|
Figure 2. Trajectories and segments.Representative trajectory of a cargo obtained during a simulation of the model with Nf = Nb = 2 showing a processive trend in the forward direction (red trace) followed by a reversion (dashed rectangle) and, finally, a motion in the backward direction (blue trace). Segmental velocities are determined from pieces of the trajectory containing 40 data points during which the cargo moves with constant velocity, such as the ones indicated with green traces. The system is symmetric and the corresponding parameters are displayed in Table 1.
|
|
Figure 3. Distributions of cargo segmental velocities.(A) Experimental results from reference [8] corresponding to motion towards MT minus-end during aggregation (light blue) and dispersion (dark-blue), and motion toward the MT plus-end during either aggregation or dispersion (red). The three curves are fits of the experimental histograms by four Gaussian functions (Eq. S1). (B) Distribution of segmental velocities obtained from simulations using the symmetric referential parameter set (RS) indicated in Table 1. The bars represent the histogram while the red curve corresponds to the fitting with Eq. S1. (C)–(I) Histograms from simulations for symmetric teams with all the parameters as in the RS excepting for those specifically indicated in the panels. In all the cases from (C) to (I), the red curve is for the RS. The normalization used for all the distributions is the same as in [8], with the maximum of the distribution set equal to 1.
|
|
Figure 4. Pulling and attachment states contribution to the velocity distribution.(A) Contribution of the pulling states to the velocity distribution for the RS studied in Figure 3B (with Nf = Nb = 2 and k = 0.02 pn/nm). (B) Ibid panel (A) for the attachment states. (C) Contribution of the pulling states to the velocity distribution for the system with Nf = Nb = 2 and k = 0.3 pn/nm shown in Figure 3G. (D) Ibid panel (C) for attachment states. For all the panels, the indications are those in panel (D). The colored curves indicate the contributions of the different states to the overall distribution (gray curves), that are labeled as “all states”.
|
|
Figure 5. Spatial distributions of motors around cargo position for different intervals of the cargo velocity.In all panels, the red and blue curves indicate the distribution of forward and backward motors, respectively. The force scales at the top axis indicate the force exerted by a motor located at the corresponding position. The vertical gray lines indicate the limit positions x−xc = ±110 nm separating regions of pulling and non-pulling motors. Panels (A), (B) and (C) show the results for the RS (k = 0.02 pN/nm) considering the velocity regions I, II and III defined in the main text, respectively. Panel (D) shows the results for the system with k = 0.3 pN/nm studied in Figure 3G considering the region 300 nm/s<v<500 nm/s. Note that, in all the cases, the data were taken from segments with positive (forward) velocity, when the cargo is moving to the right. The parameter μ, defined in the main text, is computed as the quotient between the area below the red histogram for x−xc>110 nm and the area below the blue one for x−xc<−110 nm.
|
|
Figure 6. Velocity distributions for symmetric systems at lower stall forces.Velocity distributions for symmetric systems with: (A)–(B) Fs = 4 pN, nγ = 333 and Fd = 2.4 pN (i.e. transformed from the RS using eq. S2 with λ = 2/3 and β = 0.8), (C)–(D) Fs = 2.7 pN, nγ = 225 and Fd = 1.7 pN (i.e. transformed from the RS using eq. S2 with λ = 0.45 and β = 0.57) and (E)–(F) Fs = 1.2 pN, nγ = 100 and Fd = 1.34 pN (i.e. transformed from the RS using eq. #2 with λ = 0.2 and β = 0.447). In all the cases, the values of k and ε are indicated in the panels while the remaining parameters are those in Table 1.
|
|
Figure 7. Representative reversions for systems with large and low motor stiffness.Trajectories in panels (A) and (B) correspond to the RS (stiffness k = 0.02 pN/nm), while those in panels (C) and (D) correspond to the system with k = 0.3 pN/nm studied in Figure 3G.
|
|
Figure 8. Trajectories of cargo and motors during reversions.(A) Details of the reversion shown in Figure 7A (RS, k = 0.02 pN/nm). (B) Ibid for the reversion shown in Figure 7C (system with k = 0.3 pN/nm). In both panels, the cargo trajectories are indicated by thick solid lines while different red and blue symbols are used for forward and backward motors, respectively. The limits for pulling (positions xc(t)±x0) are indicated by dotted lines. The bottom panels indicate the numbers of attached and pulling motors of both species as function of time during the reversion. The vertical dotted segments indicate relevant detachment events for the reversions.
|
|
Figure 9. Modeling aggregation and dispersion regimes.Histograms of segmental velocities obtained from simulations with non-symmetric motor teams considering: (A) Parameter Set 1: πf = 3.5/s, πb = 2.2/s, εf = 0.08/s and εb = 0.07/s. (B) Parameter Set 2: πf = 1.8/s, πb = 2.2/s, εf = εb = 0.08/s. The remaining parameters of Set 1 and Set 2 are those of the RS (Table 1), excepting for the value of the viscosity for which now we consider nγ = 300. Panel (C) shows the fits by Eq. S1 of the positive and negative velocity branches of the histograms in panels (A) and (B), computed as explained in Information S3. The magenta and the dark-blue curves are the fits of the forward and backward branches of the distribution in panel (A), respectively. The red and the light-blue curve are the fits of the forward and backward branches of the distribution in panel (B), respectively. The parameters of the fits in panel (C) are shown in Information S3.
|
|
Figure 10. Velocity distributions for systems with strongly asymmetric stall forces.Segmental velocity distributions from simulations considering Fsf∼5 Fsb with Nf = 1 and Nb = 5. The red and blue curves represent the distribution for forward and backward velocity segments, respectively. A and C correspond to low motor linker stiffness (kf = kb = 0.02 pN/nm) while B and D consider kf = kb = 0.1 pN/nm. The remaining parameters are the following: A–B: Fsf = 5 pN, Fsb = 1.1 pN, Fdf = 3 pN, Fdb = 1 pN, Πf = 5/s, Πb = 1.6/s, nγ = 100. C–D: Fsf = 5.5 pN, Fsb = 1.1 pN, Fdf = 3 pN, Fdb = 1.5 pN, v0f = v0b = 1000 nm/s, Πf = 5/s, Πb = 1.6/s, nγ = 100. The rest of parameters are displayed in Table 1.
|
|
Figure 1. Scheme of the model.Example of an instantaneous configuration of the system with Nf = Nb = 2. The cargo is represented by a grey bubble. Forward motors (kinesin) are represented in red, while backward motors (dynein) in blue. The MT has green sites separated by Δx = 8 nm where motors can engage and step. The shaded zone around the cargo position indicates the region of the MT where attached motors do not exert forces on the cargo. Motors are labeled with numbers from 1 to 4. From left to right the figure shows a backward pulling motor (labeled as 1), a backward non-pulling motor (2), a detached forward motor (3) and a forward pulling motor (4). The attachment state (see main text) is thus (nf,nb) = (1,2), while the pulling state is (qf,qb) = (1,1). Although qf = qb, the force on motor 4 is larger than that exerted on motor 1 since |x4−xc|>|x1−xc| (see Eq. 1). Thus, the cargo moves to the right.
|
