7.2.1 Structure of the model
The model proposed in this chapter aims at describing the response in nuclear NF-κB activity to the separate or combined stimulation of the TNF receptors 1 and 2. The model is specifically developed for the Kym-1 cell type, which is a human rhabdomyosarcoma cell line grown in cell culture. Kym-1 cells express both TNF receptor 1 and 2, and thus constitute an ideal experimental system for the study of TNF receptor crosstalk effects.
The structure of the model has been derived from basic knowledge of relevant proteins which are involved in the signalling network, and from literature data on their interactions.
For the NF-κB pathway downstream of the receptor complexes, we rely mainly on previous modelling efforts. The structure of this part of the model is adapted from Lipniacki et al.
(2004). Other sources are the models described in Hoffmann et al. (2002), Lipniacki et al.
(2007), and Ashall et al. (2009). For the receptor complex formation, the construction of mathematical models is not as advanced as for the NF-κB pathway. The TNFR1 complex formation has been modelled by Schliemann et al. (2007), although focusing on different adaptor proteins than considered here. For the formation of the TNFR2 complex and its signalling, no previous mathematical models are known to the author.
The structure of the model developed in this chapter is coarsely depicted in Figure 7.1.
The model is organised into four modules: the TNF receptor complex formation, the activation of the I-κB kinase (IKK), the activation and nuclear translocation of NF-κB, and the NF-κB induced gene expression.
Upon ligand binding, the TNF receptors start to recruit adaptor proteins to form the relevant signalling complexes. The TNF receptor 1 first recruits TRADD (Ermolaeva et al., 2008; Micheau and Tschopp, 2003; Pobezinskaya et al., 2008), but for simplicity, this step is not explicitly included in the model. Rather, TRADD is assumed to bind instantly, or to be already associated to the TNFR1. In the next step, the TNFR1 recruits the adaptor proteins RIP1 and TRAF2. From available biological data, it is not clear whether these adaptor proteins can only bind sequentially, and, if so, what the sequence is, or whether RIP1 and TRAF2 can independently bind to the TNFR1 under in vivo conditions. In our model, we use the hypothesis that TRAF2 is recruited to the receptor complex only after RIP1, as proposed by Festjens et al. (2007). Concerning signal transduction from this complex, there is experimental evidence that signalling from TNF
7.2 Development of a model for the anti-apoptotic TNF network
TNFR1 TNFR2
TNF receptor module
TNFR1 complex
TNFR2 complex specific ligand 2 specific
ligand 1 TNF
RIP TRAF2
IKK
module IKK A20
NFκB module
NFκB IκBα
NFκB–IκBα
NFκB IκBα
NFκB–IκBα
gene expression module IκBα production
A20 production TRAF2 production
Figure 7.1: Coarse model structure of the TNF induced NF-κB pathway.
receptor 1 towards the NF-κB pathway requires both RIP1 and TRAF2 (or TRAF5 as a substitute) to be in the receptor complex (Micheau and Tschopp, 2003; Pobezinskaya et al., 2008; Wertz et al., 2004).
The situation is much more simple for the TNF receptor 2: the receptor directly recruits TRAF2, and the thus formed complex transmits the signal towards the NF-κB pathway (Bryde, 2004; Ye and Wu, 2000). A relevant additional effect is the ubiquitination and subsequent proteasomal degradation of TRAF2 at the TNF receptor 2 complex (Li et al., 2002; Wu et al., 2005). Such observations have not been made for TRAF2 when recruited to the TNF receptor 1 complex. The fact that both receptor complexes require TRAF2 for efficient signal transduction constitutes a crosstalk between the two receptor complexes, with potentially important effects on TNF signal transduction (Bryde, 2004).
The crucial mediator between the TNF receptor complexes and the NF-κB pathway is the I-κB kinase complex (IKK) which is activated at the TNF receptor complexes by TRAF2. Active IKK then phosphorylates I-κBα, which is subsequently degraded, thus liberating NF-κB to move to the nucleus (see Hayden and Ghosh, 2008, for a recent review). For the NF-κB, IKK and part of the gene expression modules, the species and reactions to be included in the model are adapted from the previous model developed by Lipniacki et al. (2004). Modifications are made in the transcription rates, where we assume a saturated rate expression, and in the activation of IKK, where we use the structure proposed more recently by Ashall et al. (2009). Whereas Lipniacki et al. (2004) explicitly consider intermediate complexes of IKK and its substrates, we use a quasi-steady state assumption for these complexes to arrive at a Michaelis-Menten type rate law for IKK mediated I-κBα degradation (Krishna et al., 2006). An important additional inhibitor of NF-κB is the protein A20, for which we consider an inhibition of the NF-κB pathway acting directly on the level of IKK (Mauro et al., 2006).
The complete lists of species, reactions and reaction rate expressions which are used in the model are given in the appendix, Sections B.1 and B.2.
7.2.2 Setting parameter values
After defining the species and reactions involved in the model, it remains to determine the model parameters, such as reaction rate constants or total concentrations of proteins subject to a conservation law. The complete list of nominal parameter values for the pro-posed TNF signal transduction model is given in the appendix, Section B.3, for reference.
For each parameter, we also indicate how the value was obtained.
A substantial part of the parameter values could be taken from literature data, mainly from previously published models of the NF-κB signalling pathway (Ashall et al., 2009;
Hoffmann et al., 2002; Lipniacki et al., 2004, 2007), or from models of the TNFR1 complex formation and signal transduction (Schliemann, 2006; Schliemann et al., 2007). Several parameters involved in the formation of the TNF receptor signalling complexes have been measured directly by Peter Scheurich and coworkers (Eissing, 2002; Grell et al., 1998;
Schliemann, 2006).
In support of the model construction described in this chapter, the degradation kinetics of TNFR2 and TRAF2 have been measured, and the corresponding parameters have directly been computed from the measurements under the assumption of a first order decay rate (Doszczak and Scheurich, unpublished data).
7.2 Development of a model for the anti-apoptotic TNF network
Yet, a significant number of parameters remain, for which neither previously deter-mined values are available in the literature, nor values could be deterdeter-mined directly from experimental measurements. Part of these parameters do not have a significant effect on the model’s trajectories (Sinini, 2008). For such parameters, we fix arbitrary values within a biologically reasonable order of magnitude. This is indicated by the comment
“assumed” in the list of parameter values (Section B.3).
The values for the remaining part of the parameters need to be determined by identifi-cation from dynamical measurements. To this end, experiments have been conducted by the group of Peter Scheurich at the Institute of Cell Biology and Immunology (University of Stuttgart). In these experiments, Kym-1 cells in culture were subjected to specific stimulation of either the TNFR1 or the TNFR2, and the resulting dynamics in protein concentrations were measured by Western blotting (Doszczak and Scheurich, unpublished data). In total, 14 parameter values for the model have to be determined from these measurements by parameter estimation methods.
Over the last decade, large efforts have been made in parameter estimation for biochemi-cal reaction networks from dynamibiochemi-cal measurements (Balsa-Canto et al., 2008; Chou et al., 2006; Feng and Rabitz, 2004; Polisetty et al., 2006; Raffard et al., 2008; Voss et al., 2004).
Yet significant challenges remain, which are typically due to non-linearity of the models, non-convexity and even non-continuity of the employed performance functions (Ljung, 2008; Radde, 2009). Constraints from the experimental side such as large measurement uncertainties, limited measurement frequency and the difficulty to measure absolute con-centration values on a cellular scale complicate matters further. Typical restrictions for parameter estimation algorithms are that they only find local optima (Balsa-Canto et al., 2008; Raffard et al., 2008; Voss et al., 2004), are tailored to specific classes of models (Chou et al., 2006; Polisetty et al., 2006), or have a high computational cost (Feng and Rabitz, 2004). For the purpose of this thesis, the choice of nominal parameter values is not critical. In fact, we found it sufficient to adjust parameter values manually while com-paring simulation results to measurement data visually. The relation between simulation results and measurement data which has been achieved with manual parameter tuning is shown in the next section.
7.2.3 Simulation results
The comparison between model simulation results and the experimental measurements is shown in Figure 7.2. Since the experiments provide only relative, not absolute, concentra-tion data, the values are scaled to fit with the simulaconcentra-tion results or, in the case of TRAF2, the experimentally determined initial condition. The comparison indicates a good qual-itative fit of the model results to the available experimental data. There is a significant difference between experimental data and simulation results for TNFR1 stimulation after about 70 minutes. However, this might be explained by the fact that Kym-1 cells quickly activate the apoptotic pathway after TNFR1 stimulation, which has not been included in our model.
For the TNFR2 stimulation, most of the data points are fitted very well. The export of NF-κB from the nucleus between 50 and 70 minutes after stimulation seems to be slower in the model than observed in experiments. However, the timing of this export corresponds to the rise of the cytosolic I-κBα amount, and the fit for the NF-κB data cannot be
0 100 200 300 400
Figure 7.2: Simulation results for the TNF network model and comparison to experimental data. Lines: simulation results; squares: experimental data points. Errorbars on squares indicate standard deviation of experimental data. For all data points, cell cultures have been stimulated with 10mlng TNF ligand. Experimental measures have been taken to assure specificity for either TNFR1 or TNFR2, as indicated below the individual figures.
7.3 Analysis of oscillatory behaviour