UNC-104 transport properties are robust and independent of changes in its cargo binding
Amir Shee Vidur Sabharwal Sandhya P. Koushika Amitabha Nandi Debasish Chaudhuri


























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                ρ^ (2) ρ̅^ (2) ρ̃^ (k) ρ̃^ (1) ρ̃^ (0) ρ̃^ (2) ρ̊D̃σk_BF̅ F⟨⟩↑↓Σ†δ̣∂łλŁΛΓøΩωγ bû r vηξ() j j_r j_d j_dαδ̣∂⟨⟩ϵηγ#1
                1/2
                [Amir Shee and Vidur Sabharwal contributed equally to this work]Northwestern Institute on Complex Systems and ESAM, Northwestern University, Evanston, IL 60208, USAInstitute of Physics, Sachivalaya Marg, Bhubaneswar-751005, Odisha, India
                [Amir Shee and Vidur Sabharwal contributed equally to this work]Department of Biological Sciences, Tata Institute of Fundamental Research, Mumbai, IndiaDepartment of Biological Sciences, Tata Institute of Fundamental Research, Mumbai, India
                [Author for correspondence: ]amitabha@phy.iitb.ac.inDepartment of Physics, IIT Bombay, Powai, Mumbai 400076, India[Author for correspondence: ]debc@iopb.res.inInstitute of Physics, Sachivalaya Marg, Bhubaneswar-751005, Odisha, IndiaHomi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India
                
                § ABSTRACT 
                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 understood. Additionally, improper regulation of ubiquitination, a pivotal post-translational modification that affects protein degradation, activation, and localization, is associated with several neurodegenerative 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 equation 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.
                
                
                UNC-104 transport properties are robust and independent of changes in its cargo binding
                    Debasish Chaudhuri
                    September 9, 2024
                =======================================================================================
                
                
                
                
                § INTRODUCTION
                
                
                Neurons are specialized cells that constitute the nervous system and are responsible for transmitting electrical 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 <cit.> (see Fig. <ref>(a)). These microtubules are used by motor proteins (MPs) like kinesins <cit.> and dyneins <cit.> 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. <ref>(b) ) <cit.>.
                
                The number of MPs on cargo affects its processive run length. Increasing the number of kinesins on cargo extends its duration of processive movement  in vitro, but this is not always observed in vivo <cit.>.
                 
                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 bidirectional <cit.>. Modeling reveals that bidirectional movement may aid in the circulation of cargo between the cell body and distal ends of the neuron <cit.>. Such bidirectional movements can arise from either (i) a tug-of-war between opposing MPs <cit.>, or (ii) coordination between opposing motors, where dynamic switching between opposing MPs can drive bidirectional motion <cit.>. Indeed, both of these hypotheses have been validated in different contexts <cit.>. 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 <cit.>.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 <cit.>.
                If MPs are not attached to their conjugate filaments, they can either diffuse freely or hitchhike after binding to a cargo <cit.>. Note that MPs must bind to cargo and attach to conjugate filaments simultaneously to mediate active cargo transport. Three different possibilities may arise <cit.>: (i) good cargo binding but weak filament 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. <ref>(c) and may arise in different contexts. As an example, good cargo binding but weak filament binding is a typical feature of dynein, where an individual dynein molecule may not generate sufficient processivity and force for reliable cargo movement, but teams of dynein can work together to move a cargo <cit.>. On the other hand, good filament binding but weak cargo binding has been observed in the slow transport of soluble cytoplasmic proteins – they move by intermittent binding to motor proteins for directed motion, interspersed with dissociation and consequent diffusion <cit.>.
                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 <cit.>.
                Arguably, 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 homeostatic distribution of MPs and their dynamics also depend on their filament processivity and related transport properties. 
                
                Post-translational modifications (PTMs) of motor proteins (MPs) are essential for controlling their amount, activity, and cargo binding <cit.>. In particular, ubiquitination and ubiquitin-like PTMs play a key role in degrading and regulating MPs <cit.>. Disruptions in these PTMs are associated with various neurodegenerative diseases <cit.>. Additionally, changes in molecular motors and cargo transport are implicated in the development and progression of these diseases <cit.>. Therefore, understanding how ubiquitin and ubiquitin-like PTMs impact motor proteins can provide valuable insights into axonal cargo transport and highlight potential targets for therapeutic interventions in neurodegenerative diseases.
                
                Recently, using a combination of live imaging of  C. elegans neurons and theoretical analysis, we investigated how different ubiquitin-like modifications affect the cargo binding of Kinesin-3 MP UNC-104, a KIF1A ortholog <cit.>. The results suggested that MPs may bind to cargo cooperatively, and the degree of cooperative binding 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 different levels of ubiquitin-like modifications on the kinematics 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 distributions 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 experimental findings using a theoretical model. 
                
                The experiments and theory allow us to uniquely determine UNC-104's effective diffusivity and drift. 
                
                
                The rest of the manuscript is organized as follows: Sec.<ref> details the experimental system and methods; Sec.<ref> covers the cargo-binding model and data-collapse of motor protein distributions; Sec.<ref> presents the theoretical model for motor protein dynamics along axons and uses experiments to evaluate transport properties; and Sec.<ref> summarizes our findings and provides an outlook.
                                 
                
                
                
                
                
                § MATERIAL AND METHODS
                
                
                
                 §.§ Worm maintenance
                
                
                C. elegans was reared on NGM agar seeded with E. coli OP50 using standard practices <cit.>. For RNAi-mediated knockdown, NGM agar was prepared with 100 μ g μ L^-1 ampicillin and 1 mM IPTG. These plates were seeded with the appropriate dsRNA [against either control 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.
                
                
                
                 §.§ 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. 
                
                
                
                
                 §.§ 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 <cit.>. 
                
                
                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 animals. The exported data is further processed to obtain the final UNC-104 intensity profile, which is discussed in the next paragraph.
                In Fig. <ref>, we show the experimental intensity profiles for the control (wild type) along the axonal length. The normalized average bare intensity profile I_unc of UNC-104::GFP is shown in Fig. <ref>(a), and the corresponding normalized average mScarlet fluorophore profile I_mscar is shown in Fig. <ref>(b). The intensity I_unc 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 axonal 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 localization per unit volume, we take the ratio I_unc/I_mscar of the intensity I_unc of UNC-104::GFP to mScarlet intensity I_mscar, which provides insight into the distribution of UNC-104 MPs along the axonal length by mitigating variability due to axonal volume. The homeostatic profile of I_unc/I_mscar averaged over tens of cells is shown in Fig. <ref>(c).
                
                
                
                 §.§ 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 around 100 μm away from the cell body. The photobleaching 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 <cit.> by drawing a 3 pixels-wide spline fit polyline. The intensity 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. <ref>
                
                § 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 <cit.>.
                We validate this cooperative binding model through a compelling data collapse by analyzing cargo puncta of different sizes, which correspond to varying numbers of bound motor proteins. Additionally, we assess how ubiquitin-like modifications influence the extent of this cooperative binding.
                
                
                We consider the evolution for the probability P(n,t) of a cargo bound to n MPs at time t <cit.>Ṗ(n,t)   =    u_+(n-1)P(n-1,t)+u_-(n+1)P(n+1,t) 
                      -[u_+(n)+u_-(n)]P(n,t).
                
                Here, u_+(n) and u_-(n) denote the rates of MP binding and unbinding, cross-linking the cargo to the microtubule. We use a pairwise detailed balance at the steady state, u_+(n-1)P_s(n-1)=u_-(n) P_s(n) and boundary condition j_+= u_-(1) P_s(1) with j_+ a diffusion-limited rate of MP cross-linking. This leads to the exact steady state given by the recurrence relation
                
                P_s(n)= j_+u_-(n)∏_m=1^(n-1)u_+(m)u_-(m)
                with the total number of motors N=∑_n=1^∞ n P_s(n). 
                
                
                Cross-linked MPs can detach with a constant rate β so that u_-(n)=β n. The presence of attached MPs can assist in further MP cross-linking such that u_+(n)=a_++b_+n, within linear approximation. Here, a_+ denotes 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 <cit.>
                H_s(n) = A  n^ e^-μ n, 
                
                where A=j_+exp(μ),  =a_+-1 and μ=1-b_+.
                
                The normalized distribution is P_s(n)= N_n^-1 H_s(n) with N_n = ∫_0^∞ H_s(n) dn = A μ^-(+1)(1+). This leads to
                P̃_s(μ n)=(1+)μ P_s(n) = (μ n)^exp(-μ n).
                
                
                From the distribution of P_s(n), we note that n̅ = /μ denotes both the mode and mean of the distribution, and its variance is given by _n^2=2/μ^2.
                
                We compare the theoretical distribution with the experimental results obtaining ≈ 1 and μ_WT=7.33× 10^-3, μ_uba1=4.79× 10^-3 and μ_fbxb65=3.05× 10^-3 <cit.>.
                
                Using these values, we plot the full distributions P̃_s(μ n) to obtain a data collapse in Fig. <ref>. The plot of Eq. (<ref>) using a black solid line agrees well with the experimental data as shown in Fig. <ref>. The scaled quantity a_+/≈ 2 is independent of the RNAi and the cooperative binding strength (b_+/)_WT < (b_+/)_uba1 < (b_+/)_fbxb65.
                
                
                
                
                
                Moreover, Eq. (<ref>), gives the evolution of average number of MPs n(t) cross-linking the cargo to microtubule, 
                ṅ=a_++(b_+-β)n(t).
                
                Using the initial condition n(0)=n_0, we obtain 
                n(t)=n̅ - (n̅-n_0)exp(-t/τ).
                
                As before, n̅=/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 type) is τ_WT=1/βμ_WT≈ 136.43 sec considering the constant detachment rate of MPs β = 1 sec^-1 <cit.>.
                
                The relaxation time also determines the fluctuation correlation at the steady state,
                δ n(t)δ n(0) = _n^2exp(-t/τ),
                
                where _n^2=2/μ^2, the variance of steady-state  distribution P_s(n). These predictions can be tested across future experiments. 
                
                
                
                § 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 microtubules 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 _̊b(x,t) and _̊u(x,t), respectively. The bound fraction moves with an active drift velocity v_0. 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 D_0. Considering the binding and unbinding kinematics in terms of rates  and , we get
                _t _̊b + v_0 _x _̊b    =   _̊u - _̊b, 
                _t _̊u - D_0 _x^2 _̊u    =    - _̊u + _̊b.
                
                At this stage, we ignored the synthesis and degradation of MPs, which will be incorporated later. 
                The above equations can be rewritten in terms of the total MP density =̊_̊b+_̊u and the density difference m=_̊b-_̊u. The evolution of $̊ follows a conserved continuity equation. In contrast, the evolution ofm(x,t)gives_t m + v_0 _x _̊b + D_0 _x^2 _̊u = 2 (_̊u - _̊b).
                In the presence of the source term on the right-hand side of the above equation, one can perform an adiabatic elimination in the hydrodynamic limit to obtain_̊u/_̊b = /.   
                Using the processivityø = /(+)one can write_̊b=ø$̊ and _̊u=(1-ø)$̊. This leads to the conserved dynamics_t +̊ v _x -̊ D _x^2 =̊0where the effective drift velocity and diffusivity of total MP density arev = ø v_0andD=(1-ø) D_0. 
                
                Now, we incorporate the other two source terms, the synthesis and degradation processes of MPs. 
                We consider the synthesis at the cell body with rateQand a homogeneous degradation with rate. Thus, the  evolution of total concentration can be expressed as (see Fig. <ref>),
                
                _t (̊x,t) =- v _x +̊ D _x^2 +̊Q (̣x) - .̊
                
                
                The Dirac-delta functionδ(x)ensures that the motor proteins are synthesized near the cell body (x=0). Remarkably, the above equation is an example of a stochastic resetting process that attracted tremendous recent interest in statistical physics <cit.>. However, unlike the typical stochastic resetting examples, in the present context, the number of particles is not exactly conserved. The mean value of the total number of MPsn(t)=∫ dx ρ(x,t)in the steady staten̅ = Q/γis determined by the synthesis and degradation ratesQand. 
                
                
                
                
                 §.§ Steady state distribution
                
                We consider a reflective boundary condition atx=Lleading to-D_x(̊x,t)|_x=L+v (̊x,t)|_x=L=0. Thus, Eq. (<ref>) has the following steady-state solution (see Appendix <ref> for a detailed derivation):
                _̊s(x) =Q ł_vł e^x/λ_v2Dsinh(L/ł)[e^-(L-x)/ł(ł_v-ł)+e^(L-x)/ł(ł_v+ł)].
                
                Here two effective length scalesλ_v=2l_vandλ=l_γ/[1+(l_γ/ł_v)^2]^1/2determine the steady-state profile. In these expressions, we used the characteristic length scalesl_v=D/vandl_γ=√(D/γ). 
                
                
                A Laplace transform method can be employed to solve the dynamical equation Eq. (<ref>) in the Laplace space. While an inverse transform to a closed-form expression could not be obtained, this solution allows us to determine the relaxation times towards the steady state. The slowest mode of time-scale^-1is determined by the degradation ratealone. The relaxation times towards steady-state for other relatively faster modes are^-1[1+l_^2 ( 1ł_v^2 + π^2 n^2L^2) ]^-1forn=1,2,3,…; see Appendix <ref>.   
                
                
                
                
                The steady-state expression given by Eq. (<ref>) may be used to fit the experimental profiles. However, we note that there are four independent parameters (Q,D,v, and ) which makes this comparison difficult. We note that ≈ 10^-4s^-1  as known from earlier studies for KIF1A <cit.>. We further eliminate Q using the total number of MPs
                𝒩 = ∫_0^L_̊s(x)dx=Qλ^2λ_v^2/D(λ_v^2-λ^2),
                
                to obtain a normalized steady-state distribution of MPs expressed in terms of the density profile ^̊𝒩_s(x) = _̊s(x)/ N and get
                ^̊𝒩_s(x) =(λ_v^2-λ^2)e^x/λ_v/2λλ_vsinh(L/λ)[e^-(L-x)/ł(ł_v-ł)+e^(L-x)/ł(ł_v+ł)].
                
                Eq. (<ref>) has only two unknowns, namely D and v. First, we estimate D independently from the FRAP experiments, which are discussed in the next section. Second, we fit normalized experimental distribution using Eq. (<ref>) to get a single fit parameter v with known values of D. Finally, we estimate values of Q utilizing the fitting of non-normalized experimental distribution using Eq. (<ref>) with known values of D and v.
                
                
                
                
                
                
                 §.§ FRAP Analysis
                
                Although a numerical solution of Eq. (<ref>) for the evolution of the density profile can be obtained, the equation 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 profile 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 theoretical approach presented in Eq. (<ref>).
                
                For this purpose, we consider the scaled evolution with respect to the theoretical steady-state, ϕ(x,t) = (̊x,t)/_̊s(x), over a small domain corresponding to the FRAP window. This follows,
                _t ϕ(x,t)    =     -v _x ϕ(x,t)  + D _x^2 ϕ(x,t) ,
                
                locally, where we began by ignoring the synthesis and degradation terms for simplicity. This leads to a uniform steady state ϕ_s(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(x_0,0)    =    0  for  -a≤ x_0 ≤ a , 
                   =    1   for  |x_0| > a.
                
                 The Greens function corresponding to Eq. (<ref>) is   
                G(x-x_0,t) = 1√(4π D t) e^-(x-x_0-v t)^2/4 D t. This, 
                along with the initial condition in Eq. (<ref>),  leads to the solution  
                
                ϕ(x,t)    =   [2- erf(a+vt-x√(4Dt))- erf(a-vt+x√(4Dt))] ,
                
                where erf(x)=2√(π)∫_0^x e^-s^2ds. Now, bringing back the synthesis and degradation terms, the same analysis leads to a solution
                      ϕ(x,t) 
                      = e^- t/2[2- erf(a+vt-x√(4Dt))- erf(a-vt+x√(4Dt))] 
                      +QΘ(t)/ρ_0(1-e^- t),
                
                where _̊0=_̊s(x=0). However, since the degradation rate  is known to be small, the intensity decay due to it over the short FRAP duration is negligible, and Eq. (<ref>) reduces to Eq. (<ref>) apart from an additive constant Q/_̊0. The recovery of the profile ϕ(x,t) is mediated by effective diffusion D, degradation , and effective drift v. 
                
                
                
                
                
                We utilize the experimental intensity evolution during FRAP, after dividing by the homeostatic intensity profile, to estimate diffusion. We extract the diffusion coefficient by directly fitting Eq. (<ref>) to the evolution of this intensity profile. Finally, by averaging over many realizations, i.e., different cells, we obtain the diffusion coefficient D (Table <ref>).
                
                While performing such fitting, one can neglect the small directed displacement v t of the local density profile due to drift over the relatively short recovery time t compared to the extent of the FRAP window w=2a, assuming v t ≪ w. However, as it turns out, a strong intensity fluctuation makes it difficult to estimate D reliably through this procedure. 
                
                To reduce such systematic error, we use the evolution of an average intensity over a small window near the minimum intensity spot at the beginning of FRAP (Fig. <ref>(a)). Setting x=0, v t ≪ a, and t≈ 0, Eq. (<ref>) gives
                ϕ(0,t) = [ 1 -  erfa√(4Dt)].
                
                We estimate the diffusion constant D by computing the half recovery time t_1/2 of the center of the FRAP region defined over a small window of size ϵ=4 μm, where  ϵ≪ 2a (considering ϵ window size as 10% of FRAP window size 2a) with 2a≈ 40  μm. 
                
                
                Thus, ϕ(0,t_1/2)=(1/2) ϕ(0,0) (see Fig. <ref>(b)), we find the values of D using 
                
                D=0.275(2a)^2t_1/2.
                In our experiments, the recovery intensity turns out to be less reliable at longer times due to large intensity fluctuations, and even determining _1/2 can be difficult for some experimental data. For them, we use ϕ(0,t_1/4)=(1/4) ϕ(0,0)
                (see Fig. <ref>(b)) and the corresponding relation 
                
                D=0.094 (2a)^2t_1/4.
                
                
                
                
                We have estimated the effective diffusion coefficient D using both t_1/2 and t_1/4 and found that diffusion coefficients were comparable (see Fig. <ref>(c) and Table <ref> in Appendix <ref>). 
                Using the estimated D values for the WT (both via t_1/2 and t_1/4), we compute ϕ(x,t) both numerically (solving Eq. (<ref>)) as well as analytically (using Eq. (<ref>)), 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. <ref>(a) (see Supplemental Material Movie-1 <cit.> for a comparison of the complete time evolution from experiment, numerical solutions and analytic expression). We also compare the theoretical time trajectories for ϕ(0,t) with the corresponding FRAP experiment; see Fig. <ref>(b). We note that for this specific case, ϕ(0,t_1/4) 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 D values estimated using t_1/4 are statistically more reliable than the estimate using t_1/2, while for the WT and uba-1 knockdown, the statistics are equivalent (see Appendix. <ref> for details). The corresponding distributions of D values, estimated using t_1/4, across different cells are compared between the WT and the two knockdown experiments in Fig. <ref>(c) (see also the first row of Table <ref>). Comparing the values of D between the WT and the knockdown experiments, we notice that the relative variation in the value of D across experiments is Δ D=(D_WT-D_KD)/D_WT≤ 10%, where WT and KD stand for wild type and RNAi treated ubiquitin-like knockdowns respectively. 
                
                The FRAP analysis thus suggests that cells with altered ubiquitin-like knockdowns of either uba-1 or fbxb-65 do not affect the MP's diffusional transport properties. Interestingly, due to the exponential form of the steady state profile _̊s(x), a FRAP performed on top of this profile can lead to an apparent emergent retrograde bias in the recovery profile (see Appendix <ref> and Movie-2 in <cit.>). 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 <ref>.
                
                
                
                
                
                
                 §.§ Active transport is crucial in determining the steady-state intensity profiles
                
                With the diffusion coefficients D and the degradation rate  known, we can now determine the effective motor speed v from the experimental steady-state distribution of the UNC-104 motors along the axon. To make a comparative 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. <ref>, where we corrected for the local volume variability of axons by scaling the experimentally obtained homeostatic intensity profiles for the MPs by the corresponding mScarlet fluorophore profiles for a given cell. This density further averaged over n=15 cells (see Fig. <ref>(c)) gives a read-out for the steady-state local density profile _̊s(x). Finally, we normalize the distribution _̊s(x) to obtain ^̊𝒩_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=ø v_0. The velocity v_0 of individual MPs obtained from live imaging of UNC-104::GFP-decorated puncta are estimated to be ∼ 1 μm/s <cit.>. Assuming that the individual motor protein properties do not change, the unknown parameter v can, therefore, be treated as a read-out for the value of processivity ø. Thus, to estimate ø (via v), we further normalize the experimental steady-state distributions with respect to the spatially integrated intensity and compare them to Eq. (<ref>). The results are shown in Fig. <ref>(a), <ref>(b), and <ref>(c), which display a good fit in all the cases. Note that here we use only a single parameter fit to v and hence obtain ø. The values of the processivity ø obtained from these fits are reported in Table <ref>.
                Several intriguing observations emerge from the fit results. First, it is evident that the transport parameters D and v remain largely unchanged by ubiquitin-like knockdowns, maintaining similar approximate values across all cases, including the wild type.
                Secondly, the processivity (ø≈ 0.05) is low, indicating that only 5% of available motor proteins participate in transport. 
                
                This also indicates that the intrinsic diffusivity D_0=D/(1-ø) is mainly due to the detached fraction of MPs and is approximately equal to the measured values. 
                
                
                Additionally, we note that the integrated value of the MP density 𝒩=∫_0^L_̊s(x) (see Eq. (<ref>)) remains almost unchanged. Since this term provides a read-out for the source term Q, we conclude that Q is invariant to uba-1 and fbxb-65 knockdown cells [Q_WT≈ 0.035, Q_uba-1≈ 0.035, Q_fbxb-65≈ 0.033.].
                
                
                
                
                
                
                The processivity ø turns out to be the most important parameter regulating the steady-state distribution. From the expression in Eq. (<ref>), we note that if ø=0, the length scale of the steady state exponential distribution is set by l_ and attains the maximum at the source (if l_<L) and the minimum at the posterior end (see Fig. <ref>(a)). Upon slowly increasing ø, the minimum position of _̊s(x) starts moving from x=L to lower values of x (see Fig. <ref>(b)–(d) ) until at ø=0.05, the ρ^𝒩_s(x) profile is already reversed and the length-scale is set by the competition between l_ and l_v (see Fig. <ref>(c)). 
                
                This profile is similar to our experimental observation. It gets even sharper at higher ø, with most of the MPs accumulating near the distal end; see Fig. <ref>(d). Moreover, as the inset of Fig. <ref>(d) shows, 
                the minimum of the profile shifts back to positive values at larger ø. The inversion of the density profile with the maximum shifting from proximal to distal end with increasing processivity is further illustrated using a heat-map in Fig. <ref>(e). Fig. <ref>(f) shows the non-monotonic variation of the minimum density and the location of this minimum with increasing ø.
                
                
                
                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 dependent on the effective processivity, remains completely unaffected.
                
                
                
                § DISCUSSION
                
                Directed axonal transport of synaptic vesicles by MPs is crucial for the proper functioning of an organism, failure of which is associated with neurodegenerative disorders <cit.>. 
                 
                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 modifications can affect the motor's intrinsic transport properties
                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 transport 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 corresponding axonal volume showed a nice collapse of the data from different RNAi knockdowns on each other, indicating that the homeostatic distribution remains unchanged under ubiquitin-like knockdowns.
                
                
                We proposed a theoretical model which clearly explains the nature of such a homeostatic profile. Synthesis, degradation, and transport, both directed drift and diffusion, 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 D from the FRAP experiments. Using the known D values, we fitted the steady-state profiles to obtain estimates for the effective drift velocity v, which is a readout for the processivity ø=v/v_0. Our analysis clearly shows that both the diffusivity and the processivity remain unchanged under ubiquitin-like knockdowns. 
                
                
                
                
                Since these kinesin-3 MPs can hitchhike on retrogradely moving cargo carried by dynein MPs, the kinetic properties measured using the overall fluorescent intensity from UNC-104 not only depend on their active motion along microtubules but also include the impact of such retrograde flux.
                We find that ubiquitin-like knockdowns increase cooperative cargo binding, thereby potentially increasing the impact of the hitchhiked retrograde motion. Consistent with this, we previously observed a higher occurrence of UNC-104 retrograde movement under ubiquitin-like knockdowns <cit.>. Surprisingly, the effective kinetic properties of UNC-104 remain unchanged under the same ubiquitin-like knockdowns. Such independence suggests a subtle regulation nullifying the effect of potentially increased retrograde flux. Note that the cargo-bound MPs may support each other's microtubule 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 analysis; 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 comments. DC thanks Sanjib Sabhapandit and Fernando Peruani for useful discussions. DC acknowledges research grants from the Department of Atomic Energy (OM no. 1603/2/2020/IoP/R&D-II/15028) and Science and Engineering Research Board (SERB), India (MTR/2019/000750) and thanks the International Centre for Theoretical Sciences (ICTS-TIFR), Bangalore, for an Associateship. AN acknowledges SERB India (MTR/2023/000507) for financial support and thanks the Max-Planck Institute for the Physics of Complex 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.
                
                
                
                
                § STEADY-STATE DISTRIBUTIONS
                
                We recast the governing equation (<ref>) in terms of current J=-(D_x - v )$̊ and considering uniform degradation(x) =constant,
                _t =    -_x J -,
                
                with the boundary condition of total current atx=0,J_x=0=QwhereQis the constant source and total current atx=L,J_x=L=0.
                
                The steady-state limit (_t =̊ 0) gives
                _x^2 _̊s - 1l_v_x _̊s - 1l_^2_̊s    =    0
                
                where the constantsl_v = D/vandl_=√(D/). Substituting_̊s ∝ e^ xin the Eq. (<ref>) leads to the quadratic equation^2-/l_v-1/l_^2=0with solution = [l_± (4l_v^2 +l_^2)^1/2]/2l_v l_. Thus, steady-state density,
                _̊s (x)    =    e^x/2l_v[𝒜 e^x√(4l_v^2 + l_^2)2l_v l_  + ℬ e^-x√(4l_v^2 + l_^2)2l_v l_]
                
                We recast again withł_v = 2 l_vandł = 2l_v l_ / √(4l_v^2 + l_^2)
                =l_ (1+l_^2/ł_v^2)^-1/2. Thus,
                _̊s (x)    =    e^x/ł_v[𝒜 e^x/ł  + ℬ e^-x/ł]
                
                Now, the total current atx=0,J_x=0=Qlead to
                
                (ł -ł_v)𝒜 + (ł +ł_v) ℬ   =   Qłł_v D ,
                
                and the total current atx=L,J_x=L=0lead to
                
                (ł -ł_v) e^L/ł𝒜 + (ł +ł_v) e^-L/łℬ   =    0 .
                
                Now, solving Eqs. (<ref>) and (<ref>), we get
                𝒜   =   Q łł_v D (ł_v - ł  ) (e^2L/ł -1) , 
                ℬ   =   Q łł_v e^2L/łD (ł + ł_v ) (e^2L/ł - 1) .
                Substituting𝒜andℬback in the Eq. (<ref>), we get
                _̊s (x) = Q  ł_v łD (e^2L/ł -1) e^x/ł_v[e^x/ł(ł_v - ł) + e^(2L-x)/ł(ł_v + ł  )] ,
                
                which can be rewritten as:
                _̊s(x) =Q ł_vł e^x/λ_v2Dsinh(L/ł)[e^-(L-x)/ł(ł_v-ł)+e^(L-x)/ł(ł_v+ł)].
                
                This is precisely Eq. (<ref>) of the main text.
                
                
                
                § LAPLACE TRANSFORM, SINGULARITIES AND SLOW DYNAMICS
                
                The equation of motion for the density can be directly solved using Laplace transform̊̃(x,s) = ∫_0^t dt' e^-s t'(̊x,t')to get
                
                ̊̃(x,s)    =   Q  ł_v ł(s)  e^x/ł_vs D (e^2L/ł(s) -1)[e^x/ł(s)(ł_v - ł(s)) + e^(2L-x)/ł(s)(ł_v + ł(s)  )] 
                
                whereł(s)=l_(1+l_^2ł_v^2+s)^-1/2. It is easy to see that the above expression gives the steady-state result_̊s(x)= lim_s→ 0 s ̊̃(x,s).
                
                The time dependence is given by the inverse Laplace transform,(̊x,t)= 12 π i∫_c-i∞^c+i∞ ds  e^s t ̊̃(x,s)withc>0so 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 ofł(s)in the numerator,̊̃(x,s)has a branch point ats_b = - (1+l_^2ł_v^2). Moreover, it has simple poles at(e^2L/ł(s) -1)=0; using1=e^i  2nπthis gives simple poles ats_n = s_b - (ℓ_L)^2  n^2 π^2withn=0,1,2,…, all lying on the real axis and to the left ofs_b.  
                
                Further, as can be seen from Eq. (<ref>),̊̃(x,s)has a simple pole ats=0, and ats^* = s_b + ℓ_^2 ł_v^2 =-<0(usingł_v = ±ł(s)) betweens=0ands=s_b.
                
                Thus, the solution is
                (̊x,t)   =   _̊s(x) + _̊1(x)e^s^* t +  ∑_n=0^∞_̊n(x) e^s_n t, 
                
                   +    e^s_b t L^-1̊̃(x, s+s_b)
                
                so that_̊1(x) = lim_s→ s^* (s-s^*) ̊̃(x,s), etc. 
                The last term is the inverse Laplace transform of the frequency-shifted function, arising due to the branch point ats_b<0.
                The slowest mode of evolution islim_t→∞[(̊x,t) - _̊s(x)] →_̊1(x)e^- tis controlled by the degradation rate. The reason for this becomes immediately clear by noting from Eq. (<ref>) that the total quantity of MPs N=∫ dx (̊x)evolves asd  N/dt= Q- N.  
                
                
                
                § COMPARISON OF DIFFUSIVITIES FROM FRAP ANALYSIS
                
                In Fig. <ref>, we show the scatter plot of diffusivities estimated using botht_1/2(Eq. (<ref>)) andt_1/4(Eq. (<ref>)) 
                for control (WT) along with uba-1 and fbxb-65 knockdown cells. 
                
                
                The corresponding comparison for the mean values of effective diffusivities with their standard errors is shown in Table <ref>. 
                
                For both control (WT) and uba-1 knockdown, D is estimated by averaging over 22 independent experimental realizations. However, for the case of fbxb-65 knockdown, D measured via t_1/4 is averaged over all 23 realizations, whereas the one via t_1/2 was averaged over 9 realizations. This is because the D estimate via t_1/2 was relatively unreliable in 14 cases due to large intensity fluctuations.
                
                § FURTHER INSIGHTS FROM THE MODEL: APPARENT RETROGRADE MOTION UNDER FRAP
                
                To characterize the intensity recovery dynamics observed in the FRAP experiments, we numerically integrate Eq. (<ref>) 
                for an initial condition that is equivalent to the density profile just after FRAP. This is constructed as follows: starting with the steady-state profile defined in [0,L], we choose a small window of size w=2a=Δ L around the center of the profile where the local density is set to zero (Fig. <ref>). The parameters are chosen corresponding to the WT case: we use the transport coefficients D and v from Table <ref> and the degradation rate γ=10^-4sec^-1. We use the typical axon length L=274.74 μm  and FRAP window size w=2a=Δ L=39.25 μm. To integrate Eq. (<ref>), we use time step dt=0.0001 sec, set the source term Q=0.035 sec^-1 at x=0 and use a reflecting boundary at x=L. The resultant MP distribution is finally normalized over the axonal length so that ∫_0^L ρ^ N(x) dx = 1.
                
                  
                In Fig. <ref>(a)-(d), the recovery of the density profile after FRAP is shown at four different times and the corresponding local currents J^N(x,t) are shown in Fig. <ref>(e)-(f) respectively. The approach toward the steady-state profile (shown by the dotted lines in Fig. <ref>(a)-(d)) is evident from 
                these figures, and at long times (see Fig. <ref>(d)), the MP distribution almost overlaps with the steady-state profile. The recovery is further evident from the time evolution of local current J^𝒩(x,t). Just after the FRAP, J^𝒩(x,t) exhibit large positive and negative peaks corresponding to the two edges of the FRAP region (see Fig. <ref>(e)). As time progresses, due to the recovery in depleted density, the peaks start receding until they disappear at long times (see Fig. <ref>(f)-(h)) when J^𝒩(x,t) approach to the steady-state value. It is important to note that at steady state, J^𝒩_s(x) is not a constant due to the presence of a non-zero source Q at one end. We also focus on the relaxation of the minimum density value ρ^𝒩_min (see Fig. <ref>(i)) and its location x_min (see Fig. <ref>(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 x_ min before vanishing captures an apparent retrograde motion of the location of FRAP minimum against the direction of MP flux. Similar behavior 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 approximate exponential steady-state profile led to the apparent retrograde motion of the FRAP center in the current example. This is further demonstrated in Movie-2 presented in the Supplemental Material <cit.>, 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. 
                
                
                
                
                
                
                
                
                
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                 [Sup()]Supply2024
                  @noop titleSee supplemental material at [publisher will
                  insert url] for description of simulation movies.Stop
                 [Note1()]Note1
                  noteQ_WT≈ 0.035, Q_uba-1≈ 0.035,
                  Q_fbxb-65≈ 0.033.Stop
