Application of the model to a single isolated cell

Before investigating cell islands, we consider a single cell case. Implementing the model with a single cell will include only two of the three mechanisms from the energy function, namely the area constraint and the perimeter contraction. The interaction between the two mechanisms in the single cell size and shape can be observed.

In the simulations a single cell’s size and shape is determined by the interaction between the perimeter contraction parameter, Γ, area constraint parameter, 𝑘𝑘, and the preferred area, 𝐴𝐴_𝑝𝑝. We

understand that parameter 𝑘𝑘 maintains the cell area close to the preferred area and parameter Γ

attempts to reduce the cell’s perimeter to zero. This means that increasing 𝑘𝑘 will make it more

difficult for the cell area to differ from the preferred area and increasing Γ will decrease the cells perimeter and size.

The equilibrium state of the single cell can be characterised by the cell area and perimeter. The simulation results are compared to an optimisation solution resulting from the optimisation problem described below.

We assume all cells have a shape categorised by the cell area and perimeter. The cell perimeter squared cannot fall below a particular area, i.e. 𝐴𝐴 ≤ 𝑔𝑔𝐿𝐿2. The value of 𝑔𝑔 is the ratio of area and perimeter squared, 𝐴𝐴 𝐿𝐿⁄ , of a defined shape, referred to as the minimum shape in Chapter 2 2, and occurs when 𝐴𝐴 = 𝑔𝑔𝐿𝐿2. In an unrestrictive lattice the minimum shape is represented by a circle; however, in a restrictive lattice the cell is instead represented by a polygon. Due to the selection of hexagonal pixels the polygon is a “hexagon”, see Figure 2.4, to replicate the simulations, and the ratio is given as 𝑔𝑔 = 1 48⁄ .

Only the interaction between the cell area constraint and cell perimeter contraction is expressed in the energy function for a single cell,

𝐸𝐸(𝐴𝐴, 𝐿𝐿) =𝑘𝑘

2 �𝐴𝐴 − 𝐴𝐴𝑝𝑝�

2

+Γ

2 𝐿𝐿2. ( 3.2 )

Additionally, simple conditions are applied, namely that the cell area and perimeter are non-

negative and the inequality discussed previously between the cell area and perimeter, i.e. 𝐴𝐴 ≤ 𝑔𝑔𝐿𝐿2. An optimisation problem can be expressed with the energy Function (3.2) and conditions giving the Lagrangian function

𝐸𝐸∗_{(𝐴𝐴, 𝐿𝐿, 𝜆𝜆}

1, 𝜆𝜆2, 𝜆𝜆3) =𝑘𝑘_{2 �𝐴𝐴 − 𝐴𝐴}𝑝𝑝�2+Γ_{2 𝐿𝐿}2

+𝜆𝜆1(𝐴𝐴 − 𝑔𝑔𝐿𝐿2) + 𝜆𝜆2(−𝐴𝐴) + 𝜆𝜆3(−𝐿𝐿),

( 3.3 )

with the use of Karush-Kuhn-Tucker (KKT) conditions (Lange, 2004). The KKT conditions are a generalisation of the equality conditions used with Lagrangian multipliers to incorporate the

inequalities of the problem. The Lagrangian Function (3.3) with these conditions results in the simultaneous equations 𝜕𝜕𝐸𝐸∗_{(𝐴𝐴, 𝐿𝐿, 𝜆𝜆} 1, 𝜆𝜆2, 𝜆𝜆3) 𝜕𝜕𝐴𝐴 = 𝑘𝑘�𝐴𝐴 − 𝐴𝐴𝑝𝑝� + 𝜆𝜆1− 𝜆𝜆2 = 0, 𝜕𝜕𝐸𝐸∗_{(𝐴𝐴, 𝐿𝐿, 𝜆𝜆} 1, 𝜆𝜆2, 𝜆𝜆3) 𝜕𝜕𝐿𝐿 = 𝛤𝛤𝐿𝐿 − 2𝜆𝜆1𝑔𝑔𝐿𝐿 − 𝜆𝜆3 = 0, 𝜆𝜆1(𝐴𝐴 − 𝑔𝑔𝐿𝐿2) = 0, 𝜆𝜆2(−𝐴𝐴) = 0 and 𝜆𝜆3(−𝐿𝐿) = 0.

The results of these simulations equations need to satisfy the inequalities of 𝜆𝜆1 ≥ 0, 𝜆𝜆2 ≥ 0, 𝜆𝜆3 ≥ 0, (𝐴𝐴 − 𝑔𝑔𝐿𝐿2) ≤ 0, −𝐴𝐴 ≤ 0 and − 𝐿𝐿 ≤ 0.

In this problem there are eight different cases including (𝜆𝜆₁ = 0, 𝜆𝜆₂ = 0, 𝜆𝜆₃ = 0), (𝜆𝜆₁ > 0, 𝜆𝜆2 = 0, 𝜆𝜆3 = 0), (𝜆𝜆1 = 0, 𝜆𝜆2 ≥ 0, 𝜆𝜆3 = 0), (𝜆𝜆1 = 0, 𝜆𝜆2 = 0, 𝜆𝜆3 ≥ 0), (𝜆𝜆1 ≥ 0, 𝜆𝜆2 ≥ 0, 𝜆𝜆3 = 0),

(𝜆𝜆1= 0, 𝜆𝜆2 ≥ 0, 𝜆𝜆3 ≥ 0), (𝜆𝜆1 ≥ 0, 𝜆𝜆2 = 0, 𝜆𝜆3 ≥ 0) and (𝜆𝜆1≥ 0, 𝜆𝜆2 ≥ 0, 𝜆𝜆3 ≥ 0). The only solution

that satisfies the equations and inequalities is from the case (𝜆𝜆₁ ≥ 0, 𝜆𝜆₂ = 0, 𝜆𝜆₃ = 0) and gives the solution of the cell area,

𝐴𝐴1 = 𝐴𝐴𝑝𝑝−_{2𝑔𝑔𝑘𝑘}Γ , ( 3.4 )

and the solution of the cell perimeter,

𝐿𝐿1 = �𝐴𝐴_{𝑔𝑔 =}�𝐴𝐴_{𝑔𝑔 −}𝑝𝑝 _2𝑔𝑔Γ_{2𝑘𝑘 .} ( 3.5 )

These solutions show that the area of a single cell, 𝐴𝐴₁, and perimeter of a single cell, 𝐿𝐿₁,

decrease with the perimeter contraction parameter Γ, see Figure 3.2a and 3.2c, respectively, and

increase with the area constraint 𝑘𝑘, see Figures 3.2b and 3.2d, respectively. The cell can disappear

when the area and perimeter equal zero for both the simulation and optimisation solutions, shown in Figure 3.3, when 𝐴𝐴_𝑝𝑝 ≤ Γ (2𝑔𝑔𝑘𝑘)⁄ .

The simulation results and the mathematical optimisation solutions above are qualitatively similar. However, a few differences between them are a result of the restrictions of the discretised lattice structure of the simulations. The main differences are observed in Figure 3.2c and 3.2d,

showing the cell perimeter against the parameters 𝑘𝑘 and Γ, respectively. They show differences

between the simulations and optimisation solution occur at comparatively small values of the

contraction parameter of those values tested, such as Γ = 1. With “low” values of contraction the

deviation of the cell’s perimeter requires less energy, and therefore the noise in the system distorts the cell’s shape more readily than a cell with a larger contraction parameter. This trend is also

replicated in the ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against 𝑘𝑘 and Γ, shown in Figure 3.2e and 3.2f, respectively. The 2

equilibrium results of the cell simulations and 𝑔𝑔 = 1 48⁄ , representing the “hexagonal” shape of the

cell on the lattice (shown on the figure as a black line), display a large difference between them for

“small” values of the contraction, such as Γ = 1. The values of 𝑔𝑔 from the simulation are smaller

than 1 48⁄ , representing distorted shapes. These differences are similar to the differences in the perimeters because the ratio is calculated with the perimeter, 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ . 2

Furthermore, there is a difference between the simulation and the optimisation solution cell area, observed in Figure 3.3a, which is caused by the lattice and smaller preferred area. For

simulations with an area constraint parameter 𝑘𝑘 = 3 and perimeter contraction parameter greater

than 5, the simulation cell area results are greater than the optimisation solutions. The optimisation

solutions tend towards zero while the simulation cell area increases as Γ increases. As the cell

becomes smaller in the simulations there are situations where the decrease in the cell perimeter requires the cell to initially increase its perimeter. This means the cell can become fixed at a larger perimeter with larger perimeter contraction coefficient values and no longer decreases its perimeter to achieve the calculated equilibrium. This artefact can be eliminated by increasing the amount of noise, or for larger preferred area values as shown in Figure 3.2. However, the artefact of the cell perimeter, discussed above, is still observable in both situations.

(a) (d)

(b) (e)

(c) (f)

Figure 3.2. Plots, with the preferred area 𝐴𝐴_𝑝𝑝 = 500, for a single cell simulated on a

hexagonal pixel grid with the CPM showing the (a) mean cell area with the geometric solution from Equation (3.5), (b) mean cell perimeter with the geometric solution from Equation (3.4) and (c) ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against the perimeter contraction, Γ, for different area 2

constraint parameters, 𝑘𝑘; and the (d) mean cell area with the geometric solution from

Equation (3.5), (e) mean cell perimeter with the geometric solution from Equation (3.4) and (f) ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against the area constraint parameter, 𝑘𝑘, with different perimeter 2

(a) (d)

(b) (e)

(c) (f)

Figure 3.3. Plots, with the preferred area 𝐴𝐴_𝑝𝑝 = 100, for a single cell simulated on a

hexagonal pixel grid with the CPM showing the (a) mean cell area with the geometric solution from Equation (3.5), (b) mean cell perimeter with the geometric solution from Equation (3.4) and (c) ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against the perimeter contraction, Γ, for different area 2

constraint parameters, 𝑘𝑘; and the (d) mean cell area with the geometric solution from

Equation (3.5), (e) mean cell perimeter with the geometric solution from Equation (3.4) and (f) ratio 𝑔𝑔 = 𝐴𝐴 𝐿𝐿⁄ against the area constraint parameter, 𝑘𝑘, with different perimeter 2

contractions, Γ. Each point is averaged over the last 50 of 1000 iterations and the

3.4 IMPLEMENTATION OF THE CPM FOR TWO OR MORE