Quantitative Analysis of the Interactions of Metal Complexes and Amphiphilic Systems: Calorimetric, Spectroscopic and Theoretical Aspects

Metals and metal-based compounds have many implications in biological systems. They are involved in cellular functions, employed in the formation of metal-based drugs and present as pollutants in aqueous systems, with toxic effects for living organisms. Amphiphilic molecules also play important roles in the above bio-related fields as models of membranes, nanocarriers for drug delivery and bioremediating agents. Despite the interest in complex systems involving both metal species and surfactant aggregates, there is still insufficient knowledge regarding the quantitative aspects at the basis of their binding interactions, which are crucial for extensive comprehension of their behavior in solution. Only a few papers have reported quantitative analyses of the thermodynamic, kinetic, speciation and binding features of metal-based compounds and amphiphilic aggregates, and no literature review has yet addressed the quantitative study of these complexes. Here, we summarize and critically discuss the recent contributions to the quantitative investigation of the interactions of metal-based systems with assemblies made of amphiphilic molecules by calorimetric, spectrophotometric and computational techniques, emphasizing the unique picture and parameters that such an analytical approach may provide, to support a deep understanding and beneficial use of these systems for several applications.


Introduction
Metals play essential roles in life processes since they are involved in cellular and subcellular functions.
Potassium and sodium ions are the most abundant cations in the human body [1]. Sodium is at a higher concentration in the extracellular media of mammal organisms, while potassium exhibits a comparable concentration inside the cells. Other ions in the alkali metal series, such as lithium, rubidium and cesium, are present only at micromolar concentrations in the human body but are of medical/physiological relevance due to their curative or toxic properties depending on the concentration [2]. The interest in understanding the effect of metal ions on biological systems has attracted countless studies in this field for a long time. Cations such as potassium and sodium, which significantly reduce the electrostatic repulsion forces between the nucleotide phosphate groups that are negatively charged, allow the structure of the double-helix of DNA to be maintained and are crucial in determining the fine details of the helix geometry. cles, liposomes, etc.), water-insoluble drugs can be easily solubilized and thus loaded for transport and targeted release [23].
The complexity of natural membranes requires models for their investigation at the molecular level, and the most commonly used membrane-mimicking systems are aqueous solutions of surfactants able to self-aggregate in micelles or vesicles. Surfactants have been employed in the removal of heavy metals by using different strategies such as soil-washing/extraction/desorption and phytoremediation [24]. Ultra-filtration methods, which take advantage of membrane pretreatment with surfactants/biosurfactants, have been successfully proposed for metal ion removal [25].
Since metal ions have multiple implications for biological systems, ranging from their involvement in cellular functions to their presence as pollutants in aqueous environments, and surfactant-based aggregates play a key role as models of biological membranes, nanocarriers, bioremediating agents, etc., an in-depth comprehension of the quantitative aspects related to metal ion-amphiphilic system interactions is crucial to understand the mechanisms and driving forces of the binding/recognition equilibria. However, despite a significant number of efforts to understand the metals or metal complexes partitioning micellar systems, the primary focus has remained of a qualitative nature. There is a general lack of quantitative data and thorough examination/discussion of such interactions and binding features, especially in terms of the thermodynamic stability, energetics of the reaction, equilibria in aqueous solution and speciation in biological fluids.
In this review, we summarize the relevant contributions in the literature to the quantitative investigation of the interactions of biologically relevant metal-based systems with assemblies made of amphiphilic molecules by analyzing results obtained by calorimetric, spectroscopic and computational techniques, to outline the current research on this interesting but not yet extensively explored topic.

Calorimetric Investigation of the Interactions of Metal-Based Compounds with Amphiphilic Systems
Calorimetry has evolved significantly since its early beginning, and with its most recent technological evolution, has become a unique technique for monitoring chemical or biological events due to both its high sensitivity and the evidence that almost all chemical processes are associated with adsorption (or release) of heat [26]. One of the most common calorimetric techniques is titration calorimetry, in which the system's composition is changed by adding one or more components and measuring the heat exchange. These experiments are typically run under isothermal conditions, and thus the technique is usually referred to as isothermal titration calorimetry (ITC). The main advantage of ITC is providing direct access to all the information needed to elucidate the thermodynamic profile of binding or dissociation reactions in solution, thus allowing us to obtain both the equilibrium constant and enthalpy of reaction values [27].
While ITC has primarily been used to investigate the thermodynamics of interactions between biological macromolecules and small molecules [28,29] or host-guest complex formation [30][31][32], it is becoming increasingly common for measuring binding interactions between assemblies of amphiphilic molecules (micelles, vesicles, etc.) and small molecules, metal ions or inorganic nanoparticles [33][34][35].

Solution Thermodynamics of Drug-Delivery Systems
The interest in functional drug-delivery systems has promoted the development of proper amphiphilic-based aggregates to be used for effective drug administration. Although micelle-based carriers for drug delivery have been widely described, quantitative characterization of the solution thermodynamics of these systems has rarely been addressed.
In a typical calorimetric experiment, an aqueous solution of the drug is titrated into a solution of the surfactant in a micellar form. Calorimetric data are usually analyzed by the so-called "one-site binding model" [41,42], in which, after the ith injection, the total heat content (Q) of the solution in the sample cell (with an active cell volume V 0 ) at fractional saturation Θ is determined from where M t is the total concentration of the surfactant, n is the number of available binding/partitioning sites on the host molecule and ∆H is the molar heat of the drug partitioning.
For an injection volume dV i , the heat released ∆Q(i) from ith injection is given by Equation (2) A least-squares fit of the chosen model to the experimental data points allows for obtaining the binding/partitioning constant (K), enthalpy (∆H) and stoichiometry (n).
Quantitative characterization of the interactions between calixarene-based micelles and model drugs (antibiotics and anticancer agents) was also reported when analyzing data obtained by ITC measurements through a simplified model, which assumes the formation of a 1:1 species between the guest and the micellar aggregate [43,44]. The net heats of the reaction were analyzed by HypCal [45], software that allows for the simultaneous determination of the standard enthalpy and binding constant values, which has been specifically designed to treat data obtained from ITC instruments operating in the overfilled mode. The thermodynamic parameters are obtained by optimizing the agreement between observed and calculated reaction heats. Optimization is performed by non-linear least-squares analysis, minimizing the objective function (U): where Q obs is the observed heat for a given reaction step, corrected for dilution (blank) effects, while Q calc is calculated as: where δn is the change in the number of moles of a reaction product and ∆H is the molar formation enthalpy of the reaction product. The sum is carried out over all the reaction steps; the squared residuals (Q obs − Q calc ) 2 are summed over all the titration points. Stability constant values and thermodynamic parameters are obtained by simultaneously analyzing calorimetric data obtained from different titrations.
A few papers described just the thermodynamic characterization of systems involving metal complexes and amphiphilic-based aggregates using calorimetry. Cheng et al. [46] combined low-molecular-weight fucoidan (LMWF 8775 , a sulfated polysaccharide), thermolysin-hydrolyzed protamine peptide (a cell-penetrating peptide, TPP 1880 ) and Gd-DTPA (a magnetic resonance imaging, MRI, contrast agent) in a nanosystem for in vitro MR imaging of the inflammatory endothelial cell model. The authors employed nano-ITC to investigate the interaction between TPP 1880 and LMWF 8775 , identifying two reaction stages attributed to the formation of TPP 1880 /LMWF 8775 ion pairs and the complex coacervation of TPP 1880 /LMWF 8775 ion pairs, respectively. Two association constant and enthalpy values were determined, indicating that the binding affinity of the TPP 1880 /LMWF 8775 ion pair that formed at the first step was stronger than that for the second event. The n values were found to be 1.73 and 13.2 for the first and second reaction steps, respectively. The first value represents the stoichiometric TPP 1880 /LMWF 8775 ratio required to form the TPP 1880 /LMWF 8775 ion pair and indicates that TPP 1880 binds to LMWF 8775 in a non-specific mode (multiple binding sites). The stoichiometry number (n) of the complex coacervation of TPP 1880 /LMWF 8775 ion pairs (the second reaction stage) is explained by assuming that the base groups in TPP 1880 were shielded during the process of self-assembly into the complex coacervates.
Cai et al. [47] reported on a drug-delivery system in which the drug is amphiphilic itself, i.e., it is able to spontaneously self-assemble into nano-aggregates that serve as both carriers and cargos. They choose 10-hydroxycamptothecine (HCPT) and cisplatin to develop a dual anticancer drug assembly by synthetizing a HCPT-peptide (HP) amphiphile ( Figure 1a). Cisplatin was chosen since it is able to form complexes with biopolymers and peptides containing carboxylic acid groups [48]. In their pioneering work, before examining the effect both in vitro and in vivo of the synthesized supramolecular nanostructure, the authors investigated the interaction between the amphiphilic drug HP and cisplatin through ITC by titrating a cisplatin solution into the peptide solution. The calorimetric data and the binding plot were analyzed using the one-site binding model, finding that HP was able to bind and suggesting that the two anticancer drugs could be taken up simultaneously by cells (Figure 1b). to investigate the interaction between TPP1880 and LMWF8775, identifying two reaction stages attributed to the formation of TPP1880/LMWF8775 ion pairs and the complex coacervation of TPP1880/LMWF8775 ion pairs, respectively. Two association constant and enthalpy values were determined, indicating that the binding affinity of the TPP1880/LMWF8775 ion pair that formed at the first step was stronger than that for the second event. The n values were found to be 1.73 and 13.2 for the first and second reaction steps, respectively. The first value represents the stoichiometric TPP1880/LMWF8775 ratio required to form the TPP1880/LMWF8775 ion pair and indicates that TPP1880 binds to LMWF8775 in a non-specific mode (multiple binding sites). The stoichiometry number (n) of the complex coacervation of TPP1880/LMWF8775 ion pairs (the second reaction stage) is explained by assuming that the base groups in TPP1880 were shielded during the process of self-assembly into the complex coacervates.
Cai et al. [47] reported on a drug-delivery system in which the drug is amphiphilic itself, i.e., it is able to spontaneously self-assemble into nano-aggregates that serve as both carriers and cargos. They choose 10-hydroxycamptothecine (HCPT) and cisplatin to develop a dual anticancer drug assembly by synthetizing a HCPT-peptide (HP) amphiphile ( Figure 1a). Cisplatin was chosen since it is able to form complexes with biopolymers and peptides containing carboxylic acid groups [48]. In their pioneering work, before examining the effect both in vitro and in vivo of the synthesized supramolecular nanostructure, the authors investigated the interaction between the amphiphilic drug HP and cisplatin through ITC by titrating a cisplatin solution into the peptide solution. The calorimetric data and the binding plot were analyzed using the one-site binding model, finding that HP was able to bind and suggesting that the two anticancer drugs could be taken up simultaneously by cells (Figure 1b).  In general terms, although the interactions and binding processes involved in these multi-component micellar-based systems are intrinsically complex, making quantitative characterization hard to be accomplished, a thorough understanding of these phenomena may be obtained from the analysis of thermodynamic data from isothermal titration calorimetry, which provides key parameters for assessing the binding events and driving forces of the molecular recognition equilibria in solution by directly measuring the heat of the reaction.

Metal Ions' Binding with Model Membranes
The interaction of metal ions with membranes has attracted significant interest in recent decades, mostly focusing on characterizing ion binding to lipid bilayers and on the corresponding change in the membrane surface potential [49][50][51]. Indeed, micelles and vesicles have become a topic of great interest to chemists and biochemists due to their similarity to biological membranes and globular proteins.
The first systematic study dealing with the binding equilibria of alkali metal chlorides with lipid vesicles, studying palmitoyloleoylphosphatidylcholine (POPC) at physiological salt concentrations, was reported by Dimova's group in 2010 [52]. The ITC measurements were carried out by titrating a solution of POPC large unilamellar vesicles (LUVs) with alkali metal chloride solutions. The calorimetric data allowed the group to determine the binding enthalpy by using Equation (5): where δn b lip is the change in the moles of lipid bound by the cations after the injection. Assuming a simple partition equilibrium of 1:1 ion-to-lipid complex formation, the apparent binding constant K was defined as where C lip is the total concentration of lipids available for binding and the difference lip is the concentration of unbound lipids. The authors then assumed that the concentration of the unbound ions was approximately equal to the total ion concentration C ion . Combining the two equations and inserting the derivative with respect to the total electrolyte concentration C ion , they obtained Equation (7): Fitting the data through Equation (7), they obtained the apparent binding constant K and the reaction molar enthalpy ∆H. The thermodynamic parameters showed that the binding of ions to phosphocholine bilayers is an entropically driven process, but enthalpically unfavored. The gain in entropy was attributed to the release of several water molecules from the hydration shell of the ions and the dehydration of the lipid membrane. The authors also observed that the (endothermic) reaction enthalpies increase in absolute value when the size of the bare cation decreases, following the Hofmeister series.
Maity et al. [53] investigated the affinity of various monovalent alkali cations with the negatively charged dioleoylphosphatidylcholine (DOPC) and dioleoylphosphatidylglycerol (DOPG) molecules used for vesicles' preparation. ITC data were analyzed by using the partition model, and the binding heats for all alkali metal ions to vesicles were evaluated. Comparison of the results with some reference measurements revealed that, besides ion binding, other forces were contributing to the ion-membrane interaction process. The observed entropy gain was attributed to the release of water molecules from the hydration shell of the ions and the hydration layer in the surroundings of the membranes. This work emphasized the significance of determining thermodynamic parameters contributing to the free Gibbs energy since the association constant values alone cannot provide information and details on the forces driving a binding process.
Calorimetric studies on the interaction of metals of biological interest with model membranes have also been reported. Kerek et al. [54] examined the interaction of Hg 2+ and Cd 2+ with different lipid structures under physiological conditions to better understand the metal-induced effects on potential targets in biomembranes. Since Hg and Cd are toxic for the central nervous system [55,56], vesicles containing different phospholipids at different compositions were prepared to create biomimetic lipid models of the human myelin sheath. Several techniques were employed to analyze the influence of both metal ions on the membrane fluidity. The initial screening identified phosphatidylethanolamine plasmalogen (PEplasmas) as the only target for Hg-induced membrane rigidity. Phosphatidylcholine plasmalogen (PCplasm) was studied as well to compare two enol/ether lipids with different headgroup architectures (choline vs. ethanolamine). The metal-lipid interaction was investigated by ITC titrations. The interaction of Hg with both PC and PE plasmalogens was found to be enthalpically favored, with a more favorable entropic contribution in the case of PEplasm. Despite the identical vinyl ether bonds in both plasmalogens, the difference in the choline and ethanolamine headgroups was significant as Hg bound PEplasm more strongly than PCplasm, thus revealing specific metal-lipid interactions that may help to understand the toxicology of these systems.
Garidel et al. [57] studied the binding reaction of Mg 2+ , Ca 2+ and Sr 2+ with 1, 2dimyristoyl-sn-glycero-3-phosphatidic acid (DMPA) vesicles, used as model membranes, to assess ion-specific lipid-divalent cation interactions. ITC experiments were conducted in the "batch" mode, where the DMPA vesicle suspension was injected into a highly concentrated aqueous solution of the metal ion, which ensured the presence of a large excess of M 2+ throughout the experiment. The binding constants' values were large enough that almost complete saturation of DMPA vesicles with the cations was achieved after addition into the solution containing excess divalent cations. This approach was employed since a titration of the divalent cation solution into a vesicle suspension would lead immediately to aggregation and precipitation of the vesicles, making it impossible to record a complete titration curve, whereas in the "batch" mode, the total reaction enthalpy was measured. The calorimetric experiments were performed in the temperature range between 8 and 70 • C, which allowed the change in the heat capacity (∆C p ) to be determined for the binding process. This parameter describes the hydration changes caused by electrostatic interactions at the bilayer-water interface.
The above cited works investigated the direct interactions between metal ions and membrane models. However, many processes of biological significance implicate the interaction between metals and proteins that are within biological membranes. In this case, one has to consider both the interaction between the metal ion and the protein and the role played by the biological membranes in the whole process.
In this framework, Li's group explored the interactions between human copper transporters (hCtr1 and hCtr2) and silver ions; members of the Ctr class of copper transporters are crucial for mediating cellular import of silver, although the mechanisms involved in the absorption of silver into the body are poorly understood. Studies were carried out using ITC in the presence of sodium dodecyl sulphate (SDS) micelles as biomimetic membrane systems. In their first paper [58], the authors investigated a 20-residue peptide from the extracellular second methionine-rich motif (M2-motif) inclusive sequence of hCtr1 (the segment from Gly34 to Asn53) in the presence of AgNO 3 . Calorimetric experiments were carried out by titrating a peptide solubilized in an SDS micellar solution with AgNO 3 (dissolved in an SDS micellar solution). The effect of substituting alanine with methionine in the peptide was also examined. As a comparison, titrations were also conducted by using aqueous solutions of the reagents only (without SDS micelles) to unveil the role played by the biological membrane. The thermodynamic parameters, obtained by fitting the experimental data using the one-site binding model, indicated that the binding of the micelle-bound peptide is different from that of the peptide in the solution state. The binding affinity of the peptide in SDS micelles was found to be larger than that in water. Furthermore, the association process of the micelle-bound peptide with Ag + was less exothermic than that of the peptide in water, compensated by less negative entropy change, indicating that more hydrophobic interactions are involved in the complexation of a micelle-bound peptide with Ag + . The solution thermodynamics yielded by calorimetry clearly indicated that the interaction between the extracellular domain of hCtr1 and the membrane surface is fundamental in the binding structure and binding process of the peptide with the ligands.
Subsequently, they described the binding of the second transmembrane domain (TMD2) of hCtr1, hCtr1−TMD2, to Ag + in SDS micelles used as membrane-mimicking agents [59]. ITC experiments ( Figure 2) carried out in the presence of different forms of the peptide, in which methionines were converted to leucines, allowed the authors to gain insights into the roles of the methionine residues, both in the peptide assembling and in the binding of Ag + . The complexation processes were driven by enthalpy changes in the case of the peptides containing methionine residues. In contrast, the entropic contribution was found to drive the process in the absence of methionine, emphasizing the role of the peptide structure for the binding of the metal ion.

Micelles/Vesicles for Nanoparticles' Stabilization
Binding processes also occur, and need to be quantitatively analyzed, when surfactants are employed in the stabilization of metal nanoparticles. Panda's research group reported the synthesis and characterization of colloidal dispersions of silver nanoparticles in the presence of cationic surfactants that may work as a source of counterion (bromide o chloride) and as a stabilizing agent. They initially described the features of colloidal dispersions of silver bromide obtained by progressive addition of silver nitrate to an aque- The same group also studied hCtr2 by analyzing the binding of TMD2 of hCtr2 to silver in the presence of SDS micelles using calorimetry; the results indicated that the reaction is Biomolecules 2022, 12, 408 9 of 25 exothermic and hydrogen-bonding and van der Waals interactions are the driving forces for the binding of the peptide to the silver ion [60].
More recently, Yang et al. [61] evaluated the binding features of the copper transport protein hCtr1 by reporting the effect of Cu + in the interaction between hCtr1 and dipalmitoyl-phosphatidylcholine(DPPC) liposomes. Calorimetric experiments were carried out by titrating a solution of a peptide from hCtr1 (with or without Cu + salts) into a liposome solution and data were fitted through a single-site binding model. Significant enthalpy changes occur only in the presence of Cu + , and this event is even more pronounced when two equivalent cuprous ions are added. The results revealed that Cu + binding enhances the hCtr1 interaction with cell membranes.
Of note, in the reviewed papers, two approaches were employed to study the interactions of amphiphiles mimicking cellular membranes with selected peptides and metal ions. In one case, the micellar solution was used as a "medium" in which metal ions and peptides were dissolved to simulate the cellular environment [58][59][60]. In these experiments, the interactions measured were those occurring between the metal ion (titrant) and the peptide (titrate), while the effect of the micellar aggregate was determined by carrying out experiments both in the presence and absence of the amphiphilic system. In the other case, the interaction between the peptide and the surfactant-based aggregates (e.g., liposomes) was directly monitored by titrating a solution of liposomes with a peptide solution [61]. The effect of the metal ions was evaluated by conducting experiments in which the peptide solution either contained or did not contain metal ions. In our mind, both approaches are apt for investigating these rather complex systems; the choice may depend on which kind of interaction needs primary quantitative determination.

Micelles/Vesicles for Nanoparticles' Stabilization
Binding processes also occur, and need to be quantitatively analyzed, when surfactants are employed in the stabilization of metal nanoparticles. Panda's research group reported the synthesis and characterization of colloidal dispersions of silver nanoparticles in the presence of cationic surfactants that may work as a source of counterion (bromide o chloride) and as a stabilizing agent. They initially described the features of colloidal dispersions of silver bromide obtained by progressive addition of silver nitrate to an aqueous solution of hexadecyltrimethylammonium bromide (CTAB), which acted both as a stabilizer and as the source of bromide ions [62].
Later, a nanocolloidal dispersion of silver chloride was synthesized in an aqueous medium using three cationic surfactants: hexadecyltrimethylammonium chloride (CTAC), hexadecylpyridinium chloride (CPC) and benzyldimethylhexadecylammonium chloride (BDHAC) as the source for the chloride ions and as the stabilizing agents [63]. The three cationic surfactants were used to further investigate the stability dependence of the synthesized silver chloride colloidal nanoparticles on the nature of the surfactant head group. The colloidal dispersions of silver nanoparticles were stabilized by the layered structured and positively charged surfactant assemblies ( Figure 3). In these works, ITC calorimetry was employed to measure the enthalpy changes for the formation of nanoparticles, obtained by titrating a solution of silver nitrate into a cationic surfactant aqueous solution; the process was always found to be enthalpically favored.
Wang et al. [64] examined the binding of surfactants with different alkyl and head groups, i.e., sodium oleate (SO), sodium laurate (SL), sodium dodecyl sulfate (SDS) and sodium dodecylphosphonate (SDP), with iron oxide nanoparticles and evaluated the effects on the colloidal stability. Surfactants with longer alkyl tails displayed larger apparent binding enthalpies; different surfactant head groups also yielded varying affinities for iron oxide nanoparticles. The amphiphiles that displayed larger apparent binding enthalpies (endothermic or exothermic) were found to form stable suspensions of iron oxide nanoparticles, suggesting that large ∆H values are associated with a larger amount of surfactant bound to the surface of the particles. Wang et al. [64] examined the binding of surfactants with different alkyl and head groups, i.e., sodium oleate (SO), sodium laurate (SL), sodium dodecyl sulfate (SDS) and sodium dodecylphosphonate (SDP), with iron oxide nanoparticles and evaluated the effects on the colloidal stability. Surfactants with longer alkyl tails displayed larger apparent binding enthalpies; different surfactant head groups also yielded varying affinities for iron oxide nanoparticles. The amphiphiles that displayed larger apparent binding enthalpies (endothermic or exothermic) were found to form stable suspensions of iron oxide nanoparticles, suggesting that large ΔH values are associated with a larger amount of surfactant bound to the surface of the particles.
The binding curve obtained for the sodium oleate/iron oxide system was analyzed by using Freundlich, Langmuir and Langmuir-Freundlich isotherms, and the latter displaying the best fit due to the combination of the non-homogenous adsorption features of the Freundlich with the finite monolayer adsorption of the Langmuir equation.

Spectroscopic Analysis of Metal Complexes in Amphiphilic Media
Investigation of systems containing metal-ligand complexes in micellar/vesicular media may be conducted using plenty of different techniques, among which we may cite fluorescence [65] or atomic force microscopy [66]. However, since often at least the ligand possesses some absorbance in the UV-Vis range, simple spectrophotometry can play a crucial role in this examination, and spectrophotometric data can also be used to analyze the mechanism of complex formation in amphiphilic media, both from the thermodynamic and kinetic points of view. Sometimes, one of the species may emit light and a fluorescence signal may be recorded. Note that metal ions may be also directly coordinated to the surfactant head, leading to the so-called metallo-surfactants (or metal surfactant complexes), which are a recent alternative to conventional surfactants possessing unique properties, opening the door to different applications [67,68]; however, these topics are out of the focus of the present section devoted to metal-ligand equilibria in amphiphilic media. The binding curve obtained for the sodium oleate/iron oxide system was analyzed by using Freundlich, Langmuir and Langmuir-Freundlich isotherms, and the latter displaying the best fit due to the combination of the non-homogenous adsorption features of the Freundlich with the finite monolayer adsorption of the Langmuir equation.

Spectroscopic Analysis of Metal Complexes in Amphiphilic Media
Investigation of systems containing metal-ligand complexes in micellar/vesicular media may be conducted using plenty of different techniques, among which we may cite fluorescence [65] or atomic force microscopy [66]. However, since often at least the ligand possesses some absorbance in the UV-Vis range, simple spectrophotometry can play a crucial role in this examination, and spectrophotometric data can also be used to analyze the mechanism of complex formation in amphiphilic media, both from the thermodynamic and kinetic points of view. Sometimes, one of the species may emit light and a fluorescence signal may be recorded. Note that metal ions may be also directly coordinated to the surfactant head, leading to the so-called metallo-surfactants (or metal surfactant complexes), which are a recent alternative to conventional surfactants possessing unique properties, opening the door to different applications [67,68]; however, these topics are out of the focus of the present section devoted to metal-ligand equilibria in amphiphilic media.

Interaction of Metal Ions with Lipid Membrane Mimicking Systems
The presence of physiological metal ions may play an important role in modulating the efficacy of bio-active compounds. A system that was the object of intense study is daptomycin/Ca 2+ . Daptomycin is a cyclic polypeptide that has been in clinical use since 2003 to treat severe infections by Gram-positive drug-resistant bacteria. It was found that the activity of this important antibiotic depends critically on the presence of calcium ions. Recent work showed that the Ca 2+ -daptomycin complex interacts specifically with cell envelope precursors in the presence of an anionic phospholipid, forming a tripartite complex and explaining the specificity of daptomycin for bacterial cells [69]. Ca 2+ is necessary for daptomycin-binding to the lipid membrane, mimicking cell walls that perturb the latter to a significant extent, causing the formation of domains and major reorganization essential to the antibiotic efficacy [70]. In a detailed kinetic study, the fluorescence resonance energy transfer (FRET) between the intrinsic Kyn residue of daptomycin and a fluorescent tag incorporated in the lipid membrane was used to monitor peptide-binding and dissociation from the vesicle-mimicking cell walls [71]. The kinetic model used to fit the data is shown in Figure 4a. Here, D is the daptomycin monomer, D Ca is Ca 2+ -bound daptomycin, (D Ca ) 2 is a Ca 2+ -bound daptomycin dimer, D Ca L Ca is the daptomycin monomer bound to a Ca 2+ -complexed lipid and L Ca is the concentration of Ca 2+ -complexed lipid. The detailed kinetic study (using a stopped-flow technique with fluorescence detection) over different experimental conditions and reactant contents enabled the mechanistic (both equilibrium and kinetic) aspects of this complex system to be determined. On the whole, these experiments were able to describe the early events that occur in the binding of daptomycin to lipid bilayers and revealed the high affinity for the Ca 2+ -complexed lipid.
Recent work showed that the Ca 2+ -daptomycin complex interacts specifically with cell envelope precursors in the presence of an anionic phospholipid, forming a tripartite complex and explaining the specificity of daptomycin for bacterial cells [69]. Ca 2+ is necessary for daptomycin-binding to the lipid membrane, mimicking cell walls that perturb the latter to a significant extent, causing the formation of domains and major reorganization essential to the antibiotic efficacy [70]. In a detailed kinetic study, the fluorescence resonance energy transfer (FRET) between the intrinsic Kyn residue of daptomycin and a fluorescent tag incorporated in the lipid membrane was used to monitor peptide-binding and dissociation from the vesicle-mimicking cell walls [71]. The kinetic model used to fit the data is shown in Figure 4a. Here, D is the daptomycin monomer, DCa is Ca 2+ -bound daptomycin, (DCa)2 is a Ca 2+ -bound daptomycin dimer, DCaLCa is the daptomycin monomer bound to a Ca 2+ -complexed lipid and LCa is the concentration of Ca 2+ -complexed lipid. The detailed kinetic study (using a stopped-flow technique with fluorescence detection) over different experimental conditions and reactant contents enabled the mechanistic (both equilibrium and kinetic) aspects of this complex system to be determined. On the whole, these experiments were able to describe the early events that occur in the binding of daptomycin to lipid bilayers and revealed the high affinity for the Ca 2+ -complexed lipid.  Kinetic studies on the behavior of biologically relevant cations may also concern redox reactions. The reduction of Cr 6+ to Cr 3+ was examined in parallel with studies on the micellar effects on the lysis of organic systems [72,73]. Cr 6+ is a known carcinogen, still present as a pollutant coming from industrial wastes. Its reduction occurs in the cell cytoplasm. The cited work [74] used micelles to mimic the cellular membranes as a model to gain information on the process occurring in the body. Spectrophotometric kinetic experiments were carried out by monitoring the decay of oxidant [Cr 6+ ] at 415 nm at different time intervals. They discerned the details of the mechanism of the reduction reaction by 1-butanol in the presence of a cationic (N-cetyl pyridinium chloride, CPC), an anionic (sodium dodecylsulfate, SDS) or a neutral (TritonX-100) surfactant. In all cases, monomeric Cr 6+ undergoes redox decomposition through two-electron transfer at the ratedetermining step and a kinetic law of the first order on both reactants and the second order on the [H + ] ion. The accelerating or decelerating effects of the surfactants were discussed on the basis of the partitioning of the reactants over different parts of the micellar phase (Gouy-Chapman or Stern layer).

Metal Complexes Speciation in Amphiphilic Media
Amphiphilic systems may profoundly alter the equilibria of a metal complex formation, thus producing speciation profiles that differ from those commonly known and applied in water. Researchers need to carefully take into account that both the species distribution and protonation profile on the micellar surface may change, in particular, when the amphiphilic system is bearing some charge. For instance, metallochromic indicators based on azoligands such as pyridine-2-azo-p-dimethylaniline (PADA) or 4-(2-pyridylazo)resorcinol (PAR) were used to tune the affinity for the micellar aggregate: the ligand is attracted by the hydrophobic core of the aggregate and drives the accumulation of the metal ion it coordinates. However, the ligand will have apparent acid dissociation constants, which differ from that in water. Indeed, PADA is a ligand with a pK A2 (PADAH + PADA + H + ) of 4.5 ± 0.2 in aqueous solution, which is shifted to an apparent pK A2 of 6.7 ± 0.2 in the presence of negatively charged micellar SDS; neutral PAR protonates with pK A1 = 2.5 and pK A2 = 5.5 in water and apparent pK A1 = 4.0 and pK A2 = 7.0 in SDS [75]. In the presence of positively charged micelles by dodecyltrimethylammonium chloride (DTAC), due to the opposite effect on the local concentration of H + ions, the induced shift on the pK A is also reversed: for PADA pK A2 = 2.6 in DTAC [76].
Thus, the apparent pK A shift (Equation (8)) is mainly associated with the higher/lower local concentration of hydrogen ions at the micelle surface, which disfavors/favors the acid dissociation process of the dye, respectively. Owing to the difference in proton concentration, any comparison between data in a micellar media solution and water should be made at an "equivalent" hydrogen ion concentration using Equation (9) [77] pH(micelle) = pH(measured) − ∆pK A Note that Equation (9) should be applied only when the system is such that the reaction totally occurs on the micelle surface. Finding the conditions where this occurs significantly simplifies the intricate analysis of these complex systems. Conversely, if this is not the case, the overall data/numbers evaluated will be the weighted sum of the contributions of water and micellar behaviors (see, for instance, Equation (13) below). Also, the discussion above on ∆pK A amplitude/sign considers how the charged micellar surface affects [H + ]. However, different and even opposite effects on ∆pK A with respect to the electrostatic hypothesis may be found if the ligand is buried in the micellar core; this is particularly true when the ligand is highly hydrophobic [78].
The ∆pK A aspects may be analyzed more in the detail to obtain information on the surface potential experienced by the acid-base ligand and on its position on the surfactant aggregate. The relationship between proton activity on a surface (a H+ S ) of some potential ψ (mV) and that in the bulk water (a H+ W ) is aggregate. The relationship between proton activity on a surface (a H+ S) of some potential ψ (mV) and that in the bulk water (a H+ W) is where F is the Faraday constant, R the gas constant and T the temperature. Equation (10) was derived by Hartley and Roe on the basis of Boltzmann's law of charged particle distribution in the presence of an electrically charged surface; note that ψ will have the same sign as the micelle charge [79]. Equation (10) can be further developed into Equation (11) where pKA i is used instead of pKA W to account also for any intrinsic effect arising from changes in the environment. Under the hypothesis that the contribution of the local environment does not change, regardless of the surface charge, we may state pKA i = pKA 0 where pKA 0 is the pKA of the probe measured in the presence of uncharged micelles (such as, for instance, Triton X). Given this theoretical basis, spectrophotometric acid-base titrations may be repeated in water or in the presence of differently charges species. This procedure where F is the Faraday constant, R the gas constant and T the temperature. Equation (10) was derived by Hartley and Roe on the basis of Boltzmann's law of charged particle distribution in the presence of an electrically charged surface; note that ψ will have the same sign as the micelle charge [79]. Equation (10) can be further developed into Equation (11) pK S A − pK i A = − Fψ 2.3RT (11) where pK A i is used instead of pK A W to account also for any intrinsic effect arising from changes in the environment. Under the hypothesis that the contribution of the local environment does not change, regardless of the surface charge, we may state pK A i = pK A 0 where pK A 0 is the pK A of the probe measured in the presence of uncharged micelles (such as, for instance, Triton X). Given this theoretical basis, spectrophotometric acid-base titrations may be repeated in water or in the presence of differently charges species. This procedure enables the calculation of pK A S − pK A 0 and thus of ψ in some charged micellar mediums. The comparison of the ψ value found with that of the inner potential (Ψ) and the zeta potential (ζ) of the micelle enables us to gain information on the positioning of the deprotonation sites of the acid-base ligand, which may be located either in the inner core or at the outer surface of the double layer, as well as protruding out of the surface toward the bulk water.
In addition to the pK A studies, UV-Vis may also be the technique of choice in analysis of the metal-complex formation. As a first test, the simple repetition of metal-ligand titrations in water or micellar media immediately gives information on the main features of the system. Dramatically different spectral signatures unequivocally indicate reactions at the micellar phase ( Figure 5).
where F is the Faraday constant, R the gas constant and T the temperature. Equatio was derived by Hartley and Roe on the basis of Boltzmann's law of charged particl tribution in the presence of an electrically charged surface; note that ψ will have the sign as the micelle charge [79]. Equation (10) can be further developed into Equation where pKA i is used instead of pKA W to account also for any intrinsic effect arising changes in the environment. Under the hypothesis that the contribution of the local ronment does not change, regardless of the surface charge, we may state pKA i = pKA 0 w pKA 0 is the pKA of the probe measured in the presence of uncharged micelles (such a instance, Triton X). Given this theoretical basis, spectrophotometric acid-base titra may be repeated in water or in the presence of differently charges species. This proc enables the calculation of pKA S − pKA 0 and thus of ψ in some charged micellar med The comparison of the ψ value found with that of the inner potential () and the potential (ζ) of the micelle enables us to gain information on the positioning of the d tonation sites of the acid-base ligand, which may be located either in the inner core the outer surface of the double layer, as well as protruding out of the surface towar bulk water. In addition to the pKA studies, UV-Vis may also be the technique of choice in an of the metal-complex formation. As a first test, the simple repetition of metal-ligand tions in water or micellar media immediately gives information on the main featu the system. Dramatically different spectral signatures unequivocally indicate reactio the micellar phase ( Figure 5). Ultrafiltration experiments coupled with the spectrophotometric determinati the ligand content may be used to further confirm the extent of retention of this rea on the surfactant (Figure 6). Ultrafiltration experiments coupled with the spectrophotometric determination of the ligand content may be used to further confirm the extent of retention of this reactant on the surfactant (Figure 6). Figure 6 shows that at a low pH, the ligand is 100% retained and thus that data will refer to a system reacting at this phase only, whilst this is no longer true at a higher pH. On the whole, the pH-range conditions for mechanistic studies on the solution equilibria have to be carefully tuned to possibly avoid intricate deconvolution problems from different contributions.
Spectrophotometric titrations (or spectroscopic kinetic experiments) may be carried out under different conditions of the microemulsion composition, H + concentration and even pressure or salt content. The latter option is needed when we seek to unveil the speciation for metal ions that may contain, for instance, chloride-coordinated forms. Fastreaction kinetic experiments (using the stopped-flow technique) [80,81] are also precious to discern the detailed reaction mechanisms, such as those in the examples reported in Figure 7.  Figure 6 shows that at a low pH, the ligand is 100% retained and thus that data will refer to a system reacting at this phase only, whilst this is no longer true at a higher pH. On the whole, the pH-range conditions for mechanistic studies on the solution equilibria have to be carefully tuned to possibly avoid intricate deconvolution problems from different contributions.
Spectrophotometric titrations (or spectroscopic kinetic experiments) may be carried out under different conditions of the microemulsion composition, H + concentration and even pressure or salt content. The latter option is needed when we seek to unveil the speciation for metal ions that may contain, for instance, chloride-coordinated forms. Fast-reaction kinetic experiments (using the stopped-flow technique) [80,81] are also precious to discern the detailed reaction mechanisms, such as those    Figure 6 shows that at a low pH, the ligand is 100% retained and thus that data will refer to a system reacting at this phase only, whilst this is no longer true at a higher pH. On the whole, the pH-range conditions for mechanistic studies on the solution equilibria have to be carefully tuned to possibly avoid intricate deconvolution problems from different contributions.
Spectrophotometric titrations (or spectroscopic kinetic experiments) may be carried out under different conditions of the microemulsion composition, H + concentration and even pressure or salt content. The latter option is needed when we seek to unveil the speciation for metal ions that may contain, for instance, chloride-coordinated forms. Fast-reaction kinetic experiments (using the stopped-flow technique) [80,81] are also precious to discern the detailed reaction mechanisms, such as those   Figure 7a adapted with permission from reference [77]. Copyright 2008 American Chemical Society. Figure 7b adapted with permission from reference [80]. Copyright 2016 Elsevier.
The particular case of pressure dependence needs peculiar apparatuses, which enable us to make changes over a quite wide range (until 1500 atm) [82]. The value of the activation volume (∆V # ) can be obtained from the slope of a plot of the natural logarithm of the rate constant over pressure; these values are crucial to unveil the fine details of the metal/ligand interaction in the different media and decide between an associative or dissociative metal ion-binding mechanism (for Ni 2+ /PADA system, see, for instance, reference [83]). Note that the same study highlights that some of the most common buffers (e.g., TRIS, tris(hydroxymethyl)aminomethane) may not be inert toward the reaction (TRIS coordinates Ni 2+ ).
The dependence on the experimental conditions of the overall/apparent thermodynamic parameters quantitatively enables the numerical evaluation of the binding and rate constants and qualitatively indicates the system speciation in the micellar medium.
Studies on Ni 2+ /PADA binding in isooctane/polyoxyethylenglycol dodecyl ether/water microemulsions (polyoxyethylenglycol dodecyl ether = Brij30, non-ionic surfactant) demonstrated that the metal ion binding to the ligand was influenced by the microemulsion composition [84]. The analysis of the apparent binding constant dependence on the composition provided the distribution constants of the reactants and the real complexation constant at the interface. The results (the complex is less stable in microemulsion), once a change in the reaction mechanism is discarded, are discussed in the frame of a more efficient hydration of the nickel hexahydrate coordination complex due to the interaction between the polar head group of the surfactant and the interfacial water. Kinetic analysis of the same system further examined the changes in the rate constants as a balance between two opposite effects: the lower electrophilic character for water at the interface of the less polar Brij30 and the nucleophilicity increase due to hydrogen bonds breaking with respect to bulk [85].
In the case of the Ga 3+ /8 These studies shed light on the differences in the speciation profiles for both the ligand and the metal ion when switching from water to the micellar phase. This strongly affects the behavior and the main properties of the system and is crucial to be known for any practical application.
The micellar medium can also strongly alter the main reacting species in the case of the metal ion. For the Pd 2+ /PADA system, in the studied pH range, PADAH + , PADAH2 2+ , PdCl4 2− and Pd(H2O)Cl3 − are the main species in water, whereas neutral PADA and a new reaction path involving the hydroxo species PdCl3OH 2− need to be involved in the reaction mechanism in DTAC micellar solutions [76]. In the case of gold, a study of the Au 3+ /PADA system showed that the aquo-species Au(H2O)Cl3 and Au(OH)3(H2O), which are not involved in the binding process in pure water, play an important role in the H2O/SDS medium [77]. A study of the reaction mechanism of the same Au 3+ /PADA system in the These studies shed light on the differences in the speciation profiles for both the ligand and the metal ion when switching from water to the micellar phase. This strongly affects the behavior and the main properties of the system and is crucial to be known for any practical application.
The micellar medium can also strongly alter the main reacting species in the case of the metal ion. For the Pd 2+ /PADA system, in the studied pH range, PADAH + , PADAH 2 2+ , PdCl 4 2− and Pd(H 2 O)Cl 3 − are the main species in water, whereas neutral PADA and a new reaction path involving the hydroxo species PdCl 3 OH 2− need to be involved in the reaction mechanism in DTAC micellar solutions [76]. In the case of gold, a study of the Au 3+ /PADA system showed that the aquo-species Au(H 2 O)Cl 3 and Au(OH) 3 (H 2 O), which are not involved in the binding process in pure water, play an important role in the H 2 O/SDS medium [77]. A study of the reaction mechanism of the same Au 3+ /PADA system in the H 2 O/DTAC medium demonstrated that Au(H 2 O)Cl 3 and Au(H 2 O) 2 Cl 2 + play the major roles here [87]. In the case of the Ga 3+ /8-hydroxyquinoline system, the Ga 3+ ion is stabilized and the GaHL bound form appears in SDS media (GaL in water) [86].
Determination of the speciation on the water/micellar pseudo phase also enables us to finetune the reactivity of the system in the presence of metal ion mixtures. For instance, Au 3+ may be efficiently extracted, recovered and separated from Cu 2+ by using a stepby-step procedure based on either DTAC or SDS thanks to the tendency of gold to form negatively charged species [88]. This also holds for the Pd 2+ /Pt 2+ mixture [89]. In addition, a similar MEUF (micellar-enhanced ultrafiltration) approach may be used to simultaneously remove metal ions and organic contaminants from wastewater. For instance, Cd 2+ (ca. 90%) and phenol (ca. 40%) may be efficiently extracted by a carefully calibrated mixture of Triton X-100 and SDS [90]. A similar study on the Cd 2+ /phenol system in Brij35/SDS mixtures evidenced the possibility of optimizing the extraction procedure until achieving more than 90% recovery of both species [91].
These studies may be much trickier than they appear as it is not straightforward that the reactions will occur either in the micelle or in the bulk; they could occur in both phases [92]. Suitable thermodynamic and/or kinetic models will be needed to correctly analyze these complicated equilibria [93]. The pseudo-phase ion-exchange (PIE) model is a widely applied extension of the pseudo-phase model (PP) and supposes that: (i) water and micelle are two separate phases in which the reactants are distributed with very fast kinetics; (ii) the reaction in the micellar pseudo phase occurs mainly at the micelle surface; (iii) charged inert ions may compete at the charged micellar surface but the association degree of counter ions remains constant. Sometimes, it has been recommended that, when counter ions are very hydrophilic, PIE is not totally correct and mass-action models are necessary [73].
Under conditions where the reactants do distribute over the two phases, the binding constants measured experimentally by spectrophotometric data become apparent values. Any apparent binding (or kinetic) constant obtained also becomes a function of the re-tention% of the reactants. The apparent equilibrium constant, K app , will be related to the binding reaction, as usually written for a one-phase-only system (Equation (12) charged inert ions may compete at the charged micellar surface but the association degree of counter ions remains constant. Sometimes, it has been recommended that, when counter ions are very hydrophilic, PIE is not totally correct and mass-action models are necessary [73]. Under conditions where the reactants do distribute over the two phases, the binding constants measured experimentally by spectrophotometric data become apparent values. Any apparent binding (or kinetic) constant obtained also becomes a function of the reten-tion% of the reactants. The apparent equilibrium constant, Kapp, will be related to the binding reaction, as usually written for a one-phase-only system (Equation (12)) Mf + Lf  MLT (12) where Mf and Lf are the total concentrations of uncomplexed metal and ligand, while MLT is the total concentration of complex which, in this simplified example, is supposed to be at a 1:1 stoichiometric ratio. Thus, in the presence of surfactant, the expression for Kapp becomes where the subscripts W and S denote the concentrations of the reactants in the aqueous and surfactant pseudo-phases, respectively. The extent of adsorption on the micelles (R) may be evaluated by ultrafiltration experiments where, after separation of retentate and permeate, a spectrum of the permeate is compared with that of the initial solution (see Figure 5). Then, Equation (14) is used to calculate the retention coefficient Ri for each i th species of the system = (14) where Ci W and Ci S are the molar concentrations in water and in the surfactant, respectively. On the basis of these latter two equations, it can be demonstrated that Equation (15) holds; a similar procedure can be applied to the kinetic constants [94].
ML T (12) where M f and L f are the total concentrations of uncomplexed metal and ligand, while ML T is the total concentration of complex which, in this simplified example, is supposed to be at a 1:1 stoichiometric ratio. Thus, in the presence of surfactant, the expression for K app becomes where the subscripts W and S denote the concentrations of the reactants in the aqueous and surfactant pseudo-phases, respectively. The extent of adsorption on the micelles (R) may be evaluated by ultrafiltration experiments where, after separation of retentate and permeate, a spectrum of the permeate is compared with that of the initial solution (see Figure 5). Then, Equation (14) is used to calculate the retention coefficient R i for each ith species of the system where C i W and C i S are the molar concentrations in water and in the surfactant, respectively. On the basis of these latter two equations, it can be demonstrated that Equation (15) holds; a similar procedure can be applied to the kinetic constants [94].
Such a procedure was applied, for instance, in the case of the study of Ni 2+ /hydroxamic acid systems in the presence of SDS. Analysis of the binding mechanism, obtained by carefully splitting the contributions, provided a picture where the Ni 2+ + HL NiHL 2+ step occurs in both pseudo-phases, whereas the Ni 2+ + L − NiL + step in water is replaced by NiOH + + HL NiL + when in SDS [94]. A similar procedure was applied in the case of the Cu 2+ /tartaric acid systems in the presence of non-ionic Brij 58 micelles [95]; this detailed thermodynamic study indicated that the structures of the complexes incorporated in the micelles and the species-distribution diagrams are modified if compared to water and allowed to model the evolution of the extraction yields of Cu 2+ ions by the micelles as a function of pH.
What has been already discussed for [H + ] concentration changes in the micellar pseudo-phase also holds for the reactants. Consequently, a further main point is that the presence of a micellar/vesicular system offers a limited place where species may be condensed, obtaining local concentrations that are much higher than those in the bulk solvent. This produces the so-called "catalytic effect", which refers to catalysis only in the sense that the rate constant for the association path is enhanced by the concentration increase on the micelle surface [75]. The effect on the dissociation rate constant is often much smaller, and therefore, the apparent association constant will increase. This effect is maximized just above the critical aggregation concentration (or critical micellar concentration, cmc). If the surfactant concentration rises, the number of micellar reactors progressively increases. Thus, the reactants are diluted and the rate and equilibrium constant enhancements are lost and leveled off. To obtain the best enhancement effect, negatively charged reactors (SDS) are used for positively charged metal ions (for instance, Cd 2+ or Ni 2+ ) [75]; however, since the metal center may be present as a negative species [76,87], conversely, a positive micelle (e.g., DTAC) becomes the best choice. The "catalytic effect" was also found to occur for the Ni 2+ /PADA system in the presence of zwitterionic micelles formed with carboxybetaines; nevertheless, it does not occur for zwitterionic micelles by sulfobetaines because metal ions are not appreciably adsorbed by them [96]. The effect may also be lost if the species are compartmentalized in different parts of the amphiphilic system: again, if the ligand is highly hydrophobic it may be preferentially located in the core of the micelle, far from the surface. For instance, the partition coefficient for 1-pyridyl-2-azo-2-naphthol (PAN) between the micelle surface and the core was found to be K PAN = [PAN] s /[PAN] c = 0.05 where "s" represents the micelle surface site (reactive) and "c" is the micelle core (unreactive) [97]. Finally, even a so-called "anti-catalytic effect" may be observed when the micellar surface bears the same charge as the metal species (for instance, gold aquo/chloro complexes in SDS) [77].
Catalytic or anti-catalytic effects can also occur in the presence of more complicated micro-heterogeneous ternary environments. For instance, the kinetics of alkaline hydrolysis of Fe 2+ /phenanthroline metal complexes changes were studied in the presence of surfactants/guar-gum mixtures [98]. The observed pseudo-first-order rate-constant (k obs ) values were obtained from the decrease in the absorbance at 510 nm versus the time under pseudo-first-order conditions ([OH − ] >> [complex]). Longer hydrophobic chains better catalyze the hydrolysis reaction (the surfactant was CH 3 -[CH2] n -CH 2 -N(CH 3 ) 3 + , cetyltrimethylammonium/CTAB/n = 14 > tetradecyltrimethylammonium/TTAB/n = 12 > dodecyltrimethylammonium/DTAB/n = 10). However, the binding to guar gum by the surfactant also follows the order CTAB > TTAB > DTAB. On the whole, it was found that the rate constant is k 1 (DTAB-guar gum) >> k 1 (TTAB-guar gum) >> k 1 (CTAB-guar gum) Biomolecules 2022, 12, 408 18 of 25 >> k 1 (aqueous medium) >> k 1 (in aqueous gum medium). These data were discussed on the basis of gum-amphiphile interactions as a function of the hydrophobic chain length of amphiphiles ( Figure 9).
Catalytic or anti-catalytic effects can also occur in the presence of more complicated micro-heterogeneous ternary environments. For instance, the kinetics of alkaline hydrolysis of Fe 2+ /phenanthroline metal complexes changes were studied in the presence of surfactants/guar-gum mixtures [98]. The observed pseudo-first-order rate-constant (kobs) values were obtained from the decrease in the absorbance at 510 nm versus the time under pseudo-first-order conditions ([OH -] >> [complex]). Longer hydrophobic chains better catalyze the hydrolysis reaction (the surfactant was CH3-[CH2]n-CH2-N(CH3 ) 3 + , cetyltrimethylammonium/CTAB/n = 14 > tetradecyltrimethylammonium/TTAB/n = 12 > dodecyltrimethylammonium/DTAB/n = 10). However, the binding to guar gum by the surfactant also follows the order CTAB > TTAB > DTAB. On the whole, it was found that the rate constant is k1(DTAB-guar gum) >> k1(TTAB-guar gum) >> k1(CTAB-guar gum) >> k1(aqueous medium) >> k1(in aqueous gum medium). These data were discussed on the basis of gum-amphiphile interactions as a function of the hydrophobic chain length of amphiphiles ( Figure 9).

Computational Approaches to Examine Metal-Based Compound/Amphiphile Interactions
The interaction of metal ions and metal compounds with amphiphilic systems has also been tackled by computational chemistry approaches, with the aim to shed light on the atomic-level binding sites and mechanisms involved and to provide support to improve properties such as solubility and stability in aqueous solutions. However, the literature search performed highlights that, despite the usefulness of such approaches, they have not yet been largely used by specialists in this research field and very few references can be found.
Typically, for such computational investigations, molecular dynamics (MD) simulations, using empirical force fields, are used to model the formation of large amphiphilic aggregates, involving thousands of atoms and the presence of an explicit water solvent, and to check how the presence of small molecules, metal ions or metal compounds, in our case, perturbs the overall system, also highlighting the preferential interactions between aggregates and metal compounds. Quite often, MD simulations have been used as a support for the structural and dynamic interpretation of experimental data, such as from NMR spectroscopy or neutron-scattering structural investigations.
Furthermore, quantum chemistry approaches, essentially using density functional theory (DFT) calculations, have been used to investigate the structure of isolated amphiphilic molecules or their small aggregates, to obtain information on their polarity properties and on their binding sites and binding strength with metal compounds.

DFT Calculations
Singh recently reported a short review focused on the applications of DFT for drug design based on the investigation of drug-biomembrane interaction, aimed at proposing structure changes to improve biomembrane crossing and to enhance the drug efficacy profile [99]. Arslan et al. [100] obtained, by DFT calculations, the octanol-water partition coefficients of drug-like molecules, with the aim to compare the Gibbs solvation free energies of potential drug candidates with experimentally available LogP values. In detail, the n-octanol/water partition coefficient, LogP or log K ow , describes the drug hydrophobicity and membrane permeability, a theoretically determined basic property for drug design, using the combinations of three functionals, two basis sets, an implicit salvation method with one explicit water molecule and performing a conformational search by using the semi-empirical PM3 method. The results obtained provide suggestions on the methodology that better reproduces the experimental data. DFT calculations have also been exploited by Popovic-Nikolic et al. [101], to theoretically investigate the ionization of three angiotensin II receptor blockers, also known as sartans, two ampholytes and one diacid molecules, all in aqueous solution and in the presence of surfactant micelle models. pK a values have been calculated using implicit salvation methods and compared with the corresponding experimental data. Moreover, structural information on the binding to the micelle surfaces was provided for the equilibrium forms of the selected sartan molecules. In this study, however, the micelles were not explicitly considered. The interaction of sartans with micelles was estimated by the Connolly accessible area descriptor, defined as the surface area made by the solvent sphere, which influences the hydrophobic interactions of a molecule. Finally, Oliver et al. [102] used dispersion-corrected DFT calculations to study the noncovalent binding of three emerging contaminants with two dipalmitoylphosphatidylcholine molecules, used as a model of a phospholipid membrane, to determine the binding energies and the structures of the contaminant-phospholipid complexes, with the aim to estimate the degree of penetration of the guest across the lipid bilayer and to predict the potential bioavailability and toxicity ( Figure 10). Solvent effects were mimicked by using both discrete water molecules in the polar head of the surfactant molecule and an implicit solvation model.

MD Simulations
John et al. [103] investigated cation-induced lipid membrane movement. In detail, a lipid membrane model system, composed of a differently charged free-floating bilayer, in the presence and absence of Ca 2+ and Na + , adjacent to a self-assembled monolayer (SAM), was investigated by neutron reflectivity and quartz crystal microbalance measurements. The experimental results were interpreted with the support of MD simulations to determine the cation and water interfacial distribution in the membrane model.
With the aim to explore the ability of lipids and proteins to shape realistic membrane models, Pezeshkian and Marrink [104] used both coarse-grained MD and all-atom MD simulations, which were combined to examine the dynamic properties of realistic membrane shapes, thus obtaining information on protein-lipid interactions related to cellular morphologies of entire organelles. Kinnun et al. [105] modelled small-angle neutron scattering (SANS) data by MD simulations to study the supramolecular structures of cell membranes. Interestingly, their investigation allowed the researchers to determine the location of cholesterol in bilayer membrane models. Tsukanov et al. [106] studied the interaction of bimetallic Ag-Cu nanoparticles (Cu70Ag30, Ag70Cu30 and Cu70Ag30O4) with phospholipid and lipopolysaccharide membranes by MD simulations. In this study, DFT calculations were also used to estimate the composition-dependent surface-charge density of the Ag-Cu nanoparticle. Finally, Pujol-Giménez et al. [107] used a combination of molecular dynamic simulations and pKa calculations to discover relevant residues in the human divalent metal transporter 1 (hDMT1) protein, a divalent metal ion transporter that contributes to H + /Fe 2+ coupled transport.

MD Simulations
John et al. [103] investigated cation-induced lipid membrane movement. In detail, a lipid membrane model system, composed of a differently charged free-floating bilayer, in the presence and absence of Ca 2+ and Na + , adjacent to a self-assembled monolayer (SAM), was investigated by neutron reflectivity and quartz crystal microbalance measurements. The experimental results were interpreted with the support of MD simulations to determine the cation and water interfacial distribution in the membrane model.
With the aim to explore the ability of lipids and proteins to shape realistic membrane models, Pezeshkian and Marrink [104] used both coarse-grained MD and all-atom MD simulations, which were combined to examine the dynamic properties of realistic membrane shapes, thus obtaining information on protein-lipid interactions related to cellular morphologies of entire organelles. Kinnun et al. [105] modelled small-angle neutron scattering (SANS) data by MD simulations to study the supramolecular structures of cell membranes. Interestingly, their investigation allowed the researchers to determine the location of cholesterol in bilayer membrane models. Tsukanov et al. [106] studied the interaction of bimetallic Ag-Cu nanoparticles (Cu 70 Ag 30 , Ag 70 Cu 30 and Cu 70 Ag 30 O 4 ) with phospholipid and lipopolysaccharide membranes by MD simulations. In this study, DFT calculations were also used to estimate the composition-dependent surface-charge density of the Ag-Cu nanoparticle. Finally, Pujol-Giménez et al. [107] used a combination of molecular dynamic simulations and pKa calculations to discover relevant residues in the human divalent metal transporter 1 (hDMT1) protein, a divalent metal ion transporter that contributes to H + /Fe 2+ coupled transport.

Conclusions
In this review, we summarized the literature contributions to the quantitative investigation of the binding processes involving metal-based compounds and surfactants. Different techniques and methods have been discussed including isothermal titration calorimetry, spectrophotometry, DFT calculations and MD simulations.
We found only a few papers reporting the binding process of metal complexes and amphiphilic-based aggregates through isothermal titration calorimetry experiments, despite the fact crucial information on the thermodynamic profiles of the species in solution may be easily obtained by this technique. Calorimetry was also used to explore the direct interaction of metal ions with models of biomembranes made of amphiphilic molecules, thus obtaining quantitative data on the driving forces involved in the binding events. We also analyzed the spectrophotometric investigations on the metal-ligand equilibria in amphiphilic media to determine both thermodynamic and kinetic parameters in solution.
Since the application of surfactants in metal complexation and extraction is a widely reported topic, the accurate differences observed in the speciation profiles of both the ligands and the metal ions from simple water to the micellar phase should be taken into account. Finally, we reviewed some works on computational chemistry approaches to study the interaction of metal ions with amphiphilic systems that investigate, at the atomic level, the binding sites and the mechanisms involved in the recognition processes.
More detailed studies will be beneficial to completely characterize the complex multicomponent systems examined in this work since only a thorough quantitative analysis of data from different techniques/approaches may provide key insights into the binding processes, affinities, species, driving forces and interactions concerning metal-based compounds and amphiphilic molecules. The ultimate goal is to better understand the mechanisms of phenomena involving biologically relevant metal ions and to promote the design and development of metal/surfactants-based systems that may efficiently operate in drug delivery as well as in water remediation.