Automated Analysis of Single-Molecule Photobleaching Data by Statistical Modeling of Spot Populations

The number of fluorophores within a molecule complex can be revealed by single-molecule photobleaching imaging. A widely applied strategy to analyze intensity traces over time is the quantification of photobleaching step counts. However, several factors can limit and bias the detection of photobleaching steps, including noise, high numbers of fluorophores, and the possibility that several photobleaching events occur almost simultaneously. In this study, we propose a new approach, to our knowledge, to determine the fluorophore number that correlates the intensity decay of a population of molecule complexes with the decay of the number of visible complexes. We validated our approach using single and fourfold Atto-labeled DNA strands. As an example we estimated the subunit stoichiometry of soluble CD95L using GFP fusion proteins. To assess the precision of our method we performed in silico experiments showing that the estimates are not biased for experimentally observed intensity fluctuations and that the relative precision remains constant with increasing number of fluorophores. In case of fractional fluorescent labeling, our simulations predicted that the fluorophore number estimate corresponds to the product of the true fluorophore number with the labeling fraction. Our method, denoted by spot number and intensity correlation (SONIC), is fully automated, robust to noise, and does not require the counting of photobleaching events.     

Non-equilibrium voltage noise generated by ion transport through pores

In this paper, we describe a systematic approach to the theoretical analysis of non-equilibrium voltage noise that arises from ions moving through pores in membranes. We assume that an ion must cross one or two barriers in the pore in order to move from one side of the membrane to the other. In our analysis, we consider the following factors: a) surface charge as a variable in the kinetic equations, b) linearization of the kinetic equations, c) master equation approach to fluctuations. To analyze the voltage noise arising from ion movement through a two barrier (i.e., one binding site) pore, we included the effects of ions in the channel's interior on the voltage noise. The current clamp is considered as a white noise generating additional noise in the system. In contrast to what is found for current noise, at low frequencies the voltage noise intensity is reduced by increasing voltage across the membrane. With this approach, we demonstrate explicity for the examples treated that, apart from additional noise generated by the current clamp, the non-equilibrium voltage fluctuations can be related to the current fluctuations by the complex admittance.     

Correction of Systematic Bias in Single Molecule Photobleaching Measurements

Single molecule photobleaching is a powerful technique to measure the number of fluorescent units in subresolution molecular complexes, such as in toxic protein oligomers associated with amyloid diseases. However, photobleaching can occur before the sample is appropriately placed and focused. Such “prebleaching” can introduce a strong systematic bias toward smaller oligomers. Quantitative correction of prebleaching is known to be an ill-posed problem, limiting the utility of the technique. Here, we provide an experimental solution to improve its reliability. We chemically construct multimeric standards to estimate the prebleaching probability, B. We show that B can be used as a constraint to reliably correct the statistics obtained from a known distribution of standard oligomers. Finally, we apply this method to the data obtained from a heterogeneous oligomeric solution of human islet amyloid polypeptide. Our results show that photobleaching can critically skew the estimation of oligomeric distributions, so that low abundance monomers display a much higher apparent abundance. In summary, any inference from photobleaching experiments with B > 0.1 is likely to be unreliable, but our method can be used to quantitatively correct possible errors.     

Protein-level fluctuation correlation at the microcolony level and its application to the Vibrio harveyi quorum-sensing circuit

Gene expression is stochastic, and noise that arises from the stochastic nature of biochemical reactions propagates through active regulatory links. Thus, correlations in gene-expression noise can provide information about regulatory links. We present what to our knowledge is a new approach to measure and interpret such correlated fluctuations at the level of single microcolonies, which derive from single cells. We demonstrated this approach mathematically using stochastic modeling, and applied it to experimental time-lapse fluorescence microscopy data. Specifically, we investigated the relationships among LuxO, LuxR, and the small regulatory RNA qrr4 in the model quorum-sensing bacterium Vibrio harveyi. Our results show that LuxR positively regulates the qrr4 promoter. Under our conditions, we find that qrr regulation weakly depends on total LuxO levels and that LuxO autorepression is saturated. We also find evidence that the fluctuations in LuxO levels are dominated by intrinsic noise. We furthermore propose LuxO and LuxR interact at all autoinducer levels via an unknown mechanism. Of importance, our new method of evaluating correlations at the microcolony level is unaffected by partition noise at cell division. Moreover, the method is first-order accurate and requires less effort for data analysis than single-cell-based approaches. This new correlation approach can be applied to other systems to aid analysis of gene regulatory circuits.     

Using FCS to accurately measure protein concentration in the presence of noise and photobleaching

Quantitative cell biology requires precise and accurate concentration measurements, resolved both in space and time. Fluorescence correlation spectroscopy (FCS) has been held as a promising technique to perform such measurements because the fluorescence fluctuations it relies on are directly dependent on the absolute number of fluorophores in the detection volume. However, the most interesting applications are in cells, where autofluorescence and confinement result in strong background noise and important levels of photobleaching. Both noise and photobleaching introduce systematic bias in FCS concentration measurements and need to be corrected for. Here, we propose to make use of the photobleaching inevitably occurring in confined environments to perform series of FCS measurements at different fluorophore concentration, which we show allows a precise in situ measurement of both background noise and molecular brightness. Such a measurement can then be used as a calibration to transform confocal intensity images into concentration maps. The power of this approach is first illustrated with in vitro measurements using different dye solutions, then its applicability for in vivo measurements is demonstrated in Drosophila embryos for a model nuclear protein and for two morphogens, Bicoid and Capicua.     

Process noise: an explanation for the fluctuations in the immune response during acute viral infection

The parameters of the immune response dynamics are usually estimated by the use of deterministic ordinary differential equations that relate data trends to parameter values. Since the physical basis of the response is stochastic, we are investigating the intensity of the data fluctuations resulting from the intrinsic response stochasticity, the so-called process noise. Dealing with the CD8+ T-cell responses of virus-infected mice, we find that the process noise influence cannot be neglected and we propose a parameter estimation approach that includes the process noise stochastic fluctuations. We show that the variations in data can be explained completely by the process noise. This explanation is an alternative to the one resulting from standard modeling approaches which say that the difference among individual immune responses is the consequence of the difference in parameter values.     

Cellular growth and division in the Gillespie algorithm

Recent experimental studies elucidating the importance of noise in gene regulation have ignited widespread interest in Gillespie's stochastic simulation technique for biochemical networks. We formulate modifications to the Gillespie algorithm which are necessary to correctly simulate chemical reactions with time-dependent reaction rates. We concentrate on time dependence of kinetic rates arising from the periodic process of growth and division of the cellular volume, and demonstrate that a careful re-derivation of the Gillespie algorithm is important when all stochastically simulated reactions have rates slower or comparable to the cellular growth rate. For an unregulated single-gene system, we illustrate our findings using recently proposed hybrid simulation techniques, and systematically compare our algorithm with analytic results obtained from the chemical master equation.     

HAPLOFREQ--estimating haplotype frequencies efficiently.

A commonly used tool in disease association studies is the search for discrepancies between the haplotype distribution in the case and control populations. In order to find this discrepancy, the haplotypes frequency in each of the populations is estimated from the genotypes. We present a new method HAPLOFREQ to estimate haplotype frequencies over a short genomic region given the genotypes or haplotypes with missing data or sequencing errors. Our approach incorporates a maximum likelihood model based on a simple random generative model which assumes that the genotypes are independently sampled from the population. We first show that if the phased haplotypes are given, possibly with missing data, we can estimate the frequency of the haplotypes in the population by finding the global optimum of the likelihood function in polynomial time. If the haplotypes are not phased, finding the maximum value of the likelihood function is NP-hard. In this case, we define an alternative likelihood function which can be thought of as a relaxed likelihood function. We show that the maximum relaxed likelihood can be found in polynomial time and that the optimal solution of the relaxed likelihood approaches asymptotically to the haplotype frequencies in the population. In contrast to previous approaches, our algorithms are guaranteed to converge in polynomial time to a global maximum of the different likelihood functions. We compared the performance of our algorithm to the widely used program PHASE, and we found that our estimates are at least 10% more accurate than PHASE and about ten times faster than PHASE. Our techniques involve new algorithms in convex optimization. These algorithms may be of independent interest. Particularly, they may be helpful in other maximum likelihood problems arising from survey sampling.     

The role of noise in a predator–prey model with Allee effect

The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator–prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator–prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.     

Continuous photobleaching in vesicles and living cells: a measure of diffusion and compartmentation

We present a comprehensive and analytical treatment of continuous photobleaching in a compartment, under single photon excitation. In the very short time regime (t<0.1 ms), the diffusion does not play any role. After a transition (or short time regime), one enters in the long time regime (t>0.1-5 s), for which the diffusion and the photobleaching balance each other. In this long time regime, the diffusion is either fast (i.e., the photobleaching probability of a molecule diffusing through the laser beam is low) so that the photobleaching rate is independent of the diffusion constant and dependent only of the laser power, or the diffusion is slow (i.e., the photobleaching probability is high) and the photobleaching rate is mainly dependent on the diffusion constant. We illustrate our theory by using giant unilamellar vesicles ranging from approximately 10 to 100 microm in diameter, loaded with molecules of various diffusion constants (from 20 to 300 microm2/s) and various photobleaching cross sections, illuminated under laser powers between 3 and 100 microW. We also demonstrated that information about compartmentation can be obtained by this method in living cells expressing enhanced green fluorescent proteins or that were loaded with small FITC-dextrans. Our quantitative approach shows that molecules freely diffusing in a cellular compartment do experience a continuous photobleaching. We provide a generic theoretical framework that should be taken into account when studying, under confocal microscopy, molecular interactions, permeability, etc.     

Bayesian inference for improved single molecule fluorescence tracking

Single molecule tracking is widely used to monitor the change in position of lipids and proteins in living cells. In many experiments in which molecules are tagged with a single or small number of fluorophores, the signal/noise ratio may be limiting, the number of molecules is not known, and fluorophore blinking and photobleaching can occur. All these factors make accurate tracking over long trajectories difficult and hence there is still a pressing need to develop better algorithms to extract the maximum information from a sequence of fluorescence images. We describe here a Bayesian-based inference approach, based on a trans-dimensional sequential Monte Carlo method that utilizes both the spatial and temporal information present in the image sequences. We show, using model data, where the real trajectory of the molecule is known, that our method allows accurate tracking of molecules over long trajectories even with low signal/noise ratio and in the presence of fluorescence blinking and photobleaching. The method is then applied to real experimental data.     

Listening to the noise: random fluctuations reveal gene network parameters

The cellular environment is abuzz with noise originating from the inherent random motion of reacting molecules in the living cell. In this noisy environment, clonal cell populations show cell‐to‐cell variability that can manifest significant phenotypic differences. Noise‐induced stochastic fluctuations in cellular constituents can be measured and their statistics quantified. We show that these random fluctuations carry within them valuable information about the underlying genetic network. Far from being a nuisance, the ever‐present cellular noise acts as a rich source of excitation that, when processed through a gene network, carries its distinctive fingerprint that encodes a wealth of information about that network. We show that in some cases the analysis of these random fluctuations enables the full identification of network parameters, including those that may otherwise be difficult to measure. This establishes a potentially powerful approach for the identification of gene networks and offers a new window into the workings of these networks.     

Deciphering interactions used by the influenza virus NS1 protein to silence the host antiviral sensor protein RIG-I using a bacterial reverse two-hybrid system

The majority of biological processes are controlled and regulated by an intricate network of thousands of interacting proteins. Identifying and understanding the key components of these protein networks, especially those that play a critical role in disease, is a challenge that promises to dramatically alter our current approach to healthcare. To facilitate this process, we have developed a method for the rapid construction of a chromosomally integrated, bacterial reverse two-hybrid system (RTHS) that enables the identification of interacting protein partners. Chromosomal integration of the RTHS enables stable protein expression, free of plasmid copy-number effects, as well as eliminating false positives arising from plasmid ejection. We have utilized this approach to identify the interactions used by the influenza virus NS1 protein to silence the host's antiviral defences.     

From single-cell genetic architecture to cell population dynamics: quantitatively decomposing the effects of different population heterogeneity sources for a genetic network with positive feedback architecture

Phenotypic cell-to-cell variability or cell population heterogeneity originates from two fundamentally different sources: unequal partitioning of cellular material at cell division and stochastic fluctuations associated with intracellular reactions. We developed a mathematical and computational framework that can quantitatively isolate both heterogeneity sources and applied it to a genetic network with positive feedback architecture. The framework consists of three vastly different mathematical formulations: a), a continuum model, which completely neglects population heterogeneity; b), a deterministic cell population balance model, which accounts for population heterogeneity originating only from unequal partitioning at cell division; and c), a fully stochastic model accommodating both sources of population heterogeneity. The framework enables the quantitative decomposition of the effects of the different population heterogeneity sources on system behavior. Our results indicate the importance of cell population heterogeneity in accurately predicting even average population properties. Moreover, we find that unequal partitioning at cell division and sharp division rates shrink the region of the parameter space where the population exhibits bistable behavior, a characteristic feature of networks with positive feedback architecture. In addition, intrinsic noise at the single-cell level due to slow operator fluctuations and small numbers of molecules further contributes toward the shrinkage of the bistability regime at the cell population level. Finally, the effect of intrinsic noise at the cell population level was found to be markedly different than at the single-cell level, emphasizing the importance of simulating entire cell populations and not just individual cells to understand the complex interplay between single-cell genetic architecture and behavior at the cell population level.     

Accessing molecular dynamics in cells by fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) analyzes spontaneous fluctuations in the fluorescence emission of small molecular ensembles, thus providing information about a multitude of parameters, such as concentrations, molecular mobility and dynamics of fluorescently labeled molecules. Performed within diffraction-limited confocal volume elements, FCS provides an attractive alternative to photobleaching recovery methods for determining intracellular mobility parameters of very low quantities of fluorophores. Due to its high sensitivity sufficient for single molecule detection, the method is subject to certain artifact hazards that must be carefully controlled, such as photobleaching and intramolecular dynamics, which introduce fluorescence flickering. Furthermore, if molecular mobility is to be probed, nonspecific interactions of the labeling dye with cellular structures can introduce systematic errors. In cytosolic measurements, lipophilic dyes, such as certain rhodamines that bind to intracellular membranes, should be avoided. To study free diffusion, genetically encoded fluorescent labels such as green fluorescent protein (GFP) or DsRed are preferable since they are less likely to nonspecifically interact with cellular substructures.     

Sample Preparation and Imaging Conditions Affect mEos3.2 Photophysics in Fission Yeast Cells

Photoconvertible fluorescent proteins (PCFPs) are widely used in super-resolution microscopy and studies of cellular dynamics. However, our understanding of their photophysics is still limited, hampering their quantitative application. For example, we do not know the optimal sample preparation methods or imaging conditions to count protein molecules fused to PCFPs by single-molecule localization microscopy in live and fixed cells. We also do not know how the behavior of PCFPs in live cells compares with fixed cells. Therefore, we investigated how formaldehyde fixation influences the photophysical properties of the popular green-to-red PCFP mEos3.2 in fission yeast cells under a wide range of imaging conditions. We estimated photophysical parameters by fitting a three-state model of photoconversion and photobleaching to the time course of fluorescence signal per yeast cell expressing mEos3.2. We discovered that formaldehyde fixation makes the fluorescence signal, photoconversion rate, and photobleaching rate of mEos3.2 sensitive to the buffer conditions likely by permeabilizing the yeast cell membrane. Under some imaging conditions, the time-integrated mEos3.2 signal per yeast cell is similar in live cells and fixed cells imaged in buffer at pH 8.5 with 1 mM DTT, indicating that light chemical fixation does not destroy mEos3.2 molecules. We also discovered that 405-nm irradiation drove some red-state mEos3.2 molecules to enter an intermediate dark state, which can be converted back to the red fluorescent state by 561-nm illumination. Our findings provide a guide to quantitatively compare conditions for imaging mEos3.2-tagged molecules in yeast cells. Our imaging assay and mathematical model are easy to implement and provide a simple quantitative approach to measure the time-integrated signal and the photoconversion and photobleaching rates of fluorescent proteins in cells.     

Shared kinase fluctuations between two enzymatic reactions

Kinases serve crucial roles in many cellular signaling pathways that process and transfer information. When signaling kinases phosphorylate two targets, these can serve as branch points that distribute information among two pathways. Responses to stimuli transmitted by activated kinases show high levels of cell-to-cell variation that influence cellular function. We ask how fluctuations around a steady state, due to kinase fluctuations and intrinsic noise, are distributed between two reactions with substrates phosphorylated by a shared kinase. We develop the formalism to answer this question and, for a realistic set of biological constants, we illustrate various features of fluctuations and relaxation times to a steady state. We find that the steady-state response determines the size and range in enzyme concentration of phosphorylated substrate fluctuations, and that the choice of an operating point can have a large impact on how shared kinase noise is distributed among two available pathways.