Limited processivity of single motors improves overall transport flux of self-assembled motor-cargo complexes
Keshav B. Patel, Shengtan Mao, M. Gregory Forest, Samuel K. Lai, Jay M. Newby
LLimited processivity of single motors improves overall transport fluxof self-assembled motor-cargo complexes
Keshav B. Patel , Shengtan Mao , M. Gregory Forest , Samuel K. Lai , and Jay M.Newby UNC/NCSU Joint Department of Biomedical Engineering, University of North Carolina at Chapel Hill Department of Mathematics, University of North Carolina at Chapel Hill Department of Applied Physical Sciences, University of North Carolina at Chapel Hill Division of Pharmacoengineering and Molecular Pharmaceutics, Eshelman School of Pharmacy, University ofNorth Carolina at Chapel Hill Department of Microbiology and Immunology, University of North Carolina at Chapel Hill, Chapel Hill Department of Mathematical and Statistical Sciences, University of Alberta
Abstract
Single kinesin molecular motors can processively move along a microtubule (MT) a few micrometerson average before dissociating. However, cellular length scales over which transport occurs are severalhundred microns and more. Why seemingly unreliable motors are used to transport cellular cargo remainspoorly understood. We propose a new theory for how low processivity, the average length of a single boutof directed motion, can enhance cellular transport when motors and cargoes must first diffusively selfassemble into complexes. We employ stochastic modeling to determine the effect of processivity on overallcargo transport flux. We show that, under a wide range of physiologically relevant conditions, possessing“infinite” processivity does not maximize flux along MTs. Rather, we find that low processivity i.e., weakbinding of motors to MTs, is optimal. These results shed light on the relationship between processivityand transport efficiency and offer a new theory for the physiological benefits of low motor processivity.
Molecular motors function though a repeating sequence of reactions called a chemo-mechanical cycle. Dueto their size and scale of force generation, thermal fluctuations significantly influence single motor dynamics,making their activity and motion stochastic. A motor’s processivity is defined as the average distance thatdirected motion along a cytoskeletal filament is consistently engaged through the motor’s chemo-mechanicalcycle. Disengagement from the filament is said to occur if the motor diffuses away from the filament into thecytoplasm and must randomly re-encounter a filament before directed motion can resume. The processivityof single kinesin-1 (KIF5) motors, commonly seen in microtubule (MT) transport, has been measured to be1 − µ m [], while cargo transport distances, particularly in neurons, can be 10 − µ m or more. Anotherway of describing processivity is with the “duty ratio” (denoted by 1 − α in this paper), which is defined asthe average fraction of time an independent motor spends engaged to a filament. Hence, the processivity isdefined as the product of the duty ratio and the average motor velocity.Processivity of the kinesin motor KIF5 was shown to be tunable through genetic modification of the necklinker; mutations that added positive charge increased processivity as much as 4-fold, while adding negativecharge decreased processivity [14]. Even though mutations to the neck linker are capable of improvingprocessivity, the neck coiled-coil regions are highly conserved. Taken together, these observations suggestthat limited processivity is important for cell function [14]. If processivity can be increased as much as 4-foldwith a simple single-nucleotide mutation, why does KIF5 have processivity in the range of a few microns whendistances typical of active transport are often 10 −
100 times longer? The purpose of this paper is to proposea new theory to answer the following question: is there a mechanistic purpose for limiting processivity ofsingle motors for cellular transport? 1 a r X i v : . [ q - b i o . S C ] F e b ransport of self-assembled motor-cargo complexes has been shown to benefit from multiple-motor teams[3]. A team of motors collectively has much higher processivity than individual motors. While single motorsmay detach from MTs quickly, only one motor in the team needs to be engaged to maintain cargo transport.Mathematical modeling shows that the mean time for the cargo to fully detach is an exponentially increasingfunction of the total number of motors in the team [9]. However, these observations fail to explain why singlemotor processivity is limited when increasing it would dramatically improve performance of a multi-motorteam.Several theories for the role of limited processivity in active transport have been proposed. First, lowerprocessivity may enhance the ability of motors to navigate obstructions and barriers [14, 3]. Second, lowerprocessivity of single motors may enhance their ability to effectively coordinate their chemo-mechanicalcycles within a multi-motor team [4]. Third, processivity may be tuned to achieve specific cargo and motordistributions across different cytoskeletal substrates [13].In this paper, we propose a new theory (which is not mutually exclusive to those previously proposed),namely, that limited processivity can improve the overall flux of cargoes when cargoes and motors must firstdiffusively self assemble into complexes before engaging in active transport. Our conclusions are motivatedby a recent theory for how antibodies trap virus in mucosal barriers by accumulating multiple antibodieson the viral surface that crosslink it to elements of the mucus polymer network [12]. Our analysis showsthat the improvement of cargo transport flux from lowering motor processivity is due to several competingfactors. Lower processivity increases the concentration of freely diffusing motors, which increases the rateof assembling multi-motor complexes from individual diffusive motors and cargoes, while simultaneouslyincreasing the average number of motors per cargo complex.The paper is organized as follows. First, we introduce the stochastic model and discuss the underlyingassumptions. We then use a quasi-steady state (QSS) reduction of the model in two stages. We presentresults using a Monte-Carlo solution to the full process, a numerical solutions of the partially reducedChapman–Kolmogorov equation, and an analytical formula obtained from the full QSS reduction. Molecular motor transport can take place in a wide range of geometries e.g., spherical domains in transport ofviral proteins to to the nucleus and branched domains found in neuronal dendrites [11, 2] and fungal syncytia.At the simplest level, motor transport occurs along linear filaments. It is sufficient for our purposes to assumea one dimensional track of length L . In our scenario, cargo begin at the negative end of the microtubule(MT) and travel down towards the positive end.The proposed theory depends on the following physical assumptions. Cargoes and motors diffusivelyassemble into complexes in the cytoplasm before active transport of cargo can occur. Single motors (eitherfree or in the cargo complex) stochastically bind to and unbind from MTs. For simplicity, we assume thatwhen motors are bound to the MT, they move at a constant velocity ( ∼ µ m s − ). Multiple motors canbind to and unbind from a cargo, either in the cytoplasm or on MTs. Cargoes diffuse in the cytoplasm if nomotors are simultaneously bound to the cargo and MT.We assume cargoes to be large (relative to motors), spherical packages that each have N independentbinding sites for motors. We define the following time dependent quantities. Let n be the number of motorsbound to a cargo and s be the number of these n motors that are also bound to a MT—in other words, thenumber of cargo-MT crosslinks.Although the binding of motors and cargo is complex and requires multiple cofactors, we simplify thereaction to assume that the two are able to bind to each other directly. Binding and unbinding of motorsand cargo occur with rates k on and k off respectively; the fraction of time motors spend unbound is givenby K = k off k off + k on . We assume that motors are sparsely distributed on the MT and thus it is unlikely for amotor bound to the MT to bind to a cargo that is already traveling along the MT (they can still encountera free diffusive cargo), meaning the state change ( n, s ) → ( n + 1 , s + 1) is forbidden for s >
0. Moreover, amotor bound to the MT cannot diffusively explore the cytoplasm and therefore has a reduced binding rate(denoted as k bon ) to diffusing cargo. By the Smoluchowski encounter rate, the reduction in the binding rateis k bon k on ∝ D cargo D cargo + D motor .Motors bind to and unbind from a MT with rates m on and m off respectively. Because the MT is essentially2igure 1: Binding diagram for reactions considered.immobile, the binding rate between a motor of a free cargo and MT is reduced compared to the binding rateof free motors to a MT. In other words, we have that m firston m on ∝ D cargo D motor . When bound to a MT, the cargo movesat the same rate as a bound motor, and we assume the motor is unaffected by the cargo (i.e., we neglectmechanical interactions between the cargo and motors).Assuming the background concentration of motors is at steady state, the fraction of motors free from theMT at any given time or position is given by α = m off m off + m on .We use the following relations for binding constants: k on = α ( D cargo + D motor ) γ and k bon = (1 − α ) D cargo γ ,where γ is a scaling parameter related to the effective binding distance between motors and cargo. In thiswork, it is taken to be 3.57 to scale k on to the appropriate value in Table 1 when α = 1 and small D cargo .See Table 1 for a list of experimental values and ranges and Fig. 1 for a summary of the reactionsconsidered.Parameter Interpretation Value(s) References N number of motor binding sites on cargo 5 −
10 [5] v trans velocity of motor along MT . − µ m s − [1, 8] D cargo diffusivity of transported particle 10 − − µ m s − NA D motor diffusivity of molecular motor protein 1 . µ m s − [6] k on binding rate between motors and cargo 5 Hz [10] k off unbinding rate between motors and cargo 1 Hz [7]Table 1: Values and ranges of constants used in the modelThe stochastic process is fully defined by the following Chapman–Kolmogorov equation: ∂∂t p ( n, s, x, t ) = δ s, D cargo ∂ p∂x − (1 − δ s, ) v trans ∂p∂x + [ M s + V n,s ] p, (1)where p () is a probability density function describing cargo position at spatial position x , time t , where n and s are the number of bound motors and active crosslinks, respectively. M and V are rate matricesdescribing motor/MT binding and motor/cargo binding respectively (see (22) and (23) in the Appendix fordefinitions). The initial condition is given by p ( n, s, x,
0) = δ n,n δ s,s δ ( x ) . (2)3e will consider two sets of boundary conditions: periodic p ( n, s, , t ) = p ( n, s, L, t ) , ∂∂x p ( n, , , t ) = ∂∂x p ( n, , L, t ) , (3)and right-end absorbing ∂∂x p ( n, , , t ) = 0 , p ( n, s, L, t ) = 0 . (4) In this section we enumerate the different characteristic timescales present in the system and their effect onoverall transport efficiency. Understanding the effect of separation between various timescales informs ourphysical understanding of the system and provides accurate approximations.We define the following timescale separation: τ α (cid:28) τ K (cid:28) τ v , (5)where τ α = m on + m off is the timescale of binding between motors and MT, τ K = k on + k off is the timescaleof binding between motors and cargo, and τ v = Lv trans is the timescale of cargo transport. An additionalimplicit assumption for directed active transport is that cargo diffusion over transport length scales takesmuch longer than active transport. In other words, we assume that τ v (cid:28) τ D , where τ D = L D cargo is thetimescale of cargo diffusion.To begin, we can rewrite the full density function p () as the product of conditional probabilities and themarginal density, u ( x, t ): p ( n, s, x, t ) = ρ ( s, t | n, x ) ρ ( n, t | x ) u ( x, t ) . (6)Because binding and unbinding are independent of spatial position, the conditional probabilities are inde-pendent of x . We assume further that s and n reach equilibrium rapidly so that we can assume that themarginal distributions are approximately at quasi-steady state. Hence, we have that ρ ( s, t | n, x ) ∼ ρ ( s | n )and ρ ( n, t | x ) ∼ ρ ( n ). These are the quasi-steady state probability distributions, and by definition we havethat the net reaction rates are zero so that N (cid:88) s =0 M s ρ ( s | n ) = 0 , N (cid:88) n =0 N (cid:88) s =0 V n,s ρ ( s | n ) ρ ( n ) = 0 . (7)It can be readily verified that the quasi-steady state distributions, satisfying (7), are given by ρ ( s | n ) = D motor α n ( D motor + D cargo ) α n + D cargo , s = 0 D cargo ( ns ) (1 − α ) s α n − s ( D motor + D cargo ) α n + D cargo , < s ≤ n (8)and ρ ( n ) = N k n off (cid:18) Nn (cid:19) n − (cid:89) j =0 ( ρ (0 | j ) k on + k bon ) . (9)Applying the QSS assumption, we obtain the following reduced description of motor transport. Substi-tuting p ( n, s, x, t ) ∼ ρ ( n | s ) ρ ( n ) u ( x, t ) + O ( (cid:15) ) into (1) and summing over s and n yields ∂∂t u ( x, t ) = D ∂ u∂x − v ∂u∂x , (10)where D = D cargo N (cid:88) n =0 ρ (0 | n ) ρ ( n ) , v = v trans N (cid:88) n =0 n (cid:88) s =1 ρ ( s | n ) ρ ( n ) . (11)The reduced parameters, D and v , are the steady-state diffusivity and transport velocity respectively ofa population of cargo, and N is a normalization factor. Eqn. (10) is then solved over the entire real line4sing the Fourier Transform and truncated to ( −∞ , L ] by the Method of Images to arrive at a closed-formapproximation to the solution: u ( x, t ) = 1 √ πDt (cid:18) e − ( x − vt )24 Dt − e vLD − ( x − vt − L )24 Dt (cid:19) . (12)We then truncate the domain to [0 , L ] so that we can compare with our other scales. See Appendix for adiscussion of the error in this assumption.It is informative to examine the system at different levels of approximation. An intermediate QSSapproximation that assumes only that τ α (cid:28) τ K is obtained by condensing the system of two bindingreactions to one effective binding reaction between motors and cargo. Denote the probability that a cargo isfreely diffusing ( s = 0) given that it has n motors attached is given by β ( n ) = ρ (0 , n ), where ρ ( · , · ) is definedin (7). The effective binding and unbinding of a cargo and motor is summarized in Eq. (13): M + M n C ( N − n )( k on β ( n ) + k bon ) −−−−−−−−−−−−−−− (cid:42)(cid:41) −−−−−−−−−−−−−−− ( n + 1) k off M n +1 C (13)Where M denotes a motor and C denotes a cargo. We use this binding reaction along with the diffusiveand transport terms to define a one-dimensional partial differential equation given by Eq. (17) (See Ap-pendix). Note that the two approximations ((17) and (10)) are consistent; if we apply an additional timescaleseparation τ K (cid:28) τ V then (17) converges to (10). We are interested in exploring parameter conditions that allow for effective transport. To quantify this weexamine the probability flux under our two boundary conditions (3) and (4). For an absorbing boundary atthe end of the microtubule, we have the time-dependent flux given by J ( t ) = − D cargo N (cid:88) n =0 ∂p∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 ,x = L + v trans N (cid:88) n =1 n (cid:88) s =1 p ( n, s, L, t ) . (14)The survival probability, defined as the probability that the cargo has not exited the system at x = L beforetime t , is given by S ( t ) = (cid:90) t J ( t (cid:48) ) dt (cid:48) . (15)The median transport time T median is given implicitly by S ( T median ) = 0 .
5. For periodic boundary conditions,we examine the steady-state probability flux given by J ∞ = lim t →∞ v trans N (cid:88) n =1 n (cid:88) s =1 p ( n, s, L, t ) . (16)Based upon our QSS approximation (10), the steady state flux is approximated by J ∞ ∼ ¯ vL . Cooperation among low processivity motors optimizes cargo flux
To obtain a steady state cargoflux, we impose periodic boundary conditions and begin with cargo evenly distributed throughout the system.We allow the system to approach a steady-state distribution by stopping the simulation when the changebetween each time step is sufficiently small. We then calculate the average flux and compare it to thetheoretical maximum. Figure 2(A), which showed a significant change in transport flux with respect to α .When N >
1, we see an optimum is present around α = 0 .
1, where on average ∼
10% of motors are freelydiffusing in the cytosol at any given time. Figure 2(C) plots the location of this optimum. Increasing N , thetotal number of binding sites for motors, causes an increase in the effective transport flux for all values of α .Surprisingly, limited processivity leads to larger steady-state transport fluxes than using perfectly processive5igure 2: The steady state behavior as a function of α , the fraction of freely diffusing motors. (A)Effective transport flux, calculated using the velocity of the cargo population once net binding rate betweencargo and motors reached 0 ( D cargo = 0 . · − µ m s − ). (B) Maximum % increase in steady-state effectiveflux when motors are allowed to dissociate from MTs during transport. (C) α values at the peaks in (A) asa function of N . Solid lines denote the QSS solution and dashed lines denote the computational solution.Other parameters used for both cases are: L = 10 µ m, v trans = 1 µ m s − , k on = 5 Hz, k off = 1 Hz and D motor = 1 . µ m s − motors that do not dissociate from the MT ( α = 0), provided that cargoes possess multiple motor bindingsites (i.e., N > α = 0 and α = 0 . N (see Fig. 2(B)). Even for a relatively modest number ofmotors ( N = 3), optimizing the processivity yields a flux increase in excess of 300% over perfectly processivemotors when the diffusivity of the free motor protein is high relative to the diffusivity of the free cargo. A separation of motor binding timescales reduces cargo passage times
Cargo are assumed to nothave any motors bound to them when they begin at the negative end of the MT. As such, we expect ourbinding parameters to play a significant role in how quickly cargo reach a steady-state equilibrium with therest of the system. Here, shorter median passage times correlate with an increase in the system’s efficiencyin moving cargo. In the following studies, we impose a Robin boundary condition with all cargo beginningat position x = 0 and monitor cargo distributions up to the absorbing boundary.The time needed for cargo to equilibrate with motors also affects transit times. We use the aforementionedsetup to calculate the amount of time needed for the probability of passage to the end of the domain toreach 0 . < µ m) where diffusiondominates, binding timescales smaller than that of transport results in shorter travel times. However, this isquickly overcome, such that after approximately 2 µ m, faster binding result in shorter median passage times.In Fig. 3(B), we record the distribution of motors on the cargo as a function of position x at the mediantime. We find that cargo rapidly bind to motors as they move down the system, such that most cargo havemore than one motor and the distribution reaches quasi-steady state within a few microns.Fig. 3(C) plots the convergence to the QSS assumption. As τ α decreases (i.e. more rapid binding andunbinding as reflected by an increase in m on and m off ) we see that our solution rapidly converges to thenumerical solution, the regime that maximizes cargo flux along MT. In Fig. 3(D), we see that decreasing τ α creates the optimum in transport flux at α = 0 .
2. From these figures we conclude that short-lived bindingbetween motors and MT, as seen physiologically, is in fact a necessary condition to create an optimumtransport flux.
Freely diffusing motors bind cargo more rapidly than motors engaged on a MT
To understandthe relationship between processivity and transport, we look at the average number of motors on the cargo.In the same model setup used to create Fig. 2, we use the distribution of the number of motors bound tocargo at steady-state to calculate averages. We also use the analytic equation for the conditional probabilitygiven by (8) to calculate the average number of crosslinks.The optimum seen in Fig. 4(B) for
N > α →
0. On the other hand, maximizing the total number of6igure 3:
The effect of timescale separation on transport efficiency. (A) The median time to reachthe end of the system. (B) The distribution of motors on the cargo (at the median time) as a function ofdistance along the MT. (C, D) Convergence between the Monte Carlo simulation and computational PDEsolution as quasi-steady state assumption is taken. Constants used are: N = 10, D cargo = 0 . µ m s − , D motor = 1 . µ m s − , v trans = 1 µ m s − , α = 0 . k on = 5 Hz, k off = 1 Hz.Figure 4: The average number of motors and average number of engaged motors at steady state.
Solid lines denote the the QSS solution and dashed lines denote the computational solution. (A) Expectednumber of motors on cargo. (B) Expected number of motors bound to both cargo and MT. Same parametervalues as used in Figure 2. 7otors bound to the cargo (see Fig. 4(A)) requires low processivity ( α → We have presented a new theory to compliment the expanding literature regarding motor processivity. Thetheory’s major benefits are two-fold. First, it is based on simple, well documented motor binding kinetics,providing an easily explained mechanism. Second, the results are based on a broadly defined model thatconnects to the more general theory that fast, weak-affinity interactions of crosslinking molecules provideoptimal steady-state binding characteristics (for example, between antibodies and mucin polymers) [12, 15].Before a cargo accumulates motors and encounters a MT, the dominant mode of travel is diffusion.Diffusive motion is not efficient for transport over large distances, but it is efficient for self assembly ofcargo-motor complexes. Once a sufficient number of motors bind to the cargo and a MT is encountered,directed motion becomes dominant and diffusive motion has little effect on the remaining travel time to theend of the MT.Allowing multiple motors on a cargo exponentially increases the probability of maintaining at least onemotor engaged to the MT at all times. This is true for any fixed 0 < α < α → α →
1, there are potentially many motors bound to a cargo, butfew motors engaged to a MT. These competing factors set the stage for an optimal transport flux for anintermediate value of 0 < α < α ≈ . k bon ) between engaged motors (on a MT) and freely diffusing cargo is proportional to D cargo . The binding rate ( k on ) between freely diffusing motors and cargo is proportional to D cargo + D motor ,which is comparatively unaffected by cargo size and diffusivity when D cargo (cid:28) D motor . As a result, motor-cargo binding at α = 0 (where all motors are engaged to a MT) becomes an unlikely event for large cargoeswith small diffusivity. While not explored here, additional factors such as confinement and subdiffusion in avisco-elastic liquid can only compound this effect.Agreement between the three timescales of our model was instrumental for validating numerical codesand to highlight the convergence of our approximations. In particular, convergence between full and reducedmodels as the motor-MT binding timescales decreases shows that a single diffusion equation can describethe system under exactly the conditions that make the optimal transport phenomenon possible (i.e., theseparation of timescales). Finally, we considered the simplest geometry of active transport, namely, a 1DMT domain of length L . Future work will be necessary to elucidate the additional effects of more complicatedgeometries on the optimal transport flux phenomena introduced here. Numerical Computation of PDE
Under the quasi steady state assumption, the system can be equiva-lently modeled by examining the population of cargo using the following 1-D partial differential equation: u t ( x, t ) = D u xx − V u x + A u (17)where the vector u is an N + 1 vector of cargo concentrations and the i th component corresponds to cargowith i motors bound. D and V are diagonal matrices with D ii = β ( i ) D cargo and V ii = (1 − β ( i )) v trans . A has values on the off-diagonals corresponding to the reactions in Equation (18). In this work, (17) issolved in MATLAB and data is stored as concentrations and fluxes at each position and time. The Forward-Time Centered-Space method is used to solve the diffusion equation (Forward-space for the advection term).8oundary conditions are usually considered to be Robin and initial conditions are set as a cargo concentrationwith n = 0 at x = 0 only. The stop criterion is determined by the amount of cargo still in the system.However, for steady-state analyses we choose a periodic boundary condition and a uniformly distributedinitial condition of cargo with n = 0. The stop criterion in this setup is determined by the magnitude of thechange in the distribution of cargo concentrations with respect to n . See Fig. 5 for a comparison betweenthe computational and analytic methods. The solution is best for large v because the effect of the absorbingboundary condition at x = 0 is minimized. Therefore, for a given diffusivity and transport velocity, usingthe optimal α will create the best congruence between the two schemes. At worst (i.e. v = 0), the analyticsolution is half the value of the computational model for all x . Because the focus on this work was primarilyon the agreement between the analytic and computational v , performing a more exact calculation was notnecessary. It is possible to perform additional iterations of the Method of Images to improve the analyticsolution. This procedure can be done infinitely, creating a sequence of functions that converges to the samesolution as the computational scheme (up to machine precision).Figure 5: Comparison of Steady-state computational scheme and analytic scheme defined by Equation (12).Constants used: N = 10, L = 10 µ m, v trans = 1 µ m s − , k on = . k off = 1 Hz, D cargo = . µ m s − , and D motor = 10 µ m s − Monte Carlo Simulation
The Monte Carlo Simulation no longer assumes the binding and unbindingrates between microtubule and motors are infinite. Let m on and m off denote those finite rates. Let k on , k bon ,and k off be as before.For a single cargo, let N be its total number of binding sites. Let n denote the number of motors boundto the cargo, and let s denote the number of those bound motors that are also cross-linked. Let the state bedefined as the ordered pair ( s, n ). The state changes according to the following rates:( s, n ) ( N − n ) k on −−−−−−−−− (cid:42)(cid:41) −−−−−−−−− ( n − s + 1) k off ( s, n + 1) (18)( s, n ) δ s, ( N − n ) k bon −−−−−−−−−− (cid:42)(cid:41) −−−−−−−−− ( s + 1) k off ( s + 1 , n + 1) (19)( s, n ) g ( s )( n − s ) m on −−−−−−−−−− (cid:42)(cid:41) −−−−−−−−−− ( s + 1) m off ( s + 1 , n ) (20) g ( s ) = (cid:40) D cargo D motor , if s = 01 , else (21)With these state changes, the reaction rate matrices M s and V n,s are the following N +1 × N +1 matrices:9 s = m off . . . n D cargo D motor m on m off . . .
00 ( n − m on . . .
00 0 ( n − m on . . . 00 0 0 0... ... ... nm off (22) V n,s = k off . . . N ( k on + k bon ) 0 2 k off . . .
00 ( N − k on + δ s, k b on ) 0 . . .
00 0 ( N − k on + δ s, k b on ) . . . 00 0 0 0... ... ... nk off (23)The code first builds a square matrix where the rows and columns represent all possible states ( s, n ). Theentry in ( s , n ) row and ( s , n ) column is the rate of ( s , n ) going to ( s , n ). For the actual simulation,suppose our cargo is currently at position x , time t , and just changed to state ( s, n ). Determine the timeelapsed by t e = − λ log ( U (0 , λ is the rate to exit the current state. If s >
0, it means the cargo is cross-linked, so it is moving onlyby transport. The change in position ∆ x = vt e (25)where v is the transport velocity. If s = 0, it means the cargo is free-floating, therefore it is only moving bydiffusion. Using the equation ∆ x = (cid:112) D cargo t e N (0 ,
1) (26)we can determine the next position of the cargo. We determine the next state by using the rates matrix anda uniform random variable. The next time is given by t + t e , and the next position is given by x + ∆ x .A few things to note: we start the cargo with x = 0 , t = 0 , ( s, n ) = (0 , x = 0.If the cargo reaches the end of microtubule with length l , the simulation for this cargo is terminated.By running this simulation for a large number of cargoes, we can extract data analogous to the dataproduced from the computational and analytic solutions. As Fig. 3(C) shows, if we let the values of m on and m off tend to infinity while keeping their ratio constant, the density distribution of the cargoes along themicrotubule tends to the computational and analytic solutions at any given time. References [1] H. Bannai, T. Nakayama, M. Hattori, and K. Mikoshiba. Kinesin dependent, rapid, bi-directionaltransport of er sub-compartment in dendrites of hippocampal neurons.
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