Modeling cytokinesis of eukaryotic cells driven by the ... · tions. Thus, a detailed understanding...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. (2016); e02774Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2774

Modeling cytokinesis of eukaryotic cells driven by the actomyosincontractile ring

Jia Zhao1 and Qi Wang2,*,†

1Department of Mathematics, University of South Carolina, Columbia, SC 29028, USA2Department of Mathematics, Interdisciplinary Mathematics Institute and NanoCenter at USC, University of SouthCarolina, Columbia, SC 29028; School of Mathematical Sciences, Nankai University, Tianjin, China 300071; Beijing

Computational Science Research Center, Beijing, China 100193

SUMMARY

A three-dimensional (3D) hydrodynamic model for cytokinesis of eukaryotic cells is developed, in whichwe model dynamics of actomyosins in the cell cortex, in particular, along the cytokinetic ring formed in thecortex and in the neighborhood of the cell’s division plane explicitly. Specifically, the active force actuatedby the actomyosin’s activity along the cytokinetic ring is modeled by a surface force whose strength isproportional to the actomyosin concentration while the cell morphology is tracked by a phase field model.The model is then solved in 3D space and time using a finite difference method on graphic processing units.Dynamical morphological patterns of eukaryotic cells during cytokinesis are numerically simulated withthe model. These simulated morphological patterns agree quantitatively with experimental observations.Copyright © 2016 John Wiley & Sons, Ltd.

Received 11 August 2015; Revised 11 February 2016; Accepted 12 February 2016

KEY WORDS: cytokinesis; actomyosin; phase field model; 3D numerical simulations

1. INTRODUCTION

A cell is the fundamental unit in all living organisms because animals and plants are all made up ofcells of a wide range of varieties. The study of cells is therefore an essential part of research in lifesciences. Among many functions of a living cell, an important one is the cell’s ability to reproduce.Because of that, all living organisms can grow and continue their lives. Given its unique role playedin living organisms, cell study has been the focus of life science research for centuries. With theadvancement of experimental technologies today, more cell functions and micro-structural detailsthat regulate the functions have been uncovered, revealing an amazingly complex, microscopic uni-verse of a cell. Many cell functions are being investigated intensively in both theory and experiment.Cell reproduction is one of those functions.

In the cell reproductive cycle, a parent cell undergoes a sequence of intracellular transformationsand eventually divides into two or more offspring cells. For prokaryotic cells, the cell proliferationprocess is called binary division or binary fission. For eukaryotic cells, it is called cell mitotic pro-cess or mitosis. The late stage of the cell mitotic process for eukaryotic cells, after the nucleus hasbeen dissolved and chromosomes have been fully separated, is called cytokinesis.

Despite extensive studies on cells over the years, many important mechanisms in living cellsremain to be resolved in order to fully understand how a tiny cell works. It has been acknowledged

*Correspondence to: Wang Qi, Department of Mathematics, Interdisciplinary Mathematics Institute and NanoCenter atUSC, University of South Carolina, Columbia, SC 29028; School of Mathematical Sciences, Nankai University, Tianjin,China 300071; Beijing Computational Science Research Center, Beijing, China 100193.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd. (1 of 18) e02774

e02774 (2 of 18) J. ZHAO AND Q. WANG

Figure 1. Time line for cell mitosis.

that cell division of eukaryotes is determined by many interacting and highly coordinated mecha-nisms. Any single step goes wrong in a cell cycle may result in a catastrophe or failure, which maythen lead to an unwelcome consequence, for instance a cancer or other undesirable cell disfunc-tions. Thus, a detailed understanding on mitosis of eukaryotes, in particular, cytokinesis is essentialto understand many diseases associated with this cell type. For more details about eukaryotic cell’smitosis, readers are referred to the insightful review article on this topic in [1]. Readers can alsofind comprehensive review materials for cytokinesis of animal cells in [2, 3] and for cytokinesis ofbacteria in [4]. In addition, some works related to mechanical properties of cells during cytokinesissuch as material properties of cells and sources of stresses can be found in [5, 6]. The study pub-lished in [7] discussed the molecular requirements for cytokinesis, and the work in [8] addressedrecent advances in the mechanism of cytokinesis in animal, yeast, and plant cells.

Experimental observations have provide us with a basic phase diagram of cell mitosis (Figure 1).For eukaryotic cells, at the beginning of the mitotic process, the parent cell first duplicates its geneticsubstances and then forms a mitotic spindle consisting of microtubules [9]. Through a cascade of sig-naling processes [10], the actin and myosin molecules undergo a self-assembly process to remodelthe cell cortex, an intercellular layer rich in actin-filaments and myosin molecules located immedi-ately adjacent to the cell membrane [11]. In sync with the elongation of the mitotic spindle, moreactin and myosin molecules ascend to a ring like region in a plane roughly orthogonal to the axis ofthe mitotic spindle to form the cytokinetic ring or contractile ring in the plane transverse to the axisof the spindle. The plane is called the cleavage plane or division plane [12]. As more actomyosinmolecules are accumulated along the cytokinetic ring, a contracting force is generated, which pointsinward toward the axis of the spindle [11]. The contracting force pushes the membrane inward tocreate what is known as the cleavage furrow on the membrane [6]. The localized activation of thesmall GTPase Rho family of proteins at the cell division plane controls the position of the contrac-tile ring [13]. When Rho is specifically activated at the division plane within the cortex, it promotesactin polymerization and myosin-2 activation via Rho effector proteins. Rho-GTP promotes actinfilament assembly and myosin-2 assembly [14]. The contractile ring is a dynamic structure withinthe cortex, in which F-actin and myosin-2 are continuously assembled and disassembled to maintaina roughly constant actomyosin molecular concentration as well as a contracting force to squeeze thecell along the contractile ring. For eukaryotic cells, the positioning of contractile ring and cleavagefurrow is regulated by distribution of long astral microtubules [15]. Additional evidence has shownthat the cell geometry can influence the position of the cell division plane [16] and ultimately affectthe morphology of the offspring cells.

With the vast amount of data from experimental observations, theorists have been trying to comeup with testable hypothesis and models to decipher underlying mechanisms that control the mitoticprocess. In [17], Akiyama et al. proposed a mathematical model for cell cleavage for the sea urchinby considering chemotactic motion of the centrosomes. In [18], Poirier et al. developed a mechani-cal cell division model by a level-set approach, highlighting the furrow thinning trajectories. In [19],Rejniak et al. formulated an immersed boundary approach for modeling cell division focusing pri-marily on the mechanical aspect of cell division, which has been extended by [20] to study asingle axisymmetric cell growth and division. However, either in the immersed boundary or level-set approach, dynamics of actomyosin were not considered explicitly, instead artificial proxy forces

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2016); e02774DOI: 10.1002/cnm

MODELING AND SIMULATIONS OF CYTOKINESIS OF EUKARYOTIC CELLS (3 of 18) e02774

were employed whose origins were unknown. In [21], Turlier et al. proposed a mechanical modelfor cytokinesis, providing a physical explanation on the independence of the duration of cytokine-sis on the cell size in embryos. In [22, 23], the authors model the contractile ring induced by activeforces in the cell cortex. However, these models did not couple the dynamics of cell membrane to theunderlying cortex or cytoplasmic dynamics in full three-dimension (3D). Although they can predictthe contractile ring dynamics, they cannot do the stress or forces in the mother cell, the flow field ofcytoplasm nor the nucleus dynamics. Given the importance of cell mitosis, especially, cytokinesis,it is the time to develop a whole cell model in 3D space and time to study mechanical properties ofcells during their cytokinesis. First of all, the stress and force distributions in the mother cell duringits cytokinesis are instrumental to determine mechanical properties of the cell. In addition, the cou-pling between the actomyosin and the cell membrane is essential for cytokinesis. Thus, an artificialproxy force seems to be too simple and too naive to be used in a whole cell model for cytokinesis.

In this paper, we set out to develop a 3D hydrodynamic model to study the cell division pro-cess systematically by a phase field approach with a focus on cytokinesis. Phase field models havegained tremendous attention in the last few years, in particular, in the study of interfacial problemsin materials science and life science where complex morphological changes need to be resolved.The advantage of the phase field approach, compared with the traditional sharp interface method,including the immersed boundary method, is that the interface is naturally embedded in the modelformation such that it does not need to be tracked separately from the actual materials’ components.This can reduce the computational cost tremendously. The phase field model converges to its sharpinterface limit [24] as the thickness of the interface approaches zero. For cells, the coarse-grainedmembrane and other interfaces within a cell do not need to be very sharp, which makes the phasefield model a good choice for describing the multiphasic structure in a cell.

Our approach in modeling the cytokinetic process is to develop a mathematical model system-atically treating a cell as a multiphasic complex fluid system, where the cytoskeleton, cytoplasm,cell nucleus, and extra-cellular matrix are treated as separate components of a multiphasic, com-plex fluid mixture. Given the complexity of a cell, a viable model can only be built step-by-step byincorporating finer mechanochemical details incrementally as more detailed cellular mechanismsare uncovered and quantitatively described. The first important step in such a modeling endeavoris therefore to build a framework to simulate the morphological change of the cell membrane incytokinesis by highlighting the most dominant mechanism during the process. The most domi-nating factor that one has identified in cytokinesis is the existence of the contractile ring or thecytokinetic ring within the cortex in the neighborhood of the division plane, which is primarilyconsisted of F-actins and myosins (F-actin polymers inter-digitated by myosin molecules) calledcollectively as actomyosins in this paper, which are regulated by the Rho family of proteins. Mod-eling the detail of the cortex as well as the microtubule structures within the cell during cell mitosisremains our ultimate goal for building a mathematical model for the whole cell. However, the effec-tive force that squeezes the cell during cytokinesis is the contractile force exerted by the contractilering and the surface tension on the membrane. To highlight this mechanism in a simplistic way,we employ a proxy force in lieu of the contractile force generated by the actomyosin dynamicsin the cortex in the model and assume the strength of the active force is proportional to the con-centration of the actomyosins based on the experimental observation. We are fully committed todevelop a more sophisticated model for the cortical layer and the contractile ring within it in thenear future.

Given the complexity involved in the nucleus breaking up, migration of replicated chromosomesand reformation of offspring nuclei during cell mitosis, we employ a proxy process in the modelto grossly coarse-grain this complex dynamics. In this model, we assume the nucleus has alreadydivided, and new ones have reformed. In the next generation of models for cytokinesis, we wouldlike to replace it by an active polymer model for the spindle formation.

The governing system of equations in the model then consists of the transport equation for eachphase variable in the form of volume fractions, continuity equation, and the momentum balanceequation for the multiphasic material system. The governing partial differential equations are dis-cretized and solved using a finite difference method on graphic processing units (GPUs) in full 3D



space and time. Numerical simulations show cytokinetic morphologies that compare well with theexperimental observations qualitatively.

We organize the paper into three sections. In Section 2, we present the multi-phase hydrodynamicmodel for the whole cell in a phase field formulation. It is followed by a brief discussion on thenumerical approximation of the governing system of equations. Finally, we present some numericalsimulations and discuss the comparison with available experimental observations for cytokinesisof eukaryotes.

2. MATHEMATICAL MODEL FORMULATION

We formulate a one fluid multi-component model to study cytokinesis of an eukaryotic cell. Inthis model, we treat the cell, cell nucleus along with the extra cellular matrix (ECM) as a three-component fluid mixture in a gross approximation. In the following, we first introduce the notationsused in the model, and then, describe its various components. Because we are modeling cytokine-sis whose initial configuration is not clearly identified internally, we refer the nucleus in our modeleither as the cloud of chromosomes together with the necessary amount of cytoplasmic materi-als to be included in the new nucleus or the actual nucleus after it forms. This allows us to trackchromosome dynamics in the model.

2.1. Notations

Consider a fluid mixture in domain ! with three bulk phases: cytoplasm, nucleus and, ECM orbuffer fluid, whose volume fractions are denoted by "1, "2, and "3, respectively. We assume thefluid mixture is incompressible so that

3XiD1

"i D 1; 0 6 "i 6 1; i D 1; 2; 3: (1)

To illustrate the idea, we show a two-dimensional (2D) schematic cartoon in Figure 2. This is clearlyshown to be a globally three phase but locally two phase model. We use vi ; #i to represent thevelocity and viscosity for each phase, i D 1, 2, 3, respectively. Then, the volume averaged velocityv and viscosity # are given by

v D3XiD1

"ivi ; # D3XiD1

"i#i : (2)

Figure 2. A schematic cartoon for the phase field model, where ! represents the computational domain.Here, "1;"2;"3 represent the volume fractions of the cytoplasm, nucleus, and cell buffer, respectively. Thenucleus membrane is traced by the level set ¹"2 D 1

2;"3 D 0º D ¹"1 D 1

2;"3 D 0º, and the cell membrane

is traced by the level set ¹"3 D 12;"2 D 0º D ¹"1 D 1

2;"2 D 0º, respectively.



For interfaces defined by "i D 1=2; i D 1; 2, their unit normal vectors are denoted by n1,n2, and the mean curvatures of the interfaces are denoted by $1; $2, respectively, which can beapproximated [25] by

ni D "r"ijr"i j

; $i D r # ni $1

jr"i j

!r2"i C

2

"2"i .1 " "i /.2"i " 1/

"; i D 1; 2; (3)

with " representing the thickness of the interfaces in steady state. We note that " is a modelparameter.

Note that "1 is the volume fraction of the cytoplasm. ¹1=2 < "1 6 1º represents the cytoplasmicregion, and ¹"1 D 1=2;"2 D 0º (or equivalently ¹"3 D 1=2;"2 D 0º) represents the cell membranesurface. So, n1 is the external normal of the membrane. ¹1=2 < "2 6 1º represents the two separatedregions of nuclei, and ¹1=2 < "3 6 1º represents the region of the buffer.

For eukaryotic cells, cytokinesis begins with the formation of the cytokinetic ring, consisting ofactomyosin polymeric networks. The exact location of the cytokinetic ring is believed to be deter-mined by the chromosome spindle during the cell mitotic process. The cytokinetic ring constrictswhile maintaining roughly a constant density during the final stage of the mitotic process. If thespatial distribution of the actomyosin concentration is not homogeneous, neither is the contract-ing stress on the contract ring, which can lead to an asymmetric cell division. In order to modelsuch phenomenon, we use the concentration of actomyosins in the cell, neglecting the microstruc-ture arrangement of the actomyosin polymers. We use "a to denote the actomyosin concentrationneglecting its mass and volume in the model for the sake of simplicity.

In the model, we use c1; c2 to represent the position of the two centrosomes at each pole of thecell, respectively, where we assume the poles are located at the opposite end of the cell in this model.Given c1; c2, if we denote

c1 C c22

D .x0; y0; ´0/ D x0;c1 " c22

D .x1; y1; ´1/; (4)

the division plane is defined as follows:

L D ¹.x; y; ´/ j .x1; y1; ´1/ # .x " x0; y " y0; ´ " ´0/ D 0º : (5)

Then, the distance from any point in the cell to the division planeL can be calculated by the formula

d.x/ D minQx2Lkx " Qxk2: (6)

We next introduce the free energy for the fluid mixture system.

2.2. Free energies

The total thermodynamic free energy for this mixture system consists of two parts:

F D Fb C Fa; (7)

where Fb is the free energy due to the interaction among the different phases and Fa is the freeenergy due to the actomyosin activity.

For the interaction among the different fluid phases, we adopt the modified Ginzburg–Landaudouble-well mixing free energy coupled with the conformational entropy, given by

Fb DZ!

´%

"3XiD1

!"

2kr"ik2 C

1

""2i .1 " "i /2

"#C %0

2""22"

23

µdx; (8)

where % is the interfacial tension, %0 > 0 is a relatively large constant to penalize the co-existence ofphase "2 and "3 in case they become adjacent to each other, and " is a model parameter proportionalto the thickness of the interface. Notice that the choice of interfacial energy is not unique, one mightpropose a more general case, say



Fb DZ!

´%

""

2

3XiD1kr"ik2 C

1

"

#"21"

22 C "21"23

$#C %0

2""22"

23

µdx: (9)

We test both and find that they give the same qualitative result. In the following, we present theresults using (8).

For Fa, we propose a chemo-attractive potential for the actomyosin towards the cytoplasm

Fa DZ!

%a"a."22 C "23/dx; (10)

where %a controls chemo-attractive strength. Namely, this potential repels the actomyosin fromthe buffer fluid phase and the nucleus phase. As the result, the actomyosin is attracted to thecytoplasmic phase.

We further define the chemical potentials &i D ıFı"i

, i D 1; 2; 3, and &a D ıFı"a

for each fluidphase and the actomyosin. Given the specific form of the free energy in (8), we list the chemicalpotentials as follows:

&1 D %!""r2"1 C

2

""1.1 " "1/.1 " 2"1/

";

&2 D %!""r2"2 C

2

""2.1 " "2/.1 " 2"2/C

%0

""2"

23 C 2%a"a"2

";

&3 D %!""r2"3 C

2

""3.1 " "3/.1 " 2"3/C

%0

""22"3 C 2%a"a"3

";

&a D %a#"22 C "23

$:

(11)

2.3. Transport equations for the phase variables

The transport equation for the volume fraction or the phase variable of each phase is formulatedas follows:

@t"i Cr # .v"i / D r #

0@

3XjD1

˛ijr&j

1AC gi ; i D 1; 2; 3; (12)

where .˛ij / is the mobility matrix and gi is the reactive term for phase i. Note that the volumefractions add up to 1:

P3iD1 "i D 1 for incompressible mixtures. This along with the Onsager

reciprocal relation implies that

3XjD1

˛ij D 0; ˛ij D ˛j i : (13)

The off-diagonal mobility coefficients can be obtained from the diagonal coefficients uniquely

˛12 D1

2.˛33 " ˛11 " ˛22/; ˛13 D

1

2.˛22 " ˛11 " ˛33/; ˛23 D

1

2.˛11 " ˛22 " ˛33/: (14)

In this model, we assume the diagonal motility parameters have the following forms:

˛i i D '"i .1 " "i /; i D 1; 2; 3; (15)

with ' represents the strength of the motility parameter. This choice of mobility coefficients rendersthe transport equations for the volume fractions the singular Cahn–Hilliard type.

It follows from the incompressibility condition that

3XiD1

gi D 0: (16)



In the early stage of cell mitosis, the parent cell increases in the cytoplasmic and organelle volume(the G1 phase) as well as increases in genetic materials (the G2 phase) right before the replicationduring the S phase. This involves the change of cell size and shape. This process can be modeledphenomenologically by the following reactive kinetics:

8̂<:̂g1 D c1"1"3 " c2"1"2;g2 D c2"1"2;

g3 D "c1"1"3:(17)

In this paper, we focus on dynamics of cytokinesis exclusively. So, we assume the G1 and G2 phasehave already completed, that is, the cytoplasm and genetic substance have already been doubled, bychoosing c1 D c2 D 0. From now on, we assume the volume of the cell will not go through anychanges, that is, the total volume is conserved during cytokinesis.

2.4. Approximation of chromosome dynamics

We approximate chromosome dynamics during the mitotic process using a simplified strategybypassing the delicate details about chromosome splitting, microtubile spindle formation, and thereformation of offspring nuclei. Because chromosome dynamics during the mitotic process beforethe reformation of offspring nuclei is such a complex process, its modeling involves delicate localdynamics and detailed chromosome structures, which warrant a whole new set of studies. We willhave it added incrementally in our systematic framework in the future as our modeling endeavorprogresses. However, in this paper, we stay focused on the cytokinetic process, in which the mem-brane of the parent nucleus has already been broken and chromosomes are well-separated near thepolarized centrosomes. At the onset of cytokinetic process, the chromosomes are not yet boundedby nucleus walls. But soon after, the offspring nuclei are about to be reformed. Technically, we usephase "2 D 1 to represent the offspring nuclei as well as the material volumes occupied by the chro-mosomes together with the necessary cytoplasmic materials to be wrapped in the nuclei after theyform. We therefore refer to the chromosome and its neighboring materials to be included in the newnucleus the chromosome cloud.

2.5. Transport equation for the actomyosin concentration

For eukaryotic cells, cytokinesis begins with formation of the cytokinetic ring, consisting of acto-myosin polymers. The cytokinetic ring constricts while maintaining roughly a constant densityduring cytokinesis, which is accomplished by a constant remodeling process. This remodeling pro-cess is characterized by polymerization of F-actins near the membrane and depolymerization inthe cytoplasm.

In this paper, we denote the concentration of actomyosins by "a. We propose that actomyosinsare convected and diffused near the cell membrane due to the gradient of chemical potential with amobility proportional to its concentration. Thus, the transport equation for "a is proposed as follows:

@t"a Cr # .v"1"3"a/ D r # .'4r&a/C caı.x; x0/"1"3."a0 " "a/ " cb"a; (18)

where '4 is the motility parameter for actomyosin and the last two terms are the reactive terms,describing polymerization kinetics near the membrane and depolymerization kinetics in the cyto-plasm for actomyosins. To describe various transport effects, we propose the motility parameteras follows:

'4 D 'a"1"3S."a."a0 " "a//; (19)

where 'a is the motility strength parameter, "1"3 is to restrict the diffusion near the cell membrane,S.x/ is the smoothed function, such that S.x/ D max.x; 0/. Here, "a."a0 " "a/ is due to thejamming effect (i.e., when the concentration of actomyosins reaches a maximum level "a0, no acto-myosin transport can take place in the cortex and the maximal transport is achieved at a moderate



value of actomyosin concentration.). In this model, the cortex is limited to the region where "a > 0.The domain of the cortex is determined by the nonzero value of "a, which depends strongly on theinitial condition. Reactive dynamics in this model are implemented by the second and third termson the right-hand side in the equation, respectively, where ı.x; x0/ is a smoothed-delta function. Itemphasizes the enhanced polymerization at the cytokinetic ring during cytokinesis. In the numericalstudy, we use

ı.x; x0/ D1

(p2)e!d.x/22!2 ; S.x/ D

8<:

x; x > ""2

#1C tanh x!0:5"

0:01"

$; 0 < x < "0; x 6 0

(20)

where d.x/ is the distance function to the cytokinetic ring centered at x0. This factor ı.x0/"1"3for actomyosin remodeling represents the preference of actomyosins in the cytokinetic ring on thedivision plane near the cell membrane. The term ."a0 " "a/ provides a mechanism for actomyosinto grow with a maximum threshold value "a0. The two parameters ca and cb represent the strengthof polymerization and depolymerization process of actomyosins, respectively. The decay rate cb isassumed a constant in this paper.

2.6. Mass and momentum transport equation

In this model, we assume the fluid mixture is incompressible and very viscous. So, the massand momentum conservation is given respectively by the Stokes equation and the divergencefree condition:

² "rp Cr # .2#D/C Fe D 0;r # v D 0; (21)

with # the volume averaged viscosity defined in (2), and Fe the interfacial force,

Fe D Fs C Fc ; (22)

where Fs represents the interfacial force due to material’s inhomogeneity and Fc represents thecontractile force induced by actomyosin activities in the cortex, given by

Fs D3XiD1

&ir"i ; (23)

Fc D "4"1"3%d1"a

"a0$1"1n1; (24)

where %d1 controls the strength of the actomyosin induced contractile force, and 4"1"3 confines thecontractile force to the cell membrane. We note there we assume this contractile force is proportionalto "a linearly for simplicity. More complex dependence of the contractile fore on myosin and actinconcentration [26, 27] would be thoroughly studied in our later research.

2.7. Boundary conditions

We put the cell undergoing mitosis in a large enough box and use periodic boundary conditions inboth the x and z direction. In the y direction, we propose the no-flux boundary condition for thephase variables, that is, if we denote n the unit normal at the boundary in the y direction,

r"i # njyD0;Ly D 0; r&i # njyD0;Ly D 0; i D 1; 2; 3; r"a # njyD0;Ly D 0: (25)

For the velocity field v D .u; v; w/, we use the no-slip boundary condition,

vjyD0;Ly D 0: (26)



2.8. Non-dimensionalization and dimensionless governing equations

Here, we introduce a characteristic length scale h D 4 % 10!5 m, time scale t0 D 10s, densityscale *0 D 1:0 % 103 kg/m3, as well as the characteristic actomyosin concentration "a0, to non-dimensionalize the following variables:

Qt D t

t0; Qx D x

h; Q"a D

"a

"a0: (27)

Then, the other variables and parameters are non-dimensionalized as follows:

Qv D t0vh; Qp D pt20

*0h2;1

ReiD #i t0

h2*0; Q'i D

'i*0

t0; Q$j D $jh; i D 1; 2; 3; j D 1; 2;

Q% D % t20*0h3

; Q%0 D%0t

20

*0h2; Qci D

cih; i D 1; 2; Q'a D

'a"3a0

t0;

Q%d1 D%d1t

20

h2*0; Qca D

cat0

h; Qcb D cbt0; Q+d D

+dh; Q( D (

h; Q" D "

h:

(28)

We will drop the Q& in the following. The governing equations in dimensionless form aresummarized below:

8̂ˆ̂̂̂ˆ̂<ˆ̂̂̂ˆ̂̂:

"rp Cr #&P3

iD11Rei"i .rvCrvT /

'C

3PiD1

&ir"i " 4%d1"21"3"a$1n1 D 0;r # v D 0;

@t"i Cr # .v"i / D r #

3PjD1

˛ijr&i!C gi ; ˛i i D 'i"i .1 " "i /; i D 1; 2; 3;

@t"a Cr # .v"1"3"a/ D r # .'4r&a/C caı.x; x0/"1"3.1 " "a/ " cb"a:

(29)

3. NUMERICAL METHODS

For the coupled equations in this hydrodynamic model, we use extrapolation to decouple themwhen discretizing in time. That is, for each time step, we solve the momentum equation first usingthe extrapolated value for various other components. Then, with the updated velocity, we solve thetransport equations for the volume fractions and the actomyosin concentration one by one. We notethat, in the following, any variable with an overline .&/nC1 represents a second-order extrapolationfrom the nth and the n " 1th step, for instance, vnC1 D 2vn " vn!1.

3.1. Numerical schemes

The numerical scheme is summarized here. Give the initial profile#"01 ;"

02 ;"

03 ;"

0a ; v

0 D 0 ,s0 D p0 D 0

$, and for simplicity of notations, assume "!11 D"01 , "!12 D"02 , "!13 D"03 , "!1a D"0a ,

v!1 D v0, s!1 D s0, and p!1 D p0. After having calculated#"n1 ;"

n2 ;"

n3 ;"

na ; v

n; sn; pn$, n > 1,

we obtain#"nC11 ;"nC12 ;"nC13 ;"nC1a ; vnC1; snC1; pnC1

$by the following steps:

1. Predict unC1:8̂<̂ˆ̂:

C#unC1 " vnC1

$C 1

Rearsn Crpn " 1

Rear2unC1 D

r #&

1RenC1

&rvnC1 CrvnC1

T''C F

nC1e " 1

Rear2vnC1;

unC1jyD0;Ly D 0;

(30)

where C is a stabilizing term [28] (choosen as C D 1 in this paper), and 1RenC1 is the volume-

averaged Renolds number, Rea is the maximum Reynolds number, that is,



Figure 3. Variable locations on the three-dimensional staggered grid.

Figure 4. Two-dimensional schematic of mapping from graphic processing unit threads to mesh elements.This figure shows a two-dimensional schematic on how each threads in graphic processing unit implemen-tation maps into a respective mesh element. Here, we have 16 % 16 uniform mesh and claim 4 % 4 blocks in

a grid, where each block contains four threads.

1

ReaD min

!1

Re1;1

Re2;1

Re3

";

1

RenC1D

3XiD1

1

Rei"nC1i ; (31)

with Rei , i D 1; 2; 3; are the Reynolds number for the cytoplasm, nucleus, and ECM,respectively.

2. Project unC1:´

", nC1 D r #unC1;@ nC1@n jyD0;Ly D 0:

(32)

3. Update#vnC1; snC1; pnC1

$:8̂<:̂

vnC1 D unC1 Cr nC1;snC1 D sn " r # unC1;pnC1 D pn " C nC1 C 1

ReasnC1:

(33)

4. Update#"nC12 ;"nC13

$:

8̂<̂ˆ̂:

3"nC1i!4"n

iC"n!1

i

2#t Cr #&

vnC1"nC1i

'D r #

"3PjD2

&!nC1ij r"nC1j C ˇnC1ij rr2"nC1j

'#;

r"nC1i # n D 0; rr2"nC1i # n D 0; i D 2; 3;(34)



Table I. Dimensional parameters.

Symbol Description Value Unit Reference

#1, #3 Dynamic viscosity of cytoplasm and ECM 102 Pa # s [18]#2 Dynamics viscosity of nucleus 2 % 102 Pa # s Approximation%; %0 Surface tension strength 5:0 % 10!5 N=m [31]%a Strength for hard core interaction 2:5 % 10!8 m2s!2 Approximationca Assembly rate of actomyosin 0.1 s!1 [32]cb Disassembly rate of actomyosin 0.2 s!1 [32]' Motility parameter for each component 2 % 10!15 kg!1m3s'a Motility parameter for actomyosin 1:64 % 103 kg!3m9s%d1 Stimulating strength for cytokinesis 7:5 % 10!9 kgm!2s!2

ECM, extra cellular matrix.

Figure 5. Three-dimensional view of cell morphology at different stages of cytokinesis. The pink surfaceshows the cell membrane, and the yellow spheres show the nucleus. The parent cell splits into two off-spring cells. After the splitting, the offspring cells round up under the influence of surface tension. (a) Themorphology of the parent cell at the onset of cytokinesis. (b–d) The morphology of the parent cell duringcytokinesis. (e) The morphology of the dividing cell near the end of cytokinesis. (f) The morphology of the

two newly created offspring cells at the end of cytokinesis.

where !ij and ˇij are functions of "i ;"j obtained from (12), whose specific forms areomitted here.

5. Update "nC11 :

"nC11 D 1 " "nC12 " "nC13 : (35)

6. Update "nC1a :8̂ˆ̂̂<ˆ̂̂̂:

".1/a D "na C,t

#L#"na ; v

na

$C caı.x0/"nC11 "nC13

#1 " "na

$" cb"na

$;

".2/a D 3

4"naC 1

4".1/a C 1

4,t&L&".1/a ; vnC1a

'Ccaı.x0/"nC11 "nC13

&1 " ".1/a

'" cb".1/a

';

"nC1a D 13"

naC 2

3".2/a C 2

3,t

!L

!".2/a ;

vnaCvnC1a

2

"Ccaı.x0/"nC11 "nC13

&1"".2/a

'"cb".2/a

";

(36)



where L."a; va/ $ "r # .va"a/ is the advection term evaluated by the third-order WENOscheme [29], with

vna D "n1"n3#vn " 'a

#1 " "na

$ #%a"

n2r"n2 C %a3"n3r"n3

$$;

vnC1a D "nC11 "nC13

&vnC1 " 'a

&1 " "nC1a

' #%a"

nC12 r"nC12 C %a3"nC13 r"nC13

$':

(37)

Figure 6. Dynamics of ring contraction. (a) Red circles are experimental data from [33], indicating theaverage of eight experimentally observed ring radius as a function of time. The blue line is the prediction bythe model in this paper; (b) dynamics of distance between two cell poles .kc1 " c2k/ in different time slots.

Figure 7. Two-dimensional cross-sectional view (at ´ D 0:5) of the cell morphology during the numericalsimulation of cytokinesis at various time slots. The red line is the cell membrane and the blue line is thenucleus membrane. (a) Initial cell morphology together with the nucleus. (b)–(e) Cell morphology and the

nuclei during the cytokinetic process. (f) Off-spring cell morphologies and the nuclei after cytokinesis.



Figure 8. A comparison between experimental observations and our numerical simulations: (a) Purple urchinzygotes during first mitosis, fixed and stained for DNA (green) and phosphorylated myosin II (magenta;single confocal sections). Top row: interphase, metaphase, and anaphase. Bottom row: early, middle, andlate telophase from [34]. (b) Numerical predictions of cell mitotic dynamics: here, red curve represents cellmembrane, green curve represents the nucleus membrane, and purple curve represents the contour of the

actomyosin distribution.

Remark 1When solving the Stokes equation, we use the Gauge–Uzawa splitting strategy [30]. If we addEquations (30),(32), and (33), the scheme is

C#vnC1"vnC1

$CrpnC1C 1

Rea,vnC1Dr #

!1

RenC1

&rvnC1CrvnC1

T'"CF

nC1e " 1

Rear2vnC1;

(38)

which is second-order accuracy in time. In (30), sn, snC1 are the Gauge variables [30] introduced inthe scheme in order to solve the Stokes equation. Numerically, to avoid the singularity, we computethe unit normal by



ni $ "r"ip

kr"ik2 C "0; i D 1; 2; (39)

where we choose "0 D 10!4.

3.2. Spacial discretization and graphic processing unit implementation

For the spatial operators in the scheme, we discretize them using second-order central finite differ-ence method over a uniform spatial grid, where the velocity field is descretized at the center of themesh surface, and pressure p, phase variables "1, "2, "3 and actomyosin "a are discretized at thecenter, as shown in Figure 3. The boundary conditions are handled by ghost cells.

The fully discretized schemes are implemented on GPUs (graphics processing units) in 3D spacefor high-performance computing. To better utilize the performance of GPUs, we store all variablesin the global memory and store all parameters and mesh information (whivh do not change thesimulation) in the constant memory, such that it reduces the latency of data access.

One advantage of using GPUs is their virtual allocations of processors (we can claim as manythread as we desire, even if it is beyond the existing number of multiprocessors in the physicaldevice). Therefore, in our implementation, we allocate as many processors as the degrees of free-dom. A schematic is shown in Figure 4, where we have 16 % 16 uniform mesh, so we claim 4 % 4

Figure 9. Two-dimensional and three-dimensional view of actomyosin concentration at various stages ofcytokinesis. (a–f) 3D view of the contractile ring at five different time slots; (g–j) two-dimensional view of

contractile ring at x D 0:75 is shown with respect to the first four selected time slots.



blocks with each block containing four threads. In this situation, there is a one-to-one map from thethread to the mesh point, such that each thread calculates the respective component for the matrixvector multiplication. This strategy turns out to be very effective.

4. NUMERICAL RESULTS AND DISCUSSION

We use the numerical solver developed based on the scheme described in the previous section tosimulate cytokinesis of eukaryotic cells in 3D space and time. The computational domain is set atŒ0 1:5- % Œ0 1- % Œ0 1-. The initial cell radius is set at 0.3, and nucleus radius is 0.08, whichare 15&m and 0:4 &m in dimensional form, respectively. The parameters used in this model aresummarized in Table I , which are obtained either from the literature or based on our best guesses.In addition, we set the model parameters " D "d D 0:01 h and ( D 0:04 h. Unless stated otherwise,we will use this set of parameters in the simulations presented in the succeeding text.

We present a 3D numerical result about cytokinesis of an eukaryotic cell in Figure 5 in a seriesof selected time slots, highlighting representative morphological changes of the cell at differentstages. The simulation starts when the cleavage furrow near the division plane starts forming (shownin Figure 5(a).) In Figure 5(c,d), a bridge forms between the two departing offspring cells. Theabscission that physically splits the parent cell into two offspring cells takes place after t D 240 s.In Figure 5(f), we observe the two newly generated offspring cells reshape into the rounded mor-phology under the influence of surface tension in their own membranes. This numerical simulationin full 3D uses the hydrodynamic phase field model accounting for actomyosin dynamics withoutassuming axisymmetry in the cell. This sets the study apart from the available ones.

Figure 10. Hydrodynamic quantities during cytokinesis. This figure shows the hydrodynamic stress . D2#D and cytoplasm velocity field "1v, at a few selected time slots. (a–b) three-dimensional and zoomedview of the stress tensor . at t D 120 s is shown as ellipsoids; the stress is large in the cytokinetic ring. Theunit is Pa. As the membrane retreat at the cytokinetic ring, it drags the fluid to the axis of the spindle in theneighborhood of the division plane while pushing the cytoplasmic fluid and fluid in the ECM near the polesoutward. (c–e) Two-dimensional slices (´ D 0:5) of cytosol velocity field "1v at time t D 86; 120; 240 s,

respectively, are shown, with unit &m=s.



This simulation compares well quantitatively with the experimental observation from [33] ondynamics of cytokinetic ring contraction, as shown in Figure 6(a). The distance between two polesof the splitting cell kc1 " c2k is also shown in Figure 6(b).

To better visualize the morphological change of the cell membrane as well as the nucleus duringthe cell mitotic process, a series of 2D cross-sectional slices at x D 0:5 are plotted in Figure 7. Inparticular, the dynamics of nuclei in the offspring cells are shown explicitly.

A comparison with experimental observations of cytokinetic processes with actomyosin distribu-tion is shown in Figure 8. Our numerical results agree qualitatively well with the morphogeneticpatterns obtained from the experimental observations shown in Figure 8(a). In particular, the modelprediction of actomyosin dynamics in the contractile ring shows qualitatively agreement as well.

The new feature in the hydrodynamic model lies in the integration of actomyosin dynamics. Itenables us to show the accumulation of actomyosins along the cytokinetic ring during cytokinesis aswell as the decisive role played by the active cytoskeletal structure. Figure 9 depicts the actomyosinconcentration at various stages of cell division, in which the 2D slices show the actomyosin con-centration in colored maps. The comparison shows that actomyosins are indeed concentrated in thecytokinetic ring and play the instrumental role in forcing and later abscising the membrane to cre-ate two offspring cells. In Figure 9, the cytokinetic ring rich in actomyosins is visualized in 3D andits cross-section in 2D in a time series during the cytokinetic process. Specifically, it shows the ringwidens and eventually splits into two separate islands inside each offspring cell after cytokinesis.The 2D plots show the projection of the cytokinetic ring onto the division plane.

We note that the advantage of the 3D hydrodynamic model is its capability to couple the interiorcytoplasmic fluid flow with the ECM or the ambient fluid through the cell membrane and to predictimportant hydrodynamic quantities such as stress tensors, normal stress differences, forces, and the

Figure 11. The external force distribution. This figure shows the interfacial force (Equation (22)) distribu-tions during cytokinesis. The unit is &N . (a–c) shows two-dimensional slice (´ D 0:5) of the interfacialforce Fs at time t D 50; 100, and 150 s, respectively. The force smooths out the cell surface; (d–f) showstwo-dimensional slice (´ D 0:5) of the contractile force Fc at time t D 50; 100 and 150 s, respectively. Inparticular, in (g–j) at the divisoin plane (x D 0:75), the contractile force Fc and surface tension force Fs at

time 50 and 100 s are shown, respectively.



hydrostatic pressure throughout the domain during the cellular morphological transformation. InFigure 10, the detail of the viscous stress 2#D, average velocity v at different times, is depicted,respectively. The stress tensor at .x; t / is visualized as a 3D ellipsoidal object centered at x, whosesemiaxes signify the length of the three eigenvalues of the stress tensor. From Figure 10(b), a zoomedview of the stress tensor is shown. The stress is highly inhomogeneous in space near the contractilering on the division plane, which correlates well with the velocity field shown in Figure 10(c–e), asthe cytokinetic ring of the cell is contracting. The velocity field correlates well with the motion ofthe cell components during this dynamical process.

In addition, the distribution of the active force due to actomyosin activities and the surface tensionforce are shown in Figure 11, respectively. In Figure 11(d–f), the active forces are mainly distributedin the cortex with much higher values on the division plane than in other places, constricting the cellmembrane towards the long axis of the cell (or the axis of the spindle). This is clearly shown in the2D view of the active force in Figure 11(g–h). As it should be, the active force is distributed on theinterfaces near the cytokinetic ring: Whereas surface tension is more evenly distributed along theinterfaces, aiming to smooth them out. These two forces oppose to each other on the interface duringthe cytokinetic process. It is the competition between the surface tension and the active force onthe cytokinetic ring that ultimately contributes to the complete of cytokinesis. Figure 11(a–c),(i–j)depicts the surface tension forces respectively from two cross-sections.

5. CONCLUSIONS

In this paper, we develop a multi-phase complex fluid model to study cytokinesis of an eukaryoticcell. Actomyosin dynamics in the cortex is modeled using its concentration coupled completelywith hydrodynamics of the entire cell and its surrounding fluid. Dynamical morphological patternsof eukaryotic cells during cytokinesis are numerically simulated with the model. These simulatedpatterns agree quantitatively with experimental observations. In addition, hydrodynamic variablessuch as the velocity, hydrostatic pressure, extra stress, and interfacial force field during cytokinesiscan be predicted in the simulations. Although some of these measurements are not yet availablefor cells. Nevertheless, the model is capable of providing the baseline data to be compared withexperiments once they become available.

Cell mitosis is a complex process governed by many additional mechano-chemical factors thatare not included in the model. Nevertheless, the new model and the accompanying 3D compu-tational tool mark a step forward in developing a full 3D hydrodynamic model for cell mitosisin a hydrodynamically and thermodynamically consistent way. With the modeling framework andnumerical tools for cell mitotic dynamics„ we can further incorporate additional cellular features oradd additional cellular components/microstructures to it.

ACKNOWLEDGEMENTS

Jia Zhao is partially supported by a Dissertation Fellowship from the Office of the Provost at the Universityof South Carolina. Qi Wang is partially supported by AFOSR, NSF, and NIH through awards FA9550-12-1-0178, DMS-1200487, DMS-1517347, and 2R01GM078994-05A1 as well as a SC EPSCOR GEAR Award.The authors also thank the anonymous reviewers for their suggestive and insightful reviews, which helpedus to improve the quality of the paper.

REFERENCES

1. Scholey JM, Brust-Mascher I, Mogilner A. Cell division. Nature 2003; 422:746–752.2. Eggert US, Mitchison TJ, Field CM. Animal cytokinesis: from parts list to mechanisms. The Annual Review of

Biochemistry 2006; 75:543–566.3. Guertin DA, Trautmann S, McCollum D. Cytokinesis in eukaryotes. Microbiology and Molecular Biology Reviews

2002; 66(2):155–178.4. Errington J, Daniel RA, Scheffers D. Cytokinesis in bacteria. Microbiology and Molecular Biology Reviews 2003;

67(1):52–65.5. Mohan K, IgIesias PA, Robinson DN. Separation anxiety: stress tension and cytokinesis. Experimental Cell Research

2012; 318:1428–1434.



6. Rechl EM, Effler JC, Robinson DN. The stress and strain of cytokinesis. Trends in Microbiology 2005; 15(4):200–206.

7. Glotzer M. The molecular requirements for cytokinesis. Science 2005; 307:1735–1739.8. Barr FA, Gruneberg U. Cytokinesis: placing and making the final cut. Cell 2007; 131:847–860.9. Mogilner A, Wollman R, Civelekoglu-Scholey G, Scholey JM. Modeling mitosis. Trends in Microbiology 2006;

16(2):88–96.10. Nigg EA. Mitotic kinases as regulators of cell division and its checkpoints. Nature Reviews Molecular Cell Biology

2001; 2:21–32.11. Miller AL. The contractile ring. Current Biology 2011; 21(24):976–978.12. Canman JC, Cameron LA, Maddox PS, Straight A, Tirnauer J, Mitchison T, Fang G, Kapoor T, Salmon E.

Determining the position of the cell division plane. Nature 2003; 424:1074–1078.13. Somers WG, Saint R. A rhogef and rho family grpase-activating protein compelx links the contractile ring to cortical

microtubules at the onset of cytokinesis. Developmental Cell 2003; 4:29–39.14. Chircop M. Rho GTPases as regulators of mitosis and cytokinesis in mammalian cells. 2014 Special Focus on Rho

GTPases, February 2014; e29770. DOI: 10.4161/sgtp.29770.15. Rankin KE, Wordeman L. Long astral microtubules uncouple mitotic spindles from the cytokinetic furrow. The

Journal of Cell Biology 2010; 190(1):35–43.16. Minc N, Burgess D, Chang F. Influence of cell geometry on division-plane positioning. Cell 2011; 144:414–426.17. Akiyama M, Tero A, Kobayashi R. A mathematical model of cleavage. Journal of Theoretical Biology 2010; 264:

84–94.18. Poirier CC, Ng WP, Robinson DN, IgIesias PA. Deconvolution of the cellular force-generating subsystems that

govern cytokinesis furrow ingression. PLOS Computational Biology 2012; 8(4):e1002467.19. Rejniak KA. A single-cell approach in modeling the dynamics of tumor microregions. Mathematical Biosciences

and Engineering 2005; 2(3):643–655.20. Li Y, Yun A, Kim J. An immersed boundary method for simulating a single axisymmetric cell growth and division.

Journal of Mathematical Biology 2012; 65:653–675.21. Turlier H, Audoly B, Prost J, Joanny JF. Furrow constriction in animal cell cytokinesis. Biophysical Journal 2014;

106:114–123.22. Sain A, Inamdar MM, Julicher F. Dynamic force balances and cell shape changes during cytokinesis. Physical Review

Letters 2015; 114(04):048102.23. Zumdieck A, Lagomarsino MC, Tanase C, Kruse K, Mulder B, Dogterom M, Julicher F. Continuum description of

the cytoskeleton: ring formation in the cell cortex. Physical Review Letters 2005; 95(25):8103.24. Caginalp G, Chen X. Convergence of the phase field model to its sharp interface limit. European Journal of Applied

Mathematics 1998; 4:417–445.25. Sun Y, Beckermann C. Sharp interface tracking using the phase-field equation. Journal of Computational Physics

2007; 220:626–653.26. Borau C, Kim T, Bidone T, García-Aznar JM, Kamm RD. Dynamic mechanisms of cell rigidity sensing: insights

from a computational model of actomyosin networks. PLoS One 2012; 7(11):e49174.27. Thoresen T, Lenz M, Gardel ML. Reconstitution of contractile actomyosin bundles. Biophysical Journal 2011;

100:2698–2705.28. Shen J, Yang X. Numerical approximation of Allen–Cahn and Cahn–Hilliard equations. Discrete and Continuous

Dynamical Systems Series B 2010; 28(4):1669–1691.29. Shu CW. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Review

2009; 51(1):82–126.30. Pyo JH, Shen J. Gauge–Uzawa methods for incompressible flows with variable density. Journal of Computational

Physics 2007; 221:181–197.31. Salbreux G, Charras GT, Paluch E. Actin cortex mechanics and cellular morphogenesis. Trends in Cell Biology 2012;

22(10):536–545.32. Shao D, Levine H, Pappel W. Coupling actin flow adhesion and morphology in a computational celll motility model.

PNAS 2012; 109(18):6855.33. Zumdieck A, Kruse K, Bringmann H, Hyman AA, Julicher F. Stress generation and filament turnover during actin

ring constriction. PLoS One 2007; 2(8):e696.34. Foe VE, Dassow G. Stable and dynamic microtubules coordinately shape the myosin activation zone during

cytokinetic furrow formation. The Journal of Cell Biology 2008; 183(3):457–470.


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