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Polymer Brush-Induced Depletion Interactions and Clustering Of Polymer brush-induced depletion interactions and clustering of membrane proteins Anvy Moly Tom,1 Won Kyu Kim,1, a) and Changbong Hyeon1 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 size of brush. The depletion interaction, which is determined by the ratio of protein size to the grafting distance between brush polymers, increases linearly with brush size as long as the polymer brush height is shorter than the protein size. When the brush height exceeds the protein size, however, the depletion attraction among proteins is slightly reduced. We also explore the possibility of brush polymer-induced assembly of a large protein cluster, which can be related to one of many molecular mechanisms underlying recent experimental observations of integrin nanocluster formation and signaling. INTRODUCTION and 8 % of soluble proteins7. In the cellular environment, the depletion force is one of the fundamental forces of In 1950s, Asakura and Oosawa (AO) proposed a simple great importance. theoretical model to explain the interaction of entropic The basic principle of AO theory on rigid bodies origin between colloidal particles immersed in a solution with spherical symmetry is straightforward; however, of macromolecules1,2, which is of great relevance to our application of the idea to the repertoire of biologi- understanding of organization and dynamics in cellular cal and soft materials requires quantitative assessment environment. According to the AO theory, rigid spherical of entropy, which is nontrivial especially when crow- objects immersed in the solution of smaller hard spheres ders are characterized with non-spherical shape and/or representing the macromolecules are expected to feel fic- with polydispersity8{12 and when the system is un- titious attraction, termed depletion force. While the in- der a special boundary condition13,14. For the past teraction energy of the system remains unchanged, the decades, there has also been much interest toward un- spherical objects can be attracted to each other. Bring- derstanding of the effects of crowding in biology3,15{19, ing the large spherical objects into contact can increase which includes crowding-induced structural transitions the free volume accessible to the smaller hard spheres in disordered chiral homopolymers20,21, protein/RNA comprising the medium, and hence increasing the total folding22{29, gene regulation through DNA looping30, entropy of the hard sphere system (∆S > 0). The free genome compaction31, efficient search of proteins for tar- energy reduction due to the gain in entropy is gets on DNA32, and molecular motors33,34. Further, it is worth mentioning a series of effort to understand the 3 dynamics of active matter in the language of depletion ∆F = −T ∆S = − λ + 1 φk T; (1) HS 2 B forces35{39. Besides the examples of depletion force-induced dy- where λ is the size ratio of large to small hard spheres, namics that all occur in three dimensional space, the and φ is the volume fraction of small spheres compris- AO theory can be extended to lateral depletion ef- 1,3,4 ing the surrounding medium . For a fixed value of fects on the objects whose motion is confined in flat φ, the disparity in size between colloidal particles (large surfaces40,41. For biological membrane where the area spheres) and macromolecular depletants (small spheres), fraction of membrane-embedded proteins is as high as characterized with the parameter λ, is the key determi- 15 { 30 %, the formation of protein clusters or nano- or 5 nant of the magnitude of depletion free energy . The micro-domains42{46 is of great relevance to understanding effect of crowding environment on the aggregation of col- the regulation of biological signal transduction and cell- arXiv:2102.12979v1 [cond-mat.soft] 25 Feb 2021 loidal particles becomes substantial when λ 1. The to-cell communication. Although other physical mecha- cellular environment is highly crowded, such that 30 % nisms are still conceivable, lateral depletion interactions of cytosolic medium is filled with macromolecules, ren- between membrane embedded proteins can arise from the dering the interstitial spacing between macromolecules fluctuations of lipids40,47,48 or other polymer-like com- 6 comparable to the average size of proteins ∼ 4 nm . More ponents comprising fluid membrane49,50, contributing to specifically, this volume fraction of E. coli mixture is con- protein{protein attraction and clustering. In this con- tributed by 11 % of ribosome, 11 % of RNA polymerase, text, the formation of integrin nanodomain which en- ables cell-to-cell communications via signaling51{54, par- ticularly, the bulky glycocalyx-enhanced integrin cluster- ings and the associated signaling-induced cancer metas- a)Electronic mail: [email protected] tasis observed by Paszek et al.42 make the brush polymer- 2 FIG. 1. Brush-induced depletion interactions. (A) Illustration of brush polymers, each of which is organized into a string of blobs 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 grey sphere (see (A)). (D) Diagram to calculate the brush-induced depletion interaction between the two cylindrical objects. The 2 area inside the dashed line, corresponding to 2π[(D + ξ)=2] − Aoverlap(r) in Eq.3, is the area inaccessible to the blob of polymer brush of size ξ. The shaded region in pale red is the overlapping area of the two discs of radius (D + ξ)=2, separated by the distance r. 3=5 induced depletion interaction between membrane pro- the grafting distance (ξ) is smaller than RF = bN , i.e., teins and their clustering a topic of great relevance to ξ < RF , where RF is the Flory radius of the polymer in investigate. good solvent, each polymer reorganizes into a string of In this paper, we study the lateral depletion interac- self-avoiding blobs due to excluded volume interactions tions between rigid inclusions embedded in the mobile with the neighboring polymers, forming a polymer brush polymer brushes in 2D surface in the spirit of the AO of height H where N=g blobs of size ξ consisting of g theory in its simplest form. We compare the results from segments fill the space above the surface (Fig. 1A)55. In our simulations with our theoretical predictions. By an- this case, the grafting density σ = Nb=A, the number of alyzing the distribution of brush polymer-enhanced pro- polymer chains (Nb) grafted on an area A, is related to tein clusters obtained from our simulations, we attempt the blob size (or the grafting distance) as σ ' 1/ξ2. It to link the brush-size dependent populations of giant pro- is straightforward to show using the blob argument that tein clusters with the strength of signal transduction ob- the brush height H scales with N and σ as55,58,59 served in Paszek et al.'s measurement. H = Nσ1=3b5=3: (2) THEORY: BRUSH-INDUCED LATERAL DEPLETION INTERACTIONS Our interest is in the lateral depletion force between two cylindrical inclusions embedded in the polymer brush As illustrated in Fig. 1A, we consider flexible polymer system, when the two inclusions, constrained to move in brushes, each consisting of N + 1 monomers of size (di- xy plane, are separated by a fixed distance r (Fig. 1B, C). ameter) b. One end of individual chain is grafted to the In the presence of the cylindrical inclusions, the volume surface but is free to move. If the grafting density σ is accessible to the individual polymer chains is determined 2 55{57 large enough to satisfy σRF > 1 or equivalently if as follows, depending on r. 8 2 D+ξ <>AH − 2π 2 − Aoverlap(r) q(h; H); for D ≤ r ≤ D + ξ V (r) = 2 (3) > D+ξ :AH − 2π 2 q(h; H); for r > D + ξ: Here, Aoverlap(r) is the overlapping area between two cir- pale red in Fig. 1D, is cular discs of radius (D + ξ)=2, the region demarcated in 1=2 Z (D+ξ)=2 " 2 # D + ξ 2 Aoverlap(r) = 4 − ρ dρ. (4) r=2 2 3 This is maximized when r = D, and its value can be writ- ten in terms of the area defined by the square of graft- A ing distance, ξ2, multiplied with a dimensionless factor χ(λbr), Z 1 2 2 1=2 Aoverlap(D) = ξ (1 + λbr) (1 − x ) dx : (5) λbr 1+λbr h > H h < H | {z } ≡χ(λbr) B b2 0.01 -5 σ = where σb2=0.02 σb2=0.04 -10 σb2=0.08 F 1 2 −1 λbr p χ(λbr) = (1 + λbr) cos − λbr 1 + 2λbr βΔ -15 2 1 + λbr ( -20 π + π−2 λ + O(λ2 ); for λ 1 4p 2 br br br ' 2 2 p -25 3 λbr; for λbr 1; 0 10 20 30 40 50 60 N isp a monotonically increasing function of λbr = D/ξ ' D σ, the ratio of the diameter of the inclusions to the C -12 grafting distance (or the blob size). Next, the function -14 -16 q(h; H) ≡ HΘ(h − H) + hΘ(H − h), defined with the F -18 step function, signifies (i) q(h; H) = H when the brush βΔ -20 N=10 height (H) is shorter than the height of the inclusion N=15 -22 N=20 (h)(H < h); and (ii) q(h; H) = h when the brush is -24 N=25 grown over the inclusion (H > h) (see Fig.2A).
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