Adsorption of the Low Density Lipoproteins on the Azobenzene-containing Polymer Brush: Modelling by the Coarse-grained Molecular Dynamics

1. Introduction

Lipoproteins are the major carriers of cholesterol within a human body and, therefore, are of great importance for metabolic processes. Low-density lipoproteins (LDL) bring cholesterol from liver to cells, whereas high-density lipoproteins (HDL) carry cholesterol from a heart and other organs back to liver, where it is removed out of the body. The LDL may cause cholesterol to build up within arteries and can eventually block arteries increasing risk for heart disease and stroke [1,2,3,4,5]. The indication of the increased risk of this scenario is an abnormally high level of LDL.

Two main strategies exist to reduce the risk of such a blockade: (i) by treating the blood vessels (stenting or stents implantation), or (ii) by reducing high concentration of LDL in a blood. The latter can be seen rather as the prevential approach with two ways of realisation: (a) in-body reduction by affecting liver activity (using statins) or digesting adsorbers, and (b) hemoperfusion approach when adsorbers are located outside the body [6].

In dealing with the atherosclerosis, which is characterised by an increased level of the total LDL cholesterol, pharmacological groups of drugs are widely and primarily used, with statins occupying the first place. The second- and third-line drugs used for this purpose are ezetimibe and fibrates. Currently, drugs combining statins and ezetimibe demonstrate particular effectiveness in this respect. Intolerance of hypolipidemic drugs manifests itself via (a) unwanted symptoms perceived by patients as unacceptable, and/or (b) abnormal laboratory results that indicate an excessive risk associated with the use of hypolipidemic drugs. In both cases such drugs are withdrawn. Such patients, characterised by either absolute or relative intolerance to this type of treatment, are the first candidates for a full or partial hemoperfusion therapy.

In a course of the adsorbent-based hemoperfusion, the patient’s blood is introduced into a container with the specific adsorbent. It binds LDL selectively and, at the same time, allows the HDL and other blood components to pass through and then be re-introduced into the patient’s body. There are several types of LDL hemoperfusion adsorbents that are widely used in clinical practice, including biomacromolecules, magnetic nanoparticles, carbon nanotubes, nanohydrogels, porous beads, see Ref. [1,2,6] for more details and references therein. The main aim of research in this area is to combine (i) high efficiency of LDL removal, (ii) high selectivity which prevents removal of HDL, and (iii) efficiency of usage in terms of high reusability.

We will mention just a few studies here that address one or more of these issues. In particular, an amphiphilic polymers were used by Cheng et al. [7], and these, at particular sulfonation rate and cholesterol grafting time, demonstrated high adsorption capacity for LDL without significantly adsorption of HDL. After $2h$ hemoperfusion, the LDL levels decreased by fraction of five. To achieve binding of LDL with high affinity, the biomimetic adsorbent was developed by Yu et al. [6], which mimics the lipoprotein microemulsion present in the blood. In vitro studies revealed the LDL adsorption rate about twice as high as that of the HDL. In yet another work, the core-shell structured magnetic nanoparticles were embedded in an amphiphilic polymer layer to provide multifunctional highly selective binding for LDL particles [8]. Because of the electronegativity of the functional layer and charged surface of LDL, the nano-adsorbent demonstrated highly selective adsorption towards LDL, whereas chemical adsorption also plays a predominant role in binding of LDL. This nano-adsorbent possesses satisfactory recyclability, low cytotoxicity and hemolysis ratios [8].

An important aspect of the adsorption during hemoperfusion is the controllability of the adsorption, namely the ability to switch it “on” and “off” by means of some, preferably clean, external stimulus. This involves the concept of the so-called “smart surfaces”, i.e. the thermo- [9,10,11], magnetically- [12] and photo- [13] controllable surfaces, for more details see recent review [14]. For example, thermo-controllable surface based on PNIPAM polymer, already found numerous biomedical applications [9]. The photo-controllability of the surface properties can be achieved by incorporation some photosensitive group into it, with the azobenzene chromophore being the most widely used one [15]. Such “azobenzination” allows a number of features of the smart surface, to name: the control over adhesive properties of a surface [16], manipulation of nano objects on it [17], photo-controllable separation of a photoresponsive surfactant from the adsorbate [18], achieving photo-reversible surface polarity [19], photo-controllable orientational order affecting surface anchoring of liquid crytals [20], etc.

Recently, the advanced LDL adsorber in a form of a photo-controllable smart surface was developed characterised by high selectivity and reusability [21]. It exhibited excellent LDL adsorption capacity and could be regenerated by illumination with high efficiency, further verified by transmission electron microscopy and Fourier-transform infrared analysis. Green regeneration of the nanoadsorbent could be achieved completely through a simple photoregeneration process, and the recovery rate was still 97.9% after five regeneration experiments [21].

This experimental work sparkled our interest towards modelling the process of the LDL adsorption by such advanced photo-controllable adsorber by employing computer simulations. One should note that the native length scale of the problem, complexity and ambiguity of the LDL structure [2,22,23,24,25,26,27,28,29] prevent performing atomistic-scale simulation of such process in a foreseen future. There are a number of computer simulation studies performed mostly on a coarse-grained level that address the structure of LDL, lipid transport, receptor mutations, and other related topics [30,31,32,33,34,35,36,37,38,39].

We see the possibility to perform the coarse-grained simulations of the adsorption of the LDL particles explicitly if one uses:

• minimalistic model containing only the elements directly involved in adsorption,
• reduction of the length scale of a problem,
• artificial speed-up of the system dynamics.

In doing this we follow our previous studies on modelling photo-sensitive polymers using coarse-grained simulations [40,41,42,43,44,45,46,47]. Modelling approach is covered in detail in Section 2. Ideally, the aim of performed simulations would provide important insights on the polymer architecture of the photo-controllable adsorber that are needed to improve its efficiency. These might be further used for refining synthesis protocols. On a minimalistic side, such simulations would validate suggested type of modelling, by comparing their results against observed experimental features.

The outline of this study is as follows. In the first part of Section 2 we review available experimental data on the overall shape and internal structure of LDL and on its photo-controllable adsorption. It sets the basis for constructing the coarse-grained model for the problem of interest, covered in detail in the second part of this section. Section 3 contains the results of the computer simulation of this model mimicking adsorption of LDL under visible light, whereas Section 4 focuses on the adsorber regeneration under ultraviolet light, followed by conclusions.

2. Experimental data on LDL Structure and the Modelling Details

Let us review first some available experimental data on the typical size of the LDLs. In general, plasma LDLs are heterogeneous in terms of their size, density and lipid content, varying from one individual to another. Two following types of LDL were identified: larger, pattern A type, of average dimension larger than $25.5\mathrm{nm}$; and pattern B of a smaller average dimension less than $25.5\mathrm{nm}$[48]. The latter ones are found to be more prevalent in patients with coronary artery disease [49] with the higher risk of myocardial infarction [50] and of developing coronary disease [51,52]. The reason for this could be their reduced affinity for the respective receptor and, as the result, an increased residence time in plasma and higher probability to be oxidized at the artery walls, leading to atherosclerosis. For more detail see discussion in Ref. [53] and references therein. Therefore, the pattern B LDL of a smaller size is the main target for adsorption via hemoperfusion. The experimental measurements of their average dimensions provide the estimates from 20 to $23\mathrm{nm}$ [53].

In terms of their internal structure, the LDL particles can be interpreted as micellar complexes, macromolecular assemblies, self-organized nanoparticles or microemulsions [26,29]. A spherical three-layer model has been suggested based on the low-resolution data [24,25]. It assumes the presence of an internal core of LDL comprising cholesteryl ester and triglycerides. The core is enveloped by an outer shell of phospholipids, with their polar heads residing on the surface of LDL and their fatty acid ester tails pointing inward the LDL. About half of the external surface of the LDL is covered with apolipoprotein B-100, which form ligand recognition loops for various receptors [2,23]. In particular, the B-100 in a form of two ring-shaped structures were reported [22]. A liquid crystalline core model of LDL was also discussed, where cholesteryl ester molecules are arranged in stacks with their sterol moieties arranged side-by-side in the higher-density regions, while the fatty acyl chains extend from either side, and are found as parallel lower-density compartments [27].

Based on the complexity of their LDL internal structure, the question of the overall LDL shape turns to be rather controversial, as the spherical [25], discotic [22,26], as well full range of spherical, discotic and ellipsoidal [27] shapes all were reported. Possible explanation for these discrepancies may stem from the fact that the shape of the LDL particles is temperature-dependent due to the predominance of cholesteryl ester molecules in the particle core. As the result, at physiological temperatures the LDL appear more spherical, whereas at lower temperatures – discoidal [28]. In fact, the discotic shape indeed was predominantly found by means of cryo-electron microscopy [22,23,26,27], performed at low temperatures. One can conclude, that the exact structure of LDL is not quite exactly known in detail and it is still the matter of debates, see Ref. [29] and discussion therein.

These experimental findings lay out the basis for developing a range of moderately coarse-grained models of LDL. In general, this type of lipid simulations [31] can address main two issues. The first one is to improve one’s understanding of the internal LDL structure [30,32], addressing mutational space of the LDL receptor [37], etc. The other is to understand the physical chemistry mechanisms behind the lipid transfer [33,36,38,39]. On a more coarse-grained level, the interactions between oxidized LDL and scavenger receptors on the cell surfaces of macrophages, related to arterial stiffening, are addressed [34]. The smectic-isotropic transition inside the LDL core, related to the liquid crystalline order found there in Ref. [27], was modelled in Ref. [35]. In general, coarse-grained models allow to cover larger system sizes and longer simulation times, but at the expense of omitting some fine details of the initial system.

The modelling approach developed in this study follows the coarse-graining plan outlined in Section 1. According to statement (1.) there, the model contains only the elements directly involved in adsorption setup by Guo et al. [21]. They developed the nanoadsorbers consisting of a spherical support particle of diameter $200\mathrm{nm}$ which is functionalized by the side-chain polymers terminated by an azobenzene group. The ratio between the diameters of a nanoadsorber and that of LDL is about $10:1$. This results in relatively low curvature of a nanoadsorber comparing to the LDL size, and, therefore, one can approximate a surface of a former by a flat surface. This is done in our study. The model contains two adsorbing surfaces on both the bottom, $z=0$, and the top, $z=L_z$, walls of the simulation box with dimensions $L_x$, $L_y$ and $L_z$. By using two walls instead on a single one, one (i) avoids possible artefacts at the top free wall, and (ii) improves statistics obtained in a course of a single simulation run. The polymers are of the side-chain architecture [6] with their side chains terminated by azobenzenes [21]. This setup is shown schematically in Figure 1 (a) with the arrangement of grafting points in the sites of a square lattice, all the details are provided in the figure caption.

In a similar way, only these structure elements of the LDL particles that are relevant to their adsorption are modelled explicitly. In the case of physiological temperature, one assumes a spherical shape for the LDL core [28] with its interior filled by cholesteryl esters and triglycerides. This part of LDL is believed not to be involved in adsorption in a direct way and can be treated as an uniform spherical object with a fixed diameter. However, the outer phospholipid shell is involved in adsorption explicitly, via the interaction with the azobenzenes of a brush and, therefore, is modelled as a collection of individual phospholipid spherocylinder particles. Such model for LDL is shown in Figure 1 (b) and (c), where a spherical core is shown in dark gray and spherocylinders representing phospholipids – in brown.

Similar coarse-grained building blocks are used and well tested in a set of our previous works involving polymer brushes and decorated nanoparticles [42,43,44,46,54,55]. The relation of their length scale to that for their real counterparts can be estimated from the diameter of the LDL spherical core. It is about $2.1\mathrm{nm}$[54], which leads to the overall diameter of the LDL particle (core + shell) of about $5\mathrm{nm}$ (see estimates for the azobenzene unit length below). Therefore, we arrive at the scaling ratio about $1:5$ between the model and the real life LDL, following the statement (2.) of the coarse-graining plan enlisted in Section 1.

The monomers of backbones and of side chains are represented by soft-core spherical beads of the diameter $\sigma =0.46\mathrm{nm}$, mimicking approximately a group of three hydrocarbons each [56]. Each second bead of a backbone is as a branching point for a side chain of two spherical beads terminated by an azobenzene. The latter is modelled as a soft-core spherocylinder with the diameter of a spherical cap equal to $D=0.37\mathrm{nm}$ and the length-to-breadth radio of $L/D=3$, resulting in the total length of a spherocylinder of $D/2+L+D/2=4D\approx 1.5\mathrm{nm}$. Phospholipids are modelled by the same type of prolate particles as the azobenzenes, to simplify modelling. An important question is the packing density of phospholipids in the outer shell, and there are some indications that it is not very high [32]. If we use 100 phospholipids per each LDL, the packing fraction of their ends on the core surface is about $\eta =0.77$. This value seems reasonable, e.g. it is close to the maximum packing fraction of discs arranged on a $2D$ square lattice, $\eta =\pi /4\approx 0.79$. Due to finite curvature of a core, packing of phospholipids at the outer surface of their shell will be essentially lower, see Figure 1 (b) and (c).

After introducing main components of a model, one needs to define the effective interaction potentials between them. Two types of coarse-graining approaches can be employed here. In the bottom-up approach, one, typically, performs some pre-runs using atomic-scale model for particular chemical compound. Then, the model is coarse-grained by splitting the molecules into groups of atoms and the effective interactions between them are parametrised by matching the forces between these groups, radial distribution functions, radius of gyration, or transport coefficients. In this way, the initial chemical details of a particular compound of interest is, at least approximately, preserved, reflecting the idea of a multi-scale modelling [57,58]. The other approach focuses on universal, rather physical than chemical, aspects of the problem and employs some generic interaction potentials that are less related to a particular chemical compound, the potentials developed within the dissipative particle dynamics approach being a prominent example [59]. The approach used in the current study is some kind of mixture of the two. On one hand, we use the coarse-grained interaction potentials obtained from atomistic modelling [56,60], but their forms are rather generic and reflect the shape and the main features of interacting beads [61]. Therefore, these potentials are not specifically tuned to mimick a certain compound, but are aimed at description of universal physical features of a wider class of polymers. By using soft-core spherical and spherocylinder beads, one greatly speeds up the dynamics in a system, hence, implementing the statement (3.) of the coarse-graining plan enlisted in Section 1.

We will start from the non-bonded interactions. To simplify equations, each pair $\{i,j\}$ of the particles is characterized by a shorthand containing a set of variables, ${\mathbf{q}}_{ij}=\{{\widehat{\mathbf{e}}}_i,{\widehat{\mathbf{e}}}_j,{\mathbf{r}}_{ij}\}$, where ${\widehat{\mathbf{e}}}_i$ and ${\widehat{\mathbf{e}}}_j$ are the unit vectors defining the orientation of the respective particles in space, and ${\mathbf{r}}_{ij}$ is the vector that connects their centers of mass. For the case of spherical particles, their orientations are not defined. The Kihara type of potential, used here, implies evaluation of the closest distance, $d\left({\mathbf{q}}_{ij}\right)$, between the internal cores of two interacting particles, where the core of a spherical particle is its center, and the core of a spherocylinder is the line connecting the centers of its two spherical caps. The scaling factor ${\sigma }_{ij}$ is evaluated for the pair, where ${\sigma }_{ij}=({\sigma }_i+{\sigma }_j)/2$ for two spherical particles, ${\sigma }_{ij}=D$ for two spherocylinder particles, and ${\sigma }_{ij}=({\sigma }_i+D)/2$ for the mixed sphere-spherocylinder pair. The dimensionless closest distance between two interacting particles is defined then as $d^{\prime }\left({\mathbf{q}}_{ij}\right)=d\left({\mathbf{q}}_{ij}\right)/{\sigma }_{ij}$.

Using these notations, the general form of the pair interaction potential between ith and jth beads, that is of the soft attractive (SAP) type [61], can be written in a compact dimensionless form

$$V_{\mathrm{NB}}^{\mathrm{SAP}}\left[d^{\prime }\left({\mathbf{q}}_{ij}\right)\right]=\left\{\begin{array}{cc}U\left\{{[1-d^{\prime }\left({\mathbf{q}}_{ij}\right)]}^2-{ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)\right\},\hfill & 0\le d^{\prime }\left({\mathbf{q}}_{ij}\right)<1\hfill \\ U\{{[1-d^{\prime }\left({\mathbf{q}}_{ij}\right)]}^2-{ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right).\hfill & \\ .-\frac{1}{4{ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)}{[1-d^{\prime }\left({\mathbf{q}}_{ij}\right)]}^4\},\hfill & 1\le d^{\prime }\left({\mathbf{q}}_{ij}\right)\le d_c^{\prime }\hfill \\ 0,\hfill & d^{\prime }\left({\mathbf{q}}_{ij}\right)>d_c^{\prime }\hfill \end{array}\right.$$

(1)

where U defines repulsion strength. The dimensionless well depth of this potential

$${ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)={\left\{4\left[U_a^{\prime }-5{ϵ}_1^{\prime }{P}_2({\widehat{\mathbf{e}}}_i·{\widehat{\mathbf{e}}}_j)-5{ϵ}_2^{\prime }\left(P_2({\widehat{\mathbf{r}}}_{ij}·{\widehat{\mathbf{e}}}_i)+P_2({\widehat{\mathbf{r}}}_{ij}·{\widehat{\mathbf{e}}}_j)\right)\right]\right\}}^{-1},$$

(2)

is obtained from the condition, that both the expression (1) and its first derivative on $d^{\prime }\left({\mathbf{q}}_{ij}\right)$ turn to zero when $d^{\prime }\left({\mathbf{q}}_{ij}\right)=d_c^{\prime }$, where $d_c^{\prime }=1+\sqrt{2{ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)}$ is the cutoff separation for the potential [61]. Here ${\widehat{\mathbf{r}}}_{ij}={\mathbf{r}}_{ij}/r_{ij}$ is a unit vector along the line connecting the centers of two beads, $U_a^{\prime }$, ${ϵ}_1^{\prime }$, and ${ϵ}_2^{\prime }$ are dimensionless parameters that define the shape of the interaction potential. These are chosen to represent the “model A” of Ref. [61]. $P_2\left(x\right)=(3x^2-1)/2$ is the second Legendre polynomial.

The effective well depth, ${ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)$, influences both the shape of the attractive part and, via$d_c^{\prime }$, its range. When the parameters, contained in the expression for ${ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)$, are such, that it asymptotically reaches zero, the cutoff $d_c^{\prime }$ approaches 1 and the interval for the second line in Eq. (1) shrinks to zero. As a result, in this limit, one retrieves the soft repulsive potential (SRP) of a quadratic form

$$V_{\mathrm{NB}}^{\mathrm{SRP}}\left[d^{\prime }\left({\mathbf{q}}_{ij}\right)\right]=\left\{\begin{array}{cc}U{\left[1-d^{\prime }\left({\mathbf{q}}_{ij}\right)\right]}^2,\hfill & 0\le d^{\prime }\left({\mathbf{q}}_{ij}\right)\le 1\hfill \\ 0,\hfill & d^{\prime }\left({\mathbf{q}}_{ij}\right)>1\hfill \end{array}\right.,$$

(3)

that is used typically in the dissipative particle dynamics simulations [59]. This limit is illustrated in Figure 2 by the blue curve and is marked as ${ϵ}^{\prime }\left({\mathbf{q}}_{ij}\right)\to 0$.

Attractive potential (1) is used to model the interaction between the trans-azobenzene and a phospholipid only. The origin of their attraction in a water-like solvent lies in strong hydrophobicity of both groups, see, e.g. Ref. [62] for the estimates of a dipole moment of 4-substituted 4‘-(12-(dodecyldithio) dodecyloxy) azobenzene). For all other pair interactions, soft repulsive potential (3) is used, reflecting the nature of coarse-grained type of modelling. In this way we emphasize the role of the azobenzene-phospholipid interactions, as the key factor in the adsorption process. Other approach can also be employed, where the pairs of two trans-azobenzenes and of two phospholipids are interacting via the potential (1) as well. In this case, adsorption of LDL particles will compete against both the aggregation of LDL particles and the self-collapse of the brush. We might consider this case in the future. Strong repulsion, with the energy parameter $U^{\prime }=2U$ in Eq. (3), is introduced for the interaction of both trans-azobenzene and polymer beads with solvent, reflecting their poor solubility in water. Such type of modelling of azobenzenes has been already used in a number of previous studies [42,43,44,45,46,47].

The expressions for the total bonded interactions within the brush and within each LDL, respectively, are given as

$${\displaystyle V_{\mathrm{B}}^{\mathrm{BR}}=\sum _{k=1}^{N_{BR}}\left[\sum _{i=1}^{n_b^{\prime }}{k}_b{(l_i-l_0)}^2+\sum _{i=1}^{n_a^{\prime }}{k}_a{({\theta }_i-{\theta }_0)}^2+\sum _{i=1}^{n_z^{\prime }}{k}_z{({\zeta }_i-{\zeta }_0)}^2\right],}$$

(4)

$${\displaystyle V_{\mathrm{B}}^{\mathrm{LDL}}=\sum _{k=1}^{N_{LDL}}\left[\sum _{i=1}^{n_b^{\prime \prime }}{k}_b{(l_i-l_0)}^2+\sum _{i=1}^{n_z^{\prime \prime }}{k}_z{({\zeta }_i-{\zeta }_0)}^2\right],}$$

(5)

where $N_{BR}$ and $N_{LDL}$ are the total number of polymer chains in a brush and of LDL particles, respectively; $n_b^{\prime }$, $n_a^{\prime }$, and $n_z^{\prime }$ are the numbers of bonds, branching angles, and terminal angles in a single polymer molecule, and $n_b^{\prime \prime }$ and $n_z^{\prime \prime }$ are the numbers of bonds and terminal angles in a single LDL particle. The purpose of bonds in LDL particles is to keep each phospholipid at a given separation from the center of a core, to form the outer spherical shell. The energy term involving branching angles, ${\theta }_i$, in Eq. (4) maintains certain level of perpendicularity of side chains to a local orientation of a backbone. Similarly, correct orientations ${\zeta }_i$ of both the azobenzenes and phospholipids, with respect to the bond by which these are attached to a spherical bead, are ensured in Eqs. (4) and (5) by the energy term involving the terminal angle ${\zeta }_i$ [60,63].

According to the model description provided above, the required numbers of beads and of various energy terms in a single polymer molecule can be derived from the chosen value for the backbone length, $l_{bb}$. Namely, each polymer contains $n_{sc}=\mathrm{div}(l_{bb},2)$ side chains (where $\mathrm{div}$ denotes division of two integers), in total $n_p=l_{bb}+2n_{sc}$ spherical and $n_a=n_{sc}$ azobenzene beads. Therefore, the number of bonded interactions are given by: $n_b^{\prime }=n_p+n_a-1$, $n_a^{\prime }=n_{sc}$, and $n_z^{\prime }=n_a$. Each LDL particle consists of a spherical core particles and $n_{pl}=100$ phospholipids, therefore, $n_b^{\prime \prime }=n_z^{\prime \prime }=n_{pl}$. All force field parameters are collected in Table 1 for the sake of convenience.

We will complete this section by specifying the parameters of the simulation runs. The simulation box of the dimensions $L_x=L_y=L_y=20\mathrm{nm}$ is used, where both bottom, $z=0$, and top, $z=L_z$, walls are functionalised by $N_{BR}$ polymers each, see Figure 1 (a). Polymers of five different backbone lengths, $l_{bb}=5,6,10,16$ and 22, are considered, as well as a range of brush grafting densities

$${\rho }_g=N_{BR}{\sigma }^2/\left(L_x{L}_y\right).$$

(6)

Both characteristics affect adsorption scenario of macromolecules by a brush, e.g. for the case of peptides [64]. The arrangement of grafting points (shown in black in Figure 1 (a)) is in the sites of the square lattice to minimise inhomogeneities in the arrangement of polymers. The number of LDL particles is constant in all cases and is equal to $N_{LDL}=25$. Box interior is filled by beads of the same dimensions as the monomers of a brush, these represent a water-like solvent. The simulations are carried out with the GBMOLDD [65,66] program generalised for the case of coarse-grained soft-core potentials [54], in the $NVT$ ensemble at the bulk density $\rho =0.5\mathrm{g}/{\mathrm{cm}}^3$ and the temperature $T=480\mathrm{K}$. This choice is based on the previous findings [54], where the melt involving the spherocylinders interacting via potential (1), exhibited the order-disorder transition at about $T=505-510\mathrm{K}$. Therefore, at $T=480\mathrm{K}$, we expect strong azobenzene-phospholipid interactions. Simulation runs of the duration of $40\mathrm{ns}$ were undertaken at each polymer length and grafting density, with the time step of $\Delta t\sim 20\mathrm{fs}$.

3. Adsorption of the LDL at Normal Conditions

3.1. Density Profiles and the Probability for the Azobenzene-Phospholipid Contact

At normal conditions, under visible light, azobenzene chromophores are found in the trans isomeric state [67], and exhibit properties similar to ordinary mesogens, such as cyanobiphenyls [68]. In a water environment, trans-azobenzene is attracted to phospholipids from an outer shell of LDL due to strong hydrophobicity of both. This forms a basis for the adsorption of LDL onto azobenzene-containing ligands [21].

The process of gradual adsorption of LDL particles from the middle of channel onto the polymer brushes is illustrated by a series of snapshots given in Figure 3. The case of the polymer backbone length of $l_{bb}=10$ and the brush grafting density of ${\rho }_g=0.02$ is shown. Adsorption starts after about $1\mathrm{ns}$, see frame (a), and partial adsorption is illustrated after $10\mathrm{ns}$, see frame (b). After $40\mathrm{ns}$, almost all available LDL particles are adsorbed on the walls, see frame (c). In general, similar scenario is found for all the backbone lengths, $l_{bb}=5-22$, of brush polymers and within a broad range of their grafting densities, $0.008<{\rho }_g<0.35$. As one can see in Figure 3 (c), in adsorbed state, azobenzenes (shown in blue) penetrate deeply into the outer layer of LDL particles (shown in brown), in a manner anticipated in Ref. [21].

To quantify the effects that both the backbone length, $l_{bb}$, and the brush grafting density, ${\rho }_g$, have on adsorption process, we introduce a set of relevant characteristics. The obvious prerequisite for the adsorption is a non-zero probability to find azobenzenes and phospholipids within the same region of space, hence we turn our attention to the spatial distributions of these beads. According to the setup symmetry, we consider a set of slabs spanning in both $OX$ and $OY$ directions that are located at a certain distance z from the bottom wall. The density profiles, ${\rho }_{az}\left(z\right)$ and ${\rho }_{ph}\left(z\right)$ are built, for azobenzenes and phospholipids, respectively. The histograms for these profiles are shown in Figure 4 for the case of $l_{bb}=16$ and three grafting densities ${\rho }_g$, as indicated in the figure. Note that the area under the histogram ${\rho }_{ph}\left(z\right)$ is constant, as $N_{LDL}=25$ is the same in all cases. In contrary, the area under the histogram ${\rho }_{az}\left(z\right)$ grows upon the increase of grafting density, ${\rho }_g$, reflecting the increase of a number of azobenzene groups within a brush. The overlapping area between the histograms for ${\rho }_{az}\left(z\right)$ and ${\rho }_{ph}\left(z\right)$ is proportional to the probability to find both types of beads in the same z-segment. As it is evident from the figure, with the raise of ${\rho }_g$ from $0.019$ to $0.076$, the overlap area grows, the main effect being the increase of the number of azobenzenes in a brush. With the further raise of ${\rho }_g$ up to $0.135$, this area decays. The main effect here is the brush entering the regime of a dense brush with the consequence that the LDLs are expulsed from the brush because of the excluded volume effects.

Quantitatively, the probability of azobenzene to meet phospholipid at the same specified distance z from the bottom wall, is given by the joint probability, ${\rho }_{az}\left(z\right){\rho }_{az}\left(z\right)$. It is displayed in Figure 5 (a) for the case of $l_{bb}=16$ and a range of grafting densities ${\rho }_g$. The global probability for azobenzene and phospholipid to be found at the same z is given by

$$p={\int }_0^{L_z}{\rho }_{az}\left(z\right){\rho }_{ph}\left(z\right)dz.$$

(7)

Figure 5 (b) shows its dependence on the grafting density of a brush, ${\rho }_g$, at various lengths of a polymer backbone, $l_{bb}$. It is evident that at each backbone length $l_{bb}\ge 10$, there is a specific value for the grafting density, ${\rho }_g^{*}$, at which p reaches its maximum value, as indicated in the figure Figure 5 (b). One, however, should take into account that the number of available azobenzenes per square area of a brush is proportional not only to the chains grafting density, ${\rho }_g$, but also to their number in a single chain, $n_{sc}=\mathrm{div}(l_{bb},2)$. Therefore, it makes sense to express the probability p also in terms of the azobenzene surface density ${\rho }_{az}$

$${\rho }_{az}=N_{az}{D}^2/\left(L_x{L}_y\right)={\rho }_g{n}_{sc}{(D/\sigma )}^2,$$

(8)

where $N_{az}$ is the total number of azobenzenes on a single wall functionalized by a brush. The dependence $p\left({\rho }_{az}\right)$, shown in Figure 5 (c), is practically the same and almost linear in ${\rho }_{az}$ in the interval ${\rho }_{az}<0.3$ for all backbone lengths. Therefore, in this interval of grafting densities, the probability p of azobenzene and phospholipid to meet at the same z is governed principally by the number of available azobenzenes. At ${\rho }_{az}>0.3$, the curves saturate indicating that the additional number of azobenzenes does not increase the probability p. This is attributed to a gradual expulsion of azobenzenes from the dense brush interior, see Figure 4 (c), and inaccessibility of azobenzenes adjacent to the wall for interacting with the LDL phospholipids. Therefore, for each given backbone length, $l_{bb}$, there exists some optimal brush grafting density ${\rho }_g^{*}$, as specified in Figure 5 (b), and the further increase of ${\rho }_g$ beyond this value does not increase the probability for the azobenzene and phospholipid to meet at the same z.

3.2. Binding Energy of LDLs

A more direct and standard measure of the adsorption efficiency is the the binding energy, $E_{\mathrm{bind}}$, evaluated per single LDL particle. In our simulations we define it as the magnitude of the sum of all negative pairwise interaction energies, $V_{ij}$, between the azobenzene-phospholipid pairs $\langle i,j\rangle$, divided by the number of the LDLs particles

$$E_{\mathrm{bind}}=\frac{1}{N_{LDL}}\left|\sum _{\langle i,j\rangle }{V}_{ij}\right|,\mathrm{if}{V}_{ij}=V_{\mathrm{NB}}^{\mathrm{SAP}}\left[d^{\prime }\left({\mathbf{q}}_{ij}\right)\right]<0.$$

(9)

The expression for the azobenzene-phospholipid interaction energy it is provided in Eq. (1). $E_{\mathrm{bind}}$ is displayed, in $k_{B}T$ units, in Figure 9. Frame (a) shows it as the function of the grafting density ${\rho }_g$, whereas frame (b) – as the function of azobenzene surface density ${\rho }_{az}$.

The first thing to note here is that the dependence of $E_{\mathrm{bind}}$ has a similar shape to the probability p in Figure 5 (b), with the maximum reached at the very same characteristic respective values ${\rho }_g^{*}$. This indicates that that the magnitude of binding energy is essentially driven by a probability for azobenzenes and phospholipids to meet at the same z, given by p. On the other hand, comparing Figure 5 (c) and Figure 9 (b), one sees that $E_{\mathrm{bind}}$ reaches progressively higher values with the increase of $l_{bb}$ at the same ${\rho }_{az}$ for ${\rho }_{az}>0.1$. Hence, there is another additional factor that increases the number of azobenzene-phospholipid pairs upon the increase of the chain length which occurs at almost constant value of the probability p.

3.3. Polymer bending effect

This additional factor is the presence of specific conformations of polymer chains. It is quite obvious that, because of a finite curvature of the LDL particles, achieving the maximum number of the azobenzene-phospholipid contacts, and, thus, the largest magnitude for the binding energy, requires essential bending of the polymer chains. Such “wrap over” conformations of polymer chains are illustrated in the snapshot extracted from our simulations and shown in Figure 7.

The most probable radius of curvature $R^{*}$ for the backbone, in the case of an ideal wrap over scenario, can be estimated from the dimensions of all involved beads provided in Section 2. This results in $R^{*}\approx 3.3\mathrm{nm}$, the value is, of course, dependent on the diameter of the LDL particles. Such bend conformation demands, on one hand, sufficient flexibility of a polymer chain and, on another hand, conditions, that there is enough free volume around a chain to bend. Effective value of $R^{*}$ is expected to increase both at high brush grafting densities (because of straightening up of polymer chains), and at high concentration of adsorbed LDLs (in which case one chain may be in contact with two adjacent LDLs).

To examine the presence of bend conformations, we analyse each polymer of a brush separately. The beads comprising chosen backbone are fitted to the spherical surface using the algorithm that reduces this problem to the linear least squares problem [69]. The latter has been solved in Refs. [70,71,72] and is available in a form of an open source FORTRAN-90 package via GitHub [73]. The radius R of a spherical surface that fits the spatial distribution of backbone beads the best, provides the estimate for its radius of curvature.

We built the distributions, $f\left(R\right)$, of such radii of curvature, R, and these are shown in Figure 8 for four backbone lengths, $l_{bb}=6$, 10, 16, and 22, and a set of characteristic grafting densities, ${\rho }_g$, including optimal density ${\rho }_g^{*}$, in each case. Let us consider the case of a short backbone, $l_{bb}=6$, first, shown in Figure 8 (a). It is evident that the distributions $f\left(R\right)$ are very much the same within the broad interval of grafting densities, $0.008<{\rho }_g<0.303$, with the maximum of the distribution positioned at $R\approx R^{*}$. It is so, because the chain length is much smaller than the diameter of LDL, and such chains follow easily the curvature of the LDL surface. For the case of a longer backbone, $l_{bb}=10$, the increase of the grafting density leads to broadening of the $f\left(R\right)$ distribution and shifting the maximum position to higher values, see frame (b) of the same figure. This reflects straightening of some polymers when the brush enters the dense brush regime. The same tendency is found upon a further increase of backbone length to $l_{bb}=16$ and $l_{bb}=22$, see frames (c) and (d). Optimal densities, ${\rho }_g^{*}$, as found in Figure 5 (b) and 9 (a), are highlighted as thicker black curves. At these respective densities, a peak at $R\approx R^{*}$ is still retained, indicating the presence of the wrap over conformations of chains around the LDL particles. Further increase of the density, ${\rho }_g\ge 0.103$ for $l_{bb}=16$ and ${\rho }_g\ge 0.076$ for $l_{bb}=22$, eliminates this peak completely reflecting both straightening of polymer chains in the dense brush regime.

3.4. Concludings Remarks on Adsorption Efficiency

We can return now to the plots for the probability p, Figure 5 (b) and for the binding energy, Figure 9 (a), and relate these to the distributions for the radius of curvature, Figure 8 (b)-(d) at the same parameters of the chains. In particular, at ${\rho }_g<{\rho }_g^{*}$, the distributions $f\left(R\right)$ have well defined peaks at $R=R^{*}$ indicating perfect wrap over conformations, as shown in Figure 7. However, the density of azobenzenes in this case is low leading to smaller values for $E_{\mathrm{bind}}$ (low adsorption efficiency). On the other hand, at ${\rho }_g>{\rho }_g^{*}$, the density of azobenzenes is high, but the peak at $R=R^{*}$ in the distributions $f\left(R\right)$ disappears, indicating relatively small amount of wrap over conformations of chains which, in turn, leads to the reduction of the value of $E_{\mathrm{bind}}$ (low adsorption efficiency). Specific densities ${\rho }_g^{*}$ are attributed to a good compromise between these two factors, when the density of azobenzenes is sufficiently high and, at the same time, there is abundance of wrap over conformations of polymer chains that ensure sufficient number of the azobenzne-phospholipid close contacts. As the result of this compromise, the binding energy reaches its maximum value (high adsorption efficiency).

Having this scenario in mind, one can suggest the reason why the maximum value for binding energy increases at the increase of chain length, $l_{bb}$, see Figure 9. The shorter chains are able to wrap over the sides of LDLs only, whereas longer chains are able to wrap over the external poles of LDLs as well, thus increasing the number of interacting pairs. This is shown in a series of snapshots in Figure 9.

At $\rho >{\rho }_g^{*}$ the excluded volume effects are brought into play and the brush expulses LDLs, and, thus, $E_{\mathrm{bind}}$ decays because of reduced number of interacting azobenzene-phospholipid pairs. This arrangement of the brush and of the LDL particles is shown in Figure 10.

The findings presented in this section allow one to made following conclusions. Firstly, at each fixed length of polymer chains, given by its backbone length $l_{bb}$, the grafting density ${\rho }_g^{*}$ exists which is optimal for the LDL adsorption. The efficiency of the latter is characterised by the binding energy $E_{\mathrm{bind}}$ per one LDL, as shown in Figure 9. The increase of adsorption efficiency at low ${\rho }_g$ should be attributed to the increase of azobenzenes density in the region accessible to LDLs, whereas the decrease of adsorption efficiency at higher ${\rho }_g$ is related both to straightening of chains reducing the azobenzene-phospholipid contacts and to the excluded volume effects in a regime of a dense brush, see Figure 10 (b). Secondly, with the increase of the backbone length, $l_{bb}$, the adsorption efficiency increases until it starts to saturate, as seen in Figure 9. The initial increase is related to the growth of a number of available azobenzenes for binding, see Figure 9, whereas the saturation occurs when the brush height reaches the diameter of the LDL particle, see the same figure (c). Thirdly, an efficient adsorption requires the ability of polymer chains to bend and to wrap over the LDL particles, as shown in Figure 7 and Figure 8. However, in reality, this factor will depend strongly on the polymer chain flexibility and the relation between the chain length and the diameter of a LDL particle.

4. Adsorber Regeneration under Ultraviolet Light

Azobenzene-containing smart LDL adsorber, as suggested and tested in Ref. [21], allows its clean and efficient regeneration and reusage. This can be achieved by illumination of an adsorber by the UV light with a suitable wavelength. In this case, the azobenzenes in a polymer brush undergo the trans-to-cis photo-isomerization [74] and loose their liquid crystalline and apolar features. As the result, their interaction with phospholipids weakens and the LDL particles desorb from the polymer brush of an adsorber. The LDL particles can be washed out of the adsorber and the latter can be reused, with the reported recovery rate of $97.9\%$ after five regeneration cycles [21].

In terms of our modelling approach, described in detail in Section 2, the trans-to-cis photo-isomerization is mimicked via switching the interaction potential between the azobenzene and phospholipid spherocylinders from the attractive (1) to a repulsive (3) form.

Figure 11 shows a series of snapshots demonstrating gradual desorption of LDLs from a polymer brush of the backbone length $l_{bb}=10$ and with, optimal for adsorption, grafting density of ${\rho }_g^{*}=0.103$. Frame (a) shows the initial, adsorbed state, obtained after $40\mathrm{ns}$ under normal conditions, see Section 3. In this state the azobenzenes are in the trans-state indicated by their blue coloring. Upon switching on the UV light, azobenzenes photoisomerize into their cis-form indicated by yellow colouring, and the process of desorption starts. At its first stage, frame (b), both bottom and top layers of LDLs are pushed out of the brush but retain their layered structure. After some illumination time, indicated in the figure, the layers are destroyed and LDLs fill the bulk central region of a pore uniformly, see frame (c). Now they can be easily washed out the pore by a flow.

The issue of interest is the dynamics of desorption of LDLs under UV light, compared to their adsorption under normal conditions, covered in Section 3. For this purpose we cannot use the binding energy $E_{\mathrm{bind}}$ (6), as it instantly drops to zero when all azobenzenes switch into a cis state. However, as it was shown in Section 3, its behaviour is very similar to that for p, the probability to find the azobenzene and phospholipid at the same distance from the bottom wall, see Figure 5. We used this probability as a rough estimate for the LDL adsorption and desorption dynamics. The results are shown in Figure 12 for three backbone lengths, $l_{bb}=10$, 16 and 22 at their respective optimal brush densities, as obtained in Figure 6.

The first observation that stems from Figure 12 is that there is a saturation in the temporal behaviour of p at $l_{bb}\ge 16$, see frame (a). The second thing to mention is that the desorption dynamics at $l_{bb}=22$ is essentially slower than for two other cases, $l_{bb}=10$ and 16, see frame (b). We explain this by essential reduction of the volume of a bulk region of a pore in the case of the longest chain and, as the result, slowing down its diffusion required for the desorption. Finally, perhaps, the main result is that the rate of desorption, shown in (a), is essentially higher that that for adsorption, shown in (b).