# Stochastic Dynamic Mass Spectrometric Quantitative and Structural Analyses of Pharmaceutics and Biocides in Biota and Sewage Sludge

## Abstract

**:**

^{−1}and collision energy CE = 5–60 V are provided. Quantitative analysis of surfactants in biota via calibration equation ln[D″

_{SD}] = f(conc.) yields the exact parameter |r| = 0.9999

_{1}, examining the peaks of BAC-C12 at m/z 212.209 ± 0.1 and 211.75 ± 0.15 for tautomers of fragmentation ions. Exact parameter |r| = 1 has been obtained, correlating the theory and experiments in determining the 3D molecular structures of ions of paracetamol at m/z 152, 158, 174, 301, and 325 in biota.

## 1. Introduction

^{−1}[1,14,15]. However, quaternary ammonium surfactants are not only toxic to environmental organisms, but also exhibit the proliferation of antibiotic resistance [16]. In 2019, there were 4.95 million antibiotic-resistant pathogens around the world, which has been considered as a silent pandemic [10]. The therapeutics’ abuse is a factor contributing to antibiotic resistance. It remains not well understood, but is recognized to be among the most pressing global environmental and human health problems [17,18]. It has been found that by 2050, approximately ten million people shall die, annually, as a result of antibiotic resistance [19]. Despite this, many quats play a vital role in biochemical pathways [20]. Benzalkonium chlorides (Figure 1) belong to the group of quaternary ammonium surfactants, and are extensively utilized for various industrial and domestic purposes [10,21,22], including contact lens solutions or eye drops [23,24,25]. For this reason, BAC-C14 is the most frequently detected biocide among benzalkonium chlorides in engineered and environmental systems [16]. Their presence in wastewater upset has led to activated sludge processes [26,27]. Benzalkonium chloride derivatives in sludge are capable of biodegrading during biological wastewater treatment or adsorbing onto a biomass. Their biodegradation paths have been examined [28,29,30]. The presence of benzalkonium chloride disinfectants in the environment promotes the abundance and diversity of antibiotic resistance genes in sewage sludge microbiomes [10]. Antibiotics, as surfactants and environmental pollutants, represent a widespread concern and ecotoxicological risk [23,24,25,31,32].

_{SD}= f(D

_{QC}) has shown |r| = 0.9999

_{4}[65].

## 2. Results and Discussion

#### 2.1. Mass Spectrometric Data

#### 2.1.1. Mass Spectrometric Fragmentation Reactions of Paracetamol and Its d_{3}-Derivative

^{+}shows fragmentation path CID (m/z 303)→303, 152 [69,75]. The ammonium adduct [2M+NH

_{4}]

^{+}exhibits a low-abundance peak at m/z 320 [75]. MS analysis of PARA radical-cations has been reported [72]. Fragmentation processes, involving a radical-cation mechanism of bond cleavage, have been proposed [77]. PARA tends to stabilize not only the Cu

^{2+}adduct, but also adducts of alkali metal ions and NH

_{4}

^{+}cation. There are species of type [M+NH

_{4}]

^{+}(m/z 169), [M+Na]

^{+}(m/z 174), [M+K]

^{+}(m/z 190), [2M+NH

_{4}]

^{+}(m/z 320), [2M+Na]

^{+}(m/z 325), and [2M+K]

^{+}(m/z 341), respectively [79]. As Figures S5–S8 reveal, the abundance of peaks depends on the applied voltage, presence of formic acid, and analyte concentration. The same is true for the peak of protonated analyte [M+H]

^{+}and its major fragmentation product of N–C bond cleavage, [M-CH

_{3}CHC=O]

^{+}, at m/z 152 and 110. The data on d

_{3}-PARA show similar fragmentation patterns together with some adducts. The peak at m/z 331 belongs to [2.d

_{3}-M+Na]

^{+}. There is an observed fragmentation reaction CID (m/z 152)→152, 110, 93 [78]. MS spectra of PARA in a sludge cake and biota (Figure 3, Figures S9, S10 and S13) show competitive fragmentation mechanisms causing not only charged cations, but also cation radicals.

#### 2.1.2. Mass Spectrometric Fragmentation Reactions of Biocides

^{+}. The major fragmentation path shows the loss of 92 Da of toluene (Figure S11). Regarding the MS spectra of PARA and its d

_{3}-PARA derivative (Figure S12), depending on the experimental conditions, there are competitive fragmentation reactions causing more than one conformational and tautomeric form of product ions (Figures S9 and S10). Although the major fragmentation MS path of these surfactants is associated with the loss of the hydrophilic head of the compounds, the product ion consisting of a charged hydrophobic tail exhibits a complex conformational preference and electronic effects (Figures S13 and S14). Therefore, after examining only experimental measurands in CID-MS/MS and SRM operation modes, a lack of assignment of observable peaks to ions remains. Consider the shape of the SRM spectrum of BAC-C12 (Figure S11) and the proposed chemical 2D diagrams and electronic structures of ions at m/z 212 and 213.

#### 2.1.3. Mass Spectrometric Fragmentation Reactions of β-Blockers

^{+}cation at m/z 260 [80]. The fragmentation paths depend on CE, pH, etc. [1,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95]. With increasing CE, there is a low-abundance ion at m/z 183, due to the cleavage of the [C

_{3}H

_{9}N]

^{0}fragment and solvent water [1,80,83] (Figures S2 and S15). The peak at m/z 282 of the MS spectrum of PRO at CE = 30V is assigned to the [M+Na]

^{+}adduct. The same is true for ATE MS reactions (Figure S16). Their identical structural (2,3-dihydroxy-propyl)-isopropyl-ammonium fragment causes peaks at m/z 145, 105, 101, 83, and 64, respectively. ATE exhibits the [M+H]

^{+}cation at m/z 267. Quantitative analysis was carried out, examining the [M+H]

^{+}cation at m/z 260 and 267 of PRO and ATE in the mixture. Employment of the SRM and SIM modes leads to pairs of MS peaks at 260/261 and 267/268 (Figure S17). Classical quantitative methods look at average m/z data on MS peaks at m/z 260.5 (SRM) and 262.07 (SIM) (PRO), as well as 267.52 (SRM) and 267.82 (SIM) (ATE). The matrix affects significantly not only the m/z data compared with the results from the fragmentation paths of standard samples, but also product ions [86,87,88,89,90,91,92,93,94,95] (Figure S18).

^{+}cation in both the cases, excluding a statistically representative set of fragmentation species, there is particular importance in accounting for the molecular conformations and electronic effects of protonated antibiotics in order to assign statistically different sets of m/z data on two SRM and SIM spectra of standard samples of antibiotics and their mixtures in environmental and biological matrixes. For the purposes of 3D structural analysis and empirical demonstration of the assignment of the [M+H]

^{+}cation at m/z 260 of PRO in the complex sample matrix, there is used a statistically representative set of MS peaks at m/z 260, 283, 157, 116, and 98 found in the MS spectrum in soil [80]. The same fragmentation species have been found in the MS spectrum of PRO in biological samples [81,91]. Owing to the proposed two competitive mechanisms of formation of the MS ion at m/z 183, our study examines the correlation between MS data and theoretical quantum chemical ones looking at ions 183

_{_a}[80,81,90] and 183

_{_b}[82]. The MS ion at m/z 116 in the CID-MS/MS spectrum of PRO has been observed examining tandem MS/MS processes of ATE and studying the ions’ CID interaction of the [M+H]

^{+}cation at m/z 267 [84]. The peak at m/z 116 has been used to determine PRO in the liver, brain, and kidney thin tissue, as well [85,94]. In addition to the peak at m/z 116, ATE exhibits a set of ions depending on the experimental conditions [96,97,98,99,100]. These are peaks at m/z 225, 208, 190, 173, 162, and 145 [84,96,97,98,99,100]. Moreover, there are peaks at m/z 133, 115, and 107 [94] (Figure S19).

#### 2.2. Determination of Stochastic Dynamic Diffusion Parameters

^{+}cation at m/z 152 (Table S2). The new data on variables of PARA show lnP1 = 17.0532. Therefore, Equation (1) shows that the MS law is valid for the temporal distribution of measurands of PARA, as well. Details of the statistical parameters A

^{i}are presented in Figure S20. Calculation tasks have been discussed previously [55,56,60,61,62,63,64,65,66,67]. The latter figure illustrates the relation between the D′

_{SD}and D″

_{SD}parameters, showing |r| = 0.9995

_{3}. The deviation from |r| = 1 is a result of the error contribution of the data processing of the temporal distribution of intensity with respect to the scan time or function (I–<I>)

^{2}= f(t) fitted to the SineSqr function, thus producing statistical parameter A

^{i}(Figure S20).

#### 2.3. Quantitative Data on Biocides

_{SD}parameters are determined per span of scan time. We shall justify the view that exact relations are obtained when there are quantified fluctuations of measurands with a short span of scan time. The question that we need to address is “Which criteria determine a set of MS measurands with respect to a concrete span of scan time as the true one?”, or which methods are used in order to validate the parameters of Equation (2). In doing so, we use data on the selected reaction monitoring mode of the PARA [M+H]

^{+}ion at m/z 152 of segments of the MS method, where segment (i) has collected a full mass scan set of variables. There are examined segment raw data QC_High_SRM_SEG_CE40_i.raw (i = 1–3) [101,102] (Figure S12). Table S2 lists the output, only, of the fragmentation ion at m/z 110 for the SRM operation mode, while Tables S3–S6 list results from the CID-MS/MS spectra of PARA of its [M+H]

^{+}cation at m/z 152 depending on experimental conditions such as CE and the presence of formic acid. Chemometrics of the normality Shapiro–Wilk test together with ANOVA data in Tables S2–S6 are summarized in Tables S7 and S8. Data on quality control standard samples of a mixture of antibiotics QC_H_SRM_CE40_3 (segment 3, Table S2) reveal three groups of m/z parameters that are mutually significantly different from the perspective of chemometrics (Table S8). These are a subset of variables of segment 3, shown as QC_High_SRM_SEG_CE40_3_1, QC_High_SRM_SEG_CE40_3_2, and QC_High_SRM_SEG_CE40_3_3. The same is true for the recorded two sets of m/z variables of the same ion of segment (2) (QC_High_SRM_SEG_CE40_2_1 and QC_High_SRM_SEG_CE40_2_2). The chemometric analysis of datasets of measurands at m/z 110.1, i.e., QC_High_SRM_SEG_CE40_3_3, QC_High_SRM_SEG_CE40_2_1, and QC_High_SRM_SEG_CE40_1, is statistically equal. In other words, correlative analysis and determination of the D″

_{SD}parameters of Equation (2) within the framework of three segments of MS spectra is carried out using those sets of measurands that are statistically significantly equal, or those with values at m/z 110.065. Therefore, there are distinguished quantitatively three sets of peaks at m/z 110: 110.84

_{49}± 0.05985, 110.06

_{525}± 0.04709, and 110.23885 ± 0.04898. The number of the subset of variables is not extensive when looking at whole datasets of average values over the whole period of measurement. Results from sector (3) (QC_High_SRM_SEG_CE40_3) show a value at m/z 110.0673. In other words, our approach does not ignore measurable sets of low-abundance variables and their fluctuations.

_{3}-derivative in the presence of antibiotics or biocides reveal peaks at a range of m/z 107–115, which further complicate not only the qualitative assignment of products, but also the quantification of those single analytes in mixtures, as well as the deuterium exchange processes of d

_{3}-PARA, if any.

^{2}= 0.9309–0.985 (Figures S9, S11 and S21) using the classical approach to quantify the average total intensity of MS peaks over the whole period of measurements [1,103]. The results were data-processed via the ICIS algorithm of peak detection. The Savitzky–Golay smoothing function with baseline correction was used. The ICIS algorithm involves a trapezoidal integration approach [1,5,6,7,104]. In order to account for the effect of the smoothing function on chemometrics, herein, we examined the same relationships, but applying baseline correction and TIA (Figure 4 and Figures S22–S24). Figure S25 illustrates relationships among the concentrations of BAC-C12 and BAC-C14 in biota showing |r|= 0.9874

_{5}and 0.9922

_{3}. Despite this, the ICSI or TIA methods show low |r| parameters and high sd(yEr±) for mean values. The linear equation obtained via the direct application of TIA of BAC-C14 is y = −16.1

_{3779}± 14.6

_{133}+ 6.5

_{4102}± 0.3

_{3481}.x. The error contribution of the data-processing algorithm affects the main value of the intercept and slope of linear regression, respectively, and the correlation equations. The reliability of quantitative data is complicated when examining biota. There are competitive fragmentation reactions producing both mono-cations and cation radicals. The CID-MS spectra of PARA of freeze-dried biota show peaks at m/z 108.07 and 109.07. The wet sample shows MS ions at m/z 107.07, 110.07, and 111.1 (Figure 1, Figure 3 and Figure 4 and Figure S26). The mixtures of biocides show pairs of ions at m/z 211 and 212 (BAC-C12) and 239 and 240 (BAC-C14), instead of a single ion according to the common fragmentation scheme. The data processing of isotopologies in biota via the latter algorithms yields high sd(yEr±) values (Figures S27 and S28). Either by the ICIS or TIA algorithm, with or without baseline correction, the uncertainty of the analytical results is increased.

^{–1}. ANOVA and t-tests show (Tables S8 and S10) two sets of m/z values at 212.2

_{2211}± 0.12988 and m/z 211.6 at c = 2 ng.mL

^{−1}(BAC-C12), which are statistically significantly different. At c = 6 ng.mL

^{−1}are distinguished three sets of measurands at m/z 213, 212, and 211.5 (Table S8, Figure S10). There are three elemental compositions, molecular conformations, and electronic structures of BAC-C12 species (212.2

_{1401}± 0.1001, 212.5

_{4356}± 0.12232, and 211.7

_{86}± 0.15203.) Datasets at m/z 212.22

_{211}± 0.12988 and 212.2

_{1401}± 0.1001 of BAC-C12 ions at c = 2 and 6 ng.mL

^{−1}are statistically not significantly different. The ion at m/z 212.2 belongs to one and the same ion at two concentrations.

_{SD}parameter and assessing the relationship ln[D″

_{SD}] = f(conc.) yields |r| = 0.9999

_{1}–0.9905

_{8}, examining c = 2–80 ng.(mL)

^{−1}. Conversely, the ICIS and TIA algorithms produce |r|= 0.98924 (Figure S21). There has been obtained |r| = 0.999 when studying the same set of analytes in sludge [5]. Data on PARA and d

_{3}-PARA c = 5–400 ng.(mL)

^{–1}show r

^{2}= 0.997. Quantification of BAC-C12 and BAC-C14 yields r

^{2}= 0.987 and 0.983 [1,6,7]. The analysis of the peak at m/z 211.75 ± 0.15 of BAC-C12 ions in biota using the [M+H]

^{+}cation at m/z 304 (Tables S11 and S12 and Figure 6) produces |r| = 0.9972

_{1}and 0.9841

_{1}when employing the equation D″

_{SD}= f(conc.) Again, there is improved method performance. The analysis of BAC-C14 yields |r| = 0.9918

_{8}within concentration range c = 20–80 ng.(mL)

^{−1}.

#### 2.4. Quantitative Functions between Mass Spectrometric Stochastic Dynamic Diffusion Parameters and Theoretical Total Intensity Variables with Respect to Experimental Parameter Collision Energy

_{SD}of Equation (2). Statistical parameters A

_{D}

^{q}and A

_{I}

^{q}are functional amplitudes of the SineSqr function fitted with relation D″

_{SD}

^{q}= f(CE) and <I>

^{q}= f(CE) of q

^{th}MS fragment ion. We look at new empirical proof of the validity of Equation (4). It has been found by examining PARA measurands (Table S5). Figures S29 and S30 depict the relations of D″

_{SD}

^{q}= f(CE) and <I>

^{q}= f(CE) of MS ions of PARA at m/z 60 and 64. The theoretical <I>

_{theor}data on ions with respect to CEs and A

_{D}

^{q}, A

_{I}

^{q}parameters (Table S12) correlate with the experimental <I>

_{exp}ones and show |r|= 0.9942

_{9}and 0.96501. Since, so far, we have considered only two cases of the application of Equation (4) for the latter purposes, we are unable, currently, to assess the apparent violation of its validity.

#### 2.5. Theoretical Data

#### 2.5.1. 3D Molecular Conformations and Electronic Structures of Analytes and Energetics

_{QC}parameters of Equation (3) has been discussed [61,63,65,66]. However, we need to discuss the correlation between the 3D molecular conformation of MS ions and their energetics, thus highlighting the advantages of Equation (3), consisting of significant sensitivity and selectivity and capable of distinguishing quantitatively among molecular structures, exhibiting subtle electronic effects (Table S15). Figures S31–S33 detail the static and MD DFT results from ions in GS and TS states. Table S16 summarizes the atomic coordinates of fragmentation ions, allowing us to extract geometry parameters such as bond lengths and angles. The energy difference in the fragmentation species of surfactants such as ions 212

_{_a}and 212

_{_b}is ∆E

^{TOT}= |0.015| a.u. The difference in the energetics of molecular ion [M+H]

^{+}of PARA and d

_{3}-PARA is of the same magnitude order (∆E

^{TOT}= |0.01| a.u.). In these and many more cases of ions [60,61,63,65,66], there is provided ample proof favoring Equation (3) as a sensitive and selective tool, allowing us to distinguish among molecular species exhibiting comparable energetics. There are almost identical ∆E

^{TOT}ions of tautomers of PARA and d

_{3}-PARA (Figure S15). The examples of species provide us with real insights into the complexity of the electronic effects and dynamics of MS ions, which are unable to be tackled precisely when examining only free Gibbs energy data on the global minimum of PES (Table S17). Despite the fact that there is ∆E

^{TOT}= |0.01| a.u. of ions 152

_{_a}and 155

_{_a}for the molecular cation of PARA and d

_{3}-PARA, the difference in the D

_{QC}parameter is ∆D

_{QC}= |3.371| (Table 1).

#### 2.5.2. Determination of Quantum Chemical Diffusion Data

_{QC}parameters of Equation (3) can be found in [62]. Methodologically, we use vibrational data on MS ions at GSs and TSs. Variations and changes in the energetics of species can be examined adequately via Born–Oppenheimer MD. Table 1 summarizes the D

_{QC}parameters of the studied herein MS ions.

#### 2.6. Correlative Data on Mass Spectrometry and Quantum Chemistry

_{SD}= f(D

_{QC}). In line with our previous studies devoted to the same issue [61,63,65,66], achieving such a goal requires the assessment of the statistical significance of the mutual relationship between D

_{QC}and D″

_{SD}data on ions belonging to one and the same molecular structure. Figure 7 shows the chemometric results from PARA at m/z 152, 158, 174, 301, and 325 depending on CE (Table 1 and Table S5). There are |r| = 0.9979

_{8}at CE = 10 V and |r| = 1–0.9936

_{1}at CE = 25 V.

_{3}-PARA and propranolol (Table 1, Tables S5–S14). In the former case, there is obtained |r| = 0.9993

_{1}. Relation D″

_{SD}= f(D

_{QC}) of PRO ions at m/z 260, 157, and 116 shows |r| = 0.9916

_{1}.

## 3. Discussion

^{+}is characterized by a set of statistically significantly different m/z variables depending on the experimental conditions, but explicitly highlighting low analyte concentrations at a range of 2.5 to 25,000 ng.(mL)

^{−1}. It has been found that there are observed mass-spectrometrically two datasets of measurands at m/z 172.071

_{8}and 172.040

_{81}of the [M+H]

^{+}cation. As can be expected, the employment of classical quantitative approaches to determine the analyte concentration yields |r| = 0.9939

_{5}–0.9940

_{4}using linear calibration equation I

^{TOT}= f(conc.), where I

^{TOT}is determined via the ICIS or TIA algorithms. The decrease in method performance has been explained with the fact that, on the one hand, there are quantified isotope shapes of two different m/z quantities, which even can belong to two different analytes in complex biological samples, when there are determined unknown compounds. On the other hand, the error contribution to the mathematical data processing of MS patterns by means of the ICIS or TIA algorithms is significant, due to the large sd(yEr±) values of integration approaches. Due to these reasons, we suggest that the reliable and exact quantitative analysis of such complicated cases of fluctuations of MS measurands at very low analyte concentrations and complex matrix effects can be carried out exactly, accurately, precisely, selectively, and sensitively only via Equation (2) and the simultaneous quantitative and 3D structural analysis of analytes with respect to the experimental conditions of measurements. These combined research tasks allow us to assign exactly statistically different sets of measurable variables to corresponding molecular conformations and electronic structures of analytes. It has been found that the aforementioned MS peaks of metronidazole at m/z 172.071

_{8}and 172.040

_{81}belong to its two different tautomeric forms. The performed quantitative analysis based on two different calibration statistical equations D″

_{SD}= f(conc.) of two fragmentation peaks has resulted in |r| = 1. Turning to the results from this study in quantifying biocides BAC-C12 and BAC-C14 in biota, it can be easily shown an analogous cases of the temporal distribution and variations of MS measurands of these analytes in complex matrix samples depending on the experimental conditions, particularly highlighting a low analyte concentration as a major factor causing the observation of sets of statistically different m/z measurable variables belonging to different molecular conformations and electronic structures of analyte fragmentation species. Due to these reasons, the employment of the ICIS or TIA algorithms, quantifying the isotope shape area of function m/z = f(I), yields |r| = 0.9304–0.9856. Conversely, as the results from our analysis using Equation (2) show, there are statistically significant sets of variables of fragmentation ions at m/z 212 of CAB-C12 obtained as a result of the SRM tandem fragmentation mode of the molecular cation of analyte [M+H]

^{+}at m/z 304 (Tables S8 and S9). The statistical linear calibration models lnD″

_{SD}= f(conc.) and D″

_{SD}= f(conc.) have resulted in exact method performance, showing |r| = 0.9999

_{1}–0.9905

_{9}and 0.9972

_{1}. There are examined MS peaks of BAC-C12 at m/z 212.209 ± 0.1 and 211.75 ± 0.15. Thus, again, there is observed a highly reliable and very prominent quantitative analysis when we use Equation (2) instead of classical quantitative MS approaches based on the aforementioned algorithms. The new data presented in this paper on the MS quantitative analysis of mixtures of biocides and antibiotics in biota and sewage sludge show clearly the capability of the exact and reliable processing of complex isotope shapes of MS measurands, obtained as a result of competitive processes of tautomers and mechanisms involving the formation of cations and cation radicals, which is not only not universal, but also is beyond the capability of classical quantitative methods for the data processing of MS measurands. It is reasonable to assume, therefore, that classical automated algorithms of data processing of observable variables of such complex MS patterns are of little use when dealing quantitatively with the analysis of environmental and biological multicomponent samples of unknown analytes, very low analyte concentrations, and sample matrix effects. Furthermore, owing to the fact that the determination of analytes within the framework of the stochastic dynamic theory and model Equation (2) is carried out without the presence of IS, it is obvious that the innovative method allows researchers to determine quantitatively and structurally by mass spectrometry any unknown analyte in a complex mixture whose measurable parameters do not fit exactly with the available ISs or there is a lack of suitable internal standards. The latter remark is associated with the fact that both quantitative and 2D structural analytical methods for mass spectrometry, so far, use mainly ISs. Thus, under the so-called confirmed structure, there is understood currently (a) a reported exact mass; (b) an unequivocally determined molecular formula, and (c) a single confirmed structure, which is obtained by means of IS. However, often, environmental and biological samples contain analytes lacking suitable ISs. Therefore, even 2D structural MS analysis produces a so-called possible structure or tentative candidates of a 2D chemical diagram. On the other hand, we should distinguish between so-called 2D chemical diagrams and 3D molecular structures as well. The 2D diagrams are obtained according to the rule of the degree of unsaturation, in addition to concepts of atomic valence and oxidation states. We note the following statements: (a) “…the sum of the valences of all the bonds formed by an ion is equal to the valence of the ion”, and (b) “…the stoichiometry must be obeyed by electro neutrality principle” [106]. However, 2D structures or 2D diagrams do not tell us anything about the chemical reactivity and the chemistry of the molecules. Why? The term “molecular structure” means a generic property determined by an ensemble of atoms in a molecule [106]. However, analytical statements, claiming 3D molecular structures, should be based on electronic structural analysis, which is reliable only when there is information about the electron density maps of the ensemble of atoms in the molecules. The electron density maps are proof of the probability density distribution, which is observable experimentally. These maps determine the probability of determining electrons at infinitely small volumes and positions in the 3D space [107]. The so-called total energy is determined on the basis of the electron density maps. Therefore, any 3D molecular model or 3D molecular structure is characterized by a unique to total energy quantity. In other words, from the perspective of structural chemistry, under a 3D molecular structure, there is understood a 3D molecular conformation and corresponding electronic structures, which are unique as a whole. There is a lack of corresponding disordered structural fragments.

_{3}-derivative, particularly examining the fragmentation reactions of MS molecular ions [M+H]

^{+}and [d

_{3}-M+H]

^{+}at m/z 152, 155, 158, 174, 301, and 325, yielding the exact coefficient of linear correlation between D

_{QC}and D″

_{SD}data on CE = 25 V (Figure 7).

## 4. Materials and Methods

#### 4.1. Chemicals and analytical instrumentation

#### 4.2. Sample Preparation Methods, Samples, and Solutions

#### 4.3. Theory/Computations

^{−1}of diamante or 5.71.10

^{−4}a.u.

#### 4.4. Chemometrics

## 5. Conclusions

- (A)
- In testing the capability of Equation (2) [${D}_{SD}^{\u2033,tot}={\displaystyle \sum _{i}^{n}{D}_{SD}^{\u2033,i}=}{\displaystyle \sum _{i}^{n}{\mathrm{2.6388.10}}^{-17}\times \overline{{I}_{i}^{2}}-{\left(\overline{{I}_{i}}\right)}^{2}}$] to quantify the MS intensity of analyte ions with a short span of scan time, we contrasted the use of classical quantitative methods based on ICIS and trapezoidal integration algorithms of peak detection. The analysis of surfactants in biota via equation ln[D″
_{SD}] = f(conc.) yields |r| = 0.9999_{1}examining the peaks of BAC-C12 at m/z 212.209 ± 0.1 and 211.75 ± 0.15. - (B)
- Equation (4) [$\overline{{I}^{TOT,q}}=\frac{1}{2}\times \frac{{A}_{I}^{q}}{{A}_{D}^{q}}\times {D}_{SD}^{\u2033,tot}$] has been proven for PARA ions. The relation between <I>
_{exp}and <I>_{theor}shows|r|= 0.9942_{9}and 0.96501. - (C)
- Parameter |r| = 1 has been obtained, determining the 3D molecular structures of PARA and its ions at m/z 152, 158, 174, 301, and 325 via the assessment of relation D″
_{SD}= f(D_{QC}) in biota at CE = 25 V.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANOVA | Analysis of variance (chemometric method) |

APCI | Atmospheric pressure chemical ionization (mass spectrometric method) |

ATE | Atenolol |

BAC-C12 | Benzyl-dodecyl-dimethyl-ammonium chloride |

BAC-C14 | Benzyl-dimethyl-tetradecyl-ammonium chloride |

BAC-C16 | Benzyl-hexadecyl-dimethyl-ammonium chloride |

BAC-C18 | Benzyl-dimethyl-octadecyl-ammonium chloride |

BOMD | Born-Oppenheimer molecular dynamics |

CID | Collision-induced dissociation (mass spectrometric operation mode) |

CE | Collision energy |

DFT | Density functional theory (quantum chemical method) |

DoF | Number of points–number of parameters |

D_{QC} | Quantum chemical diffusion parameter according to Arrhenius’s theory |

D_{SD},D_{SD}^{’’} | Stochastic dynamic diffusion parameters according to our theory |

ESI | Electrospray ionization (mass spectrometric method) |

FT-ICR | Fourier transform ion cyclotron resonance (mass spectrometric method) |

GS | Ground state |

IS | Internal standard |

LMW | Low molecular weight (analyte) |

MD | Molecular dynamics |

MM | Molecular mechanics |

MS | Mass spectrometry |

MS/MS | Tandem mass spectrometric operation mode |

PARA | Paracetamol |

PES | Potential energy surface |

PRO | Propranolol |

QuEChERS | Quick, easy, cheap, effective, rugged, and safe (extraction method) |

|r| | Statistical coefficient of linear correlation (chemometrics) |

SD | Stochastic dynamics |

SIM | Selected ion monitoring (mass spectrometric operation mode) |

sd(yEr±) | Standard deviation (chemometrics) |

SRM | Selected reaction monitoring (mass spectrometric operation mode) |

se(yEr±) | Standard error (chemometrics) |

TIA | Trapezoidal integration approach |

TS | Transition state |

ν_{o}^{i} | Frequencies at a stationary state of potential energy |

ν_{s}^{i} | Frequency in a transition state of potential energy |

H^{#} | Activation enthalpy |

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**Figure 1.**Chemical diagrams of benzalkonium chlorides and pharmaceutics; exact masses’ theoretical m/z data in cationic form [M]

^{+}; theoretical mass spectrometric isotopic compositions and intensity ratios of the isotope shapes.

**Figure 2.**CID-MS/MS spectra of molecular cation [M+H]

^{+}of d

_{3}-deuterated paracetamol at m/z 155 assessing within-day and between-day variability of mass spectrometric measurands.

**Figure 3.**The total ion current (

**top**) of standard sample of wet biota; CID-MS/MS spectrum of molecular cation [M+H]

^{+}at m/z 152 of paracetamol (

**bottom**).

**Figure 4.**SRM spectra of analyte BAC-C12 in biota, using analyte molecular cation [M]

^{+}at m/z 304 at concentrations c = 2, 6, 10, and 20 ng.(mL)

^{–1}; baseline correction of isotopologies; data on trapezoidal integration approach (see also Figures S22–S24).

**Figure 5.**Linear functional relations ln[D″

_{SD}] = f(conc.) and D″

_{SD}= f(conc.) of fragmentation ions at m/z 212.2

_{096}and 240.2

_{247}of SRM spectra of molecular cation [M]

^{+}at m/z 304 and 332 of surfactants BAC-C12 and BAC-C14 in biota; chemometrics; the coefficient of correlation is highlighted in red.

**Figure 6.**Linear functional relation D″

_{SD}= f(conc.) of fragmentation ions at m/z 211.75 and 211.78 of SRM spectra of molecular cation [M]

^{+}at m/z 304 of surfactant BAC-C12 in biota; chemometrics.

**Figure 7.**Functional relationships between D″

_{SD}[cm

^{2}.s

^{−1}] and D

_{QC}data on Equations (2) and (3) of fragmentation ions of CID-MS/MS reactions of molecular ion [M+H]

^{+}of paracetamol at CE 10 (

**A**) and 25 (

**B**) V; chemometrics.

**Table 1.**Theoretical D

_{QC}parameters of Equation (3) according to most stable analyte 3D molecular and electronic structures of species at ground and transition states and energetics (Table S15); frequencies and atomic coordinates are listed in Tables S16 and S17.

m/z | Form | D_{QC} | m/z | Form | D_{QC} |
---|---|---|---|---|---|

110 | 110 | 2689.768 | |||

152 | 152_{_a} | 101.531209 | 155 | 155_{_a} | 98.1605 |

152_{_b} | 646,318 | 155_{_b} | 3.45123 | ||

158 | 158 | 367.4416 | 161 | 161 | 2902.9217 |

174 | 174 | 3009.373 | 177 | 177 | 42,352.98 |

325 | 325 | 21.08666 | 301 | 301 | 14.141939 |

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**MDPI and ACS Style**

Ivanova, B. Stochastic Dynamic Mass Spectrometric Quantitative and Structural Analyses of Pharmaceutics and Biocides in Biota and Sewage Sludge. *Int. J. Mol. Sci.* **2023**, *24*, 6306.
https://doi.org/10.3390/ijms24076306

**AMA Style**

Ivanova B. Stochastic Dynamic Mass Spectrometric Quantitative and Structural Analyses of Pharmaceutics and Biocides in Biota and Sewage Sludge. *International Journal of Molecular Sciences*. 2023; 24(7):6306.
https://doi.org/10.3390/ijms24076306

**Chicago/Turabian Style**

Ivanova, Bojidarka. 2023. "Stochastic Dynamic Mass Spectrometric Quantitative and Structural Analyses of Pharmaceutics and Biocides in Biota and Sewage Sludge" *International Journal of Molecular Sciences* 24, no. 7: 6306.
https://doi.org/10.3390/ijms24076306