1, . . . , n T , aṡ where the integral is calculated analytically following Refs. [8,1,5]; the hydrodynamic equations evaluated for points on the head, with n = 1, . . . , n H , are written as The spatially discrete system can then be written in the form of a non-linear autonomous initial value problem, The system (S14) can be augmented at each time point to include the unknown force densities as The matrix blocks of A (A E , A K , A H , A M ) encode the elastodynamic, force-and moment-free equations (Eqs. S5-S9); the kinematic and hydrodynamic equations (Eqs. S10-S12); and active moment equations ((S13)) respectively.  D r a f t