## 3 Results
### Translating Model Dynamics from Mouse to Human  
Of course, the question of inter-species translatability is always critical for extrapolating any insights from animal models into humans. Before this question can be addressed, it is important to understand how the mouse lifespan maps onto the human lifespan, since ASD is typically diagnosed in children before 5 years of age. Instead of assuming that each mouse day has a uniform conversion to a human day throughout the animal's lifespan, it is more instructive to align the two lifespans by key developmental milestones, from \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & **Description** & **Value (Mouse)** & **Value (Human)** \\ \hline \(k_{n}\) & Rate of conversion of serotonin into ENS & 0.5 & 10.1 \\ \(n_{out}\) & Rate of normal ENS pruning (to have the system & 0.1 & 0.99 \\  & reach equilibrium) & & \\ \(S_{0}\) & Serotonin synthesis rate in the gut & 0.2 & 1 \\ \(k_{dpg}\) & Gut serotonin degradation rate & 1.386 & 1.386 \\ \(k_{sym}\) & \(S_{0}*k_{\text{deg}}\) & 0.2772 & 1.386 \\ \hline \multirow{3}{*}{_sert_} & \multicolumn{3}{c}{0 if knockout (KO)} \\  & Rate of sert-mediated serotonin uptake & 0.24 if normal & 0.4 for norm \\  & & 1.1 if SERT+ & \\ \hline \(x_{i}\) & Half-maximal concentration of serotonin needed & 0.19 & 0.9 \\  & for gut serotonin-mediated ENS growth & & \\ \(s_{out}\) & Serotonin clearance from the gut not through sert & 0.0035 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values used in simulations for both mouse and human dynamics. Figure 1: Impact of relative levels of SERT expression on (**a**) simulated serotonin in the gut and (**b**) size of the ENS relative to WT. At t = 50 days, WT mice had 100% ENS, with SERT KO mice at 120% of ENS, and SERT+ mice at 60% ENS relative to WT. gestation, to weaning, to puberty, to adulthood, and into old age, as has been done by [PERSON] et al. [16]. Figure 2 visualizes the analysis done in [16], aligning mouse and human lifespans by developmental milestones. [PERSON] et al. [13] report data that was collected in mice at 6-8 weeks, which, according to the mapping in Figure 2, corresponds to humans of about 11-13 years of age. It is unknown if these animals exhibited ASD-like behaviors around week 5, which would be closer to the human age of around 5 years old, by which time we would already observe ASD-like behavior, although it is possible. If these relationships indeed translate, and if the relative size of the ENS is indeed correlated with an ASD-like phenotype, then the differences in the ENS would be observed by age 5. The model for projected human analysis is therefore re-parametrized to capture similar relationships between ENS size for non-ASD (parametrized to 100%) vs. ASD children by t = 5 years, with the understanding that it is possible for ASD symptoms to manifest much earlier. Furthermore, interestingly, [PERSON] et al. [17] reported differences in the serotonin synthesis capacity in ASD vs. non-ASD children, showing that for non-autistic children, serotonin synthesis capacity increased more sharply and to higher values in early life but declined towards adult values after 5 years of age, while in autistic children, it increased more gradually until 15 years of age but did not decline as much as it did for non-ASD children. Within the frameworks of the proposed model, these differences can be captured as a term for serotonin production, with the assumption that at least the pattern of relative differences in serotonin synthesis capacity reported in [PERSON] et al. [17] mirrors that of serotonin synthesis in the gut, a hypothesis that requires further experimental validation. A mathematical expression to describe change in serotonin synthesis for ASD and non-ASD children is described in Appendix B. The projected serotonin dynamics over time for ASD and non-ASD children using this function are shown in Figure 3.  
Figure 2: Aligning human and mouse lifespans by developmental milestones; the figure visualizes analysis done by [PERSON] et al.