# Role of molecular turnover in dynamic deformation of a three-dimensional cellular membrane

## Abstract

In cells, the molecular constituents of membranes are dynamically turned over by transportation from one membrane to another. This molecular turnover causes the membrane to shrink or expand by sensing the stress state within the cell, changing its morphology. At present, little is known as to how this turnover regulates the dynamic deformation of cellular membranes. In this study, we propose a new physical model by which molecular turnover is coupled with three-dimensional membrane deformation to explore mechanosensing roles of turnover in cellular membrane deformations. In particular, as an example of microscopic machinery, based on a coarse-graining description, we suppose that molecular turnover depends on the local membrane strain. Using the proposed model, we demonstrate computational simulations of a single vesicle. The results show that molecular turnover adaptively facilitates vesicle deformation, owing to its stress dependence; while the vesicle drastically expands in the case with low bending rigidity, it shrinks in that with high bending rigidity. Moreover, localized active tension on the membrane causes cellular migration by driving the directional transport of molecules within the cell. These results illustrate the use of the proposed model as well as the role of turnover in the dynamic deformations of cellular membranes.

## Keywords

Cytomembrane Molecular turnover Triangular mesh model Mechanosensing Coarse-grained modeling Multiscale simulation## 1 Introduction

This turnover can be regulated by membrane-associated proteins (Peters et al. 2016; McMahon and Gallop 2005; Kozlov et al. 2010). On a molecular scale, these proteins are collectively localized on the membrane and form complexes to pinch off parts of the membrane as liposomes. These liposomes are transported to another membrane and fused through the activities of membrane-associated proteins, such as SNAREs (Grant and Donaldson 2009). Notably, these proteins are known to play the role of mechanosensors, owing to the dependence of adhesion upon the membrane-stress state (Kozlov et al. 2010). Therefore, understanding the effects of turnover upon membrane deformation requires analyzing the feedback from the membrane-stress state on the turnover.

Several computational methods have been proposed to analyze the dynamics of cellular membranes. At a molecular level, molecular dynamics methods have been often used, whereby individual atoms are expressed as particles. Because the membrane dynamics are realistically expressed on the scale of individual lipid molecules, turnover in these methods physically results from molecular interactions (van der Ploeg and Berendsen 1982; Heller et al. 1993; Chiu et al. 1995). On the contrary, at a continuum level, several coarse-graining models have been proposed, ignoring the degrees of freedom of individual lipid molecules (Gompper and Kroll 1997; Ho and Baumgärtner 1990; Boal and Rao 1992; Gompper and Kroll 2004; Zhao and Kindt 2005). In particular, triangular mesh models have often been used to analyze the macroscopic dynamics of organelles and cytomembranes (Noguchi and Gompper 2005b; Ramakrishnan et al. 2013, 2015), and also applied to the analyses of cellular mechanotransduction (Atilgan and Sun 2007; Powers et al. 2012, 2014). In these models, membrane morphology is expressed by a triangular meshwork and membrane fluidity is successfully expressed by dynamically remodeling the meshwork topology (Gompper and Kroll 1998). Thus, modeling the membrane turnover on a triangular mesh model yields a powerful tool for investigating the dynamics of cellular membranes.

In this study, we propose a new computational model for simulating the turnover-dependent dynamics of three-dimensional cellular membranes. Firstly, we propose topological operations on a triangular network to express the plastic extension and shrinkage of a membrane. Secondly, we propose stochastic descriptions of molecular transport that depend upon the membrane-stress state. Thirdly, using the proposed model, we demonstrate computational simulations of several membrane dynamics and explore the effects of turnover upon membrane deformation. Finally, we discuss the applicability of the proposed model and report new findings on the effects of turnover upon membrane deformations.

## 2 Multiscale modeling of turnover-dependent membrane dynamics

### 2.1 Description of three-dimensional membrane deformation

The membrane shape is expressed by a triangular meshwork (Fig. 1a), whereby the membrane surface is expressed by a patch of triangles (Fig. 1b, c).

*i*th vertex by \(\varvec{r}_{i}\), the vertex motion obeys the over-dumped Langevin equation as follows:

*i*th vertex. Scalar \(\eta \) is a friction coefficient between the membrane and its microenvironment. The right-hand side of Eq. (1) denotes the energetic force acting on the

*i*th vertex, where

*U*is an effective energy function. Variable \(\varvec{w}_i\) is the Gaussian noise exerted on the

*i*th vertex that satisfies the following statistics:

*T*is the effective temperature.

*U*is given by

*i*th cell volume \(v_i\), the

*i*th triangle area \(a_i\), the mean curvature around the

*i*th vertex \(M_i\) and the mean surface area around the

*i*th vertex \(A_i\). Using these variables, \(U_\text {eff}\) is simply described by

*i*th vesicle, respectively. The second term indicates the surface elastic energy of the membrane exerted on individual triangle areas, where \(K_\text {a}\) and \( a_{\text {eq} }\) are membrane-surface elasticity and equilibrium area, respectively. The third term indicates the bending rigidity of the membrane, as exerted on individual vertices, where \(K_\text {c}\) is a membrane bending rigidity (Julicher 1996; Tsubota 2014). Variable \(M_i\) denotes the total mean curvature around the

*i*th vertex: \(M_i = \sum _{j(i)}^\text {edge} l_j \theta _j / 4\), where index

*j*(

*i*) is the

*j*th edge surrounding the

*i*th vertex. Variable \(A_i\) is the surface area around the

*i*th vertex: \(A_i = \sum _{j(i)}^\text {triangle} a_j / 3\), where index

*j*(

*i*) is the

*j*th triangle surrounding the

*i*th vertex.

### 2.2 Description of membrane fluidity and turnover

Cellular membrane has a fluidity, which causes viscous dissipation during membrane deformation. Moreover, membrane molecules are transported between the target vesicle and the reservoir comprising other vesicles within cell. In this model, we regard individual triangular elements as comprising a constant number of membrane molecules. Namely, in the model, the molecules composing the target vesicle is explicitly expressed as the triangular elements, whereas we implicitly express the molecules within the reservoir as a variable. Then, the fluidity can be expressed by a rearrangement of the triangular network (Gompper and Kroll 1998; Noguchi and Gompper 2004, 2005a). Moreover, the turnover can be regarded as a conversion between the triangular elements of the membrane and the number of molecules within the reservoir. Namely, the vesicle size increases when molecules are transferred to and decreases when transferred away from its surface. Therefore, in the model, the fluidity and turnover are expressed from two standpoints: topology and mechanics.

#### 2.2.1 Spatiotemporal scales of our scope

To express membrane fluidity and turnover, we argue spatiotemporal scales of our scope.

#### 2.2.2 Expression of membrane fluidity

*i*th edge, represented by \(P_{\text {f} i}\), is given as the following probability:

*i*th edge. Notably, \(\tau _\text {f}\) reflects the magnitude of membrane viscosity (Noguchi and Gompper 2004, 2005a).

#### 2.2.3 Expression of membrane turnover

From a topological viewpoint, the turnover is modeled differently for the increase and decrease in the vesicle size. The increase in the vesicle size is expressed by the local network extension: splitting a couple of neighboring triangles (Fig. 2b). The decrease in the vesicle is expressed by the local network shrinkage: merging a couple of neighboring triangles with their surroundings (Fig. 2c). In these process, the position of a new vertex is determined as the center of the edge shared by the neighboring triangles.

*i*th edge, represented by \(\Delta _i E\), as well as the chemical potential of the reservoir, represented by \(\mu _\text {r}\). The frequency of the split and merge around the

*i*th edge, represented by \(P_{\text {t} i}\), is given by the following probability:

## 3 Computational simulation of turnover-dependent membrane dynamics

### 3.1 Introducing turnover behavior into the proposed model

In order to analyze effects of turnover on vesicle dynamics, we simply model the mechanosensing regulation using \(\Delta _i E\) and \(\mu _\text {r}\) in Eq. (16).

*i*th edge, represented by \(\left\langle a_i \right\rangle \), we define \(\Delta _i E\) as follows:

To simply define \(\mu _\text {r}\), we consider an isolated system composed of the target vesicle and reservoir. Because membrane transport is actively driven by membrane-associated proteins, the active fluctuation of the turnover is much larger than the thermal fluctuation. Hence, by ignoring osmotic pressure, we focus on the active fluctuation around equilibrium.

### 3.2 Non-dimensionalization and parameter setting

To solve Eq. (1), parameter values are normalized to have unit length (*l*), unit time (\(\tau \)), unit number of molecules (*m*), and unit energy (\(k_BT\)). Here, *l*, \(\tau \), and *m* are set as \(l=\left( a_{\text {eq}} \right) ^\frac{1}{2}\), \(\tau = 0.1 \eta a_{\text {eq}} / k_BT\), and \(m=m_\text {u}\). Hereafter, physical parameters are described as dimensionless values. In case where a specific membrane is focused upon, the physical parameters employed in the simulations can be determined based on those measured by experiments. By assuming the system temperature to be 310 K, unit energy \(k_BT\) becomes \(4.3 \times 10^{-20}\) J. Based on this, the values of \(K_\text {c}\) employed in this study correspond to \(4.3 \times 10^{-20}\)–\(4.3 \times 10^{-19}\) J. These values have the similar range of those of the dimyristoylphosphatidylcholine (DMPC), the plant thylakoid lipid digalactosyldiacylglycerol (DGDG), and other general lipid membranes (Duwe et al. 1990; Engelhardt et al. 1985; Schneider et al. 1984; Mutz and Helfrich 1990; Evans and Rawicz 1990; Kummrow and Helfrich 1991).

To establish that the proposed model successfully recapitulates turnover-dependent membrane dynamics, several parameters are varied, such as \(K_\text {c}\) and \(\tau _\text {t}\). The state under the initial condition is set as a single vesicle composed of 1000 triangles, which are equilibrated under the condition \(K_\text {c} = 10\). The equilibrium volume of the vesicle is set to \(v_\text {eq} = 2527\), which corresponds to the volume of the vesicle with the area \(1000 a_\text {eq}\) and sphericity 0.85. The setting of the geometric constraint \(l_{\text {rep}}\) in Eq. (5) is described in Appendix A. Moreover, to set physical parameters, the force balance among individual energy terms in Eqs. (5) and (6) is taken into account, as described in Appendix B. Numerical implementation and calculation is described in Appendix C. All model parameters are shown in Table 1.

### 3.3 Proposed model successfully recapitulates turnover-dependent membrane dynamics

To establish whether the proposed model successfully recapitulates turnover-dependent membrane dynamics, we simulate vesicle dynamics in case with and without membrane turnover (\(\tau _\text {t} = 1.0\) and \(+\infty \)) (Fig. 3). In the case without membrane turnover, the vesicle is slightly deformed by fluctuations while maintaining its surface area (Fig. 3a, c). On the other hand, in the case with turnover, the vesicle is significantly deformed as its surface expands (Fig. 3b, c). Hence, the large deformation is permitted by the surface-area extension (Fig. 3c). Moreover, for a long timescale, the total surface area and number of molecules reached the plateau (Fig. 3c, d). This tendency seems independent on the values of the bending rigidity \(K_\text {c}\) and transport instability \(M_\text {inst}\); meanwhile, we could not observe the plateau in the time range of our simulations in case with large \(M_\text {inst}\). Importantly, the extension is caused by the increase in the number of molecules within the vesicle (Fig. 3d), but not by elastic deformation. In the process of this expansion, the number of molecules within the vesicle gradually increases in a stochastic manner (Fig. 3e). These results suggested that the proposed model successfully recapitulates the turnover-dependent membrane dynamics.

### 3.4 Molecular turnover adaptively facilitates vesicle deformation

Next, to investigate the effects of molecular turnover upon membrane deformation, we analyze the effects of the bending rigidity \(K_\text {c}\) and the instability modulus of the number of molecules within vesicle \(M_\text {inst}\) (Fig. 4). To focus on the fundamental effects of turnover on membrane dynamics, we set \(U_\text {act} = 0\) in Eq. (4) by assuming a simple membrane behavior.

Model parameters

Symbol | Value | Description |
---|---|---|

| ||

\(K_\text {v}\) | \(1.0 \times 10^4\) | Volume elasticity of vesicle |

\(K_\text {a}\) | 50 | Surface elasticity of membrane |

\(K_\text {c}\) | 1.0–10 | Bending rigidity of membrane |

\(\kappa _\text {act}\) | 1.0–3.0 | Active surface energy on membrane |

\(v_\text {eq}\) | 2527 | Equilibrium volume of vesicle |

\(\tau _\text {f}\) | \(1.0 \times 10^{-2}\) | Characteristic time of membrane fluidity |

\(\tau _\text {t}\) | 1.0, \(+\infty \) | Characteristic time of membrane turnover |

\(M_\text {eq}\) | \(1.0 \times 10^3\) | Equilibrium number of molecules within vesicle |

\(M_\text {inst}\) | 10–\(1.0 \times 10^3\) | Instability of the number of molecules within vesicle |

\(\epsilon _\text {t}\) | \(1.0 \times 10^{-1}\) | Energetic cost of molecular turnover |

\(\gamma _\text {t}\) | \(5.0 \times 10^{-2}\) | Critical strain for energetic reduction of molecular turnover |

| ||

\(K_\text {r}\) | \(1.0 \times 10^3\) | Repulsive modulus |

\(\Delta t_\text {M}\) | \(1.0 \times 10^{-3}\) | Time step of numerical integration |

To analyze effects of \(K_\text {c}\) on membrane shape, we measured the averaged local Gaussian curvature over every vertex, which is estimated from a set of the surrounding vertices. Notably, the averaged local Gaussian curvature cannot be conserved in the defiance of the Gauss–Bonnet theorem but dynamically vary. This is because the curvature is locally defined at individual vertices, whose number dynamically varies by turnover. Interestingly, the dependence of the averaged local Gaussian curvature on \(M_\text {inst}\) changes directions at three areas; positive in cases \(K_\text {c} \lesssim 2\), negative in cases \(2 \lesssim K_\text {c} \lesssim 20\) and positive \(20 \lesssim K_\text {c}\) (Fig. 4b). On the other hand, the number of membrane molecules is inversely proportional to \(K_\text {c}\) independent on \(M_\text {inst}\). These behaviors can be simply explained by the geometric constraint imposed by the volume–area balance; the surface must be finely folded in cases with large area (\(K_\text {c} \lesssim 2\)), laminarly flatted in cases with middle area (\(2 \lesssim K_\text {c} \lesssim 20\)) and smoothly spherical in cases with small area (\(20 \lesssim K_\text {c}\)). Therefore, the resulting membrane morphologies are regulated by the number of membrane molecules through turnover.

The turnover is dependent on \(K_\text {c}\) because it reduces the local residual stress generated by the global force balance: in the case with low \(K_\text {c}\), the membrane-surface area tends to expand because of the thermal fluctuation force. As \(K_\text {c}\) increases, the membrane-surface area tends to decrease to minimize its bending energy. Therefore, the strain of the membrane-surface area \(\left\langle a_i \right\rangle / a_{\text {eq}}\) is inversely proportional to the bending rigidity, \(K_\text {c}\). By sensing local stress as Eq. (17), the molecular turnover is biased to cause expansion or shrinkage. These results suggest that the turnover serves to adaptively facilitate membrane deformation depending upon the membrane-stress state.

### 3.5 Molecular turnover permits autonomous cell migration

*x*-axis and the vector from the center of the vesicle to the center of the

*i*th triangle. This function is similar to that used in expressing the active energy on cells during collective migration (Sato et al. 2015).

As a result, in case without turnover, the velocity of migration drastically decreases (Fig. 5a, c). On the other hand, in case with turnover, the cell dynamically migrates along the *x*-axis (Fig. 5b, c). This is because the active energy in Eq. (20) drives the transport of molecules from the rear to the front through the turnover (Fig. 5d). This result suggests that cells can migrate even without any traction force on membrane, with turnover playing a crucial role in driving migration. Notably, while the migration can be observed in the wide range of \(K_\text {c}\) and \(M_\text {inst}\), \(K_\text {c}\) must be high to maintain the spherical cell shape as described in Appendix E.

Notably, because the force from the energy in Eq. (20) is internal within cell but not external, the force is balanced within cell at each time. On the other hand, because \(\phi _i\) dynamically varies with time under membrane deformation, *U* becomes non-conservative so as to set the system into non-equilibrium. Such non-conservative energy function has been known as to generate active cell movements (Sato et al. 2015). Moreover, the stress-dependency of the turnover in Eq. (16) breaks the detail balance of molecular transport. Thus, despite the force balance within cell, this model can generate cell migration in a physically consistent manner.

In biological systems, there are a lot of types of cell migration such as single and collective cell migration in wound healing, morphogenesis and cancer invasion (Friedl and Wolf 2003). Importantly, while mechanism of the resulting process is a kind of Brownian ratchet, it differs from the mechanism of the well-known single-cell migration. In this mechanism, the front extension of the cell initiates migration: It extends ahead by cytoskeletal polymerization and the rear follows it by cytoskeletal contraction. On the contrary, in our model, localized active tension at the rear initiates migration by directionally transporting molecules to the front. Notably, our model does not conflict with the mechanism of the single-cell migration. Therefore, turnover may contribute to cellular migration as a main or secondary process.

## 4 Discussion

In this study, by making several physical assumptions, we integrally formulated the fluidity and turnover of a cellular membrane in a physically consistent manner. In the computational simulations, the cellular vesicles actively deform by the turnover of the membrane molecules (Fig. 3). During deformations, the triangular property was maintained to satisfy the modeling assumptions as in Appendix D. Moreover, based on the dependence of turnover upon the stress state, vesicle morphology drastically differed with the bending rigidity (Fig. 4); a smooth sphere was obtained under high rigidity, and a lobate vesicle was obtained under low rigidity. Furthermore, based on directional molecular transport by turnover, the localized active tension on the membrane drove the cellular migration of the vesicle (Fig. 5). From these results, the proposed model successfully recapitulated the turnover-dependent dynamics of three-dimensional cellular membranes.

In the computational simulations, the turnover behavior described in Eq. (16) was simply provided under the following assumptions: (i) that the energetic cost for molecular transport, \(\Delta _i E\), depended on the local strain of the membrane area, as shown in Eq. (17) and (ii) that the chemical potential \(\mu _\text {t}\) stabilized the number of molecules in the reservoir, as shown in Eq. (18). In general, these assumptions were not necessary for the proposed model. We emphasize that the functions of \(\Delta _i E\) and \(\mu _\text {r}\) can be entirely arbitrary.

Although the cellular mechanical property was expressed simply in these simulations, as described in Eq. (6), it can be expressed in more detail. For example, in cells with rich lining cytoskeletons, cell membranes could have non-negligible longitudinal and transverse elasticity, which may be important for their deformations (Heinrich et al. 2001). Moreover, during blebbing, the actin cytoskeleton lining membrane is locally broken, which causes a spatiotemporal inhomogeneity in membrane rigidity (Manakova et al. 2016). These details can be reflected by choosing a proper effective energy function, \(U_\text {eff}\) in Eq. (6).

In principle, detailed expressions for turnover at a molecular scale are limited in the proposed model; for example, the adhesion process of membrane-associated proteins cannot be directly expressed. Such behaviors at the molecular scale must be instead coarse-grained into those at the minimum scale of the proposed model. For example, the dependence of protein adhesion upon lipid behavior can be expressed by \(\Delta _i E\) as a dependence of the turnover upon membrane strain. Moreover, the biased adhesion of proteins to lipids can be expressed by \(\mu _\text {r}\) as directional molecular transport. Thus, by designing \(\Delta _i E\) and \(\mu _\text {r}\), the proposed model can be applied to various behaviors of turnover.

## 5 Conclusion

To analyze the effect of molecular turnover on three-dimensional deformations of cellular membranes, we proposed a new computational model for simulating the turnover-dependent dynamics of three-dimensional cellular membranes. The proposed model successfully simulate turnover-dependent membrane dynamics, and suggested the roles of turnover to drive the adaptive deformation and directional migration of vesicles. These results illustrate the importance of turnover in the dynamic deformations of cellular membranes in addition to the use of the proposed model for exploring general effects of molecular turnover on cellular dynamics.

## Notes

### Acknowledgements

We appreciate Dr. Masashi Tachikawa (RIKEN, Japan) and Prof. Ken-ichi Tsubota (Chiba University, Japan) for their valuable comments.

**Funding** This work was partially supported by JSPS KAKENHI Grant No. 15K14534, Japan. This work was partially supported by JST/PRESTO Grant No. JPMJPR16F3, Japan.

### Compliance with ethical standards

### Conflict of interest

There is no financial and personal relationship with other people or organizations that could inappropriately influence this work.

## Supplementary material

Supplementary material 1 (mp4 3148 KB)

Supplementary material 2 (mp4 6125 KB)

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