Integration Of Differential Rate Equations Research Articles

Probing of the substrate binding domain of lactose permease by a proton pulse

The lactose permease of Escherichia coli coupled proton transfer across the bacterial inner membrane with the uptake of β-galactosides. In the present study we have used the cysteine-less C148 mutant that was selectively labeled by fluorescein maleimide on the C148 residue, which is an active component of the substrate transporting cavity. Measurements of the protonation dynamics of the bound pH indicator in the time resolved domain allowed us to probe the binding site by a free diffusing proton. The measured signal was reconstructed by numeric integration of differential rate equations that comply with the detailed balance principle and account for all proton transfer reactions taking place in the reaction mixture. This analysis yields the rate constants and p K values of all residues participating in the fast proton transfer reaction between the bulk and the protein’s surface, revealing the exposed residues that react with free protons in a diffusion controlled reaction and how they transfer protons among themselves. The magnitudes of these rate constants were finally evaluated by comparison with the rate predicted by the Debye–Smoluchowski equation. The analysis of the kinetic and p K values indicated that the protein–fluorescein adduct assumes two conformation states. One is dominant above pH 7.4, while the other exists only below 7.1. In the high pH range, the enzyme assumes a constrained configuration and the rate constant of the reaction of a free diffusing proton with the bound dye is 10 times slower than a diffusion controlled reaction. In this state, the carboxylate moiety of residue E126 is in close proximity to the dye and exchanges a proton with it at a very fast rate. Below pH 7.1, the substrate binding domain is in a relaxed configuration and freely accessed by bulk protons, and the rate of proton exchange between the dye and E126 is 100 000 times slower. The relevance of these observations to the catalytic cycle is discussed.

Mechanism of proton entry into the cytoplasmic section of the proton-conducting channel of bacteriorhodopsin.

Bacteriorhodopsin is the light-driven proton-pumping protein of Halobacterium salinarum that extracts protons from the well-buffered cytoplasmic space within the time limits set by the photocycle turnover. The specific mechanism of the proton uptake by the cytoplasmic surface of the protein was investigated in this study by the laser-induced proton pulse technique. The purple membrane preparations were labeled by fluorescein at two residues (36 or 38) of the cytoplasmic surface of the protein, sites that are close to the orifice of the proton-conducting channel. The membranes were pulsed by protons discharged from photoexcited pyranine [Nachliel, E., Gutman, M., Kiryati, S., and Dencher, N.A. (1996) Proc. Nat Acad. Sci. U.S.A. 93, 10747-10752). The reaction of the discharged protons with the pyranine anion and the fluorescein was measured with sub-microsecond resolution. The experimental signals were reconstructed through numeric integration of differential rate equations which quantitated the rates of all proton transfer reactions between all reactants present in the system. The interaction of protons with the orifice of the cytoplasmic channel is enhanced by the exposed carboxylates of the protein. A cluster of three carboxylates acts as a strong proton attractor site while one carboxylate, identified as D36, acts as a mediator that delivers the proton to the channel. The combination of these reactions render the surface of the protein with properties of a proton-collecting antenna. The size of the collecting area is less than that of the protein's surface.

Experimental design for kinetic analysis of protein-protein interactions with surface plasmon resonance biosensors

The reaction between antibody immobilized to surfaces, with and without a dextran matrix, and antigen in solution was studied using surface plasmon resonance detection. The use of a reference surface made it possible to identify conditions where a response related to changes in matrix conformation could be ignored. It was possible therefore to improve data quality by separating signals related to binding events from signals due to differences in refractive index between sample and running buffer. When antigen was injected over antibody immobilized to surfaces with and without dextran matrix the binding curves were virtually superimposable. Consequently, no binding artifacts associated with the dextran matrix were observed. Sets of binding curves obtained with different antigen concentrations were analyzed using numerical integration of differential rate equations and global fitting. When data was inconsistent with a one to one reaction it was possible to obtain good fits to an entire data set assuming several other reaction schemes including parallel, competitive and two-state reactions. Thus data analysis alone was not sufficient to discriminate between different reaction schemes. In contrast several reaction schemes could be ruled out with simple experiments; the duration of antigen injection, and reanalysis of antigen recovered in fractions from the antibody surface. In view of these findings experimental design appears to be the key to successful interaction analysis.

The mechanism of monensin-mediated cation exchange based on real time measurements

Monensin is an ionophore that supports an electroneutral ion exchange across the lipid bilayer. Because of this, under steady-state conditions, no electric signals accompany its reactions. Using the Laser Induced Proton Pulse as a synchronizing event we selectively acidify one face of a black lipid membrane impregnated by monensin. The short perturbation temporarily upsets the acid-base equilibrium on one face of the membrane, causing a transient cycle of ion exchange. Under such conditions the molecular events could be discerned as a transient electric polarization of the membrane lasting approx. 200 μs. The proton-driven chemical reactions that lead to the electric signals had been reconstructed by numeric integration of differential rate equations which constitute a maximalistic description of the multi equilibria nature of the system (Gutman, M. and Nachliel, E. (1989) Electrochim. Acta 34, 1801–1806). The analysis of the reactions reveals that the ionic selectivity of the monensin (H + > Na + > K +) is due to more than one term. Besides the well established different affinity for the various cations, the selectivity is also derived from a large difference in the rates of cross membranal diffusivities (MoH > MoNa > MoK), which have never been detected before. (v) Quantitative analysis of the membrane's crossing rates of the three neutral complexes reveals a major role of the membranal dipolar field in regulating ion transport. The diffusion of MoH, which has no dipole moment, is hindered only by the viscose drag. On the other hand, the dipolar complexes (MoNa and MoK) are delayed by dipole-dipole interaction with the membrane. (vi) Comparison of the calculated dipoles with those estimated for the crystalline conformation of the [MoNa(H 2O) 2] and [MoK(H 2O) 2] complexes reveals that the MoNa may exist in the membrane at its crystal configuration, while the MoK definitely attains a structure having a dipole moment larger than in the crystal.

Kinetics and mechanism of autocatalytic oxidation of Fe(phen)2+3 and Fe(bpy)2+3 by nitric acid

The kinetics of the title reactions has been studied and the investigation of the oxidation of the iron(II) aqua ion by nitric acid has been extended. All three reactions are autocatalytic in nature. Oxidation of Fe(aq)2+ is practically complete at [HNO3] > 0.5 mol dm–3. In the case of the two other complexes the process is reversible; the conversion in the equilibrium state depends on the initial concentrations of reactants. Values of the equilibrium constants have been determined from experimental data and compared to those calculated from standard electrochemical potentials. The autocatalytic S-shaped kinetic curves have been measured by means of spectrophotometry and potentiometry and are characterised using the time coordinate (ti) and the slope (rmax) of the concentration vs. time curves at the inflection point. The general mechanism of autocatalytic nitric acid oxidations has been applied in order to fit single experimental curves and all concentration dependences. The validity of the proposed mechanism is supported by the agreement between experiments and the results of the numerical integration of differential rate equations derived from the mechanism. An interesting local maximum and minimum are found in the tivs.[Fe2+]0 curves both experimentally and numerically.