This is an uncorrected proof.
The PI3K/AKT signaling pathway plays a role in most cellular functions linked to cancer progression, including cell growth, proliferation, cell survival, tissue invasion and angiogenesis. It is generally recognized that hyperactive PI3K/AKT are oncogenic due to their boost to cell survival, cell cycle entry and growth-promoting metabolism. That said, the dynamics of PI3K and AKT1 during cell cycle progression are highly nonlinear. In addition to negative feedback that curtails their activity, protein expression of PI3K subunits has been shown to oscillate in dividing cells. The low-PI3K/low-AKT1 phase of these oscillations is required for cytokinesis, indicating that oncogenic PI3K may directly contribute to genome duplication. To explore this, we construct a Boolean model of growth factor signaling that can reproduce PI3K oscillations and link them to cell cycle progression and apoptosis. The resulting modular model reproduces hyperactive PI3K-driven cytokinesis failure and genome duplication and predicts the molecular drivers responsible for these failures by linking hyperactive PI3K to mis-regulation of Polo-like kinase 1 (Plk1) expression late in G2. To do this, our model captures the role of Plk1 in cell cycle progression and accurately reproduces multiple effects of its loss: G2 arrest, mitotic catastrophe, chromosome mis-segregation / aneuploidy due to premature anaphase, and cytokinesis failure leading to genome duplication, depending on the timing of Plk1 inhibition along the cell cycle. Finally, we offer testable predictions on the molecular drivers of PI3K oscillations, the timing of these oscillations with respect to division, and the role of altered Plk1 and FoxO activity in genome-level defects caused by hyperactive PI3K. Our model is an important starting point for the predictive modeling of cell fate decisions that include AKT1-driven senescence, as well as the non-intuitive effects of drugs that interfere with mitosis.
Complex diseases such as cancer often alter more than one facet of a cell’s function. In addition to breakdown in individual functions, cancer progression leads to unhealthy combinations of cellular behaviors. For example, cancer cells rely on non-physiological combinations of cell functions drawn from an arsenal that includes proliferation, resistance to apoptosis, migration, and blood vessel recruitment. These functions are all critical to health or development, often in a different tissue than that of the tumor. Building predictive models that reproduce this coordination of functions could greatly boost our ability to combat complex disease. Here, we develop a large network model of the processes that control a mammalian cell’s life and death. Our model reproduces a non-intuitive oscillation in a key cell division pathway (PI3K/AKT1), along with the cell-cycle altering effect of its oncogenic activation. To do this, we incorporate the role of Polo-like kinase 1 (mitotic driver, chemotherapy target) and model mitotic failure when Plk1 is blocked. Finally, we offer testable predictions on the unexplored drivers of PI3K oscillations, their timing with respect to division, and the mechanism by which hyperactive PI3K leads to genome-level defects. Thus, our work can aid development of powerful models that cover most processes that go awry when cells transition into malignancy.
Citation: Sizek H, Hamel A, Deritei D, Campbell S, Ravasz Regan E (2019) Boolean model of growth signaling, cell cycle and apoptosis predicts the molecular mechanism of aberrant cell cycle progression driven by hyperactive PI3K. PLoS Comput Biol 15(3): e1006402. https://doi.org/10.1371/journal.pcbi.1006402
Editor: Jeffrey J. Saucerman, University of Virginia, UNITED STATES
Received: July 23, 2018; Accepted: February 12, 2019; Published: March 15, 2019
Copyright: © 2019 Sizek et al. This 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.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: EER was partially supported by the National Institutes of Health/National Heart, Lung, and Blood Institute grant HL119322 (URL: https://www.nhlbi.nih.gov). EER, HS and AH were supported by internal funding from the College of Wooster, primary in the form of travel and conference support. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Mammalian cells require extracellular growth signals to divide and specific survival signals to avoid programmed cell death (apoptosis) . The pathways leading to proliferation, quiescent survival or apoptosis are not fully independent; rather, they have a large degree of crosstalk. For example, most pathways activated by mitogenic signals such as PI3K → AKT1 and MAPK signaling also promote survival [2,3]. Moreover, regulatory proteins required for normal cell cycle progression such as E2F1, Myc and cyclin-dependent kinases (CDKs) can promote apoptosis as well [4,5]. Conversely, cell cycle inhibitors such as p16INK4a can enhance survival . As several of our most intractable diseases—cancer, cardiovascular problems and cellular aging-related complications—all involve dysregulation of these processes [7,8], creating predictive models to characterize them has been an ongoing focus for computational and systems biology. Approaches that couple computational modeling with experimental validation have made impressive strides in deciphering the networks in charge of cell cycle progression [9–11] and apoptosis [12–15], as well as the mechanisms of cell cycle arrest in response to stressors such as DNA damage [16–20]. Building on these efforts, our collective focus is increasingly shifting from models that describe individual functions towards ones that successfully integrate several aspects of cellular behavior [21–28]. These integrated models aim to predict the context-dependent outcomes of the crosstalk between different subsystem of large signaling networks, along with the knock-on effects of perturbing one subsystem on others. Furthering this effort, here we offer a comprehensive model of the nonlinear dynamics of PI3K → AKT1 ⊣ FoxO signaling coupled to the cell cycle. Our model can reproduce non-intuitive phenotypic effects of oncogenic PI3K , and offer testable predictions about the molecular mechanism responsible for them.
The canonical PI3K → AKT1 pathway is a major relay for growth and survival signals (Fig 1A) , as phosphorylated AKT1 has more than a hundred known direct targets [31,32]. First, AKT1 promotes the cell growth required for division and tissue growth, primarily by activating the mTORC1 signaling complex. Provided that the cell is not experiencing amino acid or energy deprivation, mTORC1 aids cell cycle commitment (Fig 1A, box 1), and orchestrates changes in cellular growth metabolism by increasing protein synthesis, lipid and nucleotide metabolism, and mitochondrial biogenesis [33,34]. Second, AKT1 inhibits GSK3β, counteracting its destabilizing effects on cell cycle-promoting and anti-apoptotic genes (Fig 1A, box 2) . Third, AKT1 aids cell cycle entry and survival by translocating the FoxO family of transcription factors out of the nucleus, thus decreasing cell cycle inhibitor and pro-apoptotic gene expression (Fig 1A, box 3) . Fourth, AKT1 phosphorylates the pro-apoptotic BAD, blocking its mitochondrial localization (Fig 1A, box 4) .
(A) Feed-forward network of interactions from growth receptors to PI3K and AKT1 (detailed description of molecular mechanisms in Methods & Model). Box 1: AKT1 activates the mTORC1 pathway, driving volume growth; box 2: AKT1 blocks the GSK3β pathway responsible for dampening cell cycle entry and survival signaling; box 3: AKT1 blocks the FoxO transcription factors that drive expression of anti-proliferative and pro-apoptotic genes; box 4: AKT1 promotes cell survival by keeping the pro-apoptotic protein BAD in check. (B) The mTORC1 pathway feeds back to dampen PI3K → AKT1 signaling by mediating the degradation of insulin receptor substrates (red arrows), aiding the cytoplasmic translocation of PTEN (purple arrows) and dampening mTORC2 activation (orange arrows).
The array of signaling events described above all point to a coherent role of AKT1 in promoting survival and proliferation. There is mounting evidence, however, that PI3K → AKT1 activity during cell cycle progression is more complex [31,32]. Overactive AKT1 in cancer cells has been associated with driving cells into senescence (an aging cell state characterized by permanent cell cycle arrest) [38,39]. More intriguing are studies showing that active FoxO3 and/or FoxO1 not only block cell cycle entry but are paradoxically required for its subsequent completion. A study by Alvarez et al has shown that the attenuation of the PI3K → AKT1 pathway after restriction point passage was required for FoxO3 activity in G2, which in turn aided the completion of cytokinesis . To explain their observations, the authors showed that FoxO3 upregulates the expression of the mitotic cyclin B and polo-like kinase 1 (Plk1), thus promoting the G2/M transition and progression out of telophase. Furthermore, work by Yuan et al also implicated FoxO1 in G2-phase Plk1 regulation . This effect, however, may be short lived, as both FoxO factors are inhibited by Plk1 phosphorylation [41,42]. To capture these subtleties in a model, we first need to understand the mechanisms that generate a short-lived AKT1 pulse.
There are several known feedback mechanisms that can explain the pulse-like spike and subsequent attenuation of AKT1 following growth factor stimulation (Fig 1B). Most of these involve mTORC1, and several are specific to insulin and IGF1 signaling . For example, mTORC1 is known to mediate the degradation of insulin receptor substrates IRS1/2 required for insulin and IGF1 signaling (Fig 1B, red arrows) . In addition, inhibition of FoxO transcription downstream of AKT1 leads to attenuated transcription of insulin and IGF1 receptors . To further complicate the picture, activation of the mTORC1 target S6K sets in motion two growth receptor-independent negative feedback loops. First, S6K can attenuate mTORC2 activity required for full AKT1 activation (Fig 1B, orange arrows) [42,43]. Second, S6K can promote nuclear export of the PI3K inhibition PTEN (Fig 1B, purple arrows) . Together, these mechanisms are thought to dampen AKT1 activation following an initial peak and regulate the homeostatic maintenance of AKT1 under ongoing growth stimulation.
In addition to feedback downstream of AKT1, a study by Yuan et al has demonstrated that even before AKT1 signaling downstream of growth receptors has a chance to engage, only a relatively small subpopulation of cells (~30%) are responsive to these signals to begin with . The remaining cells do not suffer from lack of receptor activation or lack of AKT1 protein; rather, they have very low levels of the PI3K subunit p110 at the time of stimulation. More surprising was their finding that as AKT1 peaked in the responding population of cells, the initially high p110 underwent rapid degradation. This effect essentially co-occurred with AKT1 activation; a process too rapid for the feedback mechanisms downstream of AKT1 to mediate. Moreover, the feedback detailed above acts on AKT1 phosphorylation and/or PI3K activity, not on their protein expression. Yuan et al showed that the cycle of rapid p110 degradation and subsequent re-synthesis was mandatory for sustaining proliferation, as clonal populations with degradation-resistant p110 and sustained peak AKT1 activity entered senescence at a high rate. Importantly, their findings were not restricted to a single growth receptor, pointing to a general, yet unrecognized set of feedback loops driving the expression cycle of p110. Finally, their work indicated that p110 heterogeneity in quiescent cells is strongly influenced by local cell density. Another study confirms that the catalytic p110 subunit of PI3K is indeed rapidly degraded upon growth stimulation in two additional cell lines, and that it re-accumulates slowly (~2 hours) . This work points to the NEDDL4 ubiquitin ligase as the driver of p110 degradation . In addition, AKT phosphorylation has been shown to exhibit at least two clear peaks before the end of S-phase in cells entering the cell cycle from quiescence , an effect observable despite rapid de-synchronization of the cell culture. These studies, however, do not address the molecular mechanisms that trigger p110 degradation specifically in response to growth factor signaling, its subsequent re-synthesis, or the way it’s oscillations interface with cell cycle control.
Current computational models of the regulation of mammalian cell life and death do not account for dynamic p110 expression . Models that incorporate feedback on AKT1 activity typically focus on the intricacies of the mTORC1 / mTORC2 crosstalk  or the effects of negative feedback on the strength of AKT1 signaling [49,50], but do not encompass the full cell cycle. Here, we put forth a large Boolean model of the regulatory interactions driving dynamic growth factor signaling, cell cycle progression and apoptosis. We built the model by bringing together several separately published, disconnected pieces of evidence regarding p110 protein and mRNA regulation [47,51,52]. We then linked the resulting growth signaling layer to an updated Boolean cell cycle model , as well as the molecular network responsible for survival vs. apoptosis. The resulting Boolean model reproduces the cell-cycle dependent role of PI3K, AKT1 and FoxO proteins [29,46] by linking them to Plk1 regulation in G2. As expected, it generates straightforward behaviors such as lack of cell cycle commitment in the absence of high p110 expression , or G1 shortening in the presence of hyperactive PI3K / AKT1. The novelty and value of our model, however, stems from its ability to reproduce more intricate, non-intuitive phenotypic outcomes. First, our model reproduces the path to apoptosis in the event of a mitotic catastrophe . Second, our model generates four distinct cell fates in response to Plk1 inhibition, depending on the timing of Plk1 loss : i) G2 arrest , ii) mitotic catastrophe [54,56–58], iii) premature anaphase and chromosome mis-segregation leading to aneuploidy , and iv) failure to complete cytokinesis following telophase [60–62], which can lead to genome duplication . Third, our model can replicate failure of cytokinesis and accumulation of binucleate telophase cells driven by hyperactive PI3K / AKT1 or by FoxO3 inhibition .
Our model’s ability to accurately reproduce a range of cell fates triggered by altered PI3K, AKT1, FoxO3 or Plk1 activity leads to several experimentally testable predictions. Namely, we predict 1) the molecular mechanisms of p110 degradation in response to high PI3K activation, and the transcriptional driver of its re-synthesis; 2) that the degradation / re-synthesis cycle of p110 occurs at least twice per division cycle (along with the molecular mechanism for their phase-locking); 3) that cell cycle defects in response to PI3K / AKT1 over-activation or FoxO3 knockdown are generally due to a loss of Plk1 in telophase; 4) loss of strong growth signaling in p110-overexpressing cells allows for normal cell cycle completion; and finally, 5) that cells in which p110 is inhibited after the start of DNA synthesis can still pre-commit to another cell cycle in the presence of saturating growth stimulation.
In order to build a mechanistic model of the dynamics of growths signaling and its influence on cell cycle progression and apoptosis, we turned to Boolean modeling . Using a modular approach proposed in , we first collected key growth signaling pathways driving cell cycle commitment in a Growth Signaling module responsible for the dynamics of PI3K, AKT1, MAPK and mTORC. Next, we identified key regulatory subsystems that control cell cycle progression, such as the Restriction Switch driving the initial commitment to DNA synthesis , the Phase Switch driving cell cycle progression from G2 to M and back to G1  (expanded from  to account for the mitotic role of Plk1 ), and a regulatory switch that tracks the licensing and firing of replication origins. Finally, we synthesized several published models of the survival vs. apoptosis decision into an Apoptotic Switch. These modules are tied together into an 87-node network by direct regulatory crosstalk, as well as a few nodes that represent cellular processes we do not track in molecular detail (e.g., DNA Replication, mitotic spindle assembly or cytokinesis). Following a detailed description of our model, we show that it faithfully reproduced quiescent, apoptotic and dividing cell phenotypes, and that its behavior is robust under synchronous or asynchronous update. To understand the role of dynamic PI3K signaling in healthy cell cycle progression, we then explore the consequences of Plk1 inhibition at different points along the cell cycle and show that the non-intuitive consequences of PI3K hyperactivation are explained by mild Plk1 inhibition in G2/M.
Modeling the dynamic regulation of p110 expression during growth factor signaling
In order to build a Growth Signaling module that incorporates the molecular drivers of p110 dynamics, we turned to the literature in search of mechanisms that can drive rapid p110 degradation and gradual re-synthesis (Fig 2A). Both the free and p85-bound versions of the PIK3CA (p110α) subunit of PI3K have been shown to undergo proteasome-dependent degradation triggered by the E3 ubiquitin ligase NEDDL4 . The activity of NEDDL4, in turn, requires Ca2+ and inositol trisphosphate (IP3) . This led us to hypothesize that the ability of NEDDL4 to ubiquitinate p110 spikes in response to sudden growth factor stimulation. Namely, growth receptors activate phospholipase C γ (PLCγ), an enzyme that generates IP3 from membrane-bound PIP2. IP3 diffuses to the endoplasmic reticulum, where it triggers Ca2+ release into the cytosol . Thus, IP3 and Ca2+ are available to activate NEDDL4 within minutes of receptor activation, leading to rapid p110 degradation. As membrane tethering of PLCγ requires PIP3—a product of active PI3K [65,66], the cascade leading to the polyubiquitination of p110 can only occur in cells that express high levels of p110 when growth signals arrive (as observed by Yuan et al ). To summarize, we posit that strong PI3K activation initiates a negative feedback loop leading to its own degradation, independently of its effect on AKT1 (Fig 2A, red links).
(A) Degradation of the PI3K subunit p110 may be driven by the PLCγ-dependent activation of the NEDDL4 ubiquitin ligase (red links); re-synthesis of p110 may be driven by FoxO3, which re-enters the nucleus following p110 degradation, as AKT1 activity falls (orange link). (B) Growth Signaling Module of our Boolean model, including the degradation/re-synthesis circuit in control of p110 expression (left, dark green), basal PI3K/AKT signaling (middle), downstream effectors of AKT1 (mTORC1 signaling, GSK3 & FoxO1, bottom), and the MAPK cascade (right). Black →: activation; red ⊣: inhibition; thick red links: p110 degradation; thick orange loop: p110 re-synthesis. (C) Periphery: sequence of network states along the synchronous limit cycle of the core PI3K circuit. Orange/blue borders: ON (expressed and/or active) / OFF (not expressed and/or inactive) node. Middle: state transition graph of the general asynchronous model (one random node updated per timestep; sampled for 10,000 steps), yielding a complex limit cycle that follows the synchronous cycle. Node size: visitation frequency; label: most similar synchronous cycle state; node color: overlap of similar synchronous cycle state (one minus normalized Hamming distance); layout: Kamada-Kawai algorithm (NetworkX , Python). (D) Overlap of states along a general asynchronous update trajectory (y axis) with each attractor state along the synchronous limit cycle (x axis). Time-step: update of a single random node.
Next, we turned to the mechanism of p110 re-synthesis. Studies of the p110α promoter indicate that this gene is positively regulated by FoxO3 . We hypothesized that reactivation of FoxO3 in the G2 phase of the cell cycle, after the initial AKT1 activation subsides, is the driving force behind p110 re-synthesis (Fig 2A, orange link). To integrate these negative feedback loops with the canonical PI3K / AKT signaling cascade activated by growth receptors, we introduced separate Boolean nodes to track basal vs. peak PI3K and AKT activity (Fig 2B; Boolean gates: S1A Table). Our model can thus distinguish between survival signaling in a low growth factor environment (where basal PI3K and AKT are ON) and peak PI3K/AKT activation following the arrival of a strong mitogenic stimulus. Complemented by a linear MAPK cascade and mTORC1/2 signaling, this non-linear PI3K/AKT axis dominates the behavior of the resulting Boolean Growth Signaling module (Fig 2B).
Modeling the two feedback loops controlling p110 expression in isolation shows that they generate a sustained, robust oscillation (Fig 2C), even though our model does not account for the fact that p110 degradation is significantly faster than its re-synthesis. This oscillation is the only attractor state of the small module regardless of Boolean update. As Fig 2C indicates, the synchronous attractor cycle clearly maps onto the cyclic succession of complex attractor states of the general asynchronous model (Fig 2C, weighted, directed network in the middle). In addition to never leaving the complex attractor shown on Fig 2C, asynchronous time series repeatedly walk through cycles of states that resemble the synchronous limit cycle (Fig 2D). Within the context of the larger Growth Signaling module, this oscillation only occurs under ongoing high growth factor stimulation.
Modeling cell cycle commitment, the licensing of replication origins and the survival/apoptosis switch
In order to investigate the downstream consequences of an oscillating Growth Signaling module, we next updated our previously published cell cycle model  and extended it with an apoptotic switch (described in detail in Methods & Models).
Core cell cycle switches.
First, we modeled the switch-like restriction point control guarding cell cycle entry by reusing the p21-positive version of our published Restriction Switch (Fig 3, blue subgraph & box; Boolean gates: S1B Table) . In isolation this module has two stable states corresponding to cell states before and after restriction point passage. Second, we expanded our published tri-stable Phase Switch driving mitotic entry and exit to account for the regulation and key functions of Polo kinase 1 (Plk1) (Fig 3A, purple subgraph & box; Boolean gates: S1C Table) . Plk1 is activated in early G2 by the FoxM1 transcription factor . In addition, decreased Plk1 expression in the absence of FoxO3  and/or FoxO1  during G2 connects Plk1 availability to the dynamics of PI3K → AKT1 ⊣ FoxO signaling. The updated Phase Switch retains three stable point attractors, matching the activity pattern of this network in G0/G1, at the G2 checkpoint, and at the Spindle Assembly Checkpoint (SAC). Third, we built a small switch that tracks the assembly, licensing and firing of replication origins (Fig 3A, brown subgraph & box; Boolean gates: S1D Table). This two-state switch reproduces the stability of assembled Pre-Origin of Replication Complexes; its stable states correspond to unlicensed and licensed origins. Fourth, we accounted for the progression and completion of cell cycle processes not modeled in molecular detail (Fig 3B, orange nodes; Boolean gates: S1E Table). The Replication and 4N_DNA nodes track DNA synthesys ; the unattached kinetochore node (U_Kinetochore) denotes incomplete mitotic spindle assembly and ongoing metaphase, while the attached kinetochore node (A_Kinetochore) marks completion of the mitotic spindle. Finally, key regulators of the coupling between regulatory switches and cell cycle processes, such as S-phase checkpoint signaling (Chk1), the unattached kinetochore sensor Mad2, and a marker of contractile ring assembly and cytokinesis (Ect2) further link the modules.
(A) Stable attractor states of isolated regulatory switches. Blue / light brown / purple / dark red boxes: stable states of the Restriction / Origin of Replication Licensing / Phase / Apoptotic Switch. Orange / blue node border: ON / OFF state. (B) Network representation of the Boolean model partitioned into regulatory switches and processes. Gray: inputs representing environmental factors; green: Growth Signaling; dark red: Apoptotic Switch; light brown: Origin of Replication Licensing Switch; blue: Restriction Switch; purple: Phase Switch; orange: cell cycle processes and molecules that bridge between the multi-stable modules. Black →: activation; red ⊣: inhibition. (C) Cell phenotypes predicted for every combination of no/low/high growth-factor (x axis) and Trail exposure (y axis). The network-wide ON/OFF states of each attractor and the molecular signatures that define their phenotypes are detailed in S2 Table. Blue fragmented cell: apoptotic states (#1–6); gray elongated cell: quiescent/non-dividing states (#7–8); cell with mitotic spindle: cell undergoing repeated cycles (#9). Yellow circle around nucleus: 4N DNA content; double-/single-headed arrows between cells: reversible/ irreversible phenotypic transitions in response to changing environments; green arrow: change in growth factor levels; red: change in Trail exposure. Image credits: apoptotic cell ; quiescent cell: https://en.wikipedia.org/wiki/Cell_culture#/media/File:HeLa_cells_stained_with_Hoechst_33258.jpg; mitotic spindle: https://en.wikipedia.org/wiki/Cell_division#/media/File:Kinetochore.jpg.
The apoptotic switch.
To account for the apoptotic effects of growth factor withdrawal and death due to mitotic failure, we synthesized published models of apoptotic commitment to create a detailed Boolean regulatory switch (Fig 3B, dark red subgraph & box; Boolean gates: S1F Table) [12–15,69–71]. This switch has two stable states corresponding to survival and apoptosis, and it is flipped when extrinsic signals from death receptors, or intrinsic signals due to mitotic failure trigger Mitochondrial Outer Membrane Permeabilization (MOMP), leading to the activation of executioner Caspase 3 . While the positive feedback loops that stabilize apoptosis are common to most published models, the signals that trigger mitotic catastrophe have not yet been modeled. To do this we incorporated Caspase 2 activation during prolonged or perturbed metaphase [53,72]. Literature indicates that normal mitotic progression is a balancing act on the part of Cyclin B/Cdk1 and Plk1. On one hand, both kinases phosphorylate and inhibit the anti-apoptotic BCL2/BCL-XL proteins, priming cells for apoptosis [73–75]. On the other hand, Cyclin B/Cdk1 inhibits Caspase 2, keeping cells alive as long as mitosis is not stalled . In addition to the loss of Cdk1 activity, metaphase cells also undergo Caspase 2 mediated apoptosis in the absence of Plk1 . Our model captures this balance of pro- and anti-apoptotic signals, such that loss of Cdk1 or Plk1 activity before cells clear the spindle assembly checkpoint can push them into mitotic catastrophe.
The network of linked regulatory models reproduces environment-dependent proliferation, quiescence, and/or apoptosis
Linked together, the modules generate a dense 87-node Boolean model with 375 links (Fig 3B). The synchronous dynamics of the full model is heavily constrained by the switch-like behavior of its modules, as evidenced by the small number of tightly coordinated behaviors (phenotypes) it generates. Indeed, when the state space of the network is sampled extensively using noisy synchronous update (Methods & Model–Mapping the attractor landscape of large Boolean networks using synchronous update), every attractor corresponds to a distinct cellular phenotype. These attractors are characterized in detail in S2 Table, along with key molecular signatures that allow us to match them to specific phenotypes. Fig 3C summarizes them according to the extracellular environment each phenotype occurs in; namely, the absence / low abundance / high abundance of growth factors (x axis on Fig 3C) combined with the presence / absence of the apoptotic signal Trail (y axis). Table 1 matches cell phenotypes generated by our model to experimentally documented cell behaviors in multiple cell types. As expected, irreversible apoptosis is stable in every environment. Moreover, the ongoing presence of saturating Trail (i.e., Trail input is ON 100% of the time) destabilizes every other cell state, leaving apoptosis as the only stable option [79–82]. Similarly, the complete absence of growth / survival signals also leads to apoptosis [83–85]. In contrast, low levels of growth signaling support quiescent cell states, and our model identifies two distinct forms. First is a healthy cell state with 2N DNA content (Fig 3C, elongated cell with blue nucleus on). Second, our model also produces a G0-like state representing cells that have failed to complete mitosis or cytokinesis in the past, now stuck with a 4N DNA content (Fig 3C, elongated cell with yellow circle around nucleus). Finally, exposure to high levels of growth factor results in a cyclic attractor representing continuously cycling cells (Fig 3C, mitotic cell).
Our modular approach allows us to attribute discrete transitions cells undergo to the dynamics of isolated regulatory switches, apparent in the activation patter of the interlinked modules under synchronous update. For example, the sequence of molecular changes that occur within our modules transitioning from a quiescent state into the cell cycle reveals the higher-order logic by which regulatory switches toggle each other (Fig 4). First, cell cycle entry involves the activation of the Growth Signaling Module. While the basally active parts of this module remain on, we see a cascade leading to MAPK signaling (Fig 4, upstream PI3K cycle). This part of the module remains stably ON in a high (saturating) growth factor environment. In contrast, the part of the module responsible for cyclic PI3K / AKT1 activation enters an oscillating pattern, as expected from the limit cycle on Fig 2C. Thus, our integrated model of growth signaling and cell cycle progression can reproduce the experimentally documented but unexplained oscillations in PI3K expression and AKT1 activity . Next, cyclic AKT1 activity triggers downstream oscillations in mTORC1 signaling and GSK3β. As these AKT1 targets are subject to feedback from the rest of the network, they do not directly mimic the dynamics of PI3K and AKT1 (see Methods & Model). Full activation of the Growth Signaling module then toggles the Restriction Switch into a state representing restriction point passage (later we observe this switch partially, but not fully reset between each cycle). Around the same time, we observe licensing of replication origins (Origin Licensing Switch), subsequently reset by the firing of replication origins in S-phase. Now committed, the cell toggles through replication, G2, mitosis and cytokinesis under the control of the Phase Switch (see Cell cycle processes). In contrast, the Apoptotic Switch only experiences minor perturbations, without being flipped.
Dynamics of regulatory molecule expression / activity during cell cycle entry from G0, showcasing the phase-locking of PI3K oscillations to the cell cycle. X-axis: time-steps; y-axis: nodes of the model organized in modules; orange/blue: ON/OFF; white boxes & arrows: first two peaks of AKT1 activation with respect to DNA replication; black dashed lines: cytokinesis; lime arrows: first AKT1-high pulse in each division cycle.
Our model’s dynamics is robust to fluctuations in signal propagation and reproduces the cell cycle with synchronous and asynchronous update
To test whether the orderly progression through the cell cycle is robust to random fluctuations in signal arrival time as they propagate through the network, we tested the model’s behavior under random order asynchronous update (Methods & Model–Boolean Modeling Framework) . As fixed-point attractors of a Boolean model remain the same regardless of update , we focused on the cell cycle. As S1 Fig shows, a fully random update order does not abolish the model’s capacity to execute a correct cell cycle sequence, but it does introduce several non-biological behaviors. First, the signals that couple successful DNA replication to the establishment of a G2 state are lost under a subset of update orders, leading to G2 → G1 reset followed by a new cell cycle (endoreduplication). Second, the signals that drive cytokinesis can also be disrupted by certain update orders. Third, the balance of pro- and anti-apoptotic signals during metaphase can tip in favor of apoptosis, as if the cell experienced mitotic catastrophe. Interestingly, all three cell cycle errors are observed in vitro in cells experiencing knockdown or overexpression of a variety of cell cycle regulators [16,54,88]. Thus, we conclude that the asynchronous model with random update order mimics the occasional short-term loss of regulators rather than the robust cycling of healthy cells.
In order to create a restricted random order that forbids asynchronous state transitions resulting from these non-physiological breaks in signal transduction, we identified genes and processes that deviated from their expected activity every time a particular error occurred and created an asynchronous version of the model with biased random update (Methods & Model–Boolean Modeling Framework). To do this, we placed a small subset of nodes at the start or end of each update order, depending on their activation status (11 nodes; list and rationale in S3 Table). Using this biased update our model repeatedly and correctly executes the cell cycle, in spite of the asynchronous update (Fig 5). Our update bias did not completely eliminate endoreduplication from G2 and apoptosis (S2 Fig), but the incidence of these errors decreased drastically. As these errors do occasionally occur in wild-type cells [16,82], we choose not to further restrict our update order to eliminate them. Rather, we measured the rate at which the two update schemes produce normal cell cycle events vs. different errors via a series of simulations at varying levels of growth factor and Trail stimulation. We did this by setting GFH or Trail ON with probability p in each time-step, OFF otherwise. As Fig 5B indicates, the asynchronous model with biased update shows a similar response to growth factors and Trail as the synchronous model. Moreover, the incidence of apoptosis or endoreduplication after G2 is significantly lower than under random update, and lack of cytokinesis all but disappears.
(A) Dynamics of regulatory molecule expression / activity during cell cycle entry from G0 using biased asynchronous update. X-axis: time-steps; y-axis: nodes organized in modules; orange/blue: ON/OFF. (B) Occurrence rate of normal cell cycle completion (mustard), G2 → G1 reset followed by genome duplication (purple), aberrant mitosis followed by genome duplication (turquoise), failed cytokinesis followed by genome duplication (blue) and apoptosis (dark red) per 100 timesteps, shown as stacked bar charts for increasing growth factor / Trail exposure (left/right) with random order asynchronous / biased asynchronous / synchronous update (top/middle/bottom).
The apoptotic fixed-point is reachable from cell cycle under both random-order and biased asynchronous update, indicating that the cell cycle is not, strictly speaking, a complex attractor . Nevertheless, starting an asynchronous time series from any state along the synchronous cell cycle attractor results in long time-courses featuring repeated (if occasionally incorrect) cycles (S4 Fig). This indicates that the system’s state space has a metastable region that traps its dynamics in a way that resembles a complex attractor. To test whether this metastable collection of states is also a cycle, we sampled the state transition graph of the asynchronous model with both update schemes by starting 10 independent time courses of 1000 steps from each state along the synchronous cell cycle. In order to sample the metastable basin rather than the path to the apoptotic attractor, we prematurely interrupted each run if it reached a fixed point. We then overlayed all observed states and transitions, visualizing the largest strongly connected component (S5 Fig, left). To test whether these state transition graphs are consistent with robust execution of the cell cycle, we classified each state as representing G1, S, G2, metaphase, anaphase, telophase and cytokinesis depending on the ON/OFF state of key processes (S6 Fig). Instead of a cycle, however, the resulting network revealed distinct regions of state-space representing G1, S and G2, then a few highly restricted and often-visited paths through anaphase and cytokinesis. Thus, asynchronous update indicates that there may be widespread molecular heterogeneity in G1, S and G2, but most of the network we model locks into a few unique states during anaphase.
It is worth noting that our model features two internal oscillators, the core cell cycle and the PI3K degradation / re-synthesis cycle. As Figs 5 and S1 Fig indicate, these two cycles are not completely phase-locked under asynchronous update. As the cell cycle proceeds, the small PI3K oscillator and the downstream mTORC1 pathway can be found in nearly any state. The sole exception is anaphase, where the two cycles appear to sync up. To show that the heterogeneity is chiefly within the growth pathway, we projected the state transition graph of each asynchronous model onto a subspace where each network state represents a unique ON/OFF state within the core cell cycle modules (Restriction SW, Origin of Replicaton SW, Phase SW and Cell cycle processes), regardless of the state of all other nodes. This process collapsed the complex state transition graph of the biased model onto a clear cyclic flow of transitions, representing normal cell cycle progression (S5 Fig, bottom right). In contrast, the random asynchronous model’s dynamics has a loop corresponding to the cell cycle, but it is dominated by prominent “backward” transitions representing endoreduplication from G2 (S5 Fig, top right).
To further test our model against published experimental data, we compared its least intuitive dynamical behaviors to experimental observations (Table 2) and described them in detail in S1A–S1D Text. To summarize, both our synchronous and biased asynchronous model reproduces the cyclic degradation and re-synthesis of p110, leading to oscillating AKT1 signaling (Figs 4 and 5). In cells entering the cell cycle from quiescence, this oscillatory behavior generates two distinct phospho-AKT peaks before cells complete DNA synthesis (Fig 4, S7 Fig). Furthermore, cells that lack high p110 protein expression fail to enter the cell cycle in response to growth factors (S8 Fig). Our models also reproduce the bifurcation of fates in cells cycling in non-saturating growth environments. Namely, a large fraction of cells were shown to pass the restriction point before cytokinesis (in late G2/M of the previous cycle), while the remainder reset into a G0-like state and commits to the next cycle again after a highly variable time-window (S9 and S10 Figs) . Finally, a comparison of 34 model knockout and 11 overexpression phenotypes to experimentally manipulated cell behaviors indicate that our model can faithfully reproduce in vitro cell behavior under a wide range of genetic manipulations (S4 Table).
Attenuated Plk1 expression in anaphase phenocopies the cell cycle defects of PI3K and AKT1 overexpression
Experimental data indicates that hyperactive PI3K and/or AKT1 in G2 leads to an enrichment of binucleated cells stuck in telophase [29,40]. Studies that document these errors point to the loss of FoxO3 and/or FoxO1 activity in G2 (a consequence of hyperactive AKT), two transcription factors that positively regulate the expression of mitotic cyclin B, as well as polo-like kinase 1 (Plk1). Cyclin B accumulation is only required for metaphase entry (a process that appears normal in cells with hyperactive PI3K/Akt); its activity is not required for cytokinesis. Plk1, in contrast, plays distinct roles at every phase of mitosis and cytokinesis . Thus, we hypothesized that telophase enrichment in cells with hyperactive PI3K/Akt may be due to compromised Plk1 expression in G2 or early mitosis [29,40], and that partial knockdown of Plk1 in our model phenocopies this error.
To test this, our previously published Phase Switch  required a revision to incorporate the complex regulatory role of Plk1 (Fig 3A). Experimental evidence indicates that Plk1 is upregulated in G2 by the FoxM1 transcription factor (also newly added). While the combinatorial regulation of Plk1 by FoxM1, FoxO3 and FoxO1 has not been investigated, experiments clearly show that Plk1 remains active until late telophase [60,61]. That said, Plk1 protein level drop dramatically in anaphase due to proteasomal degradation by APC/CCdh1 [60,61]. It is the availability of the remaining Plk1 pool, responsible for the assembly of a contractile ring, that appears compromised in the absence of FoxO activity in G2 . To capture this within a Boolean framework, we accounted for the role of FoxO factors in creating an increased pool of Plk1 by introducing two Boolean nodes to track Plk1 activity (S11 Fig, S1E Text). Thus, the Plk1 node represents the active enzyme required for mitotic entry, normal mitotic progression and anaphase completion. In contrast, Plk1H = ON represents the short-lived accumulation of a large enough Plk1 pool to survive APC/CCdh1 mediated degradation past anaphase, and aid the assembly of a contractile ring by recruiting the RhoA GEF protein Ect2 .
Next, we tested whether our model can accurately account for all known roles of Plk1 during cell cycle progression. To this end, we first modeled the inhibition of Plk1 at different points along the cell cycle using synchronous update . As Fig 6 shows, Plk1 inhibition in our model reproduces four distinct, experimentally documented phenotypic outcomes, depending on the precise timing of Plk1 inhibition during the cell cycle (Table 3). First, loss of Plk1 before prometaphase (i.e., before robust Cdc25C & Cdk1 activation) results in G2 arrest (Fig 6A). Second, complete of loss Plk1 at the prometaphase /metaphase transition or early metaphase leads to prolonged arrest and mitotic catastrophe (Fig 6B). Third, our model predicts that Plk1 loss in late metaphase can trigger permute anaphase rather than mitotic catastrophe, leading to chromosome mis-segregation and aneuploidy (Fig 6C). This occurs when Plk1 and CyclinB / Cdk1 are both available to phosphorylate the APC/C subunit of the Anaphase Promoting Complex , leading to Cyclin A degradation . The loss of this key APC/CCdh1 inhibitor, together with the subsequent loss of Cdk1 activity in the absence of Plk1, results in premature APC/CCdh1 activation. APC/CCdh1 disassembles the incomplete mitotic spindle, allowing a narrow escape from apoptosis and instead leading to chromosome mis-segregation and premature telophase. Lack of Plk1 past this point guarantees that cytokinesis does not follow. Making matters worse, our model shows that continued growth factor signaling can lead to a new round DNA synthesis (Fig 6C). Fourth, Plk1 inhibition a time-step later leads to normal anaphase mediated by APC/CCdc20 (Fig 6D). As long as Plk1 inhibition starts before APC/CCdh1 activation, however, cytokinesis still fails (Fig 6D, lime green line).
(A-D) Top: (A) Molecular mechanism leading to G2 arrest via Plk1 knockdown before the start of prometaphase due to a lack of Cdk1 activation; (B) mitotic catastrophe and apoptosis via Plk1 knockdown in prometaphase or early metaphase due to concurrent Casp2 activation and deactivation of the antiapoptotic BCL2 family; (C) aberrant (premature) anaphase and no cytokinesis via Plk1 knockdown later in metaphase due to premature APC/CCdh1 activation, and (D) normal anaphase but no cytokinesis via Plk1 knockdown post SAC passage due to loss of Plk1H in telophase. Orange/blue background: higher/lower than normal activity; gradient background: premature node transition; no background: other relevant node / process; →: activation; ⊣: inhibition. Bottom: Dynamics of expression / activity of Phase Switch, Cell cycle processes and Apoptotic Switch nodes in cells exposed to Plk1 inhibition at different stages of the cell cycle. X-axis: time-steps; y-axis: nodes of the model organized in modules; orange: ON (expressed and/or active); blue: OFF (not expressed and/or inactive); black: OFF, forcibly inhibited. Black dashed line: timing of Plk1 inhibition; white pathways: processes that initiate apoptosis (B), premature anaphase (C), or failed cytokinesis (D); red box & bar: telophase/G1 in the absence of cytokinesis, followed the next round of DNA synthesis; lime green line: point of normal APC/CCdh1 activation, marking the end of the Plk1 inhibition window that can compromise cytokinesis (D); only relevant module activity is shown (full dynamics available in S1 File).
Given that our model adeptly captures four distinct effects of Plk1 inhibition, next we asked whether partial loss of Plk1 can phenocopy the effects of hyperactive PI3K/ AKT1. We modeled partial knockdown of Plk1 by running stochastic simulations in different growth conditions, where we forced the OFF state of Plk1 in every time-step with a fixed probability and allowed the node to obey its normal regulation when not forced (Fig 7; Methods & Model–Modeling non-saturating growth factor stimulation and partial knockdown / overexpression within a Boolean framework). Our simulations indicate that the dominant failure mode in a population of cells depends on the strength of Plk1 inhibition. When Plk1 inhibition is very strong (but not complete), cells often start mitosis but do not complete it. This leads to increased mitotic length, often followed by apoptosis (Figs 7 and S12A). In contrast, aberrant mitosis leading to aneuploidy is more common at moderate Plk1 inhibition (peak at 60%), though apoptosis is still more likely (Figs 7 and S12A). The most common cell fate at this point, however, is normal mitosis followed by prolonged telophase (S12B Fig) and failure to undergo cytokinesis. This remains the prominent failure mode at moderate-to-weak Plk1 inhibition levels (peak at 30%; Figs 7 and S12A). Performing the same series of in silico experiment using biased asynchronous update lead to qualitatively similar results (Fig 7), with the caveat that the asynchronous model occasionally skip mitosis altogether–an effect that does not change with Plk1 inhibition. In summary, weak Plk1 inhibition in our model phenocopies the experimentally documented effects of hyperactive PI3K and/or AKT–in line with the hypothesis that the cause of weakened Plk1 expression is lack of some (i.e., FoxO3 and FoxO1) but not all transcriptional Plk1 activators in G2 (FoxM1 remains active).
Stacked bar charts showing the relative occurrence of normal cell cycle completion (mustard), G2 → G1 reset followed by genome duplication (purple), aberrant mitosis followed by genome duplication (turquoise), failed cytokinesis followed by genome duplication (blue) and apoptosis (dark red) relative to the rate of cell cycle in wild-type cells (black dashed line) at growth factor exposure of 40%, 60% and 80% in Plk1-deficient cells using synchronous (top) and biased asynchronous update (bottom).
Cyclic degradation of PI3K is required for normal cell cycle progression
To test whether our model accurately links non-degradable p110 to altered Plk1 expression leading to failure cytokinesis, we ran in silico experiments in which we kept the p110H node forcibly ON, starting at different points along the cell cycle (Fig 8). As expected, expression of a non-degradable p110 leads to high sustained PI3K and AKT activity. Loss of FoxO3/FoxO1 during G2 and M prevents Plk1 levels from accumulating enough to outlive APC/CCdh1-mediated depletion (Plk1H does not turn on; Fig 8A). The result is failure to undergo cytokinesis (Fig 8A, red box), matching the experimentally documented enrichment of telophase cells in the presence of overactive PI3K, AKT, or inactive FoxO3 . In addition to telophase enrichment, our model shows genome reduplication in the resulting bi-nucleated cells, also supported by experimental evidence (Table 3). Intriguingly, our model predicts that the loss of high growth factors during G2 or M allows these cells to compete cytokinesis (Fig 8A, second cycle). This occurs because high p110 protein expression alone is not sufficient for high AKT1 activation; it also requires ongoing growth signaling and active Ras . Thus, loss of strong growth stimulation allows the re-entry of FoxO3 into the nucleus, leading to Plk1 expression and cell cycle completion.
(A) Top: Molecular mechanism leading to the failure of cytokinesis in the presence of non-degradable p110H. Blue background: lower than normal activity; no background: other relevant node / process; →: activation; ⊣: inhibition. Bottom: Dynamics of regulatory molecule activity during the transition from cell cycle to telophase, then genome duplication upon expression of non-degradable p110 (yellow). X-axis: time-steps; y-axis: nodes of the model organized in modules; orange/blue: ON/OFF; yellow: ON, forcibly expressed; white arrows & nodes: factors driving Plk1 expression and lack of Plk1H accumulation; red arrows & box: failure of cytokinesis followed by G1 in bi-nucleated cells; only relevant module activity is shown (full dynamics available in S1 File). (B-C) Relative occurrence of normal cell cycle completion (mustard) vs. genome duplication following failed cytokinesis (blue) relative to the rate of cell cycle in wild-type cells (stacked bar charts) at 80% growth factor exposure in cells with non-degradable p110H (top), non-degradable p110H + active PI3KH (middle) and hyperactive AKTH (bottom). Modeled using synchronous (B) and biased asynchronous update (C).
Synchronous update allows us to track the molecular consequences of locking p110 into a high-expression state, but it has several drawbacks. Most importantly, it assumes the presence of a saturating growth factor environment and 100% FoxO inhibition downstream of PI3K/AKT1. To test whether our results hold in the presence of intrinsic or extrinsic fluctuations such as moderate growth factor availability and incomplete hyperactivation of p110H, we tracked the fate of cells in a variety of non-saturating growth factor environments. In each environment, we tested the effect of incomplete p110H, p110H + PI3KH, or AKT1H over-expression by stochastically forcing the ON state of these nodes with a fixed probability (Figs 8B and S13). These results also point to a high prevalence of cells that cannot exit telophase in near-saturating growth environments (blue bars on Fig 8B). As hyperactive AKT1H in the model is forced ON regardless of growth signaling, it drives both an increase in proliferation and the failure to exit telophase even in low growth factor environments (S13B Fig).
As the mediators of cell cycle progression errors in hyperactive PI3K/AKT1 are thought to be FoxO factors, we next showed that partial inhibition of FoxO3 phenocopies hyperactive PI3K and/or AKT1 in our synchronous model (S13E Fig, Table 3). That said, strong FoxO3 inhibition also slows / stops re-synthesis of p110, leading to a lengthened cell cycle (documented in cancer cells and tumors in vivo ; Table 3). The subsequent weakening of AKT1 signaling counterbalances the loss of FoxO3, weakening its effects. In line with this, our biased asynchronous update results do not show an increase in cytokinesis failure with FoxO3 knockdown (Figs 8C and S13E).
In addition to reproducing the effects of hyperactive PI3K/AKT1, our model offers several experimentally testable predictions: 1) We predict that the observed cycle of p110 degradation and re-synthesis is driven by the network in Fig 2A. As a result, knockdown of PLCγ, NEDDL4, or the chelation of intracellular Ca2+ is expected to lead to sustained high p110 protein expression in vitro. 2) In addition to an increase in cell cycle length, we predict PLCγ knockdown to enriche for telophase cells that fail to complete cytokinesis. 3) Continuously cycling cells execute at least two rounds of PI3K activation and destruction for each round of division (Figs 4 and 5). 4) Loss of saturating growth signals in G2 allows p110-overexpressing to complete a normal cell cycle (S10 Fig). 5) Once committed to a cell cycle, saturating growth stimulation allows cells to keep cycling even if p110 levels drop later in the cycle, and pre-commitment in p110-inhibited cells is driven by mitotic mTORC1 aiding the re-activation of Myc (S10 Fig).
In this study we developed a detailed modular Boolean model of the regulatory pathways driving growth factor signaling, cell cycle progression and apoptosis (Fig 3B). While there are several published models with a similar coverage of cellular behaviors [15,21,23,25,26], the focus of our study was to capture the dynamical behavior of the PI3K → AKT1 signaling axis driving cell growth. To this end, we proposed a mechanism capable of driving the experimentally documented oscillations of PI3K protein expression [46,47], explored the importance of high and low PI3K activity during different phases of the cell cycle, and showed that our model can offer mechanistic insight into the cellular effects of hyperactive PI3K (failure of cytokinesis). To do this, we identified two negative feedback loops potentially responsible for driving PI3K dynamics. The first loop is triggered by high growth factor signaling and high PI3K activity, and it involves PLCγ-mediated activation of the NEDDL4 ubiquitin ligase , known to target the p110 subunit of PI3K for degradation . The second loop involves the loss of AKT-mediated FoxO3 inhibition as PI3K activity drops, allowing FoxO3 to drive the re-expression of p110 . As these two pathways were key to our model’s ability to reproduce the effects of hyperactive PI3K and AKT1 on cell cycle progression, they represent its two most significant predictions.
Linking PI3K oscillations to the rhythm of cell division required an update of our previously published Phase Switch  to include the multifunctional Plk1 protein required for all phases of mitosis and cytokinesis . According to our model, during normal cell cycle progression the low-PI3K / low-AKT phase of the PI3K oscillations lead to nuclear re-entry of FoxO3 and FoxO1, which aid the accumulation of Plk1 and are required for cytokinesis. In addition, we predict that the inhibitory influence of Plk1 on FoxO3  helps lock PI3K oscillations to the cell cycle (Fig 4). That said, a strict phase-locking is only enforced at the metaphase / anaphase transition, as evidenced by the behavior of the asynchronous version of our model (Figs 5 and S5 Fig). In addition to offering testable predictions of the mechanisms behind PI3K oscillation summarized in Fig 9, our model is the first to account for the multifaceted role of Plk1 in cell cycle progression (Fig 6). Namely, we were able to reproduce G2 arrest in the complete absence of Plk1 , mitotic catastrophe in response to Plk1 removal in metaphase [54,56–58], the potential for premature APC/CCdh1 activation and chromosome mis-segregation , as well as failure to carry out cytokinesis in the absence of a Plk1 pool that survives APC/CCdh1-mediated destruction [60–62].
(A) Degradation and re-synthesis of the PI3K subunit p110 is driven by PLCγ-dependent activation of the NEDDL4 (red links) and FoxO3 (orange link), respectively. During G2, loss of strong PI3K / AKT signaling is required for nuclear translocation of FoxO3 and/or FoxO1, which aids Plk1 accumulation to levels that can outlast its degradation in anaphase (modeled via the Plk1H node). During telophase this remaining pool of Plk1 localizes to the central spindle and promotes the assembly of a contractile ring by recruiting the RhoA GEF Ect2. Red nodes: key pathway linking PI3K and AKT dynamics to Plk1 and cytokinesis. (B) Plk1 inhibition at different points along the cell cycle leads to four distinct failure modes. Image credits: https://commons.wikimedia.org/wiki/File:Mitosis_cells_ sequence.svg.
A limitation of our current model stems from uncertainties in the experimental literature on the connection between Plk1 and FoxO factors. As we detailed in Results, the combinatorial regulation of Plk1 by FoxM1, FoxO3 and FoxO1 is not characterized. It is not clear whether these factors cooperate or independently augment Plk1 expression. Moreover, our assumption that either FoxO factor alone can boost Plk1 sufficiently to survive until telophase has not been tested in vitro. Thus, the logic gates connecting Plk1 and the FoxO factors may need a revision in light of additional data. That said, aspects of Plk1 regulation that guarantee its loss in telophase but not earlier in cells with hyperactive PI3K/AKT1 requires key elements of our regulatory logic to remain intact .
Throughout this work we used synchronous and asynchronous Boolean modeling in parallel, allowing us to leverage the advantages and mitigate the drawbacks of each update scheme. A key advantage of synchronous update is that the dynamics it generates is entirely deterministic . This allowed us to probe the effects of inhibiting nodes at specific times along a dynamical trajectory such as the cell cycle, and predict distinct phenotypic outcomes depending on the precise timing of inhibition. For example, using synchronous update to model Plk1 inhibition along the cell cycle point to a time-sensitive sequence of failure modes: G2 arrest, mitotic catastrophe and aberrant anaphase, followed by normal anaphase but failed cytokinesis (Fig 6). The power of these simulations is that they reveal distinct ways in which the molecular balance of Plk1, Cdk1/Cyclin B, premature activation of APC/CCdh1, and pro-apoptotic factors accumulated by mitotic delay can be tipped (Fig 9). In the presence of molecular noise in vitro, however, we expect Plk1 knockdown to generate a mix of these cell cycle errors. Indeed, experiments indicate that mitotic death and aberrant anaphase co-occur in Plk1-inhibited cells . To reproduce this, we simulated the partial stochastic inhibition of Plk1, resulting in a changing mix of errors with both synchronous and asynchronous update (Fig 7). Our success with the latter is especially helpful for showing that the four failure modes are not artifacts of non-biological synergies in signal arrival, a pitfall of synchronous update.
Comparing cell cycle progression with the two update schemes revealed that asynchronous update introduces a stochasticity in cell cycle entry, observed in several mammalian cell lines [90,97,98]. This is similar to the behavior of the synchronous model in non-saturating environments, and it is largely due to the fact that the PI3K/AKT1 cycle does not stay in sync with cell cycle progression for most of the cycle. As a result, the ability of AKT1 to relay growth signals to the Restriction Switch in late G2 / early metaphase, and thus pre-commit cells to another division, remains stochastic under asynchronous update even in saturating growth environments. Cycling cells in vitro are likely somewhere in between; less random in their ability to re-commit during G2/M than the asynchronous model, but not deterministic either. In addition to cell cycle commitment, overly noisy signal propagation is likely responsible for the asynchronous model’s results in cells with hyperactive AKTH and low FoxO3 (Figs 8C and S13E). In contrast to simulations with synchronous update, the fraction of cells that failed to complete cytokinesis under asynchronous update was small. Here too, we think that synchronous update underestimates, while asynchronous update over-estimates the rate of cell cycle errors. In summary, our parallel use of the synchronous and asynchronous Boolean frameworks helped us uncover subtle inter-dependencies in the dynamics of our coupled regulatory modules, but also guaranteed that our results are robust with respect to noise in signal propagation and do not depend on non-biological synchrony of parallel signals with a variety of speeds.
An intriguing model prediction related to the coupling of the cell cycle and the PI3K cycle leverages our model’s ability to reproduce pre-commitment to another cycle at the G2/M transition (Fig 5) , a feature inherited from our previous cell cycle model . Even though high p110 expression is required for cell cycle entry from quiescence (S8 Fig) , we predict that under saturating growth factor conditions high p110 / PI3K is not required for pre-commitment to another cycle (S10 Fig). This is surprising, as pre-commitment at the G2/M boundary normally coincides with the AKT-high portion of the second PI3K cycle. Our model suggests that the stabilization of Myc in p110-low cells occurs in pro-metaphase in spite of low AKT and high GSK3β, owing to increased activity of mTORC1 driven by Cdk1/Cyclin B, specifically in the presence of GSK3β . Experimental validation of these predictions is an important step toward understanding the complex interplay of factors that control Myc expression and pre-commitment to another cycle at the G2/M boundary. The same experimental setup that showed the existence of pre-committed cells could accomplish this  by probing the effect of simultaneous MEK and PI3K inhibition on pre-commitment. According to our prediction, this dual inhibition would reduce but not eliminate the fraction of cells that finish their current cycle following MEK and PI3K inhibition, then complete another.
Our modeling results on partial Plk1 inhibition offer a cautionary note on targeting Plk1 as a tumor-suppressive strategy. On one hand, Plk1 inhibition does significantly limit proliferation due to G2 arrest (Fig 6A) and promotes apoptosis in cells that escape from G2 via mitotic failure (Fig 6B). On the other hand, our model predicts that an ill-timed, short-lived pulse of Plk1 inhibition can lead to faulty anaphase, chromosome mis-segregation resulting in aneuploidy, and subsequent genome duplication (Fig 6C). According to our model, weak Plk1 inhibition in individual cells is especially dangerous in this regard (Fig 7). Thus, it is possible that cancer therapy based on Plk1 inhibition  could increase genomic instability in cells that survive it. A propensity for aneuploidy and genome duplication was indeed observed in Plk1-inhibited cells , but also in a mouse model harboring the oncogenic mutation in the alpha subunit of PI3-Kinase . The molecular mechanisms behind the latter were never explained. Our model not only reproduces these outcomes (Table 3), but also points to the ill-timed loss of Plk1 as the likely culprit (Fig 6).
Looking ahead, our current model lays the groundwork for modeling the mechanisms of AKT1-induced senescence [38,39,100]. An extended version of our model with DNA damage-induced G2 arrest in cells with hyperactive AKT1 would express most key drivers of senescence (i.e., mTORC1, RB, p53 and p21). Thus, an important next goal is to complement our model with a DNA damage signaling module [16–18,101,102], then build the regulatory switch that locks in and maintains senescence [20,24,103–105]. The predictive power of our model could be further expanded by revising our Growth Signaling and Apoptosis modules to leverage more detailed computational models of MAPK signaling , as well as the apoptosis/necrosis decision . Finally, building a contact inhibition module to capture the connection between cell-cell contacts and p110 expression could pave the way towards modeling interacting epithelial cell communities . We thus see our current model as a seed for more powerful models of the processes that go awry when healthy cells transition into malignancy.
Methods and model
Boolean modeling framework
To capture the complex combinatorial logic by which the 87 molecular species and cellular processes in our model interact, we used a Boolean network modeling formalism . Boolean models approximate the activity of regulatory molecular species as ON (active; orange outlines on Fig 3A) or OFF (inactive; blue outlines on Fig 3A) , and focus on the combinatorial logic by which multiple regulatory inputs work together. This requires specifying the ON/OFF response of each node for every combination of the states of its inputs. The resulting Boolean functions (gates) can be represented as truth tables (input-output tables that specify every response of a node explicitly), or via the Boolean logic operators AND, OR, and NOT (S1 Table). Once the Boolean gate of each node is specified, the time-dependent dynamics of the whole network can be simulated from an arbitrary initial condition . Since the expression / activity of the molecules is discrete, time also proceeds in discrete steps in which nodes can change their ON/OFF state.
During the construction of our model we used synchronous update , a scheme in which every node of the network updates, synchronously, in every time-step. The advantage of this scheme is that it renders the dynamics of the system completely deterministic. This allowed us to account for the precise role of each molecular species at every causal step along a biological process and compare it to experimental data. By building a synchronous Boolean model first, we could follow the molecular causes of behaviors that deviated from known cell dynamics, and to fix the model to better match experimental evidence.
Asynchronous update changes the state of one node at a time and uses this new state as it updates its targets. In general asynchronous update, nodes are chosen randomly in each time-step regardless of the last time they were updated (Fig 3C). In contrast, random order asynchronous models update every node in every time-step, but they do so sequentially in a random order re-shuffled before every step (S1 Fig). Asynchronous update schemes are favored in biological modeling, as they simulate the unfolding of the same regulatory process along a large number of slightly different paths, each with different likelihood , mimicking a type of stochasticity present in vitro . Moreover, asynchronous update eliminates potential artifacts of synchronous modeling; behaviors that rely on perfect and deterministic coordination of parallel signals–a condition that cells rarely satisfy. That said, they can also generate biologically non-realistic sequences of molecular events by failing to follow up on the effects of short-lived signals that live cells reliably respond to (Figs 5 and S1). To mitigate this, we used a hybrid framework where the update order of some, but not all nodes is not random (termed the biased asynchronous model, Fig 5). As most nodes in our model are controlled by slow as well as fast processes, setting their update frequency was not a viable strategy. Instead we choose to update 11 of the 87 nodes either first or last, depending on their correct state, as detailed in S3 Table.
The state space of a Boolean regulatory network
In a Boolean representation, a regulator network can have 2N possible expression / activity profiles, where N represents the number of molecules in the model. Starting a time-series from most of these 2N states reveals that they are not stable, in that several regulatory nodes immediately change their ON/OFF state as dictated by the ON/OFF state of their inputs. Allowing the network’s dynamics to proceed from an unstable state will lead to a sequence of expression/activity changes that can cascade through the network. Eventually, every such cascade must end in two ways, regardless of update: 1) a stable state in which all Boolean rules are satisfied (called a point attractor), or 2) a more complex set of states that a) repeat in an exact cycle termed a limit cycle attractor under synchronous update, or b) repeat in a more stochastic sequence of states called a complex attractor that traps the dynamics under asynchronous update. The latter can also represent a rhythmic, repeating series of state-changes, but this is not guaranteed. The collection of sequential state-changes running from each of the 2N model states to the model’s attractor states or cycles can be represented as a directed network of states, termed the state transition graph.
Under synchronous update the model’s dynamics leads to a single attractor from each unstable state. The collection of all the paths leading to the same final state creates a subgraph of the state transition graph, and represents the attractor basin of the final attractor state . Conceptually, this attractor basin can be thought of as a valley in the pseudo-energy landscape of the model . As most network states are unstable and lead, in time, to an attractor, biologically relevant robust phenotypes of the model are expected to correspond to its attractor states . Moreover, rhythmic biological behavior such as that of a continuously cycling cell is expected to map onto a limit cycle attractor.
Under asynchronous update, some unstable states can lead to different attractors with different probabilities depending on update order, while others can only lead to a single attractor—making the definition of attractor basins less straightforward. In addition, regions of the state transition graph can act as metastable “valleys” (S5 Fig). These represent state collections that trap the dynamics of the system for long periods of time, but it is not strictly speaking impossible for the system to escape to a proper attractor. Indeed, the cell cycle in our asynchronous models is such a metastable “pseudo-attractor”.
Reproducing our modeling results
To simulate the dynamics of our Boolean model and work through key methods, see “SI_notebook.ipynb” in S2 File, a Jupyter Notebook in Python (https://jupyter.org). The code in this notebook uses BooleanNet  and NetworkX . S3 File contains BooleanNet model files, including the full model (“PI3K_cell_cycle_apoptosis”). To convert these files to commonly used formats used by other packages, see (http://colomoto.org/biolqm/doc/formats.html).
For synchronous update, see S2 File– 1.a for the PI3K oscillator and S2 File– 3 for the full model. To run the PI3K oscillator module using general asynchronous update see S2 File– 1.b; for the full model with random order asynchronous and biased asynchronous update see S2 File– 4.a-c. To sample the full state space of the individual network modules, see S2 File– 2; to sample and visualize the general asynchronous state transition graph of the PI3K oscillator, see S2 File– 1.c; to map the cell cycle pseudo-attractor of the full model, see S2 File– 4. Both files are available as a package at https://github.com/deriteidavid/cell_cycle_apoptosis_Sizek_etal_PloSCompBio_2019.
Mapping the attractor landscape of large Boolean networks using synchronous update
In order to generate a comprehensive picture of all the attractor basins of the model, we use a stochastic state space sampling procedure adapted from , as described in . To this end, we first implemented a noisy version of synchronous Boolean dynamics, in which each regulatory node is affected by a small amount of noise in every time-step. The noise is implemented as a small probability pn = 0.02 that each node generates the incorrect output, rather than the one dictated by its inputs . This noisy dynamics sets up a Markov process, guaranteeing that the system can spontaneously visits any state (not just the attractors) with non-zero long-term probability [112,115]. We used the noisy dynamics to aid our sampling procedure by starting the network from a random initial condition and simulating a time-course of Nseries = 20 noisy time-steps. As the model generates this noisy dynamical trajectory, the algorithm pauses at each state it visits to perform two checks. First, it finds the attractor basin this state would fall into if the dynamics were to continue in a deterministic fashion. Second, it scans the immediate neighborhood of this state by enumerating every state the system could reach from the current one via a single node-state flip and identifying their attractor membership (via deterministic dynamics). This allows the algorithm to access parts of the state space the noisy dynamics might never go near, and to find even small basins relatively fast. As a result, the algorithm is quite slow on random Boolean networks with large numbers of small basins. Our model’s robust phenotype-representing attractor basins, by contrast, are typically large and thus rapidly found. The full algorithm descried in  tracks the visitation probability of each state, basin and transition (not used here). The only update to the algorithm since  involves partitioning the full state space of the model into sub-spaces corresponding to each unique environmental node state-combination and sampling each subspace from Nrnd = 500 random initial conditions.
Automated isolation of a subnetwork from a larger (multi-switch) Boolean network
In order to automatically model the dynamical behavior of any isolated subgraph (Fig 3A), we have previously developed an algorithm that defines the Boolean gates of nodes when they lose some of their incoming connections . The main goal of this algorithm was to optimally preserve the regulation of a node by its remaining inputs. Briefly, whenever a subset of inputs is removed from a Boolean gate, the algorithm assumes that they are frozen into either an ON or an OFF state. To best preserve the dynamical influence of the remaining nodes, it finds one of the 2k possible combinations of frozen inputs such that: a) all remaining input nodes are functional (i.e., they are able to impact the output in some way), and b) the entropy of the remaining Boolean gate fragment, HG = - p · log(p)- (1 - p) · log(1 - p), is as large as possible (p is the fraction of OFF-outputs). For easy reproducibility of our module networks, S3 File includes a BooleanNet model file for each module.
Boolean network modules representing distinct cellular regulatory functions
Growth factor signaling.
To build a dynamic PI3K → AKT1 signaling module (S1A Table), we first incorporated the canonical PI3K → AKT1 pathways shown in Fig 1A (reviewed in [31,32]). Namely, we modeled growth factor stimulation leading to receptor tyrosine kinase (RTK) activation, which in turn activates the PI3K enzyme at the plasma membrane . PI3K produces PIP3, which then recruits both the PDK1 kinase and its phosphorylation target, AKT1. PDK1 then phosphorylates AKT1 at one of its two key sites, T308. To account for basal growth factor signaling that promotes cell survival versus the strong but short-lived activation of AKT1 in response to high growth factor stimulation, we tracked the activity of both PI3K and AKT1 via two Boolean nodes (PI3K and AKTB for basal activity; PI3KH and AKTH for peak activation). Next, we added key downstream targets of AKT1: 1) AKT1 activates mTORC1 signaling by inhibitory phosphorylation of TSC2, responsible for the inactivation of the Rheb GTPase [33,34]. mTORC1 targets S6K and the translation initiation factor eIF4E (held in check by 4E-BP1 in the absence of active mTORC1), both of which aid cell cycle commitment  (Fig 1A, box 1). 2) AKT1-mediated inhibition of GSK3β counteracts the destabilizing effects of this kinase on several cell cycle-promoting genes such as Myc and cyclin D1, as well as the anti-apoptotic BCL-2 family member MCL-1 (Fig 1A, box 2) . 3) AKT1 aids cell cycle entry and cell survival by translocating FoxO factors out of the nucleus. FoxO targets include apoptotic proteins such as BIM, and cell cycle inhibitors such as p27Kip1 and p21Cip1 (Fig 1A, box 3) . 4) AKT1 phosphorylates the pro-apoptotic BCL-2 family member BAD, leading to its translocation and sequestration from the mitochondrial membrane (Fig 1A, box 4) .
In addition, our model incorporates receptor-independent feedback mechanisms known to temper PI3K/AKT1 signaling; namely the effects of S6K on mTORC2 inhibition and PTEN translocation to the cytosol [45,117]. The two feedback loops that control high p110 expression (Fig 2) are detailed in Results (module available as the “PI3K” BooleanNet file in S3 File). Finally, the Growth Signaling Module includes a linear MAPK cascade that is i) required for maximal PI3K activity , ii) aids the activation of mTORC1 and the inhibitor of FoxO3 as well as GSK3β [118–120], iii) drives the expression of Myc during cell cycle entry , and iv) contributes to survival signaling .
The restriction switch and the mitotic phase switch.
To model the switch-like restriction point control guarding cell cycle entry, we used the p21-positive version of our previously published Restriction Switch (Fig 3A and 3B; S1B Table; module available as the “Restriction_Switch” BooleanNet file in S3 File) . This switch has two stable states in isolation, representing cells before and after they pass the restriction point. Next, we expanded our previously published Phase Switch  to account for the critical role of the FoxO target Plk1 (Fig 3A and 3B; S1C Table; module available as the “Phase_Switch” BooleanNet file in S3 File). Plk1 is required for normal cell cycle progression, as animal cells do not assemble a bipolar spindle in its absence . It is activated in early G2 by the FoxM1 transcription factor [123,124]. FoxM1 transitional activity is also essential of the completion of the mitotic program, above and beyond its effect on Plk1 [125,126]. FoxM1 levels are kept low by auto-repression in quiescent and G1 cells. During the S/G2 transition this inhibition is relieved by Cdk2 phosphorylation (in complex with cyclin E or A) , and later by active Cdk1 . Once active, FoxM1 regulates several of proteins involved in cell cycle progression, of which our model includes Cdc25A, Cdc25B, Plk1 and cyclin B (we do not explicitly include the activity of the Skp1/cullin/F-box protein complex active at the G1/S transition; thus its FoxM1 target components are also missing from our model) . In addition to FoxM1, accumulation of sufficient Plk1 to drive cytokinesis (modeled by the Plk1H node) also requires at least one FoxO factor [29,40]. Finally, Plk1 degradation is driven by APC/CCdh1 .
To model the downstream effects of Plk1, we first incorporated positive feedback on its own transcriptional activator, FoxM1. Cdk1-phosphorylated FoxM1 was shown to bind Plk1, and further phosphorylation by Plk1 is required for full transcriptional activity during the G2/M transition . To model this, FoxM1 requires either Cyclin E or Cyclin A-bound Cdk2 activity, or the simultaneous presence of Cyclin B/ Cdk1 and Plk1 (S1C Table). Second, we added a Plk1 requirement to the activation of Cdc25C, as the absence of Plk1 prevents nuclear localization of Cdc25C during prophase , and Plk1-mediated Cdc25C phosphorylation is required for its activity . Third, Plk1 cooperates with active Cdk1/Cyclin B complexes to target the APC/C inhibitor Emi1 for destruction . This ensures that Emi1 is no longer present to interfere with APC/CCdc20 activation when cells clear the spindle assembly checkpoint .
A few of Plk1’s effects extend outside the new Phase Switch. First, localization of Plk1 to unattached kinetochores during metaphase is required for the formation of stable microtubule attachments . To model this we require active Plk1 for the completion of the mitotic spindle, represented by the Attached Kinetochores (A_Kinetochores) node. Second, Plk1 is targeted to the spindle midzone during and after anaphase, where it helps recruit Ect2 to the central spindle . Ect2 is a RhoGEF that aids the accumulation of GTP-bound RhoA , and thus the formation of the contractile ring . To account for the role of Plk1 at the central spindle, past the point where a large fraction of Plk1 is degraded, we included a Plk1H dependent Ect2 node required for the cytokinesis step of our model (marked by de-activation of the 4N_DNA node). Third, Plk1 feeds back to tag FoxO3 and FoxO1 for export from the nucleus [41,42]. The addition of Plk1, FoxM1 and Emi1 to our previously published Phase Switch resulted in the module on Fig 3A. Its three stable states represent the robust expression patterns seen in cells in G0/G1, during G2 (before passage of the DNA damage checkpoint), and at the spindle assembly checkpoint (SAC).
The origin licensing switch.
To model the dynamics of origin of replication licensing, we built a small bistable regulatory switch that tracks the assembly and firing of replication origins (Fig 3A and 3B; S1D Table; module available as the “Origin_Licensing_Switch” BooleanNet file in S3 File) [133,134]. First, the assembly of a functional replication complex requires DNA binding of Origin of Replication (ORC) proteins, marking the origins of replication along mammalian chromosomes . ORC proteins then recruit Cdc6 (transcribed in late G1 by E2F1) , and the resulting ORC/Cdc6 complexes further recruit Cdt1 (also an E2F1 target) . Next, the heterohexameric complex of MCM2–7 proteins binds, completing the pre-replication complex (Pre-ORC) . Once assembled, the complex remains stable and protected from disassembly until Cdk2-dependent phosphorylation of Cdc6 triggers the firing of replication origins, and dissociation of Cdc6 . At this point, a replication bubble is formed by the helicase action of the MCM complex, and the Pre-ORC falls apart . In addition to the degradation of phosphorylated Cdc6 , mammalian Cdt1 and ORC proteins are also degraded at this point, likely due to Cdk-dependent phosphorylation and ubiquitination .
To model the stability of the assembled Pre-ORC, we included a series of positive feedback links from the Pre-ORC node (representing the assembled complex waiting to fire) to its components, ORC, Cdc6 and Cdt1. As a result, the isolated module has two stable states; ON and OFF (Fig 3A). As the cell cycle progresses, this switch is toggled ON when E2F1 activates its components (as long as geminin does not block Cdt1, and Plk1 does not sequester Cdc6 to the spindle pole or the central spindle) [141,142]. Conversely, it is toggled OFF by Cdk2-mediated destruction of Cdc6, and the start of DNA synthesis at each origin. The firing of all origins in a mammalian cell, however, does not occur in one instant . The handoff of origin firing from early to late-replicating genes is accompanied by a handoff of CyclinE/Cdk2 to CyclinA/Cdk2 complexes . This ongoing process is not trivial to represent in the context of a Boolean model, where the cell-wide availability of licensed ORCs is tracked by a single Boolean node (namely Pre-ORC). In order to make sure that the turning OFF of this node in our model represents the firing of all origins required for successful replication, we placed Cdc6 under the inhibitory control of Cyclin A. In addition, Pre-ORC is turned off by the completion of DNA synthesis, marked by the appearance of 4N_DNA in the context of a Replication node that is still ON.
Cellular processes during cell cycle progression.
The above three switches control cell cycle passage by triggering the processes of DNA replication, spindle assembly and cytokinesis (Fig 3B, orange nodes; S1E Table). To model this, we included the Replication and 4N_DNA nodes from our published cell cycle model , an unattached kinetochore node (U_Kinetochore) to denote incomplete mitotic spindle assembly, and attached kinetochore (A_Kinetochore) to mark completion of the mitotic spindle. These process-nodes are accompanied by key regulators of the coupling between the regulatory switches and the processes themselves. Namely, ATR and CHK1 are activated during replication to monitor the completion of DNA synthesis by blocking the G2/M transition , Mad2 is a SAC protein that blocks anaphase entry before the mitotic spindle is complete , active Ect2 marks ongoing cytokinesis , while CAD (Caspase Activated DNAase) fragments DNA in apoptotic cells . Finally, Plk1H represents a sufficiently large Plk1 pool to briefly outlive APC/CCdh1-mediated destruction, and aid cytokinesis. These cell cycle processes, in turn, feedback to influence the control switches. For example, completion of the mitotic spindle (marked by A_Kinetochore) blocks Mad2, thus relieving the inhibition of APC/CCdc20 and flipping the Phase Switch from SAC to G0/G1.
The apoptotic switch.
To accurately capture events that can kill cells in the absence of DNA damage—i.e., complete growth factor withdrawal, extrinsic apoptotic signals or mitotic catastrophe, we built on previously published models of apoptotic commitment to create a detailed Boolean version of this regulatory switch (Fig 3A and 3B; S1E Table; module available as the “Apoptotic_Switch” BooleanNet file in S3 File) [12–15,69–71]. Briefly, the switch is flipped when extrinsic signals from death receptors (S3B Fig), intrinsic signals such as loss of survival signaling (S3C Fig), or mitotic delays (12B Fig) trigger Mitochondrial Outer Membrane Permeabilization (MOMP) . MOMP occurs when the oligomerization of the mitochondrial membrane pore forming BAK/BAX proteins is triggered, releasing cytochrome C and SMAC from mitochondria to the cytosol . These proteins form the Apoptosome , a platform that aids Caspase 9 activation followed by Caspase 3 cleavage . In addition, SMAC deactivates the final check on executioner Caspase 3 activity, the Inhibitor of Apoptosis (IAP) proteins . Once active, Caspase 3 initiates the destruction of a wide range of proteins , activates DNA-fragmentation by releasing the Caspase Activated DNAase (CAD) , and contributes to the switch-like functioning of the apoptotic machinery via Caspase 6-mediated positive feedback that leads to further cleavage of initiator caspases . The resulting module has two stable states, corresponding to survival and apoptosis (Fig 3A).
Our model accounts for three distinct ways in which apoptosis is triggered. First, the extrinsic, receptor-mediated route (S3B Fig) is initiated by Caspase 8 activation at death receptors , leading to the cleavage of tBID . This triggers BAK/BAX oligomerization, leading to MOMP . Caspase 8 also contributes to the direct activation of the executioner Caspase 3 . Experimental evidence suggests, however, that MOMP is not only involved, but marks the moment of irreversible commitment to apoptosis . The second mechanism we modeled is the loss of survival signals (S3C Fig). This triggers MOMP primarily via the loss of BAD phosphorylation by AKT1, ERK or S6K [37,153,154]. Hypo-phosphorylated BAD blocks the antiapoptotic BCL-2 family proteins (BCL2, BCL-XL, MCL-1), which normally keep BIM and BIK (inducers of mitochondrial membrane pore formation), in check . Once pore-forming BAX and/or BAK oligomerize, apoptosis proceeds as described above. Finally, a third path to apoptosis in our model is triggered by mitotic catastrophe marked by prolonged SAC arrest (Fig 9B) . Loss of CyclinB / Cdk1 or Plk1 function before the completion of the mitotic spindle triggers Caspase 2 activation [57,76], with a similar effect to that of Caspase 8, namely BIK and BID activation leading to MOMP.
Modeling non-saturating growth factor stimulation and partial knockdown / overexpression within a Boolean framework
To generate model predictions in non-saturating growth factor conditions, we ran time courses of T = 50,000 or 500,000 time-steps in which the GFH input node was randomly toggled ON/OFF in each time-step with a tunable ON-probability pHigh_GF (Fig 5B) . The ongoing simulations tracked the number of cell cycles completed without error (black cycle on S6 Fig), the number of genome duplication even from G2 (orange transition on S6 Fig), the number of premature metaphase-anaphase transitions that did not involve completion of the mitotic spindle followed by genome duplication (green transition on S6 Fig), the number of genome duplication events in the absence of a cytokinesis step between telophase and the next S-phase (red transition on S6 Fig), and the number of apoptotic events (purple transition on S6 Fig & other apoptotic events). Time courses that resulted in apoptosis before time T were restarted until a minimum of T steps of live-cell dynamics were sampled. In addition, the simulation tracked the average length of G1, S, G2, metaphase and telophase (the time cells spent with 2 nuclei, even if the cell cycle control network reset to G0/G1).
To generate model predictions with incomplete knockdown or overexpression of a target molecule, we combined the non-saturating stochastic growth factor inputs described above with a similar stochastic locking of the target molecule OFF or ON with a tunable probability pKD (knockdown) or pOE (over-expression), respectively. In time-steps where the molecule was not locked ON or OFF, it followed the internal Boolean regulatory influences of the rest of the network as if it was unperturbed. To run sample time courses, see S3 File– 1.a; to sample cell cycle errors see S3 File– 1.b.
S1 Fig. Random order asynchronous update often generates cell cycle progression errors.
Dynamics of regulatory molecule activity during cell cycle entry from G0 using random order asynchronous update (example time-course chosen to illustrate errors). X-axis: time-steps; y-axis: nodes organized in modules; orange/blue: ON/OFF. Black arrows: robust PI3K oscillations; white box: normal cell cycle; white circles: common cell cycle progression errors (labeled).
S2 Fig. Biased order asynchronous update occasionally generates cell cycle progression errors.
Dynamics of regulatory molecule activity during cell cycle entry from G0 using random order asynchronous update (example time-courses chosen to illustrate errors). X-axis: time-steps; y-axis: nodes organized in modules; orange/blue: ON/OFF; white circles: cell cycle progression errors.
S3 Fig. Trail exposure and growth factor withdrawal induce apoptosis.
(A-C) Top: Molecular mechanism leading to apoptosis in response to Trail (A-B) and growth factor withdrawal (C). Red background: extracellular signal; orange/blue background: higher/lower than normal activity; gradient background: premature node transition; no background: other relevant node / process; →: activation; ⊣: inhibition. Bottom: Dynamics of regulatory molecule activity in response to Trail exposure in cycling (A) / quiescent (B) cells (synchronous update), or in response to complete growth factor withdrawal (biased asynchronous update, average of 1000 runs) (C). X-axis: time-steps; y-axis: nodes of the model organized in modules; orange/blue color saturation: percentage of cells in which a node is ON/OFF in each time-step; only relevant module activity is shown (full dynamics available in S1 File).
S4 Fig. Random order update with bias generates state series that resemble the state sequence within the synchronous cell cycle.
Overlap of states along a random order vs. biased random order asynchronous update trajectory (y axis) with attractor states of the synchronous cell cycle (x axis). Time-step: one randomized update round.
S5 Fig. Heterogeneity of microstates in G1, S and G2 is due to a lack of phase-locking between the core cell cycle oscillator and the PI3K cycle.
(A) State transition graph of the random order (top) vs. biased random order (bottom) asynchronous models, sampled for 10 independent runs of 1000 time-steps starting from each of the 21 synchronous cell cycle attractor states (cut short if the model reached apoptosis). The largest strongly connected component of each resulting state transition graph representing the cell cycle pseudo-attractor was visualized using the Kamada-Kawai algorithm (NetworkX , Python). (B) Projection of each state transition graph onto the sub-space defined by the expression of core cell cycle modules (bottom). Nodes: collection of all states that have identical core cell cycle node activity but differ in the activity of nodes in other modules such as Growth Signaling, illustrated by linked black circles from (A) to (B); Node color: cell cycle phase best approximated by each sampled state; node size: state visitation count; node label: most similar synchronous cell cycle state; black loop (top) & black cycle (bottom): areas of the projected state transition graph with a cyclic pattern of transitions that match the cell cycle; orange arrow (top): direct G2 → S transition (endo-reduplication); orange box (bottom): G0-like pause in the G1 phase of the cell cycle, forming a distinct module apart from the G1 states of cells that pre-commit in their previous cycle.
S6 Fig. Graphical illustration of the algorithm that tracks correct vs. erroneous cell cycle progression.
White boxes along the cycle: activity of nodes monitored to determine the model’s cell cycle phase; black arrows: state transitions along a normal cycle; colored arrows & labels: transitions that represent errors in cell cycle progression.
S7 Fig. Two distinct peaks of AKT activity are detectable before the first G2 in quiescent cell populations entering the cell cycle.
Biased asynchronous dynamics of regulatory molecule activity in response to high growth factor stimulation in a population of 1000 cells. Orange/blue color saturation: percentage of cells in which a node is ON/OFF in each time-step; white boxes: first two peaks of high AKTH activity, observable before the cells loose synchrony of cell cycle progression; white arrows: AKTH (two peaks) and 4N_DNA (fraction of cells that finished DNA synthesis).
S8 Fig. High p110 expression in G0 is required for cell cycle entry.
(A) Top: Synchronous dynamics of regulatory molecule activity during the transition from G0 to early G1, with p110 inhibition (black) before vs. after Cyclin D and E2F1 activation. X-axis: time-steps; y-axis: nodes of the model organized in modules; orange/blue: ON/OFF; black: OFF, inhibited; only relevant module activity is shown (full dynamics available in S1 File). Bottom: Molecular mechanism leading to cell cycle commitment in response to GFH, before and after restriction point passage. Black background: p110H inhibition; orange/blue background: high/low activity; gradient background: nodes in transition; →: activation; ⊣: inhibition. (B) Number of normal divisions competed in 100 time-steps (top) and average G1 length (bottom) as a function of p110H inhibition at varying growth environments (synchronous update). pHigh_GF ∈ [20%, 40%, …, 100%]; sampling: 500,000 time-steps. (C) Stacked bar charts showing the relative occurrence of normal cell cycle completion (mustard), G2 → G1 reset followed by genome duplication (purple), aberrant mitosis followed by genome duplication (turquoise), failed cytokinesis followed by genome duplication (blue) and apoptosis (dark red) as a function of p110H inhibition, relative to the cell cycle rate in wild-type cells (black dashed line) at pHigh_GF = 95% (biased asynchronous update).
S9 Fig. Restriction point passage for cells entering the cell cycle from quiescence is in late G1, while cycling cells can pre-commit in late G2 of the previous cycle.
(A-B) Top: Molecular mechanism leading to cell cycle commitment in response to GFH, before and after restriction point passage in quiescent (A) and cycling (B) cells, showing the failure (left) or success (right) of locking in the Myc ⇆ E2F1 and Myc ⇆ mTORC1 feedback loops in (A), or the Myc ⇆ E2F1 loop in the presence/absence of GSK3-β and Cyclin A in (B). Orange/blue background: high/low activity; gradient background: nodes in transition; →: activation; ⊣: inhibition; solid/dashed red arrows: key interactions impacting / not yet impacting the outcome. Bottom: (A) Synchronous dynamics of regulatory molecule activity in response to 6 (left) or 7 (right) time-steps of high growth factor stimulation in quiescent cells. White arrows & nodes: factors driving cell cycle commitment in late G1; dashed / solid lime green arrow: lack of / presence of feedback from E2F1 to mTORC1. (B) Synchronous dynamics of regulatory molecule activity in response to high growth factor withdrawal cycling cells during G2, before (left) and after (right) pre-commitment to another division. X-axis: time-steps; y-axis: nodes of the model organized in modules (showing relevant modules); orange / blue: ON / OFF; white arrows & nodes: factors driving cell cycle commitment in late G2; dashed / solid lime green arrow: E2F1 inhibition (left) / lack of inhibition (right) by Cyclin A; only relevant module activity is shown shown (full dynamics available in S1 File).
S10 Fig. High p110 expression is not required for pre-commitment to another cell cycle in saturating growth environments.
(A) Synchronous dynamics of regulatory molecule activity in response to p110H knockdown past the point of commitment from G0 to the first cycle. Lime green nodes & arrows: pre-commitment is not driven by E2F1 reactivation following Cyclin A degradation; rather, Cdk1/Cyclin B-mediated activation of mTORC1 → eIF4E is required to stabilize Myc in spite of the presence of GSK3β. Dark red nodes & arrows: in the absence of high AKT1, ERK is required for two additional time-steps compared to wild-type cells, in order to stabilize the E2F1 ⇄ Myc feedback loop; only relevant module activity is shown shown (full dynamics available in S1 File). (B) Molecular mechanism responsible for pre-commitment, before and after restriction point passage in prophase, showing the failure (top) or success (bottom) of locking in the Myc ⇆ E2F1 feedback loop in the absence/presence (top/bottom) of CyclinB/Cdk1-activated mTORC1 signaling. Black background: p110H inhibition; Orange/blue background: high/low activity; gradient background: nodes in transition; →: activation; ⊣: inhibition; solid/dashed arrows: key interactions impacting / not yet impacting the outcome.
S11 Fig. Modeling Plk1 activity and persistence.
Regulatory network surrounding Plk1 expression, enzyme activity and the accumulation of a Plk1H pool driven by FoxO3 or FoxO1. Red nodes: two Boolean nodes representing Plk1 activity and accumulation; Blue nodes: inputs of the two Plk1 nodes. Black arrows: regulation and maintenance of Plk1 expression, activity and persistence; green arrows: feedback on FoxO factors from Plk1, and its downstream target Cyclin B/Cdk1.
S12 Fig. The strength of Plk1 inhibition sets the relative prominence of cell cycle failure modes.
(A) Number of normal divisions (first panel), mitotic catastrophe (second panel), aberrant mitosis with genome doubling (third panel) and failed cytokinesis with genome doubling (fourth panel) per 100 time-steps as a function of Plk1 inhibition in varying growth environments (synchronous update). (B) Average time spent in G1 (first panel), G2 (second panel), metaphase (third panel) and telophase (binucleated cells in G1) (fourth panel) as a function of Plk1 inhibition in varying growth environments. pHigh_GF ∈ [20%, 40%, …, 100%]; sampling: 500,000 time-steps (synchronous update).
S13 Fig. Partial FoxO3 inhibition phenocopies the effects of non-degradable p110H, leading to a mild enrichment of telophase cells.
(A-D) Number of normal divisions competed in 100 time-steps (A), average G1 length (B), number of divisions with failed cytokinesis in 100 time-steps (C), and average telophase length (D) as a function of the rate of forced p110H (first panel), p110H + PI3KH (second panel), AKTH activation (third panel) and FoxO3 inhibition (fourth panel) in varying growth environments (synchronous update). (E) Stacked bar charts showing relative occurrence of normal cell cycle completion (mustard), G2 → G1 reset followed by genome duplication (purple), aberrant mitosis followed by genome duplication (turquoise), failed cytokinesis followed by genome duplication (blue) and apoptosis (dark red) as a function of FoxO3 inhibition, relative to the cell cycle rate in wild-type cells (black dashed line) at pHigh_GF = 80% modeled with synchronous (left) and biased order asynchronous update (right). Sampling: 50,000 time-steps.
S1 Text. Supplementary text detailing modeling results and validation that do not fall within the main focus of the study, but lend further credibility to the accuracy of the model.
(A) Dynamics of AKT1 during the cell cycle; (B) High p110 expression in G0 is required for cell cycle entry; (C) Context-dependent timing of R-point passage and predicts pre-commitment in p110-deficient cells; (D) Pre-commitment in p110-deficient cells; (E) Assumptions for constructing the regulatory logic of Plk1 and Plk1H.
S1 Table. Description and experimental support for the model’s Boolean regulatory logic.
Explanation and literature support for each individual link and regulatory logic gate in the model. (A) Growth signaling; (B) Restriction switch; (C) Phase switch; (D) Origin of replication licensing; (E) Cell cycle processes; (F) Apoptotic switch.
S2 Table. Attractors of the synchronous Boolean model.
(A) Expression profile of synchronous model attractor states, numbered to match Fig 3; orange/blue: ON/OFF. (B) explanation of the molecular signatures allowing us to match them to cellular phenotypes.
S3 Table. Biased update order and rationale.
Details and logic of the biased update order required to accurately reproduce cell cycle progression, including figures (last column) that summarize relevant regulatory feed-forward and feedback loops susceptible to non-biological signal propagation under fully asynchronous update, mitigated by the early/late update bias on the nodes listed in the table. Black/red/green arrows: feed-forward / negative feedback / positive feedback; node color: module membership according to Fig 3; translucent nodes: updated in random order. Time traces under each network show the order of biased update among these nodes during normal cell cycle progression; dashed horizontal line: time-step (update-round) boundary; orange/blue: ON/OFF; black up/down arrows: timestep in which nodes turn ON/OFF.
S4 Table. Knockout and overexertion predictions compared to experimental data.
Rows: independent in silico knockout / over-expression experiment (downward/upward arrows), performed in stochastic non-saturating environments indicated in column 3. Figure panels (columns 5–6): changes to normal cell cycle and/or apoptosis as a function of inhibition / overexpression strength (x-axis). Each stacked bar graph shows the relative occurrence of normal cell cycle completion (mustard), G2 → G1 reset followed by genome duplication (purple), aberrant mitosis followed by genome duplication (turquoise), failed cytokinesis followed by genome duplication (blue) and apoptosis (dark red) as a function of node inhibition/overexpression, relative to the cell cycle rate in wild-type cells (black dashed line) with synchronous (left) and biased order asynchronous update (right). Sampling: 50,000 time-steps.
S1 File. Additional simulation data.
S2 File. Jupyter Notebook in Python.
Jupyter Notebook in Python that uses the BooleanNet software package to simulate synchronous and asynchronous versions of our model and reproduce our key results (“SI_notebook.ipynb”; available at https://github.com/deriteidavid/cell_cycle_apoptosis_Sizek_etal_PloSCompBio_2019).
First and foremost, we thank Réka Albert for her insightful comments and guidance throughout the process of comparing the dynamics of our Boolean model under synchronous vs. asynchronous update, as well as for encouraging DD (currently in her laboratory) to put significant effort into the simulations and comparative analysis of these dynamics. We thank Peter Regan for his feedback on the clarity of our manuscript.
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