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# Actin dynamics drive cell-like membrane deformation

## Abstract

Cell membrane deformations are crucial for proper cell function. Specialized protein assemblies initiate inward or outward membrane deformations that the cell uses respectively to uptake external substances or probe the environment. The assembly and dynamics of the actin cytoskeleton are involved in this process, although their detailed role remains controversial. We show here that a dynamic, branched actin network is sufficient to initiate both inward and outward membrane deformation. The polymerization of a dense actin network at the membrane of liposomes produces inward membrane bending at low tension, while outward deformations are robustly generated regardless of tension. Our results shed light on the mechanism cells use to internalize material, both in mammalian cells, where actin polymerization forces are required when membrane tension is increased, and in yeast, where those forces are necessary to overcome the opposing turgor pressure. By combining experimental observations with physical modelling, we propose a mechanism that explains how membrane tension and the architecture of the actin network regulate cell-like membrane deformations.

## Main

Many cell functions rely on the ability of cells to change their shape. The deformation of the cell membrane is produced by the activity of various proteins that curve the membrane inwards or outwards, by exerting pulling and pushing forces or by imposing membrane curvature via structural effects. When cells take up external material, it is often associated with membrane invaginations followed by vesicle transport. This process is called endocytosis. Such inward deformation of the cell membrane can be initiated by specific proteins, such as clathrin, which coat the membrane and impose geometrical constraints that bend the membrane inwards. In this view, the action of the actin cytoskeleton, a filamentous network that forms at the membrane, is crucial only at a later stage for membrane elongation. Nevertheless, correlation methods revealed unambiguously that, in yeast, membrane bending is not triggered by the presence of coat proteins, but by a dynamic actin network formed at the membrane through the Arp2/3 complex branching agent1,2,3. In mammalian cells, clathrin-mediated endocytosis requires the involvement of actin if the plasma membrane is tense (for example, following osmotic swelling or mechanical stretching4). However, the exact mechanism of membrane deformation in this process is still poorly understood. Strikingly, the same type of branched actin network is able to bend the membrane the other way in, creating outward-pointing membrane deformations, called dendritic filopodia. These structures are precursors of dendritic spines in neurons, and essential for signal transmission5. Dendritic filopodia differ from conventional filopodia, localized at the leading edge of the cell, where actin filaments are parallel. Whereas the pioneering work of Liu et al.6 already established how thin filopodia form by bundling actin filaments, the production of a dendritic filopodia-like membrane protrusion containing a branched actin network has never been investigated.

How the same branched actin structure can be responsible for the initiation of filopodia, which are outward-pointing membrane deformations, as well as endocytic invaginations that deform the membrane inward is what we want to address in this paper. Such a question is difficult to investigate in cells that contain redundant mechanisms for cell deformation. Actin dynamics triggered at a liposome membrane provide a control on experimental parameters such as membrane composition, curvature and tension, and allow the specific role of actin dynamics to be addressed. We unambiguously show that the same branched actin network is able to produce both endocytosis-like and dendritic filopodia-like deformations. With a theoretical model, we predict under which conditions the stress exerted on the membrane will lead to inward- and/or outward-pointing membrane deformations. Combining experiments and theory allows us to decipher how the interplay between membrane tension, actin dynamics and actin network structure produces inward or outward membrane deformations.

## Membrane tubes and spikes

Liposomes are covered with an activator of the Arp2/3 complex, pVCA, the proline-rich domain–verprolin homology–central–acidic sequence from human WASP, which is purified with a streptavidin tag, and that we call hereafter S-pVCA. A branched actin network grows at their surface when placed in a mixture containing monomeric actin, profilin the Arp2/3 complex and capping protein (CP) (‘reference condition’, Methods and Fig. 1a). Strikingly, the membrane of liposomes is not smooth, but instead displays a rugged profile: membrane tubes, hereafter called ‘tubes’, radiate from the liposome surface and extend into the actin network (Fig. 1b), even when comet formation has occurred7,8 (Supplementary Fig. 1a). The initiation of these tubes is reminiscent of the early stage of endocytosis. Interestingly, some liposomes display another type of membrane deformation, characterized by a conical shape (hereafter referred to as ‘spikes’) that points towards the liposome interior (Fig. 1b); these spikes are reminiscent of dendritic filopodia structures in cells. Some of the liposomes carry both tubes and spikes, while others are ‘undetermined’, as no membrane deformation is visually detectable (Fig. 1b). Spikes have a wide base of a few micrometres and a length that spans at least half of the liposome diameter. In contrast, tubes are thin, with a diameter under the resolution limit of optical microscopy (less than a few hundred nanometres). When membrane tension is unaffected, 63.0% of liposomes display only tubes, 2.3% only spikes, while 6.1% of liposomes carry a mix of both, and 28.6% are undetermined (Fig. 1c, non-deflated liposomes). To examine how membrane tension affects the occurrence of tubes and spikes, liposomes are deflated by a hyper-osmotic shock (Methods) before actin polymerization is triggered. This treatment leads to a huge increase in the number of liposomes displaying spikes: 65.0% of deflated liposomes display spikes (with or without tubes), compared to 8.4% in non-deflated conditions (Fig. 1c, P < 0.0001). Yet, the frequency with which tubes (with or without spikes) are observed is essentially unaffected: 69.1% for non-deflated liposomes compared to 74.8% for deflated liposomes (not significant, P = 0.24 > 0.05, Supplementary Fig. 1b). An increase in membrane tension by a hypo-osmotic treatment (Methods) does not change the occurrence of tubes and spikes significantly (Supplementary Fig. 1c).

Membrane tubes and spikes exclusively rely on the presence of the actin network, as they disappear when the network is destructed7 (Fig. 1d,e and Methods). A possible effect of membrane pre-curvature induced by pVCA attachment to the membrane is ruled out (Supplementary Information and Supplementary Fig. 2).

## Characterization of tubes

To assess where new actin monomers are incorporated during tube growth, we incorporate differently labelled monomers (green) after 20 min (Methods). As previously observed for actin networks growing around polystyrene beads9,10, new monomers insert at the liposome surface (Fig. 2a). Strikingly, new (green) monomers are also observed within the already grown (red) actin network (Fig. 2a), indicating new actin incorporation on the sides of membrane tubes (tubes are evidenced by phase-contrast imaging, Fig. 2a, top). This observation is confirmed by the localization, along tubes and at the liposome surface, of S-pVCA (Fig. 2b), the Arp2/3 complex (Fig. 2c) and free barbed ends (Supplementary Fig. 3). Moreover, the presence of the Arp2/3 complex everywhere in the whole volume of the actin network demonstrates its dentritic nature (Fig. 2c).

We find that the average length of the longest tubes increases linearly with network thickness (Fig. 3a,b). In fact, maximal tube length roughly equals the thickness of the actin network, independently of membrane tension (Fig. 3b, slope 0.89 ± 0.04), albeit deflated liposomes produce a smaller actin cortex. Moreover, we find that tubes grow simultaneously with the actin network (Fig. 3c,d and Supplementary Fig. 4). Tubes shorter than the network thickness are also present, as evidenced by confocal microscopy (Supplementary Fig. 5a).

The origin of the accumulation of membrane fluorescence detected at the tip of some of the longer tubes is unclear. We observe that S-pVCA forms aggregates on membranes and sticks membranes together, even in the absence of actin (Supplementary Fig. 6). It is possible that small vesicles are attached via S-pVCA to the membrane before polymerization starts and are pushed outward by actin growth. However, the presence of different tube lengths (Supplementary Fig. 5) rules out the possibility that tubes could be formed only by pre-existing attached vesicles.

## Characterization of spikes

We find that new actin is incorporated at the tips of the spikes as well as at the sides (Fig. 4a), consistent with the localization of S-pVCA (Fig. 4b). Spikes are filled with the Arp2/3 complex and CP (Fig. 4c and Supplementary Fig. 7), characteristic of a branched network. A clump of actin is observable at the base of the spikes (Fig. 4d). The thickness of the clump bears no clear correlation with the length of the spikes (Supplementary Fig. 8a), but slightly correlates with their width (Supplementary Fig. 8b). Spikes initially elongate with time until polymerization slows down; the basal width of spikes, however, remains roughly constant over time (Fig. 4e and Supplementary Fig. 8c).

## Effect of network mesh size and membrane tension

Lowering the Arp2/3 complex or CP concentrations could, in principle, result in loosening the network, but fails to form a cohesive thick enough (>500 nm) network11. Using the property of profilin to inhibit branching and therefore loosen the actin network12, we obtain a visible, thick, network comparable to reference conditions (Supplementary Fig. 9a and Methods). We find that the occurrence of tubes is reduced in these conditions (74.8% of liposomes display tubes when profilin is in excess compared to 91.4% in reference conditions, Supplementary Fig. 9b, P < 0.0001). Strikingly, decreasing membrane tension in loosened network conditions significantly increases the presence of tubes and spikes (Supplementary Fig. 9b, P < 0.0001).

## Theoretical models for spikes and tubes

The appearance of large-scale membrane deformations (spikes) driven by a uniformly polymerizing actin network is rationalized using analytical modelling and numerical finite-element calculations (Methods). The actin network behaves as a viscoelastic material with an elastic behaviour at short time and a viscous behaviour at long time due to network rearrangement, the crossover time being on the order of 1–10 s (refs. 13,14,15). We focus on the viscous behaviour as the growth of the network occurs on timescales of tens of minutes.

We model the growth of the actin network with a uniform actin polymerization velocity v g normal to the liposome membrane (motivated by Fig. 4a) and solve the hydrodynamic force balance equation at low Reynolds number (the ‘Stokes equation’) (Methods). Actin polymerization on a flat membrane results in a uniform actin flow that does not generate any mechanical stress. Small perturbations of membrane shape modulate the actin velocity field and generate viscous stress on the membrane. For a periodic deformation (Fig. 5a, left), the actin stress varies as the square of the deformation amplitude (Methods) in agreement with actin growth on a curved surface13,16. For a localized (Gaussian) membrane perturbation $$u\left( x \right) = A{\rm e}^{ - (x/b)^2}$$ with amplitude A and width b (Fig. 5a, right), we calculate the pressure and velocity fields in the actin layer numerically (Fig. 5b). Velocity gradients in the growing actin layer, generated by the deformed surface, induce a normal pushing force at the centre of the perturbation, and pulling forces at the periphery of the perturbation (Fig. 5c), that amount to a zero net force when integrated over the deformation area. This contrasts with existing models of filopodia formation, which usually consider bundled actin filaments exerting a net pushing force on the membrane that do not precisely address the force balance within the actin network6,17,18. Here, we do not a priori distinguish the detailed structure of the actin network at the membrane from the one in the protrusion, treating the actin network as a continuum.

A scaling analysis of the Stokes equation, confirmed by our numerical calculation, leads to a normal stress at the centre of the perturbation (x = 0) that scales as σ nn ≈ −ηA 2 b −3 v g, where η is the viscosity of the actin layer (Supplementary Fig. 10a,b). An intuitive understanding of this scaling behaviour is given in the Supplementary Information.

The normal stress σ nn is balanced by the membrane elastic restoring stress19 $$\sigma _{\rm {memb}} = -\gamma C + \kappa \,\partial _{\rm s}^2C$$ , where γ is the membrane tension, κ is the bending rigidity, C is the membrane curvature (~A/b 2) and ∂s the curvilinear derivative (~1/b). Considering that b is larger than the characteristic length $$\gamma = \sqrt {\kappa /\gamma }$$ , the stress is dominated by membrane tension. The balance of actin polymerization and membrane stresses defines a threshold amplitude A* = γb/(ηv g). When the amplitude of the perturbation is smaller than this threshold (A < A*) the membrane stress dominates and the perturbation relaxes. Above the threshold (A > A*) the force exerted by the network is dominant and the instability develops. We now evaluate whether such a perturbation could be reached by thermal fluctuations characterized by the Boltzmann constant k B and the temperature T. The average membrane thermal roughness at length scales larger than the actin mesh size ξ, characterized by the average of the gradient of the membrane shape h, is given by $$< |\nabla h|^2 > \sim \frac{{k_{\rm B}T}}{{4{\rm \pi} \kappa }}\log \left( {\left( {\frac{{2{\rm \pi} \lambda }}{\xi }} \right)^2 + 1} \right)\) (ref. 19). Identifying$$< |\nabla h|^2 >\) with (A/b)2 (provided λ and ξ are on the same order), spikes are predicted below a threshold tension: $$\gamma \ast \approx \eta v_{\rm g}\sqrt {k_{\rm B}T/(4{\rm \pi} \kappa )}$$ . Evaluating actin network viscosity η as the product of the elastic modulus (E) times the viscoelastic relaxation time (τ ve): η ≈  ve ≈ 104 Pa s (with E ≈ 104 Pa (ref. 20) and τ ve ≈ 1 s (refs. 14,15)), κ ≈ 10 k B T and v g ≈ 10−9 m s−1 (Fig. 3d, note that this velocity is lower than the polymerization of a single actin filament because the network grows under stress16), we find γ* ≈ 10−6 N m−1. This value is in the range of membrane tension for non-deflated liposomes21, but is larger than the tension of deflated liposomes, leading to the prediction that deflated liposomes are prone to the formation of spikes, in agreement with our experimental results (Fig. 1c). Spike initiation also depends on the structure of the actin network through the value of the network viscosity η. Using the relationship22 η ≈ k B Tl p τ ve/ξ 4, with l p being the persistence length of the actin filament (~10 μm)23, we find the following condition for spike initiation:

$$\gamma \xi ^4 < k_{\rm B}Tl_{\rm p}v_{\rm g}\tau _{ve}\sqrt {\frac{{k_{\rm B}T}}{{2{\rm \pi} \kappa }}}$$
(1)

In contrast to ‘thin’ spike-like protrusions6, the spikes we consider here are formed by the growth of a branched network with a uniform polymerization along the liposome membrane (Fig. 4). The compressive stress resulting from actin polymerization (shown in Fig. 5b) explains that spikes are much wider than the ones previously observed6, and that they grow faster than the surrounding actin layer (Supplementary Information and Fig. 4).

The initiation of membrane tubes in the reference condition requires a pulling force at the tip of the tube larger than $$f_{\rm{tube}} = 2{\rm \pi} \sqrt {2\kappa \gamma } \approx 2\,{\rm {pN}}$$ (refs. 24,25; Fig. 5d with the above estimates). The tube radius ( $$r_{\rm{tube}} = \sqrt {\kappa /(2\gamma )} \approx 20\,{\rm{nm}}$$ ) is smaller than the size of the actin mesh through which it is pulled. This situation differs from spikes where the flow of the actin network is enslaved to the shape of the membrane, thus generating a wider deformation. In our case, tube pulling requires physical attachment of the actin to the membrane through the activator pVCA26.

### Reporting Summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

Journal peer review information: Nature Physics thanks Allen Liu and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Change history

• ### 26 April 2019

In the Supplementary Information initially published online for this Article, Supplementary Figs. 1, 2, 4, 5, 8–11 were corrupted; these have now been corrected.

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## Acknowledgements

We acknowledge A. Kawska at IlluScientia.com for the figures. We thank J. Pernier for suggesting the excess profiling experiment for loosening the actin network. This work was supported by the French Agence Nationale pour la Recherche (ANR), grants ANR 09BLAN0283 and ANR 12BSV5001401, by Fondation pour la Recherche Médicale, grant DEQ20120323737, by the LabEx CelTisPhyBio postdoctoral fellowship (M.L.), no. ANR-10-LBX-0038 part of the IDEX PSL NANR- 10-IDEX-0001-02 PSL, by Marie Curie Integration Grant PCIG12-GA-2012-334053, ‘Investissements d’Avenir’ LabEx PALM (ANR-10-LABX-0039-PALM), ANR grant ANR-15-CE13-0004-03 and ERC Starting Grant 677532. Our groups belong to the CNRS consortium CellTiss. This work was supported by grants from the French National Research Agency through the ‘Investments for the Future’ (France-BioImaging, ANR-10-INSB-04), the PICT-IBiSA Institut Curie (Paris, France).

## Author information

### Author notes

1. These authors contributed equally: Camille Simon, Rémy Kusters, Valentina Caorsi.

2. These authors jointly supervised this work: Pierre Sens, Cécile Sykes.

### Affiliations

1. #### Laboratoire Physico Chimie Curie, Institut Curie, PSL Research University, CNRS UMR168, Paris, France

• Camille Simon
• , Rémy Kusters
• , Valentina Caorsi
• , Antoine Allard
• , Majdouline Abou-Ghali
• , John Manzi
• , Aurélie Di Cicco
• , Daniel Lévy
• , Jean-François Joanny
• , Julie Plastino
• , Pierre Sens
•  & Cécile Sykes
2. #### Sorbonne Universités, UPMC Univ. Paris 06, Paris, France

• Camille Simon
• , Rémy Kusters
• , Valentina Caorsi
• , Antoine Allard
• , Majdouline Abou-Ghali
• , John Manzi
• , Aurélie Di Cicco
• , Daniel Lévy
• , Jean-François Joanny
• , Julie Plastino
• , Pierre Sens
•  & Cécile Sykes
3. #### LAMBE, Université Evry, CNRS, CEA, Université Paris-Saclay, Evry, France

• Antoine Allard
•  & Clément Campillo
4. #### LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Orsay, France

• Martin Lenz
5. #### ESPCI-Paris, Paris, France

• Jean-François Joanny

### Contributions

C.S., R.K. and V.C. have equal contributions. C.S. and V.C. performed experiments and analysed data. R.K. performed the development of theoretical models. A.A. M.A.-G., J.M., A.D.C., D.L., C.C. and J.P. contributed to experimental data; M.L. and J.-F.J. contributed to the development of the model; P.S. and C.S. designed the research. All authors contributed to writing the paper.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Pierre Sens or Cécile Sykes.

## Supplementary Information

1. ### Supplementary Information

Supplementary References 1–14 and Supplementary Figures 1–11

### DOI

https://doi.org/10.1038/s41567-019-0464-1