Emergence of flagellar beating from the collective behavior of individual ATP-powered dyneins

Flagella are hair-like projections from the surface of eukaryotic cells, and they play an important role in many cellular functions, such as cell-motility. The beating of flagella is enabled by their internal architecture, the axoneme, and is powered by a dense distribution of motor proteins, dyneins. The dyneins deliver the required mechanical work through the hydrolysis of ATP. Although the dynein-ATP cycle, the axoneme microstructure, and the flagellar-beating kinematics are well studied, their integration into a coherent picture of ATP-powered flagellar beating is still lacking. Here we show that a time-delayed negative-work-based switching mechanism is able to convert the individual sliding action of hundreds of dyneins into a regular overall beating pattern leading to propulsion. We developed a computational model based on a minimal representation of the axoneme consisting of two representative doublet microtubules connected by nexin links. The relative sliding of the microtubules is incorporated by modeling two groups of ATP-powered dyneins, each responsible for sliding in opposite directions. A time-delayed switching mechanism is postulated, which is key in converting the local individual sliding action of multiple dyneins into global beating. Our results demonstrate that an overall nonreciprocal beating pattern can emerge with time due to the spatial and temporal coordination of the individual dyneins. These findings provide insights in the fundamental working mechanism of axonemal dyneins and could possibly open new research directions in the field of flagellar motility.

Figure 1

(a) Structure of an axonemal dynein comprising three basic units: a head, stem, and stalk [3]. Presence of the dyneins (outer arms) at a regular spacing along the microtubules; see the lower panel [18]. (b) An electron micrograph of an axoneme cross-section showing the $9\times 2+2$ configuration [19]. (c) Flagellar beating of a typical spermatozoon. The three video-fields (A-C) are 200 ms apart and the scale bar is $10\mu \mathrm{m}$ [20]. Figures are reproduced with permission of the copyright owner(s).

Figure 2

ATP hydrolysis cycle, ATP $\rightleftharpoons \mathrm{ADP}+{\mathrm{P}}_{\mathrm{i}}$, governing the power (and recovery) stroke of a dynein molecule leading to a conformational change and sliding of the attached microtubule.

Figure 3

During planar flagellar beating the dyneins are able to coordinate their operation along the axoneme length as well as at the three-dimensional axoneme cross-section. The unidirectional motion of the dyneins (denoted by the dots) are directed toward the basal end (minus end) of the microtubule, which is indicated by the small arrows. However, this action leads to sliding of the neighboring (attached) microtubule toward the distal end of the axoneme, which is indicated by the bigger arrows [13]. The figure is adopted from Ref. [13] and is reproduced with permission of the copyright owner.

Figure 4

(a) A representative computational model of the axoneme structure where two microtubules are connected via nexin links. Two groups (A and B) of dyneins are present at a regular spacing along the microtubules, each responsible for sliding in opposite directions. The dynein activity is shown by the vertical arrows, where the length of an arrow indicates the magnitude of the respective dynein force. Note that the actual dynein force is tangential to the axoneme. (b) The dynein force $\left(F\right)$—responsible for the relative sliding of the microtubules—is transferred to the neutral axis of the beam (representing the microtubules in the finite element model) and equilibrium moments are introduced, given by $M=0.5F(d+l_{\mathrm{nexin}})$, such that the loading system imposed by the dynein satisfies force and moment equilibrium. The dyneins are periodic force generators by linearly increasing the applied force (until the stall force $F_{\mathrm{stall}}$ is reached) during the power stroke (i.e., active period $t_{\mathrm{a}}$) followed by a recovery (“waiting”) period $t_{\mathrm{w}}$ in which the applied force remains zero. The dynein has an input duty ratio, $f_{\mathrm{input}}=t_{\mathrm{a}}/t_{\mathrm{cycle}}$, where $t_{\mathrm{cycle}}=t_{\mathrm{a}}+t_{\mathrm{w}}$, with the time periods depending on the ATP hydrolysis cycle [23, 24].

Figure 5

Steady-state response in the absence of a switching mechanism for the reference case. (a) Two groups of dyneins contribute to the axoneme deformation, where the normalized tip deflection, $\delta /L$ (left vertical axis), is indicated by the sold line, and the net number of active dyneins working at a time, $\mathrm{\Delta }N=N_A-N_B$ (right vertical axis), is denoted by the symbols. Representative steady-state cycles are shown as a function of normalized time, $\mathrm{\Delta }t/t_{\mathrm{cycle}}$, with $\mathrm{\Delta }t$ representing the time elapsed during steady-state beating. (b) The dynein activity along the axonemal length with time is shown by the vertical arrows, where the length of an arrow indicates the magnitude of the respective dynein force (that is actually oriented tangential to the axoneme). Upward and downward arrows refer to the forces generated by the dyneins of group-A and group-B, respectively. The deflection amplitude remains low as the dyneins of the two different groups are competing with each other at a given cross-section.

Figure 6

Axoneme deflection and dynein activity with time for the reference case with switching for $\tau =0$, i.e., instantaneous deactivation of the dyneins at the occurrence of negative-work. (a) The normalized tip deflection, $\delta /L$ (left vertical axis), is indicated by the sold line, and the net number of active dyneins working at a time, $\mathrm{\Delta }N=N_A-N_B$ (right vertical axis), is denoted by the symbols. (b) The dynein activity along the axonemal length with time is shown by the vertical arrows. Note, however, that the actual dynein force is oriented tangential to the axoneme. The large deflection of the axoneme is due to temporal regulation of the dynein activity, i.e., at any given cross-section only one group of dynein (either A or B) can be active at a given time instance.

Figure 7

Steady-state response for the reference case with switching for $\tau /t_a=0.2$. (a) The effective number of active dyneins working at a time, $\mathrm{\Delta }N=N_A-N_B$, is shown in comparison with the number of active dyneins of group A $\left(N_A\right)$ and group B $\left(N_B\right)$. The normalized tip deflection $\left(\delta /L\right)$ indicates the period of oscillations to be $t_{\mathrm{cycle}}$, where representative steady-state cycles are shown as a function of $\mathrm{\Delta }t/t_{\mathrm{cycle}}$. (b) The dynein activity along the axonemal length with time is shown by the vertical arrows indicating the dynein coordination (temporal as well as spatial) causing an evolution of local bend regimes in the microtubules and their propagation with time. Note that the dynein force is oriented tangential to the axoneme.

Figure 8

Swimming response with switching for the reference case with $\tau /t_a=0.2$. (a) The swimming displacement $\left(x\right)$ with time indicating transient and steady-state swimming for six different realizations. (b) Axoneme deflection and dynein activity with time for a representative realization, which indicates that the steady-state beating emerges with time once the dynein activity stabilizes due to interference of the switching mechanism.

Figure 9

Steady-state swimming response with a time-delayed deactivation of the dyneins (based on negative work) for the reference case. (a) Normalized swimming velocity as a function of the normalized time delay for $0\le \tau \le 1$. The swimming velocity increases sharply at the onset of wave propagation due to the emergence of dynein coordination. The value of $\tau /t_a=0$ corresponds to instantaneous deactivation leading to reciprocal beating with large deflections, while $\tau /t_a=1$ corresponds to the case in the absence of a switching mechanism. The error bars indicate the standard deviation of the results obtained by choosing ten different realizations. The filled symbol for $\tau /t_a=0.2$ is for a partially uniform distribution of the dynein initiation time discussed in the text. (b) Snapshots of steady-state swimming motion for $\tau /t_a=0.2$ with each snapshot (denoted by the numbers) separated by $t_{\mathrm{cycle}}/5$. The curvature remains nearly constant as it propagates along the axoneme.

Figure 10

Influence of $F_{\mathrm{n}}$ on the flagellar waveform and swimming velocity for $\tau /t_a=0.2$ and $f_{\mathrm{input}}=0.5$ for different values of $D_{\mathrm{n}}$ (left panel). For a constant $D_{\mathrm{n}}$ value, $F_{\mathrm{n}}$ is observed to control the wave amplitude and the wave length (see the insets in the right panel), which results in a decrease in the swimming velocity. The insets in the left panel indicate the waveforms for four cases for increasing values of $D_{\mathrm{n}}$ and $F_{\mathrm{n}}$. The thick solid lines are drawn only as a guide to the eye.

Figure 11

Influence of $D_{\mathrm{n}}$ on the flagellar waveform and swimming velocity for $\tau /t_a=0.2$ and $f_{\mathrm{input}}=0.5$ for different values of $F_{\mathrm{n}}$. For a constant $F_{\mathrm{n}}$ value, $D_{\mathrm{n}}$ is observed to control the wave amplitude (see the insets), which leads to a corresponding increase in the swimming velocity. The thick solid lines are quadratic fits to the data.

Figure 12

Influence of $f_{\mathrm{input}}$ on the swimming velocity for a constant cycle time $t_{\mathrm{cycle}}$.

Figure 13

A representative finite element with the corresponding nodal degrees of freedom $\left(u,v,\varphi \right)$ subjected to the external traction forces $\left(T_u,T_v\right)$ due to the surrounding fluid.

Figure 14

(a) Influence of the fluid viscosity and the step size on the stepping behavior of the dyneins subjected to the restraining action of the attached spring representing the experimental conditions for optical-trap nanometry (see the inset). Note that the motion of the bead is stalled when the applied force of the dynein is in an equilibrium with the restoring force of the spring, which appears to be independent of the step-size and fluid viscosity. (b) The time history of the resulting applied force by the dynein and resulting stepping behavior for ${\delta }_{\mathrm{step}}=8\text{nm}$ and $\mu ={\mu }_0=0.48$ mPa.s with the assumption that one dynein is constantly interacting with the neighboring microtubule, where the thin line represents the dynein force.

Figure 15

(a) Influence of the microtubule length (or number of dyneins, $N_0$) on the resulting sliding displacement due to the dynein activity $\left(\mu ={\mu }_0\right)$. To represent the experimental conditions related to the microtubules-motility assay dyneins are placed with a constant interspacing (96 nm) over the substrate, which limits the maximum number of dyneins $\left(N_0\right)$ that can interact with a microtubule of length $L$ as shown in the inset. (b) The sliding velocity with length of the microtubule, which also quantifies the number of dyneins (possibly) interacting with the microtubules. The sliding velocities are normalized such that the maximum sliding velocity is $U_{\mathrm{MT}}^{*}=1$ for $\mu ={\mu }_0$. The data points are obtained via simulations, where dyneins have a low duty ratio $f_{\mathrm{dynein}}=0.15$ and a step size of 8 nm is used. The solid lines are the fitted trend that is in agreement with the statistical expression relating the number of dyneins and their duty ratio to an expected sliding velocity of the microtubule (see the main text).