George H. Weiss

Theory of single-molecule fluorescence spectroscopy of two-state systems

We derive an explicit expression for the probability density of the amplitudes of the biexponential fluorescence decay of a two-state system observed in single-molecule experiments of finite duration T. The shape of the probability density of the amplitudes depends on (k1+k2)T, where the ki are the interconversion rate constants. It approaches a delta function in the limit T→∞. When the interconversion rates in the ground and excited states differ, the single-molecule and bulk experiments will give different results even in the limit of infinite duration of the experiment.

Effects of anisotropic optical properties on photon migration in structured tissues

It is often adequate to model photon migration in human tissue in terms of isotropic diffusion or random walk models. A nearly universal assumption in earlier analyses is that anisotropic tissue optical properties are satisfactorily modelled by using a transport-corrected scattering coefficient which then allows one to use isotropic diffusion-like models. In the present paper we introduce a formalism, based on the continuous-time random walk, which explicitly allows the diffusion coefficients to differ along the three axes. The corrections necessitated by this form of anisotropy are analysed in the case of continuous-wave and time-resolved measurements and for both reflectance and transmission modes. An alternate model can be developed in terms of a continuous-time random walk in which the times between successive jumps differ along the three axes, but is not included here.

Diffusion in the presence of periodically spaced permeable membranes

The diffusion of molecules in biological tissues and some other microheterogeneous systems is affected by the presence of permeable barriers. This leads to the slowdown of diffusion at long times as compared to barrier-free diffusion. At short times the effect of barriers is weak. In consequence, the diffusion coefficient D(t) decreases as a function of time. We derive an exact solution for the Laplace transform of D(t) for diffusion in a space separated into layers by equally spaced, parallel identical planes of arbitrary permeability. Additionally, we give an approximation to D(t) which is reasonably accurate over the whole range of the partition permeability from zero (the case of isolated layers) to infinity (the case of no barriers).

Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface

The theory of ligand binding to receptors on a cell surface suggested by Berg and Purcell and generalized by Zwanzig and Szabo uses the assumption that receptors are circular absorbing disks on an otherwise reflecting sphere. One of the key ingredients of this theory is a solution for the rate constant for ligand binding to a single circular receptor on a reflecting plane. We give an exact solution for the rate constant for binding to a single elliptic receptor and an approximate solution for binding to a single receptor of more general shape. The latter was tested by Brownian dynamics simulations. We found that the approximate formula predicted the rate constant with better than 10% accuracy for all studied receptor shapes. Using our solutions one can find the rate constant for ligand binding to a cell covered by N noncircular receptors by means of the Zwanzig-Szabo formula.

Reaction dynamics on a thermally fluctuating potential

This paper analyzes the kinetics of escape of a particle over a barrier fluctuating between two states, the fluctuations being produced by thermal noise. By this we mean that the jump rates for transitions between the two states are position-dependent, satisfying detailed balance at any point along the reaction coordinate. The fast-fluctuation limit can be analyzed in terms of the potential of mean force, and for high barriers the survival probability is found to be a single exponential. In the slow-fluctuation regime the survival probability is a linear combination of two exponentials. In the case of a linear potential the slow-fluctuation solution describes the kinetics, as obtained from simulations, quite well over the entire range of the jump rates between the two states. Our analysis suggests that this is true for more general forms of the potential. Further, for a thermally fluctuating potential the mean lifetime is shown to decrease monotonically as the jump rate increases. This is in contrast to t...

Photon migration in turbid media with anisotropic optical properties

We analyse properties of photon migration in reflectance measurements made on a semi-infinite medium bounded by a plane, in which optical parameters may vary in directions neither parallel to, nor perpendicular to the bounding plane. Our aim in doing this is to develop the formulae necessary to deduce parameters of directionality from both time-gated and continuous wave measurements. The mathematical development is based on a diffusion picture, in which the bounding plane is regarded as being totally absorbing so that all photons reaching the surface contribute to the reflectance.

Equilibration in two chambers connected by a capillary

A problem common to biophysics, chemical engineering, physical chemistry, and physiology relates to a description of the kinetics of particle transport between two or more chambers. In this paper we analyze the case of two chambers connected by a cylindrical capillary. We derive general solutions for the Laplace transforms of the relaxation functions describing the equilibration of particles between the two chambers and the capillary. These solutions show how the relaxation functions depend on geometric parameters (volumes of the two chambers, the length and radius of the capillary) as well as diffusion coefficients in the three compartments. The general solutions are used to analyze the reduction to single-exponential kinetics which describes equilibration of the particles when the capillary is not too long. When all of the diffusion constants are equal we derive simple expressions for the average relaxation times. Brownian dynamics simulations were run to check the accuracy of approximations used to der...

Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries.

There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRWs in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.

A measure of photon penetration into tissue in diffusion models

In application of laser-based techniques to estimate optical parameters of tissue it is necessary to know the parts of the tissue that have been sampled by the photons. Towards this end it is desirable to define an average depth of a photon trajectory. This has been done earlier for lattice random walk models of photon trajectories where one simply counts the number of visits to a given set of sites. Here we calculate the equivalent quantity in the diffusion picture by equating the average depth of penetration to the local time of a diffusion process. We show that for a semi-infinite medium bounded by a plane the average depth probed by a photon trajectory in a continuous-wave (cw) experiment has the same functional form as that calculated from a random walk model. The local time, or average depth is shown to take a very simple form in the case of a time-gated experiment. We extend our results to the cw transillumination experiment.

Asymmetry of protein peaks in capillary zone electrophoresis: effect of starting zone length and presence of polymer.

The asymmetry of R-phycoerythrin (M(r) = 240,000) peaks in capillary zone electrophoresis measured as In[(tm-t1)/(t2-tm)], where tm, t1 and t2 are migration times of the peak mode and at the intersection of the peak width at half-height with the ascending and descending limbs, respectively, was found to undergo a transition from negative to positive values with increasing starting zone length. The transition is compatible with a mathematical model of peak dispersion which assumes that an interaction of protein with the capillary walls governs the evolution of the peak during capillary zone electrophoresis. Models assuming a final peak shape defined solely by longitudinal diffusion, or by a heterogeneity with regard to mobility or by a conductivity difference between analyte zone and background electrolyte, have failed to give rise to a change in the sign of peak asymmetry when the starting zone length is varied. The presence of polyethylene glycol in the buffer within a concentration range up to 4% does not appreciably affect the peak asymmetry regardless of whether the concentration regime is dilute or semi-dilute. Above 4% of polyethylene glycol, the asymmetry becomes nearly independent of starting zone length, and progressively negative with increasing polymer concentration. The concentration range at which the transition from negative to positive asymmetry disappears coincides with that at which the average mesh size of the polymer network falls below the size of the protein.