Cooperative Regulation of PIP5K by β-Arrestin–GPCR Complexes at Clathrin-Coated Pits Demonstrated by Mathematical Modeling

1. Introduction

Phosphoinositides are critical regulators of signaling, trafficking, and cytoskeletal dynamics although they are minor constituents of cellular membranes [1]. Among them, phosphatidylinositol 4,5-bisphosphate (PI(4,5)P₂ or PIP₂) plays a central role in clathrin-mediated endocytosis by recruiting adaptor proteins, clathrin, and actin regulators [2,3,4,5]. Acute perturbation of PIP₂ levels demonstrates its indispensability: optogenetic or enzymatic depletion of PIP₂ blocks clathrin-coated pit (CCP) initiation and receptor internalization [6,7,8]. These studies establish PIP₂ not only as a structural lipid but also as a dynamic regulator of receptor trafficking.

G-protein–coupled receptors (GPCRs) comprise the largest family of cell-surface receptors and mediate responses to diverse extracellular cues. Upon activation, G_q-coupled GPCRs stimulate phospholipase Cβ (PLCβ), which hydrolyzes PIP₂ into diacylglycerol (DAG) and inositol trisphosphate (IP₃), thereby reducing plasma membrane PIP₂ levels [9,10]. Yet GPCRs such as protease-activated receptor 2 (PAR2) undergo clathrin-dependent endocytosis for desensitization and trafficking, a process that requires PIP₂. This apparent contradiction — that G_q signaling consumes PIP₂ while GPCR endocytosis depends on it — represents a long-standing paradox in GPCR biology [11,12].

CCPs are increasingly recognized as more than passive vesicle carriers. CCPs serve as multifunctional signaling microdomains that integrate receptor trafficking with downstream signal transduction [13,14]. For example, they provide a spatial platform for MAPK activation downstream of GPCRs [15]. β-arrestins are key to this dual role. They not only terminate G-protein signaling but also scaffold MAPK components and link receptors to clathrin and AP-2 [16]. Live-cell and single-molecule imaging studies have shown that β-arrestin–receptor complexes display dynamic lifetimes and can persist within CCPs, where they regulate both receptor endocytosis and signal propagation [17,18,19].

The previous work provided evidence that β-arrestins contribute directly to the regeneration of PIP₂ at the plasma membrane following PAR2 activation [20]. Using lipid biosensors, I found that β-arrestin knockdown impaired PIP₂ recovery. Further experimental results suggested that phosphatidylinositol 4-phosphate 5-kinase (PIP5K), the kinase that synthesizes PIP₂ from PI(4)P, is recruited to β-arrestin-receptor-clathrin complexes, linking receptor signaling to localized lipid metabolism [21]. These findings offered a potential solution to the PIP₂ paradox. However, several observations remain unexplained: the degree of PIP5K hyperactivation was modest (2–3 fold), and β-arrestin knockdown produced a hump-like trajectory in PIP₂ recovery inconsistent with linear recruitment models.

In the present study, I tested the hypothesis that PIP5K activity is cooperatively modulated within CCPs using mathematical modeling. I proposed that CCPs can provide spatial confinement and clustering of receptor–β-arrestin complexes that amplify cooperatively the PIP5K activity. Through mathematical modeling, I showed that transient, weak protein–protein interactions at CCPs can be sufficient to reproduce the nonlinear kinetics of PIP₂ recovery, the hump phenotype of β-arrestin knockdown, and the modest hyperactivation of PIP5K. These results advance a conceptual framework in which both stable and transient protein interactions at CCPs govern lipid metabolism, thereby ensuring that GPCR signaling and endocytosis remain tightly coupled and reversible.

2. Materials and Methods

2.1. Model Overview

A mathematical model was developed to describe cooperative regulation of PIP5K activity during recovery of PIP₂ following activation of G_q-coupled PAR2. The model builds upon our previous framework [20] in which PIP₂ hydrolysis and resynthesis kinetics were represented as coupled differential equations describing the balance between phospholipase C (PLC)–mediated degradation and PIP5K-dependent synthesis. The current model introduces additional parameters to account for cooperative enzyme activation and the distinct contributions of newly forming versus preexisting CCPs prior to endocytosis of receptor by dynamin action.

Model Assumptions: 1) Activation of PAR2 by 100 μM activating peptide (AP) induces rapid PIP₂ hydrolysis by PLC, leading to transient depletion of the lipid pool at the plasma membrane. 2) PIP5K activity recovers PIP₂ levels with two distinct components: one associated with newly forming CCPs and another with preexisting CCPs. 3) β-arrestin–dependent recruitment of PIP5K to CCPs modulates enzyme activity cooperatively, described by Hill-type equations with coefficients n and m, representing newly forming and preexisting CCPs, respectively. 4) The kinetic equations assume well-mixed conditions and constant total enzyme concentration. Spatial heterogeneity and stochastic fluctuations were not explicitly modeled but are implicitly reflected in the cooperative parameters.

2.2. Mathematical Formulation

The cooperative activation term was expressed as:

$$\text{Stim\_5K\_ max}(\mathrm{n},\mathrm{m},\mathrm{a},\mathrm{b})=\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{l}+\mathrm{a}\cdot {\displaystyle \frac{[\mathrm{R}\mathrm{L}\mathrm{P}\mathrm{A}\mathrm{A}\mathrm{P}\mathrm{C}{]}^{\mathrm{n}}}{[\mathrm{R}\mathrm{L}\mathrm{P}\mathrm{A}\mathrm{A}\mathrm{P}\mathrm{C}{]}^{\mathrm{n}}+{\mathrm{K}}_{\mathrm{a}}^{\mathrm{n}}}}+\mathrm{b}\cdot {\displaystyle \frac{[\text{RLPA\_CCP}{]}^{\mathrm{m}}}{[\text{RLPA\_CCP}{]}^{\mathrm{m}}+{\mathrm{K}}_{\mathrm{b}}^{\mathrm{m}}}},$$

where, RLPAAPC represents the ligand (L)–bound, phosphorylated (P) GPCR (R) complexed with β-arrestin (A) that nucleates newly forming CCPs through adaptor (AP)–clathrin (C) interactions. RLPA_CCP represents the ligand (L)-bound, phosphorylated (P) GPCR (R) complexed with β-arrestin (A) associated with a preexisting CCP. a and b are weighting constants describing the fractional contributions of newly forming and preexisting CCPs, respectively. K_a and K_b are apparent dissociation constants for each process. n and m are Hill coefficients indicating the degree of cooperativity for each component.

2.3. Parameter Selection

Parameter values were based on previously reported measurements and adjusted to match experimentally observed kinetics of PIP₂ depletion and recovery after PAR2 activation [20,21]. The Hill coefficients (n, m) were varied between 1 and 6 to explore the effects of cooperativity. Weighting constants (a, b) were set to evaluate the relative contributions of newly forming and preexisting CCPs. All parameters and equations for reactions were described in Table 1 and Table 2.

Simulation and Analysis: All simulations were performed in Virtual Cell. Plots were generated using the Origin pro (OriginLab Corp. MA, USA) for figures.

β-Arrestin Knockdown Simulations: To simulate β-arrestin knockdown, the amount of cytosolic β-arrestin was reduced by 20%, corresponding to the experimentally observed decrease in PIP₂ recovery following β-arrestin depletion. This reduction delayed PIP₂ recovery and produced the characteristic “hump” phenotype in the simulated time course.

3. Results

Using a simple diagram, I summarized previous findings [21] as shown in Figure 1A. According to a new simulation result, one modification was localization of PIP5K among the β-arrestin-GPCR complexes at the CCP to describe two major findings: 1) modest hyperactivation of PIP5K after desensitization of PAR2 and 2) hump formation of PIP₂ level while β-arrestin was knocked down using siRNA.

To build upon our previous findings, we first examined the dynamics of PIP₂ following PAR2 activation. Using a PIP₂-sensitive PH probe, we observed that basal PIP₂ levels after desensitization of PAR2 exhibited an overshoot [21]. Recovery of PIP₂ was markedly impaired when β-arrestin expression was reduced by siRNA, producing a pronounced hump-like trajectory. This data suggested that an additional regulatory mechanism must be incorporated.

Interestingly, PIP5K activity was enhanced only 2–3 fold following PAR2 activation, a degree of hyperactivation that is difficult to reconcile with a simple conformational change mechanism which makes order of magnitude different activity of enzymes. This observation led us to hypothesize that the spatial organization of CCPs could play a critical role in modulating lipid kinase activity. Consistent with this idea, we found that PAR2 associated with both preexisting and newly forming CCPs. However, PAR2 was confined to the periphery of preexisting CCPs, whereas newly forming CCPs dynamically recruited PAR2–β-arrestin complexes together with endocytic machinery. These findings pointed out to de novo CCP formation as a major driver of PIP5K activation.

To capture these features, we developed a revised mathematical model (Figure 1B).

Unlike my previous framework, which treated PIP5K stimulation as a constant, the updated model incorporated dynamic terms reflecting CCP formation. The second and third terms of the equation represent contributions from newly forming and preexisting CCPs, respectively. I further propose that cooperative activation of PIP5K arises within the confined environment of newly forming CCPs rather than through direct conformational changes. Cooperative effects were modeled using Hill equations, with coefficients (n and m) representing newly forming and preexisting CCPs. Parameter tuning allowed the model to reproduce the modest 2–3-fold increase in kinase activity observed experimentally. As the Hill coefficient increased, basal PIP₂ levels rose and reached a plateau (Figure 1C, inset).

The model also recapitulated the hump phenotype under conditions of β-arrestin knockdown. At high Hill coefficients (n and m = 6), PIP₂ recovery remained incomplete and exhibited a hump-like trajectory (Figure 2A). Correspondingly, PIP5K activation was delayed and sustained at higher levels due to cooperative activation of the kinase (Figure 2B). Knockdown of β-arrestins further prolonged this delay, producing a slow recovery phase followed by accelerated restoration of PIP₂. These results suggest that the hump arises from delayed, cooperative clustering of PIP5K mediated by residual β-arrestin–dependent CCP formation.

We next dissected the relative contributions of preexisting and newly forming CCPs by varying constants a and b (Figure 3A). When a = 0, eliminating the contribution of newly forming CCPs, PIP₂ recovery was substantially reduced, more so than when b = 0. This indicates that newly forming CCPs play the dominant role in PIP₂ homeostasis during desensitization of PAR2. Moreover, the delayed activation pattern of PIP5K disappeared when a = 0 but persisted when b = 0 (Figure 3B), further highlighting the central role of newly forming CCPs in shaping lipid kinase dynamics. As shown in Figure 3C and D, the PIP₂ hump and PIP5K activity were reproduced by b = 0 which reflects that only newly forming CCP works for the recovery, suggesting that in the condition, newly forming CCP can be a key locus of the PIP₂ recovery. However, it cannot be ruled out the contribution of preexisting CCP due to low degree of PIP5K activation and PIP₂ recovery.

Interestingly, constraining m = 1 to reflect the lack of cooperative PAR2–β-arrestin clustering within preexisting CCPs had some effect on either PIP₂ recovery or PIP5K activation kinetics (Figure 4). These results indicate that cooperativity associated with newly forming CCPs is the primary contributor to PIP₂ regeneration, but not full contributor as previously mentioned, suggesting that the contribution between preexisting and newly forming CCP on the cooperative behavior of the kinase would be balanced by the Hill coefficients (n and m) and weighting constants (a and b) in the individual cases.

4. Discussion

This study identifies a cooperative mechanism by which transient β-arrestin–dependent activation of PIP5K restores PIP₂ levels at the plasma membrane following activation of G_q-coupled receptors such as PAR2. Through mathematical modeling, we show that even modest increases in PIP5K activity—amplified by cooperative interactions within CCPs—are sufficient to regenerate PIP₂ and maintain receptor signaling competence. This mechanism provides a molecular explanation for the long-standing paradox of how G_qPCRs can undergo endocytosis despite hydrolysis of their essential substrate, PIP₂.

4.1. Cooperative PIP5K Activation as the Key Regulatory Principle

My model demonstrates that the recovery of PIP₂ cannot be explained by linear enzyme activation alone. The inclusion of Hill-type cooperative behavior in PIP5K activity, mediated by β-arrestin–GPCR complexes, reproduces the experimentally observed “hump” pattern of PIP₂ recovery following β-arrestin knockdown. The Hill coefficients (n and m) describe the degree of cooperativity within newly forming and preexisting CCPs, respectively. High cooperativity (n & m ≈ 6) in CCPs accounts for both the overshoot in PIP₂ recovery and the delayed activation kinetics seen experimentally. Importantly, these Hill coefficients should be viewed as phenomenological parameters that capture emergent cooperative behavior rather than literal molecular stoichiometry. Such cooperativity likely arises from multiple weak and transient interactions among β-arrestin, GPCRs, and endocytic adaptors within spatially confined CCP microdomains. How about effect of contribution constants a (newly forming CCP) and b (preexisting CCP) on the PIP₂ recovery? Probably, the relative contribution can be determined by ratio of the newly forming CCP and preexisting CCP depending on cell cycle and classes of G_qPCR.

4.2. Functional Difference Between Newly Forming and Preexisting CCPs

By systematically varying contribution parameters (a and b), our simulations revealed that newly forming CCPs are the primary drivers of delayed PIP₂ regeneration with nonlinearity, whereas preexisting CCPs play a marginal role for the dynamics. Newly forming CCPs provide an environment where β-arrestin–receptor complexes can nucleate adaptor proteins and clathrin assembly, enabling cooperative clustering of PIP5K which makes delayed kinetics. In contrast, preexisting CCPs are diffusion-limited and recruit the GPCR-β-arrestin complexes at the edge of the CCPs, restricting the formation of cooperative assemblies which make linear saturation kinetics. This distinction explains why β-arrestin–dependent clustering at newly forming CCPs produces robust recovery kinetics, whereas preexisting CCPs cannot fully compensate under knockdown conditions.

4.3. Broader Implications of Transient and Weak Interactions

Beyond this specific system, our findings highlight the broader principle that weak and transient protein–protein interactions can collectively generate robust signaling outcomes. Although individual β-arrestin–PIP5K interactions are short-lived, their repeated and cooperative engagement within confined CCPs amplifies enzymatic activity and ensures lipid homeostasis. This concept aligns with emerging views that weak, dynamic assemblies—rather than stable complexes—underlie efficient information transfer in signaling networks. Similar cooperative mechanisms may regulate other lipid kinases or scaffolded enzymes operating in nanoscale membrane domains.

Transient protein–protein interactions can induce weak or short-lived conformational changes in kinases that enhance their catalytic output without fully stabilizing the active conformation. Such partial structural rearrangements, often involving local loop motions or small domain shifts, can transiently align catalytic residues or promote substrate accessibility, thereby producing a modest but physiologically significant increase in activity [22]. This contrasts with stable conformational changes, such as strong scaffold binding, which lock the enzyme into a fully active state and yield maximal catalytic efficiency [23]. The transiently induced state thus represents a dynamic intermediate between inactive and fully active conformations—sufficient to fine-tune signaling output while maintaining reversibility and rapid responsiveness to fluctuating stimuli. In this framework, transient binding events act as kinetic modulators that bias the conformational ensemble toward catalytically permissive states without requiring full structural rearrangement, allowing cells to modulate kinase activity in a graded, energy-efficient manner.

4.4. Potential Crosstalk of PAR2–PIP₂–PIP5K Signaling with Oncogenic KRAS

Our findings describing transient β-arrestin–dependent activation of PIP5K and cooperative recovery of PIP₂ at the plasma membrane have potential implications beyond receptor endocytosis. Increasing evidence suggests that PAR2-mediated lipid signaling intersects with KRAS activation and phosphoinositide 3-kinase (PI3K)–Akt pathways, all of which play central roles in cancer progression. Protease activated receptors are aberrantly expressed in many cancers, including breast, pancreatic, and colorectal tumors, where it enhances proliferation, migration, and epithelial–mesenchymal transition [24,25]. KRAS requires stable association with the inner leaflet of the plasma membrane via electrostatic interactions between its C-terminal polybasic domain and negatively charged phosphoinositide, particularly phosphatidylserine (PS) and PIP₂ [26]. Thus, transient PIP₂ regeneration following PAR2 activation may reinforce KRAS membrane anchoring and preserve its signaling-competent conformation, providing a mechanistic bridge between protease sensing, lipid metabolism, and Ras activation. Once activated, KRAS recruits and activates PI3K, which phosphorylates PIP₂ to generate PIP₃, leading to Akt recruitment toward the plasma membrane and the initiation of downstream survival signaling [27,28]. Through this cooperative loop, PAR2 signaling could synergize with oncogenic KRAS to sustain PI3K–Akt activity, enhancing tumor cell proliferation, motility, and resistance to stress.

Taken together, the PAR2–PIP₂–KRAS–PI3K–Akt axis may represent a molecular hub of cooperative signaling that links receptor desensitization, lipid regeneration, and oncogenic amplification. The cooperative PIP5K activation described in our model provides a plausible kinetic framework for how weak, transient β-arrestin–PIP5K interactions can amplify lipid-dependent signaling in cancer contexts. While each interaction is short-lived, their cumulative effect within spatially confined domains produces a sustained output—analogous to how stochastic, diffusion-limited binding can yield emergent cooperativity. In tumor cells, where PIP5K and PAR2 are often overexpressed, this mechanism could lower the activation threshold for KRAS–PI3K–Akt signaling, enabling persistent survival and invasive behavior even in fluctuating extracellular conditions.

4.5. Limitations and Outlook

While the present model successfully reproduces key kinetic features of PIP₂ recovery, it simplifies several aspects of the system. It does not explicitly account for protein diffusion barriers, PIP5K isoform specificity, or dynamic exchange of PIP5Ks within CCPs. Future experimental studies employing single-molecule imaging, optogenetic control of PIP5K, and real-time FRET biosensors could quantitatively test these predictions and reveal the spatiotemporal organization of these transient complexes. Incorporating stochastic spatial or agent-based models will further refine our understanding of how nanoscale cooperativity translates into large-scale lipid regulation.

5. Conclusions

In conclusion, our results demonstrate that transient β-arrestin–dependent activation of PIP5K, though modest in magnitude, is sufficient to restore PIP₂ levels through cooperative regulation within CCPs. This cooperative mechanism not only explains the paradox of PIP₂-dependent endocytosis during G_qPCR activation but also establishes a broader paradigm: cellular homeostasis and signaling fidelity emerge from the collective effects of dynamic, weak interactions that fine-tune enzymatic and trafficking processes.