2.1. Stable cell line generation

We engineered the MDA-MB-231 breast cancer cells used in this work to express a stable histone-2B (H2B) nuclear marker (mCherry), along with kinase translocation reporters [32] for the Akt and ERK kinases, as described previously [33,34]. We refer to these cells as pHAEP cells. Only the nuclear marker was used for tracking cells.

2.2. Cell culture

We cultured MDA-MB-231 pHAEP cells in Dulbecco’s Modified Eagle Medium (DMEM) with 10% fetal bovine serum (FBS). We passaged cells at a ratio when they were approximately 90% confluent. For imaging experiments, we cultured cells in imaging media, consisting of fluorobrite phenol red-free medium supplemented by variable FBS (depending on desired experimental conditions), 1X penicillin/streptomycin, 1X GlutaMAX, and 1X sodium pyruvate. Sodium pyruvate was added as an antioxidant to reduce imaging-induced stress.

2.3. Scratch assay and live cell microscopy

For scratch assay experiments, we seeded 50,000 MDA-MB-231 pHAEP cells in 1 ml imaging media in a 24-well glass-bottom imaging plate. We grew cells to full confluency (approximately 36 hours) before starting the scratch assay. For the scratch assay, we manually scratched each cell monolayer using a p200 pipette tip. Immediately after scratching, we washed the cells with 1 ml warm phosphate buffered saline (PBS) and then added 1 ml of warm imaging media containing experimental treatments.

We imaged cells using an EVOS M7000 fluorescent microscope with on-stage incubator. After scratching each well, we placed the well plate into the prepared EVOS incubator. The incubator was maintained at 37^°C, 5% CO₂, and humidity. For time-lapse imaging, we captured fields centered on the scratch near the center of each well. We captured fluorescence from mCherry to position individual cells over time. We acquired images every 20 minutes over 48 hours in all wells in the well plate. We imaged one region of the scratch per well. We quantified wound closure using MATLAB. We stitched the wound images together, drew a line at each side of the wound at 0 hours and 48 hours, and then calculated the distance between the lines.

2.4. Automated image processing

We processed fluorescent images of the MDA-MB-231 pHAEP cells as described previously [35]. Image processing was performed using MATLAB. Briefly, we first thresholded the nuclear images using an adaptive thresholding method. After identifying nuclear pixels, we extracted the centroid pixel of each distinct nuclear object. In contrast to previous work, we did not track each cell between frames, because we were interested in tracking the evolution of the density field, not individual cells. After automated image processing, we extracted cell density fields (number) from each well in the experiment. (Here, is the two-dimensional position vector of a point.) To do so, we applied spatial binning to the cell positions. Bin sizes between 50 and 100 were used. We smoothed in space and time. We applied a moving average filter with a window of 150 to spatial data and a window of 3 hour and 40 minutes to temporal data. Thus smoothing the data alleviates numerical noise in the calculation of derivatives.

The bin sizes were chosen empirically to ensure that each bin contained a sufficient number of cells to meaningfully represent cell density and continuum behavior. This approach naturally led to a noisy field with discrete jumps, as cells moved between bins, necessitating additional filtering to smooth the data field for PDE inference. The filtering window sizes were determined using synthetic data with added Gaussian noise, where we identified the smallest window size that effectively reduced noise while preserving spatial and temporal variations in the data. Synthetic data generation was guided by the expected noise levels in experimental data to ensure robustness.

2.5. Wound healing quantification

We quantified wound closure by identifying the distance between wound edges at the first, and last time instants in the experiment. By aligning the wound edges with the x₂−axis, we estimated the position of the front of each wound as a line x₁ = d(t) at time t and calculated the distance in the x₁ direction (perpendicular to the wound) between each wound front. Then, we calculated wound closure as:

(1)

3.1. Variational formulation of the advection diffusion reaction problem

The advection-diffusion-reaction equation represents transport of cell density, and is encoded in the operator, . Mechanistically, the advection represents the directed motion of the cells, diffusion represents their random motion and reaction models cell proliferation/death. The strong form of the advection-diffusion-reaction equation is written as follows:

(5)

where is the diffusivity, is the advective speed, is a unit vector perpendicular to the wound, and r is the cell reaction rate. The statement of the Initial and Boundary Value Problem (IBVP) includes the initial density , Dirichlet and Neumann boundary conditions on and , respectively, where and :

(6)

The Dirichlet boundary condition specifies the cell density, and the Neumann boundary condition specifies the cell flux on the respective boundaries. Here, n is the unit outward normal vector on the boundary .

There exist models that have considered density-dependent effects on migration, for instance representing cells interacting with each other, and on cell proliferation, for instance, modeling its saturation at high cell density [11,36]. We incorporate such mechanisms in our model by considering parameters that are functions of cell density, D(C), , and r(C).

The weak form of Eq (5) is:

(7)

for all w in a suitable functional space, which we detail below. We seek the solution in the Sobolev space , which consists of functions that are square-integrable and have square-integrable first-order weak derivatives. The field w is a weighting function and belongs to a subspace of defined as .

3.2. The finite element form

The weak form of the advection-diffusion-reaction equation, given in Eq (7), can be discretized using the finite element method (FEM). The domain Ω is partitioned into elements, . The unknown field is replaced by a finite-dimensional approximation , defined over each element using a linear combination of basis functions:

(8)

where are the time-dependent coefficients at finite element nodes , and the spatial dependence of is represented by the finite element basis functions (also known as shape functions). The span of these basis functions defines a finite dimensional H¹ space: . There are many ways to define these basis functions but for our study we will choose piecewise linear basis functions for one- and two-dimensional problems, the latter using triangular elements [37]. Also within the Galerkin approach, we consider finite dimensional weighting functions, which we write as where N_j are the basis functions of V^h that vanish on . Substituting these finite-dimensional approximations C^h and w^h for C and w, respectively, in (7) and invoking its validity for all we obtain the following residual vector:

(9)

where is the vector of finite element basis functions, and the second integral on the right imposes the Neumann boundary condition. The field C^h is used to estimate the spatial derivatives in (9), as and the time derivative as where the is estimated using backward Euler scheme on the time-discretized data point. The residual vector is a finite-dimensional version of the residual of the weak form of the PDE.

3.3. Inference of the advection diffusion reaction system

We infer the advection-diffusion-reaction system by posing it as a standard optimization problem using the residual derived in the previous section and the cell density field from wound healing experiments (see Sect 2.4). The density field extracted from the data is written by interpolation of its values d_i(t) by positioning finite element nodes such that . This allows the use of (8) with the nodal values d_i(t) defining the density field . We impose Dirichlet boundary conditions on all edges. Therefore, the Neumann boundary integral of (9) does not feature in the following discussion.

We consider the following ansatz for the parameters D, and r:

(10a)

(10b)

(10c)

where C is replaced by C^h from Eq (8) during inference. The constant reaction term was omitted, since it represents cell proliferation (r > 0) or death (r < 0) in a region with zero cell density, both of which are unphysical. We included advection terms, even though they have not previously been required to model wound closure, to demonstrate the utility of our method as a model discovery tool, and because advection has been used in other cases to model cell migration tracked with live-cell microscopy [16,17]. We also omitted delay differential terms for modeling the advection diffusion reaction system from the main focus of this work, even though they have been shown to provide slightly better fits to data in the past. [13] Although delay differential terms can be treated within the Variational System Identification framework, they pose significant challenges for PDE-constrained optimization, which is a critical subsequent component of our workflow. The combination of delay differential equations with PDE-constrained optimization remains an open area of computational research beyond the scope of this communication. Instead, in S2 Appendix we present a restricted delay model within the Variational System Identification framework, and show that its performance is comparable to that of the non-delay treatment.

We can now rewrite the residual equations equation as a matrix-vector problem with the following form, where A represents the standard finite element assembly operation that maps local element contributions to the global system of equations [37].

(11)

(12)

(13)

Here, represents the vector of basis functions for where is the number of nodes in the problem, and is a matrix where the columns represent the residual vector terms corresponding to diffusive, advective and reaction operators in weak form from Eq (9). The rows correspond to the components of the basis function vector concatenated over timesteps. We have,

(14a)

(14b)

(14c)

Here, the integration over the element domain is numerically approximated using the Gauss-Legendre quadrature [37]. We finally state the inference problem for the optimal parameters as the following minimization problem with a quadratic cost defined as the Euclidean norm :

(15)

In Variational System Identification, spatial derivatives on the data field are limited to first order, despite originating from a second-order differential equation. In the weak form, derivatives that act on the trial solution in the strong form are transferred to the weighting functions, reducing the need for direct computation of higher-order derivatives. Since noise propagation in numerical differentiation scales as for a d-order derivative, transitioning from a second-order derivative in the strong form to a first-order derivative in the weak form reduces noise amplification by . This significantly improves the reliability of estimated parameters in the presence of noisy data.

In the interest of model parsimony, we seek to estimate the most significant terms given ansatz Eq (10) for the parameters using stepwise regression [24,25,27]. Since an increase in parsimony comes at the cost of an attendant growth in the loss between model iterates, we adopt a search across surviving terms and select for elimination the candidate that, when excluded from the basis, leads to the minimal growth in the loss of the reduced optimization problem from Eq (15). This results in a parameter vector that is sparse in the sense that most of its components for .

3.4. PDE constrained optimization for model refinement

Following Variational System Identification that identifies the parsimonious governing PDE structure, PDE-constrained optimization is employed to refine the numerical values of the coefficients through adjoint-based gradient computations, further improving model fit. The optimization problem statement is:

(16a)

(16b)

(16c)

Here, is a finite element interpolant field that is different from the data-derived field C^h introduced in equation 8. Thus, the optimization problem in (16) involves solution of the forward model, which can be computationally expensive and require exploring regions of parameter space that are numerically unstable, making it unsuitable for large models. However, if a parsimonious model has been identified, then the approach in (16) can be applied to refine the PDE parameters starting from their values inferred by Variational System Identification as initial iterates. We solve the forward model in (16b) using experimentally observed initial conditions, generating a predicted cell density field, which enables computation of an associated loss in (16a). However that minimization problem requires computing a variational derivative through the chain rule which also involves evaluating the derivative of with respect to . We complete this step via adjoint-based gradient optimization with the BFGS solver. PDE-constrained optimization with adjoint-based gradient computation does not change the selected operators but generates a new set of parameter values, which fit the data better than the those obtained by Variational System Identification with stepwise regression.

3.5. Computational framework for inference

The numerical examples presented in this work have been posed and solved in two dimensions by the finite element method implemented on the FEniCS platform [38,39]. The 1D forward solutions presented in this work were carried out on a uniform mesh with 42 piecewise linear elements. The 2D solutions were obtained on a rectangular domain using a structured mesh with 6478 linear triangular elements. The nonlinear optimization for minimizing (15) was carried out using the Newton solver in the FEniCS package. The PDE constrained optimization (16) was implemented in the dolfin-adjoint package [40,41]. In the computational studies presented in the next section, for the nonlinear PDE solver, we used a Newton solver with absolute and relative error tolerance stopping criteria of 1e–8 and 1e–9 respectively. For dolfin-adjoint we used ‘L-BFGS-B’ with tolerance of 1e–9.

It is also worth noting the possible non-uniqueness of the identified model. In general, the ability to uniquely recover PDE parameters depends on the richness of the data, particularly the initial and boundary conditions as well as the spatiotemporal resolution of observations. Given a specific choice of initial conditions, the available simulation data may not contain sufficient information to uniquely determine all mechanisms in the PDE, potentially leading to non-uniqueness in the inferred parameters. In our previous work [42], we demonstrated that while this inference technique can successfully recover a PDE model whose solution closely matches the ground-truth simulation, the inferred parameters may differ from the true values. This suggests that while the model captures the essential dynamics of the system, multiple parameter sets may yield similar macroscopic behavior. In such cases, it is important to assess whether the inferred parameters are biophysically reasonable and consistent across different experimental conditions. In Sect 4.2, we will use the sensitivity plots to determine the degeneracy in the inferred model. From a more abstract perspective, however, it is also possible for non-uniqueness to be intrinsic to the model itself, regardless of data quality or resolution, as highlighted in recent works on structural identifiability of PDE models [43]. These studies distinguish between globally and locally structurally identifiable models, and our framework, which is linear in parameters, falls under this setting. While our VSI formulation guarantees a unique global minimum under a positive definite quadratic form, subsequent PDE-constrained optimization explores only local neighborhoods and may therefore overlook other admissible solutions.

4.1. Variational system identification of 1-D wound healing dynamics from data

We first tested Variational System Identification on a previously reported scratch assay dataset. The dataset, collected by Jin et al., [11] consists of scratch assays performed on PC-3 prostate cancer cells with varying initial densities prior to the scratch. Different extents of confluence were achieved by varying the initial seeding between 10,000 and 20,000 cells in steps of 2000. The high cell densities that were achieved in this experiment justified the consideration of continuum advection-diffusion-reaction PDEs to model these data. For each of the resulting six datasets, cell density was measured every 12hr for 48hr and averaged across three replicates. Next, the density was averaged along the (x₂) direction parallel to the scratch, yielding a one-dimensional density field (along x₁) for each initial density.

The 1D cell density profiles over time are shown in Fig 2. The scratch, which is approximately long, is clear in the initial density profiles. For all initial seeding with 14000 or more cells, the center of the scratch (around ) is occupied by cells after 36 hours. Furthermore, there is an increase in cell density at the scratch edge (near ) over time for all initial seeding densities, indicating that the cells continue growing during the experiment.

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Fig 2. Experimental wound healing data (symbols), the model inferred by Jin et al. [11] (dashed curves), and the model inferred by Variational System Identification and refined by PDE-constrained optimization (solid curves).

For the 18000 initial density data, the 4-term model was used. Inferred model parameters appear in Table 1.

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Table 1. Model terms learnt by Variational System Identification and PDE-constrained parameter optimization for each experimental condition presented in Jin et al.

As in the text, C is the cell number density measured in the units of cells/.

https://doi.org/10.1371/journal.pcbi.1013607.t001

Next, we used our computational pipeline to find parsimonious models for each of the six scratch assay datasets corresponding to different initial cell densities. First, we used the data to generate diffusivity, advective velocity, and reaction rates that are polynomial functions of density up to second order (quadratic in (10)). Then, we used VSI to identify models from the generated bases. Fig 3 shows that for all cases initial cell densities, the loss does not decrease for models having more than 3 terms. In every case, the three-term model excluded advection. We next performed PDE-constrained optimization of the 3-term model for each of the six cases with different initial number of cells seeded, with the additional physical constraint of positive diffusivities. The diffusivity in each resulting optimized model has a constant term, and either a linear or quadratic dependence on local density, C. All the inferred reaction functions have linear density dependence, except the 4-term model for 18,000 cells (see Table 1). For the case of 18,000 cells, we observed minimal improvement in the cost function Eq (16c) from the PDE-constrained optimization. The model, both before and after refinement, is presented in Table 1 under the label 18,000 (3 terms). Furthermore, we noticed that in the 3-term model, the partial derivatives of the cost function with respect to the diffusivity coefficients remained low, indicating that the available data was not sufficiently sensitive to the diffusive mechanism of random cell motion. To address this issue, we expanded the model to include four terms–incorporating both linear and quadratic dependence on the local cell density. This expansion led to a significant reduction in the cost function during PDE-constrained optimization. The refined model, presented in Table 1 under the label 18,000 (4 terms), demonstrates a strong agreement between the model solution and the experimental data. The diffusivity values after refinement by PDE-constrained optimization have a constant term and a positive linear or quadratic dependence on the local cell density, C. This suggests a mechanism of intercellular communication between the cells, which is likely to be contact-mediated, and drives the cells to further explore their immediate vicinity by random walks. However, given that the maximum cell density satisfies (Fig 2), this density dependence is weak. The constant diffusivity term has a roughly increasing trend with the initial number of cells seeded (except for the outlier at 18,000 cells that interrupts this increasing trend, Table 1). The trend for higher diffusivity with increase in the number of cells initially seeded is consistent with the increase of diffusivity with local density C. The proposed mechanism of contact-mediated random walk explorations of the environment is also a reasonable explanation for both aspects of increasing diffusivity. Similarly, refinement of the inference results by PDE-constrained optimization yields a reaction (proliferation) term with a weak linear dependence on local cell density C for all except the 18,000 cell seeding case. Note that the final form of the reaction (proliferation) with the 4-term 18,000 cell model has a negative quadratic dependence on density that results in a decrease of proliferation at higher densities.

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Fig 3. Elbow curve illustrating the Variational System Identification (VSI) loss at each step of stepwise regression.

At each step, the loss is computed as the minimum residual norm over the space of all admissible parameters for the current model.

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We used these refined models to run a forward simulation from the initial conditions of each dataset with the different numbers of initial cells seeded. The model predictions and experimental observations appear in Fig 2, where we also present the forward simulation of the reaction-diffusion model using the parameters provided by Jin et.al. [11]. We draw the reader’s attention to the qualitative match between the forward solutions and the dynamics observed in the dataset at early times and the quantitative match at later times. We found that for four of the six initial seeding densities, our models have root mean squared errors (RMSEs) that are significantly lower than the RMSE reported by Jin et al. and are similar to their results for the other two cases (Fig 4). These results suggest that our inference framework of Variational System Identification and PDE-constrained parameter optimization can be used to infer models for cell migration dynamics that are competitive with previous methods such as those by Jin et al, which fit a model based on prior knowledge about the system. In this work, we considered a polynomial cell density dependence in the diffusivity for modeling nonlinear diffusion, but at higher cell densities, more rigorous models, such as Maxwell-Stefan diffusion would be more appropriate [42]. In such formulations, phenomenological extensions can be constructed by treating void space as an additional species, which effectively yields a single-independent-component representation of nonlinear diffusion.

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Fig 4. Root mean squared error (RMSE) calculated between experimental data and reaction-diffusion models from Jin et al. (blue), and a model obtained by PDE-constrained optimization following Variational System Identification (red) for each case with different initial number of seeded cells.

For the 18000 initial density data, the 4-term model was used. The model parameters inferred in this work are shown in Table 1.

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4.2. Model-based quantification of the effect of inhibitor levels on migration dynamics in 2-dimensional scratch assays

After demonstrating that Variational System Identification and PDE-constrained optimization could be used to learn parsimonious advection-diffusion-reaction models for cell migration data in the literature, we used the approach on our own cell density data gathered from fluorescence microscopy experiments. We performed scratch assays using MDA-MB-231 breast cancer cells and tracked scratch closure over time using live-cell fluorescence microscopy Fig 5. As seen in Fig 6A, the cell densities achieved are comparable to those observed by Jin et al., [11] and admit treatment of the problem using continuum advection-diffusion-reaction PDEs.

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Fig 5. Fractional wound closure Eq (1) calculated for a range of trametinib concentrations and an untreated control (labeled NT), with N = 4 wells for all conditions.

Individual datapoints are shown in red, and the mean +/- standard deviation is shown by the bars and error bars, respectively.

https://doi.org/10.1371/journal.pcbi.1013607.g005

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Fig 6. (A) Comparison of experimental data and predictions of a model inferred by Variational System Identification followed by PDE-constrained parameter optimization under the same initial conditions.

The experimental condition is with no treatment. (B) Variational System Identification loss as a function of the number of terms in the model inferred for the untreated and 100 nM trametinib conditions. (C-D) PDE-constrained and optimized diffusivity (C) and reaction (cell proliferation) (D) terms as functions of density. Diffusivity is constant, while reaction/proliferation is a combination of linear and quadratic terms. The horizontal axis ranges were chosen to represent the densities present in the experimental data.

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We used this assay to compare how trametinib, a MEK kinase inhibitor, affects cell migration. MEK kinase is known to regulate cell migration and proliferation [44]. Our past study showed that trametinib reduces both random and directed motion in chemotaxis [18]. We captured wound-closure dynamics for cells exposed to 5 trametinib concentrations from 10 μM to 100 nM, and for an untreated control. We observed that untreated cells were able to almost close the wound after 24 hours, while 100 nM trametinib was enough to significantly inhibit wound closure. Higher trametinib concentrations did not further inhibit wound closure (Fig 5).

We next applied our PDE inference pipeline to learn models for the observed wound closure dynamics. We used the same library of candidate operators, including constant, linear, and quadratic terms for diffusion and advection, and linear and quadratic terms for reaction. Four replicates of the scratch assay were obtained for each treatment condition (five trametinib concentrations and one with no treatment). The fractional wound closures across the replicates for each treatment condition had standard deviations less than 15% of the mean (Fig 5). Guided by this reproducibility, the PDE parameters were inferred using the data aggregated across replicates, generating a single model for each condition. We aggregated the data by minimizing the VSI loss function (equations 15 and 16) across all 4 replicates. The PDE-constrained minimization procedure results in further refinement of losses as shown in the Fig 7. The average loss and individual losses for each replicate are presented in Fig 8, indicating consistent losses across replicates. This verifies that the inferred PDE model does not exhibit sensitivity to initial conditions, such as distribution of cells, which would vary between the replicates. Finally, we ran the inferred model for each condition starting from the experimentally measured initial cell density (Fig 6A). We observed that the Variational System Identification loss function is essentially constant for two-term models including diffusion and reaction terms, and only starts to increase for models that lack reaction terms (Fig 6B). Based on the these results, we adopted a three-term model. For both the untreated condition and 100 nM trametinib, these terms corresponded to constant (density-independent) diffusivity, and reaction terms that are linear and quadratic in cell density. Table 2 shows the coefficients inferred initially by Variational System Identification in the process of delineating mechanisms, and their subsequent refinement by PDE-constrained optimization. After model refinement, our inferred models show that diffusivity decreases by approximately 44%, from in the untreated cells to , in 100 nM trametinib, and remains comparable for higher levels of the inhibitor. We also found a non-linear functional dependence of cell proliferation (reaction in the PDE) rate on cell density: positive at low cell densities and negative above a critical density; i.e., the model predicts cell death sets in (Fig 6D). This inferred mechanism could be interpreted as a crowding-induced death in the cell population. However, no cell death was observed in our experimental data. The models inferred with and without trametinib suggest that even 100 nM trametinib causes a decrease in random cell migration and cell proliferation. Notably, the effect of trametinib treatment quickly saturates, as is reflected in the coefficients for constant diffusivity and linear/quadratic reaction terms in Table 2. The RMSEs between the forward predictions of the model and the data are presented in Fig 8, and are essentially comparable across trametinib concentrations and for the untreated case.

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Fig 7. Loss minimization during PDE-constrained optimization process.

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Fig 8. RMSE evaluated between the forward prediction of VSI+Optim models (Variational System Identification and PDE-constrained parameter optimization) presented in Table 2.

The red dots represent the RMSE for each replicate, and the bar represents the mean value.

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Table 2. Model terms inferred by Variational System Identification and PDE-constrained parameter optimization for each experimental condition shown in Fig 3.

As in the text, C is the cell number density measured in the units of cells/.

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To assess the sensitivity and robustness of the inferred parameters in the 3-term model, we analyze the contour plots of the loss function (16a) as a function of parameter variations in Fig 9. In each plot, one parameter is held fixed at its inferred value, while the remaining two are systematically varied. The first two columns reveal a shallow minimum in the diffusion parameter direction, suggesting that a broad range of admissible diffusive values can yield similarly low losses, indicating lower sensitivity to diffusion estimates. The third column highlights an affine relationship between the coefficients of the linear and quadratic reaction terms, showing a lack of sensitivity along this direction. It can be shown that the ratio is equal to the carrying capacity (the maximum cell density) used in previous work [11]. The linear relationship between C₂ and C₁ shows that while the carrying capacity is sharply estimated, the growth rate exhibits greater uncertainty.

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Fig 9. Sensitivity analysis of the inferred parameters in the 3-term model: .

Contour plots show the normalized loss function from Eq (16a) as a function of parameter variations. Each column represents a case where one parameter– C₂ (left), C₁ (middle), or D₀ (right)–is held fixed at its inferred value, while the remaining two parameters are varied in a neighborhood around their inferred values: , , and , respectively. Each row corresponds to one of the six experimental conditions. Contours illustrate the sensitivity of the loss function to changes in the inferred parameters, where sharp minima indicate high sensitivity (well-constrained parameters), and broader regions suggest lower sensitivity (potential parameter degeneracy).

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We note that our current analysis uses point estimates of the inferred parameters, which does not provide a complete picture of the underlying parameter distributions. If the posterior distribution of a parameter is broad or flat, then comparing single values, whether they are maximum likelihood estimates (MLEs), maximum a posteriori (MAP) estimates, or other point summaries, can obscure the true level of uncertainty. A more comprehensive characterization in terms of posterior means, variances, correlations, and higher moments would provide a deeper understanding of how drug treatments influence the parameters.

A full Bayesian treatment using Markov Chain Monte Carlo would represent a rigorous approach to obtain such posterior distributions. However, for PDE models with repeated forward evaluations, especially in the presence of the high variability suggested by our sensitivity plots, this becomes computationally expensive, requiring a large number of proposed points to adequately explore the parameter space. In place of such a Bayesian treatment, which we are pursuing in related work, we have performed a post-inference variability analysis of the inferred parameters using a local Gaussian approximation to the posterior in S1 Appendix. As that analysis shows, the standard deviation on the inferred diffusivity is of the order of the reported parameter values. This large uncertainty reflects the nature of random motion and the second-order Laplacian of the concentration which amplifies noise in the data as we have shown previously. [24] Given the analysis in S1 Appendix, the % decrease in diffusivity upon application of 100 nM trametinib that we report in Table 2 corresponds to one parameter set that explains the data, but other combinations with smaller differences in diffusivity also could be admissible models.