Full details of the original MDCK cell culture experiments are provided in 11. Briefly, MDCK cells were triturated into single-cell suspensions in type I collagen gel. Cells were grown for 7–10 d until cysts with lumina formed. For immunofluorescence staining of cysts, samples were incubated with primary antibodies overnight, followed by an overnight incubation with fluorescent dye-labeled secondary antibodies. To quantitate cyst polarity, cysts were stained for gp135 (apical surface), β-catenin (basolateral surface) and nuclei, and then visualized using a confocal microscope.

ISEA1 and ISEA2 are discrete event 12, agent-based 13 systems that comprise the core analogue and system-level components for experimentation and analysis (Figure 2). Because ISEA2 is based on ISEA1, both share a common design, and their experiment features overlap significantly (discussed below). Before moving forward with model refinement and experimentation, implementation redundancies of ISEA1 and ISEA2 were removed. We revised the existing framework to enable simulation of multiple, somewhat different CELL analogue types. ISEA1 was ported and revalidated within the new framework prior to ISEA2 development. To clearly distinguish ISEA components and processes from their in vitro counterparts, hereafter we use small caps when referring the former.

We created system-level components including EXPERIMENT MANAGER, OBSERVER, and CULTURE graphical user interface (GUI) to enable semi-automated experimentation and analysis. EXPERIMENT MANAGER, the top-level system component, is an agent that provides experiment protocol functions and specifications. The specifications define the mode of experimentation and the system's parameter vector. Experiments can be conducted in default, visual, or batch modes. Batch mode enables automatic construction and execution of multiple experiments, as well as processing and analysis of recorded measurements. Based on user-defined specifications, EXPERIMENT MANAGER automatically generates a set of parameter files and executes a batch of experiments, each corresponding to a different parameter file. After completion of all experiments, basic analytic operations collect and summarize data. OBSERVER is responsible primarily for recording measurements. At the end of every simulation cycle, OBSERVER scans the CULTURE internals and performs measurements. The measurements are recorded as time series vectors. At simulation's end, data are written to a set of files for analytic processing by EXPERIMENT MANAGER. CULTURE GUI provides a visualization console, which can be used interactively to start or pause a simulation and to access live states of CULTURE grid content. Using CULTURE GUI functionalities, OBSERVER can capture time-lapse CULTURE images and store them in multiple formats for post-processing.

Detailed descriptions of ISEA1 design features, and development methods, are available in 9. ISEA2 design uses similar features, which have been refined to meet study requirements. An abridged description follows. The referent in vitro cell culture was conceptually abstracted into four components: cells, media containing matrix (matrix hereafter), matrix-free media (free space hereafter), and a space to contain them. Discrete software objects with eponymous names represent those four essential cell culture components: CELL, MATRIX, FREE SPACE, and CULTURE. MATRIX and FREE SPACE are passive objects. A MATRIX object maps to a cell-sized volume of extracellular matrix (ECM). A FREE SPACE object maps to a similarly sized volume of material that is essentially free of cells and matrix elements. FREE SPACE also represents luminal space and non-matrix material in pockets enclosed by cells. The latter are called LUMINAL SPACE when distinction from FREE SPACE is useful. CELLS are quasi-autonomous agents (as agents, they can schedule their own events; they follow their own agenda). They use a set of rules or decision logic to interact with their local environment. A CULTURE is an agent that maps abstractly to a cell culture within one well of a multi-well culture plate. The CULTURE uses a standard two-dimensional (2D) hexagonal grid to provide the space in which its objects reside. The grid has toroidal topologies. For simplicity, each grid position is occupied by one object. That condition can be changed when the need arises.

There is a direct link between the choice of level of detail—granularity—and the list of targeted attributes. Granularity is the extent to which a larger entity is subdivided. There is also a direct link between required mechanistic detail and granularity. We can discover that a cell always (or almost always) executes a particular move when confronted with a specific situation without knowing (or needing to represent) details of how the move was accomplished. Our goal has been to first discover plausible cell-level mechanistic details that account for a variety of targeted attributes; cell size is thus a logical granularity level. We can then explore more detailed (fine-grained) explanations for how a particular mechanistic detail was enabled, because a coarse-grained component can be replaced by a finer-grained component when that is needed. A more coarse-grained mechanism that can account for targeted attributes is preferred over a more detailed mechanism because the coarse-grained mechanism is simpler. The parsimony guideline is to prefer the simpler explanation of the facts (the targeted attributes).

A CULTURE has base methods that are called automatically at a simulation's start and end. The start function initializes the grid and CULTURE components, CELLS, MATRIX, and FREE SPACE. Simulation starts upon completion of that process. As execution advances, the event schedule is stepped for a number of simulation cycles or until a stop signal is produced. At simulation's end, the CULTURE finish function closes open files and clears the system.

Simulation time advances discretely, and is maintained by a master event schedule. Event ordering within a simulation cycle is pseudo-random. Having objects update pseudo-randomly simulates the parallel operation of cells in culture and the nondeterminism fundamental to living systems, while building in a controllable degree of uncertainty. Within a simulation cycle, each CELL in pseudo-random order is given an opportunity to interact with adjacent objects in its environment and, if required, undertake an action. Every CELL uses the same step function. A set of CELL axioms (Figure 3) determines all CELL actions. A CELL selects just one axiom and corresponding action during each simulation cycle.

An agent has rules and protocols for interacting with external components. Rules can take any form. We elected to have all rules take the form of axioms. We use the term "axiom" to reinforce an idea that our computational model is a mathematical, formal system and that analogue execution is a form of deduction from the original axioms or assumptions explicitly programmed into the model. An axiom specifies a precondition and corresponding action. We specified what we judged to be a minimal set of action options: replace an adjacent non-CELL object with a CELL copy, DIE (vanish) and leave behind a LUMINAL SPACE, create MATRIX, destroy an adjacent non-CELL object and move to that location leaving behind a LUMINAL SPACE, POLARIZE, DEPOLARIZE, and do nothing. For any precondition, only one action option was executed.

ISEA1 had eleven axiomatic operating principles that enabled the analogue to validate against its initial targeted attributes. For convenience, the final ISEA1 axioms are summarized as follows. The precondition applies to the six objects adjacent to each CELL.

1. All neighbors are CELLS: DIE (delete self) and leave behind a LUMINAL SPACE.

2. All neighbors are LUMINAL SPACE: DIE and leave behind a LUMINAL SPACE.

3. All neighbors are MATRIX: replace a randomly selected MATRIX with a CELL copy.

4. Neighbors comprise one CELL and LUMINAL SPACES: add MATRIX between self and the adjoining CELL.

5. Neighbors comprise at least two CELLS and LUMINAL SPACES, but no MATRIX: DIE (undergo ANOIKIS) and leave behind a LUMINAL SPACE.

6. Neighbors comprise at least one CELL and MATRIX: create a CELL copy; the copy replaces any MATRIX that maximizes its number of CELL neighbors.

7. Neighbors comprise at least two LUMINAL SPACES and MATRIX: create a CELL copy; the copy replaces any LUMINAL SPACE that adjoins MATRIX.

8. Neighbors comprise CELLS, MATRIX, and at least two adjacent LUMINAL SPACES: create a CELL copy; the copy replaces any LUMINAL SPACE neighbor that adjoins MATRIX and LUMINAL SPACE.

9. Two CELL neighbors are separated on one side by MATRIX and on the other side by LUMINAL SPACE: POLARIZE.

10. A POLARIZED CELL has noncontiguous MATRIX neighbors: revert to NONPOLARIZED CELL state.

11. None of the preceding preconditions has been met: do nothing; CELL mandates achieved.

Detailed descriptions of supporting biological evidence and assumptions made for ISEA1 CELL axioms are provided in 9. Briefly, CELL DEATH axioms (Axioms 1, 2, and 5) were based on a general biological principle that cells, such as epithelial cells, undergo a process of cell death within some interval after detaching from ECM 1415. That behavior is observed in MDCK cell cultures 216. Axiom 4, which dictates MATRIX deposition between two adjacent CELLS, was specified based on observations that some matrix is produced de novo between two adhering MDCK cells in suspension culture 17. A CELL DIVISION axiom, Axiom 3, follows from experimental observations that, when embedded in matrix, single MDCK cells proliferate 1116. Other CELL DIVISION axioms, Axioms 6, 7, and 8, follow from a similar, general principle that epithelial cells proliferate when they adhere to ECM and tend do so in arrangements that maximize intercellular contact 1819. CELL POLARIZATION axioms, Axioms 9 and 10, reflect in vitro observations on MDCK cell polarity 218. Axiom 11 applied when the CELL achieved mandates that map to the three-surfaces principle articulated in 118.

Starting with the ISEA1 axioms, we devised, tested, and iteratively refined candidate axioms to enable the CELLS to consistently develop CYSTS with smooth margins and a convex shape (in the hexagonal grid representation), while validating for the targeted attributes described in 9. At each step, variations of an axiom were tested, and those that moved the analogue closer to validation were selected for further refinement. In its validated form, ISEA2 used Axioms 1–10 from ISEA1 without change. However, ISEA1's Axiom 11 was replaced by the following two axioms.

11. Neither the preceding nor the following preconditions have been met: do nothing; CELL mandates achieved.

12. A POLARIZED CELL confirms that Axiom 9 precondition is met and has only one MATRIX neighbor: the POLARIZED CELL deletes the adjacent MATRIX, moves to its location, and leaves behind a LUMINAL SPACE.

The revised axioms diagrammed in Figure 3 and summarized in Table 1 represent what we determined as a minimal change that was required for final validation. Revisions that were more elaborate also enabled those ISEAs to achieve the target attributes. However, they were rejected because they were not parsimonious. The final validation required that > 98% of the CYSTS formed during 50 simulation cycles in 100 Monte Carlo simulations must have a roundish, convex shape (visually inspected). We determined by visual inspection that convex CYSTS had no dimples or irregular margins. Manual inspection of the ISEA CYSTS sufficed for this study's purposes. However, we will need algorithmic metrics to expedite and automate analyzing and quantifying CYST convexity.

We implemented a method to disrupt selectively the operation of individual CELL axioms. We added a parameter, p, for each axiom. It controlled the probability of the decision-making CELL electing to follow the axiom when its precondition applied. Parameter values ranged from 0 to 1 inclusively. A parameter value = 1 corresponded to 100% adherence. Setting it to zero completely blocked the prescribed action and, as specified, dictated an alternate action. An additional control was added to allow the CELL to draw a pseudo-random number (PRN) from the standard uniform distribution at each decision point. The axiom's prescribed action was followed only when the PRN was ≤ the probability threshold set by its parameter.

We considered, and used when applicable, alternative actions that map to plausible in vitro cell actions occurring in a dysregulated state (Table 1). Axioms 1, 2, and 5 governed CELL DEATH; a reasonable alternative was to remain ALIVE (i.e., do nothing). Axiom 3 dictated non-directional CELL DIVISION; its alternate action was to do nothing (i.e., prevent REPLICATION). We also assigned the alternate action of 'do nothing' to Axiom 4 (MATRIX production). Several dysregulated action options were available for Axiom 6 (directed CELL DIVISION). One was to do nothing, effectively suppressing CELL DIVISION. Another was DISORIENTED CELL DIVISION, positing the CELL copy in a random direction without regard for the number of CELL neighbors. We elected to use the latter, for which adequate, supportive biological information is available 20212223. Axiom 7, which dictated CELL DIVISION, had available the same alternative action options. Axiom 8 (CELL DIVISION or POLARIZATION) had a precondition comprising all three component types (CELL, MATRIX, and LUMINAL SPACE), which presented many plausible action options. One option was preventing CELL DIVISION; another was to allow the CELL to DIVIDE non-directionally as described above. Another option was to initiate POLARIZATION. The remaining axioms, Axioms 9–12, posed a similar problem of having many plausible action options. Because no wet-lab experimental insight was available to narrow the options, we elected to defer investigation of those axioms until more information becomes available.

The following describes design and execution of ISEA1 and ISEA2 simulation experiments. First, the top-level system component, EXPERIMENT MANAGER, was initialized. Next, EXPERIMENT MANAGER created a new CULTURE and filled its grid with MATRIX. The grid width and height were set to 100. CULTURE initialized a PRN generator with a seed set to the system's clock. A new seed was used to initialize the CULTURE'S PRN generator at the start of each simulation. Pseudo-random seeds were generated from the CULTURE'S PRN generator to initialize those used by CELLS. Following CULTURE grid setup, one CELL was placed at the center of the CULTURE grid, replacing an existing MATRIX object. The simulation started when the initialization of the CULTURE contents was completed. Each simulation experiment comprised 100 Monte Carlo (MC) runs. Each MC run was executed for 50 simulation cycles. At simulation's end, the recorded measurements were written to files and the CULTURE was destroyed. A new CULTURE was created for each repetition.

The model framework was implemented using MASON, a multi-agent, discrete event simulation library, coded in Java 24. Batch simulation experiments were performed on a small-scale Beowulf cluster system. For model development, testing, and analysis, we used personal computers. Computer codes and project files are available at http://biosystems.ucsf.edu/research_epimorph.html.

In MDCK cultures, apoptosis contributes centrally to lumen formation 2. Cells in the inner region of the developing structure undergo anoikis. We speculated that if ISEA CELL actions have MDCK counterparts, then the two analogues would exhibit (predict) LUMEN filling when ANOIKIS is compromised. We simulated the condition by disrupting application of Axiom 5. So doing caused aberrant growth (Figure 5) and changed CELL activity patterns (Figure 6). Growth rates increased nonlinearly with increasing dysregulation. ISEA1 was more sensitive to dysregulation at mid-range p of 0.4 and 0.6 than was ISEA2. No marked differences were noted at other tested levels. CELL population measurements after 50 simulation cycles reflected changes in growth (Figure 5C). ISEA2 (vs ISEA1) produced structures having fewer CELLS.

Visual assessment of sample images showed that the CULTURE morphology became irregular with increased dysregulation (Figure 7). Relative to ISEA2, irregularities were more pronounced when ISEA1's Axiom 5 was dysregulated. When CELL creation events outpaced DEATH, small, inverted CYSTS formed and stabilized (through POLARIZATION) within LUMENS. As dysregulation increased, surface irregularities postponed POLARIZATION enabling further CELL creation events and surface expansion. For ISEA2, other factors contributed to LUMEN clearing. The convexity drive (Axiom 12) enabled surface CELLS to POLARIZE sooner. It also retarded inverted CYST formation by CELLS trapped within LUMENS. Trapped CELLS were thus more likely to satisfy the precondition of Axiom 5, even when Axiom 5 was partially dysregulated.

Figure 6 shows how changes in CELL activity patterns accompanied morphology changes for two levels of Axiom 5 dysregulation. ANOIKIS dysregulation changed the occurrence frequencies of axiom preconditions. That change resulted in increased CELL creation events for both ISEA1 and ISEA2. Interestingly, for p = 0.8 and 0.6, those changes led to a net increase in CELL DEATH events. For ISEA1, many of the additional CELL creation events occurred along the CYST's outer edge, whereas for ISEA2, many of the additional CELL creation and DEATH events occurred within the LUMEN. The CELL creation events within LUMENS were enabled by the Axiom 4 action: create MATRIX between two CELLS. Blocking Axiom 4 use blocks almost all CELL creation events within LUMENS and promotes LUMEN clearance (not shown).

Oriented cell division is central to multicellular morphogenesis 252627. Matrix contact and cell adhesions play an important role in determining the orientation of the division axis in vitro 2829. Similar to its in vitro counterpart, CELL creation from Axiom 6 was oriented (not random). We dysregulated Axiom 6 by allowing the decision-making CELL to place a new CELL in a randomly selected MATRIX location, rather than selecting one that maximizes CELL contact.

We ran simulations with Axiom 6's p ranging from 0 to 1, and recorded changes in CULTURE growth and morphology along with CELL activity patterns. The overall results are shown in Figure 8. CULTURE growth rate and CELL count after 50 simulation cycles increased monotonically with Axiom 6 dysregulation. The changes were less dramatic than those observed following Axiom 5 dysregulation, and there were marked differences between dysregulated ISEA1 and ISEA2 CULTURE growth. ISEA2 was less susceptible to disoriented placement of a newly created CELL. Mean CELL count in ISEA2 CULTURES was always smaller than that for ISEA1 at every tested dysregulation level.

Dysregulating Axiom 6 using p = 0.8 and 0.6 increased CELL DEATH and CELL PROLIFERATION activities of ISEA2 less than ISEA1 (Figure 6). CELL DEATH events were offset by an approximately equal number of CELL creation events, and that was consistent with the observation that LUMEN-entrapped CELLS underwent cycles of CELL creation and DEATH.

Inspection of Figure 7C, D shows that the morphological irregularities resulting from a given degree of Axiom 6 dysregulation were less pronounced than from a corresponding degree of Axiom 5 dysregulation. For ISEA1, the morphology change produced by a degree of Axiom 6 dysregulation was very similar to that caused by a lesser degree of Axiom 5 dysregulation. ISEA1 structures produced using dysregulated Axiom 6 contained a larger fraction of POLARIZED CELLS than did corresponding Axiom 5 dysregulated structures, and so the former changed more slowly as simulations progressed. For ISEA2, because all CELL DEATH axioms were always followed, there was less LUMEN filling when Axiom 6 was dysregulated, compared to when Axiom 5 was disrupted to the same degree. As noted above, ISEA2 LUMEN filling was enabled by Axiom 4. Blocking it severely restrained and often eliminated formation of INTRALUMINAL CELL CLUSTERS.

Figure 9 presents dynamic phenotype: the normalized frequency of axiom use by both ISEA1 and ISEA2. The CYSTOGENESIS mechanism at any stage in the process is the set of all events occurring within that interval. It is clear from Figure 9 that there is no specific CYSTOGENESIS mechanism. From start to the end of a simulation or until a stable structure forms, the mechanism evolves. How it evolves is a feature of that analogue's dynamic phenotype. Use patterns were similar for those axioms common to both analogues and that were used most frequently (1, 3, 5, 6, 9, and 11). Major differences were evident only for the less frequently used axioms (2, 4, 7, 8, and 10). As noted earlier, enabling CELL movement (Axiom 12) had an unanticipated consequence: it enabled the occasional formation of long-lived, small islands of CELLS within a LUMEN. Once a unit of MATRIX was formed, CELLS within a LUMEN could move and that gave rise to preconditions for creation of new CELLS as well as CELL DEATH. The process can continue for an extended interval and that accounts for the very low frequency of use of Axioms 2, 4, 7, and 8 by ISEA2. Note that when CELLS are trapped within an otherwise stable CYST, those INTRALUMINAL events are the only events. For that simulation, their relative use frequencies are large, and it is those values that are averaged with the values from other simulations, which are typically zero.

If nutrient levels within lumens are less than outside the cyst, then intraluminal cell division may not be sustainable. Furthermore, under 3D culture conditions, there is no direct evidence of matrix production by MDCK cells trapped within early-stage lumens during cystogenesis. It is noteworthy that by simulation cycle 50, when Axiom 4 is blocked, ISEA2's use frequency of axioms 2, 7, 8 and 10 drops to zero (not shown): ISEA2's axiom frequency of use pattern becomes similar to that of ISEA1.

Axiom dysregulation changed dynamic phenotype. Additional records for dysregulating Axioms 5 and 6 are provided in additional file 1: Supplemental Material for both ISEA1 and ISEA2. Because trends are similar for ISEA1 and ISEA2, we present in Figures 10 and 11 selected results for ISEA2. Figure 10 shows ISEA2 axiom use frequencies for Axiom 5 p = 0.8 and 0.6. The major consequence was reduction in Axiom 11 usage (do nothing: mandates achieved). That decline was mirrored by the rise in Axiom 5* (dysregulated action) usage, which remained relatively constant after five simulation cycles. In parallel, the use patterns for all other axioms changed relative to their p = 1 patterns. Even though only Axiom 5 was disrupted occasionally, all ISEA2 operating principles were impacted to some extent: the entire dynamic phenotype changed. However, the morphological consequences for p = 0.8 were difficult to detect: except for a tendency to be larger, most stabilized CYSTS were indistinguishable from those formed when p = 1. The potential morphological consequences of relaxing Axiom 5's p by 20% were thwarted by small shifts in the use frequencies of all other axioms. This observation suggests that the networked nature of ISEA2 axiom usage acts to buffer the consequences of small disruptions of any one operating principle.

Both the morphological and dynamic phenotypic consequences of Axiom 6 dysregulation were less dramatic than those of Axiom 5. They were also less dramatic in ISEA2 than in ISEA1. Reducing p led to larger structures that eventually stabilized (Figure 8B) and to more CELLS being trapped within occasional LUMENS (Figure 7D). Comparison of Figures 10 and 11 reveals that the influence of Axiom 6 disruption was also less significant than that of disrupting Axiom 5 to the same degree. For p = 0.8 and 0.6, the activities of CELLS trapped in LUMENS were primarily responsible for increased axiom use after about 20 simulation cycles. When Axiom 4 was blocked (not shown), those axiom use frequencies diminished considerably making an increased CYST size the primary consequence of Axiom 6 disruption.