Weak carbohydrate-carbohydrate interactions in membrane adhesion are fuzzy and generic
Batuhan Kav, Andrea Grafmüller, Emanuel Schneck, Thomas R. Weikl
WWeak carbohydrate-carbohydrate interactions in membrane adhesion arefuzzy and generic
Batuhan Kav, a Andrea Grafmüller, a Emanuel Schneck, b , c and Thomas R. Weikl a Carbohydrates such as the trisaccharide motif Le X are key constituents of cell surfaces. Despite intense research, the interactions be-tween carbohydrates of apposing cells or membranes are not well understood. In this article, we investigate carbohydrate-carbohydrateinteractions in membrane adhesion as well as in solution with extensive atomistic molecular dynamics simulations that exceed the sim-ulation times of previous studies by orders of magnitude. For Le X , we obtain association constants of soluble carbohydrates, adhesionenergies of lipid-anchored carbohydrates, and maximally sustained forces of carbohydrate complexes in membrane adhesion that are ingood agreement with experimental results in the literature. Our simulations thus appear to provide a realistic, detailed picture of Le X –Le X interactions in solution and during membrane adhesion. In this picture, the Le X –Le X interactions are fuzzy, i.e. Le X pairs interact ina large variety of short-lived, bound conformations. For the synthetic tetrasaccharide Lac 2, which is composed of two lactose units, weobserve similarly fuzzy interactions and obtain association constants of both soluble and lipid-anchored variants that are comparable tothe corresponding association constants of Le X . The fuzzy, weak carbohydrate-carbohydrate interactions quantified in our simulationsthus appear to be a generic feature of small, neutral carbohydrates such as Le X and Lac 2. Introduction
Carbohydrates are omnipresent at cell surfaces as constituentsof glycolipids and glycoproteins . During cell adhesion, thesecarbohydrates get in touch with proteins and carbohydrates onapposing cell surfaces. While specific interactions between car-bohydrates and proteins are known to play important roles incell adhesion events, the role of carbohydrate-carbohydrate in-teractions in these events is less clear . About three decadesago, homophilic carbohydrate-carbohydrate interactions of thetrisaccharide Lewis X (Le X ) have been reported to be involvedin embryonal cell compaction and aggregation , and interac-tions between long carbohydrate chains have been linked to thespecies-specific aggregation of marine sponges . In the followingdecades, carbohydrate-carbohydrate interactions in adhesion havebeen investigated in a variety of reconstituted or synthetic systemsincluding nanoparticles and surfaces functionalized with carbo-hydrates , atomic force microscopy setups , and recon-stituted vesicles or membranes containing glycolipids.While some carbohydrate-carbohydrate interactions have been re-ported to be strong , interactions of small, neutral carbo-hydrates are typically considered to be weak . However, thebinding association constants, in particular at membrane inter-faces, and the structural binding mechanisms are often not known.In this article, we present detailed results from atomistic molec-ular dynamics simulations of carbohydrate-carbohydrate interac-tions in membrane adhesion and in solution for Le X and the syn-thetic saccharide Lac 2, which is composed of two lactose units (see Fig. 1). Our simulations employ a recent carbohydrate forcefield that allows a more faithful representation of carbohydrate-carbohydrate interactions and exceed the times and system a Max Planck Institute of Colloids and Interfaces, Department of Theory and Bio-Systems,Am Mühlenberg 1, 14476 Potsdam, Germany. b Max Planck Institute of Colloids and Interfaces, Department of Biomaterials, Am Müh-lenberg 1, 14476 Potsdam, Germany. b Technische Universität Darmstadt, Physics Department, Hochschulstraçe 12, 64289Darmstadt, Germany. sizes in previous simulation studies of carbohydrate-carbohydrateinteractions in solution by orders of magnitude. Le X hasbeen investigated extensively as a model system for carbohydrate-carbohydrate interactions , and experimental dataavailable from these investigations are central to corroborate oursimulation results. In our Le X glycolipids, the Le X trisaccharide isconnected via a lactose disaccharide and a glycerol linker to lipidstails (see Fig. 1). In our Lac 2 glycolipids, the Le X trisaccharide isreplaced by another lactose disaccharide, which allows to comparethe carbohydrate-carbohydrate interactions of Le X to those of thecommon saccharide lactose. From simulations of soluble pairs ofLe X and Lac 2, we obtain association constants K a of the order of M − , which agrees with a K a value of Le X derived from weak affin-ity chromatography experiments . From simulations of pairs ofLe X and Lac 2 glycolipids at apposing membrane surfaces, we ob-tain comparable association constants K trans for the Le X and Lac 2glycolipids that strongly decrease with increasing membrane sep-aration. For the membrane separation and thermal roughness ofmembrane multilayers with mol% Le X glycolipids measured inneutron scattering experiments , we determine an adhesion en-ergy per area of the order of µ J/m from our K trans values, inagreement with the adhesion energy per area reported for vesiclesthat contain mol% of Le X glycolipids . The average force onbound Le X glycolipid complexes determined in our simulations in-creases with increasing membrane separation up to a maximumvalue of about pN, which agrees with the Le X –Le X unbind-ing force obtained from atomic force microscopy experiments .The agreement with experimental results indicates that our simu-lations provide a realistic, detailed picture of weak carbohydrate-carbohydrate interactions in solution as well as in membrane ad-hesion. A striking feature is that the carbohydrate-carbohydrateinteractions are fuzzy, i.e. both soluble and lipid-anchored variantsof Le X and Lac 2 interact in our simulations via a large variety ofdiverse, bound conformations.1 a r X i v : . [ q - b i o . B M ] A ug HO OOHOHOH OHO OOHOH O OOHOHOH HO OOHOH O O OOOO OHOHOOH OHOOHHO OHOH O AcNH O O OOOOHO OHOOHo OHOH O OH OHO OOHOHOH OHO OOHOH O OOHOHOH OHO OCH3OHOHOO OHOHOOH OHOOHHO OHOH O AcNH OCH3 galactose fucose N-acetylglucosamine soluble Le X galactose galactoseglucose glucosegalactose glucose soluble Lac 2lipid-anchored Le X lipid-anchored Lac 2 galactose N-acetylglucosaminefucosegalactose galactoseglucose glucose linker Fig. 1
Structures of the soluble and lipid-anchored saccharides investigated in our simulations.
Results
Interactions of soluble carbohydrates
We first consider the interaction of two Le X trisaccharides in so-lution and compare this Le X –Le X pair interaction to the inter-action of two Lac 2 tetrasaccharides, which are composed oftwo lactose units (see Fig. 1). Standard carbohydrate forcefields lead to osmotic pressures for solutions of neutral carbohy-drates that are systematically too low compared to experimen-tal values. This underestimation of the osmotic pressure of thecarbohydrate solutions results from an overestimation of attrac-tive carbohydrate-carbohydrate interactions . To avoid un-realistically attractive carbohydrate-carbohydrate interactions, wehave used the GLYCAM06 TIP5POSMOr14 force field, in which the vander Waals parameters for saccharide-saccharide interactions of thestandard force field GLYCAM06 have been reparametrized to cor-rectly reproduce experimentally measured osmotic pressures .The GLYCAM06 TIP5POSMOr14 force field employs the TIP5P water modelbecause this water model leads to more reliable carbohydrate-carbohydrate interactions in GLYCAM06, compared to the standardTIP3P water model . Using graphics processing units (GPUs)and the software AMBER GPU , we have generated simula-tion trajectories with a length of . µ s for two Le X molecules in aperiodic simulation box of volume V = . , and 40 trajecto-ries with a length of µ s or close to µ s for two Lac 2 molecules ina simulation box of volume V = . , at the simulation tem-perature 30 ◦ C. Our total simulation times are µ s for the Le X pair and 39.5 µ s for the Lac 2 pair, which greatly exceed the totalsimulation times up to ns in previous simulation studies ofLe X -Le X pair interactions in solution and the total simulationtime of a few ns for pair interactions of trisaccharide epitopes frommarine sponges .In our simulations, we observe thousands of interaction eventsin which the two Le X molecules or the two Lac 2 molecules are in contact. These interaction events are separated by longer orshorter trajectory parts in which the two molecules are not in con-tact. Figs. 2(a) and (b) display pair conformations of Le X andLac 2 in which the two molecules exhibit at least 20 or 50 con-tacts of non-hydrogen atoms, respectively. The shown pair con-formations are randomly selected from the simulation frames ofour trajectories. One of the carbohydrate molecules is aligned inthe pair conformations and represented in blue colors, while theother molecule is represented in red/yellow colors. The clouds ofred/yellow molecules around the aligned blue molecules in theseconformations illustrate that the carbohydrate-carbohydrate inter-actions are ‘fuzzy’ , i.e. the two molecules interact in broadensembles of conformations, rather than via a single binding con-formation. For both Le X and Lac 2, the ensembles of conformationswith at least 50 contacts are narrower than the ensembles of con-formations with at least 20 contacts. In conformations with at least50 contacts, the two Le X molecules tend to stack above each otherin different orientations, and the two Lac 2 molecules tend to alignparallel or anti-parallel. However, the probability distributions ofcontact numbers in Fig. 2(c) illustrate that pair conformations with50 or more contacts of non-hydrogen atoms are rather rare and nottypical. The probability distributions decrease monotonously withincreasing number of contacts.The interaction events of the two Le X molecules or the twoLac 2 molecules can be characterized by their lifetime and by themaximum number of contacts of the events. These interaction Table 1
Association constants K a in units of M − for different cutoffs n c forthe contact number of binding events n c = n c = n c = Le X . ± . . ± . . ± . Lac 2 . ± . . ± . . ± .
20 40 60 8000.1%0.2%0.3%0.4%0.5% 0 10 20 30 40 50 600246810 0 0.5 1.0 1.5 2.000.51.01.52.02.53.03.5 r ad i a l d i s t r i bu t i on f un c t i on g distance r [ nm ] maximum number of contacts of interaction event a v e r age li f e t i m e [ n s ] number of contacts p r obab ili t y (a) soluble Le X : bound complexes≥ 20 contacts ≥ 50 contacts ≥ 20 contacts ≥ 50 contacts(b) soluble Lac 2: bound complexes Lac 2Le X (c) (d) (e) Lac 2Le X Lac 2Le X Fig. 2 (a) and (b) Randomly selected pair conformations of two Le X and two Lac 2 molecules with at least 20 or at least 50 contacts between non-hydrogen atoms within a distance less than . nm, respectively. One of the molecules is aligned in the 50 pair conformations and represented inblue colors, while the other molecule is represented in red/yellow colors. In the aligned Le X molecules, fucose is represented in dark blue, galactosein light blue, and N-acytylglucosamine in cyan. In the other Le X molecules, these monosaccharide units are represented in red, orange, and yellow,respectively. In the aligned Lac 2 molecules, the terminal galactose is represented in dark blue, the adjacent glucose in light blue, and the remaininggalactose and glucose in cyan. In the other Lac 2 molecules, these monosaccharides are shown in red, orange, and yellow. (c) Probability distributionsof the number of contacts between non-hydrogen atoms obtained from our simulations of two soluble Le X or two soluble Lac 2 molecules. (d) Averagelifetime of interaction events as a function of the maximum number of contacts of the interaction events. Interaction events are consecutive stretches ofsimulation frames at intervals of . ns with nonzero contacts of the two molecules. The error bars represent the standard deviations of the observedlifetimes. (e) Radial distribution functions g ( r ) of two soluble Le X or Lac 2 molecules with center-of-mass distance r . events are obtained from our simulation trajectories as consecu-tive stretches of frames at intervals of . ns with nonzero contactsof the two molecules. Fig. 2(d) shows that the average lifetimeof the interaction events increases with the maximum number ofcontacts observed during the event. With average lifetimes in thenanoseconds range, the interactions of the two Le X or the two Lac2 molecules are rather short-lived. Nonetheless, the radial distri-bution functions in Fig. 2(e) indicate that the interactions are at-tractive. The maxima of the radial distribution functions at center-of-mass distances of about . nm for Le X and . nm for Lac 2are significantly larger than the value 1 for a non-interacting idealsolution.Quantifying the attractive interactions of the two Le X or two Lac2 molecules requires distinguishing bound and unbound states.This distinction is somewhat arbitrary because of the fuzzy in-teractions of the carbohydrates. The probability distributionsof carboyhydrate-carbohydrate contact numbers in Fig. 2(c) aremonotonously decreasing and, thus, not bimodal as required fora clear distinction of two states. Table 1 presents association con-stants of two Le X or two Lac 2 molecules calculated for differ-ent cutoffs n c of the maximum number of contacts of interactionevents. In these calculations, only interaction events with a maxi-mum number of contacts larger or equal to the cutoff n c are takento be binding events. The probability P b that the two Le X or two Lac 2 molecules are bound has been determined from the total du-ration of the binding events, and the association constants from K a = V P b / P u where P u = − P b is the probability that the moleculesare unbound, and V is the volume of the simulation box. The K a values in Table 1 slightly decrease with increasing contact cutoff n c for binding events. For Le X , a K a value of 10 M − has been ob-tained from weak affinity chromatography experiments , whichis of the same order of magnitude as the values derived from oursimulations. Interactions of lipid-anchored carbohydrates
To investigate the interactions of two lipid-anchored Le X or twolipid-anchored Lac 2 molecules, we have performed simulations ofLe X and Lac 2 glycolipids embedded in POPC lipid membranes.Our Le X and Lac 2 glycolipids have the same lipid tails as POPC,and carbohydrate tips that are connected to these lipid tails by aglycerol linker group (see Fig. 1). The carbohydrate tip of the Le X glycolipid consists of the Le X trisaccharide and an additional lac-tose disaccharide as spacer between Le X and the glycerol linker.The Lac 2 glycolipid has the linear Lac 2 tetrasaccharide as car-bohydrate tip. The force field of our simulations combines theGLYCAM06 TIP5POSMOr14 carbohydrate force field for the TIP5P wa-ter model with the AMBER Lipid14 force field for lipid mem- embrane separation 5.5 nm 6.0 nm 6.5 nm 7.0 nm unbound Le X bound Le X Le X c o m p l e x e s Fig. 3 (top) Membrane conformations with two unbound or bound Le X glycolipids. The Le X glycolipids are anchored in the different monolayers ofthe membrane and interact because of the periodic boundary conditions of the simulation box. The height of the simulation box corresponds to themembrane separation from bilayer midplane to midplane. Each membrane monolayer contains 35 POPC lipids, which have the same lipid tails asthe Le X glycolipids. The fucose and galactose at the branched tip of the Le X glycolipids are represented in red and orange, and the remaining threemonosaccharide units in yellow. (bottom) 50 randomly selected complexes of the carboyhydrate tips of the Le X glycolipids at different membraneseparations. The selected complexes exhibit at least 10 contacts between non-hydrogen atoms of the two carbohydate tips. The carbohydrate tip ofthe lower Le X glycolipid is aligned in the 50 complexes and represented in blue colors, while the carbohydrate tip of the upper glycolipid is representedin red/yellow colors. The Le X motif of the carbohydrate tips are represented in the same colors as in Fig. 2(a). The lactose disaccharides of thecarbohydrate tips, which are located between the Le X trisaccharide and the linker group of the glycolipid, are represented in light blue and light yellow,respectively. number of contacts p r obab ili t y maximum number of contacts of binding event a v e r age li f e t i m e [ n s ] membrane separation 5.5 nmmembrane separation 6.0 nmmembrane separation 6.5 nmLe X binding : (a)(b) Fig. 4 (a) Probability distributions of the number of contacts between twoLe X glycolipids at different membrane separations. (b) Average lifetime ofinteraction events at different membrane separations as a function of themaximum contact number of the events. branes. Because simulations of AMBER Lipid14 POPC membranesin TIP5P water lead to an unreasonably small area per lipid, wehave rescaled the Lennard-Jones interactions between the TIP5Pwater molecules and the lipid headgroup atoms to obtain the samearea per lipid as in standard AMBER Lipid14 simulations with theTIP3P water model (see Methods).We quantify the interactions of two Le X or two Lac 2 glycolipidsat apposing membrane surfaces in a system that consists of a sin-gle lipid bilayer with one glycolipid anchored in each monolayer(see Fig. 3). In this system, the two glycolipids in the differentmonolayers interact due to the periodic boundary conditions of thesimulation box, and the separation of the membrane monolayerscan be adjusted by varying the number of water molecules in thesimulation box. The values for the membrane separation l givenin Fig. 3 correspond to the separation from membrane midplane tomembrane midplane and, thus, to the height of the simulation box.At each membrane separation, we have generated 10 trajectorieswith a length of µ s for the Le X system and a length of µ s for theLac 2 system at the temperature ◦ C. The total simulation timesat each membrane separation thus are µ s and µ s for the Le X and Lac 2 systems, respectively. The membranes contain in eachmonolayer 35 lipids besides the single glycolipid and have an area A of . nm . The height of the simulation box l increases withthe number of water molecules n w as l (cid:39) . + . n w nm . The thickness of the water layer in the simulations thus is about l − . nm.The interactions of the glycolipids strongly depend on the mem-brane separation. For the membrane separations l = . , . , . ,and . nm, 50 randomly selected complexes of the Le X glycol-ipid tips with at least 10 contacts of non-hydrogen atoms are dis-played at the bottom of Fig. 3. The carbohydrate tip of the lowerLe X glycolipid is aligned in the 50 complexes and represented inblue colors, while the carbohydrate tip of the upper glycolipid isrepresented in red/yellow colors. The clouds of red/yellow carbo-hydrates illustrate that the interactions of lipid-anchored Le X arefuzzy, similar to soluble Le X and Lac 2 (see Fig. 2). The over-lap of the cloud of the upper, red/yellow carbohydrates with thelower, blue carbohydrate decreases with increasing membrane sep-aration. At the membrane separation . nm, the Le X glycolipidsinteract via their entire carbohydrate tips. At the separation . nm, the interactions are limited to the Le X trisaccharide of the gly-colipid tip, and at the membrane separations . nm and . nm,the interactions are further restricted to the galactose and fucosemonosaccharides at the branched end of the Le X glycolipid. Thedecrease of interactions with increasing separation is also reflectedin the probability distributions of contact numbers shown in Fig.4(a) and in the average lifetime of the interaction events for dif-ferent maximum numbers of contacts in Fig. 4(b). At the smallestmembrane separation . nm, complexes of Le X glycolipids can ex-hibit up to 60 and more contacts of non-hydrogen atoms (see Fig.4(a)), and average lifetimes up to 50 ns for interaction events witha maximum number of 60 contacts (see inset of Fig. 4(b)), whichare about one order of magnitude larger than the average lifetimesfor interaction events of soluble Le X molecules with the same max-imum number of contacts. At the membrane separations . and . nm, the overall contact numbers and lifetimes of interactionevents are significantly smaller.Analogous to soluble carbohydrates, the binding associationconstants K trans = AP b / ( − P b ) of the glycolipids in the differentmembrane monolayers can be determined from the probability P b that the two Le X or two Lac 2 glycolipids are bound. The bindingconstants shown in Fig. 5 are calculated for binding events witha maximum number of at least n c = contacts of non-hydrogenatoms. For the larger binding cutoff n c = , the K trans values of thetwo Le X glycolipids are about 10% smaller than the values in Fig. 5at the membrane separations . and . nm, and the values of theLac 2 glycolipids are about 15% smaller at these separations. The K trans values decrease with increasing membrane separation. Formembrane separations larger than about . nm, the glycolipidscannot form contacts.The binding constant K trans can be related to membrane adhe-sion energies, which have been measured for membrane vesiclesthat contain mol% of Le X glycolipids . For two apposing,large membrane surfaces of area A that contain a total number of N t glycolipids, the free energy difference for forming the n th bondof the glycolipids is (see Methods) ∆ G n = − k B T ln [ K trans ( N t − n + ) / nA ] (1)The free energy differences ∆ G n are negative and, thus, favourable, .5 6.0 6.5 7.0 7.50123456 b i nd i ng c on s t an t K t r an s [ n m ] membrane separation [ nm ] Le X Lac 2
Fig. 5
Binding constant K trans of two Le X and two Lac 2 glycolipids ver-sus membrane separation, calculated for binding events with a maximumnumber of at least n c = contacts of non-hydrogen atoms. from bond 1 until the equilibrium number n eq of bonds. For bondnumbers n > n eq , the free energy difference ∆ G n is positive and,thus, unfavorable for binding. The adhesion free energy g ad perarea now can be calculated by summing up the free energy differ-ences ∆ G n from bond 1 to bond n eq : g ad = n eq ∑ ∆ G n / A (2)For an area per lipid of . nm measured in our simulations,the area of a membrane surface that contains N t glycolipids at aconcentration mol% is A (cid:39) . N t nm . From Eqs. (1) and (2)and the values of K trans for the Le X glycolipids in Fig. 5, we obtainthe adhesion free energies g ad = ± , ± , ± , and ± µ J/m at the membrane separations l = . , . , . , and . nm respectively. For lipid vesicles that contain mol% of Le X glycolipids, an adhesion free energy per area of ± µ J/m hasbeen reported , which is comparable to the adhesion free energyobtained from our simulations with membrane separation . nm. Forces on lipid-anchored carbohydrates in trans-direction
The binding of glycolipids in our simulations is associated with de-viations of the glycolipids relative to the surrounding lipids. Thesedeviations in the trans-direction perpendicular to the membranesurface result from forces on bound glycolipid complexes. Fig. 6(a)illustrates distributions of trans-deviations between the center ofmass of the linker group of a Le X glycolipid (see Fig. 1) and thecenter of mass of all lipid head groups in the same monolayer asthe glycolipid. The trans-deviations d are calculated from the sim-ulation frames of our trajectories at intervals of . ns. We obtaintwo values of d per simulation frame for the two glycolipids rel-ative to the monolayer in which they are embedded. An increasein d indicates glycolipid motion away from the membrane mid-plane. With increasing membrane separation, the distributions forbound Le X glycolipids deviate more and more from the distribu-tion for unbound Le X , which reflects increasing forces. The distri-bution of trans-deviations d of unbound Le X glycolipids shown inFig. 6(a) is calculated from our simulation trajectories at the mem-brane separation . nm, at which Le X bonds do not occur, and can be approximated by a Gaussian distribution exp [ − V ( d ) / k B T ] with V ( d ) = k ( d − d u ) . The trans-deviations d of unbound Le X gly-colipids thus can be described by a harmonic potential V ( d ) withforce constant k and mean extension d u , which can be determinedfrom the standard deviation σ and mean ¯ d of the Gaussian as k = k B T / σ = ± / nm and d u = ¯ d = − . ± . nm. The dis-tributions of trans-deviations of bound Le X glycolipids in Fig. 6(a)are calculated from our simulation trajectories at the membraneseparations . , . , . , and . nm, for binding events with a max-imum number of at least n c = contacts of non-hydrogen atoms.The average force f = k ( d b − d u ) on bound Le X glycolipids at themembrane separations l = . , . , . , and . nm then can becalculated from the difference between the mean trans-deviations d b = − . ± . , − . ± . , − . ± . , and − . ± . nmof the bound glycolipids at these membrane separations and themean trans-deviation d u of the unbound glycolipids. The force f on bound Le X glycolipids increases with increasing membrane sep-aration up to a value of . ± . pN at the separation . nm (seeFig. 6(b)). This maximal force value agrees with the unbindingforce ± pN of two Le X molecules obtained from atomic forcemicroscopy experiments . For bound Lac 2 glycolipids, we obtaina maximal force of . ± . pN at the separation . nm, whichis about of the same magnitude as the maximal force sustained bythe Le X complexes.The forces on bound Le X glycolipids lead to an adhesion pres-sure between the membranes. Fig. 7 illustrates the adhesion pres-sure p of membranes that contain 10 mol% of Le X glycolipids asa function of the membrane separation. The adhesion pressureis estimated as p = P b f / A where P b is the probability that a Le X glycolipid is bound at the concentration 10 mol%, f is the aver-age force on the bound glycolipid, and A (cid:39) . nm is the averagemembrane area of membrane patch with a single glycolipid at thisconcentration (see above). The negative pressure values for mem-brane separations l of 7.0 nm and smaller, at which the glycolipidscan bind, indicate membrane attraction. From integration of thepressure profile along the dashed interpolation line shown in Fig.7, we obtain adhesion energies g ad = (cid:82) l ∞ p ( l (cid:48) ) d l (cid:48) (cid:39) µ J/m for l = . nm and g ad (cid:39) µ J/m for l = . nm. These adhesion en-ergies per area agree with values g ad = ± µ J/m and ± µ J/m obtained directly from the binding constants K trans at themembranes separations l = . and . nm (see above), which in-dicates that average forces f on bound Le X glycolipids of Fig. 6(b)are consistent with the binding constants K trans shown in Fig. 5. Discussion and Conclusions
The membranes in our simulation systems are essentially planarbecause of the small size of the membranes, and because the gly-colipid in one monolayer interacts with the glycolipid in the othermonolayer across the periodic boundary of the simulation box.In larger, experimental systems, in contrast, the membranes ex-hibit thermally excited shape fluctuations, which lead to a stericrepulsion between adjacent membranes . During membrane ad-hesion, this steric repulsion needs to be overcome by attractiveinteractions . The average separation and thermal roughness ofthe adhering membranes is determined by the the interplay of theattractive interactions and the steric repulsion . From neutron p r obab ili t y den s i t y [ n m - ] f o r c e [ p N ] sep. 5.5 nmsep. 6.0 nmsep. 6.5 nmsep. 7.0 nmbound Le X : sep. 8.0 nmunbound Le X :trans-deviation of Le X linker [ nm ] membrane separation [ nm ] unbound Le X bound Le X (a)(b) Fig. 6 (a) Distributions of trans-deviations of Le X linker groups relativeto the surrounding lipids. The trans-deviations are calculated as the differ-ence between the center of mass of Le X glycolipid linker group (see Fig. 1)and the center of mass of all lipid head groups in the same monolayer asthe glycolipid. These trans-deviations of Le X in the direction perpendicularto the membrane plane are determined from the simulation trajectories ofthe system illustrated in Fig. 3. (b) Forces on bound and unbound Le X glycolipids at the different membrane separations. The trans-deviationsand forces of bound glycolipids are obtained from the simulation frames ofbinding events with a maximum number of at least n c = contacts of non-hydrogen atoms. Deviations to force values obtained for the cutoff n c = are smaller than the error bars. Forces an unbound glycolipids are calcu-lated from simulation frames with zero contacts between the glycolipids. scattering experiments of DPPC membrane multilayers that con-tain 10 mol% of Le X glycolipids , an average membrane sep-aration of ¯ l = . ± . nm and a relative membrane roughnessof ξ ⊥ = . ± . nm has been obtained. ∗ Because of the pe-riodicity of the membrane multilayers, the distribution of the lo-cal membrane separations l between adjacent membranes canbe approximated by the symmetric Gaussian distribution P ( l ) (cid:39) exp (cid:2) − ( l − ¯ l ) / ξ ⊥ (cid:3) / ( √ πξ ⊥ ) with mean ¯ l and standard deviation ξ ⊥ . The average membrane separation ¯ l obtained from neutronscattering is larger than the membrane separations at which theLe X glycolipids interact in our simulations. Trans-binding of theglycolipids therefore requires local membrane separations of thefluctuating membranes that are smaller than the average separa- ∗ The relative membrane roughness follows from Eq. (2) of Ref. 23 as ξ ⊥ = (cid:112) g ( ) with parameter values given in Table 2. Here, g ( r ) is the membrane displacementcorrelation function g ( r ) of adjacent membranes in the multilayer. p r e ss u r e [ N / m ] membrane separation [ nm ]
10 mol% Le X Fig. 7
Adhesion pressure p of membranes with 10 mol% of Le X glycol-ipids obtained for the force values f on bound Le X of Fig. 6(b). The dashedinterpolation line is added as a guide for the eye and used to estimate ad-hesion energies via integration (see text). In this integration, the pressure p is taken to be zero at separations l ≥ . nm. tion of the membranes. The average adhesion energy per areaof adjacent membranes can be estimated as ¯ g ad = (cid:82) g ad ( l ) P ( l ) d l ,where g ad ( l ) is the adhesion energy as a function of the local mem-brane separation l . From the four values of g ad ( l ) at the membraneseparations l = . , . , . , and . nm determined in the section"Interactions of lipid-anchored carbohydrates", we obtain the esti-mate ¯ g ad = ± µ J/m for the average separation ¯ l and relativemembrane roughness ξ ⊥ of the neutron scattering experiments.This estimate of the average adhesion energy per area is compa-rable in magnitude to the adhesion free energy per area of ± µ J/m reported for adhering membrane vesicles that contain 10mol% of Le X glycolipids . The Le X glycolipids embedded in thevesicles have the same carbohydrate tip as the Le X glycolipids ofthe neutron scattering experiments and of our simulations. How-ever, the carbohydrate tip of the vesicle system is connected to aceramide, which contains a different linker between the carbohy-drate tip and the lipid tails. Another difference is that the neutronscattering experiments have been performed at the temperature60 ◦ C to ensure that the DPPC membranes in these experiments arefluid . The Le X glycolipids of our simulations differ from those ofthe neutron scattering experiments only in the lipid tails. We havefocused on POPC membranes and corresponding glycolipid tails tobe able to run simulations of fluid membranes at the temperature30 ◦ C, which is close to the calibration temperature of the forcefields. In principle, membrane tension suppresses shape fluctua-tions of the membranes and can lead to stronger adhesion. How-ever, the suppression of fluctuations occurs only on lateral lengthscales larger than the characteristic length (cid:112) κ / σ , which adoptsvalues between 100 and 400 nm for typical membrane tensions σ of a few µ N / m and typical membrane bending rigidities κ between and k B T . . These values are significantlylarger than the lateral correlation length ξ (cid:107) of membranes adher-ing via Le X glycolipids, which is only a few nanometers for a rel-ative membrane roughness ξ ⊥ of about . nm . On these smalllength scales, the membrane shape fluctuations are dominated bythe bending energy of the membranes and the adhesion energiesof the glycolipids, and are not affected by membrane tension. he fuzzy interactions and comparable magnitude of the associ-ation constants of Le X and Lac 2 obtained in our simulations indi-cates that the interactions of small, neutral carbohydrates such asLe X and Lac 2 are rather generic and not dependent on specific,structural aspects of the carbohydrates. The good agreement toexperimental results for the association constant of soluble Le X ,adhesion energies of membranes with Le X glycolipids , and max-imally sustained forces of Le X complexes suggests that our sim-ulations provide a realistic, detailed picture of weak carbohydrate-carbohydrate interactions in solution as well as in membrane adhe-sion. The fuzzy binding reduces the loss of rotational and transla-tional entropy of the molecules during binding , because bindingcan occur for a large variety of different relative orientations of thesaccharides, in contrast to e.g. binding via specific hydrogen-bondpatterns as suggested previously for Le X based on simulations onshort timescales up to 40 ns . The fuzzy binding results froma subtle interplay between the rotational and translational entropyof the saccharides and the van der Waals, hydrogen bond, and hy-drophobic interactions of the saccharides in the various bindingconformations.We have investigated the binding of Le X in the absence ofCa + . Several groups have reported that Le X binding depends onCa + , whereas other groups have observed no de-pendence on Ca + . As pointed out by Kunze et al. , the Ca + concentration used by most groups are of the order of 10 mM and,thus, greatly beyond physiological Ca + concentrations. In vesi-cle adhesion experiments, Kunze et al. observed a rather smallincrease of the number of bound vesicles for a physiological Ca + concentration of 0.9 mM, compared to experiments in the absenceof Ca + . However, a strong increase of the number of bound vesi-cles in the experiments occurred for a Ca + concentration of 10mM. In atomic force microscopy experiments of Le X unbinding ,in contrast, the same unbinding force of about ± pN has beenobtained both in the absence of Ca + and for a Ca + concentra-tion of 10 mM. Overall, these experimental results suggest thatthe binding of Le X is not strongly affected at least by physiologicalconcentrations of Ca + . Methods
Simulations of soluble carbohydratesSystem setup – We have used the GLYCAM06
TIP5POSMOr14 carbohy-drate force field in our simulations of soluble pairs of Le X and Lac 2 in water. Initial structures of the Le X trisaccharides andLac 2 tetrasacchardies were created with the Glycam CarbohydrateBuilder program and solvated in truncated octahedral simula-tion boxes with TIP5P water molecules for the Le X pair andwith TIP5P water molecules for the Lac 2 pair. In the initialconformations, the two saccharides were placed in the simulationboxes such that they were not in contact. We have subsequentlyminimized the simulation systems in minimization steps ofsteepest decent and additional steps of the conjugent gradi-ent algorithm. The systems were then heated from the tempera-ture 0 K to 303 K at constant volume in integration timesteps of fs with temperature control by a Langevin thermostat with collision frequency γ = 1.0 ps − . Production runs – After equilibration for ns at K, we havegenerated independent trajectories for the Le X pair and tra-jectories for the Lac 2 pair with a fs integration step in AMBER 14and 16 GPU using the Monte-Carlo barostat and a Langevinthermostat with collision frequency γ = 1.0 ps − to keep the tem-perature at K and the pressure at bar. On these trajectories,the lengths of bonds that contain hydrogens were restrained withthe SHAKE algorithm , non-bonded interactions were trun-cated at a cutoff value of 1 nm, and the Particle Mesh Ewald al-gorithm (PME) was used to treat all electrostatic interactions.The simulation trajectories for Le X pair have a length of . µ s,and the trajectories for the Lac 2 pair have a length of 1 µ s orclose to 1 µ s. The total simulation times of these trajectories are µ s for Le X system and . µ s for the Lac 2 system. Analysis of trajectories – We have identified interactions eventsof the two Le X or two Lac 2 molecules along the simulation tra-jectories as consecutive stretches of simulation frames at intervalsof . ns with nonzero contacts of the molecules. These interac-tion events are separated by stretches of simulation frames withzero contacts and can be characterized by their lifetime and bythe maximum number of contacts during the events. The contactsare defined as contacts between non-hydrogen atoms of the twomolecules within a distance of less than . nm. We consider in-teraction events with a maximum number of contacts that is largeror equal to a cutoff number n c as binding events. For the cutoffnumbers n c = , , and , we have obtained , , and binding events of the two Le X molecules on all trajectories,and , , and binding events of the two Lac 2 molecules.We have thus observed dozens of binding and unbinding events oneach trajectory, with binding and unbinding times that are signifi-cantly smaller than the trajectory lengths (see also Fig. 2(d)). Toensure independence from the initial, unbound conformations ofthe trajectories, we have discarded the first ns on all trajec-tories in our calculations of the binding probablity P b of the twomolecules, which is defined as the fraction of simulation framesbelonging to binding events. We have calculated P b for each tra-jectory and have determined the overall value and error of P b asmean and error of the mean of the values for all trajectories. Theassociation constants K a reported in Table 1 were calculated fromthese binding probabilities via the relation K a = V P b / ( − P b ) where V is the simulation box volume . The errors of K a are calculatedby error propagation from the errors of P b . The errors of the prob-ability distributions and radial distribution functions in Fig. 2(c)and (e) are calculated as error of the mean of the correspondingquantities for the individual trajectories. Simulations of lipid-anchored saccharidesSystem setup – We have generated the initial structures of thePOPC lipid membranes with the CHARMM-GUI program . Forour simulations with glycolipids, one lipid in each monolayer hasbeen replaced by a Le X or Lac 2 glycolipid, which have the samelipids tails as POPC (see Fig. 1). Following Ref. 43, we have per-formed the initial minimization and equilibration steps of all mem-brane systems as follows: We have first performed a minimizationof the water molecules for fixed lipids and glycolipids in min- .0 1.1 1.2 1.3 1.4 1.50.550.600.650.70 a r ea pe r li p i d [ n m ] scaling factor α TIP3Pexperiments
Fig. 8
Area per lipid for a Lipid14 POPC membrane in TIP5P water as afunction of the scaling factor α for well-depth of the Lennard-Jones interac-tions between TIP5P water molecule and the lipid head group atoms. Thedashed horizontal lines represent the area per lipid values of POPC mem-branes for Lipid14 in TIP3P water and from experiments . The dashedline through the data points is a guide for the eye. Errors or the simulationdata are smaller than the point sizes. The error of the experimental valueis indicated by they shaded region. The temperature of the simulationsand experiments is 30 ◦ C. imization steps of steepest descent and subsequent steps ofthe conjugent gradient algorithm. The lipids and glycolipids havebeen fixed by harmonic constraints with a force constant of kcal mol − Å − in this minimization. We have next removed theharmonic constraints, and have repeated the same minimizationsteps for the complete systems. The subsequent heating of the sys-tems has been performed in three steps: (1) heating from K to
K at constant volume with harmonic constraints on lipids andglycolipids with a force constant of kcal mol − Å − ; (2) heat-ing from K to
K with a reduced force constant of kcalmol − Å − of the harmonic constraints on lipids and glycolipids;and (3) heating from K to
K at constant pressure and amembrane tension of zero with the same harmonic constraints asin the second step using a semi-isotropic pressure coupling and theBerendsen barostat with a pressure relaxation time of ps. Eachheating step consist of MD integration steps of length fswith temperature control by a Langevin thermostat with a collisionfrequency of . ps − . Rescaling of Lennard-Jones interactions between water andlipid headgroups – We have used the GLYCAM06
TIP5POSMOr14 car-bohydrate force field for the carbohydrates and the AMBERLipid14 force field for the lipids of our MD simulations of POPCmembranes with glycolipids. Simulations of AMBER Lipid14 POPCmembranes in TIP5P water lead to an unreasonably small area perlipid of . ± . (see Fig. 8) and to density profiles that de-viate significantly from profiles obtained from simulations in thestandard TIP3P water model (see Fig. 9), which has been used inthe parametrization of the AMBER Lipid 14 force field . We havetherefore rescaled the well depth of the Lennard-Jones interactionsbetween the TIP5P water molecules and the Lipid14 lipid head-group atoms by a scaling factor α in order to obtain the same areaper lipid as in simulations of POPC membranes with TIP3P water.We chose to rescale the Lennard-Jones interactions between wa- − − distance from bilayer center [ nm ] waterlipid heads TIP3PTIP5P lipid tails e l e c t r on den s i t y [ n m - ] Fig. 9
Electron density profiles for a Lipid14 POPC membrane in TIP3Pand TIP5P water at the temperature 303 K. The membrane is composedof 128 lipids. ter and lipid headgroups because the density profiles of AMBERLipid14 POPC membranes in TIP5P water show a smaller over-lap between the water and lipid head group regions, compared toTIP3P water (see Fig. 9). This smaller overlap likely results fromweaker Lennard-Jones interactions, and not from different atomsizes, because the Lennard-Jones radius . of the TIP5P oxygenatom is in fact smaller than the radius . of the TIP3P oxygenatom. Therefore, we have only rescaled the Lennard-Jones well-depth ε for the interaction between TIP5P water and the lipid headgroup atoms by a scaling factor α .Fig. 8) illustrates simulation results for the area per lipid as afunction of the scaling factor α . The membranes in these sim-ulations consists of 128 POPC lipids, and the number of watermolecules is 6400. For each value of α , we have generated independent trajectories of length ns with semi-isotropic pres-sure coupling at a membrane tension of zero and a temperatureof 303 K using the same barostat and thermostat settings as inthe last heating step of the system setup (see above). We havedetermined the area per lipid from the last 100 ns of these trajec-tories, with errors calculated as error of the mean of the values forthe individual trajectories. The value α = . leads to an area perlipid in TIP5P simulations that is close to the area per lipid both inTIP3P simulations and in experiments (see Fig. 8). We have there-fore used α = . in our simulations of lipid-anchored saccharides.For this value of α , the density profile of AMBER Lipid14 POPCmembranes in TIP5P water (not shown) is practically identical todensity profile in TIP3P water, and the membrane thickness d m andlateral diffusion coefficient D of the lipids are identical within er-rors or close to the values obtained in TIP3P simulations (see Table2). We have determined the bilayer thickness as the distance be-tween the electron density peaks of the lipid head groups, and thelateral diffusion constant from the relation D = MSD ( t ) / ( t ) where MSD ( t ) is the mean-squared-displacement of a lipid molecule attime t . To obtain MSD ( t ) , we have first removed the center ofmass motion of each leaflet to eliminate the ‘caterpillar effect’ and have divided our trajectories into ns fragments. We havethen calculated MSD ( t ) from the MSD profiles of single lipids byaveraging over all lipids and all trajectory fragments. The diffu- able 2 Membrane thickness d m and lipid diffusion coefficient D from sim-ulations with TIP5P water for different values of the scaling factor α , fromsimulations with TIP3P water, and from experiments on POPC lipid mem-branes α d m [ nm ] D [ µ m / s ] . ± .
03 3 . ± . . ± .
01 4 . ± . . ± .
01 5 . ± . . ± .
01 5 . ± . . ± .
01 6 . ± . . ± .
01 7 . ± . TIP3P . ± .
01 5 . ± . exp. . . sion coefficients in Table 2 are calculated from linear fits in thetime intervals from t = ns to ns in which MSD ( t ) approachesa constant slope. Production runs – The membranes of our simulations with twoLe X or two Lac 2 glycolipids are composed of 35 POPC lipids andone glycolipid in each monolayer. By varying the number of watermolecules in the simulation box, we have created several mem-brane systems that differ in simulation box height. In our sim-ulations with Le X glycolipids, we have obtained the average boxheights l = . , . , . , . , . , and . nm for the numbers , , , , , and of TIP5P water molecules,respectively. In our simulations with Lac 2 glycolipids, we haveobtained the average box heights l = . , . , . , . , and . nm for the numbers , , , , and of watermolecules. The height l of the rectangular simulation box corre-sponds to the separation from membrane midplane to membranemidplane across the periodic boundary of the box in the directionperpendicular to the membrane. After equilibration for ns, wehave produced independent trajectories for each system withthe software AMBER 16 GPU . The trajectories have a lengthof 3 µ s for the Le X systems and a length of 1 µ s for the Lac 2 sys-tems. We have regulated the simulation temperature of K us-ing a Langevin thermostat with a collision frequency of 5.0 ps − ,and have employed a semi-isotropic pressure coupling with a pres-sure of 1 bar in all directions, which corresponds to a membranetension of zero. We have used the Berendsen barostat with re-laxation time τ = ps for the pressure regulation because of thestability of the semi-isotropic pressure coupling in AMBER 16 GPUin combination with this barostat. For large systems as consideredhere, the weak-coupling scheme of the Berendsen barostat can beexpected to lead to results that are essentially equivalent to otherbarostats . We have constrained the bond lengths for hydrogenatoms with the SHAKE algorithm and have used an integra-tion timestep of fs in all simulations. A cutoff length of . nmwas used in calculating the non-bonded interactions with the Par-ticle Mesh Ewald (PME) algorithm . Analysis of trajectories – We have identified interactions eventsbetween the carbohydrate tips of the two Le X or two Lac 2 glycol-ipids in the same way as described above for the soluble saccha-rides. For two Le X glycolipids, we have obtained , , , binding events with a maximum contact number of at least n c = on the trajectories at the membrane separations . , . , . , and . nm, respectively. For two Lac 2 glycolipids, we haveobtained , , , and such binding events on the trajecto-ries at the corresponding membrane separations.To ensure independence from the initial conformation of the tra-jectories, we have discarded the first 10% of each trajectory in ourcalculations of the binding probablity P b of the two molecules. Inanalogy to soluble carbohydrates, we have determined P b and itserror as mean and error of the mean of the values for the 10 trajec-tories at a given membrane separation. The binding constant thenfollows as K trans = AP b / ( − P b ) where A is the membrane ara . Wehave calculated the errors of the probability distributions in Figs.4(a) and 6(a) and of the forces in Fig. 6(b) as error of the mean ofthe corresponding quantities for the individual trajectories. Calculation of adhesion free energies from trans-binding con-stants of membrane-anchored molecules
The binding constant K trans of molecules anchored to two apposingmembrane surfaces and of area A is related to the on- andoff-rate constants of these molecules via K trans = k on / k off (3)If the total numbers of the molecules at the two surfaces are N and N , up to n ≤ min ( N , N ) trans-bonds can be formed. The effectiverate for going from a state with n − trans-bonds to a state with n bonds is k + = k on ( N − n + )( N − n + ) / A (4)and the effective rate for going back from n bonds to n − is k − = nk off (5)The condition of detailed balance implies P n − k + = k − P n (6)where P n is the equilibrium probability of the state with n trans-bonds. The free-energy difference ∆ G n between the states with n and n − bonds is related to the equilibrium probabilities via exp [ − ∆ G n / k B T ] = P n / P n − (7)From these equations, we obtain ∆ G n = − k B T ln (cid:20) K trans ( N − n + )( N − n + ) nA (cid:21) (8)The adhesion free energy g ad per area then can be calculated bysumming up the free energy differences ∆ G n from bond 1 to bond n eq where n eq is the equilibrium number of bonds at which ∆ G n changes sign (see Eq. 2). Conflicts of interest
There are no conflicts to declare.
Acknowledgements
Financial support of the International Max Planck Research School(IMPRS) on Multiscale Bio-Systems and by the German ResearchFoundation (DFG) via
Emmy Noether grant SCHN 1396/1 is grate- OTES AND REFERENCES NOTES AND REFERENCES fully acknowledged. We would like to thank Mark Santer for help-ful discussions.
Notes and references
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