Emergent Transport Properties of Molecular Motor Ensemble Affected by Single Motor Mutations
Shreyas Bhaban, Donatello Materassi, Mingang Li, Thomas Hays, Murti Salapaka
EEmergent Transport Properties of Molecular Motor EnsembleAffected by Single Motor Mutations
Shreyas Bhaban ∗ , Donatello Materassi , Mingang Li , Thomas Hays , and MurtiSalapaka Department of Electrical Engineering, University of Minnesota-Twin Cities, Minneapolis MN 55455, USA Department of Electrical Engineering and Computer Science,University of Tennessee-Knoxville, Knoxville,TN 37996, USA Department of Genetics, Cell Biology, and Development,University of Minnesota-Twin Cities, MinneapolisMN 55455, USA
Abstract
Intracellular transport is an essential function in eucaryotic cells, facilitated by motor proteins - pro-teins converting chemical energy into kinetic energy. It is understood that motor proteins work inteams enabling unidirectional and bidirectional transport of intracellular cargo over long distances.Disruptions of the underlying transport mechanisms, often caused by mutations that alter single mo-tor characteristics, are known to cause neurodegenerative diseases. For example, phosphorylation ofkinesin motor domain at the serine residue is implicated in Huntington’s disease, with a recent study ofphosphorylated and phosphomimetic serine residues indicating lowered single motor stalling forces. Inthis article we report the effects of mutations of this nature on transport properties of cargo carried bymultiple wild-type and mutant motors. Results indicate that mutants with altered stall forces mightdetermine the average velocity and run-length even when they are outnumbered by wild type motorsin the ensemble. It is shown that mutants gain a competitive advantage and lead to an increase in theexpected run-length when the load on the cargo is in the vicinity of the mutant’s stalling force or amultiple of its stalling force.A separate contribution of this article is the development of a semi-analytic method to analyzetransport of cargo by multiple motors of multiple types. The technique determines transition ratesbetween various relative configurations of motors carrying the cargo using the transition rates betweenvarious absolute configurations. This enables an exact computation of biologically relevant quantitieslike average velocity and run-length. It can also be used to introduce alterations of various singlemotor parameters to model a mutation and to deduce effects of such alterations on the transport ofa common cargo by multiple motors. Our method is easily implementable and we provide a softwarepackage for general use. ∗ [email protected] ntroduction Motor proteins- kinesin, dynein and myosins- are nanoscale machines that are the main effectors ofintracellular transport. They play a critical role in the growth and sustenance of healthy cells by en-abling a transport of intracellular cargo over networks of microtubules [1]. Disruption of the functionsperformed by these molecular motors is linked to neurodegenerative diseases such as Huntington’s,Parkinson’s and Alzheimer’s Disease [2, 3], muscular disorders such as heart disease, uterine com-plications and high blood pressure. The mechano-chemical behavior of a single motor moving alongthe microtubule substrate to transport a cargo is relatively well understood [4]. However in vivo itis known that multiple motors work in teams to transport a common cargo [5]; how multiple motor-proteins coordinate to transport a common cargo is not well understood [5, 6, 7]. Many studies employa probabilistic description of the behavior of a single motor protein to construct models that describehow multiple motors transport a common cargo [5, 8, 9, 10]. In [11], Gross and coworkers employedMonte-Carlo simulation studies built on a model of a single kinesin in to explore how multiple identicalkinesin motors might interact to transport a cargo against a hindering load force. Their work indicates,counter intuitively, the existence of a form of strain-gating, where the motors of an ensemble shareloads unequally enabling cargo transport over longer distances. Xu and coworkers examined the effectsof ATP concentration on the transport of cargo carried by single and two motors in [12]. At decreasedlevels of ATP concentration, the velocities of cargoes transported by single and two motor proteinsdecreases. Coincidentally at decreased ATP concentration there was an an appreciable increase in therun-length of cargoes transported by two motors, while no such effect was seen in the case of transportby one motor. The authors proposed that the increased run-length observed in the presence of twomotors results from the lowered dissociation of each motor from the microtubule at decreased ATPconcentrations and the increased probability that the cargo stayed bound to the microtubule. Studiessuch as [10, 13] using probabilistic models of single motors have also predicted in that an ensemble ofkinesin motors is a robust system and the robustness increases under high loads [14].The study of cargo transport by a heterogeneous ensemble of motor proteins composed of bothwild type and mutant motors is important to inform our understanding of how mutant motors impactintracellular transport and lead to an onser of diseases. Recent studies have implied that alterations inthe kinesin-1 motor domain may have a role in impaired axonal transport. Phosphorylation of a mam-mal kinesin motor domain by kinase c-Jun N-terminal kinase-3 (JNK3) at a conserved serine residue(Ser-176 in A and C isoforms and Ser-175 in B isoform) is implicated in Huntington’s disease [15].However the mechanisms affected by Ser-175 phosphorylation are not well understood. An experimen-tal study by Selvin and coworkers in [16] reported that a negative charge at Ser-175, acquired throughmutation or phosphorylation, leads to a lower stall force and decreased velocity under external loadsof 1 pN or more, while leaving the ATPase, microtubule-binding affnity and processivity unchanged.Using a semi-analytical method, we reveal surprising emergent transport behaviors arising whena cargo is transported by multiple motor proteins, some of which are mutated and some are not.In particular we analyze the impact of Ser-175 kinesin mutation such as those reported in [16] oncargo transport in the presence of wild type motors. The detailed investigation made possible byour method leads us to hypothesize that under certain conditions, proteins moor the cargo to themicrotubule and prevent it from being lost. While these mooring proteins do not contribute to themotion of the cargo, they enhance the probability of attachment of other cargo-bound motor proteinsto the microtubule that subsequently contributes to an increase in average cargo displacement. Theactivation of mooring mechanism depends on a number of external factors such as load force and ATPconcentration. However, it is also determined by intrinsic properties of the motor protein such as thestall force of the individual motors. Remarkably, mutant motors that have stalling forces matched tothe external load force can act dominantly and determine emergent transport properties such as longer2un-length, even when they are outnumbered in the ensemble by wild-type motors. Such mechanismscould potentially point to how diseased states emerge and progress coincident with the accumulationof the mutant motor species.A separate contribution of the article is a semi-analytical method for determining the probabilitydistribution of various configurations of a cargo carried by multiple number and types of motor-protein.A detailed experimental study of the various modalities of transport by multiple motor proteins (ho-mologous or otherwise) requires significant instrumental resolution than what is needed to investigatesingle motor behavior. As a consequence, observing the transport dynamics of multiple motors is ex-perimentally challenging [17, 18]. It is further compounded by the combinatorial complexity introducedby the multitude of scenarios possible when many motors and motor types participate in transport.Such challenges motivate the use of analytical and computational tools. The mean-field approach in[8, 10] makes use of simplifying assumptions, such as equal load sharing among all motors, to achieveanalytical results, thereby sacrificing significant detail for computational benefits. The Monte Carloapproach in [11], provides better fidelity where complex models can be employed; however, the accu-racy of results depend on the number of iterations and on the rarity of the events that occur. Unlikethe Monte-Carlo simulations or any currently implemented simulation method, our Master Equationbased method [19] analytically solves for the probability distributions of all possible scenarios at anytime point. The methodology is uniquely powerful and enables the calculation of various biologicallyrelevant quantities such as average velocity and run-length, for reasonably sized ensembles and withhigh accuracy while using lesser computational resources and time. The underlying concepts behindour methodology are motivated by earlier work reported in [13]. The key enabling concept is thatof ‘relative configuration’ of motor proteins, determining the transition probabilities between relativeconfigurations from the transition probabilities of the absolute configuration space and subsequentlydetermining the biologically relevant quantities from the relative configuration space. The computa-tion engine is implemented using MATLAB and can be used to simulate cargo transport by any twounidirectional species.Our method provides a general platform to study the transport of cargo by multiple motors of twodifferent types where each type of motor protein can be individually characterized by a probabilisticmodel describing its stepping, detachment and attachment rates. For this article, the technique hasbeen utilized to introduce alterations of various parameters from the nominal ones to model a mutationof the serine residue and compute the effect of such a mutation on cargo carried by a mixture ofwild-type and mutant motor-proteins. In summary, we developed a simulation engine to study thetransport of cargo by multiple motor proteins with distinct properties that in concert can exhibitemergent transport behaviors.
Results and Discussion
In this study, we investigate the impact of a previously reported kinesin mutation on the transport of amotor ensemble and its attached cargo. The mutation, located within the motor domain of the kinesin,mimics the phosphorylated state of Ser-175 [16]. The motor domain phosphomimetic mutation doesnot affect the ATPase, microtubule-binding affinity or processivity of the motor, but does reduce thestall force and velocity of kinesin under a load force. The in vitro phosphorylation of Ser-175 for a fulllength kinesin similarly reduces stall force and velocity of the motor. Both the mutant and in virtophosphorylated kinesin showed no other aberrant single motor behavior.Here we use computational modelling to analyze the impact of the Ser-175 mutant kinesin on aheterogenous ensemble of motors and its transported cargo. The wild-type and mutant kinesin motorshave a different stalling force; the wild type kinesin has a stall force of F s = 6 pN and the mutated3inesin has a reduced stall force of ¯ F s = 5 . pN (see [16] for more detail). In our analysis we consideredcargoes transported by the following motor ensembles : cargo with two wild-type (WW) motors , onewild-type and one mutant (WM) motors, two mutant (MM) motors , three wild-type (WWW) motors, two wild-type and one mutant (WWM) motors, one wild-type and two mutant (WMM)motors, andthree mutant (MMM) motors. How External Loading Conditions Affect Behavior of Heterogenous MotorEnsemble
Motor configuration when engaged to the microtubule approaches steady state
For cargoes carried by multiple motor proteins, transport occurs as each motor steps along themicrotubule lattice, sequentially binding, translocating and releasing from the microtubule in a welldefined mechanochemical cycle [5, 9, 20, 21]. Eventually, the condition where no motors are attachedto the microtubule will be reached and the cargo will diffuse away from the microtubule. Thus it isevident that the only steady state is the condition when none of the motors of the ensemble carryingthe cargo remain attached to the microtubule. Indeed, in all the cases considered where there are threeor less motors on the cargo (i.e. M ≤
3) it is seen that for any specified initial probability distributionof motors engaged to the microtubule, the steady-state probability distribution has no motors attachedto the microtubule i.e. cargo is eventually lost. Figure 1(a) shows the exactly computed probabilitydistribution for the WM case, where with increasing time the probability of no motors attached (redcurve) increases and eventually reaches one.Given that the steady-state probability distribution is the trivial case of no motors attached,nothing much of value can be derived regarding the motor behavior from the trivial steady-stateprobability distribution. However the conditional probability distribution of the various configurationsof motor proteins carrying a common cargo, given that at least one motor is attached to the microtubule (i.e. when the cargo is not lost), is of significant interest. Here it is apriori not evident whether theconditional probability distribution has a steady state.Remarkably, the results show that under a constant load force on the cargo the conditional prob-ability distribution of the various motor configurations reaches a steady state. The steady statedistribution is independent of the initial probability distribution chosen (see for example, Figure 1(b)for the WM case). This indicates that the cargo-motor ensemble consisting of motors with differentstalling forces is a robust system which, starting from any arbitrary initial condition, behaves accordingto a fixed distribution after some transient period. Furthermore, the existence of such a steady-statedistribution, termed here as the steady-state conditional pdf , allows the computation of most of thevariables of biological interest.
Load forces on cargo affect conditional steady state distribution of motor ensemble
The conditional steady-state distribution is used to analyze transport of cargo with two attachedmotors and assumes that at least one motor is attached (Figure 2). Figure 2 shows the probabilitiesfor the number of wild-type and mutated motor proteins that remain attached under varying loadforces. The analysis indicates that when the cargo has two wild type motors attached (WW case),the probability that both motors remain attached to the microtubule peaks when load force is closeto 12 pN . This equals twice the stalling force of a single wild type motor; indicating a relationshipbetween number of motors attached (obtained using the steady state distribution) and applied load.Thus at these loads there exists a propensity to retain and not loose the cargo. Similarly, in the4
200 2 4 6 800.20.40.60.81
Motors engagedto the microtubuleTime (sec) P r ob a b ili t y
120 2 4 6 800.20.40.60.81
Motors engagedto the microtubuleTime (sec) P r ob a b ili t y (a) (b) Figure 1:
Probability of motors being bound to the microtubule at time t under load forceof . pN for the WM case. In (a) the probability of one bound motor (blue) and two boundmotors (green) approaches 0 as time t → ∞ . Probability of no bound motors (red) (i.e. the cargodissociates from the microtubule) approaches 1 as t → ∞ . In (b) the probability of one bound motor(blue) and two bound motors (red) under the condition that cargo is not lost, reaches a steady state.MM case (where the cargo is attached to two mutant motors) the probability that both motors areattached to the microtubule is maximum near 11 pN which is twice the stalling force of the mutatedmotor. The WM ensemble displays lower peaks for the dual attachment of motors at both load forcesof 11 pN and 12 pN .It is evident that for ensembles containing two motors, the conditional steady state distributionsof the number of motors attached is not significantly affected until the load force on the cargo is closeto twice the stalling force values of the wild -type or mutant motor (2 F s = 12 pN or 2 ¯ F s = 11 pN ). Asimilar observation is made for ensembles with three motors and for load force values close to twiceand thrice the stalling force of a single motor or 2 F s , 2 ¯ F s , 3 F s (= 18 pN ) and 3 ¯ F s (= 16 . pN ), pointingto a mechanism of equal sharing of loads in these load force ranges. Thus a cooperative mode betweenan ensemble of motors is encouraged when the load forces are close to the stall-force or multiples of thestall forces. For example, if a WW ensemble is subjected to load force close to 2 F s = 12 pN , unequalload sharing would subject one of the motors to a force greater than its stalling force F s = 6 pN . Thiswill lead to its detachment and eventual transfer of the entire load onto the remaining motor. This, inturn, would gradually lead to the remaining single motor being detached from the microtubule, andthe resultant loss of cargo attachment to the microtubule substrate. On the other hand, an equal loadsharing will stall both the motors but ensure the retention of the cargo. A similar inference can bemade for cargo transport by three motors and load forces approaching the values F s , ¯ F s , 2 F s or 2 ¯ F s (figure not shown).Note that, it is possible when the load force is close to twice the stalling force, the cargo detachesfrom the microtubule with a high probability. In such a case, the previously mentioned conclusionsreached about the probabilities by applying the condition that the cargo is not lost (i.e. at least onemotor remains engaged to the microtubule) will not carry much significance. We then utilized the5 Load force in pN WWWMMM P r ob a b ili t y o f exac t l y t w o m o t o r s b e i ng a tt ac h e d Figure 2:
Probability of exactly two motors attached to the microtubule for ensemblesWW, WM and MM. probabilities obtained without the conditioning to reach similar conclusions. Figures 3 and 4 showthat the time taken for the cargo to be lost (i.e. time taken for the probability of detachment of thecargo to be close to one) is higher when the load force is close to but less than the appropriate stallingforce or twice the appropriate stalling force value. The time taken for detachment is lesser for a loadforce of an intermediate value (compare with Figure 5). The figures show that if the load force ismatched to an appropriate stall force or its multiples (for example, close to 2 F s for WW or 2 ¯ F s forMM ensemble), the probability of detachment of the cargo from the microtubule becomes appreciableonly after a long time has passed. Thus, the conclusions reached using the conditional distributionare also supported by results inferred from the entire configuration space with no restriction that thecargo remains attached. It implies that the study on the condition that ‘the cargo is not lost’ providesimportant insights.In conclusion, load force on the cargo has a significant effect on the steady state distribution ofmotor ensembles and can provide an insight into how multiple motor ensembles function under varyingloads. In particular, when a cargo is subjected to load forces that are multiples of the stalling force forindividual motors, a cooperative behaviour in the form of equal load sharing is observed. This avertsa scenario where unequal load sharing would subject some of the motors to loads greater than theirstall forces, leading to their detachment and the eventual loss of cargo from the microtubule. Suchcooperative behavior possibly helps retain the cargo. It is thus more probable that cargoes attachedto a higher number of wild-type motors are retained under load forces close to multiples of the stallingforce of a wild-type motor, making the presence of more mutants disadvantageous in these load forceregimes. Hindering Load Force on the Cargo can Tune Multiple Motor Travel
The knowledge of the steady state conditional probability distributions allows one to compute biolog-ically relevant quantities such as average velocity and run-length. The results obtained for ensembles6
Time (sec) P r ob a b ili t y o f t h e ca r go b e i ng l o s t WW−5.8 pNWM−5.8 pNMM−5.8 pNWW−11.8 pNWM−11.8 pNMM−11.8 pN
Figure 3:
Probability of cargo being lost for ensembles WW, WM and MM at load forcesclose to but less than F s and F s ( F s = 6 pN ). containing two and three motors for varying load forces are reported. Average velocity typically reduces with increasing number of mutant motors, but variesas load forces approach multiples of stall forces
The effect of load force on the average velocity of a cargo is described in Figures 6 and 7 for a totalof two and three attached motors, respectively. It is evident that for any given ensemble, average cargovelocity mostly reduces with increase in the hindering load force on the cargo. It is further evidentfrom Figure 6 that for a given load force on the cargo, the average velocities are highest for the WWensemble, followed by WM and the lowest for MM ensemble. For the same load force and number ofmotors in the ensemble, it follows that the average velocity decreases with increased participation ofmutant motors (which have a lower stall force). A similar conclusion is reached when three motors areattached to the cargo (see Figure 7). Thus, for a fixed load force and total number of motors, averagecargo velocity reduces with increasing number of mutant motors in the ensemble.However, for some motors ensembles the trend of decreasing cargo velocity at elevated load forcesis not observed. Figure 8 shows the average velocity for load forces between 10 . pN and 12 . pN . Inthe case of WWW ensemble, as the load forces increase to 12 pN (which is equal to 2 F s ) the averagevelocity increases with increasing load force. Our semi-analytic method allows for a fine analysis of suchcounter-intuitive observations where it is possible to extract the precise motor ensemble configurationsthat contribute to the observed effect.Upon analysis it is observed that as the load force approaches the appropriate multiples of stallingforce (2 F s or 2 ¯ F s in Figure 8), detachment of motors from the microtubule becomes less likely and theattached motors tend to cluster close together on the microtubule. This is similar to an observationmade in [11, 13] where under high loads, the motors carrying a common cargo tend to remain clusteredtogether while under low loads they tend to spread apart. Since the detachments are less likely,backward transitions of the cargo induced by single motor detachment are less frequent. Moreover,if a detachment were to occur, the magnitude of load induced backward transition of the cargo isless due to the clustering of motors under high loads. On average, this contributes to an increase in7 Time (sec) P r ob a b ili t y o f t h e ca r go b e i ng l o s t WW−5.3 pNWM−5.3 pNMM−5.3 pNWW−10.8 pNWM−10.8 pNMM−10.8 pN
Figure 4:
Probability of cargo being lost for ensembles WW, WM and MM at load forcesclose to but less than F s and F s ( F s = 5 . pN ). average velocity despite increasing load. This explains the increase in average velocities of the threemotor ensembles at load forces approaching 12 pN .Interestingly, the increase in average velocities at load forces approaching 2 F s = 12 pN is moreprominent in the WWW ensemble than in MMM ensemble, indicating that it is the detachment of thewild-type motors in the ensemble that becomes less unlikely. Thus, as the number of wild type motorsincreases, detachment events at load forces close to 2 F s = 12 pN become less likely. Similarly, atload forces approaching 2 ¯ F s = 11 pN the average cargo velocity increases with increasing load force;with the increase becoming more prominent for ensembles having more numbers of mutant motors(i.e. higher for MMM ensemble than WWW ensemble).Significantly, this observation suggests that by tuning the load forces on a cargo to values near themultiples of the stall force, one could counter a decreased velocity resulting from an increase in loadforce or the presence of a higher number of mutant motors.An important feature of the semi-analytic method to be emphasized is the ease with which detectionof rare events such as those described above are made possible. Monte Carlo based methodology, forexample, would not only take significant computations to simulate rare events with a high degree ofconfidence, but would also be unable to determine the incidence causes of rare events. Thus our modelallows the user to easily unearth the cause of a rare event.In conclusion, load forces approaching multiples of the appropriate stall force for a cargo decreasethe probability of detachment of associated motors from the microtubule. Since detachment of motorsin a multiple motor ensemble can lead to backward motion of the cargo (because kinesin doesn’tactively step backward i.e. towards minus end), a decrease in the detachment probability leads tohigher forward motion of cargo on average, leading to an increase in average velocity even thoughthe load increases to approach multiples of stalling forces. If load forces are close to multiples of F s (or ¯ F s ) detachment of wild-type (or mutant) motors in the ensemble decreases and cargoes withhigher wild-type (or mutant) motors demonstrate greater increase in average velocity. Cargo run-lengths impacted by heterogenity in the motor ensemble Time (sec) P r ob a b ili t y o f t h e ca r go b e i ng l o s t WW−3 pNWM−3 pNMM−3 pNWW−8 pNWM−8 pNMM−8 pN
Figure 5:
Probability of cargo being lost for ensembles WW, WM and MM at load forces pN and pN The dependence of run-length on the load force when the cargo has two and three motors attachedto it is shown in Figures 9 and 10 respectively. A general trend is that the average run-length reduceswith an increase in the antagonistic load force on the cargo. Moreover, for a fixed number of motorsin an ensemble, at most values of load force, the run-length is reduced with an increasing number ofmutant motors in the ensemble (Figure 9 inset(A) and Figure 10 inset(A)).Surprisingly, under a load force approaching 6 pN (Figure 9 inset(B)), there is a surge in theaverage distance traveled by the cargo, peaking close to 6 pN and falling for larger load forces. Theaverage run-length when load force is close to 6 pN increases with the number of wild-type motorsand decreases with the number of mutant motors (note that, at load force equal to 5 . pN , ensembleWW travels ≈ nm , WM travels ≈ nm while MM travels ≈ nm ). Similar behavior isseen in Figure 10 insert(B). Note that 6 pN is the stalling force F s of the wild-type motor.Similarly, when the load force is close to 12 pN (Figure 10, inset C), the average run-length of acargo is higher when the number of wild-type motors is greater that the number of mutant motors inthe ensemble. Note that the overall run-length values for the three motor ensemble are lower here,since the load force on the cargo is higher (compare 12 pN , Figure 10 inset(C) versus 6 pN , Figure 10inset(B)). Also note that 12 pN is twice the wild-type motor stalling force F s .In Figure 9 inset(B), when the load force is close to 5 . pN , a surge in the run-length is observed,peaking close to 5 . pN and falling for larger load forces. However, the order of increase in run-lengthseen here is opposite to that seen near 6 pN , i.e. the run-length is higher for larger number of mutantmotors and is lower for a larger number of wild-type motors in the ensemble. Similar behavior is seenin Figure 10 inset(B). Note that 5 . pN is the stalling force ¯ F s of the mutant motor.In Figure 10 inset(C), for a load force close to 11 pN , the average run-length is higher when alarger number of mutant motors is present in the ensemble. Note that 11 pN is twice the mutantmotor stalling force ¯ F s .To explain the reason behind such an observation, we examine the run-length observed for hetero-geneous motor ensembles (e.g. WM). Using the conditional steady state probability distributions forthe scenario where the load force is close to the wild-type stalling force, we determined the probabilityfor the wild-type and mutant motor remaining attached to the microtubule (Figure 11). It is evident9 Load force on the cargo in pN A ve r a g e V e l o c i t y i n n m / sec WWWMMM
Figure 6:
Average velocity with varying load forces on the cargo for ensembles WW, WMand MM. that the probability of the wild-type motor remaining engaged with the microtubule lattice is highwhen the load force is close to 6 pN . It can thus be inferred that under this condition, the cargoremains bound to the microtubule with a higher probability. In consequence, the detached motor hasa greater opportuinity to reattach to the microtubule, overcoming the stalled condition of the leadingmotor and contributing to the increase in run-length. Thus the run-length under these conditions willbe proportional to the number of wild-type motors, since it is the wild-type motors that will stall,tethering the cargo to the microtubule lattice and favoring its continued translocation. The peak inthe run-length observed in Figure 7 inset(C) for a load force close to 12 pN can similarly be explained.As the two leading wild-type motors stall and remain attached to the microtubule, the third motorwill reattach, overcome the stalled condition and promote an extension of the run-length.The same load dependency of cargo run-lengths is also observed for mutant motors with reducedstall forces. At the reduced stall force, a mutant motor exhibits an increased probability for remainingattached to the microtubule lattice, preventing the cargo from diffusing away from the microtubule andincreasing the runlength (Figure 11, see peak in green trace at 5 . pN ). The tethering by the stalledmutant motor increases the probability that the disengaged wild type motor will reattach, overcomethe stalled mutant motor and extend the cargo runlength. The same effect also explains the positivecorrelation between run-length and number of mutant motors at load forces close to ¯ F s and multiplesof ¯ F s , as the probability of having mutant motors that are stalled increases making the loss of cargomore unlikely.Significantly, in these load force ranges and for a fixed number of motors attached to the cargo,the cargo with more mutant motors attached will travel further than other cargoes with fewer or nomutant motors attached. Remarkably, the run-length of the cargo is predicted by the single motorcharacteristics, namely the stalling force, of the mutant motor. ATP Concentration can Tune Multiple Motor Travel
It is known that kinesin hydrolyzes one ATP per step [22]. Thus, the rate at which kinesinsteps depends on its rate of ATP hydrolysis. It was shown experimentally in [12] that under noload conditions, average velocity for ensembles having one and two motors decreases with decreasing10
Load force on the cargo in pN A ve r a g e V e l o c i t y i n n m / sec WWWWWMWMMMMM
Figure 7:
Average velocity with varying load forces on the cargo for ensembles WWW,WWM, WMM and MMM.
ATP concentration. It was also shown that run-length was not appreciably affected by changingATP concentrations when the cargo had only one kinesin attached. However, when the cargo hastwo kinesin motors attached, run-length was demonstrated to have a strong negative correlation withsingle motor velocity (and single motor velocity was shown to have a positive correlation with ATPconcentration). Our semi-analytic method was used to assess the conclusions reached in [12] and tofurther understand how the presence of mutant motors (with a different stalling force ¯ F s ) can impactthe ATP dependence of run length. The altering of this behavior due to the presence of more thantwo motors in the ensemble is also studied. Average run-length for multiple motor ensemble negatively correlates to ATP concen-tration when restricted to certain regimes
We first considered the affect of ATP concentration on run-length of a cargo containing only asingle wild type motor (Figure 12). Our analysis shows that as ATP concentration increases, thedistance traveled by the cargo also increases monotonically until ATP concentration saturates. Whenthe cargo carried by a single wild-type motor is subjected to 0 . pN load force, the effect of ATPconcentration on run-length is not substantial for concentrations above 10 µM (611 nm at 10 µM ATP and 786 nm at 2 mM ATP). Thus run-length of a cargo with a single wild-type motor is notsubstantially affected by ATP concentration, agreeing with the trend observed in [12].However, the effect of ATP concentration is more pronounced for ensembles containing more thanone motor. For ensembles comprised of multiple motors, decreased ATP concentrations under cer-tain conditions leads to increased average run-length of the cargo. For example, in both the WW(Fig 13) and WWW (Figure (S3Fig)) ensembles, we observe that with reducing ATP concentration,the run-length first increases, peaks at a certain value and then decreases. The average velocity ofcargo transport will decrease with decreasing ATP concentration (see Figure(S1Fig) for WW andFigure(S2Fig) for WWW), indicating that the increased run-length is not due to increasing velocitybut instead reflects the increased probability that the cargo remains attached to the microtubule fora longer time. As is evident from Figure 14 for WW ensemble, not only does the probability of at-11 . . . . . . Load force on the cargo in pN A ve r a g e V e l o c i t y i n n m / sec WWWWWMWMMMMM
Figure 8:
Figure showing variation of average velocity with load force on the cargo forensembles containing motors, for load forces between . pN and . pN . tachment increase in the presence of two motors by comparison to one motor, but the probability thatthe motors remain attached to the microtubule also increases as the ATP concentration is decreased.A similar argument is provided for the experimental observations made by Xu and team in [12] forensemble WW, that reported a negative correlation between single-motor velocities and run-length ofcargo transport by ensemble WW.Furthermore, the results also predict a negative correlation will persist until a certain minimumATP concentration, below which the average run-length starts decreasing. To summarize, the run-length cannot continue to increase as ATP concentration falls, since at a certain threshold the lowATP concentrations will hinder ATP hydrolysis and the stepping of the motor domains along themicrotubule lattice. At this limiting ATP concentration, regardless of how long the cargo remainsattached, the motors will no longer be able to translocate along the microtubule lattice and runlengthwill decrease.In conclusion, the ATP concentration in certain regimes negatively correlates with cargo run-length,as long as there are more than one motor attached to the cargo. Such behavior observed for a twomotor ensemble agrees with results reported in [12]. The heterogeneity of motors within an ensemble does not alter the effect of ATP con-centration on run-length
To analyze the impact of ATP concentration on the behavior of ensembles with multiple motors,we characterized the average run-length and velocity for ensembles with two motors (Figure 13 andFigure(S1Fig)) and three motors (Figure(S3Fig) and Figure(S2Fig)).Ensembles with mutant motors also exhibited a similar trend of increasing run-length as ATPconcentration was reduced under fixed load forces (Figure 15, WM ensemble). When compared withother two-motor ensembles (Figure 13, load force is 0 . pN ), it is seen that at constant load force andATP concentrations, the cargoes carried by ensembles WW, WM and MM exhibit almost the same12 A ve r a g e R un l e ng t h i n n m WWWMMM (B)(A) 2.2 pN 5.5 pN 6 pN Figure 9:
Average Run-length under different load forces on a cargo with ensembles oftwo motors, wild type and mutant (WW, WM and MM). average run-length.Thus, if the ATP concentration and number of motors within the ensemble is constant and if theload force does not approach the stall force (or multiples of stall force) for either wild type or mutant(i.e. F s or ¯ F s ), then the average run-length will not be affected by the number of mutant motors inthe ensemble. Similar conclusions hold for ensembles with three motors.When load forces are close to, but less than, the stalling force F s for the same number of motors inthe ensemble, then irrespective of the ATP concentration the ensembles with more wild-type motorswill dominate cargo transport and exhibit longer run-lengths. In contrast, if load forces are close to,but less than ¯ F s , then ensembles with more mutant motors will dominate cargo transport and exhibitlonger run-lengths. Conclusions
Single motor mutation influences the emergent properties of cargo transport
We developed a simulation engine to study the transport properties of a cargo carried by multiplemotor proteins of different types. To demonstrate the efficacy of our methodology we capitalized ona previous study by Selvin and coworkers [16] which characterized a kinesin motor that exhibits areduced stall force. The study reported that placing a negative charge at the Ser-175 position of thekinesin motor domain, through either mutation or phosphorylation, reduces the stalling force of themotor and decreases its average velocity under external loads. We incorporate this into our methodto enable an analysis of the behavior of ensembles comprising of several wild-type and mutant kinesinmotors, where the mutant motors have lower stalling forces. The analysis provides support for the13 Load force on the cargo in pN A ve r a g e R un l e ng t h i n n m WWWWWMWMMMMM (A) (B) (C)2.2 pN 5.5 pN 6 pN 11 pN12 pN Figure 10:
Average Run-length under different load forces on a cargo with ensembles ofthree motors, wild type and mutant (WWW, WWM, WMM and MMM). following conclusions :1. A mutation that modulates stalling force of the motors impacts the emergent transport propertiesof a molecular motor ensemble as follows :(a) For a cargo bound to a fixed number of motors and under a constant load force away fromthe mutant motor stalling force ¯ F s (or its multiples), the average velocity and run-lengthdecreases with an increase in the number of mutant motors in the ensemble. A similartrend is observed for a load force close to the wild-type motor’s stalling force F s (or itsmultiples).Such a behavior is expected, since cargoes carried by ensembles having a higher proportionof mutant motors (that have a reduced stalling force) would traverse shorter run-lengths onaverage. Under these conditions, ensembles with a higher mutant population will eventuallylag behind similar-sized ensembles with a lower mutant population.(b) Remarkably, for a cargo subjected to load forces close to the mutant motor’s stalling force ¯ F s (or its multiples), the average run-length increases with an increase in the number of mutantmotors in the ensemble. We hypothesize the existence of a mooring mechanism , wheremutant motors tether the cargo to the microtubule, providing the other detached motorswith a greater opportunity to reattach to the mirotubule and move the cargo forward. Itconsequently leads to a higher average run-length of the cargo.Thus, surprisingly under these conditions, ensembles with a higher population of mutantsgain an aggressive edge over similar sized ensembles with a lower mutant population.2. ATP concentration modulates cargo transport by multi-motor ensembles as follows :14 Load force on the cargo in pN P r ob a b ili t y P(Wild-type motor engaged to microtubule)P(Mutant motor engaged to microtubule)
Figure 11:
Probability of wild-type and mutant motor being attached for ensemble WM. (a) ATP concentration in specific ranges negatively correlates with average run-length of thecargo under a constant load force, if the cargo has more than one motor attached. Thisfinding agrees with the experimental observation made by Xu and coworkers for a 2-motorensemble in [12].(b) For a fixed number of motors bound to the cargo, the effect of ATP concentration onrun-length is independent of the ratio of wild-type and mutant motors.3. While we used a kinesin phosphorylation mutant [16] to explore the consequences of a multi-motor ensemble containing motors of lower stall forces, similar analysis and the computationalmethodology can be utilized to obtain insights on the effect of different types of motors modeledby different sets of parameters on the transport of a common cargo. Moreover, mutations thatimpact other single motor parameters, including on/off rates, directionality, and elasticity, couldbe evaluated for impacts on ensemble behavior.It is likely that the aberrant behavior caused due to the presence of mutant motors with lowerstalling forces, when occurring over multitudes of ensembles, can contribute to the disruption of cargotraffic, impediment of neuronal function and the emergence of neurodegeneration.
Simulation engine developed for evaluating the impact of single motor pa-rameters on behavior of motor ensembles
The computational methodology proposed by Gross and coworkers in [11] employed Monte Carlosimulations to offer several insights regarding transport properties of molecular motor ensembles. Inthis article, we have developed a novel, semi-analytical approach to study the effect of single motormutations on the behavior of motor ensembles and cargo transport. This methodology allows us tocompute exactly, the biologically relevant quantities such as average run-length and velocity for anensemble of motors using experimental parameters derived from single motor experiments. The results15 −4 ATP concentration in molar A ve r a g e R un l e ng t h i n n m Figure 12:
Effect of ATP concentration on run-length for ensemble containing one wild-type motor, for various hindering load forces on the cargo. for ensembles WW and WWW (i.e. all wild-type motors) are in exact quantitative agreement withthose obtained via Monte Carlo simulations as presented in [11] as well as [13] , fully validating themethodology. Unlike Monte Carlo approaches, this model is computationally less extensive and itsefficiency is independent of the number of iterations. The model also offers ways to investigate rareevents and can be extended to any species of motor proteins, given the knowledge of their individualparameters. It is also easily possible to interrogate specific single motor parameters and to determinetheir effect on ensemble behavior, thus making it a useful tool to study the contribution of other singlemotor mutations as well as post translational modifications to the transport behavior of cargoes.
Methods
The Master Equation based methodology used to obtain the aforementioned results is described in thissection. This method is used to study emergent properties of an ensemble of multiple motors of twotypes, that can each take a step on, detach from or reattach onto the microtubule. The knowledge oftransition probabilities of stepping, detachment and attachment enable the determination of transitionrates between various absolute configurations of the motors, allowing for the calculation of transitionrates between the corresponding relative configurations . These rates enable the calculation of theprobability distribution of the various ensemble configurations, thereby facilitating the computationof several biologically relevant quantities such as average velocity, run-length and number of attachedmotors.We begin with the construction of the relative state space, along with calculations necessary toarrive at several biologically relavent quantities. Then, a general methodology to obtain the transitionrates between absolute configurations given the knowledge of the probability rates of stepping, detach-ment and reattachment to the microtubule for a wild-type motor( P S , P D , P A ) and probability ratesfor a mutant motor( ¯ P S , ¯ P D , ¯ P A ) is presented. Finally, the model used to determine the probabilityrates for kinesin motor proteins is detailed. 16 −4 ATP concentration in molar A ve r a g e R un l e ng t h i n n m WWWMMM
Figure 13:
Effect of ATP concentration on run-length for ensembles WW, WM and MMagainst load force of . pN . Construction of Relative Configuration Space
Consider a cargo that is carried by both wild-type and mutant motor proteins on a microtubule.The microtubule is modeled as directed linear lattice formed by equally sized dimers with dimen-sion d . Here the k th dimer is located at location ¯ a k = kd and indexed by the set of integers I = { ..., − , − , , , , ... } . Each motor protein bound to the cargo can attach, take a forward stepor detach from the microtubule. The absolute configuration Ω := (cid:26) Ω h,k Ω d,k (cid:27) k ∈ I of motor-protein ar-rangement on the microtubule specifies the number, Ω h,k , of wild-type motor proteins and the number,Ω d,k , of mutant motor proteins at the k th location on the microtubule.For example, the absolute configuration of the ensemble of motors illustrated in Figure 16 is givenby Ω = [ · · · Ω − Ω Ω Ω Ω Ω Ω Ω · · · ] = (cid:20) · · · · · ·· · · · · · (cid:21) . The relative configuration of an ensemble of motors is represented using a string of three symbols. Givenan absolute configuration we first identify the rearguard motor which is the motor that is attached tothe microtubule and lags behind all the other motors on the microtubule. Using the location of thereargurad motor as a reference, the relative configuration ϑ is obtained as a string of three symbols‘ M h ’, ‘ M d ’ and ‘ | ’, where ‘ M h ’ and ‘ M d ’ denote wild-type and mutant motors respectively, with ‘ | ’denoting a separator that distinguishes different microtubule locations. The motor located the farthestfrom the rearguard motor on the microtubule is identified as the vanguard motor .For example, the relative configuration of the ensemble in Figure 17(a) is the string ‘ | M h M d || M d || M h | ’.The configuration that results after the furthermost mutant motor in Figure 17(a) takes a step is shownin Figure 17(b) which has a relative configuration given by ‘ | M h M d ||| M d | M h | ’.Both the mutant as well as the wild-type motor proteins are characterized via their own setof stepping, attachment, and detachment probabilities(for wild-type ( P S , P D , P A ) and for mutant17 −4 ATP concentration in molar P r ob a b ili t y P(1 motor attached to the microtubule)P(2 motor attached to the microtubule)
Figure 14:
Effect of ATP concentration on run-length for ensemble WW. ( ¯ P S , ¯ P D , ¯ P A )). The individual motors for both species are modeled as hookean springs when stretchedthat offer no resistance when compressed. A single motor is assumed to have a linkage rest length L and spring stiffness constant K e . Motors of both the species are assumed to not step backward andare bound to the cargo particle irreversibly. More complex models of motor-proteins can be easilyaccommodated. It is further assumed that the there exists a force F stall called the stalling force , whereif the force on the motor protein F ≥ F stall then the motor does not take a forward step and steppingprobability is zero [21]. The stalling force can be that is also used to estimate how many motors arecarrying the cargo. For an ensemble of wild-type and mutant motors carrying a cargo, the followingresult holds: Result 1 : Given an ensemble of M molecular motors attached to a common cargo that is subjectedto a load force F load , the distance between the rearguard and the vanguard motor is bound by n = max (cid:26) ( M + 1) max ( F s , ¯ F s ) − F load K e + d, F load K e (cid:27) + 2 L (1) where F s is the minimum load force for which the stepping probability of the wild-type motor proteinbecomes zero ( i.e. the stalling force for the wild-type motor protein), ¯ F s is the minimum load forcefor which the stepping probability of the mutant motor protein becomes zero (i.e. the stalling force forthe mutated motor protein), L is the rest length of the motor linkage, K e is the linkage stiffness and d is the step-size of the motor. A detailed derivation is provided in the Supplementary Information Text S1.It is to be noted that the absolute configuration space admits infinitely many representations asthere is always a small probability of finding the cargo at any location on the microtubule. However,the above result concludes, that given a stall force for both wild-type and mutant motors the relativeconfiguration space is finite, since there are no motors beyond n units away from the rearguard motorin any relative configuration. 18 −4 A ve r a g e R un l e ng t h i n n m ATP concentration in Molar 0.2 pN0.6 pN1 pN
Figure 15:
Effect of ATP concentration on run-length for ensemble WM, for various hin-dering load forces on the cargo.Transition Probabilities between Relative Configurations
Next, we determine the probability of transitioning from a relative configuration to another from thetransition probabilities in the absolute configuration space.It is evident that an absolute configuration can be associated with a unique relative configuration.For example, the relative configuration associated with Figure 16 is | M h || M h M d || M d || M h | . Thus theabsolute configuration Ω = (cid:20) · · · · · ·· · · · · · (cid:21) is mapped to the relative configuration | M h || M h M d || M d || M h | . We denote the projection operator Υthat maps an absolute configuration Ω to a corresponding relative configuration by Υ(Ω).The probability that the absolute configuration is Ω (cid:48) at time t + ∆ t conditioned on an initialconfiguration of Ω at t is represented as P Ω (Ω (cid:48) , t +∆ t | Ω , t ). It is assumed that the transition probability P Ω (Ω (cid:48) , Ω) between the two absolute configurations Ω and Ω (cid:48) is given by ν Ω (Ω (cid:48) , Ω)∆ t for a small timeinterval ∆ t , where the notation ν Ω (Ω (cid:48) , Ω) represents the probability rate of transition between Ω andΩ (cid:48) . The underlying assumption that the rate ν Ω (Ω (cid:48) , Ω) is independent of the time instant t holds true,since the ν Ω only depends upon the initial configuration and the type of transition (motor stepping,detachment or attachment) from Ω to Ω (cid:48) .In a similar manner, the transition probability that the relative configuration is ϑ (cid:48) at time t + ∆ t given that it is ϑ at time t is denoted by P ϑ ( ϑ (cid:48) , t +∆ t | ϑ, t ). The transition probability P ϑ ( ϑ, ϑ (cid:48) ) betweentwo relative configurations ϑ and ϑ (cid:48) is expressed as ν ϑ ( ϑ (cid:48) , ϑ )∆ t where ν ϑ ( ϑ (cid:48) , ϑ ) denotes the probabilityrate of transition between ϑ and ϑ (cid:48) .The knowledge of the transition rate ν Ω in the absolute configuration space enables the followingresult : 19 utant Wild-type Figure 16:
Locations of wild-type and mutant motors on a section of the microtubulelattice.
The microtubule is modelled as a directed linear lattice ¯ a k = ¯ a + kd , where ¯ a k is the positionof the k th location. If all the motors carry a common cargo i.e. the wild-type motor at a is therearguard motor, the string representation is | M h || M h M d || M d || M h | Result 2 : The rate of transition between relative configurations ϑ and ϑ (cid:48) is given by : ν ϑ ( ϑ (cid:48) , ϑ ) = (cid:88) ≤ β ≤(cid:100) nd (cid:101) ν Ω ( τ β Ω (cid:48) , Ω) (2) where Ω is any absolute configuration that satisfies Υ(Ω) = ϑ , Ω (cid:48) is any absolute configuration thatsatisfies Υ(Ω (cid:48) ) = ϑ (cid:48) , ( τ β Ω (cid:48) ) is an absolute configuration obtained after linearly shifting all the motorsin Ω (cid:48) by β locations on the microtubule towards the right and d is the dimension of a single microtubuledimer A detailed derivation is provided in Supplementary Information Text S2.As n is finite for a finite number of molecular motors attached to the cargo (from (1)), () involvesonly a finite number of computations.The transition rates between absolute configurations are obtainable using the chemical kinetics ofthe motor protein as it steps, detaches or attaches to the microtubule. The rate ν ϑ ( ϑ (cid:48) , ϑ ) is obtainableusing Result 2 and the knowledge of the rates ν Ω between corresponding absolute configurations. Results 1 and 2 together imply that given a finite number of molecular motors carrying a commoncargo, the relative configuration space is finite. If H is the set of all the possible relative configurations,the knowledge of transition rates from () can be used to show that the probability P ϑ ( ϑ, t ) of the relativeconfiguration being ϑ at time t satisfies the master equation, ∂∂t P ϑ ( ϑ, t ) = (cid:88) ϑ (cid:48) ∈ H ν ϑ ( ϑ, ϑ (cid:48) ) P ϑ ( ϑ (cid:48) , t ) − P ϑ ( ϑ, t ) (cid:88) ϑ (cid:48) ∈ H ν ϑ ( ϑ (cid:48) , ϑ ) . (3) Evolution of Probability Distribution of Relative Configuration
Solution for the master equation (3) determines the time evolution of probabilities of all the relativeconfigurations. Consider a ordering of all relative configurations given by ϑ , . . . , ϑ N (where N isthe finite number of relative configurations) where the probability of finding the motors in a relativeconfiguration ν i at time t is denoted by P i ( t ) and let P ( t ) = [ P ( t ) , . . . , P N ( t )] T . Using the expressionof transition rates ν ϑ ( ϑ j , ϑ (cid:48) i ) and (3), it can be shown that the dynamics of the model describing the20 argo (b) Cargo (a)
Mutant Wild-type
Figure 17:
Locations of wild-type and mutant motors on a section of the micro-tubule lattice.
The string representation for the two configurations is ( a ) | M h M d || M d || M h | and( b ) | M h M d ||| M d | M h | vector P ( t ) is given by : ddt P ( t ) = Γ P ( t ) , where Γ ∈ R N × N is a sparse stochastic matrix which is determined by the transition rates ν ϑ ( ϑ j , ϑ i )(Γ ji = ν ϑ ( ϑ j , ϑ i ) if i (cid:54) = j , Γ ii = 1 − (cid:80) i (cid:54) = j ν ϑ ( ϑ j , ϑ i )). Given a specified initial probability vector P ( t ) , it follows that P ( t ) = e Γ( t − t ) P ( t ) . (4)When specified for kinesin-1 motors and realistic values of number of motors ( M ≤
5) and systemparameters, the dimension of Γ is N × N where N lies between 10 and 10 , making the evaluationof e Γ( t − t ) manageable using a standard computer(the results in this article are obtained using Intelquad core i5 processor, 3.4 GHz, RAM 8 GB). The software is easily implementable using platformslike MATLAB and is faster and computationally more efficient than Monte-Carlo based approaches.Computer clusters can be used to manage more complex scenarios involving larger number of motors. Calculating Biologically Relevant Quantities
Once the probability vector P ( t ) is known, expressions of several biologically relevant quantities canbe obtained in the following manner. Average Number of Engaged Motors (Wild-type/Mutant) m h ( t ) of wild-type motors attached to the microtubule is given by < m h ( t ) > = N (cid:88) i =1 m h ( ϑ i ) P i ( t ) , (5)where m h ( ϑ i ) represents the number of wild-type motors in the relative configuration ϑ i or the numberof M h symbols in the representation ϑ i . In a similar manner, the average number of mutant motors m d ( t ), attached to the microtubule is given by < m d ( t ) > = N (cid:88) i =1 m d ( ϑ i ) P i ( t ) , (6)where m d ( ϑ i ) represents the number of mutant motors in the configuration ϑ i or the number of M d symbols in the representation ϑ i . Average Velocity and Average Run-length
Average velocity v ( t ) of the cargo being carried by M motors is determined as v ( t ) = (cid:88) ϑ ∈ H (cid:88) ϑ (cid:48) ∈ H d avg ( ϑ (cid:48) , ϑ ) ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t ) , (7)where d avg ( ϑ (cid:48) , ϑ ) is the expected change in cargo position, when the initial and final relative configu-rations at t and t + ∆ t are restricted to being ϑ and ϑ (cid:48) respectively and is given by d avg ( ϑ (cid:48) , ϑ ) = 1 ν ϑ ( ϑ (cid:48) , ϑ ) (cid:88) ≤ β ≤ n max d ( ρ β Ω (cid:48) , Ω) ν Ω ( ρ β Ω (cid:48) , Ω) . (8)The expression d (Ω (cid:48) , Ω) in (8) is the change in cargo equilibrium position when the absolute config-uration changes from Ω to Ω (cid:48) . The detailed formulation of (7) and (8) is provided in SupplementaryInformation Text S3.Average run-length is then calculated by summing the average velocity over time
Average Runlength = (cid:90) + ∞ v ( t ) dt (9) Transition rates between absolute configurations
In this section we present a general scheme for determining transition rates between absolute con-figurations. We begin with a structural model for single motor protein that consists of motor head,stalk and cargo binding tail domain. The linkage between the motor-heads and tail for single motoris modeled as a hookean spring when stretched, that has a rest length L . It offers no resistance whencompressed [11]. The motor heads move along the microtubules exerting a force F on a cargo that isexpressed as a function of its length L by, F ( L ) = K e ( L − L ) if L ≥ L , | L | < L ,K e ( L + L ) if L ≤ − L .Z eq is the mean position of the cargo that is its equilibrium position determined by the forcesexerted by the motors on the cargo through their linkages and the load force F load on the cargo. The22ean cargo position for a fixed F load > Z eq = Z eq (Ω).If the cargo position is assumed to follow a truncated Gaussian distribution Θ( z ) with variance σ , itsprobability density Θ( z ) for | z | < σ is given by,Θ( z ) = ( e − z σ ) / (2 (cid:90) σ e − z σ dz ) . The effect of thermal noise can be incorporated by determining the steady state variance σ of thecargo position.A transition to another configuration Ω (cid:48) occurs if either the wild type or mutant motor at alocation ¯ a k steps forward to ¯ a k +1 , detaches from the location ¯ a k or reattaches to the location ¯ a k on the microtubule. By representing Ω (cid:48) as Ω + S , S is a sequence that corresponds to the type oftransition(step, detach or attach) and the type of motor(wild-type or mutant) that has transitioned.The transition rate from Ω to Ω (cid:48) is determined by averaging the associated probability rate over theposition of the cargo.The model of a single motor-protein is specified via the probability P S ( F ) of the motor takinga step, the detachment probability, P D ( F ), of the motor detaching from the microtubule, and theprobability of attachment P A of an unattached motor-protien to the microtubule, per second. Here F is the force acting on the motor which is considered positive if it is directed opposite to the motorstepping direction (e.g. kinesin forward stepping is towards the mictorubule + end). Here in order tocalculate the transition rates between absolute configurations it is assumed that the probability ratesof step, detachment and attachment are known; later we illustrate a way to compute these probabilitiesfor kinesin motors. Transition Rate for Stepping
The stepping transition of a wild-type motor from the location ¯ a k to ¯ a k +1 is represented asΩ ST EP h −−−−−→ Ω + S ( step ) h,k , i.e. Ω = (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) ST EP h −−−−−→ (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) + (cid:20) · · · − · · ·· · · · · · (cid:21) . As the transition Ω
ST EP h −−−−−→ Ω + S ( step ) h,k occurs if any of the Ω h,k wild-type motors located at ¯ a k step forward to the position ¯ a k +1 , the associated transition rate for stepping, ν Ω (Ω + S ( step ) h,k , Ω) isdetermined by averaging over the position of the cargo as, ν Ω (Ω + S ( step ) h,k , Ω) = Ω h,k (cid:90) Z eq (Ω)+3 σZ eq (Ω) − σ P S ( F ( z − ¯ a k ))Θ( z − Z eq (Ω)) dz where Ω h,k is the number of wild-type motors located at ¯ a k and P S is the probability of a wildtype motor taking a step from the location ¯ a k on the microtubule per second.The stepping transition of a mutant motor from the location ¯ a k to ¯ a k +1 is represented asΩ ST EP d −−−−−→ Ω + S ( step ) d,k , i.e. Ω = (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) ST EP d −−−−−→ · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) + (cid:20) · · · · · ·· · · − · · · (cid:21) . As the transition Ω
ST EP d −−−−−→ Ω + S ( step ) d,k occurs if any of the Ω d,k mutant motors located at ¯ a k stepforward to the position ¯ a k +1 , the associated transition rate for stepping, ν Ω (Ω+ S ( step ) d,k , Ω) is determinedby averaging over the position of the cargo as, ν Ω (Ω + S ( step ) d,k , Ω) = Ω d,k (cid:90) Z eq (Ω)+3 σZ eq (Ω) − σ ¯ P S ( F ( z − ¯ a k ))Θ( z − Z eq (Ω)) dz where Ω d,k is the number of mutant motors located at ¯ a k and ¯ P S is the probability of a mutant motortaking a step from the location ¯ a k on the microtubule per second. Transition Rate for Attachment/Detachment
The attachment/detachment transition of wild-type motor at location ¯ a k is represented asΩ AT T h /DET h −−−−−−−−→ Ω ± S ( att ) h,k i.e.Ω = (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) AT T h /DET h −−−−−−−−→ (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) ± (cid:20) · · · · · ·· · · · · · (cid:21) , where the plus sign is for attachment and minus sign is for detachment.As the transition Ω AT T h /DET h −−−−−−−−→ Ω ± S ( att ) h,k occurs if any of the Ω h,k wild-type motors located at¯ a k detach from the microtubule, the associated transition rate for detachment, ν Ω (Ω − S ( att ) h,k , Ω) iscalculated by averaging over the position of the cargo as, ν Ω (Ω − S ( att ) h,k , Ω) = Ω h,k (cid:90) Z eq (Ω)+3 σZ eq (Ω) − σ P D ( F ( z − ¯ a k ))Θ( z − Z eq (Ω)) dz where Ω h,k is the number of wild-type motors located at ¯ a k and P D is the probability of a wildtype motor detaching from the location ¯ a k on the microtubule per second.The attachment/detachment transition of mutant motor at location ¯ a k is represented asΩ AT T d /DET d −−−−−−−−→ Ω ± S ( att ) d,k i.e.Ω = (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) AT T d /DET d −−−−−−−−→ (cid:20) · · · Ω h,k Ω h,k +1 · · ·· · · Ω d,k Ω d,k +1 · · · (cid:21) ± (cid:20) · · · · · ·· · · · · · (cid:21) . As the transition Ω
AT T d /DET d −−−−−−−−→ Ω ± S ( att ) d,k occurs if any of the Ω d,k mutant motors located at¯ a k detach from the microtubule, the associated transition rate for detachment, ν Ω (Ω − S ( att ) d,k , Ω) isdetermined by averaging over the position of the cargo as,24 Ω (Ω − S ( att ) d,k , Ω) = Ω d,k (cid:90) Z eq (Ω)+3 σZ eq (Ω) − σ ¯ P D ( F ( z − ¯ a k ))Θ( z − Z eq (Ω)) dz where Ω d,k is the number of mutant motors located at ¯ a k and ¯ P D is the probability of a mutant motordetaching from the location ¯ a k on the microtubule per second.For both the wild-type and mutant motors, a constant probability rate of reattachment to themicrotubule, P A and ¯ P A , are assumed. If a motor is linked to the cargo, it is assumed to attach to themicrotubule without stretching its linkage. Thereby, the only locations where the motor reattachmentcan occur are located within a distance L of the cargo. Probability Rates for Kinesin
The results put forth in this article correspond to an instantiation of our methodology for kinesin motorprotein. The probability rates of stepping, detachment and attachment of a single kinesin motor aredetermined using several available studies [5, 10, 20, 21].
Probability of Stepping, per second
Kinesin takes a step on the microtubule by hydrolyzing an ATP molecule [22] in the following manner: M + AT P k on (cid:10) k off M AT P k cat −−→ M + ADP + p i + ∆ E, where ∆ E is the energy released. From [23], the ATP hydrolysis rate predicted using Michaelis-Mentendynamics relates to the probability rate stepping for a wild-type kinesin by assuming that the freemotor head binds to the microtubule location with defined probability η . P S is then expressed as P S = k cat [ AT P ][ AT P ] + k m η, (10)where k m = k cat + k off k on . From [11], the force F exerted by the cargo on the motor affects motor dynamics by modifying η as, η ( F ) = F = 0 , − ( FF s ) if 0 < F < F s , . (11)Furthermore [11] assumes that the force F influences the kinetics of ATP hydrolysis by affecting k off in the following manner : k off ( F ) = k ,off e FδlKbT , where k ,off is the backward reaction rate of hydrolysis when F = 0, δ l is an experimentally deter-minable parameter, T is the temperature and K b is the Boltzmann constant. Thus, the probabilityrate of stepping for a wild-type motor under a constant force F is given by, P S ( F ) = k cat [ AT P ][ AT P ] + k cat + k off ( F ) k on η ( f ) . (12)25n the case of a mutant motor protein with an altered stalling force ¯ F s , the probability rate ofstepping under a force F is given by,¯ P S ( F ) = k cat [ AT P ][ AT P ] + k cat + k off ( F ) k on ¯ η ( F ) , (13)where the probability ¯ η that the free mutant motor head binds to the microtubule location is writtenas ¯ η ( F ) = F = 01 − ( F ¯ F s ) if 0 < F < ¯ F s Probability of Detachment, per second
Kinesin motors are processive species that take a certain number of steps during their ATP drivenmovement along microtubule before dissociating from it. The processivity (denoted by L ) representshow far the motor can move, on average, before its detachment from microtubule. From definition of L in [24], L = d [ AT P ] Ae − F δ l K b T [ AT P ] + B (1 + A ) e − F δ l K b T , where A , B and δ l are experimentally determinable parameters. The relation between the probabilityrate of stepping P S and detachment P D is given by, P S ( F ) P D ( F ) = L d . (15)As long as the wild-type motor is not stalled i.e. F < F s , the probability rate of detachment is givenby, P D ( F ) = [ AT P ] + B (1 + A ) e − F δ l K b T [ AT P ] Ae − F δ l K b T P S ( F ) . (16)When the load force on the wild-type motor equals to or exceeds its stalling force i.e. for F ≥ F s , aconstant detachment rate is assumed in [11] as, P D ( F ) = P back = 2 /sec. For a mutant motor that is not stalled i.e.
F < ¯ F s , the probability rate of detachment is given by,¯ P D ( F ) = [ AT P ] + B (1 + A ) e − F δ l K b T [ AT P ] Ae − F δ l K b T ¯ P S ( F ) . (17)When the load force on the mutant motor equals to or exceeds its stalling force i.e. for F ≥ ¯ F s , theconstant detachment rate is assumed to be¯ P D ( F ) = P back = 2 /sec. Probability of Attachment, per second
Probability of attachment is experimentally found to be P A ≈ /sec [25, 26]. In this article it isassumed that the probability of attachment for a mutant motor, ¯ P A ≈ /sec .26 umerical parameters for Kinesin The numerical parameters considered for wild-type motors in this article are the same as that used in[12], all of which have been experimentally verified. Specifically k on = 2 . M − s − , k off = 55 s − , k cat = 105 s − , F s = 6 pN , d = 8 nm , d l = 1 . nm , δ l = 1 . nm , A = 107, B = 0 . µM , K el = 0 . pN/nm and T = 300 K . All the parameters for mutant motors are the same, except for itsstalling force which is taken as ¯ F s = 5 . pN . Acknowledgments
The authors declare no competing interests. The authors acknowledge the support of University ofMinnesota Supercomputing Institute (MSI) for lending their resources.
References [1] Karel Svoboda, Christoph F Schmidt, Bruce J Schnapp, Steven M Block, et al.
Direct observa-tion of kinesin stepping by optical trapping interferometry . Nature , 365(6448):721–727,1993.[2] Michael A Welte, Steven P Gross, Marya Postner, Steven M Block, and Eric F Wieschaus.
Developmental regulation of vesicle transport in Drosophila embryos: forces andkinetics . Cell , 92(4):547–557, 1998.[3] A Ashkin, Karin Sch¨utze, JM Dziedzic, Ursula Euteneuer, and Manfred Schliwa.
Force gen-eration of organelle transport measured in vivo by an infrared laser trap . Nature ,1990.[4] Lorin S Milescu, Ahmet Yildiz, Paul R Selvin, and Frederick Sachs.
Maximum likelihood es-timation of molecular motor kinetics from staircase dwell-time sequences . BiophysicalJournal , 91(4):1156–1168, 2006.[5] Stefan Klumpp, Corina Keller, Florian Berger, and Reinhard Lipowsky.
Molecular Motors:Cooperative Phenomena of Multiple Molecular Motors . In
Multiscale Modeling in Biome-chanics and Mechanobiology , pages 27–61. Springer, 2015.[6] Comert Kural, Hwajin Kim, Sheyum Syed, Gohta Goshima, Vladimir I Gelfand, and Paul RSelvin.
Kinesin and dynein move a peroxisome in vivo: a tug-of-war or coordinatedmovement?
Science , 308(5727):1469–1472, 2005.[7] DB Hill, MJ Plaza, K Bonin, and G Holzwarth.
Fast vesicle transport in PC12 neurites:velocities and forces . European Biophysics Journal , 33(7):623–632, 2004.[8] Stefan Klumpp and Reinhard Lipowsky.
Cooperative cargo transport by several molecularmotors . Proceedings of the National Academy of Sciences of the United States of America ,102(48):17284–17289, 2005.[9] Filippo Posta, Maria R DOrsogna, and Tom Chou.
Enhancement of cargo processivityby cooperating molecular motors . Physical Chemistry Chemical Physics , 11(24):4851–4860,2009. 2710] Ambarish Kunwar and Alexander Mogilner.
Robust transport by multiple motors withnonlinear force–velocity relations and stochastic load sharing . Physical Biology ,7(1):016012, 2010.[11] Ambarish Kunwar, Michael Vershinin, Jing Xu, and Steven P Gross.
Stepping, strain gating,and an unexpected force-velocity curve for multiple-motor-based transport . CurrentBiology , 18(16):1173–1183, 2008.[12] Jing Xu, Zhanyong Shu, Stephen J King, and Steven P Gross.
Tuning multiple motor travelvia single motor velocity . Traffic , 13(9):1198–1205, 2012.[13] Donatello Materassi, Subhrajit Roychowdhury, Thomas Hays, and Murti Salapaka.
An exactapproach for studying cargo transport by an ensemble of molecular motors . BMCBiophysics , 6(1):14, 2013.[14] Woochul Nam and Bogdan I Epureanu. Highly loaded behavior of kinesins increases the robustnessof transport under high resisting loads.
PLoS Comput Biol , 11(3):e1003981, 2015.[15] Gerardo A Morfini, Yi-Mei You, Sarah L Pollema, Agnieszka Kaminska, Katherine Liu, KatsujiYoshioka, Benny Bj¨orkblom, Eleanor T Coffey, Carolina Bagnato, David Han, et al.
Pathogenichuntingtin inhibits fast axonal transport by activating JNK3 and phosphorylatingkinesin . Nature neuroscience , 12(7):864–871, 2009.[16] Hannah A DeBerg, Benjamin H Blehm, Janet Sheung, Andrew R Thompson, Carol S Bookwalter,Seyed F Torabi, Trina A Schroer, Christopher L Berger, Yi Lu, Kathleen M Trybus, et al.
Motordomain phosphorylation modulates kinesin-1 transport . Journal of Biological Chemistry ,288(45):32612–32621, 2013.[17] Subhrajit Roychowdhury, Shreyas Bhaban, Srinivasa Salapaka, and Murti Salapaka. Design of aconstant force clamp and estimation of molecular motor motion using modern control approach.In
American Control Conference (ACC), 2013 , pages 1525–1530. IEEE, 2013.[18] Subhrajit Roychowdhury, Tanuj Aggarwal, Srinivasa Salapaka, and Murti V Salapaka. Highbandwidth optical force clamp for investigation of molecular motor motion.
Applied PhysicsLetters , 103(15):153703, 2013.[19] C Gardiner.
Stochastic methods: a handbook for the natural and social sciences 4thed.(2009) .[20] RE Lee DeVille and Eric Vanden-Eijnden.
Regularity and synchrony in motor proteins . Bulletin of Mathematical Biology , 70(2):484–516, 2008.[21] Karel Svoboda and Steven M Block.
Force and velocity measured for single kinesinmolecules . Cell , 77(5):773–784, 1994.[22] Mark J Schnitzer and Steven M Block.
Kinesin hydrolyses one ATP per 8-nm step . Nature ,388(6640):386–390, 1997.[23] Edgar Meyh¨ofer and Jonathon Howard.
The force generated by a single kinesin moleculeagainst an elastic load . Proceedings of the National Academy of Sciences , 92(2):574–578, 1995.[24] Mark J Schnitzer, Koen Visscher, and Steven M Block.
Force production by single kinesinmotors . Nature Cell Biology , 2(10):718–723, 2000.2825] Janina Beeg, Stefan Klumpp, Rumiana Dimova, Ruben Serral Gracia, Eberhard Unger, andReinhard Lipowsky.
Transport of beads by several kinesin motors . Biophysical Journal ,94(2):532–541, 2008.[26] C´ecile Leduc, Otger Camp`as, Konstantin B Zeldovich, Aur´elien Roux, Pascale Jolimaitre, LineBourel-Bonnet, Bruno Goud, Jean-Fran¸cois Joanny, Patricia Bassereau, and Jacques Prost.
Co-operative extraction of membrane nanotubes by molecular motors . Proceedings of theNational Academy of Sciences of the United States of America , 101(49):17096–17101, 2004.
Supporting Information
Supplementary Information Text S1
Here, the validity of Result 1 is established.
Result 1 : Given an ensemble of M molecular motors attached to a common cargo that is subjected toa load force F load , the distance between the rearguard and the vanguard motor is bound by n = max (cid:26) ( M + 1) max ( F s , ¯ F s ) − F load K e + d, F load K e (cid:27) + 2 L where F s is the minimum load force for which the stepping probability of the wild-type motor proteinbecomes zero ( i.e. the stalling force for the wild-type motor protein), ¯ F s is the minimum load forcefor which the stepping probability of the mutant motor protein becomes zero (i.e. the stalling force forthe mutated motor protein), L is the rest length of the motor linkage and K e is the linkage stiffness. Consider an ensemble of wild-type and mutant motors bound to a cargo with the following assumptions: • There are a total of M (wild-type and mutant) motors bound to the cargo. The wild-type motorstalling force is F s and the mutant motor stalling force is ¯ F s . • The motor linkages have a rest length of L and stiffness K e . The linkages are modelled ashookean springs when stretched that offer no resistance when compressed. • There exists a constant load force F load on the cargo. • Z eq is the mean position of the cargo, which is the equilibrium position determined by the forcesexerted on the cargo by the motors through their linkages and the load force F load • Any unattached motor bound to the cargo can attach to only those locations on the microtubulethat are within L distance of the cargo position.For the discussion below, it is assumed that there is at least one motor on the cargo opposing itsmotion (for other cases, similar arguments can be used). Suppose there be f motors assisting the cargomotion and r motors opposing it ( f + r ≤ M ). Let the locations of the f motors on the microtubuleassisting the cargo motion be Z +1 , Z +1 , ..., Z + f . The linkage length of a motor located at Z + i is given by L + i = Z + i − Z eq and the force it exerts on the cargo is F + i = K e ( L + i − L ). If the motors are locatedin such a way that 0 < L ≤ L +1 ≤ L +2 ≤ ... ≤ L + f , then F +1 ≤ F +2 ≤ ... ≤ F + f and the total forceexerted on the cargo in the forward direction is F fwd = (cid:80) fi =1 F + i .29et the locations of the r motors on the microtubule opposing the cargo motion be Z − , Z − , ..., Z − r .The linkage length of a motor located at Z − j is given by L − j = Z eq − Z − j and the force it exerts onthe cargo is F − j = K e ( L − j − L ). If the motors are located in such a way that 0 < L ≤ L − ≤ L − ≤ ... ≤ L − r , then F − ≤ F − ≤ ... ≤ F − r and the total force exerted on the cargo in the backwarddirection is F back = (cid:80) rj =1 F − j . The separation between the vanguard and rearguard motor can befound as S = L + f + L − r where L + f and L − r are the linkage lengths of the vanguard and rearguard motorrespectively.Suppose the vanguard motor (thereby all the motors) is not stalled. Then F fwd < M max ( F s , ¯ F s ).If the vanguard motor is a wild-type motor, F ( L + f ) < F s thus L + f < F s K e + L and if it is a mutantmotor, F ( L + f ) < ¯ F s thus L + f < ¯ F s K e + L . In general L + f < max ( F s , ¯ F s ) K e + L . At equilibrium, F back = F fwd − F load , thus (cid:80) rj =1 F − j = F fwd − F load which implies that F ( L − r ) + (cid:80) r − j =1 F − j = F fwd − F load .It follows that F ( L − r ) ≤ F fwd − F load . Substituting for F ( L − r ) it follows that K e ( L − r − L ) Transition rates between relative configurations Here the validity of Result 2 is established. Result 2 : The rate of transition between relative configurations ϑ and ϑ (cid:48) is given by ν ϑ ( ϑ (cid:48) , ϑ ) = (cid:88) ≤ β ≤(cid:100) nd (cid:101) ν Ω ( τ β Ω (cid:48) , Ω) , where Υ(Ω) = ϑ , Υ(Ω (cid:48) ) = ϑ (cid:48) , ( τ β Ω (cid:48) ) is an absolute configuration obtained after linearly shifting allthe motors in Ω by β locations on the microtubule towards the right and d is the periodicity of themicrotubule lattice Let Ω and Ω (cid:48) be a pair of absolute configurations such that Υ(Ω) = ϑ and Υ(Ω (cid:48) ) = ϑ (cid:48) . It isassumed that the system is translation invariant with its stochastic behavior unaffected if the cargo30nd the ensemble of motors bound to it shift to a new location along the microtubule. The translationinvariance property enables the construction of a set Ω( ϑ ) of all absolute configurations that have thesame relative configuration ϑ as Ω( ϑ ) = { τ β Ω : β ∈ I } , where I is the set of integers. In a similarmanner, Ω( ϑ (cid:48) ) = { τ β (cid:48) Ω (cid:48) : β (cid:48) ∈ I } Using the definition of transition probability, P ϑ ( ϑ (cid:48) , t + ∆ t | ϑ, t ) can be expressed as follows : P ϑ ( ϑ (cid:48) , t + ∆ t | ϑ, t ) = P ϑ ( ϑ (cid:48) , t + ∆ t, ϑ, t ) P ϑ ( ϑ, t )= (cid:88) Ω ∈ Ω( ϑ ) (cid:88) Ω (cid:48) ∈ Ω( ϑ (cid:48) ) P Ω (Ω (cid:48) , t + ∆ t, Ω , t ) (cid:80) Ω ∈ Ω( ϑ ) P Ω (Ω , t )= 1 (cid:80) β P Ω ( τ β Ω , t ) (cid:88) β (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t, τ β Ω , t )= 1 (cid:80) β P Ω ( τ β Ω , t ) (cid:88) β (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t | τ β Ω , t ) P Ω ( τ β Ω , t )= 1 (cid:80) β P Ω ( τ β Ω , t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t | τ β Ω , t )= 1 (cid:80) β P Ω ( τ β Ω , t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) P Ω ( τ ( β (cid:48) − β ) Ω (cid:48) , t + ∆ t | Ω , t )= 1 (cid:80) β P Ω ( τ β Ω , t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t | Ω , t )= (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t | Ω , t )= (cid:88) β (cid:48) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω)∆ t. In the third equality, (cid:80) β P Ω ( τ β Ω , t ) is performed over all shifts β of an absolute configuration Ωsuch that its projection is the relative configuration ϑ i.e. Υ(Ω) = ϑ . Similarly while determining (cid:88) β (cid:88) β (cid:48) P Ω ( τ β (cid:48) Ω (cid:48) , t + ∆ t, τ β Ω , t ), the absolute configuration Ω (cid:48) satisfies Υ(Ω (cid:48) ) = ϑ (cid:48) .In the sixth equality, translation invariance property is applied wherein both the absolute config-urations at t and t + ∆ t are shifted by β places to the left (via the operation τ − β ). For the seventhequality, the set { τ ( β (cid:48) − β ) } = { τ β (cid:48) } , since β (cid:48) is any integer and β is fixed. Moreover, since Ω has beenarbitrarily chosen with the only condition that Υ(Ω) = ϑ , this will hold for every Ω ∈ Ω( ϑ ). Usingthe definition that P ϑ ( ϑ (cid:48) , t + ∆ t | ϑ, t ) = ν ϑ ( ϑ (cid:48) , ϑ )∆ t , ν ϑ ( ϑ (cid:48) , ϑ )∆ t = (cid:80) β (cid:48) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω)∆ t and thus, ν ϑ ( ϑ (cid:48) , ϑ ) = (cid:88) β (cid:48) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω) . Since for a finite number of motors bound to a cargo the maximum distance between the vanguardand rearguard motor is finite (and equal to (cid:100) nd (cid:101) locations on the microtubule), ν ϑ ( ϑ (cid:48) , ϑ ) = (cid:88) ≤ β ≤(cid:100) nd (cid:101) ν Ω ( τ β Ω (cid:48) , Ω) . upplementary Information Text S3 Average velocity of the cargo Here, average velocity of the cargo is determined by undertaking the following steps : Step I : The expected change in cargo position when the initial and final relative configurations at t and t + ∆ t are restricted to being ϑ and ϑ (cid:48) respectively, denoted by d avg ( ϑ (cid:48) , ϑ ), is obtained. Step II : The expected change in the cargo position in time ∆ t , denoted as D ∆ t is obtained byremoving the restriction of the relative configurations being ϑ and ϑ (cid:48) at t and t + ∆ t . Theaverage velocity of the cargo at time t follows as, v ( t ) = D ∆ t ∆ t To obtain d avg ( ϑ (cid:48) , ϑ ) the following steps are undertaken :1. The change in cargo equilibrium position, d (Ω (cid:48) , Ω) = Z eq (Ω (cid:48) ) − Z eq (Ω) , is determined for every possible transition from an initial absolute configuration Ω at time t toa final absolute configuration Ω (cid:48) at time t + ∆ t , where Ω and Ω (cid:48) are chosen such that Υ(Ω) = ϑ and Υ(Ω (cid:48) ) = ϑ (cid:48) .2. The probability P (Ω (cid:48) , t + ∆ t, Ω , t | ϑ (cid:48) , t + ∆ t, ϑ, t ) of transitioning from Ω to Ω (cid:48) in time ∆ t isdetermined for every such pair of absolute configurations (Ω , Ω (cid:48) ), conditioned on the fact thatthe relative configuration also transitions from ϑ to ϑ (cid:48) in the same time ∆ t .3. A weighted sum of d (Ω (cid:48) , Ω) with the weights given by the probabilities P (Ω (cid:48) t + ∆ t, Ω , t | ϑ (cid:48) , t +∆ t, ϑ, t ) is obtained.Starting with any pair of absolute configurations Ω and Ω (cid:48) satisfying the conditions Υ ( e ) (Ω) = ϑ and Υ ( e ) (Ω (cid:48) ) = ϑ (cid:48) , the expected change in cargo position d avg ( ϑ (cid:48) , ϑ ), when the initial and final relativeconfigurations at t and t + ∆ t are restricted to being ϑ and ϑ (cid:48) respectively, is given by32 avg ( ϑ (cid:48) , ϑ ) : = (cid:88) Ω ∈ Ω( ϑ ) (cid:88) Ω (cid:48) ∈ Ω( ϑ (cid:48) ) d (Ω (cid:48) , Ω) P (Ω (cid:48) , t + ∆ t, Ω , t | ϑ (cid:48) , t + ∆ t, ϑ, t )= (cid:88) β (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , τ β Ω) P ( τ β (cid:48) Ω (cid:48) , t + ∆ t, τ β Ω , t | ϑ (cid:48) , t + ∆ t, ϑ, t )= (cid:88) β (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , τ β Ω) P ( τ β (cid:48) Ω (cid:48) , t + ∆ t, τ β Ω , t, ϑ (cid:48) , t + ∆ t, ϑ, t ) P ( ϑ (cid:48) , t + ∆ t, ϑ, t )= (cid:88) β (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , τ β Ω) P ( τ β (cid:48) Ω (cid:48) , t + ∆ t, τ β Ω , t ) P ( ϑ (cid:48) , t + ∆ t, ϑ, t )= (cid:88) β (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , τ β Ω) ν Ω ( τ β (cid:48) Ω (cid:48) , τ β Ω) P Ω ( τ β Ω , t ) ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t )= 1 ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , τ β Ω) ν Ω ( τ β (cid:48) Ω (cid:48) , τ β Ω)= 1 ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) d ( τ ( β (cid:48) − β ) Ω (cid:48) , Ω) ν Ω ( τ ( β (cid:48) − β ) Ω (cid:48) , Ω)= 1 ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t ) (cid:88) β P Ω ( τ β Ω , t ) (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , Ω) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω)= 1 ν ϑ ( ϑ (cid:48) , ϑ ) (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , Ω) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω) . In the seventh equality, translation invariance property is applied wherein both the absolute configu-rations at t and t + ∆ t are shifted by β places to the left ( via. the operation τ − β ). For the eighthequality, the set { τ ( β (cid:48) − β ) } = { τ β (cid:48) } , since β (cid:48) is any integer and β is fixed. The result is identical forany choice of absolute configuration Ω that satisfies Υ ( e ) (Ω) = ϑ . Thus, d avg ( ϑ (cid:48) , ϑ ) = 1 ν ϑ ( ϑ (cid:48) , ϑ ) (cid:88) β (cid:48) d ( τ β (cid:48) Ω (cid:48) , Ω) ν Ω ( τ β (cid:48) Ω (cid:48) , Ω) . After obtaining the expected value d avg ( ϑ (cid:48) , ϑ ), the expected change D ∆ t in cargo position in time ∆ t is determined to be, D ∆ t = (cid:88) ϑ ∈ H (cid:88) ϑ (cid:48) ∈ H d avg ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ (cid:48) , t + ∆ t, ϑ, t ) , = (cid:88) ϑ ∈ H (cid:88) ϑ (cid:48) ∈ H d avg ( ϑ (cid:48) , ϑ ) ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t )(∆ t ) . where H is the set of all possible relative configurations. It enables the calculation of average velocityas, v ( t ) = D ∆ t ∆ t = (cid:88) ϑ ∈ H (cid:88) ϑ (cid:48) ∈ H d avg ( ϑ (cid:48) , ϑ ) ν ϑ ( ϑ (cid:48) , ϑ ) P ϑ ( ϑ, t ) . upplementary Figures ATP concentration in Molar A ve r a g e V e l o c i t y i n n m / sec WWWMMM Figure 18: ( S1Fig ) Effect of ATP concentration on average velocity for 2-motor ensembles WW, WMand MM against load force of 0 . pN −3 ATP concentration in Molar A ve r a g e V e l o c i t y i n n m / sec WWWWWMWMMMMM Figure 19: ( S2Fig ) Effect of ATP concentration on average velocity for 3-motor ensembles WWW,WWM, WMM and MMM against load force of 0 . pN ATP concentration in Molar A ve r a g e R un l e ng t h i n n m WWWWWMWMMMMM Figure 20: ( S3Fig ) Effect of ATP concentration on average runlength for 3-motor ensembles WWW,WWM, WMM and MMM against load force of 0 . pNpN