Polymer brush-induced depletion interactions and clustering of membrane proteins
PPolymer brush-induced depletion interactions and clustering of membraneproteins
Anvy Moly Tom, Won Kyu Kim, a) and Changbong Hyeon Korea Institute for Advanced Study, Seoul 02455, Korea (Dated: 26 February 2021)
We investigate the effect of mobile polymer brushes on proteins embedded in biological membranes by em-ploying both Asakura-Oosawa type of theoretical model and coarse-grained molecular dynamics simulations.The brush polymer-induced depletion attraction between proteins changes non-monotonically with the sizeof brush. The depletion interaction, which is determined by the ratio of protein size to the grafting distancebetween brush polymers, increases linearly with brush size as long as the polymer brush height is shorter thanthe protein size. When the brush height exceeds the protein size, however, the depletion attraction amongproteins is slightly reduced. We also explore the possibility of brush polymer-induced assembly of a largeprotein cluster, which can be related to one of many molecular mechanisms underlying recent experimentalobservations of integrin nanocluster formation and signaling.
INTRODUCTION
In 1950s, Asakura and Oosawa (AO) proposed a simpletheoretical model to explain the interaction of entropicorigin between colloidal particles immersed in a solutionof macromolecules , which is of great relevance to ourunderstanding of organization and dynamics in cellularenvironment. According to the AO theory, rigid sphericalobjects immersed in the solution of smaller hard spheresrepresenting the macromolecules are expected to feel fic-titious attraction, termed depletion force. While the in-teraction energy of the system remains unchanged, thespherical objects can be attracted to each other. Bring-ing the large spherical objects into contact can increasethe free volume accessible to the smaller hard spherescomprising the medium, and hence increasing the totalentropy of the hard sphere system (∆
S > F HS = − T ∆ S = − (cid:18) λ + 1 (cid:19) φk B T, (1)where λ is the size ratio of large to small hard spheres,and φ is the volume fraction of small spheres compris-ing the surrounding medium . For a fixed value of φ , the disparity in size between colloidal particles (largespheres) and macromolecular depletants (small spheres),characterized with the parameter λ , is the key determi-nant of the magnitude of depletion free energy . Theeffect of crowding environment on the aggregation of col-loidal particles becomes substantial when λ (cid:29)
1. Thecellular environment is highly crowded, such that 30 %of cytosolic medium is filled with macromolecules, ren-dering the interstitial spacing between macromoleculescomparable to the average size of proteins ∼ . Morespecifically, this volume fraction of E. coli mixture is con-tributed by 11 % of ribosome, 11 % of RNA polymerase, a) Electronic mail: [email protected] and 8 % of soluble proteins . In the cellular environment,the depletion force is one of the fundamental forces ofgreat importance.The basic principle of AO theory on rigid bodieswith spherical symmetry is straightforward; however,application of the idea to the repertoire of biologi-cal and soft materials requires quantitative assessmentof entropy , which is nontrivial especially when crow-ders are characterized with non-spherical shape and/orwith polydispersity and when the system is un-der a special boundary condition . For the pastdecades, there has also been much interest toward un-derstanding of the effects of crowding in biology ,which includes crowding-induced structural transitionsin disordered chiral homopolymers , protein/RNAfolding , gene regulation through DNA looping ,genome compaction , efficient search of proteins for tar-gets on DNA , and molecular motors . Further, itis worth mentioning a series of effort to understand thedynamics of active matter in the language of depletionforces .Besides the examples of depletion force-induced dy-namics that all occur in three dimensional space, theAO theory can be extended to lateral depletion ef-fects on the objects whose motion is confined in flatsurfaces . For biological membrane where the areafraction of membrane-embedded proteins is as high as15 – 30 %, the formation of protein clusters or nano- ormicro-domains is of great relevance to understandingthe regulation of biological signal transduction and cell-to-cell communication. Although other physical mecha-nisms are still conceivable, lateral depletion interactionsbetween membrane embedded proteins can arise from thefluctuations of lipids or other polymer-like com-ponents comprising fluid membrane , contributing toprotein–protein attraction and clustering. In this con-text, the formation of integrin nanodomain which en-ables cell-to-cell communications via signaling , par-ticularly, the bulky glycocalyx-enhanced integrin cluster-ings and the associated signaling-induced cancer metas-tasis observed by Paszek et al. make the brush polymer- a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1. Brush-induced depletion interactions. (A) Illustration of brush polymers, each of which is organized into a string ofblobs of size ξ above the surface. (B) Two cylindrical inclusions (red) separated by distance r surrounded by brush polymers(grey). (C) Top view of (B). The lateral dimension of brush polymer ξ corresponds to the size of a blob depicted with greysphere (see (A)). (D) Diagram to calculate the brush-induced depletion interaction between the two cylindrical objects. Thearea inside the dashed line, corresponding to 2 π [( D + ξ ) / − A overlap ( r ) in Eq.3, is the area inaccessible to the blob of polymerbrush of size ξ . The shaded region in pale red is the overlapping area of the two discs of radius ( D + ξ ) /
2, separated by thedistance r . induced depletion interaction between membrane pro-teins and their clustering a topic of great relevance toinvestigate.In this paper, we study the lateral depletion interac-tions between rigid inclusions embedded in the mobilepolymer brushes in 2D surface in the spirit of the AOtheory in its simplest form. We compare the results fromour simulations with our theoretical predictions. By an-alyzing the distribution of brush polymer-enhanced pro-tein clusters obtained from our simulations, we attemptto link the brush-size dependent populations of giant pro-tein clusters with the strength of signal transduction ob-served in Paszek et al. ’s measurement. THEORY: BRUSH-INDUCED LATERAL DEPLETIONINTERACTIONS
As illustrated in Fig. 1A, we consider flexible polymerbrushes, each consisting of N + 1 monomers of size (di-ameter) b . One end of individual chain is grafted to thesurface but is free to move. If the grafting density σ islarge enough to satisfy σR F > or equivalently if the grafting distance ( ξ ) is smaller than R F = bN / , i.e., ξ < R F , where R F is the Flory radius of the polymer ingood solvent, each polymer reorganizes into a string ofself-avoiding blobs due to excluded volume interactionswith the neighboring polymers, forming a polymer brushof height H where N/g blobs of size ξ consisting of g segments fill the space above the surface (Fig. 1A) . Inthis case, the grafting density σ = N b /A , the number ofpolymer chains ( N b ) grafted on an area A , is related tothe blob size (or the grafting distance) as σ (cid:39) /ξ . Itis straightforward to show using the blob argument thatthe brush height H scales with N and σ as H = N σ / b / . (2)Our interest is in the lateral depletion force betweentwo cylindrical inclusions embedded in the polymer brushsystem, when the two inclusions, constrained to move in xy plane, are separated by a fixed distance r (Fig. 1B, C).In the presence of the cylindrical inclusions, the volumeaccessible to the individual polymer chains is determinedas follows, depending on r . V ( r ) = AH − (cid:20) π (cid:16) D + ξ (cid:17) − A overlap ( r ) (cid:21) q ( h, H ) , for D ≤ r ≤ D + ξAH − π (cid:16) D + ξ (cid:17) q ( h, H ) , for r > D + ξ. (3)Here, A overlap ( r ) is the overlapping area between two cir-cular discs of radius ( D + ξ ) /
2, the region demarcated in pale red in Fig. 1D, is A overlap ( r ) = 4 (cid:90) ( D + ξ ) / r/ (cid:34)(cid:18) D + ξ (cid:19) − ρ (cid:35) / d ρ. (4)This is maximized when r = D , and its value can be writ-ten in terms of the area defined by the square of graft-ing distance, ξ , multiplied with a dimensionless factor χ ( λ br ), A overlap ( D ) = ξ (1 + λ br ) (cid:90) λ br1+ λ br (1 − x ) / d x (cid:124) (cid:123)(cid:122) (cid:125) ≡ χ ( λ br ) . (5)where χ ( λ br ) = 12 (cid:20) (1 + λ br ) cos − (cid:18) λ br λ br (cid:19) − λ br (cid:112) λ br (cid:21) (cid:39) (cid:40) π + π − λ br + O ( λ ) , for λ br (cid:28) √ √ λ br , for λ br (cid:29) , is a monotonically increasing function of λ br = D/ξ (cid:39) D √ σ , the ratio of the diameter of the inclusions to thegrafting distance (or the blob size). Next, the function q ( h, H ) ≡ H Θ( h − H ) + h Θ( H − h ), defined with thestep function, signifies (i) q ( h, H ) = H when the brushheight ( H ) is shorter than the height of the inclusion( h ) ( H < h ); and (ii) q ( h, H ) = h when the brush isgrown over the inclusion ( H > h ) (see Fig.2A). It is as-sumed that when
H > h the volume above the inclu-sions, A × ( H − h ), is fully accessible to the polymerchains, which is a reasonable assumption when H (cid:29) h .Furthermore, under an assumption of no correlation be-tween the polymer chains, the partition function for thebrush system in the presence of the 2D inclusions sep-arated by r is Z ( r ) = [ V ( r )] N b × ( N +1) , where N b is thenumber of polymers consisting the brush. The thermo-dynamic equilibrium is attained by maximizing the to-tal entropy of the system or minimizing the free energy βF ( r ) = − log Z ( r ) = − N b ( N + 1) log V ( r ). The gain infree energy due to depletion attraction can be obtained bytaking the difference after and before the inclusions are infull contact with each other as β ∆ F = βF ( D ) − βF ( r ≥ D + ξ ) (see Appendix A for an alternative derivation us-ing the depletion force): − β ∆ F = N b ( N + 1) log V ( D ) V ( r ≥ D + ξ )= N b ( N + 1) log A overlap ( D ) q ( h, H ) AH − π (cid:16) D + ξ (cid:17) q ( h, H ) ≈ N b ( N + 1) ξ χ ( λ br ) q ( h, H ) AH = ( N + 1) χ ( λ br ) q ( h, H ) H = (cid:40) ( N + 1) χ ( λ br ) , for h > H ( N + 1) χ ( λ br ) hH , for h < H, (6)where a large volume ( AH (cid:29)
1) was assumed for thebrush system, with A overlap ( D ) = ξ χ ( λ br ) and σξ (cid:39) BC h > H h < H A σ b = σ b = σ b = σ b = - - - - - N β Δ F N = N = N = N = - - - - - - - σ b β Δ F FIG. 2. (A) Two different cases of brush-induced depletion in-teraction: h > H (left), and h < H (right). (B), (C) Free en-ergy gain due to brush-induced depletion interaction. Eq. (6)was calculated as a function of N for varying σ (B), and asa function of grafting density ( σ ) for varying N (C), witha cylindrical inclusion at fixed diameter D = 5 b and height h = 5 b .
1. Eq. (6) suggests that N and λ br (or σ ) are the keyparameters that determine the free energy gain upon thebrush-induced clustering.According to Eq. (6) plotted against N in Fig. 2B, thebrush induced depletion interaction, quantified in termsof stability gain − β ∆ F increases linearly with polymerlength ( − β ∆ F ∝ N ) when the brush is kept shorter thanthe height of the inclusion ( H < h ). However, as soon asthe brush height exceeds the inclusion height (
H > h ),the free energy gain is reduced. When
H > h , the sameamount of accessible volume A ( H − h ) is added regard-less of the state of the two inclusions, increasing boththe volume V ( D ) and V ( r ≥ D + ξ ) accessible for brushpolymers. This leads to the reduction of − β ∆ F . Thefactor h/H that appears in the last line of Eq.6 quanti-fies the extent of this reduction in free energy gain (seeAppendix B for further clarification).For H (cid:29) h , the free energy gain converges to − β ∆ F ∼ χ ( λ br ) hσ / b / < χ ( λ br ) N, (7)where the inequality holds because of h < H = A N =5 N =10 N =15 B FIG. 3. (A) A snapshot of simulations. The spheres (red)and polymers (grey) represent membrane proteins and brushpolymers grafted on the 2D surface, respectively. (B) Lateralview of simulations for different brush sizes ( N = 5, 10, and15). N σ / b / . Also, in the limit of H (cid:29) h , it can be shownthat − β ∆ F ∼ σ − / h , which explains the σ -dependentlimit of β ∆ F at large N in Fig.2B. The crossover pointof polymer length N ∗ changes with the grafting densityas N ∗ (cid:39) hσ − / b − / .There is a crossover in the stability gain as well whenthe grafting density ( σ ) is increased (Fig. 2C). The de-pletion free energy scales with σ as − β ∆ F ∼ (cid:40) ( N + 1) σ / , for σ < σ ∗ N +1 N σ − / , for σ > σ ∗ , (8)with the crossover grafting density σ ∗ b (cid:39) ( h/N b ) . NUMERICAL RESULTSA. Model
The system is defined by N b brush polymers compris-ing the brush, and M membrane proteins embedded inthe brush on the 2D surface (Fig. 3). The center ofthe protein, modeled as a sphere whose diameter (orvdW radius) is D = 5 a , is constrained on the surfaceat z = D/
2, with a harmonic potential, to move onlyin parallel to the surface. The individual polymer con-sisting of N segments (or N + 1 monomers) is modeledusing an energy potential for a bead-spring chain withself-avoidance. Each monomer with diameter a is con-nected via the harmonic potential, V s ( r i,i +1 ) = k s r i,i +1 − b ) , (9)where k s = 3000 k B T /a is the spring constant and b = 2 / a is the equilibrium bond length. Similarly to the protein, the first monomers of the chain, grafted tothe surface at z = a/
2, are free to move in the xy plain,but constrained in the z direction via a harmonic poten-tial. Any non-grafted monomer whose distance from thegrafting surface is z ≤ a is repelled by the Lennard-Jones(LJ) potential truncated at z = a , V surfLJ ( z ) = (cid:40) k B T (cid:104)(cid:0) az (cid:1) − (cid:0) az (cid:1) (cid:105) , for z ≤ a , for z > a. (10)Both intra-chain and inter-chain monomer–monomer in-teractions as well as protein–monomer and protein–protein interactions are modeled with LJ potential. V αβ LJ ( r ij ) = (cid:15) αβ (cid:20)(cid:16) d αβ r ij (cid:17) − (cid:16) d αβ r ij (cid:17) (cid:21) , for r ij ≤ r c , for r ij > r c . (11)Here, α and β denote different particle types, α, β ∈{ m , P } , with m and P standing for monomer and pro-tein. r ij is the distance between particles i and j , (cid:15) αβ isthe strength of the interaction, and d αβ (= ( d α + d β ) / α and β . We have chosen β(cid:15) αβ = 1 . d P = 5 a , d m = a ; r c = 2 . × d PP , d mP , and d mm are the values of cut-off distance for protein-protein,monomer-protein, and monomer-monomer pairs, respec-tively. As a result, monomer–protein and monomer–monomer interactions are purely repulsive; and theprotein–protein interactions in the absence of polymerbrush are effectively under Θ-solvent condition to yielda nearly vanishing second virial coefficient.The simulation box has a dimension of L x = L y = 200 a and L z = ( N + 1) b + ∆ with ∆ = 5 a , where a isthe basic length unit of our simulations. The systemis periodic along the x and y directions and finite inthe z direction. With the fixed number of proteins M = 400, the area fraction of the membrane proteinsis φ P = π ( D/ M/ ( L x L y ) = 0 .
2, which corresponds tothe surface density, σ P = 0 . /a . The φ P is related with σ P as φ P = σ P × π ( D/ . The grafting density of brushpolymer is calculated using σ = N b / ( L x L y − π ( D/ M ).In the simulations, σa is varied between 0.05 and 0.09. B. Simulations
For the efficient sampling of the configurations of thepolymer brush system including proteins, we used thelow-friction Langevin dynamics to integrate the equationof motion . m ¨ x i = − γ ˙ x i − ∂ x i V ( { r k } ) + η i ( t ) , (12)where m is the mass of i -th particle. The characteristictime of the equation is set τ = ( ma /(cid:15) ) / with the char-acteristic energy scale of inter-particle interaction (cid:15) = σ a = σ a = σ a = σ a = - - - - N Δ B / a FIG. 4. The measure of the brush polymer-induced protein–protein interaction, ∆ B = B − B ref2 , as a function of thepolymer brush size ( N ) for different grafting densities ( σ ).The data point at N = 0 is for the protein-only referencesystem. k B T specified in the energy potential V ( { r k } ). Then,the friction constant is set to γ = 0 . m/τ . The lastterm η i ( t ) acting on the i -th particle ( i ∈ { m , P } ) is theGaussian white noise with zero mean, (cid:104) η i ( t ) (cid:105) = 0, satis-fying the fluctuation dissipation theorem, (cid:104) η i ( t ) η j ( t (cid:48) ) (cid:105) =2 γk B T δ ij δ ( t − t (cid:48) ). The equation of motion (Eq. (12))was integrated using the velocity-Verlet algorithm withthe integration time step δt = 0 . τ . After thepre-equilibration that fully randomizes the initial config-urations of the system, the production runs of 4 × time steps were performed and collected for the statisti-cal analysis. C. Second virial coefficient
The radial distribution function g ( r ) between themembrane proteins (Fig. S1) is associated with the sec-ond virial coefficient and is calculated for different set ofparameters of brush size ( N ) and grafting density ( σ ) asfollows. B = 12 (cid:90) (1 − e − βu ( r ) )d r (cid:39) π (cid:90) ∞ (1 − g ( r )) r d r. (13)We denote the second virial coefficient of a protein-only system as B ref2 , and assess the depletion interac-tion in terms of ∆ B = B − B ref2 , which can be re-lated to the depletion induced free energy stabilizationas β ∆ F ∼ ∆ B σ P σ . To simplify our interpretation ofthe simulation result, we have chosen the parameters forthe protein–protein interaction to yield B ref2 (cid:39) σ ) and brush size ( N );however, this trend is saturated or even inverted when the brush size is greater than a certain value (Fig. 4). Thenon-monotonic dependence of the depletion interaction(∆ B ) on N becomes more pronounced at high graftingdensity. Fig. 4 shows that the depletion effect for σa =0 .
09 is maximized at N = N ∗ (cid:39)
10, at which the brushheight ( H ) becomes comparable to the size of protein,( D ). This behavior is in agreement with the theoreticalprediction of crossover at h (cid:39) H = N ∗ σ / b / (Fig. 2B).With h = 5 a , σa = 0 .
09, and b = 2 / a , we obtain N ∗ = hσ − / b − / (cid:39) . D. Brush-induced protein clustering
One of the goals of this study is to identify the condi-tion that yields a large sized protein clustering. To thisend, we analyze the snapshots of simulations to calculatethe cluster size distribution. We consider that two mem-brane proteins form a cluster of size two if the distancebetween them is less than the distance criterion of 6 a ,which can be extended to identify a cluster of size m .Although the mean cluster size obtained from the sim-ulation results is small ( (cid:104) c (cid:105) (cid:104) = (cid:82) c ≥ cP ( c )d c (cid:105) = 2 − P ( c )s display long tails signifying the presence of largeclusters (Fig.5). Deviation of P ( c ) from that of theprotein-only reference system ( P ref ( c )) is observed at c (cid:38) c ∗ ≈
10 (Fig.5). With an assumption that the inten-sity of downstream signal ( S ) is proportional to the sizeof a cluster ( c > c ∗ ), which is greater than c ∗ , weightedby the population ( P ( c )), we evaluate the signal relayedfrom the protein clusters using S ( N, σ ) ∝ (cid:90) c ≥ c ∗ cP ( c ; N, σ )d c, (14)with c ∗ = 10. The signal intensity calculated for varyinggrafting densities (Fig. 6) demonstrates a sigmoidal in-crease of S as a function of brush size ( N ) up to N ≤ N ∗ ,beyond which S decreases, suggestive of shrinking clustersize, reflecting the decrease of | ∆ B | . The mid-point of S ( N ) shifts to a smaller N from N (cid:39) N (cid:39) σ increases from σa = 0 .
05 to 0.09.
DISCUSSION
The AO theory extended to the brush system (Eq.6)differs from the hard sphere systems with two types (largeand small spheres) in three dimensions (Eq.1) in severalaspects: (i) One of the key parameters λ br (= D/ξ ) isthe ratio of inclusion size ( D ) to blob size ( ξ , graftingdistance), whereas λ (= R L /R S ) is the ratio of large tosmall sphere sizes, R L and R S . The blob size ( ξ (cid:39) bg ν ),equivalent to the grafting distance, is decided, indepen-dently from the size ( b ) of monomers, via the adapta-tion of polymer configuration. The term χ ( λ br ), which is A B σ a = σ a = σ a = σ a = - - - - - - c P ( c ) N = σ a = σ a = σ a = σ a = - - - - - - c N = σ a = σ a = σ a = σ a = - - - - - - c N = FIG. 5. Cluster size distribution. (A) A snapshot from simulation carried out with N = 10, σ P a = 0 .
01, and σa = 0 .
09. (B)The cluster size distribution, P ( c ), with σ P a = 0 .
01 for varying brush sizes ( N = 5, 10, 15) and grafting densities ( σ ). Thedashed lines represent P ref ( c ), the cluster size distribution for the protein-only system. (cid:1) σ a = σ a = σ a = σ a = N S ( N ) / S ( ) c * = (cid:2) FIG. 6. The intensity of signaling S ( N ; σ ) normalized by S ( N = 0) (circle) is calculated based on Eq. (14), as a functionof the brush size ( N ) for different grafting densities ( σ ) withthe threshold cluster size c ∗ = 10. a key determinant of the depletion free energy, is maxi-mized for a larger λ br value under the condition of H < h ;(ii) | β ∆ F HS | ∼ λ , whereas | β ∆ F | ∼ √ λ br for λ br (cid:29) β ∆ F HS , the depletion free energy of thehard sphere system, depends linearly on the volume frac-tion of crowders φ (Eq.1), the dependence of area frac-tion of brush polymer (or grafting density, σ ) is given as β ∆ F ∼ λ / ∼ σ / for σ < σ ∗ (Eq.8). (iv) The non-monotonic dependence of depletion free energy on thebrush size N is unique to the brush-induced depletioninteraction (see Appendix B); such feature is absent inthe hard sphere systems in three dimensions.The general consensus on the protein clusters on cellsurface is that the size of membrane protein assembliesis on the order of ∼
100 nm . On the plasma mem-brane of T-cells, CD4 proteins form clusters of size vary-ing from 50 to 300 nm . The size of clusters formed bySNARE-protein syntaxin is 50 – 60 nm, containing 50 –75 molecules . Compared with the quantitative knowl-edge on nanodomains of membrane proteins, the size of protein clusters implicated in Fig. 5A is smaller. Besidesthe brush polymer enhanced assembly of protein cluster,one can consider other physical mechanisms that increasethe effective attraction between proteins, such as inter-protein helix-helix interactions , protein sorting viahydrophobic mismatch , membrane curvature ,and thermal Casimir-like long-range force resulting frommembrane undulation . Upon increasing the LJ po-tential parameter from β(cid:15) PP = 1 to β(cid:15) PP = 2, which in-creases the direct protein–protein interaction drastically(Fig. S2), the contribution of the tail part of P ( c ) be-comes significant, and a host of large and stable pro-tein clusters are more frequently found (Fig. 7). For β(cid:15) PP = 2, the protein cluster size could be as large as m ≈ (cid:1) βϵ = βϵ = - - - - - - m P ( m ) N = (cid:2) β ϵ =1 β ϵ =2 (cid:1) βϵ = βϵ = - - - - - - m P ( m ) N = (cid:2) β ϵ =1 β ϵ =2 βϵ PP = 1 βϵ PP = 2 βϵ PP = βϵ PP = - - - - - - c P ( c ) FIG. 7. The cluster size distribution, P ( c ), for β(cid:15) PP = 1 and2. The two panels shown on the right are the snapshots ofsimulations at β(cid:15) PP = 1 (top) and 2 (bottom). CONCLUDING REMARKS
We have studied polymer brush-induced entropic forcein a system of rigid bodies constrained to move on thesurface. Both of our theory and simulation results showthat the depletion free energy is non-monotonic functionof brush height ( H ), which is determined by the brushsize ( N ) and surface grafting density ( σ ). Our theoreti-cal argument explaining the features of lateral depletionforce is based on the AO theory, which takes only thevolume accessible to individual brush polymers into con-sideration to calculate the depletion free energy in termsof geometrical factors ( N and λ br ), but ignores the ef-fects of correlation between the brush polymers. Despitethe simplicity of our theoretical argument, the main fea-tures of brush-induced depletion interaction observed inthe simulation results are well captured.Our study confirms the depletion force induced assem-bly of protein clusters, although the size of protein do-mains is slightly smaller than that estimated from mea-surements. Given that the brush-induced depletion in-teraction considered here is merely one of many phys-ical mechanisms of protein–protein attraction, of greatsignificance is the semi-quantitative agreement with ex-perimentally observed size of nano-domains. Our studyreiterates that the entropic force, which is brush-induced,is of fundamental importance in cell membrane environ-ment. APPENDIXA. Depletion force
The brush-induced 2D depletion force acting on thetwo objects is βf ( r ) = − ( ∂βF/∂r ) β , βf ( r ) = N b ( N + 1) A (cid:48) overlap ( r ) q ( h, H ) AH − (cid:20) π (cid:16) D + ξ (cid:17) − A overlap ( r ) (cid:21) q ( h, H ) , (A1)for D ≤ r ≤ D + ξ and βf ( r ) = 0 for r > D + ξ . For verylarge system ( A (cid:29) π (( D + ξ ) / ), the denominator ofEq. (A1) is dominated by the term AH , and the depletionforce for D ≤ r ≤ D + ξ simplifies to βf ( r ) = − σ ( N + 1) (cid:34)(cid:18) D + ξ (cid:19) − (cid:16) r (cid:17) (cid:35) / q ( h, H ) H , (A2)where the grafting density of polymer brush σ = N b /A was used. For r > D + ξ , βf ( r ) = 0. It is noteworthythat the depletion force is always attractive ( f ( r ) < D ≤ r ≤ D + ξ .The free energy gain upon aggregation or the workneeded to separate the two inclusions in the brush system apart beyond the distance D + ξ is obtained by integratingthe depletion force from r = D to r = D + ξ , which yieldsthe expression identical to Eq.6. B. Non-monotonicity of depletion free energy gain withincreasing brush polymer size ( H ) Here, we clarify how the non-monotonic change of − β ∆ F arises with increasing H , starting from the ex-pression of the free energy gain ( − β ∆ F ) given in thefirst line of Eq.6. − β ∆ F ∼ N b Hσ / log V ( D ) V ( r ≥ D + ξ ) . (B1)To begin, we define a c the area occupied by the inclusionswhen they are in contact, and a the area occupied by theinclusions when they are separated beyond r = D + ξ .Other parameters N b , H , h , and A are already defined inthe main text. Below we use the condition that the over-lapping area A overlap ( D ) = a − a c ≡ δa is small comparedto A ( δa/A (cid:28) H < h , − β ∆ F ∼ N b Hσ / log ( A − a c ) h ( A − a ) h = N b Hσ / log (cid:20) δaA − a (cid:21) ≈ N b σ / (cid:18) H − a/A (cid:19) δaA ≈ σ / (cid:18) H − a/A (cid:19) χ ( λ br ) (B2)where δa = ξ χ ( λ br ), N b /A = σ , and σξ (cid:39) H < h , − β ∆ F increase linearly with H .(ii) For H ≥ h , − β ∆ F ∼ N b Hσ / log ( AH − a c h )( AH − ah )= N b Hσ / log (cid:20) δa × hAH − ah (cid:21) ≈ N b σ / (cid:18) h − ah/AH (cid:19) δaA = 1 σ / (cid:18) h − ah/AH (cid:19) χ ( λ br ) (B3)Thus, for H ≥ h , − β ∆ F decreases with H from − β ∆ F = σ / (cid:16) h − a/A (cid:17) χ ( λ br ), which isthe maximum value of − β ∆ F , and converges to( h/σ / ) χ ( λ br ) when H/h (cid:29) ACKNOWLEDGMENTS
This study was supported by KIAS Individual GrantsCG076001 (W.K.K.) and CG035003 (C.H.). We thankthe Center for Advanced Computation in KIAS for pro-viding computing resources.
REFERENCES S. Asakura and F. Oosawa, J. Chem. Phys. , 1255 (1954). S. Asakura and F. Oosawa, J. Polym. Sci. , 183 (1958). D. Marrenduzzo, K. Finan, and P. R. Cook, J. Cell. Biol. ,681 (2006). C. Jeon, C. Hyeon, Y. Jung, and B.-Y. Ha, Soft Matter , 9786(2016). H. Kang, P. A. Pincus, C. Hyeon, and D. Thirumalai, Phys. Rev.Lett. , 068303 (2015). R. Phillips, J. Kondev, J. Theriot, N. Orme, and H. Garcia,
Physical Biology of the Cell (Garland Science New York, 2009). E. Roberts, A. Magis, J. O. Ortiz, W. Baumeister, and Z. Luthey-Schulten, PLoS Comput. Biol. , e1002010 (2011). L. Onsager, Ann. NY Acad. Sci. , 627 (1949). M. Dijkstra and D. Frenkel, Phys. Rev. Lett. , 298 (1994). Y. Mao, M. Cates, and H. Lekkerkerker, Phys. Rev. Lett. ,4548 (1995). Y. Mao, M. Cates, and H. Lekkerkerker, J. Chem. Phys. ,3721 (1997). H. Kang, N. M. Toan, C. Hyeon, and D. Thirumalai, J. Am.Chem. Soc. , 10970 (2015). P. D. Kaplan, J. L. Rouke, A. G. Yodh, and D. J. Pine, Phys.Rev. Lett. , 582 (1994). A. D. Dinsmore, D. T. Wong, P. Nelson, and A. G. Yodh, Phys.Rev. Lett. , 409 (1998). A. P. Minton, Curr. Opin. Struct. Biol. , 34 (2000). S. R. McGuffee and A. H. Elcock, PLoS Comput Biol , e1000694(2010). R. Ellis, Trends Biochem. Sci. , 597 (2001), ISSN 0968-0004. L. Sapir and D. Harries, Current opinion in colloid & interfacescience , 3 (2015). B. van den Berg, R. J. Ellis, and C. M. Dobson, EMBO J. ,6927 (1999). Y. Snir and R. D. Kamien, Science , 1067 (2005). A. Kudlay, M. S. Cheung, and D. Thirumalai, Phys. Rev. Lett. , 118101 (2009). H. X. Zhou, G. Rivas, and A. P. Minton, Annu. Rev. Biophys. , 375 (2008). A. Elcock, Curr. Opin. Struct. Biol. , 196 (2010). M. S. Cheung, D. Klimov, and D. Thirumalai, Proc. Natl. Acad.Sci. U. S. A. , 4753 (2005). D. L. Pincus, C. Hyeon, and D. Thirumalai, J. Am. Chem. Soc. , 7364 (2008). D. Kilburn, J. H. Roh, L. Guo, R. M. Briber, and S. A. Woodson,J. Am. Chem. Soc. , 8690 (2010). N. Denesyuk and D. Thirumalai, J. Am. Chem. Soc. , 11858(2011). A. G. Gasic, M. M. Boob, M. B. Prigozhin, D. Homouz, C. M.Daugherty, M. Gruebele, and M. S. Cheung, Phys. Rev. X ,041035 (2019). A. Soranno, I. Koenig, M. B. Borgia, H. Hofmann, F. Zosel,D. Nettels, and B. Schuler, Proc. Natl. Acad. Sci. U. S. A. ,4874 (2014). G.-W. Li, O. G. Berg, and J. Elf, Nature Phys. , 294 (2009). J. S. Kim, V. Backman, and I. Szleifer, Phys. Rev. Lett. ,168102 (2011). C. A. Brackley, M. E. Cates, and D. Marenduzzo, Phys. Rev.Lett. , 108101 (2013). B. J. Reddy, S. Tripathy, M. Vershinin, M. E. Tanenbaum, J. Xu,M. Mattson-Hoss, K. Arabi, D. Chapman, T. Doolin, C. Hyeon,et al., Traffic , 658 (2017). G. Nettesheim, I. Nabti, C. U. Murade, G. R. Jaffe, S. J. King,and G. T. Shubeita, Nature Physics , 1144 (2020). J. Dzubiella, H. L¨owen, and C. Likos, Phys. Rev. Lett. , 248301(2003). L. Angelani, C. Maggi, M. Bernardini, A. Rizzo, andR. Di Leonardo, Phys. Rev. Lett. , 138302 (2011). T. Sanchez, D. T. Chen, S. J. DeCamp, M. Heymann, andZ. Dogic, Nature , 431 (2012). R. Ni, M. A. C. Stuart, and P. G. Bolhuis, Phys. Rev. Lett. ,018302 (2015). L. Huber, R. Suzuki, T. Kr¨uger, E. Frey, and A. Bausch, Science , 255 (2018). T. Sintes and A. Baumg¨artner, Biophys. J. , 2251 (1997). K. Suda, A. Suematsu, and R. Akiyama, arXiv preprintarXiv:2011.06232 (2020). M. J. Paszek, C. C. DuFort, O. Rossier, R. Bainer, J. K.Mouw, K. Godula, J. E. Hudak, J. N. Lakins, A. C. Wijekoon,L. Cassereau, et al., Nature , 319 (2014). G. J. Bakker, C. Eich, J. A. Torreno-Pina, R. Diez-Ahedo,G. Perez-Samper, T. S. van Zanten, C. G. Figdor, A. Cambi,and M. F. Garcia-Parajo, Proc. Natl. Acad. Sci. U. S. A. ,4869 (2012). C. Selhuber-Unkel, M. L´opez-Garc´ıa, H. Kessler, and J. P. Spatz,Biophys. J. , 5424 (2008). T. S. van Zanten, J. G´omez, C. Manzo, A. Cambi, J. Buceta,R. Reigada, and M. F. Garcia-Parajo, Proc. Natl. Acad. Sci. U.S. A. , 15437 (2010). M. F. Garcia-Parajo, A. Cambi, J. A. Torreno-Pina, N. Thomp-son, and K. Jacobson, J. Cell. Sci. , 4995 (2014). O. Soubias, W. E. Teague Jr, K. G. Hines, and K. Gawrisch,Biophys. J. , 1125 (2015). A. Kusumi and J. S. Hyde, Biochemistry , 5978 (1982). J. U. Kim and B. O’Shaughnessy, Macromolecules , 413(2006). R. K. Spencer and B.-Y. Ha, Macromolecules , 1304–1313(2021). M. J. Williams, P. E. Hughes, T. E. O’Toole, and M. H. Ginsberg,Trends in cell biology , 109 (1994). L. Kornberg, H. S. Earp, J. T. Parsons, M. Schaller, and R. Ju-liano, J. Biol. Chem. , 23439 (1992). M. J. Paszek, D. Boettiger, V. M. Weaver, and D. A. Hammer,PLoS Comput Biol , e1000604 (2009). B. Cheng, W. Wan, G. Huang, Y. Li, G. M. Genin, M. R. Mofrad,T. J. Lu, F. Xu, and M. Lin, Sci. Adv. , eaax1909 (2020). P. G. de Gennes, Macromolecules , 1069 (1980). L. Liu, P. A. Pincus, and C. Hyeon, Macromolecules , 1579(2017). L. Liu and C. Hyeon, J. Chem. Phys. , 163302 (2018). S. Alexander, J. Phys. , 983 (1977). M. Rubinstein, R. H. Colby, et al.,
Polymer physics , vol. 23 (Ox-ford university press New York, 2003). T. Veitshans, D. Klimov, and D. Thirumalai, Folding Des. , 1(1997). C. Hyeon and D. Thirumalai, J. Am. Chem. Soc. , 1538(2008). T. Lang and S. O. Rizzoli, Physiology , 116 (2010). F. Baumgart, A. M. Arnold, K. Leskovar, K. Staszek, M. F¨olser,J. Weghuber, H. Stockinger, and G. J. Sch¨utz, Nature methods , 661 (2016). T. Lukeˇs, D. Glatzov´a, Z. Kv´ıˇcalov´a, F. Levet, A. Benda,S. Letschert, M. Sauer, T. Brdiˇcka, T. Lasser, and M. Cebecauer,Nat. Commun. , 1 (2017). J. J. Sieber, K. I. Willig, R. Heintzmann, S. W. Hell, and T. Lang,Biophys. J. , 2843 (2006). N. Ben-Tal and B. Honig, Biophys. J. , 3046 (1996). J. H. Lorent and I. Levental, Chemistry and physics of lipids , 23 (2015). V. Anbazhagan and D. Schneider, Biochimica Et BiophysicaActa (BBA)-Biomembranes , 1899 (2010). U. Schmidt, G. Guigas, and M. Weiss, Phys. Rev. Lett. ,128104 (2008). D. Milovanovic, A. Honigmann, S. Koike, F. G¨ottfert,G. P¨ahler, M. Junius, S. M¨ullar, U. Diederichsen, A. Janshoff,H. Grubm¨uller, et al., Nat. Commun. , 1 (2015). B. West, F. L. Brown, and F. Schmid, Biophys. J. , 101 (2009). H. T. McMahon and J. L. Gallop, Nature , 590 (2005). B. J. Reynwar, G. Illya, V. A. Harmandaris, M. M. M¨uller,K. Kremer, and M. Deserno, Nature , 461 (2007). M. Goulian, R. Bruinsma, and P. Pincus, EPL (Europhysics Let-ters) , 145 (1993). J.-M. Park and T. Lubensky, J. Phys. I , 1217 (1996). B. B. Machta, S. L. Veatch, and J. P. Sethna, Phys. Rev. Lett. , 138101 (2012). FIG. S1. The radial distribution function, g ( r ), betweenthe proteins for different brush sizes ( N ) with σa = 0 . σ P a = 0 .
01 and β(cid:15) PP = 1. - - - - βϵ PP B , N = / a FIG. S2. The second virial coefficient, B ,N =0 , for theprotein-only systems as a function of the interaction strength β(cid:15) PP between the proteins with σ P a = 0 .
01, where B ref2 isdepicted at β(cid:15) PP = 1. σ a = σ a = σ a = σ a = N 〈 H 〉 / a FIG. S3. The mean brush height, (cid:104) H (cid:105) , as a function of thebrush size ( N ) for different grafting densities ( σ ), shown inlog-log scales. The solid lines depict the scaling relation H = Nσ /3
01, where B ref2 isdepicted at β(cid:15) PP = 1. σ a = σ a = σ a = σ a = N 〈 H 〉 / a FIG. S3. The mean brush height, (cid:104) H (cid:105) , as a function of thebrush size ( N ) for different grafting densities ( σ ), shown inlog-log scales. The solid lines depict the scaling relation H = Nσ /3 b /3