Excess area dependent scaling behavior of nano-sized membrane tethers

Abstract

Thermal fluctuations in cell membranes manifest as an excess area ([Formula: see text]) which governs a multitude of physical process at the sub-micron scale. We present a theoretical framework, based on an in silico tether pulling method, which may be used to reliably estimate [Formula: see text] in live cells. We perform our simulations in two different thermodynamic ensembles: (i) the constant projected area and (ii) the constant frame tension ensembles and show the equivalence of our results in the two. The tether forces estimated from our simulations compare well with our experimental measurements for tethers extracted from ruptured GUVs and HeLa cells. We demonstrate the significance and validity of our method by showing that all our calculations performed in the initial tether formation regime (i.e. when the length of the tether is comparable to its radius) along with experiments of tether extraction in 15 different cell types collapse onto two unified scaling relationships mapping tether force, tether radius, bending stiffness κ, and membrane tension σ. We show that [Formula: see text] is an important determinant of the radius of the extracted tether, which is equal to the characteristic length [Formula: see text] for [Formula: see text], and is equal to [Formula: see text] for [Formula: see text]. We also find that the estimated excess area follows a linear scaling behavior that only depends on the true value of [Formula: see text] for the membrane, based on which we propose a self-consistent technique to estimate the range of excess membrane areas in a cell.

Figure 1

Equilibrium values of $\langle {\mathcal{A}}_{\text{ex}}\rangle$ as a function of $\langle \mathcal{A}\rangle$ in the constant N-σ-τ-T ensemble. Data shown for five different values of τ=−412, −206, 0, 206 and 412 μN/m, for a membrane with κ = 20 k_BT and σ = 0.

Figure 2

(a) Representative equilibrium conformation of a membrane with κ = 20 k_BT and ${\mathcal{A}}_{\text{ex}}~40\mathrm{\%}$. The set of biased vertices at the tip ({X_T}) and at the base ({X_B}) are explicitly marked as spheres — R_T and R_B denote their respective center of masses. {X_T} is the set of all vertices within a region of size ℛ_bead. The conformation of the membrane with a fully developed tether for ℒ_t=400 nm are shown in panels (b) and (c). The snapshot in panel (b) corresponds to a constant $N-\sigma -{\mathcal{A}}_{\text{patch}}-T$ ensemble, while that in (c) corresponds to a constant N-σ-τ-T ensemble. l_t and ℛ_t, the length and radius of the membrane tether, and the membrane dimension ℒ_patch are also marked.

Figure 3

(a) Representative conformations of a membrane with κ = 20 k_BT and ${\mathcal{A}}_{\text{ex}}~40\mathrm{\%}$ as a function of ℒ_t. Panels (b) and (c) show the computed values of the tether length l_t, and radius ℛ_t, respectively, as a function of ℒ_t. These quantities are computed as described in the methods section. The shaded regions mark the three regimes for initial tether formation (i.e., for l_t/ℛ_t < 5) namely, regime 1: suppression of undulations, regime 2: formation of tethers, and regime 3: extrusion of tethers at a constant radius. The boxed numbers in the top panel denote the regimes to which the configurations correspond to.

Figure 4

Dependence of the tether radius on the size of the biasing region. (a) Representative conformations of tethers, for ℒ_t=300 nm, extracted using beads with ℛ_bead = 25, 50, 75, and 100 nm, from a membrane with κ = 20 k_BT and ${\mathcal{A}}_{\text{ex}}~10\mathrm{\%}$. Panels (b) and (c) show the computed values of ℛ_t, as a function of ℛ_bead, for κ = 20, 40, and 160 k_B T for ${\mathcal{A}}_{\text{ex}}=10$ and 40%, respectively.

Figure 5

(a) The potential of mean force ${\mathcal{W}}_{\mathrm{t}}$ and the tether force ℱ_t, as a function of the tether length l_t, for a membrane with κ = 20 k_BT and ${\mathcal{A}}_{\text{ex}}~40\mathrm{\%}$. In the top panel, ${\mathcal{W}}_{\mathrm{t}}$ shows a linear scaling in regimes 1 and 3, which are represented by the functions ℱ₁l_t and ℱ₂l_t, respectively. The lower panel compares values of ℱ_t estimated from direct numerical differentiation of ${\mathcal{W}}_{\mathrm{t}}$ (symbols) to that obtained from the scaling relations (lines). (b) Force displacement curves for experimental tether pulling assay using ruptured GUVs (top panel) and HeLa cells (lower panel) – the inset shows a transition between regions of constant force. The illustration in the top panel shows the state of the membrane tether at various stages of the experiment. The vertical deflection of the AFM tip is measure of the tether force ℱ_t and its separation from the sample is a measure of the tether length l_t.

Figure 6

The potential of mean force ${\mathcal{W}}_{\mathrm{t}}$ as a function of the tether length l_t, extracted with ℛ_bead = 50 nm, from membranes with ℒ_patch = 0.51 μm and 1.02 μm, and excess areas ${\mathcal{A}}_{\text{ex}}=10\mathrm{\%}$ and 40%. Data for κ= 20 k_BT are shown in panel (a) and that for κ = 40 k_BT is shown in panel (b).

Figure 7

Comparison of tether characteristics in the constant N-σ-τ-T ensemble for four different membrane patches, denoted A1-A4. Panels (a), (b) and (c) show the area $(\mathcal{A})$, projected area $({\mathcal{A}}_{\text{patch}})$ and excess area $({\mathcal{A}}_{\text{ex}})$ as a function of the reaction coordinate ℒ_t. The measured values of tether length l_t and tether radius ℛ_t are shown in panel (d) as filled and open symbols, respectively. The potential of mean forces and the tether forces are shown in panels (e) and (f) as a function of the tether length l_t.

Figure 8

Comparison of tether characteristics in the constant $N-\sigma -{\mathcal{A}}_{\text{patch}}-T$ and N-σ-τ-T ensembles. In panel (a), the tether length and radius are shown as open and filled symbols, respectively. The potential of mean forces (panel (e)) and tether force (panel (f)) are plotted as a function of the tether length l_t. Data shown for a membrane with κ = 20 k_BT.

Figure 9

(a) Six model membrane systems, denoted M1–M6, with specified values of ${\mathcal{A}}_{\text{ex}}$ and κ. For any system M_i (i = 1 ⋯ 6), M_i1, M_i2, and M_i3 correspond to tethers extracted with ℛ_bead = 25, 50, and 75 nm, respectively. The values of ${\mathcal{W}}_{\mathrm{t}}$, ℱ_t, and ℛ_t for all the systems are shown in panels (b), (c), and (d), respectively.

Figure 10

Validity of the scaling relations for κ and σ for data from simulations (M1–M6, shown as open symbols) and experiments (C1–C15, shown as filled symbols). Panel (a) shows the relation κ/α = 1/2π and panel (b) shows the scaling relation σ/Γ = 1/4π, and the corresponding correlation coefficients for systems M1-M6 are found to be r² = 0.846 and r² = 0.952, respectively. The dotted lines in panels (a) and (b) correspond to 1/2π and 1/4π respectively.

Figure 11

(a) A plot of α vs Γ for M1–M6, for different values of ℛ_bead, show data clustering in an excess area dependent fashion. (b) $\mathcal{G}(\alpha )$, the analytical estimates for the membrane excess area for M1–M6, computed using Equation (2). The dotted line denotes a scaling of the form G/α, with G ∼ 1107.

Figure 12

(a) Scaling plot of η vs α for systems M1–M6 for four different values of ℛ_bead. The dotted lines, show representative scaling relations of the form η = mα + 1, for small, intermediate, and large ${\mathcal{A}}_{\text{ex}}$ regimes. (b) A plot of the slope m as a function of ${\mathcal{A}}_{\text{ex}}$ and the dotted lines denote the best linear fit to the data. Fitting $m({\mathcal{A}}_{\text{ex}})=K{\mathcal{A}}_{\text{ex}}$we find the value of K = 0.00085.