Three rule-based model families for HSCs dynamics

We integrated information based on literature, to describe HSC dynamics during liver fibrosis and its reversion however the lack of some information led us to propose hypotheses and finally to develop three different families of models. As shown in Fig 1A, the three families share common processes involving the different states of HSCs regulated by TGFβ1. In normal liver, HSCs are maintained in a non-proliferative quiescent state (qHSC) and store vitamin A. Upon liver injury (virus, toxic, etc.), the inflammatory response leads to the production of TGFβ1 which activates qHSCs into an activated state (aHSC). The fully activated HSCs are characterized by a myofibroblast (MFB) phenotype. Upon removal of liver injury, i.e in the absence of TGFβ1 in the models, MFBs are either eliminated through apoptosis and senescence pathways (apop_sene_MFB) or reverted to an inactivated HSC state (iHSC) close to but different from the quiescent state. Upon a new liver injury, i.e in the presence of TGFβ1 in the models, the iHSCs are reactivated (react_HSCs). In the same way that HSCs transform into MFBs, react_HSCs transform into reactivated MFBs (react_MFBs) however information is lacking about the fate of these cells. To overcome this issue, we developed two families of models in which the react_MFBs are either i) completely eliminated by the apoptosis/senescence pathways but not by inactivation, this family of models called reactMFB-wo-inactivation (wo for without) is described in Fig 1A₁), or ii) eliminated in the same way as MFBs, i.e. by the apoptosis/senescence pathways and/or the inactivation pathway, this family of models called reactMFB-with-inactivation is described in Fig 1A₂).

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Fig 1. Three models to describe HSC dynamics.

A1 Schematic representation of models reactMFB-wo-inactivation; A2 Models reactMFB-with-inactivation and A3 Models iHSC-reversion-to-qHSC. (B) Schematic representation of Kappa-modeled processes in all models. HSCs are agents characterized by seven cell physiological states (qHSC, aHSC, MFB, iHSC, apop_sene_MFB, react-HSC, and react_MFB). Rules describe the processes: (1) qHSCs are activated into aHSCs, (2) aHSCs are transformed into MFBs, (3) MFBs are transformed into apop_sene_MFBs to be eliminated, (4) MFBs are inactivated into iHSCs, (5) iHSCs are reactivated into react_HSCs upon TGFβ1 stimulation, (6) react_HSCs are transformed into react_MFBs. Similarly to MFBs, react_MFBs are eliminated by apoptosis and senescence (3) or by inactivation (4). (7) aHSCs and react_HSCs proliferate, (8) MFBs and react_MFBs proliferate, (9) aHSCs and react_HSCs produce COL1, (10) MFBs and react_MFBs produce COL1, (11) apop_sene_MFBs are degraded, (12) iHSCs are degraded. (13) qHSCs self-renewal. The effect of TGFβ1 is represented by gray arrows, it induces the activation of qHSCs, the reactivation of iHSCs, and the production of COL1 in aHSCs, react_HSCs, MFBs and react_MFBs. The reversion of iHSCs into qHSCs are represented by a dotted arrow. (C) Kappa contact map. HSCs are agents with three sites: cell_state (qHSC, aHSC, MFB, iHSC, react_HSC, react_MFB, apop_sene_MFB), TGFB1_binding (free or bound) and TGFBR (membrane, internalized or degraded).

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Similarly to react_MFB, information is lacking about the fate of iHSCs. While the activation of iHSCs by TGFβ1 has been described using in vivo and in vitro experiments [24, 25], no data have been reported about the fate of iHSCs in the absence of TGFβ1. However, Kisseleva et al. [24] showed a decrease in the iHSC population during the reversion process. We therefore developed a third family of models in which, in the absence of TGFβ1, iHSCs are either eliminated or return to a quiescent state, thereby participating in the repopulation of the injured liver with new qHSCs. This potential new source of qHSCs needs to be balanced with the self-renewal of native qHSCs in order to restore HSC homeostasis in the repaired liver. This models in which iHSCs revert to a quiescent state constitutes the third family called iHSC-reversion-to-qHSC which is illustrated in Fig 1A₃.

Finally, we developed three families of models: reactMFB-wo-inactivation, reactMFB-with-inactivation, iHSC-reversion-to-qHSC. The biological processes of all the models have been integrated in Fig 1B, where the interactions between TGFβ1 and aHSC, react_HSC, MFB and react_MFB cells leading to the induction of their proliferation and the production of COL1 are indicated.

The three families of models were built using the Kappa rule-based language (see Methods section) and simulations were performed using the KaSim tool [26] based on the Gillespie stochastic simulation algorithm (SSA) [27]. In these Kappa models, the hepatic stellate cells are the agents which are characterized by three sites (Fig 1C) which are: i) the cell_state of HSCs (qHSC, aHSC, MFB, iHSC, react_HSC, react_MFB, apop_sene_MFB), ii) the TGFB1_binding that can be free or bound and iii) the state of the TGFβ1 receptor (TGFBR) that can be localized at the membrane, internalized or degraded. In addition to these agents, we introduced counters to scale the intermediate steps between cell states, and we used tokens to describe TGFβ1 and COL1 quantities instead of detailing the behavior of each molecule, thereby highly reducing the computational cost (see Methods section).

The reactMFB-wo-inactivation models contain 75 rules and 41 parameters. A rule for react_MFB inactivation and two parameters controlling the proportion between inactivated and eliminated react_MFB have been added to obtain the reactMFB-with-inactivation models, containing 76 rules and 43 parameters. An additional rule for iHSC reversion and two other parameters controlling the proportion between reverted and eliminated iHSC have been added to obtain the iHSC-reversion-to-qHSC models. These rules are grouped into 7 biological processes: 1) qHSC renewal, 2) binding of TGFβ1 to HSCs, 3) TGFBR dynamics, 4) HSC activation and differentiation, 5) HSC proliferation, 6) COL1 production and elimination and 7) MFB inactivation (detailed in Methods section).

Inactivation loops of react_MFBs are essential to maintain COL1 accumulation during chronic liver injury

To identify the families of models that predict HSC behaviors and collagen accumulation that fits with the experimental data from Kiesseleva et al. [24], we conducted simulation studies using the protocol described by the authors for a mouse model of liver fibrosis and reversion, i.e. twice-weekly CCl4 (TGFβ1 in the model) injections for 8 weeks and an overall reversion period of six months. Observations reported by Kisseleva et al. [24] included a 1.43-fold increase in cells expressing α-SMA and a ∼ 14-fold increase in the percentage of COL1 deposits at 8 weeks and a return to initial values at 6 months after the last injection. In addition, the authors assessed iHSCs/qHSCs ratios and identified a 50/50 distribution at one month and a total number of HSCs similar to initial conditions at six months of reversion. As shown in Fig 2, most qHSCs (98%) were immediately activated upon the first stimulation of TGFβ1 across all models, a small number (2±1%) of qHSCs persisting during the stimulation period. Activation of qHSCs led to the production of aHSCs that differentiated into MFBs. These cells express α-SMA protein, and we tracked α-SMA positive cells by summing the occurrences of aHSCs, MFBs, react_HSCs and react_MFBs. Note that even if the number of qHSCs and iHSCs is low, they can be activated or reactivated by TGFβ1 during repeated stimulation for two months. In line with this observation, Rosenthal et al. [28] recently suggested that iHSCs remain at a low level even during the activation phase. Comparison of the models showed that the react_MFB inactivation loop greatly affects model behavior. When inactivation of react_MFBs was not allowed (models reactMFB-wo-inactivation)(Fig 2A), we observed a lower amount of α-SMA expressing cells (0.8-fold±0.02) and of COL1 deposits (9.5-fold±0.16) than expected at 8 weeks, with amounts even decreasing before stimulation was stopped. Conversely, when inactivation of react_MFBs was allowed (models reactMFB-with-inactivation)(Fig 2B, 2C and 2D) we observed an accumulation of α-SMA expressing cells and COL1 deposits that varied according to the percentage of react_MFB inactivation. While 5% allows the model to fit with experimental observations (red circle in Fig 2), increasing the percentage to 10 and 50% led to higher amounts of α-SMA expressing cells (1.6-±0.05 and 5-fold±0.03, respectively) and COL1 deposits (16-±0.36 and 40-fold±0.16, respectively) at 8 weeks. Only the model with 5% of react_MFB inactivation led to a 50/50 distribution of the iHSCs/qHSCs ratio at one month and a return to the initial total number of cells at six months of reversion, as described in the experimental data [24].

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Fig 2. Time-course analyses of cells and COL1 in models without or with react_MFB inactivation.

A series of simulations was performed using conditions of stimulation reported in Kisseleva et al. [24]. We used 10,000 TGFβ1 molecules per cell and 16 stimulations (twice a week) during 2 months. A1 Models reactMFB-wo-inactivation. B1 Models reactMFB-with-inactivation in which 5% of react_MFBs are inactivated. C1 Models reactMFB-with-inactivation in which 10% of react_MFBs are inactivated. D1 Models reactMFB-with-inactivation in which 50% of react_MFBs are inactivated. Stimulation of cells with TGFβ1 started on day 4 to allow the model to equilibrate. A2,B2,C2 and D2 display the variation in the number of cell occurrences for qHSCs, aHSCs, MFBs, iHSCs, react_HSCs, react_MFBs and α-SMA positive cells. The number of α-SMA cells is the sum of the number of aHSCs, MFBs, react_HSCs and react_MFBs. Simulations are expressed as the mean of 10 replicates. A3, B3, C3 and D3 display the variation of COL1% area. Simulations are expressed as arbitrary unit (a.u) and all 10 replicates are represented. The red circles indicate the biological observations reported in [24]. From left to right (A2,B2,C2 and D2), the first circle is for an 1.43-fold increase in cells expressing α-SMA at 8 weeks, the second circle is for an equal proportion of iHSCs and qHSCs at one month of recovery and the third circle is for an equal number of cells at 6 months and of cells at T0. The circle for COL1 simulations (A3, B3, C3 and D3) is for a ∼ 14-fold increase in the percentage of COL1 deposits at 8 weeks.

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Based on the models reactMFB-with-inactivation parameterized with 5% of react_MFB inactivation, we next evaluated the impact of a potential reversion of iHSCs to a quiescence state. As described in these new models iHSC-reversion-to-qHSC, we observed that regardless of the percentage of reversible cells, the overall behavior of the other model components was not affected (see S1 Fig).This may be due mainly to the slow dynamics of iHSCs elimination and quiescent cell renewal, which limit the impact of reversion. We have discarded this model, which was developed on the hypothesis of iHSC reversion but without any supporting experimental observations.

Taken together, our data demonstrate that inactivation of react_MFBs is required to sustain COL1 accumulation by maintaining the level of α-SMA expressing cells, thus validating the models reactMFB-with-inactivation. Simulation analyses allowed us to determine that a 5% ratio between reac_MFB inactivation and apoptosis_senescence pathways reflected the cells and COL1 behaviors seen experimentally. Regarding iHSCs, our observations suggest that complete reversion of iHSCs to a quiescent state is a potential pathway to eliminate iHSC in the absence of TGFβ1. Since the outcome generated by the models iHSC-reversion-to-qHSC, in which iHSCs can revert to qHSCs was similar to the one obtained in the models reactMFB-with-inactivation (S1 Fig) we chose to retain the latter for the rest of the analyses.

Collagen I accumulation is sensitive to the dynamics of TGFβ1 stimulation

Our models are calibrated using experimental data from a mouse model of CCl4-induced fibrosis [24], a reproducible experimental model widely used to study fibrosis and repair [29]. Although the protocols vary somewhat in terms of the number, frequency and modalities of CCl4 injections, these experimental models induce similar liver fibrosis. Other experimental models of induction of liver fibrosis in rodents have been developed, including injection of other chemical agents (such as thioacetamide (TAA) and dimethylnitrosamine (DMN)), high-fat diets and surgical interventions (such as bile duct ligation) [30, 31]. Whatever the etiology, induced damages lead to TGFβ1-dependent activation of HSCs, but with variable dynamics. It can happen very quickly in a model induced by CCl4 or slower in a model induced by a fatty diet. To explore the sensitivity of our validated model reactMFB-with-inactivation, we varied the concentration of TGFβ1 and the number and periodicity of stimulations. As shown in Fig 3, the model was highly sensitive to the amount of TGFβ1, but this sensitivity varied with the number of stimuli and periodicity, particularly for TGFβ1 values lower than 10,000 molecules per cell. Beyond that, the number of molecules exceeded the number of receptors per cell, and the model became saturated, as shown by the superposition of kinetic curves at 10,000 (green) and 100,000 (yellow) molecules per cell. At lower TGFβ1 levels, collagen accumulation was much more subtle. When stimulations were spaced sufficiently far apart, collagen levels returned to baseline and, remarkably, we observed a much stronger induction upon second stimulation, particularly at doses of 10,000 and 100,000 molecules per cell (Fig 3A, periodicity 30 and 60 days). This induction of collagen 1 was less pronounced at low doses, but increasing the number of stimuli (Fig 3B, periodicity 180 hours and 15 days) showed that collagen continues to accumulate with each new addition of TGFβ1 at the dose of 1000 molecules per cell, whereas it decreased for doses 10,000 and 100,000 molecules per cell. This effect remained visible for a larger number of stimulations (Fig 3C, periodicity 180h and 15 days and Fig 3D, periodicity 90h and 180h).

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Fig 3. Time course analysis of COL1 as a function of the variation in TGFβ1 concentration, number and periodicity of stimuli.

Results are expressed as the number of occurrences of COL1 according to the number of TGFβ1 stimuli: (A) 2, (B) 4, (C) 8, (D) 16, (E) 32, the periodicity which varies from 22.5 hours to 60 days and the concentration of TGFβ1: 10 (Red), 100 (Blue), 1,000 (Purple), 10,000 (Green) and 100,000 (Yellow) molecules per cell. The first stimulation occurs at day 4. Simulations are expressed as the mean of 5 iterations. P, Periodicity.

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Model predictions validated by new experimental data in mice

To evaluate the model predictions, we analysed COL1 accumulation in a mouse model of CCl4 induced liver fibrosis. For the quantification of collagen deposits, we used second harmonic generation (SHG) microscopy which we have previously demonstrated to quantify fibrillar collagen in human [32] and mouse [33] livers. Note that the global evaluation of collagen deposits using Sirius red [24] and SHG microscopy may overestimate the contribution of HSCs since other cells can contribute to liver fibrosis such as portal fibroblasts and bone marrow-derived cells [34]. However, HSCs have recently been shown to be the exclusive source of myofibroblasts in CCl4-treated liver [35] which is why we chose data from CCl4-induced liver fibrosis to calibrate our model. As shown in Fig 4A, we compared the signal quantification data obtained by SHG microscopy in our experimental model with the simulation data. Our experimental data showed maximum accumulation of collagen after 6 weeks and stabilization at a lower level at 8 and 10 weeks after treatment (Fig 4A₁). In tissue samples from mice that have undergone the reversion protocol, the SHG signal shows a decrease in COL1 at 8 weeks after CCl4 removal (Fig 4A₂). For the simulation studies, TGFβ1 parameters were adapted to the experimental protocol, i.e. three TGFβ1 stimuli in the first week and one stimulus per week for ten weeks. CCl4 induces rapid hepatocyte damages leading to inflammation and TGFβ1 production that we modeled using saturating TGFβ1 quantities (10,000 molecules/cell). As shown in Fig 4B₁, the predicted curves for COL1 accumulation reached a maximum value similar to the experimental data, then at 8 and 10 weeks, the stabilization plateau was slightly higher in our simulation. Regarding the reversion protocol, the predicted curves for COL1 are in full agreement with the experimental data (Fig 4B₂).

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Fig 4. Comparison of model predictions and experimental data obtained in a mouse model of CCl4 induced liver fibrosis.

A1 Representative SHG microscopy images of collagen in mouse livers after 2, 4, 8 and 10 weeks of CCl4 treatment. A2 For the reversion experiments, treatment was stopped at week 4. A3 Collagen quantification was performed with ImageJ and results are expressed as percentage of SHG signal relative to total area (Mean ± SD). B1, Model simulations were performed using parameters corresponding to the CCl4 injection protocol (three TGFβ1 stimuli in the first week and one stimulus per week for ten weeks). Collagen accumulation was plotted and experimental data (from A1) are indicated with black bars. B2, Model simulations were performed using parameters corresponding to the reversion protocol (three TGFβ1 stimuli in the first week and one stimulus per week for 4 weeks). Collagen accumulation was plotted and experimental data (from A1) are indicated with blue bars.

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We also validated the prediction of COL1 dynamics using experimental data from other laboratories, including liver fibrosis models induced by dimethyl_nitrosamine [36] and thioacetamide [37]. Results are presented in (S2A and S2B Fig).

iHSC accumulation is associated with human liver fibrosis

A major observation by Kisseleva et al. [24] is that the fibrotic response is exacerbated in mice that have undergone an initial chronic aggression followed by a reversion period allowing a return to a healthy tissue phenotype. This was associated with the presence of inactivated stellate cells (iHSCs) which display an enhanced response to TGFβ1 when stimulated in cell culture, leading to faster and greater production of COL1 [24, 25, 38]. These iHSCs would therefore still be present in the “repaired” liver after six months of reversion. To take this into account, we calibrated our models reactMFB-with-inactivation on Kisseleva’s observations et al. [24] after the second aggression by introducing the hypothesis of a slow elimination of iHSCs. However, there is no experimental data describing the fate of these cells during aggression/reversion cycles. Using the relapse protocol described by Kisseleva et al. [24], we studied the behavior of iHSCs throughout injury-repair cycles. This protocol includes chronic injuries induced by eight CCl4 injections over one month, followed by a six-month recovery period, and then a new injury similar to the first. We repeated this cycle three times (Fig 5).

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Fig 5. iHSCs accumulate during injury-repair cycles.

Simulations were performed using repeated cycles comprising a first injury cycle (8 injections twice a week for 1 month), followed by a 6-month recovery period and a second injury cycle as previously described by Kisseleva et al. [24]. Simulations are expressed as the mean of 50 replicates for cells (A), and all 50 replicates have been represented for COL1 (B) and iHSCs (C). Overlaps are shown as grey area.

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Our simulations show that the dynamics of α-SMA expressing cells and amounts of COL1 were consistent with Kisseleva’s et al. [24] experimental observations, i.e. higher levels of these two markers after the second aggression compared to the first one (Fig 5A and 5B, cycles 1, 2). Importantly, the model allowed us to detail the dynamics of iHSCs that was not experimentally evaluated, and we observed an increased number of iHSCs (Fig 5C, cycles 1, 2). When the aggression/reversion cycles were repeated, we showed an accumulation of iHSCs that was associated with that of α-SMA expressing cells and COL1 (Fig 5A, 5B and 5C, cycles 3, 4). The increased number of iHSCs may result from the balance between the higher number of α-SMA expressing cells leading to the inactivation of more iHSCs and the half-life of iHSCs. On the basis of these observations, the model predicts a progressive accumulation of iHSCs during cycles of aggression/reversion, suggesting a critical role for iHSCs in the dynamics of fibrosis, with accumulated iHSCs constituting a source of COL1 overproducing cells during reactivation.

To validate this prediction, we searched for iHSC accumulation in samples from patients with liver fibrosis. To this end, we analyzed RNAseq data from a large multicenter study comprising 206 histologically characterized liver samples from patients with non-alcoholic fatty liver disease (NAFLD) including 102 patients with non-alcoholic steatohepatitis (NASH) associated with liver fibrosis [39]. Patients were divided in two groups: with low (F0-F1, n = 34) and high (F3-F4, n = 68) fibrosis grades. To identify iHSCs in human liver samples, we used an iHSC gene expression signature reported by Rosenthal et al. [28] in a mouse model of non-alcoholic steatohepatitis (NASH). This signature contained 39 genes that were significantly differentially expressed in iHSCs when compared to both qHSCs and aHSCs. In addition, we used 10 iHSC gene markers identified in a mouse model of CCl4-induced liver fibrosis [24] and validated in an in vitro reversion model of human activated cells [38]. The list of these genes is detailed in S1 Table.

As shown in Fig 6A, gene set enrichment analyses (GSEA) using the signature identified by Rosenthal et al. [28] showed that iHSC genes were collectively enriched in the high fibrosis group compared to the low fibrosis group (normalized enriched score = 1.38, p-value = 0.097). We performed a new analysis, adding the 10 marker genes identified by Kisseleva et al. and El Taghdouini et al. [24, 38] to the list of genes identified by Rosenthal et al. [28]. As shown in Fig 6B, this additional set of genes increased the statistical power of the GSEA analysis (normalized enriched score = 1.57, p-value = 0.048), suggesting that the expression of this iHSC-specific gene set is increased in patients with high fibrosis. These observations are in line with the increase in the number of iHSCs predicted by our model, during the progression of hepatic fibrosis.

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Fig 6. iHSC accumulation is associated with liver fibrosis.

Gene set enrichment analyses of iHSC gene expression was performed in low fibrosis (F0-F1, n = 34) and high fibrosis (F3-F4, n = 68) samples from patients with non-alcoholic steatohepatitis (NASH) ([39]). A) Analysis performed with the iHSC gene signature identified by Rosenthal et al. [28]. B) Analysis performed with the iHSC gene signature identified by Rosenthal [28], supplemented by ten iHSC markers identified in [24, 38]. Results are presented in the form of correlation profiles between gene lists, enrichment scores (ES) which indicate to which phenotype (low or high fibrosis) the gene set is most positively correlated and heat maps visualizing clusters and gene distribution.

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Kappa, a language adapted to multiscale modelling

While ordinary differential equations (ODEs)-based models have long been used to model biological dynamics, rule-based models (RBM) are much more appropriate for dealing with networks with combinatorial multi-state interactions as discussed by Chylek et al. [42]. Indeed, ODE approaches require the implementation of an equation for each species. Agent-based models (ABM), while more appropriate for bottom-up approaches than ODEs, require a description of all possible behaviors of agents. Common ABM approaches are based on the object-oriented paradigm [43] or process calculi [44].

In the case of our model, the single agent has 4 sites, some of which have 8 or 14 different states. Using an ABM approach would have exploded the number of rules (∼ 300) in the model to describe the 39 possible agent states. RBM approaches were therefore the most appropriate to represent the known interactions operating on stellate cells during fibrosis development and reversion. Based on site graph rewriting, BioNetGen [45] and Kappa [21, 22] are the two major languages used for RBM implementation and simulation.

Kappa formalism has been widely used for modeling biological reaction networks such as cell signaling [46], gene regulation [47], epigenetic regulation [48], repressilator system [49] and DNA repair [50]. Using Kappa language, our team previously developed a model to describe the regulatory role of extracellular matrix networks in TGFβ1 activation [51]. The transition from molecular to cell-molecule interaction models has been a new challenge, and we have recently demonstrated the applicability of the Kappa language in multi-scale modeling [52]. However, to overcome the explosion in memory demand due to the declaration of molecules and cells as agents, we have used tokens and counters. The way tokens are stored in memory and the absence of explicit bond with agents considerably reduces the computational cost associated with the existence of different scales in the model. This involves the use of hybrid rules between agent and token whose applicability has been demonstrated in our model. Counters allowed a more detailed description of the activation and differentiation states of HSCs by creating intermediate steps and cell behaviors depending on their activation/differentiation stage, that better reflects experimental observations. Moreover, counters allow to model phenomena that the duration of which is more predictable than chemical interactions. With time-exponentially distributed event, there is only one parameter: standard deviation which is fully characterised by the average time. Erlang distributions provides a mean to narrow the distribution hence leading to events which are more predictable (and which corresponds from a mathematical point of views to the composition of several atomic events following time-exponential distributions).

In short, counters and tokens are key to multiscale modelling in Kappa. Tokens enable a more compact representation of agent populations, counters enable a more compact representation of site populations, with both facilitating the search and abstraction of potential rule applications. To our knowledge, beside a model of pulmonary viral infection by William Waites [53], our study is the first one where counters and tokens have been used in a Kappa model.

A unique fibrosis model for different etiologies

While the role of TGFβ1 in HSC activation is well established, our model showed how its concentration, the number of stimuli and their periodicity influence the dynamics of liver fibrosis. These observations underline the adaptability of the model to the different pathophysiological conditions associated with the development of fibrosis. In murine models of liver fibrosis induced by chemical compounds such as CCl4, the toxic effect is massive, with strong hepatocyte necrosis, exacerbated inflammatory response and a saturating release of TGFβ1. In other models, such as diet-induced fibrosis, the immune response is different and fibrosis progresses more slowly. Sensitive to small variations in TGFβ1, our model reproduces the collagen accumulation observed in these different fibrosis mouse models. Similarly, the response to liver injury in human is etiology-dependent and may affect the dynamics of HSC activation. In contrast to chemical models, diet-induced fibrosis models developed to mimic the progression of NAFLD to non-alcoholic steatohepatitis (NASH) are characterized by a lower level of fibrosis. In addition, the dynamics of fibrosis are highly variable depending on the rodent strain and diet composition. Moreover, the optimal model of steatohepatitis leading to fibrosis (choline_deficient, amino_acid defined diet) is poorly reversible [54]. To overcome this heterogeneity, we chose a reversible diet model [55] which we compared with model predictions. We set the parameters of TGFβ1 to low levels (4500 molecules/cell, 6 stimuli and a periodicity of 14 days). As shown in supplementary Fig S2-C, the model captured the increase in COL1 but reversion dynamics were too fast, suggesting that for this condition the model requires additional regulators.

A model to capture stellate cell heterogeneity and microenvironment remodeling

Recent single-cell studies have demonstrated the great heterogeneity of HSCs and MFBs during the development of fibrosis [17, 28, 56]. These works also show that, in addition to a diversity of phenotypes, the dynamics of HSCs during their activation and differentiation process are far from linear, as shown by pseudotime analyses [57, 58]. Our approach manages to model this heterogeneity by introducing intermediate steps in the activation and differentiation process, and allowing cells in each of these states to proliferate, progress through the activation process and produce COL1. The implementation of these rules combined with the SSA algorithm (Stochastic Simulation Algorithm) means that, at each point in the simulation, there are different phenotypes and a ‘unique’ history is kept for each agent.

However, modelling HSC dynamics does not capture the full complexity of the microenvironment that governs these dynamics such as the multitude of events regulated by immune cells. Other multiscale models include the role of Kupffer cells in HSC activation via TNF-α [13, 15] or antagonistic regulation by M1 and M2 macrophages [16]. Similarly, it is difficult to include all the components involved in extracellular matrix remodeling. This process is finely regulated by the balance between the production of matrix metalloproteinases (MMPs) and their inhibitors, Tissue Inhibitor of metalloproteinases (TIMPs), both secreted by HSCs, MFBs, but also by immune cells involved in the resolution of fibrosis [59–61]. In their model, Friedman & Hao [16] reduced ECM remodeling to the sole contribution of macrophages and did not take into account the known role of HSCs at all. To overcome this molecular and cellular complexity while keeping remodeling process into our own model, we introduced tokens to represent low and high remodeling COL1. In this way we modeled both collagen accumulation during TGFβ1 stimulation and its regression upon TGFβ1 withdrawal.

Inactivation of reactivated MFBs, a critical factor in the progression of fibrosis

The process of inactivating HSC-derived MFBs has been presented as a key element in the reversion of fibrosis enabling the elimination of ∼ 50% of MFBs [24, 25]. The authors have shown that the inactivated cells can be reactivated both in vivo and in vitro in response to a new stimulus, and that they are much more reactive and fibrogenic than initially activated cells [24, 38]. However, the fate of these reactivated cells remained unknown, leading us to hypothesize a potential re-inactivation of these reactivated cells. Our results showed that the inactivation of these cells was essential to maintain COL1 accumulation in chronic lesions, but that this inactivation had to be limited. Indeed, our models have shown that 5% of reactivated MFBs should be inactivated, experimental observations are no longer respected when using higher percentages of reactivated MFBs. This loop of inactivation and reactivation induces a change in the MFB population. As lesions accumulate, cells inactivated between aggressions are reactivated, resulting in a population of reactivated MFBs that becomes the majority and more fibrogenic cells. In support of these predictions, single-cell sequencing data identified these ECM-producing MFB phenotypes with increase fibrogenic activity [28].

Reversion of inactivated stellate cells to a quiescent state as an alternative to cellular elimination

Although the existence of iHSCs has been supported by several studies, their behavior is not yet well understood. These cells have a phenotype close to that of quiescent cells, but retain the memory of their previous activation [24, 38]. However, the fate of iHSCs in the absence of reactivation stimuli remains unknown. The development of our models enabled us to observe that the reversion of these iHSCs to a quiescent state, regardless of the percentage of cells affected, did not impact the overall behavior of the other components. This may be due mainly to the time required for the elimination of iHSCs and the renewal of quiescent cells, which is particularly slow in our models. This dynamic limits the impact of reversion. It is also possible that these cells have a faster elimination dynamic, but that this is compensated by a proliferation capacity, thus slowing down their elimination and increasing the impact of potential reversion. As it stands, our models suggest that the complete reversion of iHSCs to a quiescent state is a potential means of eliminating them in the absence of TGFβ1. Experimental follow-up of these cells over reversion times of more than one month would provide more precise information on their fate.

Inactivated HSCs as new markers of fibrosis progression?

The accumulation of iHSCs during fibrosis is a very important concept from a clinical perspective, as the progression of fibrosis is not at all linear. iHSCs within tissues could act as a kind of memory of an initial injury, making the tissue more sensitive to the next injury. Of course, this memory facilitates rapid repair, but the recurrence of these repair episodes, if too close together, accelerates the progression of fibrosis. Although our model is not fully adapted to follow the dynamics of the mouse NASH model due to the slow release of diet-induced TGFβ1, we showed that the observation of iHSC accumulation is supported by the enrichment of iHSC gene signature in NASH patients with increased fibrosis.

Identifying inactivated cells in vivo would provide a genuine marker of patient history. However, this identification remains complex due to the intermediate phenotype of these cells, between quiescent and activated state. There is as yet no established phenotypic signature for inactivated cells, and gene expression signatures vary according to the mouse models used (NASH, CCl4). These phenotypic signatures depend both on the time of reversion and on the methods used to identify them, depending on whether their expression is compared with that of quiescent cells and/or activated cells. Recent advances in single-cell sequencing will enable us to refine these signatures by characterizing more specific groups of genes.

Kappa syntax

Kappa is a site graph rewriting language [62–65] that uses a chemistry-inspired syntax to transparently describe the interactions between component occurrences. The syntax used by Kappa is that of site graphs, using rules to describe the behavior of agents over time. An agent describes species (e.g. cells, proteins) and defines the characteristics of these species. Rules define both the interactions between agents and their behavior. Agents are described by one or more sites which can have different states. These states allow agents to establish links between themselves. A rule describes a process that sets a condition (left-hand member) and an event (right-hand member). All the elements at the left of the → are the elements needed for the rule to be applied, the elements at the right are what will be produced. In the following example, two agents A and B interact to form a complex AB. (1)

The agents A and B and their respective sites are written as: Agent_name(site_name{state_value}[binding_state]). The agent A has its site x in a state p and its site y free ([.]), and the second agent B has it site y free. A and B will bind together via their site y. The symbol @ is followed by the expression that defines the rate of each potential application of the rule in the state of the system. That is to say the probability that a potential event effectively occurs within an infinitesimal interval of time.

Simulating models in which agents have to be explicitly bound to other agents can be computationally expensive. Instead, such bounds can be described implicitly. To this aim, the binding state of the binding site in an agent A can be encoded as an internal state which specifies whether the site if free, or bound. In the latter case, the site to which this site is bound to is not specified. Then the amount/quantity of agent B remaining is encoded by the means of a token. Consequently, Tokens constitute a pool of variables enabling the description of abundant molecular species in a continuous manner. Their use allows us to resolve the memory limitation caused by declaring molecular species as an agent. If a token such as ATP is added to the rule 1, the following rule is obtained (rule 2), where the binding of A and B requires ATP. What stands between | and @ defines the quantity of token to be deleted or added (2) when the rule is applied. Because the binding of A and B depends on the amount of ATP, the rate of the rule has to be modified to include the quantity of ATP, written as |ATP|. (2)

Models in which agents have numerous states require many rules to describe the changes between these states. Instead of describing all these states using numerous rules, it may be sufficient to know the number of steps to follow a process, requiring only a few rules, by increasing the step index. With counters, we can use discrete intervals to describe a special type of site which can be increased or decreased in the rules. In the following example (rule 3), A has a site called n, which is a counter that counts the number of bonds that can be formed by A during its lifetime in the system. This new rule allows A to bind to B as long as the value of n for A is greater or equal to one. When a bond is formed, the value of n decreases by one. (3) However, the current syntax of Kappa cannot allow to test whether the value of a counter is inferior to a given value (i.e., the “<” symbol is not supported). To overcome this limitation, it is necessary to use two counters whose sum remains constant. This syntactic limitation is overcome in the new version of Kappa (which internalises the use of pairs of counters to deal with less than inequality tests).

Modeling HSC dynamics

Using Kappa rules, we described HSC dynamics by including various biological processes such as the activation of quiescent HSCs (qHSC) by TGFβ1 leading to activated HSCs (aHSC), the transdifferentiation of aHSCs into Myofribroblasts (MFB), the elimination of MFBs after removal of TGFβ1 either by apoptosis and senescence pathways (apop_sene_MFBs) or by reverting to an inactivated state (iHSC), the reactivation of iHSCs upon new TGFβ1 stimulation leading to reactivated HSCs (react_HSC) and the transdifferentiation of react_HSCs into reactivated MFBs (react_MFB). We also used rules to describe cell proliferation and production of collagen (except for qHSCs and iHSCs). In these models, HSCs are agents characterized by 3 sites: i) the cell state of HSCs (qHSC, aHSC, MFB, iHSC, react_HSC, react_MFB, apop_sene_MFBs), ii) the TGFB1 binding (free or bound) and iii) the state of the TGFβ1 receptor (TGFBR) that can be localized to the membrane, internalized or degraded. We introduced a pair of counters (called intermediary-step and control_counter) to scale the intermediate steps between cell physiological states, and we used 4 tokens to describe TGFβ1 and collagen molecules. The resulting families of models contained 75 to 77 rules and 37 to 41 parameters, depending on families. All rules are available in the GitHub repository: https://github.com/MBougueon/HSC_model_Kappa. We detailed here the representative rules of the biological processes. These rules are given in graphic form for easier reading.

qHSC renewal.

qHSC renewal is modeled using two rules (Fig 7). These rules take into account the fact that the site TGFB1_binding on qHSCs must be free, since binding of TGFβ1 induces activation into proliferating HSC.

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Fig 7. Renewal of quiescent HSCs.

An agent HSC with its site cell_state in a state quiescent and its site TGFB1_binding in a state free disappears (degradation). Conversely, the second rule describes the creation of this same agent. The rate of application of each rule is shown on the right.

https://doi.org/10.1371/journal.pcbi.1011858.g007

TGFβ1 binding to cells.

To describe the interaction of TGFβ1 with HSCs, 7 different rules are used according to the cell state. Fig 8 shows the binding rule of TGFβ1 to qHSCs. Only the site cell_state of the agent differs between these rules, except for MFBs and react_MFBs, where the rule rate is decreased. The fact that each agent can bind TGFβ1 introduces competition between agents for the binding of TGFβ1.

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Fig 8. Binding of TGFβ1.

Binding of TGFβ1 (as a token TGFB1_free) on the site cell_state of an HSC agent in a quiescent state, its TGFB1_binding site is in a free state and its TGFBR site is in a Membrane state. When the rule is fired, the value of the token TGFB1_free is decreased and the state of the site TGFB1_binding changes to bound and the state of the site TGFBR changes to Internalized.

https://doi.org/10.1371/journal.pcbi.1011858.g008

Turnover of TGFBR.

The endocytic pathways regulating TGFBR receptor signaling and turnover has been previously described by Di Guglielmo et al. [66] and modeled by Zi et al. [67]. During this process, TGFβ1 binding to TGFBR2 induces the recruitment of TGFBR1, leading to the activation of signaling pathways that regulate TGFβ1-dependent processes such as COL1 expression and cell proliferation. The trafficking of the TGFBR1/TGFBR2 complex between membrane and cytosol takes 30 min and is not affected by TGFβ1 binding. In the present study, we used TGFBR to represent the TGFBR1-TGFBR2 complex and we did not describe the intracellular signaling mechanisms. Upon TGFβ1 binding, the receptor TGFBR is internalized to transduce signals and can be either recycled (Fig 9A) or degraded (Fig 9B). To reduce model and time of calculation, the basal turnover of the receptor is not included in the absence of TGFβ1. To model TGFBR synthesis and localization to the membrane, a rule similar to the rule for degradation is used, changing the site TGFBR from the state degraded to membrane.

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Fig 9. Degradation and recycling of the TGFβ1 receptor.

(A) The TGFBR site in an Internalized state changes into a Membrane state for the recycling rule, (B) and into a Degraded state for the degradation rule.

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HSC activation and differentiation.

During activation and differentiation, cells have the potential to proliferate and produce COL1 (in variable quantities depending on their activation and differentiation state). However, as each agent can only undertake one action at a time, different rules to describe cell activation, differentiation, proliferation and COL1 production are used.

To model HSC activation and differentiation into MFBs, counters are used ranging from 0 to 14, corresponding to the 14 days reported in in vitro experiments (0 to 7 corresponding to the change from qHSC to aHSC; 8 to 14 corresponding to the changes from aHSC to MFB). On one hand, two rules describe the change of cell state (qHSC to aHSC and aHSC to MFB) and on other hand, two rules describe the step-by-step dynamics by increasing counter intermediate_step and decreasing counter control_counter.

Increasing the counter intermediate_step allows us to follow the activation and differentiation processes. The decrease of the counter control_counter ensures that the value of the counter intermediate_step remains in the range from 0 to 14 (needed because the “<” symbol is not supported in Kappa). Fig 10 describes the dynamics of the activation, defined in 7 steps, corresponding to the seven days of the transition from qHSC to aHSC, with the implementation of a single rule. For the reactivation process, rules similar to those of the activation process were used.

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Fig 10. HSC activation process.

The rule specifies that as long as an agent HSC, whose site cell_state is in the state activated has its counter control_counter with a value lower than or equal to eight then, the value of its counter control_counter will be decremented by one and that of its counter intermediate_step will be incremented by one. This simple rule describes the seven stages of activation.

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Cell proliferation.

A proliferation rule is written for each counter value. If an HSC agent proliferates, all the sites of the newly created agent will have the same value as those of the parent cell, except for TGFBR. Fig 11 shows an example of proliferation rules for the agent HSC whose site cell_state is in the activated state and its counter intermediate_step is equal to 7. The activation or differentiation steps do not affect the proliferation rate of agents and aHSCs and react_HSCs have additional rules for proliferation upon TGFβ1 binding. There are 30 rules to describe the proliferation process. The value of state, intermediate_step and control_counter is different between each rule to allow all cells to proliferate. Among these rules, 11 allow agents whose site cell_state is in the aHSC or react_HSC states to proliferate upon TGFβ1 stimulation by requiring the site TGFB1_binding to be in a state bound, then transforming this site into a state free after the rule is fired.

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Fig 11. Example of proliferation rules.

For an agent HSC whose site cell_state is in the state activated and its counter control_counter equal to 7.

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Collagen 1 turnover.

The amount of COL1 in liver fibrosis and reversion results from the balance between production and degradation. Upon TGFβ1 stimulation, activated HSCs and MFBs are the major cell producing COL1 while apop_sene_MFBs contribute to the degradation of COL1 during fibrosis reversion [68, 69]. To overcome the complex molecular mechanisms involved in regulation of COL1 degradation, COL1 was described using two different tokens: one for high remodeling (COL1_remodeling_high) and another for low remodeling (COL1_remodeling_low). These two tokens allow the description of the low degradation rate of COL1 during HSC activation and the high degradation rate during the process of reversion. COL1 dynamics is described using 12 rules: 8 for the production of COL1_remodeling_high by aHSCs, MFBs, react_HSCs and react_MFBs in the presence and absence of TGFβ1; 2 for the degradation of each token (COL1_remodeling_low being degraded at a slower rate than COL1_remodeling_high) and 2 to enable the two tokens to be linked (Fig 12A). The transformation of COL1_remodeling_high to COL1_remodeling_low depends on the number of COL1-producing cells (aHSCs, react_HSCs, MFBs and react_MFBs) which are also characterized by expression of α-smooth muscle actin (αSMA).

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Fig 12. Example of the rules involved in Collagen 1 turnover.

A), Decrease of collagen degradation, the value of token COL1_remodeling_high is decreased and that of COL1_remodeling_low is increased; B), Production of collagen by activated HSCs, the rate of application of the rule is proportional to the value of counter intermediate_step. This enables the cells furthest advanced in the activation process to produce more collagen than those just activated.

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The more cell agents are activated (or reactivated) and differentiated into MFBs (or react_MFBs), the greater the production of COL1_remodeling_high. A total of 8 rules describe the production of COL1. The state and the variable defining the amount of COL1_remodeling_high is modified according to cells (Fig 12B). Of these 8 rules, 4 are TGFβ1-free and 4 are TGFβ1-induced. For the latter 4, the cells must have the site TGFB1_binding in a state bound and will be transformed into free after the production of COL1_remodeling_high.

MFBs inactivation.

After TGFβ1 removal, i.e. in the absence of TGFβ1 stimulation in the model, MFBs and react_MFBs are eliminated through the apoptosis and senescence pathways (Fig 13A). An additional rule describes the process of elimination of apop_sene_MFB. The dynamics of the inactivation process is described in two steps using two rules (Fig 13B). The first rule changes the MFBs to an inactivated cell state (iHSC) and decrements the counter value named intermediate_step to 0. The second rule increments the value of this same counters by 1 and increments the value of the control_counters by 13.

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Fig 13. Rules for eliminating MFBs by apoptosis/senescence and inactivation.

A), The apoptosis/senescence rule consists of changing the site cellular_state of an agent HSC from a state MFB to apoptosis_senescence. B), For inactivation, two rules are necessary, the first changes the site cellular_state of an agent HSC from a state MFB to inactivated while reducing the value of the counter intermediate_step by 14. The second rule increases the values of control_counters by 13 and intermediate_step by 1.

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Calibration and parameters estimation

For all the biological processes included in the families of models, 13 parameters were obtained directly from the literature, 9 were calculated and 22 parameters were estimated from biological observations by comparing model simulations with experimental data, running hundreds of simulations (S2 Table). Special attention was paid to calibrate these 22 parameters because of the undocumented information about them, but also because of the interdependence between the parameters and the stochastic approach used.

To calibrate the models, we used a block-based parameter estimation method, i.e. a set of parameters for each subsystem (block) was estimated sequentially. Each set was defined in terms of a specific biological process such as cell proliferation, cell inactivation, collagen production and degradation. Block estimation reduced the impact of interdependence and enabled us to find a set of parameters corresponding to biological observations. As the evaluation of collagen was the experimental model best described, we first calibrated the set of parameters related to on collagen-producing cells. Next, we identified the set of parameters related to react-MFB inactivation which strongly affects model dynamics. The third set of parameters focused on iHSC and qHSC dynamics. The final set of parameters focused on collagen itself, encompassing its production and degradation. Following this approach, the estimation of the first set of parameters excluded the family of models called reactMFB-wo-inactivation, as shown in Fig 2A. Finally, we performed a global verification to determine whether all the parameters matched the experimental observations (visual inspection).

In addition, the behavior of iHSCs remains unknown, and we developed 3 different families of models taking into account different hypotheses: 1)model family reactMFB-wo-inactivation where reactivated MFBs cannot be inactivated, 2) model family reactMFB-with-inactivation where reactivated MFBs can be inactivated, 3) model family iHSC-reversion-to-qHSC where iHSCs can revert into qHSCs, in the latter model family the react_MFBs can be inactivated. Here, we described the parameters based on biological observations. Note that the rule application rate is drawn according to an exponential distribution whose parameter is equal to the sum of the propensities of all the potential events in the system state. The rates of the rules must therefore be defined as the value T_1/2 of the biological process they describe.

Experimental validation using a mouse model of CCl4-induced liver fibrosis

To validate the predictions of the model, collagen I was quantified during the course of fibrosis and reversion using a mouse model of CCl4 induced liver fibrosis. As previously described [77], seven-week-old female C57Bl/6 mice were treated with oral administration of CCl4 (Sigma-Aldrich, St. Louis, MO, USA) diluted in olive oil. A first dose of 2.4 g/kg of mouse weight was administered to mice three days before starting weekly treatment with a 1.6 g/kg dose for ten weeks. Control mice were treated with the vehicle only (olive oil). Mice were sacrificed at 24 hrs after the last CCl4 dose. For fibrosis reversal experiments, mice were injected for 4 weeks with CCl4 and sacrificed at 8 days (1 week of recovery), or 15 days (2 weeks of recovery) after the last injection. Each group (control, fibrosis and reversion) at each time of sacrifice contained 5 mice. Collagen quantification was performed by SHG microscopy on 20 μm frozen tissue sections as previously described [78].

Computational validation using Gene set enrichment analysis of RNA-Seq data from patients with liver fibrosis

The enrichment for iHSC gene expression signature in human fibrotic samples was performed using Gene Set Enrichment Analysis (GSEA) [79, 80]. We used the iHSC gene expression signature previously identified by Rosenthal et al. [28] in a mouse model of non-alcoholic steatohepatitis (NASH) and we added iHSC markers previously identified in a mouse model of CCl4 induced liver fibrosis [24] and an in vitro reversion model of human activated HSCs [38]. The list of genes is detailed in supplementary S1 Table. For clinical samples, RNAseq data from patients with non-alcoholic fatty liver disease (NAFLD) [39] were used. Note that three genes (KRT20, GABRA3 and GSTT1) were not identified in the RNAseq data due to a low expression level. Liver samples with fibrosis were selected and separated in two groupss: with low (F0-F1, n = 34) and high (F3-F4, n = 68) fibrosis grade. RNAseq data were filtered using the BIOMART database [81] to select genes encoding proteins and data were normalized using DESeq2 [82] with a threshold of 100 reads. The filtration steps reduced the number of genes from 21,595 to 16,021.