Spatial Organization of the Cytoskeleton enhances Cargo Delivery to Specific Target Areas on the Plasma Membrane of Spherical Cells

Intracellular transport is vital for the proper functioning and survival of a cell. Cargo (proteins, vesicles, organelles, etc.) is transferred from its place of creation to its target locations via molecular motor assisted transport along cytoskeletal filaments. The transport efficiency is strongly affected by the spatial organization of the cytoskeleton, which constitutes an inhomogeneous, complex network. In cells with a centrosome microtubules grow radially from the central microtubule organizing center towards the cell periphery whereas actin filaments form a dense meshwork, the actin cortex, underneath the cell membrane with a broad range of orientations. The emerging ballistic motion along filaments is frequently interrupted due to constricting intersection nodes or cycles of detachment and reattachment processes in the crowded cytoplasm. In order to investigate the efficiency of search strategies established by the cell's specific spatial organization of the cytoskeleton we formulate a random velocity model with intermittent arrest states. With extensive computer simulations we analyze the dependence of the mean first passage times for narrow escape problems on the structural characteristics of the cytoskeleton, the motor properties and the fraction of time spent in each state. We find that an inhomogeneous architecture with a small width of the actin cortex constitutes an efficient intracellular search strategy.


Introduction
The accurate delivery of various cargoes is of great importance for maintaining the correct function of cells and organisms. Particles, such as vesicles, proteins, organelles, have to be transported to their specific destinations. In order to enable this cargo transfer, cells are equipped with a complex filament network and specialized motor proteins. The cytoskeleton serves as tracks for molecular motors. They convert the energy provided by adenosine triphosphate hydrolysis into active motion along the cytoskeletal filaments, while they simultaneously bind to cargo [1,2]. In addition to intracellular transport, the dynamic cytoskeleton and its associated motors also stabilize the cell shape, adjust it to different environmental circumstances, and drive cell motility or division [3].
The two main constituents of the cytoskeleton involved in intracellular transport are the polarized microtubules and actin filaments. In cells with a centrosome, the rigid microtubules grow radially from the central microtubule organizing center (MTOC) towards the cell periphery. In conjunction with the associated motor proteins kinesin and dynein, microtubules manage fast long-range transport between the cell center and periphery. In contrast to microtubules, which spread through the whole cell, actin filaments are mostly accumulated in a random fashion underneath the plasma membrane and construct the so called actin cortex [3]. Myosin motors operate on actin filaments and are therefore specialized for lateral transport in the cell periphery. Consequently, the cytoskeletal structure is very inhomogeneous and characterized by a thin actin cortical layer [4,5].
The 'saltatory' transport [6] by molecular motors is a cooperative mechanism. Several motors of diverse species are simultaneously attached to one cargo [7][8][9][10]. This enables a frequent exchange between actin and microtubule based transport, which is necessary for specific search problems. A prominent example of collaborative transport on actin and microtubule networks is the motion of pigment granules in fish and frog melanophores [11,12]. The activity level of the particular motor species, and thus the share in cooperation, is regulated by cell signaling [10,13]. In fish and frog, pigment granules are accumulated near the nucleus by extracellular stimuli transduced via PKA (protein kinase A) [11,12]. The motor activity is thereby controlled via the level of cAMP (cyclic adenosine monophosphate). While low concentrations promote the action of dyneins, intermediate values stimulate myosin motors and high amounts of cAMP activate kinesins [12]. Moreover, the infection of cells by adenoviruses triggers signaling through PKA and MAP (mitogen activated protein kinase), which enhances transport to the nucleus [14]; and the net transport of lipid droplets in Drosophila is directed to the cell center (periphery) by absence (presence) of the transacting factor Halo [10,15].
Another aspect of intracellular transport is its intermittent nature. Molecular motors perform two phases of motility. Periods of directed active motion along cytoskeletal filaments interfere with effectively stationary states [16]. Intersection nodes of the cytoskeleton cause the molecular motors to pause until they either manage to squeeze through the constriction and pass it on the same filament or switch to the crossing track and thus change the direction of the transported cargo [7,[17][18][19]. Motor proteins also detach of the filaments out of chemical reasons. In the cytoplasm the cargoes experience subdiffusive dynamics due to crowding effects [20]. The displacement is limited to the vicinity of the detachment site and negligible compared to the one of the active motion phase. Hence, detachment and reattachment processes effectively contribute to waiting times. However, cargo particles preferentially change their direction of motion at cytoskeletal intersections, which constitute motion barriers. The mean distance between two intersections, the mesh size of a network, is typically smaller than the processive run length of a single molecular motor [21].
The spatial organization of the cytoskeleton as well as the activity of the different motor species and their behavior at network intersections establish a typical stochastic movement pattern of intracellular cargo, which suggests a random walk description [16,22]. The narrow escape problem describes a common search process, where the target destination is represented by a specific, small region on the plasma membrane of a cell. Typical examples involve secretion processes in cytotoxic T lymphocytes (CTL) which play a key role in immune response and defeat tumorigenic or virus-infected cells. When in contact with an antigen-presenting cell, CTL form a connection with a diameter of the order of microns to it, which is called immunological synapse. Lytic granules are actively transported and released at the immunological synapse. They contain perforin and granzymes which induce apoptosis of the pathological cell. In order to prevent unintended damage of neighboring cells, the release of lytic granules is strictly confined to the immunological synapse. Thus, the transport of lytic granules towards the immunological synapse constitutes a narrow escape problem to be solved by CTL [23][24][25][26]. Moreover, the outgrowth of dendrites or axons from neurons [27,28] as well as repair mechanisms for corruptions of a cell's plasma membrane [29,30] require the target-oriented transport of mitochondria and vesicles.
In prior work on the narrow escape problem, Schuss et al analytically investigated first passage properties of a purely diffusive searcher [37,38], while Bénichou et al identified benefits of search efficiency to small targets on the surface of spherical domains by intermittent phases of surface-mediated and bulk diffusion [39]. Ballistic motion along the cytoskeleton and the different characteristics of microtubule-and actin-based transport was taken into account by Slepchenko et al in order to model the spreading of pigment granules in melanophores [32]. They were able to determine the switching rate between microtubules and actin filaments by fitting their theoretical results to experimental data of aggregation and dispersion of pigment granules in fish melanophores, whose cytoskeleton appears to be rather homogeneous. In order to study first passage properties of cargo transport from the nucleus to the complete plasma membrane, Ando et al recently considered an inhomogeneous network distribution in which the cytoskeleton is confined to a delimited shell [40]. Within a continuum model of increased bulk diffusion, they found that the transit time can be significantly reduced when the cytoskeletal shell is placed close to the nucleus. Moreover, they explicitly modeled cytoskeletal networks and investigated the impact of number, length and polarity of filaments as well as detachment and reattachment processes by considering intermittent phases of ballistic transport and cytoplasmic diffusion. Via extensive computer simulations, they showed, inter alia, that an outward-directed network polarity expectedly improves transit times while the actual distribution of filament orientations is less effective. Nonetheless, transport in cells comprises an intricate interplay between motor performance and spatial organization of the cytoskeleton, which is generally inhomogeneous itself. The description of this interaction is a challenging theoretical task and we still lack precise knowledge about how cells adapt to various transport tasks and especially to narrow escape problems.
In the following, we present a coarse grained model of intracellular transport by considering the effective movement between network nodes, while discarding the single steps of individual motors at the molecular level. We introduce a random velocity model with intermittent arrest states where the dynamic cytoskeleton is implicitly modeled by probability density functions for network orientation and mesh size. The proposed model allows the study of diverse transport tasks. Here we focus on the narrow escape problem and address the effects of the interplay between inhomogeneous cytoskeletal architecture and motor performance on the search efficiency to small targets alongside the membrane of spherical cells.

Model
General random velocity model Cells establish specific search strategies for cargo transport by alterations of the cytoskeletal organization and regulation of the motor behavior at network intersections. In order to study the efficiency of spatially inhomogeneous search strategies we formulate a random velocity model in continuous space and time composed of two states of motility: (i) a ballistic motion state at constant speed = v 1, which corresponds to active transport by molecular motors in between two successive intersections of the filamentous network and (ii) a waiting state, which is associated to pauses at intersection nodes of the cytoskeleton. The swaps from one state to another are arranged via constant but generally asymmetric transition rates  k m w for a switch from motion to waiting and  k w m for an inverse transition, see figure 1(b). These lead to exponentially distributed time periods t m , t w spent in each state of motility and mean residence times of  k 1 m w ,  k 1 w m for the motion and waiting state, respectively, which is biologically consistent as active lifetimes of cargo particles are exponentially distributed [31]. The event rates are directly connected to biologically tractable properties of the cytoskeleton and the motor proteins. The mesh size ℓ of the underlying cytoskeletal network, which reflects the typical distance between two consecutive intersections, defines the rate =  ℓ k v m w for a transition from the ballistic motion to the pausing state. Whereas the event rate  k w m is determined by the characteristic waiting time at an intersection node.
Whenever a particle has reached a network intersection and paused, it may either keep moving processively along the same filament with probability p or it may change to a crossing track with probability -( ) p 1 , as sketched in figure 1(d). This provides a typical timescale - - ] pk w m 1 ) of changing (remaining on) the filament subsequent to a waiting period. With regard to the rotational symmetry of a cell, the new direction q f a = + rot is always chosen with respect to the radial direction f = ( ) y x tan . The rotation angle a rot is drawn from a distribution a ( ) f rot which is characteristic for the underlying cytoskeletal network. Due to the inhomogeneous structure of the cytoskeleton, a ( ) f rot depends on the location of the particle inside the cell.

Model geometry
Our model system is designed according to the inhomogeneous internal organization of a cell, see figure 1(a). We assume a circular confined geometry of radius R membrane , which displays the plasma membrane and will be fixed in the following . The cytoplasm is split into an interior region, where only microtubules are present, and a periphery, which is dominated by the actin cortex but may also be pervaded by microtubules. The width of the actin cortical layer is denoted by δ, so that an internal margin of radius emerges. Throughout this article, the target destination of the cargo is assumed to be a narrow escape hole in the plasma membrane with opening angle a target .

Random velocity model in the interior
The interior of a cell is controlled by the radial network of microtubules with its associated kinesin and dynein motors. Since they manage fast long-range transport inside a cell, the rate to switch from the motion to the waiting state =  k 0 m w i is fixed to zero for simplicity. This leads to uninterrupted radial movement along the internal microtubule network. However, the particle may be forced into the waiting state due to confinement events, as dyneins are assumed to stop at the central MTOC. Hence the transition rate to the motion state is set to a non-zero value  k w m i .

Random velocity model in the periphery
Due to the complex network structure of the periphery the searcher frequently encounters intersection nodes at rate  k m w p and switches to the waiting state. Subsequently, the particle may either keep moving along the previously used track at rate  p k w m p or it may change to a crossing filament at rate - which specifies the peripheral environment in terms of the filament orientation ( a ( ) f i rot ) and motor species activity (q i ). The probabilities q K , q D , q M correspond to directional changes induced by kinesin, dynein and myosin, respectively, with . With regard to the radial orientation of microtubules and the directionality of the motors, the rotation angle distributions associated to kinesins and dyneins are delta peaked a d a In case of a directional change initiated by myosins, the rotation angle distribution is assumed to be either uniform or cut-off-gaussian, which takes into account the randomness of actin networks where m s > > 0, 0 and  denotes the normalization constant, see figure 1(c). In conclusion, the peripheral network is characterized by a mean mesh size  k 1 m w , its structure is reflected by the rotation angle distributions a ( ) f i rot and the motor activity is defined via q i .

Confinement events
The spatial geometry of the model system imposes various confinement events, which are further specified in the following. At the onset, each searcher is assumed to start its walk in the center of the cell, where it is linked to a kinesin and runs in a uniformly distributed initial direction. As soon as a particle encounters the outer membrane margin at radius R membrane it will switch into the pausing state, since it is assumed to detach of the filament, and check for the target zone a target . If it is found, the walk will be terminated, otherwise the particle will wait at rate - ( ) p k 1 w m p and the rotation angle distribution will be restricted to allowed values. The same holds in the case that a cargo transported by myosin hits the internal margin R internal , which is created by the structural inhomogeneity of the cytoskeleton. Crossovers of the internal margin by kinesins or dyneins, happen uninterruptedly under a change of the characteristic event rates for interior and periphery. Whenever a dynein coupled particle reaches the MTOC in the center of the cell, it will wait with rate  k w m i before it will change to kinesin motion in a uniformly distributed direction. Consequently, the mean waiting time (MWT) at confinement events is not necessarily the same as the one at network intersections in the bulk. In general it will be larger.

Results
In the following, we perform extensive Monte Carlo simulations in order to analyze the dependence of the search efficiency to narrow escapes alongside a cell's membrane on the spatial organization of the cytoskeleton as well as the motor performance at network intersections. For that purpose we define the mean first passage time (MFPT) to a target as the ensemble average over first passage events of5 10 5 independent realizations of the walk, in which all cargoes are initially located at the center of the cell and start moving in an uniformly distributed direction . At first, we will consider search strategies, where the motors operate on homogeneous cytoskeletal networks. Then, we will focus on the inhomogeneous architecture and elaborate the influence of the actin cortical width on the mean first passage properties of narrow escape problems.

Homogeneous search strategy
Here, we neglect the inhomogeneous structure of the cytoskeleton and assume that it spreads through the whole cell in a homogeneous manner. Since we aim to isolate the impact of the cytoskeletal architecture, we ignore the motor's processivity ( = p 0) and waiting processes ( = ¥  k w m ). Hence at each network intersection the particle immediately changes its direction according to a a a a = + In the case of = q 0 M myosin motors are deactivated and the transport is managed by kinesin and dynein on a radial network of microtubules, while = q 1 M leads to a pure actin mesh with myosins and either uniformly or gaussian distributed filaments. Intermediate values of q M correspond to cooperative transport on homogeneous combinations of microtubules and actin filaments with active kinesins, dyneins and myosins. Sample trajectories which implicitly reflect the network structure are given in figure 2 for uniformly random actin orientations.
In order to investigate the influence of the network composition and motor activity, we measure the MFPT in dependence of the target size a target for different event rates  k m w and various probabilities q M . As expected, the MFPT increases monotonically with decreasing opening angle a target , see  [40]. Remarkably, the search for small targets does also benefit from outward-directed actin filaments. While inward-directed (m p > 2) actin polarities drive the cargo towards the cell center, which lowers the probability to reach the membrane and detect the target, outward-directed (m p < 2) actin networks push the particle towards the plasma membrane, as indicated by the sample trajectories in  outward-directed actin polarities, while detours to the center are necessary for transport along microtubules in order to change radial direction and reach the target.

Inhomogeneous search strategy
In case of a homogeneous cytoskeleton, the most efficient search strategy is an uninterrupted motion on a pure actin network without any directional changes in the bulk. Such a motion scheme is sufficient for large membranous targets, but generally fails for narrow escape problems. May an inhomogeneous search strategy, like it is found in living cells, be the key to the efficient detection of small departure zones on the plasma membrane? Guided by this question, we check for the influence of the actin cortical width δ on the effectiveness of the narrow escape search problem. . This back-and-forth pattern significantly hinders both the hitting with the membrane as well as the retraction to the central MTOC, which is necessary in order to change radial direction and detect the membranous target. Consequently, the MFPT monotonically increases with growing cortical width δ, because a larger peripheral area increases the interruption by back-and-forth motion. In the case of ¹ q 0 M , the minimum of the MFPT for d Î ( ) 0; 1 is most prominent for = q 1 M , i.e. a peripheral motion dominated by transport along actin filaments, achieved by a high activity level of myosins, is most efficient. Contrarily, a high activity of dyneins would be favourable for virus trafficking or aggregation of pigment granules near the nucleus [12,32]. But for small membranous targets, inhomogeneous networks lead to a considerable gain of efficiency in comparison to the corresponding homogeneous limits d  1 and d  0.  While the MFPT is continuous in the limit d  1, it diverges for d  0 and ¹ q 0 M . Consequently, a comparison to MFPT d = ( ) 0 is not directly possible and we include this limit, which we studied in figure 2, by the black dashed line in figure 4 and following figures. The divergence of the MFPT for d  0 originates from a motility restriction by a narrow cortex. Cargo particles which are transported by myosins get localized for narrow actin cortices, since the resulting prompt collisions with the two margins at radii R internal and R membrane inhibit substantially large displacements. Thus, the actin cortex can also act as a transport barrier. The results shown in figure 4 visualize that an increased chance of directional alterations by increase of  k m w p , leads to a loss of search efficiency. This directly suggests a profit by a processive behavior of motor proteins at intersection nodes. Furthermore, a decrease in the opening angle a target provokes a more prominent minimum of the MFPT at lower cortical widths δ, compare figures 4(a) and (b).  The MFPT is composed of the total mean motion time (MMT) and the total MWT which the cargo experiences in the course of the search. Figure 5(b) shows that the MMT displays pronounced minima and is indeed independent of the rate  k w m and thus fully determined by the mesh size of the network given via  k m w . In contrast to that, the MWT, given in figure 5(c), depends on both rates  k m w and  k w m . Since the rate  k w m determines the MWT per arrest state, it influences the MWT only via a multiplicative factor, as obvious by the shift in  figure 5(c)) and thus a favour of homogeneous cytoskeleton. Larger rates  k m w , and hence biologically relevant mesh sizes, conserve the gain of search efficiency by inhomogeneous cytoskeletal organizations, since the number of waiting periods exhibits a minimum for d Î ( ) 0; 1 (inset of figure 5(c)).

Influence of the rotation angle distribution of the actin cortex
The polarity of the actin filaments in the cortex is typically random. Hence we have previously investigated uniform rotation angle distributions f M u . However, actin filaments may align to the radial microtubule network [34] and for instance the protein complex Arp2/3 induces a formation of actin branches at a distinct angle (»  70 ) compared to the parent filament [35,36]. In general, such mechanisms influence the orientational distribution of the actin filaments.
Here, we study the impact of the expectation value and width of a cut-off-gaussian rotation angle distribution f M g (as given in equation (7)) on the search efficiency to narrow membranous targets. Inward-directed actin polarities (m p > 2) drive the cargo towards the cell center, which lowers the probability to reach the membrane and detect the target. As evident from the sample trajectory in figure 6(a), decreasing the actin cortical width δ draws the particle nearer to the cell's boundary. Consequently, a thin actin cortex is essential in order to improve the search efficiency to membranous targets as manifested by the prominent minimum of the MFPT for m p > 2 in figure 6. Lateral actin orientations (m p = 2) result in an equal chance for outward-and inward-directed motion of the cargo. Hence the behavior of the MFPT is similar to the one of uniformly random actin networks defined by f u M and an inhomogeneous cytoskeletal structure with a thin actin cortex δ generally advances the search efficiency. To the contrary, for outward-pointing actin filaments (m p < 2) the particle is predominantly moving in close proximity Δ of the cell's boundary, as sketched for the sample trajectory in figure 6(a). For that reason the cortex width δ does not significantly influence the MFPT as long as it is out of range of Δ and the homogeneous limit d = 1 is most efficient, as evident from figure 6. This limit is even more efficient than a homogeneous microtubule network due to the outward-directed network structure, as found in figure 2. In general, a larger standard deviation σ broadens the distribution of actin filaments and randomizes the search. Consequently, the behavior of the MFPT converges to the uniform case f u M for increasing σ and an inhomogeneous cytoskeleton generally improves the search of small membranous targets again.
Influence of the motor processivity Molecular motors do not necessarily change their direction of motion at each intersection node they encounter. A main feature of molecular motors is their processivity. Motor driven cargoes may also overcome the barrier opposed by network intersections and keep moving ballistically along the same track, which has been neglected so far ( = p 0). Here, we would like to investigate the impact of the motor processivity p on the search efficiency to narrow target zones. For that purpose, we assume = find that a higher processivity systematically improves the search efficiency. This benefit is emphasized in figure 7(c). The MFPT for fixed actin cortical widths decreases monotonically in dependence on the processivity p and is most efficient for = p 1. Remarkably, the introduction of waiting processes results in the development of an optimal processivity ¹ p 1, as presented in figure 8(a)  . For a fixed cortical width δ, figure 8(b) shows that the mean number of waiting periods decreases with p due to the overall gain in search efficiency by an enhanced processivity p. However the MWT drastically  1. Figure 8(c) visualizes, that the MFPT is dominated by the MWT for small rates  k w m . Consequently, the MFPT exhibits a minimum for ¹ p 1 in contrast to the MMT, which is minimal for = p 1 (compare to figure 7(c)). Due to inevitable waiting processes, it may be more efficient to change directions with a specific probabilityp 1 rather than transport by completely processive motors ( = p 1). A specific processivity p is also reported in biological systems [7,[17][18][19].

Discussion
With the aid of a random velocity model with intermittent arrest states, we studied the first passage properties of intracellular narrow escape problems. Via extensive computer simulations we are able to systematically analyze the influence of the cytoskeletal structure as well as the motor performance on the search efficiency to small target zones on the plasma membrane of a cell.
For a spatially homogeneous cytoskeleton, the MFPT diverges in the limit a , which underlines the benefit of motor activity regulation by outer stimuli. Homogeneous motion pattern are sufficient for large target sizes, but actually fail in the biologically relevant case of narrow escape problems.
By varying the width of the actin cortical layer, we elaborate the impact of the cytoskeletal inhomogeneity on the search efficiency to narrow escapes. Remarkably, we find that a cell can optimize the detection time by regulation of the motor performance and convenient alterations of the spatial organization of the cytoskeleton. An inhomogeneous architecture with a thin actin cortex constitutes an efficient intracellular search strategy for narrow targets and generally leads to a considerable gain of efficiency in comparison to the homogeneous pendant. A confinement of the search to a thin shell below the plasma membrane, where the target area is located, saves time as it prevents extensive excursions to the cell interior-but the shell must not be too thin because otherwise the motion of the searcher gets localized and a time loss due to many stops occurs. Consequently, the MFPT diverges in the limit of d  0, which outlines that the actin cortex can act as a transport barrier or functional gateway [41].
Molecular motors do not necessarily change their direction of motion at each filamentous intersection, they may also overcome the constriction and remain on the same track, a property referred to as processivity. We find that an increased motor processivity systematically improves the search efficiency in the case of instantaneous directional changes on networks of a mesh size defined by  k m w p . Due to waiting processes an optimal value of the motor processivity ¹ p 1 emerges, which minimizes the detection of membranous targets. Specific probabilities to overcome constricting filament crossings are also reported during intracellular transport [7,[17][18][19]. So far we have investigated a model for intracellular search strategies in two-dimensional cells. However, a generalization of our approach to three-dimensional spherical cells is straightforward and the former principles stay valid. If we assume a cell radius of 5 μm, which is consistent with the typical size of a CTL, the opening angle of a a = = ( ) 0.02 0.2 target target leads to an arc length of m ḿ = ( ) 0.02 5 m 100 nm 1 m . For instance the diameter of an immunological synapse is of the order of microns [23][24][25][26]. We further assume that motors move processively at intersections with probability = p 0.5 and their velocity typically is about 1μm s -1 . A transition rate to the waiting state of 10 s −1 thus leads to a mesh size of 100 nm, which is biologically reasonable [4,5]. Under these conditions, in 3D. This is in good qualitative agreement to biological data [4,5,42].
In summary, our model indicates that the spatial organization of the cytoskeleton of spherical cells with a centrosome minimizes the characteristic time necessary to detect small targets on the cell membrane by random intermittent search processes along the cytoskeletal filaments (see also [43,44] for similar findings in a model with intermittent diffusive search). The minimization is achieved by a small width of the actin cortical layer and by regulation of the motor activity and behavior at network intersections. Remarkably a thin actin cortex is also more economic than distributing cytoskeletal filaments in all directions over the cell body. Thus, our work outlines that a thin confinement of the actin cortex, besides its advantages concerning cell stability or motility, also serves as an efficient key to intracellular narrow escape problems.