Steady State Cargo Transport Modalities of Molecular Motor Ensembles Emerge from Single Motor Behavior
Shreyas Bhaban, James Melbourne, Saurav Talukdar, Murti V. Salapaka
SSteady State Cargo Transport Modalities of Molecular Motor Ensembles Emerge fromSingle Motor Behavior
Shreyas Bhaban, James Melbourne, Saurav Talukdar, and Murti V. Salapaka
University of Minnesota, Minneapolis, USA.
Transport of intracellular cargo is often mediated by teams of molecular motors that function ina chaotic environment and varying conditions. We show that the motors have unique steady statebehavior which enables transport modalities that are robust . Under reduced ATP concentrations,multi-motor configurations are preferred over single motors. Higher load force drives motors tocluster, but very high loads compel them to separate in a manner that promotes immediate cargomovement once the load reduces. These inferences, backed by analytical guarantees, provide uniqueinsights into the coordination strategies adopted by motors.
Introduction:
Intracellular cargo such as vesicles, fil-aments and organelles are transported inside the cell bymolecular motors like kinesin, dynein and myosin. Themotors convert chemical energy to mechanical energythrough ATP hydrolysis [1], while operating individuallyas well as in homologous or heterologous teams [2, 3]. Nu-merous experimental studies have analyzed multi-motorensembles and have demonstrated the benefits of work-ing in teams, such as enhanced run-lengths [4, 5] androbustness in a turbulent environment [6]. On the an-alytical side, there are multiple studies that investigatecargo transport by two [7, 8] or more than two [6] mo-tors and analyze average velocity, run-length and dis-tributions of bound motors [9] that shed light on teambehavior. Several of these studies utilize a probabilisticdescription of single motor behavior to build models thatdescribe transport of cargo by a team [6, 9–11]. Thesemodels coupled by Monte-Carlo (MC) methods offer nu-merous benefits in terms of simulating behavior of teamsof motors that help guide tedious experiments [4]. WhileMonte-Carlo simulations have proven helpful, they sufferfrom (i) the inability to detect modalities of transportthat are rare, (ii) a lack of deductive capability of un-derstanding a specific mode seen in a realization of theMC simulation easily and (iii) the inability to provideinsights into the asymptotic behavior of the team of mo-tors. On the other hand simplified models that encap-sulate the dynamics of team of motors provide insightsanalytically, but these models lie on the other end of thespectrum to MC simulations where a detailed descriptionis not addressed.In this article we adopt a semi-analytic methodology[12] that, while capturing the detailed description of MCmethods, is able to provide conclusions on asymptoticbehavior and a deductive capability that is lacking inMC methods. The method utilizes a finite dimensionalreduction of a Markov model to enable an exact calcu-lation of the probability distribution function of the rel-ative behavior of motors in a team, through a computa-tionally efficient semi-analytic approach. We prove (fora generalized case of a team of finite motors) that therelative configurations of the motors in the team whiletransporting cargo, have a unique and non-trivial steadystate distribution. This implies that the system of mul- tiple motors carrying a cargo, is a highly robust system where irrespective of the initial orientation, the motorsin the team assume a unique steady state distribution.Moreover, the distribution which is dependent on exter-nal environmental factors is determinable; thereby pro-viding a means to reach conclusions on how the team ofmotors overcomes an adverse environment. Here the ef-ficacy of the approach is demonstrated by investigatinghow teams deal with changing ATP concentrations andexternal load forces by examining two and three motorensembles. Key analytical results reported here are that,as ATP concentrations are lowered, teams tend to favourmulti-motor configurations over single motor configura-tions; with the cargo more probable to be careied bymore than one motor than only a single motor. The im-plications are that the average run-length of multi-motorensembles is increased even though average ensemble ve-locity decreases as the ATP concentration reduces. It isknown that reduced ATP concentration lead to reducedsingle motor velocity, which is hypothesized as a reasonfor increased runlengths when cargo is transported by twomotors [13]. Such a mechanism also provides a possibleregulating mechanism by which transport occurs by anensemble containing a mixture of different motors [14].The approach in the article also establishes that whenthe cargo is subjected to different load forces, motorsadopt a form of cooperation by clustering together in or-der to handle increasing load forces. The propensity tocluster increases with load force. However the trend (ofcloser clustering to handle higher load forces) does nothold for all loads. Indeed, for very high values of loadforces (possible when cargo encounters obstacles alongthe path of travel), the teams abandon clustering. Theyinstead resort to spreading out in a manner where the en-tirety of the load falls on as few motors as possible, withthe rest of the motors assuming configurations where theyare subjected to forces that are near the maximum ca-pacity (or ’stall-force’) of a single motor. We hypothesizethat if the load force can be shared in a manner that pre-vents any of the motors from being loaded beyond stallforce, clustering is preferred as it aids motion. In thecase that the load force is very high where even with eq-uitable load sharing the forces on each motor is beyond itsstall force, the motor ensemble abandons clustering and a r X i v : . [ q - b i o . S C ] J u l 𝑙 " 𝑎 " 𝑎 $% 𝑎 $& 𝑎 $’ 𝑎 ( 𝑎 ’ 𝑎 & 𝑎 % 𝑎 ) 𝑑 + 𝐹 -./0 𝑙 " FIG. 1. (a) Schematic of structure of the molecular motorKinesin-I (b) Representation of an ensemble of three motorscarrying a cargo. F load is the load force opposing the cargo.The absolute configuration is Z = [ ... ... ] and relative configuration is M ||| M ||||| M , where M denotesmotor location on the MT and | denotes the MT locations. prefers configurations where most of the load is taken by afew motors while other motors are only loaded to be nearstall and are primed to take steps on a small reduction ofload force. Thus separation of motors at very high loadsis preferred in order to enable higher probability of im-mediate cargo translation in the forward direction, oncethe high forces (e.g. due to obstacle in path of transport)have passed and the high load phenomenon has subsided.In this article, we begin with a brief overview of thesemi-analytic model and introduce terminology to estab-lish the finite-dimensional model. We then establish theexistence and uniqueness of the steady state distribu-tion of various configurations of the motors in a team,by instantiating the model for molecular motors involv-ing stepping, detachment and reattachment probabilities.We then analyze the impact of changing ATP concentra-tions and load forces on ensembles of multiple Kinesin-Imotors by studying their effect on the steady state dis-tribution. We leverage the detailed information providedby the complete knowledge of the distribution of motorconfigurations to give an explanation for the observedbehaviors of motor ensembles. Semi-analytic Model:
A single motor is modeled asa hookean spring when stretched, with a rest length l ( nm ) and stiffness constant K e ( pN/nm ), that offers noresistance when compressed [4] (e.g. see schematic forKinesin-I motor in Fig. 1(a)). The stalling force, F S ,characterizes the maximum load bearing capacity of themotor. We assume that a team or ensemble has ¯ m mo-tors attached to the cargo, and the cargo has a constantload force, F load , acting against its motion under fixedATP concentration. The motors traverse on the micro-tubule (MT) filement, which is modeled as a sequenceof equally spaced dimers a q = a + qds , where a q is thelocation of q th dimer and d s is the dimer length. Float-ing motors attached to the cargo can only reattach tolocations on the MT that are within l distance of thecargo. Based on [12], the locations of the motors on theMT are represented by Z := { z q } , q ∈ I where z q motors are located on the a thq location of the MT and I is theset of integers (e.g. see Fig. 1(b)). A motor in the en-semble steps, detaches or reattaches to the MT, changingthe configuration from Z to Z (cid:48) . If the probability rate oftransition λ Z ( Z (cid:48) , Z ) from Z to Z (cid:48) is known, we can definean infinite dimensional Markov model, similar to [16, 17].Here, the probability of going from Z to Z (cid:48) in time ∆ t is represented by λ Z ( Z (cid:48) , Z )∆ t . Thereby, the probabil-ity that the configuration is Z (cid:48) at t given that it was Z at initial time t , is represented by P Z ( Z, t | ¯ Z, t ) , andsatisfies the probability Master Equation where the ratesare given by λ Z . However, as the typical MT filament ismuch longer ( µm ) than the average run-lengths of mo-tors ( nm ), it effectively makes Z a bi-infinite sequenceand the subsequent master equation intractable.The issue is resolved in [12] by showing that themaximum distance between the forward most (van-guard) and rear most (rearguard) motor is bounded,thus enabling a projection of the infinite dimensionalmodel to a finite dimensional model and preserving theMarkov property. Thus, instead of an absolute configu-ration Z we define a relative configuration σ that cap-tures the locations of the motors in an ensemble rela-tive to the rearguard motor (see Fig. 1(b)). Subse-quently, in the finite dimensional model, λ σ ( σ (cid:48) , σ ) de-notes the transition rates between the relative config-urations σ and σ (cid:48) . In a manner similar to the ab-solute configurations, the probability P σ ( σ, t ) of being σ at time t obeys the Master Equation, ∂∂t P σ ( σ, t ) = (cid:88) σ (cid:48) ∈ S λ σ ( σ, σ (cid:48) ) P σ ( σ (cid:48) , t ) − P σ ( σ, t ) (cid:88) σ (cid:48) ∈ S λ σ ( σ (cid:48) , σ ) , where ¯ S = { σ , . . . , σ n , σ n +1 , . . . σ ˜ n } is the set of finite relativeconfigurations. For a finite ¯ m and F load , the space ofrelative configurations is finite [12]. Let the probabilityvector be P ( t ) = [ P ( t ) , . . . , P ˜ n ( t )] where each element P i ( t ) is the probability of the system being in the i th relative configuration, σ i , at time t . The probability vec-tor P ( t ) satisfies the master equation ddt P ( t ) = ¯Γ P ( t ) ,where ¯Γ ∈ R ˜ n × ˜ n is defined by the transition rates λ σ ( σ j , σ i ) and is impacted by external conditions suchas F load and ATP concentration. Solving for P ( t ) , weget P ( t ) = e ¯Γ ( t − t ) P ( t ) = ¯J P ( t ) , where P ( t ) is aninitial probability vector and ¯J ∈ R ˜ n × ˜ n . Unique Steady State Distribution:
By propagatingthe above master equation for known models of molecu-lar motors with stepping, detachment and reattachmentprobabilities (such as, for example, Kinesin-I), it is seenthat the steady state distribution P ss reflects the con-dition of cargo being permanently disengaged from theMT. Thus, if σ n +1 = φ denotes the relative configurationwhere no motors are attached to the MT which charac-terizes the state where the cargo is permanently lost thenthe steady state distribution P ( ss ) is such that P ( ss ) n +1 = 1 and P ( ss ) j = 0 if j (cid:54) = n + 1 where P n +1 ( t ) = 1 . This isan intuitive yet a trivial and uninformative result, sinceloss of cargo is an inevitable eventual outcome of trans-port of cargo [4]. We consider cargo transport under ameaningful condition that a fixed number of motors, say m > , remain attached to the MT. Such a condition-ing enables an analysis of the mechanisms employed bythe motor teams while the cargo is being attached to theMT. When conditioned on the ensemble having at least m motors attached to the MT, the new state space reducesfrom ¯ S to S = { σ , . . . , σ n } , where S contains all the rel-ative configurations with at least m motors attached tothe MT. The probability vector P ( t ) = [ P ( t ) , . . . , P n ( t )] now satisfies the master equation ddt P ( t ) = Γ P ( t ) , where Γ ∈ R n × n is the transition matrix after conditioning onat least m motors attached to the MT and is defined bythe transition rates λ σ ( σ j , σ i ) . Solving for P ( t ) , we get P ( t ) = e Γ ( t − t ) P ( t ) = J P ( t ) , where P ( t ) is an initialprobability vector and J ∈ R n × n . The distribution P ( t ) contains information regarding the manner in which mul-tiple motors transport a common cargo. It is not evidentapriori what the nature of P ( t ) would be like; whetherit has a unique steady state and what the nature of thesteady state is. By using the underlying model for molec-ular motors [4] and a constant load force on the cargo,we utilized properties of the underlying Markov model toprove the following : Consider a Markov Model with a state space S , tran-sition matrix J for an ensemble of ¯ m motors carrying acommon cargo; with at least m ∈ [0 , ¯ m − motors alwaysremaining attached to the MT and the cargo subjected toconstant load force F L [15]. Then, the associated Markovchain has a unique steady state distribution P ss . See supplementary material [15] section I for proof. Itimplies that the ensemble of molecular motors carryinga common cargo is a highly robust system that behavesin a fixed manner after an initial transient period haspassed. Furthermore, the steady state distribution P ss isindependent of the initial distribution P ( t ) at the time t , indicating that no matter how the motors are orientedprior to the initiation of the cargo transport, they preferto align themselves according to a fixed distribution thatis dependent on external conditions.The steady state distribution P ss is obtained by solv-ing P ss = J P ss . In [15] section II it is seen that the thevalue of P ( t ) obtained by propagating P ( t ) = J P ( t ) ap-proaches P ss obtained by solving P ss = J P ss . Thus bypropagating P ( t ) = J P ( t ) we can not only convenientlyobtain an estimate for P ss but can also analyze the dy-namics of the process of orientation of the motors in theensemble as time progresses, thus making the long termbehavior of the ensemble tractable. Analysis of Steady State Distributions
The steadystate distribution P ss can be utilized to compute impor-tant biological quantities governing intracellular traffic,such as average cargo velocity, run-length and averagenumber of engaged motors. An instantiation for an en-semble of finite Kinesin-I (two and three-motor) yieldsquantitative and qualitative properties that are in goodagreement with existing studies (see [15] section III, IVfor details). The agreement between results obtained us-ing the analytic model and existing literature justifies the usage of the analytic framework to arrive at conclusionsabout the steady state dynamics of the ensemble andhow it responds to changing external conditions. Subse-quently, we utilized the computational model to quantifythe effect of external conditions on the behavior of themultiple motors, which we observe is captured by thesteady state P ss . In particular, we analyze the effect ofvarying ATP concentration and load forces on transportof cargo by multi-motor ensembles. Effect of ATP concentration:
Fig. 2(a-d) shows theimpact of ATP concentration on two and three motor en-sembles. We analyzed the probability of motor beingattached to the MT (denoted by p mot ) and the proba-bility of more than motor being attached to the MT(equal to − p mot ). For both two and three motor en-sembles, with reducing ATP concentration, the respec-tive p mot reduces and (1 − p mot ) increases. Changes inload force on the cargo have no impact on these observedtrends. A direct impact of this behavior can be observedon the variation of average velocity and run-length ofmulti-motor ensembles with ATP concentration. It isseen ([15] section IV) that, with reducing ATP concen-trations, the average velocity for multi-motor ensembles reduces but the runlength increases . The increase in run-length can be attributed to the fact that, even thoughthe velocity has diminished with reducing ATP concen-tration, the probability of more than 1 motor remainingattached to the MT increases. Here, the cargo has ahigher probability of remaining engaged to the MT. Thisenables the second motor (and third motor for three mo-tor ensemble) to contribute to the cargo motion, allow-ing for a higher probability of the cargo being linked tothe MT during the course of cargo travel. It enhancesthe overall distance covered by the multi-motor ensem-ble and corroborates experimental observations such as[13], while providing an explanation based on steady stateprobability distributions of the multi-motor ensembles. .The increase in the probability of more than one mo-tor attached to the MT can be attributed to reductionin single motor detachment rates with ATP concentra-tion, based on the single motor detachment model [4]used for the analysis in this article. Our analysis pre-dicts that for multi-motor ensembles, the probability ofcargo remaining attached through more than 1 motor in-creases with reduced ATP concentrations contributing toincreased cargo run-length. Effect of load force:
The simulation model is furtherutilized to analyze the impact of varying load forces onthe steady state probabilities of relative configurations oftwo and three motor ensembles, as shown in Fig. 2. Itis seen that at very small values of load forces Fig. 2(e) and (h), a significant majority of configurations havelow probabilities with little variation. No one relativearrangement is particularly favored, with the variation inprobabilities being very gradual. This indicates that themotors do not prefer any particular relative configurationat these values of load forces. An intuitive explaination isthat at such low loads, since equal or unequal load sharing
Relative Configuration Index Relative Configuration IndexRelative Configuration Index Relative Configuration Index P r o b a b ili t y P r o b a b ili t y P r o b a b ili t y P r o b a b ili t y (a) (b) (c) (d) Relative Configuration Index 𝐹 𝑙𝑜𝑎𝑑 = 0.01 𝑝𝑁 P r o b a b ili t y (e) 𝐹 𝑙𝑜𝑎𝑑 = 5 𝑝𝑁 Relative Configuration Index P r o b a b ili t y (f) 𝐹 𝑙𝑜𝑎𝑑 = 25 𝑝𝑁 Relative Configuration Index P r o b a b ili t y (g) 𝐹 𝑙𝑜𝑎𝑑 = 0.01 𝑝𝑁 P r o b a b ili t y Relative Configuration Index (h) 𝐹 𝑙𝑜𝑎𝑑 = 10 𝑝𝑁 P r o b a b ili t y Relative Configuration Index (i) 𝐹 𝑙𝑜𝑎𝑑 = 50 𝑝𝑁 P r o b a b ili t y Relative Configuration Index (j) (k)
Effect of ATP concentration Effect of Load Force - + - + 𝑀|||||𝑀
𝑀𝑀|||||||||||𝑀 𝐹 𝑙𝑜𝑎𝑑 = 25 𝑝𝑁 𝐹 𝑙𝑜𝑎𝑑 = 50 𝑝𝑁 (l) (m) FIG. 2. Variation of probability distribution function of two and three motor ensembles with ATP concentration cause insignificant variations in the load forces balancedby each motor in the ensemble, there is no advantage toadhere to a specific orientation. Thus in these regimesmotors tend to spread out more evenly while transportingthe common cargo.However, as the load force is increased, certain config-urations become more probable. As is seen in Fig. 2 (f)for two motor ensemble with F load = 5 pN , the relativeconfigurations M M and M | M are more probable, witha combined . share of the total probability (as com-pared to . if all configurations were equally likely).For a three motor ensemble with F load = 10 pN (Fig.2 (i)), the configurations M M M and M | M M becomemore probable, with a . share of the total proba-bility (as compared to . if all configurations wereequally likely). A common attribute connecting theseconfigurations is that these represent arrangements wheremotors are clustered together, with little to no separationbetween their relative locations on the MT.Let’s define probability of clustered states as the sumof probabilities of the clustered states for the two andthree motor ensembles respectively. As is seen in Fig.2(k), not only is clustering favored at the load forces un-der consideration, but the probability of clustered statestends to increase with load forces. A possible explanationfor the preference to clustering is that, clustered statesimply motor extensions of similar magnitudes. Thus insuch configurations, the load force is shared more equallythan in other configurations where the motors are farapart and not clustered together. As load force F load is increased, if non-clustered states are preferred then itwould lead to more unequal load sharing between the mo-tors of the ensemble. In such configurations one or moremotors may have to handle a higher percentage of F load ,bringing the value closer to or above the stalling force F s ( pN in this case). For example, for two motor ensem-ble with F load = 10 pN , in a non-clustered configuration M || M the rearguard motor takes . pN while vanguardbalances . pN load force. This will place the vanguardmotor beyond stall, making it unable to step forwardand thus inhibiting the overall cargo propagation. Thus, with increasing load forces, it is disadvantageous to pre-fer unequal load sharing i.e. configurations with a higherseparation between the motors on the MT. Thereby, it isbeneficial to prefer a more equal load sharing i.e. clus-tering, as F load on the cargo increases, as is evidenced byFig. 2(f),(i) and (k).However, the trend of ’more clustering’ with ’higher F load ’ does not hold for higher magnitudes of load forces.In Fig. 2(g),(j) it is seen that beyond a certain valueof load force, there is an abrupt departure from a pref-erence to clustering of motors. In case of two motorensemble with F load = 25 pN (Fig.2(g)), the clusteredstates of M M and
M M M are improbable while M ||||| M (Fig.2(l)) is the most probable ( share of total prob-ability). For a three motor ensemble with F load = 50 pN (Fig. 2(j)), M M ||||||||||| M (Fig.2(m)) is most probableamong the three motor configurations ( . share oftotal probability). An analysis of the load sharing inthese configurations reveals non-intuitive insights.In the two motor case with F load = 25 pN , therearguard motor in the most preferred configuration of M ||||| M handles . pN while the vanguard handles therest . pN load force. For three motor ensemble with F load = 50 pN , the rearguard motors in the most pre-ferred configuration of M M ||||||||||| M each handle . pN of the load force while the vanguard handles the rest . pN force. A possible advantage of such relative con-figurations is as follows. Here, stalling force for the mo-tors used is F s = 6 pN ; when the load on the motor isbeyond F s , the motor is unable to take a forward step.At these high loads, a common property among the mostpreferred configurations is that there is one vanguard mo-tor while the rest of the motors are all rearguard motorson the same location on the MT. The vanguard and rear-guard motors are spread out such that all the rearguardmotors are loaded beyond but near stall force F s and thevanguard motor bears the remaining load, whose value iswell beyond the stalling force F s .Such a high loading event relates to an infrequent yet apossibly debilitating phenomenon inside the cells, such asan unanticipated obstacle along the cargo travel path oran encounter with an oppositely directed cargo. Occur-rences like these are most likely sudden events that thatdo not lead to a progressive loading of the load force onthe cargo, but an unexpected spike. Based on our anal-ysis, the preferred orientation is such that as few motorsas possible (i.e. a single vanguard motor) are loaded wellbeyond stall force. In such an arrangement, once thetransient loading event goes away and F load starts to sub-side, it takes little reduction in F load to reduce the forceon each of the rearguard motors, subsequently enablingthem to take a step and propel the cargo forward (sinceforce on each motor now falls below F s ). It is in this ar-rangement that the maximum number of motors can bemobilized by the least reduction in F load , while keepingthe load force on the one vanguard motor as low as pos-sible. In contrast, for configurations where the rearguardand vanguard motors are closer (like a clustered config-uration preferred for lower F load ), the non-vanguard mo-tors would be taking a load higher than F s and therewould be less number of motors loaded just above theirstall force. Thus a larger reduction of load force wouldbe necessary to enable the same numbers of motors tobe able to step forward, as in the previous configura-tion. Another possible advantage is that, for the samereduction in F load , the preferred orientation enables themaximum number of motors (i.e. all but one vanguard)to be able to walk while maintaining the load force on the vanguard motor as low as possible. Thus, this ori-entation ensures that the least amount of reduction inthe high load force is needed to enable the resumption ofmotor motion and forward propagation of cargo towardsthe destination. Summary:
Using a semi-analytic Markov model toanalyze intracellular cargo transport by teams of finitemotors, we prove that the motors orient themselves ac-cording to unique steady state distributions irrespectiveof the initial orientation. It demonstrates the robust-ness of the intracellular transport mechanism. Analyz-ing how the distribution is impacted by external factorsreveals interesting coordination mechanisms. For a teamof multiple Kinesin-I motors at reduced ATP concentra-tions, more motors prefer to remain attached to the mi-crotubule. This contributes to enhanced ensemble run-length despite despite reduced velocitiy. Furthermore,an increase in hindering load force on the cargo is tack-led by ’clustering’ together. However, at very high loadforces the motors abandon clustering and adopt config-urations that prefer anchoring the cargo and immediatecargo translocation once the large loading event has sub-sided. Our approach provides unique insights into theteam behaviors of molecular motors, with the analyt-ical approach suitable to study transport mechanismsadapted by teams of other motors, such as dynein, myosinor even heterologous ensembles. [1] M. J. Schnitzer and S. M. Block, Nature , 386 (1997).[2] C. Kural, H. Kim, S. Syed, G. Goshima, V. I. Gelfand,and P. R. Selvin, Science , 1469 (2005).[3] C. Leduc, F. Ruhnow, J. Howard, and S. Diez, Pro-ceedings of the National Academy of Sciences , 10847(2007).[4] A. Kunwar, M. Vershinin, J. Xu, and S. P. Gross, Cur-rent biology , 1173 (2008).[5] M. J. Müller, S. Klumpp, and R. Lipowsky, Biophysicaljournal , 2610 (2010).[6] A. Kunwar and A. Mogilner, Physical biology , 016012(2010).[7] F. Berger, C. Keller, S. Klumpp, and R. Lipowsky, Phys-ical review letters , 208101 (2012).[8] J. W. Driver, A. R. Rogers, D. K. Jamison, R. K. Das,A. B. Kolomeisky, and M. R. Diehl, Physical ChemistryChemical Physics , 10398 (2010).[9] S. Klumpp and R. Lipowsky, Proceedings of the NationalAcademy of Sciences of the United States of America , 17284 (2005).[10] S. Klumpp, C. Keller, F. Berger, and R. Lipowsky, in Multiscale Modeling in Biomechanics and Mechanobiol-ogy (Springer, 2015) pp. 27–61.[11] F. Posta, M. R. DâĂŹOrsogna, and T. Chou, PhysicalChemistry Chemical Physics , 4851 (2009).[12] D. Materassi, S. Roychowdhury, T. Hays, and M. Sala-paka, BMC biophysics , 14 (2013).[13] J. Xu, Z. Shu, S. J. King, and S. P. Gross, Traffic ,1198 (2012).[14] X. Pan, G. Ou, G. Civelekoglu-Scholey, O. E. Blacque,N. F. Endres, L. Tao, A. Mogilner, M. R. Leroux, R. D.Vale, and J. M. Scholey, The Journal of cell biology ,1035 (2006).[15] S. material, (2018).[16] J. L. Doob, Transactions of the American MathematicalSociety , 455 (1945).[17] D. T. Gillespie, The journal of physical chemistry81