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. 2020 Oct 23;82(11):141.
doi: 10.1007/s11538-020-00820-0.

Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors

Youngmin Park  1 , Thomas G Fai  2   3
Affiliations

Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors

Youngmin Park et al. Bull Math Biol. .

Abstract

We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We reduce the lubrication model proposed in Fai et al. (Phys Rev Fluids 2:113601, 2017) to a fast-slow system, yielding an analytically and numerically tractable equation equivalent to the original model in the overdamped limit. The model's key parameters include: (1) the ratio of motors that prefer to push toward the head of the dendritic spine to the motors that prefer to push in the opposite direction, and (2) the viscous drag exerted on the vesicle by the spine constriction. We perform a numerical bifurcation analysis in these parameters and find that steady-state vesicle velocities appear and disappear through several saddle-node bifurcations. This process allows us to identify the region of parameter space in which multiple stable velocities exist. We show by direct calculations that there can only be unidirectional motion for sufficiently close vesicle-to-spine diameter ratios. Our analysis predicts the critical vesicle-to-spine diameter ratio, at which there is a transition from unidirectional to bidirectional motion, consistent with experimental observations of vesicle trajectories in the literature.

Keywords: Cell physiology; Dendritic spines; Motor transport; Neurophysiology; Vesicle transport.

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Conflict of interest statement

Conflicts of interest/Competing interests: The authors declare no conflicts of interest.

Figures

Fig. C1
Fig. C1
A comparison of bifurcations in the lubrication and symmetric kinesin models. A: Example force-velocity curves from the lubrication model with different values of viscous drag. Here we use the parameters π1 = 1, π3 = 1, π4 = 4.7, π5 = 0.1, ϕ = 0.5. B: Corresponding one-parameter bifurcation. Multistability terminates through saddle-node (SN) bifurcations C: Phase line curves in the symmetric kinesin model. D: One-parameter bifurcation diagram of the symmetric kinesin model in ζ. For moderate values of viscous drag ζ, there are two stable motor states. As the drag increases, the system undergoes a pitchfork bifurcation. Beyond the pitchfork, only equal numbers of motors from both species are attached, and therefore only the zero velocity is stable. The parameters used in the kinesin motor model are κ0 = 0.5, γ = 3, K = 35.
Fig. 1
Fig. 1
Idealized dendritic spine and molecular motors. A: Three dimensional spine geometry with the vesicle (black sphere) shown at the center of the constriction. The black arrow shows the direction towards the spine head Rp = 0.96, Rc = 1.22. See Figure 2A for additional details on the geometry. B: Transverse cross-section. C: Vertical cross-section with molecular motors. Blue: upwards-preferred motors. Red: downwards-preferred motors.
Fig. 2
Fig. 2
Constriction geometry and resulting dynamics for different parameter sets. A: The initial spine diameter (6μm) decreases to the neck radius Rc = 1.22μm. The vesicle (black circle, radius Rp = 0.96μm) begins at the base of the channel (dashed vertical gray line) and moves in the direction of the arrow for initial condition 1. B, C: Resulting velocity U (μm/s) and position Z (μm) plotted over time (s) for two different initial conditions (U0, Z0) = (0.43μm/s, s, −5μm) (black) and (−0.3μm/s, −5μm) (red). We use the parameters ϕ = 0.57, π1 = 1, π3 = 1, π4 = 4.7, π5 = 0.1, π6 = 10, F0 = 50. D, E, F: same information as A, B, C, but with parameters Rc = 2.15μm, Rp = 1.5μm, ϕ = 0.54, π1 = 1, π3 = 1, π4 = 4.7, π5 = 0.02, π6 = 10, F0 = 200. Initial spine diameter is 6μm. The two initial conditions are (U0, Z0) = (0.17μm/s, −5μm) (black) and (−0.1μm/s, 0μm) (red). Simulation parameters ε = 1, dt=0.02, integrated numerically (see Appendix A.1).
Fig. 3
Fig. 3
Microscopic motor dynamics of a downwards-preferred motor. The x-axis represents the Z coordinate of the motor head relative to its base. A, B: when the vesicle (gray) moves in the preferred direction of a motor (red hatched), the motor attaches with a rate α and detaches with a rate β. C, D: when the vesicle moves in the non-preferred direction of a motor, the vesicle attaches with a rate α and detaches with rate β, but has an additional mechanism of detachment when the motor extends past Z = B.
Fig. 4
Fig. 4
Example force-velocity curves. A, B, C: as parameters ϕ and ζ vary, the underlying force-velocity curves and viscous drag forces change. Blue dashed: proportion of the motor force from upwards-pushing motors. Red dashed: proportion of the motor force from downwards-pushing motors. Purple: total force from molecular motors. Gray: total force from viscous drag. Black dots indicate intersections between the motor forces and viscous drag. D, E, F: respective total force including viscous drag, i.e., plots of ϕF−A(U)+(1 − ϕ)FA(U) − ζU. Black dots indicate force-balance (equilibria), and arrows indicate stability. The changing numbers of equilibria as a function of parameters indicate the loss or gain of multistability. We use the parameters π1 = 1, π3 = 1, π4 = 4.7, π5 = 0.1, π6 = 10
Fig. 5
Fig. 5
The mapping between the bifurcation diagram and critical manifold through the viscous drag function. A: An example of the critical manifold in the phase space of U and Z. Black arrowheads denote the direction of motion on the slow manifold. Gray dashed arrows indicate the direction of motion in the fast system. B: An example of a one-parameter bifurcation diagram. Steady-states U are plotted as a function of ζ. C: The relationship between viscous drag ζ and position Z. The dimensional positions Z = −5μm through Z = 0μm, we expect the critical manifold to resemble a version of the bifurcation diagram given by the mapping between drag ζ and position Z. Beyond the constriction, from dimensional positions Z = 0μm through Z = 5μm, the critical manifold resembles a reflected version of the bifurcation curve. A–C: Parameters as in Figure 2A–C. D–F: Parameters as in Figure 2D–F.
Fig. 6
Fig. 6
Two parameter bifurcation diagram in ϕ and ζ. Saddle-node (SN) bifurcations are shown in (D) as colored branches with a unique color and symbol. Numbers in (D) indicate the total number of fixed points in the corresponding region of parameter space. Subplots A, B, C, E, F, show one-parameter slices of the two-parameter diagram. Saddle-nodes are labeled with the corresponding branch color and symbol. The critical vesicle-to-spine diameter ratio at the cusps is roughly 2μm/3μm.
Fig. 7
Fig. 7
Cusp bifurcations as a function of π4 and π5. A: For each ζ > 0,cusp bifurcations exist along a set of π4, π5. Example level curves are plotted for ζ = 0, 4.3 × 10−5, 1.3 × 10−4, 2.2 × 10−4. We take 4 representative pairs of π4, π5 labeled B–E and show the corresponding two-parameter bifurcation diagrams in B–E. The point labeled with ⋆ corresponds to Figure 6.
Fig. 8
Fig. 8
Time-lapse images of recycling endosomes adapted from (da Silva et al., 2015) and available under the CC BY NC ND license. A: a recycling endosome translocates through a thin spine in four time-lapse images (left). A kymograph is shown to the right. The vesicle is roughly 1μm in diameter, and the distance between the vesicle and neck wall is at most 0.1μm. B: a recycling endosome translocates into a stubby spine in a series of time-lapse images, with the associated kymograph on the right. The vesicle is roughly 0.5μm in diameter, the distance between the vesicle and neck wall is roughly 0.15μm, and the distance between the vesicle and spine wall is roughly 0.5μm. All scale bars 2μm. C: approximate ranges of drag (red transparent) superimposed on the two-parameter diagram from Figure 6 with the drag ζ plotted on a log scale. Labels A and B correspond to Panels A and B.
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References

    1. Acheson David J. Elementary fluid dynamics, 1991.
    1. Adrian Max, Kusters Remy, Wierenga Corette J, Storm Cornelis, Hoogenraad Casper C, and Kapitein Lukas C. Barriers in the brain: resolving dendritic spine morphology and compartmentalization. Frontiers in Neuroanatomy, 8:142, 2014. - PMC - PubMed
    1. Allard Jun, Doumic Marie, Mogilner Alex, and Oelz Dietmar. Bidirectional sliding of two parallel microtubules generated by multiple identical motors. Journal of Mathematical Biology, pages 1–24, 2019. - PMC - PubMed
    1. Arellano Jon I, Benavides-Piccione Ruth, DeFelipe Javier, and Yuste Rafael. Ultrastructure of dendritic spines: correlation between synaptic and spine morphologies. Frontiers in Neuroscience, 1:10, 2007. - PMC - PubMed
    1. Bagnall Josephine Shaw, Byun Sangwon, Begum Shahinoor, Miyamoto David T, Hecht Vivian C, Maheswaran Shyamala, Stott Shannon L, Toner Mehmet, Hynes Richard O, and Manalis Scott R. Deformability of tumor cells versus blood cells. Scientific Reports, 5:18542, 2015. - PMC - PubMed

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