Mechanical Principles Governing the Shapes of Dendritic Spines

Abstract

Dendritic spines are small, bulbous protrusions along the dendrites of neurons and are sites of excitatory postsynaptic activity. The morphology of spines has been implicated in their function in synaptic plasticity and their shapes have been well-characterized, but the potential mechanics underlying their shape development and maintenance have not yet been fully understood. In this work, we explore the mechanical principles that could underlie specific shapes using a minimal biophysical model of membrane-actin interactions. Using this model, we first identify the possible force regimes that give rise to the classic spine shapes-stubby, filopodia, thin, and mushroom-shaped spines. We also use this model to investigate how the spine neck might be stabilized using periodic rings of actin or associated proteins. Finally, we use this model to predict that the cooperation between force generation and ring structures can regulate the energy landscape of spine shapes across a wide range of tensions. Thus, our study provides insights into how mechanical aspects of actin-mediated force generation and tension can play critical roles in spine shape maintenance.

Keywords: dendritic spines; deviatoric curvature; lipid bilayer; membrane-actin interactions; tension.

Figure 1

Modeling of forces relevant to spine shape. (A) Schematic depiction of different shape categories of dendritic spines (Reprinted with permission from SynapseWeb, Kristen M. Harris, PI, http://synapseweb.clm.utexas.edu/). The inset shows a schematic of a tubular neck with a radius r and a spontaneous deviatoric curvature D_m along the total neck length l. (B) The surface parametrization of the membrane geometry in axisymmetric coordinates. s is the arclength, n is the unit normal vector to the membrane surface, and a_s is the unit tangent vector in the direction of arclength. We assume that the actin filaments can apply axial (F_z) or normal (F_n) forces to the membrane surface. We assume that there is a large membrane reservoir with a fixed area, and we focused on the local region of the membrane under tension λ, as indicated by the dotted box.

Figure 2

Formation of stubby and filopodia shaped spines with a localized axial force. (A) Linear relationship between the magnitude of axial force and tension in a small stubby-shaped membrane deformation (Derényi et al., ; Powers et al., 2002). The dashed line is the fitted curve (F_z = 0.688λ) with R² = 0.9967. (B) Linear relationship between the magnitude of axial force and the height of the stubby spine for a fixed tension (Derényi et al., ; Powers et al., 2002). The dashed line is the fitted curve (F_z = 16.82L) with R² = 0.9896. (C) A stubby-shaped spine with a total length L = 0.44 μm is formed with F_z = 7.5 pN applied along the blue area (λ = 10 pN/μm). (D) Neck radius of a filopodium as a function of tension (r $=\sqrt{\kappa /(2\lambda )}$) (Derényi et al., 2002). (E) The magnitude of axial force needed to form a filopodium as a function of tension (${\text{F}}_{\text{z}}=2\pi \sqrt{2\kappa \lambda }$) (Derényi et al., 2002). (F) A filopodium-shaped protrusion with a total length L = 5 μm and neck radius r = 0.2 μm is formed with F_z = 11.2 pN applied along the spherical cap of the filopodium, which is shown in blue (λ = 9 pN/μm).

Figure 3

Formation of thin-shaped spines with localized normal force density along the spine head. (A) Neck radius of a thin-shaped spine as a function of tension (r $=\sqrt{\kappa /(2\lambda )}$) (Derényi et al., 2002). (B) Linear relationship between the magnitude of normal force density needed to form a thin-shaped spine and the tension. Here, the area of the applied force is set at A_force = 0.44 μm². The red squares represent the results obtained from simulation and the dashed line is the derived analytical solution (${\text{f}}_{\text{n}}=4\lambda \sqrt{\pi /A_{\text{force}}}$, Equation (7)). (C) The magnitude of a normal force density needed to form a thin-shaped spine as a function of the area of the spine head. The tension is set at λ = 36 pN/μm. The red squares represent the results obtained from our simulations and the dashed line is the derived analytical solution ($({\text{f}}_{\text{n}}=4\lambda \sqrt{\pi /A_{\text{force}}}$, Equation (7). (D) A thin-shaped spine with a total length L = 0.98 μm, neck radius r = 0.05 μm, and head volume V = 0.033 μm³ is formed with f_n = 382.23 pN/μm² applied along the head of spine which is shown in red (λ = 36 pN/μm and A_force = 0.44 μm²).

Figure 4

Formation of mushroom-shaped spines with localized normal forces along the spine head and PSD. (A) A mushroom-shaped spine with a total length L = 1.51 μm, neck radius r = 0.1 μm, head volume V = 0.25 μm³, and area of PSD/area of head = 0.2 is formed with f_n = 84.04 pN/μm² applied along the head of spine (red domain) and f_n,PSD = 334.88 pN/μm² applied along the PSD (gray domain) (λ = 9 pN/μm). (B) The non-monotonic behavior of the volume of a mushroom-shaped spine head when increasing tension. Three different shapes of mushroom-shaped spines are shown for low, intermediate, and high tensions. With increasing magnitude of tension, the mushroom-shaped spine head flattens. (C) The magnitude of normal force densities in the spine head (red squares) and in PSD (gray squares) increases with increasing tension.

Figure 5

Effective tension including spontaneous deviatoric curvature regulates the neck radius and the magnitude of axial force in a tubular membrane. (A) Analytical solution for the neck radius of a tubular membrane as a function of spontaneous deviatoric curvature and tension ($\text{r}=\sqrt{\kappa /(2(\lambda +\kappa {\text{D}}_{\text{m}}^2))}$, Supplementary Equation 37). (B) The neck radius obtained from numerical solutions as a function of effective tension ($\lambda +\kappa {\text{D}}_{\text{m}}^2$). Here, for fixed three different tensions, we varied the effective tension by changing the spontaneous deviatoric curvature between $0<{\text{D}}_{\text{m}}<30\text{μ}{\text{m}}^{-1}$. The radii of the membrane necks collapse onto a single curve for different tensions. (C) Analytical solution for the magnitude of an axial force needed to maintain a tubular protrusion as a function of spontaneous deviatoric curvature and tension (${\text{F}}_{\text{z}}=2\pi (\sqrt{2\kappa (\lambda +\kappa {\text{D}}_{\text{m}}^2)}-\kappa {\text{D}}_{\text{m}})$, Supplementary Equation 37). The axial force needed to maintain a tubular protrusion has a local minimum along the red line where λ =$\kappa {\text{D}}_{\text{m}}^2$ (Supplementary Equation 39). (D) The effective axial force (F_z+2πκD_m) obtained from numerical solutions as a function of effective tension ($\lambda +\kappa {\text{D}}_{\text{m}}^2$). Here, for fixed three different tensions, we varied the effective tension by changing the spontaneous deviatoric curvature between $0<{\text{D}}_{\text{m}}<30\text{μ}{\text{m}}^{-1}$. Effective axial forces collapse onto a single curve for different tensions.

Figure 6

Formation of thin and mushroom shaped spines with a combination of forces and spontaneous deviatoric curvature. (A) Formation of a thin-shaped spine by applying a uniform normal force density along the spine head (left) vs. applying a uniform normal force density along the head and spontaneous deviatoric curvature (purple region) along the spine neck (right) (λ = 10 pN/μm). (B) Formation of a thin-shaped spine by applying an axial force along the spherical cap (blue region) and spontaneous deviatoric curvature along the spine neck (purple region), λ = 10 pN/μm. All thin spines in (A,B) have a neck radius r ~ 0.05 μm and head volume V ~ 0.033 μm³. (C) Formation of a mushroom-shaped spine by applying a non-uniform normal force density along the spine head (left) vs. applying a non-uniform normal force density along the head and spontaneous deviatoric curvature along the spine neck (purple region), (right), λ = 5.5 pN/μm. The formed mushroom spine with normal force densities f_n = 57.14 pN/μm² and f_n,PSD = 154 pN/μm² and deviatoric curvature D_m = 1.8 μm⁻¹ has a neck radius r ~ 0.1 μm and head volume V ~ 0.27 μm³. (D) The magnitude of a normal force density that is required to form a thin-shaped spine with and without spontaneous deviatoric curvature as a function of effective tension and tension, respectively. (E) The magnitude of an axial force that is required to form a thin-shaped spine with and without spontaneous deviatoric curvature as a function of effective tension and tension, respectively. (F) The magnitude of normal force densities in the spine head and in PSD that is required to form a mushroom spine with and without spontaneous deviatoric curvature as a function of effective tension and tension, respectively.

Figure 7

Characterizing different shapes of dendritic spines based on the mechanical model. (A) Stubby spines can be formed with an axial force and in a wide range of tensions. (B) An axial force is sufficient to form a long filopodial spine. (C) A thin-shaped spine can be formed with three different mechanisms; (1) a uniform normal force density along the spine head, (2) a uniform normal force density along the spine head and spontaneous deviatoric curvature along the neck, and (3) a uniform axial force density along the spine head and spontaneous deviatoric curvature along the neck. In the bar plot, the total energy of the system is shown for three different mechanisms. The total energy of the system for the second and third mechanisms with spontaneous deviatoric curvature is much less than the energy for the first mechanism with just a normal force. (D) A mushroom-shaped spine can be formed with two different mechanisms; (1) a non-uniform normal force density along the spine head and PSD region and (2) a non-uniform normal force density along the spine head and PSD region plus a spontaneous deviatoric curvature along the spine neck. The resulting mushroom spine with a combination of normal forces and spontaneous deviatoric curvature has lower energy compared to the spine that is formed with just normal forces (bar graph).