Data analysis methods for synthetic polymer mass spectrometry: autocorrelation

Journal of Research of the National Institute of Standards and Technology, Jan, 2002 by William E. Wallace, Charles M. Guttman

Autocorrelation is shown to be useful in describing the periodic patterns found in high- resolution mass spectra of synthetic polymers. Examples of this usefulness are described for a simple linear homopolymer to demonstrate the method fundamentals, a condensation polymer to demonstrate its utility in understanding complex spectra with multiple repeating patterns on different mass scales, and a condensation copolymer to demonstrate how it can elegantly and efficiently reveal unexpected phenomena. It is shown that using autocorrelation to determine where the signal devolves into noise can be useful in determining molecular mass distributions of synthetic polymers, a primary focus of the NIST synthetic polymer mass spectrometry effort. The appendices describe some of the effects of transformation from time to mass space when time-of-flight mass separation is used, as well as the effects of non-trivial baselines on the autocorrelation function.

Key words: autocorrelation; correlation function; data analysis methods; informatics; mass spectrometry; polymer; time series.

1. Introduction

The advent of rapid, high-resolution, broad-mass-range mass spectrometry has revolutionized synthetic polymer single-chain characterization (1). Along with this new measurement technology has come a flood of high-quality mass spectral data of an exceedingly complex nature. It is not unusual for synthetic polymer mass spectra to contain hundreds of separate peaks even when excluding those simply derived from naturally-occurring isotope distributions. Automated data analysis methods are needed in order to make full and timely use of the data.

Time series analysis, which first came to fore with the publication of Norbert Wiener's seminal text Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (2) in 1949, has proved invaluable in many fields of data analysis. Weiner's text represents the first complete exposition of the study of operations on time series, including auto-correlation and cross-correlation. In the intervening years these correlation methods have been applied to many types of mass spectral data for many purposes (3-5). Owens has reviewed the use of correlation functions in mass spectroscopy, in particular, the use of autocorrelation and crosscorrelation as applied to ion fragments in order to identify small organic molecules in standard libraries (6). Hercules and coworkers have used autocorrelation of isotope distributions as a method to optimize automated data collection (7). Here we discuss the application to synthetic polymer mass spectra for the purpose of efficiently extracting inf ormation from complex data.

First we define the mass autocorrelation and show how to treat the data properly for its use. Then we present autocorrelation for a spectrum of a simple polyethylene oxide homopolymer to establish the fundamentals. Following that we present data on two more complicated structures, specifically two silsesquioxanes produced by condensation polymerization (8) in which the mass spectra can be related directly to the polymer architecture. Finally, we apply autocorrelation to the issue of quantitation in polymer mass spectrometry using the example of polybutacliene.

2. The Mass Autocorrelation Function

We define the mass autocorrelation function as

G(L) = [summation over (i)]S([m.sub.i])S([m.sub.i L])/[summation over (i)]S([m.sub.i])S([m.sub.i]) (1)

where S([m.sub.i]) is signal at mass [m.sub.i] taken on equal intervals of mass, [delta]m. Equal intervals of mass are used because most correlation algorithms, and the closely related field of fast Fourier transforms (FFT), require the signal to be evenly spaced points on the scale of interest.

Time-of-flight (TOF) mass separation (9) is the technique most often applied to synthetic polymers due to their high molecular masses, typically in excess of 1000 u and often much greater (into the 100 000 u range and beyond). No other mass separation technique can reach such high masses. The TOF signal, s([t.sub.i]), is collected on equal intervals of time. The transformation from this time-base signal s([t.sub.i]) to a mass-base signal S([m.sub.i]) involves both an interpolation and a change of the signal itself by a Jacobean transform. The mathematics to affect this transformation is discussed in Appendix A.

3. Example 1: A Simple Linear Homopolymer

The most obvious use of mass autocorrelation function is to get an accurate representation of the repeat unit of the polymer. This can be difficult in a spectrum with noise where identification of peak position will inevitably lack precision and lead to inaccuracies in calculating the repeat unit mass. Figure 1 shows the mass spectrum for a low-molecular-mass polyethylene oxide (repeat unit: [-[CH.sub.2]-[CH.sub.2]-O-]); while Fig. 2 is its autocorrelation function with different values of [delta]m. Data were obtained by matrix-assisted laser desorption/ionization (MALDI) TOF mass spectrometry (10, 11). Before autocorrelation a baseline was pulled off the data in time space and the data was subsequently transformed from time space to mass space by the partial integration method described in Appendix A. The autocorrelation clearly shows the 44.03 u repeat unit of polyethylene oxide with a precision difficult to match by simply picking adjacent peaks and calculating a mass difference.