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. It shall be interesting then to investigate whether this type of description can be extended to describe directly the movement of cargo driven by motors to understand some of its characteristics observed in experiments. |
Cargo transport by motors happens at the cellular environment where simple diffusion of vesicles or nutrients is severely limited by the presence of innumerous structures inside the cytoplasm. Besides, it is known that virus particles can take advantage of the existing transport mechanisms using molecular motors as car... |
. Therefore, by studying the properties of a model that leads to quantitative predictions on the movement of both motor and cargo might be helpful to understand mechanisms to prevent cell infection and/or to design more efficient drug carriers |
We have already worked on this problem to investigate the movement of cargo in connection to the short-time behavior of the motor density profile defined in the context of the continuum limit of an ASEP with periodic boundary conditions |
. Here, we present an alternative to describe the long-time regime (steady-state) of the movement of cargo using a discrete version of the model. The elementary dynamical processes, that is, the processes at the level of individual particles moving on a defined one-dimensional lattice with periodic boundary conditions ... |
. This kind of model can also be used to study the properties of the density profile in the presence of defects The important point we want to notice is that, up to the present, all kinds of particles in the ASEP models presented in the literature are, in all cases, provided with their own (intrinsic) dynamics. As a ru... |
[EQUATION] that is, either particles of first [MATH] and second [MATH] classes can move by their own if the target neighboring site is unoccupied [MATH] . They can also move if interacting with other particles by interchanging occupancy sites as in [MATH] |
[MATH] . This is also the case of another ASEP-like model for which the particles with different dynamics simulate the presence of cars [MATH] and trucks [MATH] on a same traffic road |
. For this model, the direction of the intrinsic movement of particle [MATH] is opposed to that in the previous case: [EQUATION] |
As we want to describe the movement of cargo driven by motors using an ASEP-like model, we need first to think on possible ways cargo may use the motors to move and up to what extent the presence of cargo can affect the intrinsic dynamics of motors. The first idea that occurred to us is that the elementary processes in... |
Based on the above considerations we present in Section 2 what we conceived as a minimum model that is able to account explicitly for the dynamics of both cargo and motors expected as a result from their mutual interactions. This is presented as an ASEP model that incorporates a few common characteristics of this biolo... |
, these states are conveniently represented as products of certain non-commuting matrices which can be used to calculate the properties of interest as the average cargo velocity discussed in Sec.3. The analytical results obtained in this way are used in Section 4 to discuss the phenomenological consequences of the mode... |
An ASEP model for motors and cargo There are a few proposals in the literature to characterize the origins and the role of the components involved in the cargo/motor interactions at a microscopic level |
. Data suggest that such interactions are mediated by certain proteins - dynactin is the most studied - but apparently there is consensus about their short-range nature as a general characteristic. In terms of the scales involved, this is equivalent to say that such cargo/motor interactions happen by ”direct contact” a... |
, there is limited information on the typical times each motor, or group of motors, remains attached to cargo in the course of its movement. Therefore, in building up our model, we avoid details of the processes associated to such microscopic interactions and simply account for these as stochastic processes. We suppose... |
The ASEP model considered here consists on a collection of [MATH] motors, and [MATH] cargoes, referred in the following as particles of type 1 and particles of type 2 respectively, distributed among the [MATH] sites of a one dimensional cyclic lattice. Each particle occupies a single site and [MATH] sites remain empty ... |
[MATH] As usual, in this kind of description the stochastic dynamics is defined through a Poissonian process taking place at the lattice such that at each time interval [MATH] , a pair of consecutive sites [MATH] and [MATH] is selected at random and the system is updated depending on whether it is possible to exchange ... |
[EQUATION] where the pair of sites [MATH] is being represented by the values assumed by the variables [MATH] According to these rules a cargo is allowed to move only if ”assisted” by a motor at a neighbor site. We see this as a possibility to describe the fact that the movement of cargo is conditioned to that of motors... |
Here, we are concerned with the kinematics of the cargo at long-time regimes. For this, we use the matrix-approach to represent any configurations of the system of [MATH] sites by a product of [MATH] matrices, so that the probability of occurrence of a particular configuration [MATH] is given by |
[EQUATION] where the Kronecker delta symbols select the correct occupancy and [EQUATION] is the normalization. The sum extends over all allowed configurations for which [MATH] and [MATH] . In the product above, site [MATH] is represented by matrix [MATH] if it is occupied by a motor (particle 1), by matrix [MATH] if it... |
In the stationary state, the probabilities [MATH] for all configurations [MATH] satisfy the condition [EQUATION] where [MATH] is the rate at which the exchange of particles occur between neighboring sites so that all nonzero terms in the above sum are those for which configurations [MATH] and [MATH] differ from each ot... |
As an example, we consider for [MATH] [MATH] [MATH] the following configuration [MATH] . There is just one way to leave this configuration that is through the process: [MATH] with [MATH] On the other hand, there are two ways to reach this configuration, either by exchanging particle positions (i) by the process [MATH] ... |
[EQUATION] As a general rule, the main difficulty in using this method to determine the probabilities [MATH] for each configuration [MATH] is to find the algebra, if any, that must be satisfied by the corresponding matrices of a given ASEP model in order to satisfy condition ( ) for a given dynamics as in ( In the pres... |
[EQUATION] for [EQUATION] then, Eq. ( ) is satisfied. This can be tested using explicit examples, as the one considered in ( ). Using ( to evaluate the traces, one can easily check that the identity holds trivially. In the following, we study properties of this model that are of interest for examining the consequences ... |
Average cargo velocity For [MATH] i.e. just one cargo, in the presence of [MATH] motors distributed along a cyclic lattice of [MATH] sites, we consider [MATH] to ensure that at least one site in the system remains empty. In this case, the average cargo velocity [MATH] at steady state is expressed as |
[EQUATION] where the sums in the numerator extends over all configurations of [MATH] motors distributed among [MATH] sites. The first sum in the RHS accounts for all configurations in which the site at the immediate right of cargo is occupied by a motor. The second sum accounts for all configurations in which there is ... |
can be written as [EQUATION] where the sum extends over all configurations of [MATH] motors distributed among [MATH] sites. In order to make reference to the above traces over products of matrices, it is convenient to introduce the functions [MATH] to indicate the configurations having the n-uple [MATH] fixed, the corr... |
[EQUATION] where [EQUATION] for [MATH] and analogous definitions for [MATH] and [MATH] Alternatively, due to the cyclic property of the trace, [MATH] |
can also be calculated from [EQUATION] These two expressions ( 13 ) and ( 14 ) are equivalent and both will be used below, at convenience. Notice that we can rewrite the sum over configurations in [MATH] as |
[EQUATION] where the first (second) sum on the RHS extends over all configurations of [MATH] motors with the triplet [MATH] fixed [MATH] motors with the triplet [MATH] |
fixed) distributed among [MATH] lattice sites. Making use of the decompositions in ( 13 ) and ( 15 ), the average cargo velocity ( 12 ) is expressed as |
[EQUATION] We compute [MATH] as it is expressed in Eq. ( 14 ). A convenient way to perform the calculations indicated above is to replace the sums over site variables [MATH] by sums over blocks defined by the integers [MATH] and [MATH] for [MATH] (see for example, ref. |
). In this representation, [EQUATION] with [MATH] [EQUATION] with [MATH] [EQUATION] with [MATH] and [MATH] [EQUATION] with [MATH] and [MATH] |
From the algebra in ( ), it follows that [MATH] This identity is needed in the evaluation of the above traces for general configurations of the variables [MATH] . The results are |
[EQUATION] Notice that [MATH] and [MATH] are functions of the size [MATH] of the block, i.e. of the number of particles of type [MATH] (motors) that precede particle [MATH] (cargo). Because of this, the evaluation of the respective sums over [MATH] and [MATH] in configurations of the type [MATH] (or [MATH] excluding [M... |
[EQUATION] and [EQUATION] [MATH] and [MATH] correspond to configurations that do not present motors behind the cargo. In the sum over configurations of the type [MATH] |
one must account for the number of ways to distribute [MATH] motors into [MATH] sites (from the total of [MATH] sites, there must be excluded [MATH] , one to fix the cargo and the other to fix an empty site). Then, |
[EQUATION] and [EQUATION] We now proceed by computing the sum over integer [MATH] in ( 22 and ( 23 ). 3.1 Aproximate expression for the average velocity of the cargo in the limit of very large systems |
Our intention is to obtain an expression for the average velocity [MATH] of cargo in the limit for which both the number of sites and the number of motors (which are conserved by the dynamics) are taken very large, that is [MATH] and [MATH] These limits are supposed to be taken in such a way to ensure that the ratio be... |
[EQUATION] converges to a defined density of motors [MATH] , such that [MATH] These same limits have already been considered in Ref. |
to calculate the average velocity of trucks in a related traffic problem. Here, we proceed along the same lines sketched by these authors. |
First, we use Stirling formula [MATH] for very large [MATH] , to approximate the combinatorial coefficients in ( 24 ) and ( 25 ). It results |
[EQUATION] and [EQUATION] In order to evaluate the sums [MATH] and [MATH] in ( 22 ) and ( 23 ), we follow the procedure used in Ref. |
There, sums involving factorials of this kind are approximated by integrals and the asymptotic regimes are obtained using Laplace’s method |
. Considering then the limit of very large systems and defining [MATH] we calculate [EQUATION] where we have defined the function [MATH] of a single variable [MATH] as |
[EQUATION] and used the fact that in the specified limit, the sum in [MATH] converges to the integral as [MATH] Now observe that [MATH] has a maximum at |
[EQUATION] so in order to apply Laplace’s method in the present case, one must distinguish between two possibilities, namely if [MATH] [MATH] ) then, [MATH] |
[MATH] i.e. [MATH] belongs to the integration interval. In this case, Laplace’s method gives [EQUATION] or if [MATH] either for [MATH] or [MATH] , then [MATH] Therefore, in this case [MATH] does not belong to the integration interval. Since [MATH] is a monotone decreasing function of [MATH] the integral is dominated by... |
[EQUATION] Analogously, the asymptotic behavior of the sum [MATH] 23 must be analyzed according to the range of [MATH] if [MATH] |
[MATH] then [EQUATION] or if [MATH] either [MATH] or [MATH] then [EQUATION] From these results, one concludes that if [MATH] then both [MATH] and [MATH] are the dominant factors in the expression for [MATH] |
both in the numerator and in the denominator. In this regime, [EQUATION] On the other hand, if [MATH] , all factors in the expression ( 16 ) for [MATH] are of the same order of magnitude and then, |
[EQUATION] The consequences of these results to the phenomenology of cargo movement will be analyzed in the next section. Discussion and concluding remarks |
The aim of the present work is to study the consequences of introducing cargo into certain lattice models where there is also present a set of biased molecular motors interacting through excluded-volume interactions. We consider an ASEP-like model specially designed to account for both kinds of particles. Therefore, th... |
We look for the probabilities associated to the configurations of the system at the stationary state which are represented by products of certain noncomuting matrices |
. Using this representation, we were able to make quantitative predictions on the average properties that characterize the movement of cargo. We focus on the computation of the average velocity of cargo [MATH] whose behavior predicted by the model suggests that the system displays a phase transition under variation of ... |
the function [MATH] displays a change in its behavior at values of [MATH] for which [MATH] as [MATH] becomes independent of [MATH] |
In order to interpret these results, it shall be easier first to discuss on the kind of movement one would expect for cargo in the context of the considered ASEP. The mechanisms in ( ) that define its elementary movements within each unit interval of time correspond to those of exchanging positions with a neighbor moto... |
The dependence of [MATH] on [MATH] in Fig.2 confirms these expectations showing that [MATH] assumes only negative values, at all ranges of parameters. One could expect, in principle, that at high values of [MATH] |
there would be a balance between a tendency for maintenance of motors in front of the cargo, as excluded-volume become more important and eventually would be responsible for expressive motor accumulation at cargo’s front. So, in principle, one could think that for sufficient high values of [MATH] there would be a chanc... |
understood by recognizing the formation of an ”infinite” (macroscopic) cluster of motors behind the cargo with which it can always exchange positions. The formation of such infinite cluster would be a consequence of the phase transition (of condensation type) predicted for this system. The average velocity in this regi... |
We can then summarize these results by saying that the single cargo in this system of many motors develops, a backwards movement at any value of the hopping rates [MATH] and [MATH] , or density [MATH] The magnitude of such velocity, however, is highly dependent on [MATH] and becomes constant at such values of [MATH] gr... |
Actually, data from Drosophila embryos show that cargo velocity presents distinct behaviors depending on the stage of embryo development. It remains to investigate whether these changes could be associated to corresponding changes in the density of motors available at each of these stages. To our knowledge, there is li... |
Acknowledgments We acknowledge the financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). Figure Caption |
Fig.1 - A configuration of the one-dimensional (discrete) ASEP model for interacting motors (gray) and cargo (black). Each particle, occupies a single site at each instant of time. The non-occupied (empty) sites are represented by line segments. The random processes are such that each motor is allowed to hop at rate [M... |
Fig.2 - Average velocity of cargo as a function of motor density [MATH] , at various values of parameter [MATH] , for fixed [MATH] and [MATH] as indicated. The predicted phase transition is illustrated by the change in the behavior of [MATH] that assume a constant value for [MATH] |
# Source: arxiv 0810.1568 # Title: Effects of the chemomechanical stepping cycle on the traffic of molecular motors # Sections: all # Downloaded: 2026-03-03T05:14:11.290190+00:00 |
Effects of the chemomechanical stepping cycle on the traffic of molecular motors Abstract We discuss effects of the stepping kinetics of molecular motors on their traffic behavior on crowded filaments using a simple two-state chemomechanical cycle. While the general traffic behavior is quite robust with respect to the ... |
pacs: 87.16.Nn, 05.60.-k, 05.40.-a Introduction Molecular motors power the transport of various kinds of cargoes within cells by directed stepping movements along filaments of the cytoskeleton Howard ( 2001 . The similarities (and differences) compared to highway traffic Klumpp et al. ( 2007 and the fact that cells are... |
Lipowsky et al. ( 2001 ); Klumpp and Lipowsky ( 2003 ); Kruse and Sekimoto ( 2002 ); Parmeggiani et al. ( 2003 ); Evans et al. ( 2003 ); Klumpp and Lipowsky ( 2004a ); Klein et al. ( 2005 ); Nishinari et al. ( 2005 |
Most studies of molecular motor traffic have modeled the stochastic stepping of an individual motor by a single Poissonian step, which is modified in dense traffic by an exclusion rule similar to the well-studied asymmetric simple exclusion process (ASEP) |
MacDonald et al. ( 1968 ); Krug ( 1991 ); Derrida et al. ( 1993 ); Schütz and Domany ( 1993 If the site to which a motor attempts to step is not accessible, because another motor (or any other kind of obstacle) is bound there, the step is rejected. Stepping of a molecular motor is however a complex process and consists... |
Nishinari et al. ( 2005 ); Wang et al. ( 2005 ); Greulich et al. ( 2007 However, the main feature of molecular motor traffic visible in experiments |
Leduc et al. ( 2004 ); Konzack et al. ( 2005 ); Nishinari et al. ( 2005 the emergence of traffic jam-like density profiles, with a region of high density of bound motors separated from a low-density region by a sharp interface |
Lipowsky et al. ( 2001 ); Klumpp et al. ( 2005 ); Parmeggiani et al. ( 2003 as well as the basic features of the phase diagrams for transport in open systems Klumpp and Lipowsky ( 2003 ); Parmeggiani et al. ( 2003 are very robust with respect to such extensions of the simplest models. |
In this paper, we discuss observable effects of the stepping kinetics on the traffic behavior. To be explicit we consider the simplest possible stepping cycle, which consists of transitions between two motor conformations or internal states of the motor, with only one transition involving movement of the motor to the n... |
II Model without motor binding/unbinding We first consider the case without exchange of motors with a reservoir, i.e. we neglect that fact that bound motors unbind from their filamentous track and unbound motors bind to it. In the simplest model, the motor cycles between two conformational states, state 1 and 2, which ... |
Fig. (a) shows the motor current as a function of the motor density [MATH] on the filament as obtained from simulations. Here and in the following plots, we normalize the motor current [MATH] and the motor velocity [MATH] by the single motor velocity [MATH] , which we consider as a known quantity given by experimental ... |
[MATH] , symmetric around [MATH] . If however the conformational transition cannot be neglected, this symmetry is lost and the maximum of the current is shifted towards higher densities. At the same time, the value of the maximal current increases. The absence of particle-hole symmetry, i.e. symmetry upon interchanging... |
While the current is the most important characteristics of a traffic system from a theoretical point of view, it is not easily accessible in experiments. A quantity that can be measured directly is the motor velocity, which can be determined by tracking individual labeled motors in a background of unlabeled motors Seit... |
The density-dependence of the motor current and the motor velocity can be described analytically using a mean-field approximation: all traffic effects can be subsumed into an effective rate for the actual stepping by replacing the rate [MATH] by [MATH] . Then the probabilities [MATH] and [MATH] that the motor is in con... |
[EQUATION] and thus to [EQUATION] and [EQUATION] These expressions are plotted in Fig. (c) and (d), again normalized by the single motor velocity [MATH] . These plots show qualitatively the same behavior as the simulation data in Fig. (a) and (b). However, except for the limiting case [MATH] which corresponds to the us... |
[MATH] in Fig. are close to this limit. III Model with binding/unbinding kinetics Next, we consider the effect of motor binding to and unbinding from the filament [Fig. (b)]. Even the simplest model with only two internal states of the motor allows many different implementations of binding and unbinding: unbinding may ... |
[MATH] , where [MATH] are the second-order binding rates and [MATH] is the concentration of unbound motors. For simplicity, we restrict the discussion to the two cases, where unbinding occurs predominantly in one of the two states. Furthermore, our simulations indicate that whether newly bound motors are in state 1 or ... |
Results from simulations that include binding and unbinding of motors are shown in Fig. . The solid and dashed curves in Fig. correspond to cases where unbinding occurs predominantly from state 1, with [MATH] and [MATH] |
en2 and only from state 2, with [MATH] and [MATH] respectively. In both cases, the unbinding rates are chosen to match a run length of 100 steps for the single motor. For processive motors, including motor binding and unbinding has only a small effect on the density-dependence of the motor current [Fig. (a)] and the mo... |
Closely connected with the unbinding rate is the run length or processivity, i.e. the number of steps a motor takes before detaching from the filament, which is shown in Fig. (d) as a function of the motor density on the filament. For simple Poissonian steppers, the unbinding rate is independent of the motor density, b... |
IV Experimental relevance We have discussed the effects of the chemomechanical stepping cycle of a molecular motor on the traffic of many molecular motors using a simple two-state chemical cycle. For this model we have determined observable quantities, in particular the motor velocity and the average run length, which ... |
Our analysis shows that the more complex stepping kinetics tend to diminish the slowing down of motors due to traffic congestion, but this effect is not very large. It is possible that this effect contributes to the surprisingly small decrease of the velocity observed in the experiment of Seitz and Surrey |
Seitz and Surrey ( 2006 . In this experiment, the velocity of kinesin 1 motors remained almost constant up to unbound motor concentrations that resulted in nominal bound motor densities of [MATH] 0.3-0.5 and reduced the motor binding rate about two-fold. The interpretation of this experiment is however complicated by a... |
A more pronounced effect of the cycle is seen in the density-dependence of the run length and the effective unbinding rate. Depending on the state from which unbinding from the filament predominantly occurs, the effective unbinding rate may both increase or decrease with increasing traffic density, while the run length... |
Seitz and Surrey ( 2006 also suggests that kinesins may be bound in different ways in the two experiments. In general, even the equilibrium binding of dimeric motors to filaments (in the absence of ATP, i.e. without active movements) may exhibit a range of stoichiometries and rather complex dynamics Frey and Vilfan ( 2... |
Finally, the main effect of crowded molecular motor traffic is the emergence of a traffic-jam-like domain of high motor density at the end of a filament |
Lipowsky et al. ( 2001 ); Klumpp et al. ( 2005 ); Parmeggiani et al. ( 2003 as observed in several experiments Leduc et al. ( 2004 ); Konzack et al. ( 2005 ); Nishinari et al. ( 2005 If the filament is in contact with a large reservoir of unbound motors, the length of the high density domain or traffic jam is of the or... |
Seitz and Surrey ( 2006 , one may expect to observe much longer traffic jams. Extending the arguments given in ref. Klumpp et al. ( 2007 , one obtains the estimate that the jam length is larger than the run length by a factor [MATH] |
Appendix The model we have discussed in this article represents the simplest possible chemomechanical cycle of a molecular motor with two subsequent transitions that are treated as irreversible. This minimal model of a stepping cycle provides a good description of the dynamics of kinesin motors under typical experiment... |
Surprisingly, the main difference to the simple model is already obtained when the reverse transition for the transition from state 1 to state 2 is included: In the original model with irreversible transitions, a motor in dense traffic often waits in state 2 until the site in front becomes available. If the reverse tra... |
Finally we also simulated the case where a motor in state 2 may make either a forward step or (with a smaller rate) a backward step, which provides the simplest model where forward and backward steps occur along different reaction pathways, a situation suggested for kinesin by both experiments and modeling Carter and C... |
Acknowledgements. The authors thank A. Seitz and T. Surrey for discussions of their experiments. SK was supported by Deutsche Forschungsgemeinschaft (grants KL818/1-1 and 1-2) and by the National Science Foundation through the Center for Theoretical Biological Physics (grants PHY-0216576 and 0225630). |
# Source: arxiv 0810.1909 # Title: Mapping of 2+1-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas model of dimer # Sections: all # Downloaded: 2026-03-02T08:58:06.617876+00:00 |
Mapping of 2+1-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas model of dimers Abstract We show that a [MATH] dimensional discrete surface growth model exhibiting KPZ class scaling can be mapped onto a two dimensional conserved lattice gas model of directed dimers. In case of KPZ height anisotropy the ... |
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