# Accounting for the mechanical response of the cell membrane during the uptake of random nanoparticles

Sarah Iaquinta, Shahram Khazaie, Frédéric Jacquemin, Sylvain Fréour

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Sarah Iaquinta, Shahram Khazaie, Frédéric Jacquemin, Sylvain Fréour. Accounting for the mechanical response of the cell membrane during the uptake of random nanoparticles. Journal of Theoretical, Computational and Applied Mechanics, 2024, pp.1-16. 10.46298/jtcam.12489. hal-04266009v2

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Submitted on 18 Sep 2024

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Distributed under a Creative Commons Attribution 4.0 International License
**Identifiers**

DOI 10.46298/jtcam.12489

HAL hal-04266009v2

**History**

Received Oct 31, 2023

Accepted Feb 21, 2024

Published Sep 15, 2024

**Associate Editor**

Anna PANDOLFI

**Reviewers**

Anonymous

Anonymous

**Open Review**

HAL hal-04576826

**Supplementary Material**

Software

SWHID [SoftwareHeritage]

Data

DOI 10.5281/zenodo.12519966

**Licence**

CC BY 4.0

©The Authors

# Accounting for the adaptation of cell membrane adhesion during the uptake of random nanoparticles

Sarah IAQUINTA, Shahram KHAZAIE, Sylvain FRÉOUR, and Frédéric JACQUEMIN

Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, F-44600 Saint-Nazaire, France

**Problem** In order to improve the efficiency of the delivery of cancer treatments to cancer cells, the cellular uptake of nanoparticles (NPs), used as drug delivery systems, is numerically investigated through a mechanical approach. An overall challenge of cancer therapy is to optimize the NP's mechanical and geometrical properties, focusing on cell-NP adhesion, membrane tension and NP's aspect ratio, to enhance their entry into cancer cells while avoiding benign ones. In previous studies, the mechanical properties are modeled as constant during the process of cellular uptake. However, recent observations of the displacement of the membrane's constituents towards the region in the cell membrane where the uptake of the NPs takes place show that the adhesion between the NP and the membrane vary during this process.

**Reason for writing** The important contribution of adhesion to the wrapping process is already well documented in literature. It is therefore crucial to model this parameter properly as the conclusions made with a constant adhesion model may not be accurate compared to reality.

**Methodology** Based on the existing knowledge on the reaction of membrane constituents to interaction with NPs, a 3-parameter sigmoidal function, accounting for the delay, amplitude, and speed of the reaction, has been used to model the evolution of adhesion. A variance-based sensitivity analysis has then been performed in order to quantify the influence of these parameters on the outputs of the model.

**Results** It was found that the introduction of a variable adhesion tends to alter the predictions of endocytosis of NPs. The contribution of the amplitude and delay is respectively 0.32 and 0.43 times as important as that of the NP's aspect ratio, which is the prominent parameter. The influence of the slope of the transition is the least important parameter and does not appear to contribute to endocytosis.

**Implications** Hence, models of the cellular uptake of NPs should use a variable, instead of constant, adhesion in order a representative as possible of the behavior of the cell membrane. The predictions are different from those obtained using a model with constant adhesion.

**Keywords:** cellular uptake, adhesion, mechano-adaptation, sensitivity analysis, meta-modeling

## 1 Introduction

Cancer, being a major disease worldwide (18 million new cases in 2020 (Ferlay et al. 2021)), has made oncology an expanding research field. Part of this research focuses on developing treatments (Sudhakar 2009). This study focuses on non-local cancers, i.e. those which have spread from one organ to the rest of the organism, or which are located in tissues like blood. This kind of cancer must be treated with therapies like chemotherapy (Nygren 2001; E. Chu 2017), hormonotherapy (Hellerstedt et al. 2002; Murphy et al. 2012) or immunotherapy (Schuster et al. 2006). More recently, targeted therapies (Gerber 2008) have emerged. They consist in using vectors, e.g. nanoparticle (NPs), in which the anti-cancer agent is placed (Briolay et al. 2021). The vector is coated with specific constituents, called ligands, that aim at targeting cancer cells. Once the cell is reached, several steps remain: the NP needs to adhere to the membrane and to be fully wrapped by the latter to ensure the cellular internalization. The wrapping of the NP takes place via a mechanical process, involving the bending of the membrane around the NP, and the adhesion between the NP and the membrane. Previous results showed that the wrapping is favored by an important capability of the membrane to bend and to adhere to the NP, along with a small membrane tension (Iaquinta et al. 2022). Once the vector is fully
**Abbreviations**

CI confidence interval

MC Monte Carlo

NP nanoparticle

PCE polynomial chaos expansion

qMC quasi Monte Carlo

**Symbols**

$\Delta E$ variation of potential energy

$\gamma$ adhesion

$\kappa$ bending rigidity

$\overline{\Delta E}$ normalized variation of potential energy

$\overline{\Gamma}$ random normalized adhesion

$\overline{\gamma}$ normalized adhesion

$\Gamma_A$ random ratio between final and initial normalized adhesion

$\overline{\gamma}_A$ ratio between final and initial normalized adhesion

$\Gamma_D$ random delay of the transition of adhesion

$\overline{\gamma}_D$ delay of the transition of adhesion

$\Gamma_S$ random curvature parameter for the transition of adhesion

$\overline{\gamma}_S$ curvature parameter for the transition of adhesion

$\overline{\Sigma}$ random normalized tension

$\overline{\sigma}$ normalized tension

$\overline{\bar{r}}$ aspect ratio of the NP

$\phi_i$ angle between the tangent and horizontal in region i

$\psi_j$ proportion of wrapping phase j

$\sigma$ tension

$a$ relative radius of the NP

$f$ wrapping degree

$f_{inf}$ inflection point of $\overline{\gamma}(f)$

$l_i$ length of region i

$N$ number of samples

$p$ circumference of the NP

$Q_2$ predictability factor

$S_i$ Sobol first order index with respect to input parameter i

$s_i$ arclength in region i

$ST_i$ total Sobol index with respect to input parameter i

wrapped, the agent enters the cell, causing its death or making its reproduction impossible. Unfortunately, the ligands may sometimes bond to receptors that are located in the membrane of healthy cells. As such, it is necessary to improve the precision of the targeting technique, in order to protect healthy cells from these agents. Moreover, discrepancies in the mechanical properties of healthy and cancer cells have been observed experimentally. Cancer cells are for instance less stiff compared to their healthy counterparts (Lekka et al. 2012; Lin et al. 2015; Hall 2009; Suresh 2007). In addition, they also adhere less to the extra-cellular medium (Yang et al. 2013; Haley et al. 2008; Kuo et al. 2018; Kanyo et al. 2020). It could therefore be theoretically possible to control the uptake of NPs by cells of different types based on a mechanical differentiation. Hence, experimental studies have been conducted in order to observe the effect of the properties of NPs on their entry into cells of different types (Canton et al. 2012; Wiegand et al. 2020; Schmid et al. 2014). However, such studies have several limitations, such as the experimental facilities, that do not enable to assess the mechanical properties of the investigated cells with accuracy and reproducibility (Rigato 2015; Vasir et al. 2008; Y.-S. Chu et al. 2005; Evans et al. 1987; Lekka et al. 2012). To overcome such practical challenges and to guide experimental work, models for the cellular uptake of NPs, focusing of the crucial wrapping step, have also been developed (S. Zhang et al. 2015). The main approaches are commonly used accordingly to the scale at which the problem is investigated and the chosen discretization. For instance, molecular dynamics approaches involve the representation of molecules or groups of molecules with their chemical interactions, while models at the scale of the NP involve a homogenized modeling of the NP and the membrane (S. Zhang et al. 2015). In this paper, a model at the scale of the NP (in opposition to those at the scale of the constituents of the cell membrane), is used. This kind of approach enables to simplify the problem by modeling the membrane as a thin line when its thickness, usually below 10 nm (Zhao et al. 2014), is small compared to the NP. To ensure this hypothesis is true, only NPs whose radius is larger than 100 nm are considered. Models at the scale of the NP have mostly been developed by Yi, Shi, et al. (2011) and were already presented in our previous article in a stochastic framework (Iaquinta et al. 2022). For these models to provide the most reliable predictions of the internalization of a NP, the input parameters must be defined accurately. As such, in this article, we present a model that accounts for the mechanical reaction of the membrane, triggered by the interaction with the NP. Accounting for this phenomenon in such a model is thus a novelty, and the consequences of this enrichment of the model are quantified via a variance-based sensitivity analysis. The objective of this article is to demonstrate the importance of accounting for variations in adhesion during wrapping.

This paper is organized as follows: Section 2 introduces the numerical model and Section 3 presents the adaptation of the membrane and the way it has been incorporated into the model. The effect of this new feature is investigated in Section 4, where sensitivity analyses on the internalization of circular and elliptic NPs are conducted. In Section 6, conclusions are drawn on the influence of the mechano-adaptation of the membrane on the predictions of the model.
## 2 Existing model

In (Iaquinta et al. 2022), we presented our model to investigate the wrapping of a rigid elliptic NP, whose circumference $p$ is set to $200\pi$ nm (the circumference of a circular NP whose radius is 100 nm), by the membrane of a cell whose diameter is about 10 µm. These objects are schematically represented in Figure 1(a). The hypothesis of a rigid NP is made to reduce the number of parameters and focus on those relative to the cell membrane. The difference in the scales of the NP and that of the cell enables us to represent the cell membrane as a single thin line, instead of modeling each constituent. With this hypothesis, the behavior of the cell membrane is described by its bending rigidity $\kappa$, the adhesion $\gamma$ between the cell and the NP, along with the membrane tension $\sigma$. The system, formed by the NP and the membrane, is therefore investigated through the variation of its potential energy, defined by the Canham-Helfrich Halmitonian (Seifert and Lipowsky 1990; Seifert 1991; Deserno and Bickel 2003; Deserno 2004; Helfrich 1973), as a function of the bending energy $\Delta E_b$ of the membrane, the energy related to its stretching $\Delta E_\sigma$, and that due to its adhesion with the NP, $\Delta E_\gamma$. The variation of the potential energy of the system reads

$$ \Delta E = \underbrace{\frac{\kappa}{2} \int_{0}^{l_2} \dot{\phi}_{2r}^2 ds_{2r}}_{\Delta E_b} + \underbrace{\frac{\kappa}{2} \int_{0}^{l_2} \dot{\phi}_{2l}^2 ds_{2l}}_{\Delta E_b} + \underbrace{\frac{\kappa}{2} \int_{0}^{l_3} \dot{\phi}_{3}^2 ds_{3}}_{\Delta E_\gamma} - \underbrace{\gamma l_3 + \sigma(2l_2 + l_3 - r_{2r}(l_2) + r_{2l}(l_2))}_{\Delta E_\sigma}, \quad (1) $$

where $s_i$ is the arclength, defined between 0 and $l_i$, $\phi_i$ is the angle between the tangent and the horizontal, as illustrated in Figure 1(b). The indices $i \in \{2r, 2l, 3\}$ correspond to the different regions of the system. The regions $2r$ and $2l$ stand for the free membrane, located at the right and left sides of the NP, and 3 represents for the contact region between the NP and the membrane.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Considerations for the model. (a) Scale of the model. (b) System of coordinates used to describe the wrapping of the NP by the cell membrane.</figcaption></figure>

Using the hypothesis of symmetry between the two sides of the membrane yields $l_{2r} = l_{2l} = l_2$ and $\phi_{2r} + \phi_{2l} = 2\pi$. As such, Equation (1) simplifies to

$$ \Delta E = \underbrace{\kappa \int_{0}^{l_2} \dot{\phi}_{2r}^2 ds_{2r}}_{\Delta E_b} + \underbrace{\frac{\kappa}{2} \int_{0}^{l_3} \dot{\phi}_{3}^2 ds_{3}}_{\Delta E_\gamma} - \underbrace{\gamma l_3 + \sigma(2l_2 + l_3 - r_{2r}(l_2) + r_{2l}(l_2))}_{\Delta E_\sigma}. \quad (2) $$

Following (Yi, Shi, et al. 2011), $\Delta E$ is adimensionalized in order to obtain an expression that is independent of the size of the NP. As such, $\bar{\Delta E}$ is introduced as $\bar{\Delta E} = 2a\Delta E\kappa^{-1}$, along with $\bar{\gamma} = 2a^2\gamma\kappa^{-1}$ and $\bar{\sigma} = 2a^2\sigma\kappa^{-1}$, where $a$ is the relative radius of the NP, defined as the ratio between the circumference $p$ of the NP and $2\pi$. Finally, $\bar{\Delta E}$ reads

$$ \bar{\Delta E} = \underbrace{\frac{a}{4} \int_{0}^{l_3} \dot{\phi}_{3}^2 ds_{3}}_{\bar{\Delta E}_b} + \underbrace{\frac{a}{2} \int_{0}^{l_2} \dot{\phi}_{2r}^2 ds_{2r}}_{\bar{\Delta E}_\gamma} - \underbrace{\frac{1}{4a} \bar{\gamma} l_3}_{\bar{\Delta E}_\sigma} + \underbrace{\frac{1}{4a} \bar{\sigma} (2l_2 + l_3 - r_{2r}(l_2) + r_{2l}(l_2))}_{\bar{\Delta E}_\sigma}. \quad (3) $$
Using the expression of $\overline{\Delta E}$, the system is at a stable equilibrium as $\overline{\Delta E}$ reaches its first local minimum in terms of the wrapping degree $f$ of the NP, defined as $f = l_3p^{-1}$. An illustration of the minima and equilibrium position of the system, based on the evolution of $\overline{\Delta E}$ in terms of $f$, is provided in Figure 2(a). The wrapping degree at which the equilibrium is reached is henceforth denoted by $\tilde{f}$. Once $\tilde{f}$ is determined, the shape of the system is observed and the phase is therefore defined. If $\tilde{f} < 0.2$, the NP is in phase 1, i.e. no wrapping. The NP is in phase 3 (full wrapping) if the two sides of the free membrane have merged above the NP. The NP is in phase 2 (partial wrapping), otherwise. The definition of these phases is illustrated in Figure 2(b).

<figure><img src="image_3.png" /><figcaption>**Figure 2** Definition of the equilibrium. (a) $\overline{\Delta E}$ in terms of the wrapping degree $f$, for $(\bar{y}, \bar{\sigma}, \bar{r}) = (6, 2, 0.3)$. The single circles correspond to the minima of energy, while the double circle stands for the equilibrium (first local minimum). (b) Description of the wrapping phases.</figcaption></figure>

# 3 Mechanical adaptation of the membrane

## 3.1 Description of the phenomenon

In this section, we present the observations, made from the literature, that led us to face the need for modeling the mechanical response of the membrane after contact with the NP.

### 3.1.1 Membrane tension

The cell contour is constituted of several irregularities, mostly invaginations, and protuberances, as illustrated in Figure 3(a). These are the so-called membrane reservoirs (Staykova et al. 2013; Kosmalska et al. 2015; Ferguson et al. 2017; Sinha et al. 2011; Rädler et al. 1995), that are unfolded (see Figure 3(b)) when the membrane is stretched to wrap the NP. As a consequence, the membrane tension does not increase during the wrapping of the NP, yielding $\overline{\sigma}(f) := \overline{\sigma}_0$, that will hereinafter be denoted as $\overline{\sigma}$.

<figure><img src="image_4.png" /><figcaption>**Figure 3** Cell membrane geometry. (a) Simplified illustration of the nonsmooth shape of a cell. (b) Observation of a reservoir on a stretched membrane, (top) during and (bottom) after unfolding, reproduced from (Kosmalska et al. 2015).</figcaption></figure>
### 3.1.2 NP-membrane adhesion

NP-membrane adhesion has two main sources: the interactions between the membrane receptor and the ligands on the surface of NP, and interactions between other molecules from the NP and the membrane, such as van der Waals, electrostatic bonds or hydrophobic interactions (S. Zhang et al. 2015; R. Zhang et al. 2019). The movement of membrane receptors has been observed and studied in the case of the interaction with a NP (Yi and Gao 2017; Decuzzi et al. 2008) and also for the adhesion of a cell to a substrate (Yi and Gao 2017; Freund 2004; Serpelloni et al. 2021). A schematic illustration of this phenomenon is provided in Figure 4.

<figure><img src="image_5.png" /><figcaption>**Figure 4** Displacement of the receptors to the contact region during the wrapping of the NP.</figcaption></figure>

Furthermore, the constituents of the membrane are reorganized laterally. As such, proteins and lipids may reach the contact zone during wrapping and increase the adhesion with the NP, as the wrapping degree increases (Cherry 1975; Cooper et al. 2007; McCloskey et al. 1984; Serpelloni et al. 2021). The hypothesis of a constant adhesion parameter, used in the existing model, does consequently not apply and $\bar{\gamma}$ needs to be modeled as a function of the wrapping degree. The nature of this function is discussed in Section 3.2.

### 3.2 Model of the mechanical adaptation of the membrane

Based on the previous assumptions, the NP-membrane adhesion, $\bar{\gamma}$, is considered as a function of the wrapping degree $f$, leading to $\bar{\gamma} := \bar{\gamma}(f)$. The nature of the function, that will be proposed in this work, is determined according to conclusions from the experimental studies reported in the literature. Hence, one can infer that the adhesion tends to increase during the wrapping process until reaching a final value, which corresponds to the stage when all the possible bonds between constituents from the NP and the membrane are formed (Yuan et al. 2010; Yi and Gao 2017; Freund 2004). It is however unclear if the constituents responsible for adhesion (specific receptors or ligands, proteins and lipids) start moving to the contact region as soon as the NP approaches the membrane, or if there is a delay, that would correspond to the information transmission to the rest of the membrane. We assume that this process contributes to an increase in adhesion and therefore we expect the adhesion to monotonically increase with respect to the wrapping degree $f$, starting from an initial minimum value and reaching a maximum final value. As such, $\bar{\gamma}(f)$ is modeled using a three-parameter sigmoidal function of $f$. Indeed, sigmoids have already been used in biology for the modeling of measures of nerve activity in terms of the arterial pressure (Head et al. 1987; Dorward et al. 1985; Ricketts et al. 1999). They are also commonly used in other fields of mechanics to model the diffusion phenomenon (Obeid et al. 2018), which may be similar to the behavior of the constituents of the membrane along its circumference. The variation of adhesion reads

$$ \bar{\gamma}(f) = \frac{\bar{\gamma}_0(\bar{\gamma}_A - 1)}{1 + \exp[-2\bar{\gamma}_S(f - f_{inf})]} + \bar{\gamma}_0 \quad (4) $$

where $f_{inf}$ is the inflection point, defined in terms of the delay $\bar{\gamma}_D$ as $f_{inf} = 0.5 + \bar{\gamma}_D$, while $\bar{\gamma}_A$ represents the amplitude of the transition and $\bar{\gamma}_S$ is the curvature parameter, which is independent of the aforementioned parameters and is used to control the slope of $\bar{\gamma}$ at the inflection point. In terms of receptor movement, $\bar{\gamma}_D$ is the delay in wrapping degree after which the receptors start moving, $\bar{\gamma}_A$ is the ratio between the final and initial quantity of receptors in the interaction region and $\bar{\gamma}_S$ is the displacement velocity of the receptors, in terms of wrapping degree. The initial value of adhesion, i.e. $\bar{\gamma}(f \to 0)$, is denoted by $\bar{\gamma}_0$. These parameters are detailed in Table 1 and their contributions to $\bar{\gamma}(f)$ are schematically illustrated in Figure 5.
<table><caption>**Table 1** Parameters of the sigmoid functions.</caption><thead><tr><th>Parameter</th><th>Definition</th><th>Range</th></tr></thead><tbody><tr><td>γ̃<sub>A</sub></td><td>Ratio between γ̃(f = 1) and γ̃<sub>0</sub></td><td>[1, 6]</td></tr><tr><td>γ̃<sub>D</sub></td><td>Delay of the transition, compared to f = 0.5</td><td>[-0.2, 0.2]</td></tr><tr><td>γ̃<sub>S</sub></td><td>Curvature parameter</td><td>[10, 50]</td></tr></tbody></table>

The lower bound of the domain of definition of $\bar{\gamma}_A$ is set to 1 in order to have an increasing function, while the upper bound has been set as being approximately equal to the amplitude of the domain of definition of $\bar{\gamma}_0$, which was defined based on previous works (Iaquinta et al. 2022; Yi, Shi, et al. 2011; S. Zhang et al. 2015) as the interval [1, 8]. It is worth noting that the particular configuration where $\bar{\gamma}_A = 1$ corresponds to a passive membrane, in which case the parameters $\bar{\gamma}_D$ and $\bar{\gamma}_S$ have no influence since the first term of the right-hand side of Equation (4) vanishes, yielding $\bar{\gamma}(f) = \bar{\gamma}_0$. The domain of definition of $\bar{\gamma}_D$ is determined using mathematical constraints. Indeed, as the mid value of the transition from $\bar{\gamma}_0$ to $\bar{\gamma}(f \to 1)$ is reached at $f_{inf} = 0.5 + \bar{\gamma}_D$, $\bar{\gamma}_D$ should vary in [-0.5, 0.5]. To avoid numerical singularities and a too-early or late transition, we chose to set a smaller interval, i.e. $\bar{\gamma}_D \in [-0.2, 0.2]$. Finally, the domain of $\bar{\gamma}_S$ was set to represent a reasonable range of values of curvatures, while ensuring that the boundary conditions $\bar{\gamma}(0) = \bar{\gamma}_0$ and $\bar{\gamma}(1) = \bar{\gamma}_0\bar{\gamma}_A$ are respected. Note that the curvature parameter $\bar{\gamma}_S$ is used to evaluate the slope of $\bar{\gamma}(f)$ at the inflection point $f_{inf}$ that is a function of $\bar{\gamma}_S$, $\bar{\gamma}_0$, and $\bar{\gamma}_A$.

<figure><img src="image_6.png" /><figcaption>**Figure 5** Effect of the parameters of an increasing sigmoid function: evolution of $\bar{\gamma}(f)$ for (a) $\bar{\gamma}_A \in \{1, 2, 3, 4\}$, (b) $\bar{\gamma}_D \in \{-0.2, -0.1, 0, 0.1, 0.2\}$ and (c) $\bar{\gamma}_S \in [0, 500]$. We set $(\bar{\gamma}_A, \bar{\gamma}_D, \bar{\gamma}_S) = (2, 0, 50)$ and $(\bar{\gamma}_0, \bar{\sigma}) = (1, 2)$ for all cases, except when stated otherwise in the graphs. Note that for the particular case, where $\bar{\gamma}_S = 0$, $\bar{\gamma}$ is independent of $f$ and equals $\bar{\gamma}_0(\bar{\gamma}_A + 1)/2 = 1.5$, wherein none of the boundary conditions are satisfied.</figcaption></figure>

# 4 Influence of the mechano-adaptation of the membrane on the predictions of the model

## 4.1 Method

### 4.1.1 Definition of the Qol

Once the function describing the evolution of $\bar{\gamma}$ is set, it is necessary to determine whether, and to which extent, the modeling of $\bar{\gamma}$ as a function instead of a constant, influences the results of the model. For this purpose, a global sensitivity analysis is conducted by calculating the Sobol indices. In this study, our Quantity of Interest (QoI) is $\psi_3$, the proportion of cells that reach the full wrapping phase. The latter is computed following several steps. (1) for each tuple $(\bar{\gamma}_0, \bar{\sigma})$, taken within the domain of definition of the variables, the variation $\Delta E$ in terms of $f$ is computed to identify the equilibrium position of the NP and thus (2) the equilibrium phase. The latter is used to (3) build the phase diagram, in which each region corresponds to an equilibrium phase $i \in \{1, 2, 3\}$. Finally, the phase proportions $\psi_i$, $i \in \{1, 2, 3\}$ are computed as the ratio of tuples $(\bar{\gamma}_0, \bar{\sigma})$ that yield the phase $i$ over the total number of tuples. This number of tuples was set to
<figure><img src="image_7.png" /><figcaption>**Figure 6** Description of the flux of calculation for obtaining the phase proportions $\psi_i(i \in \{1, 2, 3\})$ given the input parameters $(\bar{\gamma}_A, \bar{\gamma}_D, \bar{\gamma}_S, \bar{r})$.</figcaption></figure>

280 after a convergence study on $\psi_i, i \in \{1, 2, 3\}$. An example of phase diagram is provided in Figure 7(a). Note that this phase diagram was built with more than 280 points for the purpose of the illustration. For each tuple of $(\bar{\gamma}_A, \bar{\gamma}_D, \bar{\gamma}_S, \bar{r})$, a phase diagram is built, from which $\psi_3$ is extracted. The flux of calculation is illustrated in Figure 6. Note that all the phase diagrams are built using a regular sampling, with the same amount of points and the same domains of definition for $\bar{\gamma}_0$ and $\bar{\sigma}$.

An illustration of the evolution of the phase proportions in the case of a mechano-adaptive membrane is provided in Figure 7(b).

<figure><img src="image_8.png" /><figcaption>**Figure 7** Wrapping phases. (a) Example of phase diagram, obtained with $(\bar{\gamma}_A, \bar{\gamma}_D, \bar{\gamma}_S, \bar{r}) = (1, 0, 10, 1)$ and displayed with 1120 points for the quality of the illustration. (b) Comparison of the evolution of the phase proportions in terms of the aspect ratio $\bar{r}$ of the NP, between a passive membrane (dotted line) and a mechano-adaptive membrane $(\bar{\gamma}_A, \bar{\gamma}_D, \bar{\gamma}_S) = (2, 0, 50)$.</figcaption></figure>

### 4.1.2 Sampling of the input parameters

The Sobol indices are functions of the conditional variance of the QoI (see (Abramov et al. 1993) for more details). As such, the latter needs to be estimated. Such an estimation may require thousands of Monte Carlo (MC) samples in order to reach convergence (Iooss et al. 2015). In this article, we consider that the convergence of the Sobol indices is reached when the range of their 95% confidence intervals (CI) is smaller than the threshold of 0.05. This criterion was proposed by Sarrazin et al. (2016), as it enables us to investigate the convergence of each index separately, contrary to other criteria that investigate the convergence of the sum of the sensitivity indices (Vanrolleghem et al. 2015) or that of the most important parameter (Herman et al. 2013). Considering the computational cost of our model, using it to generate that number of samples is not feasible (about 32 days to build 1000 samples, with the available computational resources). In
order to reduce the number of samples necessary to reach convergence, the dataset is constructed using the quasi Monte Carlo sampling technique, built with Sobol's sequences. In addition, a surrogate model needs to be used to evaluate the QoI faster than the model itself. For this purpose, the Kriging metamodel, also known as Gaussian process, has been constructed, using the open-source library OpenTURNS. Technical details on Kriging and its implementation for this model have been presented in (Iaquinta et al. 2022) and general information on this metamodel can be found in (Cressie 1990; Sacks et al. 1989; Stein 2012; Santner et al. 2013; Rasmussen 2004; De Lozzo et al. 2016; Marrel et al. 2008). The development of this tool will be presented in the following sections. In order to understand the effect of the parameters of the sigmoid on $\psi_3$, the sensitivity analysis will first be conducted on circular NPs in Section 4.2. Then, the same study will be conducted on elliptic NPs in Section 4.3 to observe the effect of these parameters, when combined with $\bar{r}$.

## 4.2 Uptake of circular NPs

In this case, the model has three input parameters: $\bar{\gamma}_A$, $\bar{\gamma}_D$, and $\bar{\gamma}_S$. A dataset, containing $2^{10} = 1024$ input samples of the random uniform variables $\bar{\Gamma}_A$, $\bar{\Gamma}_D$, and $\bar{\Gamma}_S$, was used to build the dataset containing the values of $\Psi_3$, necessary for the construction of the metamodel. Note that uniform distributions have been used to model these variables based on the maximum entropy principle (Jaynes 1957) since the only available information is the extreme values of each parameter. An estimation of the probability density function (PDF) of $\Psi_3$, based on this dataset, is represented in Figure 8(a). In order to determine the minimal amount of data that is necessary for testing the predictions of the metamodels, the representativeness of the dataset is investigated by computing the normalized absolute gradient of the mean and the standard deviation of $\Psi_3$. The latter, for a function $y$ depending on a variable $x$, is defined as $|y(x + 1) - y(x)|/|y(x)|$. A subdataset is considered to be representative of the behavior of the dataset when both of these first and second-order statistics are smaller than the threshold criterion, defined as $10^{-2}$. According to Figures 8(b) and 8(c), the test dataset should contain at least 144 samples. The remaining dataset will therefore be used to construct the metamodels.

<figure><img src="image_9.png" /><figcaption>**Figure 8** Observations on the $\Psi_3$ dataset in the case of circular NPs. (a) Estimation of the PDF of $\Psi_3$ based on the dataset. (b) Absolute gradient of the mean of $\Psi_3$, in terms of the number of samples. (c) Absolute gradient of the standard deviation of $\Psi_3$, in terms of the number of samples.</figcaption></figure>

The Kriging metamodel has been built using a constant trend function along with a squared exponential (Gaussian) correlation function. The predictions obtained with the metamodel are compared to the values contained in the test dataset in Figure 9(a). To quantify the accuracy of the metamodel, the predictability factor $Q_2$ is computed. The latter is defined as

$$Q_2 = 1 - \frac{\sum_{i=1}^{N} (Y_i - \hat{Y}_i)^2}{N \text{Var}(\hat{Y})} \qquad (5)$$

where $Y_i$ and $\hat{Y}_i$ are the i-th true value in the test dataset and the corresponding prediction of the QoI, respectively. The metamodel yielded $Q_2 = 0.99$. Figure 9(b) compares the estimation of the PDF of the predictions generated via Kriging to that of the model, using the responses of the metamodel to $10^5$ MC-based input samples. In conclusion, Kriging can be used to
perform the sensitivity analysis, which will be presented in the following, as it provides accurate approximations of the model.

<figure><img src="image_10.png" /><figcaption>**Figure 9** Kriging predictions. (a) Predicted vs true values obtained with Kriging. (b) Comparison of the PDFs estimated from the predictions of this metamodel to that obtained from the model.</figcaption></figure>

The Sobol sensitivity indices have been estimated with the Saltelli (Saltelli 2002), Mauntz-Kucherenko (Sobol' et al. 2007), Martinez (Baudin et al. 2016) and Jansen (Jansen 1999) algorithms, implemented in OpenTURNS. Once the convergence is reached, the Sobol indices become independent of the algorithm. The range of the 95% CIs, in terms of the number of estimations, are depicted in Figure 10.

<figure><img src="image_11.png" /><figcaption>**Figure 10** Convergence of the Sobol indices. (a) First order Sobol indices. (b) Total Sobol indices. Ranges of the 95% confidence intervals in terms of the number of estimations of the metamodel, computed using the Mauntz-Kucherenko algorithm. The black dashed lines correspond to a threshold of 0.05.</figcaption></figure>

The converged Sobol indices are summarized in Table 2. These values reveal that the most important variable is the delay of the transition, $\bar{\gamma}_D$, with a total index of 0.64. Its first order index is 0.5, implying that the Sobol index relative to the interactions of $\bar{\gamma}_D$ with $\bar{\gamma}_A$ and $\bar{\gamma}_S$ are 0.64 – 0.50 = 0.14. Furthermore, since the interactions of $\bar{\gamma}_S$ are negligible (the first and total indices are close), $\bar{\gamma}_D$ interacts mostly with $\bar{\gamma}_A$, which is the amplitude of the transition. The latter is the second most influential parameter on the variance of $\Psi_3$ ($ST_{\bar{\gamma}_A} = 0.43$) and the interactions of $\bar{\gamma}_D$ with $\bar{\gamma}_A$ and $\bar{\gamma}_S$ contribute by $100 \times 0.14/0.43 = 33\%$ to the effect of $\bar{\gamma}_A$ on $\Psi_3$. These results lead to the conclusion that $\bar{\gamma}_D$ is the most important parameter, followed by $\bar{\gamma}_A$, which contributes to the output almost twice as less as $\bar{\gamma}_D$. The interactions between these two variables also contribute to the variance of the output. Last, $ST_{\bar{\gamma}_S} = 0.10$, which, even if it is small, is not negligible compared to the order of magnitude of the contribution of the other parameters.

## 4.3 Uptake of elliptic NPs

In this section, the study conducted previously is applied to elliptic NPs, whose aspect ratio $\bar{r}$ is defined as the ratio between the semi-minor and semi-major axes. Hence, $\bar{r}$ is smaller (resp.
<table><caption>**Table 2** First and total Sobol indices, calculated using $10^5$ estimations of the metamodel.</caption><thead><tr><th>Index</th><th>Parameter</th><th>Estimation</th></tr></thead><tbody><tr><td rowspan="3">S<sub>i</sub></td><td>Y&#773;<sub>A</sub></td><td>0.28</td></tr><tr><td>Y&#773;<sub>D</sub></td><td>0.50</td></tr><tr><td>Y&#773;<sub>S</sub></td><td>0.07</td></tr><tr><td rowspan="3">ST<sub>i</sub></td><td>Y&#773;<sub>A</sub></td><td>0.43</td></tr><tr><td>Y&#773;<sub>D</sub></td><td>0.64</td></tr><tr><td>Y&#773;<sub>S</sub></td><td>0.10</td></tr></tbody></table>

larger) than 1 for vertical (resp. horizontal) NPs, and the particular case where $\bar{r} = 1$ stands for circular NPs. In this study, we investigate NPs whose aspect ratios range from 1/6 to 6. The random variable $\bar{R}$ is added to the set of input parameters. As such, the problem contains four input parameters and the QoI is still $\Psi_3$. The distribution of $\bar{R}$ is built so that half of the dataset contains horizontal NPs ($1 < \bar{R} < 6$), and the remaining half of the values of $\bar{R}$ are the inverse of the aspect ratios of the horizontal ellipses. Hence, the distribution of $\bar{R}$ for $\bar{R} < 1$ should be the inverse of the uniform distribution used to maximize the entropy for the distribution of the horizontal NPs, leading to the PDF

$$f_R(x) = \begin{cases} \frac{1}{2} \frac{1}{6-1} \frac{1}{x^2} = \frac{1}{10} \frac{1}{x^2} & \text{for } x \in [1/6, 1] \\ \frac{1}{2} \frac{1}{6-1} = \frac{1}{10} & \text{for } x \in ]1, 6] \\ 0 & \text{otherwise.} \end{cases} \qquad (6)$$

The distribution of $\Psi_3$ is depicted in Figure 11(a). According to the absolute normalized gradient of the mean and standard deviation of $\Psi_3$, represented respectively in Figures 11(b) and 11(c), the test dataset requires to contain at least 365 samples. One can note that more samples are necessary compared to the previous case, which is due to the addition of the input parameter $\bar{R}$.

<figure><img src="image_12.png" /><figcaption>**Figure 11** Observations on the $psi_3$ dataset in the case of elliptic NPs. (a) Estimation of the PDF of $\Psi_3$ based on the dataset. (b) Absolute gradient of the mean of $\Psi_3$, in terms of the number of samples. (c) Absolute gradient of the standard deviation of $\Psi_3$, in terms of the number of samples.</figcaption></figure>

Here again, the Kriging metamodel is built in order to generate estimations of the model, necessary to further evaluate the Sobol indices. The Kriging metamodel was constructed with the same hyperparameters as those used in the previous case, yielding $Q_2 = 0.92$, based on the prediction presented in Figure 12(a). An estimation of the PDF of the predictions of $\Psi_3$ using $10^5$ MC-based generated input samples is represented in Figure 12(b). According to these results, the predictions from Kriging can be used to estimate the Sobol indices, since they are able to reproduce the behavior of the model. However, it is worth noting that this metamodel yields some inaccurate predictions, that may influence the precision of the Sobol indices, computed in the following.

The convergence of the Sobol indices, in terms of the number of estimations, is depicted in Figure 13. The convergence is ensured when at least $2 \times 10^4$ estimations are used. The converged values are summarized in Table 3.

This study highlights that $\bar{r}$ is the most influential parameter, with the largest total Sobol index, $ST_{\bar{r}} = 0.72$, while the curvature parameter $\bar{\gamma}_S$ is the less influential parameter with the
<figure><img src="image_13.png" /><figcaption>**Figure 12** Kriging predictions. (a) Predicted vs true values. (b) Comparison of the PDFs estimated via this metamodel.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 13** Convergence of the Sobol indices. (a) First order Sobol indices. (b) Total Sobol indices. Ranges of the 95% confidence intervals in terms of the number of estimations of the metamodel, computed using the Mauntz-Kucherenko algorithm. The black dashed lines correspond to a threshold of 0.05.</figcaption></figure>

lowest total Sobol index, $ST_{\bar{Y}_S} = 0.07$. Its first-order index is almost zero ($S_{\bar{Y}_S} = 0.02$), meaning that its influence on $\Psi_3$, is primarily due to interactions with other parameters $\bar{Y}_A, \bar{Y}_D$ and $\bar{r}$. A similar observation is made for the amplitude of the transition, $\bar{Y}_A$, whose first-order Sobol index is small ($S_{\bar{Y}_A} = 0.09$). The parameter with the largest total Sobol index, after $\bar{r}$, is the transition delay $\bar{Y}_D$, with $ST_{\bar{Y}_D} = 0.33$. It is thus the second most influential parameter. Still, its first-order index ($S_{\bar{Y}_D} = 0.11$) is small compared to the total index, meaning that the effect of $\bar{Y}_D$ is mainly due to its interactions with the remaining parameters. The curvature parameter and the amplitude of the transition, $\bar{Y}_S$ and $\bar{Y}_A$ being of small importance, one can infer that the influence of $\bar{Y}_D$ on $\Psi_3$ is mostly due to interactions with the aspect ratio $\bar{r}$ of the NP. One can also note that the ranking of importance between the parameters $\bar{Y}_A, \bar{Y}_D$, and $\bar{Y}_S$ is the same as the one obtained for circular NPs ($\bar{r} = 1$), as illustrated in Figure 14.

# 5 Discussion

The Kriging metamodel used to generate estimations of the model in the case of circular NPs yielded accurate predictions ($Q_2 = 0.99$). The same model, applied to elliptic NPs, could not be approximated with the same accuracy via Kriging ($Q_2 = 0.92$), showing that $\bar{r}$ induced nonlinearities that could not be handled by this metamodel. Alternative surrogate modeling approaches could have been tested, e.g. Polynomial Chaos Expansion (PCE), PCE-Kriging, and artificial neural networks, that may provide different and potentially better predictions. Hence, it would be of great interest to investigate these methods to improve the accuracy of the estimations used to compute the Sobol indices. Furthermore, PCE could be used to analytically calculate the Sobol indices based on the expansion coefficients (Sudret 2008). In addition, the output of Kriging could be post-processed to clip the predictions within the domain of definition of the QoI. As such, each estimation smaller than the lower bound would be replaced by the bound that has not
<table><caption>**Table 3** First and total Sobol indices, calculated using $10^5$ estimations of the metamodel.</caption><thead><tr><th>Index</th><th>Parameter</th><th>Estimation</th></tr></thead><tbody><tr><td rowspan="4">S<sub>i</sub></td><td>Y&#773;<sub>A</sub></td><td>0.07</td></tr><tr><td>Y&#773;<sub>D</sub></td><td>0.18</td></tr><tr><td>Y&#773;<sub>S</sub></td><td>0.02</td></tr><tr><td>r&#773;</td><td>0.47</td></tr><tr><td rowspan="4">ST<sub>i</sub></td><td>Y&#773;<sub>A</sub></td><td>0.25</td></tr><tr><td>Y&#773;<sub>D</sub></td><td>0.34</td></tr><tr><td>Y&#773;<sub>S</sub></td><td>0.09</td></tr><tr><td>r&#773;</td><td>0.79</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>**Figure 14** Distribution of the total Sobol indices obtained in (a) the investigation of the influence of $\bar{Y}_A$, $\bar{Y}_D$ and $\bar{Y}_S$ on the uptake of a circular NP and in (b) the comparison of the influence of $\bar{Y}_A$, $\bar{Y}_D$ and $\bar{Y}_S$ to that of the aspect ratio $\bar{r}$ of an elliptic NP.</figcaption></figure>

been respected, which would compensate the under-estimations. Furthermore, little is known about the input parameters, especially those of the sigmoid used to describe the variation of the adhesion between the NP and the cell membrane during the wrapping process. Hence, the domain of definition of these parameters was set following inferences based on observations reported in the literature, along with mathematical considerations. Given that the domain of definition of a variable affects the sensitivity analyses (Cousin et al. 2019), conducting a study in which the bounds vary could consequently allow us to quantify the dependence of the results of the sensitivity analyses on the domains of definition of the parameters. A similar remark can be made concerning the influence of the statistical distribution of these parameters on the sensitivity analysis, as it is also likely to alter the results. Furthermore, the model investigated in this article focuses on wrapping, which only one step of endocytosis. The prior and additional steps (eg clearance and exocytosis) also need to be considered to consider the likelihood of cellular internalization of a nanoparticle. The major contribution of the NP's aspect ratio matches experimental observations from the literature. For instance, Champion, Katare, et al. (2007); Champion and Mitragotri (2006) highlighted that the entry of a NP within macrophages, whose adhesion has not been controlled, is dictated by the NP's aspect ratio.

## 6 Conclusion

In this article, we have presented an enrichment of the existing model of the cellular uptake of rigid elliptic NPs, at the scale of the NP, by accounting for the mechanical adaptation of the membrane. This phenomenon was described using a sigmoidal variation of the NP-membrane adhesion, in terms of the wrapping degree of the membrane. To quantify the influence of the parameters that have been introduced, sensitivity analyses have been conducted, in the case of the uptake of circular and elliptic NPs. They showed that the aspect ratio of the NP influences $\Psi_3$ as much as the parameters related to the variation of $\bar{Y}$, among which the delay $\bar{Y}_D$ of the transition is the most important, followed by its amplitude $\bar{Y}_A$, while the curvature parameter $\bar{Y}_S$ can be considered as non-influential. The mechanical adaptation of the membrane plays therefore an important role in the predictions of the model. The values of $\bar{Y}_D$ and $\bar{Y}_A$ should consequently be precisely determined, based on experimental investigations, to obtain accurate predictions of cellular internalization of NPs. In addition, the results presented in this article revealed that the
aspect ratio of the NP is the most important parameter in the case of an adaptive membrane, which was also the case when the adaptation of the membrane was not considered (Iaquinta et al. 2022). Consequently, efforts should first be made on the precision and repeatability of the manufacturing of NPs. Second, additional investigations should be performed to accurately measure the transition delay and the amplitude of the variation of the adhesion between the cell membrane and the NP.

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**Authors' contributions** Author 1 carried out most of the study, performed numerical simulations, and drafted the manuscript. Author 2 helped with implementation and numerical issues. All authors developed the methodology, conceived of the study, and participated in its design, coordination, and critical review of the manuscript. All authors read and approved the final manuscript.

**Supplementary Material** The Python code containing the model of the cellular uptake of rigid circular and elliptic NPs, along with the routines to obtain $\psi_3$ is available at the permalink 10.5281/zenodo.12519966. It also contains the data and methods used to implement and validate Kriging metamodel, as well as algorithms used to compute the Sobol indices.

**Acknowledgements** The authors are very thankful to Dr. Christophe BLANQUART and Prof. Elena ISHOW for their excellent long-term collaboration and support.

**Funding** i-Site NExT, Région Pays de la Loire and CNRS (French National Centre for Scientific Research) through the ENAMEL project, Grant/Award Number: ANR-16-IDEX-0007.

**Competing interests** The authors declare that they have no competing interests.

**Journal's Note** JTCAM remains neutral with regard to the content of the publication and institutional affiliations.