8A Length-regulation by intraflagellar transport
In Sect. 8.1.3, we briefly described a mechanism for flagellar (or ciliary) length
control, based on the idea that a fixed number of IFT particles are present inside
the flagellum (or cilium). As the flagellum grows in length, each IFT particle has
to travel a longer distance to deliver tubulin at the tip of the flagellum, resulting
in a balance between assembly and disassembly at a critical flagellar length. However, recent photobleaching studies have shown that IFT particles move in and out
of the flagellum, suggesting that the influx of IFTs is regulated [3]. Trains of IFTs
enter the flagellum through the flagellar pore, a membrane-spanning structure at
the base of the flagellum that may be homologous to the nuclear pore (see Sect.
7.3.2). There is also a microtubule-organizing center known as the basal body, analogous to the centriole (see Sect. 8.2), which, anchors the flagellar microtubules at
the plasma membrane and integrates them with the cytoplasmic microtubules. IFT
proteins dock around the basal body and assemble into trains prior to entering the
flagellum [1]. It appears that the rate at which IFT trains enter the flagellum depends
on the amount of docked IFTs in the basal body, with faster growing flagellar having
more localized IFTs [2, 4]. This suggests that there is some length-dependent mechanism for regulating the accumulation of IFT particles at the basal body (and possibly the loading of cargo to docked IFTs [4]). Ludington et al. [3] compare several
different mathematical models for regulating IFT accumulation. Here, we describe
a diffusion-based model that fits well with the experimental photobleaching data. It
is based on the hypothesis that a RanGTP gradient regulates IFT particle flux into
the flagellum in an analogous fashion to nuclear transport through the nuclear pore
complex (see Sect. 7.3.2).
Diffusion-based model
Consider a cylindrical flagellum of length L and cross-sectional are A. Suppose that
RanGTP is produced at a rate s at the tip (x = L), resulting in a concentration gradient, see Fig. 8A.1. Assume that cytoplasmic RanGTP concentration is negligible
and k is the flow rate through the pore at x = 0. Then the RanGTP concentration per
unit volume C(x,t) evolves as
∂ 2C
∂C
=D 2
∂t
∂x
gC,
x 2 [0, L],
(8A.1)
where g is a degradation rate. The boundary conditions are
D
∂C
= kC,
∂x
x = 0;
D
∂C
= s,
∂x
x = L.
(8A.2)
Integrating equation (8A.1) with respect to x and using the boundary conditions
gives
1
k+
basal binding site
IFT particle
k-
RanGTP
σ
Fig. 8A.1: RanGTP regulated injection of IFT particles at the base of a flagellum.
dR
= s kC(0,t) gR
dt
where R(t) is the total number of RanGTP molecules per unit area,
R(t) =
Z L
0
C(x,t)dx.
(8A.3)
p
If we assume that diffusion is fast so that the characteristic length D/g
L, then
C(x,t) is approximately uniform and we can take C(0,t) ⇡ R(t)/L. Therefore,
dR
=s
dt
k
R
L
gR
(8A.4)
Equation (8A.4) has the steady-state solution
R=
sL
,
gL + k
so that the concentration at the basal pore is
C0 = C0 (L) =
s
.
gL + k
(8A.5)
Suppose that there are N binding sites for IFT particles in the basal body, and the
concentration of IFTs within the cytoplasm is B. Denoting the binding/unbinding
rates by k± with k+ = k+C0 (L). The kinetic equation for the number N(t) of bound
IFTs is then
dN
= k+ B[N N(t)] k N(t),
(8A.6)
dt
which has the steady-state solution
2
N0 =
k+ B
N.
k+ B + k
It is now assumed that the length-dependent concentration C0 (L) of RanGTP at the
basal body regulates the binding rate of IFT particles so that k+ = k+C0 (L) and
N0 =
k+C0 (L)B
N.
k+C0 (L)B + k
(8A.7)
Under the adiabatic approximation that the growth-rate of the flagellum is much
slower than the various kinetic processes, we can treat N0 as a constant. The final
component of the model is to take the injection times of the IFT particles to be exponentially distributed with rate µ0 = l N0 . The mean flux of IFTs into the flagellum is
thus a monotonically decreasing function of L, see Fig. 8A.2(a). The critical length
is then determined by the balance between the influx and the length-independent
rate of disassembly.
It also follows that the number M(t) of particles injected into flagellum over
the interval [0,t] is given by a Poisson process. That is, setting Qm (t) = P(M(t) =
m|M(0) = 0) we have (see Sect. 3.6)
dQm
= µ0 [Qm 1 (t)
dt
which has the solution
Number of IFTs
IFT flux [# s-1]
Qm (t) =
(µ0t)m
e
m!
Qm (t)],
(8A.8)
µ0 t
(8A.9)
,
(a)
200
0
0
2
4
6
8
flagellar length [μm]
10
12
2
4
6
8
flagellar length [μm]
10
12
(b)
500
0
0
Fig. 8A.2: RanGTP regulated injection of IFT particles at the base of a flagellum. Parameter values
were fitted using experimental data. Typical values used were as follows [3]: l = 29/s, N = 100
1000, t = 1s, v = 2µ m/s, s = 5 20/s, g = 10 400/s, k = 5 25µ m/s.
3
It immediately follows that
hM(t)i = µ0t,
var[M(t)] = µ0t.
Suppose that each injected particle remains in the flagellum a time T = 2L/v +
t before being removed, where v is the arithmetic mean of the anterograde and
retrograde speeds of each IFT particle, and t is the time spent at the tip. It follows
that the number of particles in the flagellum at time t, t > T , is
F(t) = M(t)
M(t
T ).
(8A.10)
In particular,
hF(t)i = µ0 (t
T )) = µ0 T = l
(t
k+C0 (L)BN
(2L/v + t).
k+C0 (L)B + k
(8A.11)
This yields an expression for how the total IFT protein in the flagellum varies with
length L, see Fig. 8A.2(b), which can then be compared to photobleaching data.
We can also estimate the variance in the number of IFT particles according to the
following:
var[F(t)] = hM(t)2 + M(t
T )2
2M(t)M(t
= var[M(t)] + var[M(t T )]
2hM(t)M(t T )i
= µ0 (2t
T ) + 2µ02t(t
T)
T )i
µ02 T 2
µ02 T 2 + µ02t 2 + µ02 (t
2[µ02 (t
T )t + µ0 (t
T )2
T )] = µ0 T.
We have used the fact that for a Poisson process with rate µ0 , the autocorrelation
function is
hM(t)M(t
T )i = h[M(t)
= h[M(t)
= µ02 (t
M(t
T )]M(t
M(t
T )]ihM(t
T )t + µ0 (t
T ).
T )i + hM(t
T )2 i
T )i + var[M(t
T )] + hM(t
T )i2
Supplementary references
1. Deane, J. A., D. G. Cole, ., J. L. Rosenbaum: Localization of intraflagellar transport protein
IFT52 identifies basal body transitional fibers as the docking site for IFT particles. Curr. Biol.
11, 1586-1590 (2001).
2. Ludington, W.B. et al.: Avalanche-like behavior in ciliary import. Proc. Natl. Acad. Sci. U.S.A.
110, 3925-3930 (2013).
3. Ludington, W.B. et al.: A systematic comparison of mathematical models for inherent measurement of ciliary length: how a cell can measure length and volume. Biophys. J. 108, 1361-1379
(2015).
4. Wren, K. N., J. M. Craft, D. Tritschler, A. Schauer, D. K. Patel, E. F. Smith, M. E. Porter, P.
Kner, and K. F. Lechtreck.: A differential cargo-loading model of ciliary length regulation by
IFT. Curr. Biol. 23, 2463-2471 (2013).
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