Ionic shielding of electrostatic interactions of fibrinogen during aggregate formation: a molecular dynamics study

Abstract

The drying-based process of enzyme-free self-assembly of fibrinogen to fabricate fibrillar biomaterials is highly dependent on the type and relative amount of used salts, and is thus often rationalised on the basis of electrostatic double-layer interactions. However, using mass-loss and turbidity data, we show here that the critical salt concentration thresholds above which rapid fiber growth occurs is well above the limit of DLVO theory, pointing towards complete electrostatic shielding beyond a tight ionic Stern layer. We further show, by means of mass-loss SEM-EDS analysis, selective retention of sodium cations, particularly when in association with phosphate anions, whereas chloride ions are largely removed after washing the formed fibres. Using the fibrinogen D domain (Fg-D) as a representative protein model, we perform all-atom molecular dynamics simulations to understand the composition and structure of the Stern layer in chloride and phosphate salt environments promoting fibre formation. We find that both Na⁺ ions and monohydrogen and dihydrogen phosphate anions display strong and persistent binding to charged basic and acidic residues, respectively, forming a tightly bound Stern layer. In stark contrast, Cl⁻ ions exhibit only transient interactions with the protein, maintaining a highly diffusive behaviour. The immobilization of ions in the Stern layer is due to the ion-chelation ability of neighbouring amino acids, concomitant with the formation of extended and poorly diffusing hydration structures. As a result, the electrostatic potential at the layer's edge is strongly modulated or even inverted depending on the local protein/salt/water features. Under these conditions, the protein–protein binding ability is not dependent on long-range electrostatic double-layer interactions, but on the precise matching of mutually facing and interpenetrating Stern-layer regions. These findings provide a physicochemical basis for designing enzyme-free fibrinogen materials with controllable fibrillar architectures and offer new opportunities for biomaterial development using tailored ionic environments.

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Design, System, Application

A multiscale investigation: how does the chosen buffer system influences the view of a protein into the surrounding solution? Buffer solutions are used daily in lab processes for proteins. In this work, we combine evaporation-driven self-assembly experiments with atomistic molecular dynamics simulations to elucidate how chloride- and phosphate-based salts modulate fibrinogen aggregation and nanofiber formation. Our results reveal ion-specific effects on the electrical double layer, solvent structuring, and electrostatic screening at the fibrinogen D domain, providing a mechanistic understanding of how tailored ionic environments can be used to design enzyme-free fibrinogen-based biomaterials with controllable fibrillar architectures.

1 Introduction

Fibrinogen is a 340 kDa plasma glycoprotein that plays a central role in hemostasis, wound repair, and tissue regeneration. Upon thrombin-mediated removal of fibrinopeptides, fibrinogen undergoes a hierarchical assembly process to form fibrin, a crosslinked fibrous network that stabilizes the blood clot and provides provisional extracellular matrix cues.¹ Structurally, fibrinogen is a symmetric homodimer composed of Aα, Bβ, and γ chains held together by 29 disulfide bonds.² Its polymerization is governed by a series of well-defined interactions, including knob–hole binding, longitudinal γD–γD alignment, and lateral contacts among β- and γ-chain domains.^3,4 Mutations disrupting these interaction sites such as substitutions at γ375 frequently impair fibrin network formation, highlighting the importance of the D domain for intermolecular assembly.⁴

Besides protein–protein interactions, ionic species critically influence fibrinogen and fibrin assembly.⁵ Calcium ions are known to accelerate fibrin formation by stabilizing FpB–β-chain interactions, and fibrinogen contains multiple Ca²⁺-binding sites, including three high-affinity positions within each D domain.³ These sites lie near key interaction regions in the D domain and implicate ionic screening as an important factor in regulating fibrinogen association.^1,3,6 Fig. 1 shows a representative structure of the fibrinogen molecule with all biologically relevant zones. Understanding such ion-mediated effects is increasingly relevant given the emerging use of fibrinogen as a biomaterial for fabricating biocompatible, patient-specific scaffolds.^7–10 Thrombin-free routes to fibril formation are particularly attractive because residual thrombin activity poses clinical risks, including thrombosis.^10–12 Consequently, several salt-based, substrate-induced, and non-denaturing approaches have been proposed to generate fibrinogen fibers without enzymatic activation.¹³

Fig. 1 Representative molecular structure of fibrinogen. The homodimeric structure is shown in NewCartoon representation. On the right side, the three monomer chains are shown in shades of red. The inter- and intra-disulfide bonds are highlighted in vdW representation with a vdW radius of 1 Å.² The monomer chain on the left side is shown in shades of gray. The positions of the γ and β holes^3,4 are highlighted as rose vdW surfaces and the four Ca²⁺ binding sites (s1–s4)³ in blue. The fibrinopeptides FpA and FpB are illustrated as dark green, and the extensions of the knobs (polymerization zones) EA and EB are highlighted in magenta.²

A broad range of experimental studies demonstrates that fibrinogen assembly in non-denaturing salts depends strongly on pH, ionic strength, temperature, protein concentration, and the identity of the ions.^{10–12,14–21} Recent experimental work further demonstrated that fibrinogen self-assembly follows a strongly anion-specific pathway: phosphate-containing salts promote ordered nanofibrous assemblies with sigmoidal nucleation–elongation kinetics, whereas chloride-containing salts predominantly induce disordered macroporous aggregation.²² Earlier studies have already shown that fibers resembling thrombin-induced fibrils can form in phosphate-containing systems.¹⁷ Gollwitzer et al. found that fibers resembling thrombin-induced fibrils can be formed at pH 6.9 with phosphate buffer at an ionic strength of 0.08.¹⁸ Stewart et al. also found the formation of fibers with the anion protamine sulfate at pH 7.4.^19,20 The isoelectric point of fibrinogen is 5.8,^23,24 which means that at these fiber-forming pH values, fibrinogen has a negative charge. The authors assumed that aggregation into fibers is due to lateral and end-to-end interactions after screening of the charges by ions. Another study showed the aggregation of bovine fibrinogen dissolved in either PBS or Tris buffer by metal ions. Three categories of salts were defined based on their ability to aggregate: 1) CaCl₂, MgCl₂ and Al₂(SO₄)₃ do not form fibers; 2) FeSO₄, CuSO₄, COCl₂ and NiCl₂ can form aggregates at concentrations ranging from 0 to 10 mM; 3) HgCl₂, ZnCl₂, LaCl₃ and Cr₂(SO₄)₃ can form aggregates at concentrations ranging from 0 to 1000 μM.²¹ Aggregation was suggested to take place because of intermolecular ionic bonding between fibrinogen molecules after binding of metal cations to the carboxyl groups of fibrinogen.²¹ However, the morphology of the aggregates was not investigated in this early work. Hämisch et al. observed the formation of globular aggregates, which were assumed to form branched networks triggered by a drop in ionic strength of NaCl in PBS buffer.¹⁰ At higher ionic strengths, they observed that the assembly process slowed down, which could be due to the strong electrostatic screening or the blocking of specific interactions by screened ionic species.¹⁰ Later, Saha et al. proved that these aggregates can form branched networks upon drying.¹¹ Hense et al.¹² observed that oxygen-containing anions with strong kosmotropicity can generate fibers in solution at slightly acidic pH at a low temperature of around 5 °C. The dominant role of the complex kosmotropic anions is in line with the observation of Stapelfeldt et al.,¹⁴ who found fiber formation to be induced by drying fibrinogen in concentrated salt solutions containing Na-PO₄ or PBS. Conversely, they observed the formation of fibers at pH > 7 only at room temperature.¹⁴ Overall, the fiber diameters and pore sizes of fibrinogen nanofibers assembled with PBS are well suited to support the interaction with different cell types including blood platelets, fibroblasts, keratinocytes and endothelial cells.^25–27 In order to control specific cell interactions, it is therefore important to understand how different ions influence the self-assembly of fibrinogen into fibers and the resulting scaffold architecture.

The influence of cationic species on the fibrillogenesis of fibrinogen has also been discussed in the literature. On the one hand, Stamboroski et al.¹⁷ have shown that the presence of monovalent ions in PBS buffer leads to porous networks of fibrinogen fibers, while with divalent cations such as calcium and magnesium, no formation of aggregates or fibers could be observed.²⁸ Atomistic simulations of Fg-D revealed the formation of direct contacts of the monovalent cations Na⁺ and K⁺ by a disruption of their hydration shells, whereas divalent cations like Mg²⁺ ions interact indirectly only via their complete hydration shell.¹⁷ On the other hand, Hense et al. found that CaCl₂ can induce fiber-formation.¹⁵ However, this could be due to the presence of cross-linking factor XIII in their samples, which is activated by Ca²⁺. The same group found that fibers can be formed by combining Mg²⁺ with the kosmotropic sulfate ion, but not with Cl⁻ ions or nitrates,¹⁶ which points towards the role of the anion kosmotropicity in the assembly process.

In summary, certain monovalent and divalent cations induce aggregation, possibly through interactions with carboxylate residues,²¹ while kosmotropic anions tend to promote dehydration-driven condensation in the presence of monovalent cations.^15,17 In contrast, chaotropic ions often inhibit or destabilize the assembly process.²⁸ These experimental observations reveal a complex landscape of ion-specific effects on the fibrinogen structure and intermolecular association, yet the underlying molecular mechanisms remain poorly understood.

Classical DLVO theory^29–31 and phenomenological Hofmeister trends^32,33 alone provide only partial explanations for ion-specific protein aggregation because they neglect heterogeneous surface charge distributions, dielectric heterogeneity, ion adsorption, and many-body ion correlations. In particular, ion–water and ion–ion correlations at the molecular level^34,35 become pronounced at high salt concentrations, when the Debye length approaches the typical molecular dimension.^24,31 Hofmeister trends vary with pH, ionic strength, and the distribution of charged patches on the protein surface, and deviations from “classical” ordering are frequently observed.^34,36 Therefore, theories that fully capture the local solvent structure, the specific ion–protein binding, and the dynamics of hydration-shells of adsorbed ions are needed. Here, contemporary theoretical frameworks for polyelectrolytes and biomolecular electrostatics already extend substantially beyond classical mean-field screening by incorporating finite ion size, counterion condensation, dipolar interactions, adsorption thermodynamics, and correlation-driven effects.^37–40

In this work, we employ atomistic simulations of the fibrinogen D-domain (Fg-D), guided by experimental observations at salt concentrations above 0.1 M. At this salt concentration and higher, the Debye screening length approaches the molecular dimension (below 1 nm), leading to substantial screening of long-range electrostatic interactions. Under these conditions, protein–protein interactions are increasingly governed by the composition and structure of the interfacial ion layer, particularly the Stern layer formed near the protein surface. Variations in Stern-layer composition and ion-specific interfacial organization can consequently modulate aggregation pathways, favoring either ordered assembly or disordered aggregation, as experimentally observed for phosphate- and chloride-containing fibrinogen systems.²² Here, we investigate how chloride- and phosphate-based salt environments influence near-surface ion structuring and the resulting protein–protein interaction behavior. By integrating turbidity kinetics, elemental composition analysis, and high-resolution imaging with atomistic insights at the molecular level into ion residence times, mobility, and the electrical double-layer structure, we identify ion-specific mechanisms that govern fibrinogen aggregation and nanofiber formation in non-denaturing environments. Our findings provide a physicochemical foundation for tuning fibrinogen-based biomaterials via controlled ionic conditions, with implications for the design of enzyme-free assembly protocols and advanced fibrillar architectures.

2 Methods

2.1 Experiments

2.1.1 Fibrinogen solution and nanofiber process. Fibrinogen was the research material derived from human blood plasma provided by Biotest AG (Dreieich, Germany) that was dissolved in sterile water. Fibrinogen stock solutions were dialyzed overnight against 10 mM Tris buffer (Carl Roth, Karlsruhe, Germany) using 14 kDa cutoff cellulose membrane dialysis tubing (Sigma) to remove low molecular weight compounds. Prior to fiber assembly, round glass slides (VWR, Darmstadt, Germany) with a diameter of 15 mm were cleaned in air plasma for two minutes using a plasma cleaner (Diener Electronic GmbH, Ebhausen, Germany). To study the combined effect of phosphate and chloride ions on fibrinogen fiber assembly, 100 mM Na-PO₄ and 100 mM NaCl were dissolved in deionized water from a TKA water purification system (Thermo Fisher Scientific, Schwerte, Germany). Subsequently, 5 mg mL⁻¹ fibrinogen was added to the solution, which was then dried overnight in a humidity chamber (Memmert, Schwabach, Germany) at 25 °C and 30% relative humidity. After drying, the samples were crosslinked for 2 h by placing them in a Petri dish containing one microliter of 37% formaldehyde solution (FA, AppliChem GmbH, Germany) per cm³, covered with Parafilm. Afterwards, the samples were aired in a fume hood for 30 min and rinsed with 1 ml deionized water for 15 min (exchanging the water every 5 min).

2.1.2 Turbidity and mass loss analysis. To simultaneously assess solvent evaporation and fibrinogen retention during drying, mass-loss experiments were conducted in parallel, following a previously established protocol.¹⁷ Fibrinogen–salt mixtures prepared as described above were pipetted into well plates and placed in a UV/vis spectrophotometer for hourly absorbance readings at 330 nm, without disturbing the samples. After each reading, the plates were briefly covered to prevent moisture uptake and weighed using an analytical balance (Precisa Instruments, Diekiton, Switzerland) to determine the remaining mass. The initial mass was recorded before drying began. The progressive mass loss over time was attributed to water evaporation, enabling estimation of the residual salt content in the dried fibrinogen matrix. The process continued until the mass stabilized, indicating complete drying. All experiments were conducted at room temperature under ambient humidity.

2.1.3 SEM analysis. The nanoscale morphology of dried, cross-linked and washed fibrinogen fibers prepared from mixed-salt solutions with starting concentrations of 100 mM Na-PO₄ and 100 mM NaCl was characterized with scanning electron microscopy (SEM). Unless specified otherwise, all samples were sputter-coated with a 7 nm layer of gold using an EM ACE600 high-vacuum sputter coater (Leica Microsystems, Wetzlar, Germany) prior to imaging. SEM analysis was carried out with a Quattro S device (Thermo Fisher Scientific, Waltham, USA) operated at an acceleration voltage of 5 kV, using a secondary electron detector.

2.1.4 EDS analysis. Energy-dispersive X-ray spectroscopy (EDS) was used to analyze the elemental composition of dried fibrinogen fibers prepared from mixed-salt solutions. The composition of crosslinked and washed fibrinogen fibers was compared to the composition of fibrinogen fibers analyzed directly after drying. All samples were analyzed without sputter coating and were imaged in low-vacuum mode (75 Pa) as described in the procedure by Riedel et al.⁴¹ EDS measurements were performed with the EDS detector of the Quattro S system. The accelerating voltage was set to 10 kV, and the working distance was maintained at 10 mm. Spectra were collected from selected sample regions through mapping, and elemental quantification was performed.

2.2 Simulations

The size of the entire symmetrical fibrinogen dimer is approximately 2964 amino acids.⁴² The fibrinopeptides of the Aα and Bβ strands prevent fiber formation in vivo.²³ The outer D domains (Fg-D) contain calcium-binding sites known from the literature.³ As the longitudinal γD–γD alignment and lateral contacts among β- and γ-chain domains^3,4 are known to govern the polymerization, in this work we limit our simulations to Fg-D as a representative system for the protein–ion interactions relevant to enzyme-free fiber formation.^14,17

2.2.1 Protein model. The crystal structure of the human Fg-D domain was obtained from the Protein Data Bank (PDB) with the PDB code 1LT9.⁴³ Fg-D is a monomer with three hetero chains α (residues 126–190), β (residues 161–458), and γ (residues 96–394) and contains eight disulfide bonds (five intrachain, three interchain), as shown in Fig. 1. All simulations were performed at pH 7, with the protonation states of the titratable residues determined using the H++ 4.0 webserver.^44–46 Fg-D contains a total of 175 amino acids with pH-dependent protonation states for (glutamic acid (GLU), aspartic acid (ASP), histidine (HIS), arginine (Arg), and lysine (LYS)). At pH 7, as predicted by the H++ webserver, all Lys and Arg residues are protonated (positively charged) with pK_a values above 7 (Lys: 10.67,⁴⁷ Arg: 12.10 (ref. 47)). HIS residues, having a pK_a value of 6.04,⁴⁷ closer to pH 7, have most of the residues deprotonated and thus neutral at pH 7 except for HIS-γ340, which remains protonated. The amino acids ASP and GLU are consequently deprotonated (negatively charged), consistent with their pK_a values below 7 (Asp: 3.71,⁴⁷ Glu: 4.15 (ref. 47)). The careful description of the protonation states is essential for the investigations presented in this study. In particular, a significant amount of the negatively charged amino acids in the D domain is known to form the characteristic calcium binding sites ion_s1 to ion_s4, as listed in Table 1.^48–50 The overall net charge of the D domain is slightly negative (−1e; with e corresponding to the elementary electron charge). To examine whether ion binding affects ionization, as observed in poly-acid systems,^51–53 protonation states of titratable amino acids were computed using the H++ server, which employs a continuum electrostatic framework at different salt concentrations (0.001 M, 0.1 M, and 0.375 M). However, this approach yielded no change in the protonation states of titratable amino acids with increased salt concentration. In contrast, when the input structures included the captured Stern layer consisting of ions and water molecules, analysis at pH 7 resulted in different effective total charges: −3e at 0.001 M, +1e at 0.1 M, and +4e at 0.375 M NaCl concentrations, accompanied by changes in the protonation states of the protein.

Table 1 Calcium binding sites on the Fg-D domain^48–50

Ion_s1	Ion_s2	Ion_s3a	Ion_s3b	Ion_s4
a Backbone involved amino acids.
Asp(γ318)	Asp(β381)	Glu(γ132)	Glu(γ132)	Asp(γ294)
Asp(γ320)	Asp(β383)	Asp(β261)	Asp(β261)	Gly(γ296)^a
Phe(γ322)^a	Trp(β385)^a	Glu(β397)	Gly(β263)^a	Asp(γ298)^a
Gly(γ324)		Asp(β398)		Asp(γ301)

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2.2.2 Simulation details. Molecular dynamics (MD) simulations were performed using the GROMACS software.⁵⁴

The non-polarised AMBER14 force field was used to describe the bonding and non-bonding parameters, and the TIP3P model was used for the water solvent. All GROMACS input files were created with the CHARMM-GUI solution builder,^55–57 and the parameters for phosphate anions were taken from Kashefolgheta and Vila Verde.⁵⁸ At pH 7, the phosphate ion exists in two charge states, −1e (dihydrogen phosphate, DHP) and −2e (monohydrogen phosphate, MHP). In mixed-salt systems, the fraction of MHP and DHP was calculated using the Henderson–Hasselbach equation. All simulation cells listed in Table 2 contain the Fg-D monomer. The solvent fractions of the systems are generated with random starting positions for the ions (phosphate ions are placed using the ‘gmx insert’ GROMACS command) and an explicit TIP3P water solvent. This reveals reasonable statistics for ionic adsorption sites over the entire simulations.

Table 2 Overview of all simulated Fg-D monomer systems

Type	System	Salt	Conc.	Ions	Water
a Experimental starting salt concentration.b Onset of turbidity.c Constant turbidity profile, x: aqueous ionic systems are used to calculate diffusion coefficient references.
ECC	A.1	NaCl	0.1 M	Na^+: 190, Cl^−: 189	99 234
	A.375	NaCl	0.375 M	Na^+: 711, Cl^−: 710	98 192
	A.75	NaCl	0.75 M	Na^+: 1319, Cl^−: 1318	86 019
	A1.125	NaCl	1.125 M	Na^+: 1978, Cl^−: 1977	82 156
	A2.0	NaCl	2 M	Na^+: 3515, Cl^−: 3514	73 554
	B.1	NaMHP	0.1 M	Na^+: 339, MHP: 169	92 788
	B.25	NaMHP	0.25 M	Na^+: 845, MHP: 422	91 375
	B1.125	NaMHP	1.125 M	Na^+: 3795, MHP: 1897	83 093
	C.1	NaDHP	0.1 M	Na^+: 190, DHP: 189	92 789
	C.25	NaDHP	0.25 M	Na^+: 422, DHP: 421	91 681
	C1.125	NaDHP	1.125 M	Na^+: 1898, DHP: 1897	84 681
	D.1^a	Mixed	0.1 M	Na^+: 403, Cl^−: 169, MHP: 64, DHP: 105	92 482
	D.375^b	Mixed	0.375 M	Na^+: 1493, Cl^−: 632, MHP: 236, DHP: 388	89 238
	D1.125^c	Mixed	1.125 M	Na^+: 4511, Cl^−: 1897, MHP: 725, DHP: 1163	80 278
Diffusion	Ax	NaCl	0.1 M	Na^+: 189, Cl^−: 189	102 291
	Bx	NaMHP	0.1 M	Na^+: 338, Cl^−: 169	106 565
	Cx	NaDHP	0.1 M	Na^+: 189, Cl^−: 189	106 588
	Dx	Mixed	0.1 M	Na^+: 402, Cl^−: 169, MHP: 64, DHP: 105	107 029

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First, all input systems were energy-minimized using the steepest descent algorithm until the forces converged to <1000 kJ mol⁻¹ nm⁻¹ to remove steric clashes. This was followed by NVT equilibrations at a constant temperature of 300 K, maintained using the V-rescale thermostat for 10 ns with position restraints on the backbone and sidechain atoms of the protein. Subsequently, NPT equilibrations were conducted at a constant temperature of 300 K and a pressure of 1 bar for 10 ns using the Berendsen barostat. The final production simulations were performed in the NVT ensemble for 100 ns, except for the aqueous ionic systems without any protein, for which diffusion coefficient references were used. These systems were also simulated for 10 ns. All analyses and graphical representations of the MD trajectories and snapshots were performed using the Visual Molecular Dynamics (VMD)⁵⁹ and MDAnalysis^60,61 tools. The radial distribution functions (RDFs) of ions and water-oxygen around the Fg-D protein were calculated using VMD. The electrostatic potential was calculated using the APBS webserver.⁶²

2.2.3 Electronic screening of ionic interactions. In our fixed-charge MD simulations, electronic polarization is absent⁶³ and therefore the screening of electrostatic interactions due to the deformation of the electron clouds is neglected.^29,64,65 This absence of electronic screening leads to overestimated direct interactions between full integer charges by a factor of the order of the typical high-frequency dielectric constant of organic materials, ∈_el = 1.78.⁶⁵ As a result, non-polarizable force fields exaggerate the strength of salt bridges in proteins and ion–ion binding in solution.^66–68 An easy fix is provided by the so-called electronic continuum correction (ECC) in molecular dynamics,⁶⁴ in which the net charges of charged amino acids and salt ions are scaled by , leading to reduced (correctly screened) Coulomb interactions.^63,65 In the present work, we have employed the ECC approach in all simulations, given the important role played by quantitatively correct ion–ion and ion–protein interactions. The used Lennard–Jones parameters and charges of the ions used in the different simulated systems (A, B, C, and D, see Table 2) are summarized in Table 3. The scaled ECC charges of negative and positive amino acids are taken from Duboue et al.⁶⁴ and listed in the SI, part I Tables S1 and S2. With these sets of parameters, the first peak distances in the radial distribution functions between the ions and the O atoms of water are in good agreement with literature values, as reported in the SI, part I Table S3. The results of control simulations performed without down-scaling the charges are presented in the SI, part II.

Table 3 Lennard–Jones parameter set for ECC-scaled simulations

System	Ion	Charge	σ (Å)	∈ (kJ mol^−1)
A	Na^+ (ref. 64 and 69)	+0.75	2.115	0.5443
A	Cl^− (ref. 64 and 69)	−0.75	4.1	0.492
B, C, and D	Na^+ (ref. 70)	+0.75	2.439	0.365
D	Cl^− (ref. 70)	−0.75	4.477	0.148

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2.2.4 Protein–protein interaction analysis. The molecular nature of the specific protein–protein interactions remains currently unknown in our experiments during random aggregation or fiber assembly. Quantitative analysis of the influence of the salt type and salt concentration on the protein–protein interactions will help to understand the underlying process. In this work, we have chosen selected Fg-D dimer crystal structures (PDB code: 1FZE),⁵⁰ which correspond to a state of formed fibers without knob–hole interactions. These dimer conformations could be particularly relevant as they capture the aggregation pathway similar to our experimental conditions, where thrombin-mediated exposure of knobs does not occur, and thus no knob–hole binding takes place. The 1FZE structure reveals three distinct modes of protein–protein association: end-to-end association involving the respective γ-chains at the residues 270–300 and two lateral associations involving γ–γ and β–β domain interfaces shown in Fig. 2.^4,50 The end-to-end association observed in both, the thrombin-induced fibrin fiber structures and the ligand-free pathway, suggests a conserved structural motif during fiber formation. In contrast to that, the lateral associations involving γ–γ and β–β domain interfaces in 1FZE involve a different face of the domain compared to those in knob–hole crystallised structures, indicating a different aggregation pathway.⁴ The interacting amino acids of the three different conformations are listed in the SI, part I Table S4. Given that our experimental conditions involve ligand-free aggregation without knob–hole interactions, we selected all three distinct conformations of 1FZE for a quantitative protein–protein interaction analysis.

Fig. 2 Conformational states of the fibrinogen double-Fg-D domain structure (PDB: 1FZE). (a) End-to-end association via the γ-chains. Nontransparent domains are used for umbrella sampling simulations. (b) γ–γ lateral association. (c) β–β lateral association. These three states represent distinct protein–protein interaction modes observed in the ligand-free conformation and were used as starting structures for umbrella sampling simulations.

2.2.4.1 Static interaction analysis with implicit solvent. The electrostatic interaction energy between two proteins at varying distances, accounting for the Stern layer, was calculated by solving the linearized Poisson–Boltzmann equation using the APBS webserver. This analysis was performed for NaCl at concentrations of 0.001 M, 0.1 M, and 0.375 M, Na-MHP at 0.001 M, 0.1 M, and 0.25 M, and Na-DHP at 0.001 M, 0.1 M, and 0.25 M. We followed the approach of the correction of surface near dielectric constants on protein surfaces 71–73 described in section 2.2.5.4. For each system, the explicit ions and solvent molecules within the first hydration shell of the protein defined as the Stern layer were included into the solute fraction with ∈_in. The ions were assigned to the positions at the protein's surface based on their residence time. An ion residing at a given site for more than 10% of the simulation time was considered as part of the Stern layer. The ion charges were scaled according to their residence time at the protein surface. Additionally, the radius of the hydrogen atoms in the TIP3P water model was adjusted to 0.25 Å. For both the phosphate systems MHP and DHP, the ions were treated as single interaction centers similar to chloride ions. To account for their multipoint nature, an effective van der Waals radius of 5 Å was assigned. Accordingly, the ion radius was defined in the APBS. The two Stern layer proteins were aligned to the three distinct conformational states of the 1FZE protein, as shown in Fig. 2. For each conformational state, a series of inputs with increasing inter-protein distances were generated. One protein was kept fixed, while the second was translated in increments of 2 Å until the center-of-mass distance between the two proteins exceeded 5 nm. The electrostatic interaction energy was computed as the difference between the total energy of the two-protein system and the sum of the individual energies of the isolated proteins: E_elec = E_total(1 + 2) − E_single(1) − E_single(2).
2.2.4.2 Umbrella sampling simulations. Following the standard building (section 2.2.1) and equilibration protocol (section 2.2.2), the three different conformations of Fg-D dimers illustrated in Fig. 2 were simulated at physiological salt concentration (150 mM NaCl) first. After equilibration, simulations in the NVT ensemble were performed for 100 ns without additional restraints. The resulting dimer contact conformations were extracted as starting points for steered MD (SMD) simulations, which reveal the series of windows for umbrella sampling simulations. To ensure the absence of artificial mirror interactions, the simulation box size was chosen long enough to extend the dimension of half of the pulling distance (5 nm center-of-mass (COM) distance along the pulling axis). For each SMD simulation an energy minimization, a 10 ns NVT and a 10 ns NPT equilibration was performed. Here, position restraints were applied to both proteins. During the pulling phase, one protein was fixed with restraints on its backbone, while the other protein was pulled along a one-dimensional distance restraint. Further restraints on the backbone of the pulled protein were applied to prevent the rotation and translation in the other two axis. The pulling was performed for 5 ns with a spring constant of 10 000 kJ mol⁻¹ nm⁻² and a pulling rate of 0.001 nm ps⁻¹ (1 nm ns⁻¹). This resulted in an end protein distance of 10 nm without remarkable interaction any longer. The overview of all the systems are shown in Table 4. From the SMD trajectories, snapshots were taken to generate the initial configurations for umbrella sampling (US) windows. An asymmetric distribution of sampling windows was used, a window spacing of 0.04 nm up to 1 nm COM distance, using a spring constant of 10 000 kJ mol⁻¹ nm⁻², and a coarser window spacing of 0.1 nm beyond 1 nm, with a spring constant of 1000 kJ mol⁻¹ nm⁻². Each US window was shortly equilibrated within an NPT ensemble for 100 ps followed by 20 ns of NVT production MD. The histograms and potential of mean force (PMF) profiles were calculated using the weighted histogram analysis method (WHAM) as implemented in GROMACS.

Table 4 Overview of all simulated Fg-D dimer systems

State	System	Salt	Conc.	Ions	Water
End–end	Ae.001	NaCl	0.001 M	Na^+: 8, Cl^−: 2	92 147
	Ae.1	NaCl	0.1 M	Na^+: 181, Cl^−: 175	91 801
	Ae.375	NaCl	0.375 M	Na^+: 662, Cl^−: 656	90 839
	Be.1	Na-PO4	0.1 M	Na^+: 235, MHP: 63, DHP: 103	91 294
	Be.375	Na-PO4	0.375 M	Na^+: 852, MHP: 232, DHP: 382	88 970
γ–γ	Ag.001	NaCl	0.001 M	Na^+: 6, Cl^−: 4	225 123
	Ag.1	NaCl	0.1 M	Na^+: 427, Cl^−: 425	224 281
	Ag.375	NaCl	0.375 M	Na^+: 1596, Cl^−: 1594	221 943
	Bg.1	Na-PO4	0.1 M	Na^+: 561, MHP: 154, DHP: 251	223 041
	Bg.375	Na-PO4	0.375 M	Na^+: 2068, MHP: 567, DHP: 932	217 377
β–β	Ab.001	NaCl	0.001 M	Na^+: 7, Cl^−: 5	252 329
	Ab.1	NaCl	0.1 M	Na^+: 478, Cl^−: 476	251 387
	Ab.375	NaCl	0.375 M	Na^+: 1788, Cl^−: 1786	248 767
	Bb.1	Na-PO4	0.1 M	Na^+: 630, MHP: 173, DHP: 282	249 990
	Bb.375	Na-PO4	0.375 M	Na^+: 2315, MHP: 634, DHP: 1045	243 585

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2.2.5 Analysis protocol of the simulations.
2.2.5.1 Ion distributions around the protein. To analyze the ion distribution around the protein, we calculated the minimum distance between each ion and any heavy atom of the protein (hydrogen excluded) for all frames in the trajectory. The distances were binned with a resolution of 0.1 Å for each frame. The bin counts were averaged across all frames and normalized by the total number of water molecules in the system to obtain a dimensionless density. Equal-distance bins were counted cumulatively to assess the total number of ions within a given proximity to the protein surface. This method tracks the distance along the nearest approach to the protein atoms, capturing near-surface interactions that account for the irregular surface shape. The additional analysis for oxygen atoms or water molecules (O_wat) provides reference density profiles of the hydration layers at the protein's surface, which are used to discuss the relative positioning of the ions therein.⁷⁴
2.2.5.2 Ion residence time. The minimum distance between each ion and each heavy atom of the protein was calculated for all frames in the MD trajectories. For each ion, the residence time was calculated as the percentage of time in which an ion remains within the specified distance limit (cutoff).
2.2.5.3 Diffusion coefficient. The self-diffusion coefficients of ions and water at different bin distances from the protein were computed from their mean-square displacements (MSDs) during the MD runs. The MSDs were calculated by averaging over all particles and all time origins:^75–77Here τ is the time lag, N is the total number of particles, r_i(t) is the position vector of particle i at time t, t₀ denotes the time origin, and 〈·〉_t0 represents the averaging over all possible time origins. The MSDs were calculated for all ions (for phosphates, the position of the P atom was tracked) and O_wat located within predefined bin distances of 2.4 to 4.0 in intervals of 0.2 Å and 4.5 to 5.5 in intervals of 0.5 Å from the protein's heavy atoms. At each time origin, all ions (or O_wat) within a given bin distance were selected. For each selection, the MSD was computed over a 1 ns interval (10 frames saved every 100 ps), averaging over the ions within the bin distance and performing multiple time-origin averages within each block. This procedure was repeated for consecutive 1 ns blocks throughout 100 ns unwrapped trajectories. Bins without an ion count during a block resulted in a corresponding MSD window marked as none and excluded from averaging. The number of valid MSD windows contributing to the average in each bin was recorded. At the end, MSD values were averaged over all valid windows as a function of lag time. The MSD calculations were carried out using the MDAnalysis.analysis.msd module. Using the MSD, the diffusion coefficient DPBC was obtained from the Einstein relation (eqn (2)) in each distance:where d = 3 is the system dimensionality. More in detail, the calculation of DPBC involved fitting the linear regimes of the averaged MSD vs. lag-time curves using ordinary least-squares regressions, and dividing the so-obtained linear slope by the correct factor of 2d. The goodness of the fit was assessed using the coefficient of determination (R², the square of the Pearson correlation coefficient). The computed DPBC values depend on the finite system size, typically increasing with the length of the simulation box.⁷⁶ To account for this effect and estimate the diffusion coefficient at infinite system size, the Yeh–Hummer correction⁷⁸ was applied:Here, D_∞ is the diffusion coefficient at infinite system size, DPBC is the diffusion coefficient computed from the simulation, k_B is the Boltzmann constant, T is the temperature (T = 300 K), ξ = 2.837 is a dimensionless constant, and η is the shear viscosity of the solvent. Here, we used the viscosity of pure water, η = 8.93 × 10⁻⁴ kg ms⁻¹.⁷⁹ The TIP3P water model underestimates the viscosity by 63% less than the experimental viscosity value of pure water.^80,81 At high salt concentrations, the viscosity of water increases significantly, reaching η = 10.7 × 10⁻⁴ kg ms⁻¹.⁸² Despite the significant differences in viscosity of water as a function of the salt concentration and the choice of the water model, the normalized diffusion coefficients remain largely unchanged as shown in the SI, part I Fig. S1. This might be due to the large simulation box size (14.6 × 14.6 × 14.6 nm³), which effectively suppresses finite size effects. As a result, the influence of viscosity variations from the water model or salt concentration is negligible in the diffusion behavior.
2.2.5.4 Electrostatic potential. The electrostatic potential at the protein surface was calculated following a multi-step procedure: first, a set of bins with equal distances to the diffusion coefficient was defined. For each frame of the trajectory, a PDB file was generated by selecting all ion and water atoms within the chosen bin distance from any heavy atom of the protein. For phosphate, the entire phosphate molecule was included as soon as any oxygen atom was found within the bin distance. Subsequently, PQR files were generated from these PDB structures using atomic radii from the AMBER14 force field. For the H atoms, which have a zero radius in the TIP3P water model, a value of 0.25 Å was assigned. The linearized Poisson–Boltzmann equation was solved using the APBS webserver to compute the electrostatic potential. The salt concentration in APBS was set to match the simulation conditions. The solvent dielectric constant (∈_out) was adjusted accordingly^83,84 and is listed in Table 5. The definition of the solute dielectric constant (∈_in) in our systems is less trivial. On plane negatively charged metal surfaces, studies have shown that the permittivity within the first hydration layer can be as low as ∈ ≈ 6, while in the second hydration layer it can increase to approximately 30.⁸⁵ In the present study, the dielectric permittivity near protein surfaces is known to be more heterogeneous.⁷¹ Compared to perfectly planar metal surfaces, proteins reveal a very heterogeneous surface topography, which is further reinforced by charged amino acid side chains, which might reach the Stern layer like tentacles. This could result in surface near permittivity constants ranging from 4 to 25.⁷¹ Simonson et al.⁷¹ proposed three strategies to overcome the estimated errors of the surface near constants. In the first one, the charged groups of the amino acid side chains are placed into the (∈_out) region.⁸⁶ The second approach introduced a position-dependent dielectric constant at the interfacial region⁸⁷ and the third approach introduced explicit solvent models into the low-dielectric medium.^72,73 Although a distance-dependent scaling of dielectric permittivity within these hydration layers may be more appropriate, the corresponding dielectric scaling for the present systems is not known. In our recent studies, the rigid water molecules and monovalent cations 17 extend up to approximately 5 Å from the protein surface, corresponding to the first and second hydration layers. Therefore, the third approach was used throughout the calculations. We counted all Stern layer atoms, namely water and ions, as part of the solute fraction, with a constant solute dielectric value of ∈_in = 2. Consequently, the resulting electrostatic analysis should be interpreted with appropriate caution. APBS gives the potential on a three-dimensional grid. To evaluate the potential at the protein surface, the MSMS program⁸⁸ was employed to generate surface points in Cartesian coordinates, with a density of 10 points per Ų. The APBS multi-value utility was then used to interpolate the electrostatic potential onto these surface points. Where needed, the average surface potential was calculated for each frame, and the results were further averaged across all frames of the trajectory.

Table 5 Dielectric constants used for APBS based on experimental values for NaCl salt solutions^83,84

Salt concentration (M)	∈
0.1	78.54
0.25	78.54
0.375	78.54
0.75	74
1.125	70
2	60

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3 Results and discussion

3.1 Mass-loss analysis and nanofiber formation

Scanning electron microscopy (SEM) analysis of fibrinogen dried in mixed-salt solutions containing 100 mM NaCl and 100 mM Na-PO₄, followed by crosslinking and washing, revealed a well-defined and interconnected nanofibrous network (Fig. 3a). Since drying continuously increased the salt concentration, the evolving ionic strength progressively enhanced the electrostatic screening of the protein during the drying process.

Fig. 3 SEM image of self-assembled fibrinogen nanofibers and mass-loss experiment. (a) SEM image showing self-assembled fibrinogen nanofibers prepared from mixed-salt solutions, followed by drying, FA vapor crosslinking, and washing. SEM imaging of the surface revealed a detailed nanofibrous surface with well-defined and interconnected fibers. (b) Mass-loss behavior during nanofiber formation of fibrinogen. The black line represents the time-dependent turbidity, and the blue line represents the salt concentration. The dashed horizontal line marks the point at which Na₂HPO₄ reaches its saturation limit.

To relate these structural transitions to changes in salt concentration, we monitored the mass loss of fibrinogen–salt mixtures over 20 h and quantified the corresponding changes in turbidity and salt concentration (Fig. 3b). The time-dependent turbidity remained constant during the initial lag phase (0.17 to 0.19 arb.u.), indicating a clear solution and the absence of bulk aggregation. During this period, the total salt concentration of the Na-PO₄/NaCl mixture remained close to the total starting concentration of 0.2 M. At approximately 9 h, the turbidity began to gradually increase to 0.40 arb.u., coinciding with an increase in the salt concentration to 0.28 M. A sharp transition occurred at 13 h, with turbidity increasing rapidly from 0.40 to 0.82 arb.u. This coincided with the total salt concentration exceeding 1 M and rising steeply to 3 M. Complete drying was reached by 15 h, with stabilization of the turbidity at 0.91 arb.u. and a total salt concentration of around 3.7 M. Overall, the slow increase in turbidity between a total salt concentration of 0.28 and 1.0 M, followed by a sharp rise above 1.0 M, is linked to two inflection points in the turbidity profile of the drying fibrinogen–salt mixture. This trend is consistent with our previous Aq analysis.^17,28 It should be noted that the reported concentration threshold refers to the initial bulk salt concentration estimated from mass-loss measurements, while the local ionic strength may increase and become spatially heterogeneous during evaporation. In evaporation-driven systems, local enrichment effects, particularly near interfaces or within partially dried regions, can lead to transiently elevated local concentrations relative to the bulk average.⁸⁹ In addition, phosphate speciation (H₂PO₄⁻ and HPO₄²⁻) may shift during drying due to concentration changes and local pH variations, further contributing to a dynamically evolving ionic environment. However, given the small sample volume and the relatively fast ion diffusion compared to the evaporation rate, significant concentration gradients are expected to remain limited during the early stages of aggregation. Therefore, the observed threshold of 1 M should be interpreted as an effective bulk indicator of aggregation onset, rather than a sharply defined local thermodynamic transition.

Therefore, this drying behavior may reflect different kinetic stages of aggregation, in which early aggregate formation precedes rapid fibrillar growth, as reported for collagen and fibrin systems.^90–93 However, in the absence of in situ structural probes, the assignment of turbidity inflection points to specific mechanistic steps (e.g., nucleation and elongation) remains uncertain. Thus, this interpretation is phenomenological and reflects changes in aggregation kinetics rather than direct evidence of distinct molecular assembly stages. Such effects will be analysed and discussed more in depth in forthcoming work.

3.2 Elemental composition and morphology of fibrinogen networks under mixed-salt conditions

To identify which ions participate in or promote the assembly of fibrinogen nanofibers in the presence of mixed-salt solutions, we performed energy-dispersive X-ray spectroscopy (EDS) with dried fibrinogen samples formed in equimolar mixtures of NaCl and Na-PO₄ that were crosslinked and washed. The corresponding EDS spectra confirmed the presence of carbon, nitrogen, oxygen, sodium, phosphorus, sulfur, and chloride within the fibrous fibrinogen network (Table 6). Normalization to nitrogen revealed the highest presence of sodium in the washed fibers, followed by phosphorus and chloride. This observation is consistent with our previous results for fibrinogen in PBS and buffer systems of individual salts obtained by X-ray photoelectron spectroscopy where the strongest ion retention was observed for Na⁺ cations.^17,28 These atomic concentrations confirm that the fibrous matrix is primarily proteinaceous and that most components of the mixed-salts, particularly NaCl, are washed out whereas sodium – possibly from the phosphate salts – was incorporated in the protein fibers.

Table 6 Elemental composition of fibrinogen–salt precipitates. 5 mg ml⁻¹ fibrinogen was assembled on plasma-cleaned glass slides with a solution containing 100 mM NaCl and 100 mM Na-PO₄, followed by drying, crosslinking and washing. All atomic concentrations were obtained from SEM-EDS analysis in low-vacuum mode and are given in atomic percent (at%) for [N] and as normalized ratios to [N] for all other elements (*). The normalized ratios represent the average of five different positions per sample

[N]	[C]*	[O]*	[S]*	[Na]*	[P]*	[Cl]*
14.3	2.51	3.02	0.03	0.31	0.09	0.02

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To find out whether any of the two salts could be detected in the formed fibrinogen nanofibers, we performed combined SEM-EDS analysis of uncross-linked and unwashed samples. Interestingly, this analysis revealed a dense nanofibrous fibrinogen matrix interspersed with large faceted surface crystals and smaller embedded crystals (see the SEM image in Fig. 4a). Localized EDS spectra taken from two individual points within the fibrinogen nanofibers showed atomic percentages dominated by protein-derived elements with clearly detectable amounts of sodium and phosphorus and negligible amounts of chlorine (Fig. 4b, points 1 and 1*). Subsequent EDS mapping of two large and well-defined crystals showed that they were enriched in phosphorus and sodium but nearly devoid of chlorine, consistent with the composition of disodium hydrogen phosphate (Na₂HPO₄) (Fig. 4c, points 2 and 2*), which is in agreement with our previous findings.⁴¹ In contrast, the smaller embedded crystals displayed strong sodium and chlorine signals with negligible phosphorus, identifying them as NaCl crystals (Fig. 4d, points 3 and 3*).

Fig. 4 SEM imaging and localized EDS point analysis of 5 mg mL⁻¹ fibrinogen samples dried in the presence of 100 mM Na-PO₄ and 100 mM NaCl, which were not crosslinked or washed after drying. (a) SEM image showing the nanofibrous network with embedded large surface crystals. (b) EDS profile for points 1 and 1*, taken from the nanofibrous region, showing protein-derived elements (C, O, N) with minor residual salt contributions of Na, P and Cl. (c) EDS spectrum for points 2 and 2*, located on the large crystals showing strong P, O, and Na signals characteristic of Na₂HPO₄ besides the protein-derived elements. (d) EDS spectrum for points 3 and 3*, located on smaller salt crystals within the network, displaying dominant Cl and Na peaks consistent with NaCl alongside protein-based elements.

Overall, these results show that both phosphate and chloride salts crystallize during fibrinogen fiber assembly. Distinct Na₂HPO₄ and NaCl crystals are observed, with sodium phosphate forming larger crystals and sodium chloride forming smaller crystals, reflecting differences in crystallization kinetics. As evaporation proceeds, nucleation of one crystalline phase alters the composition of the remaining solution. However, turbidity measurements indicate that fibrinogen aggregation begins before extensive salt crystallization, suggesting that the primary assembly occurs while both ions remain largely dissolved.¹⁷ Consequently, early-stage aggregation is governed mainly by ion–protein interactions in the liquid phase, whereas later-stage crystallization and phase separation progressively modify the local ionic environment. Importantly, the two salt species do not contribute equally: phosphate ions, which show stronger retention and interaction with the fibrinogen network, likely play a dominant role in stabilizing aggregates, whereas chloride ions primarily contribute to transient electrostatic screening and are largely removed during washing. This is supported by EDS analysis, which shows comparable phosphorus and chlorine signals in unwashed samples (cf. Fig. 4c and d), but preferential loss of chloride after washing (Table 6), consistent with stronger phosphate–protein interactions.

In our previous experimental work, fibrinogen dried in NaCl solutions formed disordered aggregates without noticeable fiber formation, whereas drying in phosphate-based salt environments promoted the formation of ordered fibrillar assemblies.^13,17 These observations suggest that although both salt systems induce protein aggregation, the resulting aggregation pathways and supramolecular organization differ significantly depending on the ionic environment. Notably, the initial salt concentrations used in our experiments, as well as the concentrations reached during the drying process, were in the range of 0.1 M to 1 M. At these concentrations, the Debye screening lengths are approximately 0.96 nm, 0.6 nm, and 0.3 nm for 0.1 M, 0.375 M, and 1 M salt concentrations in 1 : 1 aqueous salt solutions, respectively.²⁹ In this regime, long-range electrostatic interactions are strongly screened, allowing proteins to approach each other at molecular length scales where overlap and restructuring of the Stern layers become significant. Under such conditions, protein–protein interactions become increasingly governed by local ion organization and Stern-layer composition near the protein surface. These interfacial ion-specific effects can ultimately modulate protein association pathways, leading to either disordered aggregation or ordered fibrillar assembly. To rationalize these effects in general for proteins in ionic aqueous solutions, we will focus on the small part of fibrinogen protein (Fg-D) as a representative interaction region and employ atomistic simulations to investigate how chloride- and phosphate-based salts differentially modulate the near-surface ion structure and protein association behavior.

3.3 Density distribution analysis

To understand how different salt ions interact with fibrinogen at the molecular level, we studied the specific protein–ion interaction in extensive all-atom MD simulations of the Fg-D domain monomer in contact with both individual and mixed salt systems. The generated datasets were used for all subsequent monomer analyses in this work. To characterize how ions and water molecules interact with the Fg-D domain, we performed a set of complementary analyses that quantify their spatial distribution, residence dynamics, and preferential binding patterns. These analyses collectively reveal how different salt types and concentrations modulate ion–protein interactions and how these interactions evolve from the Stern layer to the diffuse layer as the ionic strength increases.

The RDF analyses of all ions and water around the protein in the different salt systems and salt concentrations are shown in Fig. 5a–d. The sodium chloride and MHP systems reveal two distinct peaks within the distance of 6 Å from the protein. In the sodium chloride system, both peaks are attributed to sodium displacement, whereas in the MHP systems, the second peak is formed by phosphate ions too. After that, the enrichment of ions and water converges smoothly to g(r) → 1 after 35 Å. In the system containing various sodium DHP concentrations, the first peak is formed by sodium ions only, analogous to the MHP system. DHP forms two further density peaks with maxima at distances of about 5 Å and 10 Å. The enrichment of ions converges to g(r) → 1 at remarkably higher distances of about 50 Å. These differences in peak formations and the longer range of the DHP containing systems is a very first indication of an ion specific local distribution at the protein surface. However, all systems reach a bulk phase and we can attribute all interface observations to individual surface effects without mirror contributions.

Fig. 5 (a–d) Radial distribution functions of water and ions around the protein for systems A to D. For each of these systems, water and ion specific normalized density profiles and cumulative ion count profiles as a function of minimum distance to any atom of the Fg-D domain are plotted for (e–h) O_wat and ions for systems A to D, (i–l) Na⁺ for systems A to D, (m) Cl⁻ for system A, (n) MHP for system B, (o) DHP for system C, and (p) anionic species for system D. The left y-axis shows the normalised density (solid lines) scaled by the total number of water molecules in the system, while the right y-axis displays the cumulative ion count (dashed lines). Shades of red correspond to control water-oxygen atom O_wat, blue to Na⁺ ions, yellow to Cl⁻, dark green to MHP-P atoms and light green to DHP-P (purple for oxygen not attached to hydrogen (O), red for oxygen attached to hydrogen (O*)). Increasing colour intensity represents higher salt concentrations. The dotted vertical lines indicate the peak maximum and are colored according to the above-mentioned colors for O_wat (red), Na⁺ (blue), Cl⁻ (yellow), MHP-P (dark green), and DHP-P (light green).

The significant enrichment of ions within the first two distribution peaks hints for a highly ordered structure near the protein surface, namely the formation of an electrical double layer at the non-uniform charged protein surface. In the following, we will focus on this range. We first examined the spatial distribution of all ions relative to the Fg-D domain in the different salt systems and concentrations listed in Table 2. Normalized density profiles (Fig. 5) and cumulative ion counts were evaluated as a function of the minimum distance from any protein atom. These profiles reveal the preferred locations of the Na⁺, Cl⁻, and phosphate species (MHP and DHP) and how their distributions shift with ionic strength. The O_wat–protein profiles are plotted as references (Fig. 5, red). Across all systems, O_wat profiles exhibit a characteristic first density peak at 2.7 Å to 2.9 Å, followed by a broader shoulder-extending peak at 3.4 Å to 3.5 Å The tail of the first peak shortens with increasing salt concentration, indicating reduced water accumulation near the protein surface under high ionic strength. This can be confirmed by the concentration-dependent decrease of cumulative counts for water. Sodium ions consistently show the closest approach to the protein surface (Fig. 5, blue). A sharp Na⁺ peak occurs at 2.2 Å to 2.4 Å. Na⁺–O_wat distances of this magnitude are reported for fully solvated ions and their first hydration shell.^94,95 This reflects direct protein–ion contact, by replacement of water molecules from the hydration shell. A broader second peak at 4.4 Å to 4.5 Å corresponds to ions that interact indirectly through their hydration shells and correlates with the water shoulder region, marking the onset of the diffuse layer. Chloride ions exhibit a distinct two-peak structure in NaCl systems (Fig. 5, yellow). A broad first peak appears at 3.3 Å, followed by a shoulder near 3.8 Å, suggesting a gradual transition between hydration-mediated interactions near the surface and more distant interactions. The maximum alignment between Cl⁻ and the water shoulder region indicates that most chloride ions interact with the protein indirectly. Phosphate ions (MHP and DHP) display more complex distributions as a result of their multi-oxygen structures (Fig. 5, MHP (dark) and DHP (light green)). Phosphate atoms as the center of mass (COM) of complex phosphate ions show density maxima at 3.6 Å to 3.8 Å, indicating that phosphate species remain slightly further away from the protein surface than Na⁺ but slightly closer than Cl⁻. Analysis of the individual oxygen atoms in phosphate ions exhibit subcomponents near 3.1 Å (MHP) and 2.8 Å (DHP) for hydroxyl oxygen atoms. For negatively charged oxygen atoms, peak maxima at 3.4 Å (MHP) and 3.2 Å (DHP) can be observed, reflecting differences in hydrogen-bonding propensity between protonated and non-protonated oxygens and a closer position of DHP than MHP.

Comparing all systems, the trends observed in individual salts (systems A to C) are consistently reproduced in mixed-salts (systems D). The number of ions in the corresponding peaks increases with ion concentration, whereas the position or spatial distribution of the peaks is comparable for all with ionic strength. Phosphate ions consistently occupy positions between the Na⁺-rich Stern layer and the Cl⁻-rich diffuse layer, indicating intermediate affinity for the protein surface. These trends highlight salt-dependent structuring. Na⁺ engages in direct interactions, while divalent cations were previously found to adsorb indirectly via their hydration shell.¹⁷ The same system-specific and concentration-dependent behavior can be drawn for simple monovalent anions in comparison with oxygen-containing complex anions, as recently discussed in our experimental study.¹⁷

3.3.1 Ion residence time. Based on the identified density peaks in Fig. 5, we next quantified how long each ion type remains within the first (Fig. 6) and second (SI, part I Fig. S2) shell region of the protein, as described in section 2.2.5.2. This analysis provides insights into the stability of interacting ions and distinguishes between transient and persistent binding. Residence time classifications (<1%, 1–10%, >10% of simulation time) reveal clear distinctions between ion types and salt concentrations. Across all systems, water and most ions fall predominantly within the 1–10% regime in both the first and second shells, reflecting short-lived interactions easily disrupted by thermal fluctuations. Chloride ions exhibit the shortest residence times, being consistently in the <1% and <10% regimes with a ratio of about 60% to 40% in contact with Fg-D. The proportion of short-lived Cl⁻ interactions below <1% decreases slightly with rising concentration, accompanied by a corresponding increase in the 1–10% regime, consistent with increased crowding near the protein at high ionic strength. In contrast, phosphate ions show markedly longer residence times. Both MHP-P and DHP-P exhibit substantial populations in the >10% regime, with DHP-P displaying the highest propensity for stable binding. This is consistent across salt concentrations and indicates stronger and more persistent attractive interactions between phosphate anions and positively charged patches on Fg-D. Sodium ions display intermediate behavior. Although most Na⁺ interactions fall within the <10% regime, a subset of Na⁺ ions exhibit prolonged binding at specific sites, especially at elevated salt concentrations.

Fig. 6 Residence time of first-shell region forming ions as a function of salt concentration. Shades of red correspond to O_wat, blue to Na⁺ ions, yellow to Cl⁻, dark green to MHP, and light green to DHP. Increasing color intensity represents higher salt concentrations. The dashed horizontal grey line marks the 50% residence time. (a–d) O_wat for systems A to D, (e–h) Na⁺ for systems A to D, (i) Cl⁻ for system A, (j) MHP-P for system B, (k) DHP-P for system C, and (l) anionic species for system D.

The analysis of the residence time of ions in the second shell is illustrated in the SI, part I Fig. S2. A shift to lower residence times of all ions and water molecules is indicative of a significantly higher mobility compared to the first-shell species. Except for a few individual water molecules, all ions and molecules are limited to maximum residence times below 10% of the total simulation time. While Na⁺ and Cl⁻ ions in the individual-salt systems A show equal mobilities in the second shell, in the mixed-salt systems D, the mobility of Cl⁻ is clearly higher, shifted to lower residence times compared to Na⁺ ions. This implies an indirect influence of phosphate ions on the mobility of the second shell species.

3.3.2 Immobile ions. Special attention is now devoted to what we define as “immobile ions”, namely those ions that remain in contact with the protein within the first hydration shell for >50% of the simulation time. Inspection of their spatial locations (Fig. 7(a–d)) shows that immobile ions cluster near known calcium-binding sites (ion_s1–ion_s4), consistent with previous reports of high-affinity calcium-binding regions on Fg-D.^48–50 In NaCl systems (A), immobile ions are primarily Na⁺, reflecting their strong attraction to negatively charged residues at Ca²⁺-binding sites.^48–50 In phosphate systems, immobile ions additionally include MHP-P and DHP-P, where DHP exhibits the greatest degree of immobilization. The presence of immobile phosphates at several positively charged surface patches supports their role in modulating fibrinogen assembly under conditions of high salt concentration. The complex phosphate ions can form multiple hydrogen bonds and therefore might act as a bridge, chelated by amino acid side chains from different proteins during fiber assembly.^96,97 In contrast, no immobile ions appeared in the second-shell region, confirming that persistent binding occurs exclusively through direct protein contacts (SI, part I Fig. S2). All immobile ions likely contribute to neutralization of local charges, reduced electrostatic repulsion, and ultimately to enhanced protein–protein attraction during aggregation.

Fig. 7 Snapshots of immobile ions within the first-shell region (residence time >50%) and ion occupancy maps showing highly occupied regions on the Fg-D domain. (a–d) Snapshots of immobile ions on the Fg-D domain for systems A to D, respectively. The Fg-D domain is shown in NewCartoon representation with different shades of grey for the three chains (α, β, γ). Water molecules are shown in vdW representation with a radius of 1 Å (oxygen: red; hydrogen: white). Na⁺ ions are shown in vdW representation with a radius of 2.6 Å in blue. MHP and DHP are shown with phosphorus atoms in vdW representation (radii: 2 Å; MHP: dark green; DHP: light green) and other atoms in CPK representation (radii: 6.1 Å; oxygen: red; hydrogen: white). The numbering 1–4 is w.r.t Ca²⁺ binding sites. (a) Immobile O_wat and Na⁺ at different concentrations for system A, (b) immobile O_wat, Na⁺, and MHP at different salt concentrations for system B, (c) immobile O_wat, Na⁺, and DHP at different salt concentrations for system C, (d) immobile O_wat, Na⁺, MHP and DHP at different salt concentrations for system D, (e–h) ion occupancy maps generated and visualised using VMD showing the preferred binding regions of ions in the first- and second-shell regions. The Fg-D domain is shown in NewCartoon representation (grey for α, β, and γ chains), with acidic residues in red and basic residues in blue. Occupancy maps are shown as isosurfaces (isovalue: 0.1), with colors corresponding to specific ions: Na⁺ (blue), Cl⁻ (yellow), MHP-P (dark green), and DHP-P (light green). Second-shell occupancy is shown in transparent. (d) Na⁺ and Cl⁻ for system A, (e) Na⁺ and MHP-P for system B, (f) Na⁺ and DHP-P for system C, (h) Na⁺ and anionic species for system D. (i–l) Snapshots of the ion occupancy map and electrostatic potential for high-salt concentration systems. The Fg-D domain is shown in NewCartoon representation with different shades of grey for the three chains. Ion occupancy maps are displayed as isosurfaces (isovalue: 0.1), with colors corresponding to specific ions: Na⁺ (blue), Cl⁻ (yellow), MHP-P (dark green), and DHP-P (light green). Electrostatic potential, calculated using APBS at a salt concentration of 0.01 M, is shown as an isosurface at isovalue ±1k_BT/e, with negative potential in red and positive potential in blue. (i) System A2.0, (j) system B1.125, (k) system C1.125 and (l) system D1.125.

3.3.3 Occupancy maps. To visualize persistent ion interactions from the protein's perspective, we constructed three-dimensional occupancy maps. To this aim, a 3D grid with a grid resolution of 1.0 Å is created. Each point in the grid is assigned a value of 1 within the van-der-Waals-radius of an ion (Na⁺: 1.36 Å, Cl⁻: 2.27 Å, both MHP-P and DHPP: 1.5 Å) and to 0 without ion occupancy. All distinct values averaged over the simulation time are visualized as an isosurface with an isovalue of 0.1 (Fig. 7(e–h)). These maps highlight regions of high ion–protein proximity, regardless of whether the same ion resides there over time. For Na⁺, the occupancy maps show consistent binding at several negatively charged residues for all salt types, with ion_s1 and ion_s2 being the most frequently occupied. At higher concentrations, additional surface patches become populated, reflecting the increased density of Na⁺ near the surface. In contrast, anions dominate the occupancy map at high salt concentrations only, enriched on positively charged regions of Fg-D. DHP-P shows the broadest and most intense occupancy patterns, followed by MHP-P, while chloride exhibits far weaker occupancy and only becomes frequent at ≥1.125 M in NaCl systems. In particular, in mixed-salt systems, Cl⁻ occupancy is nearly absent, indicating competitive exclusion by phosphate. Overlaying occupancy maps with the electrostatic potential calculated with APBS (Fig. 7(i–l)) reveals that regions of high occupancy align with strong surface potential extrema, emphasizing the central role of electrostatics in directing the localization of the ion. Collectively, the occupancy analysis, together with residence time and density distributions, provides a coherent picture of ion-specific structuring at the protein interface.

3.4 Electrical double layer

The combined density, residence time, and occupancy analyses reveal how ions organize around Fg-D, but do not directly indicate how this structuring modifies the effective electrostatic environment during protein–protein interactions. To address this, we examined ion diffusion (as a measure of local mobility) and the distance-dependent electrostatic potential surrounding the protein. Together, these analyses define the boundaries of the Stern and diffuse layers and quantify the extent of electrostatic screening in each salt system.

3.4.1 Diffusion coefficient. To assess ion mobility near the protein surface, diffusion coefficients were computed at several distances from the protein (Fig. 8). In addition, reference D_sim0 bulk values were calculated for aqueous ionic simulation systems (Ax to Dx) at 0.1 M concentration with our charge scaled parameter set. The resulting reference D_sim0 bulk values are listed in Table 7 in direct comparison with experimental values and calculated values. We note that the TIP3P water model exhibits a diffusion coefficient value of around 6.1 × 10⁻⁵ cm² s⁻¹.¹⁰⁰ This is almost three times higher than the experimental diffusion coefficient of water at infinite dilution 2.3 × 10⁻⁵ cm² s⁻¹.¹⁰¹ Consistently, our calculated D_sim0 for water is in good agreement with the overall diffusion coefficient of the TIP3P water model and varies between 6.0 and 6.2 × 10⁻⁵ cm² s⁻¹ in our simulations in aqueous salt concentrations of 0.1 M. Also, the diffusion coefficients of Na⁺ and Cl⁻ reported in the MD literature (D_lit) are typically larger than the corresponding experimental values (D_exp). We obtained even slightly larger values (Table 7). This is not unexpected given that their parameter sets are tuned to reproduce interaction strengths rather than diffusivities, especially after down-scaling of the net charges according to the ECC approach. Therefore, we focus on the diffusion coefficients normalized against the D_sim0 bulk values of our protein-free reference simulations for systems A to D.

Fig. 8 Normalized density distribution and normalized diffusion coefficient (D_norm) shown as a function of distance from the Fg-D surface. The left y-axis shows the normalized density for systems A to D, using the same color scheme as in Fig. 5. The right y-axis shows D_norm, colored in shades of gray, with higher intensity representing higher salt concentrations. Dots represent points with a good linear regression fit, while triangles indicate data with a poor linear regression fit. Symbol filling corresponds to the percentage of windows used for MSD calculation, as indicated by the color bar. (a–d) O_wat for systems A to D, (e–h) Na⁺ for systems A to D, (i) Cl⁻ for system A, (j) MHP-P for system B, (k) DHP-P for system C, and (l) anionic species for system D.

Table 7 Diffusion coefficient values (D × 10⁻⁵ cm² s⁻¹) after finite periodic box size effect correction at 0.1 M concentration for the different systems mentioned in Table 2, literature values of MD simulations (D_lit) and experimental diffusion coefficients at infinite dilution (D_exp)⁹⁸

System	Ion	Dsim0	Dlit	Dexp
Ax	Na^+	4.3	2.28 (ref. 99)	1.334 (ref. 98)
Bx, Cx	Na^+	4.2
Dx	Na^+	4.1
Ax	Cl^−	5.0	2.90 (ref. 99)	2.032 (ref. 98)
Dx	Cl^−	4.7
Bx	MHP	2.7	—	0.759 (ref. 98)
Dx	MHP	2.7
Cx	DHP	3.3	—	0.959 (ref. 98)
Dx	DHP	3.1
Ax	TIP3P	6.2	6.1 (ref. 100)	2.3 (ref. 101)
Bx, Cx	TIP3P	6.1	6.1 (ref. 100)	2.3 (ref. 101)
Dx	TIP3P	6.0

---

The normalized diffusion coefficients are shown together with the profiles of salt concentrations along the distance from the protein in Fig. 8. The ion density distributions from Fig. 5 are plotted in the background, in order to directly correlate changes in diffusion to the specific properties of adsorption and residence of ions. The diffusion data are color-coded on the basis of the percentage of MSD windows in which ions are captured at each distance. The shape of the marker indicates whether the linear fit to the MSD is of good quality (r² > = 0.8) or not. The MSD windows provide insight into whether the ions are persistently present or only appear randomly during certain time frames. In all systems and for all concentrations, the reference density peak of O_wat is located at 2.6 Å. In analogy to the electrical double layer theory, the diffusion coefficient at this distance (within the Stern layer) is significantly reduced to 20% of the bulk values for all salt-specific systems at all concentrations. This indicates strong interactions with the protein and within the other water molecules in densely-packed layers in protein proximity. The diffusion coefficients increase in the region between the first and the second hydration shells, and reach plateau values at distances of about 4 Å to 5 Å from the protein surface. This behavior is consistent for all systems, but the value of the plateau depends on the salt concentration in the individual salt systems. At lower salt concentrations, the water diffusion coefficients converge close to their bulk values. At higher concentrations, they reach only about 80% of the bulk value. The ion/water RDFs around the proteins (Fig. 5) reveal an enrichment of ions around the protein surface up to 5 nm. Our D value is not yet completely converged to bulk values within the investigated 1 nm. This effect is more dominant for the MHP and DHP systems. We assume a slight increase over the long range of 5 nm.

Sodium ions exhibit very low diffusion coefficients of approximately 10% of the bulk values in the region of the first density peak. In the second shell region, the accuracy of the calculated diffusion values strongly depends on the salt concentration (the number of ions in the simulation box). At large salt concentrations, the scattering of the computed values is significantly lower and the MSD window is shifted towards >80%. Between the first and second shell regions, a local maximum diffusion value is observed for lower salt concentrations, which could be the result of an unfavoured position between two density maxima. Nevertheless, for sodium ions, a diffusion plateau of around 80% of the bulk value is reached after about 4 Å for all systems. This implies that the second shell of sodium ion density has no further local impact on the diffusion coefficient, in line with a weak interaction with the protein.

The anionic species reveal several differences in the variation of the diffusion coefficients along the distance from the protein. Chloride ions show a rapid increase to diffusion bulk values as soon as they are at a distance of 3.5 Å from the protein, which is consistent with a lower protein affinity. In contrast, MHP and DHP reveal significantly reduced diffusion coefficients (20% of the bulk values) within the first density peak at a distance of 3.4 Å. The diffusion coefficients slowly increase and reach about 50% of the bulk value only after 4.5 Å.

Since the scattering of the Na⁺ ion diffusion coefficient is significantly higher in phosphate systems than in Cl⁻ systems, we checked the influence of phosphate ions on the electrical double layer of the protein for longer ranges. The SI, part I Fig. S3(a–c), reports a converged value of diffusion for MHP at around 70% of its bulk diffusion. In contrast, the profile of the diffusion coefficient of DHP still reveals a slight slope after 10 Å. This effect is also pronounced in the mixed-salt system and suggests that, due to their size and very rigid bonding, phosphate ions have a large influence on the order structure of the water layers and the mobility of other ions far into the solution phase.

In summary, our simulations reveal that the water mobility decreases near the first density peak, reflecting highly structured hydration layers. The Na⁺ mobility decreases significantly near the protein surface, particularly within the first density peak. This reduced mobility reflects strong electrostatic attraction to acidic residues. Cl⁻ mobility remains relatively high, consistent with its weaker surface affinity. Phosphate ions exhibit the lowest mobility near the surface, consistent with long residence times and persistent binding. At distances beyond 4.5 Å, all ions and water except phosphate approach bulk-like diffusion, marking the boundary of the diffuse layer. DHP, in particular, shows markedly slow diffusion up to 10 Å distance from the protein, and also perturbs the cationic diffusion well into the bulk. These differences found in our simulations can explain the observation of the EDS measurements, where only Na⁺ and phosphate ions are found to be embedded in the fibrinogen fibers (section 3.2). The Stern layer consists of low-mobility Na⁺ and phosphate ions directly interacting with the protein, whereas Cl⁻ and loosely structured water dominate the diffuse layer. This direct interaction close enough with regard to distance and long enough with regard to the time scale enables the embedding of Na⁺ and phosphate ions into the fibers, whereas other ions like Cl⁻ are washed away.

3.4.2 Electrostatic potential landscape. To conclude our study, we calculated the electrostatic potential surrounding the protein at increasing distances to quantify how ion distributions translate into effective electrostatic screening (Fig. 9) During the aggregation of fibrinogen into fibers, the screening of the electrostatic potential by salts around the protein influences how the proteins perceive their mutual potential landscapes and determines their interaction mode. Fig. 9 shows the profile of the averaged electrostatic potential at different distances from the protein. Again, the corresponding ion density distributions are plotted in the background to correlate the potential with the ionic clouds.

Fig. 9 Normalized density distribution (N.D) and averaged electrostatic potential coefficient (〈ψ₀〉) for four systems at different salt concentrations, shown as a function of distance from the protein surface. The left y-axis shows the normalized density for systems 1–4. The dotted vertical lines indicate the peak maximum using the same color scheme as in Fig. 5. The right y-axis shows 〈ψ₀〉, colored in shades of gray, with higher intensity representing higher salt concentrations. The dashed horizontal line marks the zero electrostatic potential value. (a) System A, (b) system B, (c) system C, and (d) system D.

In NaCl systems, the potential near 2.4 Å is strongly positive due to the abundance of Na⁺. The magnitude increases with salt concentration. At 3.2 Å and beyond, the diffuse cloud of chloride ions progressively neutralizes the positive field. Full screening occurs by about 5 Å. In NaMHP systems, the near-surface potential is similarly positive, but a rapid inversion to negative values occurs already at about 3.2 Å, where phosphate ions begin to influence the Stern layer. The negative potential persists until 4 Å, after which it gradually approaches zero. In NaDHP systems, the potential is already near zero at 2.4 Å to 2.8 Å and becomes strongly negative by 3.0 Å, reflecting the high affinity of DHP for near-surface regions. Mixed-salt systems resemble NaDHP in a short range, with a slight positive potential at 2.8 Å followed by a pronounced negative region up to 4.0 Å. Explicit-solvent snapshots (SI, part I Fig. S4) illustrate the population of water and ions in specific potential shells around the protein for the mixed-salt system D at the lowest and highest salt concentration. For positive potential values in close proximity to the protein (Fig. 9, distance 2.4 to 3), a mix of water, sodium and few phosphate ions is visible. The inversion from positive to negative potential values is induced by the increase of phosphate ions in these shells. The following increase of the electrostatic potential can be attributed to a higher presence of sodium ions. Finally, the convergence to bulk values starts with the additional presence of Cl⁻ and is finalized by a stepwise reduction of all ionic species to bulk concentration. The calculated electrostatic potentials should therefore be interpreted as qualitative spatial trends rather than quantitatively exact local electrostatic energies, particularly under highly concentrated electrolyte conditions. Notably, in the phosphate-containing systems, the electrostatic potential profiles do not converge to zero even at a distance of 6 Å (Fig. 9 and SI, part I Fig. S3). This is consistent with the observed profile of diffusion coefficients, which reach a value of only 50% of the bulk value at this distance (see above). In the SI, part I Fig. S3(d–f), the analyses for the phosphate-containing systems B, C and D are expanded up to 10 Å. They clearly show a significantly longer range of influence with regard to ordering and mobility for phosphate-containing systems than for NaCl systems. In all simulated systems, at shorter distances from the protein, the averaged electrostatic potential increases with increasing salt concentration. However, the snapshots and iso-value surfaces shown in the SI, part I Fig. S4, reveal that high salt concentrations are associated with a stronger electronic screening at both positive and negative surface charge patches. Overall, these results show that NaCl screens the protein at larger distances, while phosphate salts screen the protein earlier and more strongly in the first shell of high ion density, with DHP producing the most pronounced charge inversion close to the surface. Indeed, investigations of Dumetz et al. showed that mutual interactions between proteins (quantified from calculations of the second virial coefficients) differ significantly in high-concentrated solutions of NaCl, ammonium sulfate or potassium phosphate salts.¹⁰² This demonstrates that the electrical double layer around fibrinogen is highly sensitive to the anion identity and that kosmotropic phosphate ions induce stronger and longer-ranged solvent structuring and screening than chloride ions.

3.5 Protein–protein interaction analysis

To date, we have investigated the influence of specific salt conditions on the formation of the electrical double layer on protein Fg-D monomer surfaces, especially the formation of the Stern layer composition. From our experimental studies, we know that protein aggregation takes place at salt concentrations higher than 0.1 M, where long-range electrostatic interactions are expected to be strongly screened. To examine the extent of this screening on protein–protein interactions at close-contact separations, we performed Fg-D dimer interaction analyses for three distinct conformational states crystallized under low salt conditions.⁵⁰

3.5.1 Static interaction profiles. In a first step, we quantified the protein–protein interactions with a static approach in an implicit solvent environment. According to Antosiewicz et al. and Chan et al.,^72,73 we used a solute fraction of protein and explicit Stern layer molecules for the calculation of the electrostatic interaction energy between two proteins via APBS.⁶² Fig. 10a–c and j–l show the electrostatic interaction profiles calculated between the two Fg-D domains under various salt conditions as a function of the COM distance for the three conformational states end–end, γ–γ, and β–β as listed in Table 4. On the basis of molecular snapshots of the solute fraction, the energy profiles are divided into three regions, separated by dashed lines: (1) the proteins would be in direct contact, vdW interaction dominates; (2) the Stern layer regions of both protein domains are in direct contact, the explicit solvent and ions from the two proteins begin to overlap; and (3) the diffuse layer (or Gouy–Chapman layer), where no explicit atomic overlap occurs and the system is dominated by implicit long-range electrostatics. In all three conformations, the first and the second region in this static approach is highly repulsive due to direct atomic clashes of the whole solute fraction. In this approach, we would like to focus on the third region after the second dashed line, which corresponds to the diffuse layer and the bulk zone dominated by long-range electrostatic interactions. In the chloride-based systems, the γ–γ and β–β conformations (Fig. 10b and c) reveal a clear dependence on the salt concentration. The diffuse layer zone follows the DLVO theory with an exponential increase of the interaction energy with decreasing COM distance for low salt conditions. A screening of the electrostatic repulsion for increasing salt concentrations, with an energy converging rapidly to zero, is clearly evident. In the phosphate-based systems, the γ–γ and β–β conformations (Fig. 10k–l) exhibit highly repulsive behavior in the diffuse-layer region closest to the Stern-layer regime, particularly at increasing salt concentrations. As discussed in section 2.2.4.1, phosphate ions were treated as single interaction centers in the calculations. However, the Stern layer was found to contain MHP and DHP species (Fig. 7), unlike the chloride-based systems. Consequently, hydrogen-bonding interactions and chelation effects involving phosphate species and amino acid residues may not be fully represented within the APBS framework. Instead, these interactions are predominantly repulsive as a consequence of an interaction radius of 5 Å. Therefore, for the phosphate-based systems, the interaction profiles were analyzed at distances of at least 1 nm beyond the defined Stern-layer regime. At these distances, a clear dependence on salt concentration is observed, with progressively stronger electrostatic screening at increasing salt concentrations. In contrast, the end–end conformation (Fig. 10a and j), in both the chloride- and phosphate-based systems, exhibits nearly zero interaction energy for all salt concentrations, indicating a favorable interaction insensitive to salt concentration. Interestingly, this conformation is the physiological one in ligand-based in vivo fibrin aggregation, where molecular interaction is stabilized by knob–hole interactions.⁴⁹ The salt concentration sensitivity of the other two conformations could give rise to a hint at a modified aggregation process under our ligand-free experimental conditions. Altogether, for the γ–γ and the β–β conformations long-range electrostatic repulsion of 4k_BT is observed at low salt concentrations. The value decreases to approximately 2k_BT for salt concentrations higher than 0.1 M, indicating that long-range electrostatic interactions are effectively screened under these conditions. Since the experimental studies were performed at concentrations above 0.1 M, these results suggest that the aggregation pathway is governed predominantly by the Stern-layer composition and interfacial ion organization rather than by long-range electrostatic interactions. However, the quantitative analysis works better for pure implicit systems and has to be read with some care, neglecting the relaxation of all explicit Stern layer molecules. For a quantitative analysis of the dimer interactions for all three different conformations and at different salt concentrations for both phosphate and chloride-based systems, we performed umbrella sampling simulations and plotted the potential mean force (PMF) between the two protein domains.

Fig. 10 Electrostatic interaction profiles and potential mean force (PMF) profiles are shown as a function of center-of-mass (COM) distance for end-to-end, γ–γ, and β–β Fg-D dimer conformations. Vertical dashed lines distinguish zones of (1) direct protein interaction, (2) merged Stern layers and (3) drift into diffuse contact zones up to bulk. For the phosphate systems, the additional grey dashed line resembles the corrected zone (2) in the static approach as a consequence of the increased radius of the solute fraction. The subfigures a to i resemble the sodium chloride systems with molecular snapshots g to i of the equilibrated dimer contact structures. The subfigures j to r show the sodium phosphate systems (MHP, solid lines and DHP dashed lines) with molecular snapshots p to r of the equilibrated dimer structures. The color code for the salt species and concentrations is adopted from Fig. 5.

3.5.2 PMF analysis. In Fig. 10, PMF profiles for the three Fg-D dimer conformations are plotted. We have performed umbrella sampling simulations for very low salt concentrations (0.001 M), medium (0.1 M), and up to high salt concentrations (0.375 M). Contact frequency histograms for the varying protein distances and time-resolved PMF profiles as a proof of convergence are plotted in the SI, part I Fig. S5 and S6. The PMF profiles calculated with intervals of 500 ps are shifted by 200 ps and consistently demonstrate convergence with increasing simulation time. After about 13 ns, the profiles are converged up to the total simulation time of 20 ns for all corresponding windows. Therefore, the PMF profiles of 13.2 ns to 20 ns were averaged using interpolation across center-of-mass (COM) distances and are presented in Fig. 10d–f (chloride) and m–o (phosphate), for salt concentrations ranging from 0.001 M to 0.375 M. All profiles reveal a significant slope in the energy profile in close dimer contact. This slope converges to stable plateaus after small oscillations. The PMF profile is divided into three regions: (1) strong gain of energy due to direct protein contacts, (2) small oscillations due to merging of the two Stern layers and (3) a plateau towards bulk diffusion with no significant interaction. The definition of the first region is drawn on the basis of minimum distance plots between all heavy atoms of the proteins over the COM distances (SI, part I Fig. S7). All values around 3.0 Å or lower (SI, part I Fig. S7) can be attributed to direct protein contact. This could be observed for the end-to-end conformation until 4 nm COM distance, for γ–γ until 4.7 nm and for β–β until 4.2 nm. Since, in this case, the protein structure is allowed to relax, the highly repulsive behavior observed in the static interaction profiles at the protein–protein contact and the Stern-layer region is not evident, but rather an attractive interaction can be seen. For the end-to-end configuration, interaction energies of about 38k_BT are observed. The γ–γ and β–β conformations show slightly higher interaction energies of 48k_BT and 50k_BT. The reason is illustrated in the molecular snapshots in Fig. 10g–i and p–r. The gain of energy for all interaction profiles can be attributed to direct protein–protein interactions of individual amino acid side chains without any ions involved, even though some of them are located between the protein domains in contact. A complete list of the interacting amino acids is plotted in the SI, part I Table S4. All of them remained stable over the entire simulation time. Despite the observation in the static interaction profiles, the corresponding umbrella sampling energy profiles are consistent not only for each salt concentration but across the salt species. As the energy profiles are calculated with restrained distance windows, the relaxation of all ions and water surrounding the two protein domains occurs in the converging phase of our MDs and electrostatic repulsion while approaching will be neglected. Only in very close contact, within distances being smaller than the sum of the two Stern layer thicknesses, the small oscillations reveal the influence of a rigid Stern layer merging process with small deviation at increasing salt concentration.

Conclusions

In this study, we demonstrated a nuanced interplay of ion-specific interactions, solvent structuring, and electrostatic screening of the Fg-D domain under high salt conditions. In silico results of dimer simulations reveal that protein–protein binding ability is not dependent on long-range electrostatic double layer interactions, but on the precise matching of mutually facing and interpenetrating Stern-layer regions at the molecular level. These results can be used to rationalize our experiments of fibrinogen under mixed-salt conditions which show that a drying-induced increase in salt concentration must exceed a critical threshold to trigger fibrinogen nanofiber assembly. The selective retention of sodium in fibrinogen fibers, particularly in association with phosphate, alongside the removal of chloride, highlights distinct and non-equivalent roles of different salts in governing nanofiber assembly during the drying process. Subsequent atomistic simulations of the fibrinogen D domain with information from experiments about the salt concentration provide a mechanistic basis for these observations. Na⁺ cations form a tightly bound Stern layer through direct interactions with acidic residues, whereas chloride anions largely occupy the diffuse layer and display transient surface affinity. In contrast, monohydrogen and dihydrogen phosphate anions exhibit pronounced and persistent binding in positively charged surface patches, particularly near known Ca²⁺-binding sites persistently occupied by Na⁺. These anions significantly reduce local ion mobility, extend the structured solvent region, and induce strong modulation of the electrostatic potential near the protein interface. As a result, phosphate-containing systems achieve electrostatic neutralization at much shorter distances than NaCl, providing a mechanistic explanation for their stronger propensity to induce fibrinogen aggregation in solution. Together, our results establish that the identity and hydration characteristics of anions, especially phosphate species, govern the reorganization of the electrical double layer and thereby control the propensity of fibrinogen molecules to overcome electrostatic repulsion to consequently form aggregates or fibers. By elucidating how specific ions tune near-surface charge compensation and the local solvent structure, this work provides a physicochemical foundation for enzyme-free strategies to direct fibrinogen assembly. These insights have important implications for the design of fibrinogen-based biomaterials. Adjusting the ionic composition and concentration will offer a straightforward route to modulate fibrillogenesis of fibrinogen and to tailor the morphology and network connectivity without relying on enzymatic activation. Such strategies may enable the development of safer, customizable fibrillar scaffolds for regenerative medicine and support new approaches to protein-based material fabrication where controlled ionic microenvironments act as powerful regulators of supramolecular architectures. Quantitative analysis of protein–protein interaction via implicit solvent and PMF analysis suggests that the long-range electrostatics are screened at experimental concentrations higher than 0.1 M and the composition of the Stern layer defines the aggregation pathway. Together with our recent experimental observations of anion-dependent fibrillogenesis and scaffold architecture formation,²² the present findings establish a multiscale framework linking ion-specific interfacial electrostatics to fibrinogen biomaterial assembly. Nevertheless, the present analysis is limited to the Fg-D domain and should be extended to the full fibrinogen system, including disordered regions such as fibrinopeptides and the αC domain. These intrinsically disordered regions may additionally influence structural organization in different salt environments. In addition, permanent adsorption of simple and complex ions can induce pK shifts to the charged amino acid side chains. Preliminary tests revealed a pK sensitivity of the Ca²⁺ binding sites, especially the S1 and S3 sites. This aspect will be further addressed in future studies. To further bridge the gap of single molecular all atom simulations and in vitro experiments of the Fg fiber formation observed in phosphate-based systems, future work employing coarse-grained simulations on the full fibrinogen suspensions will be performed using the molecular-level insights obtained from the present study.

Author contributions

Aparna Sai Malisetty: simulation parameter set and protocol development, molecular dynamics simulations, analysis design and programming, writing – original draft. Antoine Eyram Kwame: experiments and analysis, writing – original draft. Lucio Colombi Ciacchi: supervision, interpretation of results, writing – review & editing. Dorothea Brüggemann: conceptualization of the experimental study, supervision, interpretation of results, writing – review & editing. Susan Köppen-Hannemann: conceptualization of the simulation study, supervision, interpretation of results, writing – review & editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data for this article are available at Zenodo https://doi.org/10.5281/zenodo.17780806. This entry includes all input and simulation setups, trajectories with a reduced sampling frequency by a factor of ten and all scripts used for the data analysis.

Supplementary information (SI): two SI files are provided. SI (part I) contains further details of the ECC approach, additional results of specific amino acid contacts, longer range investigations of the phosphate ion containing systems and additional information about the umbrella sampling approach (selection of windows, convergence profiles). SI (part II) contains reference results of the respective unscaled simulations (no ECC approach) analogous to the plots in the main manuscript. See DOI: https://doi.org/10.1039/d5me00229j.

Acknowledgements

DB and AEK thank Biotest AG for providing highly purified fibrinogen. ASM, LCC and SKH are grateful for provision of computational resources by the North German Supercomputing Alliance (HLRN) under project ID hbb00003. All authors acknowledge funding of the German Research Council (DFG) under grant no. 462381005, “SAL-FIB” and grant no. 514140860.

Notes and references

1. A. M. Kusova, A. E. Sitnitsky and Y. F. Zuev, Langmuir, 2021, 37, 10394–10401.
2. M. W. Mosesson, J. Thromb. Haemostasis, 2005, 3, 1894–1904.
3. J. Hermans and J. McDonagh, Semin. Thromb. Hemostasis, 1982, 8, 11–24.
4. Z. Yang, I. Mochalkin and R. F. Doolittle, Proc. Natl. Acad. Sci. U. S. A., 2000, 97, 14156–14161.
5. A. P. Laudano and R. F. Doolittle, Science, 1981, 212, 457–459.
6. G. Marguerie, G. Chagniel and M. Suscillon, Biochim. Biophys. Acta, 1977, 490, 94–103.
7. T. A. Ahmed, E. V. Dare and M. Hincke, Tissue Eng., Part B, 2008, 14, 199–215.
8. T. Rajangam and S. S. An, Int. J. Nanomed., 2013, 8, 3641–3662.
9. S. Li, X. Dan, H. Chen, T. Li, B. Liu, Y. Ju, Y. Li, L. Lei and X. Fan, Bioact. Mater., 2024, 40, 597–623.
10. B. Hämisch, A. Büngeler, C. Kielar, A. Keller, O. Strube and K. Huber, Langmuir, 2019, 35, 12113–12122.
11. S. Saha, A. Büngeler, D. Hense, O. I. Strube and K. Huber, Langmuir, 2024, 40, 4152–4163.
12. D. Hense, A. Büngeler, F. Kollmann, M. Hanke, A. Orive, A. Keller, G. Grundmeier, K. Huber and O. I. Strube, Biomacromolecules, 2021, 22, 4084–4094.
13. S. Stamboroski, A. Joshi, P.-L. M. Noeske, S. Köppen and D. Brüggemann, Macromol. Biosci., 2021, 21, e2000412.
14. K. Stapelfeldt, S. Stamboroski, P. Mednikova and D. Brüggemann, Biofabrication, 2019, 11, 025010.
15. D. Hense and O. I. Strube, Gels, 2023, 9, 175.
16. D. Hense and O. I. Strube, Gels, 2023, 9, 892.
17. S. Stamboroski, A. S. Malisetty, K. Boateng, J. Lierath, J. Aniol, P. Schiffels, P.-L. M. Noeske, L. Colombi Ciacchi, S. Köppen and D. Brüggemann, Biomacromolecules, 2025, 6755–6772.
18. R. Gollwitzer, W. Bode and H. E. Karges, Thromb. Res., 1983, 29, 41–53.
19. G. J. Stewart and S. Niewiarowski, Biochim. Biophys. Acta, 1969, 194, 462–469.
20. G. Köppel, Die Umwandlung des Fibrinogens in Fibrin : elektronenmikroskopische Untersuchungen zur Funktionsmorphologie des Fibrinogens, des Fibrins und der Thrombozyten beim spontanen Gerinnungsablauf im menschlichen Normalblut einschließlich der Retraktion des Koagulums, Schattauer, Stuttgart, 1962.
21. F. S. Steven, M. M. Griffin, B. S. Brown and T. P. Hulley, Int. J. Biol. Macromol., 1982, 4, 367–369.
22. A. E. Kwame, A. S. Malisetty, M. Maas, L. Colombi Ciacchi, S. Köppen-Hannemann and D. Brüggemann, Macromol. Biosci., 2026, 26(6), e70203.
23. Z. Adamczyk, A. Bratek-Skicki, P. Dąbrowska and M. Nattich-Rak, Langmuir, 2012, 28, 474–485.
24. T. S. Tsapikouni, S. Allen and Y. F. Missirlis, Biointerphases, 2008, 3, 1–8.
25. M. Kenny, S. Stamboroski, R. Taher, D. Brüggemann and I. Schoen, Adv. Healthcare Mater., 2023, 11, e2200249.
26. A. Joshi, T. Nuntapramote and D. Brüggemann, ACS Omega, 2023, 8, 8650–8663.
27. S. L. Meermeyer, A. Joshi, C. von Kaisenberg, D. Brüggemann and C. Lee-Thedieck, Adv. Healthcare Mater., 2025, e03449.
28. S. Stamboroski, K. Boateng, J. Lierath, T. Kowalik, K. Thiel, S. Köppen, P.-L. M. Noeske and D. Brüggemann, Biomacromolecules, 2021, 22, 4642–4658.
29. J. N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 2nd edn, 1991.
30. S. Salgın, U. Salgın and S. Bahadır, Int. J. Electrochem. Sci., 2012, 7, 12404–12414.
31. A. M. Smith, A. A. Lee and S. Perkin, J. Phys. Chem. Lett., 2016, 7, 2157–2163.
32. F. Hofmeister, Arch. Exp. Pathol. Pharmakol., 1888, 24, 247–260.
33. B. Kang, H. Tang, Z. Zhao and S. Song, ACS Omega, 2020, 5, 6229–6239.
34. H. I. Okur, J. Hladílková, K. B. Rembert, Y. Cho, J. Heyda, J. Dzubiella, P. S. Cremer and P. Jungwirth, J. Phys. Chem. B, 2017, 121, 1997–2014.
35. O. Matsarskaia, F. Roosen-Runge and F. Schreiber, ChemPhysChem, 2020, 21, 1742–1767.
36. H. Zhao, J. Chem. Technol. Biotechnol., 2016, 91, 25–50.
37. S. Ghosh and A. Kundagrami, J. Chem. Phys., 2024, 160, 084909.
38. M. Muthukumar, J. Chem. Phys., 2004, 120, 9343–9350.
39. N. Malikova, S. Čebašek, V. Glenisson, D. Bhowmik, G. Carrot and V. Vlachy, Phys. Chem. Chem. Phys., 2012, 14, 12898–12904.
40. Y. D. Gordievskaya, Y. A. Budkov and E. Y. Kramarenko, Soft Matter, 2018, 14, 3232–3235.
41. J. Riedel, L. Dierker, A. E. Kwame, K. N. Wagner, J. J. Unterholzner, J.-H. Dirks and D. Brüggemann, Adv. Eng. Mater., 2025, e202502218.
42. A. Henschen, F. Lottspeich, M. Kehl and C. Southan, Ann. N. Y. Acad. Sci., 1983, 408, 28–43.
43. M. S. Kostelansky, L. Betts, O. V. Gorkun and S. T. Lord, Biochemistry, 2002, 41, 12124–12132.
44. R. Anandakrishnan, B. Aguilar and A. V. Onufriev, Nucleic Acids Res., 2012, 40, W537–W541.
45. J. Myers, G. Grothaus, S. Narayanan and A. Onufriev, Proteins: Struct., Funct., Bioinf., 2006, 63, 928–938.
46. J. C. Gordon, J. B. Myers, T. Folta, V. Shoja, L. S. Heath and A. Onufriev, Nucleic Acids Res., 2005, 33, W368–W371.
47. Properties of Amino Acids, in CRC Handbook of Chemistry and Physics, Internet Version 2005, ed. D. R. Lide, CRC Press, Boca Raton, FL, 2005.
48. V. C. Yee, K. P. Pratt, H. C. Côté, I. L. Trong, D. W. Chung, E. W. Davie, R. E. Stenkamp and D. C. Teller, Structure, 1997, 5, 125–138.
49. S. J. Everse, G. Spraggon, L. Veerapandian, M. Riley and R. F. Doolittle, Biochemistry, 1998, 37, 8637–8642.
50. S. J. Everse, G. Spraggon, L. Veerapandian and R. F. Doolittle, Biochemistry, 1999, 38, 2941–2946.
51. S. Ghosh, A. Chowdhury, D. T. Tomares, B. Schuler, A. Kundagrami and R. V. Pappu, J. Chem. Phys., 2025, 163, 194908.
52. D. Beyer, C. Holm and Z.-G. Wang, J. Phys. Chem. B, 2026, 130, 4325–4332.
53. M. J. Fossat, A. E. Posey and R. V. Pappu, ChemPhysChem, 2023, 24, e202200746.
54. M. Abraham, A. Alekseenko, C. Berg, C. Blau, E. Briand, M. Doijade, S. Fleischmann, B. Gapsys, G. Garg, S. Gorelov, G. Gouaillardet, A. Gray, M. E. Irrgang, F. Jalaypour, J. Jordan, C. Junghans, P. Kanduri, S. Keller, C. Kutzner, J. A. Lemkul, M. Lundborg, P. Merz, V. Miletić, D. Morozov, S. Páll, R. Schluz, M. Shirts, A. Shvetsov, B. Soproni, D. van der Spoel, P. Turner, C. Uphoff, A. Villa, S. Wingbermühle, A. Zhmurov, P. Bauer, B. Hess and E. Lindahl, GROMACS 2023.3 Manual, Zenodo, 3rd edn, 2023.
55. J. Sunhwan, T. Kim, V. G. Iyer and W. Im, J. Comput. Chem., 2008, 29, 1859–1865.
56. J. Lee, X. Cheng, J. M. Swalis, M. S. Yeom, P. K. Eastman, J. A. Lemkul, S. Wei, J. Buckner, J. C. Jeong, Y. Qi, S. Jo, V. S. Pande, D. A. Case, C. L. Brooks III, A. D. MacKerell Jr., J. B. Klauda and W. Im, J. Chem. Theory Comput., 2016, 12, 405–413.
57. J. Lee, M. Hitzenberger, M. Rieger, N. R. Kern, M. Zacharias and W. Im, J. Chem. Phys., 2020, 153, 035103.
58. S. Kashefolgheta and A. Vila Verde, Phys. Chem. Chem. Phys., 2017, 19, 20593–20607.
59. W. Humphrey, A. Dalke and K. Schulten, J. Mol. Graphics, 1996, 14, 33–38.
60. N. Michaud-Agrawal, E. J. Denning, T. B. Woolf and O. Beckstein, J. Comput. Chem., 2011, 32, 2319–2327.
61. R. J. Gowers, M. Linke, J. Barnoud, T. J. E. Reddy, M. N. Melo, S. L. Seyler, J. Domański, D. L. Dotson, S. Buchoux, I. M. Kenney and O. Beckstein, Proceedings of the 15th Python in Science Conference, 2016, pp. 98–105.
62. E. Jurrus, D. Engel, K. Star, K. Monson, J. Brandi, L. E. Felberg, D. H. Brookes, L. Wilson, J. Chen, K. Liles, M. Chun, P. Li, D. W. Gohara, T. Dolinsky, R. Konecny, D. R. Koes, J. E. Nielsen, T. Head-Gordon, W. Geng, R. Krasny, G. W. Wei, M. J. Holst, J. A. McCammon and N. A. Baker, Protein Sci., 2018, 27, 112–128.
63. I. V. Leontyev and A. A. Stuchebrukhov, J. Chem. Phys., 2009, 130, 085102.
64. E. Duboué-Dijon, M. Javanainen, P. Delcroix, P. Jungwirth and H. Martinez-Seara, J. Chem. Phys., 2020, 153, 050901.
65. I. V. Leontyev and A. A. Stuchebrukhov, J. Chem. Theory Comput., 2010, 6, 1498–1508.
66. A. Catte, M. Girych, M. Javanainen, C. Loison, J. Melcr, M. S. Miettinen, L. Monticelli, J. Määttä, V. S. Oganesyan, O. H. S. Ollila, J. Tynkkynen and S. Vilov, Phys. Chem. Chem. Phys., 2016, 18, 32560–32569.
67. D. A. Tolmachev, O. S. Boyko, N. V. Lukasheva, H. Martinez-Seara and M. Karttunen, J. Chem. Theory Comput., 2020, 16, 677–687.
68. J. Melcr and J.-P. Piquemal, Front. Mol. Biosci., 2019, 6, 143.
69. M. Kohagen, P. E. Mason and P. Jungwirth, J. Phys. Chem. B, 2016, 120, 1454–1460.
70. I. S. Joung and T. E. Cheatham, J. Phys. Chem. B, 2008, 112, 9020–9041.
71. T. Simonson and C. L. Brooks, J. Am. Chem. Soc., 1996, 118, 8452–8458.
72. S. L. Chan and C. Lim, J. Phys. Chem., 1994, 98, 692–695.
73. J. Antosiewicz, J. A. McCammon and M. K. Gilson, J. Mol. Biol., 1994, 238, 415–436.
74. G. Gonella, E. H. G. Backus, Y. Nagata, D. J. Bonthuis, P. Loche, A. Schlaich, R. R. Netz, A. Kühnle, I. T. McCrum, M. T. M. Koper, M. Wolf, B. Winter, G. Meijer, R. K. Campen and M. Bonn, Nat. Rev. Chem., 2021, 5, 466–485.
75. M. J. Saxton, Biophys. J., 1997, 72, 1744–1753.
76. E. J. Maginn, R. A. Messerly, D. J. Carlson, D. R. Roe and J. R. Elliot, Living J. Comp. Mol. Sci., 2018, 1, 6324.
77. H. Kumar, C. Dasgupta and P. K. Maiti, RSC Adv., 2015, 5, 1893–1901.
78. I.-C. Yeh and G. Hummer, J. Phys. Chem. B, 2004, 108, 15873–15879.
79. K. R. Harris and L. A. Woolf, J. Chem. Eng. Data, 2004, 49, 1064–1069.
80. Y. Song and L. L. Dai, Mol. Simul., 2010, 36, 560–567.
81. M. Lai, M. Kalweit and D. Drikakis, Mol. Simul., 2010, 36, 801–804.
82. Z. Hai-Lang and H. Shi-Jun, J. Chem. Eng. Data, 1996, 41, 516–520.
83. J. B. Hasted, D. M. Ritson and C. H. Collie, J. Chem. Phys., 1948, 16, 1–21.
84. R. A. X. Persson, Phys. Chem. Chem. Phys., 2017, 19, 1982–1987.
85. H. Butt, K. Graf and M. Kappl, in The Electric Double Layer, Wiley, 2006, ch. 4, pp. 45–59.
86. M. Delepierre, C. M. Dobson, M. Karplus, F. M. Poulsen, D. J. States and R. E. Wedin, J. Mol. Biol., 1987, 197, 111–122.
87. G. Pack, G. Garrett, L. Wong and G. Lamm, Biophys. J., 1993, 65, 1363–1370.
88. M. F. Sanner, A. J. Olson and J. C. Spehner, Biopolymers, 1996, 38, 305–320.
89. S. Lee, A. M. Tiara, G. Cho and J. Lee, Nanomaterials, 2022, 12, 2600.
90. I. A. Parfentjev, M.-L. Johnson and E. E. Cliffton, Arch. Biochem. Biophys., 1953, 46, 470–480.
91. D. Q. Tran, N. Stelflug, A. Hall, T. Nallan Chakravarthula and N. J. Alves, Biomolecules, 2022, 12, 1864.
92. M. E. Carr and J. Hermans, Macromolecules, 1978, 11, 46–50.
93. F. H. Silver and D. E. Birk, Collagen Relat. Res., 1983, 3, 393–405.
94. O. C. Gagné and F. C. Hawthorne, Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater., 2016, 72, 602–625.
95. K. Kiyohara and Y. Kawai, J. Chem. Phys., 2019, 151, 104704.
96. S. Saurabh, Q. Zhang, J. M. Seddon, J. R. Lu, C. Kalonia and F. Bresme, Mol. Pharmaceutics, 2024, 21, 1285–1299.
97. M. Kulke, M. Uhrhan, N. Geist, D. Brüggemann, B. Ohler, W. Langel and S. Köppen, J. Chem. Inf. Model., 2019, 59, 4383–4392.
98. D. R. Lide, Ionic Conductivity and Diffusion at Infinite Dilution, in CRC Handbook of Chemistry and Physics, Internet Version 2005, CRC Press, Boca Raton, FL, 2005.
99. I. S. Joung and T. E. Cheatham, J. Phys. Chem. B, 2009, 113, 13279–13290.
100. I. V. Leontyev and A. A. Stuchebrukhov, J. Chem. Theory Comput., 2012, 8, 3207–3216.
101. K. Krynicki, C. D. Green and D. W. Sawyer, Faraday Discuss. Chem. Soc., 1978, 66, 199.
102. A. C. Dumetz, A. M. Snellinger-O'brien, E. W. Kaler and A. M. Lenhoff, Protein Sci., 2007, 16, 1867–1877.

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