A work on nonsubstituted cyclodextrins does illustrate persuasively the applicability of our innovative stochastic dynamic formulas connecting among measurable outcome intensity, analyte concentration in solution, the temperature and molecular properties to quantify and determine 3D structurally analytes. They bridge the gap between theory and experiment, in developing highly selective, sensitive, accurate and precise methods for quantification and exact 3D structural analytes by mass spectrometry.

## TABLE OF CONTENT

PREFACE

## Acknowledgments

## ABBREVIATIONS

1. INTRODUCTION

2. EXPERIMENTAL

2.1. Materials and methods

2.2. Sample preparation for ESI- and APCI-MS measurements

2.3. Determination of statistical parameters *accuracy* and *precision*

2.4. Determination of statistical parameters *repeatability* and *reproducibility*

2.5. Chemometrics

2.6. Theory/computations

2.6.1. Stochastic dynamic theory and model formulas

2.6.2. Quantum chemical computations

2.7. Experimental design

3. RESULTS

3.1. Figures of merit

3.2. Mass spectrometric data

3.2.1. Assignment of fragment ions of randomly acetylated Ac-b- and Ac-g-cyclodextrins

3.2.1.1. Fragment ions within low *m/z* -values

3.2.1.2. Fragment ions within high *m/z* -values

3.2.1.2.1. Self-associates of nonsubstituted cyclodextrins

3.2.1.2.2. Self-associates of randomly acetylated cyclodextrins

3.2.2. Determination of mass spectrometric diffusion parameters and correlative analysis with the quantum chemical diffusion data

3.2.3. Temperature dependency of the stochastic dynamic diffusion parameters

3.2.4. Functional relationship of the stochastic dynamic diffusion parameters and the statistical parameters within the framework of the empirical modification of the characteristic function diffusion

4. DISCUSSION

CONCLUSION

REFERENCES

APPENDIX A (Chemometrics)

APPENDIX B (Experimental mass spectrometric data)

APPENDIX C (Theoretical quantum chemical data)

## PREFACE

A work on nonsubstituted cyclodextrins does illustrate persuasively the applicability of our innovative stochastic dynamic formulas connecting among measurable outcome *intensity*, analyte concentration in solution, the *temperature* and molecular properties to quantify and determine 3D structurally analytes. They bridge the gap between theory and experiment, in developing highly selective, sensitive, accurate and precise methods for quantification and exact 3D structural analytes by *mass spectrometry*. In this work, we will explore the same theoretical framework considering significantly more complex macromolecular objects of randomly acetylated derivatives of b- and g-cyclodextrins as well as their noncovalent bond interacting self-associates (*m/z* 1400–1900.) The relationship between statistical parameter Ai of the SineSqr fitting of experimental relationship (I–<I>)2 = *f* (t) and diffusion parameter is tested (Ai = (m.DiSD)/{–(ln((kB.T)/m)3.(2.T.Dt.kB)}.) An approximation to Ai simplifies further our basic equation yielding to formula: D’SD » 1.3194.10-17.(<I2> – <I>2) is also tested. The experimental proof of these model equations is presented, as well.

## Acknowledgments

The authors thank the *Deutsche Forschungsgemeinschaft* for the grant 255/22-1; the *Alexander von Humboldt Stiftung*; the *Deutscher Akademischer Austausch Dienst* for grant within priority program *Stability Pact South-Eastern Europe*; central instrumental laboratories for structural analysis at Dortmund University of Technology, and analytical, respectively, computational laboratory clusters at the Institute of Environmental Research, therein.

The contribution was carefully carried out. Nevertheless, authors and publisher do not warrant the information to be free of errors. It is being published in English aiming at a widest access to the scientific information. English is not native language of the authors; thus, stylistic rough edges might occur. The authors hope for understanding of the reader.

## Conflicts of interest

Michael Spiteller has received research grant (255/22-1, DFG); Bojidarka Ivanova has received research grant (255/22-1, DFG).

## Address correspondence to the authors:

Lehrstuhl für Analytische Chemie, Institut für Umweltforschung, Fakultät für Chemie und Chemische Biologie, Universität Dortmund, Otto-Hahn-Straße 6, 44221 Dortmund, Nordrhein-Westfalen, Deutschland.

**Keywords:**

Mass spectrometry; diffusion; quantum chemistry; stochastic dynamics; acetylated cyclodextrins

## ABBREVIATIONS

Abbildung in dieser Leseprobe nicht enthalten

## 1. INTRODUCTION

Substantial progress has been made over decades in developing the chemistry of the cyclodextrins. A considerable amount of research on ~ 11000 derivatives (Rezanka, 2019) focuses on their application. These oligosaccharides are non-toxic and easily biodegradable naturally occurring products. Important application of CDs, among others, is to area of *medicinal chemistry*; *molecular drugs-design* of new therapeutics; *pharmacy*; *nanomedicine*, *nanotherapeutics*, and more. The host-guest inclusion complexes of CDs and drugs improve the solubility of the therapeutics and enhanced their bioavailability (Szabo et al. 2017; Shown et al. 2010; Sutyagin et al. 2002; Inoue et al. 2000; Shipilov et al. 2015; Ide et al. 2010; Sakairi, Wang, & Kuzuhara, 1995; Molina et al. 2012; Brown et al. 1993; Xiao et al. 2010; Ajisaka et al. 2000; Edunov et al. 2012; Takeo, Ueraura, & Mitoh, 1988; Xiao et al. 2016; Lian et al. 2014; Yoshikiyo, Matsui, & Yamamoto, 2015; Xue et al. 2015; Khan et al. 1998; Damager, et al. 2010; Inoue et al. 2000; Jiang et al. 2014; Mako, Racicot, & Levine, 2019; Dsouza, Pischel, & Nau, 2011.) Depending on the type of the chemically substituted CBs, there is gained a broad spectrum of solubility of the interacting ensembles of medications. The chemical modification affects on the CDs’ capability of forming inclusion complexes. It also affects on their catalytic activity and many other physico-chemical properties (Bjerre & Bols, 2010; Szabo et al. 2017; Shown et al. 2010.)

Also, supramolecular noncovalent self-associates of CDs have been used to study macromolecular models of highly organized systems such as enzymes, biological membranes, ribosomes and more (Risti et al. 2011; Ham et al. 2013; Harada et al. 2009; Xue et al. 2015; Bayley, & Martin, 2000 ; Thota, Urner, & Haag, 2016; Delbianco et al. 2016; Liu, Zhang, & Wang, 2015.) Besides, CDs have been used as supramolecular chiral selectors to capillary electrophoresis in the *analytical practice* of chiral separation methods (Yu, & Quirino, 2019; Haegele, Hubner, & Schmid, 2019; Stavrou et al. 2017; Ikai, & Okamoto, 2009; Juvancz et al. 2008; Adly, Antwi, & Ghanem, 2016; Astray et al. 2009.)

The CDs have been applied to *agro-chemistry* and *food industry*, as well (Martina et al. 2010.) Overall, enormous research effort has been concentrated on synthesis of chemically substituted CDs in order to model physico-chemical properties and application. However, there are two major *statistical* and *steric* problems from the perspective of the synthetic chemistry (Boger, Corcoran, & Lehn, 1978.) The OH-groups can be derived depending on the type of reagents and experimental conditions. The chemistry of native CDs has been chiefly associated with OH-groups at 2-, 3- and 6-positions **.** They show subtle difference in chemical reactivity. Attempt at overcoming this problem by developing of selective synthetic protocols has been outlined in the literature as shown above. However, often the synthesis is laborious or has resulted in losses in yields. Nevertheless, synthesis of randomly substituted CBs has been acknowledged (Pumera, Jelinek, & Jindrich, 2001; Chankvetadze et al. 2003; Legrand et al. 2009.) More than one OH-groups is chemically substituted. The preferred position for a substitution depends on the type of the incorporated functional group. Favourable is the 6-position. The 3-position is least accessible from the reagents for an electrophilic substitution. CDs of the latter type have been obtained *via* a preliminary modification of the macromolecules leading to suitable orientation of the OH-group at 3-position. The OH-group of the secondary side (at 2-position) is more chemically reactive comparing with the former OH-group (Rezanka, 2019; Tian et al. 2000; Martin et al. 1995; Tiwari et al. 2016; Anderson et al. 2015.)

In the light of these facts, it goes without saying that quantitative and 3D structural analyses of mixtures of randomly acetylated derivatives represent significant analytical challenge. It can be tackled difficult using the largest part of the available to the analytical practice instrumentation for quantitative and structural analyses, even looking at robust instrumental methods such as the nuclear magnetic resonance, mass spectrometry and single crystal X-ray diffraction, respectively. Therefore, a large number of phenomena of the chemical reactivity and the correlation between molecular structure and properties of CDs are not well understood even looking at nonsubstituted native CDs.

At this point the reader may go onto ask: Why the mass spectrometry does not provide unambiguous quantification of substituted CDs owing to the fact that the method has: *(i)* Ultra-high resolving power *via* mass analyzers such as orbitrap, multipass time of flight, or Fourier transform ICR providing the highest resolution of experimental outcome (Xian, Hendrickson, & Marshall, 2012; Kaiser et al. 2013.) It has been simultaneously determined 126 264 elemental compositions using 9.4T ICR analysis of a single sample (Krajewski, Rodgers, & Marshall, 2017; Kaiser et al. 2011.) The MS resolving power currently achieved is > 1 400 000 at 500 Da (Hur et al. 2018) at analytical concentration of analytes 1.0–0.1 mg.(mL)-1 (Ruddy et al. 2018.) The highest achieved currently broadband MS resolving power and mass accuracy of the available mass analyzers at 21 T (Hendrickson et al. 2015) is 300 000 at mass-to-charge (*m/z*) 400 and spectral acquisition rate of 1 Hz. The analyte concentration has been < 150 ppb. 2 000 000 resolving power has been achieved determining proteins for a detection period of 12 s; *(ii)* The mass spectrometry has capability of studying LMW analytes and macromolecules; (*iii)* it allows direct assay of solid or liquid samples without a sample pretreatment (Raesaenen et al. 2014); *(iv)* it has imaging technique to special and chemical information about organs and whole bodies, thus bridging the gap between interdisciplinary research fields, including the aforementioned ones (Leon et al. 2019; Semaan et al. 2012; Pu et al. 2020; Seeley & Caprioli, 2011; Norris & Caprioli, 2011; Prentice, Chumbley, & Caprioli, 2016;) (*v)* there is continuous flow monitoring of chemical reactions my mass spectrometry, allowing to study both at an industrial and laboratory scales thermodynamics, kinetics and diffusion of the processes; besides, application to thermodynamics and kinetics studies of environmental processes, which crucially contributes to the field of the E-monitoring and analysis (Loren et al. 2019; Wleklinski et al. 2016); *(vi)* portable design and capability of miniaturization of the scheme of the apparatus (Li et al. 2014; Snyder et al. 2016); *(vii)* direct assay of complex mixtures of analytes (Chen et al. 2013; Cole, 2010); *(viii) in vivo* diagnostics (Ferreira et al. 2019), and so on.

Due to the complexity of the CDs samples, as aforementioned, it is not enough to concentrate only on the MS instrumental characteristics. Despite, enormous effort on establishment of MS based protocols for quantification in biological samples. There are only few methods capable of simultaneous quantification of such analytes. Furthermore, they are time-consuming; require large quantities of solvents and expensive sample pretreatment procedure based mainly on solid-phase extraction, which is a routine approach to extract analytes from biological or environmental samples. Besides, often, there is a lack of isotope labeled internal standards of substituted CDs.

Therefore, it is of paramount importance to develop, elaborate and validate simple, fast, easy, selective, sensitive, accurate, precise, and reproducible analytical method for a highly reliable quantification of CDs in complex samples. Furthermore, it should be applicable to different MS methods and should operate without isotope labeling internal standards.

From the perspective of the *analytical chemistry* the pursuing of such goal represents a challenging research task. The method should ensure quality and comparability of the corresponding analytical results in accordance with the Council Directive 96/23/EC of 29 April 1996 and its implementation (2002/657/EC) concerning the analytical method performances and corresponding the interpretation of the analytical results of 12 August 2002 in front of the European Union 96. The MS based protocol validated according to the latter Directive, of analysis of LMW in complex CB mixture has achieved concentration limit of detection 0.3 μg.kg-1 at a reliability of |r| = 0.9806–0.9976 (Douny et al. 2013.)

This work is an adequate response to the latter problem dealing not only with highly accurate, selective, sensitive and precise quantitative analysis, but also with MS based 3D structural determination. In context of employment of MS for multidimensional structural analysis, the reader needs to be aware that, in spite of, the irreplaceable MS application to the analytical practice, it is mainly associated with quantitative analysis. The MS based structural analysis is chiefly limited to 2D molecular structure of the analytes. Thus, it is very important to see that our study does not provide only MS quantification of randomly substituted CDs, but also it provides 3D structural information, which is far beyond the current capability of the available mass spectrometric protocols.

At this point, one might draw a conclusion that such claim has to be justified soundly in the light of what we already know, as aforementioned, about the capability of the mass spectrometry for quantitative and structural analyses. In particular, the MS based structural analysis is at most inchoate subarea of the *analytical chemistry* (Song & Spezia, 2018; Veenstra, 1999.)

Also, it might be said that even if one admits that our quantitative determination is precise, accurate and sensitive, the MS selectivity determining structurally similar oligomers and polymers like the analytes of interest studied in this paper, the MS methods show low selectivity. Therefore, the reader might maintain that our claim shown above does not go enough persuasively.

However, the reader needs to be aware that the statements have to be judged in the light of our innovative quantitative *stochastic dynamic* concept and model equations which we have developed more recently as powerful method for exact quantification of the MS variable *intensity* (Ivanova, & Spiteller, 2014, 2019a–c, 2020.)

As we shall see from the content of the work, for the first time in the literature, we have applied successfully this concept to determine 3D structurally and quantitatively so complex molecular objects such as randomly acetylated CDs, addressing persuasively important questions: *(i)* How our formulas contribute to quantify highly selectively structurally similar compounds such as the equally chemically modified CDs; and *(ii)* how can be determined exactly 3D molecular and electronic structures of such molecules, when they are characterized by weak changes of molecular conformations and subtle electronic effects?

Through, this paper we maintain that the outcomes from the analyses provide persuasive experimental proof of not only validity of the presented in the next sections relations, but also their capability of addressing the latter questions. There is considerable evidence that the formulas are applicable to quantify fragment processes and determine multidimensional molecular structures under different MS conditions toward ionization methods. These are ESI, APCI, CID and MALDI methods.

Besides, the *stochastic dynamic diffusion parameters* (DSD), which we have introduced within the framework of the SD theory are linearly connected with the so-called *quantum chemical diffusion parameter* (DQC) according to the Arrhenius’s theory. The latter point extends crucially the MS capability of providing exact 3D structural data on analytes in condense phases. The DQC parameter reflects unique 3D molecular and electronic structures of a molecule. The analysis in the following sections demonstrates convincingly that the relations are capable to analyse highly precisely, accurately, sensitively and selectively even very complex molecular objects such as chemically substituted CDs in mixture.

Therefore, in contrast to the current view about method performances (Council Directive 96/23/EC; Douny et al. 2013) of the mass spectrometry and its capability of providing 3D-structural information (Song & Spezia, 2018; Veenstra, 1999) we stand that: *(i)* The quantification of the MS variable *intensity* by DSD parameter according to our innovative formulas improves significantly the method performances achieving excellent-to-exact chemometric parameters correlating between theory and experiment; and *(ii)* it provides exact 3D molecular structure of analyte when the diffusion parameters according to our concept are correlated with quantum chemical diffusion parameters according to Arrhenius’s theory.

Therefore, our formulas fill three different types of gap: (A) First, there are improved remarkably the method performances; (B) second, there is increased the MS capability of providing 3D structural information; and (C) the error contributions due to mathematical fitting of the experimental temporal relations of the intensity by means of SineSqr function by writing the new derivative equation D’SD » 1.3194.10-17.(<I2> – <I>2) are overcome, respectively. These statements imply that we shall look in the work not only for corroborative empirical evidence supporting for our claims, thus, showing that the formulas are well-corroborated and are empirically testable, but also for the experimental proof of formula D’SD » 1.3194.10-17.(<I2> – <I>2) of the diffusion parameter within the same theoretical framework. We shall sketch the proofs in validity of two of our formulas reported, so far. Besides, we shall try to justify persuasively new formulas derived in order to increase the explanatory power of the theory and its application to resolve complex quantitative and 3D structural problems taking into consideration affects on different molecular and environmental factors on the MS variable- *intensity* and 3D molecular conformation of the analytes.

To sum up, this work studies quantitatively and 3D structurally by means of high resolution ESI and APCI mass spectrometry as well as high accuracy static and MD methods of the computational chemistry of mixtures of randomly acetylated b- and g-CDs (**Figure A1**, **Appendix A**) by means of our innovative *stochastic dynamic* concept and model equations. A new model relationship between diffusion parameter and MS intensity (D’SD » 1.3194.10-17.(<I2> – <I>2)) is presented and evidenced experimentally, as well.

## 2. EXPERIMRNTAL

### 2.1. Materials and methods

Mass spectrometric measurements were carried out by TSQ 7000 instrument (Thermo Fisher Inc., Rockville, MD, USA). A triple quadruple mass spectrometer (TSQ 7000 Thermo Electron, Dreieich, Germany) equipped with an ESI 2 source were used to ESI- and APCI-MS measurements. A LTQ Orbitrap XL (Thermo Fisher Inc.) was used. A combination of mass detectors was used. The data were saved as individual files. The MS intensities were processed by QualBrowser 2.7, ProteoWizard 3.0.11565.0 (2017) and AMDIS 2.71 (2012) program packages. **Table A1** shows experimental conditions of the measurements. The chromatograms were obtained by Gynkotek (Germering, Germany) HPLC instrument, equipped with a preparative Kromasil 100 C18 column (250×20 mm, 7 μm; Eka Chemicals, Bohus, Sweden) and a UV detector set at 250 nm. The MS are correlated with UV-VIS ones (**Figure A2.**)

The cyclodextrins (b- and g-CDs) were Sigma-Alrdich products. The acetylated derivatives were obtained according to (Ivanova, & Spiteller, 2014.) The analytes studied in this work are: b-cyclodextrin (caraway; cycloheptaamylose; cyclomaltoheptaose; schardinger β-dextrin;) monoacetyl-β-cyclodextrin; triacetyl-β-cyclodextrin (β-Cyclodextrin heneicosaacetate); heptakis(6-O-acetyl)-β-cyclodextrin; g-cyclodextrin (cyclomaltooctaose, cyclooctaamylose, schardinger γ-dextrin;) octakis(6-O-acetyl)-g-cyclodextrin; monoacetyl-g-cyclodextrin; undecakis(6-O-acetyl)-g-cyclodextrin; tridecakis(6-O-acetyl)-g-cyclodextrin; and pentadecakis(6-O-acetyl)-g-cyclodextrin, respectively.

### 2.2. Sample preparation for ESI- and APCI-MS measurements

Standard solutions of Ac-CDs were prepared in CH3CN:CH3OH in multiplicate (samples E1–E3 of Ac-b- and samples E1–E3 of Ac-g-CDs.). The solutions were prepared daily by diluting CDs in mobile phase (25:75 CH3CN:H2O *v/v*). They were homogenized for 10 min at 1500 r.min-1 with shaker. The solutions stored in the dark at –4oC. Aliquots of all samples were measured by ESI- and APCI-MS methods.

### 2.3. Determination of statistical parameters accuracy and precision

They have been evaluated using data on multiplication of CDs in solvent mixture CH3OH:CH3CN at molar ratios 1:1 at concentration 2 mg.(mL)-1 according to (Taylor, 1987).

### 2.4. Determination of statistical parameters repeatability and reproducibility

Three independent measurements of two replicated samples compared with the data of standard samples are processes by chemometrics. The convention (Taylor, 1987) is used to calculate the statistical parameters. The formulas: 2.(2)1/2.SD1 and 2.(2)1/2.SD2 are used. The SD1 and SD2 denote short and long-time standard deviations. The span of time is t = 264 h.

### 2.5. Chemometrics

The software **R4Cal Open Office STATISTICs for Windows 7 was used.** The statistical significance was checked by *t* -test. The model fit was determined upon by F-test. The **ANOVA test was also used. The nonlinear fitting of experimental MS data was carried out by means of searching method based on** Levenberg-Marquardt algorithm (Taylor, 1987; Otto, 2017; Apache OpenOffice; Madsen, Nielsen, & Tingleff, 2004; Miller, Miller, & Miller, 2018; Kelley, 2009).

### 2.6. Theory/computations

#### 2.6.1. Stochastic dynamic theory and model formulas

Let us begin this subsection by looking at our model equations, which we have derived more recently ¾ formulas (1)–(3) ¾ on the base on SD methods and our empirical modification of the characteristic function *diffusion* D(x,t) in the forward Chapman-Kolmogorov or forward Fokker-Planck equation (4) (Ivanova, & Spiteller, 2019a–c.)

Abbildung in dieser Leseprobe nicht enthalten

Our basic SD concept behind the derivation of equations (1)–(3) is that the experimental MS variable *intensity* is treated as a *stochastic variable*. *Intensity* -values of MS peak of an ion *per* different span of scan time of a measurement are treated as *‘’possible values’’* or as a *probability distribution*. This set of observable outcomes can be described as a discrete set of variables *per* short span of scan time, instead of, over the whole time of the measurement. The values can be processed as continuous data over the latter time of a measurement or can be treated as partially continuous and partially discrete sets, respectively (Van Kampen, 2007.) This is one of the advantages of the stochastic methods, among many others, for mathematical processing of MS measurable parameters. The set of stochastic variables can be multidimensional. The probability distribution is described as density distribution (P(x)) (equation (4)) of all these ensembles of *m/z* - or *intensity* -values of ith MS peaks of an analyte. The P(x) in our case is viewed as having thermal Maxwell-Boltzmann distribution. The population of themodynamically stable MS ions obtained under single MS operation mode is approximated to latter function (Goeringer, & McLuckey, 1996.) It is agreed that under the low-field experimental conditions, the state of MS ion is approximated to thermal equilibrium. The standard Maxwell-Boltzmann distribution is normal Gaussian distribution of the kinetic energy (Hernandez, 2017.)

Within the framework of our theory of the MS outcome *intensity*, we consider the time-evolving of the intensity values with respect to different spans of scan time (Dt). Under *temporal behavior of the MS intensity* there is understood dynamics of the ensemble of values *per* span of scan time or over the whole time of a measurement. The probabilistically description of the temporal behavior of a random variable ¾ in our case of MS *intensity* – within the framework of stochastic methods can be expressed mathematically by equation (4) (See the comprehensive description of stochastic processes in Gillespie, 1996a.)

If we describe the behavior of the MS species as *homogeneous Markov processes* the characteristic functions A(x,t) (it is frequently called *drift function*) and D(x,t) in equation (4) are given by equations (5) and (6) (Gillespie, 1992, 1996a.) Any Markov process with functions A(x) = -k.x (k >0) and D(x) = D (D > 0) is called Ornstein-Uhlenbeck process. (The latter process is introduced essentially by Langevin (Gillespie, 1992, 1996a,b; McConnel, 1980). The mean and the variance of a general Ornstein-Uhlenbeck process are detailed in the latter references. Therefore, the functional forms of A(x,t) and D(x,t) define a process and are given by equations (5) and (6) (Kaznessi, 2012.)

Abbildung in dieser Leseprobe nicht enthalten

The Fokker-Planck equation and its approximation to Ornstein-Uhlenbeck process exactly determine the stochastic process or the time-evolution of a random variable. It is carried out by means of the average value of the variable *per* span of time and variance (s2.) The experimental MS *intensity* -value reflects concentration of analyte. The concentration represents mass of analyte *per* volume. The examination of temporal behavior of MS *intensity* with respect to small spans of the scan time, essentially examines dynamics of mass over a small span of the scan time of the whole time of measurement. When ions are affected on low electric field, then their drift velocity along the direction of the field can be superimposed on stochastic thermal motion (Gillespie, 1996b). The upshot of the latter assumption is that, when the mass of the analyte ions is approximated directly to the observable values of the intensity; and the temporal behavior of the velocity of the MS ions is described as a stochastic variable, then the *intensity* -outcome is also described as random variable. According to our theory we examine the temporal behavior of the variable over spans of the whole time of a measurement. The theory accounts for fluctuations of the observable intensity and *m/z* -values of peaks of an analyte ion with respect to different spans of scan time of the measurements under MS continuum or environment (**Figures A3** and **A4**.) These fluctuations are due to ion/molecular interactions or, ion/continuum interactions. Under *continuum* we understand all other molecules and ions around analyte MS ions. In general, the issue of fluctuation phenomena in the natural processes leads back to fundamental works by Einstein, which crucially contributes to their mathematical and physical expression, and respectively, description (Renn, 2005.) In Einstein’s works (Einstein, 1905, 1906) we approximate the temporal behavior of the fluctuating path of solid particles in liquid to elementary diffusion equation. The latter equation represents precisely the *forward Fokker-Planck equation* (4) (Gillespie, 1996b.) The exact numerical solution of the *Ornstein-Uhlenbeck process* approximating A(x,t) and D(x,t) in equation (4) has been shown that express mathematically the fluctuating path of solid particles in liquid (Gillespie, 1996b.) The characteristic function D(x,t) according to Einstein’s approximation is given by equation (7).

Abbildung in dieser Leseprobe nicht enthalten

In writing equations (2) and (3) we empirically modify equation (7). The Einstein’s model does not fit our experimental data perfectly in chemometric terms (Ivanova, & Spiteller, 2019c.) Equation (7) we have written as equation (8), thus yielding to excellent coefficients of correlation (|r|) between theory and experiment.

Abbildung in dieser Leseprobe nicht enthalten

In equations (2), (3) and (8) kB denotes the Boltzmann constant (kB = 1.3806.10-23 m2.kg.s-2.K-1;) T is temperature [K]; Dt – short span of time [s] of the whole time of a MS measurement; m is the molecular weight of the analyte ion or the *m/z* -value when the charge is equal to one; DSD denotes the stochastic dynamic diffusion parameter according to our theory [cm2.s-1] and CM represents the concentration of the analyte in solution.

Let us explore further another aspect of formulas (1)–(3). Treating the experimental outcome *intensity* as random variable, there is understood random numbers according to the probability distribution. The initial velocity of the MS ions which produce the corresponding intensity value as outcome as shown above and the random displacement can be assigned. In doing so, we have adopted the Box-Müller method (Gillespie, 1996a; Satoh, 2011; Coffey, Kalmykov, & Waldron, 2004.)

Since, the MS outcome intensity is treated as a stochastic variable and the function (I – < I >)2 = *f* (t) which we obtain experimentally is a nonlinear one (**Figures B5** and **B6**, **Appendix B,**) one way to connect measurable outcome and diffusion parameters with DSD parameters according to our formulas is by fitting experimental relationship with a nonlinear function such as the SineSqr function (Ivanova, & Spiteller, 2019c) (**Figure B12**.) The *curve-fitting approach* is a common tool used to fit calculated diffusion current to experimental data depending on the theoretical diffusion model. There are obtained excellent-to-absolute correlation coefficients |r| between experimental relationship (I – < I >)2 = *f* (t) and its SineSqr approximation (|r| = 0.9998–1; Ivanova, & Spiteller, 2019a–c; **Appendix B.**) The statistical parameter Ai in equation (1) is, thus, functionally connected with DSD parameter (Ivanova, & Spiteller, 2019c.) However, how do we make physical sense of the statistical parameter “Ai”? (Ivanova, & Spiteller, 2020.) We can do so doing correlation between equations (1) and (2). They are empirically testable and verifiable. It depends on experimental factors, parameters and measurable outcome intensity (Ai = *f* (T, DT, m, I).) The relationship is written as equation (9), which validity is evidenced in **subsection 3.2.4**. Also, its validity has been proven studying solvate clusters mass spectrometrically, showing coefficients of correlation |r| = 0.9973–0.9801 (Ivanova & Spiteller, 2020.) As the latter reference has shown the parameter Ai cannot be assigned to diffusion parameter DSD due to lack of physical meaning as the latter reference has discussed. Moreover, the parameter Ai is dimensionless. Because of, the Taylor row of ln(KB.T/m) (lnx = 2.{((x-1)/(x+1)) + (1/3.((x-1)/(x+1))3)} + (1/5.((x-1)/(x+1))5)+...} leads to Ai [units] (kg.m2.s-1)/(K.s.m2.kg.s-2.K-1) or it is dimensionless as mentioned lastly. Conversely, the DSD parameter as can be expected according to the same equation (9) shows DSD [units] (K.s.m2.s-2.K-1.kg)/(kg) or it has units [m2.s-1]. The units of D(x,t) according to equation (8) are [s-1].

Abbildung in dieser Leseprobe nicht enthalten

Nevertheless, excellent chemometric data reported, so far, when complex relationships (I – < I >)2 = *f* (t) are observed experimentally the curve-fitting method could yield to error contribution to the Ai-value. This affects on the superior method performances. In order to overcome such drawback ¾ despite, it could be obtained rarely ¾ we derive the following formula. Combining the idea presented as equation (9) with equation (1) obviously the DSD parameter can be approximated to DSD’ coefficient determined according to equation (10).

Abbildung in dieser Leseprobe nicht enthalten

We judge the applicability of the latter model by evaluating the correlation between DSD and D’SD parameters according to equations (1) and (10). It is also carried out in the subsection lastly mentioned.

#### 2.6.2. Quantum chemical computations

The GAUSSIAN 98, 09; Dalton2011 and Gamess-US (Frisch, et al. 1998, 2009; Dalton 2011; Gordon, Schmidt, 2005) program packages were used. The outputs were visualized by GausView03 (GausView03, 2005). DFT molecular optimization was carried out by B3PW91 method (Hay, & Wadt, 1985; Western, Casassa, & Janda, 1984; Fernandez-Ramos, 2013; Truhlar, 1991; Wang, Castillo, & Bozzelli, 2015; Alves et al. 2016). Truhlar’s functional M06-2X was used (Zhao & Truhlar, 2008a,b). The Bernys’s algorithm was employed in determining the ground state. The stationary points on the potential energy surface were determined by standard harmonic vibrational analysis. The criterion confirming minima of energy is absence of imaginary frequencies of second-derivative matrix. Dunning’s basis set cc-pVDZ and quasirelativistic effective core pseudo potentials from *Stuttgart* - *Dresden(* - *Bonn) (SDD, SDDAll, http://www.cup.uni-muenchen.de/oc/zipse/los-alamos-national-laboratory-lanl-ecps.html) were used*. The doubly polarized triple- *z -* basis sets augmented by diffuse functions aug-cc-pVDZ was also utilized. Species in solution were studied by means of explicit super molecule and “mixed” approach of micro hydration by means of PCM approaches. Large species were treated using ONIOM method. The equilibrium and transition states were confirmed using frequency analysis, as well. The MD computations were performed by *ab initio* BOMD and ADMP methods. The BOMD was carried out by M062X functional. BOMD computations with ONIOM were utilized, as well. The trajectories were integrated using Hessian-based predictor-corrector approach with Hessian updating for each step on BO PES. The step sizes were 0.3 and 0.25 amu1/2Bohr. The trajectory analysis stops when: *(a)* centres of mass of a dissociating fragment are different at 15 Bohr, or *(b)* when the number of steps exceed the given to as input parameter maximal number of points. The total energy was conserved during the computations within at least 0.1 kcalmol-1. The computations were performed by means of fixed trajectory time speed (*t* = 0.025 fs) starting from initial velocities. The velocity Verlet and Bulirsch-Stoer integration approaches were used. The Allinger’s molecular mechanics force field MM2 was utilized (Burkert & Allinger, 1982; Allinger, 1977). The low order torsion terms is accounted with higher priority rather than van der Waals interactions. The accuracy of the method comparing with experiment is 1.5 kJ.mol-1 of diamante or 5.71.10-4 a.u. The differences in heats of formation of alcohols and ethers are |0.04|–|6.02| kJ.mol-1.

### 2.7. Experimental design

Ions of types: {[(*x* -CD).(CBi)]+} (*x* = b- or g-; *i* = 1–4), where CD denotes cyclodextrin, while CB — short chain oligosaccharide; {[Acy- *x* -CD]+} (y = 3, 7, 8, 10, 11, 13 and 15;) their K+-, NH4+- adducts and solvate complexes are examined theoretically and experimentally. Characteristic MS peaks of acetylated derivatives within low mass-to-charge values at *m/z* 255, 273, 283, 300 and 324 are examined experimentally by means of ESI and APCI mass spectrometry at temperatures T = 273.15, 717.37 and 723.15 K, respectively.

## 3. RESULTS

### 3.1. Figures of merit

Looking at the theoretical concept behind the derivation of equations (1)–(3) dealing with quantification of the outcome *intensity per* span of the scan time of measurements, in addition to, the fact that equation (3) connects the latter experimental outcome with *m/z* -values of the ions; perhaps, it is self-evident that the determination of the error contribution from measurements to the experimental MS variable should be carried out not only over the whole time of a measurement, but also *per* span of the scan time. The evaluation of different random errors and other variation factors of measurable variables are carried out by chemometrics studying not only outcomes of samples measured in multiplication, but also among datasets of variables of corresponding replicates. Since, a large number of tests assume a normal distribution of the variables, we start this subsection with results from Shapiro-Wilk test assessing the normal distribution of the outcome values (Goodson, 2011; **Table A2**.) The data in the latter table and those depicted in **Figure A3** show that the *m/z* -values of the peak at *m/z* 214 appears not normal looking at the outcomes of three independent measurements of Ac-g-CD. Conversely, the results from the peak at *m/z* 158, which are tested over short spans of scan time is characterized by normal distribution. Therefore, our treatment of MS outcome *per* different span of scan time provides highly reliable quantification of experimental variables comparing with chemometric data on the whole time of measurements. In advancing our approach, we should highlight, again, that the DSD parameter provides a highly meaningful in chemometric terms quantification of experimental intensity, together with equations (2) and (3), which accounts for *m/z* -values *per* concrete span of the scan time. Besides, the results from this paper agree excellent with previous conclusions drawn from chemomerics of temporal behavior of MS ions of solvates (Ivanova, & Spiteller, 2020.)

The *m/z* -values of ion at *m/z* 214 of experiment E1 of Ac-g-CD and the fact that the distribution does not appear a normal one, lead us to a conclusion that there are different subsets of experimental outcomes belonging to the same MS peak, which should be tackled as independent datasets. **Figure A3** and **Table A3** depict the chemometric data on *m/z* -values at *m/z*: 214.08972168, 214.089736938, 214.089752197 and 214.089706421, respectively; all exhibiting not normal distribution. The W-values of samples are lower comparing with the critical W-value W0.05 = 0.803 (Goodson, 2011.) The fact that Wcalc < Wcrit0.05 assumes that at a level of confidence 95 % the distribution does not originate from a normal distribution of the variables. The ANOVA analysis indicates that these sets are significant different (**Table A4.**) The latter test was used in order to compare means of two or more datasets of variables whether the mean values differ significantly. The analysis includes test of datasets of different samples and test of datasets of measurements in multiplications. In particular, the two-way ANOVA was used to compare unequal dataset sizes. These are datasets of *m/z* -values having different abundance of the MS ions *per* span of time (**Figures A3, A4**, **B5** and **B6**.) Due to the random phenomena of producing of MS ions *per* sample and *per* measurement in multiplication, the ANOVA test yields to important information about equality of *m/z* -values, when the sizes of datasets are different. It able to account for more than one sources of random errors of measurements, as well. In such cases we test among sub-datasets within the whole time of measurements. Therefore, this paper presents results from one- and two-way between-sample and within-sample tests aiming at establishing sources of variation of mean *m/z* -values. Large *F* -values obtained as a result of ANOVA test leads to small *P* -values (**Table A4**.) In this case even P = 0. Small P-values mean that the hull hypothesis (H0) is rejected. For this reason, the decision rule in **Table A4** states that at the 0.001 level the population means are significantly different. ANOVA tests are very sensitive. Therefore, they are capable of detecting differences in the mean values of datasets of variables when real differences of the means exist (**Tables A4** and **A5**.) Importantly, the F-test of ANOVA is not so sensitive toward deviation from the normal distribution of the tested populations or variables. For this reason, we use it even when there is weak deviation from the normality of the probability distribution of the MS outcome values. The same set of exact *m/z* -values is obtained experimentally within the framework of different experiments and multiplication of measurements; and chemometric evaluation show that the means of all these sub-datasets are mutually equal. Therefore, the analysis of corresponding intensity values according to equations (1)–(3) should distinguish among several DSD parameters corresponding to the MS peak at *m/z* 214. The ANOVA test of the datasets of *m/z* -values with respect to scan times of measurements of ion at *m/z* 158 under experiments E2 and E3 of Ac-g-CD shows that they are not significantly different (**Table A5**.) The same is true for sub-datasets of *m/z* -values of Ac-b-CD. **Table A6** presents results from Shapiro-Wilk test of *m/z* -values over the time of the measurement. The datasets belonging to MS peak at *m/z* 214 exhibit that it does not originate from a normal distribution. Conversely, the analysis of data on MS ion at *m/z* 951 shows a normal distribution. At this point we will engage with an amount of discussion of the significance of the W-data according to Shapiro-Wilk test; and, looking at outcomes summarized in **Tables A3** and **A5**. A large W-value assumes a normal distribution of the variables (Goodson, 2011). Despite, the histograms in our case show an asymmetric distribution, but we have found that the distribution of *m/z* -values with respect to scan time of peak at *m/z* 214 consistents with log-normal distribution (*r* 2 = 0.8835.) The same is true for datasets of values at *m/z* 158, where the Shapiro-Wilk test indicates a normal distribution. Its consistency with log-normal distribution shows *r* 2 = 0.90227. **Figure A5** gives histograms and probability plots of *m/z* -values. In fact, there are obtained non-linear scales for the cumulative counts or the frequently distribution of the measurable outcomes. Despite, the S-shaped curves the largest part of the data lie on a line approximately, thus, supposing an assumption that the variables come from a normal distribution. As shown above, however, the coefficient of correlation is relatively low. Nevertheless, the decision about nonlinearity of the former dataset of outcomes of *m/z* 214, based on Shapiro-Wilk test we also apply ANOVA tests. Conversely, the Shapiro-Wilk test used to the dataset of values of ion at *m/z* 214 of Ac-g-CD; however, measured under E3 experimental conditions over the whole time of measurements shows a normal distribution of the variables. **Figure A6** illustrates the histograms and log-normal approximation yielding to consistency of *r* 2 = 0.9334. Since, the central concept we explore is that an accurate description of MS outcomes should be carried out *per* spans of scan time according to our model relationships, the ANOVA test of *m/z* -values over the whole time of measurements t = 0–30 mins of MS ion at *m/z* 214 of two Ac-b- and Ac-g-CDs shows that these datasets are significant different (**Table A7**.) The same test applied to datasets of *m/z* -values of MS ion at *m/z* 158 of the two acetylated CDs indicates that they are not significantly different (**Tables A8** and **A9**.) The application of the two-sample *t* -test to the datasets of values at *m/z* 158 and 214 of Ac-b- and Ac-g-CDs indicates that the populations are not significantly different *per* pair of experimental variables (**Table A10**.) The latter table contains also the *power* of the test. It represents the probability of Type II error (when the power is equal to one.) This type of error accounts for the retention of the H0 even when it is false. As can be seen the power is closes to zero. The one- and two-sample t-test were used to the former case in order to test whether or not the mean values of the variables are equal to corresponding standard value. While, the two-sample t-test is used to compare whether two datasets of variables are mutually equal. Like in the case of ANOVA tests the H0 is evaluated. The decision is presented, analogously.

### 3.2. Mass spectrometric data

#### 3.2.1. Assignment of fragment ions of randomly acetylated Ac-b- and Ac-g-cyclodextrins

##### 3.2.1.1. Fragment ions within low m/z -values

Looking at the ESI- and APCI-MS spectra of non-substituted CDs (Ivanova, & Spiteller, 2019b) and those of randomly acetylated derivatives studied in this work, there seems to be a very close match among fragment ions within low *m/z* -values *m/z* = 100–500. The MS species at *m/z* 214, 231, 245, 252, 259, 272, 274, 279, 294 and 313 are observed in the substituted CDs, as well (**Figure B1**.) However, the ions at *m/z* 255, 273, 283, 300 and 324 are typical only for the acetylated CD derivatives (**Figures B1** and **B2**.) This is an important difference in MS spectra of non- and substituted CDs allowing for their determination in mixture. Nevertheless, as have been noted (Tueting, Adden, & Mischnick, 2004; Carroll, et al. 1993; Yamagaki, & Sato, 2009) the MS fragment processes of substituted CDs exhibit a set of common paths depending on the position of the substituted OH-group. In cases when the C2–OH group is free from chemical substitution the product Yn and Bn type ions ¾ there is used the nomenclature of the fragment products of CBs according to Domon and Costello (Domon, Costello, 1988) ¾ dominate in the spectra. Despite, the complexity of the MS spectra of CDs can be increased, due to, formation of alkali metal ion adducts (Sakairi, & Kambara, 1989; Madhusudanan, 2003; Peptu et al. 2018).

The corresponding assignment of the later set of *m/z* -values to corresponding 2D and 3D chemical diagrams and molecular structures is given in **Figures 1, 2** and **Appendix C**. The correlative analysis between the *absolute total intensity* (ITOT) of the ions with respect to the thermodynamic *free Gibbs energy* parameters in gas- and condense phases is carried out. At this point it needs to be underlined, again, that behind equations (1) and (2) there is the plausibility theories describing the MS intensity as a stochastic variable (Ivanova, & Spiteller, 2019b.) Since, the MS intensity reflects concentration of an analyte ion in initial solution (Ivanova, & Spiteller, 2020a.) and in ESI continuum, examining the fluctuations, we study, in fact, dynamics of continuum at a molecular level (Graham, 2018; Sinaiski, Zaichik, 2008.) The key problem is that the dynamics of macromolecules at continuum or solution can be very complex. Therefore, both the initial solution of the macrocyclic CBs and the ESI-MS continuum of macromolecular ions and their noncovalent bond interacting associates are regarded as complex fluids or complex systems. Accounting for the size of the macromolecules, the fluctuations in motion of the molecular ions of CBs cannot be neglected. Fluctuations of medium are hypothesised even without presence of analyte ions. Furthermore, in deriving equations (1) and (2) we have adopted the Ornstein-Uhlenbeck's and Einstein's approximations to motion of particles in liquid, with a remark that we empirically have modified the diffusion characteristic function of the forward Fokker-Planck equation (Ivanova, & Spiteller, 2019c.) It may seem, therefore, obvious that the DSD parameters reflect molecular level property of analyte ions in solution or ESI continuum. This statement agrees well with MS data on solvate complexes (Ivanova, & Spiteller, 2020b.) However, correlating DSD parameter from MS experiment with DQC data according to Arrhenius's theory, there arises a question: Do the DSD parameters correlate with the DG data on GP or they correlate with the solvation *free Gibbs energy* (DGSol) parameter? Experimental results, so far, have shown coefficients of linear correlation between DSD and DGSol equal to *r* = 0.97. Moreover, the MS spectra of CBs reveal solvated noncovalent bond interacting adducts (Puzot, & Prome, 1984; Harada, Suzuki, & Kambara, 1982.) Their theoretical quantum chemical treatment requires a mixed solvation approach in GS or solution or both of these. In general, there is a strong affect on solvation processes on the binding energy of macromolecules (Biedermann & Schneider, 2016.)

Abbildung in dieser Leseprobe nicht enthalten

**Figure 1.** Experimental MS spectra of Ac-b- and Ac-g-CDs within low *m/z* -values; chemical diagrams of fragment ions (the characteristic ions of only acetylated derivatives are shown and the experimental peaks are highlighted;) the distinction among structures is carried out on the base on their energetic; theoretical *m/z* -values and experimental intensity rations of isotope subpeaks of species [%]; assignment of common ions to non- and chemically substituted CD derivatives together with their 3D structural determination can be found (Ivanova, & Spiteller, 2019b.)

In order to shed further light on ESI ionization phenomena and the nature of the MS continuum, which appear important aspects of theoretically determined DQC parameters, we look at DG parameters of CBs ions at GS and polar continuum (**Tables C1** and **C2**,) together with ITOT-values of ions at *m/z* 255, 273, 283, 300 and 324, respectively. The results from the latter tables reflect 3D molecular and electronic structures of ions corresponding to the minimum of potential energy surfaces according to both static and MD computations (**Figures C1–C6**.) Among the studied ions at *m/z* 273, most stable is cation m273_a, showing an energy difference in DE = |34.6386| a.u. comparing with cation **m273_b_a** (**Table C1**.) It is important to stress that the later ion exhibits lower DGSol value comparing with cation m273_a. It appears more stable species in solution (**Table C2**.) The upshot goes to suggest that depending on the nature of the ESI-MS continuum, there is expected different thermodynamic stability of species comparing with GS calculus. As **Figure 2** shows the correlation between ITOT values of the discussed ions of Ac-b-CD and Ac-g-CD shows *r* = |0.9736| **–** |0.9989|. Since, in course of our ongoing development of the described, herein, method we have found that frequently ITOT values correlate linearly with the *free Gibbs energy parameter* as mentioned before, we carry out analysis of acetylated CDs (**Figures A7** and **A8**.) The relations between total intensity data and DG parameters in GP and in solution show *r* = |0.6655| and |0.8772|. As the results reported, so far, the two coefficients cannot be neglected.

**Abbildung in dieser Leseprobe nicht enthalten**

**Figure 2.** Functional relationship between total intensity (ITOT) values of common peaks of Ac-b- and Ac-g-CDs; chemometrics.

##### 3.2.1.2. Fragment ions within high m/z -values

3.2.1.2.1. Self-associates of nonsubstituted cyclodextrins

Before we turn to quantitative and structural analyses of noncovalently bonded self-associates of randomly acetylated CDs we need to give detail on binding properties of nonsustituted derivatives. The motivation behind this description is that the MS fragment paths are significantly complicated in cases of mixtures of randomly acetylated CDs. They show common MS peaks at 1873, 1881 and 1930, which are also observed in MS spectra of nonsubstituted compounds (**Figure 3**.) Moreover, these ions are found in the ESI-MS spectrum of a-CD (**Figure A7.**) They are assigned to fragments ions of type {[(*x* -CD)(CBi)]+} where *x* = a-, b-, or g-, while “*i*” denotes acyclic short chain CBs. In considering the thermodynamic stability of self-associates, we distinguish between non- and inclusion complexes. Consider the data on ion at *m/z* 1881 of species {[(b-CD)(CB1)]+} and {[(g-CD)(CB2)]+}, where the CB1 and CB2 represent acyclic oligosaccharides. The inclusion complexes show lower total energy values. The hydrophobic interaction is the governing force determining the lower energy values. The difference in energy values with respect to type of the macrocyclic CDs is DETOT = |43.91| ({[(b-CD)(CB2)]+}) and |14.79| kcal.mol-1 ({[(g-CD)(CB2)]+}). The analysis of ions at *m/z* 1873, 1881 and 1930 is carried out according to 2D structures (**Figures C2–C4**.) The chemical diagrams of ions at *m/z* 1873 depending on the type of CD are also presented. Despite, the interaction modes between CDs and CBi fragments, the ionic ensembles reveal a significant deformation of the CD ring. Nevertheless, as can be expected, the later type of interactions is characterized by lower energy values. The effect of protonation on the energetics of molecular ensembles is examined looking at 6-hydroxymethyl-tetrahydro-pyran-2,3,4-triol and 6-hydroxymethyl-3,6-dihydro-2H-pyran-3-ol fragments of oligomer CBi species of interacting ensembles (See ions m1930_a and m1930_b of {[(b-CD)(CB3)]+}) The protonation at the former fragment appears thermodynamically preferred. The same is true for cations of type {[(g-CD)(CB4)]+}. The ensembles of ions of type {[(g-CD)(CB4)]+} exhibit lower energy values comparing with those of species of type {[(b-CD)(CB3)]+}. However, in both cases most stable are inclusion complexes of type m1930_b_b. The assignment of the common peak at *m/z* 1930 CDs agrees with the typical fragment path of CD showing difference in one glucosyl unit. It is important to underline why we want to ask, namely, such question about the distinction between fragment paths of b-CD, g-CD, Ac-b-CD and Ac-g-CD yielding to MS peaks at *m/z* 1930. Because of, as aforementioned, and as has been observed experimentally the cleavage of one glycosyl unit appears the major fragment path of CBs. In our case, it is obvious that a direct fragment reaction of one of the discussed units in the Ac13-b-CD should result in a peak at *m/z* 1930. Moreover, the corresponding fragment process in the Ac15-g-CD is associated with cleavage of C–O bond. The energy values of the latter ions are comparable with those of ions of type {[(x-CD)(CBi)]+} of native CDs. Due to this fact we might be in a position that further research effort is needed in order to solve this puzzling issue, dealing with governing forces determining the preferred reaction paths of substituted CDs with respect to experimental and molecular factors. Importantly, the following question requires an adequate answer: Do really the type of the substituent of native CDs chiefly governs the fragment paths? Despite, that the acetyl-group does not appear bulk substituent, it changes significantly the 3D molecular and electronic structures of the CDs. Nevertheless, as aforementioned the energy values of these systems are almost comparable with those obtained for the ensembles of interacting native CDs with short chain oligosaccharides. There is obvious a room for further debates on this issue, which, however, appears out of the scope of the current study. However, the reported in this paper results call upon to address further this topic successfully.

3.2.1.2.2. Self-associates of randomly acetylated cyclodextrins

Mass spectrometric data on randomly acetylated derivatives of native CDs have been presented (Schmitt, 2004.) The Ac-b-CD shows a completely acetylated derivative stabilized as Na+-adducts at *m/z* 2040, while the peaks at *m/z* 1284, 1578 and 1788 have been assigned to Ac3-b-, Ac10-b- and Ac15-b-CDs, respectively. Consider the cases of acetylated derivatives reported, herein, in **Figures 3**, **B2** and **B3**, respectively. Excluding from the common MS ions discussed in the previous subsections, the experimental MS spectra of randomly acetylated CDs within high *m/z* -values distinct unambiguously between the two type Ac-b-CD and Ac-g-CD derivatives. The MS peaks at *m/z* 1398 and 1449 are assigned to Ac7-b-CD derivatives, the latter of which is an ammonium adduct of the acyclic macromolecule. The fact that there are stabilized corresponding cyclic and acyclic derivatives of the CD is expectable in the light of the results reported to the last subsection; due to, as mentioned before the comparable thermodynamic stability of the both cyclic and acyclic self-associates. Like in the cases of the ions at *m/z* 1930 the MD computations result in a significant distortion of the cyclic geometry of the initial CD. The peak at *m/z* 1876 corresponds to methanol solvate ion of Ac13-derivative of g-CD ({[(Ac13-g-CD+H)(CH3OH)]+}; **Figure C5**.) Looking at the molecular structure of this ion, perhaps the most important feature is associated with the position of the proton. The protonation of the OH-group at C2-position yields to a more stable ion (m1876_b) comparing with the corresponding C3-OH2+ containing cation (m1876_a.) The difference in energy is DE = |14.41| kcal.mol-1.The MS peak at *m/z* 1840 belongs to {[(Ac11-g-CD).(K+)]} adducts, having the following theoretical MS parameters: {[C72H104KO52]+}; *m/z* 1839.5131 (100.0 %); and 1840.5165 (80.1.) The corresponding ammonium adduct {[(Ac11-g-CD).(NH4+)]} is observed at *m/z* 1819. Its theoretical MS parameters are: {[C72H108NO52]+}; *m/z* 1818.5838 (100.0 %); and 1819.5871 (80.1.)

Abbildung in dieser Leseprobe nicht enthalten

**Figure 3.** APCI-MS spectra of Ac-g-CD within *m/z* = 1800–200 values with respect to different retention times (RT); APCI-MS spectra of g-CD and b-CD according to reference (Ivanova, & Spiteller, 2019b); the common peaks of ions and characteristic peaks of acetylated derivatives are summarized.

#### 3.2.2. Determination of mass spectrometric diffusion parameters and correlative analysis with the quantum chemical diffusion data

In a nutshell, this subsection deals with quantitative determination of DSD parameters of MS ions of randomly acetylated derivatives of CDs according to equation (1). We have already provided detail on the computational steps, studying the nonsubstituted CDs (Ivanova, & Spiteller, 2019b). Herein, we only present new data on the chemically substituted derivatives (**Figures B1** – **B112**, **Tables 1, 2**, **B1** and **B2**.) One of the important outcomes, amongst other, is that the quantitative treatment of the MS parameter *intensity per* short span of the scan time *via* the DSD-parameters of common ions at *m/z* 141, 158 and 171 of Ac-b- and Ac-g-CDs yields to excellent coefficient of linear correlation |r| = 0.999939 (**Figure 4**.) Conversely, a correlative analysis of the intensity-values of these ions by means of the total intensity (ITOT) data, obtained over the whole time of the measurements yields to |r| = 0.486. Obviously enough, the employment of our innovative methodology and model equation provides highly accurate and precise treatment of the measurable outcome intensity of the analyte ions comparing with the classical approach to treat the ITOT-values over the whole spans of the experimental measurements.

**[...]**