Determining parameter impact in systems biology models via sensitivity analysis: a comparative approach

Abstract

Mathematical modeling is a powerful tool to understand biological phenomena, predict behavior, and guide experiments. Model parameters represent experimentally derived rates, and it can be challenging to understand how parameters affect the system’s output(s). Global sensitivity analysis (GSA) can help determine how uncertainty in model outcomes can be attributed to parameters. Methods like Sobol’ indices provide detailed analysis but at a high computational cost. Thus, it may be necessary to precede it by another method to reduce the parameter space. Morris Method Screening (MMS) is a qualitative analysis that can provide information about sensitivity and parameter interactions, making it a good partner for Sobol’. Partial Rank Correlation Coefficient (PRCC) may also be a viable option as it has a similar cost to MMS. This work tests whether a ranking (PRCC) or screening (MMS) method is better when used in combination with Sobol’ analysis. Using a model of the epithelial-mesenchymal transition as a test case, this work shows that PRCC-Sobol’ and MMS-Sobol’ can produce diverging results, making it difficult to analyze how biological events are informed by specific rates and quantities. However, the differences between these methods mean that these techniques may be most effective when all three are used together.

Introduction

Mathematical modeling is a powerful tool to understand biological phenomena, predict systems’ behavior, and guide future experiments. The parameters in these models represent rates and quantities present in biological systems¹. Determining how changes to these parameters affect the system’s output(s) can be complicated and requires thorough investigation. Global sensitivity analysis (GSA) is a powerful method to help determine how uncertainty in specific model outcomes, or quantities of interest (QoI) can be attributed to parameters^2–4. Within the varied parameter set, all parameters are varied at once, and GSA provides information about parameters that may have epistemic or aleatory uncertainty^3,5,6. This analysis can guide future exploration but requires a delicate balance: large parameter sets and wide parameter ranges can provide a better understanding of their model mechanisms, but analysis could be computationally expensive^3,5,7. Considering that many mathematical models describe real-world systems and results can have significant consequences shaping experimental directions, choices must be made carefully and methodically during analysis.

There are many different GSA techniques available, and the benefits of several of these are outlined in Fig. 1. One method is Partial Rank Correlation Coefficient (PRCC)⁷. PRCC is a ranking method, meaning that it puts parameters in order based on their effect on model output variability⁷. This method is usually combined with Latin Hypercube Sampling (LHS), a type of Monte Carlo sampling⁸. LHS divides the K parameter spaces into N subsections and randomly selects a value from each subunit ($N>\frac{4}{3}K$) before combining them into parameter sets ^8,9. PRCC ranks the individual parameter values and outputs and quantifies the monotonic relationship between them through the correlation coefficient (ρ)¹⁰. It has moderate computational requirements and provides correlation directionality (ρ ∈ [−1, 1])⁷. Within this range, ∣ρ∣ ≈ 1 indicates that the QOI is sensitive to the input parameter, while ρ ≈ 0 indicates non-sensitivity ¹¹. A threshold (∣ρ∣ > 0.5) is often used ^10,12–16. However, PRCC requires the outcome change monotonically when parameters are varied individually. Lack of monotonicity may result in inaccurate or misleading analysis ^7–9.

Fig. 1. The analyses performed on the model from¹⁵, as well as the pros and cons of each methodology.

LHS-PRCC was originally performed in refs. ^16,28 as a stand-alone analysis (Step 1). PRCC is a ranking method with a moderate computational cost and the resulting correlation coefficient (ρ) that can determine whether a parameter positively or negatively impacts the output. However, PRCC requires that outcomes change monotonically when each parameter is varied individually^7–10. Therefore, the PRCC results were used in Step 2 to reduce the parameter set for Sobol' analysis. Conversely, MMS is a screening method that can also be used to reduce the parameter set. Like PRCC, MMS has a moderate computational cost that does not require monotonicity, but it can only provide semi-quantitative information about nonlinearity and parameter interactions^3,7,19. In Step 3, MMS was performed on the EMT model to reduce the parameter set before Sobol' was performed. Sobol' can both rank and screen parameters to provide detailed information about parameter impacts and interactions but, unlike MMS and PRCC, it has a high computational cost^5,7,20.

LHS-PRCC is not the only method for assessing parameter impact. Morris Method Screening (MMS), which uses one-at-a-time (OAT) sampling and determines the variation in the derivative index over the parameter domain, may also be appropriate^7,17,18. Unlike PRCC, MMS is a screening method, which means that it helps determine which parameters are unidentifiable and thus have little effect on variability in the output⁷. In this methodology, a finite difference scheme approximates the derivative and the average of many samples produces semi-informative statistics about the QoI, such as the mean and standard deviation (μ^*, σ²). MMS normalizes the parameter range from [0, 1], divides it into l levels, and uses a step-size of Δ to select the parameter value^3,5,7. While MMS does not require monotonicity, it also cannot determine the exact impact of a parameter. MMS only indicates the relative importance and interaction associated with a parameter^3,7,19. Thus, it may be best to use it as a screening method to reduce the space before using more robust analysis, such as Sobol’⁵.

Sobol’ is a variance-based method that uses ANOVA-like decomposition to determine the impact of the parameters on outcome variations^5,20,21. Sobol’ can provide detailed information about the parameter-outcome relationship, such as the total parameter impact (S_T), primary impact (S₁), interaction effects (S_T − S₁), and one-one parameter interactions (S₂) ^2,5,7. While Sobol’ does not have a strict sensitivity cut-off, nor can its statistics provide directionality, it can both screen and rank parameters^7,22–24. Unless S_T = 0, the total impact is used to determine the relative importance of a parameter while the primary impact (S₁) can be used for ranking^5,7.

Unfortunately, when modeling real-world scenarios, performing Sobol’ alone is not always feasible. This issue could be due to a number of reasons, including the high computational cost^5,7,12. Thus, there is a need to find methods for making Sobol’ more tractable when analyzing these complex systems. One possibility would be to use a screening technique, such as MMS, to reduce the parameter space to a small subset that would allow for Sobol’ to be performed⁵. Another option would be to forgo Sobol’ entirely and choose a different method, such as LHS-PRCC. However, PRCC would require monotonicity and would lack information about complex interactions governing the system⁹. A third option would be to use PRCC prior to Sobol’, reducing the parameter space before obtaining information about important parameter impacts and interactions in the system^9,12.

This work seeks to understand whether a screening method, such as MMS, or a ranking method, such as PRCC, is better for parameter set reduction prior to Sobol’ when dealing with complex biological problems. To examine how MMS- and PRCC-derived subsets compare under Sobol’, a theoretical two-equation model of the epithelial-mesenchymal transition (EMT) is used as a test case¹⁵. EMT is the process by which epithelial cells lose their adhesive nature and adopt mesenchymal-associated invasiveness, allowing carcinoma cells to migrate away from a primary tumor and form metastases^25–27. This model presents a unique opportunity to gain both mathematical and biological insights. The model is small in size with only two equations, and it has been shown that PRCC can be performed^16,28. However, the model does also presents some complications. It has been hypothesized that there is a bistable switch underlying EMT: transitioned cells can maintain the mesenchymal phenotype in the absence of extracellular factors¹⁴. Multiple model input variables in this model seek to capture multiple bistable switches in response to multiple external factors, meaning that the screen-rank comparison must be carried out under several different treatment groups. Further, with 11 parameters to raise the computational cost and preliminary results highlighting some numerical instabilities, performing Sobol’ alone will not adequately capture the complicated behavior. Thus, for this model, Sobol’ will be preceded by a reduction methodology.

To perform a comprehensive inquiry into using combined sensitivity analyses in complex, multi-scale systems, as well as the biases introduced by researchers, a methodical approach was used. The model was exposed to eight different combinations of the input variables (TGF-β, cellular contact) and the parameters were varied over two different ranges (±10%, ±25% of the original model value in Table 1) to capture the heterogeneity inherent in the tumor, possible bistable behavior, and variability due to lack of certainty about a parameter value. MMS-Sobol’ and PRCC-Sobol’ were then performed on the long-term values of the of the two model variables associated with mesenchymal (Slug) and epithelial (E-cadherin) phenotypes. Ultimately, comparing a screening method to a ranking method prior to Sobol’ highlights potential issues that may arise during analysis. While Slug results show conclusive agreement on which parameters control this factor, E-cadherin results differ based on parameter range, treatment group, and methodology. Not only do MMS and PRCC select different subsets, but important parameters from PRCC directly conflict with parameters highlighted by Sobol’. These differences suggest that different sampling methods may skew the results. But this work also shows that, given the pros and cons of each method, all three are needed to derive a better picture of the dynamics underlying the system. Together, they suggest new avenues for computational and experimental exploration, as well as warnings when applying and using these analytical techniques.

Results

Figures 2, 3, 4, and 5 show Sobol’ analysis of E-cadherin (E) and Slug (S) when parameters are varied over ±10% and ±25% of their original value. If the reduced parameter set by MMS and PRCC was the same, a single set of results is shown. If the MMS and PRCC subsets differed, Sobol’ results are side-by-side. The entire bar is the total parameter effect (S_T), with primary parameter effects (S₁) shown as solid color and the interaction effects (S_T − S₁) are checkered.

Fig. 2. Sobol’ analysis for parameter impact on E-cadherin (E) when parameters are varied over ± 10% of their value in Table 1.

If the MMS- and PRCC-subsets were the same, only a single set of results is shown. The entire bar for each parameter is the total effect (S_T), with primary parameter effect (S₁) shown as solid color and the interaction effects (S_T − S₁) are checkered. MMS and PRCC differ for C = 0−2 when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ but present similar trends. For $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$, E-cadherin depends on Slug-associated parameters and α₂, k₁ grow in importance as contact is increased. Additionally, it is when C = 1,2 cells that the parameters have the highest interaction effects. For $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$, parameters have much less interaction and E-cadherin moves from Slug-associated parameters to E-cadherin-associated parameters as contact is increased. Sobol' analysis results for cells with C = 0, 1, 2, and 6 neighbors A, C, E, G without any exogenous TGF- $\beta \left(T=0\frac{\text{ng}}{\text{mL}}\right)$ and B, D, F, H with exogenous TGF- $\beta \left(T=3.64\times 10^{-6}\frac{\text{ng}}{\text{mL}}\right)$.

Fig. 3. Sobol’ analysis for parameter impact on E-cadherin (E) when parameters are varied over ± 25% of their value in Table 1.

If the MMS- and PRCC-subsets were the same, only one set of results is shown. The entire bar for each parameter is the total effect (S_T), with primary parameter effect (S₁) shown as solid color and the interaction effects (S_T − S₁) are checkered. MMS and PRCC differ for C = 0−6 cells when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and C = 0 when $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$ In the MMS-subsets when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$, α₂, k₁ importance emerges as contact is increased. For C ≥ 1 these parameter subsets have increased levels of interaction compared to ±10% ranges. For $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$, PRCC-subsets present less interaction and shift towards depending on E-cadherin-associated parameters as contact is increased. With $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$, parameters have less interaction and E-cadherin moves from Slug-associated parameters to E-cadherin-associated parameters as contact is increased. Sobol' analysis results for cells with C = 0, 1, 2, and 6 neighbors A, C, E, G without any exogenous TGF- $\beta \left(T=0\frac{\text{ng}}{\text{mL}}\right)$ and B, D, F, H with exogenous TGF- $\beta \left(T=3.64\times 10^{-6}\frac{\text{ng}}{\text{mL}}\right)$.

Fig. 4. Sobol’ analysis for parameter impact on the value of Slug (S) when parameters are varied over ± 10% of their value in Table 1.

If the MMS- and PRCC-subsets were the same, only a single set of results is shown. The entire bar for each parameter is the total effect (S_T), with primary parameter effect (S₁) shown as solid color and the interaction effects (S_T − S₁) are checkered. MMS and PRCC differ for C = 2 when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and C = 2,6 when $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$. Despite some differences between MMS- and PRCC-subsets, results show that increasing contact for both levels of TGF-β pushes Slug from β₂-dependence to α₂, k₁-dependence. Sobol' analysis results for cells with C=0, 1, 2, and 6 neighbors A, C, E, G without any exogenous TGF- $\beta \left(T=0\frac{\text{ng}}{\text{mL}}\right)$ and B, D, F, H with exogenous TGF- $\beta \left(T=3.64\times 10^{-6}\frac{\text{ng}}{\text{mL}}\right)$.

Fig. 5. Sobol’ analysis for parameter impact on the value of Slug (S) when parameters are varied over ±25% of their value in Table 1.

If the MMS- and PRCC-subsets were the same, only a single set of results is shown. The entire bar for each parameter is the total effect (S_T), with primary parameter effect (S₁) shown as solid color and the interaction effects (S_T − S₁) are checkered. MMS and PRCC differ for C = 2,6 when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and C = 2 when $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$. Despite some differences between MMS- and PRCC-subsets, results show that increasing contact for both levels of TGF-β pushes Slug from β₂-dependence to α₂, k₁-dependence. Sobol' analysis results for cells with C = 0, 1, 2, and 6 neighbors A, C, E, G without any exogenous TGF- $\beta \left(T=0\frac{\text{ng}}{\text{mL}}\right)$ and B, D, F, H with exogenous TGF- $\beta \left(T=3.64\times 10^{-6}\frac{\text{ng}}{\text{mL}}\right)$.

E-cadherin sensitivity varies in response to treatment group, parameter range, and combined methodologies

Figure 2 shows similar trends in MMS- and PRCC- subsets in systems without exogenous TGF-β ($T=0\frac{\mathrm{ng}}{\mathrm{mL}}$) when parameter range is restricted to ±10% of the original value (Figs. 2A, C, E, G). In the MMS-subsets, α₂ and other Slug-associated parameters equally impact E-cadherin. For C = 0, this includes {β₂,IC_S}, and for C = 1,2,6 cells, this includes k₁. Sobol’ analysis reveals that, for extreme levels of contact (C = 0,6 cells), the MMS-subsets have minimal interaction effects (<12% of S_T). It is the low-to-moderate contact levels (C = 1,2 cells) with high interaction between the MMS-subset parameters (>50% of S_T). Parameter S₂ values in Supplementary Figure S17 reveal that most interaction here is between α₂ and k₁. Combined PRCC-Sobol’ analysis in Fig. 2 also highlights the importance of Slug-associated parameters when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$. When C = 1 cell where the MMS- and PRCC-subsets did not agree, Sobol’ showed underlying similarities. Here, PRCC selects α₂ and two E-cadherin-associated parameters ({β₁,IC_C}) but Sobol’ analysis reveals the low impact of these E-cadherin-associated parameters. Instead, the k₁ absence results in a higher α₂-S_T and less overall interaction in the subset. Similarly, for C = 2 cells, although the PRCC-subset also included IC_C, it is α₂ and k₁ that primarily impact E-cadherin, as was predicted by MMS.

As shown in Fig. 3, the ±25% range presents a different trend when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and combined MMS-Sobol’ analysis is used. Without any contact (C = 0 cells), {α₂,β₂,IC_S} equally impact E-cadherin. When C = 1, it is still α₂ and k₁ that control E-cadherin but the importance of β₁ is now higher. Increasing cellular contact increases the parameter subset to include both Slug-associated and E-cadherin-associated parameters. In addition to α₂ and k₁ for C = 2,6 cells, MMS also selects β₁ and IC_C for C = 2 and β₁ and k₀ for C = 6. The parameters in each of these subsets also have, in general, a higher level of interaction effects than the ±10% range. Interactions now accounting for >24% of the S_T when C = 0 and >87% of the S_T values when C = 1. For C = 2 cells, interaction effects for α₂, k₁ were >75% of their respective S_T values while for C = 6 cells, α₂, k₁ interaction effects were >49% of their respective S_T values.

The PRCC-Sobol’ method with a ±25% range also produces different results for E-cadherin when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$. Increasing contact shifts the PRCC-subset from Slug-associated to E-cadherin-associated parameters. When C = 0 cells, PRCC chooses {β₂,IC_S} and Sobol’ reveals both parameters impact E-cadherin equally with little interaction (<16% of S_T). Once cellular contact is added (C ≥ 1 cells), E-cadherin only depends on E-cadherin-associated parameters. Unlike the ±10% PRCC-subsets or the ±25% MMS-subsets, PRCC does not select α₂ or k₁. Instead, both C = 1,2 show that PRCC selected {β₁,k₀,IC_C} with β₁ influencing E-cadherin through its primary effects (S₁). The interaction effects for β₁ are <16% of the S_T value for both treatment groups. Extending contact to C = 6 cells reduces the parameter set further to {β₁,k₀} with β₁ still exerting the most influence of E-cadherin and little interaction between the two parameters (<2% of S_T). The dependence of β₁ on primary effects for all three levels of contact contrasts with the α₂ − k₁ interactions present in the analysis of the MMS-subset.

Results with for $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$ present less variability. Now, both ±10% and ±25% ranges show a transition from Slug-associated parameters to E-cadherin-associated parameters as contact increases. For C = 0, E-cadherin depends on {β₂,IC_S}. While the ±10%, ±25% MMS-subsets and the ±10% PRCC-subset also selected α₂, the impact of α₂ is less than the other parameters, and PRCC ±25% eliminates α₂ from future analysis. Increasing contact to C = 1,2 results in a {β₁,k₀,IC_C} subset for both MMS and PRCC. Sobol analysis for all ranges shows IC_C controlling E-cadherin for these treatment groups. The subset is reduced to β₁ and k₀ for C = 6 cells, with the degradation rate (β₁) controlling E-cadherin in both ±10% and ±25% ranges. Additionally, for all levels of contact, parameter ranges, and methodologies, impacts under regimes with exogenous TGF-β present are due to the primary effects (S₁), rather than interactions, which account for <27% of the S_T values in Figs. 2B, D, F, H and 3B, D, F, H.

Consistent patterns across parameter ranges and treatment groups emerge for Slug sensitivity as cellular contact is increased

Unlike the E-cadherin, Figs. 4 and 5 show overarching patterns in the parameters underlying Slug. For both levels of TGF-β ($T=0,3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$) and parameter ranges (±10%, ±25%), when C = 0 neighbors, the parameter set is reduced to {α₂, β₂} and Sobol’ analysis shows that β₂ has a higher S_T value. Increasing cellular contact without exogenous TGF-β ($T=0\frac{\mathrm{ng}}{\mathrm{mL}}$), Sobol’ indicates that α₂ is the dominant parameter for Slug across both parameter ranges, a trend that continues as contact is increased. For cells with exogenous TGF-β ($T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$) across both parameter ranges, α₂ becomes the dominant parameter as contact is increased, but the system requires more contact for this to occur (C = 6). Increasing cellular contact across both levels of TGF-β and parameter ranges also highlights the growing importance of k₁. However, its emergence as an important parameter varies based on treatment group, parameter range, and methodology, and its S_T value always remains less than α₂.

Changes in parameter ranges and methodologies also showed that (C,T) = {(1,0),(2,0)} are the treatment groups with the most parameter interactions. For C = 1, the MMS-subset parameters have interaction effects comprising >51% of their S_T values under the ±10% ranges, while the parameters in C = 2 have interaction effects comprising >31% of their total S_T values. Increasing to the ±25% range reduces the interaction effects for both treatment groups, but these are still where the most interaction effects occur. For the subset in C = 1, interaction effects are 12–23% of the S_T values, while for the MMS subset of parameters with C = 2, the interaction effects are 10–37% of the S_T values. Parameters in all other treatment groups in Figs. 4 and 5 have interaction effects <5% of their S_T.

There were six treatment groups in Figs. 4 and 5 where the PRCC- and MMS-selected subsets differed. In these treatment groups, PRCC selected at least two of the parameters chosen by MMS. Each S_T for the PRCC-subset parameters are higher and, in general, there is less interaction between the parameters. But the overarching relationship between the S_T values is maintained.

Overall, while there are variations in parameter subsets, S_T values, and interaction effects across treatment groups, parameter ranges, and methodologies, analysis produces an observable trend in the parameters controlling Slug: as cellular contact is increased, the cell moves from a reliance on degradation of Slug (β₂) to having the production rate of Slug (α₂) and the rate at which Slug is suppressed by membrane-bound E-cadherin (k₁) gain importance.

Discussion

This work highlights the issues that can emerge when using common combined GSA methods over ill-defined parameter ranges in biomathematical modeling. Creating a biologically relevant mathematical model relies on well-known and accurately measured parameter ranges. A wider parameter range chosen for GSA alludes to the certainty of the parameter measurement. Here, two ranges were used to test how different levels of certainty can affect sensitivity results. Additionally, due to the size of the parameter range and numerical instabilities present for certain parameter combinations, this model cannot be subjected to Sobol’ as a whole. Instead, it is necessary to perform combined methodologies to reduce the parameter space to the most impactful parameters and then apply Sobol’ to glean important details about their impact on desired outcomes and the effects of their interactions. This work presents two possible combinations for analysis: MMS-Sobol’ and PRCC-Sobol’. MMS and PRCC have different benefits and drawbacks as a first step, but both need to be combined with another analysis, such as Sobol’. PRCC is a ranking method and uses a hard cut-off to determine sensitivity, but it can only indicate the correlation, not the magnitude of change, between an input and an output⁹. Further, the ∣ρ∣ ≥ 0.5 meant that for some cases, only one parameter was initially selected by PRCC for further analysis. MMS is a screening method, and the semi-quantitative information given about parameter-outcome relationships means a second analysis is also necessary. While different cut-offs can be used and screening can be subjective, a cut-off of $\frac{1}{2}{\mu }_{\max }^{*}$ for each treatment group was used for MMS sensitivity here. The limits in information available from PRCC and MMS make them both candidates for parameter space reduction prior to Sobol’ use. However, the results from this study comparing the use of PRCC and MMS to reduce the parameter space give mixed results. While the differences in sampling and analysis did not appear to extensively influence Slug across both the ±10% and ±25% parameter ranges, choosing MMS or PRCC for parameter space reduction did produce varying results for E-cadherin.

Figure 6 summarizes Figs. 2–5, highlighting the most important parameter for each range, treatment group, and methodology, as well as the overlap in parameters chosen by PRCC and MMS for subsequent Sobol’ analysis. Slug results showed consistency across parameter ranges, methodologies, and levels of exogenous TGF-β. The Sobol’ subsets chosen by PRCC and MMS were the same for 10 of the 16 treatment groups. Further, for both the ±10% and ±25% ranges of Slug, as cellular contact for the cell increased, the importance of the production rate of Slug (α₂) increased. Additional inspection of Figs. 2–5 shows an increase in importance of the suppression of Slug by membrane-bound E-cadherin (k₁). The agreement between MMS-Sobol’ and PRCC-Sobol’ results suggests an analytical recipe to apply moving forward: (1) reduce the parameter set via MMS to the most important parameters; (2) apply Sobol’ analysis to determine the total impact and interaction effects of each parameter; (3) use PRCC to determine whether changes in these parameters are having positive or negative impacts on Slug. The parameters selected can then be used to suggest novel computational, experimental, and analytical strategies moving forward.

Fig. 6. A summary the most impactful parameter chosen by each analysis.

Each treatment group lists the most impactful parameter given by PRCC, MMS, PRCC-Sobol' (Sobol (P)), and MMS-Sobol' (Sobol (M)). If both Sobol' subsets chose the same parameter, a single Sobol' grouping is listed. If both PRCC and MMS highlighted the same parameter, a single parameter is listed for those two analyses. If all four agreed, a single parameter is listed. Each treatment group also lists the number of parameters that overlap between the two Sobol' subsets. Overall, the importance of β₂ (low contact) and α₂ (high contact) on Slug is clear, as is the heterogeneity in E-cadherin results. Specifically, discrepancies between E-cadherin results occur in treatment groups without TGF-β. However, the Sobol' subsets for both E-cadherin and Slug all have at least one parameter in common, indicating that the methods do put forth some similar parameters for further investigation.

Unlike Slug, E-cadherin Sobol’ results had higher variability. While the MMS- and PRCC-subsets agreed for both ±10% and ±25% parameter ranges with $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$, they diverged for $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$. As shown in Fig. 6, for $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and the ±10% parameter range, MMS and PRCC chose at least 2 of the same parameters for the further Sobol’ evaluation. Initial PRCC and MMS analysis did not agree on the most impactful parameter for this parameter range for C = 0,1,2. But, the Sobol’ subsets for both methods chose the same most impactful parameter. Then, for C = 6, PRCC, MMS, and both Sobol’ subsets agree that α₂ is the most important parameter. Thus, for ± 10% with $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$, both Sobol’ subsets show a shift towards the rate of Slug production (α₂) as the most important parameter as cellular contact is increased. However, despite the agreement this summary shows, it is important to remember that the original PRCC results for C = 1,2 with $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ only highlighted the impact of IC_C. MMS and both Sobol’ subsets either did not include or discounted the impact of this parameter upon further analysis, showing divergence in the methods.

The ±25% parameter range shows further differences present between PRCC, MMS, and the two Sobol’ subsets. Similar to the ±10% range, PRCC analysis originally chose only IC_C with ∣ρ∣ > 0.5 for C = 1, 2 with $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$. Fig. 6 now shows that there is also less overlap between the parameters in the MMS- and PRCC- subsets. While there was more agreement between MMS and PRCC as to which parameter was the most impactful, this agreement did not translate to Sobol’ analysis. Sobol’ analysis highlighted E-cadherin-associated parameters for the PRCC-subset (β₁) but Slug-associated parameters for the MMS-subset (α₂, k₁) when C > 0, with a higher level of parameter interactions for both subsets.

With the variability present in the E-cadherin results, one question that arises is whether this is due to the ∣ρ∣≥0.5 cut-off used in selecting the PRCC-subset. While this threshold has been used in previous work, it is an arbitrary choice made further undefined by the inclusion of the top three parameters if two did not pass this threshold. To determine the effects of this selection methodology, additional analysis was run using ∣ρ∣ ≥ 0.25, as shown in Supplementary Figure S23. Lowering the threshold for sensitivity altered 16 of the 32 original PRCC-subsets by either including an extra parameter for analysis or losing a parameter that fell below ∣ρ∣ ≥ 0.25. Of these 16 sets, nine agreed with the MMS-subsets, leaving seven for further analysis: six E-cadherin and one Slug. Lowering of the ρ threshold changed some of the PRCC-subsets, but similar patterns emerged. More treatment groups in E-cadherin than Slug disagreed between MMS and PRCC. However, lowering the ρ threshold did not change the subsequent Sobol’ analysis. The parameters identified as important by Sobol’ in these subsets remained consistent. Thus, it is necessary to look to other areas to understand the variability present in the E-cadherin results.

The variability present in the E-cadherin results could be from several sources in the model and analysis. One such source could be the structure of the model or, specifically, the E-cadherin equation. This model is a small two-equation system with two bifurcations: a reversible bistable switch with respect to cellular contact (C) and an irreversible bistable switch with respect to TGF-β (T). The bistability present in C = 1 and C = 2 when $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ is apparent when the E-cadherin outputs are examined. Supplementary Figures S1–S16 show the distribution of E-cadherin and Slug values at t = 10,000 min for Sobol’ analysis in Figs. 2–5. These plots show how the distribution and bistable behavior changes depending on which parameters are varied within the analysis. E-cadherin distributions are different depending on whether it was an MMS- or PRCC-subset, as well as the size of the parameter range. These differences could indicate that the bifurcations present in the model are creating complexity for further analysis. Additionally, this model was created with multiple simplifications used to describe complex intracellular processes. The shifting of the parameter ranges for α₂ or k₁ for analysis could point to a need for further expansion of these terms in future iterations of the model to fully understand and capture these mechanisms. Thus, E-cadherin variability could be structural.

Another possible source of variability in the E-cadherin results could stem from the analysis performed. The ±10% and ±25% ranges chosen for the parameters in this model were chosen due to a lack of knowledge about the values. Further experiments could and should be performed to help determine the best range for these parameters before analysis is applied. It may also be prudent to conduct approximate Bayesian computation based on sequential Monte Carlo (ABC-SMC) to determine the posterior parameter distributions. ABC methods can be used to find the posterior distributions where likelihood functions are computationally expensive to estimate by comparing observed data to simulated outputs. In ABC-SMC, parameter values are sampled from the prior distribution and are transmitted through intermediate distributions until it depicts the target distribution. ABC-SMC could give better parameter estimates and intervals and provide information about parameter inferability for low computational cost²⁹. Performing ABC-SMC prior to combined sensitivity techniques, such as MMS and Sobol’ has been used before to analyze mathematical models³⁰ and may help provide better parameter ranges for this multi-bifurcation model before performing sensitivity analysis.

A final reason for the differing results in E-cadherin here could be due to the original question posed: ranking (PRCC) vs. screening (MMS). First, PRCC and MMS have different sampling methods over the parameter space and provide different information about the impact of the parameters. Second, due to the different constraints present in each method, use of either PRCC or MMS prior to Sobol’ may inadvertently cause the user to inflict their own bias. One benefit of PRCC is that ρ ∈ [−1,1], meaning that a set threshold could be chosen and applied to all analysis. Here, while the choice of ρ is arbitrary, if it applied consistently, the bias in it may be missed. In contrast, there is no set cap on μ^* in MMS, making it difficult to potentially determine what may be an appropriate cut-off for sensitivity. Lack of objectivity here can have drastic consequences in systems where parameters have similar μ^* values or large values of σ². Thus, the objectivity involved applying the same threshold value for ρ makes PRCC “user-friendly,” but could also mean information is missed that could have been provided by MMS.

While this work uses an objective methodology for both PRCC and MMS, it highlights where that objectivity fails and how insights from both methods can help with analysis. Similar to a set value of ρ in PRCC, a cut-off of $0.5{\mu }_{\max }^{*}$ was used for MMS to create a standardized system for analysis in a methodology that does not have the sensitivity parameters μ^* and σ². Conversely, the ∣ρ∣ ≥ 0.5 initially produced some PRCC subsets containing only 1 parameter, which is insufficient for Sobol’ analysis. Extending the parameter subsets to include the top 3 parameters implements flexibility where needed, as would occur in MMS. This flexibility is most likely why lowering the threshold to ∣ρ∣ ≥ 0.25 in Supplementary Figure S23 did not change the results significantly. Thus, understanding both methods and how they function can help produce a better subset for Sobol’ analysis. Further, given the variability in the results, as well as the differences in the sampling methods, rather than determining whether a screening method or a ranking method should precede Sobol’, this work shows that all three are necessary for adequate analysis. Semi-quantitative analysis via MMS can suggest parameters that PRCC may have excluded; Sobol’ can provide in-depth information about parameter impact and interaction; PRCC can determine whether these parameters are positively or negatively correlated with the QoI.

Whether the methodologies agree or diverge, what becomes clear is that sensitivity analysis must be performed following model development. Agreement between analyses can suggest specific areas as experimental targets and modeling expansion, while discrepancies can show areas that need improvement and retooling. But what is paramount to remember is that, although the model examined here is theoretical, the underlying phenomenon is not. Preventing cells within a primary tumor from transitioning is imperative in preventing metastasis. The GSA performed, in addition to other works with this model, highlights how the levels of E-cadherin and Slug present in different treatment groups is analogous to shifts in these intracellular components depending on the position in the tumor and exposure to exogenous factors. Variability in the tumor microenvironment demands methodical analysis to help propose thoughtful, directed methods of therapeutic treatment. The combined efforts of MMS, PRCC, and Sobol’ here highlight the difficulty in building accurate models and determining the appropriate analysis when the dynamics underlying real-world process are complex and multi-scale. But using these GSA techniques together will ultimately provide the most insight for continued interrogation of the models capturing biological processes. Ultimately, the results of these combined analyses play a key role in the computational-experimental feedback loop underlying biomathematical research.

Methods

The analyses performed on the model from¹⁵, as well as the pros and cons of each methodology, are outlined in Fig. 1. In Step 1, LHS-PRCC was originally performed in refs. ^16,28 as a stand-alone analysis, the results of which are detailed in Supplementary Tables S3–S6. For Step 2, the PRCC results were used to create a reduced parameter subset for Sobol’. All parameters with ∣ρ∣ ≥ 0.5 comprised the reduced subset. If less than 2 parameters were chosen with this cut-off, the three parameters with the largest ∣ρ∣ were included. Additionally, MMS was performed on the original EMT model (Supplementary Figures S1–S4) using a cut-off threshold of $\frac{1}{2}{\mu }_{\max }^{*}$ in each treatment group. The reduced subset was then used for Sobol’ analysis (Step 3). All GSA methodologies were carried out over the ranges of ±10% and ±25% of the original parameter value in Table 1. These ranges are detailed in Supplementary Table S1. The model and sensitivity methods used are discussed below.

Table 1.

Parameter definitions and values for E-cadherin-Slug model from¹⁵

Parameter Definitions for E-cadherin-Slug Model
Parameter	Definition	Assumed	Units	Source
		Value
α1	E-cadherin production	0.19355	$\frac{\mathrm{ng}}{\mathrm{mL}\cdot \min }$	Estimated
α2	Slug production	0.080	$\frac{\mathrm{ng}}{\mathrm{mL}\cdot \min }$	Estimated
β1	E-cadherin degradation	6.035	$\frac{1}{\min }$	Estimated
β2	Slug degradation	0.830	$\frac{1}{\min }$	Estimated
k0	Rate at which E-cadherin moves to the membrane due to cell-cell contact	0.1245	$\frac{\mathrm{ng}}{\mathrm{mL}\cdot \min }$	Estimated
k1	Rate of Slug suppression from membrane-bound E-cadherin activation	0.080	$\frac{\mathrm{ng}}{\mathrm{mL}\cdot \min }$	Estimated
k2	Rate of Slug upreglation via TGF-β	0.040	$\frac{\mathrm{ng}}{\mathrm{mL}\cdot \min }$	Estimated
ICS	Half maximal concentration of Slug required to inhibit E-cadherin production	0.01920	$\frac{\mathrm{ng}}{\mathrm{mL}}$	Estimated
ICE	Half maximal concentration of E-cadherin required to inhibit Slug upregulation	0.010	$\frac{\mathrm{ng}}{\mathrm{mL}}$	Estimated
ICT	Half maximal concentration of TGF-β required to activate Slug	3.64 × 10−6	$\frac{\mathrm{ng}}{\mathrm{mL}}$	Estimated
ICC	Half maximal number of cells required to draw E-cadherin to the membrane for activation	2.17	cells	Estimated
n1	Hill Coefficient	3	–	Estimated
n2	Hill Coefficient	4	–	Estimated
n3	Hill Coefficient	2	–	Estimated
n4	Hill Coefficient	3	–	Estimated

These values were estimated to produce the phenomenological and qualitative results observed in the data from²⁷.

Mathematical model of EMT

Gasior et al. developed a mathematical model to examine the changes in membrane-bound E-cadherin (E) and the transcription factor, Slug (S) and capture the bistable behavior observed in MCF7 breast carcinoma cells^15,27. The model is detailed in Equations 1–2. A schematic of this model is found in Fig. 7A, while parameter values and definitions are in Table 1¹⁵. In addition to production (α₁, α₂) and degradation (β₁, β₂), this model incorporates the influence of both pro-epithelial and pro-mesenchymal factors on the bistable switch underlying EMT. In the absence of cell-cell contact, E-cadherin can be endocytosed but, in the presence of neighboring cells, it translocates to the membrane to interact with E-cadherin on neighboring cells³¹. These interactions give epithelial cells their adhesive nature³², and the movement of E-cadherin to the membrane is represented by the term $k_0\left(\frac{{\left(\frac{C}{I{C}_C}\right)}^{n_2}}{1+{\left(\frac{C}{I{C}_C}\right)}^{n_2}}\right)$ in Equation 1, where C is cell-cell contact. Additionally, E-cadherin is anchored at the membrane on the intracellular side by members of the catenin family, such as β-catenin³². By interacting with E-cadherin at the membrane, as well as the full functioning of the GSK-3β degradation complex^32,33, free β-catenin is kept in a low quantity and is unable to translocate to the nucleus to activate production of the transcription factor Slug. Thus, by keeping β-catenin at low levels, the existence of membrane-bound E-cadherin (E) suppresses Slug activation, which is represented by the term $k_1\left(\frac{{\left(\frac{E}{I{C}_E}\right)}^{n_3}}{1+{\left(\frac{E}{I{C}_E}\right)}^{n_3}}\right)$.

Fig. 7. Intracellular signaling EMT model with bistable behavior with respect to TGF-β and cell-cell contact.

A A schematic of the EMT model in refs. ¹⁵ with a double-negative feedback loop between E-cadherin and Slug. B This model hypothesizes that there is a reversible bistable switch with respect to cell-cell contact and an irreversible bistable switch with respect to exogenous TGF-β. In systems without exogenous TGF-β ($T=0\frac{\mathrm{ng}}{\mathrm{mL}}$) present, if an epithelial cell with 6 neighbors (C = 6) loses contact and crosses L₂, it transitions to the mesenchymal phenotype. If that newly mesenchymal cell then gains enough cell-cell contact to cross L₁, it transitions back to the epithelial phenotype. In systems with C = 1 or C = 2 neighbors, the cell begins in the bistable region with the epithelial phenotype. Adding enough exogenous TGF-β forces the cells to cross L₂ and transition to the mesenchymal phenotype. Loss of TGF-β exposure for these levels of contact would not allow them to cross the L₁ threshold, leaving them as mesenchymal cells. This analysis examines 8 different treatment groups marked by black stars: two levels of TGF-β ($T=0,3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$), as indicated by the red dashed lines, and four levels of cellular contact (C = 0,1,2,6 cells), as indicated by the blue dashed lines. Cells with C = 1,2,6 neighbors begin as epithelial cells while C = 0 cells begin as mesenchymal cells in the model. C Time-course simulations for E-cadherin and Slug in cells with 1 (C = 1 cell) and 2 (C = 2 cells) neighbors both with and without exogenous TGF-β using parameter values in Table 1. Levels of E-cadherin and Slug in cells without TGF-β ($T=0\frac{\mathrm{ng}}{\mathrm{mL}}$) are shown with a dashed line. In these epithelial cells, membrane E-cadherin levels are high and Slug levels are low. With the addition of TGF-β ($T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$), E-cadherin quickly decreases and Slug quickly increases to levels associated with mesenchymal cells. Steady state values were quickly reached but all analysis was carried out at t = 10,000 min to capture long-term behavior.

When TGF-β is released from the surrounding microenvironment, it interacts with its receptors on the epithelial cell and activates an intracellular signaling cascade that upregulates the transcription factor, Slug^34,35. This activation is represented by the term $k_2\left(\frac{{\left(\frac{T}{I{C}_T}\right)}^{n_4}}{1+{\left(\frac{T}{I{C}_T}\right)}^{n_4}}\right)$ in Equation 2. When Slug is activated, it is capable of suppressing transcription of E-cadherin³⁶$\left(\frac{{\alpha }_1}{1+{\left(\frac{S}{I{C}_S}\right)}^{n_1}}\right)$, reducing the pool of E-cadherin capable of moving to the membrane to form cell-cell bonds and maintain the epithelial phenotype. Thus, the cell can then undergo the bistable switch and attain the mesenchymal phenotype.

In^15,16,28, this system assumes that cell-cell contact (C) is continuous. In a system without exogenous TGF-β, cells that have 6 neighbors (C = 6) begin as epithelial cells. If the cell were to lose enough cell-cell contact, it would transition to the mesenchymal phenotype. A cell without any contact (C = 0 cells) exists in the mesenchymal state. However, if that cell were to gain enough cellular contact back, it would transition back to the epithelial phenotype. With the addition of TGF-β, epithelial cells with 1 neighbor (C = 1 cell) will undergo a bistable switch and become mesenchymal once a threshold of exogenous TGF-β is surpassed. Cells with 2 neighbors (C = 2) can also begin as epithelial cells but are capable of attaining the mesenchymal phenotype with the addition of exogenous TGF-β. Cells with 3+ neighbors begin as epithelial cells and are unable to go a bistable switch with respect to TGF-β, no matter how much exogenous TGF-β is added¹⁵. A two-parameter bifurcation diagram of this system for cell-cell contact (C) and TGF-β (T) is shown in Fig. 7B. Time-course plots of E-cadherin and Slug for cells with C = 1,2 neighbors when exposed to exogenous TGF-β (T) are shown in 7C.

$$\frac{dE}{dt}=\frac{{\alpha }_1}{1+{\left(\frac{S}{I{C}_S}\right)}^{n_1}}+k_0\left(\frac{{\left(\frac{C}{I{C}_C}\right)}^{n_2}}{1+{\left(\frac{C}{I{C}_C}\right)}^{n_2}}\right)-{\beta }_{1}E$$ 1

$$\frac{dS}{dt}={\alpha }_2-k_1\left(\frac{{\left(\frac{E}{I{C}_E}\right)}^{n_3}}{1+{\left(\frac{E}{I{C}_E}\right)}^{n_3}}\right)+k_2\left(\frac{{\left(\frac{T}{I{C}_T}\right)}^{n_4}}{1+{\left(\frac{T}{I{C}_T}\right)}^{n_4}}\right)-{\beta }_{2}S$$ 2

Latin hypercube sampling and partial rank correlation coefficient

Gasior^16,28 used Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) to understand the sensitivity of E-cadherin (E) and Slug (S) values at t = 10,000 min to changes in the 11 parameters outlined in Equations 1–2 and Table 1^15,16,28. The model was subjected to two different levels of exogenous TGF-β ($T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ and $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$) and four different levels of cell-cell contact (C = 0,1,2,6 cells), creating 8 treatment groups over which sensitivity was examined. Parameters were varied over ranges that included ± 10% and ±25% of the original value given in Table 1^16,28. These ranges are listed in Supplementary Table S1. PRCC requires that the E-cadherin and Slug values at t = 10,000 min are monotonic when each parameter is varied individually^7–9. Additionally, while the original model does not produce negative values for E-cadherin and Slug, some values for α₂ and k₁ produced negative values when explored over the ±10% and ±25% ranges. Thus, the parameter ranges for α₂ and k₁ were shifted to accommodate monotonic behavior and prevent negative values at t = 10,000 min when parameters were varied individually.

Latin Hypercube Sampling requires that the K parameter spaces are sampled N times, where $N>\frac{4}{3}K$⁸. In this sampling methodology, the parameter space is divided into N equiprobable intervals and then each interval is sampled⁸. With 11 parameters, N > 15, but to adequately sample the space,^16,28 chose N = 10,000 samples from uniform parameter distributions. Initial conditions were established for each level of cell-cell contact, and time-course simulations were run in MATLAB for t = 10,000 min. The values of E and S were then recorded at this time point and rounded to the nearest 10⁻⁴ to preserve monotonicity.

To perform PRCC, the parameter values and E and S values at t = 10,000 min are first transformed into ranked values. The partial correlation identifies the linear relationship between each ranked input parameter and the ranked values of E and S after controlling for the effect of the other parameters. The partial correlation coefficient (ρ) ranges from ρ ∈ [−1, 1] and is calculated between the ranked values using Equation (3)³⁷.

$${\rho }_{x_{i}y}=\frac{\mathrm{Cov}(x_i-\widehat{x_i},y-\widehat{y})}{\sqrt{\mathrm{Var}(x_i-\widehat{x_i})\mathrm{Var}(y-\widehat{y})}}i=1,\dots ,K$$ 3

Here $(x_i-\widehat{x_i})$ and $(y-\widehat{y})$ are the residuals between the ranked parameter values (x_i) and ranked outputs y = {E, S} and the linear regression estimates in Equations (4, 5)³⁷.

$$\widehat{x_i}=q_0+\sum _{j=1}^K{q}_j{x}_j\forall i\ne j$$ 4

$$\widehat{y}=d_0+\sum _{j=1}^K{d}_j{x}_j\forall i\ne j$$ 5

The PRCC (ρ) between the parameters and the E and S values at t = 10,000 min was determined using the partialcorr function in MATLAB. Full analysis results from using LHS-PRCC to determine the sensitivity of E-cadherin and Slug values at t =10,000 min to the 11 model parameters from Equations 1–2 can be found in Supplementary Tables S3–S6.

For the analysis presented in Figs. 2–5, all parameters with ∣ρ∣ ≥ 0.5 comprised the reduced subset. If less than 2 parameters were chosen with this cut-off, the three parameters with the largest ∣ρ∣ were included. In these instances, three parameters were included because (1) subsequent Sobol’ requires at least two parameters for analysis and (2) many of the treatment groups had two parameters that did not clear ∣ρ∣ ≥ 0.5 but were of similar value. These parameters are marked in blue in Tables S3–S6.

As a point of comparison, a cut-off value of ∣ρ∣≥0.25 was used to produce the reduced subset for each treatment group in Tables S3–S6. These parameters are marked in yellow. Using a low threshold of ∣ρ∣ meant that at least two parameters comprised the resulting subset. If the subsequent subset was (1) different from the original subset created from the PRCC results or (2) different from the original subset created by the MMS results, new Sobol’ analysis was carried out. These results are found in Figures S23–S24.

If a parameter satisfied the original ∣ρ∣≥0.5 metric used in Figs. 2–5 and ∣ρ∣ ≥ 0.25, this parameter is marked in purple.

Morris method screening

Morris Method Screening (MMS) uses one-at-a-time (OAT) sampling to determine the sensitivity of an outcome to the parameters⁷. While MMS is a general methodology, the resulting estimates depend on assumptions of the parameter distributions⁵. The domain of each parameter X_j for j = 1,…,11 is scaled from [0,1] and divided into l levels. The jth elementary effect or the approximate derivative of an outcome (Y) in the X_j parameter direction is shown in Equation (6). The value of Δ is selected from the range $\mathrm{\Delta }\in \{\frac{1}{l-1},\dots ,1-\frac{1}{l-1}\}$^3,7.

$$E{E}_j=\frac{Y(X_1,\dots ,X_j+\mathrm{\Delta },\dots ,X_{11})-Y(X_1,\dots ,X_{11})}{\mathrm{\Delta }}$$ 6

This process was conducted for r runs where each parameter (X_j) was randomly sampled from the given parameter space. The elementary effects were then used to calculate the sensitivity measures ${\mu }_j^{*}$ and ${\sigma }_j^2$ in Equations (7)-(8).

$${\mu }_j^{*}=\frac{1}{r}\sum _{i=1}^r\mid E{E}_j^i\mid$$ 7

$${\sigma }_j^2=\frac{1}{r-1}\sum _{i=1}^r{(E{E}_j^i-{\mu }_j^{*})}^2$$ 8

MMS was carried out for all 11 parameters in Equations 1–2 and Table 1, excluding the Hill coefficients (α₁, α₂, β₁, β₂, k₀, k₁, k₂, IC_S, IC_E, IC_T, IC_C) under 8 different treatment groups (C = 0,1,2,6 cells and $T=0,3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}})$) and two different parameter ranges ($(\pm 10\%,\pm 25\%$). The ranges for each parameter prior to being scaled to [0,1] are listed in Supplementary Table S1 and are the same as those determined under LHS-PRCC for consistency.

To determine which parameters were influencing E-cadherin and Slug, a threshold of $\frac{1}{2}{\mu }_{\max }^{*}$ was used in each treatment group. This threshold was extended for E-cadherin analysis in the C = 1, 2 cells, $T=3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}}$ treatment groups in both the 10% and 25% ranges. Here, only one parameter surpassed the imposed threshold (IC_C). The threshold was also extended for E-cadherin analysis in the 10% parameter range with C = 1 cell and $T=0\frac{\mathrm{ng}}{\mathrm{mL}}$ due to a larger σ² on β₁. The full MMS results are shown in Supplementary Figures S1-S4. This work used l = 12 levels, and the total number of runs used to calculate ${\mu }_j^{*}$ and ${\sigma }_j^2$ was r = 1200. Code from Herman and Usher was used^38,39.

Sobol’ sensitivity analysis

Sobol’ sensitivity analysis uses the ANOVA decomposition to determine the effects of each parameter on the variance of the values of E-cadherin (E) and Slug (S) at t = 10,000 min. One advantage of Sobol’ is that it can provide extensive information about the impact of the parameters on the outcomes of the model, including the total impact of the parameter on the variance in E and S at t = 10,000 min (S_T), primary impact of the parameter on the output variance (S₁), the interaction effects of all parameters of interest on the output variance (S_T − S₁), and the one-one interaction effects between the chosen parameter and another parameter of interest (S₂), as outlined in Equations (9)-(11).

$$S_{Ti}=\frac{E_{X_{~i}}(V_{X_i}\left(Y\mid X_{~i}\right)}{V\left(Y\right)}$$ 9

$$S_{1i}=\frac{V_{X_i}\left(E_{X_{~i}}\right(Y\mid X_i\left)\right)}{V\left(Y\right)}$$ 10

$$S_{2_{ij}}=\frac{V_{X_{ij}}\left(E_{X_{~ij}}\right(Y\mid X_i,X_j)-V_i-V_j)}{V\left(Y\right)}$$ 11

In these equations, Y is the model output, X_i is the parameter of interest, and X_~i are all other parameters. V is the variance, and E is the expected value. Thus, S₁ is the ratio of the variance of the expected value given knowledge of the parameter to the total variance, while S₂ is the ratio of the variance of the expected value given knowledge of the chosen parameter (X_i) and the other parameter (X_j) in all combinations to the total variance^2,5,40. This study assumes a uniform distribution for each parameter, as detailed knowledge of the underlying statistics of the cellular environment is unknown. Previous work showed that these parameters are uncorrelated^16,28.

Sobol’ was carried out for both PRCC and MMS subsets under 8 different treatment groups (C = 0, 1, 2, 6 cells and $T=0,3.64\times 10^{-6}\frac{\mathrm{ng}}{\mathrm{mL}})$) and two different parameter ranges (±10%, ±25%). The ranges are listed in Supplementary Table S1 and are the same as those used under LHS-PRCC and MMS for consistency.

To determine the S_T, S₁ and S₂ values for each parameter, the number of simulations was doubled until $(S_{T{i}_{2k}}-S_{T{i}_k})/S_{T{i}_k}<0.1$ where i is the parameter, k is the number of simulations per set and n is the number of Sobol’ samples. Code from Herman and Usher was used to calculate the indices³⁸ redwhere Saltelli sampling produced the sequence of Monte Carlo parameter sets. The number of evaluations (k) was calculated in Equation (12).

$$k=n\cdot (i+2)$$ 12

Less than 10% change in the S_T value ensures that the conclusions drawn from Sobol’ analysis will be robust to the estimation error. Supplementary Tables S7–S13 list the number of evaluations for each iteration necessary for convergence, and convergence plots are shown in Supplementary Figures S5–S10. Note that for some doublings, the S_T relative change is larger than 100%, and thus the bar appears to extend past the height of the plot. A relative change of ±100% was established to allow for viewing smaller changes near or within the ±10% range, which is marked with a dashed line. All S₂ results are detailed in Supplementary Figures S17–S22.

Author contributions

K.I. Gasior conceived of and carried out all the work presented here. K.I. Gasior also wrote the entirety of this manuscript.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Code availability

All code used in this work will be made available upon reasonable request.

Competing interests

The author declares no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41540-026-00757-5.