# Dynein-Inspired Multilane Exclusion Process with Open Boundary Conditions

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## Abstract

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## 1. Introduction

## 2. Model Description and Dynamics

- (i)
**Attachment:**One unit of ATP attaches to the empty primary binding site with a fixed probability ${P}_{att}$ or to any of the available secondary sites with an also constant probability, ${S}_{att}$.- (ii)
**Detachment:**One unit of ATP detaches either from the primary site with the fixed rate ${P}_{det}$ or from one of the three secondary sites with the constant probability ${S}_{det}$.- (iii)
**ATP hydrolysis:**An occupied primary site hydrolyses ATP to ADP to generate mechanical energy that enables it to perform movement. This ATP binding induces dissociation of the motor from the microtubule. After detaching from the microtubule, the motor rearranges its structure and becomes poised for the powerstroke movement, triggered by the ATP hydrolysis. This structural change is followed by a diffusional search for the target binding site over the microtubule [20,21]. Thus, if the primary binding site is occupied, the motor attempts to hop $(4-s)$ steps, where s is the number of secondary sites holding ATP (thus, s can be either 1, 2, or 3).

#### Phase Diagram of the One-Dimensional TASEP with Hop Rate r

- (a)
**Low-density phase (LD)**: In this phase, the bulk stationary density saturates to a constant value ${\rho}_{LD}=\alpha /r$, which yields the stationary current ${J}_{LD}=\alpha (1-{\rho}_{LD})=\alpha (1-\alpha /r)$. Near the exit boundary, the density corresponds to ${\rho}_{{L}_{x}}={J}_{LD}/\beta $.- (b)
**High-density phase (HD)**: Here, the bulk stationary density is ${\rho}_{HD}=1-\beta $, thus the steady-state particle current is given by ${J}_{HD}=r\beta (1-\beta )$.- (c)
**Maximal-current phase (MC)**: The maximal particle current is obtained when the density in the bulk is $1/2$ and hence $J={J}_{max}=r/4$ for $\alpha >r/2$ and $\beta >1/2$. Near the boundaries, the density deviation from its bulk value shows power law decay, which indicates the presence of long-range correlations in the system.

## 3. Results

#### 3.1. Phase Diagram for Dynein Particles in One Dimension

#### 3.2. Phase Diagram for Dynein Particles in Two Dimensions

#### 3.3. Dynamics of Dynein Particles in Two Dimensions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**A schematic representation of the dynamics of dynein motors over a tube. We have an open longitudinal boundary through which a motor particle enters from the left edge with the rate $\alpha $, and exits on the right with the rate $\beta $. Periodic boundary conditions are employed along the transverse direction. Each particle comprises four internal sections representing the primary (red circle) and secondary (green circle) ATP binding sites. A motor particle may move in any of the three forward and transverse directions, provided the primary ATP binding site is filled. The probability of hopping in the up or down direction is the same, ${P}_{y}$, whereas the likelihood of forward motion is set to $(1-2{P}_{y})$.

**Figure 2.**Phase diagram in the influx–outflux rate space for the dynamics of dynein particles in one dimension. The phase diagram under high-load conditions closely approximates the standard TASEP phase diagram as indicated by the red dotted line. However, the maximal current region is reduced to a small portion of the phase space under the low-load condition. The insets display the approximately TASEP-like density profile. The jump length statistics in the respective phases are also shown in the insets. In the MC phase, a reduction in the stationary density is observed upon increasing the probability of long jumps; thus, ${\rho}_{MC}\approx 0.45$ for high-load and ${\rho}_{MC}\approx 0.34$ for low-load conditions as indicated by the solid green lines in the insets of Figure 2a,b, respectively. The (magenta) broken line indicates the calculated HD/MC and LD/MC transition boundaries obtained from Equation (9) in the maximal-current phase.

**Figure 3.**Two-dimensional lattice: The upper panels display the density profile in different phases for the three distinct load conditions. The lower panels provide the statistics of the number of steps taken by the dynein particles. Under high-load conditions, the count of smaller steps is significantly larger (Figure 3d), and the solid black line represents the best fit to a tangent density profile, ${\rho}_{max}(1-a\phantom{\rule{3.33333pt}{0ex}}q\mathrm{tan}\left[a(x-0.5)\right]$, for the MC phase, similar to the TASEP, as shown in Figure 3a. For both medium- and low-load conditions, the bulk density in the high-density phase is $(1-\beta )$, whereas it is proportional to the injection rate $\alpha $ in the low-density phase. The number of lanes set for these graphs is four.

**Figure 4.**The graphs depict the steady-state properties obtained from the dynamics of the dynein particles in the quasi-two-dimensional setting. The main Figure 4a shows the phase diagram with a first-order transition separating the low- and high-density phases under low- (red dashed line) and intermediate-load (green line) conditions for small transverse jump rate ${P}_{y}=0.025$. Figure 4b shows the schematic dependence of the phase diagram on the probability to select the hopping direction ${P}_{att}=0.8,\phantom{\rule{3.33333pt}{0ex}}{S}_{att}=0.2$ for low loads. The values of $\alpha $ and $\beta $ beyond which we observe the maximal-current phase for various ${P}_{y}$ are ${P}_{y}=0.1:\phantom{\rule{3.33333pt}{0ex}}\alpha >0.95,\phantom{\rule{3.33333pt}{0ex}}\beta >0.9$ (blue line), ${P}_{y}=0.15:\phantom{\rule{3.33333pt}{0ex}}\alpha >0.9,\phantom{\rule{3.33333pt}{0ex}}\beta >0.8$ (brown), ${P}_{y}=0.2:\phantom{\rule{3.33333pt}{0ex}}\alpha >0.8,\phantom{\rule{3.33333pt}{0ex}}\beta >0.75$ (green), ${P}_{y}=0.15:\phantom{\rule{3.33333pt}{0ex}}\alpha >0.75,\phantom{\rule{3.33333pt}{0ex}}\beta >0.7$ (purple). The dotted (black) line indicates the transition between the MC/LD and MC/HD phases for the one-dimensional TASEP.

**Figure 5.**The main part of the plot shows the current profiles as a function of the influx rate $\alpha $, for three different outflux rates $\beta =0.2,0.4,0.6$, under low-load conditions; the inset depicts the corresponding density profiles illustrating the discontinuous transition from the low- to the high-density state.

**Figure 6.**Single tagged dynein particles’ trajectories under high-load conditions. Particles A, B and C enter at $t=0$, $t\approx {L}_{x}$ and $t\approx {L}_{x}^{2}$, respectively. The influx and outflux rates used here are $\alpha =0.7,\phantom{\rule{3.33333pt}{0ex}}\beta =0.7$, corresponding to the low-density phase.

**Figure 8.**The time-dependent growth of the mean-square displacement (MSD) for the tagged particles’ C (in the main figure panels) and A (in the insets) for different phases on a finite quasi-two-dimensional lattice. The different time regimes are also marked; here, $L=500$, ${P}_{att}=0.8$, and ${S}_{att}=0.8$ representing a high-load, and ${S}_{att}=0.2$ for a low-load condition. The results were averaged over ${10}^{3}$ different realizations.

**Figure 9.**Dwell-time distribution for tagged dynein particles at different entry times in the quasi-two-dimensional system. The main figure corresponds to the HD phase, while the inset displays data for the LD phase. The waiting times for the low-density phase are always governed by a single exponential distribution with a density-dependent decay rate, as shown in the inset. In contrast, the main graph displays prominent double-exponential behavior characteristic of an extremely congested system. The dashed black line shows the best fit to a double exponential, $18(2{e}^{-0.09x}-{e}^{-0.32x})$.

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**MDPI and ACS Style**

Nandi, R.; Täuber, U.C.; Priyanka.
Dynein-Inspired Multilane Exclusion Process with Open Boundary Conditions. *Entropy* **2021**, *23*, 1343.
https://doi.org/10.3390/e23101343

**AMA Style**

Nandi R, Täuber UC, Priyanka.
Dynein-Inspired Multilane Exclusion Process with Open Boundary Conditions. *Entropy*. 2021; 23(10):1343.
https://doi.org/10.3390/e23101343

**Chicago/Turabian Style**

Nandi, Riya, Uwe C. Täuber, and Priyanka.
2021. "Dynein-Inspired Multilane Exclusion Process with Open Boundary Conditions" *Entropy* 23, no. 10: 1343.
https://doi.org/10.3390/e23101343