Providing positional information with active transport on dynamic microtubules

Abstract

Microtubules (MTs) are dynamic protein polymers that change their length by switching between growing and shrinking states in a process termed dynamic instability. It has been suggested that the dynamic properties of MTs are central to the organization of the eukaryotic intracellular space, and that they are involved in the control of cell morphology, but the actual mechanisms are not well understood. Here, we present a theoretical analysis in which we explore the possibility that a system of dynamic MTs and MT end-tracking molecular motors is providing specific positional information inside cells. We compute the MT length distribution for the case of MT-length-dependent switching between growing and shrinking states, and analyze the accumulation of molecular motors at the tips of growing MTs. Using these results, we show that a transport system consisting of dynamic MTs and associated motor proteins can deliver cargo proteins preferentially to specific positions within the cell. Comparing our results with experimental data in the model organism fission yeast, we propose that the suggested mechanisms could play important roles in setting length scales during cellular morphogenesis.

Figure 1

(A) Schematic drawing of a fission yeast cell. (Upper MT) Motor-mediated protein accumulation at the tip of a growing MT. (Lower MT) The MT undergoes a catastrophe and the proteins are locally delivered at the position of catastrophe. (B) Space-time-plot, showing the tip of a growing MT in fission yeast, decorated with the motor protein Tea2-GFP.

Figure 2

MT length distributions. (A) Schematic drawing of MT length (L) changes, according to the two-state model of dynamic instability. Dynamic instability is commonly characterized by four parameters (63): the growth and shrinkage velocities v_g and v_s, as well as the catastrophe and rescue frequencies f_c and f_r, which are the rates at which microtubules switch from the growing to the shrinking state and vice versa. (B) MT length distributions according to Eq. 2 for different parameter settings. Exponential: f_c = 0.5/min, f_r = 0/min, v_g = 2 μm/min, and v_s = NA. Mitotic Xenopus (Gaussian): f_c(L) = L × 0.19/μm/min, f_r(L) = 2.7/min − L × 0.21/μm/min, v_g = 10 μm/min, and v_s = 15 μm/min. Fission yeast (half-Gaussian): f_c(L) = L × 0.05/μm/min, f_r = 0/min, and v_g = 2 μm/min.

Figure 3

Motor lattice-distribution and tip-accumulation on growing MTs. (A) Schematic drawing of the model, showing binding with rate α, motor motion with velocity v_m, motor dissociation with rate β, and MT growth with velocity v_g. In addition, motors can bind to the MT tip with rate α_t or they can walk into the MT tip via the MT lattice. At the MT tip, motors dissociate with rate β_t. (B) Average number of motors per 0.2 μm of MT lattice on a growing MT, including accumulation at the MT tip (according to Eqs. 9 and 12). Different shaded values correspond to density profiles at different time points. Parameters: α = 4 μm⁻¹ min⁻¹, β = 1/min, v_m = 4 μm/min, v_g = 2 μm/min, α_t = 0/min, and β_t = 2/min. (C) Motor accumulation at MT tip according to Eq. 12. Schematic drawings correspond to panel A. (Dotted line) The value n₀, i.e., the number of motors at infinite length. (Solid curve) Motor proteins bind only to the lattice and walk into the tip: α = 10 μm⁻¹ min⁻¹, β = 0.5/min, v_m = 3 μm/min, v_g = 2μm/min, α_t = 0/min, and β_t = 2/min. (Dashed curve) Motors bind only to the tip: α = 0, β = NA, v_m = 3 μm/min, v_g = 2 μm/min, α_t = 5/min, and β_t = 0.5/min.

Figure 4

Creating positional information with dynamic MTs and motor accumulation at MT tips in fission yeast (S. pombe). (A) Motor accumulation n_tip/n₀, MT catastrophe rate f_c, and MT length distribution p(L) as used to compute cargo delivery in panel B. (B) Cargo delivery rate by a transport system composed of dynamic MTs and motor proteins, according to Eq. 15, where the MT catastrophe rate linearly increases with MT length as measured in fission yeast. Parameters: k_c = 0.05/μm/min, f_r = 0/min, v_g = 2 μm/min, v_s = NA, v_m = 10 μm/min, and β as indicated. (C) As in panel B, but for MTs of different average length, as determined by the effective parameter $\sigma =\sqrt{v_g/k_c}$ (see Eq. 15). Parameters: f_r = 0/min, v_s = NA, v_m = 10 μm/min, β = 5/min, σ₀ = 6.3 μm, and σ as indicated.

Figure 5

Distribution of motors on growing microtubules. (A–C) Motor densities on growing MTs according to Eq. 7. Shaded areas correspond to different time points (at t = 0, MTs have zero length and there are no motors bound). In all panels: α = 4 μm⁻¹ min⁻¹, β = 4 min⁻¹, and v_g = 8 μm/min. (A) The value v_m = 11 μm/min, assuming that motors reaching the end of the MT do not accumulate but dissociate. (B) The value v_m = 3 μm/min. (C) The v_m = 0 μm/min.