Global asymptotic stability of the active disassembly model of flagellar length control

Abstract

Organelle size control is a fundamental question in biology that demonstrates the fascinating ability of cells to maintain homeostasis within their highly variable environments. Theoretical models describing cellular dynamics have the potential to help elucidate the principles underlying size control. Here, we perform a detailed study of the active disassembly model proposed in Fai et al. (elife 8:e42599, 2019). We construct a hybrid system which is shown to be well-behaved throughout the domain. We rule out the possibility of oscillations arising in the model and prove global asymptotic stability in the case of two flagella by the construction of a suitable Lyapunov function. Finally, we generalize the model to the case of arbitrary flagellar number in order to study olfactory sensory neurons, which have up to twenty cilia per cell. We show that our theoretical results may be extended to this case and explore the implications of this universal mechanism of size control.

Appendix A: Flux derivation

Consider the lengths of two flagella and their evolution in time, which we denote by $L_1(t)$ and $L_2(t)$. As stated in Sect. 1, we assume a total number of molecular motors M and a total amount of tubulin T.

We now derive the dynamical equations for the flagellar lengths, closely following (Fai et al. 2019). Assembly occurs in two steps in our model, each of which follows the law of first-order chemical kinetics. First, tubulin aggregates on an IFT particle containing a single molecular motor. Second, the IFT particles are injected into the flagellum. The first step of protein aggregation on IFT particles results in particles carrying an amount of tubulin proportional to $T_f$. In the second step, i.e. the injection of IFT particles at the flagellar base, the number $M_f$ of free molecular motors determines the injection rate $J_i$ via:

$$\begin{aligned} J_i= \frac{1}{2}k_\text {on} M_f, \end{aligned}$$

(65)

where $k_\text {on}$ is the rate constant of injection and the factor of one half is due to equal probabilities of injection into either flagellum.

Because the timescale of IFT (order of seconds) is fast compared to the timescale of changes in the flagellar lengths (order of minutes), we assume a quasi-steady state in which there is no accumulation of IFT particles along the flagellum. Therefore, the rate of injection equals the arrival flux of IFT particles to the flagellar tip.

The assembly rate is assumed proportional to this arrival flux times the average amount of tubulin carried by an IFT particle, resulting in

$$\begin{aligned} \text {assembly rate} = \gamma J_i T_f = \frac{1}{2}\gamma k_\text {on} M_f T_f, \end{aligned}$$

(66)

where $\gamma$ is a constant of proportionality.

As mentioned above $T_f = T-L_1-L_2$. In order to express $M_f$ in terms of the flagellar lengths, we must incorporate the detailed motion of proteins during IFT. While, as mentioned above, IFT particles including the motors and cargo move at constant speed in the anterograde direction, it has been shown that some IFT proteins such as kinesin motors diffuse in the retrograde direction back to the flagellar base (Chien et al. 2017). We capture these different possibilities by letting $M_{b,i}$ and $M_{d_i}$ represent the numbers of motors undergoing ballistic and diffusive motion, respectively, on the $i^{\mathrm{th}}$ flagellum. By conservation of motor number, we have

$$\begin{aligned} M_f = M-M_{b,1}-M_{b,2}-M_{d,1}-M_{d,2} \end{aligned}$$

(67)

We may express the ballistic flux $J_{b,i}$ on the $i^{\mathrm{th}}$ flagellum as the product of the concentration of motors moving ballistically in the anterograde direction times their speed:

$$\begin{aligned} J_{b,i}= \frac{M_{b,i}v}{L_i}, \end{aligned}$$

(68)

where v is the IFT speed in the anterograde direction.

On the other hand, To capture this diffusive motion, we let the concentration of diffusing particles on the $i^{\mathrm{th}}$ flagellum be $c_{d,i}(x)$ for positions $x\in [0,L_i]$ along the flagellum. If the diffusive flux $J_{d,i}$ is constant, then by Fick’s law $J = -D \partial c/\partial x$ so that the concentration will be a linear function of x, and in particular

$$\begin{aligned} c_{d,i}(x) = \frac{M_{d,i}}{L_i}+\frac{J_{d,i}}{D}\left( x-L_i/2\right) , \end{aligned}$$

(69)

where we have used the fact that $\int _0^{L_i} c_{d,i} \mathrm {d}x = M_{d,i}$.

Assuming the flagellar base acts as a sink, we may apply the boundary condition $c_{d,i}(0)=0$ to (69) and express the flux in terms of the number of diffusing motors:

$$\begin{aligned} J_{d,i} = \frac{2DM_{d,i}}{L_i^2}, \end{aligned}$$

(70)

and it follows that

$$\begin{aligned} c_{d,i}(x) = \frac{J_{d,i}}{D}x. \end{aligned}$$

(71)

Since $J_i=J_{b,1}=J_{b,2}=J_{d,1}=J_{d,2}$ by the quasi-steady state assumption, we drop the subscripts and denote the constant flux by J. Plugging the expressions from (67) (68) and (70) into (66) yields

$$\begin{aligned} J = \frac{1}{2}k_\text {on}\left( M-J\frac{L_1+L_2}{2v}+J\frac{L_1^2+L_2^2}{4D}\right) , \end{aligned}$$

and upon solving for J we obtain

$$\begin{aligned} J = \frac{k_\text {on}M/2}{1+k_\text {on}\left( L_1+L_2\right) /2v+k_\text {on}\left( L_1^2+L_2^2\right) /4D}. \end{aligned}$$

(72)

Inserting this expression into (66) yields finally

$$\begin{aligned} \text {assembly rate} = \gamma J \left( T-L_1-L_2\right) , \end{aligned}$$

(73)

where J is given by (72).

For the disassembly rate, we assume there is both a basal rate of disassembly $d_0$ as well as an additional disassembly which depends on the concentration of depolymerase at the flagellar tip. For simplicity, we will assume that the depolymerase concentration at the $i^{\mathrm{th}}$ flagellum is proportional to the kinesin concentration $c_{d,i}(L_i)$. (As discussed in Fai et al. (2019), this would be the case so long as the depolymerase follows the same pattern of ballistic anterograde to diffusive retrograde motion.) Therefore, we have

$$\begin{aligned} \text {disassembly rate} = d_0+d_1 c_{d,i}(L_i) = d_0+d_1\frac{J L_i}{D}. \end{aligned}$$

(74)

Subtracting the assembly and disassembly rates yields the following system of coupled nonlinear ODE’s:

$$\begin{aligned} \frac{\mathrm {d}L_1}{\mathrm {d}t}&= \gamma J \left( T-L_1-L_2\right) -d_0-d_1\frac{J L_1}{D} \end{aligned}$$

(75)

$$\begin{aligned} \frac{\mathrm {d}L_2}{\mathrm {d}t}&= \gamma J \left( T-L_1-L_2\right) -d_0-d_1\frac{J L_2}{D}, \end{aligned}$$

(76)

where J is given by (72) above.