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* Computational Biology & Biological Physics, Department of Theoretical Physics, Lund University, Lund, Sweden; and Molecular Cell Biology, Leiden University Medical Center, Leiden, The Netherlands
Correspondence: Address reprint requests to C. Peterson, E-mail: carsen{at}thep.lu.se.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONS AND OUTLOOKAPPENDIXSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONS AND OUTLOOKAPPENDIXSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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–5)). It is used to simulate the kinetics of large signaling networks, where one cannot only rely on biological intuition. In such studies, the aim is to identify and shed light on the role of key components and modules. Furthermore, such approaches allow for predicting quantities not yet measured. Rate equation modeling involves three major steps:
A novel tool in this context is developed to deal with the optimization of parameters, simulated tempering (ST), which has previously been used to map out thermodynamical properties of protein-folding models (6,7). As with any other Monte Carlo method, ST naturally provides ensembles of solutions rather than single ones, subject to analysis by standard clustering techniques.
In this article, we apply the rate equation methodology to the Transforming Growth Factor ß (TGF-ß) pathway in endothelial cells. The members of the TGF-ß superfamily are responsible for many different biological functions, including proliferation, differentiation, apoptosis, embryonic development, and wound healing. Perturbations in the TGF-ß pathway have been detected in several human diseases, most notably in many forms of cancer, and in fibrotic diseases of the liver, the kidney, and the lung (8). This pathway is not too large for modeling, since there are a sufficient number of measurements available to infer the value of the parameters available. Neither is it small enough to use visual inspection or a simple ON/OFF language as means to draw conclusions about its dynamics and function. We compare the models both to existing data (9,10) and to novel measurements first presented here. The experiments consist of kinetic (time-course) measurements after TGF-ß stimulation under different conditions: untreated cells and three cases in which different components of the pathway have been perturbed. Two of the experiments are used to fit the model parameters and the other two are left as "blind test" experiments. In addition, we predict the response of the system when varying the ligand dosage. Thus, we develop a predictive model that is tested against existing data. Furthermore, we make testable predictions for further experiments. We also identify, among other things, a feedback loop (Smad7) as important for explaining all data sets used and for the stability of the model.
To our knowledge, this is the first time the TGF-ß pathway including regulatory aspects is approached with dynamical models. Recently, Vilar et al. (5) presented a detailed receptor model for TGF-ß signaling, and we will discuss how this model relates to our simplified receptor description.
The TGF-ß pathway in endothelial cells
The TGF-ß signaling pathway in endothelial cells (see Fig. 1 for a simplified layout) is triggered by the TGF-ß protein, which acts as a ligand, by binding to and activating a heteromeric complex of type I and type II serine/threonine kinase receptors. The type I receptor acts downstream of the type II receptor and the signal is propagated inside the cell as the activated receptor complex is internalized and binds to and phosphorylates a protein of the Smad family, called receptor-regulated Smads or R-Smads (11–13). The R-Smads include Smad1, Smad2, Smad3, Smad5, and Smad8. The phosphorylated R-Smads can form complexes with Smad4, also referred to as Co-Smad (11,12). These complexes move into the nucleus where they regulate the transcription of target genes. There is also an inhibitory effect generated by the inhibitory-Smads (I-Smads), Smad6, and Smad7 (11,12). The I-Smads negatively regulate the TGF-ß signaling pathway by binding to the receptors and compete with R-Smads for receptor interaction, by recruiting ubiquitin ligase to activated receptor complexes and thereby target the receptor for proteasomal degradation or by recruiting phosphatases (PP1-
) that inactivate the type I receptor by dephosphorylation (12–14).
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The two receptor proteins ALK1 and ALK5 give rise to two distinct pathways, which in turn induce opposite cellular functions. The TGF-ß/ALK5 pathway induces the phosphorylation of Smad2 and Smad3 whereas the TGF-ß/ALK1 pathway is responsible for the phosphorylation of Smad1 and Smad5. Moreover, ALK5 inhibits migration and proliferation while ALK1 stimulates these processes (9).
The phosphorylated R-Smads also display different behaviors in endothelial cells. It has been shown in Valdimarsdottir et al. (10) that the negative regulation of Smad1/5 is dependent on some newly synthesized protein and that Smad7 is induced by TGF-ß/ALK1 signaling but unaffected by the TGF-ß/ALK5 signaling. An interpretation of this would be that in endothelial cells, TGF-ß induced activated Smad1/5, together with Smad4, activates the production of Smad7. The effect of Smad7 on the two pathways is also different. It has been shown to inactivate the ligand-bound ALK1 receptor. It can target the activated receptor for an ubiquitin-ligase-dependent degradation (14,16). Smad7 can also recruit a phosphatase (PP1-) to the activated ALK1 receptor and thus inhibiting further phosphorylation of Smad1/5 (10). It has been shown that only high levels of Smad7 have an inhibitory effect on phosphorylated Smad2 (10). This leads to the conclusion that Smad7 negatively regulates the phosphorylation of both Smad1/5 and Smad2 but the strength of the latter interaction is much weaker.
The putative TGF-ß-induced negative feedback from Smad7 is an interesting aspect of the pathway. What is its purpose? If it is merely to shut off the ALK1 pathway, could this not be controlled by simpler means, such as in the form of creation and degradation? These are two main questions investigated in our computational analysis of the pathway.
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MATERIALS AND METHODS |
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). ). ). ). There are many possible ways to fit the model to experimental data, many of which display nonbiological behavior. To reduce the number of possible solutions, we fit to more than one set of experimental data. For detailed studies, we use Experiments I, II, and V for this calibration, whereas the others are used as blind-test experiments. In this way, the predictive power of our approach is tested. We also permute the experiments used for calibration to investigate the effects of such alterations.
Details of new measurements
Kinetics of TGF-ß3 induced Smad2 phosphorylation versus TGF-ß3 induced Smad1/5 phosphorylation
Mouse embryonic endothelial cells were stimulated with 1 ng/ml TGF-ß3 for different time points before lysis, fractionated by 6% SDS-PAGE and blotted. As a positive control, 293-cell lysate transfected with either Smad2/constitutively active ALK5 (PS2) or Smad1/constitutively active ALK1 (PS1) was used. The filters were incubated with phospho-Smad2 or phospho-Smad1 antibodies; detection was performed by enhanced chemoluminescence.
Ligands and cells and Western blot analysis
Recombinant TGF-ß3 was obtained from K. Iwata (OSI Pharmaceuticals, Melville, NY). All assays were performed with both ligands with essentially the same results. Recombinant BMP6 was a gift from Dr. K. Sampath (Curis, Cambridge, MA). Mouse embryonic endothelial cells were cultured and Western blot analysis was performed as described in Goumans et al. (9) and shown in Fig. 4 C below.
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). This detailed model is capable of describing different kinds of receptor responses to extracellular ligand concentrations depending on the situation at hand. We demonstrate that, regardless of the simplifications, our receptor model behaves strikingly similarly to the more complicated model of Vilar et al. (5), at least as long as only one ligand of the TGF-ß superfamily is present (see Supplementary Material for details). An explanation for this similarity is that, although our simplistic receptor model has far fewer parameters, it does include variants of the parameters pinpointed as the most important ones by Vilar et al. (5), which are determining the ratio of degradation of the unbound compared to the activated receptor.
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Feedback inhibition
As described above, Smad7 has an inhibitory effect on the signal. This is modeled by recruitment of the phosphatase PA (PB) to the activated ALK1 (ALK5), which leads to an inactivation of the receptor. Since an ubiquitin ligase-dependent degradation of the activated receptor leads to a similar inactivation behavior, we do not account for this process explicitly in the model.
Formalism
The reactions in Table 1 are implemented with standard rate equations using deterministic ordinary differential equations (Table 2). This assumes an ample amount of molecules involved and not to rare events. These conditions are very likely satisfied in the TGF-ß case. For all reactions we use mass action or Michaelis-Menten enzyme kinetics. The complete set of equations is given in Table 2. As an example, the equation for Smad1 concentration is given by
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for the production and degradation terms where r and l correspond to the production rate and the equilibrium level for the production/degradation terms, respectively. These equilibrium levels are also used as initial concentrations in the simulations.
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Solving the system of ordinary differential equations
The efficiency of the differential equation solver is extremely important since this is where most computational time is spent, in particular since the equations are often stiff. We use a procedure that adaptively switches between two methods to minimize the computational load:
) and initial parameter values and other details in this procedure can be found in the Appendix.
Calibration
In the optimization procedure, we estimate the parameters of the model by fitting to experimental data. After each solution to the ODEs in the iterative process, the K parameters p = (p0,...,pK–1) are adjusted such that the model should more accurately describe the experimental data. The latter consist of N discrete time points 
for each experiment. As error measure, the quadratic difference is used,
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denote model points and experimental points, respectively, and the index i denotes the different molecules (M in total). We use two experiments in the optimization procedure, and the sum of the two R values is used as error measure. To find good approximate solutions to global minima for Eq. 2, one can use Monte Carlo methods like simulated annealing (18). Here, we employ a related but more powerful method, simulated tempering (ST) (6,7), where the fictitious temperature is a dynamic variable, and the system is always kept at equilibrium for the different temperatures. Solutions are obtained by "quenching" from the lowest temperature to T = 0 corresponding to a local search. The underlying idea is to scan sizable parts of the solution space at different high temperatures and regularly visit low temperature solutions. In a sense, this optimization method corresponds to simulated annealing with multiple random starts and it yields ensembles of solutions rather than single ones. The details of the ST implementation are found in the Appendix. To further restrict the behavior of solutions included in the analysis, we select solutions from the optimization step to correctly describe the dosage Experiment V. We run the model for different dosages of TGF-ß and calculate a measure similar to R (see Appendix for details). Finally, a small subset of these solutions is removed based on an overfitted behavior. (Note that a small number—four—of the solutions display a high-order behavior in the simulations. Although these solutions do get a good R-value, the behavior does not fit the experiments well if the concentration levels are assumed to interpolate smoothly between the measurements. These solutions are removed by inspection, but could have been removed by, e.g., using a criteria of not allowing for multiple peaks. If these solutions are included in the analysis, they cluster with the group not using the feedback. The group behavior is not altered significantly, but the sensitivity and the variation in the predictions are slightly increased.)
Solution properties
To investigate properties and interior structures of the solution space, we use three different methods:
Robustness
A common method used to analyze the robustness of a system is to use the derivatives of the molecule concentrations, xi(t, p), with respect to the different parameters, p, as a direct measurement of the sensitivity of the system (20). We define a sensitivity vector according to
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONS AND OUTLOOKAPPENDIXSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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200 solutions. As can be seen from Fig. 4, these solutions fit both experiments well. Hence the parameterization form is appropriate and the optimization method efficient. Next, we select for those solutions that at the same time successfully describe the saturated behavior at different TGF-ß dosages (Experiment V). An ensemble of 38 solutions pass this filtering step (see Fig. 5), from which four are removed based on an overfitting criteria (see end of Calibration, above). The remaining 34 solutions are used for further investigations.
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Prediction
The solutions that were clustered were chosen to accurately predict the dosage experiment (Fig. 5). To further analyze the predictive power of the two defined ensembles of solutions we have performed two blind-test experiments: Cells treated with the proteasome inhibitor MG-132 (III) and phosphatase inhibitor orthovanadate (IV), respectively. In Fig. 7, the model predictions from group 1 and group 2 are shown and compared with experiments, again for levels of PSmad1 and PSmad2. As can be seen, the PSmad2 levels are not affected significantly in either of the perturbed systems as compared to the control experiment (Fig. 4 A). This behavior is accurately predicted by both groups of solutions. In the MG-132 experiment (see Fig. 7 A), the PSmad1 signal still appears transient although the peak is broadened in time. Both groups of solutions predict a transient PSmad1 signal very similar to the behavior of the control experiment in this case. This lack of broadening of the peak for all solutions is discussed in more detail below, where we do optimization on control and MG-132.
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It should be noted that these experiments are quite crude and may affect the cells in ways not feasible to include in our model, which is restricted to the molecules directly involved in the TGF-ß pathway. A much more direct experiment for model prediction would be to perturb a single specific molecule included in the model, e.g., silencing Smad7 by an siRNA knockdown. The predicted PSmad1 and PSmad2 behaviors for the two groups when Smad7 is silenced are shown in Fig. 8. This is particularly interesting since the two solution groups exhibit very different behaviors. Again, the unchanged PSmad1 behavior of group 1 shows that these solutions do not need the Smad7 feedback to achieve a transient signal. The prediction for the feedback model is dependent on the assumption that Smad7 is the I-Smad active in endothelial cells, which is based on experiments. Smad6 could potentially also be active although there is no data for Smad6 behavior in endothelial cells. In other cell types, Smad6 has been shown to be more moderately and transiently induced by TGF-ß compared to Smad7 (21,22). A fair assumption would be that if Smad6 is induced in endothelial cells its behavior would resemble the Smad7 behavior, which would lead to similar behavior for a model including Smad6 in all previous experiments but not for the Smad7 knockdown experiment. Instead, the effect of Smad7 knockdown would be less pronounced in such a feedback model.
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for the solutions in the two groups gives a p-value <10–6. The largest difference is found in the parameters governing the production and degradation of Smad1 and Smad4 (parameters p2–p5). This indicates that group 1 uses Smad1 and Smad4 production and degradation to achieve the transient PSmad1 signal instead of using the negative feedback of Smad7. It is indeed very interesting that the transient signal can be achieved by a pathway with fewer molecular players, but it appears that the drawback for the cells would be that the levels and production/degradation rates for the Smad1 and Smad4 need to be tightly regulated to achieve a robust signal behavior. In contrast to this, the group that uses the Smad7 feedback shows a low sensitivity in respect to Smad4 levels (p4,p5), and more or less no sensitivity at all to Smad1 levels (p2,p3). This latter fact, and the lack of sensitivity toward changes of the Michaelis-Menten constant in the phosphorylation step (p16), indicates that the Smad1 levels are saturated. A more detailed look at the parameter values and Smad1 levels reveals that all solutions in group 2 indeed have saturated levels of Smad1 (data not shown), which hence can be regarded as a prediction of the model using Smad7 feedback.
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)). The most sensitive parameters in group 2 are p1, p9, p15, p17, p22, and p24, and group 1 is about equally sensitive to these parameters. These parameters govern the initial ALK1 and ALK5 levels (p1, p9), as well as the rates of phosphorylation and dephosphorylation of Smad1 (p15, p17) and Smad2 (p22, p24). The early PSmad1 and PSmad2 kinetics and (at least partly) the entire PSmad signal are also dependent on these parameters. Hence, it is expected that the fitting to our kinetic PSmad1 and PSmad2 data is sensitive to these parameters. A final note is that although the ALK1 and ALK5 levels are important, the production and degradation rates are not (p0, p9). A more detailed look at the parameter levels show that these rates are low (data not shown), and it appears that it is the initial values that are important for the model to explain data.
Permuting the experiments for the calibration
To further analyze the model behavior we also permuted the experiments used for calibration. We used combinations including the control experiment in the calibration part since this is the only experiment where all the parameters are present. Also, here we applied the dose experiment as a filtering step after optimization. The two additional calibration sets used were optimization on control (I) and MG-132 (III), and on control (I) and orthovanadate (IV) experiments. The new parameter sets are presented in a PCA-plot in Fig. 10 together with the previously defined parameter sets.
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In the case of optimizing against the control and MG-132, the optimization procedure works less efficiently. Among the solutions provided by the algorithm, only a very few resulted in R < 0.01 and among those, none passed the filtering step against the dose experiment (see Appendix for details). The parameter sets from this case provided in Fig. 10 are solutions with R < 0.015, which pass the dose experiment filter. These solutions show an average behavior for the PSmad1 lying in between the experimental curves for control and MG-132, and with very small change in behavior when protein degradation is removed (see Supplementary Material, Fig. S4). None of the parameter sets use the Smad7 feedback, and therefore these provide a poor prediction of the orthovanadate experiment, while the predictive power is small for the cyclohexamide experiment since the behavior is very spread out. An interesting note is that this apparent conflict for explaining the MG-132 together with the other experiments can be used to direct improvements for the model. This is illustrated by a slight adjustment of the model perturbation for the MG-132 experiment where a decreased inactivation of the activated receptors is included (simulating reduced ubiquitin-dependent degradation), which leads to an improved behavior (see Supplementary Material, Fig. S5).
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CONCLUSIONS AND OUTLOOK |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONS AND OUTLOOKAPPENDIXSUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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In our robustness analysis we have investigated how the dynamical levels of different PSmads change for different parameter perturbations. The PSmads represent the signal through the pathway, but perhaps a more biologically relevant measure is the robustness in cell response. Hence, in the future one should augment the PSmad concentration measurements with downstream gene expression data and perform an integrated analysis. In this context, one should also include the effects from cross talk with neighboring pathways that are part of the TGF-ß family.
Very recently, a detailed model for receptor dynamics was introduced in the context of the TGF-ß pathway (5). It does not target endothelial cells specifically, but presents a detailed study of receptor dynamics including internalization and a specific inactivation of the ligand-bound receptor complex by degradation. This model is sufficient to explain a transient signal for PSmad2 after sustained TGF-ß stimulation. To relate this to our more simplistic receptor model, not explicitly including receptor recycling, we showed that our receptor model has as versatile activation pattern when a single ligand is presented to the receptor. The behavior of PSmad1 in endothelial cells when treated with cyclohexamide is to extend the signal, while the same treatment in HaCaT cells has been shown to shorten the PSmad2 signal (23). Although the detailed receptor model predicts a shortened activation at cyclohexamide treatment (see Supplementary Material, Fig. S1) in full agreement with the PSmad2 data, our full pathway model can indeed explain the PSmad1 behavior in cyclohexamide-treated endothelial cells.
From the behavior of our different solution groups, we argue for a model where there exists a feedback from TGF-ß induced Smad7 to repress the PSmad1 activation. This is based on indications from several experiments, which are all reproduced by the feedback model. Needless to say, a more distinct test of this model would be to perform a dedicated knockdown experiment for Smad7, which is currently in progress in siRNA experiments targeting Smad7. In this context, the importance of Smad6 in endothelial cells also needs to be investigated.
Our approach is not restricted to systems where all parameter values can be experimentally estimated. Rather, it allows for several solutions to solve a problem, and can account for similarities in behavior of highly conserved modules such as the TGF-ß pathway, although quantitative details differ. In this study we are confined to experimental data which has not been calibrated to units of concentration. This lack of knowledge propagates to our parameters. Also, the measurements are restricted to a few components, and we have therefore chosen a simplistic description of some of the reactions. Hence, we have focused on relevant biological behavior of the measured molecules for different conditions and not attempted to evaluate parameter values with respect to biologically reasonable ones, which would have been dependent on further assumptions. Additional experiments, which provide quantitative estimates of parameters and concentration levels, are important and will constrain the solution space for the models. On the other hand, we demonstrate that the models, can pinpoint experiments that will provide maximal information given the current knowledge, and the combination of experiments and modeling provides an effective methodology for an increased understanding of highly complex biological networks.
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APPENDIX |
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Solving the systems of ODEs
In Table 2 we show the system of ODEs used in our calculations, in which the following assumptions are made in the calibration process:
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 | (A2) |
).
Parameter estimation
For generating ensembles of solutions we use simulated tempering, where configurations are generated for different fictitious temperatures Tj and the system is allowed to move between the different Tj-values. In other words, at a given Monte Carlo step, one updates the system by swapping configurations of the systems, or alternatively trading two temperatures. The method amounts to simulating the joint probability distribution
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). The algorithm parameters gj govern the weights pj of the different temperatures, Tj. The latter are chosen according to
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R
j and put g20 = 0 and gj–1 = gj – Rj(1/Tj–1 – 1/Tj). In the next step, we perform longer simulations to obtain good estimates of the weights pj; the uniform distribution is then obtained by replacing gj with gj + ln pj (7).
The parameters are updated one at a time with pi 
rpi, where r is a multiplicative factor (r = 1.1 is used) and in 50% of the cases we set 
. At T = Tmin, r is allowed to vary freely in the range r 
[1:2] individually for each parameter, to keep the acceptance ratio above 50%. Updates are accepted according to Eq. A3. For each K number of attempted parameter updates, K being the number of parameters, we attempt one update to an adjacent temperature Tj±1 with a probability also governed by Eq. A3.
The performance of the algorithm is displayed in the table below showing the number of simulations it takes on average to find a minimum (middle panel) and the percentage of these minima having R < 0.01 (right panel) for each of the three sample permutations. These results can be compared with for example (24) where different optimization algorithms including simulated annealing are compared. The poor performance on the control+MG-132 set is discussed in the text and in the Supplementary Material (see Table 3).
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denotes the concentration of molecule i at time t given the parameters p* and an initial concentration of TGF-ß of C ng/ml. For the cutoff value 
we found = 0.05 to be appropriate.
Implementation
The calibration framework as well as the robustness analysis are implemented in C++. For the two clustering methods, K-means and hierarchical clustering, and for the PCA, we used MatLab implementations (The MathWorks, Natick, MA) corresponding to the MatLab functions dendrogram, kmeans, and princomp, respectively.
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ACKNOWLEDGEMENTS |
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This work was in part supported by the Swedish Foundation for Strategic Research through a "Senior Individual Grant" (C.P.), by the Knut and Alice Wallenberg Foundation through Swegene and the Swedish Research Council (H.J.), and by the European Community project "Angiotargeting Integrated Project" No. 504743 and the Dutch Cancer Society with grant No. NKI 2005-3371 (to P.t.D.).
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FOOTNOTES |
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Submitted on January 3, 2006; accepted for publication September 8, 2006.
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, and the x symbols refer to the experimental data points after 45 min for 0.025, 1.0, and 5.0 ng/ml, respectively. The time series show the corresponding predictions from the ensemble with standard deviations.
