Nanomaterial-cell Interactions Research Articles

Biocompatibility Evaluation of TiO2, Fe3O4, and TiO2/Fe3O4 Nanomaterials: Insights into Potential Toxic Effects in Erythrocytes and HepG2 Cells.

Nanomaterials such as titanium dioxide and magnetite are increasingly used in several fields, such as water remediation and agriculture. However, this has raised environmental concerns due to potential exposure to organisms like humans. Nanomaterials can cause adverse interactions depending on physicochemical characteristics, like size, morphology, and composition, when interacting with living beings. To ensure safe use and prevent the risk of exposure to nanomaterials, their biocompatibility must be assessed. In vitro cell cultures are beneficial for assessing nanomaterial-cell interactions due to their easy handling. The present study evaluated the biocompatibility of TiO2, Fe3O4, and TiO2/Fe3O4 nanomaterials thermally treated at 350 °C and 450 °C in erythrocytes and HepG2 cells. According to the hemolysis experiments, non-thermally treated NMs are toxic (>5% hemolysis), but their thermally treated counterparts do not present toxicity (<2%). This behavior indicates that the toxicity derives from some precursor (solvent or surfactant) used in the synthesis of the nanomaterials. All the thermally treated nanomaterials did not show hemolytic activity under different conditions, such as low-light exposure or the absence of blood plasma proteins. In contrast, non-thermally treated nanomaterials showed a high hemolytic behavior, which was reduced after the purification (washing and thermal treatment) of nanomaterials, indicating the presence of surfactant residue used during synthesis. An MTS cell viability assay shows that calcined nanomaterials do not reduce cell viability (>11%) during 24 h of exposure. On the other hand, a lactate dehydrogenase leakage assay resulted in a higher variability, indicating that several nanomaterials did not cause an increase in cell death as compared to the control. However, a holotomographic microscopy analysis reveals a high accumulation of nanomaterials in the cell structure at a low concentration (10 µg mL-1), altering cell morphology, which could lead to cell membrane damage and cell viability reduction.

Overview of Nanomaterials and Cellular Interactions

In the last three decades, there has been wide progress in nanomaterials development, and several studies are being performed to show its biological effects and cellular interaction for biomedical applications. Due to the exponential increase in nanomaterial diversity, production, and possibilities of applications in different areas, there is an important concern about its toxicity for humans, animals, and ecosystems. There is a great effort to minimize experimental assays in animals, and this is a commendable initiative. Several alternatives in vitro assays are available; however, several new protocols have been introduced to elucidate the mechanisms of cell-nanomaterial interaction. Wide and fast progress in nanotechnology has been observed. Nonetheless, the nanomaterial interaction with cells or biological systems is still not totally described. In this aspect, this paper is a brief overview of nanomaterials and cellular interactions (nano-bio interaction).

Differential Collective Cell Migratory Behaviors Modulated by Phospholipid Nanocarriers.

Phospholipid nanocarriers have been widely explored for theranostic and nanomedicine applications. These amphiphilic nanocarriers possess outstanding cargo encapsulation efficiency, high water dispersibility, and excellent biocompatibility, which render them promising for drug delivery and bioimaging applications. While the biological applications of phospholipid nanocarriers have been well documented, the fundamental aspects of the phospholipid-cell interactions beyond cytotoxicity have been less investigated. In particular, the effect of phospholipid nanocarriers on collective cell behaviors has not been elucidated. Herein, we evaluate the interactions of phospholipid nanocarriers possessing different functional groups and sizes with normal and cancerous immortalized breast epithelial cell sheets with varying metastatic potential. Specifically, we examine the impact of nanocarrier treatments on the collective migratory dynamics of these cell sheets. We observe that phospholipid nanocarriers induce differential collective cell migratory behaviors, where the migration speed of normal and cancerous breast epithelial cell sheets is retarded and accelerated, respectively. To a certain extent, the nanocarriers are able to alter the migration trajectory of the cancerous breast epithelial cells. Furthermore, phospholipid nanocarriers could modulate the stiffness of the nuclei, cytoplasm, and cell-cell junctions of the breast epithelial cell sheets, remodel their actin filament arrangement, and regulate the expressions of the actin-related proteins. We anticipate that this work will further shed light on nanomaterial-cell interactions and provide guidelines for rational and safer designs and applications of phospholipid nanocarriers for cancer theranostics and nanomedicine.

Chemically Induced pH Perturbations for Analyzing Biological Barriers Using Ion-Sensitive Field-Effect Transistors.

Potentiometric pH measurements have long been used for the bioanalysis of biofluids, tissues, and cells. A glass pH electrode and ion-sensitive field-effect transistor (ISFET) can measure the time course of pH changes in a microenvironment as a result of physiological and biological activities. However, the signal interpretation of passive pH sensing is difficult because many biological activities influence the spatiotemporal distribution of pH in the microenvironment. Moreover, time course measurement suffers from stability because of gradual drifts in signaling. To address these issues, an active method of pH sensing was developed for the analysis of the cell barrier in vitro. The microenvironmental pH is temporarily perturbed by introducing a low concentration of weak acid (NH4+) or base (CH3COO−) to cells cultured on the gate insulator of ISFET using a superfusion system. Considering the pH perturbation originates from the semi-permeability of lipid bilayer plasma membranes, induced proton dynamics are used for analyzing the biomembrane barriers against ions and hydrated species following interaction with exogenous reagents. The unique feature of the method is the sensitivity to the formation of transmembrane pores as small as a proton (H+), enabling the analysis of cell–nanomaterial interactions at the molecular level. The new modality of cell analysis using ISFET is expected to be applied to nanomedicine, drug screening, and tissue engineering.

Effects of Protein Source on Liposome Uptake by Cells: Corona Composition and Impact of the Excess Free Proteins.

Corona formation in biological fluids strongly affects nanomedicine interactions with cells. However, relatively less is known on additional effects from the free proteins in solution. Within this context, this study aims to gain a better understanding of nanomaterial-cell interactions in different biological fluids and, more specifically, to disentangle effects due to corona composition and those from the free proteins in solution. To this aim, the uptake of liposomes in medium with bovine and human serum are compared. Uptake efficiency in the two media differs strongly, as also corona composition. However, in contrast with similar studies on other nanomaterials, despite the very different corona, when the two corona-coated liposomes are exposed to cells in serum free medium, their uptake is comparable. Thus, in this case, the observed differences in uptake depend primarily on the presence and source of the free proteins. Similar results are obtained when testing the liposomes on different human cells, as well as in murine cells and in the presence of murine serum. Overall, these results show that the protein source affects nanomedicine uptake not only due to effects on corona composition, but also due to the presence and composition of the free proteins in solution.

A Finite Element-Based Coarse-Grained Model for Cell–Nanomaterial Interactions by Combining Absolute Nodal Coordinate Formula and Brownian Dynamics

Abstract A fundamental understanding of the interactions between one-dimensional nanomaterials and the cell membrane is of great importance for assessing the hazardous effects of viruses and improving the performance of drug delivery. Here, we propose a finite element-based coarse-grained model to describe the cell entry of nanomaterials based on an absolute nodal coordinate formula and Brownian dynamics. The interactions between nanoparticles and lipid membrane are described by the Lennard–Jones potential, and a contact detection algorithm is used to determine the contact region. Compared with the theoretical and published experimental results, the correctness of the model has been verified. We take two examples to test the robustness of the model: the endocytosis of nanorods grafted with polymer chains and simultaneous entry of multiple nanorods into a lipid membrane. It shows that the model can not only capture the effect of ligand–receptor binding on the penetration but also accurately characterize the cooperative or separate entry of multiple nanorods. This coarse-grained model is computationally highly efficient and will be powerful in combination with molecular dynamics simulations to provide an understanding of cell–nanomaterial interactions.

Facet-dependent antibacterial activity of Au nanocrystals

Engineered nanomaterials have attracted significantly attention as one of the most promising antimicrobial agents for against multidrug resistant infections. The toxicological responses of nanomaterials are closely related to their physicochemical properties, and establishment of a structure-activity relationship for nanomaterials at the nano-bio interface is of great significance for deep understanding antibacterial toxicity mechanisms of nanomaterials and designing safer antibacterial nanomaterials. In this study, the antibacterial behaviors of well-defined crystallographic facets of a series of Au nanocrystals, including {100}-facet cubes, {110}-facet rhombic dodecahedra, {111}-facet octahedra, {221}-facet trisoctahedra and {720}-facet concave cubes, was investigated, using the model bacteria Staphylococcus aureus. We find that Au nanocrystals display substantial facet-dependent antibacterial activities. The low-index facets of cubes, octahedra, and rhombic dodecahedra show considerable antibacterial activity, whereas the high-index facets of trisoctahedra and concave cubes remained inert under biological conditions. This result is in stark contrast to the previous paradigm that the high-index facets were considered to have higher bioactivity as compared with low-index facets. The antibacterial mechanism studies have shown that the facet-dependent antibacterial behaviors of Au nanocrystals are mainly caused by differential bacterial membrane damage as well as inhibition of cellular enzymatic activity and energy metabolism. The faceted Au nanocrystals are unique in that they do not induce generation of reactive oxygen species, as validated for most antibiotics and antimicrobial nanostructures. Our findings may provide a deeper understanding of facet-dependent toxicological responses and suggest the complexities of the nanomaterial-cell interactions, shedding some light on the development of high performance Au nanomaterials-based antibacterial therapeutics.

SERS-fluorescence-superresolution triple-mode nanoprobe based on surface enhanced Raman scattering and surface enhanced fluorescence.

Multifunctional nanoprobes play important roles in cell imaging and sensing. Here, we present a novel optical nanoprobe based on surface enhanced Raman scattering (SERS) and surface enhanced fluorescence (SEF), which can realize the SERS-fluorescence and superresolution triple-mode imaging of cancer cells. Compared with other previously reported multifunctional nanoprobes, the proposed nanoprobe holds two exquisite properties. The first one is that, in addition to normal SERS and fluorescence imaging, the nanoprobe can also be used for single molecule localization microscopy (SMLM) imaging, which helps compensate for the diffraction limited spatial resolution of normal SERS and fluorescence imaging. The second one is that, other than simple fluorescence, SEF is used in the nanoprobe to produce a stronger signal for fluorescence imaging and, more importantly, better photo-switching for SMLM imaging. In the experiment, we optimized the structure of the nanoprobe to obtain the best SEF effect. With the optimal structure, the triple-mode imaging of a breast cancer cell line (SKBR3) is realized. Since such triple-mode imaging of cancer cells has never been achieved before, we believe that the presented nanoprobe holds great potential for cancer cell targeting or the investigation of cell-nanomaterial interactions.

An in vitro assay and artificial intelligence approach to determine rate constants of nanomaterial-cell interactions

In vitro assays and simulation technologies are powerful methodologies that can inform scientists of nanomaterial (NM) distribution and fate in humans or pre-clinical species. For small molecules, less animal data is often needed because there are a multitude of in vitro screening tools and simulation-based approaches to quantify uptake and deliver data that makes extrapolation to in vivo studies feasible. Small molecule simulations work because these materials often diffuse quickly and partition after reaching equilibrium shortly after dosing, but this cannot be applied to NMs. NMs interact with cells through energy dependent pathways, often taking hours or days to become fully internalized within the cellular environment. In vitro screening tools must capture these phenomena so that cell simulations built on mechanism-based models can deliver relationships between exposure dose and mechanistic biology, that is biology representative of fundamental processes involved in NM transport by cells (e.g. membrane adsorption and subsequent internalization). Here, we developed, validated, and applied the FORECAST method, a combination of a calibrated fluorescence assay (CF) with an artificial intelligence-based cell simulation to quantify rates descriptive of the time-dependent mechanistic biological interactions between NMs and individual cells. This work is expected to provide a means of extrapolation to pre-clinical or human biodistribution with cellular level resolution for NMs starting only from in vitro data.

Fluorescent and magnetic stellate mesoporous silica for bimodal imaging and magnetic hyperthermia

There is currently a crucial need of innovative multifunctional nanoparticles combining, in one formulation, imaging and therapy capacities allowing thus an accurate diagnosis and a therapy monitored by imaging. Multimodal imaging will ensure to speed up diagnosis, and to increase its sensitivity, reliability and specificity for a better management of the disease. Combined with a therapeutic action, it will also enable to treat the disease in a specific personalized manner in feedback mode. The mastered design of such bioprobes as well as the demonstration of their efficiency are still challenges to face in nanomedicine. In this work, novel fluorescent and magnetic core–shell nanocomposites have been designed to ensure, in one nanoformulation, bimodal fluorescence and MRI imaging coupled with therapy by magnetic hyperthermia. They consist in the coating of a magnetic iron oxide (IO) core (ca.18nm diameter to ensure magnetic hyperthermia) by an original large pore stellate mesoporous silica (STMS) shell to produce uniform and mono-core magnetic core–shell nanocomposites denoted IO@STMS NPs. To confer fluorescence properties, CdSe/ZnS quantum dots (QDs) NPs were grafted inside the large pores of the IO@STMS nanocomposites. To provide biocompatibility and opsonization-resistance, a tightly-bound human serum albumin (HSA) coating is added around the nanocomposite using an original IBAM-based strategy. Cellular toxicity and non-specific cell–nanomaterial interactions allowed to determine a concentration range for safe application of these NPs. Cellular endosomes containing spontaneously-uptaken NPs displayed strong and photostable QD fluorescence signals while magnetic relaxivity measurements confirm their suitability as contrast agent for MRI. HeLa cell-uptaken NPs exposed to a magnetic field of 100kHz and 357Gauss (or 28.5kAm−1) display an outstanding 65% cell death at a very low iron concentration (1.25μgFemL−1), challenging current magnetic hyperthermia nanosystems. Furthermore, at the particularly demanding conditions of clinical use with low frequency and amplitude field (100kHz, 117Gauss or 9.3kAm−1), magnetic hyperthermia combined with the delivery of a chemotherapeutic drug, doxorubicin, allowed 46% cell death, which neither the drug nor the NPs alone yielded, evidencing thus the synergistic effect of this combined treatment.

PH optimization for high-efficiency PEGylation of gold nanorods

The use of methoxy polyethylene glycol thiol (mPEG-SH), in conjunction with additives, is one of the most straightforward, versatile, and widely used methods for modifying gold nanorods (GNRs), especially for applications in the field of biomedicines. The decreased toxicity of PEGylated GNRs is critical for investigating cell-nanomaterial interactions. Most techniques suffer from low PEGylation efficiency. Several investigations of the PEGylation process have been conducted to determine its mechanism. Herein, we systematically investigated the qualitative and quantitative effects of the pH value on GNR PEGylation efficiency. The PEGylation process was applied to modify the GNRs at pH 0.3–14. PEGylation of GNRs at pH 2.3 and 9.9 resulted in better modification efficiency than under other pH conditions, achieving average PEGylation efficiencies of 95.5% and 83.6%, respectively. In both cases, the resulting GNRs could survive centrifugation after surface modification as demonstrated by measuring their longitudinal surface plasmon resonance (LSPR) UV-Vis absorption peaks. The aggregation index correlated to the PEGylation efficiency at the various pH levels. The origin of the differences observed at different pH values was also elucidated. These findings significantly expand the potential use of the aggregation index, as well as the understanding of processes used to modify gold nanoparticles.

Toxicity and regulations of food nanomaterials

Food nanotechnology has been rapidly growing in last decade due to the unique properties of nanomaterials. Nonetheless, the presence of nanomaterials in food induces potential risks of toxicity because nanoparticles can easily cross the barriers of human anatomy, through respiratory, dermal, and gastrointestinal routes. In that respect, some countries such as India do not have strict regulations to control nanofood products. This review describes nanomaterial–cell interactions that induce toxicological responses. We discuss the toxicity of food nanomaterials; mechanisms of oxidative stress, genotoxicity and carcinogenicity; and safety regulations.

Widespread changes in transcriptome profile of human mesenchymal stem cells induced by two-dimensional nanosilicates

Two-dimensional nanomaterials, an ultrathin class of materials such as graphene, nanoclays, transition metal dichalcogenides (TMDs), and transition metal oxides (TMOs), have emerged as a new generation of materials due to their unique properties relative to macroscale counterparts. However, little is known about the transcriptome dynamics following exposure to these nanomaterials. Here, we investigate the interactions of 2D nanosilicates, a layered clay, with human mesenchymal stem cells (hMSCs) at the whole-transcriptome level by high-throughput sequencing (RNA-seq). Analysis of cell-nanosilicate interactions by monitoring changes in transcriptome profile uncovered key biophysical and biochemical cellular pathways triggered by nanosilicates. A widespread alteration of genes was observed due to nanosilicate exposure as more than 4,000 genes were differentially expressed. The change in mRNA expression levels revealed clathrin-mediated endocytosis of nanosilicates. Nanosilicate attachment to the cell membrane and subsequent cellular internalization activated stress-responsive pathways such as mitogen-activated protein kinase (MAPK), which subsequently directed hMSC differentiation toward osteogenic and chondrogenic lineages. This study provides transcriptomic insight on the role of surface-mediated cellular signaling triggered by nanomaterials and enables development of nanomaterials-based therapeutics for regenerative medicine. This approach in understanding nanomaterial-cell interactions illustrates how change in transcriptomic profile can predict downstream effects following nanomaterial treatment.

NMR Metabolomics Reveals Metabolism-Mediated Protective Effects in Liver (HepG2) Cells Exposed to Subtoxic Levels of Silver Nanoparticles.

The expansion of biomedical and therapeutic applications of silver nanoparticles (AgNPs) raises the need to further understand their biological effects on human cells. In this work, NMR metabolomics has been applied to reveal the metabolic effects of AgNPs toward human hepatoma (HepG2) cells, which are relevant with respect to nanoparticle accumulation and detoxification. Cellular responses to widely disseminated citrate-coated AgNPs (Cit30) and to emergent biogenic AgNPs prepared using an aqueous plant extract as reducing and stabilizing agent (GS30) have been compared with a view to assess the influence of nanoparticle coating on the metabolic effects produced. Subtoxic concentrations (IC5 and IC20) of both nanoparticle types caused profound changes in the cellular metabolome, suggesting adaptations in energy production processes (glucose metabolism and the phosphocreatine system), antioxidant defenses, protein degradation and lipid metabolism. These signatures were proposed to reflect mainly metabolism-mediated protective mechanisms and were found to be largely common to Cit30 and GS30 AgNPs, although differences in the magnitude of response, not captured by conventional cytotoxicity assessment, were detected. Overall, this study highlights the value of NMR metabolomics for revealing subtoxic biological effects and helping to understand cell-nanomaterial interactions.

Computational approaches to cell-nanomaterial interactions: keeping balance between therapeutic efficiency and cytotoxicity.

Owing to their unique properties, nanomaterials have been widely used in biomedicine since they have obvious inherent advantages over traditional ones. However, nanomaterials may also cause dysfunction in proteins, genes and cells, resulting in cytotoxic and genotoxic responses. Recently, more and more attention has been paid to these potential toxicities of nanomaterials, especially to the risks of nanomaterials to human health and safety. Therefore, when using nanomaterials for biomedical applications, it is of great importance to keep the balance between therapeutic efficiency and cytotoxicity (i.e., increase the therapeutic efficiency as well as decrease the potential toxicity). This requires a deeper understanding of the interactions between various types of nanomaterials and biological systems at the nano/bio interface. In this review, from the point of view of theoretical researchers, we will present the current status regarding the physical mechanism of cytotoxicity caused by nanomaterials, mainly based on recent simulation results. In addition, the strategies for minimizing the nanotoxicity naturally and artificially will also be discussed in detail. Furthermore, we should notice that toxicity is not always bad for clinical use since causing the death of specific cells is the main way of treating disease. Enhancing the targeting ability of nanomaterials to diseased cells and minimizing their side effects on normal cells will always be hugely challenging issues in nanomedicine. By combining the latest computational studies with some experimental verifications, we will provide special insights into recent advances regarding these problems, especially for the design of novel environment-responsive nanomaterials.

Correction: Computational approaches to cell-nanomaterial interactions: keeping balance between therapeutic efficiency and cytotoxicity.

Correction for 'Computational approaches to cell-nanomaterial interactions: keeping balance between therapeutic efficiency and cytotoxicity' by Hong-ming Ding et al., Nanoscale Horiz., 2018, 3, 6-27.

Nanomaterials in food and agriculture: An overview on their safety concerns and regulatory issues

ABSTRACTNanotechnology has seen exponential growth in last decade due to its unique physicochemical properties; however, the risk associated with this emerging technology has withdrawn ample attention in the past decade. Nanotoxicity is majorly contributed to the small size and large surface area of nanomaterials, which allow easy dispersion and invasion of anatomical barriers in human body. Unique physio-chemical properties of nanoparticles make the investigation of their toxic consequences intricate and challenging. This makes it important to have an in-depth knowledge of different mechanisms involved in nanomaterials's action and toxicity. Nano-toxicity has various effects on human health and diseases as they can easily enter into the humans via different routes, mainly respiratory, dermal, and gastrointestinal routes. This also limits the use of nanomaterials as therapeutic and diagnostic tools. This review focuses on the nanomaterial–cell interactions leading to toxicological responses. Different mechanisms involved in nanoparticle-mediated toxicity with the main focus on oxidative stress, genotoxic, and carcinogenic potential has also been discussed. Different methods and techniques used for the characterization of nanomaterials in food and other biological matrices have also been discussed in detail. Nano-toxicity on different organs—with the major focus on the cardiac and respiratory system—have been discussed. Conclusively, the risk management of nanotoxicity is also summarized. This review provides a better understanding of the current scenario of the nanotoxicology, disease progression due to nanomaterials, and their use in the food industry and medical therapeutics. Briefly, the required rules, regulations, and the need of policy makers has been discussed critically.

Transmission electron microscopy artifacts in characterization of the nanomaterial-cell interactions.

We investigated transmission electron microscopy artifacts obtained using standard sample preparation protocols applied to the investigation of Escherichia coli cells exposed to common nanomaterials, such as TiO2, Ag, ZnO, and MgO. While the common protocols for some nanomaterials result only in known issues of nanomaterial-independent generation of anomalous deposits due to fixation and staining, for others, there are reactions between the nanomaterial and chemicals used for post-fixation or staining. Only in the case of TiO2 do we observe only the known issues of nanomaterial-independent generation of anomalous deposits due to exceptional chemical stability of this material. For the other three nanomaterials, different artifacts are observed. For each of those, we identify causes of the observed problems and suggest alternative sample preparation protocols to avoid artifacts arising from the sample preparation, which is essential for correct interpretation of the obtained images and drawing correct conclusions on cell-nanomaterial interactions. Finally, we propose modified sample preparation and characterization protocols for comprehensive and conclusive investigations of nanomaterial-cell interactions using electron microscopy and for obtaining clear and unambiguous revelation whether the nanomaterials studied penetrate the cells or accumulate at the cell membranes. In only the case of MgO and ZnO, the unambiguous presence of Zn and Mg could be observed inside the cells.

Electrospinning of PVA/sericin nanofiber and the effect on epithelial-mesenchymal transition of A549 cells

This research aims to investigate the cell-nanomaterial interaction between epithelial-mesenchymal transition of A549 cell and electrospinning nanofibers composed of polyvinyl alcohol (PVA)/silk sericin (SS). The electrospinning of regenerated nanofiber was performed with water as a spinning solvent and glutaraldehyde as a chemical cross-linker. Solution concentration, applied voltage and spin distances as well as other parameters were optimized to generate fine nanofibers with smooth surface in good homogeneity. From the scanning electron microscopy (SEM) analysis, the nanofibers had an average diameter of 200nm. Epithelial-mesenchymal transition (EMT) is a process by which epithelial cells lose their cell polarity to become mesenchymal stem cells. This transition is affected by multiple biochemical and physical factors in cell metabolism cascade. Herein, we investigate the biophysical effect on A549 EMT by culturing cells on nanofibrous mats with different topography and composition. The cell viability was evaluated by biochemical assay and its morphology was observed with SEM. The results demonstrate that cells appropriately attached to the surface of the nanofibrous mats with extended morphology by their filopodia. Gene expression analysis was conducted by real-time PCR using multiple markers for detecting EMT: N-cadherin (NCad), Vimentin (Vim), Fibronectin (Fib) and Matrix metallopeptidase (MMP9). An increasing expression pattern was observed on NCad, Vim, Fib, with respect to a negative control as cell cultured on polystyrene dish. This result indicates the 200nm PVA/SS nanofibers may induce A549 cells to process epithelial–mesenchymal transition during the culturing.

The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate.

Mechanics of the bilayer membrane play an important role in many biological and bioengineering problems such as cell–substrate and cell–nanomaterial interactions. In this work, we study the effect of thermal fluctuation and the substrate elasticity on the cell membrane–substrate adhesion. We model the adhesion of a fluctuating membrane on an elastic substrate as a two-step reaction comprised of the out-of-plane membrane fluctuation and the receptor–ligand binding. The equilibrium closed bond ratio as a function of substrate rigidity was computed by developing a coupled Fourier space Brownian dynamics and Monte Carlo method. The simulation results show that there exists a crossover value of the substrate rigidity at which the closed bond ratio is maximal.