## 4 Discussion and Conclusions
ZA inhibits FPP synthase in the mevalonate pathway through conformational changes in the enzyme upon binding of ZA. 3. A second conformational change after the binding of the second substrate, IPP, forms a tightly bound inhibition complex that provides further stabilisation of the FPPS and ZA complex [45]. The associated conformational change is measured as an isomerization constant (Kisom) [46] and explains ZA's exceptional potency. Indeed, to inhibit FPPS, only small amounts of internalised ZA in the cytosol are needed [16]. The variation over time in the inhibition of the enzyme by ZA is not explicitly taken into account in the first model since this does not represent enzyme kinetics in detail. Conversely, the second model considers the bound/unbound conformation of the enzyme and can account for the variable time-course of its inhibition by ZA. A more precise model, focusing on the three conformational structures of FPPS (Figure 6C), could be designed, and, in theory, it would be possible to identify its parameters if specific data were available, detailing the crystal structure analysis and the MS analysis of FPPS complexed with ZA. Our simple mathematical models fill a gap in the literature about the quantitative study of the effects of ZA in the generation of IPP into tumor cells. A quantitative study would be essential to predict the most effective treatment modality in cell-based immunotherapy. While the zoledronate kinetic sub-model equations are the same for both Model 1 and Model 2, different parameter values are still obtained for the continuous and pulse experiments. In particular, it is interesting to note that cellular zoledronate is only transferred into the medium (\(\gamma(t)\)) and is not diminished by any other mechanism. The ZA degradation rate within the cell (\(k_{XZ}\)) is, in fact, estimated to be essentially zero for both models and experiments. The pre-zoledronate acid degradation rate is non-zero but assumes very small values in both Model 1 and Model 2. Another expected result is that the residue of zole-dronate in the supernatant after rinsing (\(Y^{*+}\)), which is not considered in the continuous experiment, is zero in the pulse experiment. This result involves a jump in the time course of \(Y(t)\) equal to the value that it assumes at \(t^{*}\). The parameter \(k_{YZ}\), which represents the transfer rate from compartment Z to Y, is significantly different in two experimental situations, and in the pulse experiment, is estimated at an unlikely value. For both models, the ZA kinetics in the continuous experiment follows the data very well, while the forecast trend of the \(Z(t)\) concentration in the pulse experiment fails to capture the initial peak, which is visible in the literature data (Figures 3 and 5a). Figure 6: **(A)** Snapshot of the two different assays used by [PERSON] and [PERSON], from which we have extracted data to formulate our mathematical models. On the left is shown the continuous assay in which tumor cells underwent treatment for the whole time of the experiment. On the right is shown the pulse experiment performed by both authors with two different tumor target cells (MCF7, T27D); in this case, the drug was removed after 1 h via medium replacement. In the central part of the figure, we represented zoledronic acid (ZA) pharmacodynamics and IPP into the tumor cells. All process steps are characterized by the calculated parameters in Models 1 and 2. ZA inhibits FPP synthase in the mevalonate pathway through a conformational change in the enzyme. A second conformational change, after the binding of the second substrate, forms a tightly bound inhibition complex that provides a further stabilisation of the FPPS and ZA complex, resulting in the IPP accumulation. The extracellular and intracellular compartments were defined through the cellular borderline. The ZA concentration is proportional to the observations by GS-MS in the intracellular compartment. In each well, the surnatant volume was the extracellular compartment.