The modeling of excited-state processes in photophysics can conveniently be done within the framework of compartmental analysis. In compartmental analysis substantial attention has been devoted to the study of deterministic identifiability, which verifies whether it is possible to determine the parameters of the compartmental model from error-free data. In this paper the similarity transformation approach is applied to the identifiability problem of the photophysical model for reversible intermolecular two-state excited-state processes. This method provides straightforward relations between the true and alternative sets of the system parameters. This allows one to explore directly the parameter space for identifiability. Since absolute values for the spectral parameters associated with excitation and emission are not available from time-resolved fluorescence experiments, the original similarity transformation approach to the identifiability problem was reformulated in terms of normalized spectral parameters, which are experimentally accessible. It is shown that six decay traces—measured at two coreactant concentrations and three emission wavelengths—are required for the model to be locally identifiable. Two sets of rate constants and associated spectral parameters may be found under these conditions. Enclosure in the analysis of the monoexponential decay at very low coreactant concentration results in global identifiability. The non-negativity requirement of the spectral parameters also can lead to the unique solution. If the fluorescence decays are independent of the emission wavelength, additional information about the photophysical system is necessary for identifiability.

A deterministic identifiability analysis of the kinetic model for a reversible intermolecular two-state excited-state process with species-dependent rotational diffusion described by Brownian reorientation is presented. The cases of both spherically and cylindrically symmetric rotors, with no change in the principal axes of rotation on interconversion in the latter case, are specifically considered. The identifiability analysis is carried out in terms of compartmental modeling based on the S(t) identical with I( parallel)(t)+2I( perpendicular)(t) and D(t) identical with I( parallel)(t)-I( perpendicular)(t) functions, where I( parallel)(t) and I( perpendicular)(t) are the delta-response functions for fluorescence, polarized, respectively, parallel and perpendicular to the electric vector of linearly polarized excitation. It is shown that, from polarized time-resolved fluorescence data collected at two concentrations of coreactant and three appropriately chosen emission wavelengths, (a) a unique set of rate constants for the overall excited-state process is always obtained by making use of polarized measurements and (b) the rotational diffusion constants and geometrical factors associated with the different anisotropy decay components can be uniquely determined and assigned to each species. The geometrical factors are determined by the absorption and emission transitions in the two rotating species. For spherical rotors, these factors depend directly on the relative orientations of the transition moments, while for cylindrically symmetric rotors they depend on the orientations with respect to each other and to the symmetry axis.

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The maximum-likelihood estimation method is considered. It is based on three-level Oehmelt scheme of isolated quantum system and it makes use of the theory of Markov-modulated Poison process (MMPP). This method is applied to the analysis of the photon-counting distribution resulting &om a single molecule fluorescence experiment. It is assumed that the data available for the estimation contain only the number of collected photons per time interval.Keywords: Markov-modulated Poisson process, Oehmeltl suggested a scheme of experiment to detect the quantum jumps in a single double-resonance atom involving weak and strong transition. According to the scheme, quantum jumps at the weak transition cause the fluorescence of the sb'ong transition to turn on and off abruptly. The fluorescence is off when the weak transition is excited and on when it is not Consequently, the atomic fluorescence should have the fonn of a random telegrnph signal, with alternating periods of zero ("off' periods) and non-zero ("on" periods) intensity. Thus, the quantum jumps could be directly monitored by observation of dte random telegrnph signal radiated by the strong transition.. Cook and Kimble2 followed Dehmelt's idea and developed a mathematical description of his two parameter scheme. Their main conclusion was that the durations of "off' and "on" intervals should follow exponential distributions, with different rates. Quantum jumps between excited electronic levels have been also observed as a fluorescence "blinking" in experiments involving a single molecule of different organic dyes. These molecules exhibit intersystem crossing to the long-lived triplet state Tl. The single molecule fluorescence experiments may be divided into two types. In a common single molecule experiment arriving photons of the fluorescence are collected consecutively during the defined intervals (dwell periods). This technique produces the fluorescence trace of the form of the telegraph noise3. In such a type of experiment arrival time of fluorescence photon is lost. The other type of single molecule fluorescence experiments that provide more information on the photophysical processes has been developed in recent years. In this kind of experiments the fluorescence photon arriving time is also registered4. In this paper we restrict our consideration to the flfSt type of single molecule experiments.The parameters of the stochastic process commonly estimated on the basis of experimental data3 are "off' and "on" -time duration and the number of "on"-state photon counts. In the approaches proposed by Ha et al: and Yip et al.5, each dwell period of the trace is classified as belonging to "on" or "off' period, depending on the number of emitted photons collected in it. The classification rule can use, e.g., a threshold derived from the distribution of observed counts5. Subsequently, the histograms of the durations of"on" and "off' are analyzed using separate exponential distributions.