Fig 1.
Outline of the model for spine formation and maturation.
Panel (a): Cartoon of spine initiation, elongation and maturation. From left to right: ‘stubby spines’ dendritic filopodia or thin spines; mature, mushroom-like spines. In our mathematical model, we solve the shape equation based on the energy functional 1 (see S1 File, Eq (1)). In this study we show that, at least for the purposes of the force calculations, the results of the shape equation can be reproduced using the geometries that are also displayed in this figure. Panel (b): Definition of axisymmetric coordinate system that use for our models.
Fig 2.
Outline of results for filopodium formation.
Panel (a): Cartoon of qualitative effect of increasing force whilst the amount of membrane is kept constant. Panel (b): Effect of growth (viz. membrane addition) on filopodium morphology (if the force on the membrane is kept constant). These shapes experience a vertical force of 35pN corresponding to approximately 9 polymerizing actin filaments. Membrane addition results in substantial elongation of the filopodium. Panel (c): Effect of cytoskeletal remodelling (viz. actin polymerization) on filopodium morphology if the amount of membrane is kept constant. These shapes have a surface-area of 0.68μm2. Increasing the number of polymerizing actin filaments leads to a marked change in morphology from a stubby-like morphology to a tubular shape. Panel (d): Force-extension curves of our models of dendritic filopodia for various values of the surface-area . Numbers at curves indicate the surface-area in units of μm2 whereby we used a radius of the base of the filopodia Rbase = 300 nm.
Fig 3.
Outline of results for spine maturation.
The spine bases have been left out in the renders in this figure. Panel (a): Cartoon showing qualitative effect of increasing force (viz. increasing the number of filaments in the spine-head) whilst the amount of membrane is kept constant. Increasing the number of actin filaments in the spine-head enlarges the spine-head and, at the same time, a thinning neck. Panel (b): Results of our model are combined in a three-dimensional growth-organization matrix, shown here with selected shapes. These shapes show clearly the effects of increasing the number of filaments in the head N, the total surface-area and the length of the spine-neck
. Panel (c): The minimum number of actin filaments required in the cytoskeleton for sustaining the contractile force fhead that the spine-head membrane exerts and for counteracting the expansive force fneck of the spine-neck. Band indicates typical values of the total amount of membrane
(cf. Table A in S1 File). Dashed lines indicate number of actin filaments required for counteracting fhead + fneck. Empirical data (black circles) shows reasonable agreement with our model (data taken from from [32], S1 File). Panel (d): Ratio of head and neck radii (left) and volumes (right) for a number of actin filaments N = 25, 50, 100, 150, 200 (lower to upper curves). For these plots we used Eq 2 with
. Experimental data from [13] is highlighted in gray. Panel (e): Effects of growth (membrane addition) and the number of actin filaments in the spine-head on spine morphology. In these models, we kept the total length of the spine-neck fixed. Dotted lines are a visual aid for showing how increasing the number of actin filaments in the spine-head results decreases the width of the spine-neck.
Fig 4.
Overview of effect of constrictions on the shapes of spine-necks.
Results of simulations that have been performed using the energy functional Eq (1) (solid curves) and theoretical model that treats these shapes as cylinders (dashed curves). The computations have been performed for Rneck = 45…105 nm as indicated in the figure. Black lines corresponds to Rneck = 75 nm. We used Kb = 5 × 10−19 J for these computations [11]. Top panel: The line tension τ as a function of the distance between the line tensions. Inset shows how line tension τ and ‘unduloid amplitude’ δ are defined. Bottom panel: The absolute value of the reduced ‘unduloid amplitude’,
. Inset panel shows the reduced ‘unduloid amplitude’ where it crosses
. Indicated is
, the approximate amplitude of variations in the width of the spine-neck, corresponding to τ ≈ 6.5 − 15pN.
Fig 5.
Cartoon of the proposed morphological stages in filopodium formation and spine maturation and their relation to various factors.
We propose that binding proteins, such as drebrin, α−actinin and CaMKIIβ [1], may be involved in aligning the actin filaments away from the dendritic shaft. Branching proteins are necessary for effecting the transition to a branched actin organization and the large number of actin filaments in mature spine-heads. A candidate protein for this would be Arp2/3, a protein that localizes to the spine-head [1, 5]. Various proteins that constrict the spine-neck possibly aid in the stabilization of a long, thin spine-neck. Of these stabilizing factors, a strong candidate (that definitely localizes consistently to spine-necks) is the ring-like F-actin complex [36]; another candidate is the ring-like septin-complex [34, 39, 40].