Total internal reflection fluorescence (TIRF) microscopy excites a thin evanescent field which theoretically decays exponentially. Each TIRF image is actually the projection of a 3-D volume and hence cannot alone produce an accurate localization of structures in the z-dimension, however, it provides greatly improved axial resolution for biological samples. Multiple angle-TIRF microscopy allows controlled variation of the incident angle of the illuminating laser beam, thus generating a set of images of different penetration depths with the potential to reconstruct the 3-D volume of the sample. With the ultimate goal to quantify important biological parameters of microtubules, we present a method to reconstruct 3-D position and orientation of microtubules based on multi-angle TIRF data, as well as experimental calibration of the actual decay function of the evanescent field at each angle. We validate our method using computer simulations, by creating a phantom simulating the curvilinear characteristics of microtubules and project the artificially constructed volume into a set of TIRF image for different penetration depth. The reconstructed depth information for the phantom data is shown to be accurate and robust to noise. We apply our method to microtubule TIRF images of PtK(2) cells in vivo. By comparing microtubule curvatures of the reconstruction results and several electron microscopy (EM) images of vertically sliced sample of microtubules, we find that the curvature statistics of our reconstruction agree well with the ground truth (EM data). Quantifying the distribution of microtubule curvature reveals an interesting discovery that microtubules can buckle and form local bendings of considerably small radius of curvature which is also visually spotted on the EM images, while microtubule bendings on a larger scale generally have a much larger radius and cannot bear the stress of a large curvature. The presented method has the potential to provide a reliable tool for 3-D reconstruction and tracking of microtubules.

MOTIVATION: Designing mathematical tools that can formally describe the dynamics of complex intracellular processes remains a challenge. Live cell imaging reveals changes in the cellular states, but current simple approaches extract only minimal information of a static snapshot. RESULTS: We implemented a novel approach for analyzing organelle behavior in live cell imaging data based on hidden Markov models (HMMs) and showed that it can determine the number and evolution of distinct cellular states involved in a biological process. We analyzed insulin-mediated exocytosis of single Glut4-vesicles, a process critical for blood glucose homeostasis and impaired in type II diabetes, by using total internal reflection fluorescence microscopy (TIRFM). HMM analyses of movie sequences of living cells reveal that insulin controls spatial and temporal dynamics of exocytosis via the exocyst, a putative tethering protein complex. Our studies have validated the proof-of-principle of HMM for cellular imaging and provided direct evidence for the existence of complex spatial-temporal regulation of exocytosis in non-polarized cells. We independently confirmed insulin-dependent spatial regulation by using static spatial statistics methods. CONCLUSION: We propose that HMM-based approach can be exploited in a wide avenue of cellular processes, especially those where the changes of cellular states in space and time may be highly complex and non-obvious, such as in cell polarization, signaling and developmental processes.

Analyzing the morphological appearance and the spatial distribution of large dense-core vesicles (granules) in the cell cytoplasm is central to the understanding of regulated exocytosis. This paper is concerned with the automatic detection of granules and the statistical analysis of their spatial locations in different cell groups. We model the locations of granules of a given cell as a realization of a finite spatial point process and the point patterns associated with the cell groups as replicated point patterns of different spatial point processes. First, an algorithm to segment the granules using electron microscopy images is proposed. Second, the relative locations of the granules with respect to the plasma membrane are characterized by two functional descriptors: the empirical cumulative distribution function of the distances from the granules to the plasma membrane and the density of granules within a given distance to the plasma membrane. The descriptors of the different cells for each group are compared using bootstrap procedures. Our results show that these descriptors and the testing procedure allow discriminating between control and treated cells. The application of these novel tools to studies of secretion should help in the analysis of diseases associated with dysfunctional secretion, such as diabetes.

Analyzing spatio-temporal dependencies between different types of events is highly relevant to numerous biological phenomena (e.g. signalling and trafficking) especially as advances in probes and microscopy have facilitated imaging of dynamic processes in living cells. For many types of events, the segmented areas can overlap spatially and temporally forming random clumps. In this paper, we model binary image sequences of two different event types as a realization of a bivariate temporal random set and propose a non-parametric approach to quantify spatial and spatio-temporal interrelations using the pair-correlation, cross-covariance and the Ripley IK functions. Based on these summary statistics we propose a randomization procedure to test independence between event types by applying random toroidal shifts and Monte Carlo tests. A simulation study assessed the performance of the proposed estimators and showed that these statistics capture the spatio-temporal dependencies accurately. The estimation of the spatio-temporal interval of interactions was also obtained. The method was successfully applied to analyze the interdependencies of several endocytic proteins using image sequences of living cells and validated the procedure as a new way to automatically quantify dependencies between proteins in a formal and robust manner.

Total Internal Reflection Fluorescence Microscopy (TIRFM) greatly facilitates to imaging the first steps of endocytosis, a process whereby cells traffic cargo from the cell surface to endosomes. Using TIRFM, fluorescent-tagged endocytic proteins are observed as overlapped areas forming random clumps of different sizes, shapes and durations. A common procedure to segment these objects consists of thresholding the original gray-level images to produce binary sequences in which a pixel is covered or not by a given fluorescent-tagged protein. This binary logic is not appropriate because it leaves a free tuning parameter to be set by the user which can influence on the conclusions of the statistical analysis. Instead, we have adopted a more realistic approach, in which segmented binary images are modelled as a fuzzy temporal random set. Here, we propose some measures of spatio-temporal interactions based on the fuzzy counterparts of the pair-correlation function and the Ripley K-function. We used a randomization procedure to test for independence. Our results show that this procedure will permit biologists to examine and quantify the interactions between endocytic proteins robustly.

Counting spatially and temporally overlapping events in image sequences and estimating their shape-size and duration features are important issues in some applications. We propose a stochastic model, a particular case of the nonisotropic 3D Boolean model, for performing this analysis: the temporal Boolean model. Some probabilistic properties are derived and a methodology for parameter estimation from time-lapse image sequences is proposed using an explicit treatment of the temporal dimension. We estimate the mean number of germs per unit area and time, the mean grain size and the duration distribution. A wide simulation study in order to assess the proposed estimators showed promising results. The model was applied on biological image sequences of in-vivo cells in order to estimate new parameters such as the mean number and duration distribution of endocytic events. Our results show that the proposed temporal Boolean model is effective for obtaining information about dynamic processes which exhibit short-lived, but spatially and temporally overlapping events.

Exocytosis is an essential cellular trafficking process integral to the proper distribution and function of a plethora of molecules, including transporters, receptors, and enzymes. Moreover, incorrect protein targeting can lead to pathological conditions. Recently, the application of evanescent wave microscopy has allowed us to image the final steps of exocytosis. However, spatio-temporal analysis of fusion of constitutive vesicular traffic with the plasma membrane has not been systematically performed. Also, the spatial sites and times of vesicle fusion have not yet been analyzed together. In addition, more formal tests are required in testing biological hypotheses, rather than visual inspection combined with statistical descriptives. Ripley K-functions are used to examine the joint and marginal behavior of locations and fusion times. Semiautomatic detection and mapping of constitutive fusion sites reveals spatial and temporal clustering, but no dependency between the locations and times of fusion events. Our novel approach could be translated to other studies of membrane trafficking in health and diseases such as diabetes.