Modeling the Effect of Exogenous Calcium on Keratinocyte

and HaCat Cell Proliferation and Differentiation

Using an Agent-Based Computational Paradigm

DAWN WALKER, Ph.D.,1 TAO SUN, Ph.D.,2 SHEILA MACNEIL, Ph.D.,2

and ROD SMALLWOOD, Ph.D.1

ABSTRACT

In this study we sought to develop a computational modeling paradigm in order to describe the influenceof calcium on normal and transformed keratinocyte proliferation and differentiation. Keratinocytes andHaCat cells were grown in monolayer cultures with low and physiologic calcium concentrations, andlevels of proliferation and involucrin expression were assessed. Both types of cells grew as monolayersunder a low-calcium environment, and stratified in media with physiologic levels of calcium. However,keratinocytes were more proliferative in low rather than physiologic levels of calcium, whereas theopposite was true for HaCat cells. Normal keratinocytes differentiated as calcium levels increased. HaCatcells showed little differentiation at any calcium concentration. However, while the computer simulationcould be modified to describe the effect of calcium on the growth of normal keratinocytes, our findingsdid not support the hypothesis that simply ‘‘turning off’’ the ability of HaCat cells to differentiate wouldaccount for the growth characteristics of these transformed cells. This demonstrates the application ofcomputational modeling to hypothesis testing in biological systems.

INTRODUCTION

C OMPUTATIONAL MODELS that attempt to describe the

behavior of biological tissue using a set of mathe-

matical equations fail to capture the granularity of the tis-

sue and the behavior of individual cells in the tissue. Our

primary interest is in the interactions of individual cells,

and the resulting emergent tissue structure and function.

We have developed a cell-based model of the growth

characteristics of urothelial cells in monolayer culture in

low or physiologic levels of calcium,1 which gives a qua-

litative fit to in vitro cell behavior. The agent-based model

was also used to predict wound healing in urothelial cell

cultures maintained in low and physiologic exogenous

calcium [Ca2þ]e. This simple rule-based model of cell

behavior, incorporating rules relating to contact inhibition

of proliferation and migration, was sufficient to qualita-

tively predict the calcium-dependent pattern of wound

closure observed in vitro.2

We now extend this approach to normal human kerati-

nocytes, which, unlike urothelial cells, will differentiate

in vitro if conditions permit. In the absence of permissive

levels of calcium ([Ca2þ]e levels lower than 0.5mM),

keratinocytes are unable to stratify and differentiate; rather,

they proliferate rapidly with a high growth fraction. They

can be rapidly induced to terminally differentiate by re-

storing calcium to the level normally found in standard

culture medium (1.2mM).3 When calcium is elevated to

physiologic levels in culture, cell-to-cell contact occurs

rapidly and desmosomes form within 2 h. The cells stratify

1Department of Computer Science, Kroto Research Institute, North Campus, University of Sheffield, Broad Lane, Sheffield, United

Kingdom.2Department of Engineering Materials, Kroto Research Institute, North Campus, University of Sheffield, Broad Lane, Sheffield,

United Kingdom.

TISSUE ENGINEERINGVolume 12, Number 8, 2006# Mary Ann Liebert, Inc.

1

by l–2 days and terminally differentiate with cell sloughing

by 3–4 days.3,4

The HaCat cell line is a spontaneously transformed

(following overheating of an incubator) human epithelial

cell line derived from adult skin. It is immortal (>140

passages), and has a transformed phenotype in vitro (clono-

genic on plastic and in agar) but remains nontumorigenic.

Despite their altered and unlimited growth potential, HaCat

cells have been reported to form an orderly structured and

differentiated epidermal tissue when transplanted onto nude

mice.5

The primary aim of this work was to further develop our

existing computational model in order to describe the in-

fluence of extracellular calcium on the growth and differ-

entiation of normal keratinocytes. The second aim was to

explore the potential of the model to explore a simple hypo-

thesis that we proposed to explain the very different response

to calcium of a spontaneously transformed keratinocyte cell

line (HaCats) compared to normal keratinocytes.

MATERIALS AND METHODS

Cell culture

Normal human keratinocytes (NHKs) were isolated as

described elsewhere.6–8 Defined keratinocyte-serum free

medium (SFM) (Invitrogen, UK; calcium concentration

0.09mM) supplemented with 100 IU/mL penicillin and

100 mg/mL streptomycin was used for both NHK and Ha-

Cat cell cultures. The concentration of calcium in the

medium was increased from 0.09mM to 2mM by the ad-

dition of calcium chloride.

Measurement of cell proliferation

Cell number was assessed using the MTT (3-(4,5-

Dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide)

assay (an indirect assay of total cell viability) as described

by Marcus et al.;9 total DNA was measured using Hoechst

fluorescent stain (33258; Sigma-Aldrich, UK) as done by

Higham et al.10 (this measures all cellular DNA regardless

of whether cells are alive or dead) and by using bromo-

deoxyuridine (BrdU) immunostaining, which measures ac-

tive DNA synthesis, as described in a following section.

Cells were incubated in BrdU-labeling medium (Roche

Applied Science, UK) for 4 h, fixed in ethanol:glycine fixa-

tive (3 parts 50mM glycine solution:7 parts absolute etha-

nol) and stored at �208C for 20–30min. After a thorough

wash with phosphate-buffered saline to remove all traces of

fixative, anti-BrdU monoclonal antibody (Roche Applied

Science) was added for 30min at 378C. The cells then were

incubated with fluorescein-conjugated anti-mouse second-

ary antibody for 30min. After counterstaining with 40,6-diamidino-2-phenylindole dihydrochloride (DAPI) for

5min,5 the cells were visualized using an Axon image ex-

press system (Molecular Devices, Sunnydale, CA).

Identification of keratinocytes and assessment

of differentiation

NHK or HaCat cells were immunolabeled with pancy-

tokeratin (which stains all keratinocytes) or involucrin

(which is an indicator of cornified envelope formation and

identifies differentiating cells) and visualized as described

elsewhere.11,12 The extent of involucrin expression was

then related to total DNA to determine the extent of dif-

ferentiation in a population of cells.

All the fluorescent stains were visualized using an Axon

image express system (Molecular Devices). Briefly, fluor-

escence micrographs of immunolabeled samples were taken

using epifluorescent illumination at lex¼ 495 nm, lem¼515 nm (for fluorescein isothiocyanate/pancytokeratin, in-

volucrin, BrdU visualization) and lex¼ 358 nm, lem¼461 nm (for DAPI/nuclei visualization).

The stain was then eluted by incubating the cells in 0.1M

sodium hydroxide for 1 h at 378C, transferred to a cuvette,

and quantified using a fluorescence plate reader (lex¼495 nm, lem¼ 515 nm for fluorescein isothiocyanate/

pancytokeratin, involucrin, BrdU; lex¼ 358 nm, lem¼461 nm for DNA).

Computational modeling

We have developed an ‘‘agent-based’’ paradigm using

Matlab (The Mathworks, Inc., Natick, MA), in which in-

dividual biological cells are represented as computational

entities or ‘‘software agents.’’ These agents live on a 2-

dimensional, square substrate with user-defined dimensions

and modifiable exogenous [Ca2þ]e. The user can select the

number of cells placed either randomly or in specified lo-

cations. Mean cell cycle time and migration rate are scaled

so that each time step in the model represents approxi-

mately 30min in real time. At each time step, agents are

interrogated in turn. The internal parameter state (e.g.,

position in cell cycle, flag indicating attachment to sub-

strate) and environment (number and proximity of neigh-

boring cells, [Ca2þ]e) control the execution of rules that

may change the internal parameters. These rules represent,

for instance, intercellular bonding, changes in morphology,

cell cycle progression, and arrest. If a cell divides, one

daughter cell overwrites the data structure of the parent,

and a second daughter cell is added to the simulation. A

simple ‘‘physical model’’ resolves the intercellular forces

resulting from cell movement, growth, and division in the

plane of the substrate. The general architecture of the

model is shown in Fig. 1.

The current model simulates the behavior of cells in

differing exogenous calcium ion concentrations. In parti-

cular, we have studied the effect [Ca2þ]e on proliferation.

The probability of an intercellular bond being formed be-

tween a pair of modeled cells is inversely proportional to

the separation between the edges and related to [Ca2þ]e via

a sigmoid function with the inflection point at 1.0mM,

based on the relationship between the binding activity of

2 WALKER ET AL.

E-cadherin–mediated bonds and [Ca2þ]e.13 Cells with 4 or

more intercellular bonds enter the G0 phase of the cell

cycle and become quiescent, which results in a lower

growth rate in models with higher [Ca2þ]e. Simulations in

low and physiologic [Ca2þ]e agree qualitatively with

in vitro experiments on urothelial cells.1

The first objective of the current study was to determine

what changes in the model rules and parameters for ur-

othelial cells were needed in order to represent the char-

acteristics of normal human keratinocytes; the second

was to explore the hypothesis that the aberrant behavior

of transformed human keratinocytes (HaCat cells) can

be explained by the inactivation of the differentiation

pathway.

Modeling NHKs. Table 1 summarizes the changes to the

rule sets that were necessary to represent keratinocyte, in

contrast to urothelial cell behavior. In particular, new rule

sets governing differentiation (defined in terms of in-

volucrin expression and cell cycle exit) and stratification

were needed. The general form of these rules (i.e., in-

creased involucrin expression/stratification with increased

exogenous calcium) was constructed on the basis of known

aspects of cell behavior, but the actual parameters defining

the rule sets were tuned to give population level results that

agreed with experimental data. A simpler ‘‘physical’’ al-

gorithm than that described elsewhere1,2 was used, with

cells sequentially shifted a maximum of 5 times per agent

iteration to minimize overlap with their neighbors. This

reduced the solution time without significantly affecting the

results (data not shown).

All model parameters were set to replicate those used in

the in vitro experimental work as closely as possible.

Model simulations were run with exogenous calcium con-

centrations set to 9 values between 0.05mM and 2.5mM.

In the case of the normal human keratinocyte population

growth simulations, or ‘‘virtual MTT assays,’’ the model

substrate size was set at 1�1mm and the number of cells

seeded at the start of the simulation was 400, giving an

initial seeding density of 4�104 cells per cm2.

The minimum cell cycle times in the model were set to

27 h (54 model iterations), and cell migration speed was set

to 20 mm/iteration (equivalent to 0.67 mm/min, a value de-

rived from several observations of cell migration published

in the literature). Simulations applied the agent rules and

simple physics interactions to the seeded cell populations.

At every iteration, statistics relating to the state of the cell

population (e.g., total cell number, number of cells with

bonds, number of cells in different phases of the cell cycle)

were saved.

In the case of the ‘‘virtual involucrin assays,’’ the model

protocol was matched to the in vitro procedure as closely

as possible (i.e., an initial cell population with a density

of 2�105 cells per cm2 was ‘‘expanded’’ for 7 days [350

iterations] in 0.05mM calcium, the model was saved, the

calcium level switched to 1 of the 9 values between 0.05mM

and 2.5mM, and the simulations resumed for a further 5

days [250 iterations]).

To ensure that trends predicted by the model were due to

the cellular level rules and were not significantly affected

by starting conditions or stochastic events based on random

numbers generated during the simulations, each set of mod-

els was run 3 times using different starting conditions and

random starting positions of seeded cells.

Modeling HaCat cells. This transformed cell line shows

increased proliferation compared to normal human kerati-

nocytes. It has been hypothesized that the differentiation

pathway for these cells has somehow been disabled, al-

though the precise mechanism is not known.14 We used our

computational model to examine this hypothesis. In the

HaCat simulations, cells were modeled to not express in-

volucrin, or to permanently leave the cell cycle. All other

rules remain the same as for untransformed cells.

In the case of the HaCat models, both a 0.5�0.5-mm

substrate with 100 seeded cells and a 1�1-mm substrate

with 400 seeded cells were used, giving identical initial

seeding densities to the normal human keratinocyte simu-

lations.

COMMUNICATIONRULES

For everycell agent

APOPTOSISRULES

MOTILITYRULES

BONDINGRULES

MORPHOLOGYRULES

CELL CYCLERULES

Incrementtime, t

DIFFERENTIATIONRULES

STRATIFICATIONRULES

EQUILIBRIATE CELL POSITIONSDUE TO GROWTH, MIGRATION, ETC.

FIG. 1. Flow chart illustrating model structure. Boxes with bold

outline indicate new behavior introduced to model keratinocytes.

EXOGENOUS CALCIUM ON KERATINOCYTE AND HaCat CELL PROLIFERATION 3

RESULTS

In vitro results

Effect of calcium on cell proliferation. Fig. 2 shows

DAPI (part A) and pancytokeratin (part B) immuno-

fluorescence micrographs of keratinocyte distribution

approximately 12 days after plating in defined keratinocyte-

SFM medium with low calcium (0.09mM). In physiologic

calcium (2mM), most of the cells stratified (as shown in

Fig. 2C and D) and the formation of colonies is apparent,

whereas cells are more evenly distributed in the low-

calcium medium.

Similar experiments with HaCat cells are shown in Fig.

2E–H. In the low-calcium media (Fig. 2E and F) cells are

very clearly separated, with no evidence of cell-to-cell

contact. In media with physiologic levels of calcium, there

is more evidence of cell-to-cell contact but the degree of

aggregation or stratification is clearly less than for the

normal keratinocytes if one contrasts Fig. 2G and H with

Fig. 2C and D.

TABLE 1. SUMMARY OF MODEL RULES

Rule type

Variable Urothelium rule Keratinocyte rule Justification

Bonding Cell has 1 in 3 chance of binding

to substrate per iteration.

Probability of bonding to substrate

related to involucrin expression

level of cell via a sigmoidal

relationship.

Keratinocytes are harder to seed

than urothelial cells—

approximately 50% do not

attach after 18 h and die. It is

probably the more differentiated

cells that do not attach.

Apoptosis Cells that have failed to bond to

the substrate after approximately

7 h have 1/50 chance of under-

going apoptosis per iteration.

No change. Cells that fail to attach undergo

apoptosis.

Cell cycle Cells enter cycle immediately

after bonding to substrate.

Cells start to cycle 300 iterations

after bonding to substrate.

Keratinocytes undergo a lag phase

of several days before

commencing exponential growth.

Cell cycle Minimum cell cycle time 15 h. Minimum cell cycle time increased

to 27 h.

Keratinocytes in culture proliferate

more slowly than urothelial cells.

Cell cycle Cells require 4 intercellular

bonds before becoming contact-

inhibited and entering G0.

Cells require 6 intercellular bonds

before becoming contact-

inhibited and entering G0.

Keratinocytes in higher calcium

levels stratify, and thus form

more bonds than urothelial cells.

Cells can continue to cycle even

in stratified colonies.

Differentiation None. Cells are seeded with an initial

involucrin level randomly

distributed between 0 and 1. Once

the cell has bonded to the

substrate and entered the cell

cycle, this expression level is

increased by [Ca2þ]e/500

per iteration until a maximum

level of 1.

Keratinocytes obtained from

patients will be at various stages

of differentiation. Once in cul-

ture, they express involucrin in

a calcium-dependent fashion.

Differentiation None. A cell with an involucrin expression

level of �0.5 will differentiate

and permanently leave the cell

cycle.

Unlike urothelial cells,

keratinocytes differentiate in

cell culture.

Stratification None. Cells subject to compressive forces

from at least 3 neighbors can lose

contact with the substrate, then

ascend to form a suprabasal layer.

The probability of any cell stra-

tifying is related to exogenous

calcium via a sigmoidal function

with the inflection point

at 1.5mM.

In contrast to urothelial cells, which

grow as a monolayer regardless

of exogenous calcium levels,

keratinocytes in higher calcium

stratify before confluence.

Stratification tends to occur only

in exogenous calcium levels>1mM.

4 WALKER ET AL.

Quantification of cell proliferation was assessed with

MTT, DNA, and BrdU assays. Fig. 3A shows the in vitro

MTT assay of normal human keratinocyte growth con-

trasted with that of HaCat cells (Fig. 3B) over a period

of 12–13 days in media with low or physiologic calcium.

For normal keratinocytes, after an initial lag phase of ap-

proximately 7–8 days the cell population of keratinocytes

in the low-calcium media begins to expand rapidly. Some

5 days later, the maximum cell number was reached (a total

of 12 elapsed days). However, the expansion of the cell

population in the physiologic-calcium medium was very

low even after 14 days of plating.

In contrast to the keratinocytes, HaCat cell populations

in physiologic-calcium media began to expand rapidly after

an initial lag phase of approximately 7–8 days, while the

expansion of the cell population in low-calcium medium

remained very low throughout the 12 days of study (as

illustrated in Fig. 3B).

Fig. 4A confirms these data assessing cell proliferation

by showing both total DNA and DNA synthesis levels in

cells over a range of calcium concentrations after 7–8 days

of culture. NHK proliferation was higher in low-calcium

media than in physiologic-calcium media, whereas HaCat

cell proliferation was lower in low-calcium media (Fig. 4B).

FIG. 2. Micrographs of (A) 40,6-diamidino-2-phenylindole

dihydrochloride (DAPI)–stained and (B) pancytokeratin-

immunostained normal human keratinocytes cultured in medium

with low calcium, (C) DAPI-stained and (D) pancytokeratin-

immunostained normal human keratinocytes cultured in me-

dium with physiologic calcium, (E) DAPI-stained and (F)

pancytokeratin-immunostained HaCat cells cultured in medium

with low calcium, (G) DAPI-stained and (H) pancytokeratin-

immunostained HaCat cells cultured in medium with physiologic

calcium. (Bar¼ 100 mm.) Color images available online at

www.liebertpub.com/ten.

Time in days

Tot

al li

ve c

ellc

dens

ity10

3 /m

m2

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 5 10 15

(D)Time in days

(A)

00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

MT

T O

ptic

al D

ensi

ty

5 10 15

0.09mM Ca2+

2.0mM Ca2+

0 5 10 150

MT

T O

ptic

al D

ensi

ty

Time in days

0.05

0.1

0.15

0.2

0.25

0.3

0.35(B)

0.09mM Ca2+

2.0mM Ca2+

(C)

0 5 10 15

Time in days

0

0.5

1.0

1.5

2.0

2.5

3.0

Tot

al li

ve c

ell c

dens

ity10

3 /m

m2

0.51.01.251.52.02.5

0.050.51.01.251.52.02.5

0.05

FIG. 3. Total cellular viability of (A) normal human keratino-

cytes cultured in medium with low and physiologic levels of cal-

cium and (B) HaCat cells cultured in medium with low and

physiologic levels of calcium. (C) Virtual 3-(4,5-Dimethylthiazol-

2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay (i.e., mod-

eled total live cell population) at different calcium concentrations.

Normal human keratinocytes (D) virtual MTT assay, HaCat cells.

Calcium concentration in mM(C) (D)

(A) (B)

0 0.5 1 1.5 2 2.5 32.1

2.2

2.3

2.4

2.5

2.6

2.7

0.91

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Calcium concentration in mM

BrD

U o

ptic

al d

ensi

ty x

103

DN

A o

ptic

al d

ensi

ty x

104

DNABrDU

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

Tot

al c

ell d

ensi

ty 1

03/m

m2

Ca2+ concentration (mM)

0 0.5 1 1.5 2 2.5 30.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

DN

A o

ptic

al d

ensi

ty x

104

0

1.0

2.0

3.0

4.0

5.0

6.0

BrD

U o

ptic

al d

ensi

ty x

103

DNABrDU

0 0.5 1 1.5 2 2.5 3Ca2+ concentration (mM)

0

0.5

1.0

1.5

2.0

2.0

3.0

Tot

al c

ell d

ensi

ty 1

03/m

m2

FIG. 4. Total population DNA of normal human keratinocytes

(A) and HaCat cells (B) cultured in medium with varying levels of

calcium. Virtual DNA assay (total modeled cell density vs cal-

cium concentration in mM) for normal human keratinocytes (C)

and HaCat cells (D).

EXOGENOUS CALCIUM ON KERATINOCYTE AND HaCat CELL PROLIFERATION 5

Effect of calcium on differentiation of keratinocytes and

HaCats. Figure 5A–D illustrates that NHKs produced more

involucrin in physiologic-calcium media than in low-

calcium media while involucrin expression in HaCat cells

(Fig. 6A–D) remained low whether cultured in physiologic-

calcium media or low-calcium media.

Fig. 7 shows the involucrin expression relative to DNA

content of the cultures for both keratinocytes and HaCats

cultured in media with varying levels of calcium. These

results indicate that keratinocytes in a high-calcium media

clearly produce more involucrin per cell than when cultured

in a low-calcium media (Fig. 7A). In contrast, HaCat cells

showed no evidence of an increase in differentiation with

increasing levels of calcium (Fig. 7B).

Model simulations

Effect of calcium on cell proliferation. Fig. 5E and F

shows plots of cells after simulating a time equivalent to 12

days in 0.05mM and 2.0mM calcium in the normal human

keratinocyte simulations. In low calcium (0.05mM), the

model cells are confluent and evenly spaced, with virtually

no stratification. By contrast, cells in physiologic calcium

(2.0mM) are stratified, although they have not reached

confluence. Fig. 6E and F shows the equivalent plots for the

HaCat simulations. In low calcium (Figs. 5 and 6C) the

results appear very similar. However, in physiologic cal-

cium, close inspection reveal many more cells, forming

several fully confluent layers. No green cells are visible in

Fig. 6C and D, reflecting the imposed restriction that model

HaCat cells do not express involucrin and therefore cannot

terminally differentiate.

Fig. 3C shows the results of the virtual MTT assay (i.e.,

the number of live cells against simulation time) in 9 dif-

ferent calcium environments for modeled NHKs. After an

initial lag period of approximately 1 week, during which

the live cell population decreases slightly (because of the

apoptosis of unattached cells), the cell population in the

lower-calcium NHK simulations increases rapidly, reach-

ing a maximum level after approximately 11 days. In the

physiologic-calcium NHK simulations, the rate of popula-

tion expansion is much slower, with the final live cell

number totaling approximately 35–40% of that for the

lowest calcium. Intermediate-calcium models exhibit be-

havior between that observed in the low and high condi-

tions, with the monotonic trend that increasing calcium

results in a lower growth rate.

The final total cell population (including living and dead

cells) in the different NHK simulations is shown in Fig. 4C.

There is a curvilinear relationship between cell number and

calcium, with the highest total population occurring at the

lowest calcium.

The virtual MTT and DNA results for HaCat cells are

shown in Figs. 3D and 4D. It is apparent that in contrast to

the NHK results, increased calcium is associated with an

increase in population growth. This was observed to be due

to the tendency of cells in the higher-calcium simulations to

stratify, thus effectively increasing the space available for

cell growth. At the lowest calcium level, the model growth

results for NHKs and HaCats are identical.

Effect of calcium on cell differentiation. Inspection of

Fig. 5E and F shows a significantly greater degree of dif-

ferentiation in the higher-calcium NHK model, where

80–90% of the cells are committed (shown in green), in

contrast to the lower calcium case, where only approxi-

mately 10% of cells have permanently left the cycle. In the

latter case, most of this differentiation arises from the fact

that the initial seeded cell population has an involucrin

level randomly distributed between 0 and 1, and although

many of the cells with higher involucrin levels will fail to

attach and hence undergo apoptosis (see rules described in

Table 1), some will survive and differentiate. HaCat cells

have been modeled to not express involucrin and therefore

cannot terminally differentiate; thus, no green cells are

visible in Fig. 6E and F.

Fig. 7C shows the results of the virtual involucrin assay

for NHKs (i.e., the mean involucrin expression per modeled

cell after 7 days growth in 0.05mM calcium and a further 5

days at an adjusted calcium level, similar to the in vitro

assay). It is apparent that much more involucrin is ex-

pressed at calcium levels greater than 1.0mM. Again,

modeled HaCat cells do not express involucrin, so Fig. 7D

is simply a straight line.

DISCUSSION

The specific objective of this study was to find a mini-

mum set of behavioral differences that could explain the

different growth characteristics of human skin epithelial

cells versus human bladder epithelial cells (results outlined

in a previous publication1) in various calcium environ-

ments, and, further, to examine the hypothesis that the

behavior of a keratinocyte transformed cell line could

be replicated simply by ‘‘turning off’’ the differentiation

pathway.

The 2 major adaptations made to the model in order to

model normal keratinocyte-like, as opposed to urothelial-

like, behavior were (1) inclusion of calcium-dependent

differentiation and (2) inclusion of calcium-dependent

stratification. The first rule allowed model cells to increase

the concentration of a hypothetical substance analogous to

the cytoplasmic protein involucrin, known to be associated

with the differentiation process, by a factor linearly related

to the environmental calcium concentration [Ca2þ]e, and to

irreversibly leave the cell cycle (become committed) when

an arbitrary threshold was reached (i.e., for the ith cell):

cell{i}_involucrin¼ cell{i}_involucrinþ [Ca2þ]e/500;

If (cell{i} is a proliferating cell) AND (cell{i}_involucrin>0.5)

cell{i} becomes a committed cell

6 WALKER ET AL.

The second rule allows cells subject to a small com-

pressive force exerted by 3 or more neighbors to move

upwards, with the probability of the cell’s moving being

sigmoidally related to the calcium concentration. A sig-

moidal relationship was selected because intercellular

bonding strength has previously been shown to be related to

exogenous calcium in this way,13 and these bonds are

crucial in the stratification process as they allow a cell to

‘‘climb’’ over its neighbors.

In this case, the rule is slightly more complex because it

also takes into account the involucrin expressed by the cell

(it has been hypothesized that cells lose adhesion to the

substrate as they differentiate15), and the cell radius (a cell

subjected to compressive forces from its neighbors will first

become ‘‘squashed,’’ then stratify as it loses contact with

the substrate):

If (force on cell{i}> 0.1) AND (cell{i} is in contact with at

least 3 others)

radius_probability_index¼minimum radius/cell{i}_current

radius;

differentiation_probability_index¼ cell{i}_involucrin;

FIG. 5. Micrographs of (A) 40,6-diamidino-2-phenylindole di-

hydrochloride (DAPI)-stained and (B) involucrin-immunostained

normal human keratinocytes cultured in medium with low cal-

cium, (C) DAPI-stained and (D) involucrin-immunostained nor-

mal human keratinocytes cultured in medium with high calcium.

(E and F) Plots of modeled normal human keratinocytes after

12 days of simulated time in 0.05mM calcium (E) and 2.0mM

calcium (F). Red: stem cell; blue: transit-amplifying cell; pink:

mitotic cell; green: terminally differentiated cell.

FIG. 6. Micrographs of (A) 40,6-diamidino-2-phenylindole di-

hydrochloride (DAPI)-stained and (B) involucrin-immunostained

HaCat cells cultured in medium with low calcium, (C) DAPI-

stained and (D) involucrin-immunostained HaCat cells cultured in

medium with high calcium. (E and F) Plots of modeled HaCat

cells after 12 days of simulated time in 0.05mM calcium (E) and

2.0mM calcium (F). Red: stem cell; blue: transit-amplifying cell;

pink: mitotic cell.

(A)

(C)

0 0.5 1 1.5 2 2.5 3Calcium concentration in mM

Invo

lucr

in/t

otal

cel

l no.

(D)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Calcium concentration in mM

Invo

lucr

in/t

otal

cel

l no.

30 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Calcium concentration in mM

Invo

lucr

in/D

NA

(B)

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Calcium concentration in mM

Invo

lucr

in/D

NA

FIG. 7. The effect of calcium concentration on the differentia-

tion of (A) normal human keratinocytes and (B) HaCat cells

in vitro. (C and D) Virtual involucrin assay–mean involucrin per

cell after 5 days in modified calcium normal human keratinocytes

(C) and HaCat cells (D).

EXOGENOUS CALCIUM ON KERATINOCYTE AND HaCat CELL PROLIFERATION 7

calcium_probability_index¼ tanh(pi*(env_ca� 1.5))þ 1)/

20;

If (product of probability indices> random number)

then cell{i}_layer¼ cell{i}_layerþ 1;

Inclusion of these 2 new rules has a conflicting effect on

cell population growth. The tendency of cells to differ-

entiate more quickly in higher calcium means that more

cells permanently leave the cell cycle, which tends to de-

crease the growth rate, whereas stratification increases the

effective space available for growth and thus tends to in-

crease the growth rate. In the case of the rule parameters

chosen, the differentiation effect dominates and cells grow

significantly more slowly in higher calcium levels (Fig.

3C), which agrees well with the in vitro observations.

The tendency of cells to grow in compact colonies, and

to stratify in physiologic calcium in vitro means that it is

extremely difficult to obtain a precise value for total cell

number, a parameter that it is trivial to extract from a

computational model simulation. Including results from

MTT, DNA, and BrdU assays (all indirect assessments of

cell number) gives the best estimate of the cell population

that is feasible for these cells in vitro. One indicator of the

accuracy of a computational model is its ability to replicate

the qualitative pattern of behavior seen in the in vitro ex-

periments. This was achieved for the calcium-dependent

growth, in that after approximately 2 weeks the model cell

population in the lowest calcium environment is roughly

2.5 times that in the highest calcium (a good agreement

with the MTT data). Similarly, an exact quantitative agree-

ment between modeled involucrin levels and those mea-

sured using immunofluorescence techniques in the in vitro

experiments cannot be expected, but the general pattern of

change shown in Fig. 7A (in vitro data) and C (modeled

data) is qualitatively similar. Hence we have demonstrated

that the rule changes summarized in Table 1 are sufficient

to represent keratinocyte behavior, in contrast to urothelial

cell behavior. The most important of these rules relate to

the ability of keratinocytes, but not urothelial cells, to

spontaneously stratify and differentiate in culture. How-

ever, these simple rules do not provide any information

about the mechanisms underlying these differences, which

will be related to the different genes and proteins expressed

in the 2 cell types, as well as a result of different culture

conditions routinely used in vitro. An experimental in-

vestigation with the aim of pin-pointing inherent versus

culture artifact induced differences between urothelial and

keratinocyte behavior is currently in the planning stages.

The tradeoff between the effects of differentiation and

stratification on proliferation leads to the question of

whether it is possible to have a situation in which the dif-

ferentiation pathway is somehow impaired, allowing cells

in higher calcium environments to grow more quickly than

in lower calcium. This was not observed for any of the

NHK models; for HaCat models, however, where the dif-

ferentiation rule was removed, the cell populations in 1mM

calcium and above expanded quickly, with those in the

highest calcium environments reaching up to 8 layers thick,

with no signs of leveling off at the end of the simulation

period. By contrast, at calcium levels below 1mM, little

stratification was evident, and population growth was ab-

rogated by contact inhibition.

In assembling the model rules for HaCat cells, we have

deliberately explored the assumption that this cell line has

lost the ability to differentiate and does not express in-

volucrin. However, Fig. 7B suggests that the involucrin

level in HaCat cell populations cultured in low calcium is

actually higher than levels in intermediate to high calcium.

At low calcium concentrations, however, we know that the

measurements are based on low cell numbers and thus are

unlikely to be as accurate as those based on much higher

cell numbers at higher calcium levels.

Inspection of MTT and DNA data (Figs. 3 and 4A and B)

shows that the growth of HaCat cells is compromised

compared to NHKs in vitro, particularly at low calcium le-

vels. By contrast, the model predicts that in low calcium the

growth of HaCat cells will be similar to that of NHKs,

whereas HaCat growth at higher calcium levels will actually

be enhanced by approximately a factor of 4. Hence, the

assumption we have tested in the model—that differentia-

tion is disabled in HaCat cells—is not sufficient to explain

the observed population growth characteristics. We there-

fore suggest that the current model is too simplistic to ac-

count for HaCat cells behavior. Some authors have found

deficiencies in differentiation in HaCats14 whereas others

have not.5 Interestingly, evidence exists that HaCat cells and

NHKs respond very differently to extracellular calcium. One

study investigated the effect of [Ca2þ]e on the induction of

p21 (a suppressor of cyclin-dependent kinases and hence an

inhibitor of cell proliferation) on NHKs and HaCat cells.16 It

was demonstrated that exposure of NHKs to physiologic

levels of calcium (1–1.5mM) induced an increase in in-

tracellular calcium levels, which activated S100C/A11 (a

process that initiates a cascade ultimately resulting in the

induction of p21), which slowed cell proliferation. In con-

trast HaCat cells required 5–10mM extracellular calcium

before a comparable increase in intracellular calcium with

induction of p21 occurred. This suggests that HaCat cells in

the process of spontaneous transformation have somehow

changed their response to extracellular calcium (mechanism

unknown) and hence become relatively insensitive to cal-

cium inhibition of proliferation. Our current model does not

include any differences in the cell’s ability to respond to

calcium. We also assumed that [Ca2þ]e-dependent inter-

cellular adhesion and hence contact inhibition of prolifera-

tion was similar for NHKs and HaCats, which may also

prove to be an oversimplification. Adaptation of our model

to include different parameters describing calcium-depen-

dent behavior for HaCat cells would thus be expected to

improve its accuracy.

Finally, it is hypothesized that epidermal tissue struc-

ture in vivo is regulated by the arrangement of NHKs into

8 WALKER ET AL.

epidermal proliferative units, where stem cells are located

in the center of proliferative cell clusters. Proceeding from

this, it is proposed that cells located on the cluster periphery

differentiate via the notch signaling pathway.17 It is also

possible that the heat shock transformation process dis-

rupted this cluster size control mechanism in HaCat cells,

resulting in the dysregulation of growth in physiologic

calcium levels. Our model does not currently include these

concepts, but our group is developing models of inter-

cellular signaling mechanisms. Linking the latter to the

current model, and eventually to models of key intracellular

signaling pathways, will allow us to explore these questions

in greater depth, and ultimately produce a powerful pre-

dictive tool for cell biologists and tissue engineers.

In summary, the current model, which describes the ef-

fect of extracellular calcium on NHK proliferation and dif-

ferentiation, indicates that a simple failure of HaCat cells to

differentiate at some stage is not sufficient to explain the

very different response of these cells to changing extra-

cellular calcium.

ACKNOWLEDGMENT

We gratefully acknowledge support from the Engineer-

ing Physical Sciences Research Council for Doctors Walker

and Sun for this project.

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Address reprint requests to:

Dawn Walker, Ph.D.

Department of Computer Science

Kroto Institute

University of Sheffield

South Yorkshire, United Kingdom

E-mail: d.c.walker@sheffield.ac.uk

EXOGENOUS CALCIUM ON KERATINOCYTE AND HaCat CELL PROLIFERATION 9