A systematic comparison of mathematical models for inherent measurement of ciliary length: how a cell can measure length and volume

Abstract

Cells control organelle size with great precision and accuracy to maintain optimal physiology, but the mechanisms by which they do so are largely unknown. Cilia and flagella are simple organelles in which a single measurement, length, can represent size. Maintenance of flagellar length requires an active transport process known as intraflagellar transport, and previous measurements suggest that a length-dependent feedback regulates intraflagellar transport. But the question remains: how is a length-dependent signal produced to regulate intraflagellar transport appropriately? Several conceptual models have been suggested, but testing these models quantitatively requires that they be cast in mathematical form. Here, we derive a set of mathematical models that represent the main broad classes of hypothetical size-control mechanisms currently under consideration. We use these models to predict the relation between length and intraflagellar transport, and then compare the predicted relations for each model with experimental data. We find that three models-an initial bolus formation model, an ion current model, and a diffusion-based model-show particularly good agreement with available experimental data. The initial bolus and ion current models give mathematically equivalent predictions for length control, but fluorescence recovery after photobleaching experiments rule out the initial bolus model, suggesting that either the ion current model or a diffusion-based model is more likely correct. The general biophysical principles of the ion current and diffusion-based models presented here to measure cilia and flagellar length can be generalized to measure any membrane-bound organelle volume, such as the nucleus and endoplasmic reticulum.

Figure 1

Schematics and predictions for each model. The models are described using a single flagellum on a Chlamydomonas reinhardtii cell as an example. The prediction for a cell with a short flagellum and a cell with a long flagellum is illustrated for each model. IFT trains are depicted as red dots. (A) The null model shows no regulation of IFT; that is, the frequency with which new IFT trains enter the flagellum is constant with respect to length because it only depends on the cytoplasmic concentration of IFT proteins, which is roughly constant. (B) The initial bolus model regulates IFT by assigning a set number of IFT trains to each flagellum when they are first initialized. These IFT proteins are in a closed pool that never exchanges with the cytoplasmic pool of IFT proteins. (C) The swim-speed feedback model postulates that length is sensed by a mechanosensory feedback due to channels on the flagellar membrane being stretched by the coupling force with the extracellular fluid. (D) The time-of-flight model proposes that a signal molecule attached to each IFT train that enters the flagellum measures the length because IFT trains stay longer in long flagella and thus have a higher probability of the signal molecule falling to the unexcited state. (E) The ciliary current model proposes that ion channels on the flagellar membrane regulate length by allowing a current to pass into the cell body. Longer flagella, which have more channels, let in a larger current. (F) The linear diffusion model suggests that the flagellar base senses a signal produced at the flagellar tip. The longer the flagellum, the lower is the signal sensed by the base. (G) The volumetric diffusion model extends the linear diffusion model such that signal is produced throughout the flagellum at a constant rate and can either decay in the flagellum or escape into the cytoplasm. The concentration of signal in the flagellar compartment decreases as the flagellar compartment becomes larger. To see this figure in color, go online.

Figure 2

Many length-measuring signals can exist in the cell. The length-measuring signal for each model (A, null; B, initial bolus; C, swim speed; D, time of flight; E, ciliary current; F, linear diffusion; G, volumetric diffusion) is plotted as a function of flagellar length (black lines). To show how sensitive each length signal is to small mismeasurements of length, the green shaded region shows how the signal changes if the sensed length is ±0.5 μm. The y axis scale differs among the models because they were fit to an IFT versus length data set (see Fig. 6). To see this figure in color, go online.

Figure 3

(A–N) Model predictions for the IFT protein flux into the cilium (A–G) and the total ciliary IFT protein (H–N). The equations in Table 1 were fit to IFT injection data (11). To depict how sensitive each length model is to small mismeasurements of length, the green shaded region shows how the model outputs changes if the sensed length is ±0.5 μm. (H–N) Model predictions for the total ciliary IFT protein versus ciliary length were calculated using the parameter fits from (A)–(G). To see this figure in color, go online.

Figure 4

Paralyzed flagella (pf) mutants in Chlamydomonas display abnormal length distributions. Cells of CC124 (wild-type, N = 16), CC1464 (short flagella, N = 20), pf14 (N = 21), pf15 (N = 153), pf18 (N = 62), pf19 (N = 90), pf20 (N = 54), and pf26 (N = 20) were measured for flagellar length. Paralyzed flagella from pf14, pf15, pf18, pf19, and pf20 show distributions of decreased length, whereas pf26 shows some increase in length.

Figure 5

Longer flagella beat more slowly. A total of 132 cells of wild-type (CC125) and long-flagella mutant lf1 were measured for beat frequency (number of beats per second) and flagellar length using a high-speed camera. The decrease in beat frequency for longer flagella offsets some of the length dependence of the swim-speed signal.

Figure 6

(A and B) IFT decreases as a function of flagellar length for both (A) the rate of KAP-GFP injection into the flagellar compartment versus flagellar length (N = 186 flagella; see Ludington et al. (11) for methods) and (B) basal body accumulation of KAP-GFP protein versus flagellar length data sets (N = 317 cells). In (B), Chlamydomonas cells were deflagellated by pH shock and allowed to regenerate flagella. Aliquots of the regenerating culture were taken at 15 min intervals, fixed, and imaged for GFP fluorescence at the basal body region as previously described (11). The two flagellar lengths per cell were averaged and the two basal bodies were quantified for fluorescence together. The ciliary current model was fit to each data set separately; however, different parameters were calculated for the two fits because the two sets measure different, albeit closely related, processes (see “Placing the models in a ciliary length control context” subsection).

Figure 7

The IFT protein shows FRAP. (A and B) KAP-GFP (KAP) Chlamydomonas flagella were bleached (A, red box) and the total integrated fluorescence in the box was measured over time (B, in the red box). The fluorescence intensity was normalized to the average integrated fluorescence of the two unbleached cell bodies (white arrowheads in the first panel of the montage). IFT20-GFP cells yielded similar results but with faster recovery times. To see this figure in color, go online.