Chinese Journal of Chemical Physics  2016, Vol. 29 Issue (5): 564-570

#### The article information

Huang Hao-zhi, Chen Yu-hao, Yu Wan-cheng, Luo Kai-fu

Superselective Adsorption of Multivalent Polymer Chains to a Surface with Receptors

Chinese Journal of Chemical Physics, 2016, 29(5): 564-570

http://dx.doi.org/10.1063/1674-0068/29/cjcp1603060

### Article history

Accepted on: May 17, 2016
Superselective Adsorption of Multivalent Polymer Chains to a Surface with Receptors
Huang Hao-zhi, Chen Yu-hao, Yu Wan-cheng, Luo Kai-fu
Dated: Received on March 29, 2016; Accepted on May 17, 2016
CAS Key Laboratory of Soft Matter Chemistry, Department of Polymer Science and Engineering, University of Science and Technology of China, Hefei 230026, China
Author: E-mail:ywcheng@ustc.edu.cn
Abstract: Multivalent polymer chains exhibit excellent prospect in biomedical applications by serving as therapeutic agents. Using three-dimensional (3D) Langevin dynamics simulations, we investigate adsorption behaviors of multivalent polymer chains to a surface with receptors. Multivalent polymer chains display superselective adsorption. Furthermore, the range of density of surface receptors at which a multivalent polymer chain displays a superselective behavior, narrows down for chains with higher density of ligands. Meanwhile, the optimal density of surface receptors where the highest superselectivity is achieved, decreases with increasing the density of ligands. Then, the conformational properties of bound multivalent chains are studied systematically. Interestingly, we find that the equilibrium radius of gyration Rg and its horizontal component have a maximum as a function of the density of surface receptors. The scaling exponents of Rg with the length of chain suggest that with increasing the density of surface receptors., the conformations of a bound multivalent polymer chain first fall in between those of a two-dimensional (2D) and a 3D chain, while it is slightly collapsed subsequently.
Key words: Multivalent polymers     Langevin dynamics simulations     Superselective adsorption
I. INTRODUCTION

Multivalent interactions exist widely in nature, which play a crucial role in many biological processes, including the adhesion of virus to cells, cell recognition and cell signaling [1-3]. Interactions between multiple ligands on a biological entity and multiple receptors on another one are considered to be multivalent. In contrast to weak monovalent binding, multivalent interactions offer the advantage of a multiple and thus dramatically enhanced binding on a molecular scale. Moreover, the superselectivity is an important feature of multivalency, which implies that the number of ligands that are bound to the surface increases faster than that linearly with the density of receptors [4-6].

Recently, multivalent polymers were exploited in biological and artificial systems, such as therapeutic agents, gene delivery, and supramolecular materials [7-12]. Previous experiments have shown that the ability of multivalent polymers selectively attaching to cell membranes is affected by the density of cell receptors [13], the polymer size and shape [14, 15], the flexibility of polymer backbone [16], the density of ligands [17], and the distribution of receptors on the cell surface [18, 19].

From the point of view of thermodynamics, the binding properties of multivalent polymer chains do share many similarities. Like most systems, multivalent interactions are governed by thermodynamics. An ideal equilibrium system is a result of minimizing free energy, which means low enthalpy and high entropy [20-25]. However, system with multivalent interactions should make a subtle balance between enthalpy and entropy.

While the local dynamics of multivalent polymers targeting to cell surface receptors is evidently important for practical applications, it is still obscure and rather difficult to assess experimentally. A likely reason for less investigation of self-assembly of multivalent polymers at a surface is its limited availability of sufficiently pure and suitably functionalized cells or matrixes [26]. To gain further microscopic details of this process, computer simulations can be rather useful. Compared with experiments, computer simulations could achieve accurate controls over the binding strength of multivalent interactions between polymers and surfaces, and possess an excellent tunability of the density of surface receptors.

In the present work, we use three-dimensional (3D) Langevin dynamics simulations to investigate the properties of multivalent polymer chains-surface binding systems. Since the dependence of the adsorption on the bulk concentration of ligands is not particularly sharp [27, 28], and the exact number of individual ligand-receptor interaction remains difficult to assess in experiments, we focus here on the adsorption behavior of a single multivalent polymer chain to a surface with variable densities of receptors. In our simulations, we consider a planar surface with a changeable density of homogeneously distributed surface receptors, which has been found for integrins on cancer cells [29]. In this work, we pay attention to the effects of the density of surface receptors and the density of ligands on the kinetic and conformational properties of a multivalent polymer chain. A deeper understanding on these two aspects will be helpful to design optimal synthetic multivalent polymer chains that enhance the target binding onto surface receptors.

II. MODEL AND METHODS

The 3D model geometry we considered in this work is sketched in Fig. 1, where the ligands in a multivalent polymer chain could bind to the surface receptors. The polymer chain is modeled as a bead-spring chain of Lennard-Jones (LJ) particles with the finite extension nonlinear elastic (FENE) potential. Each LJ particle represents a segment. The excluded volume effect of the polymer chain is achieved by applying the following short-range repulsive LJ potential between segments.

 (1)
 FIG. 1 Schematic representation of the binding of ligands (in red) in a multivalent polymer chain to a surface carrying multiple immobile receptors (in blue). Snapshots of the chain conformation (a) before adsorption, and (b) after adsorption. Here the length of chain is $N$=100, the density of surface receptors is $\varphi$=0. 30, and the density of ligands is $\phi$=0. 20.

Here $\sigma$ is the diameter of a segment, and $\varepsilon_0$ is the well depth of the LJ potential. In this work, we defined $\sigma$=1, $\varepsilon_0$=1. The connectivity between neighboring segments is modeled by a FENE spring:

 (2)

where $r$ is the distance between consecutive segments, $k$=30$\varepsilon_0$/$\sigma^2$ is the spring constant, and $R_0$=1. 5$\sigma$ is the maximum allowed distance between connected segments. The LJ parameters $\varepsilon_0$, $\sigma$, and the segmental mass $m$=1 fixed the energy, length, and mass scales of the system, respectively. The time scale is then given by $t_\mathrm{LJ}$=($m$$\sigma^2/\varepsilon_0)^{1/2}. The thermal energy of the system is set as k_\mathrm{B}$$T$=1. 2$\varepsilon_0$.

The homogenous surface at the $z$=0 plane with an area $S$=$L_x$$L_y=150\sigma$$\times$150$\sigma$ is a virtual wall that interacts with the chain segments through the above repulsive LJ potential. $N_{\mathrm{sr}}$ surface receptors at the $z$=1 plane are evenly distributed. Thus, the density of surface receptors is given by $\varphi$=$N_{\mathrm{sr}}$/S. $N_\mathrm{li}$ ligands in a multivalent chain interact with the surface receptors through the attractive Morse potential [30]

 (3)

Here $D_\mathrm{e}$=5$\varepsilon_0$ is the well depth of the Morse potential, $r_0$=1. 1$\sigma$ is the equilibrium distance between a ligand and a surface receptor, and $a$=7. 5/$\sigma$ is a constant that controls the width of the potential. If the distance between a ligand and a surface receptor is smaller than 2. 0$\sigma$, the ligand is considered to be adsorbed. Note that the density of ligands is denoted by $\phi$=$N_\mathrm{li}$/$N$ in the following.

The motion of a chain segment in the simulations is described by the Langevin equation:

 (4)

Here $\xi$ and $\upsilon$$_i is the friction coefficient and the velocity of a segment, respectively. -\nabla$$U_i$ and $-\xi$$\upsilon_i is the conservative, frictional forces exerted on the ith segment, respectively. F_i^R is the random force which satisfies the fluctuation-dissipation theorem [31]. The Langevin equation is integrated in time by the method proposed by Ermak and Buckholz [32]. Initially, the first segment of the chain is tethered to the center of the surface with the remaining segments being under thermal collisions described by the Langevin thermostat. During this relaxation process, the repulsive LJ potential is applied for all particle pairs. Then, the first monomer is released and the Morse potential is used to describe the attractions between ligands and surface receptors. By real-time recording the number of bound ligands onto the surface n, we could study the adsorption behavior of a multivalent chain and its conformational properties after adsorption. Typically, we average the data over 1000 independent runs with uncorrelated initial conditions. III. RESULTS AND DISCUSSION First, it is necessary to clarify the effect of the molecular weight of a multivalent polymer chain on its adsorption behavior. To this end, we investigate the adsorption properties of multivalent polymer chains of different lengths N=50, 100, 150, and 200 with the same density of ligands \phi=0. 20. As shown in Fig. 2, the fraction of bound ligands \theta=n/N_{\mathrm{li}} at three different densities of surface receptors \varphi=0. 25, 0. 30, and 0. 40 hardly depend on the length of chain N. Therefore, N is fixed at 100 in following unless stated otherwise.  FIG. 2 The fraction of bound ligands \theta as a function of the length of chain N at three different densities of surface receptors \varphi=0. 25, 0. 30, and 0. 40. Here the density of ligands is \phi=0. 20. Next, how \varphi and \phi affect the adsorption kinetics of a whole multivalent polymer chain should be illustrated. By monitoring the fraction of bound ligands in real time \theta_\mathrm{t}, we could learn how long it takes for a multivalent chain to reach the adsorption equilibrium state in which \theta_\mathrm{t} equals to the equilibrium value \theta. It is clearly shown by Fig. 3(a) that the average time for a multivalent chain to reach the adsorption equilibrium state \tau decreases firstly as \varphi increases, and then nearly keeps constant when \varphi is beyond a certain threshold value \varphi^*=0. 40. In contrast, \tau displays a monotonous decrease with an increase in \phi at \varphi=0. 40, see Fig. 3(b).  FIG. 3 Time evolution of the real-time fraction of bound ligands \theta_\mathrm{t} at (a) different densities of surface receptors \varphi with the density of ligands \phi=0. 20, and (b) different \phi with \varphi=0. 40. Here the length of chain is N=100. A. Superselectivity of the multivalent chain-surface receptors binding systems In a theoretical work of Martinez-Veracoechea et al. [28], it has been shown that compared with monovalent nanoparticles, the multivalent counterparts displayed a superselective behavior when they were bound to the surface receptors, i. e. , the fraction of bound particles grows faster than linearly with \varphi. Given that the difference in the distribution way of ligands in a spherical nanoparticle and a multivalent polymer chain, we are wondering whether the latter will present a similar superselective behavior. Figure 4(a) shows the variation of the fraction of bound ligands in a multivalent chain \theta with increasing \varphi. As expected, the multivalent chains almost leave the surface untouched at low \varphi. However, with a further increase in \varphi, \theta increases rapidly first and then approaches one progressively around a certain threshold \varphi^*=0. 40. These adsorption behaviors of multivalent chains could be understood as follows. At low \varphi, the number of surface receptors accessible to a ligand is far below one, leading to rare formation of the ligand-receptor complex. With increasing \varphi, ligands in a multivalent chain could bind to the surface receptors simultaneously. Around \varphi^*, almost all of ligands bind to the surface receptors stably.  FIG. 4 (a) The fraction of bound ligands \theta as a function of the density of surface receptors \varphi at three different densities of ligands \phi=0. 20, 0. 34, and 0. 50. (b) \varphi-dependent parameter \alpha quantifying the selectivity of the multivalent chain-surface binding systems. The dashed line signifies \alpha=1. Here the length of chain is N=100. To determine whether the multivalent chains in our simulations present superselective behaviors, we define a parameter \alpha=\Delta$$\theta$/$\Delta$$\varphi, which characterizes the relative increasing speed of \theta and \varphi. \alpha$$>$1 denotes that a multivalent chain binds to surface receptors superselectively. As shown in Fig. 4(b), multivalent chains display superselective behaviors over a wide range of $\varphi$. Interestingly, $\alpha$ shows a nonmonotonic behavior, and reaches a maximum $\alpha_{\mathrm{max}}$ with increasing $\varphi$. This phenomenon illustrates that a multivalent polymer chain could be designed to achieve superselectivity to $\varphi$ when it binds to the surface. We note that this conception has been realized in Dubacheva $et$ $al$'s recent experimental work [37].

In addition, the location of $\alpha_{\mathrm{max}}$ shifts to lower values of $\varphi$ as $\phi$ increases. As the length of chain is fixed at $N$=100 here, a higher value of $\phi$ means larger number of ligands in the multivalent chain and thus shorter ligand spacing. In other words, multivalent chains with shorter ligand spacing display the maximal superselectivity at lower $\varphi$. This is due to the simple fact that with increasing $\varphi$, the average distance between neighboring surface receptors $d_\mathrm{sr}$ shows a power-law decrease as $d_\mathrm{sr}$$\approx$$\varphi$$^{-1/2}. An increase in \varphi from 0 would lead to a significant decrease in d_\mathrm{sr}, which favors the binding of multivalent chains with higher values of \phi obviously. Meanwhile, we have also noticed that the range of \varphi where a multivalent chain displays the superselective behavior gets smaller as \phi increases. These two findings here provide a rule of thumb to design multivalent chains. For instance, multivalent chains with higher values of \phi should be used when it is expected to achieve the superselectivity at low \varphi; in contrast, multivalent chains with lower values of \phi become the only choice when it is expected to achieve the superselectivity over a wide range of \varphi. In this work, both the ligands in a multivalent chain and receptors on the surface are evenly distributed. Then, there comes a question that whether the superselective adsorption is a general property of the multivalent chain-surface receptors binding systems. We have carried out further simulations, and found that the appearance of the superselective adsorption does not depend on the sequence of ligands and the order of receptors. B. Conformational properties of bound multivalent polymer chains To gain an intuitive insight into the adsorption of a multivalent polymer chain to the surface receptors, we examined the conformational properties of the chain, which has been ignored in conventional in vitro binding assays due to the limit of experimental conditions [27]. The distance of each monomer in a multivalent polymer chain from the surface z_\mathrm{center} (when the adsorption equilibrium state is reached) is a good characterization of the chain conformations. The z_\mathrm{center}-s plot provides a direct knowledge about the conformation of a multivalent polymer chain, and the extent of its adsorption. As shown in Fig. 5(a), at a low \varphi=0. 05, all of monomers, including the ligands in the multivalent polymer chain are far away from the surface. Figure 5 (b) and (c) indicate that with increasing \varphi, most of ligands get adsorbed onto the surface receptors. Incidentally, most of ordinary monomers are confined near the surface such that a soft film is formed after the adsorption. This is in accordance with experimental observations [27, 37, 38]. It has been reported that for strong attractions, the adsorption of tethered chains to a surface can be considered to be irreversible, and the adsorption process obeys a simple zipping mechanism for linear chains [39]. Once adsorbed, there formed loops, tails and trains structures in the chain [39]. As to the adsorption of a multivalent chain, we have checked the index of attached ligands, and found that the ligands attachment could not be described by a simple zipping mechanism. It is suggested clearly by Fig. 5(b)-(c) that loops are formed in the bound multivalent polymer chains. We have counted the loop size, i. e. , the number of monomers between the adjacent bound ligands from the simulations. It is found that the loop size displays a rapid decrease with increasing \varphi, and keeps at 5 as \varphi$$\geq$$\varphi^*, see Fig. 5(d). As shown in the inset of Fig. 5(d), the average distance between adjacent bound ligands D_\mathrm{ap} shows a similar behavior as \varphi increasing with D_\mathrm{ap}$$\approx$3. 0 at $\varphi$$\geq$$\varphi^*$. A multivalent chain in this work could be considered to be consisted of $n$ subchains with 5 monomers. $D_\mathrm{ap}$$\approx3. 0 is larger than the equilibrium size of a subchain R$$\approx$5$^{\upsilon_\mathrm{3D}}$=2. 58 with $\upsilon_\mathrm{3D}$=0. 588 being the 3D Flory exponent, indicating that the subchains are slightly elongated.

 FIG. 5 The distance of each monomer in a multivalent polymer chain from the surface $z_\mathrm{center}$ when the adsorption equilibrium state is reached at (a) $\varphi$=0. 05, (b) $\varphi$=0. 20, and (c) $\varphi$=0. 40. (d) The loop size of bound multivalent polymer chains as a function of $\varphi$. The inset in (d) shows the average distance between adjacent bound ligands $D_\mathrm{ap}$. Here the length of chain is $N$=100, and the density of ligands is $\phi$=0. 20.

In order to get further informations about the chain conformations, we have calculated the equilibrium radius of gyration $R_\mathrm{g}$, its horizontal component $R_{\mathrm{g}, \parallel}$ and vertical component $R_{\mathrm{g}, \perp}$. As expected, $R_{\mathrm{g}, \perp}$ decreases rapidly at the beginning, and then approaches a constant value which is close to the thickness of the formed monolayer soft film, see Fig. 6. Interestingly, as $\varphi$ increases, there is a maximum in both $R_\mathrm{g}$ and $R_{\mathrm{g}, \parallel}$ at $\varphi$$\approx0. 30. From the perspective of statistics, a multivalent chain in solution adopts a spherical conformation with a radius of gyration R_\mathrm{g}. With the proceeding of adsorption, the chain is flattened gradually. The degree of flatness gets larger with the increasing \varphi such that a maximum in both R_\mathrm{g} and R_{\mathrm{g}, \parallel} appears. However, with a further increase in \varphi, the local density of surface receptors is high enough for the binding of all ligands. Therefore, the chain begins to shrink to maximize its conformational entropy.  FIG. 6 The equilibrium radius of gyration R_\mathrm{g}, its horizontal component R_\mathrm{g, \parallel} and vertical component R_\mathrm{g, \perp} as a function of the density of surface receptors \varphi at two different densities of ligands \phi=0. 20 and 0. 34. The length of chain is N=100. A mature theoretical framework in polymer physics about the conformations of flexible polymers upon adsorption to surfaces has been developed [40, 41]. Figure 7 shows the log-log plots of the equilibrium radius of gyration R_\mathrm{g} against the length of chain N at different densities of surface receptors \varphi. Then, the scaling exponents \upsilon of R_\mathrm{g}$$\sim$$N^{\upsilon} can be extracted. For \varphi=0. 20 and 0. 30, \upsilon_{\mathrm{3D}}$$\upsilon$$\upsilon_{\mathrm{2D}} indicates that the conformations of bound multivalent chains fall in between these of a 2D and a 3D chain. Here \upsilon_{\mathrm{2D}}=0. 75 is the 2D Flory exponent. However, as \varphi increases to 0. 40 and 0. 60, \upsilon$$\upsilon_{\mathrm{3D}}$ suggests that the bound multivalent chains adopt slightly collapsed conformations, as stated above.

 FIG. 7 Log-log plots of the equilibrium radius of gyration $R_\mathrm{g}$ against the length of chain $N$ at different densities of surface receptors $\varphi$. Here the density of ligands is $\phi$=0. 20.
IV. CONCLUSION

In this work, we have performed 3D Langevin dynamics simulations to investigate the adsorption of multivalent polymer chains to surface receptors. We show that multivalent polymer chains display superselective behaviors when they bind to the surface receptors. Furthermore, the range of the density of surface receptors $\varphi$ narrows down for chains with higher density of ligands $\phi$. Meanwhile, the optimal $\varphi$ decreases with increasing $\phi$. These results provide a rule of thumb to design multivalent polymer chains when it is expected to achieve a balance between the $\varphi$ region that achieves the superselectivity and the optimal $\varphi$ realizing the highest superselectivity.

By checking the conformations of a bound multivalent polymer chain, we find that a soft film is formed on the surface. Interestingly, the equilibrium radius of gyration $R_\mathrm{g}$ and its horizontal component $R_\mathrm{g, \parallel}$ have a maximum as a function of $\varphi$. The scaling exponents of $R_\mathrm{g}$$\sim$$N^{\upsilon}$ indicate that with increasing $\varphi$, the conformations of a bound multivalent polymer chain first fall in between these of a 2D and a 3D chain, while it is slightly collapsed subsequently.

V. ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (No. 21225421 and No. 21074126), the 973 Program of Ministry of Science and Technology of China (No. 2014CB845605), and the Fundamental Research Funds for the Central Universities (No. WK2060200020). Wan-cheng Yu gratefully acknowledges the funding support from the China Postdoctoral Science Foundation (No. 2015M581998).

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