Temperature and cholera toxin B are factors that influence formation of membrane nanotubes in RT4 and T24 urothelial cancer cell lines

Abstract

The growth of membrane nanotubes is crucial for intercellular communication in both normal development and pathological conditions. Therefore, identifying factors that influence their stability and formation are important for both basic research and in development of potential treatments of pathological states. Here we investigate the effect of cholera toxin B (CTB) and temperature on two pathological model systems: urothelial cell line RT4, as a model system of a benign tumor, and urothelial cell line T24, as a model system of a metastatic tumor. In particular, the number of intercellular membrane nanotubes (ICNs; ie, membrane nanotubes that bridge neighboring cells) was counted. In comparison with RT4 cells, we reveal a significantly higher number in the density of ICNs in T24 cells not derived from RT4 without treatments (P = 0.005), after 20 minutes at room temperature (P = 0.0007), and following CTB treatment (P = 0.000025). The binding of CTB to GM1–lipid complexes in membrane exvaginations or tips of membrane nanotubes may reduce the positive spontaneous (intrinsic) curvature of GM1–lipid complexes, which may lead to lipid mediated attractive interactions between CTB–GM1–lipid complexes, their aggregation and consequent formation of enlarged spherical tips of nanotubes. The binding of CTB to GM1 molecules in the outer membrane leaflet of membrane exvaginations and tips of membrane nanotubes may also increase the area difference between the two leaflets and in this way facilitate the growth of membrane nanotubes.

Keywords:

• cancer cells
• membrane nanotubes
• cholera toxin

Supplementary material

The model

To test the hypothesis that the initial formation of a membrane nanotube is driven by the aggregation of CTB–GM1 complexes, we constructed a model to study the effects of strong binding of CTB to GM1 molecules. The model is an extension of previous theoretical work,^20,22 and the present aim is to test whether the less positive spontaneous curvature of CTB–GM1 complexes as well as the strong binding of CTB are sufficient to drive an instability causing the growth and coalescence of membrane protrusions.

Our model is a coarse-grained model, whereby we do not describe the detail of the molecular-scale level. The minimal length-scale along the membrane that is relevant to this model is of the order of 100 nm. The model is written as a set of equations of motion for the continuum fields that describe the membrane shape and density of CTB–GM1 complexes, including the actual forces acting on the membrane, and the details of the membrane elasticity. For the sake of simplicity, the modeled membrane contour of RT4 or T24 cells is considered to be nearly flat. The bending energy part of the free energy of the model is based on the Helfrich energy form.²⁴ The free energy expression includes also the CTB–GM1 binding, configurational entropy, and direct-interaction energy:

$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_{s}n(\text{ln}(n)-1)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1)

where k is the membrane bending rigidity, n is CTB–GM1 density, H is the local mean membrane curvature, H̄ is the parameter describing the intrinsic curvature of the CTB–GM1s, n_s is the saturation density of CTB–GM1s on the membrane, σ is the membrane surface tension, α is a proportionality constant describing the effective adhesion interaction between CTB and GM1s, h = h(x) describes the shape of the contour, γ is a restoring force of cytoskeleton, J is the direct nearest-neighbor interaction energy between CTB and GM1s, and ds = d_m · dl is an element of membrane area, where d_m is the dimension of membrane perpendicular to the contour and dl is a line element along the contour. Note that k equals 4 times the k used in Helfrich.²⁴

The first term gives the bending energy due to the mismatch between the membrane curvature and the spontaneous curvature of a CTB–GM1 complex. The term −αn describes the negative adhesion energy between CTB and GM1s. The third term describes an external trapping of the membrane, which can be represented as the force of the cytoskeleton. The fourth term gives the entropic contribution due to the lateral thermal motion of the CTB–GM1 in the membrane in the limit of small n. The fifth term describes the nearest-neighbor attractive interactions between CTB and GM1s, and the sixth term prevents the sharp jumps in n.

Derivation of the curvature force on the membrane

We derive the equations of motions of the membrane contour using the derivation of the free energy [Eq. (1)$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_s\text{ln}(n)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1) ] with respect to the membrane coordinate and CTB–GM1 concentration.²¹ To take into account the drag due to viscous forces, we assume for simplicity only local friction forces,^20,21 with coefficient ξ.

The equation of motion of the membrane is

$\xi \frac{\partial \overrightarrow{r}}{\partial t}·\overrightarrow{n}=-\frac{\delta F(s,\hspace{0.17em}t)}{\delta n}$(2)

where ξ is the coefficient of the local friction force due to viscous drag of the fluids surrounding the membrane, r⃗ is the radial vector in the (x, y) coordinate system, t is time, n⃗ is the normal direction, and δF(s, t)/δn is the derivation of Eq. (1)$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_s\text{ln}(n)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1) with respect to the x and y directions. Here we consider only the changes along the y direction. The derivation of the free energy is projected to give the forces normal to the membrane contour.²¹ The following is the list of parameter values incorporated in our model: ξ = 125 s⁻¹gr, D = 0.002 μm²s⁻¹, Λ = D/k_BT, α = 0.013 gr s⁻², γ = 0.0004 grs⁻², n_s = 10 μm⁻², k = 100 k_BT, H̄ = 5 μm⁻¹, J = 0.00035 grs⁻², and σ = 0.001 grs⁻².

We next derive the forces at the membrane, by treating it as a “one-dimensional membrane”, ie, a thin strip of width w, with a bending modulus and tension coefficient. The free energy of this membrane is given in Eq. (1)$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_s\text{ln}(n)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1) , and is used to derive the local restoring forces by the usual derivation method. Since the overall contour length is not constant in our system, the derivation of the coordinates has to be taken with respect to their absolute index u along the contour, which is constant. In these terms the curvature H appearing in the Helfrich part of the free energy²⁴ is written as (standard differential geometry)

$H=\frac{\dot{x}\ddot{y}-\ddot{x}\dot{y}}{\sqrt{{\dot{x}}^2+{\dot{y}}^2}},$(3)

where the · symbol denotes differentiation with respect to the index of the point along the contour, and the free energy is

$F=w\stackrel{1}{\int }\left(\frac{1}{2}\kappa H^2+\sigma \right)\sqrt{{\dot{x}}^2+{\dot{y}}^2}\mathit{\text{du}},$(4)

where $\mathit{\text{ds}}/\mathit{\text{du}}=\dot{s}=\sqrt{{\dot{x}}^2+{\dot{y}}^2}$. The derivation of this free energy gives the forces, for example in the x-direction

$F_x=-\frac{\delta F}{\delta x}=\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}-\frac{d^2}{d{u}^2}\frac{\partial F}{\partial \ddot{x}}.$(5)

The resulting equations of motion from this derivation gives very long expressions, which are not amenable to easy analysis, although they can be used for the numerical simulations. In order to arrive at simpler expressions we will develop the terms in Eq. (5)$F_x=-\frac{\delta F}{\delta x}=\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}-\frac{d^2}{d{u}^2}\frac{\partial F}{\partial \ddot{x}}.$(5) , and simplify at the end by assuming that the arc-length separation between the nodes along the contour are all the same. This is maintained as the simulation progresses by using the spline routine to rediscretize evenly the contour as its length evolves.

The first term on the r.h.s. of Eq. (5)$F_x=-\frac{\delta F}{\delta x}=\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}-\frac{d^2}{d{u}^2}\frac{\partial F}{\partial \ddot{x}}.$(5) is

$\frac{\delta F}{\delta \dot{x}}=H^2\frac{\delta x}{\delta s}+2H\frac{\partial H}{\partial \dot{x}}\dot{s},$(6)

where $\frac{\delta \dot{x}}{\delta \dot{s}}=\dot{x}/\sqrt{{\dot{x}}^2+{\dot{y}}^2}=\frac{\partial x}{\partial s}$.

$\begin{array}{ll}\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}=\hfill & 2\dot{s}H{H}^{\prime }{x}^{\prime }+\dot{s}{H}^2{x}^{″}+2{\dot{s}}^2{H}^{\prime }\frac{\partial H}{\partial \dot{x}}\hfill \\ \hfill & +2H\frac{\partial H}{\partial \dot{x}}\ddot{s}+2H{\dot{s}}^2\frac{\partial H^{\prime }}{\partial \dot{x}},\hfill \end{array}$(7)

where ∂H/∂ẋ = ÿ⋅³ – 3x′ H/⋅. We therefore need to find an expression for ÿ (and ẍ), by using the definition of H = (x′ÿ – y′ẍ)/⋅² and: s̈ = x′ẍ + y′ ÿ. The final expressions that we get are: ÿ = s̈y′ + ⋅² x′ H, and ẍ = s̈x′ – ⋅² y′H. We will now assume that ⋅ is independent of u, so that s̈ = s⃛ = 0. The last term in Eq. (7)$\begin{array}{ll}\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}=\hfill & 2\dot{s}H{H}^{\prime }{x}^{\prime }+\dot{s}{H}^2{x}^{″}+2{\dot{s}}^2{H}^{\prime }\frac{\partial H}{\partial \dot{x}}\hfill \\ \hfill & +2H\frac{\partial H}{\partial \dot{x}}\ddot{s}+2H{\dot{s}}^2\frac{\partial H^{\prime }}{\partial \dot{x}},\hfill \end{array}$(7) becomes

$\frac{\partial H^{\prime }}{\partial \dot{x}}=-\frac{2}{\dot{s}}\frac{\partial }{\partial s}(x^{\prime }H).$(8)

The second term on the right hand side of Eq. (5)$F_x=-\frac{\delta F}{\delta x}=\frac{d}{\mathit{\text{du}}}\frac{\delta F}{\delta \dot{x}}-\frac{d^2}{d{u}^2}\frac{\partial F}{\partial \ddot{x}}.$(5) is

$\frac{\partial F}{\partial \ddot{x}}=\frac{-2H{y}^{\prime }}{\dot{s}}$(9)

and we get

$\frac{d^2}{{\mathit{\text{du}}}^2}\frac{\partial F}{\partial \ddot{x}}=-2\dot{s}\frac{{\partial }^2}{\partial s^2}(H{y}^{\prime }).$(10)

We now write H in terms of s as: H = H⃗ · n̂ = −x″ y′ + y″ x′, where: H⃗ = x″ x̂ + y″ ŷ, and n̂ = −y′ x̂ + x′ ŷ so that

$\begin{array}{c}{H}^{\prime }=-y^{\prime }{x}^{‴}+x^{\prime }{y}^{‴}\\ H^{″}=-y^{″}{x}^{‴}+x^{″}{y}^{‴}-y^{\prime }{x}^{(4)}+x^{\prime }{y}^{(4)}\end{array}$(11)

There is another force contribution from the membrane tension, giving a term of the form: F_x ∝ ⋅x″, so in the normal direction we get: F_n = ⋅σ(−y′ x″ + x′ y″) = ⋅σH, where we used the identity: x′² + y′² = 1 and therefore: 2(2(x′x″ + y′y″) = ∂(x′² + y′²)/∂s = 0.

Putting everything together, the normal force acting on the membrane due to curvature and tension is

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}F=\overrightarrow{F}·\widehat{n}=-y^{\prime }{F}_x+x^{\prime }{F}_y\\ =\dot{s}\frac{1}{2}\kappa (-2{\nabla }^{2}H-2H(y^{\prime }{y}^{‴}+x^{\prime }{x}^{‴})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-3{(\overrightarrow{H}·n)}^3)+\dot{s}\sigma H\end{array}$(12)

$\begin{array}{c}=\dot{s}\kappa \left(-{\nabla }^{2}H-\frac{1}{2}{(\overrightarrow{H}·\widehat{n})}^3\right)+\dot{s}\sigma H\\ \dot{s}\frac{1}{2}\kappa (2(y^{\prime }{x}^{(4)}-x^{\prime }{y}^{(4)})-3{(\overrightarrow{H}·n)}^3)+\dot{s}\sigma H\end{array}$(13)

where we used the identity: x′² + y′² = 1, and therefore: 2(x′x″ + y′y″) = ∂(x′² + y′²)/∂s = 0, and −(x″² + y″²) = −H²

We need forces per unit length, whereas we calculated above the forces per unit u, so we divide by ⋅ and finally get

$\kappa \left((y^{\prime }{x}^{(4)}-x^{\prime }{y}^{(4)})-\frac{3}{2}{H}^3\right)+\sigma H.$(14)

CTB–GM1s with spontaneous curvature

When there are CTB–GM1s with spontaneous curvature, the free energy.

(Eq. (4)$F=w\stackrel{1}{\int }\left(\frac{1}{2}\kappa H^2+\sigma \right)\sqrt{{\dot{x}}^2+{\dot{y}}^2}\mathit{\text{du}},$(4) ) changes to

$F=w\stackrel{1}{\int }\left(\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)\right)\sqrt{{\dot{x}}^2+{\dot{y}}^2}\mathit{\text{du}}$(15)

where n the density of CTB–GM1s along the contour, which may not be uniform. Expanding the quadratic term we get: H² − 2HH̄n + (H̄n)². The derivation of the first term was done above (all the variations are of the integrand times the ⋅ factor).

The new contributions to the forces acting on the membrane are (normal force per unit length)

$\begin{array}{ll}{F}_{\mathit{\text{spon}},n}\hfill & ={\overrightarrow{F}}_{\mathit{\text{spon}}}·\widehat{n}=-y^{\prime }{F}_{\mathit{\text{spon}},x}+x^{\prime }{F}_{\mathit{\text{spon}},y}\hfill \\ \hfill & =\frac{\kappa }{2}\left({(\overline{H}n)}^{2}H-2(\overline{H}{n}^{″})-n\overline{H}{H}^2\right)\hfill \end{array}$(16)

and

$F_{\mathit{\text{tension}},n}={\overrightarrow{F}}_{\mathit{\text{spon}}}·\widehat{n}=-\alpha nH.$(17)

Fluxes and diffusion of CTB–GM1s

The conservation equation for the CTB–GM1s along the contour becomes

$\frac{1}{\dot{s}}\frac{\partial \dot{s}n}{\partial t}=\frac{D}{\dot{s}}{\nabla }_s^2(\dot{s}n)+\frac{1}{\dot{s}}\frac{\Lambda }{n_s}{\nabla }_s(\dot{s}n){\nabla }_s\left(\frac{1}{\dot{s}}\frac{\delta F}{\delta n}\right),$(18)

where the number of CTB–GM1s in each unit contour length is N = ⋅nn_s (n_s is the saturation concentration of the CTB–GM1s), F is the energy functional of Eq. (1)$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_s\text{ln}(n)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1) , and the derivative along the contour is ∇_s = ∇_u/⋅. We therefore get

$\frac{\partial n}{\partial t}+\frac{n}{\dot{s}}\frac{\partial \dot{s}}{\partial t}=\frac{D}{\dot{s}}{\nabla }_s^2(\dot{s}n)+\frac{D}{k_B{\mathit{\text{Tn}}}_s{\dot{s}}^2}{\nabla }_u\left({n\nabla }_u\frac{\delta F}{\delta n}\right)$(19)

where e is the energy per unit length, ie, the integrand in Eq. (15)$F=w\stackrel{1}{\int }\left(\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)\right)\sqrt{{\dot{x}}^2+{\dot{y}}^2}\mathit{\text{du}}$(15) with respect to ds.

If the number of CTB–GM1s is conserved, even though we allow the membrane overall length to change, then Eq. (19)$\frac{\partial n}{\partial t}+\frac{n}{\dot{s}}\frac{\partial \dot{s}}{\partial t}=\frac{D}{\dot{s}}{\nabla }_s^2(\dot{s}n)+\frac{D}{k_B{\mathit{\text{Tn}}}_s{\dot{s}}^2}{\nabla }_u\left({n\nabla }_u\frac{\delta F}{\delta n}\right)$(19) is correct. If however there is a reservoir of membrane that allows it to change in length, then this membrane can include lipids and CTB–GM1s, so that the total number of CTB–GM1s is not conserved when the membrane length changes. In this case the change in the density due to length changes is removed, as it is assumed to be balanced by the currents into/out of the reservoir. Eq. (19)$\frac{\partial n}{\partial t}+\frac{n}{\dot{s}}\frac{\partial \dot{s}}{\partial t}=\frac{D}{\dot{s}}{\nabla }_s^2(\dot{s}n)+\frac{D}{k_B{\mathit{\text{Tn}}}_s{\dot{s}}^2}{\nabla }_u\left({n\nabla }_u\frac{\delta F}{\delta n}\right)$(19) is then modified by removing the second term on the left hand side.

In our calculations, a nonlinear tension was employed, and as a result, the length of each membrane segment changed very little, so the second term on the left hand side of Eq. (19)$\frac{\partial n}{\partial t}+\frac{n}{\dot{s}}\frac{\partial \dot{s}}{\partial t}=\frac{D}{\dot{s}}{\nabla }_s^2(\dot{s}n)+\frac{D}{k_B{\mathit{\text{Tn}}}_s{\dot{s}}^2}{\nabla }_u\left({n\nabla }_u\frac{\delta F}{\delta n}\right)$(19) was neglected.

Numerical realization of the model

Discretization of the model

Since the flat shape model represents a segment of the whole cell, we used periodic boundary conditions. Thus, the number of grid points N equals the number of discretizations. In our model, the density n of element i is given by:

$n_i=\frac{N_i}{{\Delta s}_i}.$(20)

The boundary conditions

We employed periodic boundary conditions. The first and second derivatives of the function along the x direction were calculated using the following explicit Euler method:

$\begin{array}{l}\frac{\partial x}{\partial s}=\frac{x_{n+1}-x_{n-1}}{2{\Delta s}_i}\\ \frac{{\partial }^{2}x}{\partial s^2}=\frac{x_{n-1}-2x_n+x_{n+1}}{{\Delta s}_i^2}\end{array}$(21)

where the subscripts n, n+1, n−1 represent the current, next, and previous nodes, respectively. The derivatives of the function along the y direction were calculated in a similar manner. For the calculations of derivatives of the first point, the last point was added before it. While for the calculation of derivatives of the last point, the first point was added after it.

The derivation of the free energy equation

The derivation of the free energy is projected to give the forces normal to the membrane contour.²¹ We now list the forces derived from the derivation of the free energy [Eq. (1)$\begin{array}{ll}F=\hfill & \int (\frac{1}{2}\kappa {(H-\overline{H}n)}^2+(\sigma -\alpha n)+\gamma h^2\hfill \\ \hfill & +k_B{\mathit{\text{Tn}}}_s\text{ln}(n)-{\mathit{\text{Jn}}}^2+\frac{J}{n_s}{(\nabla n)}^2)\mathit{\text{ds}},\hfill \end{array}$(1) ]²¹

$F_c=\kappa \left(-{\nabla }^{2}H+\overline{H}{\nabla }^{2}n+\frac{1}{2}{n}^2{\overline{H}}^{2}H-\frac{1}{2}{H}^3\right)$(22)

$F_t=(\sigma -\alpha n)H$(23)

$F_s=-2\gamma y$(24)

$F_e=k_B{\mathit{\text{Tn}}}_s(n\text{ln}(n)-1)H$(25)

$F_J=J\left(-n^2+\frac{1}{n_s}{(\nabla n)}^2\right)H$(26)

where F_c is the force due to the curvature energy mismatch between the membrane curvature and the spontaneous curvature of the CTB–GM1s, F_t is the membrane tension force, F_s is the spring restoring force, and F_J is the force due to the nearest-neighbor interaction of the CTB–GM1s. F_e arises from the entropy of the CTB–GM1s in the membrane, which acts to expand the length of the contour.

We now calculate the dynamics of the CTB–GM1 density, using the following conservation equation

$\frac{\partial n}{\partial t}=-\nabla ·\overrightarrow{J}=\frac{\Lambda }{n_s}{\nabla }_s\left({n\nabla }_s\frac{\delta F}{\delta n}\right)-\frac{n}{\sqrt{g}}\frac{\partial \sqrt{g}}{\partial t},$(27)

where Λ is the mobility of CTB–GM1s and J⃗ is the total current of CTB–GM1 on the membrane, which includes the following terms

$J_{\mathit{\text{att}}}=\frac{\kappa \Lambda \overline{H}}{n_s}n\nabla H$(28)

$J_{\mathit{\text{disp}}}=-\frac{\kappa \Lambda {\overline{H}}^2}{n_s}n\nabla n$(29)

$J_{\mathit{\text{agg}}}=\frac{J\Lambda }{n_s}n\nabla n+\frac{J\Lambda }{n_s^2}n{\nabla }^{3}n$(30)

$J_{\mathit{\text{diff}}}{=-D\nabla }_n$(31)

where J_att is the attraction flux resulting from the interaction between the CTB–GM1s through the membrane curvature, J_disp is the dispersion flux due to the membrane resistance to CTB–GM1 aggregation due to their membrane bending effects, J_agg is the flux due to the direct CTB–GM1 aggregation interactions, and J_diff is the usual thermal diffusion flux, which depends on the diffusion coefficient, D = Λk_BT. The last term in Eq. (27)$\frac{\partial n}{\partial t}=-\nabla ·\overrightarrow{J}=\frac{\Lambda }{n_s}{\nabla }_s\left({n\nabla }_s\frac{\delta F}{\delta n}\right)-\frac{n}{\sqrt{g}}\frac{\partial \sqrt{g}}{\partial t},$(27) arises from the covariant derivative of the density with time on a contour whose length evolves with time.²³ In this term $\sqrt{g}$ is the matrix tensor, which in our one-dimensional contour is simply the line element dl. This term ensures that the total number of CTB–GM1s is conserved as the contour length changes.

Acknowledgements

This work was supported by ARRS grants J3-9219-0381 and P2-0232-1538.

The authors are grateful to Henry Hägerstrand and Nir Gov for useful discussions.

Disclosure

The authors disclose no conflicts of interest.