Reversible Geminate Recombination Research Articles

An exact a posteriori correction for hydrogen bond population correlation functions and other reversible geminate recombinations obtained from simulations with periodic boundary conditions. Liquid water as a test case.

The kinetics of breaking and re-formation of hydrogen bonds (HBs) in liquid water is a prototype of reversible geminate recombination. HB population correlation functions (HBPCFs) are a means to study the HB kinetics. The long-time limiting behavior of HBPCFs is controlled by translatoric diffusion and shows a t-3/2 time-dependence, which can be described by analytical expressions based on the HB acceptor density and the donor-acceptor inter-diffusion coefficient. If the trajectories are not properly "unwrapped," the presence of periodic boundary conditions (PBCs) can perturb this long-time limiting behavior. Keeping the trajectories "wrapped," however, allows for a more efficient calculation of HBPCFs. We discuss the consequences of PBCs in combination with "wrapped" trajectories following from the approximations according to Luzar-Chandler and according to Starr, each deviating in a different fashion from the true long-time limiting behavior, but enveloping the unperturbed function. A simple expression is given for estimating the maximum time up to which the computed HBPCFs reliably describe the long-time limiting behavior. In addition, an exact a posteriori correction for systems with PBCs for "wrapped" trajectories is derived, which can be easily computed and which is able to fully recover the true t-3/2 long-time behavior. For comparison, HBPCFs are computed from MD simulations of TIP4P/2005 model water for varying system sizes and temperatures of 273 and 298K using this newly introduced correction. Implications for the computations of HB lifetimes and the effect of the system-size are discussed.

Structure and Dynamics of Proton Transfer in Liquid Imidazole. A Molecular Dynamics Simulation

Proton transfer (PT) via the Grotthuss mechanism in liquid imidazole (im) at 393 K is studied with molecular dynamics simulation using a reactive multistate empirical valence bond (MS-EVB) model. It is found that the proton is tightly binded to an imidazole to form an imidazolium (imH(+)), which is solvated in a distorted Eigen-like complex (im-imH(+)-im), whereas the Zundel-like complex (im-H(+)-im) is rare. PT occurs via an Eigen-Zundel-Eigen scenario for switching the identity of imH(+) from an Eigen-like complex to another, intermediated by a Zundel-like complex. Structural and dynamical analyses demonstrate that PT in imidazole can be considered as a local event with very short spatial/temporal correlation, characterized by a few "rattling" or recurrent PT events. At long time scale, the trend of the PT correlation function may be recast with the diffusion model of reversible geminate recombination toward the power-law decay. The formation of the hydrogen bonds (HBs) for the imidazole molecules between the first and second solvation shell of imH(+) is crucial to pave the PT pathway. The above features may be understood by the flexibility of the HBs in liquid imidazole, as a stable HB network is essential for the Grotthuss mechanism.

Comparison of Alternate Approaches for Reversible Geminate Recombination

This work compares various models for geminate reversible diffusion influenced reactions. The commonly utilized contact reactivity model (an extension of the Collins-Kimball radiation boundary condition) is augmented here by a volume reactivity model, which extends the celebrated Feynman-Kac equation for irreversible depletion within a reaction sphere. We obtain the exact analytic solution in Laplace space for an initially bound pair, which can dissociate, diffuse or undergo “sticky” recombination. We show that the same expression for the binding probability holds also for “mixed” reaction products. Two different derivations are pursued, yielding seemingly different expressions, which nevertheless coincide numerically. These binding probabilities and their Laplace transforms are compared graphically with those from the contact reactivity model and a previously suggested coarse grained approximation. Mathematically, all these Laplace transforms conform to a single generic equation, in which different reactionless Green's functions, g(s), are incorporated. In most of parameter space the sensitivity to g(s) is not large, so that the binding probabilities for the volume and contact reactivity models are rather similar.

Liquid Water: From Symmetry Distortions to Diffusive Motion

Water deviates from tetrahedral symmetry on different scales, creating "defects" that are important for its dynamics. In this Account, I trace the manifestations of these distortions from the isolated molecule through gas-phase clusters to the liquid phase. Unlike the common depiction, an isolated water molecule has a nonsymmetric charge distribution: although its positive charge is localized at the hydrogens, the negative charge is smeared between the lone-pair sites. This creates a "negativity track" along which a positive charge may slide. Consequently, the most facile motion within the water dimer is a reorientation of the hydrogen-bond (HB) accepting molecule (known as an "acceptor switch"), such that the donor hydrogen switches from one lone pair to the other. Liquid water exhibits asymmetry between donor and acceptor HBs. Molecular dynamics simulations show that the water oxygens accepting HBs from the central molecule are spatially localized, whereas water hydrogens donating HBs to it are distributed along the negativity track. This asymmetry is manifested in a wider acceptor- versus donor-HB distribution. There is a higher probability for a water molecule to accept one (trigonal symmetry) or three HBs than to donate one or three HBs. A simple model can explain semiquantitatively how these distributions evolve by distorting perfectly tetrahedral water. Just two reactions are required: the dissociation of a HB between a double-donor donating to a double-acceptor, D(2)···A(2), followed by a switching reaction in which a HB donor rotates its hydrogen between two double-acceptor molecules. The preponderance of D(2)···A(2) dissociation events is in line with HB "anticooperativity", whereas positive cooperativity is exhibited by conditional HB distributions: a molecule with more acceptor bonds tends to have more donor bonds and vice versa. Quantum mechanically, such an effect arises from intermolecular charge transfer, but it is observed even for fixed-charge water models. Possibly, in the liquid state this is partly a collective effect, for example, a more ordered hydration shell that enhances the probability for both acceptor and donor HBs. The activation energy for liquid water self-diffusion is considerably larger than its HB strength, pointing to the involvement of collective dynamics. The remarkable agreement between the temperature dependence of the water self-diffusion coefficient and its Debye relaxation time suggests that both share the same mechanism, likely consisting of coupled rotation and translation with collective rearrangement of the environment. The auto-correlation function of a hydrogen-bonded water molecule pair is depicted quantitatively by the solution of the diffusion equation for reversible geminate recombination, up to long times where the ubiquitous t(-3/2) power law prevails. From the model, one obtains the HB dissociation and formation rate coefficients and their temperature dependence. Both have a similar activation enthalpy, suggesting rapid formation of HBs with alternate partners, perhaps by the HB switching reaction involving the trigonal site. A detailed picture of how small fluctuations evolve into large-scale molecular motions in water remains elusive. Nonetheless, our results demonstrate how the plasticity of water can be traced to its asymmetric charge distribution, with duality between tetrahedral and trigonal ligation states.

The residence probability: single molecule fluorescence correlation spectroscopy and reversible geminate recombination

The residence probability of a freely diffusing particle within an open d-dimensional ball is calculated as a function of time, for an initial distribution that is either a spherical delta function or uniform within the sphere. The latter is equivalent to the autocorrelation function (ACF) of fluorescence correlation spectroscopy (FCS) when utilizing near-field scanning optical microscopy (NSOM) probes. Starting from the general equation for the Laplace transform of the residence probability, we solve it in Laplace space for any dimensionality, inverting it into the time domain in one- and three-dimensions. The short- and long-time asymptotic behaviors of the residence probability are derived and compared with the exact results. Approximations for the two-dimensional ACF are discussed, and a new approximation is derived for the NSOM-FCS ACF. Also of interest is an analytic expression for a three-dimensional ACF, which could be useful for two-photon FCS. Analogy with the binding probability for reversible geminate recombination suggests that more information could be extracted from the long-time tails in FCS experiments.

Kinetics of Proton Migration in Liquid Water

We have utilized multistate empirical valence bond (MS-EVB3) simulations of protonated liquid water to calculate the relative mean-square displacement (MSD) and the history-independent time correlation function, c(t), of the hydrated proton center of excess charge (CEC) with respect to the water molecule on which it has initially resided. The MSD is nonlinear for the first 15 ps, suggesting that the relative diffusion coefficient increases from a small value, D(0), at short separations to its larger bulk value, D(infinity), at large separations. With the ensuing distance-dependent diffusion coefficient, D(r), the time dependence of both the MSD and c(t) agrees quantitatively with the solution of a diffusion equation for reversible geminate recombination. This suggests that the relative motion of the CEC is not independent from the nearby water molecules, in agreement with theoretical and experimental observations that large water clusters participate in the mechanism of proton mobility.

Probing the Inner Cavities of Hydrogels by Proton Diffusion

The reversible proton dissociation and geminate recombination of a photoacid was studied in peptide-based hydrogels over a large concentration range of the self-assembled peptide nanotube (PNT) network. 8-Hydroxypyrene-1,3,6-trisulfonate trisodium salt (HPTS) was used as the photoacid. We found that the proton transfer to the solvent rate is not affected by the presence of the fibrous structures. However, the proton geminate recombination process is affected by the PNT walls. We used the photoprotolytic reversible geminate recombination model to analyze the time-resolved emission data in the hydrogel samples. Our assumptions lead to the suggestion that the hydrogels form water nanopools of the order of 100 A.

Excited-state reversible geminate recombination in two dimensions

Excited-state reversible geminate recombination with two different lifetimes and quenching is investigated in two dimensions. From the exact Green function in the Laplace domain, analytic expressions of two-dimensional survival and binding probabilities are obtained at short and long times. We find that a new pattern of kinetic transition occurs in two dimensions. The long-time effective survival probabilities show a pattern of (ln t)(-1)-->constant-->e(t) depending on the rate constants while the effective binding probabilities show t(-1)(ln t)(-2)-->t(-1)-->e(t).

Reversible geminate recombination of hydrogen-bonded water molecule pair

The (history independent) autocorrelation function for a hydrogen-bonded water molecule pair, calculated from classical molecular dynamics trajectories of liquid water, exhibits a t(-3/2) asymptotic tail. Its whole time dependence agrees quantitatively with the solution for reversible diffusion-influenced geminate recombination derived by Agmon and Weiss [J. Chem. Phys. 91, 6937 (1989)]. Agreement with diffusion theory is independent of the precise definition of the bound state. Given the water self-diffusion constant, this theory enables us to determine the dissociation and bimolecular recombination rate parameters for a water dimer. (The theory is indispensable for obtaining the bimolecular rate coefficient.) Interestingly, the activation energies obtained from the temperature dependence of these rate coefficients are similar, rather than differing by the hydrogen-bond (HB) strength. This suggests that recombination requires displacing another water molecule, which meanwhile occupied the binding site. Because these activation energies are about twice the HB strength, cleavage of two HBs may be required to allow pair separation. The autocorrelation function without the HB angular restriction yields a recombination rate coefficient that is larger than that for rebinding to all four tetrahedral water sites (with angular restrictions), suggesting the additional participation of interstitial sites. Following dissociation, the probability of the pair to be unbound but within the reaction sphere rises more slowly than expected, possibly because binding to the interstitial sites delays pair separation. An extended diffusion model, which includes an additional binding site, can account for this behavior.

Theory and Simulation of Diffusion-Controlled Michaelis−Menten Kinetics for a Static Enzyme in Solution

We develop a uniform theory for the many-particle diffusion-control effects on the Michaelis-Menten scheme in solution, based on the Gopich-Szabo relaxation-time approximation (Gopich, I. V.; Szabo, A. J. Chem. Phys. 2002, 117, 507). We extend the many-particle simulation algorithm to the Michaelis-Menten case by utilizing the Green function previously derived for excited-state reversible geminate recombination with different lifetimes (Gopich, I. V.; Agmon, N. J. Chem. Phys. 2000, 110, 10433). Running the simulation for representative parameter sets in the time domain and under steady-state conditions, we find poor agreement with classical kinetics but excellent agreement with some of the modern theories for bimolecular diffusion-influenced reactions. Our simulation algorithm can be readily extended to the biologically interesting case of dense patches of membrane-bound enzymes.

Proton Antenna Effect of the γ-Cyclodextrin Outer Surface, Measured by Excited State Proton Transfer

The reversible proton dissociation and geminate recombination of the common photoacid, 8-hydroxypyrene-1,3,6-trisulfonate (pyranine), either in dilute aqueous solution or when forming a complex with gamma-cyclodextrin (gamma-CD), has been studied by time-resolved fluorescence spectroscopy and supplemented by molecular modeling and dynamics simulations. We find that the dissociation rate of the proton from the excited molecule was decreased to about approximately 50% of its value in water, while the rate of recombination was doubled. These observations were evaluated by molecular modeling of the reactants at atomic resolution. The combination of the two methodologies indicates that the pyranine in the complex can assume more than one level of interaction with the solvent. The polysugar torus surrounding the pyranine perturbs the hydrogen bond in the dye's immediate vicinity and deforms the electrostatic potential inside the Coulomb cage, causing major deviations from a simple spheric symmetry. These observations can account for the special kinetic features measured for the complex. We suggest that this system can be used as a basic model for evaluating the mechanism of proton transfer in non-homogeneous systems, such as the surface of proteins or biomembranes.

Temperature Dependence of Excited-State Proton Transfer in Water Electrolyte Solutions and Water−Methanol Solutions

The reversible proton dissociation and geminate recombination of a photoacid is studied as a function of temperature in water electrolyte solutions and binary water-methanol mixtures, containing 0.1 and 0.2 mole fractions of methanol. 8-Hydroxypyrene-1,3,6-trisulfonate trisodium salt (HPTS) is used as the photoacid. The experimental data are analyzed by the reversible geminate recombination model. We found that the slope of the logarithm of the proton-transfer rate constant as a function of the inverse of temperature (Arrhenius plot) in the liquid phase of these samples are temperature-dependent, while in the solid phase, the slope is nearly constant. The slope of the Arrhenius plot in frozen electrolyte solution is larger than that of the water-methanol mixtures, which is about the same as in pure water. Careful examination of the time-resolved emission in ice samples shows that the fit quality using the geminate recombination model is rather poor at relatively short times. We were able to get a better fit using an inhomogeneous kinetics model assuming the proton-transfer rate consists of a distribution of rates. The model is consistent with an inhomogeneous frozen water distribution next to the photoacid.

Transition in the Temperature-Dependence of GFP Fluorescence: From Proton Wires to Proton Exit

In green fluorescent protein, photo-excitation leads to excited-state proton transfer from its chromophore, leaving behind a strongly fluorescing anion, while the proton is commonly thought to migrate internally to Glu-222. X-ray data show that the protein contains more extended hydrogen-bonded networks that can support proton migration (i.e., proton wires). Here we study the temperature-dependence of the transient fluorescence from both the acid and anionic forms up to 15ns. At low temperatures, we find that the (lifetime-corrected) fluorescence of the acidic form decays asymptotically as t−1/2, following quantitatively the solution of a one-dimensional diffusion equation for reversible geminate recombination with quenching. This indicates proton migration along the internal proton wires. A small degree of geminate proton quenching is attributed to the formation of the zwitterion by proton migration on a side-branch of the proton wire. Above 230K, the fluorescence kinetics undergo a transition, exhibiting an asymptotic t−3/2 decay, and the quenching effect disappears. We interpret these findings as evidence for a conformational change enabling the rotation of Thr-203, which eventually allows the proton to escape to the exterior solution.

Reduced parameter set determinable from geminate kinetics

By fitting time-resolved measurements to models of diffusion influenced reactions it is possible to determine only parameters with no spatial dependence. The original set of distance-dependent parameters is reduced to a set of distance-independent parameters, and only these can be determined by fitting the model to the data. The present work demonstrates this point for reversible geminate recombination in one and three dimensions, showing how the reduced set of parameters is found by utilizing dimensional analysis or known analytic solutions. It is applied to the kinetic detection of proton-wires in the green fluorescent protein.

Diffusion-influenced reversible geminate recombination in one dimension. III. Field effect on the excited-state reaction.

We obtain exact analytic solutions of the diffusion-influenced excited-state reversible geminate recombination reaction, A* + B<-->(AB)*, with two different lifetimes and quenching under the influence of a constant external field in one dimension. These fundamental solutions generalize two previous results [Kim et al., J. Chem. Phys. 111, 3791 (1999); 114, 3905 (2001)] and provide us with the insight necessary to analyze their specific relations and asymptotic kinetic transition behaviors. We find that the number of kinetic transitions can be changed due to interplay between the field strength and lifetimes. Unlike the previous works, the number of lifetime dependent transitions is found to be one or zero. On the other hand, the number of the field dependent transitions becomes two, one, or zero. We find a new pattern of kinetic transition e(t)-->t(-1/2)-->e(t) when there is only one field dependent transition.

Effect of Pressure on the Proton-Transfer Rate from a Photoacid to a Solvent. 2. 2-Naphthol-6-sulfonate in Water

The reversible proton dissociation and geminate recombination of the photoacid 2-naphthol-6-sulfonate is studied as a function of pressure in liquid water. Our time-resolved experimental data are analyzed by the reversible diffusion-influenced chemical reaction model. The proton-transfer rate increases significantly with pressure. At 10 kbar the rate increases by about a factor of 8. The main pressure effect is the decrease of the distance between the proton donor and acceptor. The pressure dependence is explained using an approximate stepwise two-coordinate proton-transfer model. The increase in rate as a function of pressure manifests the strong dependence of proton tunneling on the distance between the two heavy atoms which decreases with an increase of pressure.

Effect of Pressure on Proton Transfer Rate from a Photoacid to a Solvent. 3. 2-Naphthol and 2-Naphthol Monosulfonate Derivatives in Water

The reversible proton dissociation and geminate recombination of three photoacids, 2-naphthol, 2-naphthol-7-sulfonate, and 2-naphthol-8-sulfonate, are studied at various pressures in water. The results are compared with the results of the pressure dependence study we recently conducted for 2-naphthol-6-sulfonate in water and 5,8-dicyanonaphthol in ethanol and propanol. Our time-resolved experimental data are analyzed by the reversible diffusion-influenced chemical reaction model. The proton transfer rate increases significantly with pressure: at ∼11 kbar, the rate increases by a factor in the range of 7.5−10. The pressure dependence is explained using an approximate stepwise two coordinate proton-transfer model. The model is compared with the Landau−Zener curve-crossing proton tunneling formulation. The increase in rate, as a function of pressure, manifests, on one hand, the strong dependence of proton tunneling on the distance between the two oxygen atoms involved in the process, which decreases with an increase of pressure, and, on the other hand, the small change in the water relaxation rate as a function of pressure.

Effect of Pressure on the Proton-Transfer Rate from a Photoacid to a Solvent. 2. DCN2 in Propanol

The reversible proton dissociation and geminate recombination of photoacids is studied as a function of pressure in liquid propanol. For this purpose we used a strong photoacid, 5,8-dicyano-2-naphthol (DCN2) (pKa* ∼ −4.5 in water), capable of transferring a proton to alcohols. The time-resolved emission data are explained by the reversible diffusion-influenced chemical reaction model. At low pressure, the proton-transfer rate slightly increases with pressure whereas, at high pressure, the rate constant decreases significantly as the pressure increases. The pressure dependence is explained using an approximate stepwise two-coordinate proton-transfer model. The model is compared with the Landau−Zener curve-crossing proton tunneling formulation. Decrease of the proton-transfer rate at high-pressures reflects the solvent-controlled limit, and the increase in rate at low-pressures reflects the proton tunneling nonadiabatic limit. The results are compared with our recent studies of the pressure dependence of proton transfer from 2-naphthol-6-sulfonate (2N6S) to water and DCN2 to ethanol. Though in 2N6S−water, the proton transfer is controlled by proton tunneling, in our current work we find that, at high pressure, the solvent controls the rate of the process.

Effect of Pressure on Proton-Transfer Rate from a Photoacid to Ethanol Solution

The reversible proton dissociation and geminate recombination of photoacids is studied as a function of pressure in liquid ethanol. For this purpose, we used a strong photoacid, 5,8-dicyano-2-naphthol (DCN2) (p ≈ −4.5 in water), capable of transferring a proton to alcohols. The time-resolved experimental data are explained by the reversible diffusion-influenced chemical reaction model. At low pressure, the proton-transfer rate increases with pressure, while at high pressure, the rate constant decreases as the pressure increases. The pressure dependence is explained using an approximate stepwise two-coordinate proton-transfer model. The model is compared with the Landau−Zener curve-crossing proton-tunneling formulation. Decrease of the proton-transfer rate at high pressures denotes the adiabatic limit, while the increase in rate at low pressures denotes the nonadiabatic limit.

Evidence for a Continuous Transition from Nonadiabatic to Adiabatic Proton Transfer Dynamics in Protic Liquids

The reversible proton dissociation and geminate recombination of photoacids is studied as a function of temperature in monols, diols, and glycerol. For this purpose, we use a strong photoacid 5,8-dicyano-2-naphthol (DCN2) (p ≈ −4.5 in water), capable of transferring a proton to alcohols. The experimental data are analyzed by the Debye−Smoluchowski equation, which is solved numerically with boundary conditions to account for the reversibility of the reaction. At high temperature, the proton-transfer rate is almost temperature independent, whereas at low temperature, the rate constant has strong temperature dependence. The unusual temperature dependence is explained using Borgis−Hynes proton transfer theory, based on the Landau−Zener curve crossing formulation. The high-temperature behavior of the rate constant denotes the nonadiabatic limit, whereas the low-temperature behavior denotes the adiabatic limit. We have used an approximate expression for the proton-transfer rate, which bridges the nonadiabatic a...