Analysis of the multiparametric cell cycle data.

This chapter was originally written in 2011. The idea was to give some history of cell cycle analysis before and after flow cytometry became widely accessible; provide references to educational material for single parameter DNA content analysis, introduce and discuss multiparameter cell cycle analysis in a methodological style, and in a casual style, discuss aspects of the work over the last 40years that we have given thought, performing some experiments, but didn't publish. It feels like there is a linear progression that moves from counting cells for growth curves, to counting labeled mitotic cells by autoradiography, to DNA content analysis, to cell cycle states defined by immunofluorescence plus DNA content analysis, to extraction of cell cycle expression profiles, and finally to probability state modeling, which should be the "right" way to analyze cytometric cell cycle data. This is the sense of this chapter. In 2023, we have updated it, but the exciting, expansive aspects brought about by spectral and mass cytometry are still young and developing, and thus have not been vetted, reviewed, and presented in mature form.

Temporal Structures in Positron Spectra and Charge-Sign Effects in Galactic Cosmic Rays.

We present the precision measurements of 11 years of daily cosmic positron fluxes in the rigidity range from 1.00 to 41.9 GV based on 3.4×10^{6} positrons collected with the Alpha Magnetic Spectrometer (AMS) aboard the International Space Station. The positron fluxes show distinctly different time variations from the electron fluxes at short and long timescales. A hysteresis between the electron fluxes and the positron fluxes is observed with a significance greater than 5σ at rigidities below 8.5 GV. On the contrary, the positron fluxes and the proton fluxes show similar time variation. Remarkably, we found that positron fluxes are modulated more than proton fluxes with a significance greater than 5σ for rigidities below 7 GV. These continuous daily positron fluxes, together with AMS daily electron, proton, and helium fluxes over an 11-year solar cycle, provide unique input to the understanding of both the charge-sign and mass dependencies of cosmic rays in the heliosphere.

Properties of Cosmic-Ray Sulfur and Determination of the Composition of Primary Cosmic-Ray Carbon, Neon, Magnesium, and Sulfur: Ten-Year Results from the Alpha Magnetic Spectrometer.

We report the properties of primary cosmic-ray sulfur (S) in the rigidity range 2.15 GV to 3.0 TV based on 0.38×10^{6} sulfur nuclei collected by the Alpha Magnetic Spectrometer experiment (AMS). We observed that above 90 GV the rigidity dependence of the S flux is identical to the rigidity dependence of Ne-Mg-Si fluxes, which is different from the rigidity dependence of the He-C-O-Fe fluxes. We found that, similar to N, Na, and Al cosmic rays, over the entire rigidity range, the traditional primary cosmic rays S, Ne, Mg, and C all have sizeable secondary components, and the S, Ne, and Mg fluxes are well described by the weighted sum of the primary silicon flux and the secondary fluorine flux, and the C flux is well described by the weighted sum of the primary oxygen flux and the secondary boron flux. The primary and secondary contributions of the traditional primary cosmic-ray fluxes of C, Ne, Mg, and S (even Z elements) are distinctly different from the primary and secondary contributions of the N, Na, and Al (odd Z elements) fluxes. The abundance ratio at the source for S/Si is 0.167±0.006, for Ne/Si is 0.833±0.025, for Mg/Si is 0.994±0.029, and for C/O is 0.836±0.025. These values are determined independent of cosmic-ray propagation.

Temporal Structures in Electron Spectra and Charge Sign Effects in Galactic Cosmic Rays.

We present the precision measurements of 11 years of daily cosmic electron fluxes in the rigidity interval from 1.00 to 41.9 GV based on 2.0×10^{8} electrons collected with the Alpha Magnetic Spectrometer (AMS) aboard the International Space Station. The electron fluxes exhibit variations on multiple timescales. Recurrent electron flux variations with periods of 27 days, 13.5 days, and 9 days are observed. We find that the electron fluxes show distinctly different time variations from the proton fluxes. Remarkably, a hysteresis between the electron flux and the proton flux is observed with a significance of greater than 6σ at rigidities below 8.5 GV. Furthermore, significant structures in the electron-proton hysteresis are observed corresponding to sharp structures in both fluxes. This continuous daily electron data provide unique input to the understanding of the charge sign dependence of cosmic rays over an 11-year solar cycle.

High-Dimensional Modeling for Cytometry: Building Rock Solid Models Using GemStone™ and Verity Cen-se'™ High-Definition t-SNE Mapping.

This chapter outlines how to approach the complex tasks associated with designing models for high-dimensional cytometry data. Unlike gating approaches, modeling lends itself to automation and accounts for measurement overlap among cellular populations. Designing these models is now easier because of a new technique called high-definition t-SNE mapping. Nontrivial examples are provided that serve as a guide to create models that are consistent with data.

Sometimes simpler is better: VLog, a general but easy-to-implement log-like transform for cytometry.

The fundamental purpose of log and log-like transforms for cytometry is to make measured population variabilities as uniform as possible. The long-standing success of the log transform was its ability to stabilize linearly increasing gain-dependent uncertainties and the success of the log-like transforms is that they extend this notion to include zero and negative measurement values. This study derives and examines a transform called VLog that stabilizes the three general sources of variability: (1) gain-dependent variability, (2) photo-electron counting error, and (3) signal-independent sources of error. Somewhat surprisingly, this transform has a closed-form solution and therefore is relatively simple to implement. By including some quantitation elements in its formulation, the shape-dependent arguments, α and ß, usually do not require optimization for different datasets. The simplicity and generality of the transform may make it a useful tool for cytometry and possibly other technologies. © 2016 International Society for Advancement of Cytometry.

Probability state modeling theory.

As the technology of cytometry matures, there is mounting pressure to address two major issues with data analyses. The first issue is to develop new analysis methods for high-dimensional data that can directly reveal and quantify important characteristics associated with complex cellular biology. The other issue is to replace subjective and inaccurate gating with automated methods that objectively define subpopulations and account for population overlap due to measurement uncertainty. Probability state modeling (PSM) is a technique that addresses both of these issues. The theory and important algorithms associated with PSM are presented along with simple examples and general strategies for autonomous analyses. PSM is leveraged to better understand B-cell ontogeny in bone marrow in a companion Cytometry Part B manuscript. Three short relevant videos are available in the online supporting information for both of these papers. PSM avoids the dimensionality barrier normally associated with high-dimensionality modeling by using broadened quantile functions instead of frequency functions to represent the modulation of cellular epitopes as cells differentiate. Since modeling programs ultimately minimize or maximize one or more objective functions, they are particularly amenable to automation and, therefore, represent a viable alternative to subjective and inaccurate gating approaches.

Human B-cell and progenitor stages as determined by probability state modeling of multidimensional cytometry data.

BACKGROUND: Human progenitor and B-cell development is a highly regulated process characterized by the ordered differential expression of numerous cell-surface and intracytoplasmic antigens. This study investigates the underlying coordination of these modulations by examining a series of normal bone marrow samples with the method of probability state modeling or PSM. RESULTS: The study is divided into two sections. The first section examines B-cell stages subsequent to CD19 up-regulation. The second section assesses an earlier differentiation stage before and including CD19 up-regulation. POST-CD19 ANTIGENIC UP-REGULATION: Statistical analyses of cytometry data derived from sixteen normal bone marrow specimens revealed that B cells have at least three distinct coordinated changes, forming four stages labeled as B1, B2, B3, and B4. At the end of B1; CD34 antigen expression down-regulates with TdT while CD45, CD81, and CD20 slightly up-regulate. At the end of B2, CD45 and CD20 up-regulate. At the end of B3 and beginning of B4; CD10, CD38, and CD81 down-regulate while CD22 and CD44 up-regulate. PRE-CD19 ANTIGENIC UP-REGULATION: Statistical analysis of ten normal bone marrows revealed that there are at least two measurable coordinated changes with progenitors, forming three stages labeled as P1, P2, and P3. At the end of P1, CD38 up-regulates. At the end of P2; CD19, CD10, CD81, CD22, and CD9 up-regulate while CD44 down-regulates slightly. CONCLUSIONS: These objective results yield a clearer immunophenotypic picture of the underlying cellular mechanisms that are operating in these important developmental processes. Also, unambiguously determined stages define what is meant by "normal" B-cell development and may serve as a preliminary step for the development of highly sensitive minimum residual disease detection systems. A companion article is simultaneously being published in Cytometry Part A that will explain in further detail the theory behind PSM. Three short relevant videos are available in the online supporting information for both of these papers.

Automated analysis of flow cytometric data for measuring neutrophil CD64 expression using a multi-instrument compatible probability state model.

BACKGROUND: Leuko64™ (Trillium Diagnostics) is a flow cytometric assay that measures neutrophil CD64 expression and serves as an in vitro indicator of infection/sepsis or the presence of a systemic acute inflammatory response. Leuko64 assay currently utilizes QuantiCALC, a semiautomated software that employs cluster algorithms to define cell populations. The software reduces subjective gating decisions, resulting in interanalyst variability of <5%. We evaluated a completely automated approach to measuring neutrophil CD64 expression using GemStone™ (Verity Software House) and probability state modeling (PSM). METHODS: Four hundred and fifty-seven human blood samples were processed using the Leuko64 assay. Samples were analyzed on four different flow cytometer models: BD FACSCanto II, BD FACScan, BC Gallios/Navios, and BC FC500. A probability state model was designed to identify calibration beads and three leukocyte subpopulations based on differences in intensity levels of several parameters. PSM automatically calculates CD64 index values for each cell population using equations programmed into the model. GemStone software uses PSM that requires no operator intervention, thus totally automating data analysis and internal quality control flagging. Expert analysis with the predicate method (QuantiCALC) was performed. Interanalyst precision was evaluated for both methods of data analysis. RESULTS: PSM with GemStone correlates well with the expert manual analysis, r(2) = 0.99675 for the neutrophil CD64 index values with no intermethod bias detected. The average interanalyst imprecision for the QuantiCALC method was 1.06% (range 0.00-7.94%), which was reduced to 0.00% with the GemStone PSM. The operator-to-operator agreement in GemStone was a perfect correlation, r(2) = 1.000. CONCLUSION: Automated quantification of CD64 index values produced results that strongly correlate with expert analysis using a standard gate-based data analysis method. PSM successfully evaluated flow cytometric data generated by multiple instruments across multiple lots of the Leuko64 kit in all 457 cases. The probability-based method provides greater objectivity, higher data analysis speed, and allows for greater precision for in vitro diagnostic flow cytometric assays.

Probability state modeling of memory CD8⁺ T-cell differentiation.

Flow cytometric analysis enables the simultaneous single-cell interrogation of multiple biomarkers for phenotypic and functional identification of heterogeneous populations. Analysis of polychromatic data has become increasingly complex with more measured parameters. Furthermore, manual gating of multiple populations using standard analysis techniques can lead to errors in data interpretation and difficulties in the standardization of analyses. To characterize high-dimensional cytometric data, we demonstrate the use of probability state modeling (PSM) to visualize the differentiation of effector/memory CD8⁺ T cells. With this model, four major CD8⁺ T-cell subsets can be easily identified using the combination of three markers, CD45RA, CCR7 (CD197), and CD28, with the selection markers CD3, CD4, CD8, and side scatter (SSC). PSM enables the translation of complex multicolor flow cytometric data to pathway-specific cell subtypes, the capability of developing averaged models of healthy donor populations, and the analysis of phenotypic heterogeneity. In this report, we also illustrate the heterogeneity in memory T-cell subpopulations as branched differentiation markers that include CD127, CD62L, CD27, and CD57.

Automated quantitation of fetomaternal hemorrhage by flow cytometry for HbF-containing fetal red blood cells using probability state modeling.

BACKGROUND: Flow cytometric methods (FCMs) are the contemporary standard for fetal red blood cell (RBC) quantitation and fetomaternal hemorrhage (FMH) detection. FCM provides greater sensitivity and repeatability relative to manual microscopic Kleihauer-Betke methods. FCM assays are not totally objective, employing subjective manual gating of fetal RBCs with measureable interobserver imprecision. We investigated Probability State Modeling to automate analysis of fetal RBCs using an assay for hemoglobin F (HbF)-containing RBCs. METHODS: Two hundred human bloods were processed using the FMH QuikQuant™ assay (Trillium Diagnostics, Brewer, ME, USA). A Probability State Model (PSM) was designed to enumerate fetal RBCs by selecting the three RBCs subpopulation based on differences in intensity levels of several parameters. The GemStone™ program uses a PSM that requires no operator intervention. Routine manual analysis by experienced users was performed, along with replicate analyses for both methods. RESULTS: The PSM by GemStone™ correlates strongly with the expert manual analysis, r(2) = 0.9986. The mean absolute difference of the FMH results between GemStone™ and manual 'expert' analysis was 0.04% with no intermethod bias detected. Manual gating demonstrated coefficient of variations (CVs) of 10.6% for intra-analyst replicates and 22.6% for interanalyst imprecision. The interanalyst agreement in GemStone™ is a perfect correlation, r(2) = 1.00, and no imprecision with a 0.00% CV. CONCLUSION: Automated PSM analysis of fetal RBCs strongly correlates with expert traditional manual analysis. PSM enumerates fetal RBCs accurately with significantly greater objectivity and lower imprecision than the traditional manual gating method. Thus, PSM provides a means to markedly improve interlaboratory variance with FMH assays based upon subjective gating strategies.

Automated analysis of flow cytometric data for CD34+ stem cell enumeration using a probability state model.

BACKGROUND: Flow Cytometry is widely used for enumeration of hematopoietic stem cell (SC) levels in bone marrow, cord blood, peripheral blood, and apheresis products. The ISHAGE single-platform gating method is considered by many to be the standard for CD34+ SC enumeration. However, attempts at uniform application of this ISHAGE method have met with only partial success. We propose an automated, multivariate classification approach for SC analysis based on Probability State Modeling™ (PSM). In this study, we compare the results from automated PSM analysis with manual ISHAGE gating analysis as performed by a trained analyst. METHODS: A total of 258 samples were assayed on BD FACSCanto II flow cytometers using a stain-lyse-no-wash technique. Populations were defined using CD34, CD45, 7-AAD, and light scatter. BD TruCount™ bead tubes were used for absolute SC concentrations. A PSM was designed to classify events into beads, debris, intact-dead cells, and intact-live SC; run unattended and record results. RESULTS: The ISHAGE and PSM methods show excellent agreement in estimating the concentration of #SC/µL: slope = 1.009, r² = 0.999. Bland-Altman Analysis for the SC concentration has an average difference (bias) of 2.018 SC/µL. The 95% confidence interval is from -59.350 to 63.380 SC/µL. The operator-to-operator agreement using PSM is perfect: r² = 1.000. CONCLUSIONS: Automated PSM analysis of SC listmode data produces results that agree strongly with ISHAGE gate-based results. The PSM approach provides higher reproducibility, objectivity, and speed with accuracy at least equivalent to the ISHAGE method.

Automated analysis of GPI-deficient leukocyte flow cytometric data using GemStone™.

BACKGROUND: Flow Cytometry is the standard for the detection of glycosylphosphatidylinositol (GPI)-deficient clones in paroxysmal nocturnal hemoglobinuria (PNH) and related disorders. Although the International Clinical Cytometry Society (ICCS) and the International PNH Interest Group (IPIG) have published guidelines for PNH assays, data analysis has not been standardized. Current analyses use manual gating to enumerate PNH cells. We evaluate an automated approach to identify GPI-deficient leukocytes using a GemStone™ (Verity Software House) probability state model (PSM). METHODS: Five hundred and thirty patient samples were assayed on BD Canto II flow cytometers using a stain-lyse-wash technique. Populations were defined using CD15, CD45, CD64 and side scatter. GPI-deficient myeloid cells were recognized as FLAER-, CD24-, and dim or absent CD16. GPI-deficient monocytic cells were identified as FLAER- and CD14-. The data were not censored in any way. A PSM was designed to enumerate monocytic and myeloid cells by adjusting the peaks and line spreads of the data, and recording the normal and GPI-deficient counts. No operator adjustments were made to the automated analysis. RESULTS: By human analysis, 53 of 530 samples showed GPI-deficient clones. Automated analysis identified the same 53 clones; there were 0 false positives and 0 false negatives. Sensitivity was 100% and specificity 100%. Gating and automated results (percent GPI-deficient cells) were highly correlated: r² = 0.997 for monocytic cells, and r² = 0.999 for myeloids. Mean absolute differences were 0.94% for monocytes and 0.78% for myeloid cells. CONCLUSIONS: Automated analysis of GPI-deficient leukocytes produces results that agree strongly with gate-based results. The probability-based approach provides higher speed, objectivity, and reproducibility.

Phenotypic complexity of T regulatory subsets in patients with B-chronic lymphocytic leukemia.

Increased numbers of T regulatory (T(reg)) cells are found in B-chronic lymphocytic leukemia, but the nature and function of these T(regs) remains unclear. Detailed characterization of the T(regs) in chronic lymphocytic leukemia has not been performed and the degree of heterogeneity of among these cells has not been studied to date. Using 15-color flow cytometry we show that T(reg) cells, defined using CD4, CD25, and forkhead box P3 (FOXP3), can be divided into multiple complex subsets based on markers used for naïve, memory, and effector delineation as well as markers of T(reg) activation. Furthermore FOXP3(+) cells can be identified among CD4(+)CD25(-) as well as CD8(+)CD4(-) populations in increased proportions in patients with chronic lymphocytic leukemia compared with healthy donors. Significantly different frequencies of naïve and effector T(regs) populations are found in healthy donor controls compared with donors with chronic lymphocytic leukemia. A population of CCR7(+)CD39(+) T(regs) was significantly associated with chronic lymphocytic leukemia. This population demonstrated slightly reduced suppressive activity compared with total T(regs) or T(regs) of healthy donors. These data suggest that FOXP3-expressing cells, particularly in patients with chronic lymphocytic leukemia are much more complex for T(reg) sub-populations and transitions than previously reported. These findings demonstrate the complexity of regulation of T-cell responses in chronic lymphocytic leukemia and illustrate the use of high-dimensional analysis of cellular phenotypes in facilitating understanding of the intricacies of cellular immune responses and their dysregulation in cancer.

Breaking the dimensionality barrier.

Recent advances in biotechnology have resulted in cytometers capable of performing numerous correlated measurements of cells, often exceeding ten. In the near future, it is likely that this number will increase by fivefold and perhaps even higher. Traditional analysis strategies based on examining one measurement versus another are not suitable for high-dimensional data analysis because the number of measurement combinations expands geometrically with dimension, forming a kind of complexity barrier. This dimensionality barrier limits cytometry and other technologies from reaching their maximum potential in visualizing and analyzing important information embedded in high-dimensional data.This chapter describes efforts to break through this barrier and allow the visualization and analysis of any number of measurements with a new paradigm called Probability State Modeling (PSM). This new system creates a virtual progression variable based on probability that relates all measurements. PSM can produce a single graph that conveys more information about a sample than hundreds of traditional histograms. These PSM overlays reveal the rich interplay of phenotypic changes in cells as they differentiate. The end result is a deeper appreciation of the molecular genetic underpinnings of ontological processes in complex populations such as found in bone marrow and peripheral blood.Eventually these models will help investigators better understand normal and abnormal cellular progressions and will be a valuable general tool for the analysis and visualization of high-dimensional data.

Multiparametric analysis of apoptosis by flow cytometry.

Flow cytometry is the most widely used technology for analyzing apoptosis. The multiparametric nature of flow cytometry allows several apoptotic characteristics to be combined in a single sample, making it a powerful tool for analyzing the complex progression of apoptotic death. This chapter provides guidelines for combining caspase detection, annexin V binding, DNA dye exclusion, and other single apoptotic assays into multiparametric assays.This approach to analyzing apoptosis provides far more information than single parameter assays that provide only an ambiguous "percent apoptotic" result, given that multiple early, intermediate and late apoptotic stages can be visualized simultaneously. This multiparametric approach is also amenable to a variety of flow cytometric instrumentation, both old and new.

Intracellular copper accumulation enhances the growth of Kineococcus radiotolerans during chronic irradiation.

The actinobacterium Kineococcus radiotolerans is highly resistant to ionizing radiation, desiccation, and oxidative stress, though the underlying biochemical mechanisms are unknown. The purpose of this study was to explore a possible linkage between the uptake of transition metals and extreme resistance to ionizing radiation and oxidative stress. The effects of six different divalent cationic metals on growth were examined in the absence of ionizing radiation. None of the metals tested were stimulatory, though cobalt was inhibitory to growth. In contrast, copper supplementation dramatically increased colony formation during chronic irradiation. K. radiotolerans exhibited specific uptake and intracellular accumulation of copper, compared to only a weak response to both iron and manganese supplementation. Copper accumulation sensitized cells to hydrogen peroxide. Acute-irradiation-induced DNA damage levels were similar in the copper-loaded culture and the age-synchronized no-copper control culture, though low-molecular-weight DNA was more persistent during postirradiation recovery in the Cu-loaded culture. Still, the estimated times for genome restoration differed by only 2 h between treatments. While we cannot discount the possibility that copper fulfills an unexpectedly important biochemical role in a low-radioactivity environment, K. radiotolerans has a high capacity for intracellular copper sequestration and presumably efficiently coordinated oxidative stress defenses and detoxification systems, which confers cross-protection from the damaging effects of ionizing radiation.

Phylogenetic and functional biomakers as indicators of bacterial community responses to mixed-waste contamination.

Few studies have demonstrated changes in community structure along a contaminant plume in terms of phylogenetic, functional, and geochemical changes, and such studies are essential to understand how a microbial ecosystem responds to perturbations. Clonal libraries of multiple genes (SSU rDNA, nirK, nirS, amoA, pmoA, and dsrAB) were analyzed from groundwater samples (n = 6) that varied in contaminant levels, and 107 geochemical parameters were measured. Principal components analyses (PCA) were used to compare the relationships among the sites with respect to the biomarker (n = 785 for all sequences) distributions and the geochemical variables. A major portion of the geochemical variance measured among the samples could be accounted for by tetrachloroethene, 99Tc, No3, SO4, Al, and Th. The PCA based on the distribution of unique biomarkers resulted in different groupings compared to the geochemical analysis, but when the SSU rRNA gene libraries were directly compared (deltaC(xy) values) the sites were clustered in a similar fashion compared to geochemical measures. The PCA based upon functional gene distributions each predicted different relationships among the sites, and comparisons of Euclidean distances based upon diversity indices for all functional genes (n = 432) grouped the sites by extreme or intermediate contaminant levels. The data suggested that the sites with low and high perturbations were functionally more similar than sites with intermediate conditions, and perhaps captured the overall community structure better than a single phylogenetic biomarker. Moreover, even though the background site was phylogenetically and geochemically distinct from the acidic sites, the extreme conditions of the acidic samples might be more analogous to the limiting nutrient conditions of the background site. An understanding of microbial community-level responses within an ecological framework would provide better insight for restoration strategies at contaminated field sites.

Hyperlog-a flexible log-like transform for negative, zero, and positive valued data.

BACKGROUND: The remarkable success of cytometry over the past 30 years is largely due to its uncanny ability to display populations that vastly differ in numbers and fluorescence intensities on one scale. The log transform implemented in hardware as a log amplifier or in software normalizes signals or channels so that these populations appear as clearly discernible peaks. With the advent of multiple fluorescence cytometry, spectral crossover compensation of these signals has been necessary to properly interpret the data. Unfortunately, because compensation is a subtractive process, it can produce negative and zero valued data. The log transform is undefined for these values and, as a result, forces computer algorithms to truncate these values, creating a few problems for cytometrists. Data truncation biases displays making properly compensated data appear undercompensated; thus, enticing many operators to overcompensate their data. Also, events truncated into the first histogram channel are not normally visible with typical two-dimensional graphic displays, thus hiding a large number of events and obscuring the true proportionality of negative distributions. In addition, the log transform creates unequal binning that can dramatically distort negative population distributions. METHODS AND RESULTS: The HyperLog transform is a log-like transform that admits negative, zero, and positive values. The transform is a hybrid type of transform specifically designed for compensated data. One of its parameters allows it to smoothly transition from a logarithmic to linear type of transform that is ideal for compensated data. CONCLUSIONS: The HyperLog transform is easily implemented in computer systems and results in display systems that present compensated data in an unbiased manner.

Coupling of functional gene diversity and geochemical data from environmental samples.

Genomic techniques commonly used for assessing distributions of microorganisms in the environment often produce small sample sizes. We investigated artificial neural networks for analyzing the distributions of nitrite reductase genes (nirS and nirK) and two sets of dissimilatory sulfite reductase genes (dsrAB1 and dsrAB2) in small sample sets. Data reduction (to reduce the number of input parameters), cross-validation (to measure the generalization error), weight decay (to adjust model parameters to reduce generalization error), and importance analysis (to determine which variables had the most influence) were useful in developing and interpreting neural network models that could be used to infer relationships between geochemistry and gene distributions. A robust relationship was observed between geochemistry and the frequencies of genes that were not closely related to known dissimilatory sulfite reductase genes (dsrAB2). Uranium and sulfate appeared to be the most related to distribution of two groups of these unusual dsrAB-related genes. For the other three groups, the distributions appeared to be related to pH, nickel, nonpurgeable organic carbon, and total organic carbon. The models relating the geochemical parameters to the distributions of the nirS, nirK, and dsrAB1 genes did not generalize as well as the models for dsrAB2. The data also illustrate the danger (generating a model that has a high generalization error) of not using a validation approach in evaluating the meaningfulness of the fit of linear or nonlinear models to such small sample sizes.