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UNC-104 transport properties are robust and independent of changes in its cargo
binding
Amir Shee,1, 2, ∗Vidur Sabharwal,3, ∗Sandhya P. Koushika,3Amitabha Nandi,4, †and Debasish Chaudhuri2, 5, ‡
1Northwestern Institute on Complex Systems and ESAM,
Northwestern University, Evanston, IL 60208, USA
2Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, Odisha, India
3Department of Biological Sciences, Tata Institute of Fundamental Research, Mumbai, India
4Department of Physics, IIT Bombay, Powai, Mumbai 400076, India
5Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India
(Dated: September 5, 2024)
Cargo distribution within eukaryotic cells relies on the active transport mechanisms driven by
molecular motors. Despite their critical role, the intricate relationship between motor transport
properties and cargo binding — and its impact on motor distribution — remains inadequately under-
stood. Additionally, improper regulation of ubiquitination, a pivotal post-translational modification
that affects protein degradation, activation, and localization, is associated with several neurodegen-
erative diseases. Recent data showed that ubiquitination can alter motor-cargo binding of the
Kinesin-3 motor UNC-104/KIF1A that transports synaptic vesicles. To investigate how ubiquitin-
like modifications affect motor protein function, particularly cargo binding, transport properties,
and distribution, we utilize the PLM neuron of C. elegans as a model system. Using fluorescent
microscopy, we assess the distribution of cargo-bound UNC-104 motors along the axon and probe
their dynamics using FRAP experiments. We model cargo binding kinetics with a Master equa-
tion and motor density dynamics using a Fokker-Planck approach. Our combined experimental and
theoretical analysis reveals that ubiquitin-like knockdowns enhance UNC-104’s cooperative binding
to its cargo. However, these modifications do not affect UNC-104’s transport properties, such as
processivity and diffusivity. Thus, while ubiquitin-like modifications significantly impact the cargo-
binding of UNC-104, they do not alter its transport dynamics, keeping the homeostatic distribution
of UNC-104 unchanged.
I. INTRODUCTION
Neurons are specialized cells that constitute the ner-
vous system and are responsible for transmitting elec-
trical and chemical signals across an organism. These
cells typically have a cell body and neurites comprising a
parallel arrangement of structurally polar microtubules
with plus ends directed away from the cell body [1,2]
(see Fig. 1(a)). These microtubules are used by motor
proteins (MPs) like kinesins [3] and dyneins [4] as tracks
to transport cargo along axons. In axons, kinesins carry
cargo anterogradely away from the cell body, whereas
dyneins transport cargo retrogradely towards the cell
body (e.g., Fig. 1(b) ) [5,6].
The number of MPs on cargo affects its processive run
length. Increasing the number of kinesins on cargo ex-
tends its duration of processive movement in vitro, but
this is not always observed in vivo [7–9]. Thus, the
consequence of having more MPs on the cargo surface
in vivo remains unclear. In a neuron, optimal cargo
transport away from the cell body may be aided by the
processive motion of MPs. However, previous studies
have shown that cargo transport is frequently bidirec-
tional [10]. Modeling reveals that bidirectional movement
∗Amir Shee and Vidur Sabharwal contributed equally to this work
†Author for correspondence: amitabha@phy.iitb.ac.in
‡Author for correspondence: debc@iopb.res.in
may aid in the circulation of cargo between the cell body
and distal ends of the neuron [11]. Such bidirectional
movements can arise from either (i) a tug-of-war between
opposing MPs [12,13], or (ii) coordination between op-
posing motors, where dynamic switching between oppos-
ing MPs can drive bidirectional motion [14,15]. Indeed,
both of these hypotheses have been validated in different
contexts [16]. Thus, while processive MP movement is
required for transporting cargo, optimal cargo transport
may require precise control over MP binding to cargo
and MP activation. Indeed, MP-cargo binding and MP
activation are under the control of several pathways that
modify the MPs [17–19].
Optimal cargo distribution by MPs depends on their
binding to cargo through the cargo-binding domain and
their efficient transport, which relies on the processive
motion of the motor domain along the filament [20]. If
MPs are not attached to their conjugate filaments, they
can either diffuse freely or hitchhike after binding to a
cargo [21]. Note that MPs must bind to cargo and at-
tach to conjugate filaments simultaneously to mediate
active cargo transport. Three different possibilities may
arise [5,22–24]: (i) good cargo binding but weak fila-
ment binding, (ii) good filament binding but weak cargo
binding, and finally, (iii) processive binding to both cargo
and filament. All these possible scenarios are illustrated
in Fig. 1(c) and may arise in different contexts. As an
example, good cargo binding but weak filament bind-
ing is a typical feature of dynein, where an individual
arXiv:2409.02655v1 [physics.bio-ph] 4 Sep 2024
2
dynein molecule may not generate sufficient processiv-
ity and force for reliable cargo movement, but teams of
dynein can work together to move a cargo [6]. On the
other hand, good filament binding but weak cargo bind-
ing has been observed in the slow transport of soluble cy-
toplasmic proteins – they move by intermittent binding
to motor proteins for directed motion, interspersed with
dissociation and consequent diffusion [25,26]. Finally,
both synaptic vesicle proteins and autophagosomes bind
reliably to motor proteins that move processively along
filaments: synaptic vesicle proteins move anterogradely,
while autophagosomes move retrogradely [27,28]. Ar-
guably, the last possibility, where MPs processively bind
to both cargo and filament, provides the most reliable
transport toward the filament or axon termini but might
reduce material availability closer to the cell body. Note
that the production of new MPs near the cell body and
their subsequent degradation already ensures a possible
steady state. However, the actual shape of the homeo-
static distribution of MPs and their dynamics also de-
pend on their filament processivity and related transport
properties.
Post-translational modifications (PTMs) of motor pro-
teins (MPs) are essential for controlling their amount,
activity, and cargo binding [17,29,30]. In particular,
ubiquitination and ubiquitin-like PTMs play a key role
in degrading and regulating MPs [24,31]. Disruptions
in these PTMs are associated with various neurodegen-
erative diseases [32–34]. Additionally, changes in molec-
ular motors and cargo transport are implicated in the
development and progression of these diseases [35–38].
Therefore, understanding how ubiquitin and ubiquitin-
like PTMs impact motor proteins can provide valuable
insights into axonal cargo transport and highlight poten-
tial targets for therapeutic interventions in neurodegen-
erative diseases.
Recently, using a combination of live imaging of C.
elegans neurons and theoretical analysis, we investi-
gated how different ubiquitin-like modifications affect the
cargo binding of Kinesin-3 MP UNC-104, a KIF1A or-
tholog [31]. The results suggested that MPs may bind to
cargo cooperatively, and the degree of cooperative bind-
ing depends on the ubiquitin-like modifications.
In this paper, we utilize RNAi-mediated knockdowns
and live imaging of C. elegans to study the impact of dif-
ferent levels of ubiquitin-like modifications on the kine-
matics of the UNC-104 motor in the mechanosensory
neurons, the posterior lateral microtubule (PLM) cells.
With the help of a data collapse for cargo-bound MP dis-
tributions and theory, we characterize cooperative cargo
binding and evaluate how the modifications influence this
process. We explore the impact on the homeostatic UNC-
104 MP distribution along the axons and study the UNC-
104 dynamics using localized FRAP. We analyze the ex-
perimental findings using a theoretical model. The exper-
iments and theory allow us to uniquely determine UNC-
104’s effective diffusivity and drift.
The rest of the manuscript is organized as follows:
Sec.II details the experimental system and methods;
Sec.III covers the cargo-binding model and data-collapse
of motor protein distributions; Sec.IV presents the the-
oretical model for motor protein dynamics along axons
and uses experiments to evaluate transport properties;
and Sec.Vsummarizes our findings and provides an out-
look.
II. MATERIAL AND METHODS
A. Worm maintenance
C. elegans was reared on NGM agar seeded with E.
coli OP50 using standard practices [39]. For RNAi-
mediated knockdown, NGM agar was prepared with 100
µg µL−1ampicillin and 1 mM IPTG. These plates were
seeded with the appropriate dsRNA [against either con-
trol empty vector (referred to as wild type), uba-1 or fbxb-
65 ] expressing E. coli HT115 bacteria and incubated for
1 day at 20C before use. After 1 day, 4 C. elegans young
adults were placed and incubated at 20C. Their progeny
at the L4 stage were then used for all experiments.
B. Microscopy
To image UNC-104::GFP distribution, we used a Zeiss
LSM 880 equipped with a 63x/1.4 N.A. oil objective at a
frame size of 1024x1024 pixels leading to a pixel size of 88
nm using a high sensitivity GaAsP detector illuminated
with a 488 nm argon LASER and imaged with a spectral
filter set to a range of 493 to 555 nm. The entire neuron
length was then imaged by tiling across six regions with
a∼10% overlap at 5% LASER power at 488 nm and
2x flyback line-scan averaging. Simultaneously, soluble
mScarlet was imaged using a spectral filter from 585 to
688 nm with a 561 nm DPSS LASER at 5% power at the
same resolution. The images were automatically stitched
using the Zen software.
C. Steady state distribution of UNC-104 along the
axonal length
A line profile using a 3 pixels-wide spline fit polyline
starting from the distal end of the PLM was traced up to
the cell body, and the intensity and distance data were
exported using FIJI [40]. The exported data consists of
the intensity profile along the entire axonal length for (a)
UNC-104 tagged with GFP (UNC-104::GFP), and (b)
mScarlet, which can diffuse freely and acts as a read-out
of the axonal volume. Both (a) and (b) are obtained for
the WT and ubiquitin-like modification knockdown ani-
mals. The exported data is further processed to obtain
the final UNC-104 intensity profile, which is discussed in
the next paragraph.
3
Cargo
UNC-104
Cargo-binding domain
Head domain
Filament
Cargo
UNC-104
Cargo-binding domain
Head domain
•Low cargo-binding affinity
•Low head-binding affinity
Cargo
UNC-104
Cargo-binding domain
Head domain
•High cargo-binding affinity
•High head-binding affinity
Cargo
UNC-104
Cargo-binding domain
Head domain
•High cargo-binding affinity
•Low head-binding affinity
(c)
x= 0 x=L
Cell body Synapse Distal Tip
Process
Posterior Lateral Microtubule (PLM)
(a)
(b)
FIG. 1. Axonal transport in C. elegans (a) Schematic representation of the worm with the mechanosensory neurons indicated by
green lines. The PLM neuron is further shown, indicating the cell body, axon, synapse, and neuronal process. The location of
the cell body and the distal tip are labeled to be at x= 0 and x=L, respectively. The plus and minus signs indicate the polarity
of the aligned microtubules in the cell. (b) Schematic showing multiple cargoes with attached UNC-104 MPs, represented by
cargo-binding domain (in red) and head domain (in blue), which can bind to microtubules and walk along them in the attached
state. When cross-linked to microtubules via MPs, cargo can be transported actively along the filament; otherwise, they diffuse
detached from the filament. (c) Three possible scenarios of active cargo transport: (i) MPs with low cargo-binding affinity
and low filament-processivity form weak cross-link between cargo and filament; (ii) MPs with high cargo-binding affinity and
high filament-processivity lead to strong cross-link between cargo and filament, and (iii) MPs with high cargo-binding affinity
and low filament-processivity produce weak cargo-filament cross-link, represented by dense attachment of UNC-104 MPs to the
cargo but not to the filament.
In Fig. 2, we show the experimental intensity profiles
for the control (wild type) along the axonal length. The
normalized average bare intensity profile ⟨Iunc⟩of UNC-
104::GFP is shown in Fig. 2(a), and the corresponding
normalized average mScarlet fluorophore profile ⟨Imscar⟩
is shown in Fig. 2(b). The intensity Iunc is expected
to be proportional to the number of UNC-104 per unit
length of the axon. This can increase due to an increase
in the density of UNC-104 or an increase in the local ax-
onal volume at a fixed UNC-104 density. Note that the
fluorophore profile gives a measure of the local volume
of the cell along each axon. It varies as the width of
the axon changes from the cell body to the terminal. In
order to get the correct readout of the UNC-104 local-
ization per unit volume, we take the ratio Iunc/Imscar of
the intensity Iunc of UNC-104::GFP to mScarlet inten-
sity Imscar, which provides insight into the distribution
of UNC-104 MPs along the axonal length by mitigating
variability due to axonal volume. The homeostatic pro-
file of ⟨Iunc/Imscar ⟩averaged over tens of cells is shown
in Fig. 2(c).
D. Fluorescence Recovery After Photobleaching
(FRAP) assay
The imaging was done in an LSM 880 using a 40x/1.4
DIC M27 Oil objective at 1.5x zoom with a pixel size of
110 nm. Images were acquired every 500 ms in the ALM
0 200
Axonal Length (µm)
0.000
0.005
0.010
0.015
hIunci
(a)
0 200
Axonal Length (µm)
0.000
0.005
0.010
0.015
hImscari
(b)
0 200
Axonal Length (µm)
0
2
4
hIunc/Imscari
(c)
FIG. 2. (a) Variation of the average bare intensity pro-
file of UNC-104::GFP ⟨Iunc⟩and corresponding (b) fluo-
rophore mScarlet intensity ⟨Imscar⟩for control (WT) exper-
iments along the axonal length. (c) The relative intensity
⟨Iunc/Imscar ⟩profile along the axonal length. All data are
averaged over n= 15 cells.
4
around 100 µm away from the cell body. The photo-
bleaching was done after acquiring six pre-bleach images
using a 488 nm diode laser at a region in the frame 50
µm in length using 80% power (3 mW maximum power
at objective) with five iterations. The imaging was done
for at least 200 frames post-bleach.
Post-acquisition, movies were analyzed using FIJI [40]
by drawing a 3 pixels-wide spline fit polyline. The inten-
sity profile at each time point was exported along with
the ROI of bleaching and the bleach time point using a
custom-built Python script. The analysis of the FRAP
data using our model framework to obtain the diffusion
coefficients is discussed in Sec. IV B
III. THEORETICAL MODEL TO STUDY THE
STEADY STATE DISTRIBUTION OF
CARGO-BOUND KINESIN MOTORS
Recent theoretical analysis of cargo-bound UNC-104
motor proteins indicated a cooperative cargo binding
mechanism [31]. We validate this cooperative binding
model through a compelling data collapse by analyzing
cargo puncta of different sizes, which correspond to vary-
ing numbers of bound motor proteins. Additionally, we
assess how ubiquitin-like modifications influence the ex-
tent of this cooperative binding.
We consider the evolution for the probability P(n, t)
of a cargo bound to nMPs at time t[31,41,42]
˙
P(n, t) = u+(n−1)P(n−1, t) + u−(n+ 1)P(n+ 1, t)
−[u+(n) + u−(n)]P(n, t).(1)
Here, u+(n) and u−(n) denote the rates of MP bind-
ing and unbinding, cross-linking the cargo to the micro-
tubule. We use a pairwise detailed balance at the steady
state, u+(n−1)Ps(n−1) = u−(n)Ps(n) and boundary
condition j+=u−(1)Ps(1) with j+a diffusion-limited
rate of MP cross-linking. This leads to the exact steady
state given by the recurrence relation
Ps(n) = j+
u−(n)
(n−1)
Y
m=1
u+(m)
u−(m)(2)
with the total number of motors N=P∞
n=1 nPs(n).
Cross-linked MPs can detach with a constant rate β
so that u−(n) = βn. The presence of attached MPs
can assist in further MP cross-linking such that u+(n) =
a++b+n, within linear approximation. Here, a+de-
notes the basal attachment rate while b+quantifies the
strength of cooperative binding. Using these expressions
and expanding up to linear order in 1/N we obtain the
following closed-form expression [42]
Hs(n) = A nαe−µn,(3)
where A=j+
βexp(µ), α=a+
β−1 and µ= 1 −b+
β.
0.0 2.5 5.0 7.5 10.0 12.5 15.0
µn
0.0
0.1
0.2
0.3
0.4
˜
Ps(µn)
WT
uba-1
fbxb-65
Theory
FIG. 3. Comparison of the steady-state distribution of the
bound UNC-104 MPs between control (WT) experiments
(black circles) and the uba-1 (orange squares) and fbxb-65
(purple triangles) knockdown cells. When scaled by corre-
sponding µvalues, the abscissa of the normalized intensity
profiles shows a reasonable data collapse that agrees with the
theory Eq. (4) with α= 1 plotted using the black solid line.
The normalized distribution is Ps(n) = N−1
nHs(n)
with Nn=R∞
0Hs(n)dn =A µ−(α+1)Γ(1 + α). This leads
to
˜
Ps(µn) = Γ(1 + α)
µPs(n)=(µn)αexp(−µn).(4)
From the distribution of Ps(n), we note that ¯n=α/µ
denotes both the mode and mean of the distribution, and
its variance is given by σ2
n= 2/µ2. We compare the theo-
retical distribution with the experimental results obtain-
ing α≈1 and µW T = 7.33 ×10−3,µuba1= 4.79 ×10−3
and µfbxb65 = 3.05 ×10−3[31]. Using these values, we
plot the full distributions ˜
Ps(µn) to obtain a data col-
lapse in Fig. 3. The plot of Eq. (4) using a black solid
line agrees well with the experimental data as shown
in Fig. 3. The scaled quantity a+/β ≈2 is indepen-
dent of the RNAi and the cooperative binding strength
(b+/β)W T <(b+/β)uba1<(b+/β)f bxb65.
Moreover, Eq. (1), gives the evolution of average num-
ber of MPs ⟨n(t)⟩cross-linking the cargo to microtubule,
˙
⟨n⟩=a++ (b+−β)⟨n(t)⟩.(5)
Using the initial condition ⟨n(0)⟩=n0, we obtain
⟨n(t)⟩= ¯n−(¯n−n0) exp(−t/τ).(6)
As before, ¯n=α/µ is the steady state average, and
τ= 1/βµ is the relaxation time. This relaxation time
is controlled by cooperative binding and should show a
slower approach for uba-1 and fbxb-65 RNAi that have
reduced UNC-104 ubiquitin-like modification. We note
that the typical relaxation time for the control (wild
5
type) is τW T = 1/βµW T ≈136.43 sec considering the
constant detachment rate of MPs β= 1 sec−1[43]. The
relaxation time also determines the fluctuation correla-
tion at the steady state,
⟨δn(t)δn(0)⟩=σ2
nexp(−t/τ),(7)
where σ2
n= 2/µ2, the variance of steady-state distribu-
tion Ps(n). These predictions can be tested across future
experiments.
IV. THEORETICAL MODEL FOR EVOLUTION
OF UNC-104 DENSITY PROFILE ALONG
AXONS
The UNC-104 transport along microtubules aligned
along axons can be approximated as a one-dimensional
(1d) motion. The MPs can be either bound to micro-
tubules and perform active, directed motion hydrolyzing
ATP or diffuse in a passive manner when detached from
the microtubule. Let us denote these bound and unbound
fractions of MPs by ρb(x, t) and ρu(x, t), respectively.
The bound fraction moves with an active drift velocity
v0. Here, we ignore the stochasticity of motion and the
probabilities of back-stepping by ignoring the diffusion of
the bound fraction. In contrast, the unbound MPs can
only diffuse with diffusion constant D0. Considering the
binding and unbinding kinematics in terms of rates kon
and koff , we get
∂tρb+v0∂xρb=konρu−koff ρb,(8)
∂tρu−D0∂2
xρu=−konρu+koff ρb.(9)
At this stage, we ignored the synthesis and degrada-
tion of MPs, which will be incorporated later. The
above equations can be rewritten in terms of the to-
tal MP density ρ=ρb+ρuand the density difference
m=ρb−ρu. The evolution of ρfollows a conserved
continuity equation. In contrast, the evolution of m(x, t)
gives ∂tm+v0∂xρb+D0∂2
xρu= 2(konρu−koff ρb).In the
presence of the source term on the right-hand side of the
above equation, one can perform an adiabatic elimination
in the hydrodynamic limit to obtain ρu/ρb=koff /kon.
Using the processivity Ω = kon/(kon +koff ) one can write
ρb= Ωρand ρu= (1 −Ω)ρ. This leads to the conserved
dynamics ∂tρ+v∂xρ−D∂2
xρ= 0 where the effective drift
velocity and diffusivity of total MP density are v= Ωv0
and D= (1 −Ω)D0.
Now, we incorporate the other two source terms, the
synthesis and degradation processes of MPs. We consider
the synthesis at the cell body with rate Qand a homo-
geneous degradation with rate γ. Thus, the evolution of
total concentration can be expressed as (see Fig. 1),
∂tρ(x, t) = −v∂xρ+D∂2
xρ+Qδ(x)−γρ. (10)
The Dirac-delta function δ(x) ensures that the motor
proteins are synthesized near the cell body (x= 0). Re-
markably, the above equation is an example of a stochas-
tic resetting process that attracted tremendous recent in-
terest in statistical physics [44]. However, unlike the typi-
cal stochastic resetting examples, in the present context,
the number of particles is not exactly conserved. The
mean value of the total number of MPs n(t) = Rdxρ(x, t)
in the steady state ¯n=Q/γ is determined by the syn-
thesis and degradation rates Qand γ.
A. Steady state distribution
We consider a reflective boundary condition at x=L
leading to −D∂xρ(x, t)|x=L+vρ(x, t)|x=L= 0. Thus,
Eq. (10) has the following steady-state solution (see Ap-
pendix Afor a detailed derivation):
ρs(x) = Qλvλex/λv
2Dsinh(L/λ)he−(L−x)/λ
(λv−λ)+e(L−x)/λ
(λv+λ)i.(11)
Here two effective length scales λv= 2lvand λ=lγ/[1 +
(lγ/λv)2]1/2determine the steady-state profile. In these
expressions, we used the characteristic length scales lv=
D/v and lγ=pD/γ.
A Laplace transform method can be employed to solve
the dynamical equation Eq. (10) in the Laplace space.
While an inverse transform to a closed-form expression
could not be obtained, this solution allows us to de-
termine the relaxation times towards the steady state.
The slowest mode of time-scale γ−1is determined by
the degradation rate γalone. The relaxation times to-
wards steady-state for other relatively faster modes are
γ−1h1 + l2
γ1
λ2
v+γπ2n2
L2i−1for n= 1,2,3, . . . ; see Ap-
pendix B.
The steady-state expression given by Eq. (11) may be
used to fit the experimental profiles. However, we note
that there are four independent parameters (Q, D, v, and
γ) which makes this comparison difficult. We note that
γ≈10−4s−1as known from earlier studies for KIF1A [18,
45,46]. We further eliminate Qusing the total number
of MPs
N=ZL
0
ρs(x)dx =Qλ2λ2
v
D(λ2
v−λ2),(12)
to obtain a normalized steady-state distribution of MPs
expressed in terms of the density profile ρN
s(x) =
ρs(x)/Nand get
ρN
s(x) = (λ2
v−λ2)ex/λv
2λλvsinh (L/λ)he−(L−x)/λ
(λv−λ)+e(L−x)/λ
(λv+λ)i.
(13)
Eq. (13) has only two unknowns, namely Dand v. First,
we estimate Dindependently from the FRAP experi-
ments, which are discussed in the next section. Sec-
ond, we fit normalized experimental distribution using
Eq. (13) to get a single fit parameter vwith known val-
ues of D. Finally, we estimate values of Qutilizing the
fitting of non-normalized experimental distribution using
Eq. (11) with known values of Dand v.
6
B. FRAP Analysis
Although a numerical solution of Eq. (10) for the evo-
lution of the density profile can be obtained, the equa-
tion does not allow a simple closed-form solution. To
analyze the experimental results, we use the following
approach. In experiments, we normalize the evolution
of the UNC-104 intensity profile by the homeostatic pro-
file before photo-bleaching. The resultant profile is flat
to begin with (before photo-bleaching). We observe the
FRAP evolution with respect to the homeostatic profile.
This evolution can be analyzed by simplifying the theo-
retical approach presented in Eq. (10).
For this purpose, we consider the scaled evolution
with respect to the theoretical steady-state, ϕ(x, t) =
ρ(x, t)/ρs(x), over a small domain corresponding to the
FRAP window. This follows,
∂tϕ(x, t) = −v∂xϕ(x, t) + D∂2
xϕ(x, t),(14)
locally, where we began by ignoring the synthesis and
degradation terms for simplicity. This leads to a uniform
steady state ϕs(x) = 1 in the finite domain. To model
FRAP over a window of size 2a, we analyze the evolution
towards steady-state starting from the initial condition
ϕ(x0,0) = 0 for −a≤x0≤a,
= 1 for |x0|> a. (15)
The Greens function corresponding to Eq. (14) is G(x−
x0, t) = 1
√4πDt e−(x−x0−vt)2/4D t. This, along with the
initial condition in Eq. (15), leads to the solution
ϕ(x, t) = 1
22−erf a+vt −x
√4Dt −erf a−vt +x
√4Dt ,
(16)
where erf(x) = 2
√πRx
0e−s2ds. Now, bringing back the
synthesis and degradation terms, the same analysis leads
to a solution
ϕ(x, t)
=e−γt
22−erf a+vt −x
√4Dt −erf a−vt +x
√4Dt
+QΘ(t)
ρ0γ(1 −e−γt ),(17)
where ρ0=ρs(x= 0). However, since the degrada-
tion rate γis known to be small, the intensity decay due
to it over the short FRAP duration is negligible, and
Eq. (17) reduces to Eq. (16) apart from an additive con-
stant Q/ρ0γ. The recovery of the profile ϕ(x, t) is medi-
ated by effective diffusion D, degradation γ, and effective
drift v.
We utilize the experimental intensity evolution dur-
ing FRAP, after dividing by the homeostatic intensity
profile, to estimate diffusion. We extract the diffusion
−25 0 25
x(µm)
0.0
0.5
1.0
1.5
φ(x, t)
(a) 1 sec
31 sec
92 sec
0 50 100 150
t(sec)
0.0
0.2
0.4
0.6
0.8
φ(0, t)
t1/4t1/2
(b)
WT uba-1 fbxb-65
0
5
10
15
D(µm2sec−1)
(c)
FIG. 4. Fluorescence Recovery After Photobleaching
(FRAP): (a) The normalized intensity profile ϕ(x, t), scaled
by the steady-state intensity, during FRAP at elapsed time
1 sec (open circles), 31 sec (open squares), and 92 sec (open
triangles). Symbols represent intensity profiles in the control
(WT) cells. The solid (using Eq. (20)) and dashed (using
Eq. (19)) lines represent the plot of Eq. (16) using the fitted
diffusion coefficient D, the FRAP window size w= 2aand ig-
noring vt ≪w, to compare against experimental results. The
region in violet shade indicates the range over which pho-
tobleaching is performed. A dynamical comparison between
experiment, numerical solution, and theory is shown in the
Supplemental Material Movie-1 [47]. (b) Time evolution of
the intensity ϕ(0, t) obtained by averaging over the narrow
gray window of size ϵindicated in (a). The diffusion coeffi-
cient Dcan be calculated using half recovery time in Eq. (19)
and the quarter recovery time in Eq. (20), both calculated
from the experimental mean integrated intensity evolutions
(open green points) similar to that shown in (b). The lines
are the plot of Eq. (18) using estimated Dby Eq. (19) (dashed
black line) and by Eq. (20) (solid black line). (c) Violin plots
of the diffusion constants obtained via t1/4using Eq. (20),
showing that the diffusivities do not depend on the ubiquitin-
like knockdowns.
coefficient by directly fitting Eq. (16) to the evolution of
this intensity profile. Finally, by averaging over many
realizations, i.e., different cells, we obtain the diffusion
coefficient D(Table I). While performing such fitting,
one can neglect the small directed displacement vt of the
local density profile due to drift over the relatively short
recovery time tcompared to the extent of the FRAP
window w= 2a, assuming vt ≪w. However, as it turns
out, a strong intensity fluctuation makes it difficult to
estimate Dreliably through this procedure.
To reduce such systematic error, we use the evolution
of an average intensity over a small window near the mini-
mum intensity spot at the beginning of FRAP (Fig. 4(a)).
Setting x= 0, vt ≪a, and γt ≈0, Eq. (16) gives
ϕ(0, t) = 1−erf a
√4Dt .(18)
We estimate the diffusion constant Dby comput-
7
ing the half recovery time t1/2of the center of the
FRAP region defined over a small window of size ϵ=
4µm, where ϵ≪2a(considering ϵwindow size as
10% of FRAP window size 2a) with 2a≈40 µm. Thus,
ϕ(0, t1/2) = (1/2)ϕ(0,0) (see Fig. 4(b)), we find the val-
ues of Dusing
D= 0.275(2a)2
t1/2
.(19)
In our experiments, the recovery intensity turns out
to be less reliable at longer times due to large inten-
sity fluctuations, and even determining τ1/2can be dif-
ficult for some experimental data. For them, we use
ϕ(0, t1/4) = (1/4)ϕ(0,0) (see Fig. 4(b)) and the corre-
sponding relation
D= 0.094(2a)2
t1/4
.(20)
We have estimated the effective diffusion coefficient D
using both t1/2and t1/4and found that diffusion coef-
ficients were comparable (see Fig. 8(c) and Table II in
Appendix B). Using the estimated Dvalues for the WT
(both via t1/2and t1/4), we compute ϕ(x, t) both nu-
merically (solving Eq. (14)) as well as analytically (us-
ing Eq. (16)), neglecting the small advection. We show
a comparison of the evolution at different time points
of the FRAP experiments with these analytic estimates,
which are shown in Fig. 4(a) (see Supplemental Mate-
rial Movie-1 [47] for a comparison of the complete time
evolution from experiment, numerical solutions and an-
alytic expression). We also compare the theoretical time
trajectories for ϕ(0, t) with the corresponding FRAP ex-
periment; see Fig. 4(b). We note that for this specific
case, ϕ(0, t1/4) agrees better with the FRAP experiment
at early times (up to ≲60 sec) and deviates at large times
when the recovery intensity is unreliable. For the fbxb-
65 knockdown experiments, the Dvalues estimated using
t1/4are statistically more reliable than the estimate using
t1/2, while for the WT and uba-1 knockdown, the statis-
tics are equivalent (see Appendix. Cfor details). The
corresponding distributions of Dvalues, estimated using
t1/4, across different cells are compared between the WT
and the two knockdown experiments in Fig. 4(c) (see also
the first row of Table I). Comparing the values of Dbe-
tween the WT and the knockdown experiments, we notice
that the relative variation in the value of Dacross exper-
iments is ∆D= (DW T −DKD )/DW T ≤10%, where W T
and KD stand for wild type and RNAi treated ubiquitin-
like knockdowns respectively.
The FRAP analysis thus suggests that cells with al-
tered ubiquitin-like knockdowns of either uba-1 or fbxb-
65 do not affect the MP’s diffusional transport proper-
ties. Interestingly, due to the exponential form of the
steady state profile ρs(x), a FRAP performed on top of
this profile can lead to an apparent emergent retrograde
bias in the recovery profile (see Appendix Dand Movie-2
in [47]). This is a general physical effect that could be
misleading and is, therefore, crucial to remember while
analyzing any FRAP data. This is discussed in detail
using our FRAP analysis in Appendix D.
C. Active transport is crucial in determining the
steady-state intensity profiles
With the diffusion coefficients Dand the degradation
rate γknown, we can now determine the effective motor
speed vfrom the experimental steady-state distribution
of the UNC-104 motors along the axon. To make a com-
parative study with our 1d theory, we must ensure that
the steady-state distribution of MPs is defined purely as
a one-dimensional density profile. This is discussed in
Sec. II C, where we corrected for the local volume vari-
ability of axons by scaling the experimentally obtained
homeostatic intensity profiles for the MPs by the corre-
sponding mScarlet fluorophore profiles for a given cell.
This density further averaged over n= 15 cells (see
Fig. 2(c)) gives a read-out for the steady-state local den-
sity profile ρs(x). Finally, we normalize the distribution
ρs(x) to obtain ρN
s(x) to eliminate the source term Q
and is used for comparison with the theoretical model.
Recall the effective motor speed is defined as v= Ωv0.
The velocity v0of individual MPs obtained from live
imaging of UNC-104::GFP-decorated puncta are esti-
mated to be ∼1µm/s [31]. Assuming that the individual
motor protein properties do not change, the unknown
parameter vcan, therefore, be treated as a read-out for
the value of processivity Ω. Thus, to estimate Ω (via
v), we further normalize the experimental steady-state
distributions with respect to the spatially integrated in-
tensity and compare them to Eq. (13). The results are
shown in Fig. 5(a), 5(b), and 5(c), which display a good
fit in all the cases. Note that here we use only a single
parameter fit to vand hence obtain Ω. The values of the
processivity Ω obtained from these fits are reported in
Table I.
Several intriguing observations emerge from the fit re-
sults. First, it is evident that the transport parameters D
and vremain largely unchanged by ubiquitin-like knock-
downs, maintaining similar approximate values across all
cases, including the wild type. Secondly, the processiv-
ity (Ω ≈0.05) is low, indicating that only 5% of avail-
able motor proteins participate in transport. This also
indicates that the intrinsic diffusivity D0=D/(1 −Ω)
is mainly due to the detached fraction of MPs and is
approximately equal to the measured values. Addition-
ally, we note that the integrated value of the MP density
TABLE I. Diffusion coefficient D(µm2s−1) and processivity
Ω with their relative errors estimated from the experiments.
WT uba-1 fbxb-65
D6.41 ±7% 5.94 ±8% 6.37 ±6%
Ω 0.05 ±3% 0.05 ±3% 0.04 ±2%
8
0 100 200
x(µm)
0.000
0.005
0.010
0.015
ρN
s(x)
WT
(a)
0 100 200
x(µm)
0.000
0.005
0.010
0.015
ρN
s(x)
uba-1
(b)
0 100 200
x(µm)
0.000
0.005
0.010
0.015
ρN
s(x)
fbxb-65
(c)
FIG. 5. Steady-state distribution of UNC-104 MPs: The normalized steady-state experimental distribution (open gray points)
for control (WT) in (a), uba-1 knockdown cells in (b), and fbxb-65 knockdown cells in (c) are fitted with Eq. (13) (red dashed
line) to estimate the single fitting parameter Ω as shown in Table-I. Here we used the known value of γ(= 10−4sec−1) and
estimated diffusion coefficient Dfrom FRAP experiments (see Table-I).
N=RL
0ρs(x) (see Eq. (12)) remains almost unchanged.
Since this term provides a read-out for the source term
Q, we conclude that Qis invariant to uba-1 and fbxb-65
knockdown cells [48].
The processivity Ω turns out to be the most impor-
tant parameter regulating the steady-state distribution.
From the expression in Eq. (13), we note that if Ω = 0,
the length scale of the steady state exponential distribu-
tion is set by lγand attains the maximum at the source
(if lγ< L) and the minimum at the posterior end (see
Fig. 6(a)). Upon slowly increasing Ω, the minimum po-
sition of ρs(x) starts moving from x=Lto lower values
of x(see Fig. 6(b)–(d) ) until at Ω = 0.05, the ρN
s(x)
profile is already reversed and the length-scale is set by
the competition between lγand lv(see Fig. 6(c)). This
profile is similar to our experimental observation. It gets
even sharper at higher Ω, with most of the MPs accumu-
lating near the distal end; see Fig. 6(d). Moreover, as the
inset of Fig. 6(d) shows, the minimum of the profile shifts
back to positive values at larger Ω. The inversion of the
density profile with the maximum shifting from proximal
to distal end with increasing processivity is further illus-
trated using a heat-map in Fig. 6(e). Fig. 6(f) shows the
non-monotonic variation of the minimum density and the
location of this minimum with increasing Ω.
We can thus conclude that despite ubiquitin-like
knockdowns of MP having a drastic effect on cooperative
binding, the steady-state MP distribution regulated by
the effective diffusivity and speed, which in turn are de-
pendent on the effective processivity, remains completely
unaffected.
V. DISCUSSION
Directed axonal transport of synaptic vesicles by MPs
is crucial for the proper functioning of an organism, fail-
ure of which is associated with neurodegenerative disor-
ders [35–38]. To ensure a robust and reliable transport
of cargo, it is important to understand what controls the
steady-state distributions of the MPs and what modifi-
cations can affect the motor’s intrinsic transport proper-
ties As was shown earlier and further consolidated here
using a convincing data collapse, knockdown of the E1
activating enzyme uba-1 or the E3 ligase fbxb-65 causes
increased cooperative binding of UNC-104 to cargo. In
this work, we further show that despite such a change in
cargo binding, the same ubiquitin-like knockdowns leave
the UNC-104 distribution along the axon and their trans-
port properties, like effective drift velocity and diffusivity,
unchanged.
We quantified UNC-104 steady-state distribution in
wild-type neurons and upon RNAi knockdowns of either
uba-1 or fbxb-65. The steady-state profile shows a larger
accumulation of MPs in the distal end for all the cases.
Normalization of the raw intensity data with the corre-
sponding axonal volume showed a nice collapse of the
data from different RNAi knockdowns on each other, in-
dicating that the homeostatic distribution remains un-
changed under ubiquitin-like knockdowns.
We proposed a theoretical model which clearly ex-
plains the nature of such a homeostatic profile. Synthesis,
degradation, and transport, both directed drift and dif-
fusion, control the steady-state density profile along the
axon. Our model allowed us to estimate the drift and
diffusion utilizing the homeostatic distribution of UNC-
104 along the axons and analysts of FRAP results. We
first estimated the diffusion coefficient Dfrom the FRAP
experiments. Using the known Dvalues, we fitted the
steady-state profiles to obtain estimates for the effective
drift velocity v, which is a readout for the processivity
Ω = v/v0. Our analysis clearly shows that both the
diffusivity and the processivity remain unchanged under
ubiquitin-like knockdowns.
9
0.0 0.5 1.0
x/L
0.30
0.35
0.40
0.45
0.50
102×ρN
s
(a)
Ω=0.0
0.0 0.5 1.0
x/L
0.34
0.36
0.38
0.40
0.42
(b)
Ω=0.01
0.0 0.5 1.0
x/L
0.2
0.4
0.6
0.8
(c)
Ω=0.05
0.0 0.5 1.0
x/L
0
2
4
6
8
(d)
Ω=0.5
0.0 0.5
0.020
0.021
10−310−210−1100
Ω
0.0
0.5
1.0
xmin/L
(f)
0.0
0.2
0.4
102×ρN,min
s
FIG. 6. Normalized steady-state distribution of MPs ρN
s(x)
(see Eq. (13)) for different values of processivity Ω indicated
in figures (a)-(d). The distribution in (c) corresponds to the
WT (see Fig. 5(a)). The zoomed-in plot in the inset in (d)
highlights the minimum in the profile. (e) Color map of ρN
sas
a function of position xand processivity Ω. (f) The minimum
of the MP distribution ρN,min
s(purple solid line) and its lo-
cation on the axon at different processivities. The red points
mark the minima at zero, and the black points identify the
minima at intermediate positions. The shaded region in (f)
indicates the experimentally realized range 0.04 ≤Ω≤0.05.
Since these kinesin-3 MPs can hitchhike on retro-
gradely moving cargo carried by dynein MPs, the kinetic
properties measured using the overall fluorescent inten-
sity from UNC-104 not only depend on their active mo-
tion along microtubules but also include the impact of
such retrograde flux. We find that ubiquitin-like knock-
downs increase cooperative cargo binding, thereby poten-
tially increasing the impact of the hitchhiked retrograde
motion. Consistent with this, we previously observed a
higher occurrence of UNC-104 retrograde movement un-
der ubiquitin-like knockdowns [31]. Surprisingly, the ef-
fective kinetic properties of UNC-104 remain unchanged
under the same ubiquitin-like knockdowns. Such inde-
pendence suggests a subtle regulation nullifying the ef-
fect of potentially increased retrograde flux. Note that
the cargo-bound MPs may support each other’s micro-
tubule processivity by sheer localization on the cargo.
AUTHOR CONTRIBUTIONS
All authors contributed to the design of the research.
VS performed experiments under the supervision of SPK.
DC and AN designed the theoretical framework and anal-
ysis; AS performed numerical calculations; AS, AN, and
DC analyzed the data. All authors discussed the results
and wrote the manuscript.
DATA AVAILABILITY
All data that support the findings of this study are
included within the article (and any supplementary files).
ACKNOWLEDGEMENTS
AN and DC thank Madan Rao for insightful com-
ments. DC thanks Sanjib Sabhapandit and Fernando
Peruani for useful discussions. DC acknowledges re-
search grants from the Department of Atomic En-
ergy (OM no. 1603/2/2020/IoP/R&D-II/15028) and
Science and Engineering Research Board (SERB), In-
dia (MTR/2019/000750) and thanks the International
Centre for Theoretical Sciences (ICTS-TIFR), Banga-
lore, for an Associateship. AN acknowledges SERB
India (MTR/2023/000507) for financial support and
thanks the Max-Planck Institute for the Physics of Com-
plex Systems (MPIPKS), Dresden, for hospitality and
support during summer visit in 2024. Research in
SPK’s lab is supported by grants from DAE (OM no.
1303/2/2019/R&D-II/DAE/2079; Project identification
number RTI4003 dated 11.02.2020), PRISM (12-R&D-
IMS- 5.02-0202), and a Howard Hughes Medical Institute
International Early Career Scientist Grant (55007425).
AS acknowledges partial financial support from the John
Templeton Foundation, Grant 62213.
Appendix A: Steady-state distributions
We recast the governing equation (10) in terms of cur-
rent J=−(D∂x−v)ρand considering uniform degrada-
tion γ(x) = γconstant,
∂tρ=−∂xJ−γρ , (A1)
with the boundary condition of total current at x= 0,
Jx=0 =Qwhere Qis the constant source and total cur-
rent at x=L,Jx=L= 0.
The steady-state limit (∂tρ= 0) gives
∂2
xρs−1
lv
∂xρs−1
l2
γ
ρs= 0 (A2)
where the constants lv=D/v and lγ=pD/γ. Substi-
tuting ρs∝eαx in the Eq. (A2) leads to the quadratic
10
equation α2−α/lv−1/l2
γ= 0 with solution α=
lγ±(4l2
v+l2
γ)1/2/2lvlγ. Thus, steady-state density,
ρs(x) = ex/2lv"Ae
x√4l2
v+l2
γ
2lvlγ+Be
−x√4l2
v+l2
γ
2lvlγ#(A3)
We recast again with λv= 2lvand λ=
2lvlγ/q4l2
v+l2
γ=lγ(1 + l2
γ/λ2
v)−1/2. Thus,
ρs(x) = ex/λvhAex/λ +Be−x/λi(A4)
Now, the total current at x= 0, Jx=0 =Qlead to
(λ−λv)A+ (λ+λv)B=Qλλv
D,(A5)
and the total current at x=L,Jx=L= 0 lead to
(λ−λv)eL/λA+ (λ+λv)e−L/λ B= 0 .(A6)
Now, solving Eqs. (A5) and (A6), we get
A=Qλλv
D(λv−λ)(e2L/λ −1) ,
B=Qλλve2L/λ
D(λ+λv)(e2L/λ −1) .
Substituting Aand Bback in the Eq. (A4), we get
ρs(x) = Qλvλ
D(e2L/λ −1)ex/λvex/λ
(λv−λ)+e(2L−x)/λ
(λv+λ),
(A7)
which can be rewritten as:
ρs(x) = Qλvλex/λv
2Dsinh(L/λ)he−(L−x)/λ
(λv−λ)+e(L−x)/λ
(λv+λ)i.(A8)
This is precisely Eq. (11) of the main text.
Appendix B: Laplace transform, singularities and
slow dynamics
The equation of motion for the density can be
directly solved using Laplace transform ˜ρ(x, s) =
Rt
0dt′e−st′ρ(x, t′) to get
˜ρ(x, s) = Qλvλ(s)ex/λv
sD(e2L/λ(s)−1) ex/λ(s)
(λv−λ(s)) +e(2L−x)/λ(s)
(λv+λ(s))
(B1)
where λ(s) = lγ1 + l2
γ
λ2
v+s
γ−1/2
. It is easy to see that
the above expression gives the steady-state result ρs(x) =
lims→0s˜ρ(x, s).
The time dependence is given by the inverse Laplace
transform, ρ(x, t) = 1
2πi Rc+i∞
c−i∞ds est ˜ρ(x, s) with c > 0
so that the function remains analytic on the right half of
the complex plane. To perform the contour integration,
we first need to analyze the structure of singularities of
the integrand.
Due to the presence of λ(s) in the numerator, ˜ρ(x, s)
has a branch point at sb=−γ1 + l2
γ
λ2
v. Moreover, it
has simple poles at (e2L/λ(s)−1) = 0; using 1 = ei2nπ
this gives simple poles at sn=sb−ℓγ
L2γn2π2with
n= 0,1,2, . . . , all lying on the real axis and to the left
of sb. Further, as can be seen from Eq. (B1), ˜ρ(x, s) has
a simple pole at s= 0, and at s∗=sb+ℓ2
γγ
λ2
v=−γ < 0
(using λv=±λ(s)) between s= 0 and s=sb.
Thus, the solution is
ρ(x, t) = ρs(x) + ρ1(x)es∗t+∞
X
n=0
ρn(x)esnt,
+esbtL−1˜ρ(x, s +sb) (B2)
so that ρ1(x) = lims→s∗(s−s∗)˜ρ(x, s), etc. The last term
is the inverse Laplace transform of the frequency-shifted
function, arising due to the branch point at sb<0. The
slowest mode of evolution is limt→∞[ρ(x, t)−ρs(x)] →
ρ1(x)e−γt is controlled by the degradation rate γ. The
reason for this becomes immediately clear by noting from
Eq. (10) that the total quantity of MPs N=Rdx ρ(x)
evolves as dN/dt =Q−γN.
Appendix C: Comparison of diffusivities from FRAP
analysis
In Fig. 7, we show the scatter plot of diffusivities es-
timated using both t1/2(Eq. (19)) and t1/4(Eq. (20))
for control (WT) along with uba-1 and fbxb-65 knock-
down cells. The corresponding comparison for the mean
values of effective diffusivities with their standard errors
is shown in Table II. For both control (WT) and uba-1
knockdown, Dis estimated by averaging over 22 inde-
pendent experimental realizations. However, for the case
of fbxb-65 knockdown, Dmeasured via t1/4is averaged
over all 23 realizations, whereas the one via t1/2was aver-
aged over 9 realizations. This is because the Destimate
via t1/2was relatively unreliable in 14 cases due to large
intensity fluctuations.
TABLE II. Comparison of diffusion coefficients D(in µm2
sec−1) calculated using t1/4and t1/2for control (WT) and
cells with uba-1, and fbxb-65 knockdowns.
RNAi D(t1/4)D(t1/2)
WT 6.41 ±0.46 7.02 ±0.50
uba-1 5.94 ±0.49 6.73±0.50
fbxb-65 6.37 ±0.41 6.33 ±0.90
11
0
2
4
6
8
10
12
14
16
D(µm2sec−1)
WT uba-1 fbxb-65
(a)
t1/4
t1/2
FIG. 7. Comparisons of the measured diffusion coefficients D
for control (WT) and the two types of RNAi cells using the
quarter t1/4and half recovery times t1/2.
Appendix D: Further insights from the model:
Apparent retrograde motion under FRAP
To characterize the intensity recovery dynamics ob-
served in the FRAP experiments, we numerically inte-
grate Eq. (10) for an initial condition that is equiva-
lent to the density profile just after FRAP. This is con-
structed as follows: starting with the steady-state pro-
file defined in [0, L], we choose a small window of size
w= 2a= ∆Laround the center of the profile where
the local density is set to zero (Fig. 8). The parame-
ters are chosen corresponding to the WT case: we use
the transport coefficients Dand vfrom Table Iand the
degradation rate γ= 10−4sec−1. We use the typical
axon length L= 274.74 µm and FRAP window size
w= 2a= ∆L= 39.25 µm. To integrate Eq. (10),
we use time step dt = 0.0001 sec, set the source term
Q= 0.035 sec−1at x= 0 and use a reflecting boundary
at x=L. The resultant MP distribution is finally nor-
malized over the axonal length so that RL
0ρN(x)dx = 1.
In Fig. 8(a)-(d), the recovery of the density profile af-
ter FRAP is shown at four different times and the corre-
sponding local currents JN(x, t) are shown in Fig. 8(e)-
(f) respectively. The approach toward the steady-state
profile (shown by the dotted lines in Fig. 8(a)-(d)) is evi-
dent from these figures, and at long times (see Fig. 8(d)),
the MP distribution almost overlaps with the steady-
state profile. The recovery is further evident from the
time evolution of local current JN(x, t). Just after the
FRAP, JN(x, t) exhibit large positive and negative peaks
corresponding to the two edges of the FRAP region (see
Fig. 8(e)). As time progresses, due to the recovery in
depleted density, the peaks start receding until they dis-
appear at long times (see Fig. 8(f)-(h)) when JN(x, t) ap-
proach to the steady-state value. It is important to note
0.000
0.005
0.010
ρN
1 sec
(a)
0.000
0.005
0.010 (b)
1 min
0.000
0.005
0.010 (c)
2 min
0.000
0.005
0.010 (d)
10 min
0.0 0.5 1.0
x/L
−0.1
0.0
0.1
JN
(e)
0.0 0.5 1.0
x/L
−0.02
0.00
0.02 (f)
0.0 0.5 1.0
x/L
−0.01
0.00
0.01
0.02 (g)
0.0 0.5 1.0
x/L
0.000
0.005
0.010 (h)
0 200 400 600
t(sec)
0.000
0.001
0.002
ρN
min
(i)
0 200 400 600
t(sec)
0.0
0.2
0.4
xmin/L
(j)
FIG. 8. The spatiotemporal evolution for numerical calcu-
lation of FRAP of normalized spatial density ρN(x, t) (a-d)
and normalized current JN(x, t) (e-h) density profile of WT
at times (a),(e) t= 1 sec, (b),(f) 1 min, (c),(g) 2 min and
(d),(h) 10 min. The dashed black lines in (a)-(d) denote the
normalized steady-state density profiles (Eq. (13)) for com-
parison with the evolution. The plot of minimum density as
a function of time is shown in (i), and the position of the
minimum density is shown in (j).
that at steady state, JN
s(x) is not a constant due to the
presence of a non-zero source Qat one end. We also fo-
cus on the relaxation of the minimum density value ρN
min
(see Fig. 8(i)) and its location xmin (see Fig. 8(j)) along
the axon. This indicates that the shape of the steady-
state density profile almost recovers at around t∼5 min.
Moreover, the decrease of xmin before vanishing captures
an apparent retrograde motion of the location of FRAP
minimum against the direction of MP flux. Similar be-
havior is observed in some of our FRAP experiments.
This apparent retrograde movement is entirely due to
the non-uniform shape of the steady-state distribution to
which the perturbation under FRAP relaxes back. The
relaxation of the profile during FRAP towards the ap-
proximate exponential steady-state profile led to the ap-
parent retrograde motion of the FRAP center in the cur-
rent example. This is further demonstrated in Movie-2
presented in the Supplemental Material [47], where we
numerically show the FRAP region’s evolution with the
minimum density point labeled by a filled square. Such
apparent movement of the FRAP center is expected in
any FRAP experiment on non-homogeneous steady-state
profiles and, thus, must be taken into consideration while
analyzing and interpreting such experimental results.
12
[1] P. R. Burton and J. L. Paige, Polarity of axoplasmic mi-
crotubules in the olfactory nerve of the frog., Proceedings
of the National Academy of Sciences 78, 3269 (1981).
[2] S. R. Heidemann, J. M. Landers, and M. A. Hamborg,
Polarity orientation of axonal microtubules., The Journal
of cell biology 91, 661 (1981).
[3] R. D. Vale, T. S. Reese, and M. P. Sheetz, Identification
of a novel force-generating protein, kinesin, involved in
microtubule-based motility, Cell 42, 39 (1985).
[4] R. B. Vallee, J. S. Wall, B. M. Paschal, and H. S. Sh-
petner, Microtubule-associated protein 1c from brain is
a two-headed cytosolic dynein, Nature 332, 561 (1988).
[5] R. D. Vale, The Molecular Motor Toolbox for Intracellu-
lar Transport, Cell 112, 467 (2003).
[6] A. K. Rai, A. Rai, A. J. Ramaiya, R. Jha, and R. Mallik,
Molecular adaptations allow dynein to generate large col-
lective forces inside cells, Cell 152, 172 (2013).
[7] J. Beeg, S. Klumpp, R. Dimova, R. S. Gracia, E. Unger,
and R. Lipowsky, Transport of beads by several kinesin
motors, Biophysical journal 94, 532 (2008).
[8] G. T. Shubeita, S. L. Tran, J. Xu, M. Vershinin, S. Cer-
melli, S. L. Cotton, M. A. Welte, and S. P. Gross, Conse-
quences of motor copy number on the intracellular trans-
port of kinesin-1-driven lipid droplets, Cell 135, 1098
(2008).
[9] J. O. Wilson, A. D. Zaragoza, and J. Xu, Tuning
ensemble-averaged cargo run length via fractional change
in mean kinesin number, Physical biology 18, 046004
(2021).
[10] M. Martin, S. J. Iyadurai, A. Gassman, J. G. Gind-
hart Jr, T. S. Hays, and W. M. Saxton, Cytoplasmic
dynein, the dynactin complex, and kinesin are interde-
pendent and essential for fast axonal transport, Molecu-
lar biology of the cell 10, 3717 (1999).
[11] I. A. Kuznetsov and A. V. Kuznetsov, Bidirectional, un-
like unidirectional transport, allows transporting axonal
cargos against their concentration gradient, Journal of
Theoretical Biology 546, 111161 (2022).
[12] V. Soppina, A. K. Rai, A. J. Ramaiya, P. Barak, and
R. Mallik, Tug-of-war between dissimilar teams of mi-
crotubule motors regulates transport and fission of endo-
somes, Proceedings of the National Academy of Sciences
106, 19381 (2009).
[13] A. G. Hendricks, E. Perlson, J. L. Ross, H. W. Schroeder,
M. Tokito, and E. L. Holzbaur, Motor coordination via a
tug-of-war mechanism drives bidirectional vesicle trans-
port, Current Biology 20, 697 (2010).
[14] M. J. I. M¨uller, S. Klumpp, and R. Lipowsky, Tug-of-
war as a cooperative mechanism for bidirectional cargo
transport by molecular motors, Proceedings of the Na-
tional Academy of Sciences 105, 4609 (2008).
[15] P. D. Chowdary, D. L. Che, K. Zhang, and B. Cui, Ret-
rograde ngf axonal transport—motor coordination in the
unidirectional motility regime, Biophysical journal 108,
2691 (2015).
[16] O. Munoz and S. Klumpp, Tug-of-war and coordination
in bidirectional transport by molecular motors, The Jour-
nal of Physical Chemistry B 126, 7957 (2022).
[17] L. Guillaud, R. Wong, and N. Hirokawa, Disruption of
kif17–mint1 interaction by camkii-dependent phosphory-
lation: a molecular model of kinesin–cargo release, Na-
ture cell biology 10, 19 (2008).
[18] T. Mathieson, H. Franken, J. Kosinski, N. Kurzawa,
N. Zinn, G. Sweetman, D. Poeckel, V. S. Ratnu,
M. Schramm, I. Becher, M. Steidel, K.-M. Noh,
G. Bergamini, M. Beck, M. Bantscheff, and M. M. Sav-
itski, Systematic analysis of protein turnover in primary
cells, Nature Communications 9, 689 (2018).
[19] J. T. Kevenaar, S. Bianchi, M. van Spronsen, N. Olieric,
J. Lipka, C. P. Frias, M. Mikhaylova, M. Harterink,
N. Keijzer, P. S. Wulf, et al., Kinesin-binding protein
controls microtubule dynamics and cargo trafficking by
regulating kinesin motor activity, Current Biology 26,
849 (2016).
[20] C. S. Mitchell and R. H. Lee, Cargo distributions differen-
tiate pathological axonal transport impairments, Journal
of theoretical biology 300, 277 (2012).
[21] T. L. Blasius, N. Reed, B. M. Slepchenko, and K. J.
Verhey, Recycling of kinesin-1 motors by diffusion after
transport, PloS one 8, e76081 (2013).
[22] D. R. Klopfenstein, M. Tomishige, N. Stuur-
man, and R. D. Vale, Role of Phosphatidylinos-
itol(4,5)bisphosphate Organization in Membrane
Transport by the Unc104 Kinesin Motor, Cell 109, 347
(2002).
[23] F. S. Rizalar, M. T. Lucht, A. Petzoldt, S. Kong, J. Sun,
J. H. Vines, N. S. Telugu, S. Diecke, T. Kaas, T. Bull-
mann, C. Schmied, D. L¨owe, J. S. King, W. Cho,
S. Hallermann, D. Puchkov, S. J. Sigrist, and V. Haucke,
Phosphatidylinositol 3,5-bisphosphate facilitates axonal
vesicle transport and presynapse assembly, Science 382,
223 (2023).
[24] J. Kumar, B. C. Choudhary, R. Metpally, Q. Zheng,
M. L. Nonet, S. Ramanathan, D. R. Klopfenstein, and
S. P. Koushika, The Caenorhabditis elegans Kinesin-
3 Motor UNC-104/KIF1A Is Degraded upon Loss of
Specific Binding to Cargo, PLoS Genetics 6, e1001200
(2010).
[25] Y. Tang, D. Scott, U. Das, D. Gitler, A. Ganguly, and
S. Roy, Fast vesicle transport is required for the slow
axonal transport of synapsin, Journal of Neuroscience 33,
15362 (2013).
[26] S. Roy, M. J. Winton, M. M. Black, J. Q. Trojanowski,
and V. M.-Y. Lee, Rapid and intermittent cotransport of
slow component-b proteins, Journal of Neuroscience 27,
3131 (2007).
[27] J. Li, R. Jahn, and A. Dahlstr¨om, Rab3a, a small gtp-
binding protein, undergoes fast anterograde transport
but not retrograde transport in neurons., European jour-
nal of cell biology 67, 297 (1995).
[28] S. Maday, K. E. Wallace, and E. L. F. Holzbaur, Au-
tophagosomes initiate distally and mature during trans-
port toward the cell soma in primary neurons, Journal of
Cell Biology 196, 407 (2012).
[29] H. Matthies, R. J. Miller, and H. C. Palfrey, Calmodulin
binding to and camp-dependent phosphorylation of ki-
nesin light chains modulate kinesin atpase activity, Jour-
nal of Biological Chemistry 268, 11176 (1993).
[30] D. M. Gordon and D. M. Roof, Degradation of the kinesin
Kip1p at anaphase onset is mediated by the anaphase-
promoting complex and Cdc20p, Proceedings of the Na-
tional Academy of Sciences 98, 12515 (2001).
13
[31] V. Sabharwal, S. P. P. Boyanapalli, A. Shee, M. L.
Nonet, A. Nandi, D. Chaudhuri, and S. P. Koushika, F-
box protein FBXB-65 regulates anterograde transport of
the kinesin-3 motor UNC-104 through a PTM near its
cargo-binding PH domain, Journal of Cell Science 137,
10.1242/jcs.261553 (2024).
[32] G. Perry, R. Friedman, G. Shaw, and V. Chau, Ubiquitin
is detected in neurofibrillary tangles and senile plaque
neurites of alzheimer disease brains., Proceedings of the
National Academy of Sciences 84, 3033 (1987).
[33] S. Nakazawa, D. Oikawa, R. Ishii, T. Ayaki, H. Taka-
hashi, H. Takeda, R. Ishitani, K. Kamei, I. Takeyoshi,
H. Kawakami, et al., Linear ubiquitination is involved
in the pathogenesis of optineurin-associated amyotrophic
lateral sclerosis, Nature communications 7, 12547 (2016).
[34] J. S. Steffan, N. Agrawal, J. Pallos, E. Rockabrand, L. C.
Trotman, N. Slepko, K. Illes, T. Lukacsovich, Y.-Z. Zhu,
E. Cattaneo, et al., Sumo modification of huntingtin and
huntington’s disease pathology, Science 304, 100 (2004).
[35] S. Gunawardena, L.-S. Her, R. G. Brusch, R. A. Laymon,
I. R. Niesman, B. Gordesky-Gold, L. Sintasath, N. M.
Bonini, and L. S. Goldstein, Disruption of axonal trans-
port by loss of huntingtin or expression of pathogenic
polyq proteins in drosophila, Neuron 40, 25 (2003).
[36] B. H. LaMonte, K. E. Wallace, B. A. Holloway, S. S.
Shelly, J. Asca˜no, M. Tokito, T. Van Winkle, D. S. How-
land, and E. L. Holzbaur, Disruption of dynein/dynactin
inhibits axonal transport in motor neurons causing late-
onset progressive degeneration, Neuron 34, 715 (2002).
[37] E. Reid, M. Kloos, A. Ashley-Koch, L. Hughes, S. Bevan,
I. K. Svenson, F. L. Graham, P. C. Gaskell, A. Dearlove,
M. A. Pericak-Vance, et al., A kinesin heavy chain (kif5a)
mutation in hereditary spastic paraplegia (spg10), The
American Journal of Human Genetics 71, 1189 (2002).
[38] C. Zhao, J. Takita, Y. Tanaka, M. Setou, T. Nakagawa,
S. Takeda, H. W. Yang, S. Terada, T. Nakata, Y. Takei,
et al., Charcot-marie-tooth disease type 2a caused by
mutation in a microtubule motor kif1bβ,Cell 105, 587
(2001).
[39] S. Brenner, The Genetics Of Caenorhabditis Elegans, Ge-
netics 77, 71 (1974).
[40] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig,
M. Longair, T. Pietzsch, S. Preibisch, C. Rueden,
S. Saalfeld, B. Schmid, et al., Fiji: an open-source plat-
form for biological-image analysis, Nature methods 9, 676
(2012).
[41] C. W. Gardiner, Handbook Of Stochastic Methods
(Springer, 1985) p. 437.
[42] D. Chaudhuri, P. Borowski, and M. Zapotocky, Model of
fasciculation and sorting in mixed populations of axons,
Physical Review E 84, 021908 (2011).
[43] S. M. Block, C. L. Asbury, J. W. Shaevitz, and
M. J. Lang, Probing the kinesin reaction cycle
with a 2d optical force clamp, Proceedings of the
National Academy of Sciences 100, 2351 (2003),
https://www.pnas.org/doi/pdf/10.1073/pnas.0436709100.
[44] M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic
resetting and applications, Journal of Physics A: Mathe-
matical and Theoretical 53, 193001 (2020).
[45] L. D. Cohen, R. Zuchman, O. Sorokina, A. M¨uller,
D. C. Dieterich, J. D. Armstrong, T. Ziv, and N. E. Ziv,
Metabolic Turnover of Synaptic Proteins: Kinetics, In-
terdependencies and Implications for Synaptic Mainte-
nance, PLoS ONE 8, e63191 (2013).
[46] E. F. Fornasiero, S. Mandad, H. Wildhagen, M. Alevra,
B. Rammner, S. Keihani, F. Opazo, I. Urban, T. Is-
chebeck, M. S. Sakib, M. K. Fard, K. Kirli, T. P. Cen-
teno, R. O. Vidal, R.-U. Rahman, E. Benito, A. Fischer,
S. Dennerlein, P. Rehling, I. Feussner, S. Bonn, M. Si-
mons, H. Urlaub, and S. O. Rizzoli, Precisely measured
protein lifetimes in the mouse brain reveal differences
across tissues and subcellular fractions, Nature Commu-
nications 9, 4230 (2018).
[47] See supplemental material at [publisher will insert url]
for description of simulation movies.
[48] QW T ≈0.035, Quba−1≈0.035, Qfbxb−65 ≈0.033.
