Photon budget analysis for fluorescence lifetime imaging microscopy

Two-color interferometric scattering (iSCAT) microscopy reveals structural dynamics in discrete plasmonic molecules

Plasmonic molecules are discrete assemblies of noble metal nanoparticles (NPs) that are of interest as transducers in optical nanosensors. So far, NPs with diameters of ∼40 nm have been the preferred building blocks for plasmonic molecules intended as optical single molecule sensors due to difficulties associated with detecting smaller NPs through elastic scattering in conventional darkfield microscopy. Here, we apply 405 nm, 445 nm two-color interferometric scattering (iSCAT) microscopy to characterize polyethylene glycol (PEG) tethered dimers of 10 nm and 20 nm Ag NPs and their monomers. Dimers of both NP sizes can be discerned from their respective monomers through changes in the average iSCAT contrast. In the case of 20 nm Ag NPs, dimer formation induces a change in the sign of the iSCAT contrast, providing a characteristic signal for detecting binding events. 20 nm Ag NP dimers with 0.4 kDa and 3.4 kDa polyethylene glycol (PEG) spacers show iSCAT contrast distributions with significantly different averages on both wavelength channels. The iSCAT contrast measured for individual PEG-tethered 10 nm or 20 nm NP dimers as a function of time shows contrast fluctuations indicative of a rich structural dynamics in the assembled plasmonic molecules, which provides an additional metric to discern dimers from monomers and paves the path to a new class of interferometric plasmon rulers.

Application of Super-resolution SPEED Microscopy in the Study of Cellular Dynamics

Super-resolution imaging techniques have broken the diffraction-limited resolution of light microscopy. However, acquiring three-dimensional (3D) super-resolution information about structures and dynamic processes in live cells at high speed remains challenging. Recently, the development of high-speed single-point edge-excitation subdiffraction (SPEED) microscopy, along with its 2D-to-3D transformation algorithm, provides a practical and effective approach to achieving 3D subdiffraction-limit information in subcellular structures and organelles with rotational symmetry. One of the major benefits of SPEED microscopy is that it does not rely on complex optical components and can be implemented on a standard, inverted epifluorescence microscope, simplifying the process of sample preparation and the expertise requirement. SPEED microscopy is specifically designed to obtain 2D spatial locations of individual immobile or moving fluorescent molecules inside submicrometer biological channels or cavities at high spatiotemporal resolution. The collected data are then subjected to postlocalization 2D-to-3D transformation to obtain 3D super-resolution structural and dynamic information. In recent years, SPEED microscopy has provided significant insights into nucleocytoplasmic transport across the nuclear pore complex (NPC) and cytoplasm-cilium trafficking through the ciliary transition zone. This Review focuses on the applications of SPEED microscopy in studying the structure and function of nuclear pores.

Single-molecule counting applied to the study of GPCR oligomerization

Single-molecule counting techniques enable a precise determination of the intracellular abundance and stoichiometry of proteins and macromolecular complexes. These details are often challenging to quantitatively assess yet are essential for our understanding of cellular function. Consider G-protein-coupled receptors-an expansive class of transmembrane signaling proteins that participate in many vital physiological functions making them a popular target for drug development. While early evidence for the role of oligomerization in receptor signaling came from ensemble biochemical and biophysical assays, innovations in single-molecule measurements are now driving a paradigm shift in our understanding of its relevance. Here, we review recent developments in single-molecule counting with a focus on photobleaching step counting and the emerging technique of quantitative single-molecule localization microscopy-with a particular emphasis on the potential for these techniques to advance our understanding of the role of oligomerization in G-protein-coupled receptor signaling.

Optimizing Long-Term Live Cell Imaging

Live cell microscopy has become a common technique for exploring dynamic biological processes. When combined with fluorescent markers of cellular structures of interest, or fluorescent reporters of a biological activity of interest, live cell microscopy enables precise temporally and spatially resolved quantitation of the biological processes under investigation. However, because living cells are not normally exposed to light, live cell fluorescence imaging is significantly hindered by the effects of photodamage, which encompasses photobleaching of fluorophores and phototoxicity of the cells under observation. In this chapter, we outline several methods for optimizing and maintaining long-term imaging of live cells while simultaneously minimizing photodamage. This protocol demonstrates the intracellular trafficking of early and late endosomes following phagocytosis using both two and three dimensional imaging, but this protocol can easily be modified to image any biological process of interest in nearly any cell type.

Simple phasor-based deep neural network for fluorescence lifetime imaging microscopy

Fluorescence lifetime imaging microscopy (FLIM) is a powerful technique to probe the molecular environment of fluorophores. The analysis of FLIM images is usually performed with time consuming fitting methods. For accelerating this analysis, sophisticated deep learning architectures based on convolutional neural networks have been developed for restrained lifetime ranges but they require long training time. In this work, we present a simple neural network formed only with fully connected layers able to analyze fluorescence lifetime images. It is based on the reduction of high dimensional fluorescence intensity temporal decays into four parameters which are the phasor coordinates, the mean and amplitude-weighted lifetimes. This network called Phasor-Net has been applied for a time domain FLIM system excited with an 80 MHz laser repetition frequency, with negligible jitter and afterpulsing. Due to the restricted time interval of 12.5 ns, the training range of the lifetimes was limited between 0.2 and 3.0 ns; and the total photon number was lower than 106, as encountered in live cell imaging. From simulated biexponential decays, we demonstrate that Phasor-Net is more precise and less biased than standard fitting methods. We demonstrate also that this simple architecture gives almost comparable performance than those obtained from more sophisticated networks but with a faster training process (15 min instead of 30 min). We finally apply successfully our method to determine biexponential decays parameters for FLIM experiments in living cells expressing EGFP linked to mCherry and fused to a plasma membrane protein.

Performance and limitation estimation of a three-tap gated imaging sensor in wide field time-gated fluorescence lifetime imaging systems

In this paper, a computational performance analysis is presented of a wide-field time-gated fluorescence lifetime imaging microscope (FLIM) using practically realizable properties of the laser, sample, and a three-tap time-gated CMOS image sensor. The impact of these component-level properties on the accuracy and the precision of the measurement results are estimated and discussed based on Monte Carlo simulations. The correlation between the detector speed and the accuracy of the extracted fluorescence lifetime is studied, and the minimum required incident photoelectron number of each pixel is estimated for different detector speeds and different fluorescence lifetime measurements. In addition, the detection limits due to the dark current and the parasitic light sensitivity of the detector are also investigated. This work gives an overview of the required fluorescence emission condition as well as the required detector properties for a three-tap time-gated image sensor to achieve good FLIM data in biological applications.

An Update on Mesoporous Silica Nanoparticle Applications in Nanomedicine

The efficient and safe delivery of therapeutic drugs, proteins, and nucleic acids are essential for meaningful therapeutic benefits. The field of nanomedicine shows promising implications in the development of therapeutics by delivering diagnostic and therapeutic compounds. Nanomedicine development has led to significant advances in the design and engineering of nanocarrier systems with supra-molecular structures. Smart mesoporous silica nanoparticles (MSNs), with excellent biocompatibility, tunable physicochemical properties, and site-specific functionalization, offer efficient and high loading capacity as well as robust and targeted delivery of a variety of payloads in a controlled fashion. Such unique nanocarriers should have great potential for challenging biomedical applications, such as tissue engineering, bioimaging techniques, stem cell research, and cancer therapies. However, in vivo applications of these nanocarriers should be further validated before clinical translation. To this end, this review begins with a brief introduction of MSNs properties, targeted drug delivery, and controlled release with a particular emphasis on their most recent diagnostic and therapeutic applications.

Improved resolution in single-molecule localization microscopy using QD-PAINT

Single-molecule localization microscopy (SMLM) has allowed the observation of various molecular structures in cells beyond the diffraction limit using organic dyes. In principle, the SMLM resolution depends on the precision of photoswitching fluorophore localization, which is inversely correlated with the square root of the number of photons released from the individual fluorophores. Thus, increasing the photon number by using highly bright fluorophores, such as quantum dots (QDs), can theoretically fundamentally overcome the current resolution limit of SMLM. However, the use of QDs in SMLM has been challenging because QDs have no photoswitching property, which is essential for SMLM, and they exhibit nonspecificity and multivalency, which complicate their use in fluorescence imaging. Here, we present a method to utilize QDs in SMLM to surpass the resolution limit of the current SMLM utilizing organic dyes. We confer monovalency, specificity, and photoswitchability on QDs by steric exclusion via passivation and ligand exchange with ptDNA, PEG, and casein as well as by DNA point accumulation for imaging in nanoscale topography (DNA-PAINT) via automatic thermally driven hybridization between target-bound docking and dye-bound complementary imager strands. QDs are made monovalent and photoswitchable to enable SMLM and show substantially better photophysical properties than Cy3, with higher fluorescence intensity and an improved resolution factor. QD-PAINT displays improved spatial resolution with a narrower full width at half maximum (FWHM) than DNA-PAINT with Cy3. In summary, QD-PAINT shows great promise as a next-generation SMLM method for overcoming the limited resolution of the current SMLM.

FLIMJ: An open-source ImageJ toolkit for fluorescence lifetime image data analysis

In the field of fluorescence microscopy, there is continued demand for dynamic technologies that can exploit the complete information from every pixel of an image. One imaging technique with proven ability for yielding additional information from fluorescence imaging is Fluorescence Lifetime Imaging Microscopy (FLIM). FLIM allows for the measurement of how long a fluorophore stays in an excited energy state, and this measurement is affected by changes in its chemical microenvironment, such as proximity to other fluorophores, pH, and hydrophobic regions. This ability to provide information about the microenvironment has made FLIM a powerful tool for cellular imaging studies ranging from metabolic measurement to measuring distances between proteins. The increased use of FLIM has necessitated the development of computational tools for integrating FLIM analysis with image and data processing. To address this need, we have created FLIMJ, an ImageJ plugin and toolkit that allows for easy use and development of extensible image analysis workflows with FLIM data. Built on the FLIMLib decay curve fitting library and the ImageJ Ops framework, FLIMJ offers FLIM fitting routines with seamless integration with many other ImageJ components, and the ability to be extended to create complex FLIM analysis workflows. Building on ImageJ Ops also enables FLIMJ's routines to be used with Jupyter notebooks and integrate naturally with science-friendly programming in, e.g., Python and Groovy. We show the extensibility of FLIMJ in two analysis scenarios: lifetime-based image segmentation and image colocalization. We also validate the fitting routines by comparing them against industry FLIM analysis standards.

Wide-field time-correlated single photon counting-based fluorescence lifetime imaging microscopy

Wide-field time-correlated single photon counting detection techniques, where the position and the arrival time of the photons are recorded simultaneously using a camera, have made some advances recently. The technology and instrumentation used for this approach is employed in areas such as nuclear science, mass spectroscopy and positron emission tomography, but here, we discuss some of the wide-field TCSPC methods, for applications in fluorescence microscopy. We describe work by us and others as presented in the Ulitima fast imaging and tracking conference at the Argonne National Laboratory in September 2018, from phosphorescence lifetime imaging (PLIM) microscopy on the microsecond time scale to fluorescence lifetime imaging (FLIM) on the nanosecond time scale, and highlight some applications of these techniques.

Efficient use of single molecule time traces to resolve kinetic rates, models and uncertainties

Single molecule time traces reveal the time evolution of unsynchronized kinetic systems. Especially single molecule Förster resonance energy transfer (smFRET) provides access to enzymatically important time scales, combined with molecular distance resolution and minimal interference with the sample. Yet the kinetic analysis of smFRET time traces is complicated by experimental shortcomings-such as photo-bleaching and noise. Here we recapitulate the fundamental limits of single molecule fluorescence that render the classic, dwell-time based kinetic analysis unsuitable. In contrast, our Single Molecule Analysis of Complex Kinetic Sequences (SMACKS) considers every data point and combines the information of many short traces in one global kinetic rate model. We demonstrate the potential of SMACKS by resolving the small kinetic effects caused by different ionic strengths in the chaperone protein Hsp90. These results show an unexpected interrelation between conformational dynamics and ATPase activity in Hsp90.

Semi-autonomous real-time programmable fluorescence lifetime segmentation with a digital micromirror device

Time-correlated single-photon counting (TCSPC) is the gold standard for performing lifetime spectroscopy in biological assays. Traditional fluorescence lifetime imaging (FLIM) using laser scanning microscopes are inherently slow due to point scanning all pixels in the field-of-view. Wide-field implementations of TCSPC spectroscopy using microchannel plates benefit from particularly fast acquisition times at the expense of temporal resolution, and are fundamentally limited by photon counting rates. Here, we introduce programmable lifetime imaging (PLI), combining the advantages of wide-field imaging using total internal reflection excitation with state-of-the-art TCSPC detector technology for accurate lifetime determination in an object-oriented manner using a digital micromirror device (DMD). The fluorescent emission is projected onto the DMD to facilitate the sequential segmentation of fluorescence from individual objects in the field-of-view, allowing for both image acquisition and fluorescence lifetime determination of the assay. The sensitivity of PLI is demonstrated by manually segmenting fluorescence from fixed cell assays. We also demonstrate an automated implementation of PLI, using a camera as a feedback mechanism to segment fluorescence produced by emitting objects of interest in the imaging field-of-view, highlighting the advantages of measurement only in areas where valuable information exists. As a result, PLI is able to reduce acquisition time of fluorescence lifetime data by at least an order of magnitude compared to laser scanning implementations.

Single-Molecule Tracking Approaches to Protein Synthesis Kinetics in Living Cells

Decades of traditional biochemistry, structural approaches, and, more recently, single-molecule-based in vitro techniques have provided us with an astonishingly detailed understanding of the molecular mechanism of ribosome-catalyzed protein synthesis. However, in order to understand these details in the context of cell physiology and population biology, new techniques to probe the dynamics of molecular processes inside the cell are needed. Recent years' development in super-resolved fluorescence microscopy has revolutionized imaging of intracellular processes, and we now have the possibility to directly peek into the microcosm of biomolecules in their native environment. In this Perspective, we discuss how these methods are currently being applied and further developed to study the kinetics of protein synthesis directly inside living cells.

Homotransfer of FRET Reporters for Live Cell Imaging

Förster resonance energy transfer (FRET) between fluorophores of the same species was recognized in the early to mid-1900s, well before modern heterotransfer applications. Recently, homotransfer FRET principles have re-emerged in biosensors that incorporate genetically encoded fluorescent proteins. Homotransfer offers distinct advantages over the standard heterotransfer FRET method, some of which are related to the use of fluorescence polarization microscopy to quantify FRET between two fluorophores of identical color. These include enhanced signal-to-noise, greater compatibility with other optical sensors and modulators, and new design strategies based upon the clustering or dimerization of singly-labeled sensors. Here, we discuss the theoretical basis for measuring homotransfer using polarization microscopy, procedures for data collection and processing, and we review the existing genetically-encoded homotransfer biosensors.

The impact of temporal sampling resolution on parameter inference for biological transport models

Imaging data has become an essential tool to explore key biological questions at various scales, for example the motile behaviour of bacteria or the transport of mRNA, and it has the potential to transform our understanding of important transport mechanisms. Often these imaging studies require us to compare biological species or mutants, and to do this we need to quantitatively characterise their behaviour. Mathematical models offer a quantitative description of a system that enables us to perform this comparison, but to relate mechanistic mathematical models to imaging data, we need to estimate their parameters. In this work we study how collecting data at different temporal resolutions impacts our ability to infer parameters of biological transport models by performing exact inference for simple velocity jump process models in a Bayesian framework. The question of how best to choose the frequency with which data is collected is prominent in a host of studies because the majority of imaging technologies place constraints on the frequency with which images can be taken, and the discrete nature of observations can introduce errors into parameter estimates. In this work, we mitigate such errors by formulating the velocity jump process model within a hidden states framework. This allows us to obtain estimates of the reorientation rate and noise amplitude for noisy observations of a simple velocity jump process. We demonstrate the sensitivity of these estimates to temporal variations in the sampling resolution and extent of measurement noise. We use our methodology to provide experimental guidelines for researchers aiming to characterise motile behaviour that can be described by a velocity jump process. In particular, we consider how experimental constraints resulting in a trade-off between temporal sampling resolution and observation noise may affect parameter estimates. Finally, we demonstrate the robustness of our methodology to model misspecification, and then apply our inference framework to a dataset that was generated with the aim of understanding the localization of RNA-protein complexes.

Giant light-harvesting nanoantenna for single-molecule detection in ambient light

Here, we explore the enhancement of single molecule emission by polymeric nano-antenna that can harvest energy from thousands of donor dyes to a single acceptor. In this nano-antenna, the cationic dyes are brought together in very close proximity using bulky counterions, thus enabling ultrafast diffusion of excitation energy (≤30 fs) with minimal losses. Our 60-nm nanoparticles containing >10,000 rhodamine-based donor dyes can efficiently transfer energy to 1-2 acceptors resulting in an antenna effect of ~1,000. Therefore, single Cy5-based acceptors become 25-fold brighter than quantum dots QD655. This unprecedented amplification of the acceptor dye emission enables observation of single molecules at illumination powers (1-10 mW cm^-2) that are >10,000-fold lower than typically required in single-molecule measurements. Finally, using a basic setup, which includes a 20X air objective and a sCMOS camera, we could detect single Cy5 molecules by simply shining divergent light on the sample at powers equivalent to sunlight.

Noise-Corrected Principal Component Analysis of fluorescence lifetime imaging data

Fluorescence Lifetime Imaging (FLIM) is an attractive microscopy method in the life sciences, yielding information on the sample otherwise unavailable through intensity-based techniques. A novel Noise-Corrected Principal Component Analysis (NC-PCA) method for time-domain FLIM data is presented here. The presence and distribution of distinct microenvironments are identified at lower photon counts than previously reported, without requiring prior knowledge of their number or of the dye's decay kinetics. A noise correction based on the Poisson statistics inherent to Time-Correlated Single Photon Counting is incorporated. The approach is validated using simulated data, and further applied to experimental FLIM data of HeLa cells stained with membrane dye di-4-ANEPPDHQ. Two distinct lipid phases were resolved in the cell membranes, and the modification of the order parameters of the plasma membrane during cholesterol depletion was also detected. Noise-corrected Principal Component Analysis of FLIM data resolves distinct microenvironments in cell membranes of live HeLa cells.

Photon counting phosphorescence lifetime imaging with TimepixCam

TimepixCam is a novel fast optical imager based on an optimized silicon pixel sensor with a thin entrance window and read out by a Timepix Application Specific Integrated Circuit. The 256 × 256 pixel sensor has a time resolution of 15 ns at a sustained frame rate of 10 Hz. We used this sensor in combination with an image intensifier for wide-field time-correlated single photon counting imaging. We have characterised the photon detection capabilities of this detector system and employed it on a wide-field epifluorescence microscope to map phosphorescence decays of various iridium complexes with lifetimes of about 1 μs in 200 μm diameter polystyrene beads.

Deep imaging in scattering media with selective plane illumination microscopy

In most biological tissues, light scattering due to small differences in refractive index limits the depth of optical imaging systems. Two-photon microscopy (2PM), which significantly reduces the scattering of the excitation light, has emerged as the most common method to image deep within scattering biological tissue. This technique, however, requires high-power pulsed lasers that are both expensive and difficult to integrate into compact portable systems. Using a combination of theoretical and experimental techniques, we show that if the excitation path length can be minimized, selective plane illumination microscopy (SPIM) can image nearly as deep as 2PM without the need for a high-powered pulsed laser. Compared to other single-photon imaging techniques like epifluorescence and confocal microscopy, SPIM can image more than twice as deep in scattering media ( ? 10 times the mean scattering length). These results suggest that SPIM has the potential to provide deep imaging in scattering media in situations in which 2PM systems would be too large or costly.

A wide-field TCSPC FLIM system based on an MCP PMT with a delay-line anode

We report on the implementation of a wide-field time-correlated single photon counting (TCSPC) method for fluorescence lifetime imaging (FLIM). It is based on a 40 mm diameter crossed delay line anode detector, where the readout is performed by three standard TCSPC boards. Excitation is performed by a picosecond diode laser with 50 MHz repetition rate. The photon arrival timing is obtained directly from the microchannel plates, with an instrumental response of ∼190 to 230 ps full width at half maximum depending on the position on the photocathode. The position of the photon event is obtained from the pulse propagation time along the two delay lines, one in x and one in y. One end of a delay line is fed into the "start" input of the corresponding TCSPC board, and the other end is delayed by 40 ns and fed into the "stop" input. The time between start and stop is directly converted into position, with a resolution of 200-250 μm. The data acquisition software builds up the distribution of the photons over their spatial coordinates, x and y, and their times after the excitation pulses, typically into 512 × 512 pixels and 1024 time channels per pixel. We apply the system to fluorescence lifetime imaging of cells labelled with Alexa 488 phalloidin in an epi-fluorescence microscope and discuss the application of our approach to other fluorescence microscopy methods.

0.5 billion events per second time correlated single photon counting using CMOS SPAD arrays

We present a digital architecture for fast acquisition of time correlated single photon counting (TCSPC) events from a 32×32 complementary metal oxide semiconductor (CMOS) single photon avalanche detector (SPAD) array (Megaframe) to the computer memory. Custom firmware was written to transmit event codes from 1024-TCSPC-enabled pixels for fast transfer of TCSPC events. Our 1024-channel TCSPC system is capable of acquiring up to 0.5×10(9) TCSPC events per second with 16 histogram bins spanning a 14 ns width. Other options include 320×10(6) TCSPC events per second with 256 histogram bins spanning either a 14 or 56 ns time window. We present a wide-field fluorescence microscopy setup demonstrating fast fluorescence lifetime data acquisition. To the best of our knowledge, this is the fastest direct TCSPC transfer from a single photon counting device to the computer to date.

Photon counting imaging with an electron-bombarded CCD: towards a parallel-processing photoelectronic time-to-amplitude converter

We have used an electron-bombarded CCD for optical photon counting imaging. The photon event pulse height distribution was found to be linearly dependent on the gain voltage. We propose on this basis that a gain voltage sweep during exposure in an electron-bombarded sensor would allow photon arrival time determination with sub-frame exposure time resolution. This effectively uses an electron-bombarded sensor as a parallel-processing photoelectronic time-to-amplitude converter, or a two-dimensional photon counting streak camera. Several applications that require timing of photon arrival, including Fluorescence Lifetime Imaging Microscopy, may benefit from such an approach. A simulation of a voltage sweep performed with experimental data collected with different acceleration voltages validates the principle of this approach. Moreover, photon event centroiding was performed and a hybrid 50% Gaussian/Centre of Gravity + 50% Hyperbolic cosine centroiding algorithm was found to yield the lowest fixed pattern noise. Finally, the camera was mounted on a fluorescence microscope to image F-actin filaments stained with the fluorescent dye Alexa 488 in fixed cells.

Wide-field time-correlated single-photon counting (TCSPC) lifetime microscopy with microsecond time resolution

A 1 MHz frame rate complementary metal-oxide semiconductor (CMOS) camera was used in combination with an image intensifier for wide-field time-correlated single-photon counting (TCSPC) imaging. The system combines an ultrafast frame rate with single-photon sensitivity and was employed on a fluorescence microscope to image decays of ruthenium compound Ru(dpp) with lifetimes from around 1 to 5 μs. A submicrowatt excitation power over the whole field of view is sufficient for this approach, and compatibility with live-cell imaging was demonstrated by imaging europium-containing beads with a lifetime of 570 μs in living HeLa cells. A standard two-photon excitation scanning fluorescence lifetime imaging (FLIM) system was used to independently verify the lifetime for the europium beads. This approach brings together advantageous features for time-resolved live-cell imaging such as low excitation intensity, single-photon sensitivity, ultrafast camera frame rates, and short acquisition times.

The influence of dead time related distortions on live cell fluorescence lifetime imaging (FLIM) experiments

Recent developments in the field of fluorescence lifetime imaging microscopy (FLIM) techniques allow the use of high repetition rate light sources in live cell experiments. For light sources with a repetition rate of 20-100 MHz, the time-correlated single photon counting (TCSPC) FLIM systems suffer serious dead time related distortions, known as "inter-pulse pile-up". The objective of this paper is to present a new method to quantify the level of signal distortion in TCSPC FLIM experiments, in order to determine the most efficient laser repetition rate for different FLT ranges. Optimization of the F -value, which is the relation between the relative standard deviation (RSD) in the measured FLT to the RSD in the measured fluorescence intensity (FI), allows quantification of the level of FI signal distortion, as well as determination of the correct FLT of the measurement. It is shown that by using a very high repetition rate (80 MHz) for samples characterized by high real FLT's (4-5 ns), virtual short FLT components are added to the FLT histogram while a F -value that is higher than 1 is obtained. For samples characterized with short real FLT's, virtual long FLT components are added to the FLT histogram with the lower repetition rate (20-50 MHz), while by using a higher repetition rate (80 MHz) the "inter-pulse pile-up" is eliminated as the F -value is close to 1.