Complex numbers in chemometrics: Examples from multivariate impedance measurements on lipid monolayers

doi:10.1016/j.aca.2007.01.037

Analytica Chimica Acta

Volume 595, Issues 1–2, 9 July 2007, Pages 152-159

https://doi.org/10.1016/j.aca.2007.01.037 Get rights and content

Abstract

Electrical impedance gives multivariate complex number data as results. Two examples of multivariate electrical impedance data measured on lipid monolayers in different solutions give rise to matrices (16 × 50 and 38 × 50) of complex numbers. Multivariate data analysis by principal component analysis (PCA) or singular value decomposition (SVD) can be used for complex data and the necessary equations are given. The scores and loadings obtained are vectors of complex numbers. It is shown that the complex number PCA and SVD are better at concentrating information in a few components than the naïve juxtaposition method and that Argand diagrams can replace score and loading plots. Different concentrations of Magainin and Gramicidin A give different responses and also the role of the electrolyte medium can be studied. An interaction of Gramicidin A in the solution with the monolayer over time can be observed.

Introduction

Chemometrics has been very successful in applying multivariate data analysis, but almost exclusively by filling vectors, matrices and multiway arrays with real numbers. In electrochemistry however, some of the data are given as complex numbers. The questions that can be asked about this situation are: (1) Do complex numbers fit in with the methods of chemometrics? (2) Should the methods of chemometrics be modified or completely changed when complex numbers are used? (3) Do complex number data lead to different interpretations when used in the complex form? The paper tries to introduce the basic concepts and to give partial answers to the above questions. The examples used are based on multivariate impedance measurements on synthetic biomembranes, also called lipid monolayers, deposited on a mercury electrode. The nomenclature and symbols used in the paper come from three different fields: electrochemistry, complex number theory and linear algebra. Where necessary, ways of avoiding naming and indexing confusion are used.

Two examples are used: (1) a 16 × 50 complex matrix of impedance data for the interaction of different concentrations of magainin with a monomolecular lipid layer deposited on a mercury electrode [1] and (2) a 38 × 50 complex matrix of impedance data using different concentrations in two different electrolyte media of gramicidin A used with the same electrode system. Different aspects of the gramicidin example are given elsewhere [2].

Section snippets

Complex numbers

Complex numbers are given as: $x = a + j b$ or as $x = | x | e^{j t}$ where x is complex number x; a the real part of x; b the imaginary part of x; j: j² = −1, usually called i outside electrochemistry; |x|: (a² + b²)^1/2 the amplitude; t is the phase angle, tan(t) = b/a.

Properties of complex numbers are described in the mathematics literature [3], [4], [5], [6]. Eqs. (1) and (2) are in the standard notation used in many mathematics books.

An important graphical device is the Argand diagram, sometimes called Wessel plot in

Basic physics

An electrode system in solution conducting a sinusoidal current may be seen as a capacitor. For an ideal capacitor in vacuum the capacitance equation is: $C = \frac{ε_{0} A}{d}$ The capacitance C is proportional to the area A and inversely proportional to the distance between the plates d. Capacitance is measured in Farad (F). The permittivity of vacuum ɛ₀ is 8.85 × 10⁻¹² F m⁻¹.

When the capacitor is filled with a non-conducting (dielectric) material, the equation is changed: $C = \frac{k ε_{0} A}{d} = \frac{ε A}{d}$ where k is the relative

Data analysis

With multivariate measurements, the nomenclature definitions from linear algebra are used: x is a scalar, x is a vector, and X is a matrix. All vectors are column vectors and a row vector is the transpose of a column vector. A transposed vector x^T is a row vector. x^H is the combined transpose and complex conjugate [11]. For real number vectors: x^H = x^T. From now on only the above notation will be used. For clarity, the size of a matrix will be indicated between parentheses e.g. Z (I × K) means a

Data pretreatment

In chemometrics, data are sometimes pretreated before a multivariate analysis is carried out. Basic pretreatments are mean-centering over the columns and scaling of each column by its standard deviation. For complex number data, these operations are slightly different from those for real numbers.

The mean z (K × 1) of a complex matrix Z (I × K) is a vector of complex mean values: $z^{T} = \frac{1^{T} Z}{I}$ where 1 is a vector of ones (I × 1) mean-centering of Z is done as follows: $Z_{c} = Z - 1 z^{T}$ where Z_c is a mean-centered matrix

Experimental

A monomolecular layer of dioleoyl phophatidylcholine (DOPC) can be deposited on a mercury drop electrode and impedance measurements are made at 50 frequencies 0.1 Hz, 0.13, 0.17, 0.23, …, 49,463 and 65,000 Hz (each frequency is the previous one multiplied by 1.3141) at 5 mV rms and −400 mV DC. The reference electrode was Ag/AgCl in 3.5 M KCl and the counter electrode was Pt. The measurement cell was at 25 °C in an Argon flow. Current generating and recording electronics was from Autolab. Current and

Results and discussion

The magainin data were analyzed by PCA, without any pretreatment. For the 16 × 100 real matrix Y, components 1 and 2 explained 97% and 2.6% of the total sum of squares, respectively. For the 16 × 50 complex matrix Z, components 1 and 2 explained 99.3% and 0.5% of the total sum of squares. One may conclude for this data set that the complex number PCA is better at concentrating the information in the first components.

Fig. 1 shows the Argand diagram of the complex first component. It can be seen that

Conclusion

Complex numbers occur often in chemistry. All Fourier transform based methods produce complex numbers: infrared and Raman spectroscopy, near infrared spectroscopy, nuclear magnetic resonance. In all these methods the Fourier transform is used to transform an interferogram to a spectrum and the spectrum is basically one of complex numbers. There is however a difference with electrochemistry in that the spectra are expected to be amplitude spectra and all effort is made to remove the phase part.

Acknowledgements

Many thanks to Conor Whitehouse and Kathryn Bradley (SOMS, Leeds) for carrying out the experiments and obtaining the data on which this paper is based. The laboratory work was funded by EPSRC Grant No. GR/R67/439/01. The sixth framework EU project BBMO is acknowledged for financial support.

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Complex numbers in chemometrics: Examples from multivariate impedance measurements on lipid monolayers

Abstract

Introduction

Section snippets

Complex numbers

Basic physics

Data analysis

Data pretreatment

Experimental

Results and discussion

Conclusion

Acknowledgements

J. Electroanal. Chem.

Biophys. J.

Mathematical Analysis

Mathematical Methods for Science Students

Mathematical Methods in the Physical Sciences

Calculus, vol. II

Physics Part I and II

Bioimpedance and Bioelectricity Basics

Dielectric properties of tissues and biological materials: a critical review

Crit. Rev. Biomed. Eng.