Trajectories Of Molecules Research Articles

Molecular trajectories provide signatures of protein clustering and crowding at the oil/water interface.

Using high throughput single-molecule total internal reflection fluorescence microscopy (TIRFM), we have acquired molecular trajectories of bovine serum albumin (BSA) and hen egg white lysozyme during protein layer formation at the silicone oil-water interface. These trajectories were analyzed to determine the distribution of molecular diffusion coefficients, and for signatures of molecular crowding/caging, including subdiffusive motion and temporal anticorrelation of the instantaneous velocity vector. The evolution of these properties with aging time of the interface was compared with dynamic interfacial tension measurements. For both lysozyme and BSA, we observed an overall slowing of protein objects, the onset of both subdiffusive and anticorrelated motion (associated with crowding), and a decrease in the interfacial tension with aging time. For lysozyme, all of these phenomena occurred virtually simultaneously, consistent with a homogeneous model of layer formation that involves gradual crowding of weakly interacting proteins. For BSA, however, the slowing occurred first, followed by the signatures of crowding/caging, followed by a decrease in interfacial tension, consistent with a heterogeneous model of layer formation involving the formation of protein clusters. The application of microrheological methods to single molecule trajectories described here provides an unprecedented level of mechanistic interpretation of interfacial events that occurred over a wide range of interfacial protein coverage.

Function evaluation of the molecular scattering angle in multicomponent gaseous medium

The molecular interaction potential (2(n+3),6) for the multicomponent gaseous medium is proposed. To compute the values of a function of scattering angle, an asymptotic analysis is performed that permits the explicit expressions for this function to be obtained. The trajectories of molecules described on a set of real and complex numbers are first considered.

Transition-Path Probability as a Test of Reaction-Coordinate Quality Reveals DNA Hairpin Folding Is a One-Dimensional Diffusive Process.

Chemical reactions are typically described in terms of progress along a reaction coordinate. However, the quality of reaction coordinates for describing reaction dynamics is seldom tested experimentally. We applied a framework for gauging reaction-coordinate quality based on transition-path analysis to experimental data for the first time, looking at folding trajectories of single DNA hairpin molecules measured under tension applied by optical tweezers. The conditional probability for being on a reactive transition path was compared with the probability expected for ideal diffusion over a 1D energy landscape based on the committor function. Analyzing measurements and simulations of hairpin folding where end-to-end extension is the reaction coordinate, after accounting for instrumental effects on the analysis, we found good agreement between transition-path and committor analyses for model two-state hairpins, demonstrating that folding is well-described by 1D diffusion. This work establishes transition-path analysis as a powerful new tool for testing experimental reaction-coordinate quality.

High-Speed Single-Particle Tracking Reveals Lipid Dynamics in Heterogeneous Raft-Containing Membranes

Phase coexistence and domain formation in lipid bilayers are intriguing phenomena and play essential roles in many cell functions. In particular, coexistence of liquid-ordered and liquid-disordered phases in membranes of ternary lipid composition has drawn great attention, as it underlies the concept of “lipid rafts” in cell membranes. Most biological significance of raft domains is originated from their capability of recruiting certain molecules into the domain through interaction between lipids and proteins. However, experimental evidence of how such interaction takes place at single-molecule level has been missing because it requires high spatiotemporal resolution for observing rapid and nanoscopic motion of individual molecules in raft-containing membranes.In this work, we investigated the dynamics of single lipid molecules in raft-containing bilayer membranes by high-speed optical imaging. An advanced microscope system was developed for tracking individual small gold nanoparticles (20 nm in diameter) attaching to the head groups of single lipid molecules (through biotin-avidin conjugation). The linear scattering signal from the nanoparticles was detected by optical interference where optimal spatial localization precision down to a few nanometers can be achieved at very high speed (50,000 frames per second). Furthermore, the photostable linear scattering signal offers unlimited observation time. Using this technique, we recorded precise and long diffusion trajectories of single lipid molecules in raft-containing supported lipid bilayers (DOPC/sphingomyelin/cholesterol, 40:40:20 mol ratios). Meanwhile, the micron-sized raft domains in the membrane were observed in fluorescence by adding a trace amount of dye-labeled DOPE that marks the non-raft regions. By correlating the detailed trajectories of single probe lipids recorded at high-speed to the corresponding fluorescence image that contours the raft domains, we reveal how a lipid molecule goes across the raft boundary and how its diffusion characteristics changes due to the raft domains with unprecedented clarity.

Identifying Transport Behavior of Single-Molecule Trajectories

Models of biological diffusion-reaction systems require accurate classification of the underlying diffusive dynamics (e.g., Fickian, subdiffusive, or superdiffusive). We use a renormalization group operator to identify the anomalous (non-Fickian) diffusion behavior from a short trajectory of a single molecule. The method provides quantitative information about the underlying stochastic process, including its anomalous scaling exponent. The classification algorithm is first validated on simulated trajectories of known scaling. Then it is applied to experimental trajectories of microspheres diffusing in cytoplasm, revealing heterogeneous diffusive dynamics. The simplicity and robustness of this classification algorithm makes it an effective tool for analysis of rare stochastic events that occur in complex biological systems.

Insights into Nanoscale Electrophoresis of Single Dye Molecules in Highly Oriented Mesoporous Silica Channels

Electrophoresis is a very important tool in chemistry, biochemistry, and biology and is commonly used for the separation and analysis of molecules. Typically, electrophoresis is performed at the macroscopic level. With a view on integrating molecular separations into micro- and nanofluidic and massively parallel systems, it would be highly desirable to obtain detailed knowledge about electrophoretic separation at the nanoscale. Here, we describe an in situ single molecule fluorescence imaging approach to provide insights into the controlled movement and separation of charged molecules in highly oriented silica nanochannels under the influence of an electric field. The velocity of the charged single molecules increases with increasing electric field strength and can therefore be directly controlled by an external stimulus. On the basis of the single molecule trajectories, we propose a model for the internal structure of the host and for the diffusion of charged and uncharged species in the nanoscale channels. In order to prove that the molecules move along the aligned pores, we applied an electric field diagonally to the pores. The molecules that were located in the mesopores showed a directed movement along the mesoporous silica channels, whereas molecules next to, but outside of, the structure followed the electric field lines. Leaky defects in the side walls of the pores were indicated by molecules switching into and out of the pores. In addition, we show the application of the mesoporous host material for separation of molecules according to their mobility and charge.

Sensitive and selective plasmon ruler nanosensors for monitoring the apoptotic drug response in leukemia.

Caspases are proteases involved in cell death, where caspase-3 is the chief executioner that produces an irreversible cutting event in downstream protein substrates and whose activity is desired in the management of cancer. To determine such activity in clinically relevant samples with high signal-to-noise, plasmon rulers are ideal because they are sensitively affected by their interparticle separation without ambiguity from photobleaching or blinking effects. A plasmon ruler is a noble metal nanoparticle pair, tethered in close proximity to one another via a biomolecule, that acts through dipole–dipole interactions and results in the light scattering to increase exponentially. In contrast, a sharp decrease in intensity is observed when the pair is confronted by a large interparticle distance. To align the mechanism of protease activity with building a sensor that can report a binary signal in the presence or absence of caspase-3, we present a caspase-3 selective plasmon ruler (C3SPR) composed of a pair of Zn0.4Fe2.6O4@SiO2@Au core–shell nanoparticles connected by a caspase-3 cleavage sequence. The dielectric core (Zn0.4Fe2.6O4@SiO2)-shell (Au) geometry provided a brighter scattering intensity versus solid Au nanoparticles, and the magnetic core additionally acted as a purification handle during the plasmon ruler assembly. By monitoring the decrease in light scattering intensity per plasmon ruler, we detected caspase-3 activity at single molecule resolution across a broad dynamic range. This was observed to be as low as 100 fM of recombinant material or 10 ng of total protein from cellular lysate. By thorough analyses of single molecule trajectories, we show caspase-3 activation in a drug-treated chronic myeloid leukemia (K562) cancer system as early as 4 and 8 h with greater sensitivity (2- and 4-fold, respectively) than conventional reagents. This study provides future implications for monitoring caspase-3 as a biomarker and efficacy of drugs.

Coincidence Measurements in Dual-Color Confocal Microscopy: A Combined Single-Particle and Fluorescence Correlation Approach

In this paper we discuss how the coincident detection of mobile particles in dual-color confocal images can be improved. Optimal coincidence detection requires a careful choice of experimental conditions and image acquisition parameters in order to maximize the overlap between the two detection volumes. By measuring this overlap with fluorescence cross-correlation spectroscopy, we show in particular that a small confocal field of view is necessary in order to maintain good coincidence. Most importantly, coincidence detection also requires a dedicated image analysis strategy. Traditionally, two approaches have been adopted to assess coincidence of mobile particles: fluorescence fluctuation measurements, notably cross-correlation spectroscopy, and single particle detection. Here we propose to combine these two approaches by calculating a cross-correlation coefficient for each of the detected single particles. We show that this allows to remove accidental coincidence events from a data set, and thus to unambiguously identify particles that instead carry two different fluorophores. This strategy can help increase the available concentration range for confocal coincidence measurements and detect rare binding events. [Formula: see text]Special Issue Comments: This article about coincident detection of mobile particle in two-color confocal images is thematically related to several articles in this Special Issue, namely the review of FRET-based single-molecule fluorescence techniques by Ruedas-Rama et al.,1 the single particle detection work presented by de Keersmaecker et al.2 and the general considerations on the mathematical treatment of single molecule trajectories presented by Flomenbom3. Our study of single liposomes is also relevant to experiments involving proteins and liposomes, such as the enzyme experiments described in the review by Jørgensen and Hatzakis.4

Charge-dependent transport switching of single molecular ions in a weak polyelectrolyte multilayer.

The tunable nature of weak polyelectrolyte multilayers makes them ideal candidates for drug loading and delivery, water filtration, and separations, yet the lateral transport of charged molecules in these systems remains largely unexplored at the single molecule level. We report the direct measurement of the charge-dependent, pH-tunable, multimodal interaction of single charged molecules with a weak polyelectrolyte multilayer thin film, a 10 bilayer film of poly(acrylic acid) and poly(allylamine hydrochloride) PAA/PAH. Using fluorescence microscopy and single-molecule tracking, two modes of interaction were detected: (1) adsorption, characterized by the molecule remaining immobilized in a subresolution region and (2) diffusion trajectories characteristic of hopping (D ∼ 10–9 cm2/s). Radius of gyration evolution analysis and comparison with simulated trajectories confirmed the coexistence of the two transport modes in the same single molecule trajectories. A mechanistic explanation for the probe and condition mediated dynamics is proposed based on a combination of electrostatics and a reversible, pH-induced alteration of the nanoscopic structure of the film. Our results are in good agreement with ensemble studies conducted on similar films, confirm a previously-unobserved hopping mechanism for charged molecules in polyelectrolyte multilayers, and demonstrate that single molecule spectroscopy can offer mechanistic insight into the role of electrostatics and nanoscale tunability of transport in weak polyelectrolyte multilayers.

Surface adsorption and bulk aggregation of cyclodextrins by computational molecular dynamics simulations as a function of temperature: α-CD vs β-CD.

The structural simplicity of native cyclodextrins (CDs) contrasts with their complex behavior in the bulk of aqueous solutions, mainly when they are combined with other cosolutes. Many scientific and industrial applications based on these molecules are supported only by empirical information. The lack of fundamental knowledge, which would allow one to rationally optimize many of these applications, is notable mainly at the solution/air interface. Basic information on phenomena such as the spontaneous adsorption of native CDs or on the structure of CD aggregates in the bulk solution is really scarce. In order to fill these gaps, a detailed computational study on the adsorption and aggregation of α- and β-CDs as a function of temperature is presented here. Our simulations reproduce, at atomic resolution, the experimentally observed much higher ability of β-CD to aggregate compared to that of α-CD at 298 K, as well as their dependence on temperature. The adsorption of both individual CDs and small CD aggregates (up to 20 molecules) to the solution/air interface is found to be negligible. 0.8 μs long trajectories of single CD molecules in aqueous solution reveal that the main differences in the behavior of both CDs are their flexibility, higher for β-CD, and the occupancy of individual intramolecular hydrogen bonds that is significantly longer for the same cyclodextrin. The aggregation pattern of α- and β-CDs is followed at the hundreds of ns time scale, allowing both the spontaneous self-assembly of cyclodextrins and their redistribution along the aggregates to be observed. This is the first attempt to study the adsorption and aggregation of native cyclodextrins by atomistic molecular dynamics simulations.

The challenges of closed-loop sensor simulation

Reality is the baseline for every simulation. Consequently, every kind of simulation aims to imitate certain aspects of reality. Depending on the choice of aspects to be simulated and, thus, the choice of other aspects not to be simulated, different classes of simulations exist. Usually, implementing a simulation means reducing the level of detail or the level of abstraction to make a simulation of a complex phenomenon possible. For example, when simulating a thunderstorm, it is usually not necessary to simulate the trajectory of each single molecule within reach of that storm. Instead, one would choose a level of abstraction closer to the intended outcome of the simulation. Likewise, fight simulations are usually cockpit simulations, in some cases including full motion, which entails making use of mechanical means to move the entire cockpit in a limited way. Hence, fight simulation is usually limited to the aspects of fight that can be perceived by a pilot within the cockpit. In the following, we will use the term operator for pilot, driver, or, more generally, the person inside the simulator.

Sensitivity of Water Dynamics to Biologically Significant Surfaces of Monomeric Insulin: Role of Topology and Electrostatic Interactions

In addition to the biologically active monomer of the protein insulin circulating in human blood, the molecule also exists in dimeric and hexameric forms that are used as storage. The insulin monomer contains two distinct surfaces, namely, the dimer forming surface (DFS) and the hexamer forming surface (HFS), that are specifically designed to facilitate the formation of the dimer and the hexamer, respectively. In order to characterize the structural and dynamical behavior of interfacial water molecules near these two surfaces (DFS and HFS), we performed atomistic molecular dynamics simulations of insulin with explicit water. Dynamical characterization reveals that the structural relaxation of the hydrogen bonds formed between the residues of DFS and the interfacial water molecules is faster than those formed between water and that of the HFS. Furthermore, the residence times of water molecules in the protein hydration layer for both the DFS and HFS are found to be significantly higher than those for some of the other proteins studied so far, such as HP-36 and lysozyme. In particular, we find that more structured water molecules, with higher residence times (∼ 300-500 ps), are present near HFS than those near DFS. A significant slowing down is observed in the decay of associated rotational auto time correlation functions of O-H bond vector of water in the vicinity of HFS. The surface topography and the arrangement of amino acid residues work together to organize the water molecules in the hydration layer in order to provide them with a preferred orientation. HFS having a large polar solvent accessible surface area and a convex extensive nonpolar region, drives the surrounding water molecules to acquire predominantly an outward H-atoms directed, clathrate-like structure. In contrast, near the DFS, the surrounding water molecules acquire an inward H-atoms directed orientation owing to the flat curvature of hydrophobic surface and the interrupted hydrophilic residual alignment. We have followed escape trajectory of several such quasi-bound water molecules from both the surfaces that reveal the significant differences between the two hydration layers.

Empirical Bayes Methods Enable Advanced Population-Level Analyses of Single-Molecule FRET Experiments

Many single-molecule experiments aim to characterize biomolecular processes in terms of kinetic models that specify the rates of transition between conformational states of the biomolecule. Estimation of these rates often requires analysis of a population of molecules, in which the conformational trajectory of each molecule is represented by a noisy, time-dependent signal trajectory. Although hidden Markov models (HMMs) may be used to infer the conformational trajectories of individual molecules, estimating a consensus kinetic model from the population of inferred conformational trajectories remains a statistically difficult task, as inferred parameters vary widely within a population. Here, we demonstrate how a recently developed empirical Bayesian method for HMMs can be extended to enable a more automated and statistically principled approach to two widely occurring tasks in the analysis of single-molecule fluorescence resonance energy transfer (smFRET) experiments: 1), the characterization of changes in rates across a series of experiments performed under variable conditions; and 2), the detection of degenerate states that exhibit the same FRET efficiency but differ in their rates of transition. We apply this newly developed methodology to two studies of the bacterial ribosome, each exemplary of one of these two analysis tasks. We conclude with a discussion of model-selection techniques for determination of the appropriate number of conformational states. The code used to perform this analysis and a basic graphical user interface front end are available as open source software.

A Markov chain approach to simulate Atomic Layer Deposition chemistry and transport inside nanostructured substrates

In this work, we present a new theoretical framework to model the transport and surface chemistry under molecular (Knudsen) flow. Our approach is based on casting the transport inside nanostructures as a single-particle discrete Markov chain process. One of the advantages of this approach is that it allows us to decouple the complexity of the surface chemistry from the transport model, thus allowing its application under general surface chemistry conditions, including atomic layer deposition (ALD) and chemical vapor deposition (CVD). Our model also allows us to determine statistical information of the trajectory of individual molecules, such as the average interaction time or the number of wall collisions for molecules entering the nanostructures as well as to track the relative contributions to thin-film growth of different independent reaction pathways at each point of the feature. This offers a straightforward way of incorporating into ALD simulations non-ideal surface processes, such as parasitic CVD or surface recombination. By studying the asymptotic behavior of the Markov chain process, we were also able to establish a direct link between ballistic models, kinetic Monte Carlo simulations, and continuous models based on the use of the diffusion equation under Knudsen conditions. Finally, we show that, under certain approximations, the coverage profile inside a nanostructure under ALD conditions is controlled by the total exposure, and not by the details of the surface flux dependence with time during the exposure, as long as the reaction probabilities are pressure independent.

Localization microscopy: mapping cellular dynamics with single molecules

Resolution describes the smallest details within a sample that can be recovered by a microscope lens system. For optical microscopes detecting visible light, diffraction limits the resolution to ∼200-250 nm. In contrast, localization measures the position of an isolated object using its image. Single fluorescent molecules can be localized with an uncertainty of a few tens of nanometres, and in some cases less than one nanometre. Superresolution fluorescence localization microscopy (SRFLM) images and localizes fluorescent molecules in a sample. By controlling the visibility of the fluorescent molecules with light, it is possible to cause a sparse subset of the tags to fluoresce and be spatially separated from each other. A movie is acquired with a camera, capturing images of many sets of visible fluorescent tags over a period of time. The movie is then analysed by a computer whereby all of the single molecules are independently measured, and their positions are recorded. When the coordinates of a sufficient number of molecules are collected, an image can be rendered by plotting the coordinates of the localized molecules. The spatial resolution of these rendered images can be better than 20 nm, roughly an order of magnitude better than the diffraction limited resolution. The invention of SRFLM has led to an explosion of related techniques. Through the use of specialized optics, the fluorescent signal can be split into multiple detection channels. These channels can capture additional information such as colour (emission wavelength), orientation and three-dimensional position of the detected molecules. Measurement of the colour of the detected fluorescence can allow researchers to distinguish multiple types of fluorescent tags and to study the interaction between multiple molecules of interest. Three-dimensional imaging and determination of molecular orientations offer insight into structural organization of the sample. SRFLM is compatible with living samples and has helped to illuminate many dynamic biological processes, such as the trajectories of molecules within living cells. This review discusses the concept and process of SRFLM imaging and investigates recent advances in SRFLM functionality. Since its announcement in 2006, SRFLM has been quickly adopted and modified by many researchers to help investigate questions whose answers lie below the diffraction limit. The versatility of the SRFLM technique has great promise for improving our understanding of cell biology at the molecular level.

Electron Transfer-Based Single Molecule Fluorescence as a Probe for Nano-Environment Dynamics

Electron transfer (ET) is one of the most important elementary processes that takes place in fundamental aspects of biology, chemistry, and physics. In this review, we discuss recent research on single molecule probes based on ET. We review some applications, including the dynamics of glass-forming systems, surface binding events, interfacial ET on semiconductors, and the external field-induced dynamics of polymers. All these examples show that the ET-induced changes of fluorescence trajectory and lifetime of single molecules can be used to sensitively probe the surrounding nano-environments.

Editorial: Special Issue on Stochastic Modelling of Reaction–Diffusion Processes in Biology

Editorial: Special Issue on Stochastic Modelling of Reaction–Diffusion Processes in Biology

A Slipping Clutch in Neuronal Growth Cones Revealed by Transient Single Molecule Interactions between Flowing Actin and N-Cadherin Adhesions

The molecular clutch model proposes that the actin motile machinery generated inside cells is coupled to adhesion at the cell membrane, thus generating forces on the substrate and allowing cells to move forward. Many reports have provided evidence for clutch-like behaviors in different cell types and adhesive systems, including integrins and cadherins (Giannone et al., Trends Cell Biol 2009). However, the exact mechanisms underlying the dynamic molecular coupling between the actin retrograde flow and adhesion proteins remain elusive. We previously inferred using optical tweezers that a molecular clutch between the actin flow and N-cadherin adhesions could drive growth cone migration (Bard et al., J Neurosci 2008), but did not achieve a direct visualization of the engagement process. Here, to trigger N-cadherin homophilic adhesions at specific locations, we cultured primary neurons on micropatterned substrates comprising arrays of dots coated with purified N-cadherin, surrounded by a cytophobic background. We then tracked the trajectories of single adhesion and cytoskeletal molecules fused with photo-convertible fluorescent proteins (mEOS2), at the ventral surface of growth cones. N-cadherin and α-catenin were immobilized, while the actin retrograde flow was significantly reduced at N-cadherin coated micro-patterns, compared to non-adhesive regions. Normal actin flow on micro-patterns was restored by expression of a dominant negative N-cadherin construct inhibiting the coupling between endogenous N-cadherin and actin, demonstrating specificity of the process. Strikingly, individual actin trajectories exhibited pauses of the order of seconds selectively on N-cadherin-coated micro-patterns. Thus, at the individual molecular level, clutch engagement is characterized by transient interactions between flowing actin filaments and the immobilized N-cadherin/catenin complex. To our knowledge, this study represents the first direct demonstration of the intrinsic molecular coupling underlying the clutch process in growth cone locomotion.

Single Molecule FRET Studies of DNA Hairpin Folding

Hairpins are the simplest structures for investigating fundamental aspects of nucleic acid folding mechanisms. For hairpins exhibiting two-state behavior, all of the mechanistic information is contained in the transition path, the rare event in single molecule trajectories when the free energy barrier between folded and unfolded states is actually crossed. The first step toward observing transition paths is to determine the transition path time. Average transition path times have recently been directly determined in proteins using the Gopich/Szabo maximum likelihood method (Gopich, Szabo J. Phys. Chem. B 2009) to analyze folding/unfolding photon trajectories in single molecule FRET experiments (Chung et al. Science 2012; Chung and Eaton Nature 2013), while the only experimental study of transition paths in nucleic acids used optical tweezer measurements to determine an upper bound of 50 μs in trajectories for a variety of structures (Neupane et al. Phys. Rev. Lett. 2012). Neupane et al. also reconstructed the free energy surface for an indirect determination of average transition path times from Szabo's analytical theory for diffusive barrier crossing. We use single molecule FRET to study a fast-folding DNA hairpin with 2 A-T and 2 G-C base pairs in the stem and 21 T's in the loop immobilized on a polyethylene glycol-coated glass surface via a biotin-streptavidin-biotin linkage. The folding time for this hairpin in 500mM NaCl is 530 μs. Maximum likelihood analysis of 780 transitions in photon trajectories with an average detection rate of 750 photons/ms yields an upper bound of 4 μs for the average transition path time, compared to the value of ∼3.6±0.8 μs predicted by Neupane et al. from hairpins with 9-30 base pair stems and 4 T's in the loop. Current experiments use viscogens to slow the transition path for directly determining the actual value.

MATHEMATICAL TREATMENTS THAT SOLVE SINGLE MOLECULES

In this article, we talk about the ways that scientists can solve single molecule trajectories. Solving single molecules, that is, finding the model from the data, is complicated at least as much as measuring single molecules. We must filter the noise and take care of every step in the analysis when constructing the most accurate model from the data. Here, we present valuable solutions. Ways that solve clean discrete data are first presented. We review here our reduced dimensions forms (RDFs): unique models that are canonical forms of discrete data, and the statistical and numerical toolbox that builds a RDF from finite, clean, two-state data. We then review our most recent filter that "tackles" the noise when measuring two state noisy photon trajectories. The filter is a numerical algorithm with various special statistical treatments that is based on a general likelihood function that we have developed recently. We show the strengths of the filter (also over other approaches) and talk about its various new variants. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet, extensions are needed in "tackling" the noise from other data, e.g. continuous data. Only the combined procedures enable creating the most accurate model from noisy discrete trajectories from single molecules. These concepts and methods (with adjustments) are valuable also when solving continuous trajectories and fluorescence resonance energy transfer trajectories. We also present a set of simple methods that can help any scientist with treating the trajectory perhaps encouraging applying the involved methods. The involved methods will appear in software that we are developing now, helping therefore the experimentalists utilizing these methods on real data. Comparisons with other known methods in this field are made. [Formula: see text]Special Issue Comment: This article about mathematical treatments when solving single molecules is related to the reviews in this Special Issue about measuring enzymes67 and about FRET experiments2 and about the software QUB.6