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The guide to the expression of uncertainty in measurement (GUM) describes the law of propagation of uncertainty for linear models based on the first-order Taylor series approximation of Y = f(X1, X2, ..., XN). However, for non-linear models this framework leads to unreliable results while estimating the combined standard uncertainty of the model output [u(y)]. In such instances, it is possible to implement the method(s) described in Supplement 1 to GUM – Propagation of distributions using a Monte Carlo Method. As such, a numerical solution is essential to overcome the complexity of the analytical approach to derive the probability density functions of the output. In this paper, Monte Carlo simulations are performed with the aim of providing an insight into the analytical transformation of the probability density function (PDF) for Y = X2 where X is normally distributed and a detailed comparison of analytical and Monte Carlo approach results are provided. This paper displays how the used approach enables to find PDF of Y = X2 without the use of special functions. In addition, the singularity of the PDF and the nonsymmetric coverage interval are also discussed.

Origins of non-excitonic spectral shifts of chlorophylls that can reach -1,000 cm(-1) in pigment-protein complexes are actively debated in literature. We investigate possible shift mechanisms, basing on absorption and fluorescence measurements in large number of liquids. Transition wavelength in solvent-free state was estimated (±2 nm) for chlorophyll a (Chl a, 647 nm), Chl b (624 nm), bacteriochlorophyll a (BChl a, 752 nm), and pheophytines. The dispersive-repulsive shift is a predominating mechanism. It depends on polarizability difference between the ground and the excited state Δα and the Lorenz-Lorentz function of refractive index of solvent (n). The approximate (± 2Å(3)) increase of polarizability Δα is close to 15Å(3) for S(1) bands of Chl a, BChl a, and BPheo a, slightly larger for Chl b (18Å(3)), and less for Pheo a (11Å(3)). The effect of solvent polarity, expressed in terms of static dielectric permittivity (ε) is relatively minor, but characteristic for different pigments and transitions. Remarkably, maximum influence of ε on S(1) band of BChl a is less (-20 ± 10 cm(-1)) than that for Chl a (-50 ± 10 cm(-1)), and not correlated with dipole moment changes on excitation Δμ (∼2D and 0.1 ± 0.1D, respectively). Hydrogen bonding in protic solvents produces red shifts in Chl a (-60 cm(-1)) and BChl a (-100 cm(-1)), but not in Chl b. Second axial ligand of BChl a has no influence on the S(1) band, whereas the S(2) transition suffers a -400 to -600 cm(-1) down shift. Aromatic character of solvent is responsible for a ∼-100 cm(-1) red shift of both Q transitions in BChl a. The S(1) bands in chlorophylls are relatively insensitive with respect to dielectric properties and specific solvation. Therefore, nontrivial mechanisms, yielding large site-energy shifts are expected in photosynthetic chlorophyll-proteins.

We present, to the best of our knowledge, a new method for differential temperature measurement based on thermal sensitivity of the fluorescence lifetime of thermographic phosphors. Pairs of thermographic phosphors are excited with intensity-modulated light at frequencies ω and ω+Δω. The phase shift Δθ of the summary fluorescence intensity beat signal envelope is measured. A prototype of a fluorometric differential temperature sensor is developed, and feasibility of the method is experimentally demonstrated with a Sm2+:SrFCl crystal and the D15->F70 transition for high thermal sensitivity. The observed linear dependence between envelope phase shift Δθ and temperature difference ΔT agrees with the theoretical prediction. Sensitivity of S=-0.97°/°C was achieved. This method could also be applied to differential measurements of any parameter affecting fluorescence lifetime.

We demonstrate that DiI and Rhodamine B, which are not easily distinguishable to one-photon measurements, can be differentiated and in fact quantified in mixture via tailored two-photon excitation pulses found by a genetic algorithm (GA). A nearly three-fold difference in the ratio of two-photon fluorescence of the two dyes is achieved, without a drop in signal of the favored fluorophore. Implementing an acousto-optic interferometer, we were able to prove that the mechanism of discrimination is second-harmonic tuning by the phase-shaped pulses to the relative maxima and minima of these cross-sections.

We present a comprehensive study of the selective excitation of two-photon fluorescence from various pairs of dyes and dyes in different conjugation states with tailored pulse shapes found with a genetic algorithm (GA). We investigate a number of biologically important dyes, and include dyes conjugated to trastuzumab (Herceptin(R)) and to a poly(amidoamine) dendrimer. We consider in detail the ability of tailored pulse shaping to discriminate dyes with significant spectral overlap. Our procedure for adaptive pulse shaping includes power-law and chirp-scaling checks to prevent trivial convergences. The GA uses a multiplicative fitness parameter in a graded search method that converges on pulse shapes that not only differentiate two-photon processes, but do so in a high signal regime. We consider the results in terms of not only the absolute maximum ratio of discrimination achieved, but also present the evolutionary course of the GA and compare the improvement to a quantitative measure of the noise level. We also implement a time-domain acousto-optic measurement of two-photon excitation cross-section spectra. The results show that the ability to discriminate dyes is determined almost entirely by their differences in two-photon excitation cross section.

We demonstrate optical coherent control of the two-photon fluorescence of the blue fluorescent protein (BFP), which is of interest in investigations of protein-protein interactions. In addition to biological relevance, BFP represents an interesting target for coherent control from a chemical perspective due to its many components of highly nonexponential fluorescence decay and low quantum yield resulting from excited state isomerization. Using a genetic algorithm with a multiplicative (rather than ratiometric) fitness parameter, we are able to control the ratio of BFP fluorescence to second-harmonic generation without a considerable drop in the maximized signal. The importance of linear chirp and power-scaling on the discrimination process is investigated in detail.

The physicochemical characteristics of the green fluorescent protein (GFP), including the thermodynamic properties (entropy, enthalpy, Gibbs' free energy, heat capacity), normal mode vibrations, and atomic fluctuations, were investigated. The Gaussian 03 computational chemistry program was employed for normal mode analysis using the AMBER force field. The thermodynamic parameters and atomic fluctuations were then calculated from the vibrational eigenvalues (frequencies) and eigenvectors. The regions of highest rigidity were shown to be the beta-sheet barrel with the central alpha-helix, which bears the chromophore. The most flexible parts of the GFP molecule were the outlying loops that cover the top and bottom of the beta-barrel. This way, the balance between rigidity and flexibility is maintained, which is the optimal relationship for protein stability in terms of Gibbs' free energy. This dual-schemed structure satisfies the requirements for GFP function. In this sense, the structure of GFP resembles a nanoscale drum: a stiff cylinder with flexible vibrating end(s).

The blue fluorescent protein (BFP) is a mutant of the green fluorescent protein, where the phenolic ring of the chromophore has been replaced by imidazole cycle of histidine residue. The usability of BFP as a fluorescent marker is hampered by its low fluorescence quantum yield at room temperature. The intensity of fluorescence increases by a factor of 4.5 when the temperature is decreased from 320 K down to 225 K. The fluorescence is also enhanced by hydrostatic pressure. Both effects have been explained by shift of the equilibrium between hydrogen nonbonded and hydrogen-bonded chromophores. Our semi-empirical quantum chemical calculations show that the fluorescence quantum yield of the BFP chromophore is low due to isomerization in the electronically excited state -twisting of the bridging bond by 90°. At this twisted geometry the potential energy surfaces of ground and excited states are situated close to each other facilitating efficient nonradiative decay.

The coupling between the heme and the surrounding protein in horseradish peroxidase was studied after substituting the iron protoheme by mesoporphyrin IX to produce a sample measurable by high-resolution fluorescence spectroscopy. The inner ring phototautomerization of mesoporphyrin was used to create a variety of prosthetic group configurations that were shown to be stable at cryogenic temperatures. Due to the properties of the heme crevice, some tautomeric states are characterized by distinct spectral bands. The original band of the tautomeric form stable at room temperature (B1) and two of those produced by photobleaching (B2, B3) could be selectively studied by two techniques, i.e., energy selected fluorescence excitation and vibronic hole burning spectroscopy. The line narrowed spectra were similar in the cases of complexes B2 and B3, while both are different from that of B1. From these spectra, four characteristic vibronic lines were selected and further studied by spectral hole burning experiments....

Fluorescence spectra of green fluorescent protein (GFP) at temperature 5 K under selective laser excitation in the lowest energy absorption band remain unstructured, indicating the absence of the zero phonon lines. This is interpreted as the result of charge transfer character of the electronic transition. The laser excitation through the absorption band at 499 nm creates the photoproduct band at 506 nm. The activation temperature (50 K) of the educt band recovery has been determined by temperature cycling. For the 471 nm (educt) and the 490 nm (photoproduct) band the activation temperature value is 130 K, indicating that the latter species are separated by substantially higher energetic barriers. The mechanism of the phototransformation has been ascribed to proton transfer between the adjacent energy minima along the direction of proton acceptor and donor sites of the chromophore.

Blue fluorescent protein (BFP) is a mutant of green fluorescent protein (GFP), where the chromophore has been modified to shift the emitted fluorescence into the blue spectral region. In this study, MD calculations were performed with the GROMACS simulation package and AMBER force field to investigate the dependence of BFPs physicochemical properties on temperature and applied pressure. The MD approach enabled us to calculate the compressibility of protein itself, separately from the nontrivial contribution of the hydration shell, which is difficult to achieve experimentally. The computed compressibility of BFP (3.94 ×10−5 MPa−1) is in agreement with experimental values of globular proteins. The center-of-mass diffusion coefficient of BFP and its dependence on temperature and pressure, which plays an important role in its application as a probe for intracellular liquid viscosity measurement, was calculated and found to be in good agreement with photobleaching recovery experimental data. We have shown that decreased temperature as well as applied pressure increases the water viscosity, but the concomitant decrease of the BFP diffusion coefficient behaves differently from Stokes-Einstein formula. It is shown that the number of hydrogen bonds around the protein grows with pressure, which explains the aforementioned deviation. Pressure also reduces root mean square (RMS) fluctuations, especially those of the most flexible residues situated in the loops. The analysis of the RMS fluctuations of the backbone Cα atoms also reveals that the most rigid part of BFP is the center of the β-barrel, in accord with temperature B factors obtained from the Protein Data Bank. © 2008 Wiley Periodicals, Inc. Biopolymers 89: 1136–1143, 2008. This article was originally published online as an accepted preprint. The “Published Online” date corresponds to the preprint version. You can request a copy of the preprint by emailing the Biopolymers editorial office at biopolymers@wiley.com

Green fluorescent proteins bearing the Y66H mutation exhibit strongly blue-shifted fluorescence excitation and emission spectra. However, these blue fluorescent proteins (BFPs) have lower quantum yields of fluorescence (Phi(f) approximately 0.20), which is believed to stem from the increased conformational freedom of the smaller chromophore. We demonstrate that suppression of chromophore mobility by increasing hydrostatic pressure or by decreasing temperature can enhance the fluorescence quantum yield of these proteins without significantly affecting their absorption properties or the shape of the fluorescence spectra. Analysis of the fluorescence lifetimes in the picosecond and nanosecond regimes reveals that the enhancement of the fluorescence quantum yield is due to the inhibition of fast quenching processes. Temperature-dependent fluorescence measurements reveal two barriers ( approximately 19 and 3 kJ/mol, respectively) for the transition into nonfluorescing states. These steps are probably linked with dissociation of the hydrogen bond between the chromophore and His148 or an intervening water molecule and to the barrier for chromophore twisting in the excited state, respectively. The chromophore's hydrogen-bond equilibrium at room temperature is dominated by entropic effects, while below approximately 200 K the balance is enthalpy-driven.

The Qx-Qy splitting observed in the fluorescence excitation spectra of Mg-mesoporphyrin-IX substituted horseradish peroxidase (MgMP-HRP) and of its complex with naphthohydroxamic acid (NHA) was studied by spectral hole burning techniques. The width of a hole directly burnt in the Qy band and that of a satellite hole indirectly produced in Qy as a result of hole burning in Qx was compared. We also studied the dependence of the satellite hole in the Qy band on the burning frequency used in the Qx band. Both the directly and indirectly burnt holes were very broad in the (higher energy) Qy band. The width of the satellite hole in the Qy band was equal to the entire width of the inhomogeneously broadened band, independently from the position of hole burning in Qx. This is indicative of a clear lack of correlation between the electronic transition energies of the Qx and Qy bands. A photoproduct was produced by laser irradiation of the MgMP-HRP/NHA complex and was identified as a species with lowered Q-splitting. Conversion of the photoproduct could be achieved by thermal activation measured in temperature-cycling experiments, with a characteristic temperature of 25K. We attribute the phototransformation to a conformational change of MgMP.

A ray-tracing approach is used to demonstrate efficient application of the vectorial laws of reflection and refraction to computational optics problems. Both the full width at half-maximum (fwhm) and offset of Gaussian beams resulting from off-center reflection and refraction are calculated for spherical and paraboloidal surfaces of revolution. It is found that the magnification and displacement depend nonlinearly on the miscentering. For these geometries, the limits of accuracy of the lens approximation are examined quantitatively. In contrast to the ray-tracing solution, this paraxial approximation would predict a magnification of a beam's fwhm that is independent of miscentering, and an offset linearly proportional to the miscentering. The focusing property of paraboloidal surfaces of revolution is also derived in setting up the calculation.

We present a portable optical cataract assessment technology which measures with a circular photodetector the fraction of light scattered backwards by the human eye lens when illuminated by a laser diode. As our signal arises directly from the fundamental pathology-increased scattering in the lens-it directly assesses cataract extent and progression. Initial clinical results in undilated human eyes show device reading correlations in agreement with clinical examination and Scheimpflug photography.