Diffusion-controlled reactions in two-dimensional fluids: discussion of measurements of lateral diffusion of lipids in biological membranes

Magnetic Field Effect in Bimolecular Rate Constant of Radical Recombination

The influence of magnetic fields on chemical reactions, including biological ones, has been and still is a topical subject in the field of scientific research. Experimentally discovered and theoretically substantiated magnetic and spin effects in chemical radical reactions form the basis of research in the field of spin chemistry. In the present work, the effect of a magnetic field on the rate constant of the bimolecular spin-selective recombination of radicals in the bulk of a solution is considered theoretically for the first time, taking into account the hyperfine interaction of radical spins with their magnetic nuclei. In addition, the paramagnetic relaxation of unpaired spins of the radicals and the non-equality of their g-factors that also influence the recombination process are taken into account. It is found that the reaction rate constant can vary in magnetic field from a few to half a dozen percent, depending on the relative diffusion coefficient of radicals, which is determined by the solution viscosity. It is shown that the consideration of hyperfine interactions gives rise to the presence of resonances in the dependence of the rate constant on the magnetic field. The magnitudes of the magnetic fields of these resonances are determined by the hyperfine coupling constants and difference in the g-factors of the recombining radicals. Analytical expressions for the reaction rate constant of the bulk recombination for magnetic fields larger than hfi (hyperfine interaction) constants are obtained. In general, it is shown for the first time that accounting for hyperfine interactions of radical spins with magnetic nuclei significantly affects the dependence of the reaction rate constant of the bulk radical recombination on the magnetic field.

Magnetic field effect on recombination of radicals diffusing on a two-dimensional plane

Magnetic Field Effects (MFEs) on the recombination of radicals, which diffuse on an infinite plane, are studied theoretically. The case of spin-selective diffusion-controlled recombination of Radical Pairs (RPs) starting from a random spin state is considered assuming uniform initial distribution of the radicals. In this situation, reaction kinetics is described by a time-dependent rate coefficient K(t), which tends to zero at long times. Strong MFEs on K(t) are predicted that originate from the Δg and hyperfine driven singlet-triplet mixing in the RP. The effects of spin relaxation on the magnetic field are studied, as well as the influence of the dipole-dipole interaction between the electron spins of the RP. In the two-dimensional case, this interaction is not averaged out by diffusion and it strongly affects the MFE. The results of this work are of importance for interpreting MFEs on lipid peroxidation, a magnetosensitive process occurring on two-dimensional surfaces of cell membranes.

Two dimensional diffusion-controlled triplet-triplet annihilation kinetics

Diffusion controlled chemical reactions are usually observed in three dimensional media. In contrast, planar bimolecular reactions taking place between reagents adsorbed at a soft interface are two-dimensional and therefore cannot be studied within the same formalism. Indeed, soft interfaces allow the adsorbed species to freely diffuse in a liquid-like manner. Here, we present the first experimental observation of a diffusion-controlled reaction in an environment that is planar at the ångström scale. By means of time-resolved surface second harmonic generation, an inherently surface sensitive technique, we observed that the kinetics of the diffusion of the reagents in the plane decreases as the surface concentration of adsorbed species increases. This is of course not the case for bulk reactions where the rates always increase with the reactant concentration. Such changes in the kinetics regime were rationalised as the evolution from a regular 2D free diffusion regime to a geometry-controlled scheme.

Effective reaction rates for diffusion-limited reaction cycles

Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.

Theory of bi-molecular association dynamics in 2D for accurate model and experimental parameterization of binding rates

The dynamics of association between diffusing and reacting molecular species are routinely quantified using simple rate-equation kinetics that assume both well-mixed concentrations of species and a single rate constant for parameterizing the binding rate. In two-dimensions (2D), however, even when systems are well-mixed, the assumption of a single characteristic rate constant for describing association is not generally accurate, due to the properties of diffusional searching in dimensions d ≤ 2. Establishing rigorous bounds for discriminating between 2D reactive systems that will be accurately described by rate equations with a single rate constant, and those that will not, is critical for both modeling and experimentally parameterizing binding reactions restricted to surfaces such as cellular membranes. We show here that in regimes of intrinsic reaction rate (ka) and diffusion (D) parameters ka/D > 0.05, a single rate constant cannot be fit to the dynamics of concentrations of associating species independently of the initial conditions. Instead, a more sophisticated multi-parametric description than rate-equations is necessary to robustly characterize bimolecular reactions from experiment. Our quantitative bounds derive from our new analysis of 2D rate-behavior predicted from Smoluchowski theory. Using a recently developed single particle reaction-diffusion algorithm we extend here to 2D, we are able to test and validate the predictions of Smoluchowski theory and several other theories of reversible reaction dynamics in 2D for the first time. Finally, our results also mean that simulations of reactive systems in 2D using rate equations must be undertaken with caution when reactions have ka/D > 0.05, regardless of the simulation volume. We introduce here a simple formula for an adaptive concentration dependent rate constant for these chemical kinetics simulations which improves on existing formulas to better capture non-equilibrium reaction dynamics from dilute to dense systems.

Rate coefficients, binding probabilities, and related quantities for area reactivity models

We further develop the general theory of the area reactivity model that describes the diffusion-influenced reaction of an isolated receptor-ligand pair in terms of a generalized Feynman-Kac equation and that provides an alternative to the classical contact reactivity model. Analyzing both the irreversible and reversible reaction, we derive the equation of motion of the survival probability as well as several relationships between single pair quantities and the reactive flux at the encounter distance. Building on these relationships, we derive the equation of motion of the many-particle survival probability for irreversible pseudo-first-order reactions. Moreover, we show that the usual definition of the rate coefficient as the reactive flux is deficient in the area reactivity model. Numerical tests for our findings are provided through Brownian Dynamics simulations. We calculate exact and approximate expressions for the irreversible rate coefficient and show that this quantity behaves differently from its classical counterpart. Furthermore, we derive approximate expressions for the binding probability as well as the average lifetime of the bound state and discuss on- and off-rates in this context. Throughout our approach, we point out similarities and differences between the area reactivity model and its classical counterpart, the contact reactivity model. The presented analysis and obtained results provide a theoretical framework that will facilitate the comparison of experiment and model predictions.

Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency

Singlet exciton diffusion plays a central role in the photovoltaic conversion in organic photovoltaics (OPVs). Upon light absorption, singlet excitons are promptly generated in organic materials instead of charge carriers because the dielectric constant (εr) is small (∼3-4), which is in sharp contrast to inorganic and perovskite solar cells. In order to convert to charge carriers, excitons need to diffuse into an interface between electron donor and acceptor materials before deactivating to the ground state. Therefore, fundamental understanding of exciton diffusion dynamics is one of the most important issues to further improve OPVs. We highlight recent leading studies in this field and describe several approaches for efficient exciton harvesting at the interface in OPVs.

Theoretical description of spin-selective reactions of radical pairs diffusing in spherical 2D and 3D microreactors

In this work, we treat spin-selective recombination of a geminate radical pair (RP) in a spherical "microreactor," i.e., of a RP confined in a micelle, vesicle, or liposome. We consider the microreactor model proposed earlier, in which one of the radicals is located at the center of the micelle and the other one undergoes three-dimensional diffusion inside the micelle. In addition, we suggest a two-dimensional model, in which one of the radicals is located at the "pole" of the sphere, while the other one diffuses on the spherical surface. For this model, we have obtained a general analytical expression for the RP recombination yield in terms of the free Green function of two-dimensional diffusion motion. In turn, this Green function is expressed via the Legendre functions and thus takes account of diffusion over a restricted spherical surface and its curvature. The obtained expression allows one to calculate the RP recombination efficiency at an arbitrary magnetic field strength. We performed a comparison of the two models taking the same geometric parameters (i.e., the microreactor radius and the closest approach distance of the radicals), chemical reactivity, magnetic interactions in the RP and diffusion coefficient. Significant difference between the predictions of the two models is found, which is thus originating solely from the dimensionality effect: for different dimensionality of space, the statistics of diffusional contacts of radicals becomes different altering the reaction yield. We have calculated the magnetic field dependence of the RP reaction yield and chemically induced dynamic nuclear polarization of the reaction products at different sizes of the microreactor, exchange interaction, and spin relaxation rates. Interestingly, due to the intricate interplay of diffusional contacts of reactants and spin dynamics, the dependence of the reaction yield on the microreactor radius is non-monotonous. Our results are of importance for (i) interpreting experimental data for magnetic field effects on RP recombination in confined space and (ii) for describing kinetics of chemical reactions, which occur predominantly on the surfaces of biomembranes, i.e., lipid peroxidation reactions.

Effective reaction rates in diffusion-limited phosphorylation-dephosphorylation cycles

We investigate the kinetics of the ubiquitous phosphorylation-dephosphorylation cycle on biological membranes by means of kinetic Monte Carlo simulations on the triangular lattice. We establish the dependence of effective macroscopic reaction rate coefficients as well as the steady-state phosphorylated substrate fraction on the diffusion coefficient and concentrations of opposing enzymes: kinases and phosphatases. In the limits of zero and infinite diffusion, the numerical results agree with analytical predictions; these two limits give the lower and the upper bound for the macroscopic rate coefficients, respectively. In the zero-diffusion limit, which is important in the analysis of dense systems, phosphorylation and dephosphorylation reactions can convert only these substrates which remain in contact with opposing enzymes. In the most studied regime of nonzero but small diffusion, a contribution linearly proportional to the diffusion coefficient appears in the reaction rate. In this regime, the presence of opposing enzymes creates inhomogeneities in the (de)phosphorylated substrate distributions: The spatial correlation function shows that enzymes are surrounded by clouds of converted substrates. This effect becomes important at low enzyme concentrations, substantially lowering effective reaction rates. Effective reaction rates decrease with decreasing diffusion and this dependence is more pronounced for the less-abundant enzyme. Consequently, the steady-state fraction of phosphorylated substrates can increase or decrease with diffusion, depending on relative concentrations of both enzymes. Additionally, steady states are controlled by molecular crowders which, mostly by lowering the effective diffusion of reactants, favor the more abundant enzyme.

Growth kinetics of circular liquid domains on vesicles by diffusion-controlled coalescence

Motivated by recent experiments on multicomponent membranes, the growth kinetics of domains on vesicles are theoretically studied. It is known that the steady-state rate of coalescence cannot be obtained by taking the long-time limit of the coalescence rate when the membrane is regarded as an infinite two-dimensional (2D) system. The steady-state rate of coalescence is obtained by explicitly taking into account the spherical vesicle shape. Using the expression for the 2D diffusion coefficient obtained in the limit of small domain size, an analytical expression for domain growth kinetics is obtained when a circular shape is always maintained. For large domains, the growth kinetics are discussed by investigating the size dependence of the coalescence rate, using the expression for the diffusion coefficient of arbitrary domain size.

Quenching of triplet state fluorophores for studying diffusion-mediated reactions in lipid membranes

An approach to study bimolecular interactions in model lipid bilayers and biological membranes is introduced, exploiting the influence of membrane-associated electron spin resonance labels on the triplet state kinetics of membrane-bound fluorophores. Singlet-triplet state transitions within the dye Lissamine Rhodamine B (LRB) were studied, when free in aqueous solutions, with LRB bound to a lipid in a liposome, and in the presence of different local concentrations of the electron spin resonance label TEMPO. By monitoring the triplet state kinetics via variations in the fluorescence signal, in this study using fluorescence correlation spectroscopy, a strong fluorescence signal can be combined with the ability to monitor low-frequency molecular interactions, at timescales much longer than the fluorescence lifetimes. Both in solution and in membranes, the measured relative changes in the singlet-triplet transitions rates were found to well reflect the expected collisional frequencies between the LRB and TEMPO molecules. These collisional rates could also be monitored at local TEMPO concentrations where practically no quenching of the excited state of the fluorophores can be detected. The proposed strategy is broadly applicable, in terms of possible read-out means, types of molecular interactions that can be followed, and in what environments these interactions can be measured.

Phase-dependent lateral diffusion of α-tocopherol in DPPC liposomes monitored by fluorescence quenching

The temperature-dependent fluorescence quenching of an amphiphilic palmitoyl derivative of 2,3-diazabicyclo[2.2.2]oct-2-ene (Fluorazophore-L) by α-tocopherol (α-Toc) has been determined in liposomes composed of a saturated lipid, 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC). The mutual lateral diffusion coefficients (D(L)) were extracted according to a laterally diffusion-controlled dynamic quenching model. Three distinct temperature regimes were identified: one between 65 and 39 °C, where the lateral diffusion coefficients were in the range of 10(-7) cm(2) s(-1) and the lifetime of the probe was monoexponential in the absence of α-Toc, a second one between 39 and 30 °C, where the lateral diffusion coefficients were in the range of 10(-8) cm(2) s(-1) and the lifetime of the probe was biexponential in the absence of α-Toc, and a third one below 30 °C, in which no diffusion was detectable, suggesting D(L) < 10(-9) cm(2)s (-1). These temperature domains were assigned, supported by differential scanning calorimetry (DSC) measurements, to the liquid-crystalline, ripple, and solid-gel phases of DPPC liposomes in the presence of the two additives. The absolute values of the individual lateral diffusion coefficients (taken as (1)/(2) of the D(L) values) of the Fluorazophore-L/α-Toc (ca. 2.5 × 10(-7) cm(2) s(-1) at 52 °C) couple demonstrates that α-Toc does not diffuse at an unexpectedly high rate in comparison to the self-diffusion of DPPC (1.5 × 10(-7) cm(2) s(-1) at 52 °C). However, diffusion in DPPC liposomes is distinctly slower than that in POPC ones (e.g., D(L) = 4.9 × 10(-7) cm(2) s(-1) versus 6.4 × 10(-7) cm(2) s(-1) at 50 °C), with an activation energy of 49 ± 5 kJ mol(-1) (value for POPC: 47 ± 5 kJ mol(-1)), in the temperature range of the liquid-crystalline phase. Diffusion in the ripple phase, that is, below the main phase transition temperature, was found to be non-negligible, with an apparent activation energy of 175 ± 50 kJ mol(-1).

Diffusion in narrow domains and application to phototransduction

The mean time for a Brownian particle to find a small target inside a narrow domain is a key parameter for many chemical reactions occurring in cellular microstructures. Although current estimations are given for a large class of domains, they cannot be used for narrow domains often encountered in cellular biology, such as the synaptic cleft, narrow compartments in the outer segment of vertebrate photoreceptors, or neuron-glia contact. We compute here the mean time for a Brownian particle to hit a small target placed on the surface of a narrow cylinder. We then use this result to estimate the rate constant of cyclic-GMP (cGMP) hydrolysis by the activated enzyme phosphodiesterase (PDE) in the narrow microdomains that build up the outer segment of a rod photoreceptor. By controlling the cGMP concentration, PDE activity is at the basis of the early photoresponse chemical reaction cascade. Our approach allows us to compute the cGMP rate constant as a function of biophysical parameters.

Reappraisal of four different approaches for finding the mean reaction time in the multi-trap variant of the Adam-Delbrück problem

Adam and Delbrück argued that the dimensionality of the diffusion space determines the average lifetime of a diffusing particle confined to a region with a central trap. Doubts have often been aired as to whether their calculation is relevant to real biological systems, where the number of traps is usually much larger than unity; or whether the rate enhancement is merely a manifestation of an increase in the concentration of the traps; or whether the diverse multi-trap versions of their expression for the mean lifetime in two dimensions are trustworthy. These issues are addressed, and the long-standing problem of finding the low-density limit of trapping time in two dimensions solved, by examining previous treatments of the problem, and by carrying out simulations of two-dimensional systems in which the particles undergo a Pearsonian random walk, and the traps are distributed randomly or on a square lattice. The mean lifetimes are found to be different in the two situations, and it is concluded that the neglect of this aspect lies at the root of the conflict between some of the existing expressions for the mean lifetime. Relations expressing the mean lifetime as a function of the concentration of the traps are presented together with a discussion of their applicability.

Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review

The quantitative study of the probability of molecular encounters giving rise to a reaction in membranes is a challenging discipline. Model systems, model in the sense that they use model bilayers and model reactants, have been widely used for this purpose, but the methodologies employed for the analysis of the results obtained in experiments, and for experimental design, are so disparate that a concerned experimentalist has difficulty in deciding about the value of each approach. This review intends to examine the several approaches that can be found in the literature showing, when feasible, the weakness, strengths and limits of application of each of them. There is not, so far, a full experimental validation of the most promising theories for the analysis of reactions in two dimensions, what leaves open a large field for new research. The major challenge resides in the time range in which the processes take place, but the possibilities of the existing techniques for these studies are far from exhausted. We review also the attempts of several authors to quantitatively analyze the kinetics of reactions in biological membranes. Especially in this field, the recently developed microspectroscopies enclose a still unexplored potential.

Lysozyme effect on structural state of model membranes as revealed by pyrene excimerization studies

Steady-state measurements of pyrene fluorescence in the model bilayer membranes composed of phosphatidylcholine (PC) and its mixtures with cardiolipin (CL) have been performed to gain insight into the effect of lysozyme on molecular organization of lipid bilayer. Analysis of vibronic structure of the probe emission spectra revealed no changes in transverse distribution of pyrene monomers on varying CL contents or increasing the extent of lysozyme binding to liposomes. Excimer-to-monomer fluorescence intensity ratio has been found to reduce on lysozyme association with lipids. The magnitude of this effect increased with increasing CL content from 0 to 40 mol%. These results have been interpreted as indicating decrease in the membrane free volume on formation of both electrostatic and hydrophobic protein-lipid contacts.

Study of rare encounters in a membrane using quenching of cascade reaction between triplet and photochrome probes with nitroxide radicals

Measurements of active encounters between molecules in native membranes containing ingredients, including proteins, are of prime importance. To estimate rare encounters in a high range of rate constants (rate coefficients) and distances between interacting molecules in membranes, a cascade of photochemical reactions for molecules diffusing in multilamellar liposomes was investigated. The sensitised cascade triplet cis-trans photoisomerisation of the excited stilbene involves the use of a triplet sensitiser (Erythrosin B), a photochrome stilbene-derivative probe (4-dimethylamino-4'-aminostilbene) exhibiting the phenomenon of trans-cis photoisomerisation, and nitroxide radicals (5-doxyl stearic acid) to quench the excited triplet state of the sensitiser. Measurement of the phosphorescence lifetime of Erythrosin B and the fluorescence enhancement of the stilbene-derivative photochrome probe, at various concentrations of the nitroxide probe, made it possible to calculate the quenching rate constant k(q)= 1.1 x 10(15) cm2 M(-1) s(-1) and the rate constant of the triplet-triplet energy transfer between the sensitiser and stilbene probe k(T)= 1.0 x 10(12) cm2 M(-1) s(-1). These values, together with the data on diffusion rate constant, obtained by methods utilising various theoretical characteristic times of about seven orders of magnitude and the experimental rate constants of about five orders of magnitude, were found to be in good agreement with the advanced theory of diffusion-controlled reactions in two dimensions. Because the characteristic time of the proposed cascade method is relatively large (0.1 s), it is possible to follow rare collisions between molecules and free radicals in model and biological membranes with a very sensitive fluorescence spectroscopy technique, using a relatively low concentration of probes.

Cholesterol modulates the organization of the gammaM4 transmembrane domain of the muscle nicotinic acetylcholine receptor

A 28-mer gammaM4 peptide, obtained by solid-state synthesis and corresponding to the fourth transmembrane segment of the nicotinic acetylcholine receptor gamma-subunit, possesses a single tryptophan residue (Trp453), making it an excellent model for studying peptide-lipid interactions in membranes by fluorescence spectroscopy. The gammaM4 peptide was reconstituted with synthetic lipids (vesicles of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine, i.e., POPC) rich and poor in cholesterol and analyzed using steady-state and time-resolved fluorescence techniques. The decrease in gammaM4 intrinsic fluorescence lifetime observed upon incorporation into a cholesterol-rich lo phase could be rationalized on the basis of a dynamic self-quenching owing to the formation of peptide-rich patches in the membrane. This agrees with the low Förster type resonance energy transfer efficiency from the Trp453 residue to the fluorescent cholesterol analog, dehydroergosterol, in the lo phase. In the absence of cholesterol the gammaM4 nicotinic acetylcholine receptor peptide is randomly distributed in the POPC bilayer with its hydrophobic moiety matching the membrane thickness, whereas in the presence of cholesterol the increase in the membrane thickness and variation of the material properties favor the formation of peptide-enriched patches, i.e., interhelix interaction energy is essential for obtaining a stabilized structure. Thus, the presence of a cholesterol-rich, ordered POPC phase drives the organization of peptide-enriched patches, in which the gammaM4 peptide occupies approximately 30% of the patch area.

Dependence of M13 major coat protein oligomerization and lateral segregation on bilayer composition

M13 major coat protein was derivatized with BODIPY (n-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-yl)methyl iodoacetamide), and its aggregation was studied in 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and DOPC/1,2-dioleoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (DOPG) or 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE)/DOPG (model systems of membranes with hydrophobic thickness matching that of the protein) using photophysical methodologies (time-resolved and steady-state self-quenching, absorption, and emission spectra). It was concluded that the protein is essentially monomeric, even in the absence of anionic phospholipids. The protein was also incorporated in pure bilayers of lipids with a strong mismatch with the protein transmembrane length, 1,2-dierucoyl-sn-glycero-3-phosphocholine (DEuPC, longer lipid) and 1,2-dimyristoleoyl-sn-glycero-3-phosphocholine (DMoPC, shorter lipid), and in lipidic mixtures containing DOPC and one of these lipids. The protein was aggregated in the pure vesicles of mismatching lipid but remained essentially monomeric in the mixtures as detected from BODIPY fluorescence emission self-quenching. From fluorescence resonance energy transfer (FRET) measurements (donor-n-(iodoacetyl)aminoethyl-1-sulfonaphthylamine (IAEDANS)-labeled protein; acceptor-BODIPY labeled protein), it was concluded that in the DEuPC/DOPC and DMoPC/DOPC lipid mixtures, domains enriched in the protein and the matching lipid (DOPC) are formed.

Diffusion of α-Tocopherol in Membrane Models: Probing the Kinetics of Vitamin E Antioxidant Action by Fluorescence in Real Time

The new fluorescent membrane probe Fluorazophore-L, a lipophilic derivative of the azoalkane 2,3-diazabicyclo[2.2.2]oct-2-ene, is employed to study the quenching of alpha-tocopherol (alpha-Toc) by time-resolved fluorescence in the microheterogeneous environments of Triton XR-100 and SDS micelles, as well as POPC liposomes. Fluorazophore-L has a small nonaromatic fluorescent polar headgroup and an exceedingly long-lived fluorescence (e.g., 140 ns in aerated SDS micelles), which is efficiently quenched by alpha-Toc (3.9 x 10(9) M(-1) s(-1) in benzene). Based on solvatochromic effects and the accessibility by water-soluble quenchers, the reactive headgroup of Fluorazophore-L, along with the chromanol group of alpha-Toc, resides at the water-lipid interface, which allows for a diffusion-controlled quenching in the lipidic environments. The quenching experiments represent an immobile or stationary case; that is, interparticle probe or quencher exchange during the excited-state lifetime is insignificant. Different quenching models are used to characterize the dynamics and antioxidant action of alpha-Toc in terms of diffusion coefficients or, where applicable, rate constants. The ideal micellar quenching model is suitable to describe the fluorescence quenching in SDS micelles and affords a pseudo-unimolecular quenching rate constant of 2.4 (+/- 0.4) x 10(7) s(-1) for a single quencher per micelle along with a mean aggregation number of 63 +/- 3. In Triton micelles as well as in unilamellar POPC liposomes, a two-dimensional (lateral) diffusion model is most appropriate. The mutual lateral diffusion coefficient D(L) for alpha-Toc and Fluorazophore-L in POPC liposomes is found to be 1.8 (+/- 0.1) x 10(-7) cm(2) s(-1), about a factor of 2 larger than for mutual diffusion of POPC, but more than 1 order of magnitude lower than a previously reported value. The comparison of the different environments suggests a quenching efficiency in the order benzene >> SDS micelles > Triton micelles > POPC liposomes, in line with expectations from microviscosity. The kinetic measurements provide important benchmark values for the modeling of oxidative stress in membranes and other lipidic assemblies. The special case of small lipidic assemblies (SDS micelles), for which the net antioxidant efficacy of alpha-Toc may be lower than expected on the grounds of its diffusional behavior, is discussed.

Membrane protein diffusion sets the speed of rod phototransduction

Retinal rods signal the activation of a single receptor molecule by a photon. To ensure efficient photon capture, rods maintain about 109 copies of rhodopsin densely packed into membranous disks. But a high packing density of rhodopsin may impede other steps in phototransduction that take place on the disk membrane, by restricting the lateral movement of, and hence the rate of encounters between, the molecules involved. Although it has been suggested that lateral diffusion of proteins on the membrane sets the rate of onset of the photoresponse, it was later argued that the subsequent processing of the complexes was the main determinant of this rate. The effects of protein density on response shut-off have not been reported. Here we show that a roughly 50% reduction in protein crowding achieved by the hemizygous knockout of rhodopsin in transgenic mice accelerates the rising phases and recoveries of flash responses by about 1.7-fold in vivo. Thus, in rods the rates of both response onset and recovery are set by the diffusional encounter frequency between proteins on the disk membrane.

Molecular mechanism of lateral diffusion of py(10)-PC and free pyrene in fluid DMPC bilayers

From the study of the kinetics of the fluorescence self-quenching of pyrene in multilamellar vesicles of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) in the fluid phase we obtain the molecular diffusion parameters, diffusion coefficients, and activation energies for lateral diffusion for three probes: 1-palmitoyl-2-(1-pyrenedecanoyl)-sn-glycero-3-phosphocholine (py(10)-PC), pyrene, and 1-pyrenebutanoic acid (py-but). The experiments are done in a range of temperatures for which the reversibility of excimer formation is negligible and the probe/phospholipid ratios used are low, avoiding non-ideal mixing of the probe. The time-resolved fluorescence decays are, in all cases, accurately and consistently described by the two-dimensional (2D) formalism for bimolecular diffusion-controlled reactions. From the parameters obtained in this way we conclude that the primary step of the diffusion of the pyrene-labeled phospholipid involves the simultaneous jump of phospholipid and fluorophore moieties, and also that although in the case of py(10)-PC the pyrene from molecules pertaining to different layers may interact during the lifetime of the exited state, this is not the case for free pyrene.

Single species diffusion-influenced reaction A+A-->alphaA: validity of the smoluchowski approach

We investigate the single species diffusion-influenced reaction, A+A-->alphaA with a finite reactivity in all dimensions. The reaction model includes a pure coagulation (alpha=1) or a pure annihilation (alpha=0) model. We apply the hierarchical Smoluchowski approach to study the dimensional aspects of the fluctuation, reactivity, particle size, and alpha(0</=alpha</=1). The theoretical results are compared with those of the Monte Carlo simulations in one, two, and three regular dimensions. The simulation results reveal that the classical Smoluchowski approach is exact in the short time limit in all dimensions and in the long time limit in three dimensions. The hierarchical Smoluchowski approach is found to be numerically exact at all times in two and three dimensions. A numerical method to obtain the exact result of the annihilation for a finite reactivity in one dimension is presented. We also propose a quite accurate analytic solution for an arbitrary alpha for the infinite reactivity in one dimension.

Quantification of short term signaling by the epidermal growth factor receptor

During the past decade, our knowledge of molecular mechanisms involved in growth factor signaling has proliferated almost explosively. However, the kinetics and control of information transfer through signaling networks remain poorly understood. This paper combines experimental kinetic analysis and computational modeling of the short term pattern of cellular responses to epidermal growth factor (EGF) in isolated hepatocytes. The experimental data show transient tyrosine phosphorylation of the EGF receptor (EGFR) and transient or sustained response patterns in multiple signaling proteins targeted by EGFR. Transient responses exhibit pronounced maxima, reached within 15-30 s of EGF stimulation and followed by a decline to relatively low (quasi-steady-state) levels. In contrast to earlier suggestions, we demonstrate that the experimentally observed transients can be accounted for without requiring receptor-mediated activation of specific tyrosine phosphatases, following EGF stimulation. The kinetic model predicts how the cellular response is controlled by the relative levels and activity states of signaling proteins and under what conditions activation patterns are transient or sustained. EGFR signaling patterns appear to be robust with respect to variations in many elemental rate constants within the range of experimentally measured values. On the other hand, we specify which changes in the kinetic scheme, rate constants, and total amounts of molecular factors involved are incompatible with the experimentally observed kinetics of signal transfer. Quantitation of signaling network responses to growth factors allows us to assess how cells process information controlling their growth and differentiation.

Fluorescence quenching data interpretation in biological systems. The use of microscopic models for data analysis and interpretation of complex systems

In micro-heterogeneous media (e.g. membranes, micelles and colloidal systems), the fluorescence decay in the absence of quencher is usually intrinsically complex, e.g. due to the existence of several sub-populations with different micro-environments. In this case it is impossible to analyze data in detail (accounting for transient effects) and simpler formalisms are needed. The objective of the present work is to present and discuss such simpler formalisms. The goal is to achieve simple data analysis and meaningful, clear data interpretation in complex systems using microscopic models that consider several sub-populations of chromophores. Two points are dealt with in detail. (i) It is shown that the approximation of the transient effects by the quenching sphere-of-action model is not always possible. The quenching sphere-of-action concept can be regarded as a valuable tool, although crude, only in a limited range of experimental conditions, namely time resolution. (ii) The Stern-Volmer equation usually used for data analysis is only valid for a limited range of small and moderate equilibrium association constants, Ka, although this is frequently overlooked in the literature. Self-consistency criteria are presented for the proposed methods. The well-known downward curvature due to a fraction of fluorophores which is not accessible to the quencher is only a limiting case from a set of possible situations which result in deviations to linearity. A systematic classification of the different types of quenching is presented.

Amplification and kinetics of the activation steps in phototransduction

We can summarize our investigation of amplification in the activation steps of vertebrate phototransduction as follows. (1) A theoretical analysis of the activation steps of the cGMP cascade shows that after a brief flash of phi photoisomerizations the number of activated PDE molecules should rise as a delayed ramp with slope proportional to phi, and that, as a consequence, the cGMP-activated current should decay as a delayed Gaussian function of time (Eqn. 20). (i) Early in the response to a flash, the normalized response R(t) can be approximated as rising as 1/2 phi At2 (after a short delay), where A is the amplification constant characteristic of the individual photoreceptor. (ii) The delayed ramp behavior of PDE activation and the consequent decline of current in the form of the delayed Gaussian are confirmed by experiments in a variety of photoreceptors; the analysis thus yields estimates of the amplification constant from these diverse photoreceptors. (iii) Eqn. 20 further predicts that the response-intensity relation at any fixed time should saturate exponentially, as has been found experimentally. (2) The amplification constant A can be expressed as the product of amplification factors contributed by the individual activation steps of phototransduction, i.e., A = nu RG cGP beta sub n (Eqns. 9 and 21), where (i) nu RG is the rate of G* production per Rh*; (ii) cGP is the efficiency of the coupling between G* production and PDE* production; (iii) beta sub is the increment in hydrolytic rate constant produced by one PDE*, i.e., a single activated catalytic subunit of PDE; and (iv) n is the Hill coefficient of opening of the cGMP-activated channels. (3) The amplification factor beta sub includes the ratio kcat/Km, which characterizes the hydrolytic activity of the PDE in vivo where cG << Km. Two different analyses based upon photocurrents were developed which provide lower bounds for kcat/Km in vivo; these analyses establish that kcat/Km probably exceeds 10(7) M-1 s-1 (and is likely to be higher) in both amphibian and mammalian rods. Few biochemical studies (other than those using trypsin activation) have yielded such high values. A likely explanation of many of the relatively low biochemical estimates of kcat/Km is that Km may have been overestimated by a factor of about 4 in preparations in which stacks of disks are left intact, due to diffusion with hydrolysis in the stacks.(ABSTRACT TRUNCATED AT 400 WORDS)

G-protein cascades: gain and kinetics

G-protein cascades provide amplification in a wide variety of biological signal transducers--from hormonal and synaptic systems to the receptor cells of vision and olfaction. Through recent understanding of the molecular mechanisms involved, it is possible to construct a quantitative description of the amplification and speed of response of the cascade. The gain and kinetics can now be described in terms of physical parameters, such as enzyme activities and the densities and lateral diffusion coefficients of the proteins involved.

Concentration effects on reactions in membranes: rhodopsin and transducin

The reaction rate of two laterally-diffusing species in a biological membrane shows a maximum at some concentration of reactants, because an increase in the concentration of reactants tends to increase the reaction rate by the law of mass action but decreases the diffusion rate of the reactants. The activation of transducin by rhodopsin in the disk membrane of the rod outer segment is described in terms of a steady-state diffusion model with concentration-dependent diffusion coefficients. The optimum concentrations of reactants are obtained from contour plots of the reaction rate as a function of rhodopsin and transducin concentrations, and the sensitivity of the results to the assumed values of the variables is examined. To determine whether the observed concentrations are in fact those yielding the maximum reaction rate, several variables must be known more accurately.