Neural networks can be used generate potential energy hypersurfaces by fitting to a data set of energies at discrete geometries, as might be obtained from ab initio calculations. Prior work has shown that this method can generate accurate fits in complex systems of several dimensions. The present paper explores fundamental properties of neural network potential representations in some simple prototypes of one, two, and three dimensions. Optimal fits to the data are achieved by adjusting the network parameters using an extended Kalman filtering algorithm, which is described in detail. The examples provide insight into the relationships between the form of the function being fit, the amount of data needed for an adequate fit, and the optimal network configuration and number of neurons needed. The quality of the network interpolation is substantially improved if gradients as well as the energy are available for fitting. The fitting algorithm is effective in providing an accurate interpolation of the underlying potential function even when random noise is added to the data used in the fit.

Single molecule experiments reveal intriguing phenomenon in chemical and biological systems. Several indicators of complex dynamics, including "intensity" correlations, "event" correlations, and characteristic functions have been proposed, but extraction of information from these indicators can be difficult since these indicators only observe certain characteristics of the system. Generally, for systems that follow Poisson kinetics, all of these indicators contain similar information about the relaxation times of the system and the connections between different relaxation times, but the information is convoluted in different ways so the strength of various indicators is system specific. The paper discusses the theoretical implications and information content of various data analysis methods for single molecule experiments and demonstrates the relationships between indicators. Under certain conditions, common indicators contain all available information about systems with Poisson kinetics between degenerate states, but extraction of this information is generally not numerically feasible. The paper also discusses practical issues associated with these analyses, which motivates a numerical study based on Bayes' formula in the companion paper [J. Witkoskie and J. S. Cao, J. Chem. Phys. 121, 6373 (2004), following paper].

Theory of linear viscoelasticity of semiflexible rods in dilute solutionA semiflexible Gaussian chain model is used to determine the statistics and correlations of single-molecule fluorescence resonant energy transfer ͑FRET͒ experiments on biological polymers. The model incorporates a persistence length in a Rouse chain and describes single-chain dynamics with normal modes. The hydrodynamic interaction is included in the dynamics of the semiflexible Gaussian chain on the preaveraging level. The distribution functions of the fluorescence lifetime and the FRET efficiency provide direct measures of the chain stiffness, and their correlation functions probe the intrachain dynamics at the single-molecule level. When measured with finite time resolution, the instantaneous diffusion coefficient for FRET is much smaller in the collapsed structure than in the coiled structure, and the variation has a quadratic dependence on the donoracceptor distance. In the fast reaction limit, single-molecule FRET lifetime measurements can be used to map out the equilibrium distribution function of interfluorophore distance. As an example of microrheology, the intrinsic viscoelasticity can be extracted from single-molecule tracking of the Brownian dynamics of polymers in solution.

As discussed in the companion paper [J. B. Witkoskie and J. S. Cao, J. Chem. Phys. 121, 6361 (2004), preceding paper], quantitative extraction of information from single molecule experiments by several proposed indicators is difficult since the experiments only observe certain characteristics of the system, even though the indicators can contain all available information. This paper shows how one can circumvent the shortcomings of these indicators by combining information extracted from indicators with a numerical Bayesian statistical approach. The Bayesian approach determines the relative probability of various models reproducing the entire sequence of the single molecules trajectory, instead of binning and averaging over the data, which removes much of this information.

This paper analyzes the observed phenomenology of the fluorescence time trace of collections of quantum dots (QDs) in terms of the model parameters that characterize the fluorescence blinking statistics of single QDs. We demonstrate that the non-universal dynamics that appear in fluorescence time traces of collections of QDs at short time scales are related to the universal dynamics that appear at longer time scales. We explore how the extent of time separation between the short and long dynamics affects the transition region and the dynamics at longer time scales. We suggest a methodology to extract single QD statistical model parameters from experimental fluorescence time traces of collections of QDs. We explore theoretical time traces and their experimental analogs for three different cases that span the diverse nonuniversal dynamics that appear at short time scales.