2.1. Computational framework of protein interactions in the polarization pathway

To study the emergence of polarization due to the different protein interactions, we adapted and extended the computational framework proposed by Jilkine et al [18] that describes the kinetics, crosstalk, and diffusion of the active and inactive form of the three different Rho-GTPases (Rho, Rac and Cdc42) in yeast cells. In light of our goal to understand the establishment of apical-basal polarization in renal epithelial cells, we altered the crosstalk accordingly and we added equations to model the binding kinetics, crosstalk, and diffusion of the apical polarity Par complex and the basolateral Scribble complex. Fig 1A depicts a schematic summary of the hypothesized signaling cascade responsible for the establishment of polarization in renal epithelial cells, which we implemented in our computational model. In short, the establishment of polarity is initiated by the binding of integrins to their specific extracellular matrix (ECM) ligand that causes activation of the integrins (e.g., β1 integrin) [19]. The activation of integrins then leads to the upregulation of active Rac1 [20,21]. Rac1 is in an antagonistic relation with Rho and can also be inactivated by the Par complex. Rho is responsible for Cdc42 activation that in turn upregulates the formation of the Par complex from its individual components (Par3, Par6 and aPKC). The Par complex downregulates the formation of the Scribble complex from its individual components (Scribble, Dlg, Lgl) and vice versa.

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Fig 1. Schematic overview of the polarization model extended from [18].

(A) This scheme shows the hypothesized protein interactions that enable the establishment of polarization due to the activation of integrins. Briefly, the activation and inactivation (noted by the subscripts a and i, respectively) of integrin heterodimers (top part) influence the crosstalk between the Rho-GTPases (middle part) via upregulation of active Rac. In addition to the crosstalk between different proteins, the Rho-GTPases themselves also switch between their active an inactive forms. The bottom part displays the interaction between polarity complexes (the complex is indicated by the subscript c). (B) Illustration depicting examples of concentration profiles in the investigated 1D domain (0 ≤ x ≤ L) with the basal domain (x = 0) connected to the ECM and the apical domain (x = L) at the opposite side of the cell.

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We based the antagonistic relation between Rho and Rac on the observation that the majority of the mechanisms in epithelial cells that connect Rho and Rac were found to lead to mutual inhibition [22]. The activation of Cdc42 by Rho rests on the finding that RhoGEF CGEF-1 is suggested to activate Cdc42 during the establishment of polarization [23]. We assumed that Cdc42 could stimulate the formation of the Par complex, since activated Cdc42 is known to recruit Par6 to the apical membrane [24]. The binding between Cdc42 and Par that leads to the formation of the Par-complex has been shown to be regulated by other proteins, such as the phosphoinositides [24,25]. Since these other proteins do not affect the direct relation between Cdc42 and the Par-complex, we have chosen to only model this direct relation [24,25]. Since Rho activates Cdc42 and Cdc42 in turn activates Par, a simplification in the model could be offered by modeling a direct relation between Rho and Par and thus leaving out Cdc42. We chose to omit this simplification based on the notion that Cdc42 is highly involved in the binding of Par complex and the extent of this role is not known [24,25]. As another part of the Par-complex (Par3) has been found to inhibit Tiam-Rac signaling at the apical membrane in MDCKs, we assumed that Rac is inhibited by the Par complex [12]. Finally, the downregulation of the formation of the Par-complex by the Scribble complex and the other way around is based on the known mutual inhibition of the Par and Scribble complex [26].

Collectively, the different protein interactions were modeled via a set of twelve partial differential equations (PDEs). The approximate solution of the set of PDEs was determined using the Finite Difference Method (FDM) and obtained using explicit time integration in MATLAB (R2021a, MathWorks Inc., Natick, MA, USA). The FDM is a convenient method for solving PDEs and the use of a simple 1D geometry does not require a more complex method such as the Finite Element Method. Decomposition of a set of PDEs in the FDM was performed with a forward difference in time and with a second-order central difference for the spatial derivative (as incorporated in diffusion terms). A 1D analysis along the apical-basal axis of a renal epithelial cell, giving a 1D representation of the cell, was performed on the domain 0 ≤ x ≤ L, where x = 0 and x = L denote the basal and apical boundaries of the cell, respectively (Fig 1B). Different mesh sizes were tested (10, 30 and 100 subdomains), revealing that using 30 subdomains (31 nodes) was sufficient to reach mesh convergence. Lastly, different time increments were tested (dt = 0.01, 0.005 and 0.001 seconds), showing that a time increment of 0.005 s was sufficient to ensure accuracy of the solution.

2.2. Model equations

The activation and inactivation of integrins are described as follows: (1) (2)

Here, I_a corresponds to integrins in an activated state, while I_i describes the inactivated state of integrins. The activation and de-activation rates are represented by I_I and δ_I respectively. Since integrin activation is enabled by the ECM, we assume that integrin activation is only possible at the cell border connected to the ECM and that activated integrins are not able to diffuse due to their connection to the ECM. Inactive integrin on the other hand is known to be transported via vesicles, and diffusion is therefore included via the ∇²I_i term that scales with the diffusion coefficient [27]. It has to be noted though that the spatial constraint imposed on the active integrins also causes a spatial predisposition for the inactive integrins, as the total amount of integrins per element is constant and divided amongst both fractions.

The active and inactive forms of the Rho-GTPases (Cdc42, Rac and Rho) are described by Eqs 3–5 and 6–8, respectively: (3) (4) (5) (6) (7) (8)

Here C_a (x,t), R_a (x,t), ρ_a (x,t) represent the concentrations of active Cdc42, Rac, and Rho at position x and time t respectively and C_i (x,t), R_i (x,t), ρ_i (x,t) the concentrations of inactive Cdc42, Rac, and Rho at position x and time t. C_tot, R_tot, ρ_tot represent the total concentrations for Cdc42, Rac, and Rho. The inhibition rates are indicated by δ_C, δ_R, δ_ρ. The baseline activation rates of Cdc42, Rac and Rho, are represented by I_C, I_R, I_ρ, respectively. β_ρ, β_PR are constants that govern the rate of inhibition of Rac by the concentration of activated Rho and the concentration of the Par complex (P_C), respectively. The inhibition is described by the Hill equation with n being the Hill coefficient. β_R is the constant that governs the rate of inhibition of activated Rho by the concentration of activated Rac. Inhibition and thus a decrease of activation rate is described by the Hill equation with n being the Hill coefficient. α_I, α_ρ are constants that describe the activation of Rac by the concentration of active integrin and the concentration of active Rho respectively. and represent the second order spatial gradients that determine the diffusion of the active and inactive forms with corresponding diffusion coëfficients and . We assumed that the inactive forms located in the cytosol diffuse faster than the membrane-bound active forms, which implies that and .

We based the formation and dissociation of the polarity complexes on the same principle as the switching between the active and inactive forms of the Rho-GTPases and the integrins. In this way, the total concentration of proteins is divided into two fractions: a concentration that represents the unbound proteins (Par3, Par6 and aPKC, and Scrib, Dlg and Lgl) and a concentration that represents the bound Par and Scribble complex. This leads to the following two equations for the Par complex: (9) (10)

Here, P_un (x,t) corresponds to the collective concentration of Par3, Par6 and aPKC in an unbound state, which means that the individual values of Par3, Par6 and aPKC are not accounted for. P_C (x,t) describes the bound state of the Par complex. To distinguish the terms related to the polarization complexes from the terms related to the Rho-GTPases and integrins, we chose to describe the rates responsible for the switch between the bound and unbound state of the polarity complexes via k_on,P and k_off,P and refer to these as association and dissociation rates. The rate of formation of the Par complex as determined by the concentration of active Cdc42 is set by α_C. β_SP governs the rate of dissociation of the Par complex as a function of the concentration of the Scribble complex. Inhibition is again described by the Hill equation with n being the Hill coefficient.

The formation and dissociation of the polarity complex Scribble is described by: (11) (12)

S_un (x,t) represents the collective concentration for the Scrib, Dlg and Lgl in an unbound state, while S_C represents the bound state of the Scribble complex. The switch between the bound and unbound state corresponds to the association and dissociation rates (k_on,S, k_off,S). β_PS governs the rate of dissociation of the Scribble complex based on the concentration of the Par complex. Once more, the inhibition is governed by the Hill equation with n being the Hill coefficient.

2.3. Parameter estimates

Since many values of the parameters can be inferred from different cell lineages but not specifically from renal epithelial cells, we have based the majority of our parameter estimates on the composite set of data of various cell types used in Jilkine et al. [18]. Slight variations in these parameters did not lead to differences in the results of the model, indicating that the parameter values were in an acceptable range. Parameters describing the total concentrations of the Rho-GTPases were derived from own performed experiments, which are described in the experimental materials and methods section. For the parameters β_PR and α_ρ, which were not included in Jilkine et al. [18], values were chosen in the same order of magnitude as the other inhibition and activation constants, for example β_R and α_C respectively. The value for I_C was decreased compared to Jilkine et al. [18]. The original value was suitable for the response in the GTP-ases, which is described by the original model from Jilkine et al. [18], but was not able to describe the response of the polarity complexes properly. We tested different values of I_C within the order of magnitude given by I_R and I_ρ, and selected the value that led to larger differences in the concentration at the apical and basal pole for the polarization complexes Par and Scribble, and thus a stronger polarized distribution. The parameter values for the Rho-GTPases are summarized in Table 1.

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Table 1. Parameter values for the Rho-GTPases in the polarization model.

https://doi.org/10.1371/journal.pcbi.1012140.t001

In our experiments, the concentrations of small G-proteins in MDCKs were determined using immunoblotting (Fig 2A). For this, we compared the average band intensities for RhoA, Rac1 and Cdc42 in MDCKs cultured on transwell filters to the band intensity of the corresponding control protein with a known concentration (3 ng ml^-1) (Fig 2B). Via calculations based on Marée et al. [28], we obtained the total concentrations, i.e. C_tot, R_tot, ρ_tot of 4.4, 2.4 and 4.0 μM respectively. In short, we first calculated the amount of each Rho-GTPase per lysate volume. Next, assuming that the total lysate represents 1.5 million cells, which is based on the number of MDCKs in a confluent culture, we calculated the amount of each Rho-GTPase per cell. Based on the molecular weight (21 kDa) and approximating cells as 10 μm diameter spheres, we calculated the amount of molecules per cell. With the Avogadro number we subsequently calculated the total concentrations. The total concentration of each protein was divided into an active and an inactive fraction, which were used as the initial conditions of the model. The total concentrations and initial conditions were spatially uniform and thus imposed on each node.

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Fig 2. Quantification of total amount of Rho-GTPases.

(A) Expression levels of RhoA, Rac1 and Cdc42 in MDCKs cultured on transwell filters and expression levels of the control protein (CP) as analyzed by western blotting using protein-specific antibodies. (B) The average band intensities for the Rho-GTPases and the band intensity of the corresponding control protein were calculated (error bars represent the standard deviation, n = 3).

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The parameter values for the polarity complexes and integrin activation are summarized in Table 2. The association and dissociation rates (k_on, k_off), and the total concentration of Par complex were taken from the work of Wen et al [29]. The diffusion rates of the Par and Scribble complex were subdivided into a cytosolic diffusion rate for the unbound proteins and a membrane diffusion rate for the entire complex [30]. We assumed that Scribble exhibits the same behavior as Par and therefore we used the same set of parameter values. Additionally, integrins are present in either an activated or an inactivated conformation. We based the total concentration of integrins on the assumption of a basal surface of 100 μm² together with an integrin density of ~100 integrins per μm² [31,32]. In addition, we assumed that active integrins are not able to diffuse through the cell as they are connected to the ECM, while inactive integrins are often transported via intracellular vesicles [33]. The activation and inactivation rates of these integrins were taken from previous studies and amount to 10 μM s^-1 and 1 s^-1 respectively [34,35].

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Table 2. Parameter values for the polarity complexes and integrins in the polarization model.

https://doi.org/10.1371/journal.pcbi.1012140.t002

2.4. Initial and boundary conditions

Since we considered that active integrins exist only close to the membrane, initial values of integrins were assigned to the first five nodes (one-sixth of the domain and thus approximately 1.67 μm) with an exponential decay (Eq 13): (13)

Here, x ranges from 1 to 5, representing the first five nodes and thus 1.67 μm starting from the basal side. Homogeneously distributed initial concentrations of active Rho-GTPases and Par and Scribble complexes were applied with C_a (x,0) = 0.8 μM; ρ_a (x,0) = 1.0 μM; R_a (x,0) = 1.75 μM and P_C (x,0) = S_C (x,0) = 0.6 μM. Concentrations of the initial inactive Rho-GTPases and unbound polarity complexes were defined as the difference between the conserved total concentration and the active concentration of the protein in question. The initial spatial uniform input values for the active and inactive fractions were changed compared to Jilkine et al [18]. (C_a (x,0) = 1.00 μM; ρ_a (x,0) = 1.25 μM; R_a (x,0) = 3.00 μM), as we aimed to model renal epithelial cells instead of yeast cells. We chose the overall fraction in the GTP-bound-state at steady state to be around 30%, based on the comparison to erythrocytes and neutrophils [36,37]. We assumed that the percentage of activated Rac1 and Cdc42 is two times higher than the percentage of activated Rho (30% compared to 15%), in correspondence to previously experiments involved with other epithelial cells [38–40].

At the outer cell membrane, a no-flux boundary condition was prescribed for all diffusible substances: C = 0 at x = 0, L. Boundary conditions were imposed as the first order backward differences at the basal side and as the first order forward differences at the apical side.

2.5. Multiparametric sensitivity analysis (MPSA)

Our computational framework includes a large number of parameters, of which some have a high level of uncertainty. This specifically concerns the parameters that define interactions between different proteins, as the strength of their interactions (e.g., the strength of inbition of Rac on Rho) and significance of the individual interactions in the polarization pathway are not known. Therefore, we performed a multiparametric sensitivity analysis (MPSA) to investigate which of our parameters have the strongest influence on establishing apical-basal polarization. To this end, we selected a set of 21 parameters (I_c, δ_c, α_c, I_R, δ_R, β_R, I_ρ, δ_ρ, α_ρ, β_ρ, k_on,P, k_off,P, β_PR, β_PS k_on,S, k_off,S, β_SP, I_I, δ_I, I_tot, α_I ∈ X_V) from the total set of 38 parameters. These parameters were chosen, as they are essential constants in the equations that describe the interactions between different proteins that cannot be measured directly in experiments.

We randomly changed the 21 parameters (X_V) simultaneously and uniformly over the whole domain in ranges of 5 times lower to 5 times higher values (20–500% of the original parameter values displayed in Tables 1 and 2). High ranges for variation were chosen, as most of the parameters could not be determined experimentally or derived from previous studies. We varied these 21 parameters in the predefined range (20–500% of the original parameter value) using Latin hypercube sampling to create 3000 samples of parameters space [41–43]. A Monte Carlo sampling scheme was used to select 3000 parameter combinations to run simulations in parallel and assess the obtained output results in relation to the prescribed threshold. The simulation with the original parameter values (displayed in Tables 1 and 2) served as the reference case (X_O).

In each of the 3000 simulations, we calculated six outputs (Y) from the model: the concentrations of active integrin, Rac, Rho and Cdc42, and the concentrations of the Par-complex and Scribble-complex (I_a, R_a, ρ_a, C_a, P_C, S_C ∈ Y). For each output (n = 1, 2, …, 6) and each simulation (m = 1, 2, …, 3000), we determined the absolute difference between the concentration at the apical () and basal () side for a given set of the 21 parameters () relative to the original absolute difference (X_O) in percentages: (14)

We defined that the system achieved polarization when ΔP had a value higher than 20%. Therefore, we assigned value 1 to simulations with ΔP > 20% (‘accepted’) and value 0 to simulations with ΔP ≤ 20% (‘unaccepted’).

The influence of each parameter (p_i ∈ X_V) on the six outputs (Y) was then statistically evaluated using the Kolmogorov-Smirnov (K-S) test [44]. To perform the statistical test for each of the six outputs (Y), we considered the 21 parameters one by one. We assigned arrays of 0 and 1’s (unacceptable vs acceptable simulations) to each value of the parameter. Subsequently, the values of each parameter were sorted and cumulative distributions for the unacceptable and acceptable cases were created. For each parameter (p_i ∈ X_V), the unacceptable and acceptable cumulative distributions were normalized to the corresponding maximum values of the respective cumulative distributions. Finally, the absolute distances between the acceptable and unacceptable distribution values for each of the parameter values were calculated. An example of the distance between the acceptable and unacceptable distributions is given in Fig 3. The maximal absolute distance for each parameter and each output is then the K-S distance and is represented by: (15) where S_a and S_u are the cumulative distributions of acceptable and unacceptable cases, respectively, n stands for the output serial number (n = 1, 2, 3, …, 6) and p_i represents one parameter within the set of 21 parameters X_V. Thus, for each output the influence of each of the 21 parameters is explained by the K-S distance, where larger values indicate increased influence compared to smaller values.

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Fig 3. Example of the maximal absolute distance or K-S distance between the acceptable and unacceptable distribution.

The example portrays the K-S distance for the parameter β_R with respect to Rho output. The K-S distance is defined as the maximal absolute distance between the distribution of acceptable and unacceptable cases.

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Together with the 21 input parameters, which were selected out of total 38 input parameters (listed in Tables 1 and 2) to be subjected to the sensitivity analysis, we included an additional set of 21 random values; the so called ‘dummy’ values. With these values, we calculated cumulative distributions and K-S distances in the same manner as for the 21 input parameters to determine the threshold sensitivity criteria. Dummy values were set at a value of 1 and were varied in the same range (20–500%) as the 21 parameters, so we obtained 3000 sets of dummy values as well. Subsequently, we assigned the same arrays of 0 and 1’s (unacceptable vs acceptable simulations) to each value of the dummy values as we did for the parameters and calculated the K-S distance for each dummy value using the same method. From the final 21 distances obtained from the dummy values and for each output, the maximal distance is taken to represent the threshold for the parameters’ sensitivity. Thus, K-S distances of the parameters that are larger then the maximal K-S distance of all dummy values considered to have a significant influence on the model output.

2.6 Potential impact of the degradation or upregulation of Rho-GTPases on the induction of EMT

The importance of the Rho-GTPases for establishing polarization was simulated by individually degrading or upregulating the active part of each Rho-GTPase (C_a, R_a, ρ_a) and updating the total amount of proteins accordingly. Exponential functions were used to capture the behavior of degradation observed in the experimental results of Abi Habib et al. [45], who investigated the degradation patterns of ubiquitinated proteins by different proteasomes. Similar exponential curves were employed to model increases of the active part of each Rho-GTPase over time. Different degrees of degradation or upregulation were imposed, ranging from 0% to ±100% of the protein in question (Fig 4). The general temporal profiles for degradation and upregulation are given by: (16)

With P representing the percentage of the active Rho-GTPase (C_a, R_a, ρ_a) over time (t) in seconds, and a = -100, -80, …, 80, 100 the percentage of degradation (a < 0) or upregulation (a > 0) in steps of 20%. All degradation and upregulation simulations were performed by first running the simulations with the original concentrations of the Rho-GTPases for 100 seconds to achieve a polarized steady state for all proteins. After these 100 seconds, the degradation or upregulation of one of the Rho-GTPases was prescribed for a period of 100 seconds. In this period, the degradation/upregulation of the active part of the selected Rho-GTPase, together with updating the total amount of the selected Rho-GTPase, was performed at the end of each time step. The inactive parts were then updated according to Eqs 3–8. After the degradation or upregulation period, the concentrations of the degraded or upregulated Rho-GTPases were kept constant for an additional 100 seconds.

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Fig 4. Different temporal profiles were prescribed to describe the degradation and upregulation of each of the Rho-GTPases.

The temporal profiles ranged between no (0%) to full (-100%) degradation (A) and no (0%) to full (100%) upregulation (B) of the active component of each of the Rho-GTPases.

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3.1. The proposed computational framework recapitulates key features of apical-basal polarization in renal epithelial cells

To simulate apical-basal polarization, we developed a computational framework that describes the interactions between the different proteins hypothesized to be involved in the polarization pathway. The model predicted the distributions of the active forms of the Rho-GTPases and the distributions of the formed polarity complexes upon reaching equilibrium. The crosstalk included in our model, together with the chosen parameter values, caused the distribution of the active forms of the Rho-GTPases to evolve toward a steady-state polarized profile (Fig 5A). Active Rac accumulated at the basal membrane due to its activation by active integrins situated at the basal membrane. The sideways inhibition between Rac and Rho caused the active form of Rho to accumulate at the apical pole of the cell. Cdc42 colocalized with Rho because the activation of Cdc42 is dependent on Rho.

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Fig 5. The hypothesized interplay between intracellular proteins, resulting from the localization of active integrins at the basal membrane, recapitulates the establishment of polarization in renal epithelial cells.

Final steady-state distribution of the active forms of the Rho-GTPases in the model (A) and of the formed polarity complexes Par and Scribble (B). Total simulation time is 100 seconds.

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The polarized profile of the Rho-GTPases subsequently led to a steady-state polarized profile for the polarity complexes Par and Scribble in our model (Fig 5B). This concurs with the establishment of apical-basal polarization in renal epithelial cells, where the Par-complex accumulates at the apical domain and the Scribble-complex accumulates at the basal domain. The localization of Par at the apical membrane is caused by the fact that Cdc42 positively contributes to the formation of the Par complex. The transition point in the distribution of the concentrations of the Rho-GTPases and therefore also of the polarity complexes was centered closer to the basal domain of the cell. This is due to the strong dependence of our model on the distribution of active integrins for the induction of Rac activation. In the proposed computational model, these active integrins were assumed to be present only within the basal region that represents one sixth of the domain. This restriction to the basal region can be explained by our assumpiton that integrin activation is enabled by the ECM at the basal side and that they do not diffuse as they are connected to the ECM.

Next, we aimed at elucidating which factors in our model have the largest impact on the establishment of polarization in renal epithelial cells using the MPSA. Achieving a polarized distribution for activated integrins was observed to depend completely on the parameters involved in the switch between active and inactive integrins (I_I, δ_I) (Fig 6A), which was expected since integrins are not connected to the rest of the pathway in our model. The establishment of polarized steady-states for the Rho-GTPases were all found to be heavily influenced by the parameters that belonged to Rho and Rac (i.e., δ_R, β_R, I_ρ, δ_ρ, α_ρ, β_ρ) (Fig 6B–6D). Especially the inhibition parameters β_ρ and β_R, which define the inhibition rates for Rho and Rac as a function of Rac and Rho respectively, displayed very high values for the K-S distance, demonstrating their relevance. Additionally, we observed that the output of the Rho-GTPases was primarily influenced by the parameters affiliated with the integrins (I_I, δ_I, α_I), with the output of Rac being the most affected by these parameters (Fig 6B). Obtaining a polarized output for both polarity complexes was found to be the most sensitive to the inhibition parameter β_SP that governs the rate of dissociation of the Par complex as a function of the concentration of the Scribble complex (Fig 6E and 6F). Additionally, the polarization of the polarity complexes was sensitive to changes in all other parameters, which can be explained by the dependence of the distribution of Par and Scribble on the crosstalk between the different Rho-GTPases and by the sideways inhibition of Par and Scribble. Based on the general observation that the parameters that belonging to Rho and Rac heavily affected all outputs, we conclude that the establishment of polarization in our computational framework is mainly dependent on the strength of the mutual inhibition of Rho and Rac.

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Fig 6. Sensitivity of polarized profiles for the six output variables to model parameters.

All subfigures show the Kolmogorov-Smirnov distances (K-S distances) normalized to the maximum distance. We show the influence of 21 parameters on the polarized distributions developed for (A) activated integrin, (B) Rac, (C) Rho, (D), Cdc42, (E), Par and (F) Scribble. These 21 parameters have been grouped in color to display which of the parameters belongs to a certain protein. The horizontal red line represents the maximal K-S distance obtained from the set of dummy values. The parameters with a increased K-S distance compared to the maximal K-S distance obtained from the dummy values were considered to be influential.

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In addition to determining the most influential parameter(s), we also analyzed which one of the six outputs is most sensitive to the changes in parameters. To this end, we inspected the ΔP for all the 3000 simulations per output variable (Fig 7). The sensitivity of the output was then related to the number of simulations from the total of 3000 simulations that resulted in no polarization or only weak polarization (ΔP < 20%). In case of a high number of simulations resulting in no polarization or weak polarization, the output was considered to be sensitive to the parameters. We found that the establishment of a polarized output for active integrins depends the least on changes of our parameters, since the output has the lowest number of simulations with a ΔP lower than 20% (Fig 7A). The notion that active integrins is the least sensitive output is again as expected since integrins are not connected to the rest of the pathway. Achieving a polarized state for Scribble was determined to be the most dependent on the parameters (Fig 7E). This observation can be explained by the fact that Scribble is at the end of the pathway where its concentration is determined by the interplay of all other proteins.

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Fig 7. Establishing a polarized profile for the six output variables is not equally sensitive to the variation of parameters.

All subfigures display the frequency of the various relative changes in concentration difference between the apical and basal side. Output sensitivity to the 21 parameters is shown for (A) activated integrin, (B) Rac, (C) Rho, (D) Cdc42, (E) Par, and (F) Scribble. The first bar (0<ΔP≤20%) represents the number of simulations in which polarization was not achieved.

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3.2. The establishment of apical-basal polarization requires a minimum level of activated integrins

Previous studies have demonstrated that integrins play a crucial role in the establishment of polarization, as inhibition of different integrins led to the prevention of cyst polarization in 3D environments [46–48]. Motivated by this, we investigated how interactions between integrins and the Rho-GTPases could affect the polarized profile of the polarity complexes Par and Scribble. First, we confirmed that active integrins provide the direction of the polarization response. The active integrins interacted with Rac at one side of the cell, which triggered the subsequent interactions between the Rho-GTPases, and subsequently caused the apical membrane as marked by Par to emerge at the opposite end of the cell in our model (S1 Fig). We then investigated whether the concentration of active integrins affects the establishment of a polarized profile for Par and Scribble.

We found that when the initial concentration of active integrin was kept below 18 μM, the active integrins were not able to generate sufficient Rac activation to induce the development of a polarized profile for the Rho-GTPases (Fig 8A). As a result, the polarity complexes Par and Scribble also remained homogeneously distributed over the cell (Fig 8B). On the other hand, initial concentrations of active integrin exceeding a threshold of 18 μM activated Rac to the degree that it caused a sufficient amount of Rho and Cdc42 to accumulate at the opposite side of the cell, thereby leading to induced polarized distributions of the Rho-GTPases (Fig 8A). The accumulation of Rho and Cdc42 at the apical side of the cells is necessary to achieve the polarized distribution for the polarity complexes Par and Scribble (Fig 8B). The strength of polarization for the Rho-GTPases and the polarity complexes, which is defined as the absolute difference in concentration between the apical and basal side, was found to be independent of the active integrin activation as long as this exceeds 18 μM (Fig 8C and 8D). Initial concentrations of active integrins higher than the threshold value of 18 μM led to only slightly higher values for Rac concentrations (Fig 8C). This stabilization of the Rac concentrations can be explained by the fact that the active concentration of Rac can only reach a maximum value given by the constant total Rac concentration. These small differences in Rac concentration however do not translate into differences in the polarized distributions of Rho and Cdc42 (Fig 8C), and therefore the distributions of the polarity complexes also remain unaffected (Fig 8D). Collectively, these results imply that the establishment of polarization in renal epithelial cells requires the concentration of active integrins to exceed a certain threshold value, which in our model corresponds to 18 μM, while the strength of polarization is independent of the concentration of active integrins beyond this threshold.

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Fig 8. The concentration of active integrins affects the distribution of Rho-GTPases and the emergence of polarization.

Examples of the final steady-state distribution of the active forms of the Rho-GTPases (A) and the formed polarity complexes Par and Scribble (B) in case of different concentration of integrins. The absolute differences in concentration of the Rho-GTPases (C) and the polarity complexes (D) between the apical and basal side show that a minimum value of 18 μM active integrins is necessary to enable the emergence of polarization.

https://doi.org/10.1371/journal.pcbi.1012140.g008

3.3. Full degradation of Rho leads to a loss of polarity

During the process of EMT, epithelial cells gradually lose their tight junctions, apical-basal polarization, and cell height, while they produce more actin fibers and specifically α-smooth muscle actin (α-SMA) and obtain a spreading morphology. Although the transforming growth factor (TGF) β1 is one of the most well-known inducers of EMT in renal epithelial cells, the signaling pathways by which TGF-β1 induces EMT are less well understood [49–51]. One of the hypotheses is that TGF-β1 causes the degradation of the Rho-GTPases, subsequently activating further signaling cascades that cause the induction of EMT [52,53]. In contrast, other studies have suggested that TGF-β1 can cause an induction of EMT by upregulating the Rho-GTPases [54,55]. To explore the validity of these hypotheses and to examine which TGF-β1 mediated changes in Rho-GTPases could explain EMT, we investigate whether degradation or upregulation of certain Rho-GTPases could induce a loss of polarity within our computational framework. To this end, degradation and upregulation ranging from no (0%) to a full (100%) degradation or upregulation of each Rho-GTPase was imposed. Next to this, we performed in vitro experiments in which we induced EMT in renal epithelial cells and subsequently measured GTPase levels relative to control conditions. The changes in these relative GTPase levels were then compared to the variations in the GTPase distributions computed in our model to examine which of the three Rho-GTPases could play a role in the TGF-β1 induced EMT.

First, degradation of each of the Rho-GTPases was simulated. Partial degradation (up to 80%) of active Rac, Rho, or Cdc42 was not able to induce a loss of polarization in our model, because the other Rho-GTPases and the polarity complexes preserved their polarized distributions upon partial degradation of individual Rho-GTPases (Fig 9). Complete degradation of Rac, however, led to homogeneous distributions of Rho and Cdc42 (Fig 9A), as a certain minimum concentration of Rac at the basal side is necessary for Rho and Cdc42 to accumulate at the apical side. Nevertheless, although Rho and Cdc42 lose their polarized distributions, the concentration of Cdc42 at the apical side remains at a sufficiently high level to prevent the dissocation of the Par complex, thereby preserving the polarized distributions of both polarity complexes. On the other hand, complete degradation of Cdc42 did not affect the distributions of Rac and Rho, which was expected since Cdc42 does not influence Rac or Rho in the proposed crosstalk. The complete degradation did cause the dissociation of the Par complex because a certain concentration of Cdc42 is required to maintain the Par complex (Fig 9B). Due to the mutual inhibition between Par and Scribble, the dissociation of the Par complex throughout the cell immediately resulted in a homogeneous distribution of the Scribble complex, thus causing a loss of polarization. Finally, complete degradation of Rho was shown to lead to a homogeneous distribution and lower concentrations of Cdc42, while the distribution of Rac was hardly affected (Fig 9C). The combination of fully degraded Rho and lower, homogeneously distributed concentrations of Cdc42, in turn, caused the dissociation of the Par complex. In case of Rho degradation, Cdc42 is still present in lower concentrations and may therefore not entirely explain the dissociation of the Par complex. The Scribble complex may also be involved in the dissociation of the Par complex, due to its inhibitory effect on the formation of the Par complex. Collectively, our model indicates that full degradation of either Rho and/or Cdc42 can lead to a loss of polarity during the initiation of EMT.

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Fig 9. Full degradation of Rho and Cdc42 can both lead to loss of polarity in the model.

Final steady-state distributions of the active forms of the Rho-GTPases and of the formed polarity complexes Par and Scribble after imposing Rac degradation (A), Cdc42 degradation (B) and Rho degradation (C). Total simulation time is 300 seconds.

https://doi.org/10.1371/journal.pcbi.1012140.g009

Next, upregulation of each of the Rho-GTPases was simulated. In all cases, upregulation of the active part of the Rho-GTPases did not lead to a loss of polarized distribution of the Rho-GTPases as well as the polarity proteins Par and Scribble (Fig 10). Upregulation of Rac was shown to have no influence on the polarized distributions of all other proteins (Fig 10A). Our previous results (section 3.1) already showed that a certain minimum amount of Rac is necessary to allow for the accumulation of sufficient Rho and Cdc42 at the apical side to achieve and preserve polarization. The Rac concentration does not affect the concentrations of Rho and Cdc42 beyond this threshold value, as each GTPase satisfies mass conservation and thus the maximum amount of the active part cannot exceed the total amount in the cell. An upregulation of Cdc42 on the other hand slightly enhances the polarized distributions of the polarity proteins (Fig 10B) as a higher concentration of Cdc42 would in turn lead to an increase in the formation of the Par complex at the apical side of the cell and therefore also to the formation of the Scribble complex at the basal side of the cell. Lastly, an upregulation of Rho also leads to a slight enhancement in the polarized distributions of the polarity proteins (Fig 10C). This can be explained by the fact that an upregulation in Rho also leads to an upregulation in Cdc42, which in turn again causes a rise in the amount of the polarity proteins at the different poles of the cell. Together, our model indicates that upregulating the Rho-GTPases does not lead to a loss of polarity and can even slightly enhance the polarized state.

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Fig 10. Upregulation of the Rho-GTPases does not lead to a loss in polarity.

Final steady-state distributions of the active forms of the Rho-GTPases and of the formed polarity complexes Par and Scribble after imposing Rac upregulation (A), Cdc42 upregulation (B) and Rho upregulation (C). Total simulation time is 300 seconds.

https://doi.org/10.1371/journal.pcbi.1012140.g010

In vitro, EMT was induced in Madin Darby Canine Kidney cells (MDCKs) via overnight serum starvation after 3 days of culture on transwell filters and the subsequent addition of TGF-β1 to the culture medium in a final concentration of 10 ng ml^-1 for 1 day. MDCKs were also cultured for 4 days on transwell filters under normal conditions as a control. The visual hallmarks of polarization are the formation of tight junctions at the apical side of the lateral membrane, the localization of the apical proteins, the cortical organization of the actin cytoskeleton, the diminishment of mesenchymal markers and an increase in cell height. The MDCKs in the control condition displayed a clear polarized organization, with tight junction formation at the apical side of the membrane (Fig 11A: ZO-1) and the localization of the apical proteins at the apical membrane (Fig 11A: Podocalyxin). Control cells also showed a low expression of α-SMA (Fig 11A: α-SMA), a well-known mesenchymal marker [56]. In contrast, MDCKs exposed to TGF-β1 had no tight junctions, substantially higher levels of actin formation, no localization of the apical proteins, and they displayed an upregulation of αSMA expression: all indicators of a mesenchymal phenotype. Culture lysates were obtained after 4 days of culture to determine the expression of the Rho-GTPases after the induction of EMT. The expression levels of RhoA, Rac1 and Cdc42 together with Tubulin as a loading control, were obtained using immunoblotting (Fig 11B). Quantification of the relative band intensities shows a sharp decrease in total RhoA, a small decrease in total Cdc42, and no significant change in total Rac1 after the induction of EMT (Fig 11C). When comparing these decreases in the levels of the Rho-GTPases to our computational results, they were found to be most similar to the case of full Rho degradation. This similarity therefore suggests that degradation of Rho may be one of the key mechanisms in the TGF-β1-induced process of EMT in renal epithelial cells.

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Fig 11. Expression of Rho-GTPases with and without the induction of EMT.

(A) Samples were generally stained for F-actin (Phalloidin A488; green), nuclei (DAPI; blue), and for tight junctions (Zona Occludens 1 (ZO-1)), apical protein localization (Podocalyxin) or α-SMA expression (all in red). All scale bars are 20 μm. (B) Expression levels of RhoA, Rac1, Cdc42 and Tubulin as a loading control for three independent experiments were determined for MDCKS in the control condition as well as MDCKs exposed to TGF-β1 by western blotting using protein-specific antibodies. (C) Rho-GTPases bands were normalized to the loading control and the average relative densities for the control and for the cells exposed to TGF-β1 were calculated (error bars represent S.D., n = 3). (p-values were determined using a Student’s t-test, * p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001).

https://doi.org/10.1371/journal.pcbi.1012140.g011

5.1. Cell culture

Madin Darby Canine Kidney II cells (MDCK-II, ECACC, The Netherlands) were cultured and expanded in a standard cell culture incubator (37°C, 5% CO₂) in Eagle’s Minimum Essential Medium (EMEM; Merck, Darmstadt, Germany) supplemented with 5% fetal bovine serum (FBS; Greiner Bio-one, Alphen aan de Rijn, Netherlands), 1% penicillin/streptomycin (Invitrogen, Waltham, MA, USA), and 1% L-glutamine (Thermofisher Scientific, Waltham, MA, USA). Culture medium was refreshed every 3–4 days and the cells were passaged at 80% confluency. Cells were seeded onto transwell inserts (Nunc, Thermofisher Scientific, Waltham, MA, USA) at a density of 20,000 cells cm^-2 for immunoprecipitation and immunofluorescence analysis. To induce EMT, after three days, cells were serum starved overnight and subsequently cultured in the presence of TGF-β1 (Peprotech EC Ltd., London, UK) in the medium in a final concentration of 10 ng mL^-1 for the subsequent day. Culture of MDCKs for 4 days on transwell filters without EMT induction served as a control.

5.2. Western blot

MDCKs were seeded onto 3.1 cm transwell inserts (Thermofisher Scientific, Waltham, MA, USA) at a density of 20,000 cells cm^-2. After four days, all samples were washed with ice-cold phosphate buffered saline (PBS) and incubated in lysis buffer (Cytoskeleton, Denver, CO, USA) for 10 minutes. After scraping, lysate was collected in Eppendorf tubes and centrifuged for 1 minute at 10,000xg at 4°C. Samples were sonicated and boiled for 5 min at 98°C for analysis with SDS-Page (4–15% gradient gel, Biorad, Lunteren, the Netherlands) and subsequent Western nitrocellulose blot (Amersham). In addition to loading 27 μg of the total protein samples, 106.5 pg of control protein RhoA and Cdc42 and 22.5 pg of control protein Rac1 were loaded. Membranes were blocked with 5% BSA/5% non-skim dry milk in PBS/0.05% Tween-20 and incubated (O/N, 4°C) with primary antibodies: mouse monoclonal anti-Cdc42 (1:500, Santa-Cruz Biotechnology, Dallas, TX, USA), mouse monoclonal anti-Rac1 (1:1000, Merck, Darmstadt, Germany), mouse monoclonal anti-RhoA (1:1000, Santa-Cruz Biotechnology, Dallas, TX, USA) and rat monoclonal anti-tubulin (1:5000, Novus Biologicals, Littleton, CO, USA). Secondary antibodies applied were Goat-anti-mouse-HRP (1:10,000, Invitrogen, Waltham, MA, USA) or Goat-anti-rat-Alexa633 (1:5000, Thermofisher, Waltham, MA, USA) for 2 hours at RT. Proteins were visualized using Supersignal West Atto ECL substrate and the iBright FL1500 Imaging system (Thermofisher, Waltham, MA, USA). Band intensities were analyzed using ImageJ software [95].

5.3. Immunofluorescence staining

MDCKs were seeded onto 4.1 cm transwell inserts. After four days, all samples were fixed with 3.7% formaldehyde solution (Merck, Darmstadt, Germany) in PBS for 15 minutes at room temperature (RT) four days after seeding and washed three times with PBS. The membrane was cut out of the chamber and cut in four. Samples were subsequently permeabilized with a 0.5% Triton-X-100 (Merck, Darmstadt, Germany) in PBS for 5 minutes at RT and blocked for 1 hour at RT in PBS with 10% horse serum to prevent nonspecific antibody binding. Samples were then stained with the primary antibodies overnight at 4°C in PBS with 1% horse serum. Primary antibodies used were mouse monoclonal anti-α-SMA (1:600; Dako, Glostrup, Denmark), mouse monoclonal anti-podocalyxin (1:200; Merck, Darmstadt, Germany) or the conjugated Zona Occludens 1 antibody (1:200; Thermofisher Scientific, Waltham, MA, USA). After washing three times with PBS, the cells were incubated with secondary antibodies in PBS for 1h at RT. During secondary antibody incubation, samples were also stained with Phalloidin Atto 488 (Merck, Darmstadt, Germany), which stains for F-actin. Samples were washed again three times with PBS and subsequently incubated with a 4’-6-diamidino-2-phenylindole solution (DAPI; Merck, Darmstadt, Germany) in PBS, staining for nuclei. Cells were finally washed three times with PBS and mounted with Mowiol (Merck, Darmstadt, Germany). Images were acquired using a Leica SP8 microscope with oil 63x magnification.

5.4. Statistical analysis

All experimental data were presented as the mean ± standard deviation. Data were tested for normality using the Shapiro Wilk test. Since data were normally distributed according to the Shapiro Wilk test, comparisons between two groups were analyzed using an independent two-tailed Student’s t test (GraphPad, La Jolla, CA, USA). Differences were considered as statistically significant when P<0.05.