Sequence Similarity Between the Adsorbed Polymer Chains.

To examine whether the protein would favor certain sequences out of the available random sequences, we characterize the correlation in the monomer sequence of the adsorbed polymer chains for the given model protein. We define the order parameters to characterize the similarity between the sequences within the same batch and between those from two batches (Materials and Methods). For ϕA=0.5−0.8, the sequence similarity order parameter for the polymer chains with random sequences within individual batches Sintra ranges from 0.55 to 0.60, which means that the sequences are sufficiently uncorrelated.

Our simulation results reveal that the sequence similarity between the adsorbed chains within the individual batches is well beyond that between all of the chains and the threshold value of 0.6 for close matches. We performed simulation of 20 different batches, each with Nc=50 polymer chains and compute the sequence correlation between the adsorbed chains among the batches Sinter (Fig. 2A). The average value of Sinter is 0.78±0.006, indeed indicating that the adsorbed monomers from any two batches are strongly correlated. We also find that the average similarity order parameter across all of the values of εhp=(0.5−2.0) (Fig. 2B) is Sinter=0.73±0.008, despite the fact that the number of the adsorbed chains nads varies with εhp. As can be seen from Fig. 2C (and SI Appendix, Fig. S3), the sticky fraction of the adsorbed chains, (fA)ads, is different from that of all of the chains. All of these results strongly suggest that certain sequences are more favored to adsorb to the protein than others.

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Fig. 2.

Sequence similarity analysis. (A) Sequence similarity order parameter between the batches Sinter of the adsorbed chains for εhp=1.0. (B) Sequence similarity order parameter for the adsorbed chains Sinter for different values of εhp. εpp=0.8 and ϕA=0.7. Different polymer batches are used for each value of εhp. A value of S=0.5 corresponds to uncorrelated sequences and S≥0.6 to close matches. (C) Histograms of the sticky component fraction, fA, of all of the chains in a batch and of the adsorbed chains, (fA)ads.

Versatility.

As shown in the work by Panganiban et al. (35), random heteropolymers of the same average composition are able to retain the activity of numerous enzymes in toluene equally efficiently. This is of practical importance because it suggests that the approach is not restricted to specific proteins or solvents, which is advantageous compared with other approaches such as polymer conjugation. Consistent with experimental findings in ref. 35, our simulation results indicate that random copolymers with the same average composition covers the proteins under investigation (Fig. 3A) fairly effectively. For instance, for ϕA=0.7 surface coverage Γ varies from 0.75 to 0.95 for 1TCA and PDB ID codes 1A2Q, 1EZM, 1CEX, and 1UBQ (Fig. 3B). The variation of Γ with ϕA is dependent upon specific proteins. For 1A2Q, there exists an optimal value of ϕA for maximum surface coverage, similar to what was observed with 1TCA. For 1CEX and 1EZM, Γ is consistently high and fairly independent of ϕA.

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Fig. 3.

Encapsulation of numerous proteins. (A) CG model of the proteins under investigation: ubiquitin [Protein Data Bank (PDB) ID code 1UBQ], cutinase (PDB ID code 1CEX), pseudolysin (PDB ID code 1EZM), and subtilisin (PDB ID code 1A2Q). (B) Surface coverage as a function of ϕA for εhp=1.2 and εpp=0.8 for the model proteins under investigation. (C) Surface coverage as a function of the variance in the distances between the adsorption sites, sd2, for ϕA=0.7, εhp=1.2 and εpp=0.8.

To better correlate the ability of the random copolymers to encapsulate the proteins with the features of the protein surface, we compute the distance distribution of the protein adsorption sites P(d) (SI Appendix, Fig. S5). The variance in the distances between the adsorption sites, sr2, is given by sd2=⟨d2⟩−⟨d⟩2. Fig. 3C shows that the proteins with smaller values of sd2, i.e., with narrow distance distributions between the adsorption sites (i.e., 1UBQ, 1CEX, and 1EZM), get encapsulated well by the random copolymers for all of the average compositions tested. Meanwhile, for those with broad distance distributions between the adsorption sites, the random copolymers should have an optimal average composition so to maximize the protein surface coverage. This suggests that the distribution of the distance between the adsorption sites, in addition to the size and shape of the proteins, would be important features for designing the encapsulating random copolymers. It is not possible for the short copolymer chains examined here to make a rigorous analysis on the relationship between the average distance between sticky monomers l and the average distance between adsorbing site on the protein d because the fluctuations in both d and l (SI Appendix, Figs. S5 and S6) are very large.

Surface Coverage as a Function of Average Polymer Adsorbing Fraction and Solvent Selectivity.

Large values of ϕA and εpp favor the folding of the polymer chains into few energetically favored conformations. Consequently, the adsorption strength, εhp, should be sufficiently high to unfold the polymer chains or, in other words, to induce the dewetting–wetting transition of the polymer adsorbed layers on the protein surface. We show in Fig. 4 and SI Appendix, Fig. S7 the variation of Γ with respect to ϕA for different values of εpp and εhp. For εpp=0.3 (Fig. 4A), the solvent is good for the polymer A beads, Γ increases monotonously with ϕA for all values of the adsorption strengths εhp. For stronger attractions between the type A beads—that is, when the solvent becomes more unfavorable to those beads, εpp=0.8 (Fig. 4B) and εpp=1.2 (SI Appendix, Fig. S7)—there exists a maximum value of Γ, which monotonously increases with the adsorption strength εhp. Importantly, we observe that the maximum surface coverage shifts to lower values of ϕA as εpp increases. The presence of an optimal ϕA for maximum protein surface coverage at large enough εpp will be explained below, where we analyze the microscopic details of the adsorbed polymer layers.

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Fig. 4.

Surface coverage Γ as a function of the average adsorbing fraction ϕA for different adsorption strengths, εhp. εpp=0.3 (A) and εpp=0.8 (B). The error bars (Right) are the standard deviations from averaging over 10 uncorrelated configurations.

Multilayer Adsorption Model.

Fig. 5 shows that the surface coverage Γ varies with the adsorption strength, εhp for different values εpp. For all of the values of ϕA studied, Γ increases with εhp until approaching a saturated value Γ∞, which depends on εpp. The inflection point in Γ(εpp), which marks a transition from the weak adsorption to a strong one, shifts to higher values of the adsorption strength εhp as ϕA and εpp increase. This is because the net polymer–protein attraction should be sufficiently strong to overcome the increased attraction between the polymer beads of type A for the polymer chains to fully cover the protein.

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Fig. 5.

(A–C) Surface coverage Γ as a function of the attraction strength between sticky protein beads and polymer beads, εhp for different values of εpp. ϕA=0.5 (A), ϕA=0.7 (B), and ϕA=0.9 (C). The error bars are the standard deviations from averaging over multiple steady-state snapshots each obtained at the end of the equilibration cycle. (D) Normalized surface coverage Γ*=Γ/Γ∞ as a function of the reduced adsorption energy εhp* for different values of εpp for ϕA=0.7, where εhp*=αεhp+logK (main text). The black curve is the fitting function of the form f(x)=1/(1+exp(−x)).

The variation of surface coverage with respect to the adsorption energy (Fig. 5 A–C) is reminiscent of the S-shaped adsorption isotherms (i.e., type 5 in the International Union of Pure and Applied Chemistry classification), where the polymer beads of type A of the adsorbed chains act as secondary adsorption sites, leading to the formation of multilayers of adsorbed type A beads. Qi et al. (37) proposed an analytical model for the adsorption of water vapor adsorbing onto activated carbon, where the adsorption capacity is governed by the ratio between the actual partial and saturated partial water vapor pressures. We propose a similar model where the protein surface coverage (Γ) and adsorption energy (εhp) play similar roles to the adsorption capacity and pressure ratio, respectively:∂Γ*∂εhp∼1−Γ*Γ*,(1)where Γ*=Γ/Γ∞ and Γ∞ is the saturated surface coverage. Analogous to the model by Qi et al. (37), our model is based on the following assumptions: (i) The number of secondary adsorption sites is proportional to the amount of polymer chains adsorbed for a specified εhp; (ii) the number of primary adsorption sites is proportional to the remaining adsorption capacity, (Γ∞−Γ); and (iii) the driving force for the change in adsorption capacity with the change in adsorption energy is proportional to the product of (Γ∞−Γ) and Γ. The solution of Eq. 1 has the form of the logistic function:Γ=Γ∞1+K⁡exp(−αεhp),, (2)where α and K are the model parameters. When the datasets in Fig. 5 are fitted against the model given in Eq. 2, the data points collapse nicely into a single curve (Fig. 5D). The fitting parameter K generally increases with εpp for different values of ϕA (SI Appendix, Fig. S8A), corresponding to the shift in the inflection point of the adsorption curves to higher values of εhp, as earlier shown in Fig. 5. Conversely, the fitting parameter α is rather insensitive to εpp and ϕA (SI Appendix, Fig. S8B). The fact that the analytical model is highly consistent with our simulation results suggests that the assumptions made in our model are relevant to the underlying physics of the system of interest. Our supplementary simulations further show that the polymer/protein molar ratio at a given polymer bulk density and temperature, and, thus, the excess chemical potential of the random copolymers in bulk relative to adsorption, influences protein surface coverage up to a certain value (SI Appendix, Fig. S10). We note that as the concentration of proteins and/or polymers increases, highly heterogeneous gel-like structures (34) or macroscopically segregated polymer chains are expected, depending on the relative magnitude of the different interactions. The delicate balance of these competitive effects is evidenced in the transition in the human eye lenses that undergo macroscopic segregation from a homogeneous mixture to insoluble protein aggregates manifested as cataracts (38).

Structural Features of the Adsorbed Polymer Layers.

To elucidate the relationship between the protein surface coverage, Γ, and the average polymer composition, ϕA, adsorption energy, εhp, and solvent selectivity, εpp, we analyze in detail the structural features of the polymer chains that are in contact with the protein surface beads. First, we compute the number of adsorbed polymer chains, nads, which is proportional to the protein surface coverage, Γ, for different values of ϕA and εhp. As shown in Fig. 6A, for a given value of ϕA nads always increases with εhp, which is consistent with Γ(εhp) (Fig. 5). For a given value of εhp, the variation of nads with ϕA depends on εpp. Specifically, for εpp=0.8, nads exhibits a nonmonotonic behavior with respect to ϕA for εhp>0.5. On the one hand, nads is proportional to the energetically favored polymer–protein contacts governed by the adsorption strength. Consequently, nads, and hence Γ, increases with ϕA as expected. On the other hand, nads is associated with the entropic penalty due to the reduced number of polymer conformations in an adsorbed state compared with polymers that are far away from the protein. For sufficiently high values εpp, the polymers in solution adopt a few energetically favorable conformations to maximize the contacts between type A beads such that the entropic penalty due to adsorption becomes small.

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Fig. 6.

Structural features of the adsorbed polymer layers. (A) Number of adsorbed polymer chains, nads, as a function of ϕA for different adsorption strengths εhp when εpp=0.8. The error bars are obtained from averaging over 10 steady-state snapshots at the end of the equilibration cycles. (B) Surface area fraction of solvophilic monomers (type B) at the outer layer of the adsorbed polymers for different values of εhp and ϕA for εpp=0.8.

For larger values of ϕA, the thermodynamics of the polymer adsorption in this regime is governed mainly by the energetics of polymer–polymer and polymer–protein interactions. As a result, there exists a threshold value ϕA* (dependent on εpp) above which forming polymer–protein contacts is less energetically favorable than keeping the A beads in an aggregated state. This leads to a decrease in nads (Γ shows similar behavior; Fig. 4B and SI Appendix, Fig. S7) with ϕA for ϕA>ϕA*. The maximum value of nads, and hence of Γ, may therefore be attributed to the competition between the tendency of the polymers to maximize their energetically favored contacts and their tendency to avoid the entropic loss due to confinement to the surface. These qualitative arguments are in agreement with the results of Srebnik et al. (8), who studied adsorption of random copolymers onto a flat surface randomly decorated by adsorption sites. They found that when the polymer intersegment interaction is strong enough, the entropic penalty due to adsorption becomes comparable to, or even smaller than, the energetic gain due to the segment–surface interaction. Our CG model underestimates the complexity in polymer architecture. Thus, it excludes the effects of sequence-dependent persistent lengths as well as small side chains that interact with the protein patterns. However, for architectural details on length scales smaller than the mean distance between the adsorption sites of the proteins, which is ∼3.0 nm in our tested cases, we expect small corrections to the protein surface coverage trends found with our model.

The pair correlation function of the attractive beads on the polymer chains and those on the protein and the radial density profile of the polymer beads (SI Appendix, Fig. S9) further reveal that the adsorbed polymer chains form a multilayer structure and that the polymer layers would behave like a liquid with number density well below 0.4σ−3.

The effective interaction between the protein and the implicit solvent in the presence of the encapsulating random copolymers is characterized by the outer layers of the polymer shell and the fraction of the protein surface that is uncovered. To characterize the effective interaction between the protein and the solvent mediated by the adsorbed polymer layers, we analyze the structural property of the polymer outer layer. Using the same method to estimate the surface coverage, we compute the area fraction, Γp, occupied by the solvophilic polymer beads (i.e., those of type B) at the distance (Rmax−rc) away from the protein center of mass. Rmax is the maximum distance between the polymer beads to the protein center of mass. Fig. 6B shows the variation in Γp with respect to εhp and ϕA. For any given value of εhp, Γp decreases as ϕA increases, which is expected because the fraction of solvophilic beads decreases. For a given value of ϕA, we observe that increasing εhp leads to a slight increase in Γp, and that Γp is strongly correlated with the fraction of the solvophilic beads ϕB=1−ϕA for εhp<1.2. This is interesting because it is suggested that the phase behavior of the proteins coated by the random polymers resembles that of the random copolymers at the given ϕA.