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Saurabh A, Safar M, Fazel M, Sgouralis I, Pressé S. Single-photon smFRET: II. Application to continuous illumination. BIOPHYSICAL REPORTS 2022; 3:100087. [PMID: 36582656 PMCID: PMC9792399 DOI: 10.1016/j.bpr.2022.100087] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Revised: 11/01/2022] [Accepted: 11/21/2022] [Indexed: 12/03/2022]
Abstract
Here we adapt the Bayesian nonparametrics (BNP) framework presented in the first companion article to analyze kinetics from single-photon, single-molecule Förster resonance energy transfer (smFRET) traces generated under continuous illumination. Using our sampler, BNP-FRET, we learn the escape rates and the number of system states given a photon trace. We benchmark our method by analyzing a range of synthetic and experimental data. Particularly, we apply our method to simultaneously learn the number of system states and the corresponding kinetics for intrinsically disordered proteins using two-color FRET under varying chemical conditions. Moreover, using synthetic data, we show that our method can deduce the number of system states even when kinetics occur at timescales of interphoton intervals.
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Affiliation(s)
- Ayush Saurabh
- Center for Biological Physics, Arizona State University, Tempe, Arizona,Department of Physics, Arizona State University, Tempe, Arizona
| | - Matthew Safar
- Center for Biological Physics, Arizona State University, Tempe, Arizona,Department of Mathematics and Statistical Science, Arizona State University, Tempe, Arizona
| | - Mohamadreza Fazel
- Center for Biological Physics, Arizona State University, Tempe, Arizona,Department of Physics, Arizona State University, Tempe, Arizona
| | - Ioannis Sgouralis
- Department of Mathematics, University of Tennessee Knoxville, Knoxville, Tennessee
| | - Steve Pressé
- Center for Biological Physics, Arizona State University, Tempe, Arizona,Department of Physics, Arizona State University, Tempe, Arizona,School of Molecular Sciences, Arizona State University, Tempe, Arizona,Corresponding author
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2
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Gopich IV, Chung HS. Theory and Analysis of Single-Molecule FRET Experiments. Methods Mol Biol 2022; 2376:247-282. [PMID: 34845614 DOI: 10.1007/978-1-0716-1716-8_14] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Inter-dye distances and conformational dynamics can be studied using single-molecule FRET measurements. We consider two approaches to analyze sequences of photons with recorded photon colors and arrival times. The first approach is based on FRET efficiency histograms obtained from binned photon sequences. The experimental histograms are compared with the theoretical histograms obtained using the joint distribution of acceptor and donor photons or the Gaussian approximation. In the second approach, a photon sequence is analyzed without binning. The parameters of a model describing conformational dynamics are found by maximizing the appropriate likelihood function. The first approach is simpler, while the second one is more accurate, especially when the population of species is small and transition rates are fast. The likelihood-based analysis as well as the recoloring method has the advantage that diffusion of molecules through the laser focus can be rigorously handled.
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Affiliation(s)
- Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, USA.
| | - Hoi Sung Chung
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, USA
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3
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Gehlen MH, Foltran LS, Kienle DF, Schwartz DK. Single-Molecule Observations Provide Mechanistic Insights into Bimolecular Knoevenagel Amino Catalysis. J Phys Chem Lett 2020; 11:9714-9724. [PMID: 33136415 DOI: 10.1021/acs.jpclett.0c03030] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
While single-molecule (SM) methods have provided new insights to various catalytic processes, bimolecular reactions have been particularly challenging to study. Here, the fluorogenic Knoevenagel condensation of an aromatic aldehyde with methyl cyanoacetate promoted by surface-immobilized piperazine is quantitatively characterized using super-resolution fluorescence imaging and stochastic analysis using hidden Markov modeling (HMM). Notably, the SM results suggest that the reaction follows the iminium intermediate pathway before the formation of a fluorescent product with intramolecular charge-transfer character. Moreover, the overall process is limited by the turnover rate of the catalyst, which is involved in multiple steps along the reaction coordinate.
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Affiliation(s)
- Marcelo H Gehlen
- Department of Physical Chemistry, Institute of Chemistry of São Carlos, University of São Paulo, 13566-590 São Carlos, SP, Brazil
| | - Larissa S Foltran
- Department of Physical Chemistry, Institute of Chemistry of São Carlos, University of São Paulo, 13566-590 São Carlos, SP, Brazil
| | - Daniel F Kienle
- Department of Chemistry and Biological Engineering, University of Colorado, Boulder, Colorado 80309, United States
| | - Daniel K Schwartz
- Department of Chemistry and Biological Engineering, University of Colorado, Boulder, Colorado 80309, United States
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4
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Chung HS, Louis JM, Gopich IV. Analysis of Fluorescence Lifetime and Energy Transfer Efficiency in Single-Molecule Photon Trajectories of Fast-Folding Proteins. J Phys Chem B 2016; 120:680-99. [PMID: 26812046 DOI: 10.1021/acs.jpcb.5b11351] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
In single-molecule Förster resonance energy transfer (FRET) spectroscopy, the dynamics of molecular processes are usually determined by analyzing the fluorescence intensity of donor and acceptor dyes. Since FRET efficiency is related to fluorescence lifetimes, additional information can be extracted by analyzing fluorescence intensity and lifetime together. For fast processes where individual states are not well separated in a trajectory, it is not easy to obtain the lifetime information. Here, we present analysis methods to utilize fluorescence lifetime information from single-molecule FRET experiments, and apply these methods to three fast-folding, two-state proteins. By constructing 2D FRET efficiency-lifetime histograms, the correlation can be visualized between the FRET efficiency and fluorescence lifetimes in the presence of the submicrosecond to millisecond dynamics. We extend the previously developed method for analyzing delay times of donor photons to include acceptor delay times. To determine the kinetics and lifetime parameters accurately, we used a maximum likelihood method. We found that acceptor blinking can lead to inaccurate parameters in the donor delay time analysis. This problem can be solved by incorporating acceptor blinking into a model. While the analysis of acceptor delay times is not affected by acceptor blinking, it is more sensitive to the shape of the delay time distribution resulting from a broad conformational distribution in the unfolded state.
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Affiliation(s)
- Hoi Sung Chung
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
| | - John M Louis
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
| | - Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health , Bethesda, Maryland 20892-0520, United States
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5
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Gopich IV. Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET. J Chem Phys 2015; 142:034110. [PMID: 25612692 DOI: 10.1063/1.4904381] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Photon sequences from single-molecule Förster resonance energy transfer (FRET) experiments can be analyzed using a maximum likelihood method. Parameters of the underlying kinetic model (FRET efficiencies of the states and transition rates between conformational states) are obtained by maximizing the appropriate likelihood function. In addition, the errors (uncertainties) of the extracted parameters can be obtained from the curvature of the likelihood function at the maximum. We study the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. The standard deviations can be obtained analytically in a special case when the FRET efficiencies of the states are 0 and 1 and in the limiting cases of fast and slow conformational dynamics. These results are compared with the results of numerical simulations. The accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a "chevron" shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated.
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Affiliation(s)
- Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892, USA
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6
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Abstract
Single-molecule spectroscopy is widely used to study macromolecular dynamics. Although this technique provides unique information that cannot be obtained at the ensemble level, the possibility of studying fast molecular dynamics is limited by the number of photons detected per unit time (photon count rate), which is proportional to the illumination intensity. However, simply increasing the illumination intensity often does not help because of various photophysical and photochemical problems. In this Perspective, we show how to improve the dynamic range of single-molecule fluorescence spectroscopy at a given photon count rate by considering each and every photon and using a maximum likelihood method. For a photon trajectory with recorded photon colors and inter-photon times, the parameters of a model describing molecular dynamics are obtained by maximizing the appropriate likelihood function. We discuss various likelihood functions, their applicability, and the accuracy of the extracted parameters. The maximum likelihood method has been applied to analyze the experiments on fast two-state protein folding and to measure transition path times. Utilizing other information such as fluorescence lifetimes is discussed in the framework of two-dimensional FRET efficiency-lifetime histograms.
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Affiliation(s)
- Hoi Sung Chung
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health (NIH), Bethesda, MD, 20892-0520
| | - Irina V. Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health (NIH), Bethesda, MD, 20892-0520
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7
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Keller BG, Kobitski A, Jäschke A, Nienhaus GU, Noé F. Complex RNA folding kinetics revealed by single-molecule FRET and hidden Markov models. J Am Chem Soc 2014; 136:4534-43. [PMID: 24568646 PMCID: PMC3977575 DOI: 10.1021/ja4098719] [Citation(s) in RCA: 76] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
![]()
We
have developed a hidden Markov model and optimization procedure
for photon-based single-molecule FRET data, which takes into account
the trace-dependent background intensities. This analysis technique
reveals an unprecedented amount of detail in the folding kinetics
of the Diels–Alderase ribozyme. We find a multitude of extended
(low-FRET) and compact (high-FRET) states. Five states were consistently
and independently identified in two FRET constructs and at three Mg2+ concentrations. Structures generally tend to become more
compact upon addition of Mg2+. Some compact structures
are observed to significantly depend on Mg2+ concentration,
suggesting a tertiary fold stabilized by Mg2+ ions. One
compact structure was observed to be Mg2+-independent,
consistent with stabilization by tertiary Watson–Crick base
pairing found in the folded Diels–Alderase structure. A hierarchy
of time scales was discovered, including dynamics of 10 ms or faster,
likely due to tertiary structure fluctuations, and slow dynamics on
the seconds time scale, presumably associated with significant changes
in secondary structure. The folding pathways proceed through a series
of intermediate secondary structures. There exist both compact pathways
and more complex ones, which display tertiary unfolding, then secondary
refolding, and, subsequently, again tertiary refolding.
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Affiliation(s)
- Bettina G Keller
- Freie Universität Berlin , Institute of Chemistry and Biochemistry, Takustr. 3, 14195 Berlin, Germany
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8
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Zarrabi N, Ernst S, Verhalen B, Wilkens S, Börsch M. Analyzing conformational dynamics of single P-glycoprotein transporters by Förster resonance energy transfer using hidden Markov models. Methods 2013; 66:168-79. [PMID: 23891547 DOI: 10.1016/j.ymeth.2013.07.026] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2013] [Revised: 07/04/2013] [Accepted: 07/16/2013] [Indexed: 12/15/2022] Open
Abstract
Single-molecule Förster resonance energy (smFRET) transfer has become a powerful tool for observing conformational dynamics of biological macromolecules. Analyzing smFRET time trajectories allows to identify the state transitions occuring on reaction pathways of molecular machines. Previously, we have developed a smFRET approach to monitor movements of the two nucleotide binding domains (NBDs) of P-glycoprotein (Pgp) during ATP hydrolysis driven drug transport in solution. One limitation of this initial work was that single-molecule photon bursts were analyzed by visual inspection with manual assignment of individual FRET levels. Here a fully automated analysis of Pgp smFRET data using hidden Markov models (HMM) for transitions up to 9 conformational states is applied. We propose new estimators for HMMs to integrate the information of fluctuating intensities in confocal smFRET measurements of freely diffusing lipid bilayer bound membrane proteins in solution. HMM analysis strongly supports that under conditions of steady state turnover, conformational states with short NBD distances and short dwell times are more populated compared to conditions without nucleotide or transport substrate present.
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Affiliation(s)
- Nawid Zarrabi
- Single-Molecule Microscopy Group, Jena University Hospital, Friedrich Schiller University Jena, 07743 Jena, Germany; 3rd Institute of Physics, University of Stuttgart, 70550 Stuttgart, Germany
| | - Stefan Ernst
- Single-Molecule Microscopy Group, Jena University Hospital, Friedrich Schiller University Jena, 07743 Jena, Germany
| | - Brandy Verhalen
- Department of Biochemistry & Molecular Biology, SUNY Upstate Medical University, Syracuse, NY 13210, USA
| | - Stephan Wilkens
- Department of Biochemistry & Molecular Biology, SUNY Upstate Medical University, Syracuse, NY 13210, USA
| | - Michael Börsch
- Single-Molecule Microscopy Group, Jena University Hospital, Friedrich Schiller University Jena, 07743 Jena, Germany; 3rd Institute of Physics, University of Stuttgart, 70550 Stuttgart, Germany.
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9
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Brox D, Kiel A, Wörner SJ, Pernpointner M, Comba P, Martin B, Herten DP. Ensemble and single-molecule studies on fluorescence quenching in transition metal bipyridine-complexes. PLoS One 2013; 8:e58049. [PMID: 23483966 PMCID: PMC3587577 DOI: 10.1371/journal.pone.0058049] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2012] [Accepted: 01/29/2013] [Indexed: 11/19/2022] Open
Abstract
Beyond their use in analytical chemistry fluorescent probes continuously gain importance because of recent applications of single-molecule fluorescence spectroscopy to monitor elementary reaction steps. In this context, we characterized quenching of a fluorescent probe by different metal ions with fluorescence spectroscopy in the bulk and at the single-molecule level. We apply a quantitative model to explain deviations from existing standard models for fluorescence quenching. The model is based on a reversible transition from a bright to a dim state upon binding of the metal ion. We use the model to estimate the stability constants of complexes with different metal ions and the change of the relative quantum yield of different reporter dye labels. We found ensemble data to agree widely with results from single-molecule experiments. Our data indicates a mechanism involving close molecular contact of dye and quenching moiety which we also found in molecular dynamics simulations. We close the manuscript with a discussion of possible mechanisms based on Förster distances and electrochemical potentials which renders photo-induced electron transfer to be more likely than Förster resonance energy transfer.
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Affiliation(s)
- Dominik Brox
- Cellnetworks Cluster and Institute of Physical Chemistry, Heidelberg University, Heidelberg, Germany
| | - Alexander Kiel
- Cellnetworks Cluster and Institute of Physical Chemistry, Heidelberg University, Heidelberg, Germany
| | | | | | - Peter Comba
- Institute of Inorganic Chemistry, Heidelberg University, Heidelberg, Germany
| | - Bodo Martin
- Institute of Inorganic Chemistry, Heidelberg University, Heidelberg, Germany
| | - Dirk-Peter Herten
- Cellnetworks Cluster and Institute of Physical Chemistry, Heidelberg University, Heidelberg, Germany
- * E-mail:
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10
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Bingemann D, Allen RM. Identification of intensity ratio break points from photon arrival trajectories in ratiometric single molecule spectroscopy. Int J Mol Sci 2012; 13:7445-7465. [PMID: 22837704 PMCID: PMC3397536 DOI: 10.3390/ijms13067445] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2012] [Revised: 06/07/2012] [Accepted: 06/12/2012] [Indexed: 11/16/2022] Open
Abstract
We describe a statistical method to analyze dual-channel photon arrival trajectories from single molecule spectroscopy model-free to identify break points in the intensity ratio. Photons are binned with a short bin size to calculate the logarithm of the intensity ratio for each bin. Stochastic photon counting noise leads to a near-normal distribution of this logarithm and the standard student t-test is used to find statistically significant changes in this quantity. In stochastic simulations we determine the significance threshold for the t-test's p-value at a given level of confidence. We test the method's sensitivity and accuracy indicating that the analysis reliably locates break points with significant changes in the intensity ratio with little or no error in realistic trajectories with large numbers of small change points, while still identifying a large fraction of the frequent break points with small intensity changes. Based on these results we present an approach to estimate confidence intervals for the identified break point locations and recommend a bin size to choose for the analysis. The method proves powerful and reliable in the analysis of simulated and actual data of single molecule reorientation in a glassy matrix.
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Affiliation(s)
- Dieter Bingemann
- Department of Chemistry, Williams College, 47 Lab Campus Drive, Williamstown, MA 01267, USA
| | - Rachel M. Allen
- San Francisco Estuary Institute, Oakland, CA 94621, USA; E-Mail:
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11
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Abstract
We consider the analysis of a class of experiments in which the number of photons in consecutive time intervals is recorded. Sequence of photon counts or, alternatively, of FRET efficiencies can be studied using likelihood-based methods. For a kinetic model of the conformational dynamics and state-dependent Poisson photon statistics, the formalism to calculate the exact likelihood that this model describes such sequences of photons or FRET efficiencies is developed. Explicit analytic expressions for the likelihood function for a two-state kinetic model are provided. The important special case when conformational dynamics are so slow that at most a single transition occurs in a time bin is considered. By making a series of approximations, we eventually recover the likelihood function used in hidden Markov models. In this way, not only is insight gained into the range of validity of this procedure, but also an improved likelihood function can be obtained.
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Affiliation(s)
- Irina V Gopich
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892, USA
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12
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Boghossian AA, Zhang J, Le Floch-Yin FT, Ulissi ZW, Bojo P, Han JH, Kim JH, Arkalgud JR, Reuel NF, Braatz RD, Strano MS. The chemical dynamics of nanosensors capable of single-molecule detection. J Chem Phys 2011; 135:084124. [DOI: 10.1063/1.3606496] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Boghossian AA, Zhang J, Barone PW, Reuel NF, Kim JH, Heller DA, Ahn JH, Hilmer AJ, Rwei A, Arkalgud JR, Zhang CT, Strano MS. Near-infrared fluorescent sensors based on single-walled carbon nanotubes for life sciences applications. CHEMSUSCHEM 2011; 4:848-63. [PMID: 21751417 DOI: 10.1002/cssc.201100070] [Citation(s) in RCA: 102] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
Many properties of single-walled carbon nanotubes (SWCNTs) make them ideal candidates for sensors, particularly for biological systems. Both their fluorescence in the near-infrared range of 820-1600 nm, where absorption by biological tissues is often minimal, and their inherent photostability are desirable attributes for the design of in vitro and in vivo sensors. The mechanisms by which a target molecule can selectively alter the fluorescent emission include primarily changes in emission wavelength (i.e., solvatochromism) and intensity, including effects such as charge-transfer transition bleaching and exciton quenching. The central challenge lies in engineering the nanotube interface to be selective for the analyte of interest. In this work, we review the recent development in this area over the past few years, and describe the design rules that we have developed for detecting various analytes, ranging from stable small molecules and reactive oxygen species (ROS) or reactive nitrogen species (RNS) to macromolecules. Applications to in vivo sensor measurements using these sensors are also described. In addition, the emerging field of SWCNT-based single-molecule detection using band gap fluorescence and the recent efforts to accurately quantify and utilize this unique class of stochastic sensors are also described in this article.
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Affiliation(s)
- Ardemis A Boghossian
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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14
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Schwering M, Kiel A, Kurz A, Lymperopoulos K, Sprödefeld A, Krämer R, Herten DP. Hochauflösende Mikroskopie mit reversiblen chemischen Reaktionen. Angew Chem Int Ed Engl 2011. [DOI: 10.1002/ange.201006013] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
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15
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Schwering M, Kiel A, Kurz A, Lymperopoulos K, Sprödefeld A, Krämer R, Herten DP. Far-field nanoscopy with reversible chemical reactions. Angew Chem Int Ed Engl 2011; 50:2940-5. [PMID: 21404374 DOI: 10.1002/anie.201006013] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2010] [Indexed: 02/06/2023]
Affiliation(s)
- Michael Schwering
- Cellnetworks Cluster & Physikalisch-Chemisches Institut, Universität Heidelberg, Heidelberg, Germany
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16
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Börsch M, Wrachtrup J. Improving FRET‐Based Monitoring of Single Chemomechanical Rotary Motors at Work. Chemphyschem 2011; 12:542-53. [DOI: 10.1002/cphc.201000702] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2010] [Revised: 12/12/2010] [Indexed: 11/07/2022]
Affiliation(s)
- Michael Börsch
- 3rd Institute of Physics and Stuttgart Research Center SCOPE, University of Stuttgart, Pfaffenwaldring 57, Fax: (+49) 711‐685‐65281
| | - Jörg Wrachtrup
- 3rd Institute of Physics and Stuttgart Research Center SCOPE, University of Stuttgart, Pfaffenwaldring 57, Fax: (+49) 711‐685‐65281
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