Modelling cellular spreading and emergence of motility in the presence of curved membrane proteins and active cytoskeleton forces
JJanuary 5, 2021
Modelling cellular spreading and emergence of motility in the presence of curvedmembrane proteins and active cytoskeleton forces
Raj Kumar Sadhu , Samo Peniˇc , Aleˇs Igliˇc , and Nir S. Gov Department of Chemical and Biological Physics, Weizmann Institute, Rehovot, Israel Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia and Faculty of Medicine, University of Ljubljana, Ljubljana, Slovenia
Eukaryotic cells adhere to extracellular matrix during the normal development of the organism,forming static adhesion as well as during cell motility. We study this process by considering asimplified coarse-grained model of a vesicle that has uniform adhesion energy with a flat substrate,mobile curved membrane proteins and active forces. We find that a high concentration of curvedproteins alone increases the spreading of the vesicle, by the self-organization of the curved proteinsat the high curvature vesicle-substrate contact line, thereby reducing the bending energy penalty atthe vesicle rim. This is most significant in the regime of low bare vesicle-substrate adhesion. Whenthese curved proteins induce protrusive forces, representing the actin cytoskeleton, we find efficientspreading, in the form of sheet-like lamellipodia. Finally, the same mechanism of spreading is foundto include a minimal set of ingredients needed to give rise to motile phenotypes.
I. INTRODUCTION
The adhesion of cells to an external substrate is an essential process allowing cells to form cohesive tissues, migrateand proliferate [1]. The stages of cellular spreading over an adhesive surface have been studied experimentally [2–9],and involve an initial stage of non-specific and weak adhesion, followed usually by spreading that is driven by theformation of thin sheet-like lamellipodia. These structures form when actin polymerization is recruited to the leadingedge of the lamellipodia [10]. The actin provides both a protrusive force that pushes the membrane outwards andtraction forces that enhance the growth of adhesion complexes. When actin polymerization is inhibited, cells exhibitvery weak spreading, small adhered area [5], and strongly retract (if the drug is delivered after normal spreading [11]).While this complex process has been explored from its biological aspects, a more basic physics understanding ofthe cell spreading process is lacking [12]. There are several theoretical treatments of the cell spreading process, withvarious levels of coarse-graining and detail, starting from the simplest dynamical-scaling model [4]. Some modelsfocus on the role of actin and actin-adhesion coupling during the spreading and adhesion, but do not describe themembrane shape dynamics in detail [13–16]. Other models describe in detail the cell shape and the stress-fibers thatspan the adhered cell [17, 18], with the higher realism obtained at a price of much higher model complexity.The simpler process of vesicle adhesion, which has been explored using in-vitro systematic experiments [19–23],is amenable to theoretical physics description [23]. A large number of theoretical studies treated the coarse-grainedadhesion of a vesicle with uniform adhesion [24–26], while other studies have explored the molecular-scale adhesiondynamics [27–29], which include ligand binding/unbinding as well as diffusion and aggregation of ligands on themembrane-substrate interface (through direct and membrane-induced interactions [30, 31]).We aim here to help bridge the gap between our understanding of vesicle adhesion and the more complex process ofactive cellular spreading. We do this by exploring a simple, coarse-grained theoretical model of vesicle adhesion whichcontains two ingredients: (i) we add a fixed density of curved membrane proteins, and (ii) exert active protrusive forcesat the locations of the membrane proteins. Both of these components are motivated by experimental properties ofcells: cell membranes contain a plethora of curved membrane proteins [32, 33], and many of these curved proteins (ormembrane-bound protein complexes) are involved in the recruitment of actin polymerization activity to the membrane[34, 35]. These two ingredients have been recently shown theoretically [36] and experimentally [37, 38] to be sufficientto induce the formation of sheet-like lamellipodia protrusions in cells. We therefore set out to explore theoreticallythe role of these ingredients during active spreading of cells.
II. MODEL
We use here the same theoretical model of [36], adapted to include membrane-substrate adhesion. We consider athree-dimensional vesicle that is described by a surface of N vertices, each connected to its neighbours with bondsof length l , to form closed, dynamically triangulated, self-avoiding network, with the topology of a sphere, as shownin Fig. 1. An adhesive surface is placed near the vesicle, parallel to x-y plane and at position z = z ad (see Fig. 1). a r X i v : . [ phy s i c s . b i o - ph ] J a n The total energy of the vesicle is the sum of four contributions, (1) the local bending energy due to its curvature, (2)the energy due to binding between neighboring proteins (direct interaction energy), (3) the energy due to the activecytoskeleton force and (4) the adhesive energy due to the attractive interaction between the vesicle and the substrate.Note that the term ”curved membrane proteins” stands for any complex of such proteins and lipids (such as innanodomains) in general, that has a spontaneous curvature, and can induce local polymerization of the cortical actincytoskeleton.The bending energy can be mathematically expressed using the Helfrich expression [39] as, W b = κ (cid:90) A ( C + C − C ) dA where, C and C are principle curvatures, C is the spontaneous curvature at any position of the vesicle, and κ is thebending rigidity. The bending energy is properly discretized following the refs. [40–42]. We model the spontaneouscurvature as discrete entities that is occupied by a vertex. The spontaneous curvature of a vertex that is occupiedby curved proteins is taken to have some non-zero value C = c , and zero otherwise. In our model, we consider apositive c >
0, i.e. convex spontaneous curvature. Note that we describe here isotropic curved proteins or isotropiccurved nanodomains [43, 44].The energy due to the binding between proteins is expressed as, W d = − w (cid:88) i FIG. 1: A schematic representation of our model. (a) A three-dimensional vesicle is placed on the adhesive surface, havinguniform adhesion interaction with the vesicle throughout. The position of the adhesive surface is at z = z ad and parallel to x − y plan. (b) The range of adhesive interaction is within a distance of ∆ z above the adhesive surface. The total adhesionenergy will be E ad times the number of vertices within this interaction range, where E ad is the adhesion strength, defined asthe adhesion energy per adhered vertex. III. RESULTS In order to validate our simulation method, we first compared the steady-state shapes of adhered protein-freevesicles to those previously obtained using detailed numerical solutions [23, 48]. The very good agreement betweenthe two different methods (Fig. A-1 of appendix B) verifies the accuracy of our simulations. The equilibrium vesicleshapes minimize the energy, striking a balance between the adhesion energy that drives the spreading and the bendingenergy that resists the deformation of the membrane.Next, we explored the effects of the curved proteins and the active forces that they recruit. A. Spreading of vesicles with passive curved proteins We start by exploring the effects of passive curved proteins, without active cytoskeletal forces. Since, there is noactive force acting on the system, the system reaches an equilibrium configuration after evolving it for sufficientlylong times. We aim to understand here, how the presence of the curved proteins affects the vesicle shape, as well asthe demixing of the curved proteins.In Fig. 2, we plot snapshots of typical equilibrium adhered vesicle shape and the protein cluster-size distributionfor different values of adhesion strength ( E ad ) and the number density of proteins ( ρ = N c /N ). The background coloris showing the adhered area fraction (AAF, A ad /A ), where A ad is the area adhered on the adhesive surface and A isthe total area of the vesicle (so that the maximal possible value is A ad /A (cid:39) . ρ and small E ad , the vesicle shape is quasi-spherical, similar to adhered protein-free vesicles. The proteinsare weakly clustered and there are small patches of proteins around the whole vesicle. As ρ increases keeping E ad small,the vesicle spreads, becomes flatter (since the volume is not fixed). This spreading is driven by the aggregation of thecurved proteins at the high-curvature region [43, 49] where the adhered vesicle contacts the substrate (”contact line”,Fig.3 (a)). Due to their large convex spontaneous curvature, the aggregation of the proteins in this region lowerssignificantly the bending energy of the vesicle, facilitates stronger bending and larger spreading on the substratecompared to the protein-free vesicle (Fig.3 (b)).Note that demixing and phase-separation of membrane proteins on an adhered vesicle were considered theoretically[50], however the curvature-based demixing [51] that we discuss here was not previously treated during membraneadhesion. A simplified analytic model (see appendix C) allows us to qualitatively recover the same trends found inthe simulations. We find that the aggregation of curved proteins along the high curvature rim of the vesicle leads to amonotonous increase in the AAF and decrease of the bending energy with increasing protein density (for low ρ , Fig.A-4).As ρ increases, the vesicle flattens and surplus proteins that have no more space along the contact line formnecklace-like protein clusters around the vesicle (Fig. 2). The necklace-like structures are formed because of the ρ (%)0.511.522.53 E ad ( k B T ) FIG. 2: Typical steady state configurations of the vesicles and cluster distribution in the E ad − ρ plane for the passive case(F=0). The background color is showing the fraction of adhered area of the vesicle on the adhesive surface. The blue part inthe vesicle denotes the protein free regions and the red color denotes the curved proteins. In the inset, we show the clusterdistribution of proteins near each snapshot; x-axis is the size of the cluster and the y-axis is the corresponding frequency ofhaving the particular cluster in the ensemble. The y-axis is shown in the log-scale. We show the snapshots for E ad =0.75,1.5, 2.5 (in units of k B T ) and ρ =3.45 %, 10.36 %, 17.27 % and 24.18 %. Other parameters are: Total number of vertices, N = 1447, κ = 20 k B T and w = 1 k B T . The width of the potential, ∆ z is taken to be l min , and the spontaneous curvature ofthe curved proteins is taken to be c = 1 l − min . isotropic membrane proteins used in our simulation [36, 52]. The AAF increases with ρ as long as the shape of thevesicle remains quasi-spherical or pancake-like. However, for large ρ , the shape of the vesicle changes and the surpluscurved proteins lead to budding all over the membrane. These buds (isolated and necklace-like), deform the membraneaway from the flat shape, and give rise to a decrease of the AAF with increasing ρ (Fig. 2).As E ad is increased, the vesicle is more spread, and the natural curvature along the contact line increases. Thishas the effect of aggregating the curved proteins more strongly and the flattening effect of the curved proteins setsin at lower densities (Fig. 2). Similarly to low E ad , the AAF decreases for large ρ , and the peak in the AAF shiftsto smaller values of ρ . As E ad increases the proteins form a large cluster at the contact line, where almost all theproteins are clustered in a single cluster.To quantify the effect of the curved proteins on the vesicle spreading, we measure the ratio of AAF between vesicleswith highly curved passive proteins and protein-free vesicles (Fig.3 (c)). We find that this ratio is larger for small E ad ,where the protein-free vesicles are very weakly adhered. As expected from Fig. 2, the enhancement of spreading dueto the curved proteins has a maximum as function of ρ , with the peak shifted to lower values of ρ as E ad increases.Note that for the passive case ( F = 0), a protein-free vesicle is similar to a vesicle with flat proteins ( c = 0). Fordetails, please see appendix D. θθθθ P ( θ ) Angle ( θ ) π /2 0 π /2(a): ρ =1.72% 00.010.020.030.040.05- π /2 0 π /2(b): ρ =8.64% 00.010.020.030.04- π /2 0 π /2(c): ρ =12.09% 00.010.020.030.04- π /2 0 π /2(d): ρ =15.55% W b A ad /A 0 5 10 15 20 25 ρ (%) 0.5 1 1.5 2 2.5 3 3.5 E ad (a)(b) (c) FIG. 3: Curved proteins facilitate spreading at low adhesion energy. (a) Protein distribution along the angle θ for the passivecase with E ad = 0 . k B T and various ρ along with the snapshots. The angle ( θ ) is defined as the angle between the lineparallel to the x-y plane, passing through the COM of vesicle and the line joining the surface of the vesicle and its COM, asshown in figure. P( θ ) is the probability that there is a protein at the angle θ , averaged over the azimuthal direction, suchthat θ varies from − π/ π/ 2. (b) Bending energy as a function of AAF with and without proteins. The blue circles are forprotein-free vesicle and the red triangles are for passive case. We also show the snapshots for each case, in the minimum andthe maximum adhered state. For protein-free vesicle, we vary E ad from 0 . k B T to 4 . k B T , while for passive vesicle, we fix E ad to 0 . k B T and vary ρ from 0 . 70 % to 27 . 64 %. (c) Ratio of AAF between vesicles with highly curved passive proteins(Fig.2) and protein-free vesicles. The ratio is close approaches unity for either ρ → E ad → ∞ . B. Spreading of vesicles with active curved proteins Next, we study the active system, with active cytoskeletal forces acting outward on the proteins. To highlight theeffects of the active force, we start with a large value of F = 4 k B T /l min (Fig. 4). Despite the presence of the activeforce, we find that for sufficiently large times most systems do reach a well-defined steady state, which allows us toextract average quantities, such as the AAF and the cluster distribution. When the density ρ is small, the shapes ofthe adhered vesicle are quite different compared to the passive case (Fig. 2). However, for large ρ , the shapes arequite similar to the passive case.For small value of ρ , unlike the passive case (Fig. 2), the shape of the vesicle is highly non-rotationally-symmetric(Fig. 4). The transition into this class of shapes, for ρ to the left of the vertical dashed green line (around 5 . 25 %), isvery sharp and was also observed for free vesicles with active curved proteins (denoted by the vertical dashed red line, ρ ∼ . 26 %) [36]. At a low number of proteins, there are simply not enough proteins to complete a circular aggregatearound the rim of the adhered vesicle. Instead, the proteins settle into two opposing arc-like aggregates, which exertopposing forces on the vesicle, that therefore assumes a stretched tube-like shape. ρ (%)0.511.522.53 E ad ( k B T ) FIG. 4: Typical steady-state configuration of vesicle shape and cluster-size distribution with active forces. Similar to Fig. 2 foran active protrusive force F = 4 k B T /l min . The green dashed line-circles denotes the transition to a pancake-like shape. Thered dashed vertical line denotes the density for the pancake transition for a free vesicle (without adhesion, [36]). We estimatethese transition lines by measuring the mean cluster size, which shows a sharp jump at the critical ρ (Fig. 5(c)). Snapshotsare shown for E ad =0.75, 1.5, 2.5 (in units of k B T ) and ρ = 3 . 45 % , . 36 % , . 27 % and 24 . 18 %. All the other parametersare the same as in Fig. 2. In Fig. 5(a) we demonstrate the dynamics of the spreading in the low ρ regime. We find that often the proteinsform three or more aggregates that drive the spreading, but they coarsen over time to form the stable two-arc shapes.These two-arc shapes are mostly non-motile, but due to asymmetry between the sizes of the arc-like protein aggregateson either side, there can be a net force that leads to sliding of these vesicles in the direction of the end that has thelarger aggregate (see Fig.9(c)).For ρ values to the right of the red dashed line in Fig. 4 there is a transition to pancake-like shapes, that correspondto very efficient spreading and high AAF. For even larger protein density, there is not enough space on the outerrim of the vesicle to accommodate all the proteins, so the pancake-like shape does not remain stable. Small bud likestructure appear around the vesicle and the AAF decreases, similar to the passive case at high densities (Fig. 2).The transition between the two-arc and pancake-like shapes for different values of ρ and E ad is shown in more detailin Fig. 5(b). As E ad → 0, the transition density increases and approaches its value for a free vesicle. The pancaketransition is quantified by the mean cluster size ( (cid:104) N cl (cid:105) ), which we plot as function of ρ (Fig. 5(c)). The quantity (cid:104) N cl (cid:105) exhibits a sharp jump near the pancake transition, where one cluster contains almost all the proteins (phaseseparation). E ad ( k B T ) ρ (%) < N c l > ρ (%)E ad =0.75E ad =1.50E ad =2.5000.511.522.53 1 2 3 4 5 6 7(b) E ad ( k B T ) ρ (%) FIG. 5: Transition to pancake-like, highly-spread shape, for active curved proteins ( F = 4 k B T /l min ). (a) Snapshots of thevesicle for small density, at different instant of time. Here, we show that there are different possible metastable states in thesmall density regime. We use here ρ = 3 . 45 %, E ad = 0 . , . , . k B T ). (b) Configurations of vesicle for differentvalues of E ad and ρ . The green line with circles denotes the density for a given E ad at which the pancake-like shape is obtained.(c) Mean cluster size of proteins, (cid:104) N cl (cid:105) , as a function of ρ for different values of E ad . The sharp increase in the value of (cid:104) N cl (cid:105) shows the discontinuous transition to pancake-like shape. We also show the snapshots of vesicle for E ad = 2 . k B T for differentdensities to show how the jump in the value of (cid:104) N cl (cid:105) gives rise to the pancake-like shape transition. Next, we study the AAF as a function of F , for different values of E ad and a fixed density ρ = 10 . 36 % (Fig. 6(a)). For large E ad the force increases the AAF smoothly, as the vesicles are already spread even in the absence ofactive forces. For small E ad , the AAF shows a large increase with F , including an abrupt jump for E ad = 0 . k B T at F ∼ . k B T /l min . This jump corresponds to crossing the pancake transition line for these parameters. In Fig. 7we plot a more complete phase-diagram, of the steady-state shapes of the vesicle for low adhesion ( E ad = 0 . k B T ),as function of the proteins density and active force. The active force is seen to shift the transition to the pancake (orthe two-arc) shape, to lower values of the density, compared with the passive system (at F = 0). This shift to lowerdensities due to the force is also seen for the initiation of protein aggregation (appendix E, Fig.A-8). A ad / A F (k B T / l min ) (b) 0 5 10 15 20 25 ρ (%) 0.5 1 1.5 2 2.5 3 3.5 E ad ( k B T ) ρ (%) 0.5 1 1.5 2 2.5 3 3.5 E ad ( k B T ) FIG. 6: Variation of adhered fraction due to the active force ( F ). (a) Fraction of adhered area with F for different values of E ad for a fixed ρ (= 10 . 36 %). For small E ad , the fraction A ad /A increases with F significantly, however, as E ad increases, thevalue of A ad /A does not vary much. We also show the shape of the vesicle for E ad = 0 . k B T . (b) The ratio of adhered areaof active vesicle with c = 1 l − min to the passive vesicle with c = 0. The ratio is maximal for small E ad over a wide range of ρ .(c) The ratio of adhered area of active to passive vesicle, with spontaneous curvature c = 1 l − min . We note that the maximumincrease in the adhered area of the active vesicle over passive one is in the small ρ and small E ad region. The role of the active force in increasing the steady-state AAF is emphasized in Fig. 6 (b), where we plot its ratiowith the AAF of the protein-free vesicle. We find that the largest increase in AAF due to the active curved proteinsis for low E ad . Compared to the vesicle containing curved passive proteins, this enhanced spreading is extended tolower values of ρ due to the pancake transition (compare to Fig. 3(c)). This is emphasized in Fig. 6 (c) where we plotthe ratio of the AAF of the active and passive curved protein systems. The largest contribution of the active force isfor low E ad and ρ , where the passive proteins do not form strong aggregation at the contact line and are ineffectivein driving spreading, while the added active forces drive the pancake transition and strongly enhanced spreading.We note that for large E ad and large ρ , the AAF is also increased due to activity, compared to the passive curvedproteins (Fig. 6 (c)). This is the region where the passive proteins form large necklace-like structures which decreasethe adhered area (Fig. 2), while the active forces tend to destabilize them and therefore increase the AAF.The active forces exerted at the locations of the curved proteins may give rise to a non-zero net force. While theplanar component of this force simply pushes the vesicle on the substrate, the vertical component (along z -direction)can affect the adhesion. Since the curved proteins prefer the free (dorsal) side of the vesicle over the perfectly flatbasal side, this force tends to overall push the vesicle away from the substrate. In the regime of very low E ad andlarge ρ , the active forces exerted by the proteins can lead to lowering the AAF by partially detaching the vesicle. InFig. A-9 of appendix F, we show the behavior of vesicles where we apply an external force that balances the totalvertical component of the active forces. For a living cell, this condition corresponds to assuming that actin filamentsthat are pushing the top membrane upwards exert an equal and opposite force on the bottom membrane. We seethat except at the lowest E ad , there is no qualitative difference, compared to the previous results (Fig.4).The importance of coupling the force to curvature, is demonstrated by simulating the adhesion of a vesicle with flatactive proteins (zero spontaneous curvature, c = 0). As in previous studies [36, 38], we find the formation of longprotrusions, that are highly dynamic (Fig. A-10 of appendix G). Due to the adhesion, the long protrusions are foundto often grow along the substrate. However, when they point upwards, they lead to partial detachment of the vesicle.Clearly, active forces that are not coupled to curvature do not contribute to effective spreading and adhesion.When comparing our results with experimental observations of the shapes of adhered cells, we begin by noting thatcells undergo a much diminished spreading (or strong retraction) when actin polymerization is inhibited (adheredarea decreases by factor of ∼ E ad . In this regime, we demonstrate thatself-organization of the actin polymerization recruited by curved membrane proteins can increase the adhered area byfactors that are similar to those observed experimentally (Fig.6b,c).However, the actin polymerization in the cell does more than just provide a protrusive force, as we assumed inour model. The actin retrograde flow produces shearing forces that triggers the growth of integrin-based adhesioncomplexes [3, 8]. This suggests that the activity of actin polymerization also effectively increases E ad for the cell,compared to the actin-inhibited cell. Similarly, increased adhesion strength ( E ad ) allows for stronger mechanicalcoupling between the actin filaments and the substrate, inducing a larger effective protrusive force F [15, 53]. Theseeffects mean that when comparing our model to cell shapes, the effective actin protrusive force F and the effectivevalue of E ad are not independent of each other.Many adhered cells are found not to be circularly spread, but have a distinct spindle-like shape with usually twooppositely formed lamellipodia protrusions. This typical shape appears naturally in our model when the density of thecurved proteins is below the pancake transition value, and the adhered vesicle assumes the elongated two-arc shape(Figs. 4, 5). Note that since the critical density for the pancake transition increases for decreasing E ad (Fig.5b), weexpect that cells can transform from the pancake to the elongated two-arc shape with decreasing adhesion strength.This is indeed observed in experiments [3, 8, 54]. The morphology of two oppositely oriented lamelipodia (similar toour two-arc shapes) was observed to stretch cells, and is sometimes utilized to drive cell division [55, 56]. C. Spreading dynamics We compare our results for the spreading dynamics of a protein-free vesicle, a vesicle with passive-curved andactive-curved proteins respectively, in Fig. 8(a-c). The vesicles with proteins are shown in the interesting regime oflow E ad . We note that the active vesicle spreading is much noisier than the passive spreading. This is because in theactive case the vesicle may transiently get locally de-adhered from the substrate, which gives rise to large variations inthe measurement of the AAF (Fig. 8(c)). In Fig. A-11 of appendix H, we plot the cross-sectional shapes, side-viewsand three-dimensional shapes, of the spreading vesicles as function of time, for all three cases. Clearly, the passivesystems are observed to spread more isotropically, compared to the active system. The anisotropic spreading of theactive vesicles is quantified in Fig. 8 (f), where we see a sharp reduction in the circularity of the adhered region duringthe initial stages of spreading.Next, we plot the increase of the adhered radius ( R ad ) as function of time, which is defined as (cid:112) A ad /π , where A ad is the adhered area (Fig. 8(d-e)). We find that for the passive systems the adhered radius ( R ad ) grows with timeas ∼ t β , where the exponent β is different for the different cases. Although this plot is given in MC time-steps, anddoes not include the hydrodynamic effects of the the membrane flow and the fluid flow within and around the vesicle,the calculated dynamics of the passive vesicles resemble the experimental observations for spreading artificial vesicles ρ (%)00.511.522.533.54 F ( k B T / l m i n ) FIG. 7: Typical steady-state configurations of vesicle shape and cluster-size distribution for different F and ρ , for a given smalladhesion strength: E ad = 0 . k B T . The snapshots are shown for F =0.50, 1.50, 2.50 and 3.50 (in units of k B T /l min ), and ρ = 3 . 45 % , . 36 % , . 27 % and 24 . 18 %. The yellow dotted lines denotes the transition to a pancake-like shape. For large F , the transition to a pancake shape from a two-arc shape is very sharp, and the transition line is estimated by measuring themean cluster, as explained in Fig. 5. For small F , the transition is not very sharp. Here, in order to identify the transition line,we measure the largest cluster, and we approximate the shape to be a pancake when the largest cluster is at least 60 % of thetotal number of proteins (see Fig. A-12 of appendix I). The green dotted line represents the transition from a quasi-sphericalto a two-arc type shape. The slope of this transition line diverges as ρ → 0. This transition is also estimated by measuring thelargest cluster and the threshold value of the largest cluster is taken to be 30 % in this case, as there are two separate clustersin the two-arc shape. All the other parameters are the same as in Fig. 2. [21].For the active case (Fig. 8e), we averaged over those cases which adhered smoothly without significant events ofde-adhesion, in order to get a less noisy curve. For this case, we find that there is a slower growth in the beginning(Fig. 8d), followed by a faster growth regime. The exponent of the faster growth stage depends on E ad (Fig. 8 (e)).The slower initial growth is due to the low circularity of the active vesicle, with the protein aggregates spread overthe vesicle and pushing the membrane upwards and in uncoordinated manner. Once the proteins form the circularaggregate along the contact-line, they induce a very efficient and rapid spreading, which corresponds to the fast growth0phase.The calculated spreading dynamics for an active vesicle resemble several aspects experimentally observed in spread-ing living cells [2, 7, 57]: (i) The active vesicles exhibit an initially accelerating radial growth, followed by a growthwith almost constant velocity (Fig. 8 (c,d)). These features are observed in living cells, and do not appear for ourpassive vesicles. (ii) The initiation of the rapid spreading phase takes longer to appear for cells on substrates of loweradhesiveness, as we also find (Fig. 8 (e)). (iii) Compared to the passive system, the active vesicle initially grows moreslowly (Fig. 8 (d)), similar to experimental observations [4]. A ad / A Time (2X10 MC Steps) A ad / A Time (2X10 MC Steps) A ad / A Time (2X10 MC Steps) 10 15 20 1 10 100 1000 R ad Time (2 x 10 MC Steps)t t t t 10 15 20 1 10 100 R ad Time (2 x 10 MC Steps)t t C i r c u l a r i t y Time (2 x 10 MC Steps) (a) (b) (c)(d) (e) (f) FIG. 8: Dynamics of the spreading process. (a) Variation of adhered area with time for a protein free vesicle. Here, we use E ad = 5 . k B T . (b) Variation of adhered area with time for a passive vesicle with ρ = 13 . 82 % and E ad = 0 . k B T . (c)Variation of adhered area with time for an active vesicle with F = 4 k B T /l min , ρ = 10 . 36 % and E ad = 0 . k B T . We notethat the protein free vesicle adhere much faster than the other cases. The spreading of passive vesicle is very smooth, while theactive vesicle spreading is very noisy. Here, the unit of time (t) is 2 × MC steps for (a)-(c). (d) Variation of adhered radius( R ad ) with time, for protein-free, passive and active vesicles. The blue circle is for protein-free vesicle, the red triangles are forpassive vesicle and the green boxes are for active vesicle. We note that the growth of a protein free vesicle and passive vesicleis uniform in the beginning, and then saturates, however, for active case, there are two phases of growth in the small timeregime before R ad saturates. For protein free vesicle, R ad ( t ) ∼ t . , while for a passive vesicle, R ad ( t ) ∼ t . . Unlike theprotein-free and passive vesicles, the active vesicle exhibits two growth phases: a slower initial growth of R ad ( t ) ∼ t . followedby a faster growth R ad ( t ) ∼ t . . Here, we use E ad = 5 . k B T for protein-free vesicle, and ρ = 10 . 36 % and E ad = 0 . k B T for passive and active vesicles. For active vesicle, we use F = 4 k B T /l min . (e) Variation of R ad with time for an active vesiclewith various values of E ad : from bottom to top, star symbols are for E ad = 0 . k B T , boxes are for E ad = 0 . 50, circles are for E ad = 1 . k B T and triangles are for E ad = 2 . k B T . Note that the fast growth phase takes longer to appear for lower E ad , whilethe growth exponent of the fast growth phase increases with E ad . We vary E ad from 0 . k B T to 2 . k B T , that gives rise tovariation in the exponent from 0 . 51 to 0 . 20. Here we use ρ = 10 . 36 % and F = 4 k B T /l min . The unit of time (t) is 2 × MCsteps for (d)-(e). (f) The circularity of the vesicle shows non-monotonic variation with time. Here, we use F = 4 k B T /l min , E ad = 0 . k B T and ρ = 10 . 36 %. Other features of cell spreading are also manifested in our spreading active vesicles: Similar to the case of spreadingcells [13], we find that the circularity of the spreading active vesicles decreases sharply during the beginning of thespreading process, and recovers slowly afterwards. Furthermore, the active forces often give rise to the transientupwards detachment of the leading edge in our simulations (Fig.8 (c)), resembling the ruffles observed at the leadingedge of spreading cells [6, 9, 37, 58].Note that since we do not conserve the vesicle volume, we find that the volume strongly decreases as the vesiclesspread. We discuss this in more detail in the appendix J (Fig. A-13).1 D. Motile vesicles at low protein densities In the regime of low (curved and active) protein density, where there are not enough proteins to form a circularaggregate around the cell rim (pancake shape), we find that the vesicles can form motile shapes. By ”motile” wemean that the active proteins form a single large aggregate on one side of the vesicle, that results in an unbalancedforce that pushes the vesicle along the adhesive substrate. Such motile crescent shapes are shown for example in Figs.9 (a,c).In Fig. 9a we show the regime of active force and adhesion energy that gives rise to the motile crescent shapes, atvery low protein density ( ρ = 3 . 45 %). We find that the regime where the crescent shapes appear coexists with two-arcshapes, and the two can transiently convert into each other (see for example Fig. 9(c)). When moving, the crescentshape remains persistent by maintaining a sharp leading edge (Fig. 9(c), inset), due to the active force concentratedin a single cluster, while the rear region is less curved due to minimization of bending energy and area conservation.In Fig. 9(b) we plot the average velocity of the crescent shapes, calculated from the simulations, divided by aneffective friction coefficient that is assumed to be linear in E ad (the vesicle speed before this scaling is shown in Fig.A-14 of appendix K). Note that also the two-arc shapes are weakly motile, as the protein aggregates at each end of thecell are not identical in size and there is a small residual force (Fig. 9(d)). The crescent shapes exhibited persistentmotility with a well defined velocity in the high force regime, while for weak active forces they exhibited diffusivemotility (see Fig. A-15 of appendix L).The region of crescent shapes in Fig. 9(a) is bounded by two transition lines. The lower line denotes the line belowwhich the proteins do not form large clusters. In this regime the proteins form disordered small clusters, and thevesicle remains approximately hemispherical. Above this line, the proteins form one or two large aggregates, therebyenabling the formation of the crescent or two-arc shapes. Above the transition line the large adhesion energy orstrong active forces induce the sufficiently high curvature at the vesicle contact line, which concentrates the proteinsand drives the formation of large aggregates. This is similar to the pancake (and two-arc) transition line denoted inFig. 7, where increasing E ad corresponds to higher ρ at the contact-line region. Note that the direct protein-proteininteraction strength w plays a minor role in this transition, which can occur even for w = 0 k B T (Fig. A-16(a) ofappendix M).The upper transition line that bounds the crescent shapes regime in Fig. 9(a) can be estimated by comparing thebending and adhesion energies of a two-arc shape versus the crescent shape. Compared to the crescent shape, thetwo-arc shape has a lower adhesion energy, since the elongated cylindrical part is more weakly adhered as it is devoidof curved proteins. It is also more strongly curved compared to the circular shape of the crescent vesicle. On the otherhand, the work done by the active forces that elongate the two-arc vesicle counts as a negative energy contribution.The net difference between the two classes of shapes can be approximately written as∆ F = − N F L + 2 πRL κ R + E ad ∆ A (2)where R , L are the radius and length of the cylindrical segment of the two-arc shape, N is the number of proteinsthat pull the membrane at the two ends of the cell and ∆ A is the difference in adhered area between the two-arc andcrescent shape. We first minimize with respect to the radius R , taking the total area of the cylindrical segment to beconserved A cyl = 2 πRL . We find that R = 2 πκ/ ( N F ). We substitute this value in Eq. 2, and calculate the criticaladhesion energy at which ∆ F = 0 ∆ F = 0 → E ad = A cyl κ ∆ A (cid:18) N F π (cid:19) (3)We assumed here for simplicity that ∆ A is constant along this transition line, which is approximately obeyed by thesimulation results (Fig. A-17, appendix N). The relation in Eq. 3 appears to capture correctly the essence of thetransition between the two shapes, as shown in Fig. 9a. We fit the Eq. 3 with simulation data points, and obtain therelation E ad (cid:39) . F , that matches well with the simulation. We also estimate this prefactor of F by calculation∆ A and A cyl from simulation, and obtain a value (cid:39) . 96 close to the value obtained by fitting the curve (see A-17 fordetails).When comparing the motile shapes that our model produces, they bear similarity to the shapes of different motilecells [59]. Similar to experimental observations [60–62], we predict a maximum of the migration speed as function ofthe adhesion strength (Fig. 9b). These results indicate that the coupling of curved proteins that recruit the actinpolymerization, and adhesion, can self-organize into a spontaneously motile shape. However, these ingredients giverise to rather delicate and transient motility, which can either disappear spontaneously (Fig. 9(c)) or when confrontedwith an external perturbation (Fig. A-18, appendix O). Within the wider context of active-matter systems, our motilevesicles can be compared to recent works that have shown similar symmetry breaking that is driven by self-organizationof active elements [63].2 ad F ad F Time -10010 -10 0 10 (a) (b)(c) (d) FIG. 9: Crescent-shaped (motile) vesicle, its speed and stability. (a) Probability for a vesicle to be found in a crescent-shapedstate (background color), in the E ad − F plane for small ρ = 3 . 45 %. In the region where there is non-zero probability ofobtaining a crescent-shaped vesicle, the snapshots of a crescent-shaped vesicles are shown, otherwise a typical snapshot isshown. The yellow solid line is the analytical prediction separating the two-arc phase to a crescent-phase (Eq.3). Here, we fitthe numerical data and obtain a good fit at E ad (cid:39) . F . We also numerically calculated the prefactor and it turns out tobe (cid:39) . 96, close to the value obtained by fitting the curve (see Fig. A-17 for details). The yellow dots are the simulation pointseparating the two-arc shapes from crescent-shaped vesicle: For a given E ad , these dots represent the value of F , above whichwe do not find crescent shaped vesicles. The green dashed line separates the region of vesicles with disordered small proteinclusters (below) and the regime of clustered proteins (either two-arc or crescent shape). Here, we estimate this line by measuringthe largest cluster, and if the largest cluster is less than 30% of the total proteins, we regard it to be in the disordered state ofsmall protein clusters. The snapshots are shown for E ad = 0.50, 1.5, 2 and 3 (in units of k B T ) and F = 0.50, 2 and 4 (in unitsof k B T /l min ). (b) Speed of the crescent-shaped vesicle, scaled by E ad . We calculate the speed as the displacement of COM perMC step (divided by E ad ), and then normalize all the values by the maximum speed found for the range of parameters shownhere, where the maximum value of the speed is 0 . l min per MC step. For large E ad but small F , the crescent vesicles exhibitdiffusive behaviour (Fig.A-15). (c) Spontaneous transition from crescent-shape to two-arc shape, demonstrating the transientnature of the motile shapes. In the inset, we show the cross-sectional view of a motile crescent-shaped vesicles. The red dotis showing the location of the high protein density. (d) Examples of asymmetric two-arc shapes, which exhibit weak residualmotility. Below each shape, we also mention the ratio of smallest to largest cluster. For both (c,d) we use E ad = 3 . k B T , ρ = 3 . 45 % and F = 4 k B T /l min . IV. DISCUSSIONS We have shown here how interacting curved membrane proteins, passive and active, affect the process of vesiclespreading and adhesion. We find that large density of passive curved proteins can greatly enhance the adhesion of vesi-3cles on low adhesion substrates. Coupling the curved proteins with active protrusive forces extends this enhancementto lower densities of curved proteins. By spontaneously self-organizing curved proteins at the cell-substrate contactline, the active forces drive a shape transition into a flat geometry with high adhered area and robust spreading. Atvery low densities of curved proteins the protrusive activity can stabilize either spindle-like elongated cells, or motilecrescent shapes.Our simplified model does not contain all the complexities of a real cell, which strongly affect its final adheredshape. One such component, the network of stress-fibers, is known to determine the cell shape in many cell types[12]. In addition, the cytoskeleton and internal organs (such as the nucleus) hinder the shape changes of the cell,and exert volume constraints. Future extensions of our model can include additional components of the cell adhesionprocess. For example, we could add non-uniform adhesion that is activated closer in proximity to the curved proteins,to describe the activation of adhesion by actin retrograde flow [1, 64–66]. Nevertheless, our model describes manyfeatures of spreading cells, allowing to relate the observed cell spreading dynamics and the cell shape to the parametersof the model.Observations in living cells emphasize the central role played by actin polymerization during cell spreading andadhesion. These observations suggest that in living cells the membrane density of highly curved proteins is relativelylow, and cells are not likely to be in the regime where a high density of curved proteins alone drives the spreading andadhesion (Fig. 2). Loading the membrane with a large density of such curved proteins may be problematic for the cell,and limit its ability to dynamically control and modify its spreading and adhesion strength. Our model demonstratesthat by having a low bare adhesion, and low density of curved proteins, the cell can achieve robust and dynamicadhesion by activating the protrusive force of actin polymerization, in a highly localized and self-organized pattern.The spontaneous aggregation of the curved proteins along the cell-substrate contact line, driven by the actin-inducedforces (and attractive direct interactions between the proteins), provides a highly controllable mechanism for cellspreading and adhesion.In addition to non-motile steady-state shapes of adhered vesicles, we found that in the low ρ regime the vesiclesmay form a polarized, crescent shape, that is motile (Fig.9). This motile vesicle resembles the shapes of motilecells, that depend on adhesion [67], and demonstrates that the combination of curved proteins that recruit the actinpolymerization, and adhesion, provide a minimal set of ingredients needed for motility. However, in order to makethe polarization that drives the motility robust and persistent (as opposed to transient), cells have evolved additionalbiochemical feedbacks of various types [68–70]. Our model does not contain many components that play importantroles in cell motility, such as contractility, and more realistic treatment of the actin-adhesion coupling, such as catchand slip-bond dynamics. Our results however highlight that curvature-force coupling, with adhesion, provide the basiccoarse-grained components that can self-organize to spontaneously break the symmetry and form a motile system. V. ACKNOWLEDGEMENTS We thank Orion Weiner, Benjamin Geiger, Ronen Zaidel-Bar, Robert Insall, Sam Safran, Jeel Raval and WojciechGozdz for useful discussions. N.S.G. acknowledges that this work is made possible through the historic generosity ofthe Perlman family. N.S.G. is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysicsand this research was supported by the Israel Science Foundation (Grant No.1459/17). A.I. and S.P. acknowledge thesupport from Slovenian Research Agency (ARRS) through program No. P2-0232 and the funding from the EuropeanUnion’s Horizon 2020 - Research and Innovation Framework Programme under grant agreement No. 801338 (VES4USproject). Appendix A: Simulation details The time evolution of the vesicle in our MC simulations consists of [36]: (1) vertex movement, and (2) bond flip.In the vertex movement, a vertex is randomly chosen and attempt to move by a random length and direction withina sphere of radius s drawn around the vertex. In the bond flip movement, a single bond is chosen, which is a commonside of two neighbouring triangles. The bond connecting the two vertices in diagonal direction is cut and reestablishedbetween the other two, previously unconnected vertices. In order to satisfy self avoidance, the ratio of maximum andminimum bond length, i.e., l max /l min = 1 . s = 0 . 15 in units of l min .We use Metropolis algorithm to update our system. Any movement that increases the energy of the system by anamount ∆ E occurs with rate exp ( − ∆ E/k B T ), otherwise if the movement decreases the system energy, it occurs withrate unity. We let the system evolve according the above rule and wait till the system reaches steady state. All theaverage quantities are measured after the system reaches steady state.4In the simulations presented in this paper we use the following model parameters: Total number of vertices, N = 1447, the bending rigidity κ = 20 k B T , the protein-protein attraction strength w = 1 k B T . The width of thepotential, ∆ z is taken to be l min . Among all the N vertices, N c of them are occupied by curved membrane proteins.The spontaneous curvature at all N c vertices are taken to be c = 1 l − min , unless stated otherwise. We chose this setof parameters to be in the interesting regime where the curved proteins form aggregates, and exhibit a force-drivenphase-separation into a pancake-like shape [36]. Appendix B: Comparison of simulated and detailed numerical solutions of the adhered vesicle shape In this section, we compare the results of our MC simulations for the shapes of adhered protein-free vesicles,with detailed numerical solutions that appeared recently in Ref. [23, 48]. In the detailed numerical solutions, theparameters used are ˜ w , the scaled adhesion strength and v , the reduced volume. The parameter ˜ w is defined as,˜ w = E ad R s /κ , where E ad is the adhesion energy per unit adhered area and R s is the radius of a spherical vesicle withsame volume as the original. Since, in our model we define E ad as the adhesion energy per vertex, we properly scaledit before comparison. In [23, 48], the reduced volume v is fixed, however, in our model, we can not fix v before theadhesion and spreading dynamics. In order to access different values of v for the same ˜ w , we use the osmotic pressuredifference p = p inside − p outside , that adds one more energy term − pV to Eq. 1, where V is the total volume of thevesicle [36]. In Fig. A-1, we show the comparisons of the shapes of the vesicle from our simulation and the detailednumerical solution (as given in Fig. 5 of ref. [23]). We note that the shape is comparable to our MC simulations, andthe agreement is very reasonable, thereby validating the MC approach. (a) (b) (c) (d) -0.8-0.400.40.81.2-1.5 -1 -0.5 0 0.5 1 1.5 Z r -0.8-0.400.40.81.2-1.5 -1 -0.5 0 0.5 1 1.5 Z r -0.8-0.400.40.81.2-1.5 -1 -0.5 0 0.5 1 1.5 Z r -0.8-0.400.40.81.2-1.5 -1 -0.5 0 0.5 1 1.5 Z r FIG. A-1: Comparison of the results of MC simulation with detailed numerical solutions [23, 48]. The red circles are forsimulation results and the blue boxes are for detailed numerical solution. (a) ˜ w = 6 . v = 0 . w = 6 . v = 0 . 75, (c)˜ w = 6 . v = 0 . 85 and (d) ˜ w = 6 . v = 0 . 95. For detailed numerical solution, the data is extracted from Fig. 5 of ref. [23]using the ‘digitize image’ tool from ‘OriginLab’ software. Appendix C: Analytical model We now present an analytic calculation of the adhered shape of the vesicle, in the presence of curved proteins andactive forces. This calculation correctly describes the qualitative features that we found in the simulations, but failsquantitatively. The main failure is the use of protein-protein interactions that are good for the dilute limit (Eq.A-3),and do not capture the large density increase at the contact line as in the simulations. In regimes where the protein-protein interactions are not playing an important role, for example when the active force is large, we find quantitativeagreement (Figs.A-5,A-6 below).We assume the average shape of the vesicle consists of three parts: (1) The base area, which is having a circularshape, with radius R b , (2) the annulus part curving around the circular base, which is the part of a torus, with radiusof the tube R t , and (3) the spherical cap, which is a part of sphere, with radius R c (see Fig. A-2). θ is the anglebetween the vertical line passing through the centre of the sphere (OQ), and the line joining the centre of sphere andthe contact point of sphere and the torus (OP), as shown in Fig. A-2. We neglect any thermal fluctuation in theshape of the vesicle. Total area of the vesicle, A = A b + A t + A c (cid:39) πR b + 2 π (1 − cosθ ) R c + 2 πR t R b ( π − θ ), where A b denotes the area of the base, A t is the area of the annulus part (torus) and A c is the area of the spherical cap. Totalarea A is taken to be constant [26].Let ρ be the density of the spontaneous curvature c , ρ c is the density on cap, and ρ t is the density on the annulus,such that, ρA = ρ t A t + ρ c Ac . Here, we assume that bottom part of the vesicle does not contain proteins, which isalso seen in the simulations. Now, the adhesion energy is given by, W A = − E ad πR b , (A-1)5 θ Π - θθ R c R b R t O PQ FIG. A-2: Schematic representation of the analytical model. R c is the radius of the spherical cap, R t be the radius of theannulus part (torus), R b be the radius of the circular disk at the base. θ is the angle between the line perpendicular to theplane ‘OQ’ and the line joining the center of sphere ‘O’ and the point ‘P’, where the spherical cap and torus section meets. the bending energy is given by, W b = 12 κA t ( 1 R b + 1 R t − c ρ t ) + 12 κA c ( 2 R c − c ρ c ) , (A-2)the protein-protein nearest neighbour attraction energy is of the form, W d = − w A c ρ c + A t ρ t ) , (A-3)the energy due to active force, W F = − F R t ρ c A c − F R b ρ t A t , (A-4)and finally, the entropy, S = − A c { ρ c ln ( ρ c ) + (1 − ρ c ) ln (1 − ρ c ) } − A t { ρ t ln ( ρ t ) + (1 − ρ t ) ln (1 − ρ t ) } , (A-5)where, E ad is the adhesion energy per unit adhered area and κ is the bending rigidity. Note that the entropy isonly due to the thermal fluctuation of proteins and any entropy due to the thermal fluctuation in the vesicle shapeis not considered here. In the calculation of entropy, we assume that only the spherical cap of the vesicle and theannulus part (torus part) contain proteins while the base of the vesicle does not contain proteins, which is also seenin simulations. Since, the slope of the surface changes continuously along the angle θ , the slope at the contact pointof torus and spherical cap should be the same. This gives us a constrains, R c sinθ = R b + R t sinθ . The free energy isgiven by, F = W b + W A + W d + W F − T S (A-6)We assume the total area A to be a constant. We then minimize the free energy (Eq. A-6), in the ( R b , ρ t , θ ) plane,and express other parameters in terms of these three parameters. More explicitly, we solve the equations, ∂F∂R b = 0, ∂F∂ρ t = 0, and ∂F∂θ = 0 with the constraint ∂ F∂R b > ∂ F∂ρ t > ∂ F∂θ > 0. Among several solutions, we consider thephysical one.In the analytical model, we define E ad as the adhesion energy per unit of adhered area (having dimension ofenergy/length ), while, in the simulation, we define it as the adhesion energy per adhered number of vertex (withdimension of energy). In order to compare the simulation and analytical results, we properly scale E ad such that thedefinition becomes consistent in both the cases. The value of the parameters used here are: A = 2200 l min , which is6approximately the average area of a unstretched vesicle in our simulation, c = 1 . l − min , κ = 20 k B T , w = 1 . k B T .The value of k B T is taken to be unity.For passive case ( F = 0), we compare our analytical results for the cross-sectional shape of the vesicles withdifferent ρ in Fig. A-3. We note that the effect of increasing ρ is not very strong in the analytical model. We showthe comparison of adhered fraction, density of proteins in the curved regions ( ρ t ) and the different energies in Fig.A-4. We note that the fraction of adhered area increases with E ad similar to our simulations. With increasing E ad the vesicle becomes more and more flat and tends to the value 1 / E ad (Fig. A-4(a)). The analyticalprediction is however lower than the simulation results. We also measure the density of curved proteins in the annuluspart, ρ t . As E ad increases, ρ t also increases and tends to unity for simulations, however, the analytical prediction isvery low (Fig. A-4(b)). This indicates that for large E ad most of the proteins are aggregated in the curved annulusregion. In our simulation, we also note that for large E ad , we do have a pancake-like phase, where all the proteins areclustered in the curved region. We also measure the fraction A ad /A as a function of ρ for given E ad (Fig. A-4(c)). Wenote that similar to the simulation results, our analytical model also shows non-monotonic variation in A ad /A with ρ , however, here also, the quantitative comparison is not very well. Finally, we also show the variation of ρ t with ρ ,which also show an increase similar to simulation results (Fig. A-4(d)).We also compare the different energies in the lower panel of Fig. A-4 for simulation and analytical cases. We notethat the energy values are quite different but the maximum difference is the protein-protein interaction energy (Fig.A-4(g)), due to which, increasing ρ is not very sensitive in our analytical model. Thus, we conclude that becauseof the simplicity of the analytical model, the quantitative comparison is not so good, however, it could describe thequalitative features of our system very well. Next, we compare the results for the active case. (a) (b)(c) (d) -20-1001020-20 -10 0 10 20 ρ =1.72%-20-1001020-20 -10 0 10 20 ρ =1.72%-20-1001020-20 -10 0 10 20 ρ =1.72%-20-1001020-20 -10 0 10 20 ρ =1.72%-20-1001020-20 -10 0 10 20 ρ =1.72% -20-1001020-20 -10 0 10 20 ρ =5.18%-20-1001020-20 -10 0 10 20 ρ =5.18%-20-1001020-20 -10 0 10 20 ρ =5.18%-20-1001020-20 -10 0 10 20 ρ =5.18%-20-1001020-20 -10 0 10 20 ρ =5.18%-20-1001020-20 -10 0 10 20 ρ =8.64%-20-1001020-20 -10 0 10 20 ρ =8.64%-20-1001020-20 -10 0 10 20 ρ =8.64%-20-1001020-20 -10 0 10 20 ρ =8.64%-20-1001020-20 -10 0 10 20 ρ =8.64% -20-1001020-20 -10 0 10 20 ρ =12.09%-20-1001020-20 -10 0 10 20 ρ =12.09%-20-1001020-20 -10 0 10 20 ρ =12.09%-20-1001020-20 -10 0 10 20 ρ =12.09%-20-1001020-20 -10 0 10 20 ρ =12.09% FIG. A-3: Comparison of the shape of vesicle for simulation results and analytical results for passive case with E ad =0 . k B T /l min and various ρ . Red circles are for simulation results and blue line is for analytical results. In the active case, we compare the cross-sectional shape of the vesicle for a given ρ and different values of F in Fig.A-5. We note that the effect of increasing F is very effective in analytical model. We also compare other results inFig. A-6. We note that the adhered fraction is very close to our MC simulations in the large F region (Fig. A-6(a)).In this case also, the energy values are not very comparable (see Fig. A-6(c-g)). Appendix D: Passive vesicle with proteins having zero spontaneous curvature In Fig.A-7 we show our results for a passive vesicle with proteins having zero spontaneous curvature, i.e., c = 0.This serves as a verification of our calculation: since the proteins and the membrane are identical, we indeed find thatthe adhered area remains constant with the density of proteins, only depending on E ad . Appendix E: Linear stability analysis for budding transition and comparison with the contour for < N cl > = 2 In [36], an expression for the critical temperature ( T c ) is derived using linear stability analysis (Eq. 6 of [36]), belowwhich the proteins will start forming aggregates (buds) [71]. We use this expression and obtain the critical force F c A ad / A E ad A ad / A ρ -300-200-1000 0 0.1 0.2 0.3 W A ρ W b ρ -2000-1500-1000-5000 0 0.1 0.2 0.3 W d ρ -100001000 0 0.1 0.2 0.3 W ρ (a) (b) (c) (d)(e) (f) (g) (h) ρ t E ad ρ t ρ FIG. A-4: Results for analytical model for passive case ( F = 0) and comparison with simulation. Red circles are for simulationresults and blue line is for analytical results. (a) Fraction of adhered area with adhesion strength E ad for ρ = 10 . / E ad . (b) The density of curved proteins in the annulus part of the vesicle( ρ t ) as a function of E ad for ρ = 10 . ρ t increases and tends to unity for large E ad . Here also,the analytical prediction is smaller that the simulation results. (c) Fraction of adhered area with the protein density ρ for agiven E ad . We note that the analytical prediction also shows non-monotonic variation as seen in the simulation. (d) ρ t with ρ for given E ad . (e) The adhesion energy ( W A )) as a function of ρ . (f) The bending energy ( W b ) as a function of ρ . (g) Theprotein-protein interaction energy ( W d ) as a function of ρ . (h) The total energy ( W ) as a function of ρ . For (c) to (h), we use E ad = 0 . k B T /l min . For analytical results, we use here A = 2200 l min , which is the average area of a unstretched vesicle inour simulation, c = 1 l − min , κ = 20 k B T , w = 1 . k B T . For analytical results, k B T is taken to be unity. For MC simulations,the other parameters are same as Fig. 2. (a) (b)(c) (d) -20-1001020-20 -10 0 10 20F=0.01-20-1001020-20 -10 0 10 20F=0.01-20-1001020-20 -10 0 10 20F=0.01-20-1001020-20 -10 0 10 20F=0.01-20-1001020-20 -10 0 10 20F=0.01 -20-1001020-20 -10 0 10 20F=0.05-20-1001020-20 -10 0 10 20F=0.05-20-1001020-20 -10 0 10 20F=0.05-20-1001020-20 -10 0 10 20F=0.05-20-1001020-20 -10 0 10 20F=0.05-20-1001020-20 -10 0 10 20F=0.10-20-1001020-20 -10 0 10 20F=0.10-20-1001020-20 -10 0 10 20F=0.10-20-1001020-20 -10 0 10 20F=0.10-20-1001020-20 -10 0 10 20F=0.10 -20-1001020-20 -10 0 10 20F=0.50-20-1001020-20 -10 0 10 20F=0.50-20-1001020-20 -10 0 10 20F=0.50-20-1001020-20 -10 0 10 20F=0.50-20-1001020-20 -10 0 10 20F=0.50 FIG. A-5: Comparison of the shape of vesicle for simulation results and analytical results for active case with ρ = 10 . 36 %, E ad = 0 . k B T /l min and various F . Red circles are for simulation results and blue line is for analytical results. as a function of ρ and other variables as, F c = wl min c − /ρR ) (cid:16) k B T wρ (1 − ρ ) − (cid:17) . Here, 1 /R is the mean curvatureat the site of the curved membrane protein. In the limit R → ∞ , the expression for F c will turn out to be, F c ( R → ∞ ) → wl min c (cid:16) k B T wρ (1 − ρ ) − (cid:17) (A-1)We use the above equation to calculate the F c for our case, and compare this line with the contour, where meancluster < N cl > = 2. Since, the budding forms along the contact line with the adhesive substrate, where the proteindensity is higher ( ρ t , say) than the average ρ (see for example Fig.3a and appendix C), we use in Eq.A-1 the value of ρ t obtained from the simulations to calculate the critical force F c , and plot it as a function of the average ρ (see Fig.A-8). We note that the line is almost vertical, and in the large F regime, it is in between the pancake transition (see8 (a) (f)(b) (c) (d) (e) (g) A ad / A F 02004006008001000 0 0.1 0.2 0.3 0.4 0.5 W b F -200-150-100-500 0 0.1 0.2 0.3 0.4 0.5 W d F -2000-1500-1000-5000 0 0.1 0.2 0.3 0.4 0.5 W F F -2000-1500-1000-5000500 0 0.1 0.2 0.3 0.4 0.5 W F-250-200-150-100 0 0.1 0.2 0.3 0.4 0.5 W A F ρ t F FIG. A-6: Results for analytical model for active case and comparison with simulation. Red circles are for simulation resultsand blue line is for analytical results. (a) Fraction of adhered area as a function of F . (b) The density of curved proteins inthe annulus part of the vesicle ( ρ t ) as a function of F . (c) The adhesion energy ( W A )) as a function of F . (d) The bendingenergy ( W b ) as a function of F . (e) The protein-protein interaction energy ( W d ) as a function of F . (f) The active energy dueto cytoskeleton forces ( W F ) as a function of F (g) The total energy ( W ) as a function of F . Here we use ρ = 10 . 36 % and E ad = 0 . k B T /l min . Other parameters are same as Fig. A-4 Fig.7) and the line < N cl > = 2. Qualitatively, the shape of this analytic line, and its dependence on the density atthe contact line, describe the transition of the proteins into small aggregates. The pancake transition at larger valuesof ρ has qualitatively the same shape, but of course can not be captured by the linear stability analysis. Appendix F: Simulations with balanced total vertical force The active force may in general act in a direction, that tries to de-adhere the vesicle from the adhesive surface. Inorder to cancel this effect, we apply an external force in the vertical direction (along z -direction), when the net verticalforce acts upward (that tends to de-adhere the vesicle). We note that even after applying the external force, see Fig.A-9, there is no qualitative change in the results. The benefit of applying this external force is that we could nowexplore much smaller E ad which could not be explored before, due to the de-adhesion of vesicle from the substrate. Appendix G: Active vesicle with zero spontaneous curvature proteins In this section, we show our results for the active case with proteins having zero spontaneous curvature. The shapeof the vesicle is very dynamic in this case and changes with time. We show few snapshots of the vesicle in Fig. A-10(a).The shapes shown in this figure should not be assumed to be a steady state shape. We also show the dynamic natureof the vesicle in Fig. A-10(b), where we show the snapshot for a given density ( ρ = 10 . 36 %) at different instant oftime, for few values of E ad . In Fig. A-10 (c), we plot the fraction of adhered area with time, and also the snapshotsnear the plot. This plot also explain the dynamic nature of the vesicle. Appendix H: Spreading dynamics and time-dependent shapes of vesicles Here, we show our results for the dynamics of spreading of the vesicles, by plotting the cross-sectional shapes andsnapshots with time (Fig.A-11). We study three different cases: A protein free vesicle (Fig.A-11 (a)), a vesicle withpassive proteins (Fig.A-11 (b)) and a vesicle with active proteins (Fig.A-11 (c)). For the protein-free vesicle, we chosea large adhesion energy E ad = 5 . k B T such that the vesicle could spread significantly. For the passive vesicle, wechose a small adhesion energy E ad = 0 . k B T and ρ = 13 . 82 %, in the regime where the curved proteins dominatethe spreading dynamics. For the active case, we use F = 4 k B T /l min , E ad = 0 . k B T and a comparably smallerdensity of proteins ρ = 10 . 36 %, where the active forces dominate the spreading. For each of the three cases, we plot9 ρ (%)0.511.522.53 E ad ( k B T ) FIG. A-7: Microstate of vesicle and cluster distributions for passive case with flat proteins ( c = 0). The background color isshowing the fraction of adhered area. For the snapshots, we use ρ = 3 . 45 % , . 36 % , . 27 % , . 18 % and E ad = 0 . , . , . k B T ). The other parameters are the same as Fig. 2. (i) the cross-section of adhered area, (ii) The side view of the vesicle, and (iii) The three-dimensional view of thevesicle. Appendix I: Quantifying pancake transition by measuring the fraction of largest cluster In Fig.A-12 we show our results for the fraction of largest cluster as function of ρ for active case with differentvalues of F and a small value of E ad = 0 . k B T . We note that for large F , there is a sharp jump in the value ofthe fraction at the pancake transition (dashed vertical line in Fig. A-12 (a)), however, for small F , the fraction doesnot show any big jump. In order to quantify the pancake transition for small F , we choose a threshold value of the0 F ρ (%) pancake transition 60, above which the shape looks like a pancake. Appendix J: Adhered area-volume relation As the cells spread, their volume is observed to decrease [45, 46]. We find similar dynamics in our model simulations.The steady-state relation between the vesicle volume and adhered area, is also similar to the experimental observations,although we have explored a smaller range of values (see Fig. A-13). Note that in cells there are internal organelles,such as the large nucleus, that limit the decrease in volume and which our model does not describe. In addition,osmotic pressure, and ion fluxes also contribute to the volume control in cells [72]. Appendix K: Speed of the crescent-shaped vesicle in the E ad − F plane In Fig.A-14 we show the speed of the crescent-shaped vesicle in the E ad − F plane, without scaling by E ad (as isshown in Fig.9b). We normalize the speed by the maximum value, where the maximum value of the speed is 0 . l min per MC steps. We note that, the speed is maximum for large E ad and large F region. Appendix L: Mean square displacement of the crescent-shaped vesicle In Fig.A-15 we show the mean square displacement of the COM of the crescent-shaped vesicle with time, for E ad = 2 . k B T , ρ = 3 . 45 % and various values of F . We note that for large F , there is a finite speed which increaseswith F , however, for small F , the displacement grows diffusively. Appendix M: Effect of varying the protein-protein interaction strength w Throughout the paper we kept the value of the protein-protein interaction strength ( w ) to be fixed at w = 1 k B T .In Fig.A-16(a), we show that even if we take w = 0 k B T , the qualitative nature of the shapes of vesicles do notchange. We still obtain the two-arc shapes or the crescent-like shapes. However, increasing w to a value such that w (cid:29) F · l min changes qualitatively the shapes of the vesicle, as shown for w = 10 k B T in Fig.A-16)b. The bud-likeprotein clusters are now solid-like, and do not easily deform or break-up, and do not merge over time.1 ρ (%)0.20.40.60.811.21.41.61.82 E ad ( k B T ) FIG. A-9: Microstates and cluster distribution of vesicle for active case with balanced total vertical force. The external forceis applied to all the vesicle nodes, such that it balances the net vertical force when is acts in the upward direction ( i.e., whenit acts against the adhesion process). We show the snapshots and cluster distribution of the vesicle with E ad = 0 . , . , . k B T ) and ρ =3.45 %, 10.36 %, 17.27 % and 24.18 %. The green dashed line denotes the transition to a pancake-likeshape. The red dashed vertical line denotes the density for the pancake transition for a free vesicle (without adhesion, [36]).Compared to the calculation without applying a balancing external force (Fig.4), we do not observe any qualitative changeexcept for the fact that in this case, we could explore much smaller E ad values. Other parameters are the same as in Fig. 4. Appendix N: Difference in the adhered area of a crescent-shaped vesicle and a two-arc shaped vesicle acrossthe crescent to two-arc transition line (Fig.9(a)) The analytical calculation of the transition line (Fig. 9(a)) separating a crescent shape and two-arc shape (Eq.3)depends on the value of ∆ A , which is difference in adhered area of the crescent-shaped vesicle and the two-arc shapedvesicle across the transition line. Here, we plot the value of ∆ A across this transition line as a function of E ad asextracted from the simulations (Fig. A-17(a)). The value of ∆ A does not show any monotonic variation with E ad , andis approximately constant along the transition line (as was assumed in Eq.3). The average value of ∆ A is (cid:39) . A cyl . We also measure this2 E ad ( k B T ) ρ (%) E ad ( k B T ) Time (2 x 10 MC step) A ad / A Time (2.5 X 10 MC Steps) FIG. A-10: Results for active case ( F = 4 k B T /l min ) with proteins having zero spontaneous curvature. (a) Mi-crostate of vesicle and cluster distributions for different values of ρ and E ad . We show the snapshots for ρ =3 . 45 % , . 91 % , . 36 % , . 82 % , . 27 % and E ad = 3 , , k B T ). (b) The dynamic nature of the vesicle fora given density 10 . 36 % and E ad = 3 , , , k B T ). (c) Variation of adhered fraction with time along with snapshots.We use here ρ = 10 . 36 % and E ad = 0 . k B T . The other parameters are the same as in Fig.4. area, and note that this value is also roughly a constant along the transition line (Fig. A-17(b)). The average valueof this area turns out to be A cyl (cid:39) . 52. We assume the number of proteins at each end of the cell to be N = 25,half of the total number of proteins. Since, the unit of E ad is different in simulations and analytical calculation, weproperly scale it to make it consistent for both the cases. Using all these values, we obtain the prefactor of F in Eq.3as (cid:39) . 96 This value if very close to the value of 2 . 04 obtained by fitting the simulation points along the transitionline (Fig.9a) with the equation. Appendix O: Motile vesicle growing against an immobile barrier In Fig.A-18 we allow a motile crescent shape to hit a rigid non-movable barrier, placed perpendicular to the directionof motion of the crescent. We take a non-adhesive as well as an adhesive barrier. In both the cases, the crescent shapefinally breaks into two-arc shape. [1] Benjamin Geiger, Joachim P Spatz, and Alexander D Bershadsky. Environmental sensing through focal adhesions. 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(a) (i)(ii)(iii) t = 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20-20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 t = 50 t = 100 t = 200 t = 800 t = 3000 (b) (i)(ii)(iii) -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20-20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 t = 25 t = 50 t = 125 t = 250 t = 1250 t = 2500 (c) (i)(ii)(iii) t = 20 t = 60 t = 100 t = 400 t = 625 t = 2375 -20020 -20 0 20 -20020 -20 0 20 -20020 -20 0 20 -20020 -20 0 20 -20020 -20 0 20 -20020 -20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 -20020-20 0 20 FIG. A-11: Spreading dynamics. (a) Spreading of a protein free vesicle. Here, we choose E ad = 5 k B T . (i) The bottom viewof the adhered area with time, showing that the vesicle adhere quickly. (ii) The side view of the vesicle, showing how thecross-sectional area decreases with the adhesion of vesicles. (iii) Three-dimensional view of the vesicle with time. (b) Spreadingof passive vesicle with ρ = 13 . 82 % and E ad = 0 . k B T . Panels (i), (ii) and (iii) are the same as above. The adhered area growsuniformly from the beginning. (c) Spreading of an active vesicle, with ρ = 10 . 36 %, E ad = 0 . k B T and F = 4 k B T /l min . Inthe beginning, the growth of adhered area is slow, then accelerates and finally the vesicle takes the shape of a pancake. Theunit of time (t) is taken as 2 × MC steps. F r a c t i on o f l a r ge s t c l u s t e r ρ (%) 00.20.40.60.81 8 10 12 14 16 18 20 F r a c t i on o f l a r ge s t c l u s t e r ρ (%) F r a c t i on o f l a r ge s t c l u s t e r ρ (%) (a) (b) (c) FIG. A-12: Fraction of largest cluster with ρ for various values of F and E ad = 0 . k B T . (a) F = 2 . k B T /l min , (b) F = 1 . k B T /l min , and (c) F = 0 . k B T /l min . The vertical dashed line of (a) is showing the pancake transition, which is asharp transition. For (b) or (c), the pancake transition is not very sharp, and the shape is assumed to be a pancake when thesize of largest cluster is at least 60 % of the total number of proteins. A r ea V o l u m e Time (2X10 MC Steps) V o l u m e Area (a) (b) FIG. A-13: Area-Volume relation. (a) The variation of area and volume with time for an active vesicle. The red triangles arefor the volume and the blue circles are for the area. We use here, ρ = 10 . 36 %, E ad = 0 . k B T and F = 4 k B T /l min . (b) Thearea-volume relation in the steady state for active case with E ad = 0 . k B T and two different values of ρ . The hollow magentacircles are for ρ = 4 . 84 % and the green solid circles are for ρ = 10 . 36 %. We also show the snapshot of vesicles for both the ρ ,for their highest area. Here, we vary F in order to access different area and volume in the steady state. We note that for larger ρ (which corresponds to the pancake shape), the system can access larger area and smaller volume in comparison to lower ρ (which corresponds to the two-arc elongated shapes). ad F FIG. A-14: Speed of the crescent-shaped vesicle in the E ad − F plane. Here, we calculate the speed as the displacement of COMper MC step and then normalize all the values by the maximum speed, where the maximum value of the speed is 0 . l min per MC step. For large E ad but small F , the vesicle speed is zero but it might show diffusive behaviour as well. The speed ismaximum for large E ad and large F region. -3 -1 M S D time (10 MC steps)tt -2 M S D time (10 MC steps)tt -1 M S D time (10 MC steps)tt -4 -2 M S D time (10 MC steps)tt (a) (b)(c) (d) FIG. A-15: Mean square displacement (MSD) of the COM of vesicle. (a) F = 0 . k B T /l min , (b) F = 0 . k B T /l min , (c) F = 2 . k B T /l min , and (d) F = 4 . k B T /l min . We note that for small F , MSD shows diffusive behaviour while for large F ,the vesicle has a finite speed. The y-axis is in the unit of l min , and the typical size of a vesicle (radius of a pancake shape, say)is ∼ l min . Here, we use E ad = 2 . k B T , and ρ = 3 . 45 %. (a) w=0 (b) w=10 (i) (ii)(i) (ii) FIG. A-16: Snapshots of vesicle with various values of protein-protein interaction strength w . (a) For w = 0 k B T , we obtain thetwo-arc and crescent-like shapes as found for the case of w = 1 k B T (Fig.9). For (i), we use E ad = 2 . k B T , F = 4 . k B T /l min and ρ = 3 . 45 %, and for (ii), we use E ad = 3 . k B T , F = 2 . k B T /l min and ρ = 3 . 45 %. (b) For very large w = 10 k B T ,the vesicle forms small isolated clusters, that do not break-up or merge after long time. Here, for (i), we use E ad = 2 . k B T , F = 3 . k B T /l min and ρ = 3 . 45 %, and for (ii), we use E ad = 2 . k B T , F = 4 . k B T /l min and ρ = 3 . 45 %. ∆ A E ad A cy l E ad FIG. A-17: (a) The difference in the adhered area of a crescent-shaped vesicle and a two-arc shaped vesicle across the crescentto two-arc transition line, ∆ A (in units of l min ), as a function of E ad . We note that the value of ∆ A does not show anymonotonic variation with E ad , and is approximately constant along the transition line (as was assumed in Eq.3). (b) The areaof cylindrical part in a two-arc shape A cyl (in units of l min ), as a function of E ad . This value also seems to be roughly aconstant along the transition line. Time Time Time (a)(b)(c) FIG. A-18: Motile vesicle growing against an immobile barrier. (a) We plot the snapshots of vesicle with time for a non-adhesivebarrier. (b) The snapshots of the vesicle with time for an adhesive barrier. (c) The side view of the vesicle for the adhesivecase. We note that in both the adhesive and non-adhesive cases, the crescent-shaped vesicle breaks into two-arc shape. Here,we use ρ = 3 .