Deciphering the transport of elastic filaments by antagonistic motor proteins

Intermediate filaments are long elastic fibers that are transported by the microtubule-associated motor proteins kinesin and dynein inside the cell. How elastic filaments are efficiently transported by antagonistic motors is not well understood and is difficult to measure with current experimental techniques. Adapting the tug-of-war paradigm for vesiclelike cargos, we develop a mathematical model to describe the motion of an elastic filament punctually bound to antagonistic motors. As observed in cells, up to three modes of transport are obtained; dynein-driven retrograde, kinesin-driven anterograde fast motions, and a slow motion. Motor properties and initial conditions that depend on intracellular context regulate the transport of filaments. Filament elasticity is found to affect both the mode and the efficiency of transport. We further show that the coordination of motors along the filament emerges from the interplay between intracellular context and elastic properties of filaments.

Figure 1

An intermediate filament is represented by $N$ nodes connected by springs of rest length $\ell$ and spring constant $\alpha$. On each node, there are $N_D$ (respectively $N_K$) binding sites available for dyneins (respectively kinesins); the binding sites are pictured as colored squares (here to illustrate $N_D=N_K=2$). Magenta (dark grey) represents kinesins and cyan (light grey) represents dyneins.

Figure 2

Four regimes for each node $i$, $i\in \{1,\cdots ,N\}$. In regime I, node $i$ moves at the maximal negative velocity (no-load velocity) of dyneins, $v_i=-v_{fD}$. In regime II (respectively III), node $i$ moves at an intermediate negative (respectively positive) velocity, $-v_{fD}<v_i<0$ (respectively $0\le v_i<v_{fK}$). In regime IV, node $i$ moves at the maximal positive velocity (no-load velocity) of kinesins, $v_i=v_{fK}$.

Figure 3

Locally asymptotically stable equilibrium points in regime II and III with $N_K=N_D=16$ and $N=5$ depending on strength of motors, $|F_{s•}|/F_{d•}$. (a) With both exponential off rates up to three dynamics can be observed: anterograde motion (positive fast velocities, $v^{*}=v_{\text{III}}^{*}$), “slow” motion (small velocities, $v^{*}=v_{\text{Slow}}^{*}$), and retrograde fast motion (negative velocities, $v^{*}=v_{\text{II}}^{*}$). For both exponential off rates, anterograde and retrograde motions are always possible. (b) For mixed rates, only the retrograde motion is always possible. Only slow filaments can be observed for weak dyneins, $F_{sD}/F_{dD}<1.1$.

Figure 4

Sensitivity of locally asymptotically stable equilibrium values $v^{*}$ to values of forward and backward speeds, $v_{f•}$ and $v_{b•}$, in regimes II and III with $N_K=N_D=16$ and $N=2$. Units of all speeds and velocities are $\mu \text{m/s}$. When varying, the forward (respectively backward) speeds range in [0.06,1.5] (respectively [0.005,0.25]). The other parameter values are constant and given in Table 2. Positive fast velocities $v^{*}=v_{\text{III}}^{*}$ appear in magenta (dark grey), negative fast velocities $v^{*}=v_{\text{II}}^{*}$ in cyan (light grey), and small velocities corresponding to slow motion $v^{*}=v_{\text{Slow}}^{*}$ in purple blue (black surface near zero). Strong motors correspond to $|F_{sK}|/F_{dK}=2$ and $F_{sD}/F_{dD}=2.3$ and weak motors correspond to $|F_{sK}|/F_{dK}=F_{sD}/F_{dD}=1$.

Figure 5

Effect of initial conditions on the long run dynamics. Color represents the final velocity $v^{*}$ of filament or node, color bars indicate velocity values (units are $\mu \text{m/s}$). When initial conditions are in the magenta (dark grey) [respectively cyan (light grey)] region, the filament or node moves anterogradely (respectively retrogradely). Filament or node starting with initial conditions in the purple blue (middle grey) region is almost stalled (slow velocities). (a)–(d) $N$ uncoupled and $N$ coupled nodes starting with same initial conditions $C_{K,1}\left(0\right)=\cdots =C_{K,N}\left(0\right)$ and $C_{D,1}\left(0\right)=\cdots =C_{D,N}\left(0\right)$ have the same dynamics ($N_K=N_D=4$). As previously mentioned, slow velocities are only possible with weak motors [(b) and (d)]. In (e)–(h) light color regions are the regions of implicit coordination for a filament of two nodes predicted by coupling the dynamics of two uncoupled nodes.

Figure 6

Dynamics of two nodes with $N_K=N_D=4$: (a)–(d) uncoupled nodes, (e)–(h) coupled nodes using the base value for $\alpha$. For each case, $10000$ simulations are performed with initial conditions $C_{K_i}\left(0\right)$ and $C_{D_i}\left(0\right)$ uniformly distributed on [0,1]. Each circle represents the initial conditions of the nodes of one simulation. (a)–(d) Color of outer (respectively inner) circles represents the final velocities $v^{*}$ of node 1 (respectively node 2). (e)–(h) The color of circles is the filament final velocity $v^{*}$. Only simulations terminating with the same final velocities for both nodes are considered ($|v_1\left(t_{\text{final}}\right)-v_2\left(t_{\text{final}}\right)|<{10}^{-3}$). Roughly 1% of the simulations do not meet the criterion and are not considered.

Figure 7

Trajectories of $N=2$ node filaments with $N_K=N_D=4$ are plotted in the $C_{K,i}-C_{D,i}-v_i$ hyperplane in case 1 with implicitly coordinated nodes (a)–(d) and case 2 with nonimplicitly coordinated nodes (e)–(h). Dark grey (respectively light grey) trajectory represents node 1 (respectively 2). The bottom cyan (top magenta) surface is regime I (respectively IV) and the mesh is the switching plane between regimes II and III. Stable equilibria with negative (respectively positive) velocities are plotted as cyan squares (respectively magenta dots). The unstable equilibria are grey triangles (respectively diamonds) when in regime II (respectively III). (a)–(d) Varying $\alpha$ does not affect the direction of motion; however, the coupling improves the transport efficiency. (e)–(h) Node 1 moves retrogradely and node 2 moves anterogradely when decoupled ($\alpha =0$). Increasing the coupling results in the coordination of nodes. In (f) the filament moves retrogradely in the limiting regime with ${10}^{-1}\alpha$. At normal elasticity $\alpha$ and stiffer, the filament exhibits slow motion, the tug-of-war is not resolved [(g) and (h)].

Figure 8

Effect of strength of elastic coupling on the motion of filament of length $N=2$ with $N_K=N_D=4$ in case 1. $10000$ simulations with initial conditions $C_{K_i}\left(0\right)$ and $C_{D_i}\left(0\right)$ uniformly distributed on [0,1] are performed; for $\alpha >0$, only simulations terminating with both nodes at the same final velocity are considered $\left[\right|v_1\left(60\right)-v_2\left(60\right)|<{10}^{-3}]$. (a) and (q) The two nodes are uncoupled; outer (respectively inner) circle represents node 1 (respectively node 2). (b)–(h) Each circle is the initial conditions of a filament, the color is its final velocity. (j)–(p) Locations of changes in regions of implicit coordination resulting from elastic coupling: color of the inner (respectively outer) circle is the final velocity of nodes when uncoupled (respectively coupled). Panels (q)–(x) show the same information as in (a)–(h) for only the dynamics in nonimplicitly coordinated regions.

Figure 9

Effect of strength of elastic coupling on the final motion of a filament of length $N=2$ with $N_K=N_D=4$ in case 2. (a) and (q) The two nodes are uncoupled; outer (respectively inner) circle represents node 1 (respectively node 2). (b)–(h) Each circle represents the initial conditions of a filament, the color is its final velocity. (j)–(p) Locations of changes in implicitly coordinated regions resulting from elastic coupling: color of the inner (respectively outer) circle is the final velocity of nodes when uncoupled (respectively coupled). Panels (q)–(x) show the same information as in (a)–(h) for only the dynamics in nonimplicitly coordinated regions. Simulations for this figure have the same setup as simulations in Fig. 8.

Figure 10

Effect of strength of elastic coupling on the final motion of a filament of length $N=2$ with $N_K=N_D=4$ in case 3. (a) The two nodes are uncoupled; outer (respectively inner) circle represents node 1 (respectively node 2). (b)–(h) A circle represents the initial conditions of a filament, the color is its final velocity. (j)–(p) Locations of changes in implicitly coordinated regions resulting from elastic coupling: color of the inner (respectively outer) circle is the velocity of nodes when uncoupled (respectively coupled). Simulations for this figure have the same setup as simulations in Fig. 8.

Figure 11

Effect of strength of elastic coupling on the final motion of a filament of length $N=2$ with $N_K=N_D=4$ in case 4. (a) and (q) The two nodes are uncoupled; outer (respectively inner) circles represent node 1 (respectively node 2). (b)–(h) Initial conditions of filaments are represented as circles whose color is the filament final velocity. (j)–(p) Locations of changes in implicitly coordinated regions resulting from elastic coupling: color of the inner (respectively outer) circle is the final velocity of nodes when uncoupled (respectively coupled). Panels (q)–(x) show the same information as in (a)–(h) for only the dynamics in nonimplicitly coordinated regions. Simulations for this figure have the same setup as simulations in Fig. 8.

Figure 12

Impact of the elastic coupling on filament final velocities $v^{*}$. Proportions of nodes ($\alpha =0$) or filaments reaching a final velocity ($v_{\text{I}}^{*}$, $v_{\text{II}}^{*}$, $v_{\text{Slow}}^{*}$, $v_{\text{III}}^{*}$, or $v_{\text{IV}}^{*}$) are shown as the coupling increases. Dot size codes for the proportion value. In the color version, cyan is $v_{\text{I}}^{*}$, light cyan $v_{\text{II}}^{*}$, purple blue $v_{\text{Slow}}^{*}$, light magenta $v_{\text{III}}^{*}$, and magenta $v_{\text{IV}}^{*}$. Additionally, the transitions of $v^{*}$ between the possible equilibrium values as the coupling strength increases are shown. Each path shows how the final velocity of a given filament evolves as the coupling increases. The linewidth and greyscale indicate the proportion of filaments whose final velocity undergoes a specific pattern of transitions between equilibrium values. For readability, only paths with a frequency larger than 0.0075 are shown (if there is a unique transition pattern as the coupling increases, its frequency is 1). Proportions and transitions of $v^{*}$ are given for all filaments in (a), (d), (g), and (j), for filaments starting with initial conditions in regions of implicit and nonimplicit coordination in (b), (e), (h), and (k) and (c), (f), (i), and (l) respectively. For nonimplicitly coordinated nodes (c), (f), and (l) and $\alpha =0$, results for the node 1 are used; results for node 2 are symmetrical. Data used for (a), (d), (g), and (j) are shown in Figs. 8–11 panels (a)–(h), for (b), (e), (h), and (k) in Figs. 8–11 panels (i)–(p), and for (c), (f), (i), and (l) in Figures 8, 9, and 11 panels (q)–(x). Only simulations satisfying $|v_1\left(t_{\text{final}}\right)-v_2\left(t_{\text{final}}\right)|<{10}^{-3}$ for all positive coupling strengths are considered. Among the $10000$ simulations, for case 1, $94.3\%$ of simulations are considered, $99.6\%$ for case 2, $95.6\%$ for case 3, and $99.8\%$ for case 4.

Figure 13

Summary of the impact of the elastic coupling on filament motion. Proportions of nodes ($\alpha =0$) or filaments reaching a specific mode of motion are shown as the elastic coupling strength increases. The possible modes of motion reached by filaments are as follows: maximal dynein velocity (cyan) filaments are transported retrogradely at the maximal dynein velocity, dynein wins (light cyan) filaments are transported retrogradely, unresolved tug-of-war (purple blue) filaments are stalled in a slow retrograde or anterograde motion, kinesin wins (light magenta) filaments are transported anterogradely, and maximal kinesin velocity (magenta) filaments are transported anterogradely at the maximal kinesin velocity. Dot sizes represent proportions of filaments in different modes of motion.

Figure 14

Velocity of kinesin and dynein as a function of load force, Eqs. (A1) and (A2).

Figure 15

Same conditions (filament of length $N=2$ with $N_K=N_D=4$ in case 1) as in Fig. 8 run for longer final time $t_{\text{final}}=180\text{s}$. (a),(b) $10000$ simulations with initial conditions $C_{K_i}\left(0\right)$ and $C_{D_i}\left(0\right)$ uniformly distributed on [0,1] are performed; for ${10}^{-3}\alpha$, only simulations terminating with both nodes at the same final velocity are considered $\left[\right|v_1\left(t_{\text{final}}\right)-v_2\left(t_{\text{final}}\right)|<{10}^{-3}]$. (a),(e) For uncoupled nodes, outer (respectively inner) circle represents node 1 (respectively node 2). (d) No changes in regions of implicit coordination resulting from elastic coupling is observed. (e),(f) Dynamics in nonimplicitly coordinated regions.