Modelling the Smad activation cycle

Smad Phosphorylation: Ligand binding induces autophosphorylation of the TGFβ receptor which in turn activates its receptor serine kinase activity towards cytosolic substrates such as Smad2/3 [48] . Receptor (de)phosphorylation occurs much faster than ligand binding and dissociation [48] , thus justifying a lumped reaction step in the model (T + R ↔ TR and TRe

→ Te + Re). Most experimental analyses suggested that endosomal but not cell-surface receptors catalyse Smad2/3 phosphorylation [310,317,318,319,320,321,322,323] . For simplicity, it was thus assumed in the model that only endosomal receptor-ligand complexes catalyse Smad2 phosphorylation. Recent studies revealed that TGFβ receptors can remain active on their way to lysosomal degradation, i.e., in the late endosome [324,325,326] .

Owing to these data, the phosphorylation reaction S2 → pS2 is modelled to be additively catalysed by early and late endosomal receptors (TR_e and TR_le) with the same efficiency. An endosomal accessory protein, SARA, is thought to be required for recruitment of Smad2 to the TGFβ receptor, and thus for receptor-mediated Smad2 phosphorylation [323,327] . In the model, SARA was assumed to recruit Smad2 to the membrane with high efficiency. This implies that Smad2 binding to the receptor will essentially behave like an association between to membrane proteins. It has been shown theoretically and experimentally that co-localisation of two proteins to the membrane drastically increases the apparent affinity of protein-protein interactions by a factor of ~1000 [328,329,330,331] . Low K_M values for receptor-mediated Smad2 phosphorylation were thus allowed in the model, as further described in Table F.2.

Smad Trimerisation: Receptor-mediated phosphorylation strongly favours Smad2/3 multimerisation, and most experimental studies indicated that trimers are the predominant multimeric R-Smad species [332,333,334,335,336] . Heterotrimerisation of two Smad2/3 molecules with one Smad4 molecule was shown to be more efficient than homotrimerisation of three Smad2/3 molecules [334,335] . More specifically, the heterotrimeric complex is slightly more stable than the homotrimer in vitro [334,335,336] , but this difference might be more pronounced in vivo [337] . Smad homo- and heterotrimerisation were modelled as reversible, single-step reactions (2 pS2 + S4 ↔ (pS2)₂S4 and 3 pS2 ↔ (pS2)₃), as the concentration of the Smad dimer intermediate was shown to be negligibly small in vitro [334]

. The model was implemented such that heterotrimers preferentially formed over homotrimers (see Table F.2). Additionally, it was assumed that nuclear trimers are formed with the same kinetics as their cytosolic counterparts, because experimental studies demonstrated efficient Smad heterotrimer formation in the nucleus [338] .

Smad Import: TGFβ stimulation induces translocation of Smad2/3 and Smad4 into the nucleus. Yet, the Smad proteins shuttle continuously between the nucleus and the cytoplasm even in unstimulated cells [339] . Recent studies indicated that the import rate of Smad2/3 is not affected by TGFβ stimulation [336,339,340] . Therefore, it was assumed in the model that all Smad2 monomer and trimer species are imported with the same import rate constant (S2

→ nS2 ; pS2 → npS2; (pS2)2S4 → (npS2)2nS4 and (pS2)₃ → (npS2)3). In contrast, the import of monomeric Smad4 was modelled to occur with a different rate constant (S4 → nS4). Thus, the model reflects the experimentally established fact that the nuclear import of Smad heterotrimers is mediated by Smad2, and that it occurs independently of Smad4 [336]

.

Table 6.1 Data sets and error estimation. 1 ng/ml TGFβ ligand removal at 50 min

1 ng/ml TGFβ (SnoN siRNA) 1 ng/ml TGFβ (mSnoN mutant)

F 1 ng/ml TGFβ ligand removal at 50 min

1 ng/ml TGFβ (SnoN siRNA) 1 ng/ml TGFβ (mSnoN mutant)

F 1 ng/ml TGFβ ligand removal at 50 min

1 ng/ml TGFβ (SnoN siRNA) 1 ng/ml TGFβ (mSnoN mutant)

F 1 ng/ml TGFβ ligand removal at 50 min

1 ng/ml TGFβ (SnoN siRNA) 1 ng/ml TGFβ (mSnoN mutant)

P Of the 46 listed data sets 27 were used for fitting (F) and 13 for testing of model predictions (P). Six redundant standard stimulation data sets (grey) were used to introduce a common scaling for results from independent experiments. The relative concentrations of total Smad2, phosphorylated Smad2, total SnoN, as well as Smad4 and SnoN co-immunoprecipitated with Smad2/3 were measured by immunoblotting. The amount of processed TGFβ in the medium was determined by ELISA. Errors were either calculated from a cubic spline (Spline), based on a spline-derived linear error model (Model), estimated employing typical errors Eabs and Erel (Estimated) or calculated as the standard deviation of three independent measurements (StdDev).

The definition of the observables in the model (according to the differential equations given in Appendix F.1) is given in the left column. In order to transform the simulated time courses into relative units, a common scaling factor was introduced for each observable, that is, each box corresponds to one scaling factor (Section 6.5.2).

Smad Export: TGFβ-induced Smad2/3 translocation into the nucleus is thought to be due to a decrease in the overall Smad2/3 export rate upon stimulation [339] . FRAP measurements indicated that this decrease might be due to selective immobilisation and thus retention of nuclear Smad trimers, e.g., by to binding to DNA [339] . Smad2/3 export has been reported to occur via Smad2/3 binding to Exportin 4, which functions as a nuclear export receptor [341] . Exportin 4 preferentially binds to unphosphorylated Smad2/3 [341] , and accordingly phosphorylated Smad2/3 is exported with low efficiency [340,341] . Thus, it appears that two independent mechanisms contribute to stimulus-induced nuclear translocation of Smad2: (i) retention of Smad trimers; (ii) inefficient export of phospho-Smad2. In the model, it was therefore assumed that only unphosphorylated Smad2/3 monomers can be exported from the nucleus (nS2 → S2), and an export reaction for monomeric Smad4 was also included (nS4 → S4). Thus, the model is in accordance with experimental studies which indicated that Smad4 can only be exported in monomeric form, but not when assembled in Smad trimers [336] .

Smad Dephosphorylation: Smad2/3 dephosphorylation in the nucleus controls the amount of nuclear Smad trimers, and is therefore an important mechanism to control the magnitude and duration of Smad-mediated gene expression [342] . It is thought that the phosphorylation sites in Smad2/3 constitute the binding interface in Smad trimers [333] . In the model that Smad trimers can therefore not be attacked by phosphatases, and dephosphorylation occurs only for monomeric nuclear phospho-Smad2/3 (npS2 → nS2). Initial numerical analyses indicated that cytoplasmic Smad2/3 dephosphorylation is not required to describe experimental data obtained in primary mouse hepatocytes, so that this reaction was skipped from the model.

Protein Synthesis and Degradation: Smad2 and Smad4 protein levels are typically not affected by TGFβ treatment [338,343,344] . In accordance with these data and with the microarray results show in Fig. 6.2, it was therefore assumed in the model that Smad2 and Smad4 are continuously synthesised (→ S2 and → S4). Degradation of all Smad species was also taken into account (S2→ ; S4→ ; nS2→ ; nS4→ ; pS2→ ; npS2→ ; (pS2)2S4 → ; (npS2)₂nS4 → ; (pS2)₃ → ; (npS2)₃ → ). For simplicity, it was assumed that the stability of monomeric Smad4 does not differ in the nucleus and in the cytosol, so that the same rate constant for the corresponding reactions (S4→ ; nS4→). Experimental studies indicate that Smad2 stability in the nucleus and in the cytosol may be different [345,346,347] and that stimulus-induced proteasomal degradation of Smad2 might occur in the nucleus at least in some cells [345] . Different degradation rate constants were therefore assumed for nuclear and cytosolic Smad2 monomers, respectively (S2 → ; nS2 → ; pS2 → ; npS2 →).

Experimental evidence suggests that Smad2 recruits ubiquitin ligases to Smad2/Smad4 heterotrimers [348] . This suggests that Smad2 plays a dominant role for the degradation of Smad heterotrimers. In the model, it was thus assumed that all Smad trimer species are degraded with the same rate constant as their monomeric Smad2 counterparts ((pS2)₂S4 →

; (npS2)2nS4; (pS2)3 →; (npS2)3 →).