A Simple Model of Cell Proliferation of Bacteria Using Min Oscillation
aa r X i v : . [ n li n . AO ] A ug A Simple Model of Cell Proliferation of Bacteria Using Min Oscillation
Hidetsugu Sakaguchi and Yuka Kawasaki
Department of Applied Science for Electronics and Materials,Interdisciplinary Graduate School of Engineering Sciences,Kyushu University, Kasuga, Fukuoka 816-8580, Japan
A mathematical model of Min oscillation in Escherichia coli is numerically studied. The oscillatorystate and hysteretic transition are explained with simpler coupled differential equations. Next, wepropose a simple model of cell growth and division using the Min oscillation. The cell cycle is notconstant but exhibits fluctuation in the deterministic model. Finally, we perform direct numericalsimulation of cell assemblies composed of many cells obeying the simple growth and division model.As the cell number increases with time, the spatial distribution of cell assembly becomes morecircular, although the cells are aligned almost in the x -direction. I. INTRODUCTION
Cell growth and division are important processes of cell proliferation. The Escherichia coli (E. coli) has beenintensively studied as one of the simplest model organism. E. coli is one of prokaryotic bacteria without a cell nucleus.In the cell division of E. coli, a division septum is formed at the mid-zone of the cell. Adler et al. found E. colimutants that could not produce a septum at the mid-zone and generated minicells [1]. Later, it was found that themin proteins: MinC, MinD, and MinE play important roles to determine the mid-zone [2–4]. The min proteins tendto be localized at cell poles, which suppresses the formation of FtsZ proteins at the poles. As a result, the FtsZproteins can be formed only in the center, which leads to the formation of the septum at the mid-zone.The min proteins are not steadily localized at poles, but exhibit reciprocal oscillation between the two poles. TheMin oscillation occurs owing to the interaction between the min proteins. The concentration of MinD is low at thecenter on average. The low concentration region of MinD becomes a position where a division septum is formed. Themin proteins diffuse in the cytoplasma and some of them are adsorbed to the cytoplasmic membrane. The transferdynamics of MinD and MinE between the cytoplasma and membrane is controlled by the densities of MinE andMinD, respectively. Several authors studied theoretically the mechanism of the Min oscillation [5, 6]. Howard et al.proposed a one-dimensional reaction-diffusion equation for the densities ρ D , ρ d , ρ E , and ρ e of MinD and MinE in thecytoplasmic membrane and cytoplasma [7].The Min oscillation generates waves in spatially extended media. Chemical waves of Min oscillation have beeninvestigated in vitro. Loose et al. constructed a system of Min oscillation on a lipid membrane, and found spiralwaves on the artificial membrane [8]. Vecchiarelli et al. found a variety of patterns including burst patterns underslightly different conditions [9].We use the deterministic model equation by Howard et al. for a basic model of simplified cell proliferation, althoughseveral authors have studied the effect of fluctuations due to the small number of proteins in one cell. [10] In Section2, we show the Min oscillation in the model equation and its dynamical transition. In Section 3, we study a simplemodel of cell growth and division based on the Min oscillation. We show the cell cycle fluctuates in time even if thesystem is deterministic. In Section 4, we study a simple model of cell proliferation, and found that the form of cellassembly is approximated at an ellipse and the oblateness decreases with time. II. ONE-DIMENSIONAL MODEL OF MIN OSCILLATION
A model equation for the Min oscillation proposed by Howard et al. is expressed as ∂ρ D ∂t = − σ ρ D b ρ e + σ ρ e ρ d + D D ∂ ρ D ∂x ,∂ρ d ∂t = σ ρ D b ρ e − σ ρ e ρ d ,∂ρ E ∂t = − σ ρ D ρ E + σ ρ e b ρ D + D E D ∂ ρ E ∂x ,∂ρ e ∂t = σ ρ D ρ E − σ ρ e b ρ D , (1)
0 100 200 0 0.4 0.8 1.2 1.6 2 t x (a) (b) (cid:129)« (cid:129)« (c) FIG. 1: (a) Time evolution of the density of MinD: ρ D ( x ) + ρ d ( x ) for L = 2. (b) Long-time average of ρ D ( x ) + ρ d ( x ) at L = 2.(c) Difference between the maximum and minimum values of the long-time average of ρ D ( x ) + ρ d ( x ) as a function of L . where ρ D and ρ d express respectively the densities of MinD in the cytoplasma and membrane and ρ E and ρ e arerespectively the densities of MinE in the cytoplasma and membrane. σ i ( i = 1 , , ,
4) and b i ( i = 1 ,
4) are parametersto control the adsorption and desorption of the min proteins on the cytoplasmic membrane. The proteins diffuse onlyin the cytoplasma. The desorption of MinD in the membrane is facilitated and the adsorption of MinD is suppressedby MinE in the membrane. On the other hand, the desorption of MinE in the membrane is suppressed and theadsorption of MinE is facilitated by MinD in the cytoplasma. In this paper, we will study this equation more in detailand use it for numerical simulations of cell proliferation.In the time evolution of Eq. (1), S D = R ( ρ D ( x ) + ρ d ( x )) dx and S E = R ( ρ E ( x ) + ρ e ( x )) dx are conserved. Theunits of space and time are µ m and s. In this section, we show some numerical results of Eq. (1) at D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, and S E = 170, which arebiologically relevant values used by Howard et al. [7]. Howard already reported several numerical results of the samemodel equation, and our numerical results are some supplemental ones to their results. The system size L is changedas a control parameter. The no-flux boundary conditions are imposed at x = 0 and L .Figure 1(a) shows the time evolution of the density of MinD: ρ D ( x ) + ρ d ( x ) for L = 2. A pulse appears near thecenter and propagates to the right and then another pulse appears near the center and propagates to the left. Theseesaw type density oscillation of MinD occurs between the left and right ends. The density of MinD just at the centeris low. Figure 1(b) shows a long-time average of ρ D ( x ) + ρ d ( x ) at L = 2, which has a minimum at x = L/
2. Anotherprotein FtsZ is produced around the minimum point of MinD, and the septum is formed there, which leads to thecell division. Figure 1(c) shows the difference between the maximum and minimum values of the long-time averageof ρ D ( x ) + ρ d ( x ) as a function of L . A stationary state is stable for L ≤ .
33 and jumps to an oscillatory state at L = 1 .
34 when L is increased. The oscillatory state jumps to the stationary state at L = 1 .
18 when L is decreased.There is a weak hysteresis in the transitions. The MinD oscillation works as a signal for the cell division. AlthoughHoward et al. already showed that the Min oscillation disappear at L < . ρ D , ρ d , ρ E , and ρ e are expressed as ρ D ( x ) = X D + X D cos( πx/L ) , ρ d ( x ) = X d + X d cos( πx/L ) ,ρ E ( x ) = X E + X E cos( πx/L ) , ρ e ( x ) = X e + X e cos( πx/L ) . (2)The substitution of these expansions into Eq. (1) yields coupled ordinary differential equations for the Fourier ampli- (a) x t A (cid:13) L(cid:13) (b)
FIG. 2: (a) Time evolution of the density of MinD: ρ D ( x ) + ρ d ( x ) = X D + X d + ( X D + X d ) cos( πx/L ) for L = 2. (b) Peakamplitude of the temporal oscillation of X D ( t ) + X d ( t ) as a function of L . (a) (b) (c) FIG. 3: Time evolutions of the spatial average of ρ D for (a) L = 2, (b) L = 2 .
4, and (c) L = 3 . tudes X D ∼ X e : dX D dt = − σ X D F − σ X D b X e (1 − F /F ) + σ X e X d + 0 . σ X e X d ,dX D dt = − D D (cid:16) πL (cid:17) X D − σ X D b X e (1 − F /F ) − σ X D ( b X e ) ( F /F − F )+ σ ( X e X d + X e X d ) ,dX d dt = σ X D F + σ X D b X e (1 − F /F ) − σ X e X d − . σ X e X d ,dX d dt = 2 σ X D b X e (1 − F /F ) + 2 σ X D ( b X e ) ( F /F − F ) − σ ( X e X d + X e X d ) ,dX E dt = σ X e G + σ X e b X D (1 − G /G ) − σ ( X D X E + 0 . X D X E ) ,dX E dt = − D E (cid:16) πL (cid:17) X E + 2 σ X e b X D (1 − G /G )+2 σ X e ( b X D ) ( G /G − G ) − σ ( X D X E + X D X E ) ,dX e dt = − σ X e G − σ X e b X D (1 − G /G ) + σ ( X D X E + 0 . X D X E ) , (3) dX e dt = − σ X e b X D (1 − G /G ) − σ X e ( b X D ) ( G /G − G ) + σ ( X D X E + X D X E ) , where F = p (1 + b X e ) − ( b X e ) , F = 1 + b X e , G = p (1 + b X D ) − ( b X D ) , G = 1 + b X D .Figure 2(a) shows the time evolution of X D + X D cos( πx/L )+ X d + X d cos( πx/L ) at L = 2. The other parametersare the same as the case of Fig. 1. The seesaw type MinD oscillation appears, however, the pulse propagation is notobserved because of the two-mode approximation. Since the long-time average of X D and X d is zero, the long-time average of the MinD density is uniform in contrast to Fig. 1(b). Figure 2(b) shows the peak amplitude of (cid:131) ˇ D t(cid:13) x(cid:13) (a) (b) FIG. 4: Time evolutions of (a) ρ D ( L/
2) and profile (b) ρ D ( x ) for L = 10. X D ( t ) + X d ( t ) as a function of L . The MinD oscillation sets in at L = 1 .
336 when L is increased, and the oscillationdisappears at L = 1 .
28 when L is decreased. The critical values are slightly different from those for the partialdifferential equation Eq. (1), however, similar hysteresis is observed. The two-mode approximation is a useful modelto understand the MinD oscillation.The Min oscillation becomes more complicated as L is increased. Figure 3 shows the time evolution of the spatialaverage R L ρ D ( x ) dx/L of ρ D for (a) L = 2, (b) L = 2 .
4, and (c) L = 3 .
6. A regular oscillation is observed at L = 2.The period doubling occurs slightly below L = 2 .
3. An oscillation with a double period is observed at L = 2 . L ∼ .
5. More complicated time evolution is observed at L = 3 .
6. When L is further increased up to L = 6, multiple pulses appear. Howard et al. reported that a two-pulsestate is stable around L = 8 .
4. [7] In such a long cell, the normal cell division by septum formation at the midpointcannot occur. Figures 4(a) shows the time evolution of ρ D ( L/
2) and Fig. 4(b) shows the time evolution of the profile ρ D ( x ) at L = 10. Spatio-temporal chaos with multiple pulses appears in a large system of L = 10. Wu et al. studiedexperimentally patterns of the Min oscillation in diverse shapes such as squares, rectangles, circles, and triangle ofvarious sizes. [13] They found the Min oscillation of multiple pulses, and transitions between the two-dimensionalpatterns with different wavenumbers. III. SIMPLE MODEL OF GROWTH AND DIVISION
In this section, we study a simple model of cell growth and division based on the equation of the Min oscillation.The cell size is assumed to obey a simple linear growth law L ( t ) = L (0) + Γ t . The numerical simulation of Eq. (1)is performed using the Runge-Kutta method by discretizing the space with an interval ∆ x = L/N where N is thegrid number. In the numerical simulation, the grid interval ∆ x ( t ) is assumed to increase as ∆ x ( t ) = ∆ x (0) + γt where γ = Γ /N . The cell division is assumed to occur at the minimum point of the density of MinD. That is,we assume a simple rule that the cell division occurs at a point where the maximum of the accumulated value I ( x ) = R t (1500 − ρ D ( x ) − ρ d ( x )) dt exceeds a critical value R c . We assume that the profiles of ρ D , ρ d , ρ E , and ρ e are maintained at the cell division. The total grid point are doubled to 2 N by inserting new grid points at midpointsbetween neighboring old grid points, and the cell is split into two at a point where I ( x ) exceeds R c first. If the celldivision occurs at the midpoint, two cells of size L ( t c ) / N are created. However, theprofiles of ρ D ( x ) et al. in the divided two cells are different, because the profile of ρ D ( x ) before the cell division isneither mirror-symmetric around the midpoint nor spatially-periodic with period L/
2, that is, two cells of different ρ D ( x ) are created at the cell division. This type of numerical simulation of cell division based on the Min oscillationis not performed before.Firstly, we performed numerical simulations of the simple growth and division model by removing the latter halfpart of the divided cell. That is, the growth and division of only the former half is repeatedly simulated. Figure 5(a)shows the cell size L ( t ) at D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, S E = 170, γ = 10 − , and R c = 1 . · . The initial cell size is L (0) = 1 . N = 50.Some spatial inhomogeneity is assumed in the initial condition. The cell size L ( t ) shows a rather regular oscillation. L ( t ) grows from L ( t ) ≃ . L ( t ) ≃ .
8. Since L ( t ) is always larger that the criticalpoint of L = 1 .
34 for the Min oscillation, the Min oscillation is maintained during the growth process. The integral I ( x ) = R (1500 − ρ D ( x ) − ρ d ( x )) dt at the midpoint increases steadily and the cell splitting occurs when I goes overthe threshold R c . The profiles of ρ D , ρ d , ρ E , and ρ e are maintained, however, they are rescaled at the splitting as R ( ρ D + ρ d ) dx = S D and R ( ρ E + ρ e ) dx = S E are conserved. Figure 5(b) shows the time evolution of cell size L ( t ) at t t L(t) L(t) (a) (b)
FIG. 5: Time evolutions of cell size at (a) R c = 1 . · and (b) R c = 10 for L ( t ) at D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, S E = 170, and γ = 10 − . P (cid:13) T(cid:13) 0(cid:13)0.002(cid:13)0.004(cid:13)0.006(cid:13)0.008(cid:13)1500(cid:13) 2000(cid:13) 2500(cid:13) 3000(cid:13) 3500(cid:13) P (cid:13) T(cid:13) 0(cid:13)0.002(cid:13)0.004(cid:13)0.006(cid:13)0.008(cid:13)1500(cid:13) 2000(cid:13) 2500(cid:13) 3000(cid:13) 3500(cid:13) P (cid:13) T(cid:13) (a) (b) (c)
FIG. 6: Probability distributions of cell cycle at (a) R c = 1 . · , (b) R c = 1 . · , and (c) R c = 10 for L ( t ) at D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, S E = 170, and γ = 10 − . R c = 10 . The cell size changes randomly between 1.1 and 2.6. For L ( t ) < .
18, the oscillatory state changes into aspatially uniform state and the spatial inhomogeneity decays in time. The dynamics of cell size becomes complicatedowing to this transition. That is, if L ( t ) > .
18 just after the splitting, I ( x ) at x = L/ R c = 10 at t = t c . In this case, the cell size just before the splitting is relatively small. The cell division occurs atthe midpoint and the cell size is reduced to L ( t c ) /
2. If the cell size L ( t c ) / I ( L/
2) increases very slowly, therefore, it takes a large time for I ( L/
2) to attain the threshold R c . Then, the cell size just before the splitting is relatively large. Thus, the cell cycleis not constant but changes randomly between 1900 and 3400. That is, the cell cycle exhibits a complex dynamics inthe upper level system of growth and division, even if min proteins exhibit a stationary state or a regular limit-cycleoscillation.Figure 6 shows the probability distributions of cell cycle for (a) R c = 1 . · , (b) 1 . · , and (c) 10 . Thewidth of the cell cycle distribution is narrow at R c = 1 . · because the oscillation of cell size is almost periodic.The width of the distribution increases as R c decreases. For R c ≤ . · , I ( x ) takes a maximum at a point differentfrom the midpoint and an inhomogeneous splitting occurs. Figure 7(a) shows the time evolutions of L ( t ) and N ( t )at R c = 0 . · . L ( t ) changes between 0.92 and 2.86. When L ( t ) takes a value near L ( t ) = 2 .
86, the inhomogeneoussplitting occurs, and the grid number N changes as shown in Fig. 7(b). IV. SIMPLE MODEL OF CELL PROLIFERATION
In this section, we study a simple model of cell proliferation. That is, we perform numerical simulation of agrowing cell assembly composed of mutually interacting cells. Each cell is assumed to be a rigid rod composed of N points. For the cell growth and division, we use the same simple model as studied in the previous section. Langevintype equations of motion are assumed for the coordinates X k and Y k of the center of gravity of the k th cell andits angle Θ k from the x axis. The coordinate of the i th grid point in the k th cell is expressed as ( x i,k , y i,k ) where x i,k = X k + ∆ k ( t )( i − N/
2) cos Θ k , y i,k = Y k + ∆ k ( t )( i − N/
2) sin Θ k . The interval ∆ k between the neighboring gridpoints increases linearly as d ∆ k /dt = γ . The grid number N is fixed to be 50, since R c is set to be larger than 0 . · .Repulsive forces are assumed to work when the distance between the grid points of different cells is smaller than 0.05. L ( t ) (cid:13) t(cid:13) (a) N (cid:13) t(cid:13) (b) FIG. 7: Time evolutions of (a) L ( t ) and (b) N at R c = 0 . · . The other parameters are the same as in Fig. 5. Furthermore, white noises of variance 3 . × − are applied. That is, X k , Y k , and Θ k obey the coupled equations: dX k dt = X d i,j ≤ . x i,k − x j,l d i,j c (0 . − d i,j ) + ξ xk ( t ) ,dY k dt = X d i,j ≤ . y i,k − y j,l d i,j c (0 . − d i,j ) + ξ yk ( t ) , (4) d Θ k dt = X d i,j ≤ . g ( x i,k − X k )( y i,k − y j,l ) − ( y i,k − Y k )( x i,k − x j,l ) d i,j (0 . − d i,j ) + ξ θk ( t ) , where d i,j = p x i,k − x j,l ) + ( y i,k − y j,l ) is the distance between the two grid points composed of the k th cell and -2(cid:13)0(cid:13)2(cid:13)-8(cid:13) -4(cid:13) 0(cid:13) 4(cid:13) 8(cid:13) -2(cid:13)0(cid:13)2(cid:13)-8(cid:13) -4(cid:13) 0(cid:13) 4(cid:13) 8(cid:13) -2(cid:13)0(cid:13)2(cid:13)-8(cid:13) -4(cid:13) 0(cid:13) 4(cid:13) 8(cid:13) (a) (b) (c) FIG. 8: Three snapshots of the proliferating cells at (a) t = 10000, (b) 17500, and (c) 21000. The parameters are D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, S E = 170, γ = 10 − and R c = 10 . P (cid:131)ƒ < (cid:131) ¢ (cid:131) ƒ > (a) (b) FIG. 9: (a) Time evolution of h Θ i . (b) Probability distribution of Θ i . the l th cell, and the summation is taken only for the pairs satisfying d i,j < .
05. The parameter c is set to be 0.01.The third equation is a Langevin equation for the rotation angle Θ k , and g/c is another parameter related to themoment of inertia, which is set to be 0.2 in our numerical simulation. For the sake of simplicity, we do not considervarious effects such as the anisotropy of mobility owing to the rod structure and the change of inertia owing to thecell growth. To our best knowledge, this type of numerical simulation of proliferation of linear cells is not reportedbefore. Molecular dynamics simulation of nematic liquid crystals in thermal equilibrium has been performed by manyauthors. [14] Recently, active nematics composed of self-propelled rods is intensively studied. [15] Long-live giantnumber fluctuations are an interesting topic in the active nematics. [16] Because our linear cells are not self-propelled,our system is not active nematics. However, our system is considered to be far from equilibrium, since the cell numberincreases exponentially in time by cell division. We will show a few numerical results for this growing system. Figure S N t (a) Y t1234 0 5000 10000 15000 20000t X (b) (c) (cid:131) ˇ (d) FIG. 10: Time evolutions of (a) cell number, (b) the first principle component, (c) the second principle component, and (d)the average density ρ = N/S at R c = 10 . S N (cid:13) t(cid:13) (a) X (cid:13) t(cid:13) (b) Y (cid:13) t(cid:13) (c) (cid:131) ˇ (d) FIG. 11: Time evolutions of (a) cell number, (b) the first principle component, (c) the second principle component, and (d)the average density ρ = N/S at R c = 1 . · . t = 10000 , D D = 0 . D E = 0 . σ = 20, σ = 0 . σ = 0 . σ = 0 . b = 0 . b = 0 . S D = 3000, S E = 170, γ = 10 − and R c = 10 . The cell numbers are respectively 16, 128, and 372 for the three snapshots in Fig. 8. Roughlyspeaking, the cell assembly takes an elliptic form. The noise strength is rather small, but the noise term is necessaryfor the formation of two-dimensional cell assembly. If the noise strength is completely zero, Y k = Θ k = 0 and cellsexpand only in the x -direction. The cell number increases exponentially and the cell assembly expands only in the x -direction as the cells do not overlap with each other, which will be a very cramped growthIn our model system with small noises, cells can move in the y -direction and rotate in the θ direction. Figure 9(a)shows the time evolution of the standard deviation h Θ i of Θ k . Figure 9(b) shows the probability distribution of Θ k at t = 21000. The fluctuation of the cell direction increases with time, however, it is still rather small at t = 21000,that is, all cells are aligned almost in the x direction.Figure 10(a) shows the time evolution of the cell number N . The cell number increases roughly as e . t . Thereare two stages of cell division and cell growth. The distribution of the cell assembly is evaluated by the principlecomponent analysis. Figures 10(b) and 10(c) show the time evolution of the first and second principle componentsdenoted by X and Y . They are defined as X = (1 / { v x + v y + q ( v x − v y ) + 4 v xy } and Y = (1 / { v x + v y − q ( v x − v y ) + 4 v xy } where v x = h ( x i,k − h x i,k i ) i , v y = h ( y i,k − h y i,k i ) i , and v xy = h ( x i,k − h x i,k i )( y i,k − h y i,k i ) i .Here, h· · · i denotes the average with respect to i and k . Roughly, X and Y increase exponentially as e . t and e . t , respectively. The width along the y -direction increases more rapidly and the oblateness of the elliptical cellassemby decreases with time. The cells grow approximately in the x -direction, because they are roughly aligned inthe x -direction. If the noise term is absent, the cell assembly grows only in the x -direction. Therefore, our numericalresult that the growth of the cell assembly occurs more rapidly in the y -direction is nontrivial. This is a new findingin our numerical simulation, although the mechanism is not sufficiently understood. If the elliptic form is assumed,the area of cells is evaluated as S = π (4 / XY . Figure 10(d) shows the time evolution of the average number density ρ = N/S . The density increases in the stage of cell division and decreases in the stage of cell growth. However, theaverage density tends to increase with time. The dashed lines are 1 / (1 . · . / (2 . · . y -direction between neighboring two cells is around 0.025, because the cell size changes between1.1 and 2.6. It is about half of the interaction range 0.05. Similarly, Fig. 11 shows the time evolutions of (a) the cellnumber, (b) the first principle component, (c) the second principle component, and (d) the average density ρ = N/S at R c = 1 . · . The cell number increases approximately as e . t . The cell division occurs more synchronouslythan the case of R c = 10 shown in Fig. 10(a), because the width of the probability distribution of cell cycle is muchnarrower at R c = 1 . · . X and Y increase approximately as e . t and e . t , respectively. The growth rate isslower at R c = 1 . · than R c = 10 because the cell cycle is longer at R c = 1 . · . The dashed lines in Fig. 11(d)are 1 / (1 . · . / (2 . · . R c = 1 . · . The cell density increases with time probably because it is difficult to keepa constant density by the local repulsive interaction when the cell number expands exponentially. V. SUMMARY
We have proposed a simple model system of cell proliferation based on the Min oscillation, which describes bacterialdynamics in the wide range from molecular to multi-cellular scales.At first, we have shown a few supplemental results for the one-dimensional model of the Min oscillation proposed byHoward et al. We have found a hysteretic transition to the Min oscillation and proposed coupled ordinary differentialequations to reproduce the Min oscillation.Next, we have performed numerical simulation of a simple model of cell growth and division. The cell cycle doesnot always take a constant value even in the deterministic model. It is caused by chaotic dynamics of cell size, whichis closely related to the transition from the stationary state to oscillatory state. That is, the cell cycle changes withtime owing to the interaction of the internal dynamics and the cell growth, even if the system parameters are fixedand no external forces or noises are applied. This phenomenon might be interpreted as an example that differentphenotypes can appear even if the genetics and environment are the same. The phenotypic variation in the samegenetics and environment has been studied experimentally using E. coli [17], crayfish [18], and so on.Finally, we have performed numerical simulation of a simple model of cell proliferation, assuming repulsive interac-tion between neighboring cells. As the cell number increases, the cell assembly expands in space, and the oblatenessof the elliptical cell assembly decreases with time or the cell assembly tends to take a more circular form over time.On the other hand, if noises are completely absent, only one-directional growth occurs in the x -direction. We considerthat some entropic effect owing to weak noises might cause the more rapid growth in the y -direction and decrease ofthe oblateness. However, our numerical results are preliminary ones and the understanding of the detailed mechanismis left to future study. [1] H. I. Adler, W. D. Fischer, A. Cohen, and A. A. Hardigree, Proc. Natl. Acad. Sci , 321 (1967).[2] D. M. Raskin and P. A. J. de Boer, Proc. Natl. Acad. Sci. , 4971 (1999).[3] Z. Hu and J. Lutkenhaus, Mol. Microbiol. , 6419 (1999).[4] S. L. Rowland, X. Fu, M. A. Sayed, Y. Zhang, W. R. Cook, L. I. Rothfield, J. Bactriol. , 613 (2000).[5] C. A. Hale, H. Meinhardt and P. A. J. de Boer, EMBO J. , 1563 (2001).[6] H. Meinhardt and P. A. J. de Boer, Proc. Natl. Acad. Sci. , 14202 (2001).[7] M. Howard, A. D. Rutenberg, and S. de Vet, Phys. Rev. Lett. , 278102. (2001).[8] M. Loose, E. Fischer-Friedrich, J. Ries, K. Kruse, and P. Schwille, Science , 789 (2008).[9] A. G. Vecchiarelli, M. Li, M. Mizuuchi, L. C. Hwang, Y. Seol, K. C. Neuman, and K. Mizuuchi, Proc. Natl. Acad. Sci ,E1479 (2016).[10] M. Howard and A. D. Rutenberg, Phys. Rev. Lett. , 128102 (2003).[11] G. Meacci and K. Kruse, Phys. Biol. , 89 (2005).[12] E. Fischer-Friedrich, G. Meacci, J. Lutkenhaus,H, Chat´e, and K. Kruse, Proc. Natl. Acad. Sci. , 6134 (2010). [13] F. Wu, B. G. C. van Schie, J. E. Keymer, and C. Dekker, Nature Nanotech. , 719 (2015).[14] e.g. A. V. Komolkin, A. Laaksonen, and A. Maliniak, J. Chem. Phys. , 4103 (1994).[15] A. Doostmohammadi, J. Ign´es-Mullol, J. M. Yeomans, and F. Sagu´es, Nature Comm. , 3246 (2018).[16] V. Narayan, S. Ramaswamy, and N. Menon, Science , 105 (2007).[17] M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, Science , 1183 (2002).[18] G. Vogt, M. Huber, M. Thiemann, G. van den Boogaart, O. J. Schmitz, and C. D. Schubart, J. Exp. Biol.211