Self-consistent calculation of the membrane gap

In the following section we outline a self consistent mean field analysis of the radial membrane fluctuations. The purpose of this is to calculate the average size of the gap that exists between the spherical particle and the fluctuating membrane; in some places the gap will be smaller and others it will naturally be larger. Our approach will be to use the average gap size as an approximation for a (constant) gap size, everywhere around the sphere. In our model the average radial extent of the membrane is governed by membrane fluctuations that drive it slightly away from the surface of the enclosed spherical particle. A similar approach has proved highly successful for planar membranes [24, 25], as well as membrane tubes [26].

The membrane fluctuates about a spherical shape around the particle (bulge) except near the two “necks” where it joins smoothly onto the walls of the membrane tube, see Fig 1. In order to describe the energetics of the membrane we write the Hamiltonian H = H_E + H_S, with (1) and (2)

Here H_E is the usual Hamiltonian for membrane elasticity [25, 27], containing both surface tension (σ) and rigidity (κ) controlled terms. The mean curvature is given by c, and g is the determinant of the metric tensor det(g_ab). The steric part, H_S, contains a harmonic potential with strength that confines the size of the membrane fluctuations in a narrow region around the average membrane sphere radius . It also contains a fluctuation pressure with strength J, which controls the average radius of the membrane sphere. This term can also be used to include any hydrostatic or osmotic pressure differences between the inside and outside of the membrane, although we set these to zero for clarity in what follows. Additionally, H_S contains a term involving A which is convenient for normalisation of the steric potential. This, most general harmonic potential, will be used to model the steric interactions between our membrane and enclosed spherical particle by way of a mean-field approach. As we are dealing in this work with an almost completely enveloped spherical particle, we can approximate the energy by computing the membrane Hamiltonian over the entire sphere. An analogous treatment has proven to be remarkably successful in describing the steric repulsion between flat membranes [24, 25], as well as between membrane tubes and an enclosed rod [26]. We proceed from Eq (1) by writing and expanding the energy H to quadratic order [28] in the radial perturbation δR(ϕ, θ) about the average membrane radius . Using spherical harmonics, we write , which yields the total membrane energy as a perturbative expansion H = H₀ + δH + δ²H + … with (3) involving a kernel (4)

The first order perturbative contribution is required to vanish so that indeed represents the true average (or ground state) membrane radius [28]. This condition then implies (from Eq (3)) that the fluctuation pressure is , as required to satisfy Laplace’s law. The quadratic fluctuations in the radial displacement (around ), contribute at order δ²H, and depend on the strength of the harmonic potential in Eq (1), via the parameter which is established as follows. The presence of the enclosed solid spherical particle sterically constrains the membrane radius, R(ϕ, θ), to remain always greater than the particle radius R₀, see Fig 1. The mean squared amplitude of the fluctuations, 〈δR²〉, depends on the parameter , as can be seen from Eq (3), which we must determine self consistently. In employing a harmonic potential, controlled by the parameter , we adopt an approximate phenomenological treatment of the steric interactions.

By integrating out the membrane fluctuations, the free energy of the tube F = H₀ + ΔF can be shown to involve the correction term: (5)

We can now physically motivate an explicit choice for the parameter A as follows. We aim to calculate the free energy difference between the case when the enclosed spherical particle is present and when it is absent (and the membrane is unconstrained). In the latter case the steric harmonic potential (of strength ) vanishes, as do terms involving that appear in K_nm. Thus we choose the parameter A so that in the limit we obtain ΔF → 0 for consistency. We must then choose (6)

After we have integrated out all radial membrane fluctuations we therefore obtain (7)

We can now state quantitatively the physical condition that we wish to impose on our membrane to mimic the steric influence of the enclosed solid sphere (with radius R₀): (8)

This gives the necessary self-consistency condition for the strength of the harmonic potential given that (9)

In order to calculate the average radius for the membrane tube we merely need to minimise F by setting .

For a large sphere inside a narrow tube, the average radius of the membrane is almost equal to the radius of the enclosed spherical particle (). The steric effects of the sphere in this limit should therefore be very strong, and the strength of the self-consistent, confining, harmonic potential becomes very large (). Using , we can approximate the resultant sum over l in Eq (9) as an integral by defining a new variable ρ = l², such that: (10)

Hence in this narrow gap limit, , to leading order. Substituting this value of into Eq (7), and approximating the sum required by an integral as in Eq (10), we obtain the following result to leading order (11)

A contribution to the energy that scales as the inverse squared distance, similar to the one appearing here, is well known for planar membranes at small inter-membrane separation [24, 29–31], as well as membrane tubes in close proximity to an enclosed rod [26].

Minimising Eq (11) w.r.t. , we find to leading order: (12)

In closing, note that the distal portions of membrane tube, unaffected by the presence of the enclosed large spherical particle, can easily be shown [26] to possess a cylindrical radius of .

Fluid hydrodynamics

Here we analyse the fluid hydrodynamics around the moving particle within the tube. This will allow us to calculate the diffusion constant of the spherical particle. The low Reynolds number hydrodynamics of the fluid within the membrane tube is governed by Stokes’ equation, along with the constraint of incompressibility [32–35]: (13) with fluid velocity , hydrostatic pressure p, and viscosity μ. Utilising symmetry considerations, we work in spherical polar coordinates, and assume that u_ϕ = 0, and ∂u/∂ϕ = 0. In the ‘narrow-gap’ approximation of interest here, we can also assume that u_R = 0, such that the fluid flow in the gap can be described by the component u_θ alone. Utilising this approximation, and taking into account the geometrical setup of our problem, Eq (13) becomes: (14)

The incompressibility condition in Eq (14) can be satisfied straightforwardly by defining , such that the z component of the flow (along the tube axis) becomes simply: u_z = −u_θ sin θ = −Φ(R). Using this expression for u_θ, we find that the equation governing the fluid flow and pressure can now be re-arranged into the following form: (15)

The most general solution to Eq (15) can easily be shown to be: , and , where a, b, c, and d are constants. For convenience, we choose to work in the lab frame with respect to which the enclosed solid sphere is moving. Furthermore, we need to take into account the fluid flow in the spherical gap between the enclosed sphere and the membrane, and the membrane fluid flow in the spherically deformed membrane, the z component of which is written as u_m, such that the boundary conditions satisfied by the fluid are as follows: (16)

The membrane tube is held stationary at its end by, e.g. a conjugated bead held in an optical trap. This means that the membrane tube is not growing in length, and that the sum of the length of tube in front of the moving sphere and the length behind it must always add up to a constant, the total tube length. The sphere is assumed to be moving such that the tube length in front of the moving sphere decreases while the tube length behind it increases. The tube and its contents remain stationary everywhere except very close to the sphere, while the enclosed fluid and membrane must flow around the sphere. If the length of tube in front of the sphere is L₀ at time t = 0 then it is L(t) = L₀ − u₀t at some later time t. Thus the volume in the leading tube is and depends on time, which demands a flow around the sphere to balance volume (with a similar argument for the membrane flow).

The surface tension gradient, expressed as a difference between the tension in the leading and trailing tubes, is O(u₀), i.e. small. Because the membrane flow is effectively one dimensional, the membrane velocity is entirely determined by (i) membrane incompressibility (and geometry) and (ii) the radii of the leading and trailing tubes, from which the membrane is extracted/deposited so as to leave the membrane in both tubes stationary everywhere (except around the sphere). This means that the membrane flow can only depend on the surface tension difference via the leading and trailing tube radii, and here these are both of order r₀(1 + O(u₀)), i.e. equal to r₀ to leading order. Thus we can safely ignore membrane tension gradients set up by the flow itself, and therefore any concomitant differences in the leading or trailing tube radii.

In the lab frame, the fluid is stationary everywhere in both leading and trailing tubes. However, due to the change in membrane tube length in front of the moving sphere this means that there must be a volumetric back-flow of fluid in the gap around the sphere (), in order to avoid fluid accumulating in the leading tube. Thus, volumetric flow balance leads to: (17)

The important point to note in Eq (17) is that from the definition of the volumetric flow rate being: , the integrand in Eq (17) is independent of the angle θ.

Additionally, in the lab frame, the membrane fluid is stationary everywhere along both leading and trailing tubes. However, this means that there must also be a volumetric back-flow of membrane around the sphere (= u₀2πr₀h), with h the membrane thickness, to avoid membrane accumulating on the leading tube. Volumetric flow balance in this case therefore leads to: (18) which we evaluate in the limit where the membrane thickness h becomes vanishingly small.

By utilising all of the above boundary conditions, we can find all the integration constants required in order to compute the drag force on the membrane enclosed sphere: (19) where Δθ ≈ r₀/R₀ is the small angle subtended by the neck of the tube, see Fig 1. In terms of these constants, the fluid velocity and pressure become: (20)

Calculation of drag

The drag force f₀ is determined by the integration of the fluid stress components (T_RR, T_Rθ) over the surface of the solid sphere (R = R₀) as: (21) where T_RR = −p(θ), as given in Eq (20), and is given as: (22)

Evaluating the integrals required in Eq (21), we find to leading order, assuming : (23) where sin Δθ = r₀/R₀, and . Using the relationship f₀ = −ξu₀, and inserting in Eq (23): , from Eq (12), we can write for the drag ratio K = ξ/ξ₀: (24) where the free, geometrically unhindered, 3d bulk friction constant is given as usual by: ξ₀ = 6πμR₀ [34]. To leading order in Δθ ≪ 1, corresponding to particles larger than the tube radius, this simplifies dramatically to: (25)

Note that in Eq (25) the drag ratio depends on the membrane tension as ∼ σ^2/3. This result for the drag provides an experimentally testable prediction. Furthermore, the diffusion constant D is given by D = k_BT/ξ = D₀/K, where D₀ = k_BT/6πμR₀ [34], which leads to a diffusion constant correspondingly (much) smaller than that for free 3d bulk diffusion in the same fluid. Additionally, the algebraic expression Eq (25), explicitly demonstrates the dependence of mobility on viscosity, such that the effect of higher viscosities than pure water, because of the inclusion of cytoskeleton and cytoplasmic soluble macromolecules (e.g. crowding agents), can easily be taken into account.

For typical fluid membranes we have κ ∼ 20k_BT, and we can investigate how our main result (given by Eq (25)) behaves in the most physiologically relevant particular regimes as follows. Given that most, generic, biophysical cargo (as depicted in Fig 1) possess a range of sizes from R₀ ∼ 0.1 − 10μm [11], while most biomembranes have surface tensions between σ ∼ 10⁻⁵ − 10⁻⁴Jm⁻², we find that our drag ratio lies in the range K ∼ 2 × 10³ − 4 × 10⁶. The above, experimentally accessible, range of values therefore encompass a host of individual case studies, or particular situations, such as (in vivo [8–16]) the transport of mitochondria (∼ 0.5 − 1μm), small organelles (∼ 0.1μm), pathogens (∼ 0.1μm), viruses (∼ 0.1μm), silicon microparticles (∼ 1μm), or other large cargo (∼ 1μm) in tunneling nanotubes (TNTs), as well as (in vitro) the transport of large colloidal particles or microbeads (∼ 10μm) in membrane tether-pulling experiments [1–7], for example. In addition, via Eq (12), we can calculate using the typical parameters quoted above the expected average size of the gap that exists between the spherical particle and the fluctuating membrane to be ∼1 − 10nm. This calculated gap size compares favourably with the analogous value of ∼ 3nm obtained from experimental work on supported lipid bilayers [36].