Derivation of the Statistical Distribution of the Mass Peak Centroids of Mass Spectrometers Employing Analog-to-Digital Converters and Electron Multipliers.

The mass peak centroid is a quantity that is at the core of mass spectrometry (MS). However, despite its central status in the field, models of its statistical distribution are often chosen quite arbitrarily and without attempts at establishing a proper theoretical justification for their use. Recent work has demonstrated that for mass spectrometers employing analog-to-digital converters (ADCs) and electron multipliers, the statistical distribution of the mass peak intensity can be described via a relatively simple model derived essentially from first principles. Building on this result, the following article derives the corresponding statistical distribution for the mass peak centroids of such instruments. It is found that for increasing signal strength, the centroid distribution converges to a Gaussian distribution whose mean and variance are determined by physically meaningful parameters and which in turn determine bias and variability of the m/z measurements of the instrument. Through the introduction of the concept of "pulse-peak correlation", the model also elucidates the complicated relationship between the shape of the voltage pulses produced by the preamplifier and the mean and variance of the centroid distribution. The predictions of the model are validated with empirical data and with Monte Carlo simulations.

Derivation from first principles of the statistical distribution of the mass peak intensities of MS data.

Despite the widespread use of mass spectrometry (MS) in a broad range of disciplines, the nature of MS data remains very poorly understood, and this places important constraints on the quality of MS data analysis as well as on the effectiveness of MS instrument design. In the following, a procedure for calculating the statistical distribution of the mass peak intensity for MS instruments that use analog-to-digital converters (ADCs) and electron multipliers is presented. It is demonstrated that the physical processes underlying the data-generation process, from the generation of the ions to the signal induced at the detector, and on to the digitization of the resulting voltage pulse, result in data that can be well-approximated by a Gaussian distribution whose mean and variance are determined by physically meaningful instrumental parameters. This allows for a very precise understanding of the signal-to-noise ratio of mass peak intensities and suggests novel ways of improving it. Moreover, it is a prerequisite for being able to address virtually all data analytical problems in downstream analyses in a statistically rigorous manner. The model is validated with experimental data.

Ion manipulations in structures for lossless ion manipulations (SLIM): computational evaluation of a 90° turn and a switch.

The process of redirecting ions through 90° turns and 'tee' switches utilizing Structures for Lossless Ion Manipulations (SLIM) was evaluated at 4 Torr pressure using SIMION simulations and theoretical methods. The nature of pseudo-potential in SLIM-tee structures has also been explored. Simulations show that 100% transmission efficiency in SLIM devices can be achieved with guard electrode voltages lower than ∼10 V. The ion plume width in these conditions is ∼1.6 mm while at lower guard voltages lead to greater plume widths. Theoretical calculations show marginal loss of ion mobility resolving power (<5%) during ion turn due to the finite plume widths (i.e. race track effect). More robust SLIM designs that reduce the race track effect while maximizing ion transmission are also reported. In addition to static turns, the dynamic switching of ions into orthogonal channels was also evaluated both using SIMION ion trajectory simulations and experimentally. Simulations and theoretical calculations were in close agreement with experimental results and were used to develop more refined SLIM designs.

Efficient calculation of exact fine structure isotope patterns via the multidimensional Fourier transform.

The isotope patterns of unknown analytes provide information that can be of great value in their identification as part of a mass spectrometry experiment. Determining the range of compounds that are consistent with an empirically observed isotope pattern requires, as an initial step, the calculation of the theoretical isotope patterns of all feasible candidate formulas, and this is not a trivial mathematical task. While algorithms based on the Fourier transform have been used for almost two decades to perform such calculation efficiently, they have hitherto not been able to provide the exact sets of masses and abundances that constitute the fundamental isotope pattern. This article presents a new approach to the treatment of such calculations, which involves arranging and manipulating the isotope patterns of distinct elements as multidimensional data structures. This enables the use of the multidimensional Fourier transform to calculate isotope patterns with an accuracy that is limited only by the errors of floating point arithmetic. The algorithm is both highly efficient and very easy to implement in many programming environments. An open-source implementation of the algorithm in the R programming language will be made publicly available and is also available upon request.

Construction of confidence regions for isotopic abundance patterns in LC/MS data sets for rigorous determination of molecular formulas.

It has long been recognized that estimates of isotopic abundance patterns may be instrumental in identifying the many unknown compounds encountered when conducting untargeted metabolic profiling using liquid chromatography/mass spectrometry. While numerous methods have been developed for assigning heuristic scores to rank the degree of fit of the observed abundance patterns with theoretical ones, little work has been done to quantify the errors that are associated with the measurements made. Thus, it is generally not possible to determine, in a statistically meaningful manner, whether a given chemical formula would likely be capable of producing the observed data. In this paper, we present a method for constructing confidence regions for the isotopic abundance patterns based on the fundamental distribution of the ion arrivals. Moreover, we develop a method for doing so that makes use of the information pooled together from the measurements obtained across an entire chromatographic peak, as well as from any adducts, dimers, and fragments observed in the mass spectra. This greatly increases the statistical power, thus enabling the analyst to rule out a potentially much larger number of candidate formulas while explicitly guarding against false positives. In practice, small departures from the model assumptions are possible due to detector saturation and interferences between adjacent isotopologues. While these factors form impediments to statistical rigor, they can to a large extent be overcome by restricting the analysis to moderate ion counts and by applying robust statistical methods. Using real metabolic data, we demonstrate that the method is capable of reducing the number of candidate formulas by a substantial amount, even when no bromine or chlorine atoms are present. We argue that further developments in our ability to characterize the data mathematically could enable much more powerful statistical analyses.

A statistically rigorous test for the identification of parent-fragment pairs in LC-MS datasets.

Untargeted global metabolic profiling by liquid chromatography-mass spectrometry generates numerous signals that are due to unknown compounds and whose identification forms an important challenge. The analysis of metabolite fragmentation patterns, following collision-induced dissociation, provides a valuable tool for identification, but can be severely impeded by close chromatographic coelution of distinct metabolites. We propose a new algorithm for identifying related parent-fragment pairs and for distinguishing these from signals due to unrelated compounds. Unlike existing methods, our approach addresses the problem by means of a hypothesis test that is based on the distribution of the recorded ion counts, and thereby provides a statistically rigorous measure of the uncertainty involved in the classification problem. Because of technological constraints, the test is of primary use at low and intermediate ion counts, above which detector saturation causes substantial bias to the recorded ion count. The validity of the test is demonstrated through its application to pairs of coeluting isotopologues and to known parent-fragment pairs, which results in test statistics consistent with the null distribution. The performance of the test is compared with a commonly used Pearson correlation approach and found to be considerably better (e.g., false positive rate of 6.25%, compared with a value of 50% for the correlation for perfectly coeluting ions). Because the algorithm may be used for the analysis of high-mass compounds in addition to metabolic data, we expect it to facilitate the analysis of fragmentation patterns for a wide range of analytical problems.

Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems.

Approximate Bayesian computation (ABC) methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper, we discuss and apply an ABC method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC provides information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well-known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.

Orders of magnitude extension of the effective dynamic range of TDC-based TOFMS data through maximum likelihood estimation.

In a recent article, we derived a probability distribution that was shown to closely approximate that of the data produced by liquid chromatography time-of-flight mass spectrometry (LC/TOFMS) instruments employing time-to-digital converters (TDCs) as part of their detection system. The approach of formulating detailed and highly accurate mathematical models of LC/MS data via probability distributions that are parameterized by quantities of analytical interest does not appear to have been fully explored before. However, we believe it could lead to a statistically rigorous framework for addressing many of the data analytical problems that arise in LC/MS studies. In this article, we present new procedures for correcting for TDC saturation using such an approach and demonstrate that there is potential for significant improvements in the effective dynamic range of TDC-based mass spectrometers, which could make them much more competitive with the alternative analog-to-digital converters (ADCs). The degree of improvement depends on our ability to generate mass and chromatographic peaks that conform to known mathematical functions and our ability to accurately describe the state of the detector dead time-tasks that may be best addressed through engineering efforts.

Prospects for a statistical theory of LC/TOFMS data.

The critical importance of employing sound statistical arguments when seeking to draw inferences from inexact measurements is well-established throughout the sciences. Yet fundamental statistical methods such as hypothesis testing can currently be applied to only a small subset of the data analytical problems encountered in LC/MS experiments. The means of inference that are more generally employed are based on a variety of heuristic techniques and a largely qualitative understanding of their behavior. In this article, we attempt to move towards a more formalized approach to the analysis of LC/TOFMS data by establishing some of the core concepts required for a detailed mathematical description of the data. Using arguments that are based on the fundamental workings of the instrument, we derive and validate a probability distribution that approximates that of the empirically obtained data and on the basis of which formal statistical tests can be constructed. Unlike many existing statistical models for MS data, the one presented here aims for rigor rather than generality. Consequently, the model is closely tailored to a particular type of TOF mass spectrometer although the general approach carries over to other instrument designs. Looking ahead, we argue that further improvements in our ability to characterize the data mathematically could enable us to address a wide range of data analytical problems in a statistically rigorous manner.