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Comput Methods Biomech Biomed Engin. Author manuscript; available in PMC 2010 May 21.

Published in final edited form as:

PMCID: PMC2874252

NIHMSID: NIHMS79295

The publisher's final edited version of this article is available at Comput Methods Biomech Biomed Engin

See other articles in PMC that cite the published article.

We propose a class of microstructurally informed models for the linear elastic mechanical behavior of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behavior resulting from anisotropic filament distributions, and a power-law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modeled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behavior of the networks over a wide range of filament densities and degrees of anisotropy.

Numerous experiments have shown mechanical loading to be an important factor in the development and/or maintenance of a wide variety of tissues such as muscle, cartilage, tendon, and bone, and organs such as the heart and lung. The deformations of the cells within these tissues and organs are dictated by their mechanical behavior under loading. Thus, it comes as no surprise that cellular mechanical behavior has been implicated as an important factor in the pathology of many diseases such as osteoporosis, osteoarthritis, cancer, heart failure, and several pulmonary disorders [1].

Our understanding of the mechanical regulation of the pathologic processes involved in these diseases would be greatly enhanced if it were possible to predict the mechanical behavior of a particular cell from microscopically obtained observations. A critical component governing the mechanical behavior of adherent cells is the actin cytoskeleton, a three-dimensional network of cross-linked actin filaments (figure 1). The microstructure of the actin cytoskeleton is highly dynamic and can change dramatically in response to mechanical loading. A growing body of evidence suggests that the ability of cells to convert mechanical signals into biochemical signals depends on the actin cytoskeletal microstructure [2, 3]. Microstructurally-based models of the actin cytoskeleton would be ideal for investigating the mechanical implications of actin microstructural organisation, since representative cytoskeletal networks observed *in vitro* could be utilized. Important microstructural features, such as spatial and angular heterogeneity, could be directly accounted for, allowing investigation of underlying mechanical ‘principles’ that may be governing cytoskeletal microarchitecture.

Fluorescent image of the actin cytoskeleton of a MC3T3-E1 osteoblastic (bone) cell. Microstructurally, the actin cytoskeleton is highly heterogeneous in both the number and orientation of filaments at each point. The inset contains a brightfield image **...**

Homogenized models for the actin cytoskeleton have yet to be obtained. A homogenized model in this context is a constitutive model for a continuum capable, in some appropriate sense, of approximating the mechanical behavior of the network. The advantage of a homogenized model is that the details of the network microstructure are used to generate the model, but are not needed thereafter. Although some advanced results are available for structured networks [4], rigorous mathematical results on general network homogenization problems remain elusive. Recent and classical theoretical investigations of ‘stiff’ or ‘semi-flexible’ polymer networks have yielded important insight into the mechanics of this class of networks and have generally identified geometric properties of random networks [5, 6, 7], different elastic regimes [8], scaling behaviors, and methods for explicit calculation of the macroscopic network elastic moduli from microscopic properties [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. All of these results but [24] have been obtained for two-dimensional, isotropic networks. However, anisotropy is highly relevant for actin networks, which forms aligned bundles of actin within the cytoskeleton (both in cultured cells and *in vivo* [27]) as well as cross-linked gels [28, 29]. The direct consequence of anisotropy is that, as opposed to previous investigations that needed to solely focus on the tensile and/or shear modulus that characterize isotropic materials, a suitable homogenized model will require the calculation of the entire elastic tensor (21 independent components) to fully specify the mechanical behavior of the network.

Incomplete knowledge of microscopic network properties is a unique challenge which makes constitutive modeling of many biopolymer networks difficult. In the case of the actin cytoskeleton, although the dimensions and material properties of individual actin filaments have been measured [30], many important microscopic properties of actin cytoskeletal networks have yet to be elucidated. For example, there are a wide variety of cytoskeletal cross-linking proteins *in vivo* whose mechanical behavior need to be characterized. In addition, although it is well accepted that the cytoskeletal network can be subject to a prestress, the degree to which each filament is prestressed is not known.

We propose here a novel class of models for the homogenized linear elastic response of cross-linked polymer networks such as the actin cytoskeleton. The proposed models can be constructed based on the filament angular distribution and spatial density. We deliberately avoided making specific assumptions on whether the elasticity of the network is the result of entropic [11] or enthalpic [8, 15] contributions, the nature of the cross-links between filaments in the network, or the existence of a prestress in the network, since these are still largely under discussion. Instead, our goal was to formulate a class of models that account for some features of the microstructure of the network, and that, through a suitable validation procedure, could be tailored to represent its homogenized linear elastic response under any or many of these conditions.

The 21 elastic moduli of the model are determined by postulating an ansatz or functional form inspired from the exact expression for affinely deformed networks with anisotropic filament distributions, see e.g., [31, 32]. The model accounts for the possibly different-than-linear exponents observed in the power law dependence of the elastic moduli with filament density (see, e.g., [8, 15]), and the effect of cross-links in the Poisson ratio. There are only 7 independent parameters, which need to be calibrated from a relatively small number of simulations of fully resolved and explicitly represented networks. We showcase the performance of the model by predicting the mechanical response of finite element models of three-dimensional, anisotropic networks of elastic rods with semi-flexible cross-links. The calibrated model shows good performance over a wide range of angular distributions and spatial densities away from the vicinity of the point of calibration. The particular type of networks chosen for this example was motivated by two-dimensional analogs that have been previously adopted as possible descriptions for the actin cytoskeleton [13, 15]. We expect, however, that the homogenized class of models proposed herein will also be useful to express the effective behavior of more general network types, resulting from a future enhanced understanding of key features of the actin cytoskeleton.

Throughout, vectors are denoted by boldface lowercase latin characters, second-order tensors by boldface lowercase greek characters, and fourth-order tensors by boldface uppercase latin characters. All tensor components are referred to an orthonormal basis. All nonboldface characters are considered scalar quantities. When indicial notation is used, an index appearing twice in a term indicates sum over it in the range 1 to 3.

We propose and detail here a class of models to approximate the homogenized linear elastic response of cross-linked polymer networks such as the actin cytoskeleton. We idealize network filaments as cylindrical rods of cross-sectional area *A* with extensional, bending, and torsional stiffness. The configuration of each filament can be characterized by its midpoint position and direction in space when unstressed. We consider three-dimensional, infinite networks that are periodic with period *L* in three orthogonal directions. The unit cell of the network is then a cubic box of side *L*. A cross-link between two filaments may be formed whenever the distance between the two is within a specified distance. We denote with *ρ* the volume fraction of filaments in the network, i.e., the quotient between the total volume occupied by all filaments and the volume of the unit cell *L*^{3}. Additionally, let *ω( n)* be the angular probability density of the volume fraction, which indicates the angular distribution of the volume fraction

Motivated by homogenization results for linear periodic composite materials (see e.g., [33]), we expect the elastic energy density of the homogenized model to be well approximated by the lowest elastic energy attainable by the network under suitably imposed boundary conditions on the unit cell. For this study, the points of intersection between the network filaments and the unit cell faces are constrained to follow an affine deformation. A class of less stringent boundary conditions requiring only periodic but not necessarily affine displacements on the boundary of the unit cell are also of interest, but will not be considered here. If ** x** denotes the position vector of one such point and

Even though homogenization of the mechanical properties for general networks is still an open problem, there are some cases in which the elastic moduli are easily obtained. Consider for example the case of an initially unstressed network in which every filament has only extensional stiffness *K*_{//} (no bending or torsion). All cross-links in the network are assumed to be able to freely rotate with no energetic cost. It is easily seen that upon constraining the filament ends to follow an affine deformation _{ε}, an equilibrium configuration of the network is obtained by mapping all cross-links with _{ε} as well (in the linear elastic regime of concern here, such network deformation has the lowest energy among all those that use the same boundary conditions; it may not be unique though). Herein, we call this class of networks ‘affine networks’. As we shall detail later, the homogenized elastic moduli in this case are given by

$${C}_{ijkl}={K}_{\u2215\u2215}{\int}_{{S}^{2}}\rho \omega \left(\mathit{n}\right){n}_{i}{n}_{j}{n}_{k}{n}_{l}dS.$$

(1)

As nicely discussed in [12], this result will not strictly hold when the filament ends are not constrained to follow an affine deformation. Previous numerical studies have also shown that the linear scaling of the moduli with *ρ* is not longer valid for more complex networks, such as when bending stiffness is accounted for. Instead, power laws of the form *(ρ-ρ _{ref})^{α}* for some exponent

We are therefore motivated to posit the following form (ansatz) for the homogenized elastic moduli

$${C}_{ijkl}={\int}_{{S}^{2}}{(\rho -{\rho}_{\mathit{ref}})}^{\alpha}\omega \left(\mathit{n}\right){K}_{ijkl}\left(\mathit{n}\right)dS.$$

(2)

Here, *K _{ijkl}(n)* is a fourth-order tensor valued function of the direction vector

$${K}_{ijkl}\left(\mathit{n}\right)={K}_{\u2215\u2215}{n}_{i}{n}_{j}{n}_{k}{n}_{l}.$$

(3)

The homogenized elastic moduli in equation (2) should have major and minor symmetries for *any* choice of angular density *ω( n)*. It then follows that

A physically reasonable requirement on the elastic moduli *C _{ijkl}* is that if the entire network is rigidly rotated by a rotation

$$\begin{array}{cc}\hfill {\underset{\u2012}{C}}_{ijkl}=& {\int}_{{S}^{2}}{(\rho -{\rho}_{\mathit{ref}})}^{\alpha}\underset{\u2012}{\omega}\left(\mathit{n}\right){K}_{ijkl}\left(\mathit{n}\right)dS\hfill \\ \hfill =& {\int}_{{S}^{2}}{(\rho -{\rho}_{\mathit{ref}})}^{\alpha}\omega \left({\mathit{Q}}^{-1}\mathit{n}\right){K}_{ijkl}\left(\mathit{n}\right)dS\hfill \\ \hfill =& {\int}_{{S}^{2}}{(\rho -{\rho}_{\mathit{ref}})}^{\alpha}\omega \left(\mathit{n}\right){K}_{ijkl}\left(\mathit{Qn}\right)dS.\hfill \end{array}$$

(4)

Correspondingly, by material frame indifference the elastic moduli of the rotated network should be given by *C _{IJKL}*=

$${\underset{\u2012}{C}}_{IJKL}={Q}_{Ii}{Q}_{Jj}{Q}_{Kk}{Q}_{Ll}{\int}_{{S}^{2}}{(\rho -{\rho}_{\mathit{ref}})}^{\alpha}\omega \left(\mathit{n}\right){K}_{ijkl}\left(\mathit{n}\right)dS.$$

(5)

Equating Eqs. 4 and 5, and utilizing the fact that the resulting identity should be valid for any filament angular probability density *ω( n)*, we obtain

$${K}_{IJKL}\left(\mathit{Qn}\right)={Q}_{Ii}{Q}_{Jj}{Q}_{Kk}{Q}_{Ll}{K}_{ijkl}\left(\mathit{n}\right),$$

(6)

which precisely defines the dependence of ** K** on

$${K}_{IJKL}\left({\mathit{n}}_{0}\right)={R}_{Ii}{R}_{Jj}{R}_{Kk}{R}_{Ll}{K}_{ijkl}\left({\mathit{n}}_{0}\right),$$

(7)

or alternatively, the tensor ** K**(

$$\mathit{K}\left({\mathit{e}}_{3}\right)=\left[\begin{array}{cccccc}\hfill {K}_{1111}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{1122}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{1133}\left({\mathit{e}}_{3}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {K}_{1122}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{1111}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{1133}\left({\mathit{e}}_{3}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {K}_{1133}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{1133}\left({\mathit{e}}_{3}\right)\hfill & \hfill {K}_{3333}\left({\mathit{e}}_{3}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}({K}_{1111}\left({\mathit{e}}_{3}\right)-{K}_{1122}\left({\mathit{e}}_{3}\right))\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {K}_{1313}\left({\mathit{e}}_{3}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {K}_{1313}\left({\mathit{e}}_{3}\right)\hfill \end{array}\right].$$

(8)

Here *K*_{1111}(*e*_{3}), *K*_{1122}(*e*_{3}), *K*_{1133}(*e*_{3}), *K*_{3333}(*e*_{3}), and *K*_{1313}(*e*_{3}) are the five independent constants to be determined. Once these constants are known, the entire tensor valued function over the unit sphere ** K**(

In order to gain insight into the physical origin of the five constants which make up ** K**(

$${W}_{\text{filament}}\left(\lambda \right)=\frac{1}{2}{K}_{\u2215\u2215}{\lambda}^{2},$$

(9)

where *λ* is the axial strain. In the linear elastic regime and under an affine deformation _{ε} the value of *λ* for a filament oriented in a direction ** n** can be computed as

$$w\left(\epsilon \right)={\int}_{{S}^{2}}\rho \omega \left(\mathit{n}\right){W}_{\text{filament}}\left({n}_{i}{\epsilon}_{ij}{n}_{j}\right)dS,$$

(10)

and the homogenized elastic moduli then follow as

$${C}_{ijkl}=\frac{{\partial}^{2}w\left(\epsilon \right)}{\partial {\epsilon}_{ij}\partial {\epsilon}_{kl}}={\int}_{{S}^{2}}\rho \omega \left(\mathit{n}\right){K}_{ijkl}{n}_{i}{n}_{j}{n}_{k}{n}_{l}dS.$$

(11)

For this case, *K*_{3333}(*e*_{3})=*K*_{//}, which explicitly shows the connection between *K*_{3333}(*e*_{3}) and the extensional stiffness of the filaments. In fact, *K*_{3333}(*e*_{3}) is the only one out of the five different constants in ** K**(

These last observations provide the basis for an interesting remark about the model. The material frame indifference argument showed that the proposed ansatz is obtained by arranging a transversely isotropic material with elastic moduli ** K** in different directions. For affine networks, the cross-links do not play a role in the deformation, i.e., they can be removed and the network deforms in the same way. This is reflected in the elastic moduli

Schematic depicting a simple, non-periodic network in which pulling the vertical filament in the axial direction results in a contraction in the transverse direction. In this case, the filament transmits forces along its own direction and, through its **...**

Simple examination reveals some of the shortcomings of the proposed model. For example, consider an initially unstressed network formed by filaments whose elastic energy arises solely due to bending (no torsion or stretching) and whose cross-links are rigid, i.e., the angle between any two filaments at a cross-link cannot change. The filaments are arranged in a prismatic microstructure so that filament directions are parallel to three orthonormal vectors *e*_{1}, *e*_{2}, and *e*_{3}, while the distance between consecutive cross-links in each direction, are *L*_{1}, *L*_{2} and *L*_{3}, respectively. As we shall show next, the homogenized linear elastic moduli for this network depend on quadratic moments of *ω*(** n**) (i.e., products of the type

For the sake of simplicity, we shall compute only the shear modulus *G*=*C*_{1212}. Notice then that the periodicity of the microstructure enables us to consider a minimal unit cell with dimensions *L*_{1}, *L*_{2} and *L*_{3}, which substantially simplifies the computation. We are interested in the energy of the network when an affine deformation _{ε}, with ** ε**=

$$w\left(\gamma \right)=\frac{1}{{L}_{1}{L}_{2}{L}_{3}}\frac{24{\gamma}^{2}{K}_{\perp}}{({L}_{1}+{L}_{2})}.$$

(12)

The volume fraction of the network is obtained as *ρ*=*A(L _{1}+L_{2}+L_{3})*/

$$w\left(\gamma \right)=\frac{{\omega}_{1}{\omega}_{2}{\omega}_{3}}{{\omega}_{1}+{\omega}_{2}}\frac{24{\gamma}^{2}{K}_{\perp}}{{A}^{2}}$$

(13)

and that

$$G=\frac{{\partial}^{2}w\left(\gamma \right)}{\partial {\gamma}^{2}}={\rho}^{2}\frac{{\omega}_{1}{\omega}_{2}{\omega}_{3}}{{\omega}_{1}+{\omega}_{2}}\frac{48{K}_{\perp}}{{A}^{2}}.$$

(14)

The last equation clearly shows that the stiffness for this type of networks depends on the product between values of the angular probability distribution in different directions. Additionally, the dependence of *G* on the angular probability distribution is highly nonlinear. These two facts are in direct contradiction to the proposed model in equation (2), which explicitly contains a linear dependence on *ω*(** n**). Notice, however, that the model is able to capture the nonlinear stiffening of the network with the volume fraction.

Generally, *ρ _{ref}*,

Finite element models of three-dimensional, periodic, cross-linked networks were constructed by placing filaments of length 350nm inside cubic domains of length 400nm (figure 4). These length scales were selected since they may be relevant for image-based finite element models of the actin cytoskeleton (i.e., finite element models meshed directly from three-dimensional image data). For example, a typical pixel/voxel size in fluorescent microscopy is 0.2~0.4μm, and actin filaments are approximately in length *in vivo* [37, 38, 39]. Filament centroids were randomly sampled from a uniform distribution in the box, while distributions with different degrees of anisotropy were used for the orientations. Without loss of generality, we idealized actin filaments as elastic rods with cylindrical cross sections. Actin filaments were modeled as Euler-Bernoulli beams with linear elastic extensional and torsional stiffness, with a diameter *d*=8nm and Young modulus *E*=1.8GPa [30]. Although there are a large number of different cross-linking proteins *in vivo* with a wide variety of lengths and microscopic properties (many of which have yet to be elucidated), for simplicity, we assume filaments closer than a rod diameter to each other were cross-linked by rods having identical properties as the actin filaments, with lengths equal to the distance between the two filaments that were cross-linked. Of direct consequence is that unlike two-dimensional networks (e.g.,, [8, 15]), the rod diameter played a critical role in the network connectivity, since the probability of a fixed number of filaments to intersect asymptotically vanishes with the rod diameter [40]. We imposed periodic boundary conditions on the walls of the unit cell by constraining the points of intersection between filaments and the cubic domain boundary to follow a prescribed affine deformation, while the torques at the same points were constrained to be continuous across the two parts of the filament lying in neighboring unit cells. The total strain energy density for a given deformation ** ε** imposed on the boundary was computed using the finite element method (Abaqus, Providence, RI). Once the strain energy densities were found, the homogenized network elastic moduli were calculated as

$${C}_{ijkl}=\frac{{\partial}^{2}w}{\partial {\epsilon}_{ij}\partial {\epsilon}_{kl}}\left(0\right)\approx \frac{[w(\Delta {\delta}_{ij}+\Delta {\delta}_{kl})-w\left(\Delta {\delta}_{ij}\right)]-[w\left(\Delta {\delta}_{kl}\right)-w\left(0\right)]}{{\Delta}^{2}},$$

(15)

where Δ>0 is a small number. By using equation (15), calculating all 21 independent elastic moduli of the fully anisotropic elastic tensor required 27 analyses per network.

Seventy networks with varying volume fractions were generated, and their elastic moduli numerically computed. Physiological volume fractions of filamentous actin within different types of cells and cytoplasmic regions are still under discussion. However, the average volume fraction within bovine aortic endothelial cells has been reported to be on the order of 1% [9]. Since we expect the volume fraction to be higher than the average within actin-rich regions such as the cortex or actin bundles, we constructed networks with volume fractions ranging from *ρ*=0-0.10. Note that “dangling” filament ends (i.e., filament segments attached to a cross-link at one end but to nothing at the other) were accounted for when quantifying volume fractions (these segments do not contribute to the elastic energy of the network). We determined *ρ _{ref}* by finding the minimum volume fraction at which the elastic tensor was non-zero among all tested networks. We found

Log-log plot of elastic moduli *C*_{3333} and *C*_{1313} versus volume fraction *ρ*. The elastic moduli scale as ~*ρ*^{1.6} (black lines). Thus, a value of *α*=1.6 was used in equation (2).

Once *ρ _{ref}*,

$${\omega}_{\xi}\left(\mathit{n}\right)=\frac{(1-\xi ){S}_{00}\left(\mathit{n}\right)+\xi {S}_{10}\left(\mathit{n}\right)}{{\int}_{{S}^{2}}\left[\right(1-\xi ){S}_{00}\left(\mathit{n}\right)+\xi {S}_{10}\left(\mathit{n}\right)]dS}.$$

(16)

Here *S*_{00} denotes the isotropic distribution function over the unit sphere, while *S*_{10} stands for a dipole distribution aligned in the *e*_{3} direction (if *ϕ* is the angle between *e*_{3} and ** n**, then

The numerically determined values of *ρ _{ref}* and

Polar plots of the components of *K*_{IJ}(*n*), the 6×6 matrix representation of *K*. The plot in the *I*th row and *J*th column is the polar plot of *K*_{IJ}(*n*). Positive values are represented in gray, negative values are represented in black. The magnitude of **...**

Error in the predicted elastic moduli as a function of the degree of anisotropy ξ. A minimal dependence of the error on anisotropy is observed.

Polar plots of *C*_{ijkl}n_{i}n_{j}n_{k}n_{l} as a function of direction *n* using elastic moduli predicted by the model (*C*^{model}, left column), and generated using elastic moduli calculated using the finite element method (*C*^{exact}, right column). The top row represents an **...**

We proposed here a novel class of models for the homogenized linear elastic response of cross-linked polymer networks. The model requires determination of only seven independent parameters: five constants to construct a fourth-order tensor valued function ** K**(

We demonstrated the applicability and validity of these models by numerically determining *ρ _{ref}*,

When we calculated the eigenvalues of ** K**(

Once we numerically determined ** K**(

Overall, once calibrated the model predicted the elastic moduli for these networks over a range of volume fractions and degrees of anisotropy. In fact, the numerical results are surprisingly good, since the variation of the moduli with the anisotropy parameter was well captured, despite the fact that the model parameters were determined with isotropic networks only. More generally, we expect the proposed model, with perhaps some minor extensions, to be useful in different regimes and for other types of networks not considered herein as well, if calibrated accordingly. This, of course, has to be explicitly studied for each network type, e.g., different cross-linker microscopic properties, prestress in the network and length distribution of actin filaments, among others.

The distinguishing feature of this class of models is that they can account for filament angular distribution and volume fraction, two microstructural parameters that can be estimated from light microscopy data. For example, although individual filaments cannot be resolved in fluorescent images (a typical pixel/voxel size in fluorescent microscopy is 0.2~0.4μm, whereas actin filaments are approximately 8nm in diameter), one can approximate the number and orientations of filaments within each image voxel. Specifically, the volume fraction of filaments within each voxel can be estimated from the voxel intensity, while their orientations can be estimated from the local image texture [44, 45] to give an approximate angular distribution of filaments [44]. Thus, the approach presented here may serve to construct cell structural models that account for the anisotropic and heterogeneous distribution of actin filaments, something that has yet to be explored and validated against appropriate experiments. Despite the facts that the actin cytoskeleton is a highly dynamic structure capable of remodeling, and that cells can undergo large deformations, the linearized mechanical response studied herein may provide important insight for the interpretation of a wide class of microrheological experiments.

We would like to acknowledge grant AR45989 from the National Institutes of Health, the National Science Foundation Graduate Fellowship, and the Veterans Affairs Palo Alto Bone and Joint Center for funding.

^{1}SE: standard error, n: number of samples

*AMS Subject Classification*: 74D05; 74K10; 74S05; 74E10

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