Mathematical modelling of the role of GADD45β in the pathogenesis of multiple myeloma

Multiple myeloma (MM) is an incurable disease with relatively high morbidity and mortality rates. Great efforts were made to develop nuclear factor-kappa B (NF-κB)-targeted therapies against MM disease. However, these treatments influence MM cells as well as normal cells, inevitably causing serious side effects. Further research showed that NF-κB signalling promotes the survival of MM cells by interacting with JNK signalling through growth arrest and DNA damage-inducible beta (GADD45β), the downstream module of NF-κB signalling. The GADD45β-targeted intervention was suggested to be an effective and MM cell-specific treatment. However, the underlying mechanism through which GADD45β promotes the survival of MM cells is usually ignored in the previous models. A mathematical model of MM is built in this paper to investigate how NF-κB signalling acts along with JNK signalling through GADD45β and MKK7 to promote the survival of MM cells. The model cannot only mimic the variations in bone cells, the bone volume and MM cells with time, but it can also examine how the NF-κB pathway acts with the JNK pathway to promote the development of MM cells. In addition, the model also investigates the efficacies of GADD45β- and NF-κB-targeted treatments, suggesting that GADD45β-targeted therapy is more effective but has no apparent side effects. The simulation results match the experimental observations. It is anticipated that this model could be employed as a useful tool to initially investigate and even explore potential therapies involving the NF-κB and JNK pathways in the future.

2. Model development 2.1. Basic structure of the model Figure 1 describes the schematic diagram of the constructed mathematical model in the paper. It demonstrates the mechanisms of the NF-κB and JNK pathways and shows how the interaction between these two pathways promotes the survival of MM cells. It should be noted that the coupling between MM cells and bone cells (osteoclastic and osteoblastic lineages) is not included in figure 1, but it is included in the electronic supplementary material, Figure A1 in Appendix A for convenience, since these cells have been previously studied (e.g. [23]). Figure 1 is made up of two parts: part A and part B, which involve the NF-κB and JNK pathways, respectively.
Part A shows how the NF-κB pathway is activated and how it then upregulates anti-apoptotic genes in MM cells. In the unstimulated state, NF-κB heterodimers are always kept in the IκBα|NF-κB complex in the cytoplasm. Upon the stimulation of TNF, IKK is transformed from its neutral state (denoted as IKKn) into its active state (denoted as IKKa). IKKa can phosphorylate and ubiquitinate IκBα and then release free NF-κB heterodimers from the IκBα|NF-κB complex. The free NF-κB heterodimers enter the nucleus and regulate the transcription of IκBα, A20 and GADD45β [24]. In addition to IκBα, A20 also serves as an inhibitor of the NF-κB pathway, promoting the transformation of IKKa into its unactivated IKKi form [25]. As a downstream factor of NF-κB signalling, the newly produced GADD45β is expressed at a high level in the MM cells and interacts with MKK7 in the JNK pathway, promoting the survival of MM cells [4,26,27] (detailed information is included in part B). It should be noted that IκBβ, IκBγ and IκBε can also bind to NF-κB heterodimers and inhibit their nuclear localization signals [28]. Only IκBα is considered in our model, as NF-κB heterodimers are primarily sequestered by IκBα.
Part B describes how the overexpression of GADD45β acts with MKK7 and dysregulates the JNK pathway to facilitate the survival of MM cells. There are three cascades, JNK (a member of the mitogen-activated protein kinase (MAPK) family), MKK7 (a member of the MAPK kinase family) and MAPK kinase kinase (MAPKKK) in the JNK pathway [8,29,30]. The activated kinase at the upper level phosphorylates the kinase at the next level down the cascade. TNF stimulation promotes the phosphorylation of MAPKKK mediated by reactive oxygen species (ROS) [26,31,32]. MAPKKK_P royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 denotes phosphorylated MAPKKK. The full activation of both MKK7 and JNK requires the phosphorylation of two conserved sites [33]. MAPKKK_P promotes the single phosphorylation of MKK7 (denoted as MKK7_P) at first and then doubly phosphorylated (denoted as MKK7_PP). Similarly, MKK7_PP then facilitates the single and double phosphorylation of JNK (denoted as JNK_P and JNK_PP, respectively). JNK_PP is able to promote the apoptosis of MM cells by up-regulating pro-apoptotic genes [15]. The GADD45β produced during NF-κB signalling binds to MKK7 and inhibits its enzymatic activity, which limits the phosphorylation of JNK promoted by MKK7_PP. The decrease in JNK_PP inhibits the expression of pro-apoptotic genes and MM cell death.

Model equations
To investigate the role of GADD45β in MM pathogenesis, the mathematical model proposed in [23] is extended by including the pathways of NF-κB and JNK as well as their crosstalk. The proposed model consists of 29 ordinary differential equations (ODEs) with five state variables, namely osteoblast precursors (OBp), active osteoblasts and osteoclasts (OBa and OCa), MM and the bone volume (BV). Equations    royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 'Hill functions' are used to represent the cellular interaction via single ligand to receptor binding denoted by π functions [34], with equations (2.6) and (2.7) describing the stimulating and inhibiting functions of ligand-receptor binding. Here. L represents the concentration of the ligand, β represents maximal expression level of the promoter, n is the coefficient which regulates the steepness of the function π and k 1 and k 2 represent the dissociation constant, respectively. Both β and n are assumed to equal 1 in the model following the work of Pivonka et al. [34].   with their interaction through GADD45β. In accordance with the earlier work of [35] and [25], equations (B1)-(B11) and (C1)-(C16) in the electronic supplementary material, were constructed to simulate the JNK and NF-κB pathways. The distinct feature of these equations is that the interaction of the two pathways, which was ignored before, was included, with the addition of E GADD45b MKK7 in electronic supplementary material, equations (B10) and (C16). E GADD45b MKK7 represents the binding of GADD45β to MKK7, which decreases the amount of MKK7 available for phosphorylation into MKK7_P. This action further leads to the inhibition of the JNK pathway and promotes the survival of MM cells as discussed before. The definition of E GADD45b MKK7 is as follows:

Results
The   figure 3. The OBa : OCa ratio is a key factor in the variation of the bone volume, since OBa and OCa are for forming and resorbing bone, respectively, during the bone remodelling process. The decreased OBa : OCa ratio causes the decline in the bone volume, as shown in figure 4, which is consistent with the observed loss of the bone volume in MM patients [44]. These simulation results match the work of [23]. Figures 5 and 6 reveal the change in the GADD45β concentration after the activation of NF-κB signalling and its influence on JNK signalling. Figure 7 reveals the temporal variation in MM cells due to the inhibition of GADD45β and MKK7   royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 binding by 10% and 40%. Figures 7 and 8 suggest how GADD45β-targeted therapy inhibits the growth of MM cells and JNK_PP. Thus, the inhibition of GADD45β binding to MKK7 leads to an obvious drop of MM cells, which then leads to a new steady state at a lower level. The larger degree of inhibition results in a larger drop in MM cells. These simulation results can be confirmed by the experimental data [4]. Figure 9 shows the variation in MM cells after IKK activation is inhibited by 10% and 40%. As shown in figure 9, IKK inhibition leads to a decrease in MM cells which is accentuated with a larger degree of IKK is inhibition. Figure 10 indicates how the inhibition of GADD45β and IKK influence NF-κB.

Discussion
The proposed model in this paper aims to investigate how the interaction between NF-κB and JNK signalling via the GADD45β|MKK7 complex promotes the development of MM. The model can not only simulate the temporal variation of bone cells, the bone volume and biochemical factors involved in the NF-κB and JNK pathways, but it can also mimic the underlying mechanism in which the coupling between the two pathways inhibits apoptosis in MM cells. Additionally, the efficacies of GADD45β-and    royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 NF-κB-targeted therapies were compared based on the model simulation. This work only demonstrates the temporal variation in the GADD45β concentration after the activation of the NF-κB pathway. The variations in other biochemical factors related to NF-κB signalling were not repeated here as they have been analysed in [25] but are included in electronic supplementary material, Appendix C. As mentioned above, JNK_PP plays an essential role in the apoptosis of MM cells, and JNK_PP is able to inhibit the growth of MM cells. According to figure 8, the decrease in MM cells can be explained by the rising amount of JNK_PP after the inhibition of GADD45β binding to MKK7. NF-κB-targeted therapies are most often performed by inhibiting IKK, which is essential for NF-κB activation [8]. Based on model simulations ( figure 10), the GADD45β-targeted therapy is suggested to be more sensitive than the NF-κBtargeted therapy as it can produce a greater suppression of MM cells than IKK inhibition when each are    Figure 9. Model simulations of the variation in MM cells due to the inhibition of IKK activation by 10% and 40% at 150th hour.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 suppressed to the same degree. In addition to the higher sensitivity, GADD45β-targeted therapy has a high MM cell specificity. This approach can not only kill MM cells effectively, but it also has no effect on NF-κB levels. Therefore, it can be predicted that the side effects seen with NF-κB-targeted treatment would be avoided. However, these in silico hypotheses generated from the mathematical model described in this paper will ultimately have to be validated by further experimental data to confirm the clinical potential of GADD45β-targeting therapies. It should also be noted that the proposed simplified model does not take into account the interactions with several other proteins and other cell structures, which represents a limitation of this study. Specifically, although the differentiation of progenitors into active osteoclasts and osteoblasts contains several intermediate stages, the model only considered four osteoblastic and three osteoclastic lineages, and contained three state variables: osteoblast precursors, active osteoblasts and active osteoclasts. Additionally, IκBs, including IκBα, IκBβ, IκBγ and IκBε, can also bind to NF-κB and form a complex but we only considered IκBα.

Conclusion
This paper describes a mathematical model of MM that was used to investigate how NF-κB signalling acts to include the interplay between NF-κB and JNK signalling that was not included in the previous models. royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 7: 192152 The model not only reconstructs how the invasion of MM cells disturbs the steady state of the bone microenvironment and triggers the variation in bone cells, but it also mimics the changes in biochemical factors involved in the NF-κB and JNK pathways. In addition, the model also investigates the efficacies of GADD45β-and NF-κB-targeted treatments, suggesting that GADD45β-targeted therapy is more effective but has no apparent side effects. The simulation results match the experimental observations. This model helps to illuminate the essential function of the crosstalk between the NF-κB and JNK pathways during MM development. It is anticipated that this model could be employed as a useful tool to initially investigate and even explore potential therapies involving the NF-κB and JNK pathways. More work is required to improve the built model to take into account many others parameters such as the interactions with other protein partners, with membranes and other cell structures.
Data accessibility. Matlab code is available in the electronic supplementary material.