Time-dependent Smoluchowski equation

Noolandi and Hong (1978), Hong and Noolandi (1978a, 1978b). and Noolandi (1982) described solutions to the time-dependent Smoluchowski equation. In the long-time limit, Hong and Noolandi s results predict the recombination rate approaches a law as... 

Y. Cheng, J. K. Suen, D. Zhang, S. D. Bond, Y. Zhang, Y. Song, N. A. Baker, C. L. Baja], M. J. Holst, and J. A. McCammon. Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations. Biophys. J., 92(10) 3397-406, 2007. 

The traditional analysis of proton transfer reactions [3, 5-7] is by a kinetic scheme e.g., eq 1, and the ensuing chemical rate equations describing the time dependence of bulk concentrations. This is insufficient for explaining the long time tail. The minimal level of complexity has to involve the spatial diffusion of the proton in the electrostatic field of the anion, as depicted by the time-dependent Smoluchowski equation [15], but with a boundary condition (the back-reaction boundary condition) which describes reversible reactions [10]. 

Figure 7. Effect of various parameters on the solution of the spherically symmetric (3d), time-dependent Smoluchowski equation with back reaction [10a]. In each panel one parameter is varied while the others are held constant. In all panels, the third curve from the top is identical and corresponds to the observed HPTS signal, multiplied by exp(

When considering the features of excimer formation in somewhat greater detail, some comment on this study seems appropriate. The key issue that one should keep in mind when considering these types of mimic studies, is how the lifetime of a pyrene moiety compares to the lifetime of a free radical. In the case of polystyrene, the polymer that the group of Winnik studied, this comparison is not very favorable. One propagation step of this monomer at a moderate temperature of 30 °C (average temperature used in excimer study) takes about 0.2 to 0.3 milliseconds. Not only can one conclude that a time-dependent Smoluchowski equation... 

Bajaj, M. J. Holst, and J. A. McCammon, Biophys. J., 92,3397-3406 (2007). Finite Element Analysis of the Time-Dependent Smoluchowski Equation for Acetylcholinesterase Reaction Rate Calculations. 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

Rice [147] has noted the similarity of form between the time-dependent rate coefficients from the Smoluchowski, eqn. (19), Debye— Smoluchowski, eqn. (53), diffusion and Forster transfer equations,... 

A complete solution of eqn. (151) for the time-dependent density distribution does not appear feasible, but Hong and Noolandi [323—325] have found the long-time behaviour, as well as the steady-state solution. The mathematics are very complex since the complications encountered in the analysis of the Debye—Smoluchowski equation (Appendix A) are compounded by the applied electric field. For small electric fields and long times, the survival probability is approximately... 

The eigenmode expansion was also used to determine the time-dependent solution of the Smoluchowski equations for diverse bistable potentials.185... 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

It has further been shown (Section 4.2.6) that in the case of a one-dimensional random walk, depends on time according to the Einstein-Smoluchowski equation... 

As ahead stated, homogeneous, amorphous systems are assumed so that the stationary dishibution function is translationally invariant but anisotropic. The formal solution of the Smoluchowski equation for the time-dependent dishibution function... 

To explain the above behaviors of the fluorescence decay function and time-resolved fluorescence anisotropy of single tryptophan in proteins, a model was proposed (33), in which an angular-dependent quenching constant was introduced into a rotational analogue of Smoluchowski equation given by Favro (34) for the internal rotation of tryptophan, as expressed by [1]. Rotational motion of tryptophan in a spherical macromolecule is illustrated in Fig.9. 

Proton transfer dynamics of photoacids to the solvent have thus, being reversible in nature, been modelled using the Debye-von Smoluchowski equation for diffusion-assisted reaction dynamics in a large body of experimental work on HPTS [84—87] and naphthols [88-92], with additional studies on the temperature dependence [93-98], and the pressure dependence [99-101], as well as the effects of special media such as reverse micelles [102] or chiral environments [103]. Moreover, results modelled with the Debye-von Smoluchowski approach have also been reported for proton acceptors triggered by optical excitation (photobases) [104, 105], and for molecular compounds with both photoacid and photobase functionalities, such as lO-hydroxycamptothecin [106] and coumarin 4 [107]. It can be expected that proton diffusion also plays a role in hydroxyquinoline compounds [108-112]. Finally, proton diffusion has been suggested in the long time dynamics of green fluorescent protein [113], where the chromophore functions as a photoacid [23,114], with an initial proton release on a 3-20 ps time scale [115,116]. 

We start with the assumption that the random process is Markoflian, that is, that the probability for p to take on a certain value at a time t + St depends only on its value at t and not on the past history of the system. The Markoflian assumption is expressed by the Smoluchowski equation for the probability function tp(p(t), p(0), t)... 

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