The TGF-β/Smad signaling system decreases its activity through strong negative regulation. Several molecular mechanisms of negative regulation have been published, but the relative impact of each mechanism on the overall system is unknown. In this work, we used computational and experimental methods to assess multiple negative regulatory effects on Smad signaling in HaCaT cells. Previously reported negative regulatory effects were classified by time-scale: degradation of phosphorylated R-Smad and I-Smad-induced receptor degradation were slow-mode effects, and dephosphorylation of R-Smad was a fast-mode effect. We modeled combinations of these effects, but found no combination capable of explaining the observed dynamics of TGF-β/Smad signaling. We then proposed a negative feedback loop with upregulation of the phosphatase PPM1A. The resulting model was able to explain the dynamics of Smad signaling, under both short and long exposures to TGF-β. Consistent with this model, immuno-blots showed PPM1A levels to be significantly increased within 30 min after TGF-β stimulation. Lastly, our model was able to resolve an apparent contradiction in the published literature, concerning the dynamics of phosphorylated R-Smad degradation. We conclude that the dynamics of Smad negative regulation cannot be explained by the negative regulatory effects that had previously been modeled, and we provide evidence for a new negative feedback loop through PPM1A upregulation. This work shows that tight coupling of computational and experiments approaches can yield improved understanding of complex pathways.
Transforming Growth Factor-β (TGF-β), a regulator of cell migration and cell fate, is a pharmaceutical target for the treatment of metastatic cancer and fibrotic diseases [1]. Signal transduction from extracellular TGF-β to the cell nucleus through the Smad pathway is well documented [2]–[7]. The TGF-β ligand binds sequentially to the type II TGF-β receptor, a constitutively active kinase, and then to the type I receptor, to form a ligand-receptor complex (LRC). The type I receptor is activated by the type II receptor and then phosphorylates the R-Smads (Smad2 and Smad3) at two C-terminal serine residues. Upon phosphorylation, R-Smads form a homomeric complex or a heteromeric complex with Co-Smad (Smad4). The key outcome of the Smad cascade is the accumulation of phosphorylated R-Smad (phospho-R-Smad) in the nucleus, affecting the transcriptional regulation of many genes[7], [8]. Smad signaling is known to decrease quickly after TGF-β stimulation, causing rapid decline of phospho-R-Smad after its initial peak. The HaCaT cell line has been adopted as an experimental model system for quantifying the detailed signaling of the TGF-β/Smad system. In HaCaT cells, there is a rapid decline of phospho-R-Smad after short exposure to TGF-β stimulation (30–45 min) [9], [10], and a gradual decline after long exposure to TGF-β (6–24 hr) [9], [11]. The duration of phospho-R-Smad activation could be crucial for regulation of different genes [12]. The self-limiting behavior of Smad signalling (i. e. , negative regulation) may be caused by ligand-induced receptor inhibition [13]–[19], phospho-R-Smad dephosphorylation [9], phospho-R-Smad degradation [11], [20]–[23], or other effects. Extensive experimental evidence has documented multiple modes of negative regulation, but the relative roles and combined effects are not well understood. Previous computational models of TGF-β/Smad signaling have contributed important biological insights, but they have only simulated some selected negative regulatory effects. Vilar et al. built a model of TGF-β receptor trafficking dynamics, including ligand-induced receptor degradation, which was able to simulate some key dynamic observations such as the peak and decline of phospho-R-Smad levels [24]. Models by Klipp and co-workers extended the work of Vilar et al. to include Smad phosphorylation and nuclear translocation [25], and to include transient versus sustained Smad signaling [12]. These models used simple representations for negative regulation, and gave a strong role to receptor degradation. The model by Schmierer et al. provided important insights into the Smad nucleo-cytoplasmic shuttling [10], but the only negative regulatory effect in this model was dephosphorylation. Other modeling studies have focused on robustness and in silico perturbation analysis [26], [27]. Mathematical models have yielded important insights, but they have not represented TGF-β/Smad negative regulation with enough detail for analyzing the contributions of different negative regulatory effects, nor for evaluating alternative hypotheses. In this work, we developed a series of computational models, representing individual and combination effects of R-Smad negative regulation. Comparisons between models and observations revealed negative regulation to occur at more than one time-scale. We classified negative regulatory effects into fast-mode (5–240 min) and slow-mode (1–24 hr), depending on how quickly they act (and how quickly they equilibrate to steady state). Models then showed that at least one fast-mode and one slow-mode effect would be required for a model to fit the phospho-R-Smad dynamics in both short-exposure and long-exposure experiments. R-Smad Dephosphorylation was a fast-mode effect and it was strong enough to explain the fast-mode observations. Receptor Degradation and P-Smad Degradation were slow-mode effects, but they were too weak to explain the observed slow-mode decline. With a shortfall in explaining why R-Smad continues to decline hours after TGF-β stimulation, we sought a novel slow-mode effect. A second key finding of this work is a novel negative feedback effect, confirmed experimentally, in which the phosphatase PPM1A is upregulated after TGF-β stimulation. A final model hypothesizes how PPM1A might be upregulated with delayed activity, based on previously published molecular mechanisms for regulating PPM1A degradation [28], [29]. Another final contribution we provide is an explanation for a previous controversy about proteasomal degradation of phospho-R-Smad [9], [11], [20], [21]. Previous experiments inhibiting proteasomal degradation showed either strong effects [11], [20], [21] or no effects [9] on phospho-R-Smad levels. These seemingly contradictory trends were both mathematically consistent with our model, and the disparity could be explained by different durations of TGF-β exposure.
Smad signaling is enormously complex, as proven by a vast literature of previous work. We first considered negative regulatory effects from the published literature, for the purpose of selecting a set of effects relevant to our studies. R-Smad dynamics depend on the duration of TGF-β stimulation. When TGF-β is administered in excess (2 ng/ml) [44], [45] for 24 hrs, phospho-R-Smad peaks at about 1 hr and then decays for 24 hrs [9]. When TGF-β is administered for 30 min and then removed (by washing following by receptor inhibition with the compound SB-431542), phospho-R-Smad is eliminated within 4 hrs [9], [10]. Our first modeling studied the kinetics of the three negative regulatory effects selected from the literature review. We were curious whether they would have different kinetic implications for the system. Both short-exposure and long-exposure TGF-β treatment datasets (Figure 2) were utilized when building the models of negative regulation (Table 1, Table S2). The models were simulated to obtain the dynamics of their effects and to estimate their potential contributions to the down-regulation of phospho-R-Smad (0. 5-24 hr). Model 1, with R-Smad Dephosphorylation, was able to recapitulate the short-exposure TGF-β treatment experiment, as dephosphorylation is a fast process. This dephosphorylation model could turn off the signal once the stimulus was cut off (Figure 2A red curve), but it reached a steady state at about 1 hr and was not able to recapitulate the extended 24 hr decline of phospho-R-Smad in long-exposure TGF-β treatment (Figure 2A blue curve). Thus we describe R-Smad Dephosphorylation as a “fast-mode” effect. To explain the prolonged decline during long-exposure experiments, a complementary “slow-mode” might be provided by cumulative processes such as degradation. Model 2 combines Receptor Degradation and R-Smad Dephosphorylation. It succeeded in recapitulating the short-exposure TGF-β treatment very well, and it had moderately good agreement with the long-exposure dataset (Figure 2B). As a control, we modeled Receptor Degradation alone (Model 3), but it could not provide an early decline in the short-exposure experiment (Figure 2C). Thus, Receptor Degradation serves as a slow-mode effect as it was able to explain the gradual and protracted decline of phospho-R-Smad in the long-exposure experiment but not the steep decline of phospho-R-Smad in the short-exposure experiment. Another cumulative process of decline is P-Smad Degradation. A model with P-Smad Degradation alone (Model 4) achieved significant negative regulation for the long-exposure case (Figure 2D), because P-Smad Degradation would persist for many hours. However, Model 4 had difficulty explaining both the short-exposure and long-exposure datasets simultaneously. If P-Smad Degradation is strong, it could recapitulate the steep decline after short-exposures, and if it is weak effect, it could recapitulate the gradual decline after long-exposures. Since it cannot be both strong and weak, it cannot explain both behaviors. Note that previous experimental evidence showed that P-Smad Degradation is not responsible for fast-mode effects in short-exposure conditions [9]. Having simulated each of the three negative regulatory effects in isolation, we could conclude that no single negative regulatory effect was able to explain phospho-R-Smad dynamics. We infer that the experimentally observed levels of phospho-R-Smad arise from a combination of fast-mode and slow-mode effects (or from higher-order combinations of effects). Many models have omitted P-Smad Degradation from simulations [10], [12], [24], [25], perhaps because this effect was found to be insignificant in the experiments of Lin et al. [9]. Noting that the Lin experiments used short-exposure conditions, we asked whether P-Smad Degradation, a slow-mode effect, might have greater significance during the negative regulation induced by long-exposure treatments. Model 5 incorporated R-Smad Dephosphorylation, Receptor Degradation, and P-Smad Degradation (Figure 2E). P-Smad Degradation was significant in this model (Figure 2F-G) when its impact was measured after more than 1 hr of TGF-β treatment. We also fitted a variety of models to the short-exposure and long-exposure experiments. The cumulative difference in phospho-R-Smad between +MG132 and -MG132 was minor in the short-exposure experiment and significant in the long-exposure experiment (Figure 2H). As yet, we have no basis for knowing which type of slow-mode degradation would be most important in R-Smad signaling. We next tried to assess the relative impact of two slow-mode effects, Receptor Degradation and P-Smad Degradation, on the dynamics of phospho-R-Smad in long-exposure TGF-β treatment. The rate constant for Receptor Degradation and the rate constant for P-Smad Degradation were varied in silico (Figure 3A), showing that many ratios were equally good at fitting the observed dynamics. Several of the successful models exhibited a strong decline in T1R, the type I receptor (Figure 3B). Moreover, the degree of T1R decline was correlated with the rate of Receptor Degradation and the rate of P-Smad Degradation (Figure 3C). Thus, to quantify the relative contribution of Receptor Degradation and P-Smad Degradation in HaCaT cells, we measured T1R experimentally at 9 time points (from 15 min to 24 hr) after TGF-β stimulation (with n = 3 replications and significance determined by Student' s t-test). Surprisingly, there was no significant loss of T1R (type-I receptor) observed in experiments (Figure 3D–E), even at late time points. (As positive control, phospho-R-Smad time series concentrations were measured in Figure 4C). Previous work has already shown that T2R (type II receptor) shows no decrease after 2 ng/ml of TGF-β treatment in HaCaT cells [46]. Unchanged receptor levels indicate that Receptor Degradation is very weak in HaCaT cells. A weak role for Receptor Degradation has also been suggested by the experimental work of Clarke et al. [47]. Other forms of receptor inactivation or sequestration may occur without changing the total T1R concentration, but there is less published evidence for these possibilities (modeling analysis rejected these possibilities as well, in Text S2). Note that the set of models (Figure 3A) capable of explaining the dynamics of phospho-R-Smad decline all exhibited a negative correlation between the degree of Receptor Degradation and the degree of P-Smad Degradation (Figure 3F), suggesting that these two effects would be balanced alternatives. In light of our experimental finding that Receptor Degradation is a very weak effect, we next turned to P-Smad Degradation as the alternative slow-mode effect to explain the long-term decline of phospho-R-Smad. A model with R-Smad Dephosphorylation and P-Smad Degradation (Model 6, without Receptor Degradation) provided an excellent fit to both the long-exposure and short-exposure treatment data (Figure 4A). However, an unavoidable consequence of this model was dramatic decline of total R-Smad (Figure 4B). Previous experiments in HaCaT cells failed to observe a large fold-change of total R-Smad [9] but the amount of decline was not quantified. To clarify this potential conflict, we repeated the experimental measurement of total R-Smad levels after TGF-β treatment, using ELISA assays, a more quantitative method. Measurements of total R-Smad at 7 time points during 24 hrs of TGF-β treatment showed no significant decrease of total R-Smad (Figure 4B–C). There is an apparent conflict between the constant level of total R-Smad (observed experimentally) and the significant degradation of R-Smad induced by TGF-β (according to Model 6). Degradation might be more difficult to rule out if we consider TGF-β-stimulated degradation in combination with Endogenous Synthesis and Degradation of R-Smad. If endogenous R-Smad is synthesized in an unphosphorylated form, and targeted by Smurf2 for degradation only in its phosphorylated form, then can P-Smad Degradation explain the decline of phospho-R-Smad despite the constant levels of total R-Smad? We therefore expanded the model to include Endogenous Synthesis and Degradation of R-Smad (Model 7). However, Model 7 diverged strongly from the observed dynamics of phospho-R-Smad, when constrained to maintain a constant level of total R-Smad. To summarize these results, P-Smad Degradation can only affect the shape of the phospho-R-Smad curve if it is not balanced by synthesis, in which case it would cause an unrealistic decline in the total Smad levels. If P-Smad Degradation is balanced by Smad synthesis, then it can only affect the height but not the shape of the phospho-R-Smad curve. Therefore we can rule out strong P-Smad Degradation (not balanced by synthesis) as an explanation for the later decline in the phospho-R-Smad curve shape. We cannot rule out the presence of significant P-Smad Degradation accompanied by Smad synthesis. Hence, our model-driven experimental tests, sensitivity analysis (Text S3), and modeling analysis showed that P-Smad Degradation and Receptor Degradation were not sufficient to explain the 1–24 hr decline in phospho-R-Smad dynamics. We next sought some other negative regulatory effect that could help explain the peak and decline of phospho-R-Smad after a long exposure to TGF-β. After excluding the three well-accepted effects of Smad negative regulation, we then examined possible alternative influences at different steps along the Smad pathway, seeking quantitative consistency with the observed peak and decline of phospho-R-Smad. One scenario that could not be rejected on kinetic grounds was upregulation of PPM1A, the phosphatase targeting phospho-R-Smad. If PPM1A were to be upregulated by TGF-β signaling, this could help explain the decline of phospho-R-Smad after long exposure to TGF-β (Text S4). To test this possibility, we performed Western blots of the PPM1A protein after TGF-β treatment. HaCaT cells were treated with 2 ng/ml of TGF-β and measured after 0. 25,0. 5,1, 2,4, or 8 hr. We found that the intensity of the PPM1A western blot band increased 2. 4-fold after 1 hour of TGF-β treatment (p<0. 05, Figure 5A-B). To the best of our knowledge, this is the first study to report that TGF-β causes upregulation of the PPM1A phosphatase. The increased abundance of PPM1A after TGF-β stimulation could be due to some type of decreased degradation and/or increased production. To aid future studies in investigating how the upregulation occurs, we have constructed a hypothetical mechanism, PPM1A Stabilization, in which we speculate that PTEN may be involved. Model 8 includes PPM1A Stabilization plus all the mechanisms of Model 7 (R-Smad Dephosphorylation, P-Smad Degradation, and Endogenous Synthesis and Degradation of R-Smad). Text S5 provides a full specification of Model 8. In previous studies, Bu et al. found that PTEN can bind to PPM1A and protect it from degradation [28]. These studies of PPM1A stability occurred in fibroblasts, where TGF-β caused dissociation of PTEN and PPM1A, leading to downregulation, not upregulation of PPM1A. In other words, they found PTEN to be a negative regulator of Smad signaling, but in their fibroblasts, TGF-β decreased this negative effect causing self-perpetuation (positive feedback) rather than self-limitation (negative feedback) of the Smad signal. The binding of PTEN in response to TGF-β is known to differ between fibroblasts and HaCaT keratinocytes. Hjelmeland et al. found that in HaCaT cells, TGF-β stimulation caused formation of a PTEN-Smad complex [29], not dissociation of the PTEN complex [28]. They did not measure participation of PPM1A in that complex, but based on our analysis of the trends from [28] and [29], we propose a scaffolding role for phospho-R-Smad to promote association between PTEN and PPM1A in HaCaT cells. In other words, Model 8 speculates that TGF-β stimulation would induce PTEN association to stabilize PPM1A. This implies that there is some new or unknown mechanism upstream of PTEN, to explain why TGF-β signaling would promote PTEN-PPM1A association in one cell type and dissociation in another cell type. Model 8 assumes that PTEN and PPM1A would have a low on-rate for binding each other in HaCaT cells without phospho-R-Smad, but they would readily form a ternary complex in the presence of phospho-R-Smad. Thus, PPM1A would not be strongly stabilized in unstimulated HaCaT cells. After TGF-β stimulation, the phospho-R-Smad mediated association between PTEN and PPM1A would protect PPM1A from degradation and create negative feedback in the system. Note that Model 8 does not imply any alteration of total PTEN protein levels, merely the recruitment of PTEN by phospho-R-Smad into complexes with PPM1A. Simulations of Model 8 in Supplementary Figure S9 confirm that total PTEN levels could in theory remain constant (as observed in [29]) while levels of the PTEN-PPM1A complex could change over time. Simulations of Model 8 were consistent with all the observed dynamics for the impact of TGF-β on HaCaT cells. This model was sufficient to explain the complete dynamics of phospho-R-Smad after short or long exposures to TGF-β (Figure 5C), the dynamics of PPM1A (Figure 5B), and the unchanged levels of T1R and total R-Smad (Figure 5D-E). With the key experimental trends satisfied, we next tested Model 8 against another dataset, obtained from combination treatment with TGF-β and a chemical inhibitor MG132. Previous studies assessed P-Smad Degradation using MG132 to inhibit proteasomal degradation, but with conflicting conclusions: Massague et al. saw a strong impact, implying an important role for degradation [11], [21], while Lin et al. found negligible impact from MG132 [9]. Both protocols measured the long-term dynamics of phosphorylated Smad2, but the Lin protocol triggered phosphorylated Smad2 using a 30 min exposure to TGF-β, while the Massague protocol used a 6 h exposure. Simulations of Model 8 with MG132 inhibition of proteasomal degradation show that MG132 would have minimal impact on Smad signaling, when triggered by short exposure to TGF-β (Figure 5F). In surprising contrast, MG132 would have a strong impact on Smad signaling, when phospho-R-Smad is triggered by longer exposures to TGF-β (Figure 5G). Figure 5H compares the P-Smad2 Change calculated from Figure 5F (red curve) and Figure 5G (blue curve) with experimental data from Lin et al. (Figure 1C in [9]) (red dots) and Alarcon et al. (Figure 2G in [21]) (blue dots). The P-Smad2 Change was calculated as Eq. 1. (Eq. 1) Model 8 shows, mathematically, that the Lin observations and the Massague observations can be generated from the same system. Model 8 contains hypothetical mechanisms (e. g. , PPM1A Stabilization) and imperfect parameters (e. g. , reaction rate constants), but it suffices to prove that the seeming conflict between Lin et al. and Massague et al. is not necessarily a contradiction. In summary, the combination of several negative regulatory effects was consistent with, and sufficient to explain, the observed nuances of negative regulation and degradation in the Smad signaling system.
Several negative regulatory effects in the Smad signaling pathway have been identified and individually studied [9], [11], [13], [14], [17], [21], [32], [48]. We focused our modeling and experiments on these specific effects with published evidence. R-Smad Dephosphorylation by PPM1A is widely recognized to be a strong form of negative regulation, having significant fast-mode impact. However, the known slow-mode effects could only recapitulate phospho-R-Smad dynamics at the expense of very strong, cumulative degradation; as much as 90% decrease of T1R at 24 hr (Figure 3B), or 90% decrease in total R-Smad at 24 hr (Figure 4B). Our experimental measurements in HaCaT found that total T1R protein levels did not decline significantly (Figure 3B, 3D), nor did total R-Smad (Figure 4B). This contrasts with previous work in 293T and COS-1 cells [17], [19]. In [17], 293T cells were transfected with I-Smad which was able to induce significant receptor degradation. The significant degradation seen in [17] may be due to transfection [47] or may be due to cell line differences. Although most dynamic models of signal transduction represent an amalgam of findings from multiple cell lines, our model (and the previous models we rely on) are specific to the HaCaT cell line. Thus a discrepancy with [17] is not necessarily a flaw of our model. In light of our experimental measurement that TGF-β treatment does not cause any significant drop in total R-Smad levels, and the evidence showing no significant decline in type I or type II receptor levels, we conclude that degradation effects, if they occur in HaCaT, must be counterbalanced by endogenous synthesis. Model 7 simulated a balance of synthesis and degradation (Endogenous Synthesis and Degradation of R-Smad) such that phospho-R-Smad was degraded while unphosphorylated Smad was synthesized; this model was not able to induce the observed decline of phospho-R-Smad in long-exposure experiments. The first key contribution of our work was to conclude that degradation of R-Smad or T1R, with or without endogenous synthesis, is not sufficient to explain the slow-mode of Smad negative regulation in HaCaT cells. Degradation with synthesis remains a plausible effect of negative regulation, but it must occur alongside other effects. Figure 6 shows the relative contributions of different negative regulatory effects in our final model (Model 8): R-Smad Dephosphorylation was crucial for maintaining a limited level of phospho-R-Smad (compare red versus yellow curves); PPM1A Stabilization was capable of explaining the decline after the peak of phospho-R-Smad (compare yellow versus green curves); and P-Smad Degradation could further adjust the absolute level of phospho-R-Smad (compare green versus blue curves). The second key contribution of our work was the discovery of a novel feedback loop in which the PPM1A protein is significantly upregulated after TGF-β treatment. Feedback loops have crucial importance in dynamical systems because they create nonlinear responses and permit self-regulation (by converting a directed subgraph into a connected subgraph). In our study, the new feedback loop via PPM1A was significant enough to allow the model to finally explain the observed trends of phospho-R-Smad decline after TGF-β treatment (Figure 6, yellow versus green curves). Because PTEN is known to stabilize PPM1A against degradation [28], we built a model to illustrate hypothetical dynamics of PTEN-induced PPM1A sequestration, including delayed enzymatic activity for PPM1A. Note that in previous experiments, the influence of PTEN served as a positive feedback loop (PTEN-induced stabilization was inhibited by TGF-β [28]), not negative feedback. HaCaT cells are an accepted model system for understanding how epithelial cells respond to TGF-β; and it will be interesting for future work to test which cell types utilize PPM1A regulation for negative feedback. Model 8 shows that PPM1A Stabilization, with delayed nuclear import, was sufficient to reconcile the early upregulation of PPM1A total protein with later decline of phospho-R-Smad. Our theoretical model could be useful for the design of experiments to determine how the upregulation actually occurs. Future work should test whether PTEN stabilizes and/or sequesters PPM1A in HaCaT after TGF-β treatment, as illustrated in Model 8. Our model would recommend testing for PPM1A-PTEN binding at 30 min–1 hr to catch the peak interaction, but testing for increased PPM1A activity at 4 hr, significantly later than the upregulation. Careful examination of a broader set of previous work reveals some issues that appear to be discrepancies. The steepness of phospho-R-Smad decline in HaCaT appears to differ slightly between the experiments of Massague and colleagues in [11], [21] versus the experiments of Lin et al. [9], which are similar to our results (Figure 4C) and similar to the results of [12]. One possible explanation is a difference in the effective concentrations of TGF-β. TGF-β has a very short half-life, and the dissolving conditions, such as carrier protein concentration, can alter the effective concentration of TGF-β. Previous authors did not report how their TGF-β was dissolved, but we found that dissolving TGF-β without carrier protein led to a steeper decline of phospho-R-Smad, similar to Massague et al. [11], [21] (data not shown). We believe this slight discrepancy in slope is a technicality of the experiments and not fundamental to the pathway analysis. Recent work has shown the importance of TGF-β depletion as a determinant of Smad signaling kinetics, for cells treated with low doses of TGF-β (10pM and 25pM) [47]. Our work did not emphasize low-dose contexts, but our models are consistent with observed TGF-β depletion behaviors. Figure S6 (Text S6) shows simulations of our final model, Model 8 except with lower doses of TGF-β treatment. Smad signaling was indeed dominated by TGF-β scarcity. When the Smad system was externally limited by TGF-β availability, self-limiting mechanisms and negative regulatory effects were not apparent. Negative self-regulation of the Smad system was strongly apparent in treatments with 2 ng/ml (80pM) of TGF-β, which is the dose studied in most previous experimental and computational studies of Smad dynamics. After successfully predicting PPM1A upregulation and achieving recapitulation of the available datasets, our final contribution was to address an existing controversy about the role of proteasomal degradation in Smad signaling. We discovered that an apparent conflict about the role of degradation was in fact a mutually consistent set of trajectories that can both emerge from a single model. Degradation is intuitively understood to be a cumulative effect seen in long-term observations, but in this case the duration of observation was irrelevant, and the crucial variable for degradation was the duration of the TGF-β stimulus. MG132 (an inhibitor of proteasomal degradation) caused negligible change in pSmad2 levels (at 1,2, 4,6 hr), in a system triggered with 30 min exposure to TGF-β, but MG132 caused a significant change in pSmad2 levels (at 1,2, 4,6 hr), in a system triggered with long exposure to TGF-β. In other words, the importance of degradation in Smad signaling depended not on the time point at which pSmad2 was measured, but rather on the duration with which the Smad system had been induced. The consistency between the two experiments can be rationalized in retrospect because degradation depends on the area under the curve, which is large in systems with prolonged stimulus, and very small in systems with short stimulus. However, the consistency between Lin et al. and Massague et al. was not apparent prior to modeling, and mathematical inference of kinetic implications is dramatically different from the interpretations provided by the previous authors. Computational modeling of any biochemical pathway involves several caveats and approximations, particularly when the system is as complex as Smad signaling. For our modeling of the Smad system, many interaction partners and post-translational modifications have been neglected, and some highly complex processes have been described as two-species reactions with simple mass action kinetics. Few of the rate constants have been determined from direct experiments and therefore, many parameters have been estimated by optimizing the fit between the model and the available datasets. Despite these limitations, we believe mathematical modeling provides valuable insights. Our modeling provides a consistent, quantitative, and fine-grained integration of available information about the negative regulation of phospho-R-Smad, both from published literature and from our experiments. Our combination of modeling and experiments showed that previous negative regulatory effects such as Receptor Degradation have a minor effect, and led us to introduce a negative feedback loop with upregulation of PPM1A. Modeling can make additional predictions (e. g. , future experiments should test for peak perturbation of PPM1A binding and activity). Also, modeling has provided a new and non-obvious interpretation for the effects of MG132 treatment. When interpreting the biological meaning of observed kinetics, informal intuition can unwittingly lead to flawed conclusions. Our modeling of Smad signaling may in the future be useful to other researchers interpreting data, designing experiments, or strategizing therapeutic perturbations.
TGF-β signaling pathway regulates a variety of cellular responses, such as differentiation, migration and apoptosis. Phosphorylated R-Smad, the central signaling protein in this pathway, exhibits self-limiting behaviors: it not only decreases quickly after TGF-β is removed, but it also decreases slowly when TGF-β remains abundant. These two self-limiting behaviors are important to understand clearly because diseases such as cancer and fibrosis might benefit from treatments to decrease Smad signaling. Several negative regulatory effects have been reported previously, and we studied the dynamics of these effects with computational modeling. Analyzing the timing of negative regulation revealed that the three most widely accepted effects were not sufficient to explain the observed declines. After considering and excluding several alternative models, we arrived at a model in which TGF-β upregulated the phosphatase PPM1A. We tested for PPM1A upregulation in cell culture experiments. In addition, our model was able to explain why different durations of TGF-β exposure could cause seemingly opposite results about the importance of Smad degradation.
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The Self-Limiting Dynamics of TGF-β Signaling In Silico and In Vitro, with Negative Feedback through PPM1A Upregulation