plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USAPCOMPBIOL-D-10-0012910.1371/journal.pcbi.1002030Research ArticleBiologyAnatomy and physiologyComputational biologyPhysiologyComputational BiologyMDCK Cystogenesis Driven by Cell Stabilization within Computational
AnaloguesMechanisms of MDCK CystogenesisEngelbergJesse A.1DattaAnirban2MostovKeith E.2HuntC. Anthony13*UCSF/UC Berkeley Joint Graduate Group in Bioengineering, University of
California, San Francisco, California, United States of AmericaDepartment of Anatomy, University of California, San Francisco,
California, United States of AmericaDepartment of Bioengineering and Therapeutic Sciences, University of
California, San Francisco, California, United States of AmericaMcCullochAndrew D.EditorUniversity of California San Diego, United States of America* E-mail: a.hunt@ucsf.edu
Conceived and designed the experiments: JAE AD CAH KEM. Performed the
experiments: JAE AD. Analyzed the data: JAE AD CAH. Contributed
reagents/materials/analysis tools: JAE AD CAH KEM. Wrote the paper: JAE CAH.
Performed in vitro experiments: AD. Analyzed results of in vitro
experiments: JAE AD. Engineered the software: JAE. Designed ISMA
experiments: JAE CAH. Executed ISMA experiments: JAE. Analyzed simulation
results of ISMA experiments: JAE CAH KEM AD.
The authors have declared that no competing interests exist.
4201174201174e10020302010201024220112011Engelberg et alThis is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
The study of epithelial morphogenesis is fundamental to increasing our
understanding of organ function and disease. Great progress has been made
through study of culture systems such as Madin-Darby canine kidney (MDCK) cells,
but many aspects of even simple morphogenesis remain unclear. For example, are
specific cell actions tightly coupled to the characteristics of the cell's
environment or are they more often cell state dependent? How does the single
lumen, single cell layer cyst consistently emerge from a variety of cell
actions? To improve insight, we instantiated in silico analogues that used
hypothesized cell behavior mechanisms to mimic MDCK cystogenesis. We tested them
through in vitro experimentation and quantitative validation. We observed novel
growth patterns, including a cell behavior shift that began around day five of
growth. We created agent-oriented analogues that used the cellular Potts model
along with an Iterative Refinement protocol. Following several refinements, we
achieved a degree of validation for two separate mechanisms. Both survived
falsification and achieved prespecified measures of similarity to cell culture
properties. In silico components and mechanisms mapped to in vitro counterparts.
In silico, the axis of cell division significantly affects lumen number without
changing cell number or cyst size. Reducing the amount of in silico luminal cell
death had limited effect on cystogenesis. Simulations provide an observable
theory for cystogenesis based on hypothesized, cell-level operating
principles.
Author Summary
Epithelial cells perform essential functions throughout the body, acting as both
barrier and transporter and allowing an organism to survive and thrive in varied
environments. Although the details of many processes that occur within
individual cells are well understood, we still lack a thorough understanding of
how cells coordinate their behaviors to create complex tissues. In order to
achieve deeper insight, we created a list of targeted attributes and plausible
rules for the growth of multicellular cysts formed by Madin-Darby canine kidney
(MDCK) cells grown in vitro. We then designed in silico analogues of MDCK
cystogenesis using object-oriented programming. In silico components (such as
the cells and lumens) and their behaviors directly mapped to in vitro components
and mechanisms. We conducted in vitro experiments to generate data that would
validate or falsify the in silico analogues and then iteratively refined the
analogues to mimic that data. Cells in vitro begin to stabilize at around the
fifth day even as cysts continue to expand. The in silico system mirrored that
behavior and others, achieving new insights. For example, luminal cell death is
not strictly required for cystogenesis, and cell division orientation is very
important for normal cyst growth.
This research was supported in part by the CDH Research Foundation (CAH), the
Alternatives Research & Development Foundation (CAH), and NIH 5R01 DK074398
and 5P01 AI53194 (KEM). The funders had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript.Introduction
Epithelial morphogenesis is fundamental to the development and functional
specialization of tissues and organs. Tight regulation of tissue size, shape and
polarization is critical for normal organ development and function. Disruption of
these regulatory mechanisms leads to an array of diseases including autosomal
dominant polycystic kidney disease, stenosis, and cancer. Epithelial cells, such as
Madin-Darby canine kidney (MDCK) cells, cultured in a 3D matrix of natural basement
membrane components, can recapitulate in vitro many of the in vivo growth
characteristics of epithelial organs. They are thus valuable model systems for
studying the cellular mechanisms of in vivo epithelial morphogenesis. Their
phenotypic simplicity coupled with accumulated knowledge of their molecular biology
provide excellent case studies for gleaning needed insight into how molecular events
and environmental feedback pathways at subcellular levels lead to cell- and
cyst-level phenotype. These model systems lend themselves to computational analysis
and modeling as the means to gain that insight and improve our understanding of
organogenesis.
To achieve that goal, we must first develop explanatory and easily challenged
computational, mechanistic models. In biological research, explanatory mechanistic
models generally precede predictive mechanistic models. The operating principles of
explanatory mechanistic models of the type described herein are hypotheses about how
we think phenomena are generated. The models are part of frameworks for generating
and testing mechanistic hypotheses, as described in [1], [2].
While many aspects of MDCK cyst formation are well understood, quantitative data for
cystogenesis has been lacking. The most recent computational models [1]-[4] relied on
previously published quantitative data that described a few aspects of MDCK cyst
growth in collagen cultures [5]. There is limited data available on the dynamics of cell
number, cyst and lumen size, and mean cell size in Matrigel cultures. That caused
previous models to assume that cell size remains constant. The presented data
demonstrate that cell size varies during the course of cyst growth.
An objective of the project was to couple in vitro and in silico model systems to
achieve a deeper understanding of cell behavior during MDCK cystogenesis within 3D
Matrigel cultures. Of specific interest were the roles played by, and the timing of
polarization, apoptosis, and lumen expansion. In order to improve our understanding
of the link between individual cell behavior and cystogenesis, we proceeded in
parallel on two fronts. We undertook new in vitro experiments designed to provide a
more temporally and spatially fine-grained record of cell-level events during the
first ten days of MDCK cystogenesis. These experiments and their results are
described in this report. A thorough quantitative analysis of these results revealed
a third stage of cyst growth after cyst initiation and lumen creation and expansion.
That stage was characterized by the presence of a new cell state marked by a
decrease in cell division rate and cessation of the decrease in cell size observed
in previous stages. We refer to a cell in that state as being
“stabilized”.
We also developed and iteratively refined abstract, spatially fine-grained,
multi-attribute, mechanistic, in silico, MDCK cell analogues (ISMAs) capable of
cystogenesis. To create and validate ISMAs, we merged two modeling techniques while
introducing several novel features. Following rounds of iterative mechanism
refinement (including falsification and validation), time-dependent measures of
several in silico cystogenesis phenomena, including sizes of cells, cysts, and
lumens, cell number, and lumen number, became quantitatively indistinguishable from
corresponding in vitro measures. The process led to two successful ISMAs that had
similar operating principles but relied on different mechanistic hypotheses for how
cells stabilized. In one, cells relied on information about the lumen. In the other,
transition to the stabilized state was a simple timed event. Independent in vitro
experiments [6],
which used molecular interventions to alter the axis of cell division in two
different ways, provided data that challenged ISMA mechanisms and the predictions of
the cystogenic consequences of such interventions. ISMA mechanisms survived the
falsification challenge: measures of cystogenesis during simulation experiments
mimicking both interventions were quantitatively similar to in vitro data. This
further supported our hypothesis that the cause-and-effect relationships
(mechanisms) occurring within ISMAs during in silico cystogenesis (and thus their
morphogenic agenda) have in vitro counterparts, both in the presence and absence of
mechanistic interventions. By challenging these in silico mechanisms we better
understand their in vitro cellular counterparts.
ResultsQuantitative in vitro results
In order to study the process of cyst development in detail, MDCK cells were
grown and observed in 3D Matrigel culture for one to ten days and analyzed
quantitatively each day. As shown in Figure 1, cysts developed in a manner
consistent with previous observations [7]-[9]. A suspension of
mostly single MDCK cells divided to form small clusters during the first 24
hours. Most cells polarized (defined by podocalyxin localization at the nascent
apical surface of the cell) during the first two days of growth and all cells
polarized by day 3. Cysts developed single (11 of 20) or multiple (9 of 20)
lumens by the end of day 2. Most cyst cross-sections appeared circular. The
deviation from a circle ranged between 2 and 5%.
10.1371/journal.pcbi.1002030.g001
In vitro MDCK cyst cross-sections.
Culture conditions were as described in the text. Confocal images were
recorded on the indicated day during cystogenesis. Colors reflect
component staining as follows: red: actin; green: gp135/podocalyxn;
yellow: red and green colocated; blue: nuclein; black: Matrigel. ML: a
multi-lumen cyst. The arrow indicates a second, small lumen. Not SLSL:
this single lumen cyst does not have a single layer of cells. The arrow
indicates a cell not in contact with lumen.
We measured and recorded cyst and lumen area and perimeter, cell number, the
number of single and multiple lumen cysts, and the number of single-lumen,
single-(cell) layer (SLSL) cysts. Results are graphed in Figures 2 and 3. We calculated mean cell area and the ratio
of total cellular area to total cyst area.
10.1371/journal.pcbi.1002030.g002
Quantitative measures of in vitro and in silico cystogenesis.
Mean values and standard deviations for (A) cell number per cyst, (B)
cyst and lumen area, (C) mean individual cell area and (D) ratio:
cellular to cyst area. Blue: in vitro data taken each day for ten days
from 20 cysts. Red: data taken from 50 cysts over ten
days using the parameter values in Table 2. Gray boxes: noted changes in
behavior. Blue lines: slope of in vitro growth illustrating changes in
rate. SSM1: Self -Similarity Measure of in vitro growth; SSM1 indicates
the percentage of in vitro values each day that fell within
±25% of the mean in vitro value for that day. SM1:
Similarity Measure for ISMA growth. SM1 indicates the percentage of ISMA
values each day that fell within ±25% of the mean in vitro
value for that day. The target was that SM1>0.5 for nine of ten
days. When the target was met, we posited that ISMA
measures were experimentally indistinguishable from in vitro measures.
Gray SM values did not achieve targeted values.
10.1371/journal.pcbi.1002030.g003
Percentage of cysts with different numbers of lumens.
(A) Percentage of cysts that have single (solid circle) or multiple (open
circle) lumens. (B) Percentage of SLSL (single-layer, single-lumen)
cysts. Blue: in vitro data for 20 cysts taken each day for ten days.
Red: in silico data for 50 cysts using parameters values from
Table 2.
Black: mean and standard deviation for “normal” MDCK cysts
observed by Zheng et al. [6] as described in the text. Solid lines
represent continuous growth of ISMA cysts. Dotted lines
represent discrete growth of MDCK cysts.
Cell number increased exponentially through day 5. It slowed and increased at a
constant rate after day 6. Coincident with that shift, the variance in cell
number per cyst increased (Figure
2A). Cyst and lumen area increased monotonically (Figure 2B). Mean cell size
decreased at a constant rate through day 6 (Figure 2C) and then leveled off at roughly
the same time that cell division slowed. Mean cell size increased slightly
following the shift. Cell size variance was smallest on days 5–8. We did
not find a strong correlation between mean cell size and other cyst
measurements, including cell number, lumen size, lumen number, or lumen
perimeter/cell number. The ratio of total cellular area to cyst area (Figure 2D) indicated that the
portion of cyst occupied by cells decreased as cysts expanded (and thus the
portion occupied by lumen increased). The ratio decreased quite steeply between
days 5 and 6 with very little overlap; the majority of cysts at day 5 had a
ratio higher than 0.6 and the majority of cysts at day 6 had a ratio lower than
0.6. These observations taken together indicated a shift in cell behavior
occurred at approximately day 5 (referred to hereafter as simply the shift). The
data also supports the idea that cell compression during lumen expansion may be
a factor triggering cell entry into the stabilized state.
Lumen percentages in vitro
During the first day of growth, some cysts developed lumens, while others had no
visible lumen. From days 2-10 all cysts had at least one lumen (Figure 3A). Multiple lumens
appeared in a number of cysts, but their frequency decreased over time. Previous
studies [6]
considered cysts to be “normal” if they contained a single layer of
actin and apical membrane markers surrounding a single lumen. We distinguished
between single-layer, single-lumen (SLSL) cysts, in which all cells contact both
extracellular matrix and lumen; cysts with a single lumen where some cells did
not touch the extracellular matrix or the lumen; and cysts with multiple lumens
(Figure 1). After day 2,
the percentage of SLSL cysts ranged between 55% and 85% (Figure 3B), in rough agreement
with the 80% of cysts observed by Zheng et al. to be “normal”
[6]. In
cases where single-lumen cysts did not have a single layer of cells, usually
only one or two cells did not contact the lumen or extracellular matrix. These
data indicate that the percentage of cysts with multiple lumens decreases over
time, likely as smaller lumens merge together into larger. It is possible that a
few cysts might increase their lumen number over time even as mean lumen number
decreased, but that behavior would only be observed using time-lapse microscopy
of individual cysts.
ISMAs capable of cystogenesis
In order to create and validate ISMAs, we used a number of modeling techniques
and approaches, detailed in Methods. To
avoid confusion between in vitro and ISMA components and mechanisms with similar
names, we use smallcaps when referring to the latter. Following the Iterative Refinement
Protocol (IR Protocol) led to two specifications of cell behavior that
achieved all targeted attributes in Table 1 and all prespecified Similarity
Measures (SMs; described below). They are the lumen stabilized ISMA (LS ISMA)
and the timed stabilization ISMA (TS ISMA). There are only three cell
states: unpolarized, polarized, and stabilized. Both LS and TS
ISMAs have a common morphogenic agenda. It is a consequence of their operating
principles, which are a networked consequence of cell state and
micromechanisms. The latter are primarily axiom-dependent, and the axioms, in
turn, depend on particular local and temporal conditions. The axioms are
placeholders for even more fine-grained micromechanisms.
10.1371/journal.pcbi.1002030.t001
Targeted attributes and specifications.
1.
A: An initial small cluster of 1-4
cells divides and increases in cell number.
S: The ISMA begins with 2-4
cells, which divide after
cycleCounter reaches 0.
2.
A: All cells polarize by the second
day of growth.
S: Cells change state to
polarized after polarCounter
reaches zero.
3.
A: One or more lumens develop by the
second day of growth.
S: Cells within
cysts form lumens after cellspolarize.
4.
A: A multilayer of cells separates
multiple lumens.
S: Cells only form
lumens when they and their neighbors do not already
contact lumen. After a lumen has formed, all
neighboring cells contact a single lumen.
5.
A: Cells can undergo apoptosis whether
or not they contact the extracellular matrix.
S: Cellsdie with specified probability. That value is larger
for cells not in contact with matrix.
6.
A: The increase in cell number over
time is similar that shown in Figure 2, leveling off at day
6.
S: When lumen size reaches a
critical value, a mechanism causes cells to
stabilize.
7.
A: The increase in cyst size over time
is similar to that shown in Figure 2.
S: Cyst size is a function of
cell area, cell number, and lumen
size.
8.
A: The increase in lumen size over
time is similar to that shown in Figure 2.
S: Lumen size is a function
of cell number, cyst perimeter, cell
stretch, and time.
9.
A: Mean cell area decreases over time
as shown in Figure
2, and levels off at day 6.
S: Cells have distinct
mechanisms for (effectively) calculating TA before and after
stabilization.
10.
A: The decrease in the ratio of
cellular to cyst area over time is similar to that in Figure 2,
decreasing faster during days 2-6.
S: Cell area, lumen
size, and cyst size must be measurable and if these
quantities validate, then so must the ratio of cellular
to cyst area.
11.
A: The percentage of single-lumen,
multiple lumen, and SLSL cysts each day is similar to that in
Figure
3.
S: When cells lack
lumen contact, they can create new lumens.
Lumen creation occurs at the site of previous
celldivision. Lumens can expand and merge.
Cells that have stabilized cannot create a new
lumen.
12.
A: The percentage of cysts with
apoptotic cells each day is similar to that observed in [9].
S: Cells shrink after
beginning to die. The percentage of cysts with
dyingcells is calculated as in vitro.
13.
A: When the orientation of the cell
axis of division is disrupted or reversed, the percentage of
normal cysts is reduced as observed in [6].
S: Cells orient their axis of
division toward the center of prior
division or toward the center of the
lumen. Axis orientation can be randomized and
reversed.
MDCK cells and cysts are the referent. The model system is called an
in silico MDCK analogue (ISMA). A: a targeted
attribute; S: an ISMA specification. All listed
attributes were achieved. The early version of the ISMA achieved TAs
1-4, but was falsified by the quantitative data. The refined ISMA
achieved all TAs except 11, which was achieved by both the LS and
the TS ISMAs.
The only difference between the LS and TS ISMAs is the mechanism used by
polarizedcells to shift to the stabilized state. Within the LS ISMA,
polarizedcells use information about the lumen to decide when to
stabilize. Within the TS ISMA, transition to the stabilized state is a simple
timed event (each cell used its own internal clock). We did not
discover any in vitro observations that would provide a basis for selecting one
micromechanism over the other.
Cell operating principles require each cell to have knowledge
of its internal state and immediate environment, including the size of the
neighboring lumen (for the LS ISMA). Celldivision is based on factors other than cell size. Early in
the process, cyst size can be independent of lumen size. The
orientation of celldivision is extremely important in influencing the formation and number
of lumens within a cyst.
We explored alternative mechanistic variations, but failed to find others of
comparable simplicity capable of achieving all targeted attributes and
prespecified SMs. For simplicity we present and discuss measures from LS ISMA
simulations within the text (Figures 2 and 3)
and provide the same simulation measures for TS ISMAs in Figures S1
and S2.
Results from earlier ISMA that were falsified because they failed to achieve one
or more SMs are also discussed.
Quantitative results in silico
ISMA cysts were similar to cysts grown within Matrigel (Figure 4). Cysts
began with 1-3 cells at day 0. Cellspolarized and formed lumens within the first two days
(Figure 3 and Video S1).
Lumens and cysts expanded at a rate indistinguishable from
that observed in vitro. In general, a cyst formed with a single lumen
surrounded by a single layer of polarizedcells (Figure 4 and
Video
S1). Occasionally multiple lumens formed, each separated by
an independent layer of cells, such that no cell contacted
more than one lumen (Figure 4 and Video S2). The ISMA successfully achieved all
qualitative and quantitative targeted attributes listed in Table 1.
10.1371/journal.pcbi.1002030.g004
In silico MDCK analogue cyst cross sections.
Note that a regular hexagon in hexagonal space maps to a circle in
continuous space. Images are from a single simulation run using
parameter settings from Table 2. Cells are unpolarized (green),
polarized (gray) or stabilized (orange).
Cell-cell and cell-matrix
borders are red; cell-lumen borders are yellow;
lumens are blue. Lower right panel: shown is a
multi-lumencyst. Not SLSL: this single lumencyst does not have a single layer of cells. The arrow
indicates two cells not in contact with lumen.
ISMA cell number also exhibited two growth phases, with the rate of
celldivision decreasing at day 6 (Figure 2). Lumen and cyst
size increased at rates similar to those observed in vitro, but standard
deviations were smaller. Cell size also decreased at a rate comparable
to in vitro, and its standard deviations were also smaller. As indicated by the
values of Similarity Measure 1 (discussed below) in Figures 1 and 2, ISMAs produced quantitative results
similar to in vitro values. ISMAs were executed using the parameter settings in
Table 2, and
cyst and lumen area were scaled by 2.25
µm2 and perimeter by 0.75 μm.
10.1371/journal.pcbi.1002030.t002
Primary ISMA parameters.
Parameter
Description
Default value
Range used
wedgeArea
W: target area of unpolarizedcells and ideal wedge area for polarizedcells
82 grid points
30-150
lambdaArea
Multiplier controls how quickly cells
change size to reach their individual target areas
5 grid points
0.5-20
stableTargetArea
Target area of stabilized cells
48 grid points
30-150
cellCycle
Used to calculate cycleCounter,
the number of simulation cycles before a celldivides
70 simulation cycles
20-100
lambdaPerim
Multiplier controlling how quickly cells
change size to reach their target perimeter
2.5
0.5-10
polarDelay
Used to calculate polarCounter,
the number of simulation cycles elapsing before an
unpolarizedcellpolarizes
42 simulation cycles
0-400
shiftDelay
In the TS ISMA, used to calculate
shiftCounter, the number of simulation
cycles elapsing before a polarizedcell stabilizes
140,000 simulation cycles
0-300
doublingArea
When divided by 2, the minimal area a cell
must have to divide
41 grid points
20-100
divisionReg
How the axis of division is
calculated
1
0, 1, 2, 3
multiplier
Used to calculate target perimeter of
cells
0.6
0-1
lumenGrowthRate
Multiplier controlling rate of lumen
expansion
0.003
0-1
deathRateLumen
Likelihood of cells to die when
not touching matrix
0.02
0-1
deathRateEpi
Likelihood of cells to die when
touching matrix
0.0004
0-1
clusterProb
Probability initial two cells will set
cycleCounter to zero at simulation cycle
1
0.8
0-1
lgrSubtract
Multiplied by cell stretch to reduce
lumen expansion
27
0-300
dyingShrinkRate
Amount subtracted from target area of
dyingcells each simulation cycle
9 grid points
0-100
stableRatio
Critical lumen size (multiplied by 1000)
at which cells will stabilize
0.5 grid points
0.1-1
stableCycleDelay
(1 – x) = probability
a stabilized cell will decrement
cycleCounter
0.85
0-1
Parameters critical to the operation of the ISMA are listed along
with descriptions, default value used for simulation, and the range
of values explored. To switch between the LS ISMA and the TS ISMA
the values of shiftDelay and
stableRatio are changed from 140,000 and 0.5 to
200 and 1000. All units are relational (e.g., grid points instead of
µM, simulation cycles instead of hours).
Lumen percentages in silico
Simulations produced single and multiple lumencysts at frequencies comparable to those observed in vitro (Figure 3A), though the
percentage of cysts with single lumens was slightly higher
than observed in vitro. The percentage of SLSL cysts (Figure 3B) leveled off between
days 2 and 6 and then increased steadily to day 10 as lumens merged.
Cells that stabilized were not allowed to create new
lumens, but could contribute to lumen expansion. If this
restriction were to be removed and cells were allowed to create new
lumens after they stabilized, the percentage of SLSL cysts
might remain steady or decrease.
Similarity measures
To provide a validation target for ISMA cystogenesis and to compare ISMA
and in vitro results, we developed SMs [10], which quantified the
similarity within and between the in silico and in vitro data. We posit that, if
in silico data satisfies the SMs, then that data would be indistinguishable from
data produced by a repeated in vitro experiment.
SM1 compared results from individual simulations to in vitro mean values,
indicating the similarity of in silico and in vitro results. SM1 is the
percentage of in silico observations that fell within±25% of the
mean in vitro value for a given measure. SM1 values are listed in Figure 2. To survive
falsification, >50% of simulations must achieve the SM1 target for
nine of ten days, as detailed in Methods. For example, the ±25% range for in vitro cell
number at day 3 was 6.7 to 11.1 with a mean of 8.9. Seventy-two percent of
simulations had cell numbers within that range at day 3. SM1
values for cell number, cyst size, mean cell area,
and the ratio of cellular to cyst area exceeded 50% at
all days, so a degree of validation was achieved. The SM1 value for
lumen size exceeded the 50% cutoff for nine of ten
days, although the values were lower.
To facilitate assessing SM1 values and comparing in vitro and in silico data, we
specified and used Self-Similarity Measure 1 (SSM1). It measured the similarity
between the in vitro mean value and individual in vitro values and thus how
closely grouped around the mean the individual in vitro values were. Similar to
SM1, SSM1 is the percentage of individual in vitro cyst measures each day that
fall within a specified range. SSM1 can be used to evaluate corresponding SM1
values. Large SSM1 values are a characteristic of measures having a small
variance. Values of SSM1 were larger than the target for all measures except
lumen size, indicating that lumen size in vitro varied more extensively about
the mean than other quantities.
SM1 did not consider the variance of the data. To address variance, we specified
SM2. It compared the coefficient of variance of in silico and in vitro
experiments. SM2 measured the absolute value of the difference between the in
vitro and in silico coefficient of variance each day. ISMAs survived
falsification if SM2<0.15 for nine of ten days (strong validation)
or <0.25 for eight of ten days (medium validation). The current ISMA
achieved strong validation for cell number, mean cell area,
and the ratio of cellular to cyst area (Table S1). It achieved medium validation for
cyst size and lumen size, comparable to SSM1 values.
Cell death
When MDCK cells can polarize well, they do not need apoptosis to form cysts with
lumens [9]. Consequently, cell death is relatively uncommon
during in vitro MDCK cyst development [9]: on a given day, no
more than 15% of cysts had one or more apoptotic cells within the lumen
and no more than 10% of cysts had one or more apoptotic cells with matrix
contact. Celldeath did occur during ISMA executions, but at slightly lower
frequencies than observed in vitro (Figure 5). In Methods, we
specified that the average duration between a cell initiating
death and being removed from the simulation to be ten simulation
cycles, which maps to five hours. The actual in vitro duration will affect the
number of visible apoptotic cells observed each day. When we caused
cells to shrink somewhat slower, the celldeath values in Figure
5B increased. The experimental results provided in Figure S3
demonstrate that decreasing the value of dyingShrinkRate from 9
to 4.5 increased the mean duration of celldeath (from 4.6 to 7.4 hours) and increased the percentage of
dyingcells. It is noteworthy that all validation targets were achieved
without requiring stabilized cells to die more frequently than
polarizedcells. Based on current knowledge, the ISMA accurately mimics in vitro
quantitative data, but the duration of apoptosis within MDCK cells in vitro has
not been quantitatively established. In order to be certain about the role
played by cell death, time-lapse movies using a caspase-3-GFP will be
required.
10.1371/journal.pcbi.1002030.g005
Percentage of cysts with dying cells.
(A) In vitro data reproduced from [9]. (B) ISMA
data from 50 cysts over ten days. Blue bars:
percentage of cysts observed to have apoptotic cells without matrix
contact. Red bars: percentage of cysts observed to have apoptotic cells
with matrix contact.
Altering cell division orientation in silico dramatically alters cyst
morphology
After the ISMAs achieved the above, targeted attributes, Zheng et al. [6] reported
measuring the consequences of disrupting cell division orientation on MDCK cyst
morphology. Knocking down LGN, which plays a role in spindle orientation during
cell division, caused cell division orientation to become random instead of
aligning with the axis perpendicular to the cellular plane. The frequency of
“normal” cysts decreased from roughly 80% to 20-30%.
We added those observations to our targeted attributes list and then explored
the degree to which cyst morphology following a comparable ISMA
intervention would mimic the in vitro results, thus surviving the challenge. We
altered celldivision so that all cells divided with a random orientation.
The results (Figure 6A) were
similar to those of Zheng et al. The altered ISMA produced less than 20%
SLSL cysts and more than 30% multi-lumencysts at days 2 through 9. Additional details are available in
Figure
S4.
10.1371/journal.pcbi.1002030.g006
Percentage of ISMA cysts with varied lumen number
when the axis of celldivision is abnormal.
Shown are the percentages of cysts that have single (solid red
circles) or multiple (open red circles) lumens when the axis
division is (A) random or (B) reversed (rotated 90°)
along with the percentage of cysts that are SLSL (purple
circles) when the axis of celldivision is (A) random or (B) reversed. Black (A and B): mean
and standard deviation for “normal” MDCK cysts observed by
Zheng et al. [6]. The in vitro control data are shown in Figure 3.
In a second experiment, Zheng et al. targeted LGN to the apical membrane. So
doing rotated the axis of division by 90°, thus reversing cell division
orientation. The procedure reduced the frequency of normal cysts to roughly
10%. We conducted a similar experiment by modifying ISMAs so that the
axis of division was parallel, rather than perpendicular to the
lumen edge. That intervention produced SLSL cysts less
than 10% of the time (Figures 6B and S5). ISMAs survived both challenges; in both cases,
altering the orientation of celldivision decreased the percentage of single lumen and SLSL
cysts to a degree similar to that observed within in vitro
experiments.
In silico cyst growth with no luminal cell death
Cell death contributes to cystogenesis, but it remains unclear to what extent it
is essential. In order to explore the consequences of decreased celldeath frequency, we executed simulations in which we reduced
deathRateLumen from 0.02 to 0.0. We did not alter the
probability of celldeath in cells contacting matrix. We noted no
significant difference in cell number during the first six
days of growth, but during days 7 through 10 mean CELL number was
10-15% higher than observed during control ISMA growth (Figure 7A). The observed
standard deviations also increased. We observed a smaller percentage of SLSL
cysts than in control simulations, especially during days
6 to 10 (Figure 7B). Values
for cyst area, lumen area, cell size, and the ratio
of cellular to cyst area were similar to control values (Figure S6),
while the percentage of single lumencysts decreased slightly (Figure S7).
10.1371/journal.pcbi.1002030.g007
Cystogenesis measures with no luminalcelldeath.
ISMA simulations executed with the parameter values from Table 2 except that
luminalcelldeath was not allowed. (A) Red: mean values and standard
deviations for cell number per cyst. Blue: in vitro
control data from Figure
2A. (B) Percentage of SLSL cysts.
Simulated cyst growth with delayed cell polarization
Delayed cell polarization is believed to contribute to the differences in cyst
growth in Matrigel and collagen [9], although it is
possible that a lower initial rate of cell clustering and a slower growth rate
might be factors as well. To explore the effect of delayed polarization
on ISMA cystogenesis, we increased the value of
polarDelay from 42 (equivalent to 21 hours) to 130
(equivalent to 65 hours). Relative to controls, cell number increased
at an equivalent rate during the first six days, but was larger during
days 7–10 (Figure 8A). Cellpolarization (data not shown) and lumen formation occurred
later than in controls (Figure
8B). The area taken up by cells remained roughly constant,
but the delay in lumen formation and resulting smaller lumens
caused the ratio of cellular area to total cyst area to be
significantly larger than control values during days 2-8 (Figure S8).
Not surprisingly, there were fewer single and multiple lumencysts during the first three days. When lumen
formation began, however, it often resulted in multiple lumens
(>80% for days 4–6); SLSL cysts were observed
infrequently. As lumens expanded and merged during the later stages of
growth, the frequency of SLSL cysts increased. The percentage of
dyingcells not contacting the matrix was significantly larger at
days 4–10, indicating that many of these cellsdied as lumen expansion occurred (data not shown). Some of
these in silico results reflect those observed within growth in collagen, but it
seems unlikely that delayed cell polarization in vitro is solely responsible for
those differences.
10.1371/journal.pcbi.1002030.g008
Cystogenesis measures when cellpolarization was delayed.
ISMA simulations executed with the parameters values from Table 2 except that
cellpolarization was delayed as described in the text. Left: mean
values and standard deviations for cell number per
cyst (top panel) and ratio of cellular to
cyst area (bottom panel). Right: Percentage of
cysts with single, multiple, and SLSL lumens. Designations
and symbols are the same as in Figures 2 and 3.
DiscussionIn vitro observations
Observations reported herein about in vitro MDCK cystogenesis are consistent with
those made previously [6], [9], [11]. There is
no evidence of behavioral differences between cells within single and multiple
lumen cysts. We could not establish a causative connection between the slowing
of cell division and the change in cell size. The evidence indicates that
initial lumen expansion is somewhat isochoric: early lumen expansion is
primarily a consequence of cell shrinkage. After an interval of lumen expansion
and cell shrinkage lasting about six days, cell behavior changes: cell size
stabilizes and cells begin to stretch as the lumen continues to expand (Figure 1); cell division slows
dramatically; the expanding lumen becomes the primary driver of cyst size; and
the variance in both cell area and cyst size increases.
Iterative process
Iteratively constructed ISMAs quantitatively mimicked a targeted set of in vitro
data and cell behaviors. Measures of ISMA cystogenesis matched
corresponding measures of MDCK cystogenesis over ten days (Figures 2, 3, & 5). The pathways and proteins that
play influential roles in cell behavior during MDCK cystogenesis are objects of
active research and are increasingly well understood. However, knowledge of how
specific cell actions and events are choreographed during cystogenesis is still
limited. The latter knowledge is needed to begin establishing causal linkages
between molecular level events and systemic phenotype.
Previous analogues [2], [3] used a simple representation of a cell: each
cell occupied a single 2D hexagonal grid space. They were falsified
when we added qualitative observations about changes in cell size and shape to
our targeted attributes list (Table 1). To mimic these newly targeted attributes, we needed
cells to be more fine-grained. To generate the current ISMA, we
began with an in silico analogue that had achieved a degree of validation and
then conducted in vitro experiments designed to challenge and possibly falsify
it. We then reengineered the in silico system to reflect, explore, and challenge
new insight provided by the fresh in vitro data. We engineered new analogues
using the cellular Potts model (CPM), which provided several capabilities,
including enabling cell size and shape change. To slow the increase in
cell number after day 6, we introduced a stable
cell state.
We envision the above in silico-wet-lab cycle continuing indefinitely. It is
straightforward to explore the consequences of in silico mechanistic
interventions. If these interventions result in altered system behaviors
(predictions), it may suggest new in vitro experiments designed to test them.
Examples include the effect of delayed polarization on cyst phenotype, the lack
of noticeable changes when cell death is inhibited, and the causal link between
lumen size and cell stabilization. Furthermore, we expect a change in cell state
(cell stabilization at day 6) to be accompanied by measurable changes in gene
expression profiles and biochemical signaling.
Improved analogue
The ISMA illustrated in Figure
4 achieved all targeted attributes. It was preceded by two earlier
versions. These ISMAs differed in the mechanism used to initiate cell
stabilization. We hypothesized that in vitro cells might use knowledge of their
internal geometry to sense their perceived stretch and subsequently stabilize.
One early analogue, the geometrical mechanism ISMA (GM ISMA), directly tested
this hypothesis; each cell used measures of its area and geometry to
determine when to shift to the stabilized state. To achieve a degree of
validation required the use of an axiom specifying that stabilized
cells would be more likely than polarizedcells to die when not in contact with matrix. This
axiom was implemented in order to decrease the number of cells within
the lumen and thus increase the number of SLSL cysts. The GM
ISMA was falsified when targeted SMs for the percentage of single
lumen, multiple lumen, and SLSL cysts were
strengthened to those achieved in Figure 2 (Figures S9 and S10). It
was falsified because the time at which cells stabilized was too
variable; some cells stabilized early, others much later, resulting in
very few SLSL cysts (data not shown).
A second version, called the timed stabilization ISMA (TS ISMA), used an internal
clock to signal cell stabilization, resulting in a uniform
stabilization time and reducing the variance in cyst size. The TS ISMA
survived falsification (Figure S1), providing evidence that
stabilization time influences SLSL cyst percentages. The GM ISMA axiom
specifying that stabilized cells would be more likely than
polarizedcells to die when not in contact with matrix was not
needed. The TS ISMA was capable of generating high percentages of SLSL cysts
even without this axiom, and so the axiom was removed in that and subsequent
ISMAs.
Although the TS ISMA survived falsification, we were not aware of any in vitro
evidence suggesting existence of an equivalent internal clock-based mechanism.
If such a mechanism does exist, it might be molecularly equivalent to that of
cell polarization. Genes that regulate cellular senescence can suppress the cell
cycle, and the sirtuin protein SIRT1 is involved in cellular senescence [12], [13]. It is
possible a cell-autonomous timing mechanism could exist that depends on the
regulation of SIRT1 and its downstream targets, as detailed in Supporting Text S1. We
hypothesized that a mechanism that used the geometry of the lumen
instead of the geometry of individual cells to signal cell
stabilization might bridge that gap and still produce a low variance in
stabilization times. We developed the lumen stabilized ISMA (LS ISMA) described
within this report to test that hypothesis and discovered that in addition to
surviving falsification (Figure
2) it generated stabilization variance between the GM and TS ISMAs.
We can surmise a mapping between the lumen-based stabilization
mechanism and a functionally equivalent in vitro mechanism in which apical
sensory input to each cell provides it with information that correlates to lumen
size. Current evidence supports the hypothesis that cells in the cyst wall can
sense lumen size. One mechanism utilizes the tension generated at the luminal
membrane by membrane stretching. This tensional information is transduced by the
subapical F-actin network, which acts both as a scaffold for maintaining luminal
integrity, as well as a region for aggregation of recycling endosomes that
regulate the protein and lipid composition of the apical plasma membrane. Thus,
regulators of this F-actin network can regulate lumen and cyst size. Potential
molecular mechanisms are detailed in Supporting Text
S1.
We should seek additional, in silico mechanisms that are equally effective in
enabling ISMAs to achieve validation targets. Given phenomena, what hypothetical
generators (and measures) might generate them? Studying an inverse mapping
requires multiple, seemingly plausible hypotheses, which then compete against
each other during simulation experiments as done here. After falsification and
validation using the IR Protocol, those that survive spawn additional, more
refined hypotheses. Having multiple mechanistic options for realizing the same
behaviors may be biomimetic in that it marginally increases system robustness.
An example of a potential additional in silico mechanism is one that uses
time-dependent dynamic parameters, which might assist in the exploration of
finer-grained, intracellular molecular behaviors.
ISMAs currently contain a small number of parameters that can have implicitly
dynamic values (such as the time that elapses between celldivision events). They change when cells change state. In
general, however, all parameters are fixed for the duration of the simulation.
Expanding the set of targeted attributes may force consideration of time varying
parameter values. If, for example, in vitro data were targeted that demonstrated
the build-up of certain proteins along the plasma membrane, dynamic variables
could be implemented that controlled the amount of the protein
counterpart within the analogue.
Challenging ISMA predictions
ISMAs had already achieved all targeted attribute when the work of Zheng et al.
[6] was
published. Results from their studies provided an independent challenge to ISMA
mechanisms and their robustness. The simulation results in Figure 6 are a consequence of two different
simulation interventions: making the cell axis of division
random (Figure 6A) and
reversing the cell axis of division (rotating it
90°)(Figure 6B).
These predictions are fully consistent with the in vitro results of Zheng et al.
As previously stated, they defined a normal cyst as one with actin staining at
the apical cell surfaces surrounding a single lumen. Included in that definition
are our SLSL cysts and cysts with a single lumen. In
Zheng et al., when cell division was randomized, the percentage of cysts with
single lumens at day 4 dropped from 81.9% to 21.5%, a different of
60.4%. In ISMA simulations, when divisionReg was changed
from 1 (ordered division) to 0 (random division) the
percentage of cysts with a single lumen dropped from
94% to 46%, a difference of 48%, which is quite similar to
the decrease observed in vitro (Figure 6A). As seen in Figure 6B, when the axis of division was reversed, the percentage of
cysts with a single lumen dropped from 81.9% to 11.5%, a
difference of 70.4%. Within the ISMA, when divisionReg
was changed from 1 to 3 (reversed division), the percentage of cysts
with a single lumen dropped from 94% to 14%, a difference
of 80%. In addition, the in silico results provide a prediction of in
vitro behavior that could be challenged through in vitro experimentation. When
division is reversed within the LS ISMA (Figure S5A)
cell number continues to increase after day 6, most likely because
the numerous small lumens do not reach a sufficient size to cause
cell stabilization. In strong contrast, when division is
reversed within the TS ISMA (Figure S11-2) cell number stops
increasing at day 5 and remains stable thereafter. Future experiments of the
type conducted by Zheng et al. that quantify cytogenesis over longer intervals
would provide evidence supporting one or the other mechanistic hypothesis.
Cell-level and intracellular events
A cell-level event is one that is visible at the current level of
resolution. An event that maps to an intracellular process (referred to as
intracellular) can occur without causing a visible change; it is
below the current level of resolution. Of the events listed in Table 3, the two marked
(*) only exist within the in silico system and have no specific in vitro
counterpart.
10.1371/journal.pcbi.1002030.t003
Cell and intracellular events that can occur within
a simulation cycle.
Cell-Level Events
Map to Intracellular Events
Cell state (& color) changes
MCell point assignment*
Celldivision
CellCycle updating at simulation
cycle 1
Lumen creation
Cell initiates dying
Lumen merging through TJ
reorganization
Death advances; cell TA
decreases
Lumen expansion through TJ
reorganization
Polarity counter (polarCounter)
begins
Lumen expansion without TJ
reorganization
Decrement cycleCounter
Isolated point engulfed*
Decrement polarCounter
Cell perimeter (but not TJs) changes
Decrement shiftCounter (TS ISMA
only)
Dying complete: cell
disappears
Compute matrix and lumen contact
length, A, TA, & TP
Matrix removal
Compute G for a potential index change
Cells change shape
*This event exists only within the ISMA system and has no
specific cystogenesis counterpart.
All cell events produce a visible change within the ISMA
visualization. Events that map to intracellular events result in a
change within a cell, but do not produce a visible change
within the ISMA. Cell-level events map to equivalent events
between in vitro MDCK cells, lumen, and matrix, while
intracellular events map to events (less well
understood) within in vitro MDCK cells.
Beyond simply modeling cystogenesis, a purpose of this research has been to
instantiate an in silico system in which cells, matrix, and
lumen have in vitro counterparts, and when executed the ISMA
produces a variety of measurable phenomena that quantitatively mimic MDCK
cystogenesis. At the systemic level, we have excellent cystogenesis similarity
over ten days for multiple measures (Figures 3-5). Further analogue improvement will, following additional cycles
of the IR Protocol, allow intracellular events to become concretized
and increasingly fine-grained, thus enabling quantitative in silico-to-in vitro
mappings at multiple levels.
All specified events were necessary and essential for achieving targeted SMs. For
cell-level events, the mappings are clear: they are direct and
quantifiable. Intracellular events, axioms, and protocols are below the
current level of resolution. There is no requirement that a specific
intracellular event, axiom, or protocol has a cell-level
counterpart. We simply hypothesize that the set of
intracellular events, axioms, and protocols—a
cell's operating principles—has an in vitro
counterpart, as illustrated in Figure 9. For some intracellular events, conceptual
mappings are clear. Examples include cell initiates dying,
death advances, and decrement polarCounter. For
others, conceptual mappings are less clear. Examples include decrement
shiftCounter (in the TS ISMA), compute TP, and compute G.
The expectation is that, in moving forward, as axioms are replaced by concrete,
interacting components (see [14] and the future experiments
subsection below) clear mappings will be easier to establish and quantify.
10.1371/journal.pcbi.1002030.g009
ISMA-to-in vitro cell culture mappings.
Left: MDCK cell cultures are the referent wet-lab systems. During
experiments, cells draw on genetically controlled operating principles,
and cystogenesis is the result. Influential mechanistic details are
reflected in the collected data. Right: an abstract mechanistic
description, a set of targeted attributes, and specifications paired to
those attributes direct analogue design. Software components are
designed, specified, coded, verified, and assembled guided by that
mechanistic description. The product of the process is a collection of
abstract mechanisms rendered in software. A clear mapping is intended
between ISMA cells, their axioms and operating principles, and
MDCK cell and intracellular details. Relative
similarity is controlled in part by parameterizations. Importantly, that
mapping can be concretized iteratively. Compilation and source code
execution gives rise to a working ISMA. Its dynamics are intended to
represent abstractly corresponding dynamics (both observed in movies and
believed to occur) within cultures during ten-day experiments. That
mapping can also be concretized iteratively. Measures of
cystogenesis provide time series data that are intended to
be quantitatively similar (according to prespecified criteria) to
corresponding measures of MDCK cell cystogenesis. Achieving increasingly
stringent SMs provides degrees of validation.
A good example of a project in which intracellular events are
incorporated and to some degree mapped back to those in vitro, is the IBCell
model [15],
[16]. It
is a biomechanical model of MCF-10A cell cystogenesis in which proteins on the
outer cell membrane and the extracellular matrix are specifically simulated. The
IBCell model successfully reproduced some aspects of cystogenesis, but it
remains unclear whether the intracellular details are necessary or
could be replaced by coarse-grained components. The quantitative data used to
validate the model lacked the level of resolution necessary to falsify
intracellular mechanisms.
Cell death and the timing of cell polarization
Surprisingly, cysts with little or no cell death can still be
well organized with a single lumen. Reducing celldeath rates (Figure
7) altered cystogenesis details only marginally, primarily
because celldeath frequency was already low (Figure 5B). Lin et al. [17] hypothesized that apoptosis is
crucial for lumen formation in MDCK cysts, but they reached that conclusion
based on observations of cystogenesis in collagen culture only.
Martín-Belmonte et al. [9] observed that
apoptosis within Matrigel cultures is less frequent than within collagen
cultures. Within ISMA simulations, earlier lumen formation results in
more organized cyst growth and fewer cells that die
after losing contact with the matrix once lumens have formed.
It is possible that apoptosis acts simply as a cleanup mechanism within MDCK
cysts, but the degree to which it is utilized depends on the environment, the
rate of cell growth, and the timing of polarization. Our experiments reducing
the rate of celldeath showed that although the rate of celldeath within cysts during growth is normally quite low,
celldeath still contributes to controlling cell number and
maintaining SLSL cysts. It is possible that environmental adjustments
may provide conditions in which MDCK cell cystogenesis produces normal SLSL
cysts without requiring cell death, as occurs in human alveolar type II
epithelial cells [18], [19].
Relative to cystogenesis in Matrigel, cells grown in collagen produce smaller
cysts with fewer cells and delayed polarization. That delay might play a role in
formation of smaller cysts. However, ISMA experiments showed that delaying
polarization (Figure
8) increased cell number and decreased the percentage of
cysts with single lumens. We take those observations as
strong evidence that delayed cell polarization alone is insufficient to account
for that difference in cystogenesis within collagen and Matrigel cultures.
Future in vitro experiments
As illustrated in Figure 9, a
goal is to build, expand, and validate in silico mechanistic networks that map
to plausible causal linkages between intracellular details and features of MDCK
cell phenotype in culture. A prerequisite is to have cells capable of
achieving increasingly fine-grained and expanding coverage of MDCK cell,
cluster, and cyst behaviors under different conditions. Advances in imaging
technology have made doing so easier. Similar coverage will be needed of
intracellular (subcellular) dynamics, including the behaviors of cell components
under different conditions. We anticipate that studies of in vitro MDCK cell
cystogenesis using high-resolution, time-lapse microscopy will reveal new
behavioral details at each level. Recent studies have employed confocal
time-lapse microscopy to understand lumen formation, but only imaged cells for
eight hours [20]. Ewald et al. [21] set the standard for
long-term time-lapse microscopy in their work on the elongation of mouse mammary
ducts, in which they captured individual images every 15 minutes for five days,
using high-sensitivity cameras to avoid phototoxicity.
There is ample evidence that tension within the extracellular matrix influences
epithelial cell behaviors [22], [23]. Paszek et al. [22] demonstrated that
increasing matrix stiffness resulted in tumorigenic behavior in MCF-10A cells.
It seems reasonable to expect changes in MDCK cell, cluster, and/or cyst
behaviors as Matrigel stiffness, density, and additives are changed. Experiments
similar to those within [22] conducted with MDCK cells and for longer durations
are needed to expand ISMA coverage of MDCK phenotype in important ways.
Although the underlying in vitro molecular mechanisms to which the TS and LS ISMA
map remain unclear, in vitro experiments may indicate one mechanism as being
more plausible. Careful analysis of images generated through time-lapse
microscopy is expected to be informative. If the elapsed time between individual
cyst polarization and stabilization of division rate or mean cell size are
similar between cysts, that would be supportive of an internal clock mechanism.
However, if the interval varied between cysts, that would falsify such a
mechanism. If mean lumen size when division rate and cell size have stabilized
are similar between cysts, that would support the shift mechanism based on lumen
size. Experiments are suggested in Supporting Text S1 to
begin identifying potential molecular counterparts to TS and LS ISMA
mechanisms.
Future in silico experiments
Five directions for in silico experiments present themselves. The first two
require seeking contradictory or supportive literature evidence of in silico
experiments. 1) Exploring the consequences of parameter changes will provide
insight into ISMA's mechanism-phenotype relationships for which there may
be biological counterparts [1]. A full suite of parameter change experiments was
conducted using the LS-ISMA; results are presented in Figure S11.
One example is to explore the consequences of changing
deathRateEpi and deathRateLumen (Figures 7 and S11),
including setting both to 0. Another is to vary lumenGrowthRate
to explore the effect of increased or decreased lumen expansion on in
silico cystogenesis (see Figure S11). Addition of any of several
compounds to the culture media in vitro will stimulate cyst expansion. Examples
include cholera toxin and forskolin. 2) Modify axioms and operating principles
to simulate targeted mechanistic interventions. One example (see Results) is to modify the way in which
cells calculate their axis of division. Another is to
modify how matrix is represented in order to explore consequences of
altered matrix properties on cystogenesis. Currently,
matrix is simply a grid space state. Matrix could be represented
using a CPM “cell” that offers resistance to cell
advancement. So doing opens the door to exploration of a variety of
matrix-cell interactions that could map to proteins
altering local matrix properties. 3) Systematically expand the targeted
attributes while keeping cells atomic. Movies, such as Video S1 from
[9]
along with the current literature, contain examples of many behaviors beyond the
scope of the current ISMAs. Adding any one of the following to the list of
targeted attributes will falsify the current ISMAs. At the cell level: when
cells undergo mitosis, they enlarge temporarily and then return to a smaller
size; some cells (and cysts) move around during the early stages of
cystogenesis; some cells migrate toward each other and cluster together before
initiating division; typically, when cells die in contact with matrix, they are
flushed into the luminal space where they shrink and disappear. At the cyst
level: cysts spin. The process was described in [20] and recently modeled in
[24].
Cyst growth may have an additional later stage characterized by significantly
slowed expansion, rather than continuing to grow steadily as predicted by the
ISMA. The dynamics of lumen merging are more complex than the merging events
that occur during simulations. Also, lumens change shape and move within cysts
during the initial stages of growth.
4) Increase realism by transforming cells from atomic to composite
objects. The axioms used by cells are placeholders for more
fine-grained micromechanisms. The latter can be instantiated in future ISMA
descendents. Before we can turn our attention to intracellular processes, we
need new ISMAs in which cells are composite (and eventually
hierarchical) analogues that can achieve essentially the same, targeted SMs as
the current ISMAs (Figures 2
and 3). Previous reports
[14], [19], [25] explained
that an in silico analogue (such as the current ISMA) that quantitatively mimics
many cell-level phenomena can be used to begin the sequential process of
drilling down and establishing plausible, causal linkages between phenotype and
molecular level details. Using cross-model validation procedures, the atomic
cell is replaced by a composite cell where phenomenal
axioms are replaced by concrete micromechanisms involving interacting objects
that map to subcellular processes and/or components in the referent. 5) Once we
have the preceding composite cells, we can expand the list of targeted
attributes to include subcellular and intracellular behaviors. Alternatively,
expanding the list of targeted attributes can require transforming
cells from atomic to composite objects. Examples of subcellular and
intracellular behaviors include the amount and location of polarization
proteins, organelle movement, the organization of the mitotic spindle, formation
of a pre-apical patch, location-dependent lipid compartments within the
membrane, etc. During cell polarization (as detailed in [8]), PTEN moves to the
apical membrane, where it converts PIP3 to PIP2, which binds to Anx2 and assists
in the recruitment of Cdc42 to the apical membrane. The task at this stage,
while adhering to a strong parsimony guideline, is to add new mechanisms and
details that enable validation against the new, targeted attributes, while
retaining all of those mechanisms and behaviors that enabled validation during
earlier cycles of the IR Protocol. So doing will enable the in silico
exploration, falsification, and validation of increasingly complex in vitro MDCK
cell behaviors, which will ultimately correlate to in vivo phenotypes of
developing epithelial organs.
We hypothesize that the local cause-and-effect relationships (mechanisms)
occurring in ISMAs during execution, and thus their morphogenic agenda, have in
vitro counterparts. Challenging these alternative hypotheses can be a focus for
future in vitro experiments and ISMA refinements.
Summary
Through careful application of the IR Protocol, analogues of MDCK cystogenesis in
cultures (ISMAs) were developed, falsified, refined, and validated against
novel, multi-attribute quantitative data. ISMAs were based on software
specifications that enabled in silico behaviors during simulation to achieve
degrees of validation: to be mapped quantitatively to measures of cystogenesis
(targeted attributes). Those specifications also enabled hypothesizing that ISMA
operating principles, axioms, components, events, and mechanisms have in vitro
counterparts. Predictions of substantive mechanistic changes were verified by
independent experiments. ISMAs were used to explore and test hypotheses about
cell and cyst dynamics. The above, coupled in vitro and in
silico experiments led to four insights. 1) The axis of celldivision significantly affects lumen number without changing
cell number or cyst size. 2) Reducing the amount of
luminalcelldeath had limited effect on cystogenesis. 3) Later stages of
cystogenesis, marked by a decrease in the rate of cell division and cessation of
the decrease in mean cell size, can be explained by the presence of a new cell
state (called stabilized), which differs in a few key behaviors. 4) The same,
multi-attribute phenotype can be a consequence of two fundamentally different
mechanisms that, in silico, only alter the mechanism of cell
stabilization. By providing a new way of thinking about cystogenesis, ISMA
simulations have provided an impetus to explore novel aspects of epithelial
morphogenesis.
MethodsIn vitro methods
A single cell suspension of MDCK cells was plated in duplicate on a layer of
100% Matrigel basement membrane (BD Biosciences) in the presence of
2% Matrigel in the media. Cysts were allowed to grow for the indicated
duration then fixed with 4% paraformaldehyde. The cells were then stained
as described in [11], [26]. Briefly, cells were stained with a monoclonal
antibody against gp135/podocalyxn, and a polyclonal antibody against
β-Catenin. F-actin and nuclei were stained with Alexa-labeled phalloidin and
Hoechst 33342 respectively. Each day, 20 cysts from the duplicate plates were
selected at random and imaged using a Zeiss 510 laser scanning confocal
microscope (Carl Zeiss Inc.). Images were acquired sequentially in four separate
channels.
Cell number was determined by counting the nuclei, when visible, and actin
borders when not. Cyst and lumen perimeter were traced using ImageJ and the size
of the cyst and lumen within each cross section was calculated using the analyze
tool. Cellular area was found by subtracting lumen area from cyst area; mean
cell area was found by dividing cellular area by the number of cells; and the
ratio of cellular area to cyst area was found by dividing cellular area by cyst
area. Standard deviations and Similarity Measure values (defined in Results) were calculated using R. The number
of lumens in each cyst was found by counting the discrete spaces within the cyst
bordered by gp135/podocalyxn and actin.
The data generated by the in vitro experiments was quantitatively consistent with
results from previous studies [6], [9], [27], as well as
being internally consistent. The goal of conducting the in vitro experiments was
to provide a particular quantitative perspective on MDCK cystogenesis. We sought
an abstract mechanistic explanation of one set of cytogenic trajectories.
Repeated in vitro experiments using a different batch of cells could result in
distinct cytogenic trajectories, which might not be explained by the current
ISMAs. Understanding and simulating such different trajectories is outside the
scope of this project.
ISMA uses
An early task in any modeling effort is to state near- and long-term uses; one
must then strive to follow a model development path intended to achieve those
uses. When dealing with biology, having explanatory mechanistic models
necessarily precedes having predictive mechanistic models. This project is an
important, early step in developing explanatory mechanistic models of
cystogenesis. A truly useful explanatory mechanistic model is one in which we
can observe putative cause-effect events at several layers as they unfold. Given
those considerations, we envisioned six near-term ISMA uses. 1) Instantiate and
challenge hypotheses about mechanisms of cystogenesis by MDCK cells under
different culture conditions. 2) Make it easy to follow mechanistic processes
and trace cause-effect relationships. 3) Achieve measures of
cystogenesis during ISMA executions of increasingly autonomous
cells that are quantitatively similar to referent measures (i.e.,
they achieve targeted SMs). 4) Achieve increasing overlap of an MDCK cell
culture's phenotype by an ISMA phenotype. 5) For validated ISMAs, explore
the consequences of mechanistic interventions on measures of
cystogenesis. 6) Expose possible gaps in our knowledge of MDCK cell
cystogenesis. Implicit in these uses is the ability of ISMA behaviors under
different conditions to stand as predictions of MDCK cell and cyst behaviors
under comparable conditions.
The preceding are prerequisites for achieving six long-term ISMA uses. 1) Enable
replacing ISMA operating principles with concrete mechanisms composed of
interacting components. So doing is required to enable hierarchical linkage of
molecular level details with specific phenotypic attributes. 2) Execute in
silico experiments that test the effect on ISMA cystogenesis of
simulated chemical and genetic interventions that affect cell
behaviors. 3) Enable continuous refinement of increasingly trustable, complex,
biomimetic mechanisms that stand as plausible explanations for increasingly
large sets of multi-attribute, multi-source wet-lab data. 4) Represent
uncertainty at multiple levels, including uncertainty in mechanistic hypotheses;
provide plausible representations of sources of variability in referent data and
phenomena. 5) Enable straightforward redeployment and adaptation of ISMA
components to represent other cell types and their behaviors; examples include
MCF-10A and primary mouse breast organoids. 6) Enable concrete translations
between in vitro knowledge and epithelial diseases such as autosomal dominant
polycystic kidney disease and cancer.
In silico methods
Components and mechanisms mapped as closely as possible to components and
mechanisms in the referent system. ISMAs were composed of cells,
luminal space, and extracellularmatrix. We set parameters such as the rate of celldivision and the initial size of cells to map to quantities
within the in vitro system. Simulation began with 2-4 cells (to mimic
the observed number of initial cells in vitro) on a 2D 100×100 hexagonal
grid. Cells expanded in size and divided using the CompuCell3D [28] cellular
Potts model architecture and customized code. Each cell occupied
multiple locations on a hexagonal grid, thus allowing cells to expand,
divide, change shape, and move in a realistic manner (Rejniak et
al. [16] used
an alternative method for enabling cell shape change). We coupled that with
features of the agent-oriented modeling approach used successfully by [14], [29]-[31].
Each cycle, cells stepped through the same decision flow (Figures 10 and S12); they
applied the operating principles described below to change shape,
divide, change state, create lumens, and die.
Logic design and implementation was constrained by the specifications in Table 1. Note that
cells are atomic objects: they have no internal parts. All of their
micromechanisms are in the form of axioms. Some axioms add behavior variability
to ISMAs, as noted in Table S2.
10.1371/journal.pcbi.1002030.g010
Key features of ISMA logic and decision control flow.
During a simulation cycle, each cell steps through five logic
modules sequentially to decide which actions to take based on its local
environment and internal state. A lumen's target
area is adjusted; lumens can merge with each other.
Cells that are not dying may begin to do so.
Cells adjust their area based on their state and the state
of neighboring cells; they stabilize if the lumen has
reached a critical size. Cells can create new lumens.
Under specified conditions they can divide to form new cells.
Future versions of ISMA logic may randomize action control in order to
simulate the parallel nature of event occurrence both within MDCK
cultures and within each cell. See Figure
S12 for complete details of the logic within each of the five
modules.
Except as noted, simulations ran using the parameter values in Table 2. A simulated
day mapped to an in vitro day and consisted of 48 simulation
cycles, equivalent to 30 minutes per cycle. Drawing on several years of prior
experience experimenting on MDCK cultures, we specified that when SM1 (defined
in Results) >0.5 for nine of ten days,
the results can be considered to be within the range of experimental and
biological variability. Specifically, when SM1 was achieved, simulation results
were taken to be experimentally indistinguishable from values obtained from an
independently repeated in vitro experiment. Empirical parameter tuning was used
to obtain frequencies of SLSL cysts comparable to that observed in
vitro. When SM targets were not achieved, that specific mechanism was falsified.
SMs also allowed for ISMA validation and falsification when new attributes were
added to the target list (discussed below).
Iterative Refinement Protocol
The Iterative Refinement Protocol (IR Protocol), described in [14], [19], [25], [29],
provided the foundation of our methods. Based on the results of prior
experiments and literature review, we selected an initial group of qualitative
attributes to target and simulate (the first few in Table 1). We implemented a simple ISMA that
reproduced them, thus achieving an initial degree of validation. We then added
new data, expanding the set of targeted attributes. So doing falsified the
simple analogue. That judgment was based on observation (for qualitative
attributes) and values of the prespecified SMs (for quantitative attributes).
The manner in which the first analogue was falsified informed us how to develop
an improved version that would survive falsification. During subsequent cycles,
we added new data or features from Table 1 to the targeted set. So doing often
resulted in falsification of the then-current ISMA. On some occasions, it was
clear that an incrementally more fine-grained set of mechanisms and/or
components would be needed to achieve the specified SMs. On other occasions, we
undertook an empirical search of parameter space in search of new sets of
parameter values that would reestablish validation. When that search failed, new
mechanisms, sometimes more fine-grained, were developed. That iterative process
ended with the attributes in Table 1 and the corresponding in silico specifications.
The IR Protocol has a number of benefits. Chief among them is that once an ISMA
is validated against targeted data, additional data can be added and the
analogue reengineered without invalidating existing mechanisms. The new data
will falsify the current ISMA by design, but a successful revision will survive
falsification by both new and existing data. Because in silico components and
mechanisms map to their in vitro equivalents, it is often the case that only a
subset of ISMA components and/or operating principles must be modified to mimic
both new and original phenomena. Examples include adding a new cell
state and replacing one axiom with two more specific axioms. Because of the
networked nature of all mechanistic details, each ISMA change requires some
retuning of the parameterizations of several already existing (unmodified) ISMA
features.
The IR Protocol consists of the following steps: first, specify a list of
targeted attributes, which forms the basis for experimental hypotheses. Devise a
specification that maps in silico components and operating principles to cell
culture counterparts. The operating principles are expected to enable
cells to exhibit behavior that is closely analogous to that
observed in vitro. Implement the analogue in code and execute it to deduce
predictions about the in silico and in vitro system. As stated in [29],
analogue execution is a form of deduction, where the behavior of the analogue
follows logically from the premises embodied by its initial conditions and input
data. In some cases, this deduction will yield obviously invalid results, which
falsifies the current list of operating principles and prompts the modification
of mechanistic hypotheses. Once the analogue cannot be falsified by data
specific to the current list of targeted attributes, add one or more new,
targeted attributes and repeat the IR Protocol.
The process facilitates mechanism exploration, leading toward deeper insight into
biological counterparts. Undertaking a series of tightly coupled in silico and
in vitro experiments further increases the confidence that the results of ISMA
intervention experiments can stand as useful predictions of MDCK counterparts.
When there is sufficient ISMA and MDCK cystogenesis similarity, we hypothesize
there is corresponding mechanistic similarity. Consequently, results of ISMA
intervention experiments will stand as predictions of in vitro phenomena
following corresponding in vitro interventions. Some of those predictions will
merit in vitro follow-up.
Agent-oriented approach
An advantage of using targeted attributes and specifications is the flexibility
of their implementation. We chose to implement the ISMAs using an agent-oriented
approach as explained below and described in [25], but their key aspects
include object-orientation, component mapping, spatial orientation, relational
grounding and striving for component autonomy. Agent-oriented models are
frequently implemented using object-oriented programming techniques, which allow
the designer to create individual computational objects corresponding to agents
and components within the specification. Components and mechanisms are mapped to
analogous components and mechanisms within the referent. So doing makes
translating in vitro and in silico observations back and forth more intuitive
and less complex. Individual agents can serve as analogues for in vitro
components. Agents are quasi-autonomous and they possess their own internal
control flow and execute actions independent of enclosing agents. Grounding is
defined as the units, dimensions, and/or objects to which a variable or model
constituent refers. When grounding is relational, variables, parameters, and I/O
are in units defined by other model components. When grounding is absolute,
variables, parameters, and I/O are in real-world units like seconds and
µg/ml. One advantage of using an agent-oriented approach with relational
grounding [25]
is that fewer assumptions are required to create or validate the ISMA, and those
that are must be clearly specified.
The ISMA contains five agents:
The experiment agent calls the MDCK plug-in agent and the Potts
agent.
The MDCK plug-in agent cycles through cell agents each
simulation cycle.
The Potts agent executes the index change step: pseudorandom index change
attempts and energy calculations.
The cell agents change their state and perform other
actions.
The screenshot agent, called in a separate thread, records a screen shot
at the end of the execution of the simulation cycle.
The cellular Potts model
ISMAs were developed using the CompuCell3D (CC3D) architecture [32], [33], an
implementation of the Glazier-Graner-Hogeweg [34] or cellular Potts model
(CPM). A CPM “cell” is not limited to a one-to-one correspondence
between objects and grid locations. The CPM extends cellular automata so that
each grid location contains an index specifying which simulation object contains
that location. A CPM with 100 grid locations can contain anywhere from 1 to 100
cells. This modification allows simulations to address
cell size, shape change, and cell-cell adhesion.
During a simulation cycle, the Potts agent calls a pseudorandom index change
algorithm that randomly selects a user-specified number of locations and
evaluates whether each will remain indexed to its current cell or
change to be indexed to another cell. If the location remains indexed
to the current cell, the grid remains unchanged. When a location's
index changes, that location and the “energy” of the system are
updated.
To calculate whether a location changes index from one cell to another,
ΔG is calculated; it is the change in “energy” if that location
changes its index to the new cell. An acceptance function generates a
probability p based on the value of ΔG, and then checks if
the pseudorandom number r[0,1]<p.
When r<p, the change is accepted and the
location is assigned to the new cell, and if not the change is
rejected. When accepted, the energy of the system changes.
It calculates the value of Gnew and Gold using a
Hamiltonian equation:
Each of these terms is calculated through a separate equation, detailed
below.
Surface area and perimeter
The energy calculation for EnergySurface depends on
LambdaArea (λA) and the difference between
the target surface area (TA) and the current surface area (A):
The larger LambdaArea is the more changes in TA will affect the
overall energy of the system and the faster these changes will be reconciled.
LambdaArea for cells is a user-set parameter,
while for lumen it is fixed at 20 to represent the large outward force
of the expanding lumen.
The calculation of EnergyPerimeter is similar:
Adhesion, connectivity, and tight junctions
The “energy” of adhesion depends on the cell type and its
location. For location (i, j), the energy is the sum of values calculated
between (i, j) and all neighboring points residing in separate cells.
If, for example, two of the six neighboring points reside in another
cell, then the energy of adhesion would be
2·X1–2, where X1–2 is a parameter
controlling the adhesion energy between cells of type 1 and type 2.
Separate adhesion energy parameters are specified for each pair of cell
types (Table
S3).
The “energy” of connectivity is generally 0, but if changing the
cell index of a location results in a location being isolated from
the rest of the cell, an energy penalty is assessed by setting
EnergyConnectivity to be very large. As a result,
cells cannot split into pieces except when they undergo
celldivision.
In addition to maintaining connectivity between all points in a cell, an
ISMA maintains integrity between tightjunctions, preventing them from being remodeled in the index change
step during a simulation cycle. If the ISMA detects that the change in a point
would result in a tightjunction being remodeled, it assesses an energy penalty by setting
EnergyConnectivityi to be very large. A
detailed explanation of tight junction remodeling is provided in Text
S1.
CompuCell3D and custom code
CC3D is designed from a system-based perspective. Each simulation cycle, each
aspect of the system is executed, from the index change step that selects random
points, to the plug-ins that update aspects of the system. CC3D was not designed
from an agent-oriented perspective, so it was necessary to expand it to gain
required capabilities. MCell objects were added to
cell objects to create a bi-directional mapping between individual
points and the cells that contained them. These objects and their
control flow were executed in sequence by the MDCK plug-in to grant full agency
to cells, which previously only executed after a location within the
cell boundary changed its index. Every simulation cycle all points
in the grid are surveyed to assess which cell they are indexed to and a
reference is stored in an MCell object corresponding to that
cell, as shown in Figure 11. Thereafter that MCell can be queried to
find out what points are located within its corresponding cell
object.
10.1371/journal.pcbi.1002030.g011
MCell point assignment flow chart.
An MCell point has no specific cystogenesis counterpart.
Once per simulation cycle, each point is assigned to the
MCell agent associated with the cell
enclosing that point. MCell point lists are initialized
during each simulation cycle. Additionally, surrounding cells
engulf isolated points.
The version of CompuCell3D used to develop this project has been superseded (see
Text
S1) by the current available version. The capabilities provided by
the current version were not judged necessary for the ISMA, especially due to
the significant addition of custom code. The project was not adapted to the
updated version.
Cells compute their target size using a value of ideal area
As shown in Results, we observed that prior
to cells stabilizing in vitro, their size correlated with the size of the cyst
and its cell number. We hypothesized that operation of yet-to-be identified
micromechanisms provides each cell with a target size. We speculated that a cell
might use information such as the tension between it and neighboring cells,
lumen pressure, and the ratio of lumen and matrix contact area in order to
update its target size. To mimic the decrease in mean cell area observed in
vitro, we developed and used an algorithm that is a placeholder for
yet-to-be-designed, concrete micromechanisms that can be implemented in a future
ISMA. Each individual cell adjusted its size and shape so that a target
area W, the projected wedge area (a wedge that includes the portion of the
perimeter in contact with matrix and terminates at the cyst
center), would move toward or equal an ideal value. The parameter
wedgeArea was a value based on the early
(pre-stabilization) 169 µm2 area observed in vitro. An ISMA
calculated W using the following formula:
A is the area of the cell, M is one-half the number in cell
grid edges in contact with matrix, and L is one-half the number in cell
grid edges in contact with lumen. This formula assumes that
cysts are somewhat circular. Variations in actual cell
size caused by non-circular cysts resulted in variance in cell
area similar to that observed in vitro. The cell subtracted W from
wedgeArea and set its target change in area to the
resulting value (with a final maximum value of wedgeArea). Use
of this algorithm during early simulation cycles caused mean cell area
to decrease and cyst area to increase, mimicking observed in vitro data
(Figure 2C).
Once cells stabilized, they no longer used the above equation. Instead,
cells strove to maintain an area that increased only slightly as
contact with the lumen increased. We speculated that cells within cysts
in vitro must maintain a minimal cell height even as they are stretched by the
expanding lumen. We specified that ISMAs use a similar guideline.
Cells compute a target perimeter
From in vitro observations, it seems likely that cells have genetic and
environmentally imposed targets for the areas occupied by different surfaces
(cell-cell interfaces, basal, and apical). We specified that 2D cells
have a target perimeter value (TP) that is computed using the
cell's current area. For simplicity, we specified that a
cell compute TP using the perimeter P of a circle having an area A
equal to its own:
K is a scaling factor and multiplier is user-specified. Two
cells having identical areas will have identical TP values, so if
one has a larger P the difference between P and TP will also be larger, causing
that cell to move toward circularity faster.
Cell polarization and stabilization
The value of polarCounter was set to equal a pseudorandom value
r[polarDelay · 0.75,
polarDelay · 1.25] when a cell first
contacted matrix. Thereafter, it decreased by one each simulation
cycle. Upon reaching 0, cell state changed from unpolarized to
polarized. Consequently, polarCounter is the
cell's counterpart to a cell, having established
matrix contact, changing and moving around its components in a process that ends
when tight junctions have formed and the apical surface is isolated and
complete.
A correlation was observed between mean cell size and the rate of cell division
in vitro, but a causal link was not apparent. Individual cells may sense the
area of matrix contact in part through β1-integrin signaling [26]. They may sense
the area of lateral cell-cell contact in part using catenins and cadherins [35]. That
information may influence whether a cell divides or not. As stated in Discussion, tension transduced by the
subapical F-actin network could allow cells to sense the size of the lumen. Such
information supported our decision to use lumen size as a signal for
cell stabilization. Each simulation cycle, a cell
bordering matrix and lumen queried the lumen for its
size. When that value ÷ 1000 was greater than the parameter
stableRatio, the cell changed to the stabilized
state.
Cell division
Decrementing cycleCounter is a cell's counterpart
to moving through the phases of the cell cycle. CycleCounter is
a variable that is initialized based on cellCycle (a
user-specified parameter that controls the duration of the cell cycle)
and decremented thereafter. Cells implemented the following method of
celldivision. For the first cell and for daughter cells
after division, the value of cycleCounter was set to a
pseudorandom value
r[0.75·cellCycle,
1.25·cellCycle] and then decremented in each
cell in every simulation cycle in which the cell had an
area > doublingArea/2. When cycleCounter
reached zero, a celldivided (Figure
12), splitting its area in half on an axis, and using the parameter
divisionReg to determine the method of calculating the axis
of division.
10.1371/journal.pcbi.1002030.g012
ISMA celldivision. Celldivision depends on the cell neighborhood.
Single isolated cells (top) that have not divided have
no midbody and divide with a random axis of
division. When cellsdivide they find their centroid and store it as the
midbody of their daughter cells. Cells
that have previously divided and have a midbody
utilize it for subsequent divisions. For these organized
divisions (top), the axis of division is
determined by a line drawn from a cell's current centroid
to the stored midbody. Cells in contact with a
lumen will also divide in an organized fashion
(bottom), using a line between their centroid and that of the
lumen to determine the axis of division. When the
axis of division is determined, all points on one side of the
line are assigned to a new cell while all points on the other
remain assigned to the original cell.
When divisionReg = 0, cells
chose the axis of division randomly. If it was 1, cells used
oriented division, finding their axis of division as shown in
Figure 12.
Cells recorded the location of their midbody as a point.
When dividing, the cell connected the midbody and the
centroid with a line. The cell assigned all points above the line to a
new cell and all points below to the old cell. It then set the
midbody of the both cells to the centroid of the
just-divided cell. When divisionReg
= 2, cells divided randomly until they reached
the polarized state and then used oriented division.
DivisionReg = 3 specified reversed
division, where cells would find the axis of
division as stated above, but then add 90 degrees, reversing
division orientation.
After a cell divided, the value of cycleCounter for
both daughter cells was reset to a new random value as detailed above.
Its value of polarCounter did not change. The new cell
inherited all values from the parent cell, except
polarCounter, which was set to
r[0.5·polarDelay,
1.5·polarDelay] –
polarDelay + polarCounter(parent). So
doing made the newly created cell have a polarCounter
value close to but not identical to that of the parent cell.
Cell clustering and cell death
In vitro data analysis revealed that when the cultures began growing, the mean in
vitro cell number was about 2 (Table S4), indicating that a small amount of
clustering took place after the cells were plated. That was expected because in
Matrigel culture suspended cells settle on the layer of 100% Matrigel and
thus most cysts grow in the same plane. Accordingly, simulations began with a
single cell, but at simulation cycle 1, the
cycleCounter of that cell was reduced to 1,
causing it to divide during the following simulation cycle. In
addition, since in vitro cells are not always at the beginning of their cell
cycle when plated, the value of cycleCounter for the two
cells was changed to equal a pseudorandom number
r[(1 – clusterProb) x
cellCycle, cellCycle]. So doing
allowed the amount of clustering to be increased without changing the
celldivision rate, simply by increasing clusterProb.
Cell death is an important factor in MDCK cystogenesis. However, it is not clear
that it is required for cyst formation. In order to validate that cysts
did not ignore or excessively rely on celldeath for normal lumen formation, the amount of celldeath observed in silico was quantified and compared to that observed
in vitro. In vitro analysis of cell death was conducted in [9]: MDCK cysts were
cultured as in this report and fixed and stained with an antibody for activated
caspase-3 (cleaved in apoptotic cells).
Within the ISMA, a cell began dying when a pseudorandom number
r[0, 1] was less than
deathRateEpi if the cell contacted matrix
or deathRateLumen if it did not. Once a cell entered
the dying state it shrank until its area reached zero. It was then
removed from the simulation. Each day, the number of cysts
with dyingcells was recorded and the percentage calculated (Figure 5). The data was separated based on
whether cells were in contact with the matrix or not.
Drawing on literature evidence [36]-[38] and expert opinion we estimated the average time
between apoptotic bodies first being visible and a dying cell breaking up into
pieces to be roughly five hours. The value of the parameter
dyingShrinkRate specified the amount that the TA of a
dyingcell was lowered each simulation cycle (Table 2). Mean dying time ranged
from 6.5 to 13 simulation cycles, with an overall mean value of 9.2, which maps
to 4.6 hours when one simulation cycle is grounded to 30 minutes.
Lumens and their creation
Polarizedcells create a new lumen when two conditions are met. 1) The
cell contacts matrix, but is not in contact with an
existing lumen. 2) The location chosen for lumen creation is
adjacent to another polarizedcell also not in contact with an existing lumen. The point
chosen for lumen creation is the cell'smidbody (Figure
12), which was the centroid of the parent cell that previously
divided to create the current cell.
Lumen formation involves cells creating and secreting fluid. Having
cells create and release units of lumen content could
simulate that. One unit could correspond to a single grid space. Those units
could merge with other units or with an existing lumen object. However,
absent validation evidence for the other ISMA mechanisms, implementing such a
fine-grained (somewhat complicated, multi-parameter) mechanism simply because it
seems biomimetic would have been contrary to the IR Protocol's strong
parsimony guideline. We took advantage of CC3D capabilities and elected to use a
more abstract, simpler approach. There is no disadvantage in doing so because a
strength of this class of analogues is that a simple mechanism that achieves a
degree of validation can later be replaced with a more detailed and realistic
counterpart. Using cross-model validation [25], this can be done without
compromising other ISMA mechanisms that have also achieved degrees of validation
[14].
Within ISMAs, lumens are a different class of
“cell” object. Their only action options are to expand and
merge. After a lumen is created, it expands using the following axiom.
LumenGrowthRate is a user-specified parameter;
estimatedArea is the area of cells in contact with
the lumen added to the lumen's area;
totalNeighbors is the number of cells in contact
with the lumen; and lgrSubtract is a quantity based on
a user-specified parameter and the degree cells are stretched.
Cells that are more stretched have a higher
lgrSubtract value, reducing the rate of lumen
expansion. A lumen does not have a target perimeter value—its
perimeter is determined entirely by the perimeter of the cells
surrounding it. Lumens can merge when their tightjunctions are reorganized.
Tight junction maintenance
Tightjunctions (TJs) were implemented in order to simulate aspects of MDCK
lumen expansion. TJs exist where two cells contact each other
and a lumen. A TJ is two points—one in each neighboring
cell—adjacent to a point within a lumen (see and
Figure
S13). TJs control lumen expansion and merging and
prevent cells from contacting multiple lumens. At the end of a
simulation cycle, when a TJ is adjacent to a different TJ, the TJs are
reorganized and the index of the two TJ points is transferred to the neighboring
lumen. Then the two lumens, now in contact with each
other, merge together. In addition, TJs can reorganize to allow lumen
expansion. To do so, at the end of a simulation cycle all points within TJs
execute the following algorithm. A TJ point first surveys its neighboring points
to verify they are not in contact with another lumen and that they are
not in the matrix or within an unpolarizedcell. It then determines if any of its neighboring points are in
different cells but are not also TJs. To reorganize, the TJ point
computes the free energy change if its index changes to the neighboring
lumen and then uses the acceptance function to accept or reject
that change. If the change is accepted, the TJ becomes lumen, and the
neighboring point becomes a new TJ.
Scaling observations from 2D to 3D
We recorded aspects of in vitro cyst growth by obtaining cross-sectional images
taken through the center of cysts. These images were necessarily a 2D
representation of a 3D structure. Based on the symmetry observed within the
cross section, in addition to separate analysis of 3D structures, we believe
that cysts were roughly symmetrical in 3D. Using this information, we
extrapolated 3D values for total cell number, cyst volume, and lumen volume from
the measured values of cross-sectional cell number, cyst area, and lumen area.
We found that the trends observed for cell number and mean cell area held when
the system was projected into 3D. If future targeted attributes required
specific modeling in 3D, we could take advantage of the 3D capabilities of
CompuCell3D, addressing considerations raised in [39].
Data storage
The in silico system recorded data about cells and cysts into a
MySQL database as specified within Text S1.
Supporting Information
Cystogenesis measures for TS ISMA. Experiments followed the same
experimental design as described in the text. Measures (red) were taken
during cystogenesis. In vitro data are provided (blue) for
comparison. Designations and symbols are the same as in Figure 2. TS ISMA used the parameter
values in Table 2,
except for stableRatio, which was set to 1000 and
shiftDelay, which was set to 200.
(TIF)
Percent of cysts with different numbers of lumens for TS ISMA. The
experiments are the same as in Figure S1. Designations and symbols are
the same as in Figure
3.
(TIF)
Percentage of cysts with dyingcells when dyingShrinkRate was reduced. (A) In
vitro data reproduced from [9]. (B) ISMA data
from 50 cysts over ten days using parameter settings from
Table 2, except
for dyingShrinkRate, which was changed from 9 to 4.5. Blue
bars: percentage of cysts observed to have apoptotic cells without matrix
contact. Red bars: percentage of cysts observed to have apoptotic cells with
matrix contact.
(TIF)
Cystogenesis measures when the axis of cell division is
random. Experiments followed the same design as in Figure
S1. Measures, designations, and symbols are also the same as in
Figure
S1. LS ISMAs used the parameter values in Table 2, except for
divisionReg, which was set to 0.
(TIF)
Cystogenesis measures when the axis of celldivision is reversed. Experiments followed the same design as in
Figure
S1. Measures, designations, and symbols are also the same as for
Figure
S1. LS ISMAs used the parameter values in Table 2, except for
divisionReg, which was set to 3.
(TIF)
Cystogenesis measures with no luminalcelldeath. Experiments followed the same design as in Figure
S1. Measures, designations, and symbols are the same as for Figure
S1. LS ISMAs used the parameter values in Table 2, except for
deathRateLumen, which was set to 0.
(TIF)
Percent of cysts with different numbers of lumens with no
luminalcelldeath. The experiments are the same as in Figure
S4. Designations and symbols are the same as in Figure 3.
(TIF)
Cystogenesis measures when cellpolarization was delayed. Experiments followed the same design as
in Figure
S1. Measures, designations, and symbols are also the same as for
Figure
S1. LS ISMAs used the parameter values in Table 2, except for cellpolarization, which was delayed as described in the text.
(TIF)
Cystogenesis measures for GM ISMA. Experiments followed the same
design as in Figure S1 except that GM ISMAs were used. Measures,
designations, and symbols are also the same as for Figure
S1. Top: note the large variances after day 5.
(TIF)
Percent of cysts with different numbers of lumens for GM
ISMA. The experiments are the same as in Figure
S9. Designations and symbols are the same as in Figure 3.
(TIF)
Parameter sweeping results. Experiments followed the same design as in Figures 2 and 3. Designations are the
same as in Figures 2 and
3. Parameters
changed from settings in Table 2 are listed within each Figure. S11-1 to S11-5: TS ISMA.
S11-6 through S11-80 used the LS ISMA. Except for S11-1 to S11-5, all
parameters were fixed except the single parameter being varied. S11-6 to
S11-9: varied wedgeArea. S11-10 to S11-13: varied
lambdaArea. S11-14 to S11-17: varied
stableTargetArea. S11-18 to S11-23: varied
cellCycle. S11-24 to S11-27: varied
stableCycleDelay. S11-28 to S11-31: varied
lambdaPerim. S11-32 to S11-35: varied
polarDelay. S11-36 to S11-39: varied
shiftDelay with high stableRatio.
S11-40 to S11-44: varied lgrSubtract. S11-45 to S11-48:
varied doublingArea. S11-49 to S11-52: varied
multiplier. S11-53 to S11-60: varied
lumenGrowthRate. S11-61 to S11-64: varied
deathRateLumen. S11-65 to S11-68: varied
deathRateEpi. S11-69 to S11-72: varied
dyingShrinkRate. S11-73 to S11-76: varied
clusterProb. S11-77 to S11-80: varied
stableRatio.
(PDF)
Full ISMA logic and control flow. Shown are the details of the five
components of Figure
10.
(TIF)
Tightjunction reorganization. Tightjunctions (TJs) prevent cells from contacting multiple
lumens. A) TJ counting when a point is within a cell
(left) or lumen (right). B) During the index change step, index
changes that result in a different number of TJs before and after the change
will be rejected. C) When pairs of TJs are adjacent and meet requirements,
lumens will merge together. D) TJ reorganization cannot occur
if it will result in a cell contacting multiple lumens. E)
Allowed TJ reorganization results in lumen expansion.
(TIF)
SM2 values for the ISMA. An SM2 value is the absolute value of the
coefficient of variance (for a specific measure) subtracted from the in
vitro coefficient of variance. Values over 0.25 (black) did not achieve the
validation target described in the text. Values between 0.15 and 0.25 (gray)
achieved the moderate validation target. Values less than 0.15 (white)
achieved the strong validation target.
(DOC)
Sources of stochasticity within ISMAs. The listed events or variable
assignments provide behavior variability during cystogenesis. Note
that index changes, which are random, contribute to many cell level
events, including shape change and lumen expansion.
(DOC)
Additional ISMA parameters. Parameters used for the ISMA system and for the
adhesion plug-in are listed along with descriptions, default values used for
simulation, and parameter ranges that are expected to give normal
results.
(DOC)
Mean cell number per day for cysts grown in Matrigel. Numbers in bold italic
are measured, mean values and non-bold numbers are projected values.
Projected values were found by multiplying or dividing the measured mean
values by the scaling factor of 1.56. During the first four days of growth,
the number of cells increased by a constant factor of 1.4 to 1.8 per day,
with a value of 1.56 minimizing the percent error between projected and
measured mean values. Using that scaling factor, the number of cells at day
0 was estimated to be 2.1, indicating that some clustering took place within
the Matrigel culture. To reflect this observation, ISMAs implemented
cell clustering.
(DOC)
Supporting methods.
(PDF)
ISMA time-lapse movie of typical cyst development. This experiment
followed the experimental design described in the text and used Table 2 parameter
settings. Cells begin to polarize at 0:07, two
lumens form by 0:11 and merge at 0:15. Sample celldeath observed between 0:35 and 0:38. Cell stabilization
occurs at 0:39. Frame rate: 6 simulation cycles per second.
(MOV)
ISMA time-lapse movie of multiple lumen formation. This experiment
also followed the experimental design described in the text and used Table 2 parameter
settings. Cellpolarization begins at 0:08 and lumen formation occurs at
0:09. (Note cells without lumen contact near the top of
the cyst from 0:09 to 0:34.) Second and third lumens form
at 0:35 and 0:36, delaying stabilization of neighboring cells. Most
cells stabilize at 0:42, while cells not in contact
with primary lumen stabilize at 0:44. Lumens merge at 1:08
and 1:09. Frame rate: 6 simulation cycles per second.
(MOV)
We thank members of the BioSystems group and the Mostov laboratory (notably David
Bryant) for helpful discussions and suggestions. We thank Donald Delmar Davis for
providing computational support. Finally, we are grateful for the valuable
discussion and feedback provided by Virginia Platt. The work was abstracted in part
from a dissertation to be presented by JAE to the Graduate Division, University of
California, San Francisco, CA, in partial fulfillment of the Ph.D. degree.
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