Analysis of non-processive molecular motor transport using renewal reward theory
AANALYSIS OF NON-PROCESSIVE MOLECULAR MOTORTRANSPORT USING RENEWAL REWARD THEORY ∗ CHRISTOPHER E. MILES † , SEAN D. LAWLEY ‡ , AND
JAMES P. KEENER § Abstract.
We propose and analyze a mathematical model of cargo transport by non-processivemolecular motors. In our model, the motors change states by random discrete events (correspondingto stepping and binding/unbinding), while the cargo position follows a stochastic differential equation(SDE) that depends on the discrete states of the motors. The resulting system for the cargo positionis consequently an SDE that randomly switches according to a Markov jump process governing motordynamics. To study this system we (1) cast the cargo position in a renewal theory framework andgeneralize the renewal reward theorem and (2) decompose the continuous and discrete sources ofstochasticity and exploit a resulting pair of disparate timescales. With these mathematical tools,we obtain explicit formulas for experimentally measurable quantities, such as cargo velocity and runlength. Analyzing these formulas then yields some predictions regarding so-called non-processiveclustering, the phenomenon that a single motor cannot transport cargo, but that two or more motorscan. We find that having motor stepping, binding, and unbinding rates depend on the number ofbound motors, due to geometric effects, is necessary and sufficient to explain recent experimentaldata on non-processive motors.
Key words. molecular motors, intracellular transport, reward renewal theory, stochastic hybridsystems, switching SDEs
AMS subject classifications.
1. Introduction.
Active intracellular transport of cargo (such as organelles) iscritical to cellular function. The primary type of active transport involves molecularmotors, which alternate between epochs of active transport (discrete stepping) alonga microtubule and epochs of passive diffusion when the motors are unbound from themicrotubule. Stochastic modeling of this fundamental process has a rich and fruitfulhistory (see the review [5]).In both experimental and modeling studies, processive motors have received con-siderable attention. Processive motors are characterized by taking hundreds of stepsalong a microtubule before unbinding. In contrast, non-processive motors (such asmost members of the kinesin-14 family) take very few (1 to 5) steps before unbind-ing from a microtubule [6, 11]. Non-processive motors are crucial to a number ofcellular processes, including directing cytoskeletal filaments [42], driving microtubule-microtubule sliding during mitosis [14], and retrograde transport along microtubulesin plants [48]. Here, we focus on motor behavior during transport.Some curious properties of non-processive motor transport were found in [16].
One non-processive (Ncd) motor has extremely limited transport ability, measured byboth velocity and run length (distance traveled before detaching from a microtubule).However, two non-processive motors somehow act in unison to produce significantdirected motion, a phenomenon termed “clustering.” This observation is supportedby the subsequent studies [26, 38], where similar experiments were performed creat- ∗ Submitted to the editors September 28, 2018.
Funding:
This work was supported by NSF grant DMS-RTG 1148230. CEM and JPK werealso supported by NSF grant DMS 1515130. SDL was also supported by NSF grant DMS 1814832. † Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA([email protected]). ‡ Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA ([email protected]). § Departments of Mathematics and Bioengineering, University of Utah, Salt Lake City, UT 84112USA ([email protected]). 1 a r X i v : . [ q - b i o . S C ] S e p CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER ing a mutant of attached non-processive kinesin-14 motors, and processivity emerges.Moreover, the authors of [16] note that adding more Ncd motors beyond two furtherincreases transport ability. In contrast, one processive motor (kinesin-1) is sufficientto produce transport, and additional motors do not significantly increase transportability [16, 45]. Other interesting facets of transport by non-processive motors in-clude the emergence of processive transport in the presence of higher microtubuleconcentration [17] or opposing motors [22].In this work, we formulate and analyze a mathematical model to investigate thenatural question: how do non-processive motors cooperate to transport cargo? Ourmodel predicts that non-processive motor stepping, binding, and unbinding ratesmust depend on the number of bound motors, and that this dependence is a keymechanism driving the collective transport of non-processive motors. We note thatsuch dependence has been observed in experiments [13, 20] and in simulations ofdetailed computational models [30, 12, 35, 17], all stemming from geometric effects ofcargo/motor configuration.Non-processive motors are notoriously difficult to study experimentally, becausethey take only a few steps before detaching. For this same reason, it is not clear howto best model non-processive motors, or known if existing modeling frameworks, suchas mean-field methods [27, 9] or averaging the stepping dynamics into an effective ve-locity [36], are appropriate. Hence, our model explicitly includes the discrete binding,unbinding, and stepping dynamics of each motor, as well as the continuous tetheredmotion of the cargo.Mathematically, our model takes the form of a randomly switching stochastic dif-ferential equation (SDE), and thus merges continuous dynamics with discrete events.The continuous SDE dynamics track the cargo position, while the discrete events cor-respond to motor binding, unbinding, and stepping. Our model is thus a stochastichybrid system [4], which are often two-component processes, ( J ( t ) , X ( t )) t ≥ ∈ I × R d ,where J is a Markov jump process on a finite set I , and X evolves continuously byd X ( t ) = F J ( t ) ( X ( t )) d t + σ d W ( t ) , (1)where { F j ( x ) } j ∈I is a given finite family of vector fields, σ ≥
0, and W is a Brownianmotion. That is, X follows an SDE whose righthand side switches according to theprocess J .However, our model differs from most previous hybrid systems in some key ways.First, the set of possible continuous dynamics (e.g. possible righthand sides of (1)) forour model is infinite. Second, the new righthand side of (1) that is chosen when J jumps depends on the value of X at that jump time, although the rates dictating J are taken to be independent of X .We employ several techniques to analyze our model and make predictions regard-ing non-processive motor transport. First, we cast our model in a renewal theoryframework, and generalize the classical renewal reward theorem [47] to apply to oursetting, distinct from previous motor applications [31, 23, 24, 25, 44]. Next, we decom-pose the stochasticity in the system by averaging over the diffusion while conditioningon a realization of the jump process. This effectively turns the randomly switchingSDE into a randomly switching ordinary differential equation (ODE), and thus apiecewise deterministic Markov process [8]. Finally, we observe that for biologicallyreasonable parameter values, the relaxation rate of the continuous cargo dynamics ismuch faster than the jump rates for the discrete motor behavior. We then exploitthis timescale separation to find explicit formulas for key motor transport statistics. ON-PROCESSIVE MOTOR TRANSPORT δ k off k step k on X ( t ) Z ( t ) Fig. 1 . Schematic describing the binding, unbinding, and stepping of motors. The positions ofcargo and bound motors are X ( t ) and Z i ( t ) , respectively, both measured with respect to the principalaxis of the microtubule. The state of the motor can switch between bound or unbound, and whilebound, the motor can step, incrementing Z i ( t ) by displacement δ . The rest of the paper is organized as follows. We formulate the mathematicalmodel in section 2. In section 3, we generalize the renewal reward theorem to applyto our model. In section 4, we derive explicit formulas to evaluate motor transport.In section 5, we use the model to make biological predictions. We conclude with abrief discussion and an Appendix that collects several proofs.
2. Mathematical model.
We model the motion of a single cargo driven by M ≥ t ≥
0, the state of our model is specified by (cid:0) X ( t ) , Z ( t ) , J ( t ) (cid:1) ∈ R × R M × { u, b } M , where X ( t ) ∈ R is the location of the center of the cargo, Z ( t ) = ( Z i ( t )) Mi =1 ∈ R M gives the locations of the centers of M motors, and J ( t ) = ( J i ( t )) Mi =1 specifies if eachmotor is unbound or bound. Spatial locations are measured along the principal axisof the microtubule, which we identify with the real line.The cargo position evolves continuously in time, while the positions and states ofmotors change by discrete events, which correspond to binding to the microtubule,stepping along the microtubule, or unbinding from the microtubule. Specifically,in between these discrete motor events, X ( t ) follows an Ornstein-Uhlenbeck (OU)process centered at the average bound motor position,d X ( t ) = kγ (cid:88) i ∈ I ( t ) (cid:0) Z i ( t ) − X ( t ) (cid:1) d t + (cid:112) k B T /γ d W ( t ) . (2)Here, I ( t ) = { i : J i ( t ) = b } ⊆ { , . . . , M } gives the indices of motors that are boundat time t ≥
0, and { W ( t ) } t ≥ is a standard Brownian motion. The SDE (2) stemsfrom assuming a viscous (low Reynolds number) regime with drag coefficient γ > k > k B T /γ , where k B isBoltzmann’s constant and T is the absolute temperature.The discrete behavior of motors is as follows. Let m ( t ) ∈ { , , . . . , M } denote CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER the number of bound motors at time t ≥ m ( t ) = M (cid:88) i =1 { J i ( t ) (cid:54) = u } ∈ { , , . . . , M } , where 1 { A } denotes the indicator function on an event A . Each unbound motor bindsto the microtubule at rate k on ( m ( t )) >
0. Since unbound motors are tethered to thecargo, if an unbound motor binds at time t ≥
0, then we assume that it binds tothe track at X ( t ) (motors can bind anywhere along the microtubule, not only bindingsites). We could allow it to bind to a random position, but if the mean binding positionis X ( t ), then our results are unchanged. The position of each bound motor is fixeduntil it either steps or unbinds. Each bound motor unbinds at rate k off ( m ( t )) > k step ( m ( t )) >
0. When a motor steps, we add δ > m ( t ), but are otherwiseindependent of X ( t ). We now give a dimension-less and more precise formulation of the model described above. First, we nondimen-sionalize the model by rescaling time by the rate k off (1) and space by the inverse length δ − . Next, we note that unbound motors do not affect the cargo position. Hence, forconvenience we can take Z i ( t ) = X ( t ) if the i -th motor is unbound, meaning that wecan include unbound motors in the sum in (2) with zero contribution, and make thesum over all motors. This yields the simplified dimensionless formd X ( t ) = ε − M (cid:88) i =1 (cid:0) Z i ( t ) − X ( t ) (cid:1) d t + σ d W ( t ) , (3)where ε := k off (1) γ/k, σ := (cid:112) k B T / ( δ k off (1) γ ) , and motors bind, unbind, and step at dimensionless rates λ on ( m ) := k on ( m ) k off (1) , λ off ( m ) := k off ( m ) k off (1) , λ step ( m ) := k step ( m ) k off (1) . (4) We find it convenient for our analysis to track the number of steps taken by eachmotor before unbinding, so let us expand the state space of J ( t ) so that its components( J i ( t )) Mi =1 each take values in { u, , , , . . . } with transition rates u λ on ( m ( t )) → , j λ step ( m ( t )) → j + 1 , j λ off ( m ( t )) → u, j (cid:54) = u. (5)The components of J ( t ) are conditionally independent given m ( t ). At time t ≥ i -th motor is unbound if J i ( t ) = u , bound if J i ( t ) ≥
0, and steps when J i ( t )transitions from j to j + 1 for j ≥
0. Under these assumptions, m ( t ) is itself a Markovprocess on { , , . . . , M } with transition rates(6) 0 Mλ on (0) (cid:10) ( M − λ on (1) (cid:10) λ off (2) (cid:10) · · · (cid:10) M − λ on ( M − (cid:10) ( M − λ off ( M − M − λ on ( M − (cid:10) Mλ off ( M ) M. ON-PROCESSIVE MOTOR TRANSPORT J i (0) = u, X (0) = Z i (0) = 0 , i ∈ { , . . . , M } . The position of the i -th motor is then Z i ( t ) = (cid:0) X ( τ i ( t )) + J i ( t ) (cid:1) { J i ( t ) (cid:54) = u } + X ( t )1 { J i ( t )= u } , i ∈ { , . . . , M } , (7)where τ i ( t ) is the most recent binding time of the i -th motor, τ i ( t ) = sup { s < t : J i ( s ) = u } , i ∈ { , . . . , M } . We assume the Brownian motion W = { W ( t ) } t ≥ and the jump process J = { J ( t ) } t ≥ are independent.
3. Cargo position as a renewal reward process.
In order to analyze ourmodel, we first show that X ( t ) is a renewal reward process with partial rewards [47]and extend the classical renewal reward theorem to our case of partial rewards. Thisframework has an intuitive interpretation: the net displacement of cargo is determinedby the displacement accrued at each epoch of being bound or unbound. However, thereis a technical challenge. In the most classical setting, the reward-renewal theoremaccrues rewards at the end of each epoch and boundedness of expectation of therewards is sufficient to apply the reward-renewal theorem. In the case of partialrewards (which we have in our model), where rewards are accrued during an epoch,stronger conditions are required, which we prove are satisfied.First, define the sequence of times in which the cargo completely detaches fromthe microtubule (off) and subsequently reattaches to the microtubule (on),0 = τ = τ < τ < τ < τ < τ < . . . by τ k off := inf { t > τ k on : m ( t ) = 0 } , k ≥ ,τ k on := inf { t > τ k − : m ( t ) ≥ } , k ≥ . (8)Next, define the sequence of cargo displacements when the cargo is attached to themicrotubule (on) and detached from the microtubule (off), R k on := X ( τ k off ) − X ( τ k on ) , R k off := X ( τ k on ) − X ( τ k − ) , k ≥ , (9)and the corresponding times spent attached or detached, T k on := τ k off − τ k on , T k off := τ k on − τ k − , k ≥ . (10)It follows directly from the strong Markov property that { ( T k off + T k on , R k off + R k on ) } k ≥ is an independent and identically distributed (iid) sequence of random variables.In the language of renewal theory, { T k off + T k on } k ≥ are the interarrival times and { R k off + R k on } k ≥ are the corresponding rewards. Let N ( t ) be the renewal process thatcounts the number of arrivals before time t ≥ N ( t ) := sup { k ≥ τ k off ≤ t } . (11) CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
Define the reward function, R ( t ), and the partial reward function, Y ( t ), by R ( t ) := N ( t ) (cid:88) k =1 ( R k on + R k off ) , Y ( t ) := X ( t ) − X ( τ N ( t )off ) , (12)and observe that X ( t ) = R ( t ) + Y ( t ) . In words, R ( t ) describes rewards accrued during past epochs, and Y ( t ) is the partialreward accrued during the current epoch. We show below that E [ | R on + R off | ] < ∞ and E [ T on + T off ] < ∞ , and therefore the classical renewal reward theorem [47] ensuresthat lim t →∞ R ( t ) t = lim t →∞ E [ R ( t )] t = E [ R on ] + E [ R off ] E [ T on ] + E [ T off ] almost surely . (13)The following theorem verifies that this convergence actually holds for X ( t ). Theorem The following limit holds, lim t →∞ X ( t ) t = lim t →∞ E [ X ( t )] t = E [ R on ] + E [ R off ] E [ T on ] + E [ T off ] almost surely . (14)To prove this theorem, we need several lemmas. We collect the proofs of theselemmas in Appendix A. The first lemma bounds the probability that the partial rewardfunction Y ( t ) in (12) is large when the cargo is detached from the microtubule. Lemma Define the sequence of iid random variables { Y k off } k ≥ by Y k off := sup t ∈ [ τ k − off ,τ k on ] (cid:12)(cid:12) X ( t ) − X ( τ k off ) (cid:12)(cid:12) , k ≥ . Then for any
C > and k ≥ , we have that P ( Y k off ≥ C ) ≤ (cid:112) π/x (2 x + 1) e − x , where x = Cσ (cid:112) M λ on (0) > . (15)Similarly, the next lemma bounds the probability that the partial reward functionis large when the cargo is attached to the microtubule. Lemma Define the sequence of iid random variables { Y k on } k ≥ by Y k on := sup t ∈ [ τ k on ,τ k off ] | X ( t ) − X ( τ k on ) | , k ≥ . There exists λ > , λ > so that if k ≥ , then P ( Y k on ≥ C ) ≤ λ √ Ce − λ √ C for allsufficiently large C > . The next lemma uses Lemmas 2 and 3 to prove that the partial reward functiongets large only finitely many times.
Lemma Define the sequence of iid random variables { Y k } k ≥ by Y k := sup t ∈ [ τ k − off ,τ k off ] (cid:12)(cid:12) X ( t ) − X ( τ k off ) (cid:12)(cid:12) , k ≥ . (16) Then P (cid:16) lim K →∞ ∞ (cid:91) k = K (cid:8) Y k > √ k (cid:9)(cid:17) = 0 . ON-PROCESSIVE MOTOR TRANSPORT Y k in (16) is finite. Lemma Define { Y k } k ≥ as in (16). Then E [ Y k ] < ∞ for all k ≥ . With these lemmas in place, we are ready to prove Theorem 1.
Proof of Theorem 1.
It follows immediately from Lemma 5 that E [ | R k on + R k off | ] < ∞ . Furthermore, T k off is exponentially distributed with rate M λ on (0), and the proof ofLemma 3 shows that E [ T k on ] < E [ S ] for an exponentially distributed random variable S with some rate λ >
0. Hence, E [ T k on + T k off ] < ∞ , and thus (13) holds by a directapplication of the classical renewal reward theorem [47].Therefore, it remains to check thatlim t →∞ E (cid:2) X ( t ) − X ( τ N ( t )off ) (cid:3) t = 0 = lim t →∞ X ( t ) − X ( τ N ( t )off ) t , almost surely . (17)The first equality in (17) follows immediately from Lemma 5.To verify the second equality in (17), we note that Lemma 4 ensures thatlim sup k →∞ Y k √ k ≤ , almost surely . Therefore,lim t →∞ | X ( t ) − X ( τ N ( t )off ) | t ≤ lim t →∞ | Y N ( t ) | t ≤ lim t →∞ (cid:112) N ( t ) t = 0 , almost surely , since lim t →∞ N ( t ) t = 1 E [ T on + T off ] , almost surely , by the strong law of large numbers for renewal processes [47].Consequently, the position of the cargo does indeed satisfy a classical reward-renewalstructure with two different types of epochs: bound and unbound, each of whichaccrue some net displacement.
4. Mathematical analysis of transport ability.
With the framework of re-newal theory constructed in section 3, we are ready to analyze the transport ability ofthe model introduced in section 2. To assess the transport ability of the motor cargoensemble, we analyze the expected run length , expected run time , and asymptotic velocity . We define the run length to be the distance traveled by the cargo betweenthe first time a motor attaches to the cargo until the next time that all motors aredetached from the microtubule, which was defined precisely in (9) and denoted by R on . The run time is the corresponding time spent attached to a microtubule, whichwas defined precisely in (10) and denoted by T on . The asymptotic velocity is(18) V := lim t →∞ X ( t ) t . The velocity V includes both the time the cargo is being transported along the mi-crotubule and diffusing while unattached.Applying Theorem 1, we have that V = E [ R on ] + E [ R off ] E [ T on ] + E [ T off ] almost surely . (19) CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
Now, E [ R off ] = 0 , (20)since the cargo is freely diffusing when no motors are bound, and since motor bindingand unbinding is independent of Brownian motion { W ( t ) } t ≥ . Furthermore, when allof the M motors are unbound, each motor binds at rate λ on (0). Hence, E [ T off ] = ( M λ on (0)) − . (21)It therefore remains to calculate two of the three quantities, V , E [ R on ], and E [ T on ],since the third is given by (19). We calculate E [ T on ] first since it is the simplest, as itis a mean first passage time of a continuous-time Markov chain. As we noted in section 2.1, the number of motorsbound m ( t ) is itself the Markov process (6). To compute the expected run time, wecompute the mean time for m ( t ) to reach state m = 0 starting from m (0) = 1.Let (cid:101) Q ∈ R ( M +1) × ( M +1) be the generator of the Markov chain m ( t ) in (6). Thatis, the ( i, j )-entry of (cid:101) Q gives the rate that m ( t ) jumps from state i to state j (cid:54) = i ,and the diagonal entries are chosen so that (cid:101) Q has zero row sums. Let Q ∈ R M × M bethe matrix obtained from deleting the first row and column of (cid:101) Q . The matrix Q istridiagonal, with the m -th row containing subdiagonal, diagonal, and superdiagonalentries mλ off ( m ), − ( mλ off ( m ) + ( M − m ) λ on ( m )), ( M − m ) λ on ( m ), respectively. Theexpected run time E [ T on ] is (by Theorem 3.3.3 in [40]),(22) E [ T on ] = T t , where Q T t = − e , where ∈ R M is the vector of all 1’s and e ∈ R M is the standard basis vector. Having calculated E [ T on ] in (22), we candetermine V by determining E [ R on ] (or vice versa). Two key steps allow us to analyze V and E [ R on ]: (i) we average over the diffusive dynamics while conditioning on arealization of the jump dynamics, and (ii) we take advantage of a timescale separationbetween the relaxation rate of the cargo dynamics and the jump rate of the motordynamics. Observe that the stochasticityin the model can be separated into a continuous diffusion part and a discrete partcontrolling motor binding, unbinding, and stepping. Mathematically, the continuousdiffusion part is described by the Brownian motion W in (3), and the discrete motorstate is described by the Markov jump process J . We first average over the diffusionby defining the conditional expectations x ( t ) := E [ X ( t ) | J ] , z i ( t ) := E [ Z i ( t ) | J ] , t ≥ , i ∈ { , . . . , M } . (23)We emphasize that (23) are averages over paths of W given a realization J . Thus, { x ( t ) } t ≥ and {{ z i ( t ) } t ≥ } Mi =1 are functions of the realization J . This definition isconvenient, because while X ( t ) follows the randomly switching SDE (3), the process x ( t ) follows a randomly switching ODE, whose solution is known explicitly. Proposition For each t > , the expected cargo position x ( t ) conditioned ona realization of the jump process satisfiesdd t x ( t ) = ε − M (cid:88) i =1 (cid:0) z i ( t ) − x ( t ) (cid:1) , almost surely . (24) ON-PROCESSIVE MOTOR TRANSPORT Proof.
Using the explicit solution of an OU process, we have that X ( t ) = X ( τ ) e − θ ( t − τ ) + µ (1 − e − θ ( t − τ ) ) + M , (25)where τ is the most recent jump time of J , τ = sup (cid:8) { } ∪ { s < t : J ( s − ) (cid:54) = J ( s +) } (cid:9) ,θ = m ( τ ) ε − , µ = m ( τ ) (cid:80) i ∈ I ( τ ) Z i ( τ ), and M satisfies E [ M| J ] = 0. We have used thenotation f ( t ± ) := lim s → t ± f ( s ). Hence, taking the expectation of (25) conditionedon J yields x ( t ) = E [ X ( τ ) e − θ ( t − τ ) | J ] + E [ µ (1 − e − θ ( t − τ ) ) | J ]= e − θ ( t − τ ) E [ X ( τ ) | J ] + (1 − e − θ ( t − τ ) ) 1 m ( τ ) (cid:88) i ∈ I ( τ ) E [ Z i ( τ ) | J ] , since τ are { m ( s ) } s ≥ measurable with respect to the σ -algebra generated by J . We next make an observation of disparatetimescales. After averaging over the diffusive noise W , the model effectively dependson two timescales: the relaxation time of the continuous dynamics (24) (characterizedby the dimensionless rate ε − ) and the switching times of the discrete motor dynamics(5) (characterized by the dimensionless rates λ on , λ step , λ off ). Even for conservativeparameter estimates, the continuous timescale is much faster than the discrete switch-ing timescale. For instance, suppose a motor exerts a Hookean force with stiffness k = 0 . r = 1 µ m in cytosol with viscos-ity η equal to that of water. It follows that k/ (6 πηr ) ≈ × s − , whereas k off (1)is on the order of 10 − to 10 s − [16]. Hence, ε := k off (1) γ/k ≈ × − (cid:28) . (26)Further, λ on , λ step , λ off are roughly order one since k on , k step , k off have similar ordersof magnitude [16].Therefore, compared to the switching timescale, x ( t ) quickly relaxes to an equilib-rium between motor switches. Furthermore, we are interested in studying E [ R on ] and V , which depend on the behavior of x ( t ) over the course of several motor switches.Hence, we approximate x ( t ) by a jump process x ( t ) obtained from assuming x ( t )immediately relaxes to its equilibrium after each motor switch.More precisely, let ( x ( t ) , z ( t ) , . . . , z M ( t )) ∈ R M +1 be a J -measurable, right-continuous process, x ( t ) = x ( t +) , z i ( t ) = z i ( t +) , t ≥ , i ∈ { , . . . , M } , with x (0) = x (0) and z i (0) = z i (0), i ∈ { , . . . , M } , that evolves in the followingway. In light of (7), we define the effective motor positions by how they are modifiedthrough the jump process, binding at τ i and then incrementing from stepping, orstaying unbound at the cargo position x ( t ), z i ( t ) = (cid:0) x ( τ i ( t )) + J i ( t ) (cid:1) { J i ( t ) (cid:54) = u } + x ( t )1 { J i ( t )= u } , i ∈ { , . . . , M } . (27)Due to the assumed fast relaxation, x ( t ) only changes when a motor steps orunbinds, as newly bound motors exert no force. That is, if J i ( t +) = J i ( t − ) forall i ∈ { , . . . , M } satisfying J i ( t − ) ≥
0, then x ( t − ) = x ( t +). Otherwise, x ( t )evolves according to the following two rules, which describe how the cargo position x ( t ) changes when a motor steps or unbinds.0 CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
1. If the i -th motor steps ( J i ( t − ) = j ≥ J i ( t +) = j + 1), then x ( t +) = x ( t − ) + 1 /m ( t ).2. If the i -th motor unbinds ( J i ( t − ) = j ≥ J i ( t +) = u ), then x ( t +) = x ( t − ) + ∆ i, ( z ,...,z m ( t − ) ) , where ( z , . . . , z m ( t − ) ) gives the positions of the m ( t − ) bound motors just before time t , and∆ i, ( z ,...,z m ( t − ) ) = 1 m ( t − ) − m ( t − ) (cid:88) i (cid:48) =1 ,i (cid:48) (cid:54) = i z i (cid:48) ( t − ) − m ( t − ) m ( t − ) (cid:88) i (cid:48) =1 z i (cid:48) ( t − ) . (28)In words, if either of these events occurs, the cargo position x ( t ) relaxes to the meanposition of the motors. These two rules describe how the mean motor position changesin the two scenarios. If a single motor steps, incrementing its position by 1, the meanmotor position increases by 1 /m ( t ). If a motor unbinds, (28) is the change in themean motor position from removing that motor.It follows from these two evolution rules for x ( t ) that x ( t ) = M (cid:88) m =1 m S m ( t ) + χ ( t ) , (29)where S m ( t ) is the number of steps taken when m motors are bound before time t (each of which modifies the position by 1 /m ), and χ ( t ) accounts for changes in thecargo position that result from a motor unbinding, χ ( t ) = N off ( t ) (cid:88) k =1 ∆ j k , ( z ( s k off − ) ,...,z m ( sk off − ) ( s k off − ) , (30)where 0 = s < s < . . . is the sequence of times in which a motor unbinds, s k off := inf (cid:8) t > s k − : J i ( t − ) ≥ J i ( t ) = u for some i ∈ { , . . . , M } (cid:9) k ≥ , and N off ( t ) := sup { k ≥ s k off ≤ t } is the number of unbindings before time t ≥ j k ∈ { , . . . , M } gives the (almost surely unique) index of the motor that unbindsat time s k off . That is, j k satisfies J j k ( s k off − ) (cid:54) = J j k ( s k off ) = u .The following proposition checks that x ( t ) converges almost surely to the jumpprocess x ( t ) as ε →
0. The proof is in Appendix B.
Proposition If T ≥ is an almost surely finite stopping time with respect to { J ( t ) } t ≥ , then lim ε → x ( T ) = x ( T ) , almost surely . From this proposition, we conclude that studying the mean behavior of the cargoposition X ( t ) can ultimately be reduced to studying the jump process x ( t ), where thejumps correspond to motor stepping and unbinding events. Since ε (cid:28) X ( t ) by investigating the analogousquantities for x ( t ), R := x ( τ ) − x ( τ ) , V := lim t →∞ x ( t ) t . (31) ON-PROCESSIVE MOTOR TRANSPORT The following proposition checks that the mean run lengthof the full process X ( t ) converges to the mean run length of the jump process x ( t ) as ε → Proposition E [ R on ] → E [ R ] as ε → .Proof. By the tower property of conditional expectation (Theorem 5.1.6 in [10]),we have that E [ R on ] = E [ E [ R on | J ]] = E [ E [ X ( τ ) − X ( τ ) | J ]] = E [ x ( τ ) − x ( τ )] . Now, Proposition 7 ensures that x ( τ ) − x ( τ ) → R, almost surely as ε → . (32)Let N ≥ τ and time τ . Since motorstake steps of distance one, we have the almost sure bound, x ( τ ) − x ( τ ) ≤ N . Stepsare taken at Poisson rate m ( t ) λ step ( m ( t )) ≤ M Λ, where Λ := max m ∈{ ,...,M } λ step ( m ).Thus E [ N ] ≤ Λ M E [ τ − τ ] < ∞ . Thus, (32) and the bounded convergence theoremcomplete the proof. Let us now investigate V in (31), observing that this quantitycan be approached in two ways. The first exploits the observation that non-zero meandisplacements only occur from motor stepping, so the velocity can be interpreted asthe product of how often a step occurs with m motors and the size of the displacement.The second approach is again a reward-renewal argument, noting that the only non-zero displacements occur during epochs of bound cargo. The connection between thesetwo approaches provides explicit relationships between the velocity, run lengths, andrun times.Recalling the decomposition of the jump process x in (29), we seek to computethe expected value E [ x ( t )] = m (cid:88) m =1 m E [ S m ( t )] + E [ χ ( t )] . Using the definition of χ ( t ) in (30), we compute its expectation by summing overall possible displacements from one of m motors unbinding ∆ j, ( z ( t − ) ,...,z m ( sk off − ) ( t − ) ,which yields m (cid:88) j =1 ∆ j, ( z ,...,z m ) = 1 m − m (cid:88) j =1 (cid:16) − z j + m (cid:88) i =1 z j (cid:17) − m (cid:88) i =1 z i = 0 . Since each of the bound motors is equally likely to unbind, it follows that E [ χ ( t )] = 0.This can be interpreted as the observation that the arithmetic mean does not change inexpectation when removing a randomly (uniformly) chosen element. In other words,the effects of motors unbinding ahead of the cargo are completely offset in the meanby motors unbinding behind the cargo. Therefore, the only long-term influence on x is stepping events.Given a realization { m ( s ) } s ≥ , the number of steps taken with m motors boundbefore time t ≥ mλ step ( m ) (cid:82) t m ( s )= m d s . Hence, E [ S m ( t )] = mλ step ( m ) E (cid:104) (cid:90) t m ( s )= m d s (cid:105) . CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
Now, { m ( s ) } s ≥ is an ergodic Markov process, so the occupation measure convergesalmost surely to the stationary measure (see Theorem 3.8.1 in [40])1 t M (cid:88) n =1 (cid:90) t m ( s )= m d s → p m , almost surely as t → ∞ , where p m := lim t →∞ P ( m ( t ) = m ) is the stationary probability m motors are bound.We note that p m is the ( m + 1)-st component of the unique probability vector, p ∈ R × ( M +1) satisfying (see Theorem 3.5.2 in [40]) p (cid:101) Q = 0 , (33)where (cid:101) Q ∈ R ( M +1) × ( M +1) is the generator matrix defined in section 4.1. Since theoccupation measure is bounded above by one, the bounded convergence theorem giveslim t →∞ E [ x ( t )] t = lim t →∞ M (cid:88) m =1 λ step ( m ) E (cid:104) t (cid:90) t m ( s )= m d s (cid:105) = M (cid:88) m =1 λ step ( m ) p m . It is easy to see that the classical renewal reward theorem applies to x ( t ) so that M (cid:88) m =1 λ step ( m ) p m = lim t →∞ E [ x ( t )] t = lim t →∞ x ( t ) t = E [ R ] E [ T on ] + E [ T on ] , almost surely . Furthermore, (19), (20), (21), and Proposition 8 yieldlim ε → V = lim ε → E [ R on ]( M λ on (0)) − + E [ T on ] = E [ R ]( M λ on (0)) − + E [ T on ] = M (cid:88) m =1 λ step ( m ) p m . In summary, we now have explicit formulas for the velocity V and expected runlength E [ R on ] of X ( t ) in the small ε limit,lim ε → V = V = M (cid:88) m =1 λ step ( m ) p m (34) lim ε → E [ R on ] = E [ R ] = (cid:16)(cid:0) M λ on (0) (cid:1) − + E [ T on ] (cid:17) M (cid:88) m =1 λ step ( m ) p m , (35)where p m is given by (33) and E [ T on ] is given by (22). In Fig. 2, we compare theseformulas for E [ R ] and V with estimates of E [ R on ] and V from simulations of the fullprocess ( X ( t ) , Z ( t ) , J ( t )) (for details on our statistically exact simulation method, seesection 4.5).Furthermore, some experimental works [16, 26] measure the average run velocity, E [ R/T on ]. Now, if σ ( m ) denotes the σ -algebra generated by { m ( t ) } t ≥ , then recalling(8) and (10) and using the tower property of conditional expectation yields E (cid:104) RT on (cid:105) = E (cid:104) T on E [ R | σ ( m )] (cid:105) = E (cid:104) T M (cid:88) m =1 m E [ S m ( τ ) | σ ( m )] (cid:105) = M (cid:88) m =1 m mλ step ( m ) E (cid:104) T (cid:90) τ m ( s )= m d s (cid:105) . ON-PROCESSIVE MOTOR TRANSPORT − k on [s − ] E [ R ][ n m ] − k on [s − ] V [ n m / s ] − k off [s − ] E [ R ][ n m ] − k off [s − ] V [ n m / s ] − k step [s − ] − E [ R ][ n m ] − k step [s − ] V [ n m / s ] Fig. 2 . Expected run lengths E [ R ] and asymptotic velocities V as a function of the parameters k on , k off , k step for M = 1 , , total motors. The curves are the analytical formulas (34)-(35) forthe ε → limit, and the dots are estimates from statistically exact realizations of the full process, { ( X ( s ) , Z ( s ) , J ( s )) } ts =0 , where the ending time t is such that N ( t ) = 10 where N ( t ) is defined in(11). Unless noted otherwise, k on ( m ) = 10 (cid:2) s − (cid:3) , k step ( m ) = 20 (cid:2) s − (cid:3) , k off ( m ) = 5 (cid:2) s − (cid:3) for each m . Further, k and γ are as in (26) and k B T = 4 . pN · nm ] . Hence, it follows from (34) that V := E [ R/T on ] = V /p on , (36)where p on = (cid:80) Mm =1 p m is the stationary probability that m ( t ) ≥ M = 1 , M = 2 , and M = 3 . In this subsection, we collect explicitformulas for the run length E [ R ] and run velocity V when the total number of motorsis M = 1 , ,
3. The run time E [ T on ] and net velocity V can be easily deduced fromthese quantities using (34)-(35) but are omitted for brevity.For M = 1 total motors, the quantities are simply E [ R ] = λ step (1) , V = λ step (1) . (37)For M = 2 total motors, we find E [ R ] = λ step (1) + λ on (1) λ step (2)2 λ off (2) , V = 2 λ off (2) λ step (1) + λ on (1) λ step (2)2 λ off (2) + λ on (1) . (38)For M = 3 total motors, we find E [ R ] = λ step (1) + λ on (1)(3 λ off (3) λ step (2) + λ on (2) λ step (3))3 λ off (2) λ off (3) , V = 3 λ off (3) [ λ off (2) λ step (1) + λ on (1) λ step (2)] + λ on (1) λ on (2) λ step (3)3 λ off (3) [ λ off (2) + λ on (1)] + λ on (1) λ on (2) . (39)To put these quantities in dimensional units, recall the jump rates (4) and multiply E [ R ] by the dimensional step distance δ > V by δk off (1) > CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
To verify our predictions for the expected runlengths and velocities, we compare to statistically exact numerical simulations of thefull process ( X ( t ) , Z ( t ) , J ( t )). In a given state, we use the classical Gillespie stochasticsimulation algorithm to generate the time of the next transition for the Markov chain J ( t ) and to choose which transition occurs. For m ( t ) ≥ X ( t ) is an OU process,generically described by d X ( t ) = α [ µ − X ( t )] d t + β d W ( t ) . To update X ( t ) to the next time t + τ , we use the statistically exact method describedin [18], summarized by(40) X ( t + τ ) = e − ατ X ( t ) + (1 − e − ατ ) µ + β (cid:114) (1 − e − ατ )2 α n where n is a standard normal random variable. When m ( t ) = 0, X ( t ) is a purediffusion process with α = 0, so (40) becomes an Euler-Maruyama update. Thisprocedure generates statistically exact sample paths of X ( t ), sampled at the transitiontimes of J ( t ). We use this scheme to generate a long realization of ( X ( t ) , Z ( t ) , J ( t )),thereby providing Monte Carlo estimates for E [ R on ] and V for a given parameter set.
5. Biological application.
We now use the formulas (37)-(39) for run velocity, V , and run length, E [ R ], to explore the behavior of non-processive motors. Thebehavior of individual non-processive motors is characterized by two observations: i)short attachment times, ii) the time it takes to hydrolyze ATP (and consequently, tostep) coincides with this attachment time [6, 15, 39]. Concretely, Ncd motors in thekinesin-14 family take 1 to 5 steps before unbinding [2, 11]. In our model, λ step (1)gives the expected number of steps before unbinding, so we characterize non-processivemotors as those with λ step (1) ∈ [1 , V in our model. Specifically, the primary manifestation of coop-erativity is that V increases substantially when the total number of motors increasesfrom M = 1 to M = 2. For M ≥
2, the velocity remains relatively constant.We thus ask the question: what features are necessary to produce this behavior?Now, if the step rate is independent of the number of bound motors, m , then itfollows immediately from (34) and (36) that V is independent of M . In particular, ifthe dimensional step rate is k step ( m ) ≡ k for all m ∈ { , . . . , M } , then in dimensionalunits, V is simply δk , regardless of any other parameter values.Therefore, our model predicts that the stepping rate must depend on the numberof bound motors in order to produce the cooperative behavior seen in run velocitiesin [16, 26]. This prediction is bolstered by the simulation results of [16]. There, theauthors constructed a detailed computational model of motor transport, and they hadto improve motor stepping ability when two or more motors are bound in order forsimulations of their computational model to match experimental run velocities.The authors of [16] also describe motor cooperativity in terms of the averagedistance traveled by a cargo before all of its motors detach from a microtubule, whichis analogous to E [ R ] in our model. Namely, they find that the run length E [ R ]dramatically increases when M increases from 1 to 2. Our model can replicate thiscooperativity if and only if we allow the binding rate, k on , and/or the unbinding rate, k off , to depend on the number of bound motors, m . ON-PROCESSIVE MOTOR TRANSPORT M = 1), the authors report average run lengths of approximately 300 [nm], and theynote that this value is necessarily an overestimate since they were unable to measurevery short runs. Furthermore, this value must also be an overestimate since a singlenon-processive motor takes only a few steps per run (by definition of non-processive),and each step is approximately 7 [nm] [11].We thus assume that λ step (1) = 4, based on [2, 11] and δ = 7 [nm]. This gives E [ R ] = 28 [nm] for M = 1, which we use instead of the reported value in [16]. Wethen match the respective approximate run lengths of 1300 [nm] and 3300 [nm] for M = 1 , M = 1 , , k step (1) ≈
14 [s − ] and k step (2) ≈ k step (3) ≈
21 [s − ], and theunbinding rate k off (1) ≈ . − ], which are all within the range of previously reportedrates. The other binding/unbinding rates are not uniquely specified, but rather mustsatisfy the relations k off (2) ≈ . k on (1) and k off (3) ≈ . k on (2). Hence, if k off wereconstant in m , then k on (1) ≈
200 [s − ] and k on (2) ≈ − ].We make two observations about this result: i) the binding rate k on (1) is an orderof magnitude larger than reported values [2, 16] and ii) the binding rate decreases asthe number of bound motors increases from 1 to 2. Both of these points can beexplained by geometry. First, the value of k on (1) is enhanced because the singlebound motor tethers the unbound motors close to the microtubule, and thus allowsthose motors to bind more easily. This binding enhancement due to geometry hasprecedent in motor studies. Indeed, in a different family of kinesins, it was shown tobe critical for determining run lengths [13]. Further, it was shown to play a criticalrole in enabling dynein processivity [20], and it was posited as an explanation for whymyosin motors can become processive when processive kinesin motors are present [22].The authors in [3] report large k on values in a model of microtubule sliding driven bykinesin-14 motors and also speculate that this is due to tethering effects.This effect can also be understood in terms of rebinding. If two motors are boundand one unbinds, then that motor can rapidly rebind since the bound motor keeps itnear the microtubule. Such rebinding was the mechanism posited in [17] to explainthe processive behavior of non-processive motors along microtubule bundles. Further,rebinding is very important in enzymatic reactions [46, 19, 33, 34]. In that context, oneincorporates rebinding by using an “effective” unbinding rate, which is the intrinsicunbinding rate multiplied by the probability that the particle does not rapidly rebind[32]. Hence, this effect could be included in our model by reducing k off (2) rather than(or in addition to) increasing k on (1). Importantly, this is exactly what is implied bythe relation, k off (2) ≈ . k on (1), derived above.Second, geometric exclusion effects can explain a decrease in binding rates as thenumber of bound motors increases from 1 to 2. When more motors are bound, it ismore difficult for additional motors to bind because the range of diffusive search isreduced for unbound motors. In numerical investigations of motor transport systems,this exact effect is observed [30, 35]. Furthermore this decrease in binding rate canarise due to motors competing for binding sites, a point posited in [29]. Interestingly,these authors find that negative cooperativity has little impact on transport velocity.6 CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
The same is true in our model, as the value of V changes by less than 1 [nm/s] as k on (2) ranges from 0 to ∞ while keeping the other parameters fixed. However, we notethat the run length for M = 3 is greatly affected by k on (2), and thus this highlightsthe importance of using both run velocity and run length to study motor transport.
6. Discussion.
In this work, we formulated and analyzed a mathematical modelof transport by non-processive molecular motors. We deliberately made our modelsimple enough to enable us to extract explicit formulas for experimentally relevantquantities, yet maintain agreement with detailed computational studies. One suchsimplification is to assume the motor stepping and unbinding rates are independentof force. The justification for this assumption is that since non-processive motorstake only a few steps before unbinding (compared to hundreds of steps by processivemotors), these motors are unlikely to be stretched long distances and therefore areunlikely to generate large forces. This assumption on the stepping rate has beenmade in other models involving non-processive motors [39] and did not appear to bea necessary feature in that context, and how force affects stepping is not completelyclear [41]. However, we note that force-dependent unbinding can be an importantcharacteristic of processive motors, as kinesin-1 and kinesin-2 detach more rapidlyunder assisting load than under opposing load [1], which increases run velocity andrun length.These limitations notwithstanding, our model makes some concrete predictionsabout motor number-dependent stepping, binding, and unbinding behavior and howthese quantities contribute to transport by non-processive motors. Specifically, weobserve that a complex cooperativity mechanism appears to be a necessary ingre-dient for non-processive motor transport, and these predictions align with severalrecent experimental and computational works. Furthermore, these predictions can beinvestigated experimentally. Indeed, we hope that the work here will spur further in-vestigation into how geometry affects non-processive motor transport, especially giventhat kinesin-14 motors are known to transport a wide variety of cargo, including long,cylindrical microtubules [21, 14] and large, spherical vesicles in plants [48].
Appendix A. Proofs of Lemmas 2-5.
Proof of Lemma 2.
Between time τ k − and τ k on , the cargo is freely diffusing.Therefore, to control Y k off , we need to control the supremum of a Brownian motion.Now, for any fixed T >
C >
0, it follows from Doob’s martingale inequality(Theorem 3.8(i) in [28]) and symmetry of Brownian motion that P ( sup t ∈ [0 ,T ] | W ( t ) | ≥ C ) ≤ (cid:16) − C T (cid:17) . (41)Hence, it follows that P ( Y k off ≥ C | T k off ) ≤ (cid:16) − C σ T k off (cid:17) , almost surely . (42)Note that (42) is an average over realizations of the diffusion W for fixed realizationsof the time T k off . That is, the inequality holds for almost all realizations of T k off .Now, T k off is exponentially distributed with rate M λ on (0). Hence, the tower prop-erty of conditional expectation (see Theorem 5.1.6 in [10]) yields P ( Y k off ≥ C ) = E [ P ( Y k off ≥ C | T k off )] ≤ M λ on (0) (cid:90) ∞ exp (cid:16) − C σ t − M λ on (0) t (cid:17) d t. (43) ON-PROCESSIVE MOTOR TRANSPORT (cid:90) ∞ λe − λt e − a/t d t = 2 √ aλK (2 √ aλ ) , if a > , λ > , (44)where K ( x ) denotes the modified Bessel function of the second kind. Hence, theproof is complete after combining (43) and (44) and the following bound, K ( x ) ≤ (cid:112) π/x (1 + 1 / (2 x )) e − x , x > , which was proven in [49]. Proof of Lemma 3.
To control Y k on , we note that if t ∈ [ τ k on , τ k off ], then X ( t ) is anOU process centered at µ ( t ) = m ( t ) (cid:80) i ∈ I ( t ) Z i ( t ) with relaxation rate θ ( t ) = ε − m ( t ).Hence, after shifting time and space so that τ k on = 0 and X ( τ k on ) = 0, we have that X ( t ) = (cid:90) t µ ( s ) θ ( s ) e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d s + σ (cid:90) t e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d W ( s ) . (45)Since each bound motor takes steps of unit length at a Poisson rate, and since a motorbinds at the current cargo location, it follows that | µ ( t ) | ≤ P ( t ) + X sup ( t ) , where X sup ( t ) := sup t (cid:48) ∈ [0 ,t ] | X ( t (cid:48) ) | , (46)and { P ( s ) } s ≥ is a Poisson process with rate µ := M max m ∈{ ,...,M } λ step ( m ). Thus, | X ( t ) | ≤ σ (cid:12)(cid:12)(cid:12) (cid:90) t e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d W ( s ) (cid:12)(cid:12)(cid:12) + P ( t ) + (cid:90) t X sup ( s ) θ ( s ) e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d s, (47)where we have used the fact that (cid:90) t f ( s ) θ ( s ) e − (cid:82) t s θ ( s (cid:48) ) d s (cid:48) d s ≤ (cid:90) t f ( s ) θ ( s ) e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d s ≤ f ( t ) , ≤ t ≤ t, (48)if f is any nonnegative and nondecreasing function.Now, a straightforward calculation using integration by parts shows thatsup t (cid:48) ∈ [0 ,t ] (cid:12)(cid:12)(cid:12) (cid:90) t (cid:48) e − (cid:82) t (cid:48) s θ ( s (cid:48) ) d s (cid:48) d W ( s ) (cid:12)(cid:12)(cid:12) ≤ s ∈ [0 ,t ] | W ( s ) | . (49)Therefore, combining (47), (48), and (49) yields X sup ( t ) ≤ α ( t ) + (cid:90) t X sup ( s ) θ ( s ) e − (cid:82) ts θ ( s (cid:48) ) d s (cid:48) d s, (50)where α ( t ) := 2 σ sup s ∈ [0 ,t ] | W ( s ) | + P ( t ). Multiplying (50) by e (cid:82) t θ ( s (cid:48) ) d s (cid:48) , applyingGronwall’s inequality, and then dividing by e (cid:82) t θ ( s (cid:48) ) d s (cid:48) yields X sup ( t ) ≤ α ( t ) + (cid:90) t α ( s ) θ ( s ) d s, Since α ( t ) is nondecreasing and θ ( t ) ≤ Θ := ε − M , we obtain X sup ( t ) ≤ α ( t )(1 + Θ t ).Therefore, we have the following almost sure inequality for ζ := (1 + Θ T k on ) − , P ( Y k on ≥ C | T k on ) ≤ P (cid:0) P ( T k on ) ≥ Cζ/ (cid:12)(cid:12) T k on (cid:1) + P (cid:0) sup t ∈ [0 ,T k on ] | W ( t ) | ≥ Cζ/ (4 σ ) (cid:12)(cid:12) T k on (cid:1) . (51)8 CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
We thus need to control the distribution of T k on . Now, the Markov chain m ( t )in (6) is a finite state space birth-death process, and thus there exists [7] a uniquequasi-stationary distribution ν ∈ R M , which is a probability measure on { , . . . , M } so that if P ( m (0) = m ) = ν m for m ∈ { , . . . , M } , then P (cid:0) m ( t ) = m (cid:12)(cid:12) m ( s ) (cid:54) = 0 for all s ∈ [0 , t ] (cid:1) = ν m , m ∈ { , . . . , M } . Furthermore, it is known that the first passage time of m ( t ) to state 0 is exponentiallydistributed with some rate λ > P ( m (0) = m ) = ν m for m ∈ { , . . . , M } [37]. Now, T on is the first passage time of m ( t ) to state m = 0 starting from state m = 1, whichmust be less than the first passage time to state m = 0 starting from any otherstate. Therefore, if S is the first passage time to state m = 0 starting from thisquasi-stationary distribution, then P ( T on > T ) ≤ P ( S > T ) = 1 − e − λT , T > . Thus, since both terms in the upper bound in (51) are increasing functions of therealization T k on >
0, the tower property yields for ζ S := (1 + Θ S ) − , P ( Y k on ≥ Cζ ) = E (cid:2) P ( Y k on ≥ Cζ | T k on ) (cid:3) ≤ E (cid:2) P ( P ( S ) ≥ Cζ S / | S ) (cid:3) + E (cid:2) P ( sup t ∈ [0 ,S ] | W ( t ) | ≥ Cζ S / (4 σ ) | S ) (cid:3) . (52)Next, if P is Poisson distributed with mean µ , then Corollary 6 from [43] yields P ( P ≥ C ) ≤ e C − µ (cid:16) µ C (cid:17) C , if C ≥ µ . Noting that C < µS (1 + Θ S ) if and only if S >
Σ := √ C Θ /µ +1 − , we obtain P ( P ( S ) ≥ Cζ S / | S ) ≤ S> Σ + e C S ) − µS (cid:16) µS (1 + Θ S ) C (cid:17) C S ) S ≤ Σ . (53)Since S ∼ Exponential( λ ), taking the expectation of (53) gives E [ P ( P ( S ) ≥ Cζ S / | S )] ≤ e − λ Σ + (cid:90) Σ0 λe C s ) − ( µ + λ ) s (cid:16) µs (1 + Θ s ) C (cid:17) C s ) d s. (54)A quick calculation shows that if C ≥ (Θ ε ) − for ε := (8 eµ Θ) − / , then λe C s ) − ( µ + λ ) s (cid:16) µs (1 + Θ s ) C (cid:17) C s ) ≤ (cid:40) λ − κ √ C if s ∈ [0 , ε √ C ] ,λe − λε √ C if s ∈ [ ε √ C, Σ] , for κ := (4Θ ε ) − . Hence, if C ≥ (Θ ε ) − , then E [ P ( P ( S ) ≥ Cζ S / | S )] ≤ λe − λ Σ + Σ λ (2 − κ √ C + e − λε √ C ) . (55) ON-PROCESSIVE MOTOR TRANSPORT E (cid:2) P ( sup t ∈ [0 ,S ] | W ( t ) | ≥ Cζ S / (4 σ ) | S ) (cid:3) ≤ (cid:90) ∞ λe − λs exp (cid:110) − κC s (1 + Θ s ) (cid:111) d s, where κ = (32 σ ) − . It is straightforward to check that2 λe − λs exp (cid:110) − κC s (1 + Θ s ) (cid:111) ≤ (cid:40) λ exp (cid:110) − κ √ C Θ +2Θ C − / + C − (cid:111) if s ∈ [0 , √ C ] , λe − λs if s ∈ [ √ C, ∞ ) , Therefore, for sufficiently large C , we have that E (cid:2) P ( sup t ∈ [0 ,S ] | W ( t ) | ≥ Cζ S / (4 σ ) | S ) (cid:3) ≤ λ √ Ce − (2 κ/θ ) √ C + 2 e − λ √ C (56)Combining (52), (55), and (56) completes the proof. Proof of Lemma 4.
Since Y k ≤ Y k off + Y k on for k ≥
1, we have that ∞ (cid:88) k =1 P ( Y k > √ k ) ≤ ∞ (cid:88) k =1 P ( Y k off > √ k ) + ∞ (cid:88) k =1 P ( Y k on > √ k ) . (57)Therefore, the upper bounds established in Lemmas 2 and 3 and the integral testshow that (57) converges. Applying the Borel-Cantelli lemma (Theorem 2.3.1 in [10])completes the proof. Proof of Lemma 5.
Since Y k ≥ E [ Y k ] = (cid:82) ∞ P ( Y k >C ) d C . Using the bounds in Lemmas 2 and 3 as in the proof of Lemma 4 shows thatthis integral is finite. Appendix B. Proof of Proposition 7.
Proof.
Fix a realization J . Let K ≥ J before time T , where t is said to be a jump time if J ( t +) (cid:54) = J ( t − ).Denote these K jump times by 0 < τ < · · · < τ K < T and let τ = 0 and τ K +1 = T .For ease of notation, define the sequences x k := x ( τ k ) , x k := x ( τ k ) , z ik := z i ( τ k ) , z ik := z i ( τ k ) , m k := m ( τ k ) , for k ∈ { , , . . . , K } . Further, define the time between jumps, s k := τ k − τ k − , for k ∈ { , . . . , K } . It follows immediately from Proposition 6 that x k +1 = x k e − s k +1 /ε + µ k +1 (1 − e − s k +1 /ε ) , k ∈ { , , . . . , K } , (58)where for k ∈ { , , . . . , K + 1 } we define µ k +1 := (cid:40) m k (cid:80) i ∈ I ( τ k ) z ik if m k > ,x k if m k = 0 . (59)Furthermore, it follows from the definition of x ( t ) that for k ∈ { , , . . . , K } , x k +1 := (cid:40) m k (cid:80) i ∈ I ( τ k ) z ik if m k > ,x k if m k = 0 . (60)0 CHRISTOPHER E. MILES, SEAN D. LAWLEY, AND JAMES P. KEENER
Now, since motors take steps of size one, it follows that if k ∈ { , . . . , K } and i ∈ { , . . . , M } , then 0 ≤ z ik ≤ K + 1 and 0 ≤ x k ≤ K + 1. Hence, if k ∈ { , . . . , K } ,then (59) implies | x k − µ k +1 | < K + 1 . (61)Next, we claim that if k ∈ { , . . . , K } andmax j ∈{ ,...,k } (cid:110) | x j − x j | , max i ∈{ ,...,M } | z ij − z ij | (cid:111) < η, (62)then max (cid:110) | x k +1 − x k +1 | , max i ∈{ ,...,M } | z ik +1 − z ik +1 | (cid:111) < ( K + 1) e − s k +1 /ε + η. (63)To see this, we use (58) and (61) to obtain | x k +1 − x k +1 | = | x k e − s k +1 /ε + µ k +1 (1 − e − s k +1 /ε ) − x k +1 |≤ ( K + 1) e − s k +1 /ε + | µ k +1 − x k +1 | . Using (59) and (60), we have that | µ k +1 − x k +1 | ≤ (cid:40) m k (cid:80) i ∈ I ( τ k ) | z ik − z ik | if m k > , | x k − x k | if m k = 0 . Furthermore, it follows from (7) and (27) that | z ik +1 − z ik +1 | ≤ max j ∈{ ,...,K +1 } | x j − x j | , i ∈ { , . . . , M } . Hence, the claim (63) is verified.Define the largest time between jumps, s := max k ∈{ ,...,K } s k . Since x = x = z i = z i for i ∈ { , . . . , M } , we apply (62) and (63) iteratively to obtain | x K +1 − x K +1 | ≤ ( K + 1) e − s/ε . Taking ε → REFERENCES[1]
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