Effective behavior of cooperative and nonidentical molecular motors
aa r X i v : . [ q - b i o . S C ] A ug Effective behavior of cooperative andnonidentical molecular motors
Joseph J. Klobusicky John Fricks Peter R. Kramer
Abstract
Analytical formulas for effective drift, diffusivity, run times, and runlengths are derived for an intracellular transport system consisting of a cargoattached to two cooperative but not identical molecular motors (for example,kinesin-1 and kinesin-2) which can each attach and detach from a micro-tubule. The dynamics of the motor and cargo in each phase are governedby stochastic differential equations, and the switching rates depend on thespatial configuration of the motor and cargo. This system is analyzed in alimit where the detached motors have faster dynamics than the cargo, whichin turn has faster dynamics than the attached motors. The attachment anddetachment rates are also taken to be slow relative to the spatial dynamics.Through an application of iterated stochastic averaging to this system, andthe use of renewal-reward theory to stitch together the progress within eachswitching phase, we obtain explicit analytical expressions for the effectivedrift, diffusivity, and processivity of the motor-cargo system. Our approachaccounts in particular for jumps in motor-cargo position that occur duringattachment and detachment events, as the cargo tracking variable makes arapid adjustment due to the averaged fast scales. The asymptotic formu-las are in generally good agreement with direct stochastic simulations ofthe detailed model based on experimental parameters for various pairings ofkinesin-1 and kinesin-2 under assisting, hindering, or no load.
Dedicated to Andy Majda for his 70th birthday, with gratitude for his lastinginspiration starting from my undergraduate and graduate days on the creativedeployment of mathematical modeling and the beautiful application of analysistechniques as a lens for exploring and understanding the dynamics of physicalsystems - PRK 1
Introduction
A biological cell during its interphase requires sufficiently fast transport of or-ganelles and other compounds for its survival [1]. Transport through diffusionalone is often far too slow. To illustrate, a compound moving through pure dif-fusion in some neurons might take years to travel over the cell’s length [71]. Foreukaryotic organisms, intracellular trafficking of vesicles is instead governed bydirected transport along a network of thin filaments, such as microtubules or actin.A vesicle and the molecular compound it encloses, collectively referred to as acargo, travel along the filaments by attaching to one or several molecular motors.As an important example on which we will focus, we can consider molecular mo-tors called kinesins, which consist of two heads which attach to a microtubule, atail which attaches to the cargo, and a coiled-coil tether connecting the heads andtail [29]. But the mathematical framework to be developed can be applied to moregeneral molecular motors, including dynein and myosin.The motor-cargo attachment is generally found to be much more durable thanthe motor-microtubule attachment [28], so models of motor-cargo complexes typ-ically assume the number of motors attached to a given cargo can be treated as afixed constant N over the transport time scale of interest [93, 45, 64, 54, 53]. Butthe number of those N motors that are attached to microtubules and therefore ac-tively engaged in transport does appear to fluctuate through dynamical attachmentand detachment of the motors to and from the microtubule. In the present work,we contemplate the simplest scenario in which the motor-cargo complex is in thevicinity of a single microtubule. The state of the motor-cargo complex can thenbe classified in terms of which of its motors are attached to the microtubule, andtherefore engaged in directed motion. Our model could also be formally applica-ble for a bundle of parallel microtubules with common polarity if they are spacedsufficiently closely that the progress of the cargo is not so sensitive to what partic-ular set of microtubules the motors are attached. Experimental observations [73,Fig. S5] show that approximating the multiple motors as all attached to a singlemicrotubule could be consistent even for some situations in cell.A cargo with two motors, for example, can fluctuate between three possiblestates while remaining connected to the microtubule: two states with one attachedand one detached motor, and one state with two attached motors (Figure 1). Onefurther state has both motors detached, after which we will consider the cargo tomove away from the microtubule and terminate its run, though one could contem-plate the motors remaining weakly bound and possibly sliding along the micro-tubule [81, 64]. 2icrotubules are oriented with a + and - end, with the molecular motor kinesinalways traveling from the - to + end [1] and the molecular motor dynein travelingin the opposite direction. This can result in “tug of war” scenarios and bidirec-tional transport for ensembles including such antagonistic motors [63]. Our focushere is on cooperative transport of a cargo by possibly different motor types whicheach move, when considered in a single motor complex, in the same direction onthe microtubule, such as two different types of motors from the kinesin superfam-ily. Since we wish to contemplate the possibility of the motors being of differenttypes, motor labelling is relevant and we distinguish between the two states withone motor attached. This is in contrast to models with N identical motors attached,which can be classified more compactly in terms of simply the number of thosemotors which are currently attached to a microtubule [39]. For multiple motorcomplexes, we are interested in calculating how statistics such as run lengths andeffective speeds depend on the properties of the individual motors. Comparisonsof statistics between motor complexes are not always intuitive. For instance, ithas been observed in [25] that heterogeneous motor complexes of kinesin-1 andkinesin-2 have longer run lengths than pure systems consisting only of kinesin-1.This observation would seem to depend upon a variety of physical parameters,often interacting in complex ways. Indeed, while kinesin-2 is about half as fastas kinesin-1 and detaches more readily under load, it also appears to reattach tomicrotubules four times more quickly than kinesin-1 [25].A number of properties of some individual motors, acting in isolation, havebeen obtained from in vitro experiments with optical traps, in which a polystyrenebead, serving as a cargo, is attached to a motor, and directed along a microtubulewith an applied optical trap force. Under this setting, several motor propertiescan be measured, particularly their speed, diffusivity, and detachment rate as afunction of the applied optical trap force [83, 60, 61, 2, 48, 9, 89]. The aboveexperimental work can be used as a basis for parameterizing biophysically mech-anistic models for individual motors in theoretical models exploring their interac-tions [45, 44, 51, 38, 36, 10, 35, 57, 84, 21, 59]. For the purpose of experimentallymeasuring the interaction of molecular motors, and in particular for motivationfor and comparison against theoretical models, an important experimental devel-opment has been to use DNA origami for cargo, which specifies closely arrangedhandle sites onto which specified motor types attach [25, 35, 26]. The actual en-gagement of motors with microtubules cannot be resolved, so particularly whennot using engineered motor-attachment constructs, the number of relevant mo-tors of various types attached to an observed cargo with a particular microtubulemust typically be statistically inferred, sometimes using simple theoretical mod-3igure 1: Attachment and detachment from a microtubule for a system of twomotors durably attached to a cargo. The spatial positions along the microtubulefor the motors are denoted X ( ) , and X ( ) , and for the cargo by Z . From a statewith two attached motors (left), each motor i can detach with a rate d ( i ) depend-ing on the current spatial configuration. From a state with one attached and onedetached motor (center), the detached motor i can (re)attach with a constant rate a ( i ) , or the attached motor i ′ can also detach at a configuration-dependent rate d ( i ′ ) ,terminating the run on the microtubule. 4ls [73, 23, 53]. Our mathematical modeling framework is based on the one developed in McKin-ley, Athreya, et al [56], where cargo transport statistics were examined for coop-erative ensembles of identical motors that are treated as permanently attached tothe microtubule. The coupled spatial dynamics of the motor and cargo positionare expressed as a system of continuous stochastic differential equations (SDEs)which are viewed as a coarse-graining of discrete stepping model. The modelis posed in essentially one spatial dimension along the microtubule, neglectingtransverse fluctuations. We extend the model in [56] in two ways: 1) allowing themotors to be distinct, but still cooperative, and 2) allowing the motors to attachand detach from the microtubule. This model represents the motor-cargo systemin terms of basic parameters concerning biophysical properties of the individ-ual motors and the cargo. The cargo and detached motor dynamics are modeledas overdamped point particles responding to spring forces from the motor-cargotether and driven by stochastic terms representing thermal fluctuations. On theother hand, as in [56], the attached motors dynamics are governed by a nonlinearforce-velocity relation together with stochastic terms arising from the nonequilib-rium stepping process. Punctuating the continuous evolution of the motor-cargodynamics is the attachment or detachment of motors from the microtubule. Theseare represented in terms of Markovian jump processes, with attachments occur-ring at constant rates but the detachments occurring at rates dependent upon theforce on the motor.To characterize the effective behavior of the motor-cargo complex as a whole,we proceed through a sequence of coarse-graining steps motivated by the sepa-ration of time scales of the various physical processes. First we average over thefast dynamics of the cargo and detached motors. This can be accomplished easilyfor arbitrary number N of cooperative motors, but our further analytical progressis restricted to the case of N = N >
2. We homogenize over the spatial dynamics ofthe attached motors between attachment and detachment events. This eventuallyyields an effective Markov chain description in terms of jumps between differentstates of motor attachment, with associated durations and displacements of themotor-cargo complex within each state. By decomposing the stochastic trajectorythrough this finite state space in terms of cycles demarcated by moments whenboth motors are attached, we further coarse-grain the motor-cargo dynamics into5 renewal-reward process. With Wald’s identity, we characterize the fundamentalstatistics of the run length and run time of the motor-cargo complex in terms ofthe statistics of duration and displacement over one of these cycles, which are inturn related to the basic biophysical parameters of the model. With a law of largenumbers argument and its application to renewal-reward processes, we relate theeffective velocity and diffusivity also to the cycle statistics, and thereby in turn tothe basic biophysical parameters. We next proceed to discuss the components ofour analysis in some more detail, with reference to related work.
The reason we work with the coarse-grained SDE model for the motors is to re-duce the number of physical parameters modeling the individual motor to thosecharacterizing the force-velocity relationship, the force-detachment relationship,and a noise parameter. Moreover, the mathematical presentation is simplified byhaving the cargo and motor dynamics in a unified SDE framework. One couldof course start with a motor stepping model [10, 18, 19, 33, 93, 22, 41, 21, 36],and readily coarse-grain it to obtain the parameters for our SDE model, and theconclusions would be equivalent unless the cargo fluctuations fed back on theforce-dependent kinetics of stepping in a substantially different way than they doon the force-dependent coarse-grained velocity. We cannot rule such subtle feed-back out [30], but if it were real, one would presumably have to represent thechemomechanical stepping cycle in more detail to capture these effects than ageneric stepping model with one step per cycle. The detailed description of thestepping dynamics of even well-studied motors such as kinesin-1 is still underactive experimental investigation [2, 58, 60], while the more coarse-grained char-acterizations needed for the SDE description are more consistently establishedacross labs. For these reasons, we will proceed here with the coarse-grained SDEdescription, in somewhat the same vein as coarse-grained integrate-and-fire mod-els are often used in place of more detailed Hodgkin-Huxley models to studynetworks of interacting neurons [90, 12, 42, 32, 68, 78, 94, 66].In Section 2, we present our extended model allowing for distinct motors andattachment/detachment from the microtubule. The modification of the spatial dy-namics is presented in Subsection 2.1, resulting in a system of SDEs (1)-(4) withthe motor dynamics depending on both the motor label and whether the motoris attached or detached. In Subsection 2.2, we present our model of switchingtimes. Attachment rates are rather difficult to quantify experimentally and ap-pears to depend on the operating conditions and whether the motor is tethered6ear the microtubule by other motors [35, 26, 76], and we simply adopt the com-mon approach of modeling them as constant (as in a homogenous Poisson processmodel). The detachment rates, on the other hand, will be taken as functions ofthe applied force on the attached motor, which is itself a stochastic process in-duced by the stochastic spatial dynamics. Mathematically, then, the detachmentprocess is more akin to a Cox process [15, 14]. The class of functions we con-sider for detachment rates is general enough to include the most common casesseen in previous works, including constant [4], exponential [2, 44, 46, 39], double-exponential functions [61], and exponential functions turning over to slower lineargrowth beyond stall [45, 8] or along the assisting direction [3].Transitions between attached and detached states have been modeled as con-tinuous time Markov chains in the case of identical motors [39, 55, 86, 38] withconstant switching rates depending on the number of attached and detached mo-tors. Our model allows for nonidentical motors, which increases the state spaceof possible attachment and detachment configurations. Similar to the discrete-space models discussed in [55, 86], our transition rates depend on relative motorpositions, but now must consider motor types. Moreover, our model does notmake the popular assumption [39, 38, 35] that the cargo is always in mechan-ical equilibrium with the motors nor the further mean-field approximation thatall motors feel a perfectly shared load from the cargo. Stochastic fluctuationsof cargo and motor positions have been shown to significantly alter the mean-field predictions, at least for small teams of motors [44, 88, 45], and experimentalobservations do not support the load-sharing assumption in teams of two kinesin-1 [35]. Bouzat [10] shows how the neglect of fluctuations in the cargo position cangive incorrect modeling results for the effective force-velocity response of a singlemotor and in particular for the stall force of cooperative motors. Berger, M ¨uller,and Lipowsky [7] and Arpa˘g, Norris, et al [3] similarly find the force felt by amotor from the cargo substantially affects its effective detachment rate, and so de-parts from the mean-field dynamical description by averaging the detachment rateagainst the modeled probability distribution of the force applied by the cargo in agiven configuration of attached and detached motors. We apply a similar, but moresystematic, procedure in Subsection 4.3.1 to compute effective detachment ratesby averaging over a quasi-stationary distribution of the force felt from the cargo,as a function of the current configuration of attached motors. Uppulury, Efre-mov, et al [88] and Kunwar, Tripathy, et al [45] conducted informative simulationstudies of discrete-state stochastic stepping models that keep track of the relativepositions of identical motors, but did not develop analytical formulas relating col-lective behavior to single-motor properties. Keller, Berger, et al [36] similarly7onducted simulation studies for a pair of identical motors which resolved thechemomechanical steps, finding in particular that the transport is degraded as thestrength of the Hookean tether coupling the motors and cargo is increased. Wangand Li [93] derive an analytical formula for the effective velocity of a cooper-ative team of identical stochastic stepping motors, but don’t include attachmentand detachment effects. Li, Lipowsky, and Kierfeld [49] developed a variationof the Markov chain model of Klumpp and Lipowsky [39] for cooperation by twogroups of motors, one fast and one slow, in the context of gliding assays where thenumbers of engaged motors are considerably larger than for cargo transport, to re-late bistability in the transport of microtubules to a small ratio of detachment forcescales to the stall forces. A recent model by Miles, Lawley, and Keener [59] forcooperative molecular motor transport employs a similar framework to ours, witha stochastic differential equation for the cargo dynamics, but with step-resolvingdynamics for the motors. The model we present extends the model of [59] in al-lowing the motors to be nonidentical and allowing the detachment rates of motorsto depend on the spatial configuration of the motors.
The analysis of our model will proceed in Section 3 by an extension of the asymp-totic analysis developed in McKinley, Athreya, et al [56], based on the cargo dy-namics being taken as fast relative to the dynamics of the motors attached to themicrotubule. Extending this nondimensionalization to our setting of nonidenticalmotors which attach and detach from the microtubule in Section 3.1, we furthermotivate taking the detached motor dynamics to be faster than the cargo (due totheir relative size), and the attachment and detachment processes as slow com-pared to the dynamical time scale of attached motors. A similar hierarchy of timescales was considered in the context of the stepping of the two heads of a kinesinmotor attached to a cargo in Peskin and Oster [70], where the detached head wastaken to move quickly relative to the cargo, while the cargo dynamics were takenas fast relative to the time scale of motor stepping.We set up an asymptotic analysis which formalizes the separation of timescales we identify, with the further assumption that the parameters for differentmotor types do not vary drastically (which seems to be reasonable for kinesin-1 versus kinesin-2, for example, [2]). We proceed in Subsection 3.2 to applystochastic averaging successively over the unbound motor positions and the cargoposition to obtain effective dynamics and effective detachment rates based onlyon the positions and identities of the attached motors. The analysis thus far ap-8lies to N cooperative but possibly not identical motors, and in fact does not re-quire a time scale separation assumption between the attached motor dynamicsand attachment/detachment dynamics. In Section 4, we use this scale separationassumption for the case of N = For a fixed number of attached and detached motors, we model the transport of amotor and cargo system through a system of SDEs. Similar to McKinley, Athreya,et al [56], we coarse-grain over this discrete stepping and model this motion as aone-dimensional continuous process along the direction of the microtubule, ne-glecting transverse fluctuations which might not be so significant [21]. The lo-cations of N motors over time t are denoted by X ( i ) ( t ) ∈ ( − ∞ , ∞ ) , i = , . . . , N .All motors in the system are assumed to remain attached to a single cargo, witha position of Z ( t ) ∈ ( − ∞ , ∞ ) . During a realization of the stochastic process, amotor will change its state from time to time. We will denote the i th motor’s stateat time t as Q ( i ) ( t ) ∈ { , } , where 0 and 1 denote states of detachment or attach-ment to the microtubule, respectively. The switching between motor states will begoverned by a jump process, which is Markovian with respect to the filtration gen-erated jointly by { X ( i ) ( t ) } Ni = , Z ( t ) and { Q ( i ) ( t ) } Ni = , and with switching (jump)rates due to attachment and detachment depending on the spatial displacementsbetween the motors and cargo. For a motor system with N motors the governing equations take, for i = , . . . , N ,the autonomous form dX ( i ) ( t ) = (cid:18) µ ( i ) d ( X ( i ) ( t ) , Z ( t )) dt + q k B T / γ ( i ) m dW ( i ) ( t ) (cid:19) ( − Q ( i ) ( t ))+ (cid:16) µ ( i ) a ( X ( i ) ( t ) , Z ( t )) dt + σ ( i ) dW ( i ) ( t ) (cid:17) Q ( i ) ( t ) , (1) γ dZ ( t ) = − N ∑ j = F ( j ) ( X ( j ) ( t ) − Z ( t )) dt − F T dt + p k B T γ dW z ( t ) . (2)10ith drift coefficients µ ( i ) a , µ ( i ) d : R → R for the motors satisfying µ ( i ) a ( x , z ) = v ( i ) g ( F ( i ) ( x − z ) / F ( i ) s ) , (3) µ ( i ) d ( x , z ) = F ( i ) ( x − z ) / γ ( i ) m . (4)A table of the parameters and their roles can be found in Table 1. We next brieflysummarize the meaning of the model equations (1)-(4); more details can be foundin [56].If Q ( i ) ( t ) =
1, equation (1) describes an attached motor with position X ( i ) ( t ) . The restorative force in (3) results from the stretching of the coiled-coil tether thatconnects the motor head and tail attached to the cargo. While a nonlinear forcemodel for the tether would seem to be most appropriate [35, 88, 13], we couldnot find a clear consensus on its precise form. To reduce technical complicationsin the formulas and to keep focus on the overall structure of the mathematicalcourse-graining, in the present work, similarly to Miles, Lawley, and Keener [59],we model the force for this tether for motor i by a simple Hookean spring rela-tion F ( i ) ( y ) = − κ ( i ) y , where y is the longitudinal displacement from the cargo tothe motor, and κ ( i ) is the effective spring constant for the tether to the i th mo-tor. The value we have cited from Furuta, Furuta, et al [26] is measured frommotors attached to a DNA scaffold, but we expect the stiffness to mostly reflectthe properties of the motor tether [21]. The value is roughly consistent with the κ ≈ . g : R → R is multiplied by an un-encumbered velocity v ( i ) to produce an instantaneous expected velocity. The argu-ment of g measures the ratio between the tether force and the motor’s stall force F ( i ) s , or the opposing force needed from the cargo to anchor a motor. Positivearguments of g correspond to forces opposing or hindering the free motion ofthe motor. To agree with the definitions of v ( i ) and F ( i ) s , g must satisfy g ( ) = g ( ) =
0. Random effects are modelled by independent Brownian motions W ( i ) ( t ) , with an effective motor diffusion of ( σ ( i ) ) . In Table 1, we have usedrandomness parameters for kinesin 1 and 2 found in [92, 2] to calculate diffusiv-ities via their relation which can, for example, be found in Eqn. (47) of Krishnanand Epureanu [43]. 11hen Q ( i ) ( t ) =
0, the equation for the position of an detached motor X ( i ) ( t ) in(1) is an overdamped Langevin equation for a particle with a friction constant γ ( i ) m ,and spring constant κ ( i ) . The friction constant γ ( i ) m was computed with the Stokes-Einstein relation γ ( i ) m = π a η for a spherical object with radius a =
50 nm in water,with a fluid dynamic viscosity of η = − pN s/nm . The Brownian motions W ( i ) ( t ) are independent of each other, and also of the Brownian motion W z ( t ) driving the cargo. Finally, the constant k B T is the Boltzmann constant multipliedby temperature.The equation for cargo position Z ( t ) in (2) also follows an overdamped Langevinequation with friction constant γ (also calculated using the Stokes-Einstein lawwith a =
500 nm), but differs from that of detached motors in two ways. First,the cargo is subject to spring forces from each of the N motors. Second, we alsoaccount for a possible constant applied optical trap force F T , as in the experi-ments of [45, 61, 60] and simulations of [10, 30, 39, 44, 85, 35]. Note that thefriction constant values for the detached motors and cargo should be viewed in asomewhat notional sense, and we have not attempted to give them precise values.Fortunately as will discuss in Section 7, the effective dynamics are not sensitiveto their precise values. We model the transition between attachment configurations with varying numbersof attached and detached motors with jump processes. See Table 1 for a list oftypical attachment/detachment values. The attachment of motors is modeled bya homogeneous Poisson process, with each detached motor having an attachmentrate of a ( i ) . As in most theoretical work [39, 45, 10], we take the attachmentrates to be independent of the configuration of the motor-cargo complex, thoughwe acknowledge that Furuta, Furuta, et al [26] finds significant reduction of theattachment rate of a second motor when the cargo is under load. The attachmentrate of a motor can naturally be expected to be somewhat different when at leastone other motor on the same cargo is also attached to a microtubule than whennone are [59], as does seem to be indicated experimentally [35]. As we do notmodel the attachment of the motor-cargo complex from a state of no attachedmotors, the attachment rate we require is the one where at least one other motoron the same cargo is currently attached to a microtubule. We therefore take ourparameter value for a ( i ) in Table 1 from recent experimental measurements in thissetting [25], rather than the conventional estimate in modeling work [39] based on12ingle-kinesin attachment rates. When a motor reattaches, we simply place it atthe same position along the microtubule as it was previously in the detached state.This is a bit different than most other reattachment models in the literature,which do not precisely track the motor position in the detached state (as we do),but apply a selection rule of where the motor reattaches. For example, [36] at-taches the second motor at a location that leads to a zero tether force with thecargo, while [59] attaches the motor directly at the current cargo position. By con-trast, [26, 10, 85, 24] randomly choose attachment sites that are within a geomet-rically defined range of the motor’s attachment point to the cargo while [53, 21]preferentially reattaches a motor to the microtubule at locations that require theleast strain energy on the motor. Our approach, as well as the other random modelsjust cited, are consistent with experimental observations that a motor can reattachahead or behind the motor already attached [25].Detachment rates are determined through the function depending on the force F felt by the motor: d ( i ) ( F ) = d ( i ) ϒ ( i ) ( F / F ( i ) d ) . (5)Here, the parameter F ( i ) d is a scale force, and ϒ ( i ) satisfies ϒ ( ) =
1, so that d ( i ) isthe detachment rate under no external force. Note that since we are only modelingthe motor-cargo dynamics along the microtubule direction, our detachment ratemodel correspondingly depends only on the longitudinal force F , with a positivesign corresponding to opposing the motor’s natural direction of motion. Notethis convention of writing the relation of the detachment rate and applied forcewith positive arguments corresponding to a hindering force is opposite to howsuch relations are typically presented in recent experimental studies [60, 61, 37]but are consistent with how force-velocity relationships are typically expressed intheoretical and simulation studies [38, 10, 56]. The force F = F ( i ) ( t ) opposingthe motor i at time t consists of the force from the tether to the cargo, which is F ( i ) ( t ) = κ ( i ) ( X ( i ) ( t ) − Z ( t )) . (6)As F ( i ) ( t ) is a random process, so is the detachment rate d ( i ) ( F ( i ) ( t )) of motor i .The resulting model for detachment thus amounts to a Cox process, a generaliza-tion of a Poisson process in which the intensity function may be a random process(see Cox and Isham [15] for an introduction).We can formalize the description of the switching model through the asso-ciation of standard Poisson counting processes with each potential state transi-tion. We thereby define, with associations, the standard Poisson counting pro-13esses Y ( i ) d ( t ) to represent detachment of motor i and Y ( i ) a ( t ) to represent attach-ment of motor i . Then the dynamics of the attachment states can be written, for i = , . . . , N : dQ ( i ) ( t ) = Y ( i ) a (cid:18) a ( i ) Z t ( − Q ( i ) ( t ′ )) dt ′ (cid:19) (7) − Y ( i ) d (cid:18) Z t Q ( i ) ( t ′ ) d ( i ) ϒ ( i ) (cid:16) κ ( i ) ( X ( i ) ( t ′ ) − Z ( t ′ )) / F ( i ) d (cid:17) dt ′ (cid:19) . Together with Eqs. (1)–(4) from Subsection 2.1, we have a complete Markoviandynamical description for the motor-cargo model.A common choice of ϒ ( i ) , following the theory of Bell [6], is an exponentialfunction (see [64, 44], for instance). For simulations in Section 6, we will use themore general double exponential detachment model, which is based on observa-tions that run length is asymmetric with respect to the direction of external load[60, 61], which can be argued to improve the processivity of a team of motors [85].Detachment rates for the double exponential detachment model are given by d ( i ) ( F ) = ( d ( i ) − exp ( − F / F ( i ) d − ) F ≤ , d ( i ) + exp ( F / F ( i ) d + ) F > . (8)Here, d ( i ) + and d ( i ) − are, respectively, limits of detachment rates as the hindering(assisting) external force approaches zero. The corresponding force scales F ( i ) d + and F ( i ) d − of detachment are expressed in the literature [60, 61] in terms of char-acteristic length scales δ ( i )+ and δ ( i ) − via F ( i ) d ± = k B T / δ ( i ) . Note again our signconvention on the force is opposite to how the experimental results are presentedin [60, 61], but is consistent with the standard representation of force-velocityrelations in theoretical studies. In our one-dimensional model, we are not distin-guishing between longitudinal and transverse force components, which may haveinteresting dynamics from geometric considerations of the cargo [35].From the additivity of Poisson rates, a system with N u attached motors has aconstant total attachment rate of ∑ i : Q ( i ) = a ( i ) . Under constant detachment rates d ( i ) , i = , . . . , N , which occur when ϒ ( i ) is a constant function, switching betweenstates is a homogeneous, continuous time, finite state Markov chain, with an av-erage time τ ( S ) spent in a state Q = ( Q ( ) , . . . , Q ( N ) ) given by: τ ( S ) = ∑ Nj = ( − Q ( j ) ) a ( j ) + ∑ Nj = Q ( j ) d ( j ) . (9)14able 1: Typical values for kinesin-1 in water-like environments at saturatingATP concentrations. Values of parameters which differ for kinesin-2 (specifically,KIF3A/B) are in bold with parentheses [92, 26, 2, 56, 60, 25, 61].Parameter Description Typical values F is Motor stall force 7 pN [92] k B T Boltzmann constant by temperature 4.1 pN nm κ ( i ) Motor-cargo tether spring constant 0.25 pN/nm [26] v ( i ) Unencumbered motor velocity 790 nm/s (
500 nm/s ) [2] γ Cargo friction 1 × − pN s/nm [56] γ m , i Motor friction 1 × − pN s/nm ( σ ( i ) ) Effective motor diffusion 5000 nm /s ( /s )[92, 2] F T Optical trapping force -20 pN to 6 pN [60] a ( i ) Motor attachment rate 4/s ( ) [25] d ( i ) − Small assisting force detachment rate 9.1/s (5.6/s) [60, 61] d ( i ) + Small hindering force detachment rate 0.7/s (2.3/s) [60, 61] F ( i ) d − Assisting force detachment scale 14 pN (10 pN) [60, 61] F ( i ) d + Hindering force detachment scale 2.1 pN (2.0 pN) [60, 61]For more complicated detachment rates, the average time until either first detach-ment or attachment admits no explicit solutions. As we will see in Section 4, how-ever, under a slow switching regime, we may approximate detachment rates byconstants ¯ d ( i ) through averaging over possible motor-cargo configurations. Thisis in contrast to the quasi-steady state model studied by Bressloff and Newby [11],in which transitions between states are considered fast compared to motor veloci-ties. In this section, we prepare for the asymptotic analysis to coarse-grain the detailedmodel from Section 2 by a systematic nondimensionalization in Subsection 3.1.We thereby identify the detached motor and cargo dynamics as faster than thoseof the attached motors, and conduct in Subsection 3.2 a stochastic averaging overthe cargo and detached motor coordinates to obtain an effective description onlyinvolving the attachment state of the motors together with the spatial positions15f attached motors. This section is essentially a generalization of the analysisfrom McKinley, Athreya, et al [56] to the case of nonidentical cooperative mo-tors, together with a consideration of the coarse-grained attachment or detachmentevents.
For the purposes of reducing the number of parameters in a motor-cargo system,we perform a nondimensionalization of (1)-(4), adapting the nondimensionaliza-ton in [56] for the case of identical cooperative motors. To not only nondimen-sionalize but normalize the variables for the asymptotic reduction exploiting timescale disparities [50, 67], a reference time scale of γ / κ and a reference length scaleof p k B T / κ was taken in the nondimensionalization, where κ was the commonmotor-cargo tether spring constant. These characterize the nominal fluctuationdynamics of the cargo, so the resulting nondimensional equations are normalizedto order unity for the cargo, and manifest the relatively slow dynamics of the at-tached motors.To extend this nondimensionalization to nonidentical motors, we define¯ κ = N ∑ j = κ ( j ) / N , ˜ κ ( i ) = κ ( i ) / ¯ κ , (10)and take the average ¯ κ to define the reference units in the nondimensionalization.Under the change of coordinates ˜ t = ¯ κ t / γ , and˜ X ( i ) ( ˜ t ) = X ( i ) ( γ ˜ t / ¯ κ ) p k B T / ¯ κ , ˜ Z ( ˜ t ) = Z ( γ ˜ t / ¯ κ ) p k B T / ¯ κ , (11)equations (1) and (2) may be written in nondimensional form d ˜ X ( i ) ( ˜ t ) = (cid:18) ε ( i ) g ( s ( i ) ( ˜ X ( i ) ( ˜ t ) − ˜ Z ( ˜ t ))) d ˜ t + q ˆ ρ ( i ) ε ( i ) dW ( i ) ( ˜ t ) (cid:19) Q ( i ) ( ˜ t ) (12) + (cid:16) − ( Γ ( i ) ) − ˜ κ ( i ) ( ˜ X ( i ) ( ˜ t ) − ˜ Z ( ˜ t )) d ˜ t + ( Γ ( i ) ) − / dW ( i ) ( ˜ t ) (cid:17) ( − Q ( i ) ( ˜ t )) , ≤ i ≤ N , d ˜ Z ( ˜ t ) = N ∑ j = ˜ κ ( j ) ( ˜ X ( j ) ( ˜ t ) − ˜ Z ( ˜ t )) − ˜ F T ! d ˜ t + dW z ( ˜ t ) . (13)The nondimensional attachment and detachment rates are now ˜ a ( i ) = a ( i ) γ / ¯ κ and˜ d ( i ) = d ( i ) γ / ¯ κ . Under this nondimensionalization, the detachment rate (5) then16able 2: Nondimensional groups and typical values for kinesin-1 and kinesin-2.When only a single value is given, it is common to both. Otherwise the kinesin-1value is listed first, with the kinesin-2 value in bold text in parentheses. Attach-ment and detachment scales are taken for hindering forces.Group Definition Typical value ε ( i ) v ( i ) γ √ k B T ¯ κ × − ( × − ) s ( i ) κ ( i ) F s q k B T ¯ κ . F T F T √ k B T ¯ κ -10 to 4ˆ ρ ( i ) ( σ ( i ) ) √ ¯ κ v ( i ) √ k B T .5 ) Γ ( i ) γ im γ . u ( i ) κ ( i ) F id q k B T ¯ κ .7 ( .7 )˜ a ( i ) a ( i ) γ ¯ κ × − ( × − )˜ d ( i ) d ( i ) γ ¯ κ × − ( × − ) becomes (expressed now as a stochastic process in time):˜ d ( i ) ( ˜ t ) = ˜ d ( i ) ϒ ( u ( i ) ( ˜ X ( i ) ( ˜ t ) − ˜ Z ( ˜ t )) . (14)A listing of nondimensional parameters introduced in Eqs. (12)-(14) and theirtypical values are provided in Table 2. By comparing magnitudes of drift coefficients, we are now able to identify fast andslow variables. For systems with multiple scales, a common method of dimensionreduction averages out fast variables by considering their stationary distributionsagainst fixed values of slow variables (see Pavliotis and Stuart [69] for multiple ex-amples). We will use standard asymptotic notation for indicating the relative mag-nitudes of quantities. Informally, f ≪ g or equivalently f ∼ o ( g ) means f is muchsmaller than g , f ∼ g or equivalently f ∼ ord ( g ) means f and g are of comparablesize, f ≫ g means f is much greater than g , and f ∼ O ( g ) means f is comparableto or smaller than g . All these concepts can be given precise asymptotic defini-tions [31], which we follow though without making the formalism explicit. From17able 2 and Eqs. (12) and (13), we observe that a plausible asymptotic orderingfor the nondimensional parameters is: ˜ a ( i ) ∼ ˜ d ( i ) ≪ ε ( i ) ≪ ≪ ( Γ ( i ) ) − . That is,the dynamics for detached motors are faster than those for the cargo, which in turnare faster than those for attached motors, which in turn are faster than the attach-ment and detachment processes. We by no means claim this asymptotic orderingis well satisfied for all molecular motors, or for kinesin-1 under all conditions, butsimply that the assumptions on which our asymptotic simplification is based is atleast plausible based on the kinesin-1 data we have drawn from the literature, sum-marized in nondimensional form in Table 2. The assumption that the attachmentand detachment processes are asymptotically slow compared to attached motordynamics was also adopted and exploited in Miles, Lawley, and Keener [59]. Formixtures of motors, while we allow the attachment, detachment, and unencum-bered velocities to differ, we assume the parameters in each group are of the sameorder of magnitude in our asymptotic ordering ( ˜ a ( i ) ∼ ˜ a ( i ′ ) ∼ ˜ d ( i ) ∼ ˜ d ( i ′ ) , ε ( i ) ∼ ε ( i ′ ) for all 1 ≤ i , i ′ ≤ N ).While the focus for this paper will mainly be for two motor systems, it ispossible to write averaged formulas for both detached motor and cargo positionsunder a generic system of N motors. Fixing a time t ≥ i ,if Q ( i ) ( t ) =
0, we may regard the distribution of the detached motor ˜ X ( i ) ( t ) asapproximately that of the quasistationary distribution p ˜ X ( i ) | ˜ Z under fast detachedphase dynamics ( Q ( i ) ( t ) = Z ( ˜ t ) heldat a fixed value ˜ z . This is the Gaussian PDF p ˜ X ( i ) | ˜ Z ( ˜ x | ˜ z ) = s ˜ κ ( i ) π exp (cid:16) − ˜ κ ( i ) ( ˜ x − ˜ z ) (cid:17) , (15)and all detached motor positions are conditionally independent given the cargoposition ˜ Z ( ˜ t ) = ˜ z . Similarly, by fixing the positions of the slow attached mo-tors, an approximation of the distribution of the faster cargo position is the quasi-stationary distribution p ˜ Z | ˜ X ( a ) of (13) with the states of all motors and the positionsof the attached motors held at fixed values, which is another Gaussian of the form p ˜ Z | ˜ X ( a ) , Q ( ˜ z | x , q ) (16) = s ∑ Nj = q ( j ) ˜ κ ( j ) π exp − N ∑ j = q ( j ) ˜ κ ( j ) ! ˜ z − " ∑ Ni = q ( j ) ˜ κ ( j ) ˜ x ( j ) − ˜ F T ∑ Ni = q ( j ) ˜ κ ( j ) , where x = ( x ( ) , x ( ) , . . . , x ( N ) ) and q = ( q ( ) , q ( ) , . . . , q ( N ) ) parameterize, respec-tively the positions and states of all N motors. Note that the above formula for the18uasi-stationary distribution of the cargo does not actually depend on the positionsof the detached motors (indices i for which q ( i ) = X ( a ) , though we write the quasi-stationarydistribution formally as a function of all motor positions to keep notation sim-ple. Moreover, the averaging of the detached motors does not affect the cargodynamics to leading order, because the tether force will average to zero, and thecontribution of the force fluctuations to the cargo diffusivity are O ( Γ ( i ) ) . The av-erage position for the cargo according to Eq. (16) is a weighted average of theattached motor positions shifted by a multiple (which would be the simple inverseof the number of attached motors if they had the same tether spring constants)of the nondimensional trap force. In reality the cargo should lag a bit from thisweighted average position of the attached motors because of balancing the vis-cous drag force. Our treatment of ε ( i ) as a small parameter, however, implies thatthe cargo drag force is being treated as small compared to the force scale of thethermal fluctuations of the cargo, so this mean lag would be small compared to thestandard deviation of the cargo fluctuations and is thus neglected. We will oftenrefer to (16) in the specific case of systems with N = p ˜ Z | ˜ X ( a ) , Q ( ˜ z | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) = s ˜ κ ( ) π exp " − ˜ κ ( ) (cid:18) ˜ z − ˜ x ( ) + ˜ F T ˜ κ ( ) (cid:19) , (17) p ˜ Z | ˜ X ( a ) , Q ( ˜ z | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) = s ˜ κ ( ) π exp " − ˜ κ ( ) (cid:18) ˜ z − ˜ x ( ) + ˜ F T ˜ κ ( ) (cid:19) , and with both motors attached, p ˜ Z | ˜ X ( a ) , Q ( ˜ z | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) = r π exp − ˜ z − ˜ κ ( ) ˜ x ( ) + ˜ κ ( ) ˜ x ( ) + ˜ F T ! . (18)Our nondimensionalization set the time scale to be order unity for the cargo, sothe dynamics of the slower attached motors expressed in Eq. (12) appears weak onthis time scale ( ord ( ε ( i ) ) drift and diffusivity for attached motors i , with ε ( i ) ≪ t = ¯ t / ¯ ε , ¯ ε = N ∑ j = ε ( j ) / N , (19)19ver which the cargo now appears ord (( ¯ ε ) − ) fast and equilibrates quickly relativeto changes in attached motor position. From stochastic averaging theory (see [80,52]), we can approximate the attached motor positions on this time scale ˜ X ( i ) ( ¯ t / ¯ ε ) by an averaged stochastic process ¯ X ( i ) ( ¯ t ) satisfying the system of SDEs d ¯ X ( i ) ( ¯ t ) = ¯ g ( i ) ( { ¯ X ( i ) } Ni = ( ¯ t ) ; { Q ( i ) } Ni = ) d ¯ t + q ρ ( i ) dW ( i ) ( ¯ t ) , ρ ( i ) = ε ( i ) ¯ ε ˆ ρ ( i ) , (20)with an effective drift obtained by averaging over the quasi-stationary distributionof the cargo: ¯ g ( i ) ( x ; s ) = ε ( i ) ¯ ε Z R g ( s ( i ) ( ˜ x ( i ) − ˜ z )) p ˜ Z | ˜ X ( a ) , Q ( ˜ z | x , q ) d ˜ z . (21)Note the averaged dynamics of the attached motors in Eq. (20) are coupled di-rectly to each other, through the averaging out of the cargo variable to which theyare explicitly coupled in (12). The averaged drift ¯ g ( i ) ostensibly depends on allmotor positions, but it actually is independent of the detached motor positions be-cause the same is true of p ˜ Z | ˜ X ( a ) , Q in Eq. (16). Bouzat [10] proposes alternativelyto coarse-grain the effects of the cargo fluctuations on the effective velocity of mo-tors through an exponential time-averaging of the force felt from the cargo. Thisshould give equivalent results to the more straightforward stochastic averagingused here for the “robust” regime of averaging time scales advocated in [10].Continuing with our assumption that the detachment process is slow comparedto the time scale of motor motion ( ˜ d ( i ) ≪ ε ( i ) ), we may also obtain an averageddetachment rate ¯ d ( i ) ( x ) for each motor i (within the framework [47] of averagingfor general Markov processes):¯ d ( i ) ( x , s ) = ˜ d ( i ) ¯ ε Z R ϒ (cid:16) u ( i ) (cid:16) ˜ x ( i ) − ˜ z (cid:17)(cid:17) p ˜ Z | ˜ X ( a ) , Q ( ˜ z | x , q ) d ˜ z . (22)This formula may be viewed as a more detailed implementation of the idea givenin [7, 3] that the effective detachment rate for a motor should be computed by aver-aging the nominal force-detachment rate formula over a probability distribution offorces felt from the cargo, and that this can lead to a significant enhancement rel-ative to the evaluation of the detachment rate in terms of simply the average forcefelt from the cargo. The constant attachment rates on the longer time scale (19)are now ¯ a ( i ) = ˜ a ( i ) / ¯ ε . (23)20hen a detached motor attaches, its attachment position is taken to be drawnfrom the quasi-stationary distribution of the detached motors, given fixed positionsof the attached motors. This follows from our modeling of the attachment processas independent of spatial configuration in the present work. [26, 10, 85] placeattaching motors uniformly at random over the region of the microtubule wherethe motor-cargo tether is unstretched; this would be roughly consistent with ourapproach when the linear spring is modified to give no resistance to compressionbelow a rest length (see Appendix A). [8] places attaching motors at the closestposition on the microtubule to their attachment point to the cargo (which, withour point particle representation of the cargo, would here amount to attaching themotor at the current cargo position ˜ Z ). Because of the assumed time scale sepa-ration, the relative position of the detached motors and the cargo is independentof the relative position of the cargo and the attached motors, and these are gov-erned by the respective Gaussian quasi-stationary distributions (15) and (17). Thequasi-stationary distribution of the position of the detached motors for fixed po-sitions of the attached motors is therefore obtained by convolving these Gaussianquasi-stationary distributions, giving rise to a Gaussian distribution as well. Inthe simple case of N = p ( a ) ˜ X ( ) ( x ′ | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) = Z R p ˜ X ( i ) | ˜ Z ( x ′ | ˜ z ) p ˜ Z | ˜ X ( a ) , Q ( ˜ z | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) d ˜ z (24) = s ˜ κ ( ) ˜ κ ( ) π exp − ˜ κ ( ) ˜ κ ( ) (cid:18) x ′ − ˜ x ( ) + ˜ F ˜ κ ( ) (cid:19) ! , (25) p ( a ) ˜ X ( ) ( x ′ | ( ˜ x ( ) , ˜ x ( ) ) , ( , )) = s ˜ κ ( ) ˜ κ ( ) π exp − ˜ κ ( ) ˜ κ ( ) (cid:18) x ′ − ˜ x ( ) + ˜ F ˜ κ ( ) (cid:19) ! . (26)For the general case of N motors, the attachment position of a currently detachedmotor i would have a Gaussian probability distribution with mean ∑ Nj = q ( j ) ˜ κ ( j ) ˜ x ( j ) − ˜ F T ∑ Nj = q ( j ) ˜ κ ( j ) Switched diffusion model.
Effective dynamics of cargo with twoattached motors on time scale ˜ a ( i ) ∼ ˜ d ( i ) ≪ ε ( i ) ≪ ˜ t ≪ ≪ ( Γ ( i ) ) − long comparedto detached motor and cargo fluctuation dynamics but short compared to attachedmotor dynamics. The cargo dynamics are represented on this time scale by a cargotracking variable M , as discussed in Subsection 4.1. Left: A cargo with bothmotors attached to the microtubule. The cargo tracking variable dynamics (42)and the detachment rate ¯ d ( i ) , (45) of each motor i depends on the displacement R between the attached motors, also evolving dynamically. Middle: With only onemotor attached, the cargo tracking variable M with one attached motor evolveswith constant drift (31) and diffusivity (32). From these states, the system eitherdetaches completely (as shown in the rightmost part of the figure), with effectiverate given in (47) or returns to having two attached motors, with rates ¯ a ( i ) (48)..and variance 12 ∑ Nj = q ( j ) ˜ κ ( j ) +
12 ˜ κ ( i ) . From here on, we will focus on dynamics for a two motor ( N =
2) system duringa processive run. At any one time, either one or two motors may be attached tothe microtubule, which gives three possible states, or the run may terminate whenboth motors are simultaneously detached from the microtubule. The dynamicaldescription for N = Coarse-grained Markov chain model.
Under the assumption of slowswitching, the switched diffusion model (see Fig. 2) is further coarsened by aver-aging intermotor separation. The random duration ∆ T ϖ spent in state ϖ is expo-nentially distributed with mean determined in the usual way by the transition ratesout the state. The random displacement ∆ M ϖ for each attachment state ϖ (shownin the upper left corner of each box) is then found through (28). Detachmentrates ¯ d ∗ ( i ) , from the state of two attached motors, given by (44), are now constantscoarse-grained with respect to intermotor separation.23argo position is now represented as a probability distribution given the locationand identity of the attached motors (see Eq. (29)), with a “cargo tracking variable” M ( ¯ t ) representing the conditional mean position of the cargo. The symmetry ofthe dynamics under common spatial translation of all entities implies the effec-tive dynamics while in the state with one motor attached has constant drift andattachment/detachment rates, while these quantities in the state with two motorsattached only depends on the directed separation of the attached motors R ( ¯ t ) ≡ ¯ X ( ) ( ¯ t ) − ¯ X ( ) ( ¯ t ) . (27)In this section, we will make use of the assumption that the attachment anddetachment rates are slow relative to the attached motor dynamics ( ˜ a , ˜ d ≪ ε ),to homogenize the spatial dynamics within each attachment state and therebyfurther coarse-grain our model, based on stochastic differential equations withconfiguration-dependent detachment rates, into simple continuous-time Markovchain dynamics on the state space of attachment states, together with the cargotracking variable M ( t ) which may be thought of as an accumulated reward func-tion associated with the Markov chain. The four states of this Markov chain arelabeled by a list ϖ of the indices of the attached motors, as indicated in Figure3. The transition rates between states after homogenization of attached motordynamics are now all constant, also indicated in Figure 3.The state ϖ = /0 acts as an absorbing state terminating the processive run.In each the other three states, the cargo tracking variable undergoes a constant-coefficient drift-diffusion dynamics. Thus, a visit to a state ϖ = /0 is associated toa cargo tracking variable increment ∆ M ϖ = V ϖ ∆ T ϖ + p D ϖ ∆ W ( ∆ T ϖ ) , where ∆ T ϖ is the duration of the visit to state ϖ , V ϖ and D ϖ are, respectively, theconstant coarse-grained velocity and diffusivity in state ϖ , and ∆ W ( t ) ∼ N ( , t ) denotes the increment of the Wiener process over a time t . In other words, thecargo tracking variable increment is Gaussian conditioned on the duration ∆ T ϖ inthe state ϖ : ∆ M ϖ | ∆ T ϖ ∼ N ( V ϖ ∆ T ϖ , D ϖ ∆ T ϖ ) . (28)Here and later, the notation Y ∼ N ( µ , σ ) indicates that Y is a normally distributedrandom variable with mean µ and variance σ . The tracking variable M will alsosuffer jumps associated to transitions, corresponding to adjustments in the meancargo position under the new configuration of attached motors.24e now proceed to more precisely quantify the elements of the coarse-grainedMarkov chain description depicted in Figure 3. The cargo tracking variable M forthe motor-cargo complex in the coarse-grained representation is defined and dis-cussed in Subsection 4.1. Thereafter, formulas for the effective drift and diffusioncoefficients of the cargo tracking variable in each state ( V ϖ and D ϖ ) are presentedin Subsection 4.2. In Subsection 4.3, we describe both the transition rates of thecoarse-grained Markov chain as well as the associated jumps in the tracking vari-able at a transition. These coarse-grained quantities are expressed following thenondimensionalization from Subsection 3.1, followed by the passage to the longtime scale ¯ t = ˜ t ¯ ε , so the effective drift, diffusion, and attachment/detachment rateformulas presented would need to be multiplied by ¯ ε to give their expressions interms of the original nondimensionalization. Note that when describing effectivetransport and switching rates from various states of attachment, we will typicallylist the indices of the attached motors in the subscript (without parentheses orbrackets), and when needed, indicating by a parenthesized superscript the indexof which motor in that state is being described by the parameter. Finally, in Sub-section 4.4, we shift focus to a more event-based view, decomposing the progressof a motor-cargo complex along a microtubule in terms of cycles of detachmentand reattachment of a motor before eventual complete detachment of both mo-tors. This representation of the motor-cargo complex dynamics in terms of thesecycles will be the basis for computing the overall statistics of cargo transport inSection 5. One challenge in characterizing the progress of the motor-cargo system throughphases of attachment and detachment is the choice of a “tracking” variable thatremains well-defined in the various states of motor attachment. The cargo vari-able ˜ Z ( ˜ t ) is an obvious candidate since in experiments it is the largest and mosteasily observed object, and whose progress through space is of practical impor-tance. Mathematically, though, as shown in the dimensional analysis from Sub-section 3.1, the cargo will tend to fluctuate more rapidly than attached motors,which makes it somewhat awkward to use its instantaneous position as a trackingvariable on long time scales. We therefore introduce the variable ˜ M ( ˜ t ) for themean cargo position at time ˜ t under the quasi-stationary distribution p ˜ Z | ˜ X ( a ) , Q (16)for the cargo given the current positions of the motors attached to the microtubule25t time ˜ t : ˜ M ( ˜ t ) ≡ Z ∞ − ∞ ˜ zp ˜ Z | ˜ X ( a ) , S ( ˜ z | ˜ X ( ˜ t ) , Q ( ˜ t ))= " ∑ Nj = Q ( j ) ( ˜ t ) ˜ κ ( j ) ˜ X ( j ) ( ˜ t ) ∑ Nj = Q ( j ) ( ˜ t ) ˜ κ ( j ) − ˜ F T ∑ Nj = Q ( j ) ( ˜ t ) ˜ κ ( j ) . (29)One can check that ˜ M ( ˜ t ) is equivalent to the deterministic mechanical equilibriumof the cargo, for given attached motor positions, in the absence of stochastic fluc-tuations. Markov chain models for cargo transport with attaching and detachingmotors often model the cargo as always being exactly at this position ˜ M ( ˜ t ) offorce balance relative to the attached motors [39, 38]; our present model accountsfor cargo fluctuations about this mechanical equilibrium. Nonetheless, ˜ M is amore convenient variable for tracking the progress of the cargo through episodesof attachment and detachment. Moreover, the cargo position ˜ Z ( ˜ t ) , under our timescale separation assumptions, has a Gaussian distribution centered at ˜ M ( ˜ t ) withstandard deviation no larger than max ≤ j ≤ N {
12 ˜ κ ( j ) } (see Eq. (16)). Consequently,the long-time statistical transport properties of the cargo position ˜ Z ( ˜ t ) and thetracking variable ˜ M ( ˜ t ) are equivalent. Note below we define the tracking vari-able on the longer time scale (which is being used after averaging out the cargodynamics in Subsection 3.2) as M ( ¯ t ) = ˜ M ( ¯ t / ¯ ε ) . (30) For states where the motor with index i is attached but the other motor is detached,the effective drift is: V i = s ˜ κ ( i ) π ε ( i ) ¯ ε Z R g ( s ( i ) y ) exp ( − ˜ κ ( i ) ( y − ˜ F T / ˜ κ ( i ) ) ) dy . (31)and the effective diffusivity is D i = ρ ( i ) . (32)These formulas follow directly from Eq. (20), once we recognize from Eq. (29),that ˜ M ( ˜ t ) = ˜ X ( i ) ( ˜ t ) − ˜ F / ˜ κ ( i ) . (33)26ow we just carry over the result from Eq. (20), noting the expression (21) isconstant for the case of one attached motor.For the state where both motors are attached, the effective drift is: V , = Z R G + ( r ) p R ( r ) dr . (34)where p R ( r ) = C R exp (cid:20) ρ ( ) + ρ ( ) Z r G − ( r ′ ) dr ′ (cid:21) , − ∞ < r < ∞ , (35)with normalizing constant C R , is the stationary probability density for the displace-ment R = X ( ) − X ( ) between the two attached motors. The effective diffusivityof the motor-cargo complex in this state is: D , = Z ∞ − ∞ (cid:18) ρ ( ) + ρ ( ) (cid:19) (cid:18) Z r − ∞ ( G + ( r ′ ) − V , ) p R ( r ′ ) dr ′ (cid:19) p R ( r ) dr + ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ρ ( ) + ρ ( ) ! Z R ( G + ( r ) − V , ) p R ( r ) · rdr + ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) . The auxiliary functions referenced in these formulas are: G + ( r ) = ˜ κ ( ) G ( ) ( r ) + ˜ κ ( ) G ( ) ( r ) , (36) G − ( r ) = G ( ) ( r ) − G ( ) ( r ) . (37)The derivation of these formulas for the effective transport for the state withtwo motors attached proceeds as follows. For known motor positions x ( ) , x ( ) ,we can express the cargo-averaged drift coefficients purely in terms of the signeddisplacement r = x ( ) − x ( ) between the motors:¯ g ( ) ( x ( ) , x ( ) ; ( , )) = G ( ) ( x ( ) − x ( ) ) , (38)¯ g ( ) ( x ( ) , x ( ) ; ( , )) = G ( ) ( x ( ) − x ( ) ) , (39) G ( i ) ( r ) = ε ( i ) ¯ ε r π Z R g ( s ( i ) y ) exp − y + ( − ) i ˜ κ ( i ′ ) r − ˜ F T ! dy . (40)This representation is achieved by the change of variable y = x ( i ) − z in (21).27he tracking variable (29) in this state is, from Eq. (18),˜ M ( ˜ t ) = ( ˜ κ ( ) ˜ X ( ) ( ˜ t ) + ˜ κ ( ) ˜ X ( ) ( ˜ t ) − ˜ F T ) / , (41)Passing now to the long time scale ¯ t / ε , we can recast the cargo-averaged dynam-ics (20) for the two attached motors in terms of this tracking variable rescaled tolarge time, M ( ¯ t ) (30), and the (signed) intermotor separation R ( ¯ t ) ≡ ˜ X ( ) ( ¯ t / ¯ ε ) − ˜ X ( ) ( ¯ t / ¯ ε ) which gives dM ( ¯ t ) = G + ( R ( ¯ t )) d ¯ t + p ρ ( ) ( ˜ κ ( ) ) dW ( ) ( ¯ t ) + p ρ ( ) ( ˜ κ ( ) ) dW ( ) ( ¯ t ) , (42) dR ( ¯ t ) = G − ( R ( ¯ t )) dt + q ρ ( ) dW ( ) ( ¯ t ) − q ρ ( ) dW ( ) ( ¯ t ) . (43)Under the regime where switching is slow relative to detachment, we may simplyhomogenize the internal variable R in order to obtain the effective velocity anddiffusion for M on the long time scales when attachment or detachment occurs.As (43) does not depend on M , the stationary distribution for the process R is apotential function, given in Eq. (35). The formula (34) follows directly.The computation for effective diffusivity is more complicated. For nonidenti-cal motors, the driving terms in the stochastic differential equations in Eqs. (42)-(43) are correlated in general, unlike the identical motor case from [56].Only minor modifications are needed to generalize the derivation for effectivediffusion found in Pavliotis and Stuart [69] for the case of uncorrelated stochasticdriving, as we show in Appendix B. After presenting in Subsection 4.3.1 the effective switching rates between attach-ment states (as indicated in Figure 3), we discuss the effectively instantaneousjumps in the cargo tracking variable that occurs in the coarse-grained representa-tion when switches occur. Namely, when a motor i detaches from the two-motor-attached state (transition ( , ) → ( i ′ ) ), the tracking variable undergoes a jump ∆ M ( d ) i given by equation (53). When a motor i attaches to form the two-motor-attached state (transition ( i ′ ) → ( , ) ), the tracking variable undergoes a jump ∆ M ( a ) i , given by equation (57). 28 .3.1 Effective detachment rates From our assumptions about the detachment time scale being slower than the at-tached motor time scale ( ˜ d ( i ) ≪ ε ( i ) ) in Subsection 3.2, we can apply stochasticaveraging [47] to approximate the rate at which motor i detaches from a state withboth motors attached by averaging over the stationary distribution p R ( r ) (35) forthe motor separation R , just as we did for the effective velocity:¯ d ∗ ( i ) , = Z R ¯ d ( i ) , ( r ) p R ( r ) dr , (44)¯ d ( i ) , ( r ) = r π ˜ d ( i ) ¯ ε Z R ϒ (cid:16) u ( i ) y (cid:17) exp − (cid:18) y − (( − ) i + ˜ κ ( i ′ ) r − ˜ F T ) (cid:19) ! dy . (45)This last formula is just an expression of the detachment rate (22) solely throughthe intermotor distance r = x ( ) − x ( ) , An effective total rate of detachment fromthe state of both motors attached is then¯ d ∗ , = ¯ d ∗ ( ) , + ¯ d ∗ ( ) , . (46)In the case of a single attached motor with index i , we have a constant attach-ment rate ˜ a ( i ′ ) , and detachment rate ˜ d ( i ) ( x ( i ) , z ) that is dependent on the attachedmotor position x ( i ) and cargo position z . On the longer ¯ t = ¯ ε ˜ t time scale, the slowswitching approximation would reduce the detachment rate (22) to a constant:¯ d ∗ i = s ˜ κ ( i ) π ˜ d ( i ) ¯ ε Z R ϒ ( u ( i ) y ) exp ( − ˜ κ ( i ) ( y − ˜ F T / ˜ κ ( i ) ) ) dy . (47)and simply rescale the constant attachment rates:¯ a ∗ ( i ) ≡ ˜ a ( i ) ¯ ε . (48) When a motor detaches from a state of two attached motors, there is an immediatechange in the force balance between motors and cargo. The cargo, which is fastrelative to a single bound motor, quickly adjusts to the new motor configuration.29his results in a jump of the tracking variable position. A similar readjustmentoccurs with motor attachment. See Fig. 4 in Section 6 for simulations depictingjumps in cargo positions. In this subsection, we describe distributions of jumpsizes at these switching events. This is done under the assumption of slow switch-ing, so that we may assume the intermotor distance variable R ( ¯ t ) in the state of twoattached motors (in addition to the cargo) has achieved its stationary distribution.We focus in this subsection on the statistical behavior of the system at a ran-dom time τ d at which one of the two motors detaches ( Q ( τ − d ) = ( , ) = Q ( τ d ) ).The distribution of R ( τ − d ) just before detachment will not be the same as the sta-tionary distribution of R ( t ) due to the dependence of the detachment rate on R ( t ) .Rather, the distribution of the intermotor distance just before first detachment R ( d ) = R ( τ − d ) will be reweighted by the detachment rate, yielding the probabil-ity density: p R ( d ) ( r ) = ( ¯ d ( ) , ( r ) + ¯ d ( ) , ( r )) p R ( r ) ¯ d ∗ , . (49)This can be readily argued by considering a short time interval of length ∆ t overwhich fluctuations in R ( t ) are negligible, using Bayes’ rule to calculate the con-ditional probability of R ( t ) given that detachment occurs during the time interval,and passing to the limit ∆ t ↓ J ( d ) the index of the motor which first detaches from thetwo-motor-attached state (at the random time τ d ). From the standard theory ofcontinuous-time jump processes, P ( J ( d ) = i | R ( τ d ) = r ) = ¯ d ( i ) , ( r ) ¯ d ( ) , ( r ) + ¯ d ( ) , ( r ) , (50)and thus the unconditional probability that motor i detaches first is: p ( i ) d ≡ P ( J ( d ) = i ) = Z ∞ − ∞ ¯ d ( i ) , ( r ) ¯ d ( ) , ( r ) + ¯ d ( ) , ( r ) p R ( d ) ( r ) dr = ¯ d ∗ ( i ) , ¯ d ∗ , . (51)This is consistent with the coarse-grained description of the attachment states ofthe motors, under the slow switching approximation, having the properties of acontinuous-time Markov chain.The jump of the tracking variable M at detachment is essentially a result ofhow the mean position of cargo relies upon the number of attached motors. This30s represented by the difference between equations (33) and (41). Consider, fornow, that motor 2 detaches ( J ( d ) =
2) from the two-motor-attached state at time τ d . The jump size ∆ M ( d ) = M ( τ d ) − M ( τ − d ) will be ∆ M ( d ) = (cid:20) ¯ X ( ) ( τ − d ) − ˜ F ˜ κ ( ) (cid:21) − h ˜ κ ( ) ¯ X ( ) ( τ − d ) + ˜ κ ( ) ¯ X ( ) ( τ − d ) − ˜ F i = ˜ κ ( ) R ( τ − d ) − ˜ κ ( ) ˜ F κ ( ) , and thus have the distribution ∆ M ( d ) ∼ ˜ κ ( ) (cid:18) R ( d ) − ˜ F ˜ κ ( ) (cid:19) . where R ( d ) i is defined as a random variable with distribution equal to that of theintermotor distance R ( d ) conditioned on the event J ( d ) = i that motor i is the onewhich detaches from the state of both motors attached. Using Bayes’ rule withEqs. (50)-(51), we can derive the probability density function p R ( d ) i = p R ( r ) ¯ d ( i ) , ( r ) ¯ d ∗ ( i ) , . (52)A similar calculation shows that when motor 1 detaches ( J ( d ) = ∆ M ( d ) ∼ ˜ κ ( ) ( − R ( d ) − ˜ F ˜ κ ( ) ) . In more compact form, the jump in the tracking variable whenmotor i detaches first from the two-motor-attached state has distribution: ∆ M ( d ) i ∼ ˜ κ ( i ) (cid:18) ( − ) i R ( d ) i − ˜ F ˜ κ ( i ′ ) (cid:19) , (53) We now compute the statistics of the jumping distances at a time τ a before whichonly one motor is attached, and at which time the second motor (with index J ( a ) )attaches. Suppose first that we begin with motor 1 attached, then motor 2 attaches( J ( a ) =
2) at time τ a .From Eq. (24), detached motor 2 has a location distributed, conditional on theposition of attached motor 1, as¯ X ( ) ( τ − a ) ∼ N ( ¯ X ( ) ( τ − a ) − ˜ F T / ˜ κ ( ) , / ( ˜ κ ( ) ˜ κ ( ) )) . (54)31nder our model that the attaching motors attaches at a position governed by itsdetached spatial distribution, the jump in the central coordinate upon attachmentof motor 2 is then ∆ M ( a ) = h ˜ κ ( ) ¯ X ( ) ( τ − a ) + ˜ κ ( ) ¯ X ( ) ( τ − a ) − ˜ F T i − (cid:20) ¯ X ( ) ( τ − a ) − ˜ F T ˜ κ ( ) (cid:21) (55) ∼ N , ˜ κ ( ) κ ( ) ! . (56)More generally, when motor i is the detached motor which attaches, the trackingcoordinate jumps by an amount ∆ M ( a ) i ∼ N , ˜ κ ( i ) κ ( i ′ ) ! , (57)which is always mean zero even when the motors are nonidentical. We remark thatthe reattachment rule of Keller, Berger, et al [36], where the tether between thedetached motor and cargo is treated as exactly slack (at rest length), and this motorreattaches similarly at a location where the tether force is zero, and the cargoinstantaneously moves to a position of mechanical force balance, corresponds toa deterministic version of the rules described for our model above, using just themeans of the random reattachment position and jump in cargo tracking variable. Starting from the fully attached state ( , ) , the motor-cargo complex will undergoa random number N c of full cycles between 2-motor and 1-motor attached states(either ( , ) → ( ) → ( , ) or ( , ) → ( ) → ( , ) ) and ultimately a terminalcycle (either ( , ) → ( ) → /0 or ( , ) → ( ) → /0) ending in complete detach-ment. Thus N c is a geometric random variable with mean ( − p /0 d ) / p /0 d , where theprobability of complete detachment during an initiated cycle is defined by p /0 d = p ( ) d ¯ d ∗ ¯ a ∗ ( ) + ¯ d ∗ + ( − p ( ) d ) ¯ d ∗ ¯ a ∗ ( ) + ¯ d ∗ . (58)For each cycle (either complete or terminal), time advances by a random in-crement ∆ T c = ∆ T , + ∆ T J ( d ) ′ , (59)32here J ( d ) ′ ≡ − J ( d ) is the index of the motor remaining attached.The distributions of time ∆ T , spent in the fully attached and time ∆ T i spent inthe state with only motor i attached are exponentially distributed random variableswith the indicated means: ∆ T , ∼ Exp (( ¯ d ∗ , ) − ) , ∆ T i ∼ Exp (( ¯ a ∗ ( i ′ ) + ¯ d ∗ i ) − ) . (60)Similarly, in each cycle the tracking variable will advance by a random incre-ment ∆ M c = ∆ M , + ∆ M J ( d ) ′ + g ∆ M ( a ) J ( d ) + ∆ M ( d ) J ( d ) . (61)Here, we write g ∆ M ( a ) i to extend the random variable ∆ M ( a ) i (57) describing thejump in the cargo tracking variable upon reattachment of motor i to omit its con-tribution (with the value g ∆ M ( a ) i =
0) in the terminal cycle when motor i ′ detachesbefore motor i reattaches. The distributions for the jump in the cargo trackingvariable upon detachment of motor i , ∆ M ( d ) i , is given in Eq. (53). By the indepen-dence of residence times and the next state visited in a continuous-time Markovchain, these jumps are independent of the time spent in any state of the cycle. Onthe other hand, from the results of Subsection 4.2 for the effective velocity anddiffusivity in each attachment state, ∆ M , | ∆ T , ∼ N ( V , ∆ T , , D , ∆ T , ) , (62) ∆ M i | ∆ T i ∼ N ( V i ∆ T i , ρ ( i ) ∆ T , ) . (63)Like the distributions for ∆ T , and ∆ T i , the displacements ∆ M , and ∆ M i withinan attachment state are independent on whether the motor system eventually re-turns to a two motor attached state. For a cooperative system of two motors, we have provided in the previous sectiona coarse-grained approximation of the stochastic process governed by equations(12)-(14). These simplified equations are adequate, under the conditions of valid-ity of the asymptotic approximations, for computing effective transport propertiesof the motor-cargo complex. We begin in Subsection 5.1 by computing the pro-cessivity measures: the mean and variance of the run time and run length. Thenwe turn in Subsection 5.2 to the theoretical calculation of the effective velocity33nd diffusivity of the motor-cargo complex. The proper definition of these trans-port statistics is not entirely obvious for a motor-cargo complex that eventuallydetaches and terminates progress along the microtubule. We discuss two distinctmathematical framings of velocity and diffusivity in this context, and relate themto approaches used in analyses of previous models as well as to experimental ap-proaches. We then, in turn, compute the velocity and diffusivity under each of thetwo mathematical interpretations.The formulas in these subsections are formulated in terms of statistics of thecycles of attachment and detachment presented and derived in Subsection 5.3. Incomplicated expressions, we will sometimes have µ Y denote the mean, σ Y denotethe standard deviation, and σ Y , Y ′ denote the covariance of the random variables Y , Y ′ . We now consider the total run time T and total run length L taken by an ensembleof a cargo with two cooperative motors before complete detachment. For simplic-ity, we take the system to start with both motors are attached. Then T is just thefirst passage time of the coarse-grained Markov chain from state ( , ) to the state/0, and L is the increment in the cargo tracking variable M until absorption at thefully detached state /0. We may then write T and L as random sums of iid randomvariables { ∆ T j c } ∞ j = and { ∆ M j c } ∞ j = , where the number of complete detachment-attachment cycles N c (and therefore also the total number of cycles N c +
1) has theproperty of a Markov (stopping) time for the natural filtration generated by thesetwo sequences of random variables together with the sequence of Markov chainstates visited: T = N c + ∑ j = ∆ T j c , L = N c + ∑ j = ∆ M j c . (64)This permits us to use Wald’s identity (Th. 14.6 of DasGupta [17]) and thesecond Wald identity (Th. 2.4.5 of Ghosh, Mukhopadhyay, and Sen [27]) to ob-tain: Proposition 1. (Run length and time from cycle statistics)1. The mean run time and run length are given by µ T ≡ E [ T ] = E [ N c + ] µ ∆ T c , (65) µ L ≡ E [ L ] = E [ N c + ] µ ∆ M c . (66)34 . The variances and covariance of the run time and run length are given by σ T ≡ Var ( T ) = E [ N c + ] σ ∆ T c + Var ( N c + ) µ ∆ T c , (67) σ L ≡ Var ( L ) = E [ N c + ] σ ∆ M c + Var ( N c + )( µ ∆ M c ) , σ T , L ≡ Cov ( T , L ) = E [ N c + ] σ ∆ T c , ∆ M c + Var ( N c + ) µ ∆ T c µ ∆ M c . (68)
3. The number of complete detachment-attachment cycles N c has the followingfirst and second order statistics: E [ N c + ] = p /0 d , (69) Var [ N c + ] = − p /0 d p /0 d , (70) with the probability of complete detachment during an initiated cycle given byp /0 d = p ( ) d ¯ d ∗ ¯ a ∗ ( ) + ¯ d ∗ + ( − p ( ) d ) ¯ d ∗ ¯ a ∗ ( ) + ¯ d ∗ . (71)The statistics of N c follow directly from the discussion at the beginning ofSubsection 4.4. Explicit expressions for the other cycle statistics in Prop. 1 willbe provided in Subsection 5.3.It may seem surprising that we do not need to subdivide the calculation intothe component from the N c complete cycles and the terminal cycle, since the prob-ability distribution of the attachment jump g ∆ M ( a ) J ( d ) ′ does depend on whether it is aterminal cycle. But Wald’s identity precisely allows us to do this because, viewedjointly, the sequence of times taken, cargo motion incurred (including the jumpvariables), and attachment/detachment events within each cycle are independentand identically distributed across cycles. It is only when one conditions on the firstcycle that leads to complete detachment ( N c +
1) that the cargo motion incurred oneach cycle is no longer identically distributed. Our calculation involving Wald’sidentity eschews this conditioning step, which in fact would further complicatethe calculation due to its effect on which motor J ( d ) detaches during a cycle (seeSubsection 5.3 below). The characterization of the effective velocity and diffusivity is not so straightfor-ward for a motor-cargo complex that eventually detaches from a microtubule. One35annot directly take the long-time limit of the ratio of distance traveled to time,since the motor-cargo complex will detach at a finite time. Of course, for coop-erative motor models that explicitly model a rate for reattachment for the motorseven from the fully detached state, one can define an effective velocity and dif-fusivity in the usual way, essentially averaging progress over both phases wherethe motor-cargo complex is attached or detached from a microtubule [51, 59, 72].While such an effective velocity is meaningful for characterizing transport, it doesnot relate so naturally to experimental observations of particular cargo, which aretracked only while they appear to be attached to a microtubule. Moreover, the timeuntil reattachment could be quite long. In a model with explicit reattachment fromthe fully detached state, one could alternatively and meaningfully define a effec-tive velocity conditioned on attachment [39], but it is not clear how to similarlydefine a diffusivity conditioned on attachment.In order to describe the effective velocity and diffusivity of cargo during periodswhere at least one of its motors is attached to a microtubule, we consider two dis-tinct definitions of effective velocity and diffusivity which could be applied to anytheoretical or simulation model for motor-cargo dynamics, without reference to amodel for reattachment from a state of complete detachment:1. Pooling run times T ( j ) and run lengths L ( j ) over independent experiments j = , . . . , S , and defining the ensemble velocity and diffusivity as: V ens ≡ lim S → ∞ ∑ Sj = L ( j ) ∑ Sj = T ( j ) , (72) D ens ≡ lim S → ∞ ∑ Sj = (cid:16) L ( j ) − V ens T ( j ) (cid:17) ∑ Sj = T ( j ) . (73)2. We may alternatively censor experiments by requiring that they completea certain number of full cycles, thereby defining the long-run velocity anddiffusivity as V run ≡ lim k → ∞ lim S → ∞ ∑ Sj = L ( j ) N ( j ) c > k ∑ Sj = T ( j ) N ( j ) c > k , (74) D run ≡ lim k → ∞ lim S → ∞ ∑ Sj = (cid:16) ( L ( j ) − V run T ( j ) ) N ( j ) c > k (cid:17) ∑ Sj = T ( j ) N ( j ) c > k . (75)36hat we have defined as the ensemble velocity should coincide with the velocityconditioned on attachment in models with explicit reattachment from a completelydetached state [39]. With regard to the distinction in the definition of the long-runtransport statistics, note that a large number of cycles during a run, N ( j ) c → ∞ ,implies (almost surely) a large run time T ( j ) → ∞ . The idea here is that one mayoften only wish to take data on sufficiently long runs in experiments, or simula-tions, in order to downplay transient effects at the beginning or end of a cargorun, and better characterize the dynamics in the middle of a run. Censoring onrun time T ( j ) rather than the number of attachment/detachment cycles N ( j ) c in asimulation or experiment would be more natural, but the derivation of the theo-retical expression would be less straightforward, so we leave its consideration fora later work. The k → ∞ limit is the more important one in the above definitionsas it characterizes a “long run”. The S → ∞ limit of many experiments passingthe censoring step is unnecessary for computing the long-run velocity and onlyneeded for computing the long-run diffusivity since we only take data on the to-tal time and displacement of a run. We have left aside here the capacity, oftenexploited, to use observations of the cargo position at intermediate times within arun to infer velocity or diffusivity through, for example, fitting mean displacementand mean-squared displacement as a function of time [75, 16, 2]. More complexstatistical definitions of measured velocity and diffusivity could be accordinglyformulated. We finally remark that in the limit of small detachment rate, the long-run statistics should converge to the ensemble transport statistics since most runswill be long [34].The ensemble definition has good mathematical properties and is an idealizedmanner of estimating velocity and diffusivity in experiments, but we must bear inmind that experiments typically only measure runs of a cargo that are sufficientlylong [82, 92, 3, 26], since short runs are difficult to detect or disambiguate fromnoise. Thus, the actual experimental values might fall somewhere between theensemble and long-run definitions given above, so we will study both. The se-lection of runs to record in an experiment can be censored in other ways as well,for example those with an observable initial attachment and complete detachmentevent [5].Some stochastic simulations also compute velocity (and sometimes diffusiv-ity) using the ensemble definitions given above [62], though some simulation stud-ies compute velocities and diffusivities in each run, and then average the single-run velocities and diffusivities over the ensemble [24, 10] by averaging the ratioof run length to run time ( S ∑ Sj = L ( j ) T ( j ) ). The latter has no inherent connection to37ong-time properties, but would be approximated by the long-run definitions ifthe run times happened to be sufficiently long in some statistical sense [34]. Thisvelocity estimator, however, has infinite variance [34] while the ensemble defini-tion (72) (with a large but finite number of samples S ) enjoys the good statisticalconvergence properties afforded by the applicability of the central limit theorem.We define A as the event that a cycle ends with a return to the two-motor-attached state ( ( , ) → ( i ) → ( , ) ) rather than to the detachment of the complex( ( , ) → ( i ) → /0). The formulas for the long-run velocity and diffusivity involvefirst and second moment statistics of cycle displacements and durations, condi-tioned upon A ; we denote these conditional statistics by appending ” | A ” in thesubscript. As N c → ∞ , the contribution of the terminal cycles in V run and D run be-come negligible. From the standard law of large numbers for independent randomvariables for the case of ensemble statistics, and its version for renewal-rewardprocesses [74, 77] for the long-run statistics, we arrive at the following expres-sions: Proposition 2. (Velocity and diffusivity from cycle statistics) If all runs are ini-tialized from a state in which both motors are attached:1. The ensemble velocities and diffusivity are given byV ens = µ L µ T = µ ∆ M c µ ∆ T c , (76) D ens = V σ T + σ L − V ens σ T , L µ T (77) = V σ ∆ T c + σ ∆ M c − V ens σ ∆ T c , ∆ M c µ ∆ T c .
2. The long-run velocity and diffusivity are given byV run = µ ∆ M c | A µ ∆ T c | A , (78) D run = µ ∆ M c | A σ ∆ T c | A µ ∆ T c | A + σ ∆ M c | A µ ∆ T c | A − µ ∆ M c | A σ ∆ M , ∆ T | A µ ∆ T c | A ! (79) = V σ ∆ T c | A + σ ∆ M c | A − V run σ ∆ M , ∆ T | A µ ∆ T c | A .
38e see that the expressions for the ensemble and long-run velocity and diffu-sivity in terms of cycle statistics have a similar structure, with the latter involving aconditioning on the cycle indeed returning to a state of two attached motors ratherthan possibly terminating in complete detachment.
Corollary 1.
When the two motors in the ensemble have identical parameters,then V ens = V run and | D ens − D run | ≤ µ ∆ T c . We prove this corollary in Subsection 5.3.3. For both kinesin-1/kinesin-1 andkinesin-2/kinesin-2 ensembles which we consider in Section 6, the difference indiffusivities is less than one percent. The distinction between ensemble and long-run transport characteristics therefore appear potentially important mainly for het-erogenous ensembles.
The formulas for the effective motor-cargo transport in Subsections 5.1 and 5.2refer to statistics of durations and displacements within detachment-attachmentcycles. We now provide formulas for these cycle statistics, followed by a discus-sion of how they are derived.
Proposition 3. (Explicit expressions of cycle statistics)1. The first and second order moments of the unconditional cycle times anddisplacements have the explicit forms in terms of the effective nondimen-sional parameters defined in the coarse-grained model from Section 4: ∆ T c = d ∗ , + p ( ) d ¯ a ∗ ( ) + ¯ d ∗ + p ( ) d ¯ a ∗ ( ) + ¯ d ∗ , (80) µ ∆ M c = V , ¯ d ∗ , + (cid:18) µ R ( d ) p ( ) d ˜ κ ( ) − µ R ( d ) p ( ) d ˜ κ ( ) (cid:19) + p ( ) d V ¯ a ∗ ( ) + ¯ d ∗ (81) + p ( ) d V ¯ a ∗ ( ) + ¯ d ∗ − ˜ F T p ( ) d ˜ κ ( ) ˜ κ ( ) + p ( ) d ˜ κ ( ) ˜ κ ( ) ! , σ ∆ T c = ( ¯ d ∗ , ) + p ( ) d (cid:0) ¯ a ∗ ( ) + ¯ d ∗ (cid:1) + p ( ) d (cid:0) ¯ a ∗ ( ) + ¯ d ∗ (cid:1) (82) + p ( ) d p ( ) d (cid:18) a ∗ ( ) + ¯ d ∗ − a ∗ ( ) + ¯ d ∗ (cid:19) , σ ∆ M c = D , ¯ d ∗ , + V , ¯ d ∗ , ! (83) + ∑ i = p ( i ) d D i ′ ¯ a ∗ ( i ) + ¯ d ∗ i ′ + V i ′ ¯ a ∗ ( i ) + ¯ d ∗ i ′ ! + − p /0 d κ ( i ) ˜ κ ( i ′ ) + ( ˜ κ ( i ) ) σ R ( d ) i + p ( ) d p ( ) d ˜ κ ( ) − ˜ κ ( ) ˜ κ ( ) ˜ κ ( ) ! ˜ F + ∑ i = ( − ) i + V i ¯ a ∗ ( i ′ ) + ¯ d ∗ i + µ R ( d ) i ˜ κ ( i ) , σ ∆ T c , ∆ M c = V , ( ¯ d ∗ , ) + p ( ) d V ( ¯ a ∗ ( ) + ¯ d ∗ ) + p ( ) d V ( ¯ a ∗ ( ) + ¯ d ∗ ) (84) + p ( ) d p ( ) d (cid:18) a ∗ ( ) + ¯ d ∗ − a ∗ ( ) + ¯ d ∗ (cid:19) × ˜ κ ( ) − ˜ κ ( ) ˜ κ ( ) ˜ κ ( ) ! ˜ F + ∑ i = ( − ) i + V i ¯ a ∗ ( i ′ ) + ¯ d ∗ i + µ R ( d ) i ˜ κ ( i ) . (85) Additional notation used here is i ′ = − i (the index of the “other” motor),the probability p ( i ) d = ¯ d ∗ ( i ) , ¯ d ∗ , (51) that motor i detaches first from the state ofboth motors attached, the unconditional probability of complete detachment n a given cycle p /0 d (58) , and µ R ( d ) i , the conditional mean of R ( d ) given de-tachment of motor i, which may be computed from its probability density(52).2. Expressions of the corresponding cycle statistics which are conditioned on A , the cycle being complete and returning to a state of two motors attached,are the same as corresponding statistics given in the above equations (80)-(84), except that(a) In all instances, p ( i ) d is replaced withp ( i ) d | A = + ¯ a ( i ′ ) ¯ a ( i ) ¯ d ∗ ( i ′ ) , ¯ d ∗ ( i ) , ¯ d ∗ i ′ + ¯ a ( i ) ¯ d ∗ i + ¯ a ( i ′ ) − . (86) (b) To compute σ ∆ M c | A , , in equation (83) we replace the term p /0 d (definedin (58)) with 0. We begin by computing statistics for run lengths and times for different attachmentstates involved in a cycle. For evolution with two attached motors, we invoke thelaw of total expectation, conditioning on ∆ T , , to obtain E [ ∆ M , ] = V , ¯ d ∗ , . (87)From (62), we may use the law of total variance, to obtain Var ( ∆ M , ) = D , ¯ d ∗ , + V , ¯ d ∗ , ! . (88)Next, from the definition (61) of ∆ M c we obtain, by conditioning on possiblevalues of J ( d ) , µ ∆ M c = V , ¯ d ∗ , + p ( ) d V ¯ a ∗ ( ) + ¯ d ∗ + ˜ κ ( ) µ R ( d ) − ˜ κ ( ) ˜ F T κ ( ) (89) + p ( ) d V ¯ a ∗ ( ) + ¯ d ∗ − ˜ κ ( ) µ R ( d ) − ˜ κ ( ) ˜ F T κ ( ) , (90)41hich is equivalent to (81). A similar argument, referring to Eq. (59) and (60)yields (80).From (53), (57), and (28) we can similarly compute the following statistics: E ( ∆ M i ) = V i ¯ a ∗ ( i ′ ) + ¯ d ∗ i , Var ( ∆ M i ) = D i ¯ a ∗ ( i ′ ) + ¯ d ∗ i + (cid:18) V i ¯ a ∗ ( i ′ ) + ¯ d ∗ i (cid:19) , E ( ∆ M ( a ) i ) = , Var ( ∆ M ( a ) i ) =
14 ˜ κ ( i ) ˜ κ ( i ′ ) , (91) E ( g ∆ M ( a ) i ) = , Var ( g ∆ M ( a ) i ) = − p /0 d κ ( i ) ˜ κ ( i ′ ) , (92) E ( ∆ M ( d ) i ) = ˜ κ ( i ) (cid:18) ( − ) i µ R ( d ) i − ˜ F ˜ κ ( i ′ ) (cid:19) , Var ( ∆ M ( d ) i ) = ( ˜ κ ( i ) ) σ R ( d ) i . (93)Here, we note that the mean µ R ( d ) i and variance σ R ( d ) i of the intermotor separation R ( d ) i at the detachment time, given motor i detaches first, may be computed directlyfrom the density given in Eq. (52) . For the special case of constant detachmentrates, µ R ( d ) i = µ R and σ R ( d ) i = σ R .To find σ ∆ T c and σ ∆ M c , observe that the times spent and dynamics within thefully attached and one-motor detached states are independent, which implies σ ∆ T c = Var ( ∆ T , ) + Var ( ∆ T J ( d ) ′ ) , (94) σ ∆ M c = Var ( ∆ M , ) + Var ( ∆ M J ( d ) ′ + g ∆ M ( a ) J ( d ) + ∆ M ( d ) J ( d ) ) . From (60),
Var ( ∆ T , ) = / ( ¯ d ∗ , ) . For the second term in (94), we use the lawof total variance, conditioning on J ( d ) , to obtain Var ( ∆ T J ( d ) ′ ) = Var ( ∆ T ) p ( ) d + Var ( ∆ T ) p ( ) d + ( E [ ∆ T − ∆ T ]) p ( ) d p ( ) d , which, using (60), yields (82). A similar calculation gives (83), with attachmentand detachment jumps, due to their association with the fast dynamics of the cargoand detached motors, taken as independent of each other and of the progress ofthe cargo tracking variable during the time one motor was attached.It remains to calculate the covariance σ ∆ T c , ∆ M c , which may be decomposed asthe sum σ ∆ T c , ∆ M c = Cov ( ∆ M , , ∆ T , ) (95) + Cov ( ∆ M J ( d ) ′ + g ∆ M ( a ) J ( d ) + ∆ M ( d ) J ( d ) , ∆ T J ( d ) ′ ) (96)42he first term follows easily from the law of total covariance, conditioning on ∆ T , : Cov ( ∆ T , , ∆ M , ) = V , ( ¯ d ∗ , ) , (97)Through another application of the law of total covariance, conditioning on J ( d ) and noting the conditional independence of the jumps in the cargo tracking vari-able g ∆ M ( a ) J ( d ) and ∆ M ( d ) J ( d ) from the residence times ∆ T J ( d ) ′ , we may writeCov ( ∆ M J ( d ) ′ + g ∆ M ( a ) J ( d ) + ∆ M ( d ) J ( d ) , ∆ T J ( d ) ′ )= p ( ) d Cov ( ∆ M , ∆ T ) + p ( ) d Cov ( ∆ M , ∆ T )+ p ( ) d p ( ) d ( E [ ∆ T ] − E [ ∆ T ]) × ( E [ ∆ M + g ∆ M ( a ) + ∆ M ( d ) ] − E [ ∆ M + g ∆ M ( a ) + ∆ M ( d ) ]) . A direct calculation of each of these terms yields (84).
By the independence of residence time of a state and the next state in a continuous-time Markov chain, the conditioning upon re-entry into the state of two attachedmotors does not affect distributions of attachment and detachment times given in(60), nor the components of the cargo-tracking displacements ∆ M , , ∆ M i , ∆ M ( a ) i ,and ∆ M ( d ) i . What is affected is the probability distribution of which motor is theone to detach from the state of two attached motors.Let J ( d ) | A denote the index of the motor which detaches during a cycle, con-ditioned on the event A of next returning to a two-motor attached state rather thanto a state of complete detachment. We can compute the distribution of J ( d ) | A p ( i ) d | A : = P ( J ( d ) = i | A )= P ( A | J ( d ) = i ) P ( J ( d ) = i ) P ( A ) = ¯ a ( i ) ¯ d ∗ i ′ + ¯ a ( i ) (cid:18) ¯ d ∗ ( i ) , ¯ d ∗ , (cid:19) ¯ a ( i ) ¯ d ∗ i ′ + ¯ a ( i ) (cid:18) ¯ d ∗ ( i ) , ¯ d ∗ , (cid:19) + ¯ a ( i ′ ) ¯ d ∗ i + ¯ a ( i ′ ) (cid:18) ¯ d ∗ ( i ′ ) , ¯ d ∗ , (cid:19) = + ¯ a ( i ′ ) ¯ a ( i ) ¯ d ∗ ( i ′ ) , ¯ d ∗ ( i ) , ¯ d ∗ i ′ + ¯ a ( i ) ¯ d ∗ i + ¯ a ( i ′ ) − . Note the conditioning on returning to the state of two attached motors biases theprobability distribution for which motor detaches toward the one that is morelikely to reattach.When conditionining ∆ M c on the event A , the only variables affected are J ( d ) and g ∆ M ( a ) J ( d ) . Thus, expressions in (80)-(84) now are computed with P ( J ( d ) = i | A ) = p ( i ) d | A rather than p ( i ) d . To obtain σ ∆ M c | A , we note that g ∆ M ( a ) J ( d ) | A ∼ ∆ M ( a ) J ( d ) | A ,and carry out calculations similar to those which yield (83). From these formulas, we observe that if two motors in an ensemble have identicalparameters, it follows from symmetry that p ( ) d = p ( ) d | A = /
2, and consequently V ens = V run . Effective diffusivities D ens and D run for identical motors ensemblesare, in general, not equal due to the difference in the term involving p /0 d in equation ( ) . However, a straightforward estimate comparing the effect of this term on(77) and (79) shows that the diffusivities differ at most by 1 / ( µ ∆ T c ) . In this section, we compare theoretical and sample statistics through direct simu-lation of equations (1)-(4) for two motor ensembles. Both homogeneous (kinesin-1/kinesin-1 or kinesin-2/kinesin-2) and heterogeneous (kinesin-1/kinesin-2) en-sembles are simulated, using the parameters in Table 1. For each ensemble, weconsider optical trap forces of F T = − , , and 5 pN. We also considered two sep-arate detachment models. The first utilizes the double exponential function given44igure 4: Switching behavior for a sample path of the kinesin-1/kinesin-2 com-plex simulation under trap force F T = < t = . .
002 seconds before and after detachment. Bottom left: Mo-tor behavior at an attachment event, with kinesin-1 (thick orange) attached at alltimes shown, and kinesin-2 (thin blue) attaching near t = . .
002 seconds before andafter attachment. 45n (8). For comparison, we also use a constant detachment rate set equal to theaverage of the double-exponential detachment rate model against the stationarydistribution of the force F applied by the cargo when the motor in question is theonly one attached with no trap force. Therefore, at zero trap force, both modelshave the same nondimensional effective detachment rates ¯ d ∗ i , but the first modelhas a double exponential function describing the detachment rate as a function offorce while the second model has a force-independent detachment rate. Note that,for the constant detachment rate case, this would also be the detachment rate fromthe state with two motors attached, but would in general differ from the effectivedetachment rate ¯ d ∗ ( i ) , (44) for the double-exponential detachment rate model. Theeffective detachment rates from the state of one motor attached will be higher un-der assisting or opposing trap forces for the double-exponential detachment ratemodel relative to the constant detachment rate model. In all cases, we used con-stant values a ( i ) for attachment rates. Values of parameters related to attachmentand detachment are found in Table 1.The stochastic differential equations (1)-(2) were simulated by an Euler-Maruyamadiscretization with a time increment ∆ t = − s. The random switching was dis-cretized with respect to the same time interval, with probabilities to switch as themomentary rate multiplied by the time step. For stability and accuracy concerns,the time step was selected to be less than the time scale γ m / κ for the drift ofunattached motors (and consequently the larger time scales for attached motors,cargo, and switching dynamics). Our choice of the nondimensional force-velocitycurve g in Eq. (4) is the same used in McKinley, Athreya, et al [56], defined by g ( x ) = A − B tanh ( Cx − D ) , (98)with g ( x ) → . x → − ∞ and g ( x ) → − . x → ∞ . With the requirements g ( ) = g ( ) =
0, the constants A , B , C , and D may be uniquely determinednumerically.We show a sample path of motor and cargo positions in Fig. 4 near times ofattachment and detachment at zero trap force F T =
0. As predicted by the calcu-lation (29) (or the force balance analysis of [39, 38]), the mean cargo position is aweighted average of the position of kinesin-1 and kinesin-2 when both motors areattached. When one motor is attached, the detached motor and cargo both havemean position equal to the current position of the attached motor. Fig. 4 also sug-gests that the transient period for motors and cargo to change their relative posi-tions is small compared to times between attachment or detachment. In particular,we can interpret the jumps in the cargo position at motor detachment events aspotentially corresponding to “flyback” seen in experimental traces [87, 76, 25].46 T = − F T = F T = Simulation Theory Simulation Theory Simulation Theorykinesin-1/kinesin-1 V run ± ± ± V ens ± ± ± D run ±
200 2500 2100 ±
200 2100 3900 ±
300 3400 D ens ±
100 2500 2100 ±
100 2100 3700 ±
200 3400 E [ N c + ] . ± .
03 1.74 1 . ± .
03 1.74 1 . ± .
03 1.74 E [ T ] . ± .
01 .33 . ± . . . ± .
01 .35 E [ L ] ± ± ± kinesin 2/kinesin 2 V run ± ± ± V ens ± ± ± D run ±
100 1000 610 ±
60 630 1600 ±
100 1200 D ens ±
40 1000 630 ±
30 620 1350 ±
60 1220 E [ N c + ] . ± .
08 4.29 4 . ± .
08 4.29 4 . ± .
09 4.29 E [ T ] . ± .
01 .65 . ± . . . ± .
01 .65 E [ L ] ± ± ± kinesin-1/kinesin-2 V run ± ± ± V ens ± ± ± D run ±
400 4400 1800 ±
100 2200 2800 ±
300 2200 D ens ±
200 3700 1800 ±
100 2000 2300 ±
100 2200 E [ N c + ] . ± .
04 2.45 2 . ± .
04 2.45 2 . ± .
04 2.45 E [ T ] . ± .
01 .42 . ± . . . ± .
01 .42 E [ L ] ± ± ± Table 3:
Simulations with constant detachment rate model . The detachmentrates d ( i ) ( F ) are taken to be constants obtained by averaging the double expo-nential detachment rate model (8) against the stationary distribution of the force F when only the motor in question is attached. The columns are organized byapplied trap force F T , with positive (negative) values corresponding to hindering(assisting) forces. The theoretical values are computed according to the formulasfrom Section 5 while the simulated values are obtained from 2,000 Monte Carlosimulations conducted as described in Section 6. Units of time and distance aremeasured in seconds and nanometers, respectively. The means of for the numberof cycles (including the terminal one) N c +
1, run time T , and run length L are esti-mated with the sample mean, where intervals denote the standard error. The errorsin the ensemble and long-run velocities ( V ens and V run ) and diffusivities ( D ens and D run ) are obtained through bootstrap sampling with 1,000 bootstrap samples.47 T = − F T = F T = Simulation Theory Simulation Theory Simulation Theorykinesin-1/kinesin-1 V run ± ± ± V ens ± ± ± D run ±
200 3000 2300 ±
200 2200 3900 ±
200 3500 D ens ±
100 3000 2200 ±
100 2200 3700 ±
100 3500 E [ N c + ] . ± .
01 1.30 1 . ± .
03 1.74 1 . ± .
02 1.44 E [ T ] . ± .
01 .14 . ± . . . ± .
01 .28 E [ L ] ± ± ± kinesin-2/kinesin-2 V run ± ± ± V ens ± ± ± D run ±
100 1300 630 ±
60 690 2600 ±
200 1700 D ens ±
50 1260 650 ±
30 680 2200 ±
100 1600 E [ N c + ] . ± .
05 2.75 4 . ± .
09 4.28 1 . ± .
02 1.47 E [ T ] . ± .
01 .30 . ± . . . ± .
01 .10 E [ L ] ± ± . ± . kinesin-1/kinesin-2 V run ± ± ± V ens ± ± ± D run ±
200 5800 2200 ±
200 2600 2600 ±
200 2200 D ens ±
100 5500 2300 ±
100 2500 2600 ±
100 2300 E [ N c + ] . ± .
03 1.89 2 . ± .
05 2.73 1 . ± .
02 1.76 E [ T ] . ± .
01 .20 . ± . . . ± .
01 .20 E [ L ] ± ± ± Table 4:
Simulations with double exponential detachment rate model . Thedetachment rates d ( i ) ( F ) are given by the double exponential detachment ratemodel (8). The columns are organized by applied trap force F T , with positive(negative) values corresponding to hindering (assisting) forces. The theoreticalvalues are computed according to the formulas from Section 5 while the simulatedvalues are obtained from 2000 Monte Carlo simulations conducted as describedin Section 6. Units of time and distance are measured in seconds and nanome-ters, respectively. The means of for the number of cycles (including the terminalone) N c +
1, run time T , and run length L are estimated with the sample mean,where intervals denote the standard error. The errors in the ensemble and long-run velocities ( V ens and V run ) and diffusivities ( D ens and D run ) are obtained throughbootstrap sampling with 1,000 bootstrap samples.48or each combination of motor ensemble (kinesin-1/kinesin-1, kinesin-2/kinesin-2, or kinesin-1/kinesin-2), detachment model (constant or double exponential),and trap force strength ( F T = − , , or 5 pN), we simulated S = ,
000 runs, eachbeginning with two attached motors and cargo at identical positions along the mi-crotubule, and terminating with complete detachment from the microtubule. Allexperiments have the same initial conditions X ( ) = X ( ) = Z ( ) =
0. However,due to repositioning from force balance, we should expect the cargo to quicklyreadjust to 10 nm for F T = − −
10 nm for F T = T , run lengths L , andnumber of cycles N c , estimates are given by sample means with intervals of thestandard error. The ensemble velocities and diffusions are estimated using the fi-nite S = ,
000 version of the formulas (72) and (73), with errors estimated frombootstrapping. Specifically, from our data Θ = ( T ( j ) , L ( j ) ) j = ,..., S for the run timesand run lengths in the S = ,
000 simulations, we drew B = ,
000 bootstrap sam-ples Θ ∗ b = ( T ( j ) b , L ( j ) b ) j = ,..., S , for each b = , . . . , B , where each ( T ( j ) b , L ( j ) b ) denotesa random sample with replacement from Θ . For V run (74) and D run (75), we use asimilar procedure, with now the dataset Θ thinned to those S / =
200 data pairsassociated to the run lengths L ( j ) in the top decile.Our main concern here will be the question of adequacy of the theoreticalresults based on the asymptotic analysis relative to the Monte Carlo simulations,but we first remark on how the velocities, both theoretical and simulated, can bequite a bit faster than either of the kinesin-1 or kinesin-2 maximum speeds whenthe trap force is assisting ( F T = − F T / γ ∼ × nm / s. Of course we are only considering the cargo during times where one ofthe associated motors is attached to a microtubule, but this indicates the cargo canmove considerably more quickly than the motor speeds when one motor detaches.So what is really causing these large velocities under assisting forces is typicallythat the lagging motor (under stronger force and therefore higher detachment ratesince the cargo is typically ahead of the motors under assisting force) detaches,freeing the cargo to quickly move forward to the new quasi-equilibrium with the49emaining attached motor, and meanwhile the detached motor also quickly movesup to the cargo’s position and reattaches near the cargo’s new position, becomingnow the leading motor. These jumpy adjustments can allow the cargo to move ata large velocity, at least until the cycle is broken by the attached motor detachingbefore the detached motor reattaches.Returning to the central question of the adequacy of the theoretical asymp-totic approximation, we see from Table 3 that the simulations well support thetheoretical approximations for models with constant detachment rates. The diffu-sivities are somewhat underestimated for hindering trap forces, and overestimatedfor the kinesin-1/kinesin-2 ensemble at zero trap force. These issues carry over tothe double exponential detachment rate model in Table 4, with now a substantialoverestimation of diffusivity for the kinesin-1/kinesin-2 ensemble with assistingtrap force. These discrepancies can be traced to order of magnitude errors in someof the second moment cycle statistics (Eqs. (82) through (84)), which are appar-ently more sensitive to the non-ideal scale separation.A more fundamental discrepancy emerges for the kinesin-2/kinesin-2 ensem-ble with hindering trap force, where the mean run length but not the mean runtime is overestimated by a factor of two by the theory, and the velocity similarlyoverestimated. What appears to make this case the most problematic for the the-ory is the failure of the assumption that detachment from the state (1,2) with bothmotors attached takes place on a time scale long compared with that required forthe intermotor separation R to reach its stationary distribution (35). First of all,the nondimensional effective rate of detachment from this state, ¯ d ∗ , = .
24, is thehighest for this case out of all considered, and is therefore the least well separatedfrom the ord ( ) time scale of the relaxation dynamics of the intermotor separa-tion R . Moreover, the force scale of detachment under hindering forces (for bothkinesin-1 and kinesin-2) is about F d + = d + appears not to lead to a substantial violation of the scale separation assumption(Table 1). 50 Discussion and Conclusions
We have developed and analyzed a mathematical model for the transport of cargoby multiple, nonidentical molecular motors along a microtubule. The spatialdynamics are formulated in terms of stochastic differential equations, coarse-graining implicitly over the stepping dynamics of the motors. The process ofdetachment is modeled via a Cox process, in that the detachment rate depends onthe spatial configuration of the motor-cargo complex, which in turn is a randomprocess governed by the stochastic differential equations. Nondimensionalizationrevealed an at least nominal separation of time scales between detached motordynamics, cargo dynamics, attached motor dynamics, and attachment/detachmentprocesses. For the case of two motors attached to a cargo, we exploited this scaleseparation by successive averaging and homogenization procedures to arrive at aneffective continuous-time Markov chain for the attachment states of the two mo-tors, together with random displacements of a cargo tracking variable associatedwith each visit to a state. The cargo tracking variable is just a smoothed repre-sentation of the position of the cargo that has the same long-time dynamics. Thedisplacements of this cargo tracking variable in each state also include jumps as-sociated with attachment and detachment events where the cargo tracking variableadjusts, on a fast time scale, to the new state. We developed analytical formulas forthe effective velocity, diffusivity, and processivity of the cargo by an applicationof the law of large numbers and renewal-reward asymptotics to a decompositionof the coarse-grained Markov chain into regeneration cycles. .Miles, Lawley, and Keener [59] previously pursued in a similar spirit an analy-sis of effective transport and processivity of a cargo with multiple motors attachedvia renewal reward theory. Their procedure, as ours, relies on a separation oftime scales between the continuous dynamics of spatial motion of the motors andcargo and the attachment and detachment kinetics. Our use of the cargo trackingvariable in Subsection 4.1 to provide a representation of the cargo position evenwhen it has been explicitly removed as a fast variable is similar in spirit to thestudy in [59] of the conditional expectation of the cargo (and motor) variablesgiven the attachment/detachment state of the motors. While our methods of anal-ysis share these similarities with [59], our model and analysis does offer several This latter procedure is presented in more generality in a separate publication “Renewal re-ward perspective on linear switching diffusion systems in models of intracellular transport” byM. V Ciocanel, J. Fricks, P. R. Kramer, and S. A. McKinley
Inspection of the effective transport formulas presented in Sections 4 and 5 shows(once redimensionalized) that they do not depend on the friction coefficients γ and γ ( i ) m for the cargo and detached motor dynamics, respectively. These coeffi-cients only need to be small enough that the cargo and detached motor dynamicsare indeed fast relative to the attached motor dynamics so that our separation ofscales arguments are valid; then their precise values are not relevant. On theother hand, the properties of the motor velocities and diffusivities while attachedplay clear roles in the determination of associated statistics for two motor sys-tems. The tether spring constant κ ( i ) plays an important role in setting the forcescale of thermal fluctuations p k B T κ ( i ) , whose ratio to stall force (in the nondi-mensional parameter s ( i ) ) and to detachment force scale (in the nondimensionalparameter u ( i ) ) potentially significantly affect the effective velocities and diffusiv-ities within states (Subsection 4.2), as well as the effective detachment rates (Sub-section 4.3.1). Moreover the magnitude of the jumps of the cargo tracking variableat detachment (Subsection 4.3.2) and attachment events (Subsubsection 4.3.3) isalso sensitive to the value of the tether spring constant.These jumps, which perhaps can be associated to cargo flyback [87, 76, 25],can have a nontrivial impact on the effective transport of ensembles of motors.The mean cargo jump at detachment of a motor is typically nonzero (Eq. (93)),due to relaxation of the cargo to a new equilibrium balancing the applied trap forcewith one tether rather than two, and the preferential detachment of the leading ortrailing motor. But, at least in our model, the cargo jump has mean zero uponreattachment of the second motor (Eqs. (91) and (92)). So in principle, the motor-53argo complex can have a substantial contribution to its effective velocity from aratcheting process in which, from the state of two attached motors, one detaches,allowing the cargo to rapidly adjust by thermal diffusion to a statistical equilibriumwith the attached motor while the detached motor even more quickly equilibratesabout the cargo, then the detached motor reattaches to a relaxed configuration(with no net mean cargo position change), and the two attached motors moveagain toward a more strained configuration leading to motor detachment. Thisphenomenon caused a speedup in our model of the motor-cargo complex underan assisting trap force which in some cases exceeded the maximum single motorvelocity.We plan to examine this flyback effect with more biophysical detail in futurework. Our primary goal in this work has been to set out a mathematical framework forrelating the various physicochemical properties of dissimilar cooperative motorson their effective transport as a team. While we have endeavored to parameter-ize our models for kinesin-1 and kinesin-2 with experimentally-based values, wemust note a few issues in this parameterization that require further study beforewe can meaningfully confront our model predictions to experimental data on mul-tiple motor transport. First of all, our parameters from Table 1 are all from in vitromeasurements. As noted as well in McKinley, Athreya, et al [56], the viscosity incell is substantially higher than water, and this can make the scale separation as-sumption between cargo and attached motor dynamics (small ε ( i ) in Table 2) lesstenable. Other biophysical parameters may also have different values in cell [45],though these are even more difficult to establish than their in vitro counterparts.Thus, our focus remains for now on targeting our mathematical framework towardunderstanding and interpreting in vitro observations.At least two parameterization problems require substantial development be-fore this can be credibly attempted, though. First of all, to keep focus on thevarious coarse-graining relationships we employed, we adopted in the main textand in the simulations a simple Hookean spring model for the motor-cargo tether,with a spring constant obtained from experimental observations of motor-cargosystems with the tether pulled to its natural extension, This linear spring approx-imation is reasonable for describing small fluctuations of the extended tether, butdoes not at all model the tether well when its end-to-end separation is smaller thanits natural extension. A simple general model used in biophysical simulations of54inesin is to have a linear restoring force under extension from a rest length (vary-ing from 40 nm in [3] to 80 nm in [36] to 110 nm in [44, 24]), but no resistance tocompression; more complex nonlinear models for extension have also been con-sidered [88, 30, 20]. Using a fully Hookean model with zero rest length and theexperimentally measured linear spring constant (shown in Table 1) leads to an ab-surd conclusion that the root-mean-square extension of the tether for kinesin-1 isabout 5 nm, when of course it should be more like 70 nm. The 5 nm is really anestimate for the magnitude of the fluctuations about this rest length when forceon the tether pulls it approximately taut. During a given phase of attachment,this neglect of the rest length of the motor-cargo tether does not necessarily havea strong impact on the calculations – we can imagine the cargo is in fact just ap-proximately this rest length behind the nominal cargo position Z ( t ) and the motorswould feel generally comparable forces as in the simple Hookean model with zerorest length which we used. The big difference, though, would be on the dynamicsof the detached motors, which should be fluctuating over a distance comparableto the motor-cargo tether rest length rather than the nominal root-mean-square ex-tension of the Hookean spring model with zero rest length. This would have abig impact on where the detached motors reattach on the microtubule. Thus, theHookean spring model with zero rest length for the motor-cargo tether cannot beexpected to give useful predictions for the transport of actual molecular motors;we must at least extend it to a nonlinear model with no resistance to compressionbelow a finite rest length as in [36, 46, 85, 3, 24, 26]. In Appendix A, we indicatehow nonlinear spring models for the motor-cargo tether can be handled by ourmathematical framework – the main complication is the jumps of the cargo track-ing variable in Section 4.3 become non-Gaussian. We have here stayed with thepurely linear spring model in the main text to minimize technical complicationsand more clearly illustrate the key concepts in the mathematical coarse-grainingof the dynamics of a system of cooperative dissimilar motors. In future work, wewill adapt this framework to more biophysically accurate nonlinear spring mod-els for the motor-cargo tether, and examine through this lens various hypothesesand experimental observations regarding the dynamics of cooperative molecularmotors.A second parameterization problem is that of the double exponential force-detachment rate relation (8). Our model is based on a relation between detachmentrate and instantaneous force, while what is measured [60, 61, 37] is the relationbetween run length and applied trap force. We convert the run length to a detach-ment rate by dividing by the unloaded velocity, which is a bit crude but arguablya reasonable rough approximation, but the more serious concern is equating the55ependence on applied trap force with the dependence on instantaneous force feltby the motor via its tether to the cargo. This leads to particular peculiarities, notedin Section 6, in the absence of trap force, since the force-detachment rate modelis discontinuous at zero force. So, in our model, the motor at zero trap force isfluctuating between high detachment rates when the cargo is pulling the motorforward and low detachment rates when the cargo is pulling the motor backward,leading to an inappropriately augmented detachment rate. Rectifying this detach-ment rate model requires, in future work, a better statistical inference approach fora relation between instantaneous force and detachment rate that would replicate,under our model, the relation between run length and applied trap force reportedin [60, 61]. A further potential improvement would be to incorporate the findingsof a recent study [37] suggesting a different structure for the dependence of thedetachment rate against a truly longitudinal applied force on kinesin-1. A mathematical question for future work is to examine whether the coarse-grainedmodel developed here would change if we started with a discrete stepping model [10,18, 19, 33, 93, 22, 41, 21, 36] for the motor dynamics, rather than the coarse-grained stochastic differential equation model (1) used here. This is essentially aquestion of how the coarse-graining of a jump process model for the motor dy-namics into a stochastic differential equation interacts with the coarse-grainingprocedures developed here. Perhaps the cargo fluctuations interacting with thediscrete stepping model would give rise to a different effective dynamics (20) forthe attached motors.Another question is how our detailed analysis of effective transport of an en-semble of two cooperative motors can be extended to more general scenarios ofmultiple motors attached to a cargo. The initial coarse-graining steps over the de-tached motor and cargo dynamics in Section 3 applied to an arbitrary number N of cooperative motors, only at the cost of complexity, but the coarse-graining overthe attached motor dynamics in Section 4 relied on the ability to homogenize theattached motor dynamics for the case N = R between the motors.For N >
2, we would have N − cknowledgements: This work grew out of conversations between JF andPRK while both were long term visitors at the Isaac Newton Institute for theprogram on “Stochastic Dynamics in Biology: Numerical Methods and Appli-cations.” The authors also would like to thank Will Hancock for discussionsinforming the model development. PRK would also like to acknowledge earlydiscussions with Leonid Bogachev, Leonid Koralov, and Yuri Makhnovskii whichhelped me map out the general mathematical framework and approaches, whilewe were all supported as long-term visitors at the Zentrum f¨ur Interdisziplin¨areForschung for the program on “Stochastic Dynamics: Mathematical Theory andApplications.” Here we’ve managed to work out one of the easier cases we con-sidered. . .
Funding
The work of JF and PRK are partially supported by National Insti-tutes of Health grant R01GM122082 and PRK was partially supported by a grantfrom the Simons foundation. The work of JK is partially supported by an NationalScience Foundation RTG grant 1344962.
A Appendix: A note on nonlinear spring models
A linear model for representing the tether between the motor and the cargo is notparticularly accurate. Better theoretical tether models can involve a model whichis Hookean for extension beyond a rest length, but offers no resistance to compres-sion [36, 46, 85, 3, 24, 26], a sigmoidal stiffness dependence on force [88], or amultiple-component model for the motor-cargo tether including separate modelsfor the neck linker and stalk [30]. We may generalize the averaging results byconsidering a nonlinear spring F ( i ) ( r ) = ¯ F ( i ) Φ ′ ( i ) ( r / L ( i ) c ) ≤ i ≤ N . (99)Here Φ ( i ) ( r ) is a nondimensionalized spring potential, L ( i ) c is an appropriate lengthscale, and ¯ F ( i ) is a characteristic force magnitude for each motor index i . Wedefine κ ( i ) ≡ ¯ F ( i ) / L ( i ) c as an “effective” spring constant of the nonlinear spring;it agrees with the usual spring constant when the spring force model is purelyHookean, as in the main text.With the same nondimensionalization as before, the equations of motion be-58ome, for 1 ≤ i ≤ N , d ˜ X ( i ) ( ˜ t ) = (cid:18) ε ( i ) g ( s ( i ) ˜ κ ( i ) ( λ ( i ) ) − Φ ′ ( i ) ( λ ( i ) ( ˜ X ( i ) ( ˜ t ) − ˜ Z ( ˜ t )))) d ˜ t + q ˆ ρ ( i ) ε ( i ) dW ( i ) ( ˜ t ) (cid:19) Q ( i ) ( ˜ t ) (100) + (cid:18) − (cid:16) Γ ( i ) λ ( i ) (cid:17) − ˜ κ ( i ) Φ ′ ( i ) (cid:16) λ ( i ) (cid:16) ˜ X ( i ) ( ˜ t ) − ˜ Z ( ˜ t ) (cid:17)(cid:17) d ˜ t + ( Γ ( i ) ) − / dW ( i ) ( ˜ t ) (cid:19) ( − Q ( i ) ( ˜ t )) , d ˜ Z ( ˜ t ) = N ∑ j = ( λ ( i ) ) − ˜ κ ( i ) Φ ′ ( λ ( i ) ( ˜ X ( j ) ( ˜ t ) − ˜ Z ( ˜ t ))) − ˜ F T ! d ˜ t + dW z ( ˜ t ) , (101)where we have introduced the new nondimensional parameter λ ( i ) = p k B T / ¯ κ L ( i ) c . (102)which describes the length-scale of thermally induced variations on the tether rel-ative to the length scale of variation of the tether force law. Calculations for theaveraged drifts ¯ g ( i ) , and thus G + and G − , are similar, but now involve pairingdrift functions with non-Gaussian stationary distributions for unattached motorsand cargo (the forms for these equations are nearly identical to those found inAppendix A in McKinley, Athreya, et al [56]). For detachment jumps, no as-sumptions of distribution type are made for p R ( r ) , and therefore the calculationsfor distributions in Subsection 4.3.2 only need to be adjusted to refer to the meancargo position under nonlinear tethers.The random variable ∆ M ( a ) i describing the change of position of the cargotracking variable after motor attachment may be computed similarly as in Sub-section 4.3.3, but it will no longer be normally distributed or have mean zero ingeneral. The calculations in Section 5 otherwise go through for a nonlinear tethermodel, with only the modified contribution to the moments of the cargo trackingvariable changes at attachment and detachment jumps. B Appendix: Derivation of effective diffusion for twomotors
The following derivation for the effective diffusion of the cargo tracking variableduring a state with both motors attached to the microtubule follows the multiscale59xpansion method illustrated in Pavliotis and Stuart [69], with rigorous expositionin Veretennikov and Pardoux [91] for the unbounded state space case relevantto our application. Having computed the effective drift V , in Eq. (34) in thisstate, we pass to a diffusive scaling centered about this mean drift ¯ t → t / ε , M → ε ( M − V , t ) , with the internal configuration variable R unscaled ( R → R ). Notein this appendix, ε is just a formal small parameter used to push to long time; itis unrelated to the physically meaningful nondimensional parameters ε ( i ) and ¯ ε inthe main text. Moreover, for simplicity for calculations within this appendix, weuse the undecorated variables t , M , R to describe the dynamics under this centereddiffusive rescaling, which read: dM ( t ) = ε ( G + ( R ( t )) − V , ) dt + p ρ ( ) ( ˜ κ ( ) ) dW ( ) ( t ) + p ρ ( ) ( ˜ κ ( ) ) dW ( ) ( t ) (103) dR ( t ) = ε G − ( R ( t )) dt + ε (cid:18)q ρ ( ) dW ( ) ( t ) − q ρ ( ) dW ( ) ( t ) (cid:19) . (104)The infinitesimal generator L for (103)-(104) is defined by its action on a testfunction v = v ( m , r ) , with L v ( m , r ) = h · ∇ v + Γ : ∇∇ v . (105)Here h ( m , r ) = (( G + ( r ) − V , ) / ε , G − ( r ) / ε ) is the drift vector, and Γ is thediffusion tensor, where Γ = √ ρ ( ) ( ˜ κ ( ) ) √ ρ ( ) ( ˜ κ ( ) ) √ ρ ( ) ε − √ ρ ( ) ε √ ρ ( ) ( ˜ κ ( ) ) √ ρ ( ) ( ˜ κ ( ) ) √ ρ ( ) ε − √ ρ ( ) ε T (106) = " ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ερ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ε ρ ( ) + ρ ( ) ε . (107)We have also used notation for the Frobenius inner product for matrices, wherefor matrices A = ( a i , j ) nxm and B = ( b i , j ) nxm , we define A : B = ∑ i , j a i , j b i , j .60e may write out (105) explicitly as L v ( m , r ) = h · ∇ v + Γ : ∇∇ v (108) = ε ( G + ( r ) − V , ) v m + ε G − ( r ) v r (109) + " ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) ! v mm + ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ε v mr + ρ ( ) + ρ ( ) ε ! v rr . (110)The generator may be decomposed to match powers of ε as L = ε L + ε L + L , (111)with L = G − ( r ) ∂ r + ρ ( ) + ρ ( ) ∂ rr , (112) L = ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ∂ mr + ( G + ( r ) − V , ) ∂ m , (113) L = ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) ! ∂ mm . (114)Assuming a multiscale solution v = v + ε v + ε v + . . . for the backward Kol-mogorov equation ∂ v ∂ t = L v , (115)we match powers of orders 1 / ε , / ε , and 1 to obtain L v = , (116) − L v = L v , (117) − L v = − ∂ v ∂ t + L v + L v . (118)The first equation implies that v is only a function of m and t . From here, thesecond equation may be simplified to − L v = ( G + ( r ) − V , ) ∂ m v ( m , t ) . (119)61s the operator L only depends on r , we may express v as v ( m , r , t ) = χ ( r ) ∂ m v ( m , t ) . (120)Proceeding to the third equation of the asymptotic expansion, the Fredholm alter-native states that for (118) to have a solution, its right hand side must be orthogonalto p R ( r ) , or ∂ v ∂ t = Z R p R ( r ) L v ( m , t ) dr + Z R p R ( r ) L ( χ ( r ) ∂ m v ( m , t )) dr (121): = I + I . (122)We look at each integral in turn. First, I = Z R p R ( r ) L v ( m , t ) dr (123) = Z R p R ( r ) " ( G + ( r ) − V , ) ∂ m v ( m , t ) + ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) ! ∂ mm v ( m , t ) dr (124) = ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) ! ∂ mm v ( m , t ) . (125)The second integral may be broken up further, as I = Z R p R ( r ) L ( χ ( r ) ∂ m v ( m , t )) dr (126) = Z R p R ( r ) h ρ ( ) κ ( ) − ρ ( ) κ ( ) ! ∂ mr ( χ ( r ) ∂ m v ( m , t )) (127) + ( G + ( r ) − V , ) ∂ m ( χ ( r ) ∂ m v ( m , t )) i dr (128): = I + I . (129)The first part satisfies I = Z R p R ( r ) ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ! ∂ mr ( χ ( r ) ∂ m v ( m , t )) dr (130) = ρ ( ) ˜ κ ( ) − ρ ( ) ˜ κ ( ) ! Z R p R ( r ) ∂ r χ ( r ) dr ! ∂ mm v ( m , t ) (131): = A ∂ mm v ( m , t ) . (132)62inally, we have I = Z R p R ( r ) [( G + ( r ) − V , ) ∂ m ( χ ( r ) ∂ m v ( m , t ))] dr (133) = Z R p R ( r ) [( G + ( r ) − V , ) χ ( r )] dr ∂ mm v ( m , t ) (134): = A ∂ mm v ( m , t ) . (135)The closed form equation for v ( m , t ) is thus given by ∂ v ∂ t = ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) + A + A ! ∂ mm v ( m , t ) , (136)and is the backward Kolmogorov equation for the SDE dM ( t ) = s ρ ( ) ( ˜ κ ( ) ) + ρ ( ) ( ˜ κ ( ) ) + A + A dW ( t ) (137) ≡ p D ( , ) dW ( t ) , (138)where W ( t ) is a standard Brownian motion.Now we compute explicit expressions for constants A and A . This involvessolving the cell problem for χ , given by − G − ( r ) χ ′ ( r ) − ρ ( ) + ρ ( ) ! χ ′′ ( r ) = ˜ g + ( r ) , (139) Z R χ ( r ) p R ( r ) dr = , (140)where we define ˜ g + ( r ) = G + ( r ) − V , . If we rewrite (139), using an integrationfactor µ ( r ) , as [ µ ( r ) χ ′ ( r )] ′ = − µ ( r ) ˜ g + ( r ) (cid:18) ρ ( ) + ρ ( ) (cid:19) , (141)then it is straightforward to show that µ ( r ) is in fact equal to the stationary distri-bution p R ( r ) .Integrating out (141) leaves us with χ ′ ( r ) = − Z r − ∞ ˜ g + ( r ′ ) (cid:18) ρ ( ) + ρ ( ) (cid:19) p R ( r ′ ) dr ′ / p R ( r ) + C / p R ( r ) (142)63or some unknown integration constant C . By the subexponential growth require-ment on χ and χ ′ [91], it follows that C =
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