Biophysical model of the role of actin remodeling on dendritic spine morphology

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.


Code Quantity
Typical Scale Source N Number of actin filaments in spine-head (see "Estimate for the Number of Actin Filaments", S1 File) R base Radius of the base of the spine (viz. where the spine is connected to the dendritic membrane). This quantity was estimated on the basis of microscopy images published by (2).
∼ 300 nm (2) R neck Radius of a typical spine-neck 75 ± 30 nm (3) R head Radius of a typical spine-head 220 ± 154 nm (3) L neck Length of a typical spine-neck 0.2 − 2 µm (2) L filop. Length of a typical filopodium 0.9 − 10 µm (4) (mean ≈ 5 µm (5) Bending rigidity of lipid bilayer membrane 5 × 10 −19 J theory: constant surface-area theory: constant surface-tension empirical shows that the models in both ensembles predict the correct order of magnitude for all morphological characteristics, although significant differences are observed between the two ensembles. Experimental data on widths was taken from (2), number of filaments N filaments required to produce dendritic filopodia was estimated from EM images published in (1). Whiskers denote maximum and minimum values of the corresponding data. Note that the standard deviation in measured quantities is necessarily lower, and often much lower, than these lower and upper bounds.

Septin Complexes are Unlikely Required for Effecting the Transition to Mature Spines
Although septin-complexes are found consistently along spine-necks (8-10), they are only reported to be positioned at the base of the spine and not along the full length of the spine-neck. Our models predict that it is required to place line tensions along the full length of the spine-neck in order to constrain it, and therefore we can refute septin-complexes as being solely responsible for constraining the long, thin spine-necks. Moreover, the assembly of septins into ring-like structures has an associated time-scale in the order of minutes (11). Hence, we find that cytoskeletal remodeling-which can performed on the time-scale of fractions of a second (6)-is much more rapid than positioning these septins. From these observations combined, we hypothesize that ring-like septin-complexes or anchoring proteins are not required for the transition from filopodium to mature spine, but could plausibly aid in the stabilization of these mature, mushroom-like spines.

Estimate for the Number of Actin Filaments
We counted the number of actin filaments as 20 on ∼ 20% of the surface-area resulting in ∼ 100 filaments for one entire spine-head as published by (1). Then, noting that on the average the filaments are not oriented perpendicular to the membrane-but rather at an angle π/4, we find the effective number of actin filaments to be ∼ 100 · cos(π/4) ≈ 71. This number falls within the range for the number of polymerizing filaments N = 50 − 150 we derived from data published in (12) ( (12) has published the density of non-stationary actin molecules, which we integrated over the surface area to obtain a measure for the number of polymerizing filaments).

Standard Deviation in Spine-Neck Width
We measured the width of the spine-neck of images by (2) by fitting the intensity of the profile with Gaussian distributions along the axis of the spine-neck. We asserted that the standard deviation of these Gaussians is a measure for the width of the spine-neck. Then, we computed the relative variation in these widths. Using this method, the relative variation in the width of the spine-neck was found to be 13.5%.

Shape Equations
Taking the first variation of the Canham-Helfrich energy functional, and insisting that the first variation δE is zero under all possible infinitesimal perturbations results in a differential equation that describes stationary shapes {r(s), ψ(s)}. The stationary shapes include shapes corresponding to an energetic minimum, an energetic maximum or a saddle point in the energy functional. A seminal paper by (13) describes the higher-order variations, from which we can infer the class of stationary point. We will not discuss this technical difficulty in this publication, although we have used numerical perturbative methods to determine which shapes correspond to energetic minima. This differential equation, that we shall henceforth call the shape equation, is (14,15) where we have dropped the s-dependencies andσ ≡ σ/K b . Most publications that we have consulted make reference to second-order shape equations (14), but-in accordance with (15)-we find the thirdorder shape equation 1 to be numerically substantially more stable. The second-order shape equations, e.g. found by taking the first integral of 1, is used to find boundary conditions for ψ . This equation is (15) ψ cos ψ = − 1 2 (ψ ) 2 sin ψ − (cos ψ) 2 r ψ + (cos ψ) 2 + 1 2r 2 sin ψ +σ sin ψ − 1 2p wheref ≡ f /2πK b . Although the point force f does not show up in the shape equation 1, it does enter in the determination of the correct boundary conditions through equation 2.
In this paper, we have used the ensemble of a prescribed surface-area available to the shape (for more details and rationale, see the main text). We have used its conjugate variable, the surface-tension σ, with a shooting-and-matching technique to constrain the surface-area to a given value. Similarly, we use respectively the vertical force f , the initial curvature ψ (0) and the total arc-length S as Lagrange multipliers (and hence, shooting variables) to generate shapes with a prescribed height L, total surfacearea A and to match them to the z−axis (i.e. r(S) = 0). For more information on this numerical technique, we refer to (16)).