Reaction diffusion models for

cell motion

Jacob RyanSupervised by Prof. Scott McCue

Queensland University of Technology

Vacation Research Scholarships are funded jointly by the Department of Education and Training

and the Australian Mathematical Sciences Institute.

1 Introduction

In many biological settings, understanding the behaviour of cells under various conditions has extremely valu-

able applications to modern medicine. More specifically, the movements and reproductive behaviour of cells in

the human body during the process of wound healing or of cancer cells in a melanoma tumour could be utilised

to eventually create targeted treatments to either inhibit or assist cells migration and proliferation [2, 4].

This behaviour can be studied using a scratch assay experiment, where a monolayer of confluent cells are

placed in a dish and a portion are ’scratched’ away, simulating a wound or in general, a habitable space for cells

to occupy . Observations and various tests are then conducted to determine the behaviour of cells as they fill

the unoccupied space over the experiment’s duration [5].

Figure 1: Left: A typical scratch assay with the upper portion of cells removed [3]. Right: A scratch assay with

FUCCI applied, highlighting the reproduction phase of cells using colours [1].

More recently, Fluorescent Ubiquitination Cell-Cycle Indicator (FUCCI) technology (see fig. 1) was devel-

oped which can classify the current cell-cycle phase of individual cells in a laboratory environment. The cell-cycle

consists of 4 main phases, shown in Figure 2 where the current phase is determined by both intracellular and

extracellular factors that can be difficult to detect in the case of the G1 to S/G2/M phases [6]. Cells undergo

easier to detect morphological changes during transition from the M to G1 phase that marks when cell will pro-

liferate. Observations from scratch assay experiments can quantify the variation in population density of cells

as well as detail individual cell motility and proliferation events [7]. This data can be modelled mathematical

using discrete stochastic models which describe the activities of individual cells or averaging arguments can be

applied to derive continuum models describing the global behaviour of the cell population [7].

1

Figure 2: Diagram of the cell cycle where FUCCI allows the transition between the G1 and S/G2/M phases to

be detected as red or green fluorescence exhibited by the cell nucleus. Cells can exhibit yellow fluorescence at

the transition point between the G1 and S phases [6].

While both approaches were considered for this project, only the continuous approach was adapted to the

FUCCI model, however it is beneficial to describe the discrete approach in some detail, as it was used as a

basis to construct the continuous model. Section 2 of this report will present the fundamental arguments used

to create the stochastic random-walk model and the conversion into their continuous analogue for the case of

a single cell phase, i.e. cell-cycle phases are unknown and events are handled probabilistically. Section 3 will

demonstrate the results of these schemes while Section 4 will extend the continuum limit arguments to the

multi-phase FUCCI model. Section 5 will present conclusions and recommendations for future research.

2 Mathematical Arguments

2.1 Discrete Model

The discrete model involves representing cells as agents on a lattice and developing mathematical arguments as

to how agents can interact within predefined constraints. The scratch assay experiment detailed in Section 1 of

this report provides an adequate and conveniently simple basis for the model. Agents will be required to move

in only 2 dimensions and to eventually fill some larger, initially vacant area to accurately represent real cell

behaviour. Restricting agent migration to 2-dimensions and forbidding agents from occupying the same space

allows the space-filling requirement to be met naturally over the course of a sufficiently long simulation.

Any choice of a lattice naturally introduces error in representing the behaviour of a physical system as events

are only allowed to occur on predefined lattice sites. A hexagonal lattice was chosen to allow for movement of

distance ∆ in 6 directions around any particular point (x, y).

2

(x+ ∆2 , y +

√3∆2 )

(5)

(x, y)

(x− ∆2 , y +

√3∆2 )

(3)

(x−∆, y) (1)

(x− ∆2 , y −

√3∆2 )

(4)

(x+ ∆2 , y −

√3∆2 )

(6)

(x+ ∆, y)(2)

Figure 3: A lattice site at coordinates (x, y) surrounded by 6 equally spaced neighbour sites with distance ∆

away.

Define K(x, y) as the occupancy of a lattice site (x, y) and Kk for k = 1, . . . 6 as the occupancy of the 6 neigh-

bouring lattice sites as shown in Figure 3. K(x, y) = 1 if (x, y) is occupied at a particular instant and 0 otherwise.

Let Pm ∈ [0, 1] be the probability that a migration event occurs for an agent at a particular realisation of

the simulation. The occupancy of a lattice site will change if R ≤ Pm where R ∼ U(0, 1). k = 1 . . . 6 is then

chosen with equal probability and a movement event is allowed if Kk = 0 at that instant for the chosen value

of k, else it is aborted for the chosen agent at (x, y).

Similarly let Pp ∈ [0, 1] be the probability a proliferation event is attempted for a chosen agent. A successful

proliferation requires that a new, daughter agent be placed on an adjacent lattice site k, selected with equal

probability. The proliferation is aborted if Kk = 1 and allowed if unoccupied.

Finally denote N(t) as the number of agents present on the lattice at time t, which is increased by some

pseudo-time τ after each available agent is selected for both types of event.

Figure 4 shows an example of a simulation with constraints defined as above. While this approach is

advantageous as it allows the behaviour of individual cells to be observed at any time, it can be computationally

inefficient to implement as multiple realisations are required to account for the variance inherent in the selection

of random variables. This motivates the development of a continuum model without these shortcomings.

3

0 50 100x

05

10

y

N(0) = 266

0 50 100x

05

10

y

N(48) = 272

0 50 100x

05

10

y

N(120) = 278

0 50 100x

05

10

y

N(240) = 297

0 50 100x

05

10

y

N(600) = 338

Figure 4: Lattice snapshots of a simulation taken at times t = 0, 48, 120, 240, 600. N(t) is the number of

agents present on the lattice at time t. Parameters are constant with ∆ = 1, τ = 1, Pm = 1 and Pp = 0.001.

Periodic boundary conditions are implemented meaning that lattice sites on edges have neighbouring sites on

the opposite edge and are treated with identical rules otherwise.

2.2 Continuous Model

A continuous model can be derived from the previous discrete definitions by averaging over multiple realisations.

We define C(x, y) ∈ [0, 1] as the average occupancy of a lattice site over multiple realisations of K(x, y):

C(x, y) =1

Ns

Ns∑j=1

K(x, y)

where Ns is the number of realisations. The true value of C(x, y) is approached as Ns → ∞. A conservation

statement for δC, the change in average occupancy of a site over some time interval, τ , can be derived as:

δC

τ=Pm6τ

[6∑k=1

Ck(1− C)−6∑k=1

C(1− Ck)

]+Pp6τ

6∑k=1

Ck(1− C) (1)

where Ck denotes the average occupancy of the kth neighbour site. Pm/τ and Pp/τ become the average rate

of migration and proliferation respectively after sufficiently many realisations. Ck(1− C) denotes the scenario

when an agent is present in neighbour site k and when (x, y) is vacant, allowing a migration or proliferation

event to occur. Similarly, C(1−Ck) denotes when (x, y) is occupied and neighbour k is vacant. This scenario is

4

a negative term as the occupancy of C(x, y) decreases when an agent migrates from (x, y) to an adjacent site.

Equation (1) can be adapted into a continuous, reaction-diffusion PDE by assuming C(x, y) is a smooth

function and introducing the following Taylor series approximations around the neighbouring lattice sites as

shown in Figure 3.

C(x−∆, y) = C(x, y)−∆∂C

∂x+

∆2

2

∂2C

∂x2+O(∆3) (2a)

C(x+ ∆, y) = C(x, y) + ∆∂C

∂x+

∆2

2

∂2C

∂x2+O(∆3) (2b)

C

(x− ∆

2, y +

√3∆

2

)= C(x, y)− ∆

2

∂C

∂x+

∆2

8

∂2C

∂x2−√

3∆2

4

∂C

∂x∂y+

√3∆

2

∂C

∂y+

3∆2

8

∂2C

∂y2+O(∆3) (2c)

C

(x− ∆

2, y −

√3∆

2

)= C(x, y)− ∆

2

∂C

∂x+

∆2

8

∂2C

∂x2+

√3∆2

4

∂C

∂x∂y−√

3∆

2

∂C

∂y+

3∆2

8

∂2C

∂y2+O(∆3) (2d)

C

(x+

∆

2, y +

√3∆

2

)= C(x, y) +

∆

2

∂C

∂x+

∆2

8

∂2C

∂x2+

√3∆2

4

∂C

∂x∂y+

√3∆

2

∂C

∂y+

3∆2

8

∂2C

∂y2+O(∆3) (2e)

C

(x+

∆

2, y −

√3∆

2

)= C(x, y) +

∆

2

∂C

∂x+

∆2

8

∂2C

∂x2−√

3∆2

4

∂C

∂x∂y−√

3∆

2

∂C

∂y+

3∆2

8

∂2C

∂y2+O(∆3) (2f)

Noting that Ck terms only appear in summations over all k values, a simplified expression can be obtained by

adding each of these equations:6∑k=1

Ck = 6C +3∆2

2∇2C +O(∆3) (3)

where ∇2 denotes the laplacian operator in 2-dimensions. Substituting Equation (3) into Equation (1) and

simplifying appropriately nets:

δC

τ=Pm∆2

4τ∇2C +

PpτC(1− C) +

Pp∆2

4τ∇2C +O(∆3) . (4)

Letting ∆, τ → 0, representing instantaneous time and continuous space, gives rise to Fisher’s equation:

∂C

∂t= D∇2C + λC(1− C) (5)

where

D =Pm4

lim∆→0τ→0

(∆2

τ

), and λ =

Ppτ.

This requires the continuum limit to be defined such that ∆2/τ approaches a finite constant and that Pp = O(τ)

The third term of Equation (4) is eliminated by assuming that Pp∆2/τ = O(∆3) which was implemented by

assuming Pp � Pm. The assumption that cells proliferate at a lesser rate than they migrate is valid in many

biological contexts [4].

5

3 Comparison of Discrete and Continuous Models

3.1 Numerical Solution Methods

Fisher’s equation is a non-linear partial differential equation that is well-studied in mathematical biology [7].

In order to test the validity of the assumptions used to derive the continuous model, a numerical solution to

Equation (5) can be compared to averaged stochastic simulation data and compared.

Firstly, the problem can be simplified by considering an initial condition that nullifies the movement in the

y direction. By considering a domain where the initial agent population is placed uniformly at random in the

y direction and leaving sufficient vacant area in the x direction, the net effect of y movement will be 0 under

periodic or no flux boundary conditions. Equation (5) can therefore be simplified to

∂C

∂t= D

∂2C

∂x2+ λC(1− C) . (6)

A numerical scheme is therefore required to evaluate the non-linear source term in the one-dimensional Fisher’s

equation.

Finite difference approximations were introduced for a uniform mesh in x with node distance δx and total

number of nodes, N . A backward Euler time stepping scheme was chosen with constant time step δt. Fixed

point iteration can be used to solve for the non-linear component. The discretised form of Equation (6) is:

D

(δx)2C

(m+1)i−1,n+1 +

[λ− 1

δt− 2D

(δx)2− λC(m)

i,n+1

]C

(m+1)i,n+1 +

D

(δx)2C

(m+1)i+1,n+1 = −Ci,n

δt(7)

where n denotes the current timestep, i denotes the node and m denotes the current iterate of the chosen

fixed-point iteration scheme. This requires in a system of the form:

C(m+1) = G(C(m))

where G is some function that is updated every iterate. Equation (7) has been rearranged to be of the form:

A(C(m))C(m+1) = b

where A is an N ×N matrix and b is an N × 1 vector. Fixed point iteration therefore requires the calculation

of A−1(C(m))b at every iteration. Equation (7) results in a tridiagonal system that can be inverted efficiently

using the Thomas algorithm.

3.2 Comparison with Averaged Simulation Data

By repeating the simulations shown in Figure 4 and averaging the density of agent in the x direction, we can

compare this to a numerical solution of Equation (6).

6

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Average Density Profiles

Discrete t = 0hPDE t = 0hDiscrete t = 48hPDE t = 48hDiscrete t = 120hPDE t = 120hDiscrete t = 240hPDE t = 240hDiscrete t = 600hPDE t = 600h

(a) 10 Realisations

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Average Density Profiles

Discrete t = 0hPDE t = 0hDiscrete t = 48hPDE t = 48hDiscrete t = 120hPDE t = 120hDiscrete t = 240hPDE t = 240hDiscrete t = 600hPDE t = 600h

(b) 50 Realisations

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Average Density Profiles

Discrete t = 0hPDE t = 0hDiscrete t = 48hPDE t = 48hDiscrete t = 120hPDE t = 120hDiscrete t = 240hPDE t = 240hDiscrete t = 600hPDE t = 600h

(c) 100 Realisations

Figure 5: Average density profiles for 10, 50 and 100 repetitions of the simulation shown in Figure 4 and

compared to Equation (6) solutions. The parameters were ∆ = 1, τ = 1, Pm = 1 and Pp = 0.001. PDE

boundary conditions are C(0, t) = C(100, t) = 0. The initial condition is identical to Figure 4, given as

C(x, 0) = H(x − 40) −H(x − 60) where H(x) denotes the heaviside step function. Numerical parameters for

the PDE solution were chosen to be δx = 1, δt = 1 and a convergence tolerance for fixed-point iteration of

ε = 1× 10−5.

By visual inspection of Figure 5, Equation (6) provides a good approximation to simulated density averages.

Comparing the Figures 5a to 5c supports the assumption that C(x, y) would be smooth for a sufficiently high

number of realisations. The match does weaken for t = 600 as the chosen Dirichlet boundary condition becomes

inappropriate as agents have begun to reach the domain limits. This problem can be alleviated by imposing

no flux conditions, which maintain the tridiagonal structure of A(C(m)) or by simply extending the domain

sufficiently far from initial population.

7

4 Extension to FUCCI System

Extending the previous arguments to a model incorporating FUCCI technology is of interest as it may provide

a more realistic model of the cell cycle under appropriate choices of parameters. The cell-cycle outlined in

Figure 2 was simplified into the following, 3 phase process which will be modelled mathematically.

G1

kRYG1/S

kY GS/G2/M

kGR G1

G1

Figure 6: Simplified cell-cycle with 3 phases represented as red, yellow and green. kRY , kY G and kGR are the

rates at which an individual agent transitions from one stage to the next in sequence.

By defining a hexagonal lattice of the same type as Figure 3 and populating it with 3 the types of agents

shown in Figure 6, a coupled system can be derived to model the density of each sub-population. Define R(x, y),

Y (x, y) and G(x, y) as the average red, yellow and green agent populations at position (x, y) and let PR, PY and

PG be the migration rates of agents of each sub-population. Identical rules for agent migration between lattice

sites can be applied, taking into account that only 1 agent of any type can occupy a site. Similar proliferation

arguments can be applied by defining kRY , kY G and kGR as the transition rates from one type of agent to the

next in the cycle. It is useful to define S(x, y) as the average occupancy of a lattice site by any type of agent

i.e. S(x, y) = R(x, y) + Y (x, y) + G(x, y). This allows a convenient way to represent simply whether a site is

occupied or not on average. This leads to the following system of conservation equations:

δR =PR6

[6∑k=1

Rk(1− S)−6∑k=1

R(1− Sk)

]− kRYR+

kGR6

[6∑k=1

G(1− Sk) +

6∑k=1

Gk(1− S)

]

δY =PY6

[6∑k=1

Yk(1− S)−6∑k=1

Y (1− Sk)

]+ kRYR− kY GY

δG =PG6

[6∑k=1

Gk(1− S)−6∑k=1

G(1− Sk)

]+ kY GY −

kGR6

[6∑k=1

G(1− Sk)

]

where δR is the change in average occupancy of site (x, y) of the red agent population, likewise for δY and δG.

Sk is the average total occupancy of neighbour lattice site k, similar to the definition of Ck previously. Note that

the negative terms in these expressions indicate an event where site x, y loses an agent of a particular type. For

example when an agent transitions from red to yellow, the total occupancy, S(x, y) is unaffected however the

red population decreases, whereas the corresponding yellow population increases. This also results in the red

population of (x, y) increasing by either a green agent proliferating successfully on site (x, y) or when a green

agent deposits a daughter agent on (x, y), providing the site is available. With these conservation statements,

Taylor series approximations (see eq. (2)) can be applied for R, Y , G and subsequently S. After simplification

8

including the use of Equation (3) and the product rule to simplify the migration terms, the following expression

is obtained:

δR =PR∆2

4∇ · [(1− S)∇R+R∇S]− kRYR+ 2kGRG(1− S) +

kGR∆2

4

[(1− S)∇2G+G∇2S

]+O(∆3)

δY =PY ∆2

4∇ · [(1− S)∇Y + Y∇S]− kY GY + kRYR+O(∆3)

δG =PG∆2

4∇ · [(1− S)∇G+G∇S] + kY GG− kGRG(1− S) +

kGR∆2

4G∇2S +O(∆3)

By letting ∆ → 0 in the limit, a coupled system of nonlinear PDEs is derived. Note also that the use of

rates instead of probabilities does not effect the limiting arguments in this case, due to the rates being defined

as probabilities over some time τ .

∂R

∂t=PR∆2

4∇ · [(1− S)∇R+R∇S]− kRYR+ 2kGRG(1− S) (8a)

∂Y

∂t=PY ∆2

4∇ · [(1− S)∇Y + Y∇S]− kY GY + kRYR (8b)

∂G

∂t=PG∆2

4∇ · [(1− S)∇G+G∇S] + kY GG− kGRG(1− S) (8c)

By using a similar initial condition as previously for each agent type, the problem can be simplified into the

one-dimensional case.

∂R

∂t=PR∆2

4

∂

∂x

[(1− S)

∂R

∂x+R

∂S

∂x

]− kRYR+ 2kGRG(1− S) (9a)

∂Y

∂t=PY ∆2

4

∂

∂x

[(1− S)

∂Y

∂x+ Y

∂S

∂x

]− kY GY + kRYR (9b)

∂G

∂t=PG∆2

4

∂

∂x

[(1− S)

∂G

∂x+G

∂S

∂x

]+ kY GY − kGRG(1− S) (9c)

Equation (9) was solved using finite difference discretisation and fixed-point iteration as in Equation (7) on

each PDE. It was assumed for initial testing purposes that migration and transition rates were identical between

each sub-population, however the model incorporates any combination of parameters provided the assumptions

of the continuum limit hold.

Figure 7 displays interesting behaviour arising under this particular choice of parameters. Figure 7a and

7b have an immediate decrease in density in the centre as agents migrate in the x direction with the yellow

population lagging slightly behind the red. In contrast, Figure 7c shows that the number of agents ready to

undergo a proliferation event increases, eventually dominating the other two sub-populations. This is due to the

overcrowding of agents leading to the lack of vacant sites to deposit daughter agents within the initial cluster.

9

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Red Density Profiles

t = 0ht = 24ht = 48ht = 72ht = 96ht = 120h

(a) Red agent density

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Yellow Density Profiles

t = 0ht = 24ht = 48ht = 72ht = 96ht = 120h

(b) Yellow agent density

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Green Density Profiles

t = 0ht = 24ht = 48ht = 72ht = 96ht = 120h

(c) Green agent density

0 10 20 30 40 50 60 70 80 90 100

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Den

sity

Total Density Profiles

t = 0ht = 24ht = 48ht = 72ht = 96ht = 120h

(d) Total agent density

Figure 7: Numerical solution to Equation (9) with parameters PR = PY = PG = 1, kRY = kY G = kGR = 0.1

and ∆ = 1. Boundary conditions are Dirichlet 0 conditions for each PDE. The inital condition is 0.33 density

for each sub-population in a centre region, given as: R(x, 0) = Y (x, 0) = G(x, 0) = 0.33[H(x−40)−H(x−60)].

Numerical solver parameters were δx = 1 and δt = 1 and fixed-point convergence tolerance of ε = 1× 10−5.

5 Conclusion

This report presents the results of extending previously studied models of cell motility and proliferation to

incorporate FUCCI technology. The accuracy of the single-phase continuum model was verified against the

corresponding stochastic random walk model and then successfully extended to derive a multi-phase continuum

model. Further research will compare this system to a corresponding discrete stochastic random walk model

before finally verifying the behaviour of these systems with real scratch assay data. It is recommended that the

behaviour of the system under other parameter choices is tested or studied using dynamical systems theory to

determine the validity of the initial assumptions used to form the model. It may also be possible to incorporate

cell death into the initial formulation to derive a more realistic model.

10

Acknowledgements

I would like to thank the Australian Mathematical Sciences Institute and my supervisor Prof. Scott McCue for

the opportunity to undertake this research and the valuable experience I have gained. Furthermore, I would

like to thank Prof. Matthew Simpson and Dr Wang Jin for assistance in developing and implementing the

models presented in this report. I would also like to thank my colleague Tamara Tambayah who also undertook

an AMSI research scholarship, focusing on the discrete approach of modelling FUCCI, for her assistance in

implementing code.

References

[1] Kimberley A. Beaumont, Andrea Anfosso, Farzana Ahmed, Wolfgang Weninger, and Nikolas K. Haass.

Imaging- and flow cytometry-based analysis of cell position and the cell cycle in 3d melanoma spheroids.

Journal of Visualized Experiments : JoVE, (106):53486, 2015.

[2] Nikolas K Haass, Kimberley A Beaumont, David S Hill, Andrea Anfosso, Paulus Mrass, Marcia A Munoz,

Ichiko Kinjyo, and Wolfgang Weninger. Realtime cell cycle imaging during melanoma growth, invasion, and

drug response. Pigment Cell & Melanoma Research, 27(5):764–776, 2014.

[3] Wang Jin, Esha T Shah, Catherine J Penington, Scott W McCue, Lisa K Chopin, and Matthew J Simpson.

Reproducibility of scratch assays is affected by the initial degree of confluence: experiments, modelling and

model selection. Journal of Theoretical Biology, 390:136–145, 2016.

[4] Stuart T Johnston, Matthew J Simpson, and DL Sean McElwain. How much information can be obtained

from tracking the position of the leading edge in a scratch assay? Journal of the Royal Society Interface,

11(97):20140325, 2014.

[5] Chun-Chi Liang, Ann Y. Park, and Jun-Lin Guan. In vitro scratch assay: a convenient and inexpensive

method for analysis of cell migration in vitro. Nature Protocols, 2:329, 2007.

[6] Asako Sakaue-Sawano, Hiroshi Kurokawa, Toshifumi Morimura, Aki Hanyu, Hiroshi Hama, Hatsuki Os-

awa, Saori Kashiwagi, Kiyoko Fukami, Takaki Miyata, and Hiroyuki Miyoshi. Visualizing spatiotemporal

dynamics of multicellular cell-cycle progression. Cell, 132(3):487–498, 2008.

[7] Matthew J Simpson, Kerry A Landman, and Barry D Hughes. Cell invasion with proliferation mechanisms

motivated by time-lapse data. Physica A: Statistical Mechanics and its Applications, 389(18):3779–3790,

2010.

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