Phase-plane analysis of the totally asymmetric simple exclusion process with binding kinetics and switching between antiparallel lanes

Motor protein motion on biopolymers can be described by models related to the totally asymmetric simple exclusion process (TASEP). Inspired by experiments on the motion of kinesin-4 motors on antiparallel microtubule overlaps, we analyze a model incorporating the TASEP on two antiparallel lanes with binding kinetics and lane switching. We determine the steady-state motor density profiles using phase-plane analysis of the steady-state mean field equations and kinetic Monte Carlo simulations. We focus on the density-density phase plane, where we find an analytic solution to the mean field model. By studying the phase-space flows, we determine the model's fixed points and their changes with parameters. Phases previously identified for the single-lane model occur for low switching rate between lanes. We predict a multiple coexistence phase due to additional fixed points that appear as the switching rate increases: switching moves motors from the higher-density to the lower-density lane, causing local jamming and creating multiple domain walls. We determine the phase diagram of the model for both symmetric and general boundary conditions.

Figure 1

Schematic of the antiparallel two-lane TASEP with Langmuir kinetics and lane switching. Two lanes (green and blue) have their plus ends (indicating the direction of motor motion) oppositely oriented. The blue lane $\left(R\right)$ has plus end to the right; the green lane $\left(L\right)$ has plus end to the left. Motors (red) bind to empty lattice sites with rate $k_{\mathrm{on}}c$, where $c$ is the bulk motor concentration, and unbind with rate $k_{\mathrm{off}}$. Bound motors step toward the lane plus end with rate $v$ (if the adjacent site toward the plus end is empty) or switch to the other lane with rate $s$ (if the corresponding site on the adjacent lane is empty). At minus ends, motors are inserted at rate ${\alpha }_{R,L}v$. At plus ends, motors are removed at rate ${\beta }_{R,L}v$.

Figure 2

Flows in the ${\sigma }_R\text{−}{\sigma }_L$ phase plane for low [(a), 0.$1{\text{s}}^{-1}]$ and high [(b), 0.$5{\text{s}}^{-1}]$ switching rates. Solid lines are trajectories that pass through the fixed points, as discussed in the text. The Langmuir isotherm is labeled LI and the transition points TP. The bulk motor concentration is 200 nM, the motor speed is $5\mu \text{m}$ ${\text{s}}^{-1}$, and other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 3

Changes in trajectories and fixed points with varying switching rate. Curves with $s=0$ ${\text{s}}^{-1}$, black; $s=\frac{3(k_{\mathrm{off}}-k_{\mathrm{on}}c)}{4}\approx 0.0862$ ${\text{s}}^{-1}$, purple; $s=s_{\mathrm{high}}\approx 0.1150$ ${\text{s}}^{-1}$, blue; $s=0.5$ ${\text{s}}^{-1}$, green. The red point labels $s_{\mathrm{low}}$. The arrow indicates how the curve with $C_{\mathrm{LI}}$ changes as $s$ increases. The bulk motor concentration is $c=200$ nM and motor speed is $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 4

Effects of domain walls and finite size on density profiles. Left, determination of domain wall positions in the phase-plane trajectory and the density profile (inset). Red points show kMC simulation results. The boundary points are $({\sigma }_R,{\sigma }_L)=(-\frac{1}{2},\frac{1}{2})$ and $({\sigma }_R,{\sigma }_L)=(\frac{1}{2},-\frac{1}{2})$. The solution locally follows the flow, which cannot connect the boundary points without crossing the lines with ${\sigma }_R=0$ and ${\sigma }_L=0$ where the velocity is ill defined. Crossing these lines uses the matching condition to connect to another exact solution curve, introducing a domain wall. Inset shows the density profiles of ${\rho }_R$ (purple, dashed) and ${\rho }_L$ (brown, solid). The sharp transitions in ${\rho }_R$ and ${\rho }_L$ indicate the domain walls. There are three regions which are separated by two domain walls: regions near ends follow $C_1$, and the region at the center is described by $C_2$. Right, finite-size constraint. The blue points (longer curve) are kMC simulation results for $N=1000$, and the green (shorter curve) $N=500$. For a larger number of sites, the dimensionless motor speed is smaller, moving the solution closer to the LI. The switching rate is 0.$44{\text{s}}^{-1}$(left), 0.$1{\text{s}}^{-1}$(right), the bulk motor concentration $c=200$ nM, and the motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 5

Illustration of the four regions of the central density, as defined in the text. The switching rate is 0.$5{\text{s}}^{-1}$, the bulk motor concentration $c=200$ nM, and the motor speed is $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 6

Examples of the nonlinear phases for low switching rate. (a) Trajectories in the ${\sigma }_R\text{−}{\sigma }_L$ plane. (b) Trajectories in the $\varphi \text{−}\omega$ plane, where $\varphi ={\sigma }_R+{\sigma }_L$ and $\omega ={\sigma }_R-{\sigma }_L$, illustrating the local maxima and minima of the central density. (c) The density in lane $R,{\rho }_R\left(x\right)$. The black dashed line (flat) indicates the LI in position space, and the brown dashed line (tilted) is the transition line in position space. (d) The total density as a function of position. The black dashed line indicates the LI (its value is $2{\rho }_0$ since it is ${\rho }_R+{\rho }_L)$ in position space, and the brown dashed line is the transition line in position space (its value is $-\frac{\gamma }{2S}+1$ since it is ${\rho }_R+{\rho }_L)$. The boundary conditions are $(\alpha ,1-\beta )=(0.05,0.1)$ (red), $(0.1,0.95)$ (purple), $(0.3,0.4)$ (gray), $(0.6,0.4)$ (yellow), $(0.3,0.95)$ (green), $(0.6,0.95)$ (blue), and $(0.95,0.99)$ (cyan). The switching rate is 0.$1{\text{s}}^{-1}$, the bulk motor concentration $c=200$ nM, and the motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows in (a) indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 7

Examples of the nonlinear phases for high switching rate. (a) Trajectories in the ${\sigma }_R\text{−}{\sigma }_L$ plane. (b) Trajectories in the $\varphi \text{−}\omega$ plane, where $\varphi ={\sigma }_R+{\sigma }_L$ and $\omega ={\sigma }_R-{\sigma }_L$, illustrating the local maxima and minima of the central density. (c) The density in lane $R,{\rho }_R\left(x\right)$. The black dashed line (flat) indicates the LI in position space, and the brown dashed line (tilted) is the transition line in position space. (d) The total density as a function of position. The black dashed line indicates the LI (its value is $2{\rho }_0$ since it is ${\rho }_R+{\rho }_L)$ in position space, and the brown dashed line is the transition line in position space (its value is $-\frac{\gamma }{2S}+1$ since it is ${\rho }_R+{\rho }_L)$. The boundary conditions are $(\alpha ,1-\beta )=(0.05,0.6)$ (red), $(0.3,0.9)$ (green), $(0.3,0.95)$ (blue), and $(0.9,1.0)$ (cyan). The switching rate is 0.$5{\text{s}}^{-1}$, the bulk motor concentration $c=200$ nM, and the motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows in (a) indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 8

Example phase-plane trajectories of L and H phases. (a) L phase. The starting and ending points are $({\sigma }_R,{\sigma }_L)=(-0.45,-0.4)$ and $(-0.4,-0.45)$ (blue), $(-0.4,-0.45)$ and $(-0.45,-0.4)$ (green), $(-0.2,-0.1)$ and $(-0.1,-0.2)$ (red), and $(-0.1,-0.2)$ and $(-0.2,-0.1)$ (purple). The dashed lines show that the L phase obeys the left boundary conditions, where the positions are ${\sigma }_R=-0.45$ (blue), ${\sigma }_R=-0.4$ (green), ${\sigma }_R=-0.2$ (red), and ${\sigma }_R=-0.1$ (purple). (b) H phase. The starting and ending points are $({\sigma }_R,{\sigma }_L)=(0.3,0.45)$ and $(0.45,0.3)$ (blue), $(0.2,0.49)$ and $(0.49,0.2)$ (green). The dashed lines show that the H phase obeys the right boundary conditions, where the positions are ${\sigma }_L=0.45$ (blue) and ${\sigma }_L=0.49$ (green). The black dashed line ${\sigma }_L={\rho }_c-\frac{1}{2}\approx 0.4567$, the critical density ${\rho }_c$ (see text) of the local minimum and local maximum phases. The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 9

The phase diagrams for low [(a), 0.$1{\text{s}}^{-1}]$ and high [(b), 0.$5{\text{s}}^{-1}]$ switching rates. The phases are L (low density), M (Meissner), H (high density), LH (low-density–high-density coexistence), and LHLH (low-density–high-density–low-density–high-density coexistence). The thick green dashed line indicates where the domain wall position $x_w=0$, which separates regions with a local maximum of the central density $\left(x\right)$ or local minimum $\left(n\right)$. The red dashed lines indicate boundaries between local maximum and minimum phases. For low switching rate, the left parts of the L and LH phases are local maximum phases, and the right are local minimum phases. The upper part in the H phase is local minimum phase. For high switching rate, the left parts of the L and LH phases are local maximum phases, and the right parts are local minimum phases. The bulk motor concentration is $c=200$ nM, and the motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 10

Illustration of calculation of the LH phase boundary. The red line indicates the central density, which lies on the ${\sigma }_R={\sigma }_L$ line for symmetric boundary conditions. Backward integration from this line (black arrows) to $x=-\frac{1}{2}$ gives the thick blue and green, where the blue line corresponds to $\sigma <0$ and green line $\sigma >0$. Applying the matching condition to ${\sigma }_L$ and ${\sigma }_R$ (green arrows) gives the dashed blue and green lines, respectively. These are the phase boundaries of the LH phase. The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 11

Illustration of the effects of changing motor speed on the LH phase boundary. High motor speed ($5\mu \text{m}$ ${\text{s}}^{-1})$ shown in blue, low motor speed 0.$5\mu \text{m}$ ${\text{s}}^{-1}$ shown in green. The red line indicates the central density, which lies on the ${\sigma }_R={\sigma }_L$ line for symmetric boundary conditions. Backward integration from this line (black and purple arrows) to $x=-\frac{1}{2}$ gives the thick blue and green lines. Applying the matching condition to ${\sigma }_L$ and ${\sigma }_R$ (blue and green arrows) gives the dashed blue and green lines. These are the phase boundaries of the LH phase. The solid black lines are trajectories which pass through the Langmuir isotherm. For lower motor speed, backward-integrated line moves closer to the black line. The phase boundary of the LH and H phases is outside of the lanes. The switching rate is 0.$1{\text{s}}^{-1}$, the bulk motor concentration $c=200$ nM; other parameters are the reference values of Table 1.

Figure 12

Illustration of calculation of the LHLH phase boundary. The red line indicates the central density, which lies on the ${\sigma }_R={\sigma }_L$ line for symmetric boundary conditions. The thick black lines are the line and hyperbola determined from Eq. (21) with $C=0$. The dashed black lines are trajectories determined by backward integration from ${\sigma }_R={\sigma }_L$ line. The blue curves show the flow directions. The switching rate is 0.$5{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 13

Illustration of calculation of phase boundaries. The red line indicates the central density, which lies on the ${\sigma }_R={\sigma }_L$ line for symmetric boundary conditions. Backward integration from this line (black arrows) to $x=-\frac{1}{2}$ gives the thick blue and green lines, where the blue line corresponds to $\sigma <0$ and green line $\sigma >0$. Applying the matching condition to ${\sigma }_L$ and ${\sigma }_R$ (blue and green arrows) gives the dashed blue and green lines, respectively. These are the phase boundaries of the LH phase. The green dashed line stops around ${\sigma }_R=-0.1386$ and will not extend to ${\sigma }_R=0$ line. The switching rate is 0.$5{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and the motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 14

Illustration of calculation of phase boundaries of local maximum and minimum phases. The dashed green line is obtained from backward integration of the ${\sigma }_R={\sigma }_L$ line to $x=-\frac{1}{2}$. The green line indicates the LH phase boundary determined by the matching condition, where the right of the line is the H phase. The red line is the transition line. Since the H phase satisfies the right boundary condition, the starting point of the density profile depends on ${\sigma }_L$ value. Therefore, the phase boundary of the local maximum and minimum phase is a horizontal line that intersects the green dashed line and the transition line. The local minimum phase is above the black dashed line. The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed is $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 15

Illustration of the density profile (a) and hypothetical phase-plane trajectories (b) in the LH phase. (b) The thick purple curves indicate the density profile starting from the boundary conditions, and the dashed blue and solid red curves are two different hypothetical trajectories which locally obey Eqs. (7), (8), and (21). The parameters are $s=0.44$ ${\text{s}}^{-1},c=200$ nM, $v=5$ $\mu \text{m}$ ${\text{s}}^{-1},k_{\mathrm{on}}=2.7\times {10}^{-6}$ ${\text{nM}}^{-1}$ ${\text{s}}^{-1}$, and $k_{\mathrm{off}}=0.00169$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows in (b) indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 16

Comparison of determination of the L phase boundaries using the total binding constraint with linear approximation [green dashed curves, Eqs. (27) and (31)] and the finite-size constraint without any approximation (blue solid curves). The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1.

Figure 17

Example phase-plane trajectories and density profiles (inset) for general boundary conditions. Insets show ${\rho }_R+{\rho }_L$ (blue), ${\rho }_R$ (red), and ${\rho }_L$ (green). Red dots are kMC simulation results. The blue dot indicates the left boundary condition and the green dot the right boundary condition. (a) The blue line is ${\sigma }_R={\alpha }_R-\frac{1}{2}$, and the green line is ${\sigma }_L={\alpha }_L-\frac{1}{2}$. (b) The blue line is ${\sigma }_L=\frac{1}{2}-{\beta }_L$, and the green line is ${\sigma }_R=\frac{1}{2}-{\beta }_R$. (c) The blue line is ${\sigma }_R={\alpha }_R-\frac{1}{2}$, and the green line is ${\sigma }_R=\frac{1}{2}-{\beta }_R$. (d) The blue line is ${\sigma }_L=\frac{1}{2}-{\beta }_L$, and the green line is ${\sigma }_L={\alpha }_L-\frac{1}{2}$. The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows indicate the vector field, which has the mathematical form of Eqs. (12) and (13).

Figure 18

Example phase-plane trajectories and density profiles for general boundary conditions, illustrating how altering the right boundary condition changes the density profile. (a) Colored dots indicate the right boundary condition. Red curve satisfies all four boundary conditions; green, blue, and brown curves do not satisfy ${\beta }_R$; and the purple curve satisfies ${\alpha }_R$ and ${\beta }_L$. (b), (c) Corresponding density profiles on the R lane (b) and L lane (c). The switching rate is 0.$1{\text{s}}^{-1}$, bulk motor concentration $c=200$ nM, and motor speed $5\mu \text{m}$ ${\text{s}}^{-1}$; other parameters are the reference values of Table 1. Arrows in (a) indicate vector field, which has the mathematical form in (12) and (13).