Figure 8. Theoretical model for flagellar length control based on IFT and microtubule turnover. (A) IFT leads to length-dependent assembly rate. Equation gives assembly versus length assuming that transport is rate limiting. N number of IFT particles cycling per flagellum. , Constant reflecting the transport capacity of a single IFT particle; v, average velocity of an IFT particle; L, flagellar length. Squares represent flagellar precursor protein, for example tubulin. Circles represent IFT particles. (B) Length-dependent assembly predicted by IFT-limited transport leads to a steady-state mechanism for length regulation. (Solid line) Length-dependent assembly rate. (Dotted line) Length-independent disassembly rate. The flagellar length is the equilibrium length (arrow) at which the two rates balance. (C) Predicted kinetics of flagellar regeneration based on equation 1. ( ) Data points taken from regeneration measurements in Rosenbaum et al. (1969). (Solid line) Best fit predicted regeneration kinetics. (D) Two possible causes of shf mutants. Either increasing disassembly rate (left) or decreasing assembly rate (right) will lead to equilibration at a shorter length. Thus, reduction in IFT is predicted to cause an shf phenotype, which accounts for the results of Fig. 6 A. (Solid line) Wild-type rates; (dotted line) rate in mutant cells; ( ) new equilibrium point in mutant. (E) Two possible causes of lf mutants. Either increasing assembly rate (left) or decreasing disassembly rate (right) will lead to equilibration at a longer length. The ability of reduced turnover to cause lf phenotype accounts for the results of Fig. 6 C. (F) Turnover model predicts results of experiment in which a single flagellum is severed. Cartoon illustrates observed behavior of such cells: as the severed flagellum regenerates, the intact flagellum shortens until both reach the same length. Graph shows result of computer simulation based on length control model. (Red) Length of severed flagellum; (green) length of the unsevered flagellum. Dashed lines connect corresponding time points of simulated response with stages of observed response. (G) Simulation of experiments in which recessive lf mutants are fused with wild- type cells. (Red) lf shorten to wild-type length once the recessive mutation is complemented by wild-type cytoplasm; (green) wild-type flagella fluctuate in length, but eventually regain their normal length. Flagellar length control | Marshall and Rosenbaum 411 This length-dependent assembly follows from the mech- anism of IFT (Fig. 8 A). If transport of flagellar compo- nents by IFT is the rate-limiting step in assembly, then the assembly rate should be proportional to vN/2L where N is the number of IFT particles, L is the flagellar length, v is the average velocity of each IFT particle, and is a constant reflecting the quantity of flagellar precur- sor protein transported per IFT complex (see Appendix for derivation). This expression is based on the data of Fig. 7 C, which shows that the number N of active IFT particles moving within the flagella is independent of flagellar length. Combining length-dependent assembly with length-inde- pendent disassembly results in the situation illustrated in Fig. 8 B, in which the two rates will balance only at one unique length. In formal terms, we can combine the length- dependent assembly rate vN/2L with a length-independent disassembly rate D to find the net rate of growth or shorten- ing as a function of length, as follows: dL/dt avN/2L D. (1) We find the steady-state solution by setting dL/dt equal to zero (corresponding to the intersection of the two curves in Fig. 8 B), showing that a stable length can only be maintained when vN/2L D, which for a given set of values of , v, N, and D, will only hold for a single value of L. If L were to somehow increase beyond this equilibrium value, the first term would become less than D, dL/dt would become nega- tive, and the flagellum would shorten. Conversely, if the fla- gellum were too short, for example during regeneration, dL/dt would be positive, resulting in continued growth. Thus, the intrinsic length dependence of IFT-mediated tip assembly, combined with the length-independent tip disassembly in- ferred from resorption kinetics (Tamm, 1967), provides a simple and robust mechanism for length determination. The simplified model described here involves the assump- tion that the disassembly rate D is a constant that is inde- pendent of length and assembly rates. However, neither of these assumptions is strictly required. As long as the disas- sembly versus length curve intersects the assembly versus length curve at a single point, the model will produce a unique stable flagellar length. Thus, it is not necessary for the disassembly rate to be completely independent of length or assembly rate, although it simplifies the mathematical form of the model. In addition to providing a simple mechanism for main- taining length, the mathematical model of Eq. 1 can also ac- count for the observed kinetics of flagellar regeneration. As illustrated in Fig. 8 C, the solution of Eq. 1 fits the observed regeneration kinetics data (Rosenbaum et al., 1969) almost exactly (see Appendix for details). This model also makes specific predictions about flagellar length mutants. There are two ways to obtain an shf pheno- type (Fig. 8 D). One is to increase the disassembly rate. In this case the two curves will intersect at a shorter equilibrium length, leading to an shf phenotype. Alternatively, the assem- bly rate could be decreased, and again the two curves will in- tersect at a shorter length. This second scenario predicts that a partial reduction in IFT should lead to a shf phenotype, as confirmed in Fig. 6 A.