Thermodynamic analysis of multivalent binding of functionalized nanoparticles to membrane surface reveals the importance of membrane entropy and nanoparticle entropy in adhesion of flexible nanoparticles

Abstract

We present a quantitative model for multivalent binding of ligand-coated flexible polymeric nanoparticles (NPs) to a flexible membrane expressing receptors. The model is developed using a multiscale computational framework by coupling a continuum field model for the cell membrane with a coarse-grained model for the polymeric NPs. The NP is modeled as a self-avoiding bead-spring polymer chain, and the cell membrane is modeled as a triangulated surface using the dynamically triangulated Monte Carlo method. The nanoparticle binding affinity to a cell surface is mainly determined by the delicate balance between the enthalpic gain due to the multivalent ligand–receptor binding and the entropic penalties of various components including receptor translation, membrane undulation, and NP conformation. We have developed new methods to compute the free energy of binding, which includes these enthalpy and entropy terms. We show that the multivalent interactions between the flexible NP and the cell surface are subject to entropy–enthalpy compensation. Three different entropy contributions, namely, those due to receptor–ligand translation, NP flexibility, and membrane undulations, are all significant, although the first of these terms is the most dominant. However, both NP flexibility and membrane undulations dictate the receptor–ligand translational entropy making the entropy compensation context-specific, i.e., dependent on whether the NP is rigid or flexible, and on the state of the membrane given by the value of membrane tension or its excess area.

1. Introduction

One of the primary challenges in nano-medicine is to design ligand-coated nanoparticle (NPs) as vehicles for diagnostic imaging, drug, or gene delivery. Each functionalized NP can bind selectively to the diffusing receptors on the cell membrane surface using multivalent interactions.^1–4 The process of binding of the NPs, in which they escape to the near wall-region close enough to bind with the endothelium, can be engineered by altering the design of the NP shape, size, chemistry, and flexibility, which can modify their biological function and fate, such as long circulation and specific targeting to diseased tissues.^5–9

The effects that size and shape of rigid functionalized NPs exert on their efficient adhesion with a cell surface have been systematically investigated and analyzed both theoretically and computationally.^10–16 However, relatively few studies have examined and explored the role of NP flexibility, on adhesion and delivery.^17–22 As such, focusing on the potential benefits of flexible NPs by controlling and tuning the NP stiffness for a desired selectivity and specificity of binding to target tissues is essential and can offer a lot of exciting opportunities in design and performance in vivo. In the deformable NP category, polymeric micelles, polymeric conjugates, and polymeric nanoparticles have been investigated.²³ Among these delivery systems, polymeric NPs promise to improve current disease therapies²⁴ because of their tunable physicochemical properties, such as the size, shape, rigidity, structure, and surface chemistry. Dextran-based, and in particular the core–shell polymer-based NPs consisting of a lysozyme rich core with a dextran-rich corona have shown to be excellent candidates for drug carriers, because of their biocompatibility, low cytotoxicity/immunogenicity and their tunable mechanical properties, which can be varied by controlling the crosslinking density in the polymeric phase.²⁵

One of the main goals of modeling is to predict the binding affinities based on the knowledge of the configuration of a complex, which would be of great benefit in rational drug design. The success of targeting cell surfaces by ligand-coated NPs is mainly determined by the NP physical and biochemical properties, ligand properties (e.g., density, architecture) and cell surface characteristics (i.e., receptor size, density, and mobility), and depends on the presentation of ligands and receptors. For example, it has been shown experimentally that an increase in the size of rigid NP and number of ligands leads to a stronger avidity of NP to the cell surface.^26–34

Each ligand either present directly on the NP or tethered to the NP via a flexible linker, could potentially bind to a receptor expressed on the cell membrane. The overall binding affinity of the NP (also referred to as the binding avidity) is determined by the interplay between the enthalpic gain due to the ligand–receptor binding and entropic loss associated with conformational degrees of freedom. The degrees of freedom can include receptor translation, receptor flexure, NP configuration (especially of its flexible internal components), membrane undulation, NP translation, and NP rotation. In order to predict the overall binding affinity or avidity of a given system, it is therefore, desirable to quantify the equilibrium bound state of a ligand-coated NP (rigid or flexible) bound to a substrate (rigid or flexible) in terms of the losses in the configurational entropies and the gains in the binding enthalpies. Computational methodologies based on Monte Carlo (MC) or/and coarse-grained molecular dynamics (CGMD) protocols for the transport and adhesion of functionalized NPs to both functionalized compliant and non-compliant surfaces have been developed and extensively validated in previous works.^{13,19,20,34–42} A summary of published reports is highlighted in Table 1.

Table 1.

Binding of NPs to diffusing ICAM-1 receptors on the cell surface: model validation

NP/membrane type	Methods	Findings and citation
Rigid/rigid	MC/in vivo, in vitro, AFM	Enhanced binding at large ligand surface coverage.34
Rigid/rigid	MC/in vivo	Increased selectivity using controlled reduction of receptor surface density.35
Rigid/flexible	MC/in vivo	Binding kinetics are influenced by mechanotype & phenotype of target cells.36
Flexible/—	CGMD/in vitro	1. Computed stiffness of the NP falls in the range of moderately soft materials. 2. Model 5 in the computation best represents the properties of the polymeric NP utilized in the experiments.37
Flexible/rigid	CGMD & MC/in vivo, in vitro	Shear-enhanced binding, effects of RBC volume fraction and NP stiffness.19,20

Frenkel and co-workers^38,42,43 developed a coarse-grained model of NP adhesion and used statistical mechanics-based models and Monte Carlo simulations to determine receptor–ligand interaction parameters for selective targeting of NPs. In their coarse-grained model, the cell surface and the NP are described as an impenetrable flat surface, and a hardsphere, respectively, and most of their study considered the limit of weak bonds. They concluded that super-selective behavior requires weak single bonds, and the design construct is more selective as the bond strength decreases. They estimated the free energy of binding and the change in the configurational entropy of ligands upon binding to immobile randomly distributed receptors on a model three-dimensional cubic lattice. They discussed that each ligand in the case of mobile receptors can bind to any receptor, and as a result, the receptors could be considered as a two-dimensional ideal gas, and the binding of a ligand to either immobile or mobile receptor becomes very similar.

Liu et al.³⁴ developed a computational model based on the Metropolis Monte Carlo method and the weighted histogram analysis method to determine the binding affinities of 100 nm spherical NP functionalized with anti-intercellular adhesion molecule-1 (ICAM-1) antibody to ICAM-1 expressing endothelial cells ECs surface-mediated through receptor–ligand interactions. The cell surface was treated as a rigid flat surface with several diffusive ICAM-1. Using the proposed model, they directly compared the measured binding affinities with those computed in experiments, and the results are in excellent agreement with results obtained from in vitro, in vivo, and atomic force microscopy (AFM) experiments.^34,44 Their computational analysis showed that an increase in antibody density, providing higher avidity, results in an increase of vascular targeting of NP to lung EC which agrees remarkably well with in vivo results in mice while simultaneously providing consistent agreement with AFM force-rupture experiments. Model results were also validated through a close agreement with binding measurements of NPs in the cell culture experiment for enhanced binding at large antibody surface coverage. Their analysis also revealed that the shear stress influence diminishes for NPs coated with antibodies at a surface density above a threshold level.

Liu et al. further extended their model by including the shear flow.^34,35 Their computational model is shown to be successful in predicting the critical threshold of antibody density above which the NPs can effectively bind to endothelial cells (EC). The model also predicts the associated multivalent interactions, i.e., the average number of receptor–ligand bonds per bound NP, as a function of local biophysical conditions (i.e., wall shear stress, wall glycocalyx and surface density of receptors) and NP features (i.e., size, surface properties such as ligand density). In vivo measurements and computational analysis of binding of anti-ICAM/NPs to ECs under static and flow conditions revealed that lowering the carrier avidity using controlled reduction of anti-ICAM surface density resulted in a marked 2-fold increase in selectivity of targeting to the inflamed endothelium and improved detection of pathology using PET imaging.

McKenzie and co-workers¹³ studied the binding affinity of rigid spherical NP with immobile ligands on the particle surface bound to a substrate with mobile diffusing receptors. The authors explicitly computed various losses in receptor translational and flexural entropies, and NP translational and rotational entropies as well as gain in binding enthalpy to delineate the entropy-dominated and enthalpy-dominated regimes for NP binding on a rigid flat substrate. In their calculation, they used the spatial patterns measured concerning the center of mass of the NP to evaluate the loss of translational entropy of bound receptors on the substrate and computed the probability distribution of the polar angle of the receptor to estimate flexural entropies of the bound and unbound receptors. The translational and rotational entropies of the rigid spherical NP were estimated by computing the average area traversed by the center of mass of the NP in its bound state and the rotational volume of a bound NP from the fluctuations of its Euler angles characterizing NP orientation about its center of mass, respectively,¹³ see Suppl. Info., Sec. S5 in the cited reference. They also calculated the enthalpic contribution to the binding by quantifying the potential of mean force through the use of umbrella sampling and the weighted histogram analysis method.¹³

However, in NP–cell adhesion, cell membrane curvature and undulations can have important influences on NP binding. Ramakrishnan et al.³⁶ developed a multiscale computational framework for the cell membrane to which a functionalized rigid NP binds and focused on the roles of membrane mechanical properties and protein expression level on regulating ligand-coated rigid NP interactions with the target cells. The framework which is based on a zero-fit biophysically based multiscale model is used to predict the live-cell/tissue targeting of 100 nm anti-ICAM1 functionalized NPs in the lung, heart, liver, kidney, and spleen of mouse and human and compare their findings to in vivo experiments. Their computational analysis showed that the stiffness of the membranes, the receptor expression, the density of ligands, and membrane-associated entropic factors significantly affect the avidity of NPs. Their predictions on tissue targeting levels in different organs compared very favorably with experimental findings, suggesting that the mechanotype and phenotype of target cells may significantly influence the binding avidity of NPs; therefore, such factors should be taken into account in designing the functionalized NPs.

New classes of NPs possess unique and exquisite chemistries for which rational design principles are quite distinct from rigid and regular-shaped NPs.^19,20,37 In an earlier report, our coarse-grained models for binding of a polymeric flexible NP to a rigid planar surface were examined to explore the effect of shear rate on the NP structure, transport properties and elastic properties.³⁷ We extended the analysis to investigate how hydrodynamic effects impact margination and binding of flexible polymeric nanoparticles in the presence of blood flow.^19,20 As evident from discussion, simulations and computational studies are capable of not only complementing experiments by providing detailed information about the mechanisms behind why and how particle stiffness impacts drug delivery processes, but also suggesting design principles for the synthesis of new systems. The objective of this article is to develop a statistical mechanics-based computational framework to predict and assess the binding affinity of a ligand-coated flexible NP to an undulating cell membrane by considering the equilibrium number of bound ligand–receptor pairs and proposing novel methods to quantify the enthalpy–entropy compensation in these displaying multivalent binding interaction systems.

2. Model and methods

2.1. Mesoscale model for multivalent binding of a functionalized star polymer NP to a cell membrane

The computational platform is based on the framework of equilibrium statistical mechanics and couples continuum field models for cell membranes with coarse-grained molecular-scale models for the NP, antibodies, and target receptors. We adopt the coarse-grained approach developed in earlier works^19,20,36,37 and further extend it to model binding of ligand-coated deformable NPs to a membrane functionalized with the target receptors. Each receptor and ligand molecule is taken as flexible rods of lengths L_r and L_l, respectively. The target membrane is taken as a square patch of dimension $\mathcal{L}$ and is chosen as a set of N_v nodes that are connected by bonds to form a triangulated surface. Membrane elasticity is studied using the Helfrich Hamiltonian which describes the bending energy of the membrane surface and involves the effective bending rigidity κ is given by:

$${\mathcal{H}}_{\text{m}}=\sum _{i=1}^{N_{\text{v}}}\frac{\kappa }{2}{\left(c_{1,i}+c_{2,i}\right)}^2{\mathcal{A}}_i.$$ (1)

Here, ${\mathcal{A}}_i$ is the curvilinear area of the membrane-associated with node i, and c_1,i and c_2,i are the maximum and minimum principal curvatures at each node, respectively. Since membrane conformation has a fixed projected area, i.e., ${\mathcal{A}}_{\text{p}}={\mathcal{L}}^2$, any nonzero value of the membrane excess area ${\mathcal{A}}_{\text{ex}}$ accommodates the additional area through undulations and is given by ${\mathcal{A}}_{\text{ex}}=\left(\mathcal{A}-{\mathcal{A}}_{\text{p}}\right)/\mathcal{A}\times 100$, where, $\mathcal{A}=\sum _{i=1}^{N_{\text{m}}}{\mathcal{A}}_i$ is the total curvilinear area of the membrane. The connectivity of the membrane is not fixed, which allows nodes to change neighbors and move throughout the membrane,³⁶ in order to maintain membrane fluidity. Two different ${\mathcal{A}}_{\text{ex}}$ are explored in our studies; these reflect two different membrane tension values, namely high ${\mathcal{A}}_{\text{ex}}$ reflective of low tension and vice versa. A mapping of ${\mathcal{A}}_{\text{ex}}$ to tension is available from previously published studies.^45,46

2.1.1. Models for NPs.

We model three types of NPs; rigid, rigid-tethered, and fully flexible.

Rigid NP.

The model presented for rigid NP was taken from ref. 36. The rigid NP is a sphere with targeting ligands loaded directly on its surface at a density of N_l = 162/NP.

Rigid-tethered NP.

The model for the rigid-tethered NP is the same as that of the rigid NP, except the targeting ligands are attached to the NP surface via polymeric tethers, we model the tethers using a freely jointed chain (FJC) model by assuming a tether elasticity, discussed later.

Flexible NP.

The microstructure of lysozyme-core/dextranshell cross-linked polymer NPs was recently modeled as a star polymer with a fixed number of arms attached to the core.³⁷ The parameters were calibrated to experiments.⁴⁷ The initial microstructure is a unit star polymer with 25 arms attached to a core, with each arm is modeled as a series of four beads connected by Kuhn springs. The size of each bead in the arms is set to be the same as the core radius, i.e., a = 6.8 nm, following the experimental estimates of Coll Ferrer et al.^47,48 The stiffness of the links between beads is derived from the FJC model, i.e., $k_{\text{s}}=\frac{3k_{\text{B}}T}{N_k{b_k}^2}$ with N_k and b_k being the number of Kuhn’s segments per bead and the size of each Kuhn’s segment, respectively. All beads of the NP interact by their adjacent connected beads through a harmonic potential, and the excluded volume interactions between beads is derived from Weeks–Chandler–Anderson potential. The interaction energy of the NP is then given by:

$${\mathcal{H}}_{\text{NP}}=\underset{\langle i,j\rangle }{\sum _i\sum _j\frac{1}{2}}{k}_{\text{s}}{\left(r_{ij}-r_0\right)}^2+\sum _i\sum _{j}4\epsilon \left[{\left(\frac{\sigma }{r_{ij}}\right)}^{12}-{\left(\frac{\sigma }{r_{ij}}\right)}^6+\frac{1}{4}\right],$$ (2)

where, r_ij = |r_i — r_j| is the distance between two interacting particles j and j. The second term is a summation over all i, j pairs and enforces excluded volume. The first term is denoted by $\langle i,j\rangle$ under the summation to signify that the summation only extends to pairs of beads that are connected by springs. r₀ is the equilibrium bond distance and is set to 2a, ε = 0.7k_BT is the interaction strength and s = 2a is the radius of the excluded volume.

2.1.2. Models for receptors on membrane surface.

The receptor molecules are considered to diffuse on the membrane surface. They are represented as cylindrical beams of total length ${\mathcal{L}}_{\text{r}}$ with flexural angles θ and ϕ, measured with respect to the membrane normal. When unbound, the receptors are perpendicular to the membrane surface in their vicinity with θ = ϕ = 0. Flexural angles of a receptor i are defined with respect to its equilibrium upright position on the membrane surface which leads to an orientational dependence of the bond energy:

$${\mathcal{H}}_{\text{f}}=\sum _{i=1}^m\frac{k_f}{2}{\left(L_{\text{r}}\sin {\theta }_i\right)}^2.$$ (3)

This term is activated only when a receptor is bonded to a ligand; therefore, the summation extends from 1 to m, where m is the multivalency. Here, k_f is the flexural stiffness of the receptors, and θ_i is the flexure angle for the ith receptor participating in the multivalent interaction.

Receptor–ligand bonds.

As noted earlier, the NP is functionalized using ligand molecules specific to target receptors on the membrane surface. Specifically, we consider an engineered antibody specific for ICAM-1 receptors. The ligand molecules are then distributed on the beads of the NP, and we model their binding as detachable springs that are connected to the ICAM-1 receptors. In our model, the receptor–ligand bond energy depends on the bond length, and we employ the Bell potential to account for the binding interaction between a ligand i and a receptor j, which is a quadratic function of the bond length:

$${\mathcal{H}}_{\text{bind}}=\sum _{i=1}^m\mathrm{\Delta }{G}_0+\frac{1}{2}{k}_{\text{b}}^{\text{eff}}{\left(d_i-d^{\text{eq}}\right)}^2,$$ (4)

for $d_i\le d^{*}$ and 0 for $d_i\ge d^{*}$. Here, d_i is the distance between the tip of the ith receptor participating in the multivalent interaction and the ligand it is bound to, and d^eq is the cutoff distance for the binding interaction, i.e., d^eq = σ/2 + L_l, where L_l is the length of the ligand molecule. ΔG₀ denotes the free energy of binding of a single receptor with its ligand. d* specifies the range of distance where binding can occur and is given by $d^{*}={\left(-2\mathrm{\Delta }{G}_0/k_{\text{b}}^{e\text{ff}}\right)}^{1/2}+d^{\text{eq}}$.

In the description above, the effective stiffness of the receptor–ligand bond (i.e., $k_{\text{b}}^{e\text{ff}}$) is given by a combination of the molecular stiffness surrounding the receptor–ligand bond³⁴ and that of the tether polymer segment represented by a bead, i.e., $k_{\text{b}}^{e\text{ff}}=\frac{k^{\text{s}}\times k^{\text{ICAM-anti}\hspace{0.17em}\text{ICAM}}}{k^{\text{s}}+ k^{\text{ICAM-anti}\hspace{0.17em}\text{ICAM}}}$ with $k^{\text{ICAM-anti}\hspace{0.17em}\text{ICAM}}\simeq 1\hspace{0.17em}\text{N}\hspace{0.17em}{\text{m}}^{-1}$ being the spring constant of a receptor–ligand bond.

The total energy of a deformable NP bound to a deformable membrane with m numbers of bonds comes from four contributions: the membrane energy $\left({\mathcal{H}}_{\text{m}}\right)$, the NP energy $\left({\mathcal{H}}_{\text{NP}}\right)$, the receptor energy $\left({\mathcal{H}}_{\text{f}}\right)$, and the binding/unbinding energy $\left({\mathcal{H}}_{\text{b}}\right)$. That is,

$$\mathcal{H}={\mathcal{H}}_{ \text{m}}+{\mathcal{H}}_{ \text{NP}}+{\mathcal{H}}_{ \text{f}}+{\mathcal{H}}_{ \text{bind}}.$$ (5)

We use the Metropolis Monte Carlo (MC) method to sample configurations of the functionalized NP–membrane system at thermal equilibrium. The set of MC moves of the system in its configurational space consists of six independent trial moves: the first is the standard MC move for the membrane, where a randomly selected node on the triangulated surface is moved to a new position to simulate thermal fluctuations in the membrane.⁴⁹ The second type of trial move is the standard MC move to translate the NP to a new position through the use of equations of motion using Brownian dynamics to evolve the beads of NP. To update the bead position, we include Brownian $\left(F_i^{\text{Br}}\right)$ force and non-Brownian forces such as forces due to the intermolecular interactions from all other beads $\left(F_i^{\text{NP}}\right)$, including bead–bead interaction $\left(F_i^{\text{s}}\right)$ and bead–bead repulsion $\left(F_i^{\text{WCA}}\right)$, i.e., $F_i^{\text{NP}}=F_i^{\text{s}}+F_i^{\text{WCA}}$, and force due to the binding interaction between receptor–ligand pair $\left(F_i^{\text{bind}}\right)$ and solve:

$$\frac{\mathrm{\Delta }{\mathbf{r}}_i}{\mathrm{\Delta }t}=\frac{F_i^{\text{Br}}+F_i^{\text{NP}}+F_i^{\text{Bind}}}{{\xi }_i},$$ (6)

where ξ_i = 6πμa is the friction factor. We consider Brownian forces as white noise which yields the following expressions:

$$\langle {\mathbf{F}}_i^{\text{Br}}\left(t\right)\rangle =0$$ (7)

$$\langle {\mathbf{F}}_i^{\text{Br}}\left(t\right){\mathbf{F}}_j^{\text{Br}}\left(t^{\prime }\right)\rangle =6k_{\text{B}}T{\xi }_i{\delta }_{ij}\delta \left(t-t^{\prime }\right)\mathbf{I}.$$ (8)

Here, I is the second-order identity tensor and δ(t – t′) is the Dirac delta function. The spring restoring force and the force due to excluded volume effect are derived from the interaction energy defined in eqn (2). The binding interaction force for a receptor–ligand pair is derived from the Bell potential defined in eqn (4). We use the forward explicit Euler time integration method to discretize eqn (6) and solve for updating the positions of the beads within a time step, which is chosen and also modified during runtime such that nearly 50% of the attempted moves are accepted. The third trial move is to simulate membrane fluidity by the membrane bond-flip move, in which a randomly chosen link is cut and replaced by a new link.⁴⁹ The new link connects two initially unconnected nodes associated with the triangles sharing the cut link. The fourth and fifth types of trial moves are random diffusion and flexure of receptors, respectively.^34,50 Binding/unbinding interaction of receptor–ligand pairs is the sixth type of trial move where a bond is formed between a randomly selected receptor–ligand pair if they are previously unbound, and the bond is broken/retained (with equal probability) if the selected pair is already in the bound state. Trial moves (1)–(5) are accepted or rejected according to the standard Metropolis criterion⁵¹ in the canonical ensemble, whereas the trial move for the formation and breakage of receptor–ligand bonds is performed and accepted by a configurational bias Monte Carlo move using the Rosenbluth sampling technique.⁵² For details of the various Monte Carlo moves, the readers are referred to ESI^† and the work of Ramakrishnan et al.³⁶

2.2. Free energy analysis

One of the objectives of this work is the calculation of the free energy for NP binding to the cell surface, which is quantified and used to compute the avidity of NP binding. Both ligands and receptors have a considerable amount of translational entropy associated with their free movement, and much of these entropies are expected to be lost upon specific binding. It is also crucial to take into account another main ingredient for the flexible polymeric NP, namely entropy loss of beads of polymeric NP. Upon multivalent binding of NP to the cell surface, the competition between the losses in the configurational entropies and the gain in binding enthalpy depends on the stiffness of the NP and the membrane, the receptor expression level, and the ligand density. The enthalpy of binding is computed as $\langle \mathcal{H}\rangle$, with the ensemble averages of $\mathcal{H}$ in eqn (5) computed during the MC simulation run. We need special methods to compute the entropic terms which arise from fluctuations in the positions of the center of mass of the beads of the NP, the motion of membrane vertices, and the spatial fluctuations of all receptors. The procedures to compute the entropic terms are described in the sections outlined below.

2.2.1. Estimating the entropy of deformable NP through quasiharmonic analysis.

In the quasiharmonic analysis,^53,54 the fluctuations in the motion of the system can be approximated as independent harmonic oscillators. Upon binding to the membrane surface, the intra-NP segmental dynamics are hindered and the entropy of the NP changes. The configurational entropy of the NP can be computed from principal component analysis (PCA) of positional fluctutations recorded in the MC trajectories.^55,56 The covariance matrix for positional fluctuations of N number of beads of the NP is given by:

$$B_{ij}=\langle \left(r_i-\langle r_i\rangle \right)\left(r_j-\langle r_i\rangle \right)\rangle ,$$ (9)

where, B_ij is the symmetric covariance matrix, r_i is the coordinate of the ith degree of freedom for i = 1,…, 3N. The covariance matrix for bead position fluctuations is corrected for removal of the translational and rotational movement of the center of mass. Within the harmonic oscillator model we can then connect the eigenvalues to frequency through the equipartition theorem by solving the following equation:

$$\det \left({\mathbf{m}}^{\frac{1}{2}}\mathbf{B}{\mathbf{m}}^{\frac{1}{2}}-\frac{k_{\text{B}}T}{{\omega }^2}\mathbf{I}\right)=0.$$ (10)

Here, ω is the angular frequency and m is the mass matrix with the masses of the beads of NP on the diagonal and all off-diagonal elements equal to zero and I is the identity matrix. The quasiharmonic entropy is then calculated from the frequency obtained from eqn (10) which is given by:

$$S_{\text{qh}}=k_{\text{B}}\sum _i\left[\frac{\hslash {\omega }_i/k_{\text{B}}T}{\exp \left(\hslash {\omega }_i/k_{\text{B}}T\right)-1}-\ln \left[1-\exp \left(-\hslash {\omega }_i/k_{\text{B}}T\right)\right]\right],$$ (11)

where $\hslash$ is the Planck’s constant divided by 2π, and ω_i is the set of angular frequencies obtained by solving eqn (10). In the calculation of quasiharmonic entropy, three eigenvalues $\left(\approx 0\right)$ correspond to translation about the center of mass, three eigenvalues $\left(\approx 0\right)$ correspond to rotation about the center of mass, and the remaining 3N – 6 eigenvalues correspond to vibrational motions.

2.2.2. Estimating the entropy of deformable membrane through quasiharmonic and Fourier analyses.

When the NP is bound to an undulating membrane, it affects the membrane entropy. While previous studies have suggested analytical methods to calculate the membrane entropy⁵⁷ for simple configurations of the curvature field, we propose two different methods to compute the configurational entropy of the membrane in more general and non-trivial configurations, such as when bound to a rigid, rigid-tethered, or flexible NP through multivalent receptor–ligand bonds. In the first method, similar to NP, we use the trajectories of positional fluctuation of the vertices of the membrane and compute the entropy through the harmonic oscillator model. This method follows the procedure outlined in Section 2.2.1 and are not repeated here. We call this procedure the PCA method or the real space method.

In the second method, we perform a Fourier analysis of membrane height undulations according to its physical properties-namely, the bending rigidity (κ) and surface tension (γ). We describe the membrane shape in the Monge gauge by its height, i.e., h(x,y), which leads to the following energy functional under the assumption of small slopes:

$${\mathcal{H}}_{\text{m}}=\int \frac{\kappa }{2}{\left({\nabla }^{2}h\left(\mathbf{r}\right)\right)}^2\text{d}\mathbf{r}+\frac{\gamma }{2}\int {\left(\nabla h\left(\mathbf{r}\right)\right)}^2\text{d}\mathbf{r}.$$ (12)

Here r = (x,y) and dr = dxdy. The Fourier transform of the membrane is then defined relative to the wavevectors (q):

$$h\left(\mathbf{r}\right)=\sum _{\mathbf{q}}{h}_{\mathbf{q}}\exp \left(i\mathbf{q}\cdot \mathbf{r}\right),$$ (13)

where, q = (q_x,q_y) = 2π(n_x/L_x,n_y/L_y). The Fourier transform of eqn (12) gives the energy in the frequency space:

$${\mathcal{H}}_{\text{m}}=\frac{{\mathcal{A}}_{\text{p}}}{2}\sum _{\mathbf{q}}{\left|h_{\mathbf{q}}\right|}^2\left\{\kappa q^4+\gamma q^2\right\}.$$ (14)

Here, q = |q|.

The average energy is obtained by taking the ensemble average:

$$\langle {\mathcal{H}}_{\text{m}}\rangle =\frac{{\mathcal{A}}_{\text{p}}}{2}\sum _q{k}_{\text{s,}q}\langle {\left|h_{\mathbf{q}}\right|}^2\rangle .$$ (15)

k_s,q is the stiffness associated with the qth mode of the membrane and it is defined by applying the equipartition theorem to the energy functional as $k_{\text{s,}q}=\frac{k_{\text{B}}T}{\langle \left|{h_q}^2\right|\rangle }$.

In both methods, we use eqn (11) to compute the thermodynamic conformational entropy of the membrane. However, the frequency of each independent harmonic oscillator is calculated differently. In the first method, the 3N — 6 frequencies are obtained from the covariance matrix, i.e., eqn (10), whereas in the second method, they are calculated from wavevectors, i.e., ${\omega }_{\mathbf{q}}=\sqrt{\frac{k_{\text{B}}T}{m_{\text{v}}\langle {\left|h_{\mathbf{q}}\right|}^2\rangle }}$ with m_v being the mass of each membrane bead; the number of q vectors included in the summation to compute the entropy is 724. It is expected that both methods give the same estimate for the entropy of the membrane. To test this premise, we consider a set of conditions for the adhesion of a NP to an undulating membrane with both small and intermediate excess-area regimes, i.e., ${\mathcal{A}}_{\text{ex}}=6\%$ and 20%, and with bending rigidity of κ = 20k_BT, as shown in Fig. 1. The entropy in the real space method (or PCA method, see Fig. 1) is based on the position fluctuation of beads, whereas in the Fourier space method (see Fourier in Fig. 1) the entropy is only associated with the membrane height fluctuations. In order to reconcile the difference, we estimate the in-plane vertex entropy as follows. The link length distributions on a triangulated lattice for two different membrane excess areas $\left({\mathcal{A}}_{\text{ex}}\right)$ are depicted in Fig. 2, and show a near-constant distribution, i.e., P_i = 1/C. Given that the in-plane membrane moves are performed on links rather than vertices, and that the distributions of the link-lengths are uniform in Fig. 2, we estimate the in-plane entropy per vertex using the 1-dimensional analog of the Sackur–Tetrode equation⁵⁸ given by $\frac{3}{2}+\ln {\left(2\pi \left(m_{\text{v}}/2\right)k_{\text{B}}T/h^2\right)}^{1/2}\times {\sigma }_{Lk}/N_{Lk}$, see Table 2 for parameter values. Here, σ_Lk is the standard deviation of the link length; in our calculations, σ_Lk = 1.77 nm for ${\mathcal{A}}_{\text{ex}}=6\%$ and σ_Lk = 1.87 nm for ${\mathcal{A}}_{\text{ex}}=20\%$. The estimates for the in-plane vertex entropy yield S_Theory/k_B = 3861 for ${\mathcal{A}}_{\text{ex}}=6\%$ and S_Theory/k_B = 4009 for ${\mathcal{A}}_{\text{ex}}=20\%$, (marked as Theory in Fig. 1). In order to calculate the total entropy of the membrane in the Fourier space, we define a corrected entropy (see Corrected-Fourier in Fig. 1) by summing the entropies associated with the membrane height fluctuations (see Fourier in Fig. 1) and the in-plane vertex entropy. Results of Fig. 1 confirmed our premise that both methods, i.e., PCA and corrected Fourier, give similar estimates for the entropy of the membrane.

Fig. 1.

Comparison of the configurational entropy of the membrane calculated in real (S_PCA) and Fourier (S_Fourier) spaces. S_Theory is the in-plane vertex entropy, and S_{Corrected-Fourier} is the entropy of the membrane in the Fourier space which is corrected by including the in-plane vertex entropy. Error bars, (horizontal parallel lines inside the symbols), are smaller than the size of the symbols.

Fig. 2.

Link distribution on a triangulated surface with A_ex = 6% and A_ex = 20%. The bond length is between a lower and an upper bound on a triangulated lattice. In the Y-axis the number of links in each bin is divided by the total number of links, where the bin size is set to 0.2 nm.

Table 2.

Details of the system parameters

Property			Symbol	Value
Membrane surface area			${\mathcal{L}}^2$	0.25 mm2
Mass of each membrane vertex			mv	5.5 × 10−19 g
Number of links			NLk	7699
Number of membrane vertices			Nv	2702
Receptor length			${\mathcal{L}}_{\text{r}}$	19 nm
Number of receptors			Nr	2000
Antigen flexural rigidity			kf	7000 pN nm2
Number of ligands per NP			Nl	162
Ligand length			${\mathcal{L}}_1$	15 nm
Bending rigidity			κ	20kBT
Free energy of binding per receptor–ligand bond			ΔG0	−7.98 × 10−20 J
Flexible NP		Symbol		Value
Bead radius		a		6.8 nm
Bead mass		mb		1.44 × 10−19 g
Number of beads in an arm		Nb		4
Number of arms attached to a core		f		25
Molecular weight of dextran monomer		Mmonomer		160 Da
Molecular weight of dextran polymer		Mpolymer		71 kDa
Size of each Kuhn segment		bk		0.44 nm
Size of monomer		b		1.5 nm
Number of monomers per bead		N		108
Number of Kuhn segments per bead		Nk		246
Stiffness of spring between beads		ks		1.74329 × 10−4 J m−2
Rigid NP	Symbol			Value
Radius	Rrigid			50 nm
Stiffness	krigid			1 J m−2
Rigid-tethered NP	Symbol			Value
Radius	Rrigid-tethered			50 nm
Stiffness of the tether	kt			3.3 × 10−4 J m−2

2.2.3. Estimating the translational entropy of diffusing receptors through the Sackur–Tetrode equation.

The translational entropy of N_r receptors is computed using the Sackur–Tetrode equation.⁵⁸ In the dilute limit, when the surface coverage is low, the receptors can move freely about on the surface without interacting with each other and therefore act as a two-dimensional ideal gas. The Sackur–Tetrode equation for an ideal gas of indistinguishable molecules with mass of m is given by:

$$S_{\text{transl}}=k_{\text{B}}\left[\sum _{i=1}^{N_{\text{r}}}\text{ln}\left(\frac{2\pi m{k}_{\text{B}}T}{N_{\text{r}}{h}^2}{\sigma }_{xi}{\sigma }_{yi}\right)+2\right].$$ (16)

Here h is the Planck’s constant and σ_xi and σ_yi are the principal root-mean-square fluctuations for the center of mass of receptor i.

3. Results and discussion

3.1. Multivalency, enthalpy, & entropy of NP binding

A summary of the detailed parameter set is provided in Table 2. The simulations are performed in a 500 × 500 × 620 nm³ simulation box with periodic boundary conditions in the plane of the membrane surface. We first equilibrate the membrane and the NP independently for 5 × 10⁷ MC steps. We then place the NP near the membrane surface and allow the system to relax for 5 × 10⁸ MC steps before collecting the data. Four independent simulation trajectories are generated for each set of conditions. To estimate the error in the calculated quantities, we calculate the average value for each independent trajectory and then calculate the standard deviation over the four ensembles. The computations for a typical trajectory noted above require three CPU-weeks on a single core of a 2.7 GHz processor.

First, we show that entropic terms governing NP flexibility and membrane are comparable in magnitudes to compete with the enthalpic term for flexible NP. In Fig. 3a we have explicitly computed the enthalpic gains due to the interactions between receptors and ligands as well as bead-to-bead interactions of flexible NP, which show the dominant effect of the receptor–ligand interactions to the total enthalpic gain. We have also shown the entropic losses in the configurational entropies associated with receptors, NP, and membrane, as demonstrated in Fig. 3b. Of these, the receptor entropy is the most dominant entropic term, as this term scales with multivalency. However, both NP flexibility and membrane undulations dictate the receptor–ligand translational entropy making the entropy compensation context-specific, i.e., dependent on whether the NP is rigid or flexible, and on the state of the membrane given by the value of membrane tension or its excess area.

Fig. 3.

Components of (a) enthalpic gain and (b) entropy loss of a flexible NP bound to a flexible membrane with varying A_ex. RE, MEM in the figure denote receptor and membrane, respectively. Note that the membrane entropy is calculated using the PCA method.

Next, we compare the binding of the rigid, rigid-tethered, and flexible NP for two different membrane excess areas. Fig. 4a–f depicts ligand coated flexible, rigid, and rigid-tethered NP, respectively bound to the cell membrane with fixed value of κ = 20k_BT and varying values of A_ex that falls into small (i.e., A_ex ~ 6%) and intermediate (i.e., A_ex ~ 20%) regimes. The flexible NP adopts a pancake-like configuration to maximize the multivalency. The histogram of multivalent binding interactions formed between the ligands on the NP and the receptors expressed on the cell surface for a flexible NP are shown in Fig. 4a and b and compared to those of the rigid (Fig. 4c and d) and rigid-tethered NP (Fig. 4e and f) for different membrane excess areas. The distribution of multivalent receptor–ligand bonds is sensitive to the type of the NP, and it may be seen that for both small and intermediate A_ex regimes, the rigid and rigid-tethered NPs stabilize less than 20 and 75 multivalent receptor–ligand bonds, respectively, whereas flexible NPs with similar excess areas stabilize between 152–162 multivalent receptor–ligand bonds.

Fig. 4.

Snapshots and equilibrium distribution of the number of simultaneous receptor–ligand bonds for (a and b) flexible NP, (c and d) rigid NP, and (e and f) rigid-tethered NPs with N_l = 162/NP bound to the cell membrane with fixed κ = 20k_BT and two distinct excess areas (i.e., A_ex = 6%, 20%).

Furthermore, A_ex of the target cell membrane can significantly alter the multivalency for the adhesion of functionalized NPs as less number of multivalent receptor–ligand bonds are formed for small A_ex.

However, the binding avidity for a NP is set by the competition between the enthalpic and entropic changes upon binding. That is, multivalent binding while leading to a gain in the enthalpy also causes loss of configurational entropy, which can offset each other. Fig. 5a shows plots of the free energy, entropy, and enthalpy of binding as a function of NP type and excess membrane area. Our results show that the enthalpy–entropy compensation in the case of flexible NP is substantially more pronounced than that for rigid and rigid-tethered NPs. It is worthwhile to note that the effect of configurational entropies associated with NPs, receptors, and membrane is particularly significant for flexible NP, showing higher entropy loss upon binding interactions (Fig. 5a). We also observe more reduction in the free energy of binding of NP in the presence of membrane with the higher excess area. Furthermore, the free energy, entropy, and enthalpy of binding per receptor–ligand bond is depicted in Fig. 5b.

Fig. 5.

Plots of free energy (ΔF/k_BT), entropy (ΔS/k_B), and enthalpy of binding (ΔH/k_BT) as a function of NP type (flexible, rigid, and rigid-tethered) and membrane A_ex; (a) the total binding energy and (b) the binding energy per receptor–ligand bond.

In order to test statistical significance of the values of the individual energy components, shown in Fig. 5, as functions of NP type and membrane excess area, the one-tailed p-values (at n = 3 and α = 0.05) are calculated with Welch two sample t-test using the statistical packages in R and illustrated in Tables 3 and 4. Results for the values of the total binding energy for different type of NPs, shown in Table 3, indicate that the reported total free energy, total enthalpy, and total entropy of binding between different types of NP are significantly different with the p-values smaller than 0.05. Per receptor–ligand bond, the p-value is still lower than 0.05 for the values of enthalpy and free energy of binding, indicating a statistically significant difference between different types of NPs. However, except for the comparison between rigid-tethered and flexible NP interacting with the cell membrane at A_ex = 6% or 20%, the statistical analysis (p-values in Table 3) illustrates that the entropy loss per receptor–ligand bond is not significantly different for different types of NPs, as the p-value is found to be larger than 0.05.

Table 3.

Report of the statistical significance (p-value) of the individual energy components between different types of NPs in Fig. 5 as a function of membrane excess areas

	Aex = 6%	Aex = 20%
Free energy
Rigid vs. rigid-tethered	8.3 × 10−5	0.0008
Rigid vs. flexible	0.001	4.8 × 10−6
Rigid-tethered vs. flexible	0.003	7.4 × 10−5
Entropy
Rigid vs. rigid-tethered	0.002	0.0002
Rigid vs. flexible	3.9 × 10−5	3.2 × 10−5
Rigid-tethered vs. flexible	0.001	0.0007
Enthalpy
Rigid vs. rigid-tethered	8.28 × 10−5	0.0008
Rigid vs. flexible	0.001	4.8 × 10−6
Rigid-tethered vs. flexible	0.003	7.4 × 10−5
Free energy per bond
Rigid vs. rigid-tethered	0.0001	0.0002
Rigid vs. flexible	0.0691	4.69 × 10−5
Rigid-tethered vs. flexible	0.005	0.0006
Entropy per bond
Rigid vs. rigid-tethered	0.81	0.5
Rigid vs. flexible	0.47	0.75
Rigid-tethered vs. flexible	0.026	0.006
Enthalpy per bond
Rigid vs. rigid-tethered	0.0001	0.0002
Rigid vs. flexible	0.069	4.86 × 10−5
Rigid-tethered vs. flexible	0.005	0.0006

Table 4.

Report of the statistical significance (p-value) of the individual energy components between membrane excess areas of 6% and 20% in Fig. 5 as a function of NP type

	Flexible	Rigid	Rigid-tethered
Free energy	0.6614	0.277	0.0166
Entropy	0.1307	0.483	0.1109
Enthalpy	0.6618	0.2771	0.01655
Free energy per bond	0.113	0.616	0.007
Entropy per bond	0.5945	0.6652	0.5668
Enthalpy per bond	0.112	0.609	0.007

Similar analysis were performed between membrane excess areas of 6% and 20% for each type of NP and reported in Table 4. For rigid and flexible NPs, the p-value between membrane excess area is 0.1 or upper, indicating that the components of the total and per receptor–ligand binding energy are statistically insignificant. In the case of the rigid-tethered NP, except for the entropy of binding, both enthalpy and free energy of binding does depend on the membrane excess area as the p-value is 0.01 or lower, proving a statistically significant difference between two membrane excess areas. Examining the p-values in Table 4, we can conclude that the free energy and enthalpy (both total and per bond) show a statistically significant difference between the two excess areas for the rigid-tethered system, with all others not passing the significance test.

We note that for the given receptor–ligand binding strength, we are operating at an adhesion density of the antibodies where wrapping does not happen. In future work, we will investigate the wrapping of the NP by extending our model to study this process directly.

3.2. Membrane entropy & NP adhesion-induced curvature

In Section 2.2.2, we presented methods to calculate the entropy for the bound membrane by assuming a zero-spontaneous curvature using the Fourier analyses. However, when NP adhesion induces a prominent curvature on the membrane, the entropic analysis needs to be altered. Here, we propose a new method where the analytical calculation can be further extended by introducing the curvature field into the analysis. In this case, the Canham–Helfrich Hamiltonian^59,60 is modified to include the coupling between the NP-induced curvature and the membrane undulations, considering the intrinsic curvature field (i.e., C₀):

$${\mathcal{H}}_{\text{m}}=\int \left(\frac{\kappa }{2}{\left(2H-C_0\right)}^2+\gamma \right)\text{d}\mathcal{A}.$$ (17)

The Helfrich energy in the Monge gauge and in the presence of C₀(r) is written as:

$${\mathcal{H}}_{\text{m}}=\int \frac{\kappa }{2}{\left({\nabla }^{2}h\left(\mathbf{r}\right)-C_0\left(\mathbf{r}\right)\right)}^2\text{d}\mathbf{r}+\frac{\gamma }{2}\int {\left(\nabla h\left(\mathbf{r}\right)\right)}^2\text{d}\mathbf{r}.$$ (18)

We define the Fourier transform of the curvature field by:

$$C_0\left(\mathbf{r}\right)=\sum _{\mathbf{q}}{C}_{0,\mathbf{q}}\exp \left(i\mathbf{q}\cdot \mathbf{r}\right),$$ (19)

and assume constant tension and bending rigidity. The average energy can then be written as:

$$\langle {\mathcal{H}}_{\text{m}}\rangle =\frac{{\mathcal{A}}_{\text{p}}}{2}\sum _q\left\{\kappa \left[q^4\langle {h_k}^2\rangle +2q^2\langle h_{\mathbf{q}}{C}_{0,\mathbf{q}}\rangle +\langle {C_{0,\mathbf{q}}}^2\rangle \right]+\gamma q^2\langle {h_q}^2\rangle \right\}$$ (20)

with q = |q|.

Snapshots in Fig. 4 suggest that only rigid-tethered NP induced the negative curvature on the membrane upon adhesion. In order to quantify the curvatures, according to eqn (20), we can estimate the curvature fields (C₀) induced in NP–membrane simulations by directly analyzing the average fluctuation of the membrane in the simulation trajectories. The expected spectrum of height–height undulations of the membrane is described in eqn (20), and as a result, the NP-induced curvature field (C₀) can be coupled to the membrane height–height undulation spectrum. In this curvature-undulation coupling method,⁶¹ we first extract the midplane heights and then based on the concept of neighborhood grids or points we superimpose an evenly-spaced grid of individual two-dimensional Gaussian functions over the instantaneous position of the NPs and fit them to the heights of the membrane in a neighborhood size of 100 nm (see Appendix). We, therefore, use the so-called neighborhood method for NP-field mapping and attempt to choose the curvature that minimizes the difference between the energy and that expected from the equipartition theorem, as was first analyzed in the work of Bradley et al.⁶¹ The minimization procedure yields the bending rigidity (κ), tension (γ), and the curvature fields (C₀(r)).

We can then incorporate the estimated curvature field in the entropy calculations. Following the previous analysis, and assuming that the equipartition theorem is valid for the membrane, the Hamiltonian becomes quadratic in h_q and thus a sum of harmonic oscillators:

$$\langle {\mathcal{H}}_{\text{m}}\rangle =\frac{{\mathcal{A}}_{\text{p}}}{2}\sum _q\left\{P_q^2\langle h_q{\prime }^2\rangle +\langle G_q\rangle \right\},$$ (21)

where, ${h_q}^{\text{'}}=h_q+\frac{F_q}{P_q}$, $P_q=\sqrt{\kappa q^4+\gamma q^2}$, $F_q=\frac{\kappa q^2{C}_{0,q}}{\kappa q^4+\gamma q^2}$ and $G_q=\kappa C_{0,q}^2-\frac{{\kappa }^2{q}^4{C}_{0,q}}{\kappa q^4+\gamma q^2}$.

As a result, by differentiating ${\mathcal{H}}_{\text{m}}$ twice concerning ${h_q}^{\text{'}}$, the stiffness (rigidity) and subsequently the frequency associated with the qth mode can be defined as:⁶²

$$k_{s,q}={\mathcal{A}}_{\text{p}}\left(\kappa q^4+\gamma q^2\right)$$ (22)

$${\omega }_q=\sqrt{\frac{k_{s,q}}{m_{\text{v}}}}$$ (23)

Interestingly, if we take the second derivative of eqn (14) with respect to h_q, where the curvature field is neglected, the stiffness of the membrane takes the same form as defined in eqn (22) for the membrane with spontaneous curvature. However, the induced curvature renormalizes the values of the bending rigidity and tension, and consequently alter the membrane entropy.⁶¹ That is, the value of k_s,q, and thus the value of the membrane entropy in the presence and absence of the curvature field is expected to be different. Our entropy analysis depicted in Fig. 6 confirms this premise for the rigid-tethered NP, which induces the highest curvature field. S1 and S2 here denote the quasiharmonic entropies of the membrane in the bound state with frequencies defined in eqn (10) with and without the consideration of the curvature field, respectively. A third method of entropy calculation S3 in the figure can be defined without any fitted parameters by directly relating the height–height undulation spectrum to the frequency, i.e., ${\omega }_q=\sqrt{\frac{k_{\text{B}}T}{m_{\text{v}}{\left|h_q\right|}^2}}$. Comparing these three estimates of entropy we observe that as the curvature field is introduced, i.e., S2, the entropy estimate for all three types of NPs become closer to the value of entropy depicted as S3, which not only establishes the self-consistency into our calculation, but also shows how induced curvature can alter the entropy value.

Fig. 6.

Comparison of the configurational entropy of the membrane calculated using the Fourier method with (S1) and without (S2) consideration of the induced curvature field at A_ex = 6%. S3 in the figure, is defined without any fitted parameters by directly relating the height–height undulation spectrum to the frequency. In this study, three replicates for each model system were included in the entropy estimates. Error bars, (horizontal parallel lines inside the symbols), are smaller than the size of the symbols.

Additionally, the results of Fig. 7 show the average height profiles (left) and the curvature field (right) for three simulation replicates for each type of NP. Each plot demonstrates an overhead view of the surface, which spans 520 nm in each lateral dimension. Each row contains three simulation replicates for each distinct NP type, including rigid (A–C), rigid-tethered (D–F), and flexible (G–I) NPs. Curvature fields (right) form a square neighborhood around the average position of the NP which includes the average of 49 separate trial functions, each of which is a two-dimensional Gaussian function in curvature (C₀) with a spacing of 40 nm. The functions were obtained using a neighborhood search, as described in the Appendix. It is useful to inspect the height profiles of the NP-laden membrane because they provide a qualitative picture of how the NPs can influence the membrane shape. The ensemble-averaged snapshots of membrane heights depicted in the left panel of Fig. 7 show that, compared to the rigid and flexible NPs, a rigid-tethered NP seems to target the undulations underneath it and that focuses the background thermal undulations to the vicinity of the NPs. Red areas are above the average height with positive deflection, while blue areas have below-average heights with z < 0, and the scale bar indicates that the spontaneous deflection is larger than the average deflection. The curvature field shown in the right panel of Fig. 7 suggests that the influence of the induced curvature field increases for the rigid-tethered NP (middle row) with more red areas, which is consistent with the visualization of the average height profile (i.e., left panel of Fig. 7).

Fig. 7.

Comparison of membrane average height (z) profiles (left) and curvature field (right) in presence of rigid (top rows, A–C), rigid-tethered (middle rows, D–F), and flexible (bottom rows, G–I) NPs, with fixed bending rigidity of κ = 20k_BT and excess area of A_ex = 6%. In the average height profile, red areas have above-average heights while blue areas are below the average. The curvature field in the right half of the plot appears to be centered on NP, and red and blue areas are the strongest curvature fields where the NP is.

Fig. 8 depicts the spectra which provide the estimates for bending rigidity and tension. The left panel shows the height–height undulation spectrum (dots) including a fit (lines) provided by the observed bending rigidity (κ) and tension (σ) values according to eqn (22), which looks close to the lines. The right panel shows the observed values of the elastic Hamiltonian (H_el) among undulation modes, given by eqn (18), after optimizing the curvature fields (C₀(x,y)) depicted in Fig. 7.

Fig. 8.

Comparison of the height–height undulation spectrum (left) and the observed values of the elastic Hamiltonian (H_el) (right) in presence of rigid (red, A–C), rigid-tethered (blue, D–F), and flexible (green, G–I) NPs. The fitted spectra provided by observed bending rigidity (κ) and tension (σ) values are represented by lines in the left panel. The left panel corresponds to the S1 case, which does not include a curvature term, while the right panel represents the S2 case which includes curvature.

The data presented here show that the energy distribution for different modes (q), which looks very similar for all three types of NP, is in good agreement with the equipartition theorem, which presumes that these modes should have uniformly distributed energies of k_BT. The observed values for bending rigidity and tension for S1 (with no curvature term) and S2 (with curvature term) cases are reported in Table 5.

Table 5.

Estimated values of bending rigidity (κ) and tension (γ) for S1 case, which does not include a curvature term, and S2 case which includes curvature. Each row represents three types of NPs, including rigid (A–C), rigid-tethered (D–F), and flexible (G–I)

	A	B	C	D	E	F	G	H	I
S1
κ (kBT)	34.13	36.04	35.22	30.54	32.97	31.55	36.85	37.32	29.03
γ (kBT nm−2 s)	0.028	0.022	0.031	0.023	0.015	0.019	0.027	0.0296	0.035
S2
κ (kBT)	35.91	38.39	38.05	31.88	34.73	32.32	39.03	40.96	29.41
γ (kBT nm−2)	−0.002	−0.006	−0.004	−0.005	−0.009	−0.0009	−0.004	−0.009	0.006

The histogram of the curvature values and error estimates for the S2 case and all three types of NP is demonstrated in Fig. 9. The left panel depicts histograms of the predicted curvature values (C₀(x,y)) shown in the fields in the right portion of Fig. 7. While there is an even amount of positive and negative curvatures around the flexible NP, more positive curvature can be observed around rigid-tethered and rigid NPs. The right panel shows the mean-squared error (MSE) for the fitted curvature fields according to the objective function presented in the appendix. These error values are uniformly lower (results not pictured here) than the equivalent errors provided by setting C₀(x,y) = 0 in the energy function given by eqn (20), indicating that including a non-zero curvature field improves the interpretation of the undulations. In general, the predictions showed here identify that even a small nm-sized domain is capable of coupling a deformation field to background thermal undulations.

Fig. 9.

Histograms of the predicted curvature value (left panel) along with the mean-squared error (right panel) upon binding of the membrane to rigid (A–C), rigid-tethered (D–F) and flexible (G–I) NPs.

4. Conclusion

We have presented a statistical mechanics based computational framework for analyzing the thermodynamic behavior of the adhesion of functionalized rigid, rigid-tethered, and flexible NPs binding to a membrane surface through multivalent receptor–ligand interactions. Our coarse-grained model presented here combines the model developed in an earlier report to study the interaction of a ligand-coated NP with cell surface receptors with the model for deformable NP by accounting for the mechanical properties of the NP. The conformational state of the NP interacting with the cell membrane is defined in terms of interaction potentials and is evolved through the Metropolis Monte Carlo approach coupled with the Brownian dynamics method to sample the micro states of the NP–membrane system at thermal equilibrium.

We presented three flavors of methods to compute the membrane entropy. (1) A direct method based on the quasiharmonic analyses of fluctuation of the membrane in real space through principal component analysis. (2) Fourier method based on the decomposition of the Helfrich energy function and application of equipartition theorem. (3) Fourier method with curvature inducers, which is an extension of (2), where NP adhesion induces significant membrane curvature. We show that after application of a crucial correction that accounts for the stretch modulus of the membrane, methods (1) and (2) agree for all NPs when no significant curvature is induced upon adhesion, and they agree more closely when there is a curvature induction upon adhesion. Together these observations validate the self-consistency of our approach and also show that the computed values of the membrane entropies are comparable in magnitude to NP conformational entropy as well as NP–receptor binding enthalpy. These observations unequivocally establish the importance of entropic factors mediating multivalent NP binding.

Our results show that the binding avidity of functionalized NPs is primarily determined by the enthalpy–entropy compensation seen in these systems. We have demonstrated this feature by proposing novel methods to compute the free energy for multivalent binding and also losses in configurational entropies associated with the receptor diffusion, and configurational motion of the NP, and the membrane. The NP flexibility strongly influences NP binding, and it was shown that flexible NPs promote higher multivalency and stronger avidity, compared to their rigid spherical counterparts. Other physiological and hydrodynamic factors can be integrated with the multivalent configurational entropy and enthalpy modules we have presented here, as outlined in our recent work.²⁰

Through a proposed curvature-undulation coupling method, we also quantified the curvature fields through adhesion of three types of NP, rigid, rigid-tethered and flexible, and demonstrated that the curvature induction has a significant effect on the entropy.

Appendix

The undulation-curvature coupling method for estimating NP adhesion-induced membrane curvature

While the bending rigidity (κ) and membrane tension (γ) are the most prominent and easily-measurable terms in the Canham–Helfrich Hamiltonian (eqn (14)), the spontaneous curvature field (C₀) created by the membrane–NP interaction also contributes to this energy functional and thereby bears on the entropy of the system. To account for the spontaneous curvature in the S2 entropy measurement presented in the main text, we turn to the undulation-curvature coupling method.⁶¹ In this method, we propose a functional form for the deformation field imposed on the membrane by the NP, C₀(x,y). The parameters that determine C₀ can be independently varied, and the resulting field is convolved with the Fourier-transformed height fluctuations h_q(x,y) in order to fit the bending rigidity (κ) and surface tension (γ) according to eqn (20). In this formalism (see ref. 61), the objective function is written as the sum of squared energy deviations from the k_BT given by the equipartition theory: $L=\sum _{q\hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}{q}_{\text{cut}}}\log {\left[\frac{{\mathcal{H}}_{\text{m,}q}}{k_{\text{B}}T}\right]}^2$ with a high-frequency cutoff distance of q_cut = 0.1 nm⁻¹. The resulting fits include an estimate for the curvature depicted in Fig. 7, as well as a corrected rigidity and tension that account for the changes in the undulation spectrum which are due to the induced curvature field. Inferring the curvature field from the height fluctuations in the membrane has the advantage of predicting curvature fields which are on the order of membrane fluctuations and cannot otherwise be distinguished by straightforward geometric methods.

Previous applications of the undulation-curvature method⁶¹ predicted deformation fields induced by small peripheral membrane proteins modeled via coarse-grained molecular dynamics simulations. This application modeled each protein-induced deformation field C₀ as a single two-dimensional Gaussian function (the ‘‘trial function’’) centered at the instantaneous position of the protein. In contrast, applying this method to the larger NP systems, which have a larger spatial extent, requires a more general functional form for C₀(x,y). In this work, we have superimposed an evenly-spaced square grid of individual two-dimensional Gaussian functions over the instantaneous position of the NP. These functions have three features which determine the overall functional form: the distance between grid points which are arranged in a square lattice, the distance from the centroid of the NP that includes these grid points, and the width of the Gaussian centered at each point. We call this the ‘‘neighborhood method’’ because the collection of grid points surrounds the instantaneous position of the NP. In this work, we have selected a grid spacing of 40 nm, a Gaussian half-width of 20 nm which ensures that the individual contributes decay to nearly zero before reaching the adjacent grid point, and a square neighborhood size of 100 nm.

These parameters generate either a 7 × 7 grid of individual Gaussian functions. The strength of curvature (i.e., the height of the Gaussian at each grid point) is allowed to independently vary during the optimization procedure, along with κ and γ. A systematic study of the neighborhood size (not pictured in this work) shows that neighborhoods larger than 100 nm show decreasing curvature strengths at greater distances from the protein, indicating that the NPs form deformations within this area, and thereby justifying our use of this neighborhood size. Similarly, a finer grid spacing does not produce qualitatively different curvature estimates and comes at a high computational cost because it increases the number of free variables in the optimization.

We have employed the ‘Scipy’ optimization package with the ‘scipy.optimize.minimize’ tool and the sequential least squares programming (SLSQP) algorithm.⁶³ Membrane and NP data were collected, interpolated to a grid of 520 × 520 points and interpolated with a spacing of 10 nm and post-processed with an in-house Python code which produced the trial function C₀(x,y) at each frame and optimized the individual field strengths for each grid point along with κ and γ in order to minimize the objective function L described above. The resulting curvature fields, shown in Fig. 7 of the main text, have a static magnitude. However, their position follows the centroid of the NP over time. All plots were produced with the ‘MatPlotLib’ package.⁶⁴ Note that the optimizer fit only the energies with wavenumbers of q_cut ≤ 0.1 nm⁻¹ since frequencies above this range are subject to noise that is induced by the interpolation.