Dynein catch bond as a mediator of codependent bidirectional cellular transport

Intracellular bidirectional transport of cargo on Microtubule filaments is achieved by the collective action of oppositely directed dynein and kinesin motors. Experimental investigations probing the nature of bidirectional transport have found that in certain cases, inhibiting the activity of one type of motor results in an overall decline in the motility of the cellular cargo in both directions. This somewhat counter-intuitive observation, referred to as paradox of codependence is inconsistent with the existing paradigm of a mechanistic tug-of-war between oppositely directed motors. Existing theoretical models do not take into account a key difference in the functionality of kinesin and dynein. Unlike kinesin, dynein motors exhibit catchbonding, wherein the unbinding rates of these motors from the filaments are seen to decrease with increasing force on them. Incorporating this catchbonding behavior of dynein in a theoretical model and using experimentally relevant measures characterizing cargo transport, we show that the functional divergence of the two motors species manifests itself as an internal regulatory mechanism for bidirectional transport and resolves the paradox of codependence. Our model reproduces the key experimental features in appropriate parameter regimes and provides an unifying framework for bidirectional cargo transport.

The transport itself happens bidirectionally, wherein dynein and kinesin motors carry the cellular cargo on microtubule(MT) filaments [1,15]. While the phenomenon of bidirectional transport has been well studied experimentally under both in-vitro and in-vivo conditions for variety of different systems [1,25], the underlying mechanism by which the motors involved in bidirectional transport are able to achieve regulated long distance transport is far from clear and is a subject of much debate [1,7,[25][26][27][28][29].
A theoretical framework proposed to explain the bidirectional transport is based on the tug-of war hypothesis [1,8,15,25,28,30,31]. The basic underlying premise of this hypothesis is that the motors act independently, stochastically binding to and unbinding from the filament and mechanically interacting with each other through the cargo that they carry ( Fig. 1a) [25,28,30]. The resultant motion arises due to the competition between the oppositely directed motors with the direction of transport being determined by the stronger set of motors [28,30]. Diverse experiments have provided support for this mechanical tugof-war picture, with the observed features matching with the simulated cargo trajectories [15,28,[32][33][34][35][36]. Studies on endosomal transport in D. discoideum cells and on mitochondria in mitral cell dendrites have observed elongation of cargo when attached to both motor species, supporting the hypothesis that oppositely directed mechanical forces act simultaneously on the cargo [37,38].
While the stochastic tug-of-war model has proven to be effective in understanding many transport properties, there remain a large class of experiments whose findings are incompatible with the predictions of this model [25,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Since the oppositely directed motors of dynein and kinesin are involved in mechanical competition, the tug-of-war model predicts that inhibiting the activity of one kind of motor would lead to enhancement of motility in the opposite direction. However, experiments of dynein inhibition using genetic mutations on bidirectional axonal transport, have reported an overall decline in the motility of the cargo [7,25,42]. The mutual interplay of kinesin and dynein in activating each other during bidirectional transport was also studied through inhibition and depletion studies of peroxisome  [29] (points) and the corresponding fit (solid line) from the TFBD model [58]. transport in Drosophila melanogaster S2 cell [46]. It was observed that inhibition of kinesin motors abolishes transport towards the minus end and vice-versa. Similar results have been found in experiments involving dynein and dynactin mutations in Drosophila embryos [47].
Inhibition of dynein and dynactin produces severe impairment of plus end directed motion, providing further support for coordinated motion between the two motor species.
These apparently counterintuitive findings have been referred to as the paradox of codependence, suggesting some kind of coordination between the oppositely directed motors which has not been accounted for in the theoretical tug-of-war model [1,25]. This invites the moot question on how the this paradox can be resolved and understood in terms of the underlying mechanism which governs bidirectional transport. In this work we seek to address this issue by re-examining the theoretical tug-of-war model [28,30].
A striking difference between the single molecular behaviour of dynein and kinesin lies in their unbinding kinetics. Unlike kinesin, dynein can exhibit catchbonding, where the propensity for the dynein motors to unbind from cellular filament decreases when subjected to increased load force (Fig1b) [29,54,55]. In-vivo experiments on lipid droplets in drosophila embryos have measured the residence times -the time the cargo remains bound before detaching -of cargo driven by kinesin and dynein motors under superstall forces. While the residence time of kinesin decreases with increasing force, for dynein the behaviour is reversed and and the residence time is enhanced with increasing opposing load [55]. Similar in-vitro experiments with polystyrene beads carried by kinesins and cytoplasmic dynein have con-firmed this atypical behaviour for dynein motors [29]. Further the distribution of residence times were used to extract a characteristic detachment time and hence dissociation rates as a function of force. These experiments unambiguously show that the detachment rate of dynein increases until the stall force, followed by a catch bond regime where the detachment rate decreases with increasing force. In contrast, the detachment rate of kinesin motors increases exponentially with increasing load force -a characteristic of slip bond [27,[55][56][57].
However this complex behaviour of detachment kinetics of dynein motors has not been taken into account in the tug-of-war theoretical description and instead the detachment rates for both dynein and kinesin motors from the MT filament are modeled as a simple exponentially increasing function of force [27,28]. In our recent theoretical modeling work, accounting for catchbonding in single dynein, we have presented a Threshold Force Bond Deformation (TFBD) model to fit the experimentally observed unbinding rate of single dynein motor in constant load force (Fig. 1b) [29,58]. Further we have illustrated that even for the case of unidirectional transport of cellular cargo by many dynein motors, the collective transport properties can be significantly affected by catchbonding behaviour [58].
In this paper, we focus our attention on this crucial ingredient of catchbonding that is exhibited by the dynein motors and which is missing in the previous theoretical formulation for describing bidirectional transport. We use the TFBD model for dynein, along with the usual slip bond model for kinesin [27,28], to study the transport properties of bidirectional cargo motion. We use experimentally relevant measures to characterize the transport properties of cellular cargo : (i) average processivity, defined as the mean distance a cargo travels along a filament before detaching, and (ii) probability distributions of runtimes in the positive and negative directions and probability distributions of pause times. Using these measures we show that in an experimentally viable parameter space, catchbonding can indeed manifest itself in the transport properties of the cellular cargo exhibiting features consistent with experimental observations of mutual codependence of kinesin and dynein. Our model thus reproduces both the tug-of-war model transport characteristics and codependent transport characteristics in appropriate biological regimes.

RESULTS
We study transport of a cellular cargo with N + kinesin motors and N − dynein motors.
Each of these motors stochastically bind to a MT filament with rates π ± and unbind from the filament with rates ε ± . The instantaneous state of the cargo is expressed in terms of the number of kinesin (0 ≤ n + ≤ N + ) and dynein (0 ≤ n − ≤ N − ) motors that are attached to the filament. At any instant only the attached set of motors generate force on the cargo and are involved in its transport. For a set of similarly directed attached motors, we assume that the load force experienced by each motor is shared equally between them. We use the Stochastic Simulation Algorithm (SSA) [70,71] where the deformation energy E d sets in beyond the stall force, and is modeled by a phenomenological equation [58], The parameter α sets the strength of the catch bond, while F d− and F 0 characterise the force scales for the dissociation energy and the deformation energy respectively. This correctly reproduces the experimentally reported dissociation dynamics of a single dynein as shown in Fig. 1(b) [58]. The unbinding kinetics of kinesin exhibits usual slip behavior (unbinding rate increasing exponentially with increasing load force). The characteristic stall forces and detachment forces of kinesin are denoted by F s+ and F d+ respectively. The values for the various parameter used in the stochastic simulations are listed in Table I. Average Processivity (µm) Average Processivity (µm) As expected, in the absence of catch-bond (α = 0) ( Fig. 2(c)), there is a smooth transition at a critical N + from a regime where the cargo moves in the negative direction to one which moves in the positive direction. This critical N + is almost independent of N − in accordance with our observation in Fig. 2(a-b). In the presence of catch-bonded dynein Experimental techniques to modulate cargo processivity can also be achieved by modifying the binding/unbinding rates of the motor proteins. Dynactin mutations in Drosophila neurons affect the kinetics of dynein binding to the filament, leading to cargo stalls [42]. To investigate this, we tune the bare unbinding rate of dynein motor (ε 0− ) with the dynein catch bond switched on (Fig. 3). For a fixed set of (N + , N − ), we observe that for weak dynein Fig. 3(a)), the processivity starts to decrease from a negative value, as the bare unbinding rate is increased, as expected. However, beyond a critical ε 0− , the processivity saturates to a small positive value. Increasing ε 0− effectively weakens the propensity of dynein to stay attached to the filament. Beyond the critical ε 0− , weakening the dynein further does not lead to any increase in the run length in the positive direction, as might be expected from a conventional tug-of-war scenario, but rather the catch bond ensures that there is no change in the run length. At a slightly higher F s− , (F s− = 2pN in Fig. 3(a) This entire spectrum of behaviour can be visualised as a phase plot of the processivity in the (F s− −ε 0− ) plane ( Fig. 3(b)). These contour plots captures the richness of the processivity behaviour due to catchbonding . For instance in Fig. 3(b) for a range of stall force for dynein 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 In dictyostelium cell extracts, it has been reported that teams of four to eight dyneins and one to two kinesins are simultaneously attached to a cargo [38]. The resultant motion was observed to be minus-end directed with intermittent pauses. To understand these results in the context of our model, we fixed N + = 2 and N − = 6 ( Fig. 4). In the absence of catchbonding, the resultant motion is strongly plus-end directed, as can be seen from the trajectory plots in Fig. 4(d). The probability distributions of runtimes show that there are many more kinesin runs (Fig. 4(c)) than dynein runs ( Fig. 4(a)), and the average runtime is also higher in the case of kinesins. The pauses in this case are also of extremely short duration ( Fig. 4(b)), leading to strongly plus-end directed runs. As an aside we also note that the overall frequency of the cargo in the pause phase always adds up to 0.5 and similarly the overall frequency of the combined positive and negative run adds up to 0.5. This is simply a consequence of the conservation of probability. Due to the stall force of kinesin motor being about 5 times that of dynein and the bare binding rate of kinesin being larger than dynein (see Table I), even with N + < N − , a plus-end directed run, on average, continues for a longer time than a minus-end directed run, leading to larger average runtimes along the positive direction. When dynein catch bond is switched on, the picture changes dramatically. Minusended runs become much more frequent than plus-ended runs, while the average pause time also increases by an order of magnitude compared to the non-catchbonded case, and becomes comparable to the average minus directed runtimes. This is shown in Figs. 4(e)-(g). Load force on dynein due to attached kinesin engages the catch-bond, making it more difficult to unbind from the filament. Therefore, we see that the manifestation of catchbonding in dynein results in strong minus directed runs with longer duration pauses, as is shown in Fig. 4(h). This qualitatively agrees with the experimental observation of transport of endosomes in Dictyostelium cells [38].
In a separate set of experiments which looks at bidirectional motility behaviour of early endosomes in fungus Ustilago maydis, it has been seen that a team many kinesin motors (3)(4)(5)(6)(7)(8)(9)(10) are involved in tug-of-war with only 1 or 2 dynein motors during transport [15]. To study the ramification of catchbond in such a scenario wherein several kinesin motors are in opposition to very few dynein motors, we generate the probability distribution of pause times along with minus-end and plus-end directed runtimes for a cargo being transported by 6 kinesins and 2 dyneins (N + = 6, N − = 2). The results displayed in Fig. 5 illustrates that while in the absence of catchbonding in dynein, the resultant motion would be strongly plusend directed, with very small pause times, incorporation of catchbonding in dynein affects 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 of the peaks of the steady state probability distribution of attached motors. A peak in the probability distribution at (n * + , 0) corresponds to plus directed motion, while a peak at (0, n * − ) corresponds to minus-directed motion, and a peak at (n * + = 0, n * − = 0) corresponds to a paused or tug-of-war state. If there is a single peak in the probability landscape, then the net motion can be of three types, fast plus (+), fast minus (-) or no motion (0). Multiple peaks are also possible and introduce four more possible types of motion, fast plus with pauses (0+), fast minus with pauses (0-), fast bidirectional motion (+-), and fast bidirectional with pauses (-0+). The motion is then categorised into one of these seven classes depending on the number and nature of peaks of the probability distribution.
However, the motility diagram approach has some inherent limitations since interpreting the motion in terms of the peaks may give an incomplete picture when the number of attached motors is very few, as is the case considered here. In Fig. 6, we plot the motility diagram for α = 40k B T and 0− = π 0− = 1/s for F s− = 1pN and F s− = 7pN . We can compare these plots with the contour plots of the processivity obtained for these same parameters as shown in Fig. 2 (d) and Fig. S2 (b). As can be seen clearly, the motility diagrams do not capture the full complexity of motion in the presence of catch bonds. For weak dynein, the motility diagrams predict that for N + > 1 and N − > 2, the motion is either paused or in the fast plus with pause state. However, as the processivity contour plots make clear, there is a range of (N + , N − ) where the nonlinear effects of the dynein unbinding rate actually result in negative directed runs in this region. This can also be seen by considering the runtime probability distributions shown in Figs. 4 and 5. For example for N + = 2 and N − = 6, we have a minus directed transport interspersed with pauses, while the motility diagram simply predicts a no-motion state. Thus for the characterisation of transport characteristics in these low attached motor regimes, the use of processivity contour plots and runtime probability distributions as illustrated in this paper provide a more accurate approach than the conventional motility diagrams.

DISCUSSION
The paradox of codependent transport [25] is in direct contradiction to the standard mechanical tug-of-war [15,28,30] picture between kinesins and dyneins. In particular it has been experimentally observed that inhibiting the activity of one type of motor can actually lead to an overall decline of motility of the cellular cargo being transported by oppositely directed set of motor proteins. A missing ingredient from existing theoretical models has been the catchbonding behaviour of dyneins, where the unbinding rate of dyneins shows a non-monotonic dependence on opposing loads [29,38].
In this article we have explicitly shown how incorporation of catchbonding behaviour of dynein motors in the modeling approach for bidirectional transport is able to resolve this paradox and provide a coherent picture of apparently disparate set of experimental observations. Many of the in-vivo and in-vitro experiments have characterized the nature of bidirectional transport in terms of cargo particle trajectories [15,38]. Applying our model which includes the effect of catchbond in dynein, qualitatively reproduces the experimentally observed features of cargo trajectories during transport. We also find that the processivity characteristics of the cargo need not be in sync with the theoretical construct of motility diagrams [28,30] as illustrated by comparison of Fig.2 with Fig.7. Thus the findings of our model points to the crucial role played by catchbonding in dynein motors in bidirectional transport, highlighting its significance as an internal regulatory mechanism during transport, albeit through mechanical interaction between the motors. We specifically summarize some Diverse experiments have also indicated that mutations of conventional kinesin in Drosophila can hamper motion of cellular cargo in both directions [73][74][75]. This is consistent with the results shown in Figs. 2(b), where reducing the kinesin number can stall cargo motion completely. Interestingly, while kinesin exhibits a conventional slip bond, the cooperative force exerted by the catch bonded dynein on kinesins, and vice versa, introduce a complex interplay which results in signatures of codependent transport being observed even on varying effective kinesin numbers. For example, as shown in Fig. 2(b) for weak dynein, reducing the number of kinesin, can in certain ranges, decrease the overall motion of the cargo in the negative direction. This counterintuitive phenomenon is a direct manifestation of the dynein catch bond, through the nonlinear unbinding kinetics of dynein motors. Similar results pointing to codependent regulation of bidirectional transport has been seen in studies of peroxisome transport in Drosophila melanogaster S2 cell [46], as well as in Drosophila embryos [47]. Our tug-of-our picture then, with the introduction a mechanical regulation mediated by the dynein catch bond, presents a unified resolution of the experimental observations of codependent transport through mutations of both kinesin and dynein motor proteins. Curiously enough, these processivity characteristics also point to the sharp difference in transport characteristics of dynein with high tenacity when com-pared to weak tenacity. In the former case regulatory role of catchbonding is very weak since the typical force scale at which catchbond is activated is quite high with respect to the typical load forces experienced by the motors. It would indeed be interesting to probe further if this is the reason for the strong dynein in yeast not being involved in transport, while weak mammalian dynein are crucial to intracellular transport.
Many of the in-vivo and in-vitro experiments have characterized the nature of bidirectional transport in terms of cargo particle trajectories. Experiments on different systems have reported different number of average bound kinesins and dyneins. For instance for experiments on endosome motion in Dictyostelium cell [38] have reported that 1-2 kinesin are opposed by teams of 4-8 dyneins, and the resultant cargo trajectories are minus-end directed. On the other hand for endosome motion for Ustilago maydis, it has been observed that typically one or two dyneins oppose teams of four to six kinesins, and the resultant trajectories showed bidirectional motion [15]. In order to test the consistency of our theoretical picture with the observed set of experimental trajectories of the transported cargo we obtain the probability distribution of runtimes for plus-ended and minus-ended runs and the pause time distributions for our simulated cargo trajectories (Figs. 4 and 5), using the typical motor parameters of kinesin and dynein. Remarkably, we find that the incorporation of catchbonding in dynein qualitatively reproduces the experimentally observed features of transport. In the absence of catchbonding, the conventional tug-of-war stochastic model predicts positive-directed trajectories in these regimes, and only the nonlinear unbinding response of catch bonded dynein can explain the observed experimental trajectories. While our phenomenological model has successfully been able to qualitatively capture many of the aspects of motor driven bidirectional cargo transport, for a more quantitatively accurate and comprehensive analysis of the transport the theory needs to be refined to take in account many of the other experimental observations. First of all, some experiments have pointed to the presence of external regulatory mechanism which coordinates the action of the motors and the resultant transport of the cargo [1,8,47,[76][77][78][79][80][81][82]. For instance, it has been observed that the transport of lipid droplets in Drosophila is regulated by Klar protein [1,8]. In the absence of Klar protein transport of the droplets is severely disrupted although the motor functionality is not affected suggesting its regulatory role [1,8]. Some recent experiments have also suggested that other factors such as the interactions between multiple motors leading to clustering, and the rotational diffusion of the cargo itself can play a role in the regulatory mechanism [83,84]. Thus it requires a careful examination of the various experiments to delineate the relative importance of mechanical regulation that is mediated through catchbonding in motors and the external modes of regulating transport. Secondly, one simplification that has been made in our phenomenological model is the assumption of equal load sharing by the motors. However it has been experimentally observed that load is not shared equally between all dynein motors, rather the leading motor of a team of attached motors bears a larger part of the load, which is also demonstrated by bunching of motors at one end [54,85]. Further for dynein motors it has been observed that under load, dynein has a variable step size. For forces below the stall force, dynein modulates its step-size from 24nm for zero load forces to 8nm for forces close to the stall force [69,86,87]. Beyond the stall force, the catch bond activates reducing the dynein dissociation rate [29,86]. The aspect of variable step size has not be taken into consideration in our theoretical description.
In summary, our phenomenological model illustrates the key principle that the incorporation of a dynein catch bond can provide a mechanical explanation of the phenomenon of codependent transport by teams of opposing molecular motors. The model is able to capture the broad qualitative features observed in context of a multitude of experiments on motor driven transport within the cell. The framework proposed here encapsulates both the tug-of-war and codependent behaviour in appropriate regimes and hence provides an unifying picture of motor-driven transport.
The kinesin and dynein binding rates are assumed to be of the form where N + π 0+ (N − π 0− ) is the rate for the first kinesin (dynein) motor to bind to the MT.
The unbinding rate for kinesin is given by the expression ε + (n + , n − ) = n + ε 0+ exp[F c (n + , n − )/(n + F d+ )] while the unbinding rate for dynein is given by ε − (n + , n − ) = n − ε 0− exp[−E d (F c (n + , n − )) + F c (n + , n − )/(n − F d− )] with the catch bond deformation energy given by Here, ε 0± denotes the zero-force single motor unbinding rates, while α parameterizes the strength of the catch bond. The cooperative force felt by the motors due to the effect of the motors of the other species is given by [30] F c (n + , n − ) = n + n − F s+ F s− n − F s− v 0+ + n − F s− v 0+ (v 0+ + v 0− ) (8) and the cargo velocity is given by v c (n + , n − ) = n + F s+ − n − F s− n − F s− /v 0− + n + F s+ /v 0+ Here, v 0± denotes the velocity of kinesin (or dynein) motors, where, v F and v B are the forward and backward motor velocities. Finally the stall forces for the two motor species are denoted by F s± .
The parameters used in the study are taken from the literature, and are summarized in Table I.

B. Numerical techniques
The time trajectory of the cargo is obtained by simulation the master equation using