Minimizing Computational Time for Long-Term Three-Dimensional Dynamic Simulation of Stem Cell Adipogenesis

Rabiei, Manoochehr; Albaruni, Md Abu Sina Ibne; Danesh, Negar; Moon, Hyejin; Cho, Michael; Bowling, Alan

doi:10.1115/1.4068320

Abstract

This paper presents a high-speed approach to simulating the long-term mechanobiological development of stem cells during the adipogenesis process. A novel three-dimensional model of human bone marrow-derived mesenchymal stem cells (hMSCs) undergoing adipogenic differentiation is presented herein. The elements of the cellular model have minute masses in femtograms and dimensions in nanometers. The disproportionality between the force and mass terms of the system, yielding a multiscale dynamic model, requires the solution to be calculated in femto- and picosecond time-steps. This makes producing the two-week time history of the adipogenic differentiation process computationally infeasible with conventional methods, even with the aid of supercomputers. The scaling method, based on the method of multiple scales proposed in authors' previous works, has been shown to address these imbalances and yield fast computational time for long-term simulation of cell processes. Herein, a novel approach to the scaling formulation is proposed, and methods for choosing scaling factors are presented and examined. Employing the new formulation results in a computational time of less than 1 h and 9 min on a normal desktop computer for the simulation of the 3D cellular model for the two-week time history of the adipogenic differentiation process. This is faster than previous efforts, which modeled the cell in two dimensions.

1 Introduction

Computer simulation of the mechanobiology of cells typically depends on multiscale models, requiring large amounts of computational time to obtain even small time histories, picoseconds to microseconds, of system evolution. The multiscale nature of these systems stems from the disproportionality between the small masses and large forces in the model and the inhomogeneity of masses of the models' elements. Cellular mechanics can be described accurately in atomistic detail with the help of molecular dynamics (MD) simulations. The forces are orders of magnitude larger than the atomistic scale masses. The resulting large accelerations will require integration time-steps of femto- and picoseconds to obtain an accurate solution, resulting in large computational time [1,2]. MD simulation of protein folding is an active area of research [3]. However, obtaining time histories of seconds remains elusive even with the aid of supercomputers [4]. The MD approach is infeasible for simulating long-term biological phenomena in subcellular detail [5].

A novel scaling approach, based on the method of multiple scales [6], has been developed to reduce computational time to the extent that long-term simulations, with time histories of days and weeks, are feasible to use in the study of cellular mechanobiology. The scaling approach delineates the “slow” and “fast” dynamics of the system by defining a characteristic time unit of the system based on the progression rate of the process [7]. This decouples the computational time from the fast dynamics response inherent in the minute size of the cellular models' elements and bases it on the progression rate of the process, which can take days or weeks. The computational time is also decoupled from the inhomogeneity in the mass distribution of the model elements by ensuring appropriate dynamic behavior separately for each element at the characteristic time unit [8]. The simulations' much faster than real-time speed enables the study of the mechanobiology of stem cells beyond experimental observations [9]. This scaling approach accurately captures the change in the nucleus's morphology [7,9] and cytoskeletal structures [10]. It also addresses the inhomogeneity of masses of the models' elements [8].

This work introduces a novel, physics-based, three-dimensional multibody dynamic model of a stem cell. This model enables the study of the adipogenic differentiation process in greater detail by including the effects of focal adhesion points, surface tensions, and volumetric reactions. The proposed model includes the nucleus, cytoskeleton, lipid droplets, cellular envelope, and focal adhesion points. The cellular model uses the hydrostatic pressure difference, inside and outside, to maintain the structural integrity of the cellular membrane [11]. This contrasts with the tensegrity-based 2D cellular models, where microtubules performed this role [12].

The proposed model's added dimension, increased particle numbers, and complexity relative to previous 2D models [7,8,10] should increase the required computational time. A novel method for choosing scaling factors is presented herein to make the simulation of this complex model computationally feasible. The new approach enables a computational time of less than 1 h and 9 min to simulate the 14-day adipogenic differentiation process. This is faster than previous work, modeled in two dimensions, which required 3.5 h [10] and 1 h and 45 min [8] of computational time.

2 Adipogenesis Test Case

Adipogenic differentiation, or adipogenesis, is the process of stem cells differentiating into fat-storing cells called adipocytes. In this work, the stem cells are human bone marrow-derived mesenchymal stem cells (hMSCs). The hMSCs are induced to differentiate into adipocytes by contact with adipogenic-promoting chemical factors [13]. Successful adipogenesis is indicated through the intracellular synthesis and accumulation of lipid vesicles (e.g., lipid droplets), shown in green in panel (a) of Fig. 1. The efficacy of this seemingly biochemical process is affected by mechanical factors such as substrate stiffness [14], extracellular matrix properties [15], cellular morphology [16], and cytoskeletal organization [17,18].

Fig. 1

View large Download slide

(a) Composite image showing hMSCs in various stages of adipogenesis. F-actin microfilaments (phalloidin, red), lipid droplets (LipidTOX, green), and nucleus (DAPI, blue) are shown. (b) Image of cytoskeletal rearrangement of F-actin microfilaments during adipogenesis. (c) Schematic of morphological changes during adipogenesis including nucleus in blue, cytoskeletal microfilaments in red, and lipid droplets in green (Color version online)

Stem cells undergo several morphological changes during progression to fully differentiated lineages. These changes manifest in the nucleus [9] as well as the cellular shape and its cytoskeletal structure [18]. Actin stress fibers, thought to be the main component of the cytoskeletal structure, keep the nucleus membrane under tension [19–21]. The actin microfilaments are shown to rearrange to the cell perimeter, dissociating themselves from the nucleus [22] during adipogenesis, presumably reducing nuclear membrane tension. The rearrangement of cytoskeletal microfilaments can be seen in panel (b) of Fig. 1. A schematic of the relevant cellular structures is shown in panel (c) of Fig. 1. The adhesion and detachment of the cellular envelope to the substrate are also shown to affect the differentiation process [23].

The experimental data of nuclear area and lipid accumulation are gathered. The experiment and data acquisition are described in detail in Refs. [9] and [10].

Fig. 2

View large Download slide

Stem-cell model depicting the (a) nucleus and the lipid droplets, (b) cytoskeletal structure, (c) cellular envelope and focal adhesion, and (d) full cellular model. The cellular model elements are marked as: (1) MPs, black dots; (2) LPs, green spheres; (3) nuclear membrane sections, blue lines; (5) cytoskeletal microfilament sections, thick red lines; (7) cortex actin fiber sections, thin red lines; and (8) focal adhesion points, yellow dots. The nucleoplasm is enclosed by the nuclear membrane sections (4), and the cytoplasm is enclosed by the cortex actin fiber sections (6) (Color version online)

3 Cellular Model

The stem cell is modeled in 3D to allow a more realistic study of the dynamics of adipogenesis. The 3D model is made simple to make it easy to compare with experimental data in 2D still images. Constructing more complex models would also require better knowledge of the mechanical properties of subcellular structures, some of which are not precisely known [24]. The proposed cellular model consists of the cellular envelope, focal adhesion points, nucleus, cytoskeletal structure, and lipid droplets. The model consists of spherical particles, 1926 microparticles (MPs) and 50 lipid particles (LPs). The constituent parts of the proposed model are shown in Fig. 2.

Fig. 3

View large Download slide

Schematic of the system of two interconnected bodies

The cellular envelope consists of a cellular phospholipid bilayer membrane [25] with an attached actin cortex layer. The actin cortex layer consists of a mesh of actin fibers [26]. Since the cellular membrane phospholipid bilayer is drastically less stiff than the actin fibers in the cortex [27], its effects on the mechanics of adipogenesis are assumed to be negligible. The cellular envelope, therefore, is modeled as a mesh of single actin fiber sections with MPs at the vertices.

The cellular envelope attaches to the substrate at focal adhesion points. The mechanics of this adhesion and detachment affect the differentiation process [23]. Herein, the MPs at the bottom of the cellular envelope are the focal adhesion points. These points do not move during adipogenesis unless they detach.

Modeling the tension in the nuclear and cellular envelopes is challenging [28]. This tension requires a compression load-bearing entity to stop the cell from imploding. The cytoskeletal microfilaments may not be able to perform this role due to their narrow cross section, which causes bending and buckling in compression [11]. The cellular model can be constructed based on the tensegrity approach [12], which assigns microtubules as the main compression load-bearing entity. However, the microtubules may be unable to perform this function due to their smaller local networks [29] and lower stiffness than microfilaments [30].

The hydrostatic pressure difference between the inside and outside of the cell can withstand the tensile load of the cortex and cytoskeletal microfilaments [31]. The effects of this pressure, in conjunction with the cortex tension, in forming bulges in the cellular membrane (blebs) have been studied [32]. The proposed cellular model employs the hydrostatic pressure concept to produce volumetric reaction forces applied to the MPs in the cellular envelope. These forces oppose the tensile forces of the actin cortex and cytoskeletal microfilaments.

The nucleus is modeled as a mesh of nuclear membrane sections connecting MPs at the vertices. The nuclear membrane is presumed to be in tension [19], so the sections are modeled as ropes. The membrane stiffness has not been measured with certainty [33] but is estimated here based on the dynamic stability of the proposed model.

The cytoskeletal structure in the proposed cellular model consists of microfilaments connected to the nucleus and the cellular envelope. The rearrangement of the cytoskeletal structure and its connections is important in the adipogenesis process [10,34]. Microfilaments are formed when actin fibers bundle together. Herein, the microfilaments are modeled as five actin fibers arranged in parallel. Due to their narrow cross section of 6–7 nm [11], the microfilaments are assumed to be in tension and are modeled as ropes. The literature provides information on the rigidity of an actin fiber [35], which is used to calculate the stiffness of microfilament sections. Each microfilament section connects two MPs.

The sections comprising the cortex, nuclear membrane mesh, and cytoskeleton are modeled as massless ropes. The size and mass of the MPs are calculated to correspond with the cross section and mass of microfilament sections, respectively. Lipid droplets form and increase in size during adipogenesis [9,36]. The size and mass of the LPs are calculated based on experimental data and their density. They are distributed randomly inside the cellular envelope.

3.1 Generalized Coordinates.

Cartesian coordinates are used to describe the position of each particle in the model

\begin{matrix} x_{i} & = [\begin{matrix} x_{i} \\ y_{i} \\ z_{i} \end{matrix}] \end{matrix}

(1)

This gives the system generalized coordinates as

\begin{matrix} q & = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}] \end{matrix}

(2)

where

n = 1976

and therefore the size of

q

is 5928

\times

1. Applying Newton's second law to a particle of constant mass yields

m \ddot{x} = - (β \dot{x} + k x) + F

(3)

where m is the particle's mass, $x$ ⁠, $\dot{x}$ ⁠, and $\ddot{x}$ are the particle's generalized position, velocity, and acceleration, respectively. The term $F$ denotes the sum of all other forces on the particle. The terms $β$ and k are damping and spring coefficients.

The model for all particles results in the equations of motion of the system, which take the form

M (q) \ddot{q} = - (D (q) \dot{q} + K (q) q) + Γ_{l} = Γ

(4)

where

q

includes the generalized coordinates, and

\dot{q}

and

\ddot{q}

are generalized velocity and acceleration. The term

M (q)

is the mass matrix, and

Γ_{l}

denotes all other large forces. In this system, the mass matrix is diagonal and populated with mass of the MPs and LPs in that order

\begin{matrix} M (q) = [\begin{matrix} m_{MP} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & m_{LP} \end{matrix}] \end{matrix}

(5)

where the mass matrix is 5928 $\times$ 5298. The terms $D (q)$ and $K (q)$ are drag coefficient and elastic stiffness matrices. The generalized active forces are included in $Γ$ with a size of 5928 $\times$ 1.

3.2 Generalized Active Forces and Parameters.

There are eight generalized forces in this system. Contact and viscous damping forces apply to all particles. The combined gravitational and buoyant forces apply to all particles. Normal forces apply to any particle that touches the glass slide surface. Nuclear membrane forces are applied to MPs within the nuclear membrane. Microfilament forces apply to MPs within the microfilaments and the cell cortex. The volumetric reaction forces apply to the MPs within the nuclear membrane and cell cortex. The forces resulting from the change of mass apply to the LPs as their masses increase. All forces are described in Cartesian coordinates

Γ = [\begin{matrix} F_{1} \\ F_{2} \\ ⋮ \\ F_{n} \end{matrix}] where F_{i} = [\begin{matrix} F_{i, x} \\ F_{i, y} \\ F_{i, z} \end{matrix}]

(6)

where

Γ

⁠, the generalized active forces contain

Γ = Γ_{c} + Γ_{d} + Γ_{gb} + Γ_{n} + Γ_{mem} + Γ_{mf} + Γ_{v} + Γ_{m}

(7)

and $Γ_{c}$ denotes the contact forces, $Γ_{d}$ the viscous damping forces, $Γ_{gb}$ the combined gravitational and buoyant forces, $Γ_{n}$ the normal forces, $Γ_{mem}$ the membrane forces, $Γ_{mf}$ the microfilament forces, $Γ_{v}$ the volumetric reaction forces, and $Γ_{m}$ the forces resulting from changing mass.

3.2.1 Contact Forces.

Contact between the particles is modeled as elastic between spherical objects. This is due to the high computational expense of enforcing rigid body constraints on a system of 1976 rigid bodies. The spring stiffness of this elastic contact is somewhat arbitrary. It is set herein to ten times the stiffness of the actin microfilament to reduce the resulting penetration of rigid bodies to an imperceptible amount.

Because of their small radii, numerous MPs are typically needed to prevent the microfilaments, nuclear membrane, and cellular cortex layer from passing through each other. This approach was employed in the cellular model developed in Ref. [10] but would drastically increase the number of bodies in the proposed model. Herein, a larger contact radius is calculated for each MP to prevent pass-through without increasing the number of MPs

{Rc}_{i} = max (d_{i j}) / 2

(8)

where the term ${Rc}_{i}$ is the contact radius of MP $_{i}$ and $d_{i j}$ is the distance of the MP $_{i}$ to all the other MPs connected to it through the cytoskeletal, cortical, or nuclear membrane structures. This approach is scalable. Increasing the number of MPs will reduce the contact radius to that of the MP.

Contact is enforced between LPs and MPs, and between MPs within the nuclear membrane and all other MPs. This enforcement is optimized by using a contact detection matrix of particles likely to make contact in the next simulation step

d_{i j} < SF \times ({Rc}_{i} + {Rc}_{j}) \to (i, j) \in C_{m}

(9)

where the term $d_{i j}$ denotes the distance between centers of particles i and j at the start of the simulation step, SF is the safety factor, Rc is the contact radius of the respective particles, and $C_{m}$ is the contact detection matrix of size $n \times 2$ where each row contains the particle numbers to be checked for contact. Using the contact detection map reduces the number of distances to be checked for contact during each simulation step. The safety factor of 2 ensures that no possible contacts will be missed during the simulation step.

Finally, the contact forces are calculated between the pair of particles in the contact detection matrix as they come into contact during the simulation step. The contacts between particles are discontinuous events. This discontinuity is made continuous by employing a third-order smoothstep function (⁠

S_{3}

⁠)

{\begin{matrix} S_{3} (x) = 0 & x < 0 \\ S_{3} (x) = - 2 x^{3} + 3 x^{2} & 0 < = x < = 1 \\ S_{3} (x) = 1 & x > 1 \end{matrix}

(10)

This function allows the contact force to be calculated continuously during the simulation step

F_{c_{i j}} = - F_{c_{j i}} = - k_{c} (d_{i j} - d_{0}) \times S_{3} (100 \times (d_{0} - d_{i j}))

(11)

where $d_{i j}$ denotes the distance between particles i and j, $d_{0} = {Rc}_{i} + {Rc}_{j}$ is the minimum allowable distance between them, and $k_{c}$ is the contact stiffness value used for the elastic contact.

3.2.2 Viscous Damping.

The viscous damping forces are calculated based on the Stokes' drag force exerted on spherical bodies in a very low Reynolds number flow

F_{d_{i}} = - β_{i} {\dot{x}}_{i} β_{i} = 6 π η_{i} r_{i}

(12)

where ${\dot{x}}_{i}$ is the velocity of particle i, $β_{i}$ is the damping coefficient based on Stokes' law for that particle, $η_{i}$ is the dynamic viscosity of nucleoplasm or cytoplasm, and $r_{i}$ is the radius of that particle. The damping coefficient remains constant for MPs but increases with LPs' radii.

3.2.3 Combined Gravitational and Buoyant Forces.

Buoyant forces are a function of the displaced volume of the liquid based on Archimedes' principle. The gravitational forces are a function of the mass of the particles. Both forces can be formulated based on the volume of the particles, their density, and the density of the cyto- and nucleoplasm. Both forces also act in the same direction, so they can be combined as

F_{{gb}_{i}} = g V_{i} (ρ_{l} - ρ_{i})

(13)

where $F_{{gb}_{i}}$ is the combined gravitational and buoyant force of the particle i, $V_{i}$ and $ρ_{i}$ are its volume and density, g is the gravitational acceleration of the earth, and $ρ_{l}$ is the density of the cyto- and nucleoplasm. This force applies in the vertical direction.

3.2.4 Normal Forces.

The reaction forces applied to the particles coming in contact with the surface of the glass slide are termed normal forces. They are modeled similar to the contact forces. The discontinuity of the contact events is made continuous by implementing the same third-order smoothstep function described in Eq. . The normal forces are thus formulated

F_{n_{i}} = - k_{c} (r_{i} - h_{i}) S_{3} (100 (r_{i} - h_{i}))

(14)

where $F_{N_{i}}$ is the normal force applied to the particle i, $r_{i}$ and $h_{i}$ are its radius and center's height, and $k_{c}$ is the contact stiffness. This force applies in the vertical direction.

3.2.5 Microfilament Forces.

The cortical actin layer is modeled as rope sections of one actin filament with the stiffness value of

k_{a}

= 43.7

{gr / s}^{2}

for a length of 1

μ m

[35] per filament. The cytoskeletal microfilaments are modeled as rope sections of five actin filament bundles. Each cytoskeletal microfilament is cut into two sections, and the stiffness of the average section,

k_{mfs}

⁠, is calculated. The spring forces between the particles i and j connected in a microfilament section,

F_{{mf}_{i j}} = - F_{{mf}_{j i}}

⁠, are thus

{\begin{matrix} - k_{mfs} (ℓ_{i j} - ℓ_{0_{s_{i j}}}), & (ℓ_{i j} - ℓ_{0_{s_{i j}}}) > 0 \\ 0, & (ℓ_{i j} - ℓ_{0_{s_{i j}}}) < 0 \end{matrix}

(15)

where the term $ℓ_{i j}$ is the current length of the spring between MP $_{i}$ and MP $_{j}$ ⁠, and $ℓ_{0_{s_{i j}}}$ is its unstretched length. The force acts in the direction between the MPs.

3.2.6 Membrane Forces.

The nuclear membrane is modeled as a mesh of MPs connected with ropes. The membrane force between connected particles i and j in a nuclear membrane mesh section,

F_{{mem}_{i j}} = - F_{{mem}_{j i}}

⁠, are thus

{\begin{matrix} - k_{ms} (ℓ_{i j} - ℓ_{0_{s_{i j}}}), & (ℓ_{i j} - ℓ_{0_{s_{i j}}}) > 0 \\ 0, & (ℓ_{i j} - ℓ_{0_{s_{i j}}}) < 0 \end{matrix}

(16)

where the term $k_{m}$ denotes the stiffness of the membrane, $ℓ_{i j}$ is the length of the spring between MP $_{i}$ and MP $_{j}$ ⁠, and $ℓ_{0_{s_{i j}}}$ is its unstretched length. The force acts in the direction between the MPs.

3.2.7 Volumetric Reaction Forces.

The cell and its nucleus are, respectively, filled with cyto- and nucleoplasm. These are incompressible fluids. The conservation of mass for incompressible fluids dictates that the cell and the nucleus volume should not change except when mass exchanges occur. This incompressibility imposes a constant volume constraint on the cellular and nuclear envelopes. The volumetric reaction force enforces this constraint.

Enforcement of the volumetric constraints in absolute terms is complex [37]. Enforcing this constraint numerically is an active area of research [38]. Herein, the incompressibility of fluid is relaxed to a stiff compressibility. A hydrostatic pressure difference between the inside and outside of the cell can be calculated to maintain close-to-constant volume

P_{hp} = - k_{V} (V - V_{0})

(17)

where $P_{hp}$ is the hydrostatic pressure difference, $k_{V}$ is the volumetric spring stiffness, V is the cellular or nuclear envelope volume, and $V_{0}$ is the volume to be maintained. The cellular and nuclear volumes can change due to mass exchange through their respective envelopes. This change is modeled by the changing the value of $V_{0}$ in Eq. (17).

Volume V is calculated using Gauss's divergence theorem on the meshed surfaces of the cellular or nuclear envelopes. The reaction pressure is applied to each cellular or nuclear envelope surface and distributed equally between the particles on each surface.

The volumetric reaction forces for the particles in the cellular or nuclear envelopes are thus formulated

F_{v_{i}} = P_{hp} (\sum (A_{i, j} / 3))

(18)

where $F_{v_{i}}$ is the volumetric reaction force of particle i in the envelope, $P_{hp}$ is the hydrostatic pressure difference, and $A_{i, j}$ encompasses the normal area vector of all the triangular envelope surfaces that include particle i.

3.2.8 Forces Resulting From Changing Mass.

Lipid accumulation is modeled as the increase of the LPs' radii. This increase in radii will result in a change in the LPs' masses. The simple equation of motion described in Eq. can be modified based on Euler's first law to account for this mass change

\sum F = \frac{d}{d t} (m V) = V \dot{m} + m \dot{V}

(19)

Substituting

V = \dot{x}

⁠,

\dot{V} = \ddot{x}

⁠, and

\sum F = - β \dot{x} - k x + F

⁠, where

F

contains all other generalized forces, yields the equations of motion for the LPs as

m \ddot{x} = - (β \dot{x} + k x) + F - \dot{m} \dot{x}

(20)

Moving the term

- \dot{m} \dot{x}

⁠, representing the effect of gaining mass from the surrounding medium to the right-hand side, makes it appear as an inertia force. The forces associated with the change of mass for an LP particle i can thus be described as

F_{m_{i}} = - {\dot{m}}_{i} {\dot{x}}_{i}

(21)

4 Scaling Approach

The mathematical formulation of the scaling approach for bodies of constant mass was proposed in Refs. [39] and [40] and developed further in Ref. [7]. The formulation for a system of masses with interconnected forces and changing mass was developed in Ref. [10]. The scaling formulation for systems with inhomogeneous mass distribution was developed in Ref. [8]. The effects of the scaling approach on the dynamic response of the system were examined in Ref. [8]. The method for choosing scaling factors and its effects have not been discussed before and are presented herein. The following discussion begins with a review of the techniques discussed in the previous works [7,8,10].

4.1 Previous Scaling Approach.

The general equations of motion for a system of two particles connected with a spring are

{\begin{matrix} m_{1} {\ddot{x}}_{1} + β_{1} {\dot{x}}_{1} + k (x_{1} - x_{2}) = Σ F_{1} \\ m_{2} {\ddot{x}}_{2} + β_{2} {\dot{x}}_{2} + k (x_{2} - x_{1}) = Σ F_{2} \end{matrix}

(22)

where $Σ F$ represents the sum of all other large forces, similar in magnitude to the spring force term, acting on the body. The effects of changing mass on the equation of motion for the particles can be moved to the right-hand side as a force equivalent as discussed in Sec. 3.2.8.

The scaling process for systems with inhomogeneous mass distribution, where

m_{1} ≫ m_{2}

⁠, involves the choice of three dimensionless numbers,

ϵ_{1}

⁠,

ϵ_{2}

⁠, and

ϵ_{3}

⁠, at the characteristic time unit of the system. The choice of this time unit dictates the time scale at which the system's motion is captured in the simulation. The importance and ramifications of this time unit are explained in detail in Refs. [7] and [8], respectively. For this cellular system undergoing adipogenesis in two weeks, a characteristic time unit of 1 ks is chosen. The dimensionless numbers become

\begin{matrix} ϵ_{1} (1 ks) = max (\frac{m_{1}}{β_{1}}, \frac{m_{2}}{β_{2}}) = (\frac{m_{1}}{β_{1}}) \\ ϵ_{2} (1 ks) = max (\frac{β_{1}}{k}, \frac{β_{2}}{k}) = (\frac{β_{1}}{k}) \\ ϵ_{3} & = \frac{m_{1}}{m_{2}} \end{matrix}

(23)

where

m_{1}

and

m_{2}

are the masses of the LPs and MPs, respectively, and

β_{1}

and

β_{2}

are their damping coefficients. These dimensionless numbers are used to decompose the time variable into different scales, where

T_{i} = ϵ_{j}^{i} t

⁠. Generalized speeds, accelerations, and mass changes are expressed as multivariable asymptotic expansions [6]

\begin{matrix} \dot{x} = \frac{d x}{d t} = \sum_{i = 0}^{\infty} ϵ_{j}^{i} \frac{\partial x}{\partial T_{i}} \\ \ddot{x} = \frac{d^{2} x}{d t^{2}} = \sum_{i = 0}^{\infty} \sum_{n = 0}^{\infty} ϵ_{j}^{i} ϵ_{j}^{n} \frac{\partial^{2} x}{\partial T_{i} \partial T_{n}} \end{matrix}

(24)

Utilizing asymptotic expansions and organizing the resulting equations into powers of the dimensionless numbers, the accelerations within the selected time unit of 1 ks can be isolated. This approach allows for scaling of the equations of motion with three dimensionless scaling factors,

a_{2}

⁠,

b_{2}

⁠, and

c_{2}

⁠, where

{\begin{matrix} a_{2} > ϵ_{1} \frac{1}{1 ks} \\ b_{2} > ϵ_{2} \frac{1}{1 ks} \\ c_{2} > ϵ_{3} \end{matrix}

(25)

The scaled equations of motion take the form [8]

{\begin{matrix} c_{2} m_{1} {\ddot{x}}_{1} + a_{2} c_{2} β_{1} {\dot{x}}_{1} + a_{2} b_{2} c_{2} k (x_{1} - x_{2}) \\ = a_{2} b_{2} c_{2} Σ F_{1} \\ m_{2} {\ddot{x}}_{2} + a_{2} c_{2} β_{2} {\dot{x}}_{2} + a_{2} b_{2} c_{2} k (x_{2} - x_{1}) \\ = a_{2} b_{2} c_{2} Σ F_{2} \end{matrix}

(26)

With this formulation, the generalized active forces on all particles are in proportion with the mass terms for each body. Thus, the model yields reasonable accelerations that can be numerically integrated using a larger time-step, reducing computational time. Implementation of the scaling approach reduces the system's sensitivity to fast dynamic changes. The slow speed of adipogenesis does not require capturing of fast dynamic responses. The scaled models' simulations and experimental results are shown to align in previous publications [7,8,10].

The method for choosing scaling factors has not been discussed before. Smaller values of the scaling factors will result in faster computation by reducing the system's stiffness. However, if the scaling factors are too small, the accuracy of the results will not be maintained. The general lower limit for the values of the scaling factors is derived from Eq. (25). In Secs. 4.2 and 4.3, the methods for the optimal choice of values of the scaling factors and the analysis of their effect on the computational time and accuracy of the results are presented.

4.2 Methods for the Choice of Scaling Factor Values.

The goal of the scaling approach is to capture the slow dynamic response of the system. The characteristic time unit is the response time that delineates the slow and fast dynamics. The speed of the process is used to determine the characteristic time unit. It is shown in previous works [8,10] that the dynamics of the two-week-long adipogenesis process can be captured accurately with a characteristic time unit of 1 ks.

The characteristic time unit,

T_{ch}

⁠, determines an equivalent characteristic frequency,

f_{ch}

f_{ch} = 1 / T_{ch} rad / s

(27)

of 1 rad/(characteristic time unit). The dynamic response of the system to changes in frequencies higher than the characteristic frequency is the fast dynamic response that the scaling approach is not formulated to capture. The system's transient response, a fast-dynamics response occurring at times smaller than the characteristic time unit, may also not be captured accurately by the scaling approach. The scaling factors are chosen to accurately reproduce the system's dynamic response at frequencies below this characteristic frequency.

4.2.1 Choice of Scaling Factor Values for Double-Scaling Approach.

The unscaled equation of motion for a single particle can be written as

m \ddot{x} + β \dot{x} + k x = Σ F

(28)

With the assumption that the system is locally linear, the homogeneous part of this differential equation can be written as

\ddot{x} + 2 ζ ω_{n} \dot{x} + ω_{n}^{2} x = 0

(29)

The terms

ω_{n}

⁠, the natural frequency,

ζ

⁠, the damping ratio, and the time constant of the system,

τ

⁠, can be defined as

{\begin{matrix} ω_{n} = \sqrt{k / m} \\ ζ = β / (2 \sqrt{k m}) \\ τ = 1 / ζ ω_{n} = m / 2 β \end{matrix}

(30)

These system properties are useful to ascertain the dynamic behavior of the system. The double-scaled equation of motion for a single particle takes the form of [10]

m \ddot{x} + a_{2} β \dot{x} + a_{2} b_{2} k x = a_{2} b_{2} Σ F

(31)

Rearranging the homogeneous part of the scaled equation of motion in the same form as Eq. yields the scaled system properties as

{\begin{matrix} ω_{n_{s}} = \sqrt{a_{2} b_{2} k / m} = \sqrt{a_{2} b_{2}} ω_{n} \\ ζ_{s} = a_{2} β / (2 \sqrt{a_{2} b_{2} k m}) = \sqrt{a_{2} / b_{2}} ζ \\ τ_{s} = 1 / ζ_{s} ω_{n_{s}} = m / 2 a_{2} β = 1 / a_{2} τ \end{matrix}

(32)

The goal of the scaling method is to capture the system's dynamic response in frequencies lower than the characteristic frequency. Notice that since the system's transient response occurs at smaller times than the characteristic time unit, only the preservation of the steady-state response of the system can be attempted. Herein, the system is presumed to reach a steady-state at $4 τ$ ⁠. The contrast between the scaled and unscaled system properties shown in Eq. (32) suggests a method for preserving the system's dynamic behavior at the characteristic frequency.

The scaling factors can be chosen so that the damping ratio of the particle can be maintained

ζ_{s} = ζ \Rightarrow a_{2} = b_{2}

(33)

The effects of a single-scaling method, which cannot preserve the damping ratio, on the system's resonance are discussed in Ref. [41]. The condition in Eq. (33) preserves the damped behavior of the system and eliminates those effects, such as overshoots.

To maintain the system's steady-state response to changes with frequencies lower than the characteristic frequency, the system should reach the steady-state at the characteristic time unit of the process. Therefore, the scaled system's time constant should be

4 τ_{s} = T_{ch} \Rightarrow τ_{s} = T_{ch} / 4

(34)

The first scaling factor can be chosen to achieve the required time constant

τ_{s} = T_{ch} / 4 \Rightarrow a_{2} = 4 τ / T_{ch} = 2 m / β T_{ch}

(35)

Considering

ϵ_{1} (T_{ch}) = m / β

from Eq. , this condition for

a_{2}

can be written as

a_{2} = 2 ϵ_{1}

(36)

The double-scaling approach limits the choice of scaling factors for the system to single values for each scaling factor [10]. The conditions set in Eqs. and should be met with the lower limits of scaling factors expressed in Eq. for systems with different particle sizes. The method for choosing the values of the scaling factors of the double-scaling thus becomes

{\begin{matrix} a_{2} = max (2 ϵ_{1}, ϵ_{2}) \\ b_{2} = a_{2} \end{matrix}

(37)

The scaling factor values determined by this method preserve the damping ratios of all bodies and ensure that the system reaches a steady-state in times smaller than the characteristic time unit. This method is utilized for choosing the scaling factors in Ref. [10]. The method for the choice of scaling factor values for the triple-scaling approach is discussed next.

4.2.2 Choice of Scaling Factor Values for Triple-Scaling Approach.

Adding the third scaling factor greatly reduces the computational time for simulating systems with an inhomogeneous mass distribution. This is achieved by bringing the generalized forces for each body into proportion to their masses. The system parameters for the unscaled equation of motion for the two-body system in Eq. are

{\begin{matrix} ω_{n 1} = \sqrt{k / m_{1}} \\ ζ_{1} = β_{1} / (2 \sqrt{k m_{1}}) \\ τ_{1} = 1 / ζ_{1} ω_{n 1} = m_{1} / 2 β_{1} \end{matrix}

(38)

{\begin{matrix} ω_{n 2} = \sqrt{k / m_{2}} \\ ζ_{2} = β_{2} / (2 \sqrt{k m_{2}}) \\ τ_{2} = 1 / ζ_{2} ω_{n 2} = m_{2} / 2 β_{2} \end{matrix}

(39)

Notice that since

m_{1} ≫ m_{2}

and

β_{1} > β_{2}

⁠,

τ_{1} > τ_{2}

⁠. The system parameters of the triple-scaled equation of motion for a two-body system in Eq. are

{\begin{matrix} ω_{n 1_{s}} = \sqrt{\frac{a_{2} b_{2} k}{m_{1}}} = \sqrt{a_{2} b_{2}} ω_{n 1} \\ ζ_{1_{s}} = \frac{a_{2} β_{1}}{2 \sqrt{a_{2} b_{2} k m_{1}}} = \sqrt{\frac{a_{2}}{b_{2}}} ζ_{1} \\ τ_{1_{s}} = \frac{1}{ζ_{1_{s}} ω_{n 1_{s}}} = \frac{m_{1}}{2 a_{2} β_{1}} = \frac{1}{a_{2}} τ_{1} \end{matrix}

(40)

{\begin{matrix} ω_{n 2_{s}} = \sqrt{\frac{a_{2} b_{2} c_{2} k}{m_{2}}} = \sqrt{a_{2} b_{2} c_{2}} ω_{n 2} \\ ζ_{2_{s}} = \frac{a_{2} c_{2} β_{2}}{2 \sqrt{a_{2} b_{2} c_{2} k m_{2}}} = \sqrt{\frac{a_{2} c_{2}}{b_{2}}} ζ_{2} \\ τ_{2_{s}} = \frac{1}{ζ_{2_{s}} ω_{n 2_{s}}} = \frac{m_{2}}{2 a_{2} c_{2} β_{2}} = \frac{1}{a_{2} c_{2}} τ_{2} \end{matrix}

(41)

Setting the steady-state condition of

4 τ_{s} = T_{ch}

⁠, and considering that

c_{2} < 1

⁠,

τ_{2} > τ_{1}

⁠, and

ϵ_{1} (T_{ch}) = m_{1} / β_{1}

⁠, gives the condition for the first scaling factor,

a_{2}

a_{2} = 4 τ_{1} / T_{ch} = 2 m_{1} / β_{1} T_{ch} \Rightarrow a_{2} = 2 ϵ_{1}

(42)

It can be seen in Eqs. (40) and (41), simultaneously maintaining the damping ratios for all bodies is impossible. Only the damping ratio of one body can be maintained, necessitating a choice between the two. This choice determines the values of the other two scaling factors, $b_{2}$ and $c_{2}$ ⁠. Herein, the methods based on the two body systems are presented and examined. The two methods can be used in systems with more than two different masses since only one damping coefficient can be maintained.

4.2.3 Maintaining Damping Ratio of the Larger Body.

The damping ratio of the larger body can be maintained with the same method discussed for the choice of scaling values in the double-scaling approach. The third scaling factor cancels out in the equation of motion for the larger body resulting in

ζ_{1_{s}} = ζ_{1} \Rightarrow a_{2} = b_{2}

(43)

Now the third scaling factor value can be calculated based on the time constant of the smaller body's scaled equation of motion. Setting

4 τ_{2_{s}} = T_{ch}

yields the condition for the third scaling factor value of

m_{2} / 2 a_{2} c_{2} β_{2} = T_{ch} / 4 \Rightarrow c_{2} = 2 m_{2} / a_{2} β_{2} T_{ch}

(44)

The conditions set in Eqs. – should be enforced within the lower limits set in Eq. . Combining all required conditions produces the first method for calculating the values of the scaling factors for the triple-scaling approach. This method is formulated as

{\begin{matrix} a_{2} = max (2 ϵ_{1}, ϵ_{2}) \\ b_{2} = a_{2} \\ c_{2} = max (2 m_{2} / a_{2} β_{2} T_{ch}, ϵ_{3}) \end{matrix}

(45)

This method was used for the calculation of the scaling factor values in Ref. [8]. Implementing this method reduces the damping ratio of the smaller body

ζ_{2_{s}} = \sqrt{c_{2}} ζ_{2}

(46)

This may result in overshoots in the transient response, and resonance in the higher frequency responses of the system. These effects are examined in Ref. [8]. It is shown that using this method to calculate the scaling factor values accurately reproduces the system's dynamic behavior for frequencies lower than the characteristic frequency. The effects of the reduction in damping ratio on the system's resonance are discussed in Ref. [41].

4.2.4 Maintaining Damping Ratio of the Smaller Body.

The scaling factor values can be chosen to maintain the damping ratio of the smaller body. This novel method is employed to formulate the scaled equations of motion in this work. The process starts with the choice of

a_{2}

based on Eq. . Setting

4 τ_{2_{s}} = T_{ch}

yields the condition for the third scaling factor value,

c_{2}

⁠, of

m_{2} / 2 a_{2} c_{2} β_{2} = T_{ch} / 4 \Rightarrow c_{2} = 2 m_{2} / a_{2} β_{2} T_{ch}

(47)

The second scaling factor,

b_{2}

⁠, can be calculated such that the damping ratio of the smaller body will be maintained

ζ_{2_{s}} = ζ_{2} \to \sqrt{a_{2} c_{2} / b_{2}} = 1 \Rightarrow b_{2} = a_{2} c_{2}

(48)

The conditions set in Eqs. , , and must satisfy the lower limits for the scaling factor values set in Eq. . This method thus takes the form of

{\begin{matrix} a_{2} = max (2 ϵ_{1}, ϵ_{2}) \\ c_{2} = max (2 m_{2} / a_{2} β_{2} T_{ch}, ϵ_{3}) \\ b_{2} = a_{2} c_{2} \end{matrix}

(49)

This approach ensures that the damping ratio of the smaller body is maintained. However, it increases the damping ratio of the larger body

ζ_{1_{s}} = \sqrt{1 / c_{2}} ζ_{1}

(50)

Increasing the damping ratio does not produce underdamped dynamic behavior such as resonances [41]. Furthermore, the generalized forces will be in the same order as the inertial term for the smaller bodies. In systems with a larger number of smaller masses, such as the cellular model developed here, it is computationally beneficial to optimize the scaling factor values based on the smaller masses. These methods are compared in Sec. 4.3.

This method is employed to choose the scaling factors used in this work. The change in LPs' masses results in changes in their mechanical properties, which affects the calculation of scaling factors. The LPs' mechanical properties at the end of each simulation step are used to calculate the scaling factors at the start of the next simulation step.

4.3 Examination of Different Scaling Approaches.

Herein, an example of the effects of the proposed methods for the choice of scaling factors on a two-body system is presented to demonstrate the utility and limitations of each method. Based on Eq. , the scaled equations of motion for this system in Fig. 3, under equal and opposite sinusoidal external forces, would be

{\begin{matrix} c_{2} m_{1} {\ddot{x}}_{1} + a_{2} c_{2} β_{1} {\dot{x}}_{1} + a_{2} b_{2} c_{2} k (x_{1} - x_{2}) \\ = a_{2} b_{2} c_{2} k sin (ω t) \\ m_{2} {\ddot{x}}_{2} + a_{2} c_{2} β_{2} {\dot{x}}_{2} + a_{2} b_{2} c_{2} k (x_{2} - x_{1}) \\ = - a_{2} b_{2} c_{2} k sin (ω t) \end{matrix}

(51)

Fig. 4

View large Download slide

The phase portrait of the two body system under equal and opposite sinusoidal forces (Color version online)

where $m_{1}$ = 506.51 fg is the mass of a lipid droplet, $β_{1}$ = 1.0562 $\times 10^{7}$ fg/ms is the damping ratio for the lipid droplet, k = 5.244 $\times 10^{11}$ fg/ ${ms}^{2}$ is the stiffness of a microfilament section, $m_{2}$ = 0.0483 fg is the microparticle mass, and $β_{2}$ = 2.5918 $\times 10^{5}$ fg/ms is the damping ratio for the microparticle. The characteristic time unit (⁠ $T_{ch}$ ⁠) of 1 ms and frequency (⁠ $f_{ch}$ ⁠) of $ω$ = 1 rad/ms are chosen for this example to make the simulation of unscaled equations of motion computationally feasible.

This system is simulated for 10 ms with zero initial conditions. The solutions of different scaling approaches and methods for the choice of scaling factors are contrasted with the solution of unscaled equations of motion. The scaling factors and computational time required to obtain the solutions are included in Table 1. The results demonstrate the computational time reduction in the order of $10^{2}$ for the double-scaling approach, $10^{3}$ for the triple-scaling using the first method (maintaining damping ratio of larger body), and ${3 \times 10}^{3}$ for the triple-scaling using the second method (maintaining damping ratio of smaller body), for this simple system in a short simulation.

Table 1

Table of parameters for the two body system simulations with different methods for the choice of scaling factors

Case	$a_{2}$	$b_{2}$	$c_{2}$	Computational time (s)
Unscaled	1	1	1	63.112
Double-scaling	3.8365 $\times$ 10 $^{- 3}$	3.8365 $\times$ 10 $^{- 3}$	1	0.618
Triple-scaling method 1	3.8365 $\times$ 10 $^{- 3}$	3.8365 $\times$ 10 $^{- 3}$	3.8860 $\times$ 10 $^{- 3}$	0.061
Triple-scaling method 2	3.8365 $\times$ 10 $^{- 3}$	1.4908 $\times$ 10 $^{- 5}$	3.8860 $\times$ 10 $^{- 3}$	0.022

Case	$a_{2}$	$b_{2}$	$c_{2}$	Computational time (s)
Unscaled	1	1	1	63.112
Double-scaling	3.8365 $\times$ 10 $^{- 3}$	3.8365 $\times$ 10 $^{- 3}$	1	0.618
Triple-scaling method 1	3.8365 $\times$ 10 $^{- 3}$	3.8365 $\times$ 10 $^{- 3}$	3.8860 $\times$ 10 $^{- 3}$	0.061
Triple-scaling method 2	3.8365 $\times$ 10 $^{- 3}$	1.4908 $\times$ 10 $^{- 5}$	3.8860 $\times$ 10 $^{- 3}$	0.022

The phase portrait of the four cases is shown in Fig. 4. Examining the phase portraits shows perfect alignment between the results of the scaling methods and the unscaled results for the steady-state part of the solution. The transient responses of the scaled systems differ from the unscaled results. The method for maintaining the damping ratio of the larger body shows overshoots typical of underdamped motion. This is unexpected in systems with low Reynolds numbers. Recently, microfluidic devices were developed to investigate this phenomenon [42,43], and no discernible underdamped motion was found. In any case, the discrepancy in the transient response of the scaled and unscaled systems is outside the valid range of results of the scaling method as it is a fast dynamic response occurring in shorter times than $T_{ch}$ ⁠. The methods for the choice of scaling factors accurately reproduce steady-state solution for the system regardless of their effects on the damping ratio.

Fig. 5

View large Download slide

The Bode plot of the two body system under equal and opposite sinusoidal forces (Color version online)

The steady-state responses of the scaling approaches can be further examined in the Bode plot shown in Fig. 5. The double- and triple-scaling approaches align with the unscaled solution up to the characteristic frequency of 1 rad/ms. The double-scaling approach introduces a phase shift, and one of the triple-scaling approaches introduces a resonance at frequencies higher than the characteristic frequency. This is expected as the damping ratio reduces in the first and increases in the second triple-scaling method. The effect of the scaling method on the system's resonance is discussed in Ref. [41].

Fig. 6

View large Download slide

The comparison between simulation and experimental data corresponding with different stages of adipogenic differentiation

This analysis shows the qualities and limitations of the methods for choosing scaling factors. All methods reduce computational time while accurately reproducing the long-term solution of dynamic systems observed at times equal to the chosen characteristic time unit. This means that the scaling approach can accurately obtain the system's responses for frequencies lower than or equal to the characteristic frequency. The triple-scaling approaches are faster than the double-scaling. The second triple-scaling approach is the fastest and does not introduce any resonance at higher frequencies. These benefits are why this method is used to formulate scaled equations of motion in this work.

5 Simulation and Results

According to the literature, there are two separate phases of adipogenic differentiation: lineage orientation into pre-adipocytes [34] and adipocyte maturation [44]. The cytoskeleton rearrangement and nucleus size reduction occur in the first phase without a significant increase in lipid accumulation. Lipid droplets accumulate inside the cell during the second phase. The experimental results discussed in Ref. [9] conform to this hypothesis. A significant reduction in nucleus size is seen without a noticeable buildup of lipid droplets in the cells before day 5 of the experiment, indicating the cytoskeletal rearrangement is concluded in that time. Fast accumulation of lipid droplets starts at day 5 and continues on to the end.

The two phases of adipogenesis are reproduced in three stages in the simulation. An initial phase of cellular model stabilization is added to the two phases of adipogenesis to stabilize the cellular model by allowing the forces acting on the cell to attain equilibrium. The last two phases correspond to the observed phases of the adipogenesis process. These phases are labeled in Fig. 6. The model parameters are altered to conform the simulation to these phases in the adipogenesis. The parameters used in this simulation are included in Table 2.

Table 2

Table of all the parameters used in the cellular model simulation

Parameter	Definition	Value	Unit
$ρ_{MP}$	Microparticle density	1.0967	${kg / m}^{3}$
$ρ_{LP}$	Lipid density	915	${kg / m}^{3}$
$ρ_{l}$	Cytoplasm density	997	${kg / m}^{3}$
$η_{C}$	Cytoplasm dynamic viscosity	1.1 [45]	cp
$k_{a}$	1 $μ m$ actin fiber stiffness	43.7 [35]	${g / s}^{2}$
$k_{{mfs}_{cyt}}$	Cytoskeletal microfilaments section stiffness	449.7589	${g / s}^{2}$
$k_{{mfs}_{cor}}$	Cortex fibers section stiffness	76.4737	${g / s}^{2}$
$k_{c}$	Contact stiffness	449.7589	${g / s}^{2}$
$k_{V_{C}}$	Volumetric spring stiffness of the cell	44.9759	${g / s}^{2}$
$k_{V_{N}}$	Volumetric spring stiffness of the nucleus	89.9518	${g / s}^{2}$
$k_{ms}$	Nuclear membrane sections stiffness	19.0211	${g / s}^{2}$
r $_{MP}$	Microparticle radius	24.2625	nm
r $_{{LP}_{0}}$	Lipid droplet initial radius	138.4484	nm
r $_{{LP}_{f}}$	Lipid droplets final radius	435.1041	nm
m $_{MP}$	Microparticle mass	0.0656	$10^{- 15}$ g
m $_{{LP}_{0}}$	Lipid droplet initial mass	10.1712	$10^{- 15}$ g
m $_{{LP}_{f}}$	Lipid droplet final mass	315.7107	$10^{- 15}$ g
$ϵ_{1_{0}}$	Initial value of $ϵ_{1}$	3.5442	10 $^{- 12}$
$ϵ_{2_{0}}$	Initial value of $ϵ_{2}$	6.3809	10 $^{- 14}$
$ϵ_{3_{0}}$	Initial value of $ϵ_{3}$	6.4508	10 $^{- 3}$
$ϵ_{1_{f}}$	Final value of $ϵ_{1}$	3.5004	10 $^{- 11}$
$ϵ_{2_{f}}$	Final value of $ϵ_{2}$	2.0053	10 $^{- 13}$
$ϵ_{3_{f}}$	Final value of $ϵ_{3}$	2.0782	10 $^{- 4}$
$a_{2_{0}}$	First scaling factor initial value	2.8353	10 $^{- 11}$
$b_{2_{0}}$	Second scaling factor initial value	1.0437	10 $^{- 12}$
$c_{2_{0}}$	Third scaling factor initial value	3.6810	10 $^{- 2}$
$a_{2_{f}}$	First scaling factor final value	2.8004	10 $^{- 10}$
$b_{2_{f}}$	Second scaling factor final value	1.0437	10 $^{- 12}$
$c_{2_{f}}$	Third scaling factor final value	3.7269	10 $^{- 3}$

Parameter	Definition	Value	Unit
$ρ_{MP}$	Microparticle density	1.0967	${kg / m}^{3}$
$ρ_{LP}$	Lipid density	915	${kg / m}^{3}$
$ρ_{l}$	Cytoplasm density	997	${kg / m}^{3}$
$η_{C}$	Cytoplasm dynamic viscosity	1.1 [45]	cp
$k_{a}$	1 $μ m$ actin fiber stiffness	43.7 [35]	${g / s}^{2}$
$k_{{mfs}_{cyt}}$	Cytoskeletal microfilaments section stiffness	449.7589	${g / s}^{2}$
$k_{{mfs}_{cor}}$	Cortex fibers section stiffness	76.4737	${g / s}^{2}$
$k_{c}$	Contact stiffness	449.7589	${g / s}^{2}$
$k_{V_{C}}$	Volumetric spring stiffness of the cell	44.9759	${g / s}^{2}$
$k_{V_{N}}$	Volumetric spring stiffness of the nucleus	89.9518	${g / s}^{2}$
$k_{ms}$	Nuclear membrane sections stiffness	19.0211	${g / s}^{2}$
r $_{MP}$	Microparticle radius	24.2625	nm
r $_{{LP}_{0}}$	Lipid droplet initial radius	138.4484	nm
r $_{{LP}_{f}}$	Lipid droplets final radius	435.1041	nm
m $_{MP}$	Microparticle mass	0.0656	$10^{- 15}$ g
m $_{{LP}_{0}}$	Lipid droplet initial mass	10.1712	$10^{- 15}$ g
m $_{{LP}_{f}}$	Lipid droplet final mass	315.7107	$10^{- 15}$ g
$ϵ_{1_{0}}$	Initial value of $ϵ_{1}$	3.5442	10 $^{- 12}$
$ϵ_{2_{0}}$	Initial value of $ϵ_{2}$	6.3809	10 $^{- 14}$
$ϵ_{3_{0}}$	Initial value of $ϵ_{3}$	6.4508	10 $^{- 3}$
$ϵ_{1_{f}}$	Final value of $ϵ_{1}$	3.5004	10 $^{- 11}$
$ϵ_{2_{f}}$	Final value of $ϵ_{2}$	2.0053	10 $^{- 13}$
$ϵ_{3_{f}}$	Final value of $ϵ_{3}$	2.0782	10 $^{- 4}$
$a_{2_{0}}$	First scaling factor initial value	2.8353	10 $^{- 11}$
$b_{2_{0}}$	Second scaling factor initial value	1.0437	10 $^{- 12}$
$c_{2_{0}}$	Third scaling factor initial value	3.6810	10 $^{- 2}$
$a_{2_{f}}$	First scaling factor final value	2.8004	10 $^{- 10}$
$b_{2_{f}}$	Second scaling factor final value	1.0437	10 $^{- 12}$
$c_{2_{f}}$	Third scaling factor final value	3.7269	10 $^{- 3}$

The difference in the initial and final size of the cell, inferred from the nuclear and lipid areas in the experimental data, suggests a reduction in the cellular size throughout the adipogenesis process. This is represented in the simulation by a reduction in cellular area and volume. The cellular area reduction is achieved by detaching the focal adhesion points from the glass slide. The focal adhesion points that fall out of the expected area are detached. The expected area of the cell is reduced as a second-order polynomial function of time

A_{C_{t}} = (A_{0 C} - A_{e C}) {(1 - \frac{t}{t_{e}})}^{2}

(52)

where

A_{C_{t}}

is the expected area of the cell,

A_{0 C}

is the initial area of the cell,

A_{0 C}

is its final area, t is the elapsed time from the start of the cytoskeletal rearrangement phase, and

t_{e} = 14 days

is the duration of the adipogenesis process. The cellular volume is reduced as a third-order polynomial function of time

V_{C_{t}} = (V_{0 C} - V_{e C}) {(1 - \frac{t}{t_{e}})}^{3}

(53)

where $V_{C_{t}}$ is the time-dependent cellular volume, $V_{0 C}$ is the initial cellular volume, and $V_{e C}$ is its final volume. The term $V_{C_{t}}$ is used as the term $V_{0}$ in Eq. (17) to calculate the volumetric reaction forces of the cell.

5.1 Cellular Model Stabilization.

The cellular model is initiated with particles at rest. The nuclear membrane, cytoskeletal microfilaments, and the cortex layer are in tension [20], so the forces in the model are not at equilibrium. To balance the forces, the cellular model is allowed to stabilize to a static equilibrium. This phase starts on day 0 and continues to day 1.

A proportional-integral-derivative controller adjusts the nuclear membrane's uncertain [33] stiffness value depending on the measured nuclear area. Calculating this stiffness for the proposed cellular model is made possible by the simulation's fast computational speed. The calculated stiffness conforms with the values found in the previous works [7,8,10]. This stiffness is used throughout the simulation.

5.2 Cytoskeletal Rearrangement.

The cytoskeletal rearrangement is posited to be caused by the depolymerization of actin microfilaments attached to the nucleus, as stated in previous works [9], supported by the literature [22,34], and observed as the migration of the remaining cytoskeletal microfilaments to the cell's periphery [10]. Depolymerization is reproduced as a reduction in stiffness in nucleus-attached microfilaments and, ultimately, their elimination. In previous works [7,8,10], the relation of this change in stiffness with time was found to be a fourth-order polynomial. The simulation results of the proposed 3D model confirm this hypothesis

k_{{mfs}_{t}} = k_{mfs} {(1 - \frac{t}{t_{C}})}^{4}

(54)

where $k_{{mfs}_{t}}$ is the time-dependent stiffness of the nucleus-attached microfilament section, $k_{mfs}$ is its initial stiffness, t is the elapsed time from the start of the cytoskeletal rearrangement phase, and $t_{C} = 4 days$ is its duration. The migration of the cytoskeletal microfilaments to the periphery is achieved by the reduction of the unstretched length of the microfilament sections connecting the cytoskeletal microfilaments to the cortex.

Depolymerization of the microfilaments allows the nuclear membrane to release tension and contract size [17]. This contraction is accomplished by reducing the unstretched length of the nuclear membrane sections in the model during this phase. The reduction in nuclear tension, along with cortex and cytoskeletal microfilament tension during adipogenesis, can be seen in Fig. 7.

Fig. 7

View large Download slide

Diagram of nuclear, cortex, and cytoskeletal microfilament tensions (Color version online)

5.3 Adipocyte Maturation.

The hMSCs at the end of the cytoskeletal rearrangement are known as pre-adipocytes. The transformation of hMSC from a pre-adipocyte into an adipocyte, or fat cell, is known as adipocyte maturation. The adipocyte maturation phase is marked by a rapid increase in the size of the lipid droplets. In the presented experiment, this phase starts on day 5 and concludes at the end of the experiment on day 15.

A total of 50 LPs are included in the model. While lipid droplets exist in the cell before the start of this phase, their size is too small to model. Therefore, the LPs' initial radii are calculated such that the sum of their areas equals the observed lipid area at day 5. The simulation lipid area is fitted to the experimental data as a fourth-order polynomial function of time. This function is used to calculate the change in LPs' radii and their corresponding volumes and masses.

Lipid area increase corresponds to a slow reduction in nuclear volume. The reduction in cellular size results in collisions between the LPs, nucleus, and cellular envelope, affecting the nucleus size and morphology. This is aided by the modulation of the nuclear membrane, cytoskeletal microfilaments, and cortex fibers' unstretched lengths. Simulation snapshots are shown in Fig. 8.

Fig. 8

View large Download slide

Snapshots of the simulation of the 3D model (Color version online)

5.4 Simulation Results.

The simulation time history is recorded in 100 s steps in matlab 2023b software. The change of LPs masses during the adipocyte maturation phase will result in different values for mechanical properties used in Eq. (49). At the end of each step, the mechanical properties of the system are calculated for the next 100 s step, and new values for the scaling factors are calculated and used for the next 100 s step of the simulation. The initial and final values of the scaling factors are included in Table 2. matlab's ODE45 solver is used to perform the integration on a desktop computer with an AMD Ryzen 5 5600X processor. The relative and absolute errors of the integrator are set to $10^{- 5}$ and $10^{- 7}$ ⁠, respectively.

The comparison between the experimental and simulation data shows an average error of 0.1509 $μ m^{2}$ and a mean squared error of 0.0725 $μ m^{4}$ for the nuclear area. The close alignment of the simulation results and experimental data validates the scaling method presented herein. The lipid area shows an average error of 0.0245 $μ m^{2}$ and mean squared error of 0.9032 $μ m^{4}$ between the experimental and simulation data.

The difference between the kinetic energy and work done is the change in total energy

Δ E = T - W

(55)

The model was initiated with no initial velocity for the particles, so $Δ E$ should remain zero through the simulation. Therefore, the values of $Δ E$ ⁠, shown in Fig. 9, represent the numerical error. The cumulative numerical error is −2.0939 $\times 10^{- 4}$ in the model units of $fg \cdot μ m^{2} / {ks}^{2}$ ⁠. This is within the tolerance of the integration algorithm for such a long-term simulation.

Fig. 9

View large Download slide

Change in total energy shows a small numerical error

The total computational time required for the simulation of the 15-day adipogenesis was 4103 s which is less than 1 h and 9 min. The unscaled cellular model, where $a_{2} = b_{2} = c_{2} = 1$ ⁠, requires 60 min of computational time to produce 1 ns of time history using identical hardware and software. This difference represents a reduction in computational time on the order of 10 $^{15}$ ⁠.

6 Conclusion

This work presents a novel, physics-based, multibody dynamic 3D model of a stem cell to capture its morphological changes during the 14-day adipogenic differentiation process. The model was able to capture the mechanics of focal adhesion, cytoskeletal rearrangement, and adipocyte maturation that occur during adipogenesis. The affinity of the computational results to the experimental data establishes the validity of the suggested model.

Multibody dynamic cellular models include particles with minute mass and size relative to the forces. The ensuing discrepancy makes long-term dynamic modeling of cellular processes computationally infeasible using conventional approaches. The scaling approach presented in this work drastically reduces the computational time for such long-term simulations. The particles in the proposed model have different mechanical properties, such as size and mass. This difference necessitates careful consideration of the choice of scaling factors. Multiple methods for the choice of scaling factors are proposed, and their effects on computational time and the system's dynamic response are presented in this work.

Employing the presented novel formulation of the scaling approach drastically reduces the computational time for obtaining the time history of the 14-day-long adipogenic differentiation process. The computational time for the simulation of the proposed 3D model is less than 1 h and 9 min using a conventional desktop computer. This represents a computational time reduction in the order of 10 $^{15}$ compared to the unscaled formulation. The increased computational speed allows for investigating the mechanics of subcellular elements beyond experimental observations. These include the mechanical characteristics of the nuclear membrane, the morphological effects of the focal adhesion points, and variations in nuclear and cytoskeletal tension during adipogenesis.

Acknowledgment

The authors thank Dr. Vatsal Joshi and Dr. Andrew McColloch for their aid in this research.

Funding Data

National Institutes of Health (NIH) National Institute of Biomedical Imaging and Bio-engineering (NIBIB) (Award No. 1R15EB030842-01; Funder ID: 10.13039/100000070).

Data Availability Statement

The authors attest that all data for this study are included in the paper.

References

1.

Dror

,

R. O.

,

Dirks

,

R. M.

,

Grossman

,

J.

,

Xu

,

H.

, and

Shaw

,

D. E.

,

2012

, “

Biomolecular Simulation: A Computational Microscope for Molecular Biology

,”

Annu. Rev. Biophys.

,

41

(

1

), pp.

429

–

452

.10.1146/annurev-biophys-042910-155245

Google Scholar

Crossref

PubMed

2.

Huggins

,

D. J.

,

Biggin

,

P. C.

,

Dämgen

,

M. A.

,

Essex

,

J. W.

,

Harris

,

S. A.

,

Henchman

,

R. H.

,

Khalid

,

S.

, et al.,

2019

, “

Biomolecular Simulations: From Dynamics and Mechanisms to Computational Assays of Biological Activity

,”

Wiley Interdiscip. Rev.: Comput. Mol. Sci.

,

9

(

3

), p.

e1393

.10.1002/wcms.1393

3.

Maruyama

,

Y.

,

Igarashi

,

R.

,

Ushiku

,

Y.

, and

Mitsutake

,

A.

,

2023

, “

Analysis of Protein Folding Simulation With Moving Root Mean Square Deviation

,”

J. Chem. Inf. Model.

,

63

(

5

), pp.

1529

–

1541

.10.1021/acs.jcim.2c01444

Google Scholar

Crossref

PubMed

4.

Gershenson

,

A.

,

Gosavi

,

S.

,

Faccioli

,

P.

, and

Wintrode

,

P. L.

,

2020

, “

Successes and Challenges in Simulating the Folding of Large Proteins

,”

J. Biol. Chem.

,

295

(

1

), pp.

15

–

33

.10.1074/jbc.REV119.006794

Google Scholar

Crossref

PubMed

5.

Lane

,

T. J.

,

Shukla

,

D.

,

Beauchamp

,

K. A.

, and

Pande

,

V. S.

,

2013

, “

To Milliseconds and Beyond: Challenges in the Simulation of Protein Folding

,”

Curr. Opin. Struct. Biol.

,

23

(

1

), pp.

58

–

65

.10.1016/j.sbi.2012.11.002

Google Scholar

Crossref

PubMed

6.

Nayfeh

,

A. H.

,

1973

,

Perturbation Methods

,

Wiley

,

Weinheim, Germany

.

7.

Rabiei

,

M.

,

McColloch

,

A.

,

Rabbani

,

P.

,

Cho

,

M.

, and

Bowling

,

A.

,

2020

, “

Long Term Dynamic Simulation of a Stem Cell Nucleus

,”

ASME J. Comput. Nonlinear Dyn.

,

15

(

11

), p.

111002

.10.1115/1.4048195

Google Scholar

Crossref

8.

Rabiei

,

M.

,

Albaruni

,

M. A. S. I.

,

Joshi

,

V.

,

Cho

,

M.

, and

Bowling

,

A.

,

2024

, “

Long-Term Dynamic Simulation of Cellular Systems With Inhomogeneous Mass Distribution

,”

Multibody Syst. Dyn.

, pp.

1

–

20

.10.1007/s11044-024-10044-y

9.

McColloch

,

A.

,

Rabiei

,

M.

,

Rabbani

,

P.

,

Bowling

,

A.

, and

Cho

,

M.

,

2019

, “

Correlation Between Nuclear Morphology and Adipogenic Differentiation: Application of a Combined Experimental and Computational Modeling Approach

,”

Sci. Rep.

,

9

(

1

), p.

16381

.10.1038/s41598-019-52926-8

Google Scholar

Crossref

PubMed

10.

Rabiei

,

M.

,

Joshi

,

V.

,

Fowlds

,

K.

,

Cho

,

M.

, and

Bowling

,

A.

,

2023

, “

Long-Term Dynamic Simulation of Adipogenic Differentiation of a Human Mesenchymal Stem Cell

,”

Multibody Syst. Dyn.

,

58

(

1

), pp.

113

–

133

.10.1007/s11044-023-09888-7

Google Scholar

Crossref

PubMed

11.

Pegoraro

,

A. F.

,

Janmey

,

P.

, and

Weitz

,

D. A.

,

2017

, “

Mechanical Properties of the Cytoskeleton and Cells

,”

Cold Spring Harbor Perspect. Biol.

,

9

(

11

), p.

a022038

.10.1101/cshperspect.a022038

Google Scholar

Crossref

12.

Ingber

,

D. E.

,

2003

, “

Tensegrity I. Cell Structure and Hierarchical Systems Biology

,”

J. Cell Sci.

,

116

(

7

), pp.

1157

–

1173

13.

Chamberlain

,

G.

,

Fox

,

J.

,

Ashton

,

B.

, and

Middleton

,

J.

,

2007

, “

Concise Review: Mesenchymal Stem Cells: Their Phenotype, Differentiation Capacity, Immunological Features, and Potential for Homing

,”

Stem Cells

,

25

(

11

), pp.

2739

–

2749

.10.1634/stemcells.2007-0197

Google Scholar

Crossref

PubMed

14.

Mao

,

A. S.

,

Shin

,

J.-W.

, and

Mooney

,

D. J.

,

2016

, “

Effects of Substrate Stiffness and Cell-Cell Contact on Mesenchymal Stem Cell Differentiation

,”

Biomaterials

,

98

, pp.

184

–

191

.10.1016/j.biomaterials.2016.05.004

Google Scholar

Crossref

PubMed

15.

Engler

,

A. J.

,

Sen

,

S.

,

Sweeney

,

H. L.

, and

Discher

,

D. E.

,

2006

, “

Matrix Elasticity Directs Stem Cell Lineage Specification

,”

Cell

,

126

(

4

), pp.

677

–

689

.10.1016/j.cell.2006.06.044

Google Scholar

Crossref

PubMed

16.

Kilian

,

K. A.

,

Bugarija

,

B.

,

Lahn

,

B. T.

, and

Mrksich

,

M.

,

2010

, “

Geometric Cues for Directing the Differentiation of Mesenchymal Stem Cells

,”

Proc. Natl. Acad. Sci.

,

107

(

11

), pp.

4872

–

4877

.10.1073/pnas.0903269107

Google Scholar

Crossref

17.

Feng

,

T.

,

Szabo

,

E.

,

Dziak

,

E.

, and

Opas

,

M.

,

2010

, “

Cytoskeletal Disassembly and Cell Rounding Promotes Adipogenesis From ES Cells

,”

Stem Cell Rev. Rep.

,

6

(

1

), pp.

74

–

85

.10.1007/s12015-010-9115-8

Google Scholar

Crossref

PubMed

18.

Gao

,

L.

,

McBeath

,

R.

, and

Chen

,

C. S.

,

2010

, “

Stem Cell Shape Regulates a Chondrogenic Versus Myogenic Fate Through Rac1 and N-Cadherin

,”

Stem Cells

,

28

(

3

), pp.

564

–

572

19.

Driscoll

,

T. P.

,

Cosgrove

,

B. D.

,

Heo

,

S.-J.

,

Shurden

,

Z. E.

, and

Mauck

,

R. L.

,

2015

, “

Cytoskeletal to Nuclear Strain Transfer Regulates Yap Signaling in Mesenchymal Stem Cells

,”

Biophys. J.

,

108

(

12

), pp.

2783

–

2793

.10.1016/j.bpj.2015.05.010

Google Scholar

Crossref

PubMed

20.

Maniotis

,

A. J.

,

Chen

,

C. S.

, and

Ingber

,

D. E.

,

1997

, “

Demonstration of Mechanical Connections Between Integrins, Cytoskeletal Filaments, and Nucleoplasm That Stabilize Nuclear Structure

,”

Proc. Natl. Acad. Sci.

,

94

(

3

), pp.

849

–

854

.10.1073/pnas.94.3.849

Google Scholar

Crossref

21.

Kim

,

D.-H.

, and

Wirtz

,

D.

,

2015

, “

Cytoskeletal Tension Induces the Polarized Architecture of the Nucleus

,”

Biomaterials

,

48

, pp.

161

–

172

.10.1016/j.biomaterials.2015.01.023

Google Scholar

Crossref

PubMed

22.

Titushkin

,

I.

,

Sun

,

S.

,

Paul

,

A.

, and

Cho

,

M.

,

2013

, “

Control of Adipogenesis by Ezrin, Radixin and Moesin-Dependent Biomechanics Remodeling

,”

J. Biomech.

,

46

(

3

), pp.

521

–

526

.10.1016/j.jbiomech.2012.09.027

Google Scholar

Crossref

PubMed

23.

Mathieu

,

P. S.

, and

Loboa

,

E. G.

,

2012

, “

Cytoskeletal and Focal Adhesion Influences on Mesenchymal Stem Cell Shape, Mechanical Properties, and Differentiation Down Osteogenic, Adipogenic, and Chondrogenic Pathways

,”

Tissue Eng. Part B: Rev.

,

18

(

6

), pp.

436

–

444

.10.1089/ten.teb.2012.0014

Google Scholar

Crossref

PubMed

24.

Knoch

,

T. T.

,

Münkel

,

C. C.

, and

Langowski

,

J. J.

,

1998

, “

Three-Dimensional Organization of Chromosome Territories and the Human Cell Nucleus: About the Structure of a Self Replicating Nano Fabrication Site

,”

Proceedings of the Sixth Foresight Conference on Molecular Nanotechnology

, Santa Clara, CA, Nov. 12–15, pp. 1–24.https://repub.eur.nl/pub/78083/012-1998-TAK-SCDKFZ-Heidelberg-AbsKeyLitRef-150302.pdf

25.

Liebman

,

C.

,

McColloch

,

A.

,

Rabiei

,

M.

,

Bowling

,

A.

, and

Cho

,

M.

,

2020

, “

Mechanics of the Cell: Interaction Mechanisms and Mechanobiological Models

,”

Current Topics in Membranes

, Vol.

86

,

Elsevier

, Amsterdam, The Netherlands, pp.

143

–

184

.

Google Scholar

Crossref

26.

Chugh

,

P.

, and

Paluch

,

E. K.

,

2018

, “

The Actin Cortex at a Glance

,”

J. Cell Sci.

,

131

(

14

), p.

jcs186254

27.

Lundbæk

,

J. A.

,

Birn

,

P.

,

Girshman

,

J.

,

Hansen

,

A. J.

, and

Andersen

,

O. S.

,

1996

, “

Membrane Stiffness and Channel Function

,”

Biochemistry

,

35

(

12

), pp.

3825

–

3830

28.

Tinevez

,

J.-Y.

,

Schulze

,

U.

,

Salbreux

,

G.

,

Roensch

,

J.

,

Joanny

,

J.-F.

, and

Paluch

,

E.

,

2009

, “

Role of Cortical Tension in Bleb Growth

,”

Proc. Natl. Acad. Sci.

,

106

(

44

), pp.

18581

–

18586

.10.1073/pnas.0903353106

Google Scholar

Crossref

29.

Hawkins

,

T.

,

Mirigian

,

M.

,

Yasar

,

M. S.

, and

Ross

,

J. L.

,

2010

, “

Mechanics of Microtubules

,”

J. Biomech.

,

43

(

1

), pp.

23

–

30

.10.1016/j.jbiomech.2009.09.005

Google Scholar

Crossref

PubMed

30.

Gittes

,

F.

,

Mickey

,

B.

,

Nettleton

,

J.

, and

Howard

,

J.

,

1993

, “

Flexural Rigidity of Microtubules and Actin Filaments Measured From Thermal Fluctuations in Shape

,”

J. Cell Biol.

,

120

(

4

), pp.

923

–

934

.10.1083/jcb.120.4.923

Google Scholar

Crossref

PubMed

31.

Pawlowski

,

P. H.

,

2007

, “

Mechanokinetic Model of Cell Membrane: Theoretical Analysis of Plasmalemma Homeostasis, Growth and Division

,”

J. Theor. Biol.

,

249

(

1

), pp.

67

–

76

.10.1016/j.jtbi.2007.07.002

Google Scholar

Crossref

PubMed

32.

Sliogeryte

,

K.

,

Thorpe

,

S. D.

,

Lee

,

D. A.

,

Botto

,

L.

, and

Knight

,

M. M.

,

2014

, “

Stem Cell Differentiation Increases Membrane-Actin Adhesion Regulating Cell Blebability, Migration and Mechanics

,”

Sci. Rep.

,

4

(

1

), p.

7307

33.

Sugitate

,

T.

,

Kihara

,

T.

,

Liu

,

X.-Y.

, and

Miyake

,

J.

,

2009

, “

Mechanical Role of the Nucleus in a Cell in Terms of Elastic Modulus

,”

Curr. Appl. Phys.

,

9

(

4

), pp.

e291

–

e293

.10.1016/j.cap.2009.06.020

Google Scholar

Crossref

34.

Potolitsyna

,

E.

,

Pickering

,

S. H.

,

Bellanger

,

A.

,

Germier

,

T.

,

Collas

,

P.

, and

Briand

,

N.

,

2024

, “

Cytoskeletal Rearrangement Precedes Nucleolar Remodeling During Adipogenesis

,”

Commun. Biol.

,

7

(

1

), p.

458

.10.1038/s42003-024-06153-1

Google Scholar

Crossref

PubMed

35.

Kojima

,

H.

,

Ishijima

,

A.

, and

Yanagida

,

T.

,

1994

, “

Direct Measurement of Stiffness of Single Actin Filaments With and Without Tropomyosin by in Vitro Nanomanipulation

,”

Proc. Natl. Acad. Sci.

,

91

(

26

), pp.

12962

–

12966

.10.1073/pnas.91.26.12962

Google Scholar

Crossref

36.

Qi

,

Y.

,

Sun

,

L.

, and

Yang

,

H.

,

2017

, “

Lipid Droplet Growth and Adipocyte Development: Mechanistically Distinct Processes Connected by Phospholipids

,”

Biochim. Biophys. Acta (BBA)-Mol. Cell Biol. Lipids

,

1862

(

10

), pp.

1273

–

1283

.10.1016/j.bbalip.2017.06.016

37.

Szewc

,

K.

,

Pozorski

,

J.

, and

Minier

,

J.-P.

,

2012

, “

Analysis of the Incompressibility Constraint in the Smoothed Particle Hydrodynamics Method

,”

Int. J. Numer. Methods Eng.

,

92

(

4

), pp.

343

–

369

.10.1002/nme.4339

Google Scholar

Crossref

38.

Va

,

H.

,

Choi

,

M.-H.

, and

Hong

,

M.

,

2022

, “

Real-Time Volume Preserving Constraints for Volumetric Model on GPU

,”

Comput., Mater. Continua

,

73

(

1

), pp.

831

–

848

.10.32604/cmc.2022.029576

Google Scholar

Crossref

39.

Guy

,

A.

, and

Bowling

,

A.

,

2018

, “

A Multiscale Formulation for Reducing Computation Time in Atomistic Simulations

,”

ASME J. Comput. Nonlinear Dyn.

,

13

(

5

), p.

051002

.10.1115/1.4039489

Google Scholar

Crossref

40.

Haghshenas-Jaryani

,

M.

,

Black

,

B.

,

Ghaffari

,

S.

,

Drake

,

J.

,

Bowling

,

A.

, and

Mohanty

,

S.

,

2014

, “

Dynamics of Microscopic Objects in Optical Tweezers: Experimental Determination of Underdamped Regime and Numerical Simulation Using Multiscale Analysis

,”

Nonlinear Dyn.

,

76

(

2

), pp.

1013

–

1030

.10.1007/s11071-013-1185-0

Google Scholar

Crossref

41.

Joshi

,

V.

, and

Bowling

,

A.

,

2022

, “

Investigation of the Power Spectral Density of a Scaled Model Simulation of an Optical Tweezer

,” Proc. SPIE,

12198

, pp.

103

–

112

.10.1117/12.2633109

Google Scholar

Crossref

42.

Danesh

,

N.

,

Joshi

,

V. A.

,

Bowling

,

A.

,

Cho

,

M.

, and

Moon

,

H.

,

2023

, “

Dielectrophoretic Particle Focusing Using Axisymmetric Quadric Electrodes

,”

2023 IEEE Sensors

, Vienna, Austria, Oct. 29--Nov. 1, pp.

1

–

4

.10.1109/SENSORS56945.2023.10325193

Google Scholar

Crossref

43.

Joshi

,

V.

,

Danesh

,

N.

,

Moon

,

H.

,

Cho

,

M.

, and

Bowling

,

A.

,

2023

, “

Exploring Nanodynamics With Optical and Dielectrophoretic Trapping

,”

Proc. SPIE

,

12649

, pp.

115

–

124

.10.1117/12.2677485

44.

Moreno-Navarrete

,

J. M.

, and

Fernández-Real

,

J. M.

,

2012

, “

Adipocyte Differentiation

,”

Adipose Tissue Biology

,

M. E.

Symonds

, ed.,

Springer

,

New York

, pp.

17

–

38

.

45.

Bicknese

,

S.

,

Periasamy

,

N.

,

Shohet

,

S.

, and

Verkman

,

A.

,

1993

, “

Cytoplasmic Viscosity Near the Cell Plasma Membrane: Measurement by Evanescent Field Frequency-Domain Microfluorimetry

,”

Biophys. J.

,

65

(

3

), pp.

1272

–

1282

.10.1016/S0006-3495(93)81179-2

Google Scholar

Crossref

PubMed

Minimizing Computational Time for Long-Term Three-Dimensional Dynamic Simulation of Stem Cell Adipogenesis

Abstract

1 Introduction

2 Adipogenesis Test Case

3 Cellular Model

3.1 Generalized Coordinates.

3.2 Generalized Active Forces and Parameters.

3.2.1 Contact Forces.

3.2.2 Viscous Damping.

3.2.3 Combined Gravitational and Buoyant Forces.

3.2.4 Normal Forces.

3.2.5 Microfilament Forces.

3.2.6 Membrane Forces.

3.2.7 Volumetric Reaction Forces.

3.2.8 Forces Resulting From Changing Mass.

4 Scaling Approach

4.1 Previous Scaling Approach.

4.2 Methods for the Choice of Scaling Factor Values.

4.2.1 Choice of Scaling Factor Values for Double-Scaling Approach.

4.2.2 Choice of Scaling Factor Values for Triple-Scaling Approach.

4.2.3 Maintaining Damping Ratio of the Larger Body.

4.2.4 Maintaining Damping Ratio of the Smaller Body.

4.3 Examination of Different Scaling Approaches.

5 Simulation and Results

5.1 Cellular Model Stabilization.

5.2 Cytoskeletal Rearrangement.

5.3 Adipocyte Maturation.

5.4 Simulation Results.

6 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Minimizing Computational Time for Long-Term Three-Dimensional Dynamic Simulation of Stem Cell Adipogenesis Free

Abstract

1 Introduction

2 Adipogenesis Test Case

3 Cellular Model

3.1 Generalized Coordinates.

3.2 Generalized Active Forces and Parameters.

3.2.1 Contact Forces.

3.2.2 Viscous Damping.

3.2.3 Combined Gravitational and Buoyant Forces.

3.2.4 Normal Forces.

3.2.5 Microfilament Forces.

3.2.6 Membrane Forces.

3.2.7 Volumetric Reaction Forces.

3.2.8 Forces Resulting From Changing Mass.

4 Scaling Approach

4.1 Previous Scaling Approach.

4.2 Methods for the Choice of Scaling Factor Values.

4.2.1 Choice of Scaling Factor Values for Double-Scaling Approach.

4.2.2 Choice of Scaling Factor Values for Triple-Scaling Approach.

4.2.3 Maintaining Damping Ratio of the Larger Body.

4.2.4 Maintaining Damping Ratio of the Smaller Body.

4.3 Examination of Different Scaling Approaches.

5 Simulation and Results

5.1 Cellular Model Stabilization.

5.2 Cytoskeletal Rearrangement.

5.3 Adipocyte Maturation.

5.4 Simulation Results.

6 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only

Minimizing Computational Time for Long-Term Three-Dimensional Dynamic Simulation of Stem Cell Adipogenesis