Date Published: February 3, 2017
Publisher: Public Library of Science
Author(s): C. A. Miermans, R. P. T. Kusters, C. C. Hoogenraad, C. Storm, Ferdinando Di Cunto.
Dendritic spines are small membranous structures that protrude from the neuronal dendrite. Each spine contains a synaptic contact site that may connect its parent dendrite to the axons of neighboring neurons. Dendritic spines are markedly distinct in shape and size, and certain types of stimulation prompt spines to evolve, in fairly predictable fashion, from thin nascent morphologies to the mushroom-like shapes associated with mature spines. It is well established that the remodeling of spines is strongly dependent upon the actin cytoskeleton inside the spine. A general framework that details the precise role of actin in directing the transitions between the various spine shapes is lacking. We address this issue, and present a quantitative, model-based scenario for spine plasticity validated using realistic and physiologically relevant parameters. Our model points to a crucial role for the actin cytoskeleton. In the early stages of spine formation, the interplay between the elastic properties of the spine membrane and the protrusive forces generated in the actin cytoskeleton propels the incipient spine. In the maturation stage, actin remodeling in the form of the combined dynamics of branched and bundled actin is required to form mature, mushroom-like spines. Importantly, our model shows that constricting the spine-neck aids in the stabilization of mature spines, thus pointing to a role in stabilization and maintenance for additional factors such as ring-like F-actin structures. Taken together, our model provides unique insights into the fundamental role of actin remodeling and polymerization forces during spine formation and maturation.
A single neuron can contain hundreds to thousands of dendritic spines, actin-rich, micron-sized protrusions which project from dendritic shafts . Mature spines consist of two basic compartments: a constricted region called the neck, supporting a bulbous head containing the postsynaptic site that makes contact with the axon of a nearby neuron. Spines come in a wide range of sizes and shapes, their lengths varying between 0.2 − 2μm and their volumes between 0.001 − 1μm3. Electron microscopy (EM) studies have identified several morphological categories of spines, such as thin, filopodium-like protrusions (‘thin spines’), and spines with a large bulbous head (‘mushroom spines’) [1–5]. Different live cell-imaging techniques have demonstrated that dendritic spines are highly dynamic structures, subject to constant morphological change even after birth.
Reflecting the approximate rotational symmetry of dendritic spines, we use an axisymmetric coordinate system consisting of an angle ψ with the horizontal, an arc-length parameter s, radial coordinate r and vertical coordinate z. Based on in vivo microscopy [1, 13, 14], we fix the angle of the shape at ψ = 0 on the edges of our integration interval Rbase = 300 nm (cf. Table A in S1 File). This coordinate system is schematically displayed in Fig 1. The arc-length parameter s=0…S is used as the independent variable and r(s) and ψ(s) as the coordinates. This coordinate system fully determines the shape, and the vertical coordinate z(s) is recovered by the geometrical relation z′(s) = − sin ψ(s). The Canham-Helfrich energy functional that we use can be written [10, 11]
where Kb ≈ 500pN · nm is the bending rigidity of the membrane , 2H = ψ′(s) + sin ψ(s)/r(s) is the mean curvature  (with ψ′(s) ≡ dψ/ds), A=∫da is the surface area, σ is a surface tension which we use as a Lagrange multiplier to enforce the surface area, f is a point-force acting on the membrane, L=z(S)-z(0) is the height of the membrane, phead is a pressure exerted on the membrane and Vhead is the volume of the spine-head. The first term in this energy functional—the one containing the mean curvature 2H—represents the bending energy of the membrane, which reflects the tendency of lipid bilayers to adopt a flat shape (or spherical in the case of vesicles with nonfixed volume). We use the surface tension σ and point-force f as Lagrange multipliers to enforce specific values of the surface-area A0 and the height of the shape L0 . Within this paradigm, we interpret the surface-area, viz. amount of membrane available to the spine, as a quantity that encodes growth [17, 18]. The height of the shape reflects the cytoskeletal architecture of the spine, having a definite length. We stress that, due to the bending energy of the membrane, the point-force f acting on the membrane gives rise to a membrane deformation of a finite size [10, 19]. Thus, the singularity in the force-field does not translate into a singularity in the membrane shape. Moreover, there is experimental evidence that the filopodial force is strongly directional and orthogonal to the dendritic shaft , highlighting the importance of a point or point-like force in spine morphogenesis.
We use the Canham-Helfrich energy functional Eq (1) to model the growth of dendritic spine membranes. The growth sequence is schematically shown in Fig 1. We will show that this growth sequence can be explained qualitatively and quantitatively by simple models that incorporate the interaction of the actin cytoskeleton and the spine membrane. To that end, we will first determine how filopodia are formed by application of forces that the cytoskeleton exerts on the spine membrane. Then, we will show that the forces generated by a branched cytoskeleton, located at the top of the spine, will result in a bulbous head and a thin spine-neck. Finally, we will show that actin-membrane anchoring or ring-like molecules are another scenario for constraining a large head and long, thin neck. For the model calculations, we shall make repeated use of the physiologically relevant parameters that we have tabulated (see Table A in S1 File).
We study the physical mechanisms that determine the morphology of dendritic spines. In particular, we investigate the ability of the actin cytoskeleton to change the size and shape of spines. We find that the most striking primary features of spine growth and spine morphology can be straightforwardly understood as a consequence of the trade-off between the elastic properties of the spine membrane and the forces actively generated by the actin cytoskeleton. Specifically, we show that the initiation and formation of dendritic filopodia may be rationalized on the basis of the protrusive forces of the actin cytoskeleton. Using realistic estimates for the number of actin filaments involved, we find that the dimensions of the filopodia in our models agrees well with the observed dimensions of newly formed protrusions in the developing neuron.