From Molecular Insight to Mesoscale Membrane Remodeling: Curvature Generation by Arginine-Rich Cell-Penetrating Peptides

From Molecular Insight to Mesoscale Membrane Remodeling: Curvature Generation by Arginine-Rich Cell-Penetrating Peptides | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results From Molecular Insight to Mesoscale Membrane Remodeling: Curvature Generation by Arginine-Rich Cell-Penetrating Peptides View ORCID Profile Katarína L. Baxová , View ORCID Profile Jovi Koikkara , View ORCID Profile Christoph Allolio doi: https://doi.org/10.1101/2025.04.14.648709 Katarína L. Baxová † Institute of Organic Chemistry and Biochemistry, Czech Academy of Sciences , Flemingovo nám. 542/2, 160 00 Prague, Czech Republic Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Katarína L. Baxová Jovi Koikkara ‡ Charles University, Faculty of Mathematics and Physics, Mathematical Institute , Sokolovská 83, 186 75 Prague 8, Czech Republic Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jovi Koikkara Christoph Allolio ‡ Charles University, Faculty of Mathematics and Physics, Mathematical Institute , Sokolovská 83, 186 75 Prague 8, Czech Republic Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Christoph Allolio For correspondence: allolio{at}karlin.mff.cuni.cz Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract The enhanced cell penetration ability of arginine-rich peptides, such as nonaarginine (R 9 ), compared to their lysine-rich counterparts remains incompletely understood. Atomistic simulations reveal that R 9 binds significantly more strongly (≈ 20 kJ / mol) and penetrates deeper into anionic lipid headgroup region than its lysine equivalent. This enhanced interaction translates into a stronger induction of negative membrane curvature by R 9 . We combine these data to construct a model of peptide binding and curvature induction in fusogenic lipid mixtures and employ a multiscale simulation approach, combining atomistic molecular dynamics (MD) with mesoscopic Monte Carlo (MC) simulations, to dissect the molecular basis and morphological consequences of arginine specificity. The results show that stable membrane invaginations, as observed in studies of cell penetration, require excess membrane and are stable only for R 9 (outside of the domain of stability of the Helfrich stomatocyte). By analyzing lipid and protein sorting coupled to the membrane structure, we explain the interplay of Gaussian and mean curvature in providing a mechanistic basis for the initial membrane deformation events potentially involved in ‘Arginine Magic’ cell entry pathways. 1 Introduction Cell-penetrating peptides (CPPs) are an important means of drug delivery: They are able to transport large, polar cargo across the plasma membrane. 1 Arginine Magic denotes the ability of arginine-rich CPPs to enter cells. 2 The positively-charged peptide non-aarginine (R 9 ) is an efficient CPP, in contrast to nonalysine (K 9 ), which has the same charge. This (molecular) ion-specific effect has been studied on artificial phospholipid mixtures rich in phosphatidylethanolamine (PE) and phosphatidylserine (PS) as a model systems for cell penetration. 3 – 5 It was found that these model membrane systems selectively undergo fusion and form multilamellar structures 3 as well as cubic phases upon R 9 addition. 4 , 5 Analogous multilamellar structures have also been documented inside cells on multiple occasions. 3 , 6 , 7 On this occasion, it should be pointed out, that the actual cellular entry mechanism was reported to be dependent on heparin binding receptors, 8 that membrane proteins are necessary for passive entry even into giant plasma membrane vesicles (GPMVs) 9 and that the artificial lipid compositions susceptible to leakage do not correspond to the composition of the plasma membrane. 3 Nevertheless, the structural analogies of the PE and PS model system have held up well - even the experimentally known selectivity for arginine over lysine transfers to this model. Hence, model lipid bilayers have been extensively studied via molecular dynamics simulations. 10 – 13 R 9 has often been reported to induce membrane curvature, 3 , 4 , 14 – 16 either positive, negative or negative Gaussian in nature. In recent years, there has been a forming consensus that membrane curvature induction is a key component of the cellular entry mechanism. 16 Despite the recognized importance of curvature, the precise mechanism by which these peptides induce membrane curvature remains poorly understood, making further investigation crucial. Curvature generation by proteins has been a subject of intensive research already for several decades. 17 – 24 Some of our work focused on quantitatively extracting membrane properties in the presence of proteins and adsorbates. 25 – 27 In particular, we made predictions based on examining negative curvature generation by cationic adsorbates on negatively charged lipids. 25 , 27 These predictions also extend to pore formation. 28 In contrast to in-vivo processes, such as bacterial division, 29 the relative simplicity of the model membrane system allows for the possibility of a thoroughly, albeit sparsely, parametrized multiscale model. Most recently, Pei and coworkers have postulated a revised mechanism of entry for cell-penetrating peptides, which departs from inward budding, presumably stabilized by negative membrane curvature. 7 While previous studies have qualitatively observed R 9 -induced curvature or modeled peptide-lipid interactions, 30 a quantitative, multiscale understanding linking specific molecular binding events (R 9 vs K 9 ) to the resulting changes in membrane material properties (curvature, bending rigidity) and subsequent large-scale morphological transitions (such as budding or invagination) is lacking. Here, we revisit the molecular basis for Argine Magic in the domain of lipid binding, we quantify and explain the induction of curvature on pure phospholipid bilayers and incorporate the resulting data into a simple model for arginine binding on mixed model bilayers using large-scale molecular dynamics (MD) simulations and then incorporate it into our recently developed general Monte Carlo (MC) toolkit for membrane deformations, which couples it to the mesoscopic deformations. 31 This enables us to investigate the conditions and physical determinants of inward budding as a proposed initial step of cell penetration. 2 Methods 2.1 Molecular Dynamics Simulations A membrane bilayer containing 1024 lipids per leaflet was built by CHARMM-GUI 32 containing DOPE:DOPS:DOPC lipids (60:20:20). 64 molecules of nonaarginine or nonalysine were used along with 50 TIP3P water molecules per lipid and 150mM KCl plus additional ions to counteract peptide charges. Both the traditional CHARMM36/LJ-PME and charge-scaled prosECCo75 33 force field systems were set to run in Gromacs 34 for 1 µ s while the last 200 ns were used for analysis. Nosé-Hoover 35 , 36 temperature coupling with a 1 ps time constant for the groups of protein, membrane and water with ions alongside with semi-isotropic Parrinello-Rahman 37 pressure coupling (1 bar) ensured the isothermal-isobaric ensemble. The long-range interactions were treated by Particle-Mesh Ewald electrostatics 38 with a cutoff of 1.2 nm. Hydrogen atom bonds were constrained by LINCS. 39 The timestep for the leap-frog integration algorithm was set to 2 fs. 2.1.1 Material Properties For the computation of elastic properties, simulations of pre-equilibrated bilayers were continued for 200 ns using Gromacs 2020.3. 34 We simulated at 300 K temperature, using the same ensemble, algorithms and cutoffs as in the previous section. Long-range electrostatics were treated using the particle-mesh Ewald method. 38 Since the pressure decomposition does not support the SETTLE algorithm, 40 the triangular geometry of water was constrained using LINCS with order five. 39 No dispersion corrections or potential switching were applied. In addition, we removed the CMAP terms, as we did not yet implement a force decomposition for these, and R 9 and K 9 are unstructured. The timestep was 2 fs. 20.000 MD snapshots with velocities were taken from each simulation. The simulated systems were pure membranes, containing 14000 TIP3P water molecules and 128 lipids, in addition to 150 mM ions (and neutralizing counterions in the case of DOPS). When adding peptides, we added 14 R 9 or K 9 to DOPS and 6 R 9 or K 9 to DOPC and DOPE, respectively, for the simulation of a fully covered membrane. The simulations were run in the presence of 150 mM ions and counterions. For the stress tensor 27 , 41 and ReSiS 42 extraction simulations, we used the unscaled CHARMM36m. Further simulation details for stress-tensor computations and details for the free energy profiles are given in the Supporting Information. 2.2 Mesoscopic Monte Carlo Simulations Monte Carlo (MC) simulations to sample the equilibrium shapes of vesicles (including thermal fluctuations using the Metropolis algorithm to achieve a canonical distribution) were performed using OrganL, 31 our Dynamically Triangulated Surface (DTS) code. The model implementation for this paper will be uploaded here. The osmolyte concentration c 0 was set to 300 mM (300 mOsm) in agreement with standard PBS buffers, and the area compressibility modulus K A was set to 200 pN / nm. 43 , 44 We initialized the system with a uniform lipid composition of DOPE:DOPC:DOPS 60:20:20 and a protein coverage of ϕ p = 0.5, with lipid and protein binding parameters detailed in Table S1 of the Supporting Information (SI). At this peptide coverage (50%), all PS lipids are neutralized, along with a fraction of PE lipids, a loading chosen to align with molecular dynamics (MD) potential of mean force (PMF) calculations. The bound peptide-to-lipid (P/L) ratio is 1 : 46. While the total protein load per vesicle was not directly measured, we estimate it by combining the rapid saturation of R 9 binding on supported bilayers, 11 our computed high adsorption energies, and the empirically low R 9 -R 9 repulsion. This simulation setup was designed to directly compare the stability of competing, potentially metastable morphologies, which we categorize broadly as “spheroid” and “stomatocyte”. Given that Helfrich energy minima are often metastable, we ensure a robust identification of the global minimum by independently evolving both stomatocyte-like and prolate spheroid-like geometries and systematically comparing their energies. See section 2.5 of SI for more details. 2.2.1 Analysis To quantify global parameters in MC simulations, a thermal averaging approach was used. Specifically, the final 500 snapshots of simulations, spanning the last 500k steps, were selected for analysis. To account for the mesh’s translational and rotational degrees of freedom, individual meshes were aligned using the Iterative Closest Point 45 (ICP) point cloud registration algorithm. Local properties were computed as face-wise averages and mapped onto the corresponding faces of the ICP-generated mesh. Additionally, a bin averaging procedure was employed to analyse local parameters to preserve spatial correlations for curvature plots. See section 2.6 of the SI for more details. 3 Results and Discussion 3.1 Peptide Binding to Pure and Mixed Lipid Bilayers 3.1.1 Pure Membranes We computed potentials of mean force (PMF) of peptide binding to single-component lipid bilayers. The PMFs ( Fig. 1 ) consistently show stronger binding of R 9 over K 9 to lipid membranes, in particular to the pure DOPS bilayer, where the difference is > 20 kJ / mol (see Table 1 for the binding energy values and errors for R 9 , while the binding energy of K 9 on DOPS bilayer is −51.4 ± 2.3 kJ / mol). In particular, there is no significant binding of K 9 to the neutral PC and PE bilayers, however, the repulsion is lower for DOPE. There also appears to be a slightly higher binding affinity to DOPE for R 9 . The difference between the charged DOPS and the other lipids is mainly due to the electrostatics of binding. We used the scaled-charge Prosecco forcefield for the computation of binding free energies. 33 Charge scaling reflects the implicit electronic contribution to the susceptibility of the medium. When comparing with similar previously published work, 30 we estimate that our binding free energies are slightly lower than those expected when using an unscaled force field. Interestingly, the binding energy difference between K 9 and R 9 on PS is larger than for the other bilayers, hinting at some specific preferential interaction of R 9 with the PS headgroup. View this table: View inline View popup Download powerpoint Table 1: MD data for protein-lipid bilayer interactions. Γ and g ext are the excess number of lipids per peptide and the extremum of the first RDF peak in Fig 4 , respectively. Errors inΓ and g ext are the absolute differences between the mean (averaged over “Prosecco” and “Charmm”) and individual model values. Download figure Open in new tab Figure 1: Top panel: Free energy profile of a single R 9 binding to a lipid patch composed of DOPE (blue), DOPS (red), DOPC (green). Bottom panel: equivalent energy profiles for membranes interacting with K 9 . These binding free energies show enhanced binding of R 9 over K 9 to all simulated lipid membranes. The long-range electrostatics do not differ (due to the identical sidechain charge). This implies that if the binding energy is part of the “magic”, it must lie in the stronger binding of arginine sidechains to membranes. 3.1.2 Mixed Membranes We also report binding energies of the peptides to mixed membranes in Table 1 . As expected, binding to the mixtures is stronger than to pure neutral lipids but weaker than to the charged pure DOPS. To examine the structural factors of this binding in a general setting, we use results from our large-scale mixed-membrane simulation. The membrane contains all examined lipids at the ratios DOPE/DOPS/DOPC 0.6:0.2:0.2. Analyzing this simulation enables us to disentangle the molecular picture at the interface. In Fig. 2 , we compare the density profiles of different species at the membrane interface in the presence of the peptides. The figure also compares scaled Prosecco and unscaled CHARMM36 forcefields. Download figure Open in new tab Figure 2: Density profiles from the large-scale simulations of R 9 (top) and K 9 (bottom) centered on the membrane midplane, showing both CHARMM (full lines) and Prosecco (dashed lines) results. Density profiles of all membrane atoms (black), water (dark blue), lipid tail C-atoms (green), phosphate P atoms (light blue), proteins (red) and headgroups (orange). Density profiles of water and lipids are scaled by 0.1 for clarity. The striking difference between R 9 and K 9 lies in the far deeper binding of the arginine sidechain inside the membrane (yellow, upper part) compared to the lysine sidechains (yellow, lower part). The arginine sidechain density is collocated with the headgroup/phosphate position, whereas lysine is mainly distributed above the headgroup area. The deeper penetration of the arginine sidechain brings it closer to the locus of negative charge. This enhances the electrostatic interaction energy as well as the mutual screening of the charges. The picture is similar to the hydrophobic insertion mechanism by Kozlov et al., 22 except that in this case, the interaction is ionic and not hydrophobic and the steric component is minor. Yet, it is widely known that arginine is more hydrophobic than lysine, and we believe this is part of the explanation for why arginine is able to enter membranes more deeply. 3.1.3 Arginine-Arginine Interaction A crucial question regarding R 9 behavior is whether R 9 peptides interact strongly with each other on the membrane surface. Past studies showed that guanidinium moieties of the arginine sidechain do not repel in aqueous solution and even interact weakly. 10 In addition, aggregation of R 9 on membranes has been observed in simulation 12 and recently also in experiment. 46 In contrast to this, the binding of R 9 to membranes charged at 20% was previously found not to be cooperative. 11 To test this again, in the context of our lipid mixture, we computed two-dimensional radial distribution functions (RDF)s of a large number of R 9 molecules on a flat lipid patch (see Methods). The results are shown in Fig. 3 . These data are not consistent with a strong binding between poly-R 9 molecules, as the RDF does not exhibit major peaks but fluctuates around unity away from a depletion/exclusion zone. It seems reasonable to approximate this behavior by a simple excluded volume of each R 9 , in agreement with the data by Cremer. 11 In contrast to R 9 , K 9 molecules were found by Cremer to repel on the membrane significantly. Recent experiments, 46 suggesting aggregation by R 9 on membranes, do not furnish evidence of more than a stochastic collocation. Therefore, we suggest treating R 9 chains as noninteracting on (negatively charged) lipid membranes. Download figure Open in new tab Figure 3: Arginine aggregation: Two-dimensional RDF of R 9 -R 9 in the upper (solid lines) and lower (dashed lines) leaflet of the large membrane simulation. 3.1.4 Arginine-Lipid Demixing Having established the weak mutual-interaction of R 9 peptides, we now turn to its interaction with lipids and the crucial phenomenon of lipid demixing. We already found that R 9 strongly interacts with lipids, depending on the lipid headgroup type, in particular its charge. The long-term large-scale MD simulations of the membrane with peptides allow us to compute a realistic two-dimensional RDFs of the lipid molecular centers of mass vs R 9 at the membrane surface in Fig. 4 . The strong binding of R 9 to negatively charged PS lipids is enough to generate a significant amount of demixing This is evidenced by a high peak of R 9 under the protein accompanied with depletion of the other lipids. This depletion is larger for DOPC than for DOPE, as observed in both charge-scaled and unscaled simulations. It suggests a strong R 9 preference towards PS at the expense of PC local lipid density, while PE remains almost unaffected. The demixing can be quantified both in coordination numbers and in excess quantities. We report estimates on an excess number of lipids per peptide (Kirkwood-Buff integrals) and the peak of RDF, together with the binding energies in Table 1 . Download figure Open in new tab Figure 4: 2D-RDFs of lipids around R 9 : R 9 -DOPE RDF in blue and cyan, from CHARMM36 force field and (scaled) Prosecco simulations, respectively; R 9 -DOPS RDF in red and orange and R9-DOPC RDF in blue and cyan. The high peptide concentration in these big leaflet simulations was chosen to improve sampling, as lipids will not have to diffuse far to bind to a peptide. It, however, also means that of the 32 lipids for each protein, a significant amount will be in direct contact. This naturally limits the amount of lipid demixing, so the amount of lipid selectivity is probably not as high as in lower concentrations. Our explicit atomistic simulation results can be compared with an earlier study by Harries et al. using a Poisson-Boltzmann/ideal mixing-based model. 47 They found only weak demixing of PS in proximity of a single adsorbed K 13 molecule. The highest increase in PS was a factor of 1.5, which is not that different from our observed RDF peaks. Note that the PB model ignores the interfacial “burying” of R 9 to the membrane headgroups. 3.2 Peptide Binding Effect on Material Properties We simulated a representative set of bilayers using the unscaled CHARMM model and computed the bending moduli κ and a bilayer tilt moduli with the ReSIS method, which is based on local tilt fluctuations. 42 , 48 We previously computed these values for DOPE, DOPC and DOPS lipid membranes, but recomputed the value for DOPS to show reproducibility of previously published simulations. 27 To examine the influence on membrane bilayer properties, we put the membrane in contact with high (charge neutralizing) loads of R 9 , which we estimate to be close to saturation. 3.2.1 Material Properties To understand how R 9 and K 9 modify the material properties of lipid bilayers, we performed simulations on a representation set of bilayers. We used 6 R 9 for uncharged lipids and 14 peptides for charged membranes to achieve approximate charge neutrality. The results, including equilibrium areas per lipid A 0 , are in Table 2 . Since K 9 does not bind to the uncharged lipids, as per our PMFs, we did not simulate the corresponding membranes. The influence of the peptides on elastic properties is not large for either peptide. However, it appears that K 9 stiffens DOPS membranes, while R 9 might slightly soften them. This effect is also visible in the area per lipid and might be attributed to a surface tension contribution of K 9 , which is offset by increased chain-packing. In contrast, the entry of R 9 into the headgroup region might counteract some of the compressive effects of charge screening. View this table: View inline View popup Download powerpoint Table 2: Simulation results for bilayer mechanical properties. Bending moduli κ and spontaneous curvatures J s are monolayer quantities. Areas per lipid A 0 and tilt moduli are bilayer properties. Superscript a from 25 unscaled simulations at 303.15K. Errors are standard errors. 3.2.2 Stress Profiles To gain deeper mechanistic insight into the curvature-inducing effects of R 9 and K 9 , we computed local stress profiles for single-component lipid membranes. The largest impact on curvature is found for the pure DOPS membrane. Figure 5 illustrates the different effects of R 9 and K 9 on charged lipids. We interpret the peak around 2 nm as associated with headgroup repulsion. It is lower for R 9 than for K 9 . The whole profile is damped for R 9 , potentially also due to packing effects, but the decay of the positive electrostatic pressure in particular is faster for R 9 . The positive pressure is due to the electrostatic repulsion of the adsorbed proteins. We attribute the lower repulsion both to the electrostatic screening by headgroup binding and, to some extent, to the compactness of the peptides due to the attraction between sidechains. Download figure Open in new tab Figure 5: Comparison of the symmetrized lateral stress profiles of DOPS in the presence of R 9 (red) and K 9 (blue) and in the absence of peptides (black). On the x axis, z denotes the distance from the membrane center projected on the surface normal. 3.2.3 Curvature Generation By combining the computed stress profiles with the elastic properties, we can now quantitatively assess the curvature-generating potential of R 9 and K 9 using the established relationship 49 , 50 between the first bending moment and the product of spontaneous curvature, J s and κ : Here, integration is carried out along the membrane normal ê z . A large negative curvature generating effect is observed for R 9 on DOPS (see J s in Table 2 ). This result is similar to what was observed in the case of Ca 2+ ions 27 on the same lipid. In our previous study, Ca 2+ was shown to stabilize DOPS membrane fusion stalks as indicative of fusion. Note also that Ca 2+ experimental behavior was similar to that of CPPs in this system. 3 The effect of R 9 is significantly stronger than that at the same loading of K 9 . In particular, the increased κ on K 9 reduces the effect on J s . It is remarkable that R 9 can reduce the spontaneous curvature of un-charged DOPE. Our previous research found pure DOPE/DOPS to be very susceptible to fusion by R 9 (over K 9 ), and this effect was reduced by adding DOPC. 3 The error bar values in the curvature calculations ( Table 2 ) are standard errors, including error propagation (see Supporting Information). The J s values given here were calculated automatically by integration over the whole box, which is sometimes conducive to errors as fluctuations in the pressure far from the membrane center can drastically influence the first part of the distribution. 3.3 Effective Mesoscopic Model In order to understand the mesoscopic structural effects of the peptide and lipid specific interactions, we propose an effective continuum model. The basis of our model is the Helfrich-Canham-Evans theory. 51 – 53 This thin-shell theoretical model is a well-established way to describe membrane curvature elasticity and structure. Instead of constraints for area and volume, we introduce laterally compressible membranes and an osmotic pressure, 54 as well as regular solution type mixing terms. The total energy of the system is then given by where Here, the parameters for (Gaussian) bending rigidities κ and spontaneous curvature J S vary over the mesh but are constant on each face. Integration is performed on the faces. H denotes the mean curvature. The bilayer binding energy of the peptide F b and the mixing energy F mix, b are calculated per face and added. The volume work E pV is calculated using the vesicle volume V from work against the osmotic pressure difference between the interior and exterior of the vesicle (assuming balanced osmolarity at V 0 ) at a solution osmolarity c 0 . The area compression energy E stretch is calculated from the total area A using the bulk modulus K A and the stress-free reference area A 0 . n p , ϕ i and µ i are the local (mesh element) numbers of peptides, fractional peptide coverage and binding energy, respectively. See the Supporting Information for exact parameter values and mixing rules. We use a Metropolis Monte Carlo approach to sample the Boltzmann distribution of lipid film structures. For details on the numerical evaluation, lipid mixing dynamics and mesh evolution see. 31 3.3.1 Model Assumptions We assume the membrane to be a curvature elastic thin film, obeying the Helfrich theory; ideal lipid mixing and osmotic pressure expressions apply. The area per lipid is assumed to be unaffected by the lipid mixing. Proteins are shapeless and interact only via size exclusion and by covering lipids. Proteins modify properties only of those lipids that they “cover” (discussed below and in Section 3.1.3 ). Furthermore, long-range electrostatic as well as tilt deformation are neglected. We also neglect inertia so that all motion is due to thermal fluctuations. In our discretization, a protein will always cover exactly one triangle, there are no fractional occupations. More than one protein on each face is not allowed, as we do not know how to interpolate between populations. In contrast, in the case of lipid mixing, we use well-established mixing terms and fractional occupations. 25 , 55 These are described in detail in Section 1 of SI. These terms neglect potential differential stress and membrane asymmetry effects. 56 , 57 3.3.2 Model Parameters Our MD simulations provide us with parameters for single-component bilayers, given in Table 2 . Values for A 0 , J s and κ for pure lipids and those in contact with the peptides from this table. We use an estimate from 27 for the Gaussian bending modulus of the bilayer (see SI), as monolayer values are difficult to obtain. 58 For the binding free energies µ i of the lipids to the peptides, we utilize values within the error bars of our single-component PMF data. All model parameters are summarized in Table S1 of the SI. By using the extracted binding free energies and material properties from the MD simulations, we transfer the essential information into an effective model to examine the morphological consequences of the local changes in curvature-elastic parameters. 3.3.3 Model Validation Consider a flat membrane divided into two virtual membrane systems: one functionalized with a peptide and the other unmodified. The free energy concerning lipid and protein is, Since the membrane is flat, the bending terms will not affect the mixing. We also approximate ∀ i, j : A i = A j ⇒ ϕ i = x i , so that we can express the energy in mole fractions x i . We divide a lipid population into a free fraction M f and a lipid population M p bound to the protein with a binding energy µ p , i . The model Hamiltonian can, therefore, be expressed as: We also apply the following constraints: where M := M f + M p is the total number of lipids and x i is the mole fraction of i th lipid, both of which are fixed. Applying either condition in the Eq. (6), in conjunction with Eq. (5), inherently satisfies the complementary constraint. The resulting energy is minimized against Eq. 4 for a given M p . Since the M p value depends on the unknown peptide area, M p is an adjustable parameter. We optimize it by minimizing a ℒ 2 norm between the MD data ( Table 1 ) and the estimated { x i , p } after converting the latter to excess quantities. The technical details are in SI. For the large simulation with 1024 lipids per leaflet and 34 R 9 bound to the leaflet, the optimal area of R 9 is 14.786 nm 2 (23.126 lipids per R 9 and weighted average APL from Table 2 of SI) resulting in the excess number of lipids per peptide,Γ, as: Here, the excess lipid number per peptide,Γ, is defined as the difference between the number of bound lipids after optimization and the initial number, which is uniformly distributed according to Eq. 7 . We find that Eq. 8 is in good agreement with Table 1 . To give an idea of the molecular dimension, this area corresponds to a radius of ca. 2.2 nm, not too far from the RDF data. Next, we consider a small system of 64 lipids and only one peptide. In this case, the cover-age ratio is only 32.8%. For such a system, we predict and a binding energy of 56.69 kJ / mol, in reasonable agreement with the MD data Table 1 . Please note that in this system, demixing is predicted to be stronger than in the high peptide concentration large-scale simulation. This leads to strong binding. In this system, Monolayer curvature is predicted to be J S = −0.25 (−0.17) nm − 1 for the bound (unbound) zone, while monolayer κ = 14.16 (14.28) kT respectively. For K 9 , we use the respective µ i from Table 1 and the identical area as for R 9 . Then the K 9 binding energy prediction for the small patch is −27.7 kJ / mol, which is in good agreement with Table 1 . Note that as K 9 has a lower binding energy and has experimentally anti-cooperative behavior, 11 the model for K 9 should be considered more of an upper bound of the effect, at least at identical peptide load. Our simplified model is able to capture key aspects of peptide-lipid interactions observed in atomistic MD simulations: It reproduces preferential binding, binding free energies on lipid mixtures, and the demixing under the peptides. We use the area prediction of R 9 as input for our simulation. By its constant lipid coverage assumption, it does not describe well the fact that the protein coverage can be higher for strongly negatively charged membranes. This is to be expected for a model that does not explicitly treat electrostatics. 3.4 Mesoscopic Consequences of R 9 Specificity 3.4.1 Stability and Morphology The equilibrium shapes of vesicles governed by the Helfrich energy with uniform J s have been extensively studied. 31 , 56 , 59 Energetically stable shapes include stomatocytes, discocytes as well as prolate, and oblate spheroids. Helfrich minima are usually plotted as a function of the reduced volume: In the standard formulation, the reduced volume is computed from the total membrane area A and the enclosed volume V . It reflects the scale invariance of the Helfrich theory in the absence of spontaneous curvature. Thus, volume and area constraints are unified into one parameter. In our case, we do not have strict volume or area preservation - the values of ν 0 correspond to the osmotic pressure difference minima of the free energy ( Eq. 2 ), i.e. zero resultant osmotic pressure ( V 0 ) and the stress-free area ( A 0 ). As ν 0 is linear in the volume, it can be interpreted as a degree of filling of the vesicle. It should be emphasized that the actual values of the instantaneous mesh are different due to, e.g. the mechanical forces working against volume preservation. In the absence of spontaneous curvature, prolate configurations remain stable up to ν 0 ≈0.65, while stomatocytes are global free energy minima for ν 0 ∼0.55. In between these two values, the discocyte structure is stable. In our case, the system is not a priori scale-invariant as the individual lipids have defined intrinsic curvatures. Hence, the results are, in principle, valid only for our choice of vesicle size and osmolarity. Nevertheless, plotting as a function of ν 0 illustrates the difference to the standard Helfrich results. Here, we investigate the stabilization of competing structures in the presence of lipids and peptides, with a particular focus on the “stomatocyte” shape. This geometry is of primary interest as it represents inward budding, a process hypothesized to be the initial step of CPP entry. 7 We compare its relative energy to that of prolate/oblate spheroid-type structures in Fig. 7 . As we perform Monte Carlo simulations sampling the Boltzmann distribution, the “metastable”, i.e. higher energy state is inherently transitory: The underlying stochastic process will converge to the stationary distribution, which knows these “metastable” states only as fluctuations. Yet, these shapes persist for long enough to compute average energies, allowing for an estimate of the relative stability. To illustrate the transitory nature of unstable shapes, they are marked in grey. Download figure Open in new tab Figure 6: Representative structures with thermally averaged R 9 occupancy. (a) Transverse cross section of stomatocyte of ν 0 = 1.00, (b) prolate of ν 0 = 0.75 and (c) discocyte of ν 0 = 0.80. Download figure Open in new tab Figure 7: Model Free Energies. All energies are given relative to that of the spherical DOPE/DOPC/DOPS system at ν 0 = 1.0 without peptides. Relative stability of spheroids vs stomatocytes are given: (a) without peptides, (b) with K 9 , and (c) with R 9 . Simulation snapshots depict the corresponding reduced volume, with color maps indicating average sorting occupancy. Grey plots denote metastable states. Error bars represent the SEM with 95% CI. For example, in the control simulation without peptides ( Fig. 7a ), the stomatocyte shapes slowly transition towards prolate geometries. This behavior mirrors the predicted global minimum resulting from Helfrich energy minimization without lipids. In the absence of lipid demixing, there will be no spontaneous curvature on the mesh, as the intrinsic curvatures of monolayers of identical composition cancel. Hence, our result is in line with the established understanding within the community that ideal mixing entropy prevents lipid demixing to a large extent. Consequently, spatial curvature variations that stabilize curved membrane structures beyond the known Helfrich shapes require additional stabilizing mechanisms. 60 , 61 The energy profiles of membranes containing K 9 peptides ( Fig. 7b ) are similar to the control run in that no stable range for the stomatocyte structure is observed for ν 0 > 0.65. The presence of R 9 peptides ( Fig. 7c ) significantly enhances the stability of invaginated structures compared to (prolate) spheroids. Energetically, invaginated and prolate shapes were comparable within a reduced volume ( ν 0 ) range of 0.80 − 0.85. Below ν 0 ≈ 0.80, a clear divergence of energy profiles is observed, with stomatocyte geometries having considerably lower energy than spheroids, in contrast to what is observed in the control run or the simulation with K 9 . This is further evident in the Helfrich Energy ( Eq. 2 ) contribution in Fig. 8 , which shows a similar pattern suggesting the active role of spontaneous curvature generation. This plot also reveals that the invagination stability decreases with a reduction in excess membrane area, as osmotic pressure progressively destabilizes this geometry. A detailed energy decomposition plot is provided in Fig. S1 of the Supporting Information. Download figure Open in new tab Figure 8: Contributions of curvature elastic term (Helfrich Energies). Profiles are shown under three conditions: control (black), with K 9 (blue), and with R 9 (red). Dashed lines represent spheroid morphologies, while solid lines indicate stomatocyte morphologies. Error bars denote the SEM with 95% CI. All energies are given relative to that of the spherical DOPE/DOPC/DOPS system at ν 0 = 1.0 without peptides. Collectively, these findings demonstrate that R 9 peptides stabilize inward budding in DOPE/DOPC/DOPS vesicles, requiring a minimum excess area or hyperosmotic stress for stable invagination formation. We attribute this finding to the spontaneous curvature generation by R 9 , which must be concomitant with significant lipid demixing. 3.4.2 Curvature Sorting We investigate the underlying mechanisms by examining the spatial organization of R 9 and lipids in representative geometries. In order to quantify curvature sorting, we define the fractional excess coverage as Here, n ( i ) is the occupancy (number of lipids) on a face being the equivalent of the i th compound in the initial uniform distribution. Figure 9 displays heatmaps of the fractional excess R 9 coverage on a fixed mesh representing a typical “stomatocyte” or inward-budded vesicle with ν 0 = 0.80, calculated by first binning the instantaneous mesh data to preserve spatial correlations and averaging over the last 200 such snapshots. The fixed mesh was employed to enhance the convergence and isolate the sorting effect. However, the findings remain consistent, albeit with reduced prominence, when using an unfrozen mesh and are included in the SI Fig. 2 for completeness. Download figure Open in new tab Figure 9: Curvature Sorting. (a) Bin-averaged heatmap of excess R 9 coverage on stomatocyte at reduced volume ( ν 0 ) of 0.80. Regions A, B, and C correspond to the exterior, neck, and invagination, respectively. (b) Heatmap showing the excess coverage of R 9 as a function of DOPE and DOPS composition. The dashed line delineates the excluded region by binning out all outliers to improve data clarity. The excess coverage Γ( R 9 ) is analyzed as a function of mean curvature ( H ) and the Gaussian curvature ( K G ) in Fig. 9a . Region A, corresponding to the vesicle exterior ( H > 0, K G > 0), shows a depletion of R 9 . In contrast, Region B, representing the invagination neck ( H ∼ 0, K G < 0), and Region C, representing the interior of the bud ( H 0), exhibit a significant enrichment of R 9 . These observations underscore a strong dependence of R 9 localization on membrane curvature. This noticeable R 9 localization at the neck and invagination regions is accompanied by significant rearrangements of DOPE and DOPS lipids, as illustrated in Fig. 9b . These lipids exhibit pronounced negative curvatures upon interaction with R 9 . Such curvature-dependent sorting illustrates the mechanistic role of lipid-peptide interactions in driving the spatial organization of R 9 on complex membrane geometries: An obvious component of the stabilization of the inward bud is the induction of negative curvature by R 9 on DOPS. It is natural to expect this “couple” to sort to a location where its energy is minimized. However, in our simulations, the binding energy of peptides to lipids also mediates the induced lipid sorting. R 9 molecules have a lower mixing entropy than the lipids and cover multiple lipids by one peptide. In our model, the presence of R 9 comes without a per face internal mixing entropy. Nevertheless, the expansion of R 9 on its mesh lattice sites is associated with its own entropy, reflected by its wide distribution. This entropy emerges from generating Monte Carlo scheme, which distributes the peptides on the mesh. Yet, this “moving lipid bracket” formed by the peptide facilitates the symmetry breaking necessary for bud stabilization. This behavior, together with the curvature generation, is the mesoscopic consequence of the behavior we extracted from molecular simulations in flat geometries. Despite its enduring ability to reproduce the morphology of cell penetration, the DOPE/DOPC/DOPS model system’s effectiveness in creating these analogies remains unexplained. 4 Conclusion We investigated peptide binding, lipid demixing and curvature generation of a model system for cell penetration using atomistic molecular dynamics simulations. The chemical specificity of lipid bilayer interaction with the efficient cell-penetrating peptide R 9 over K 9 is visible both in the binding energies and in the curvature generation. R 9 binds more strongly and generates more negative curvature than K 9 for all examined lipids. In contrast to other studies, we find this effect to be unrelated to guanidinium pairing. Instead, the deeper penetration of the guanidinium sidechains into the headgroup region of the membrane is responsible for the observed differences. We systematically transferred these specific interactions into effective parameters of a continuum model. After validation of this model, we used it to elucidate the mesoscopic consequences of chemical specificity. Inward budding is believed to be the initial step of cell penetration. 7 Our mesoscopic DTS simulations show how the R 9 peptide stabilizes inward buds. These invaginations exhibit a strong tendency for R 9 localization in regions of negative mean and negative Gaussian curvature. The formation of such structures requires sufficient available membrane, corresponding to a reduced volume of approximately 0.85, thereby requiring either an excess membrane area or the application of hyperosmotic stress to induce the transition. This poses a challenge for lipid vesicle experiments (where the surface/volume ratio would have to be modified by vesicle fusion or osmotic pressure), but not for cells with rough membrane surfaces. K 9 was found to be unable to generate this type of structure, in agreement with its known inability to passively enter mammalian cells. Our findings indicate that strong binding and curvature generation are key to R 9 cell penetration. The crosslinking and membrane-aggregating effect of R 9 on membranes is the known unknown in the mechanism of CPP entry. Finally, these results do not contradict our previous investigations 3 in any way, as multilamellarity and fusion are expected to arise in the follow-on steps. Author Contributions CA designed and supervised the research. KB and CA performed MD simulations and their analysis. JK performed DTS simulations and analysis. The model was implemented by CA and JK. CA, JK and KB wrote the paper. Acknowledgement CA and JK were supported by Charles University PRIMUS grant (PRIMUS/20/SCI/015). JK wishes to thank Charles University for the UNCE Math MAC Scholarship (UNCE/24/SCI/005). KB acknowledges support from Charles University, where she is enrolled as a Ph.D. student. KB acknowledges HPCg at IOCB Prague for computational resources. Funding Charles University https://ror.org/024d6js02 PRIMUS/20/SCI/015 , UNCE/24/SCI/005 References 1. ↵ Misawa , T. Cell-Penetrating Peptides ; John Wiley & Sons, Ltd , 2023 ; Chapter 12, pp 203 – 218 . 2. ↵ Vazdar , M. ,, Heyda , J. ; Mason , P. E. ; Tesei , G. ; Allolio , C. ; Lund , M. ; Jung-wirth , P. Arginine “Magic”: Guanidinium Like-Charge Ion Pairing from Aqueous Salts to Cell Penetrating Peptides . Acc. Chem. Res . 2018 , 51 , 1455 – 1464 . OpenUrl CrossRef PubMed 3. ↵ Allolio , C. ; Magarkar , A. ; Jurkiewicz , P. ; Baxová , K. ; Javanainen , M. ; Mason , P. 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