Fluorescence fluctuation analysis for the study of interactions between oligonucleotides and polycationic polymers

The interactions between a cationic polymer, poly(2-dimethylamino)ethyl methacrylate (pDMAEMA), and negatively charged rhodamine-labeled 25-mer phosphodiester oligonucleotides (Rh-ONs) were studied by fluorescence fluctuation spectroscopy and other techniques. The composition of the pDMAEMA/Rh-ON complexes was investigated as a function of the charge ratio (+/-) by increasing the pDMAEMA concentration and keeping the Rh-ON concentration constant. We applied two different methods for analyzing the fluorescence fluctuation profiles of the pDMAEMA/Rh-ON complexes, which depended on their composition. First, we analyzed the data with the photon counting histogram (PCH) technique, which determines the molecular brightness and the concentration of fluorophores (Chen et al, 1999). A particular challenge for the data analysis is the occurrence of sudden fluorescence bursts in the fluorescence fluctuation profiles, which are linked to the appearance of multimolecular complexes (i. e. when several Rh-ONs were present in one complex). A quantitative interpretation of the analysis for the complexes remains challenging and is connected to the rarity of the fluorescent bursts, which do not provide sufficient data statistics. To specifically address the problem of the fluorescent bursts we employed a method described by Van Craenenbroeck et al. (1999). This method, applicable only when data were integrated over much longer time bins, allowed us to estimate the number of fluorescence bursts which could be considered as a relative measure of the amount of multimolecular complexes present. When monomolecular complexes were formed, i. e. at high values of the charge ratio, highly intense fluorescence peaks were not present and the interpretation of the PCH analysis was more straightforward. The molecular brightness of the species (epsilon), as revealed from PCH analysis, was greater than epsilon for the free Rh-ONs, indicating that the Rh-ONs were attached to pDMAEMA chains.     

Dual-color photon-counting histogram

We report on the development of dual-color photon-counting histogram (PCH) analysis. Dual-color PCH is an extension of regular PCH and considers the photon counts received in two detection channels instead of one. Because each detection channel records a different color, dual-color PCH distinguishes fluorescent species not only by differences in their brightness, but also according to their color. The additional discrimination by color increases the sensitivity of PCH in resolving a mixture of species considerably. Most dual-color fluorescence fluctuation experiments are performed on fluorophores with overlapping emission spectra. This overlap results in spectral cross talk between the detector channels, which reduces resolvability. Here, we demonstrate that dual-color PCH is able to resolve binary dye mixtures in the presence of cross talk from a single measurement without any additional information about the sample. We discuss the effect of sampling time on the fit parameters of dual-color PCH. Differences between dual-color fluorescence correlation spectroscopy and dual-color PCH will also be addressed. We quantitatively resolve a mixture of the two fluorescent proteins CFP and YFP, which is challenging because of the strong spectral overlap of their emission spectra. Dichroic mirrors are needed to direct the light into the two detection channels. We quantify the influence of these filters on dual-color PCH analysis and determine the optimal transition wavelength of the dichroic mirror for the CFP-YFP pair.     

Calculating confidence intervals for prediction error in microarray classification using resampling

Cross-validation based point estimates of prediction accuracy are frequently reported in microarray class prediction problems. However these point estimates can be highly variable, particularly for small sample numbers, and it would be useful to provide confidence intervals of prediction accuracy. We performed an extensive study of existing confidence interval methods and compared their performance in terms of empirical coverage and width. We developed a bootstrap case cross-validation (BCCV) resampling scheme and defined several confidence interval methods using BCCV with and without bias-correction. The widely used approach of basing confidence intervals on an independent binomial assumption of the leave-one-out cross-validation errors results in serious under-coverage of the true prediction error. Two split-sample based methods previously proposed in the literature tend to give overly conservative confidence intervals. Using BCCV resampling, the percentile confidence interval method was also found to be overly conservative without bias-correction, while the bias corrected accelerated (BCa) interval method of Efron returns substantially anti-conservative confidence intervals. We propose a simple bias reduction on the BCCV percentile interval. The method provides mildly conservative inference under all circumstances studied and outperforms the other methods in microarray applications with small to moderate sample sizes.     

Cumulant analysis in fluorescence fluctuation spectroscopy

A novel technique for the analysis of fluorescence fluctuation experiments is introduced. Fluorescence cumulant analysis (FCA) exploits the factorial cumulants of the photon counts and resolves heterogeneous samples based on differences in brightness. A simple analytical model connects the cumulants of the photon counts with the brightness epsilon and the number of molecules N in the optical observation volume for each fluorescent species. To provide the tools for a rigorous error analysis of FCA, expressions for the variance of factorial cumulants are developed and tested. We compare theory with experiment by analyzing dye mixtures and simple fluorophore solutions with FCA. A comparison of FCA with photon-counting histogram (PCH) analysis, a related technique, shows that both methods give identical results within experimental uncertainty. Both FCA and PCH are restricted to data sampling times that are short compared to the diffusion time of molecules through the observation volume of the instrument. But FCA theory, in contrast to PCH, can be extended to treat arbitrary sampling times. Here, we derive analytical expressions for the second factorial cumulant as a function of the sampling time and demonstrate that the theory successfully models fluorescence fluctuation data.     

The photon counting histogram in fluorescence fluctuation spectroscopy.

Fluorescence correlation spectroscopy (FCS) is generally used to obtain information about the number of fluorescent particles in a small volume and the diffusion coefficient from the autocorrelation function of the fluorescence signal. Here we demonstrate that photon counting histogram (PCH) analysis constitutes a novel tool for extracting quantities from fluorescence fluctuation data, i.e., the measured photon counts per molecule and the average number of molecules within the observation volume. The photon counting histogram of fluorescence fluctuation experiments, in which few molecules are present in the excitation volume, exhibits a super-Poissonian behavior. The additional broadening of the PCH compared to a Poisson distribution is due to fluorescence intensity fluctuations. For diffusing particles these intensity fluctuations are caused by an inhomogeneous excitation profile and the fluctuations in the number of particles in the observation volume. The quantitative relationship between the detected photon counts and the fluorescence intensity reaching the detector is given by Mandel's formula. Based on this equation and considering the fluorescence intensity distribution in the two-photon excitation volume, a theoretical expression for the PCH as a function of the number of molecules in the excitation volume is derived. For a single molecular species two parameters are sufficient to characterize the histogram completely, namely the average number of molecules within the observation volume and the detected photon counts per molecule per sampling time epsilon. The PCH for multiple molecular species, on the other hand, is generated by successively convoluting the photon counting distribution of each species with the others. The influence of the excitation profile upon the photon counting statistics for two relevant point spread functions (PSFs), the three-dimensional Gaussian PSF conventionally employed in confocal detection and the square of the Gaussian-Lorentzian PSF for two photon excitation, is explicitly treated. Measured photon counting distributions obtained with a two-photon excitation source agree, within experimental error with the theoretical PCHs calculated for the square of a Gaussian-Lorentzian beam profile. We demonstrate and discuss the influence of the average number of particles within the observation volume and the detected photon counts per molecule per sampling interval upon the super-Poissonian character of the photon counting distribution.     

Bayesian estimation for species identification in single-molecule fluorescence microscopy

In this article we describe a recursive Bayesian estimator for the identification of diffusing fluorophores using photon arrival-time data from a single spectral channel. We present derivations for all relevant diffusion and fluorescence models, and we use simulated diffusion trajectories and photon streams to evaluate the estimator's performance. We consider simplified estimation schemes that bin the photon counts within time intervals of fixed duration, and show that they can perform well in realistic parameter regimes. The latter results indicate the feasibility of performing identification experiments in real time. It will be straightforward to generalize our approach for use in more complicated scenarios, e.g., with multiple spectral channels or fast photophysical dynamics.     

Fluorescence fluctuation spectroscopy in the presence of immobile fluorophores

Fluorescence contributions from immobile sources present a challenge for fluorescence fluctuation spectroscopy (FFS) because the absence of signal fluctuations from stationary fluorophores leads to a biased analysis. This is especially of concern for cellular FFS studies on proteins that interact with immobile structures. Here we present a method that correctly analyzes FFS experiments in the presence of immobile sources by exploiting selective photobleaching of immobile fluorophores. The fluorescence decay due to photobleaching of the immobile species is modeled taking into account the nonuniform illumination volume. The experimentally observed decay curve serves to separate the mobile and immobile fluorescence contribution, which is used to calculate the molecular brightness from the FFS data. We experimentally verify this approach in vitro using the fluorescent protein EGFP as our immobilized species and a diffusing dye of a different color as the mobile one. For this special case, we also use an alternative method of determining the brightness by spectrally resolving the two species. By conducting a dilution study, we show that the correct parameters are obtained using either technique for a wide range of mobile fractions. To demonstrate the application of our technique in living cells, we perform experiments using the histone core protein H2B fused with EGFP expressed in COS-1 cells. We successfully recovered the brightness of the mobile fraction of H2B-EGFP.     

Spectral fluctuation of a single fluorophore conjugated to a protein molecule

We have measured the fluorescence spectra of a single fluorophore attached to a single protein molecule in aqueous solution using a total internal reflection fluorescence microscope. The most reactive cysteine residue of myosin subfragment-1 (S1) was labeled with tetramethylrhodamine. The spectral shift induced by a change in solvent from aqueous buffer to methanol in both single-molecule and bulk measurements were similar, indicating that, even at the single molecule level, the fluorescence spectrum is sensitive to microenvironmental changes of fluorophores. The time dependence of the fluorescence spectra of fluorophores attached to S1 molecules solely showed a fluctuation with time over a time scale of seconds. Because the fluorescence spectra of the same fluorophores directly conjugated to a glass surface remained constant, the spectral fluctuation observed for the fluorophores attached to S1 is most likely due to slow spontaneous conformational changes in the S1 molecule. Thus, single-molecule fluorescence spectroscopy appears to be a powerful tool to study the dynamic behavior of single biomolecules.     

Dual-color photon counting histogram analysis of mRFP1 and EGFP in living cells

We investigate the potential of dual-color photon counting histogram (PCH) analysis to resolve fluorescent protein mixtures directly inside cells. Because of their small spectral overlap, we have chosen to look at the fluorescent proteins EGFP and mRFP1. We experimentally demonstrate that dual-color PCH quantitatively resolves a mixture of EGFP and mRFP1 in cells from a single measurement. To mimic the effect of protein association, we constructed a fusion protein of EGFP and mRFP1 (denoted EGFP-mRFP1). Fluorescence resonant energy transfer within the fusion protein alters the dual-channel brightness of the fluorophores. We describe a model for fluorescence resonant energy transfer effects on the brightness and incorporate it into dual-color PCH analysis. The model is verified using fluorescence lifetime measurements. Dual-color PCH analysis demonstrated that not all of the expressed EGFP-mRFP1 fusion proteins contained a fluorescent mRFP1 molecule. Fluorescence lifetime and emission spectra measurements confirmed this surprising result. Additional experiments show that the missing fluorescent fraction of mRFP1 is consistent with a dark state population of mRFP1. We successfully resolved this mixture of fusion proteins with a single dual-color PCH measurement. These results highlight the potential of dual-color PCH to directly detect and quantify protein mixtures in living cells.     

The photon counting histogram in fluorescence fluctuation spectroscopy with non-ideal photodetectors

Fluorescence fluctuation spectroscopy utilizes the signal fluctuations of single molecules for studying biological processes. Information about the biological system is extracted from the raw data by statistical methods such as used in fluctuation correlation spectroscopy or photon counting histogram (PCH) analysis. Since detectors are never ideal, it is crucial to understand the influence of photodetectors on signal statistics to correctly interpret the experimental data. Here we focus on the effects of afterpulsing and detector dead-time on PCH statistics. We determine the dead-time and afterpulse probability for our detectors experimentally and show that afterpulsing can be neglected for most experiments. Dead-time effects on the PCH are concentration-dependent and become significant when more than one molecule is present in the excitation volume. We develop a new PCH theory that includes dead-time effects and verify it experimentally. Additionally, we derive a simple analytical expression that accurately predicts the effect of dead-time on the molecular brightness. Corrections for non-ideal detector effects extend the useful concentration range of PCH experiments and are crucial for the interpretation of titration and dilution experiments.     

Effect of concentration quenching on fluorescence recovery after photobleaching measurements.

Standard analysis of fluorescence recovery after photobleaching (FRAP) data is valid only if the quantum yield of unphotobleached fluorophores is independent of concentration, yet close molecular packing in two-dimensional systems may lead to significant fluorescence concentration quenching. Using total internal reflection fluorescence, we quantified the surface concentration dependence of the relative quantum yield of fluorescein isothiocyanate-labeled proteins adsorbed to polymeric surfaces before performing measurements of fluorescence recovery after pattern photobleaching. Adsorbed layers of FITC-labeled ribonuclease A displayed significant concentration quenching, and thus the standard FRAP analysis method was unacceptable. We present an extended FRAP analysis procedure that accounts for the changing quantum yield of diffusing fluorophores in systems that are influenced by concentration quenching. The extended analysis shows that if concentration quenching conditions prevail, there may be significant error in the transport parameters obtained from FRAP measurements by using the standard procedures.     

Simultaneous diffusion and brightness measurements and brightness profile visualization from single fluorescence fluctuation traces of GFP in living cells

Fluorescence correlation spectroscopy (FCS) and photon-counting histogram (PCH) analysis use the same experimental fluorescence intensity fluctuations, but each analytical method focuses on a different property of the signal. The time-dependent decay of the correlation of fluorescence fluctuations is measured in FCS yielding, for instance, molecular diffusion coefficients. The amplitude distribution of these fluctuations is calculated by PCH analysis yielding information about the molecular brightness of fluorescent species. Analysis of both FCS and PCH results in the molecular concentration of the sample. Using a previously described global analysis procedure we report here precise, simultaneous measurements of diffusion constants and brightness values from single fluorescence fluctuation traces of green-fluorescent protein (GFP, S65T) in the cytoplasm of Dictyostelium cells. The use of a polynomial profile in PCH analysis, describing the detected three-dimensional shape of the confocal volume, enabled us to obtain well fitting results for GFP in cells. We could visualize the polynomial profile and show its deviation from a Gaussian profile.     

Time-integrated fluorescence cumulant analysis in fluorescence fluctuation spectroscopy

We introduce a new analysis technique for fluorescence fluctuation data. Time-integrated fluorescence cumulant analysis (TIFCA) extracts information from the cumulants of the integrated fluorescence intensity. TIFCA builds on our earlier FCA theory, but in contrast to FCA or photon counting histogram (PCH) analysis is valid for arbitrary sampling times. The motivation for long sampling times lies in the improvement of the signal/noise ratio of the data. Because FCA and PCH theory are not valid in this regime, we first derive a theoretical model of cumulant functions for arbitrary sampling times. TIFCA is the first exact theory that describes the effects of sampling time on fluorescence fluctuation experiments. We calculate factorial cumulants of the photon counts for various sampling times by rebinning of the original data. Fits of the data to models determine the brightness, the occupation number, and the diffusion time of each species. To provide the tools for a rigorous error analysis of TIFCA, expressions for the variance of cumulants are developed and tested. We demonstrate that over a limited range rebinning reduces the relative error of higher order cumulants, and therefore improves the signal/noise ratio. The first four cumulant functions are explicitly calculated and are applied to simple dye systems to test the validity of TIFCA and demonstrate its ability to resolve species.     

FRET-based identification of mRNAs undergoing translation

We present proof-of-concept in vitro results demonstrating the feasibility of using single molecule fluorescence resonance energy transfer (smFRET) measurements to distinguish, in real time, between individual ribosomes programmed with several different, short mRNAs. For these measurements we use either the FRET signal generated between two tRNAs labeled with different fluorophores bound simultaneously in adjacent sites to the ribosome (tRNA-tRNA FRET) or the FRET signal generated between a labeled tRNA bound to the ribosome and a fluorescent derivative of ribosomal protein L1 (L1-tRNA FRET). With either technique, criteria were developed to identify the mRNAs, taking into account the relative activity of the mRNAs. These criteria enabled identification of the mRNA being translated by a given ribosome to within 95% confidence intervals based on the number of identified FRET traces. To upgrade the approach for natural mRNAs or more complex mixtures, the stoichiometry of labeling should be enhanced and photobleaching reduced. The potential for porting these methods into living cells is discussed.     

Fluorescence-quenching-resolved spectroscopy of proteins

A new procedure is described for using fluorescence-quenching data of tryptophan residues in proteins to resolve their fluorescence emission spectra. In this concept the Stern-Volmer quenching plot is determined at each particular emission wavelength and iterative non-linear least-squares fitting procedure allowed to resolve the steady-state emission spectra into components. The resolved components, attributed to each of tryptophan residue, can be characterized by different accessibility to the quencher. The ability to resolve fluorescence emission spectra can be improved by using different kinds of efficient quenchers, which can selectively quench the emission of exposed or both exposed and buried fluorophores. The method was used to decompose emission fluorescence spectra in two-tryptophan-containing proteins; horse liver dehydrogenase, sperm whale apomyoglobin and metalloprotease from Staphylococcus aureus. The resolved spectra of alcohol dehydrogenase and metalloprotease are in excellent agreement with those previously obtained by single-photon counting or phase methods. The method presented here is technically simple and does not require expensive instrumentation.     

Resolving heterogeneity on the single molecular level with the photon-counting histogram

The diffusion of fluorescent particles through a small, illuminated observation volume gives rise to intensity fluctuations caused by particle number fluctuations in the open observation volume and the inhomogeneous excitation-beam profile. The intensity distribution of these fluorescence fluctuations is experimentally captured by the photon-counting histogram (PCH). We recently introduced the theory of the PCH for diffusing particles (Chen et al., Biophys. J., 77:553-567), where we showed that we can uniquely describe the distribution of photon counts with only two parameters for each species: the molecular brightness of the particle and the average number of particles within the observation volume. The PCH is sensitive to the molecular brightness and thus offers the possibility to separate a mixture of fluorescent species into its constituents, based on a difference in their molecular brightness alone. This analysis is complementary to the autocorrelation function, traditionally used in fluorescence fluctuation spectroscopy, which separates a mixture of species by a difference in their diffusion coefficient. The PCH of each individual species is convoluted successively to yield the PCH of the mixture. Successful resolution of the histogram into its components is largely a matter of the signal statistics. Here, we discuss the case of two species in detail and show that a concentration for each species exists, where the signal statistics is optimal. We also discuss the influence of the absolute molecular brightness and the brightness contrast between two species on the resolvability of two species. A binary dye mixture serves as a model system to demonstrate that the molecular brightness and the concentration of each species can be resolved experimentally from a single or from several histograms. We extend our study to biomolecules, where we label proteins with a fluorescent dye and show that a brightness ratio of two can be resolved. The ability to resolve a brightness ratio of two is very important for biological applications.     

Oligomer size of the serotonin 5-hydroxytryptamine 2C (5-HT2C) receptor revealed by fluorescence correlation spectroscopy with photon counting histogram analysis: evidence for homodimers without monomers or tetramers

Fluorescence correlation spectroscopy (FCS) and photon counting histogram (PCH) are techniques with single molecule sensitivity that are well suited for examining the biophysical properties of protein complexes in living cells. In the present study, FCS and PCH were applied to determine the diffusion coefficient and oligomeric size of G-protein-coupled receptors. FCS was used to record fluctuations in fluorescence intensity arising from fluorescence-tagged 5-hydroxytryptamine 2C (5-HT(2C)) receptors diffusing within the plasma membrane of HEK293 cells and rat hippocampal neurons. Autocorrelation analysis yielded diffusion coefficients ranging from 0.8 to 1.2 μm(2)/s for fluorescence-tagged receptors. Because the molecular brightness of a fluorescent protein is directly proportional to the number of fluorescent proteins traveling together within a protein complex, it can be used to determine the oligomeric size of the protein complex. FCS and PCH analysis of fluorescence-tagged 5-HT(2C) receptors provided molecular brightness values that were twice that of GFP and YFP monomeric controls, similar to a dimeric GFP control, and unaltered by 5-HT. Bimolecular fluorescence complementation of the N- and C-terminal halves of YFP attached to 5-HT(2C) receptors was observed in endoplasmic reticulum/Golgi and plasma membranes with a brightness equal to monomeric YFP. When GFP-tagged 5-HT(2C) receptors were co-expressed with a large excess of untagged, non-fluorescent 5-HT(2C) receptors, the molecular brightness was reduced by half. PCH analysis of the FCS data were best described by a one-component dimer model without monomers or tetramers. Therefore, it is concluded that 5-HT(2C) receptors freely diffusing within the plasma membrane are dimeric.     

Analysis of multi-component fluorescence emission by phase-sensitive detection using one modulation frequency

We describe the use of phase-sensitive detection of fluorescence to resolve the lifetimes and fractional intensities from multi-component fluorescence samples, using data obtained at a single modulation frequency. Phase-sensitive spectra of the mixture are recorded at arbitrarily chosen detector phase angles. The steady-state spectrum of each component must be known. The phase-sensitive spectra are fitted, using a nonlinear least-squares algorithm, to obtain the lifetimes and fractional intensities of each fluorophore in the mixture. Simulations for two- and three-component mixtures are presented to illustrate how the resolution is affected by spectral overlap and lifetime separation. Experimentally, we resolved two- and three-component mixtures of protein-like fluorophores (N-acetyl-L-tyrosinamide, N-acetyl- L-tryptophanamide, indole and 2,3-dimethylindole) using data collected at 30 MHz. These fluorophores have closely spaced lifetimes of 1.5, 2.9, 4.5 and 4.3 ns, respectively, and display extensive spectral overlap. These results demonstrate that phase-sensitive spectra, recorded at only one modulation frequency with a standard phase fluorometer, can be used to resolve multi-component emissions.     

Molecular Brightness Characterization of EGFP In Vivo by Fluorescence Fluctuation Spectroscopy

We characterize the molecular properties of autofluorescence and transiently expressed EGFP in the nucleus and in the cytoplasm of HeLa cells by fluorescence correlation spectroscopy (FCS) and by photon counting histogram (PCH) analysis. PCH has been characterized and applied in vitro, but its potential for in vivo studies needs to be explored. Thus, this study mainly focuses on the characterization of PCH analysis in vivo. The strength of PCH lies in its ability to distinguish biomolecules by their molecular brightness value. Because the concept of molecular brightness is crucial for PCH analysis, we study the molecular brightness of EGFP and determine the statistical accuracy of its measurement under in vivo conditions. We started by characterizing the influence of autofluorescence on EGFP measurements. We found a molecular brightness of EGFP that is a factor of 10 higher than the brightness of the autofluorescence. Moment analysis demonstrates that the contribution of autofluorescence to fluorescence fluctuation experiments is negligible at EGFP concentrations of one protein per excitation volume. The molecular brightness of EGFP measured in the nucleus, the cytoplasm, and in vitro are identical and our study demonstrates that molecular brightness is a very stable and predictable quantity for cellular measurements. In addition to PCH, we also analyzed the autocorrelation function of EGFP. The diffusion coefficient of EGFP is a factor of 3 lower in vivo than compared to in vitro, and a simple diffusion process describes the autocorrelation function. We found that in the nucleus the fluorescence intensity is stable as a function of time, while measurements in the cytoplasm display fluorescence intensity drifts that complicate the data analysis. We introduce and discuss an analysis method that minimizes the influence of the intensity drifts on PCH analysis. This method allows us to recover the correct molecular brightness of EGFP even in the presence of drifts of the fluorescence intensity signal. We found the molecular brightness of EGFP to be a very robust parameter, and anticipate the use of PCH analysis for the study of oligomerization processes in vivo.     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.