Characterization of Specific Ion Effects on PI(4,5)P₂ Clustering: Molecular Dynamics Simulations and Graph-Theoretic Analysis

Abstract

Numerous cellular functions mediated by PI(4,5)P₂ (PIP₂) involve clustering of the lipid as well as co-localization with other lipids. Although the cation-mediated electrostatic interaction is regarded as the primary clustering mechanism, the ion-specific nature of the intermolecular network formation makes it challenging to characterize the clusters. Here we use all-atom molecular dynamics (MD) simulations of PIP₂ monolayers and graph-theoretic analysis to gain insight into the phenomenon. MD simulations reveal that the intermolecular interactions preferentially occur between specific cations and phosphate groups (P1, P4, and P5) of the inositol headgroup with better-matched kosmotropic/chaotropic characters consistent with the law of matching water affinities (LMWA). Ca²⁺ is strongly attracted to P4/P5, while K⁺ preferentially binds to P1; Na⁺ interacts with both P4/P5 and P1. These specific interactions lead to the characteristic clustering patterns. Specificially, the size distributions and structures of PIP₂ clusters generated by kosmotropic cations Ca²⁺ and Na⁺ are bimodal, with a combination of small and large clusters, while there is little clustering in the presence of only chaotropic K⁺; the largest clusters are obtained in systems with all three cations. The small-world network (a model with both local and long-range connections) best characterizes the clusters, followed by the random and the scale-free networks. More generally, the present results interpreted within the LMWA are consistent with the relative eukaryotic intracellular concentrations Ca²⁺ << Na⁺ < Mg²⁺ < K⁺; i.e., concentrations of Ca²⁺ and Na⁺ must be low to prevent damaging aggregation of lipids, DNA, RNA and phosphate containing proteins.

Graphical Abstract

1. INTRODUCTION

Phosphatidylinositol (4,5)-bisphosphate (PI(4,5)P₂, and henceforth referred to as PIP₂), derived from phosphorylation of phosphatidylinositol at 4 and 5 positions on the inositol ring, mediates a broad range of cellular processes.^1–4 Dysregulation of the PIP₂ pathway leads to a number of human diseases such as cancer, metabolic disorders, inflammation and neurological diseases including Alzheimer’s, Parkinson’s and epilepsy.^4–7 The products of PIP₂ hydrolysis, inositol (1,4,5)-trisphosphate (IP₃) and diacylglycerol (DAG), are key second messengers in cellular signaling pathways and the phosphatidylinositol-3,4,5-trisphosphate (PI(3,4,5)P₃), a phosphorylated derivative of PIP_2, is an integral part of the PI3K/Akt cell survival signaling pathway.^{2, 3, 8} PIP₂ regulates the activity of ion channels in the cell membrane including inward rectifier K⁺ channels, voltage-gated K⁺ channels, voltage-gated Ca²⁺ channel, and plasma membrane Ca²⁺ ATPase.⁹ Synaptic vesicle exocytosis in neurons is mediated by PIP₂ via modulation of vesicle docking.^10–12 Recently, a molecular mechanism has been proposed for general anesthesia regarding PIP₂:¹³ Inhaled anesthetics activate a TREK-1 (i.e., tandem of pore domains in a weak inward rectifying K⁺ channel (TWIK)-1-related K⁺-channel)¹⁴ through disruption of ordered lipid domains and PIP₂ regulates the sensitivity and selectivity of TREK-1 to anesthetics by influencing the activity of enzyme phospholipase D2 (PLD2).¹⁵ In addition, PIP₂ carries out its various functions via binding to proteins such as phosphatase and tensin homolog deleted on chromosome 10 (PTEN),^{16, 17} syntaxin-1A,¹¹ epsin N-terminal homology (ENTH), AP180 N-terminal homology (ANTH), Four-point-one, ezrin, radixin, moesin (FERM) and pleckstrin homology (PH) domains.^18–20

How is it possible for a single lipid to be involved in so many different functions? An emerging view is that critical modulation is provided by dynamic transitions between non-clustered and clustered structures (including co-localization with other lipids).^{1, 21} For non-clustered structures, the functions are mainly rooted in the rich chemical functionality of the phosphoinositide headgroup,^{2, 22–27} while in case of clustering collective functions emerge from a complex organization of the lipids in the cluster.^{21, 28–32} As expected, the high negative charge of PIP₂ (–4 at pH 7.0)²³ makes the clustering transitions particularly sensitive to ionic conditions;³³ i.e., stable and large PIP₂ clusters will only be able to form if the electrostatic repulsion of the negatively charged headgroups is compensated by counterions.^{20, 30, 31, 34–38} However, specific ion effects on the non-clustered and clustered structures of PIP₂ remain poorly understood. Why are some ions more effective than others and how do different ions affect the PIP₂ clustering? To what degree and how can added ions stabilize the PIP₂ clusters relative to the isolated PIP₂? Specific ion effects are often framed in terms of the Hofmeister paradigm,^39–41 whereby the strengths of action of the ions follow a well-defined order. The effects are most evident in high salt concentrations (usually above 100 mM for 1:1 electrolytes) where long-range electrostatic forces are screened, regardless of whether occurring in bulk solutions or at macromolecule/water and air/water interfaces.^{39, 42} This Hofmeister paradigm is widely used to describe the stability and structure formation of macromolecules such as proteins, deoxyribonucleic acid (DNA), and ribonucleic acid (RNA).^43–45

This study investigates the specific effects of cations in the Hofmeister series^{39, 40, 42} (i.e., Ca²⁺, Na⁺, and K⁺ with a common anion, Cl^–) on the PIP₂ clustering using all-atom molecular dynamics (MD) simulations and the graph-theoretic analysis.^46–51 The MD trajectories of PIP₂ monolayers were mapped as molecular graphs³³ and then the cluster–size distributions and the intermolecular connectivity were analyzed using graph-theoretic measures. Graph spectral analysis was employed to quantitatively measure the structural complexity of a cluster and to compare the similarities among the clusters. Furthermore, the PIP₂ networks were compared with the following complex network models:⁵² Barabási-Albert scale-free,⁵⁰ Erdös-Rényi random,⁵³ and Watts-Strogatz small-world.⁴⁸ Specific ion effects are interpreted based on Collins’s conceptual framework on electrostatic pairing preference, the law of matching water affinities (LMWA).^54–59 In essence, the LMWA holds that the strong inner sphere electrostatic pairs (i.e., contact ion-pairs, not separated by solvent) are preferentially formed between oppositely charged molecules with matching water affinities: The kosmotropic (k)-kosmotropic (k) and chaotropic (c)-chaotropic (c) pairs can build strong associations, while the k-c pairs cannot.

By way of outline, Section 2 (Methods) describes the MD simulation protocols (2.1), analysis (2.2), and the graph generation from MD trajectory (2.3). Section 3 reviews the theoretical frameworks for specific ion effects and examines the differences in water affinities among ions from the simulations (3.1) and graph-theoretic analysis (3.2). Section 4 begins with the specific ion binding patterns revealed by radial distributions from the MD trajectories (4.1), followed by the analysis of structural properties of PIP₂ clusters (4.2). Section 5 combines the Discussion and Conclusions and relates the propensity of PIP₂ clustering to the intra and extra-cellular ionic concentrations.

2. METHODS

2.1. MD simulation protocols.

Table 1 lists the 4 PIP₂ monolayer systems (S1–S4). Each system is composed two monolayers separated by a large layer of water with an equally large vacuum region between the chains (Figure 1, right). Each monolayer contains 64 PIP₂ with stearic (18:0) and arachidonic (20:4) acid chains at the 1 and 2 glycerol positions, respectively, and with singly-protonated at 4- or 5-phosphate on inositol ring in a 1:1 mixture. Figure 1 (left) shows protonation at the 4 position. The systems were fully hydrated with 15360 molecules of TIP3P water⁶⁰ and contain different chloride salts: CaCl₂ (S1), NaCl (S2), KCl (S3), and CaCl₂/NaCl/KCl (S4). Neutralizing cations for the negative charge of PIP₂ headgroup (–4 at pH 7.0)²³ and additional cation/Cl⁻ salts were added to achieve a bulk concentration of ~125 mM, preserving charge neutrality in the bulk water phase. Simulations of SAPC (1-stearoyl-2-arachidonoyl-sn-glycero-3-phosphocholine) monolayers were also carried out for comparison (see SI of details). Monolayer systems were built using CHARMM-GUI program⁶¹ and then the number of ions was adjusted using tailored CHARMM⁶² scripts.

Table 1.

Component numbers and bulk ion concentrations in MD systems.

System	Lipida	Ionb				Bulk concentration (mM)
	PIP2 (64/monolayer)	Ca2+	Na+	K+	Cl−
S1	128	298	–	–	84	125 (Ca2+)
S2	128	–	535	–	23	125 (Na+)
S3	128	–	–	535	23	125 (K+)
S4	128	249	45	45	76	41.67 (Ca2+) 41.67 (Na+) 41.67 (K+)

^aA system contains two monolayers, each consisting of 64 PIP₂ singly-protonated at either the 4 or 5 phosphate positions.

^bThe Ca²⁺-containing systems (S1 and S4) required extra Ca²⁺ and Cl^– ions to obtain ~125 mM equilibrium bulk concentrations as they created electric Ca²⁺/Cl^– double layers at the headgroup/water interfaces, where the interface concentrations exceed the bulk concentration by a factor of ~3. No double layer formation was observed for the Na⁺ (S2) and K⁺ (S3) only systems.

Figure 1.

The simulation system. (left) Structure of PI(4,5)P₂ headgroup: P1, P4, and P5 indicate phosphate groups attached to C1, C4 and C5 carbons of the inositol ring, respectively. Half of the PIP₂ are protonated at P4 (shown here), and half are protonated at P5. (right) Snapshot obtained from the last 1ns of S4 (the system containing a mixture of Ca²⁺, Na⁺, K⁺, and Cl^–). The system consists of two monolayers, vacuum phases, and an ionic solution. Blue lines indicate the periodic boundaries of the simulation box. The length of the cell normal to the interface was kept constant (h_z = 200 Å) during simulation, while the area of monolayer plane and the distance between two planes varied preserving the h_x/h_y ratio as 1. Coloring is as follows: PIP₂, grey; water, light blue; Ca²⁺, red; Na⁺, blue; K⁺, green; Cl^–, yellow.

The CHARMM program (c41b2)⁶² was used for the MD simulations. The CHARMM C36 lipid force field⁶³ with the newly developed pair-specific LJ parameters (NBFIX) of Ca²⁺ with Cl^– and with the carbonyl oxygen of the phosphate group were used.³³ The NBFIX parameters yielded good agreement with experimental osmotic pressure over a wide range of concentrations for solutions of Ca²⁺, Cl⁻, and dimethyl phosphate, and the common artifact of ion aggregation was eliminated.

The systems were minimized using the steepest descent algorithm and heated to 293.15 K over 40 ps. They were equilibrated for 0.45–0.52 μs gradually reducing ions to achieve the 125 mM bulk concentration, followed by the production runs of 0.8–1.08 μs under constant number, tangential pressure (P_t) and length normal to the surface of monolayer (h_z) (NP_th_zH)^{64, 65} with constant temperature using the Nosé-Hoover thermostat.⁶⁶ The integration time step equaled 1 fs, and coordinate sets were saved every 5 ps. Electrostatics were evaluated using particle-mesh Ewald (PME) with ca. one grid point per angstrom (Å), a sixth-order spline interpolation for the complementary error function, a κ value of 0.32, and a 12 Å real space cutoff. The van der Waals term used a standard 6–12 LJ form, with force-switched truncation over the range 8–12 Å. The SHAKE constraint method⁶⁷ was applied to all covalent bonds to hydrogen, with the default tolerance (1.0 × 10^–10 Å).

Each system was simulated at different surface tensions, i.e., γ = 40, 50, and 55 dyn/cm (except S3; K⁺ only). For S3, γ = 40, 50, and 52 dyn/cm because the system becomes unstable at γ > 52 dyn/cm, ultimately leading to the rupture of the monolayer. From the expression for surface pressure Π of monolayers, Π = γ₀ − γ, where γ₀ is the surface tension of water (72.75 at 293.15 K), the preceding surface tensions yield surface pressures of approximately 32, 22, and 17 dyn/cm respectively. Hence, a monolayer γ = 40 dyn/cm reasonably corresponds to a bilayer at equilibrium^{37, 38, 68–72} and is the primary focus of this study. The higher γ correspond to bilayers under tension and therefore higher surface area. The larger surface tensions were included to investigate clustering at larger areas, and to evaluate the area compressibility K_A from the surface tension surface area isotherm.

2.2. MD simulation analysis.

The area compressibility K_A is defined as

$$K_A=A_0{\left(\frac{d\gamma }{dA}\right)}_T$$ (1)

Where A₀ is the surface area at γ = 40 dyn/cm, and T is the absolute temperature. The slope was obtained from a linear fit over the three surface tensions. The uncertainty of the slope was estimated from the standard errors of the areas at each surface tension.

The potential of mean force (PMF), W(r), between a pair of interacting atoms (in a unit of kcal/mol), is obtained by taking the negative logarithm of the corresponding radial distribution function g(r):

$$W\left(r\right)=-k_{B}Tln\left[g\left(r\right)\right]$$ (2)

Where K_B is Boltzmann constant and r the distance between the atoms.

As evident from the time series of the area/PIP₂ (Figure S1) and the largest cluster size (Figures 6 and S2) the systems show good stability by 150 ns (following the ~500 ns of ion adjustment and equilibration). For consistency, the last 750 ns of each of the trajectories were used for analyses. The cluster size fluctuations are on the 50–100 ns time scale (Figures 6 and S2), and the trajectories were divided into 5 independent blocks of 150 ns for each monolayer (i.e., in total 10 blocks) to estimate standard errors (SE). Figure S3 shows that the smallest independent block size is approximately 150 ns for the systems containing calcium and/or sodium (S1, S2, and S4) and 50 ns for the potassium only system (S3). For simplicity and consistency, 150 ns block sizes were used to estimate standard errors of all reported averages.

Figure 6.

Time evolution of the largest cluster size for the monolayer on the left side (z < 0) at γ= 40 dyn/cm during production runs. The cluster size denotes the numbers of PIP₂ in a cluster.

Visualization and analysis of MD trajectories were carried out using VMD (Visual Molecular Dynamics) (version 1.9.3),⁷³ CHARMM (c41b2),⁶² and GNUPLOT (version 5.2)⁷⁴ programs.

2.3. Graph generation.

A graph is a pair of sets G = (P, E), where a set of vertices (nodes) P is connected by a set of edges (links) E. A graph G with n vertices can be represented by its adjacency matrix A(G) with n×n elements A_ij, whose values are as given by

$$A_{ij}=1\text{if}i\text{and}j\text{are}\text{linked},$$

$$A_{ij}\text{=}\text{0}\text{otherwise},$$

$$A_{ij}=0\text{for}i=j\left(\text{no}\text{self}-\text{loops}\right).$$

The directionality of the edges was not considered in the PIP₂ graph generation (i.e., undirected graph). The graphs were constructed based on the MD trajectories in a similar manner to the previous study:³³ PIP₂ molecules (vertices) were taken to be connected to each other by an edge if the distance between interacting oxygen atoms was less than the distance threshold of 3.3 Å. The stable and tight intermolecular interactions occur within this threshold. The list of the oxygen atoms considered in the edge formation is as follows:

1. 4 oxygens of the non-protonated P1, ‘O1^np’

2. oxygen of the hydroxyl group at C2

3. oxygen of the hydroxyl group at C3

4. 4 oxygens of the singly-protonated P4, ‘O4^p’

5. 4 oxygens of the non-protonated P4, ‘O4^np’

6. 4 oxygens of the singly-protonated P5, ‘O5^p’

7. 4 oxygens of the non-protonated P5, ‘O5^np’

8. oxygen of the hydroxyl group at C6

9. oxygen of the carbonyl group of the sn-1 acyl chain

10. oxygen of the carbonyl group of the sn-2 acyl chain

The 10 different oxygen types yield 55 (i.e., ₁₀H₂ = 55) edge types. Lipids separated in the primary cell that interact with their images within the distance threshold are taken to be connected to each other.

The creation of the adjacency matrix from the MD trajectory was aided by the open-source C and R package ChemNetworks (version 2.2).⁷⁵ Visualization and analysis of graphs were carried out using tailored Python scripts and the Python package NetworkX (version 2.1).⁷⁶

3. THEORETICAL FRAMEWORK.

3.1. Specific ion effect.

According to the law of matching water affinities (LMWA),^{39, 54, 55, 57–59} the kosmotropic (derived from the Greek noun kosmos, meaning order)/chaotropic (Greek chaos = disorder) nature is the key to understand electrostatic pairing in biological systems and this can be explained in terms of water affinities. Namely, charged molecules with high charge density are kosmotropes, binding to water molecules strongly to promote the intermolecular interactions, whereas those with low charge density are chaotropes which interact with water molecules weakly relative to the strength of water-water interactions leading to weakening of the interactions. These characteristics are apparent in the entropy changes of solutions due to the ions:^{59, 77} kosmotropes reduce the translational motion of water compared to that around another water (i.e., negative entropies relative to that for pure water) while chaotropes display positive entropy deviations. Interestingly, the order of strength is generally parallel with the Hofmeister series. More importantly, the strong inner sphere pairs are preferentially formed between oppositely charged molecules with matching water affinities. Like attracts like (i.e., kosmotrope-kosmotrope and chaotrope-chaotrope pairs) in contrast unlike molecules tend to repel each other (kosmotrope-chaotrope pair). The water affinity corresponds to charge density or thermodynamic quantities such as absolute hydration enthalpy (or free energy), the activation energy of stripping a water molecule from a molecule’s first shell, and Samoilov’s measure of hydration.^78–81

The relative charge densities of K⁺, Na⁺, and Ca²⁺ with respect to water (i.e., the density of the positively charged portion of water as 1) are approximately 0.59, 1.06, and 2.24, respectively.^{55, 82} The hydration free energy (i.e., the PMF at first minimum) and activation energy of stripping a water molecule from the first coordination shell (i.e., the difference between the first minimum and the first maximum in the PMF) follows the charge density order as reproduced in the mixed system (S4; Ca²⁺/Na⁺/K⁺) (Figure 2). This means that the kosmotropic character increases (chaotropic character decreases) in the order K⁺ < Na⁺ < Ca²⁺: Ca²⁺ is the strong kosmotrope (denoted as ‘k’), Na⁺ the weak kosmotrope (‘wk’), and K⁺ the chaotrope (‘c’). Likewise, the relative charge densities of the phosphate groups of the PIP₂ headgroup increase in the order non-protonated P1 (a phosphate diester with a net charge of −1 and a larger surface area than P4/P5) < protonated P4/P5 (a phosphate monoester; −1 net charge) < non-protonated P4/P5 (a phosphate monoester; −2); the kosmotropic character increases with the order.

Figure 2.

Potential of mean force (W(r)) between water (oxygen) and cation. It was calculated for the last 150 ns of the mixed (Ca²⁺/Na⁺/K⁺) cation system (S4) at γ = 40 dyn/cm using eq 2. The results are consistent over different γ values.

3.2. Graph-theoretic analysis.

3.2.1. Basic measures.

The degree distribution P(k) captures how vertices and edges are organized:^46–51 In general, P(k) of a random graph exhibits a Poisson distribution with a peak at the average vertex degree 〈k〉, while curves such as exponential, power–law, log–normal, or Weibull imply characteristic connections. The local clustering coefficient of each vertex c_i is defined as the ratio between the number of connections among the l_i (nearest) neighbors of a given vertex i and the maximum possible value, l_i(l_i–1)/2. It takes a value between 0 and 1, 0 if none of the neighbors of a given vertex are connected together and 1 if all the neighbors are linked. It thus captures the local connectivity (i.e., the degree to which adjacent vertices tend to cluster together) of a vertex. The average clustering coefficient (C) is the average of all c_i values in the graph.^48–51 The average path length (L) is defined as the average number of edges along the shortest paths for all possible pairs of vertices.^{47, 52} The efficacy of information transfer in networks can be measured by their L values. Most practical networks (e.g., social, telecommunication, biological, and chemical networks) have short average path lengths that are relatively independent of the network size. A PIP₂ cluster can be defined as a group of connected vertices (i.e., PIP₂ molecules) such that every vertex is connected by one or more edges to one or more vertices in the cluster. The cluster size denotes the number of vertices in the cluster. In the present analysis, two vertices were treated as connected (i.e., each vertex gains 1 degree) if they are linked by one or more edges. The largest possible cluster size in this study is 64, the number of PIP₂ in the primary cell. This eliminates the artifact of infinite sized clusters arising because of the periodic boundary conditions. The clusters could be larger if the system sizes were larger. Nevertheless, the present system sizes are sufficient to draw conclusions regarding ion effects and to compare the simulations with different kinds of graphs.

3.2.2. Graph spectral analysis.

The eigenvalues (λ) of the adjacency matrix A(G) are directly related to the graph’s topological features.^{47, 83–87} For this, the distribution of the eigenvalues (i.e., graph spectrum) is considered as a fingerprint of the graph.^{47, 83–87} A(G) of an undirected graph G (with n nodes) is symmetric, and therefore has a complete set of n real eigenvalues, λ_n ≤ λ_n-1 ≤ ⋯ ≤ λ₁.⁸⁴ If no PIP₂ molecules are connected to any other molecules (i.e., empty graph), all the eigenvalues would be zero. In the case of dimer formation, the eigenvalues are 1 and −1. If the molecules form large clusters, the graph spectrum will exhibit characteristic distributions depending on their generation rules. Graphs generated by the same process have the same spectrum—that is, if two spectra are different the generation processes are different.

Furthermore, the graph spectral entropy (GSE) of graph G, measured in ‘bits’ (i.e., Shannon entropy⁸⁸) is defined based on the graph spectrum (P(λ), represented as a histogram that is normalized to obtain the sum of the histogram values is equal to one). We divide the range of the variable (λ) into n intervals (l_k, u_k), k=1, …, n, so that

$$GSE\left(\lambda \right)=-\sum _{k=1}^n{\int }_{l_k}^{u_k}P\left(\lambda \right){\text{log}}_{2}P\left(\lambda \right)\text{d}\lambda .$$ (3)

The entropy is used as a quantitative measure for the uncertainty of a graph (i.e., the amount of information needed to describe the graph topology).^{84, 89}

The Kullback–Leibler (KL) divergence⁹⁰ is introduced based on GSE to measure the difference between two graph spectra, P₁(λ) and P₂(λ), as

$$KL\left(P_1|P_2\right)=-\sum _{k=1}^n{\int }_{l_k}^{u_k}{P}_1\left(\lambda \right){\text{log}}_2\frac{P_2\left(\lambda \right)}{P_1\left(\lambda \right)}\text{d}\lambda .$$ (4)

The KL divergence measures the relative entropy of P₁ with respect to P₂ (i.e., the information gain achieved if P₂ is used instead of P₁) and thus it can be interpreted as the quality of fitting P₁ to P₂. In the context of coding theory, it represents the expected amount of information (‘bits’) required to code samples from P₁ using code optimized for P₂ rather than the code optimized for P₁. The Jensen–Shannon (JS) divergence⁹¹ is defined based on KL divergence such that

$$JS\left(P_1,P_2\right)=\frac{1}{2}KL\left(P_1|P_M\right)+\frac{1}{2}KL\left(P_2|P_M\right)$$ (5)

where $P_M=\frac{1}{2}\left(P_1+P_2\right)$. It always has a finite value and is symmetric, i.e., JS(P₁, P₂) = JS(P₂, P₁), while the KL(P₁|P₂) is infinite when P₁ is defined in a region where P₂ cannot exist and in general KL(P₁|P₂) ≠ KL(P₂|P₁). The JS value increases as the two distributions become more different, while it is zero for the same distribution. The square root of the JS divergence is a metric often referred to as JS distance (JSD).

In the present study, the JSD was used to measure the topological similarity between PIP₂ graph spectra and to infer the mechanisms underlying PIP₂ clustering with respect to the three representative complex network models.

4. RESULTS

4.1. Cation-phosphate interactions.

Figure 3 shows representative snapshots from simulations of S1–S4 at surface tension γ=40 dyn/cm. Based on the analysis described below, the connectivity of each PIP₂ was determined. It is clear that S3 (chaotropic K⁺ only) contains the highest number of monomers and that the clusters are mostly dimers. The other systems with kosmotropic cations (Ca²⁺ and Na⁺) show large string-like clusters, lower populations of monomers, and lower surface areas (see Table S1).

Figure 3.

Molecular snapshots of the PIP₂-monolayers obtained from the last 1ns of each trajectory. The filled and unfilled circles (grey) indicate clustered and non-clustered PIP₂, respectively. The red, blue, and green dots denote Ca²⁺, Na⁺, and K⁺, respectively. The position of each circle/dot is the x-y coordinate of the C4 atom of the inositol headgroup/the center of cation on the monolayer plane of the systems (S1; Ca²⁺, S2; Na⁺, S3; K⁺, S4; Ca²⁺/Na⁺/K⁺) at γ=40 dyn/cm. Periodic boundary boxes (orange) are from the average over the equilibrated trajectories.

To begin a quantitative analysis, Figure 4 shows g(r) for cations and oxygens of the phosphate groups (i.e., ‘O1^np’, ‘O4^np’, and ‘O5^np’ denote oxygens on the non-protonated P1, P4, and P5, and the ‘O4^p’ and ‘O5^p’ are those on the singly-protonated P4 and P5, respectively). In all systems (S1–S4), the peak intensities of g(r) within a system increase in the order O1^np < O4^p ≈ O5^p < O4^np ≈ O5^np parallel with the relative charge density of the phosphate groups. This implies that the interaction is primarily driven by the thermodynamic stabilization of the negatively charged phosphate groups: Cations replace water molecules from the phosphate oxygen’s coordination shells by creating the cation-oxygen pair interactions as cations can stabilize the negative charges more effectively than water.

Figure 4.

Radial distribution function g(r) between cation and phosphate oxygen. The ‘O1^np’ denotes oxygens on the non-protonated P1, the ‘O4^p’ and ‘O4^np’ indicate those on the singly-protonated and non-protonated P4, and the ‘O5^p’ and ‘O5^np’ those for P5, respectively. The first peak of each g(r) is only shown. These were obtained by taking the averages over the last 150 ns of the systems at γ = 40 dyn/cm. The different γ values (i.e., 40, 50, and 55(52) dyn/cm) have parallel trends.

However, the specific cation interactions with respective phosphates vary depending on the relative kosmotropic/chaotropic nature of cations and phosphates (i.e., Ca²⁺ and O4^np/O5^np, kosmotrope (k); Na⁺ and O4^p/O5^p, weak kosmotrope (wk); K⁺ and O1^np, chaotrope (c)). In general, the binding affinity order for O1^np increases in the order Ca²⁺ < Na⁺ ≤ K⁺, while those for O4^np/O5^np follow K⁺ < Na⁺ < Ca²⁺. The binding affinities of O4^p/O5^p follow the trends of O4^np/O5^np, but the differences among cations are not as great as those of O4^np/O5^np. The strongest interactions occur for Ca²⁺-O4^np and Ca²⁺-O5^np pairs (i.e., the k-k pairs), while the weakest one is between Ca²⁺ and O1^np (the k-c pair). The binding affinities of the other pairs are between the two extremes.

This binding preference induces competition among the cations toward the respective phosphate in the presence of all three types of cations (S4; Ca²⁺/Na⁺/K⁺) (Figure 4, column 4). The binding affinities of Na⁺ and K⁺ for O4^np/O5^np (Figure 4, rows 4 and 5) are substantially reduced in the presence of Ca²⁺ compared to the systems containing only one type of cation (Figure 4, columns 2 and 3 for S2 (Na⁺) and S3 (K⁺), respectively), while Ca²⁺ more strongly binds to O4^np/O5^np in the presence of Na⁺ and K⁺ than in S1 (Figure 4, column 1). This implies that Ca²⁺ replaces Na⁺ and K⁺ for the O4^np/O5^np: Ca²⁺ expels Na⁺, while K⁺ is less influenced by Ca²⁺ because of the interaction distance (i.e., the distance corresponding to the maximum peak) of Na⁺ overlaps with that of Ca²⁺, in contrast to K⁺. On the other hand, Na⁺ and K⁺ are not replaced by Ca²⁺ for O1^np as their binding affinities toward O1^np are higher than that of Ca²⁺ (Figure 4, row 1). The binding affinities of cations with O4^p/O5^p (Figure 4, rows 2 and 3) are between those with the relatively strong kosmotropic (O4^np/O5^np) and chaotropic (O1^np) phosphate oxygens: the binding affinities of Ca²⁺ with O4^p/O5^p are less than those with O4^np/O5^np, exhibiting no increase compared to Ca²⁺ only system (S1) (Figure 4, column 1). The affinities of Na⁺ and K⁺ with O4^p/O5^p are reduced compared to Na⁺ (S2) and K⁺ (S3) only systems (Figure 4, columns 2 and 3, respectively), but their reductions are less than those for O4^np/O5^np.

Altogether, the strong cation-phosphate interactions preferentially occur between the molecules with better-matched kosmotropic/chaotropic character, while the poorly matched pairs fail to achieve strong interactions, consistent with the LMWA. Ca²⁺ is strongly attracted to P4/P5 (the k-k pair), while K⁺ preferentially binds to P1 (the c-c pair); Na⁺ (wk) interacts with both P4/P5 (k) and P1 (c). Furthermore, the competitive ion binding in the mixed cation system (S4; Figure 4, column 4) leads to the stronger total cation interactions for every type of phosphate (i.e., O1^np, O4^p/O5^p, and O4^np/O5^np) than the systems containing only one type of cation (S1, S2, and S3; Figure 4, columns 1–3, respectively). These specific interactions lead to the characteristic PIP₂ clustering patterns.

4.2. PI(4,5)P₂ clustering.

Figure 5 displays the size distributions, graphical representations, and intermolecular connectivities (degree and eigenvalue distributions) of the PIP₂ clusters of S1–S4 at γ = 40 dyn/cm. The graphs indicate the formation of large string-like clusters in the presence of kosmotropic cations (Ca²⁺ and Na⁺), while the clusters are significantly smaller with the chaotropic cation (K⁺). The trend is similar to the molecular snapshots of the MD trajectories in Figure 3, but a little different because unlike the MD snapshots, the nodes in the graphs are not assigned as the x-y coordinates of the monolayer plane. In addition, the clustering is more pronounced in the graphs because all the 55 different cation-mediated interactions are considered in the edge formation, while only the relative distance between neighboring C4 atoms are considered in the MD snapshots for illustration purposes. Additional graphs of the last 150 ns trajectories are displayed in Figure S4.

Figure 5.

Structural properties of PIP₂ clusters. (left) The probability distribution of PIP₂ cluster size, (middle) that of degree k, P(k) with network graph, and (right) the eigenvalue λ, P(λ). These were obtained by taking the averages and the standard errors (SE) over the 10 blocks of 150 ns (5 blocks for each monolayer) of the systems at γ= 40 dyn/cm. The distributions were normalized to obtain the sum of the values that are equal to one. Monomers are not displayed in the network graphs.

The clustering patterns are more similar between two kosmotropic cations, Ca²⁺ and Na⁺, than between the kosmotropic and chaotropic cations, Na⁺ and K⁺. For the system with chaotropic (c) K⁺ (S3), the system exhibits mostly monomers or small clusters. The probability of finding monomers (i.e., P_m; cluster size = 1) is 48.5 % (Table 2) and the distribution decays quickly where the cluster size ranged between 1 and 10 (Figure 5, column 1, row 3). The probability of finding clusters with the size ≥10 (i.e., P_≥10) for is 4.8 %. On the other hand, the kosmotropic (k) Ca²⁺ (S1) and weak kosmotropic (wk) Na⁺ (S2) containing systems have fewer probabilities of monomer than S3 (K⁺), i.e., 39.4 and 39.6 %, respectively. In addition, these cations can induce larger clusters of which the cluster size is greater than or equal to 10. The P_≥10 for S1 (Ca²⁺) and S2 (Na⁺) are 24.5 and 37.6 %, respectively. Likewise, the graph spectral entropy (GSE), the complexity measure of a cluster, is greater for Ca²⁺ (3.47 bits) and Na⁺ (3.59 bits) than for K⁺ (3.15 bits) (Table 2). The clustering is more pronounced when the system contains both kosmotropic and chaotropic cations, i.e., S4 (Ca²⁺/Na⁺/K⁺), revealing a synergistic effect: the P_m (29.7%) is reduced, while P_≥10 (55.6%) and GSE (3.63 bits) are promoted compared to the systems with one type of cation. In addition, the probability of finding cluster size ≥60 of which almost all the 64 PIP₂ molecules in the monolayer are clustered together is 51.6% (Figure 5, column 1, row 4). This result agrees with our previous simulations of monomethyl phosphate aqueous ionic solutions where the largest clusters are formed when Ca²⁺, Na⁺ and K⁺ are all present.³³ However, this synergistic effect of kosmotropic/chaotropic cations is not observed for monolayers at higher surface tensions (50 and 55 (52) dyn/cm) where the surface area is expanded (Table S1 and Figures S5 and S6).

Table 2.

Graph-theoretic indices of the systems.^a

System	Pm (%)	P≥10 (%)	GSE (bit)	<k>	C
S1 (Ca2+)	39.4±2	24.5	3.47±0.01	2.21±0.03	0.13
S2 (Na+)	39.6±2	37.6	3.59	2.88±0.05	0.34±0.01
S3 (K+)	48.5	4.8	3.15±0.01	1.36±0.01	0.05
S4 (Ca2+/Na+/K+)	29.7±3	55.6	3.63±0.01	2.92±0.04	0.24±0.01

^aP_m and P_≥10 denote the probability of finding monomers and clusters with the size ≥ 10, respectively. GSE is the graph spectral entropy of a graph, measured in ‘bits’. The <k> and C indicate average vertex degree and an average of (local) clustering coefficient c_i, respectively. Standard error (SE) values are indicated only when they are greater than or equal to 0.01. Data were obtained by taking the averages over the 10 blocks of 150 ns (5 blocks per monolayer) of the systems at γ = 40 dyn/cm.

The local clustering tendency measured by C follows the order: S3 (0.05) < S1 (0.13) < S4 (0.24) < S2 (0.34) (Table 2). Interestingly, the order of C of S2 and S4 is reversed with respect to the order of the cluster structural measures (i.e., P_≥10 and GSE). This implies that even though Na⁺ can induce strong local networks among the nearest neighbors via the binding to both P1 and P4/P5, it cannot generate global networks, of which both local and long-range edge formations among the vertices are required.

It is evident from Table 2 (<k>; average degree) and the second column of Figure 5 (P(k); degree distribution) that different cations have different effects on the PIP₂ network connectivity. The <k> of the systems containing kosmotropic cations (S1, S2, and S4) are greater than that with chaotropic cations (S3; K⁺ only). P(k) for S3 shows an exponential distribution³³ with very low probabilities for high-degree vertices (11.0 and 1.4 %, for k = 3, and 4, respectively)—that is, there are no strongly connected hubs in this system. In contrast, the systems with kosmotropic cations (S1, S2, and S4) show the right-skewed distributions with high-degree vertices, k ≥ 3 (Figure 5, column 2), which could promote the formation of large and stable clusters. These characteristic degree distributions imply that the underlying organization of vertices and edges of those systems with different cations are different.

The structural similarity between PIP₂ clusters was evaluated using Jensen–Shannon distance (JSD) (Table S2). Among the systems with only one type of cation, the JSD increases in the order JSD(S1,S2) (0.13) < JSD(S1,S3) (0.16) < JSD(S2,S3) (0.24); the shorter distance, the higher similarity. This reveals that the kosmotropic/chaotropic nature of a cation is important for determining the cluster structure. The similarity of kosmotropic Ca²⁺ (S1) with weak kosmotropic Na⁺ (wk; S2) is higher than that with chaotropic K⁺ (S3). The similarity between weak kosmotropic and chaotropic cations with the same net charges (Na⁺ and K⁺) becomes least. At γ = 40 dyn/cm, the cluster structure of the mixed cation system (S4; Ca²⁺/Na⁺/K⁺) is most similar to that of the system with Na⁺ only (S2), followed by Ca²⁺ only (S1); that for chaotropic K⁺ only (S3) is significantly lower than kosmotropes (Na⁺ and Ca²⁺): JSD(S2, S4) < JSD(S1, S4) < JSD(S3, S4). However, the order of JSD(S2, S4) and JSD(S1, S4) is reversed at γ = 50 and 55(52) dyn/cm, i.e., JSD(S1, S4) < JSD(S2, S4) < JSD(S3, S4). This is related to the Ca²⁺ ion’s characteristic role to maintain the PIP₂ network topologies against the membrane expansion forces as revealed by the comparison with the complex network models as below.⁵²

The time evolutions of the largest cluster size for the monolayer on the left side (z < 0) at γ= 40 dyn/cm are shown in Figure 6. During the last 750 ns of the production runs the cluster sizes fluctuate on the 50–100 ns timescale. The largest variation in cluster size is found for the K+ only system (sizes range from 5–44), and the smallest is for the system with all three cations present (47–64). The periods can be considered as the approximate lifetimes of PIP₂ clusters. The accurate values are difficult to calculate because the PIP₂ clusters are in dynamic equilibrium, whereby the formation and deformation of the links between PIP₂ are independently repeated during simulations. The monolayer on the right side (z > 0) of each system follows the trend (Figure S2). Figure S7 compares the largest cluster size distributions of the four systems. While increasing the system size would be expected to yield larger clusters for systems S1, S2 and S4, significant differences among these distributions are evident for the present simulations, where the maximum cluster size possible is 64. The distributions of the two monolayers for each system are quite similar indicating good sampling. Hence, the present systems sizes are sufficiently large for productive comparisons among the different ion species.

Figure 7 illustrates the topological properties of the complex networks. In the Barabási-Albert scale-free model, a graph G(n, m) of n vertices is grown by attaching new vertices each with m edges that are preferentially attached to existing vertices with a high degree—the more connected a vertex is, the more likely it is to receive new edges. This preferential attachment process leads to a (scale-free) degree distributions, i.e., P(k) = ck^−γ where c is a proportionality constant and typically 2 < γ < 3. An Erdös–Rényi model G(n, p) is constructed by connecting n vertices randomly in which each pair of vertices is connected by an edge with a given probability p independent from every other edge; p=1 leads to the complete graph in which every pair of vertices is connected by an edge, while p=0 gives the empty graph consisting of n isolated vertices with no edges. P(k) follows the binomial distribution such as $P\left(k\right)=\left(\begin{array}{c}n-1\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-1-k}$, and it gives the Poisson distribution with the small p values, $P\left(k\right)=\frac{z^k{e}^{-z}}{k!}$ (where z and e are the average numbers of events per interval and the Euler’s number, respectively) and approaches to the normal distribution $P\left(k\right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{\frac{-{\left(k-\mu \right)}^2}{2{\sigma }^2}}$ (where μ and σ are the mean and the standard deviation, respectively) with large n. In the Watts-Strogatz small-world model, a graph G(n, l, p_r) is created from a ring over n vertices wherein each vertex in the ring is joined to its l neighbors (or l−1 neighbors if l is odd). Each edge between neighbors is then rewired to another vertex that is randomly chosen with a probability p_r. When p_r = 0, the small-world graph is a ring, whereas the graph becomes an Erdös-Rényi random graph with p_r = 1. Small-world graphs are characterized by short path lengths (L)—that is, with rewiring only a short amount of edges, most vertices can be reached from every other vertex by a small number of steps, preserving high clustering coefficient (C).^{92, 93}

Figure 7.

Complex network models. (top) Barabási-Albert scale-free G(n, m), (middle) Erdös-Rényi random G(n, p), and (bottom) Watts-Strogatz small-world G(n, l, p_r) models. The first column presents the network graphs and the probability of degree k, P(k) and the second column that of eigenvalue λ, P(λ). They are normalized to obtain the sum of the histogram values to be one.

The parameters were set to n=64 (same as a PIP₂ network), m=1, p=0.03, l=2, and p_r=0.5; with that, the complexities of the model networks, measured by GSE lead to the similar values as the PIP₂ networks, i.e., ~3.4 (bits). The graphs represent one realization of the random processes, while P(k) and p(λ) are the averages of 10⁵ graphs with different random seeds (the standard errors of the mean are less than 0.0002). The P(λ) values were used to calculate JSD between the complex network (Figure 7, column 2) and the PIP₂ network (Figure 5, column 3 for γ = 40 dyn/cm).

As shown in Table 3, for all the surface tensions (γ = 40, 50, 55(52)), JSD between the PIP₂ network and complex network model increases in the order Watts-Strogatz small-world JSD(PIP₂, SW), Erdös-Rényi random JSD(PIP₂, R), and Barabási-Albert scale-free JSD(PIP₂, SF). It can be inferred from the trend that the generation process of the Watts-Strogatz small-world model, including both short- and long-range interactions among its vertices, best describes the mechanism underlying PIP₂ clustering, followed by the Erdös-Rényi random (i.e., random connection) and the Barabási-Albert scale-free (i.e., preferential attachment) models. In particular, in the presence of Ca²⁺ (S1 and S4), the JSD(PIP₂, WS) values are kept small for not only low but also high γ values. This is in agreement with the macroscopic condensation behavior of the monolayer. Under high surface tension (i.e., γ = 55 (52 for S3) dyn/cm), the areas/PIP₂ of S1 and S4 are 87.3 and 87.6, while those for S2 and S3 are 94.1 and 103.0 Ų, respectively (Table S1). Moreover, the area compressibility (K_A) for Ca²⁺ containing systems (~121 (S1) and ~89 dyn/cm (S4)) are significantly greater than those for Na⁺ and K⁺ only systems (~60 (S2) and ~56 dyn/cm (S3)), which means that in the Ca²⁺ containing systems PIP₂ molecules more strongly interact with each other, compared to the other systems (Table S1).

Table 3.

Jensen-Shannon distance between complex network model and PI(4,5)P₂ network.^a

System	γ (dyn/cm)	JSD(PIP2, SF)	JSD(PIP2, R)	JSD(PIP2, SW)
S1 (Ca2+)	40	0.25	0.21	0.13
	50	0.26	0.22	0.16
	55	0.25	0.21	0.15
S2 (Na+)	40	0.27	0.26	0.21
	50	0.28	0.27	0.24
	55	0.28	0.26	0.23
S3 (K+)	40	0.29	0.25	0.19
	50	0.35	0.32	0.29
	52	0.29	0.26	0.20
S4 (Ca2+ Na+ K+)	40	0.26	0.24	0.19
	50	0.26	0.22	0.16
	55	0.27	0.23	0.17

^aJSD(PIP₂, SF), JSD(PIP₂, R), and JSD(PIP₂, SW) indicate Jensen-Shannon distance between the PIP₂ clusters of each of systems (i.e., S1, S2, S3, and S4) and the complex network models (SF, Barabási-Albert scale-free; R, Erdös-Rényi random, and SW; Watts-Strogatz small-world). Data indicate the average and standard error (SE) over the last 10 blocks of 150 ns (5 blocks per monolayer) wherein SE values are negligible (< 0.005).

In accordance with the low JSD(PIP₂, SW), L (the average path length; the number of steps between two randomly chosen nodes) values are within the typical range of the small-world network (e.g., L = ~6 for social networks; “six degrees of separation”⁹⁴). To illustrate the topological similarities, the representative PIP₂-networks (S4; Ca²⁺/Na⁺/K⁺) are compared with the complex network models in Figure S8.

These cation-specific clustering characteristics are rooted in the interaction of PIP₂ headgroup with cations as confirmed by comparison with the simulations on SAPC monolayers. The lipid has identical (18:0–20:4) acyl chains with PIP₂, while the structure of the headgroup (i.e., phosphocholine) is simpler than the inositol PIP₂ headgroup. Unlike the PIP₂ monolayers, no significant differences in both the area/lipid and the K_A were observed upon different cation conditions (M1; Ca²⁺ only, M2; Na⁺ only; M3; K⁺ only; M4; Ca²⁺/Na⁺/K⁺) for γ = 40, 50, and 55 dyn/cm (Table S4).

5. DISCUSSION AND CONCLUSIONS

This study investigated specific ion effects on the PIP₂ clustering in monolayers using all-atom MD simulations. The analysis began with characterizing cation binding patterns. The binding of cations at the phosphate groups results in the loss of water molecules and the gain of cations in the phosphate oxygen’s coordination shells. Likewise, cations lose water molecules and gain oxygen atoms in their coordination shells, which causes the loss of thermodynamic stability (i.e., hydration enthalpy).^{95, 96} Therefore, in order to form PIP₂ clusters, the interactions between cations and phosphate groups should be strong enough to overcome the dehydration penalties and the entropic factors which drive the system towards non-clustered states. As shown in Figure 4, the strong cation-phosphate interactions preferentially occur between the molecules with better-matched kosmotropic/chaotropic characters to yield the more stable and tighter intermolecular interactions, while the poorly matched pairs fail to achieve strong interactions. The affinity order of cations for P4/P5 (the major binding sites for clustering) follows the reverse Hofmeister series, K⁺ < Na⁺ < Ca²⁺. In contrast, the order of cation binding for P1 follows Ca²⁺ < Na⁺ ≤ K⁺. Specifically, K⁺ and Na⁺ are significantly replaced by Ca²⁺ at non-protonated P4/P5, partially replaced at singly-protonated P4/P5, and not replaced at P1. This competitive cation binding is consistent with the X-ray fluorescence experiments showing that Ca²⁺ does not completely replace K⁺ in PIP₂ headgroups.⁹⁷ The simulations presented here show the explicit sites of incomplete replacement.

The preceding results show the utility of the law of matching water affinities (LMWA): kosmotropic Ca²⁺ binds chaotropic P1 oxygens to a much lesser degree compared to chaotropic K⁺ (Figure 4, row 1), despite the charge density of Ca²⁺ is more than 2 times greater than that of K⁺.^{58,55, 82} The same reasoning can be applied to another somewhat counterintuitive result: experimental evidence indicates that Mg²⁺ binds PIP₂ headgroups with a significantly lower affinity than Ca²⁺ and does not induce significant PIP₂ clustering^{97, 98} even though Mg²⁺ has a higher charge density (> 2 fold) than Ca²⁺.^{58,55, 82} In this case, Ca²⁺ and phosphates have similar water affinities. They are well matched in the LMWA and thereby form strong inner sphere complexes (i.e., contact ion-pairs). In contrast, Mg²⁺ has a higher water affinity than phosphates, and because of this mismatch they only form weak outer-sphere complexes (i.e., solvent separated ion-pairs).⁵⁸ More broadly, the eukaryotic intracellular concentrations of the cations discussed here are inversely correlated to the (water affinity) matching with respect to phosphate: Ca²⁺ (~10⁻⁴ mM) << Na⁺ (~10 mM) < Mg²⁺ (~40 mM) < K⁺ (~159 mM).^{99, 100} Phosphates are critical inorganic anions contained in RNA, DNA, lipids and some proteins. The well-matched Ca²⁺ should be maintained in low concentrations (via Ca²⁺ pumps) to avoid abnormal aggregations of these important cellular components; Na⁺ at high concentration is toxic because it forms moderately strong inner sphere pairs with the phosphates; the effects of Mg²⁺ and K⁺ are relatively minor.⁵⁸

These specific interactions between cations and phosphate groups lead to the characteristic intermolecular interactions between adjacent PIP₂ molecules as examined via graph-theoretic approaches. Kosmotropic Ca²⁺ and Na⁺ induce the formation of large clusters, while that due to K⁺ is minimal (mostly monomers and dimers). Na⁺ can induce strong local clustering via binding to both P1 and P4/P5, but it fails to create large clusters. The strongest clustering is achieved by mixtures of kosmotropic and chaotropic cations (S4) at γ = 40 dyn/cm, revealing a synergistic effect, because the competitive ion binding among the cations with different kosmotropic/chaotropic characters leads to a better screening of the negatively charged phosphate groups compared to the systems containing only one type of cation (S1, S2, and S3). The connectivities of PIP₂ clusters are revealed by P(k) profiles (Figure 5, column 2). The observation that the system with only chaotropic K⁺ (S3) exhibited only small-sized clusters could be related to the dominance of low-degree vertices. The development of large clusters of systems with kosmotropic cations, Ca²⁺ (S1 and S4) and Na⁺ (S2), are consistent with their characteristic right-skewed degree distributions with considerable amounts of high-degree vertices.

Graph spectral analysis was used to quantitatively measure the cluster structure and to infer the clustering mechanism. The graph spectrum represents a graph’s topological property and generation process and is the basis of the information-theoretic measures, the graph spectral entropy (GSE) based on Shannon entropy⁸⁸ and the Jensen-Shannon distance (JSD).⁹¹ It is an appropriate topological index in the sense that the statistical ensembles of graph spectra are invariant and reproducible for the graphs with similar generation processes, no matter what graph size or its structural complexity.^{84, 86} For this, the graph spectrum based information-theoretic measures have been using in many data analytics problems and recently resulted in several deep learning algorithms including generative adversarial networks¹⁰¹ and word2vec¹⁰². It is particularly useful when the indices for the macroscopic condensation effects (i.e., area/lipid and membrane thickness) and the basic graph-theoretic measures⁵² (i.e., degree distribution, clustering coefficient, and average path length) fail to characterize the topological features.¹⁰³ Although these measures are essential to understand the general properties of the clustering, they are neither invariant in time nor similar across graphs with similar generation processes.^{52, 84, 86, 87} They can capture the overall differences between the cation conditions or between the headgroups (i.e., inositol 4,5-bisphosphate for PIP₂ and phosphocholine for SAPC), but they fail to discriminate small structural properties within the same system (Table S1) and cannot provide the insight into the generation mechanisms at atomistic resolution.

Many real-world networks (e.g., social, telecommunication, biological, and chemical networks) often show non-trivial structural features that may not be derived from either purely regular or purely random connections among their elements.^{47, 103} They share some features of complex network models including Barabási-Albert scale-free,⁵⁰ Erdös-Rényi random,⁵³ and Watts-Strogatz small-world⁴⁸ models whose topological properties and generation mechanisms are extensively studied for the last decades.⁵² Likewise, the PIP₂ clusters can be compared with the complex network models via the information-theoretic measures. We observed that the small-world network best describes all the PIP₂ clusters, followed by the random and scale-free networks, and that Ca²⁺ is necessary to maintain small-world-like PIP₂ networks.

The PIP₂ clusters have string-like topologies with fewer connections between the strings, as captured by MD snapshots (Figure 3) and graphical representations of MD trajectories (Figures 5 and S4). A cation may mediate limited numbers of cation-bridged structures (i.e., Oxygen−Cation–Oxygen) between neighboring PIP₂ molecules, preferentially forming string-like clusters. However, cations and phosphate oxygens associated within the clusters may experience different environments in terms of the extent of direct cation-oxygen contacts, the influence of water (i.e., water-mediated indirect interactions), and the degree to which the hydration of the cation is perturbed. This electrostatic perturbation would make it possible to form a few long-range intermolecular interactions. This situation becomes more complicated and subtler when the different ions compete for the phosphate groups (i.e., S4; Ca²⁺/Na⁺/K⁺), wherein the clustering is more pronounced. Likewise, the Watts-Strogatz small-world model is based on a linear structure wherein all the vertices are in a ring and each of vertices in the ring is joined to its neighbors. By randomly rewiring only a small number of edges between neighbors (i.e., small rewiring probability, p_r), the ring-structure is deformed and the resulting string-like structures are connected to other strings. Consequently most vertices in the whole systems can be reached from every other vertex by a small number of steps.^{92, 93} However, when p_r = 1, the small-world graph becomes an Erdös-Rényi random graph. Therefore, if the tendency of the random long-range connection of the monolayer increases the similarity of PIP₂ clusters with the small-world model decreases, while that with the random model increases. In the present study, the totally random assumption (i.e., the small-world model with p_r = 1 or the Erdös-Rényi random model) exhibited less similarity than the (typical) small-world model in all cation/surface tension conditions. However, at high surface tensions (γ = 50 and 55 (52 for S3) dyn/cm), the similarities of the PIP₂ clusters with the Watts-Strogatz small-world and Erdös-Rényi random models become comparable without Ca²⁺ (i.e., S2; Na⁺ and S3; K⁺), while in the presence of Ca²⁺ (i.e., S1; Ca²⁺ and S4; Ca²⁺/Na⁺/ K⁺) the small-world similarities are higher than those of Erdös-Rényi random. This implies that Ca²⁺ would maintain the small-world-like PIP₂ networks via the strong interactions with P4/P5 (as shown in Figure 4) against the membrane expansion force.

The pure PIP₂ monolayers, each consisting of 64 lipids with an average surface areas/lipid of ~77 Ų at γ=40 dyn/cm (Table S1), simulated here are reasonable starting models for characterizing the patterns in the highly concentrated local clusters in cell membranes.^{37, 38, 72} The preceding values yield disks with diameters ~80 Å. Of course the system size is not large enough to capture the behaviors at larger length scales. For example, clusters found along high-density linear paths (> 10⁻⁵ m) on the distal ends of cells¹⁰⁴ and exhibit the skewed distributions with the average size ~80 nm in diameter.^{11, 104} However, in a large scale context, other lipids must be included in the system. There are no large regions of highly concentrated PIP₂; even in “fenced” PIP₂ regions the concentration is only ~7%,²⁹ and the clustering may be influenced by other neighboring lipids (e.g., cholesterol). Furthermore, the main observations at γ = 40 dyn/cm (ion-specific intermolecular interactions, cluster structures, and macroscopic condensations) should be rigorously tested using bilayer models. Although the surface tension at the monolayer surface may approximately correspond to a bilayer at equilibrium,^{37, 38, 68–72} the clustering behaviors at bilayer interfaces may differ from the local concentrations of PIP₂ and are influenced by the presence of peripheral or transmembrane proteins. While the systems are simple (monolayers containing only 64 PIP₂, and no other lipids and no proteins) it is still possible to consider some of the biological implications of the present simulations. As noted above, the relative intracellular concentration of cations in eukaryotic cells is Ca²⁺ << Na⁺ < Mg²⁺ < K⁺. PIP₂, which is typically ~1% of membrane lipids,^{1, 9, 23} was anticipated to be mostly in the monomeric state considering its low abundance. However, ion channels pump extracellular Ca²⁺ and Na⁺ into the cell which should trigger clustering. In lipid bilayer model systems, physiologically relevant concentrations of Ca²⁺ have been found to trigger PIP₂ clustering.³¹ K⁺, already abundant in the cell, should further stabilize these clusters as shown with system containing Ca²⁺/Na⁺/K⁺ (S4).

For decades, cations were speculated to target PIP₂ headgroups promoting the PIP₂ organization, but specific ion effects on the non-clustered and clustered structures of PIP₂ only have begun to be explored. Most existing studies on electrostatic interactions between PIP₂ and ions have been focused on the physicochemical properties of the lipid, PIP₂-mediated biological processes, or macroscopic condensation effects including variations of surface areas of PIP₂ membranes upon addition of ions, rather than mechanisms underlying the ion-specific clustering. The data presented in this study indicate that the law of matching water affinities provides substantial insight into PIP₂ clustering. Characterizing the structural properties of PIP₂ clusters using simple model membranes should provide a better understanding why some ions are more effective than others, and how different combinations of ions affect the dynamic transitions between non-clustered and clustered structures that modulate cell signaling.