Introduction

Demand for biocidal polymers as coating materials that provide protection from the invasion of microbes is increasing. Surfaces contaminated with pathogenic organisms are vectors for the spread of pathogens, and affect those most vulnerable to infection. Quaternary ammonium (QA) compounds are used in a variety of commercial applications ranging from cosmetic preservatives to hospital disinfectants and sanitizers. They have many advantages over other biocides (e.g., phenols and aldehydes), including broad-spectrum antimicrobial activity, effectiveness over a wide pH range, low human toxicity, and lack of a detectable odour1. However, QA-based antimicrobial coatings are unsustainable owing to their amphiphilic properties, which presumably accelerate their liberation from surfaces and their immediate dispersion into various organic solvents and water. Moreover, they are small molecules (<1000 Da). The development of hierarchical assemblies of QAs with chemical durability (e.g., water-resistance) is therefore a promising yet challenging field of material nanotechnology.

Ashiuchi et al.2 first developed a stoichiometric ion complex comprising equal proportions of carboxyl groups from archaeal poly-γ-l-glutamate (l-PGA) and hexadecylpyridinium cations (HDP+) derived from QA compounds used in toothpaste. They called this self-assembling material l-PGA ion complex (PGAIC). PGAIC has a broad spectrum of antimicrobial activity (against food-poisoning bacteria, a prevalent species of Candida, filamentous fungi, and Influenza viruses)2,3, and has potential as a bio-based plastic. This new material could be useful in food-related engineering and pharmaceutics. However, the solvation assays developed for PGAICs2,3 also highlight the importance of improving their durability, in particular, moderate resistance against organic solvents including alcohols (e.g., ethanol) and chloroform. We herein present a new strategy to introduce multiple non-covalent crosslinkages into the essential architecture of PGAIC to develop engineered materials with excellent chemical durability. A sustainable PGAIC material was actually synthesized using a bis-QA (quinolinium) compound used in medicinal lozenges, called dequalinium cations (DEQ2+). In this article, the potential of DEQ-bound PGAIC (PGA/DEQ) materials as durable antimicrobial coatings is compared with that of traditional HDP-bound PGAIC (PGA/HDP) materials.

Results

Development of PGA/DEQ, a novel PGAIC material with non-covalent crosslinkages

In the present study, several crosslinker candidates were used to synthesize PGAICs (Fig. 1). Diamine-type compounds (e.g., hexamethylene (viz., aliphatic; panel a) and phenylene (viz., aromatic; b) diamines) did not meet the requirements for PGAIC materials, whereas N, N’-hexamethylenebis (4-carbamoyl-1-decylpyridinium) (BDP2+, a bis-pyridinium compound) was applicable (c), as was HDP+ (a mono-pyridinium compound)2,3. DEQ2+ (a bis-quinolinium compound) was useful as well (d), but mono-quinolinium compounds (e and f) were not preferred for the formation of water-resistant complexes. This implies unexpected performance (e.g., extreme solvent resistance and sustainable functionality) in PGA/DEQ as a novel PGAIC product with multiple, non-covalent crosslinkages. The “soakage” tests for PGA/DEQ actually indicated a dramatic improvement in the resistance of PGAIC materials to all the organic solvents tested (Table 1). In contrast, another new product called PGA/BDP has yet to overcome the concerns about chemical stability that apply to traditional PGAIC products (e.g., PGA/HDP), suggesting that quinolinium-based crosslinkers are more beneficial than pyridinium-based crosslinkers in developing durable PGAIC materials.

Figure 1
figure 1

Development of poly-γ-L-glutamate (PGA) ion complexes (PGAICs) with multiple, non-covalent crosslinkages. In the reaction mixtures, the initial concentration (mg/mL) of PGA was 4.0 ± 0.2, indicating the presence of the anionic (carboxy) groups at ~31 mM, and the supply of crosslinker candidates (a), 1,6-diaminohexane; (b) p-phenylene diamine; (c) BDP2+; (d) DEQ2+; (e) quinolinium dichromate; and (f) quinoline ethiodide) was controlled so that the concentrations (mM) of their cationic (amine) groups reached the same levels as the anionic groups. The increases in turbidity virtually implied the accumulation of water-insoluble PGAICs.

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Table 1 Comparison of the solvation properties of bis-quinolinium- and mono- and bis-pyridinium-bound materials of PGAIC.
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Structural features of PGA/DEQ

We will now describe the effective synthesis and nuclear magnetic resonance (NMR) analysis of PGA/DEQ. After its synthesis, little L-PGA (<10 wt% of the initial amounts; n = 5) was detected in the aqueous phase of the reaction mixtures using published procedures4, confirming that L-PGA was almost completely converted to a water-insoluble form of PGAIC by interacting with DEQ2+. By 13C cross polarization and magic angle spinning (13C CP-MAS) NMR spectroscopy method, we successfully characterized the chemical shifts that correspond to L-PGA and DEQ2+ from the spectrum of PGA/DEQ (Fig. 2, panel a), while in particular the corresponding peaks to DEQ2+ were much broader than those of DEQ2+ itself (Supplementary Fig. 1a), suggesting development in solid-state PGA/DEQ of the molecular interaction between DEQ2+ (as smaller) and L-PGA (as higher moieties). Next, the proton signals found in b were assigned as follows: bis-quinolinium rings (chemical shifts 2 Ha and 2(Hb–He), 6.88 and 8.30–7.78; relative intensities, 1.00 and total 3.18), αCH-PGA (Hα, 4.72; 0.92), bis-quinolinyl CH2 (2 Hf, 4.55; 2.10), bis-quinolinyl CH3 (2CH3, 2.83; 3.17), βCH2-PGA (Hβ, 2.63; 1.82), γCH2-PGA (Hγ, 2.40–2.20; total 1.84), CH2-DEQ2+ (2 Hg, 1.95; 2.05), and (CH2)3-DEQ2+ (2(Hh–Hj), 1.57–1.38; total 6.18). Hence, PGA/DEQ was determined to be a stoichiometric ion complex comprising equal proportions of the carboxyl (viz., negatively-charged) groups of PGA and the quinolinium (viz., positively-charged) moieties of DEQ2+, which also demonstrates the feasibility of DEQ2+ as a potent crosslinking reagent for synthesizing durable PGAICs.

Figure 2
figure 2

(a) 13C NMR and (b) 1H NMR spectra of PGA/DEQ; the insets illustrate assigned PGA ion complex (PGAIC) structures.

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Thermal characteristics of PGA/DEQ

The calorimetric measurements of PGA/DEQ (Fig. 3) revealed that the minor and major melting points (Tms) were 60.1 °C (transition enthalpy, ΔH = 1.896 J/g) and 131.9 °C (219.1 J/g), respectively (panel a, solid line). PGA/DEQ, virtually a cross-linked polymer product, can retain its potential as a thermoplastic. We also found that the major and minor Tms of DEQ2+ were 130.7 °C (ΔH = 114.3 J/g) and 168.6 °C (8.679 J/g), respectively (a, dotted line). Interestingly, the ΔH value of PGA/DEQ (weight ratio, 1: 4) was nearly twice that of authentic DEQ2+, and was comparable with that of 100% crystalline chemically-synthesized nylons (e.g., 241 J/g of α-phase and 239 J/g of γ-phase)5. The Tm of PGA/HDP, however, was only ~60 °C (ΔH = 74.08 J/g)2: close to that of the former (or minor) Tm of PGA/DEQ. This may suggest the ubiquitous thermal manner in archaeal L-PGA-based PGAICs, while it seems likely that the former (or major) Tm of DEQ2+ mainly involves the thermal plasticity of PGA/DEQ. A recent article described the comparatively high Tm (157 °C) yet small ΔH (2.3 J/g) values of a poly-ε-L-lysine ion-complex6. PGA/DEQ is, to the best of our knowledge, one of the most heat-resistant bio-based chiral nylon (or isopeptide)7 ion complex materials developed to date. Figure 3 also showed that the decomposition (Td) of PGA/DEQ started at 262.1 °C (panel b). In contrast to DEQ2+ (dotted line), the weight of PGA/DEQ decreased in two steps during the rise in temperature (solid line), implying that the configuration of theoretical PGA/DEQ oligomers (or digests) accumulated in the late stage of decomposition resulted in a superior heat resistance to that of DEQ2+ itself.

Figure 3
figure 3

Calorimetric investigations of PGA/DEQ. Panels (a) differential scanning calorimetry (DSC) and (b) thermogravimetry (TG) profiles of PGA/DEQ (─) and authentic DEQ2+ (……).

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Characterization of microstructures in PGA/DEQ

The thermal properties of plastics essentially depend on the crystallinities of the materials, and the X-ray diffraction (XRD) analysis provides an insight into the structural features of the crystalline moieties of polymers. Figure 4 describes the wide-angle X-ray diffraction (WAXD) diagram of PGA/DEQ (panel a, solid line); a curve fitting model8,9 was further applied to separate the crystalline diffraction pattern (dotted line) from amorphous scattering (broken line). The crystallinity of PGA/DEQ was then calculated to be ~45% by dividing the integrated intensity of crystalline areas by the overall scattering intensity. It is also noteworthy that multiple (at least five) harmonic peaks were detected from 5 to 10 nm−1 of the q values in addition to the major crystalline peak at 16 nm−1. These extraordinary peaks indicate self-oriented assemblies of the crystalline moieties, specifically in the PGA/DEQ microstructure, because corresponding peaks were absent from the original microstructures of both PGA and DEQ2+ (Supplementary Fig. 2). Small-angle X-ray scattering (SAXS) analysis was performed to determine the evolved crystalline structures (Fig. 4, panel b). The profile of observed peaks revealed the presence of a peculiar harmonic, which does not correspond to any of the harmonics previously identified, including those of typical lamellar structures10,11,12. Hereafter, we will refer to the structures found in the PGA/DEQ microstructures as atypical crystalline assemblies.

Figure 4
figure 4

XRD analysis of PGA/DEQ. Panels (a) WAXD (, , crystalline diffractions; , amorphous scattering) and (b) SAXS diagrams.

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Performance of PGA/DEQ as antimicrobial coatings

As shown in Fig. 5–panel A, PGA/HDP coatings (black bars; columns b) on the plastic surfaces (a) were stable during water-soaking; however, they were readily removed by the indicated organic solvents (c and d) and concentrated salt solutions (e–g). In contrast, the PGA/DEQ coatings (white bars) on the same materials (a) had excellent durability against all the solvents (or chemicals) tested (b–g). The morphology of PGAIC-coated microfibers on these plastic surfaces was further observed using a scanning electron microscope (SEM) (Supplementary Fig. 3). The transformation of PGAIC into durable structures led to the further sustainable functionality of the antimicrobial coatings. The antimicrobial performance of the PGA/HDP coatings (Fig. 5–panel B, black bars) disappeared after soaking in the indicated solvents (c–g) other than water (b), whereas a PGA/DEQ-coated disk (Fig. 5B) resulted in the dramatic suppression of microbial proliferation (white bars) under any treatment conditions (b–g; Supplementary Fig. 4). Compared with the normal proliferation of Escherichia coli (~1.3 × 109 colony-forming units (CFU)), the log-reduction scores of PGA/HDP and PGA/DEQ coatings (after treatment) were estimated to be almost zero and approximately nine, respectively. These PGAIC coatings could, therefore, contribute to the microstructure-based polymer engineering of antimicrobial (bioactive) coatings.

Figure 5
figure 5

(A) Chemical durability and (B) functional sustainability of PGA/DEQ (▯) and PGA/HDP (▮) coatings. The basal (a) or a PGAIC-coated (b) disk was soaked into 2 mL of the indicated solvents (c, EtOH; d, CHCl3; e, 1.5%; f, 3.0%; and g, 5.0% NaCl) and incubated at room temperature while gently shaking for 30 min. This “soaking” process was repeated ten times per disk, though almost of the PGA/HDP coatings are usually removed after the first “soaking”, in particular using (c) or (g). The resulting disks were washed three times each with 2 mL of distilled water and finally dried at 60 °C for 1 h. The immobilized QAs and log-reduction scores (n = 15 each) were determined, based on the spectroscopic characteristics of BPBICs formed onsite (see also Supplementary Figs 10–12) and the enumeration of living microbial cells, respectively (see the Method section).

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Controllable removability of PGA/DEQ coatings

Further investigation on the sustainability of PGA/DEQ as antimicrobial coatings (viz., the pH-response tests) accidentally provided a possible strategy for the controlled removal (or enforced termination) of PGA/DEQ from the modified surfaces. Our data (Supplementary Fig. 5) implied the significance (or specificity) of citrate di-salts (theoretical pKa2 = 4.75). In fact, highly concentrated citrate di-salts were the best candidate mediators of PGA/DEQ removal (Fig. 6), whereas other electrochemical forms, e.g. citrate mono- (pKa1 = 3.09) and tri-salts (pKa3 = 6.41), were essentially ineffective. Furthermore, we used quantitative assays to verify that the PGA/DEQ molecules were exclusively eliminated following soaking in the concentrated di-salts (see the Methods section). Crosslinked polymer composites (e.g., thermoset matrices) are usually difficult to recycle13,14. Further, the strategies for their downstream (waste) treatment are also limited. Finding a means of controllably removing (and possibly recycling) PGA/DEQ will help to enhance its value as a functional supramolecular polymer15 for versatile prophylactic coatings (or plastics/fibres)2,3,16.

Figure 6
figure 6

Controlled removal of PGA/DEQ coatings using electrochemically different forms of citrate. The “soaking” processes and the determination of log-reduction scores (n = 3) were essentially conducted according to the investigations described in Fig. 5. The insets illustrate the predicted structure of the citrate salts used, and the approximate pH values of their solutions can be found in parentheses below. Concentrations (a), 5; (b), 50; and (c), 100 mM.

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PGAIC-immobilized surfaces carrying potent microbicidal mechanism

A poly-ε-L-lysine ion complex had a dramatically reduced antimicrobial activity compared with native poly-ε-L-lysine6, but we found that the PGAIC solutions were quite similar to solutions of the corresponding partner surfactants in an assessment of minimal inhibition concentrations (MICs)3. In the present study, the average amount of PGA/DEQ immobilized on the surfaces of a basal disk was determined to be 17.6 μg; if the PGA/DEQ coatings were all released in 5 mL of the culture medium, the final concentration had the potential to reach 3.52 ppm. Because the concentration of the liberated DEQ2+ moieties was ~22.2 nmol (via complete decomposition of the coatings), the maximal concentration was theoretically calculated to be 4.44 μM (or 2.35 ppm). Possibly, there is a microbicidal factor that operates at significantly lower doses than the MIC values for E. coli cells in the original (free) DEQ solutions (Supplementary Table 1), as we have observed the potent, sustainable microbicidal performance of PGA/DEQ-coated surfaces (Supplementary Figs 6 and 7). In the next stage of PGAIC research, it may be interesting to elucidate such peculiar and potent molecular strategies by investigating them on solid interfaces (Supplementary Fig. 8).

Discussion

Using a fine (sustainable) polymer from extremophiles, we have succeeded in developing a novel antimicrobial coating, showing the antithetical qualities of extreme durability and briefly controllable removability. This archaeal polymer, namely stereoregular PGA, possesses a peculiar functionality that enables it to bind cooperatively to various cationic compounds such as some critical metals17 and organic surfactants (Supplementary Fig. 9).

PGAICs can be transformed into versatile nanofiberplastics2 and safe dispersants to create antimicrobial surfaces, and they may be used to contribute to hygienic control and infection prophylaxis in various public facilities such as schools, hospitals, hotels, and transportation3, resulting in a decreased risk of airborne infections, contagions, and serious pneumonia, which are often fatal. Engineering advanced functional (e.g., controlled capture–killing) coatings would eventually prevent unforeseen epidemics caused by new contagions.

Methods

Materials

Archaeal PGA was isolated from a Natrialba aegyptiaca culture medium2,3. Two healthcare compounds (or cationic surfactants) suitable for PGAIC synthesis, i.e., HDP+ and DEQ2+, were purchased from the Tokyo Chemical Industry (Japan) and Sigma Co. (St. Louis, MO, USA), respectively., Salts of BDP2+ were obtained from Wako Pure Chemicals (Osaka, Japan) because it is a commercially available bis-QA (pyridinium) compound18. All other chemicals were of analytical grade.

PGAIC synthesis

PGA/DEQ was synthesized by mixing a 2 wt% L-PGA solution with four times the volume of 1 wt % DEQ2+ solution (corresponding to approximately 48 mol% of the carboxyl groups contributed by the PGA molecules). Pellets were collected by centrifugation at 12000 g for 10 min, then washed several times with distilled water at 80 °C, soaked in pure methanol (>99.8 wt% MeOH) for 1 h at room temperature to remove the accidentally liberated DEQ2+, and lyophilized. The other (traditional) products of PGAIC (e.g., PGA/HDP) were prepared according to published procedures2,3.

NMR spectroscopy

Solid-state analysis

13C CP-MAS NMR spectroscopy was performed on the lyophilized samples (50 mg each) using a Bruker DSX300 spectrometer (La Jolla, CA, USA) operated at 75.5 MHz. We scanned 40000 and 28583 times at 5.5 kHz for DEQ2+ and PGA/DEQ, respectively.

Liquid-state analysis

A lyophilized PGAIC sample (10 mg) was dissolved in 1 mL of di-deuterated formic acid (DCO2D), and analysed using a Bruker AVANCE500 spectrometer operated at 500 MHz. The NMR signals were assigned based on the chemical shifts (ppm), their relative intensities, and coupling patterns, and further comparisons with the signals of indispensable (or starting) materials: L-PGA (refer to ref.2) and DEQ2+ (see Supplementary Fig. 1).

Calorimetric measurements

DSC and TG analysis of lyophilized PGA/DEQ (~5 mg) were performed at a heating rate of 10 °C/min under a nitrogen atmosphere using Seiko DSC EXTRA6000 and SSC5200 equipment (Chiba, Japan), respectively.

XRD analysis

WAXD and SAXS measurements were used to decipher the short-range order and the large-scale structure of the crystalline regions of PGA/DEQ, respectively. WAXD was carried out on a Rigaku SmartLab system (Tokyo, Japan). Because the incident X-ray beam from a line focus tube (Cu, operating at 40 kV and 30 mA) is generally divergent, it was transformed into an intense parallel Kβ radiation-free beam using a Göbel mirror. This beam (from an incident 5° Soller slit against a 10-mm longitudinal limits slit) was achieved using an equatorial axial 0.114° Soller slit in the secondary beam path, and the scattering angle 2θ varied from 3° to 90° in reflection (θ–2θ configuration) with a step of 0.01° and a speed of 5°/min. For SAXS, we operated a Rigaku NANO-Viewer IP instrument at 40 kV and 30 mA, the flight path and the sample chamber of which were under vacuum. The Cu Kα radiation (from three pinhole slits) had a wavelength (λ) of 1.54 Å, and the sample-to-detector distance was estimated to be 368 mm when silver behenate was used as a distance calibration standard. The scattered intensity I (q) was recorded in the range 0.08 < q < 8.0 nm−1, where q is the scattering vector defined as q = (4π/λ) sin (θ). During this XRD analysis, the PGA/DEQ sample was investigated without thermal treatment.

Coating of PGA/DEQ on plastic surfaces

A recent study dealt with the performance of PGA/HDP as a widely applicable adhesive on the surfaces of versatile plastics (and other materials), in which additional chemical modifications (e.g., non-biodegradable crosslinking) and thermal treatments (e.g., heating) were omitted from the coating process3. Hereafter (in situations that require the further strengthening of PGAIC coatings)2,3, it is urgently necessary to demonstrate the potential of PGA/DEQ as an antimicrobial coating compared with commercially available (or ready-made) plastic materials (e.g., HIYEX non-woven cloths). The kinetics of PGAIC synthesis revealed the occurrence of potent cooperative interaction between PGA and the QA-type surfactants to form the stoichiometric ion complexes (Supplementary Fig. 9). In the present experiment, the direct formation of PGAIC (viz., PGA/DEQ and PGA/HDP) coatings on the material surfaces was performed using a modified layer-by-layer19 (or onsite coating) method (Supplementary Fig. 10A). First, we coated 40 μL of ~ 0.12 wt % L-PGA solution onto the surface of a disk (diameter 12 mm) of HIYEX HO-503M non-woven plastic cloth (or sheet) (Kuraray, Tokyo, Japan), and dried it at 28 °C until its weight had been reduced by almost half. The PGA-mounting disks were immersed briefly in warmed solutions of QA (1 mL/disk), i.e., DEQ2+ (1 wt%; 80 °C) and HDP+ (3 wt %; 60 °C), and dried at 60 °C for 1 h to introduce PGAIC assemblies onsite. Based on the differences in the solvation properties between the QAs and their corresponding PGAICs (Table 1), excess unbound QAs were thoroughly removed from the surfaces using methanol and dimethyl sulfoxide (DMSO), respectively (Supplementary Fig. 6A, step d). Because of the excellent water resistance of the PGAIC coatings2,3,16, the disks were soaked in distilled water five times (2 mL/disk) at 28 °C, dried again at 60 °C, and used as PGAIC-coated materials.

Scanning Electron Microscopy

A Nihon Denshi Field Emission Scanning Electron Microscope JSM 6700 F (Tokyo, Japan) was operated with an acceleration voltage of 5 kV to produce the indicated SEM images.

Investigation of surface-immobilized PGAIC coatings

Bromophenol blue (BPB; Nakarai, Kyoto, Japan) is a water-soluble (anionic) dye that enables the visualization of various quaternary ammonium compounds20,21. A BPB-staining method3 was modified to quantify QA moieties immobilized on the surfaces (Supplementary Fig. 10B). PGAIC-coated materials were immersed in 0.04 w/v % BPB solution (1 mL/disk) at 28 °C for 10 min (step e), washed five times with running water (5 mL/disk) to remove unbound BPB (step f), and dried at 60 °C. The materials were dyed blue because water-insoluble complexes comprising BPB anions and QA cations (viz., DEQ2+ and HDP+) were formed onsite (Supplementary Fig. 11). Hereafter, this blue compound will be referred to as BPB ion complex (BPBIC); it is readily soluble in methanol (Supplementary Fig. 10, step g), and the spectrophotometry of BPBIC can be used for the quantitative analysis of surface-immobilized PGAIC coatings.

Spectrophotometric assay of BPBICs

Water-insoluble BPBICs were immediately formed in a mixture comprising BPB solution (0.04 wt%; at neutral pH) and a concentrated solution that contained virtually half-molar amounts of DEQ2+ or equimolar amounts of HDP+. Pellets were collected by centrifugation, and washed six times with water at 60 °C. The lyophilized BPBICs were then dissolved in methanol (>99.5 wt %), and the spectrophotometry of the resulting solution was conducted using a GE Healthcare GeneQuant 100 instrument (Tokyo, Japan) to obtain the absorption spectra of the DEQ-bound BPBIC (BPB/DEQ) and the HDP-bound BPBIC (BPB/HDP) (Supplementary Fig. 12, panel a). The calibration curves of BPB/DEQ and BPB/HDP were then constructed based on the absorbance at 590 nm as their common maximum absorption wavelength (b and c). Because the predicted molecular masses (Da) of BPB/DEQ and BPB/HDP were 1796.6 and 974.51, respectively, the molar extinction coefficients (M−1·cm−1) of BPB/DEQ and BPB/HDP were calculated to be 8.99 × 104 and 3.44 × 104, respectively.

Assessment for antimicrobial performance

Although PGA is virtually non-toxic to humans and even pathogenic microbes, PGAICs exhibit a “broad-spectrum” antimicrobial activity with regard to Influenza viruses and fungi2,3, suggesting that the desirable performance of PGAIC is dependent on the potential of its corresponding QA moieties (e.g., HDP+ in PGA/HDP and DEQ2+ in PGA/DEQ). The antimicrobial spectrum of DEQ2+ is quite extensive (Supplementary Table 1), and is essentially the same as that of HDP+ 2,3. Herein, we focused on the observation that the anti-E. coli activities of PGAICs are often moderate compared with their other antimicrobial activities, because the efficacy of PGAIC-coated surfaces against E. coli cells is profoundly influenced by the stability (or durability) of PGAIC as a coating. It is worth noting that E. coli is an environmental (water) pollution indicator species16. In the present experiment, viable E. coli cells (~1.7 × 105 CFU) were first inoculated into 5 mL of Luria–Bertani (LB) medium22, then a PGAIC-coated disk (see above) was introduced and the cells were cultured at 37 °C for 16 h. The culture suspension (0.1 mL) was spread onto a DAIGO agar (Nihon Pharmaceutical) plate and incubated at 37 °C for 12 h to determine the number of E. coli colonies, viz. the direct plating method for the enumeration of living cells16,23,24. When >300 colonies had formed on the plate, the samples were additionally diluted by 10 times and the enumeration test was repeated. The antimicrobial performance of the PGAIC coatings was assessed using log-reduction scores, which can be calculated using the following equation: log10 x/y, where x and y correspond to the counts of viable cells in the LB cultures harbouring the original (thus non-coated) and PGAIC-coated disks, respectively.