Kinetics of the lactate dehydrogenase reaction in high-viscosity media

The effect of the medium viscosity on kinetics parameters of lactate dehydrogenase reaction was studied. The viscosity increase results in a sharp decline in the catalytic rate for both the pyruvate reduction and lactate oxidation reactions. It is shown that the catalytic step and its associated conformational motions is the only step which is considerably retarded when the viscosity increases. The reaction is not sensitive to changes in the dielectric properties of the medium. An inverse power function observed between the rate constant and viscosity cannot be explained by the theory of absolute reaction rates. However, it can easily be interpreted on the basis of the Kramers theory dealing with the transition over the activation barrier as a diffusional motion in the field of random forces. The influence of the medium's viscosity on the kinetic parameters indicates the existence of strong coupling between the dynamics of the solvent and the conformational motions of the protein molecule, which are correlated with the catalytic step.     

Effect of solution viscosity on intramolecular electron transfer in sulfite oxidase

Our previous studies have shown that the rate constant for intramolecular electron transfer (IET) between the heme and molybdenum centers of chicken liver sulfite oxidase varies from approximately 20 to 1400 s(-1) depending upon reaction conditions [Pacheco, A., Hazzard, J. T., Tollin, G., and Enemark, J. H. (1999) J. Biol. Inorg. Chem. 4, 390-401]. These two centers are linked by a flexible polypeptide loop, suggesting that conformational changes, which alter the Mo-Fe distance, may play an important role in the observed IET rates. In this study, we have investigated IET in sulfite oxidase using laser flash photolysis as a function of solution viscosity. The solution viscosity was varied over the range of 1.0-2.0 cP by addition of either polyethylene glycol 400 or sucrose. In the presence of either viscosogen, an appreciable decrease in the IET rate constant value is observed with an increase in the solvent viscosity. The IET rate constant exhibits a linear dependence on the negative 0.7th power of the viscosity. Steady-state kinetics and EPR experiments are consistent with the interpretation that viscosity, and not other properties of the added viscosogens, is responsible for the dependence of IET rates on the solvent composition. The results are consistent with the role of conformational changes on IET in sulfite oxidase, which helps to clarify the inconsistency between the large rate constant for IET between the Mo and Fe centers and the long distance (approximately 32 A) between these two metal centers observed in the crystal structure [Kisker, C., Schindelin, H., Pacheco, A., Wehbi, W., Garnett, R. M., Rajagopalan, K. V., Enemark, J. H., and Rees, D. C. (1997) Cell 91, 973-983].     

Viscosity-dependent structural fluctuations in enzyme catalysis.

The effect of viscosity on the rate of catalysis of carboxypeptidase A has been tested. By use of the tripeptide carbobenzoxy-l-alanyl-l-alanyl-l-alanine [Z(L-Ala)3] as substrate, it was shown that most of the effect on the hydrolysis rate caused by the presence of 30 or 40% methanol or glycerol in aqueous solution can be ascribed to a contribution of viscosity to the catalytic rate constant, kcat. Arrhenius plots of kcat in 30 and 40% glycerol or methanol are linear and almost parallel. When the rate constants are "corrected" for the viscosity of various media, the difference between the various Arrhenius plots is considerably reduced; it vanishes, within experimental error, when the effect of the dielectric constant of the solutions is taken into account as well. It is proposed that the viscosity of the medium can influence the rate-limiting step of the enzymic reaction, which is the rate of transitions over the energy barrier preceding product formation. According to the suggested mechanism, the enzyme--substrate complex can overcome this energy barrier by viscosity-dependent structural fluctuations. The quantitative agreement between the theory and the experimental results suggests that (a) due to the temperature dependence of the viscosity of the solution, the potential energy barrier of the reaction is about 5 kcal/mol lower than the observed activation energy and (b) information about the structural flexibility of the complex can be obtained by kinetic measurements.     

Conformational fluctuations and protein reactivity. Determination of the rate-constant spectrum and consequences in elementary biochemical processes

When a protein's active site happens to be strongly coupled with the protein structure, the rate constant of the reaction may eventually be modulated by the conformational fluctuations. Evidence for this effect has long been provided by extensive flash photolysis investigations of liganded hemoproteins and more recently of the non-heme respiratory protein hemerythrin in hydro-organic solvents. Within a given protein conformational substate, an elementary reaction step is characterized by one single free energy barrier and by a first-order rate constant, k, which changes with temperature according to an Arrhenius law. At physiological temperature and low viscosity, ultrafast conformational relaxation causes efficient averaging of the reaction rates and the protein displays exponential kinetics with an average rate constant (k). Under sufficiently general conditions, it can be shown that (k) also follows a simple Arrhenius law with 'effective' values of the pre-exponential factor Aeff and activation enthalpy Heff. It is found that Aeff strongly depends on the overall shape of the rate constant distribution and that Heff actually corresponds to the lower limit of the enthalpy of activation, i.e. the value associated with the highest possible reaction rate. The underlying distribution of rate constants can be reconstructed from a set of experiments in which the kinetics depart from an exponential, i.e. at low temperature and high viscosity. The most probable distribution of exponentials consistent with the observed kinetics of the geminate recombinations of oxygen with photodissociated hemerythrin has been determined by using a new approach, known as the maximum entropy method. The results are consistent with a single pre-exponential value and a distributed enthalpy spectrum. As expected, Heff does not coincide either with the most probable nor with the average value of the enthalpy. The most salient findings are that the probability for any protein molecule to have an enthalpy of activation equal to the effective value Heff vanishes and that Aeff differs by nearly three orders of magnitude from the true value A0. Biochemical reaction rates are actually average values, since protein reactions are measured under physiological conditions, where conformational relaxation is always fast. Our understanding of the significance of Aeff and Heff is therefore entirely dependent on the knowledge of the distribution function of the rate constants. In particular, enthalpy and entropy terms of similar reactions performed by different proteins cannot be compared as long as the distribution of the rate constants remains unknown.     

Viscosity dependence of ethidium-DNA intercalation kinetics

The kinetics of ethidium intercalation into double-stranded poly[d(G-C)] were investigated by use of repetitive pressure-jump chemical relaxation at 20 degrees C in low ionic strength (0.1 M NaCl) aqueous buffers containing either glycerol or methanol. The viscosity of the various solvents differed by more than an order of magnitude while other physical properties (e.g., dielectric constant) remained approximately constant. The single-reciprocal kinetic relaxation time (tau -1) increases linearly with DNA concentration. The observed association rate constant is lower in all organic-aqueous mixtures than in water and is inversely proportional to the viscosity. These results provide evidence for an additional step in the intercalation mechanism which is identified as an obligatory DNA conformational change preceding ethidium intercalation. From the data presented, the equilibrium constant of this local conformational change is approximately 10(-3), i.e., greatly favoring the structure incapable of intercalation. The corresponding kinetics were not directly determined; however, in order to be consistent with all of the data the forward and/or reverse rate constants of the conformational change must be larger than the rate of the intercalation reaction. Thus, it is proposed that the rate of the conformational change back to the nonintercalating B-DNA structure is greater than approximately 500 s-1, implying a rate of opening greater than approximately 0.5 s-1, in agreement with other hydrogen exchange and NMR data. The observed overall rate constant for the dissociation of ethidium is inversely proportional to the solvent density, possibly reflecting a dependence on the solvent free volume. The overall volume change of intercalation is less negative in the organic-aqueous solvent mixtures than in water.     

Nectar feeding by the hovering hawk moth Macroglossum stellatarum: intake rate as a function of viscosity and concentration of sucrose solutions

Although nectar feeding in insects has long been studied, the knowledge of the effect of nectar energy content on the ingestion dynamics separately from the viscosity of the fluid is very limited. To determine the effects of both factors on the feeding behavior of the hovering hawk moth Macroglossum stellatarum, we developed a method to independently manipulate sucrose concentrations and viscosity. The intake rate was analyzed as a function of sucrose concentration, the concentration at constant viscosity (kept constant by adding tylose, an inert polysaccharide), and of the different viscosities of a 30% weight/weight (w/w) sucrose solution (by adding different amounts of tylose). By increasing the concentration, and thus its viscosity, the solution intake rate (in microl s (-1)) decreased beyond a 20% w/w sucrose solution. For a 30% sucrose solution, the intake rate decreased with increasing viscosity. At constant viscosity, the solution intake rate decreased beyond a 30% w/w sucrose solution. However, if we considered the quantity of sucrose ingested per unit time (sucrose intake rate), the same fitted maximum was attained for both series in which the sucrose concentration changed (33.6% w/w). Results suggest that the gustatory input affects the dynamics of fluid ingestion separately from the viscosity.     

Effect of solution viscosity on intraprotein electron transfer between the FMN and heme domains in inducible nitric oxide synthase

The FMN-heme intraprotein electron transfer (IET) kinetics in a human inducible NOS (iNOS) oxygenase/FMN construct were determined by laser flash photolysis as a function of solution viscosity (1.0-3.0 cP). In the presence of ethylene glycol or sucrose, an appreciable decrease in the IET rate constant value was observed with an increase in the solution viscosity. The IET rate constant is inversely proportional to the viscosity for both viscosogens. This demonstrates that viscosity, and not other properties of the added viscosogens, causes the dependence of IET rates on the solvent concentration. The IET kinetics results indicate that the FMN-heme IET in iNOS is gated by a large conformational change of the FMN domain. The kinetics and NOS flavin fluorescence results together indicate that the docked FMN/heme state is populated transiently.     

Investigations into the rheological characteristics of bovine amniotic fluid

The rheological characteristics of bovine amniotic fluid have been studied at different shear rates. The viscosity of bovine amniotic fluid at 20°C was found to increase with time at a constant low shear rate during the measurement. Additionally, the viscosity was observed to decrease with increasing shear rate, indicating that a shear thinning behaviour of the fluid was occurring. The log-log plot of shear stress versus shear rate yielded a straight line, consistent with non-Newtonian behaviour of the fluid and characteristic of pseudoplastic liquids. The data of shear stress versus shear rate could be represented by a power law model. The treatment of amniotic fluid with cetylpyridinium chloride (CPC) resulted in the precipitation of a mixture of components, including complex sulphated polysaccharides and extracellular proteoglycans, with the viscosity of the resulting liquid similar to that of water at 20°C. In addition, the viscosity of the CPC-pretreated fluid did not increase with time at constant shear rate and remained constant with the increase in shear rate. The apparent increase in viscosity with time and the shear thinning behaviour of the amniotic fluid can thus be attributed to pseudoplastic liquid behaviour associated with the presence of structurally complex polysaccharides and extracellular proteoglycans. The implications of this fluid viscosity behaviour are discussed in terms of their impact on the operation of packed or expanded (fluidized) chromatographic bed systems when amniotic fluid biofeed-stocks are used as a source of commercially important proteins.     

Kinetics and thermodynamics of isothermal seed germination.

We have been successful in building a mathematical model that fits both the germination rate and the total number of seeds that germinate as a function of time. This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes play an important role. The model gives values for the initial concentration of two enzymes. From these initial enzyme concentrations an equilibrium constant is calculated and the thermodynamic model gives the change in enthalpy, entropy, free energy and the activation energy. A plot of the natural logarithm of the equilibrium constant as a function of the reciprocal of the absolute temperature gives two straight lines. The change of enthalpy for the process below 33 °C differs considerably to the change above 33 °C. The free energy as a function of the absolute temperature gives a straight line from which the change in entropy is calculated. The activation energy is determined from the slope of the natural logarithm of the rate constant as a function of the reciprocal of the absolute temperature.     

Dynamic aspect of kinetics of reaction catalyzed by lactate dehydrogenase

The influence of solvent viscosity on the kinetic parameters of the pyruvate reduction reaction catalyzed by lactate dehydrogenase has been investigated. The viscosity was adjusted by sucrose and glycerol solutions at concentrations from 0 to 44% and from 0 to 63%, respectively. The reaction rate decreased abruptly with an increase in viscosity. The study of different reaction stages (enzyme-substrate complex formation, catalysis, inhibitory complex decomposition, competitive inhibition by chlorine ions) revealed that the catalysis (and the related conformational changes) is the only stage (of the above mentioned) that depends markedly on the solvent viscosity. The reaction is insensitive to the changes in the dielectric properties of the solution induced by the addition of alcohols and dioxane. The observed power dependence of the rate constant on viscosity is explained in terms of Kramer's theory which considers the proton transition through the activation barrier to be a diffusion in the field of random forces. The influence of solvent viscosity on enzymic kinetics indicates a direct relation between solvent dynamics and relevant protein conformational movements.     

Effects of solvent viscosity and different guanidine salts on the kinetics of ribonuclease A chain folding

The kinetics of folding of the two forms of unfolded ribonuclease A have been measured as a function of solvent viscosity by adding either glycerol or sucrose. The aim is to find out if either reaction is rate limited by segmental motion whose rate depends on external friction. The fast folding reaction (U₂ ? N) is known to be the direct folding process, and the slow folding reaction (U₁ ? N) is known to be rate limited by an interconversion between two forms (U₁ ? U₂) which are present after unfolding in strongly denaturing conditions. No dependence on solvent viscosity is found, either for the direct folding reaction or for the interconversion reaction. Each folding reaction has also been tested to see if its rate depends on the concentration of one or more partly folded intermediates, by adding denaturants destabilize any partly folded structures. Different guanidine salts are used as denaturants to vary the denaturing effectiveness of the salt while holding the guanidinium ion concentration constant. The rates both of the direct folding reaction and of the interconversion reaction decrease in relation to the denaturing effectiveness of the salt. However, there is a basic difference between the responses of the fast and slow folding reactions to low concentrations of denaturants. Although each folding reaction produces native protein, there is an 800-fold decrease in the rate of the fast folding reaction in 1M guanidine thiocyanate and only a 13-fold decrease in the rate of the slow folding reaction. This is consistent with the fast reaction being the direct folding process and the slow reaction being rate limited by the initial conversion of the slowrefolding to the fast-refolding form. Both the lack of viscosity dependence and the effects of denaturants indicate that the formation of structure is rate limiting in the direct folding reaction, U₂ ? N. The failure to find a viscosity dependence for the interconversion reaction, U₁ ? U₂, indicates that in this reaction also friction-limited segmental motion is not the rate-limiting process. Since the U₁ ? U₂ interconversion still occurs when the polypeptide chain is completely unfolded, the surprising result is that its rate in refolding conditions depends significantly on a reaction intermediate which is “denatured” by guanidine salts.     

Kinetics of pressure inactivation at subzero and elevated temperature of lipoxygenase in crude green bean (Phaseolus vulgaris L.) extract

Lipoxygenase (LOX) in crude green bean extract was irreversibly inactivated by pressure treatments combined with subzero or elevated temperature. LOX inactivation was described accurately assuming a first-order reaction. In the entire pressure-temperature domain studied (200 to 700 MPa and -10 to 60 degrees C), an increase in pressure at constant temperature enhanced the LOX inactivation rate, whereas at constant pressure, an increase in reaction rate was obtained by either increasing or decreasing temperature at 20 degrees C. At elevated pressure, LOX exhibited the greatest stability around 20 degrees C. Also the pressure dependence of the inactivation rate constants for LOX was the highest around 20 degrees C. On the basis of the estimated LOX inactivation rate constants, an iso-rate contour diagram as a function of pressure and temperature was constructed, and an empirical mathematical model describing the combined pressure-temperature dependence of the LOX inactivation rate constants was formulated.     

Dissolving carbon dioxide in high viscous substrates to accelerate biocatalytic reactions

Solvent free biotransformation of polyglycerol-3 and lauric acid yields polyglycerol-3-laurate and water. This conversion can be catalyzed by Novozym 435. However, the performance is limited by the viscosity of polyglycerol as well as of polyglycerol-3-laurate. A decrease of viscosity by increasing reaction temperature is only possible in a certain temperature range because of the limited stability of the applied enzyme. By dissolving high dense carbon dioxide into the reaction system the viscosity could be reduced, keeping the temperature at an acceptable level at the same time. Thus the reaction rate was increased by a factor of 4 while working at a pressure of 280 bar and 60°C.     

Diffusive motions control the folding and unfolding kinetics of the apomyoglobin pH 4 molten globule intermediate

The sperm whale apomyoglobin pH 4 folding intermediate exists in two forms, Ia and Ib, that mimic transient kinetic intermediates in the folding of the native protein at pH 6. To characterize the nature of the kinetic barrier that controls the formation of the earliest intermediate Ia, we have investigated the effects of small viscogenic cosolvents on its folding and unfolding kinetics. The kinetics are measurable by stopped-flow fluorescence and follow a cooperative two-state model in the absence and presence of cosolvents. Small cosolvents stabilize Ia, but, by applying the isostability test to separate the viscogenic effect of the cosolvent from its stabilizing effect, we found that, in both folding and unfolding conditions, the apparent rate constant decreases when solvent viscosity increases. The unitary inverse dependence of the apparent rate constant on solvent viscosity indicates a diffusion-controlled reaction. This result is consistent with the hypothesis that folding of the apomyoglobin pH 4 intermediate obeys a diffusion-collision model. Additionally, the temperature dependence of the reaction rate at constant viscosity indicates that the formation of Ia is also controlled by an energy barrier. Linear free energy relationships show that the transition state of the U <==> Ia reaction is compact and buries 45% of the surface area that is buried in native apomyoglobin. We conclude that the transition state of the U <==> Ia reaction resembles that for the formation of native proteins; namely, it is dry and its compactness is closer to that of the folded (Ia) form than of the unfolded form.     

Site directed mutagenesis at position 193 of human trypsin 4 alters the rate of conformational change during activation: role of local internal viscosity in protein dynamics

Upon activation of trypsinogen four peptide segments flanked by hinge glycine residues undergo conformational changes. To test whether the degree of conformational freedom of hinge regions affects the rate of activation, we introduced amino acid side chains of different characters at one of the hinges (position 193) and studied their effects on the rate constant of the conformational change. This structural rearrangement leading to activation was triggered by a pH-jump and monitored by intrinsic fluorescence change in the stopped-flow apparatus. We found that an increase in the size of the side chain at position 193 is associated with the decrease of the reaction rate constant. To analyze the thermodynamics of the reaction, temperature dependence of the reaction rate constants was examined in a wide temperature range (5-60 degrees C) using a novel temperature-jump/stopped-flow apparatus developed in our laboratory. Our data show that the mutations do not affect the activation energy (the exponential term) of the reaction, but they significantly alter the preexponential term of the Arrhenius equation. The effect of solvent viscosity on the rate constants of the conformational change during activation of the wild type enzyme and its R193G and R193A mutants was determined and evaluated on the basis of Kramers' theory. Based on this we propose that the reaction rate of this conformational transition is regulated by the internal molecular friction, which can be specifically modulated by mutagenesis in the hinge region.     

Temperature dependence of the viscoelastic recovery of red cell membrane.

The time-dependent recovery of an elongated red cell is studied as a function of temperature. Before release, the elongated cell is in static equilibrium where external forces are balanced by surface elastic force resultants. Upon release, the cell recovers its initial shape with a time-dependent exponential behavior characteristic of a viscoelastic solid material undergoing large ("finite") deformation. The recovery process is characterized by a time constant, tc, that decreases from approximately 0.27 s at 6 degrees C to 0.06 s at 37 degrees C. From this measurement of the time constant and an independent measurement of the shear modulus of surface elasticity for red cell membrane, the value for the membrane surface viscosity as a function of temperature can be calculated.     

Hydrodynamic shear breakage of DNA

The rate of breakage of duplex DNA molecules by laminar flow through a capillary has been studied. For λb2b5c DNA (molecular wt., M = 25 × 10⁶) the point at which breakage occurs is normally distributed around the center of the molecule with a standard deviation of 12.5% of the molecular length. At constant shear stress or shear rate, the breakage rate is independent of ionic strength. Thus, shear induced local denaturation is not a rate limiting, preliminary step in breakage. In experiments at constant temperature with varying solvent viscosity (controlled by added sucrose) the breakage rate is a function of shear rate, not of shear stress. The rate of opening of hydrogenbonded circles into linear molecules by hydrodynamic shear is also shown to be a function of shear rate and not of shear stress. The breakage rate at constant shear rate is not greatly dependent on temperature. The shear rate required to achieve breakage is inversely proportional to M^1,2. The breakage rate constant, k varies as a very high power of the shear rate; at 25°C, d In k/d In Gm ～ 15; at 10°C, d In k/d In Gm ～ 26, where Gm is the maximum shear rate at the capillary wall. The unexpected result that breakage rate is mainly dependent on shear rate, not shear stress, supports a model in which the DNA molecule is distorted with a driving force which depends on the hydrodynamic shear stress, ηG, but the rate limiting step is segment diffusion into a highly extended configuration. The characteristic time to achieve this configuration is proportional to solvent viscosity, η, hence the breakage rate is dependent on ηG/η or G, the shear rate.     

Elongation of actin filaments is a diffusion-limited reaction at the barbed end and is accelerated by inert macromolecules

We used a fluorescence method to measure the rate constants for the elongation of pyrene-labeled actin filaments in a number of different solvents. The absolute values of the rate constants were established by electron microscopy. Using glycerol, sucrose, or ethylene glycol to vary the solution viscosity, the association rate constant (k+) was 10(7) M-1 s-1 viscosity-1 (in centipoise). Consequently, plots of 1/k+ versus viscosity are linear and extrapolate to near the origin as expected for a diffusion-limited reaction where the rate constant approaches infinity at zero viscosity. By electron microscopy, we found that this inhibitory effect of glycerol is almost entirely at the fast growing, barbed end. For the pointed end, plots of 1/k+ versus viscosity extrapolate to a maximum rate of about 10(6) M-1 s-1 at zero viscosity, so that elongation at the pointed is not limited by diffusion. In contrast to these small molecules, polyethylene glycol, dextran, and ovalbumin all cause a concentration (and therefore viscosity)-dependent increase in k+. At any given viscosity, their effects are similar to each other. For example, at 3 centipoise, k+ = 2.2 X 10(7) M-1 s-1. We presume that this is due to an excluded volume effect that causes an increase in the thermodynamic activity of the actin. If the proteins in the cytoplasmic matrix have a similar effect, the association reactions of actin in cells may be much faster than expected from experiments done in dilute buffers.     

Kinetics of acrylamide formation/elimination reactions as affected by water activity

The influence of water activity on the kinetics of acrylamide formation and elimination reaction was investigated using low-moisture equimolar asparagine-glucose model systems, which were heated at temperatures between 120 and 200 degrees C for variable heating times. To determine the water content corresponding to the water activities tested, a sorption moisture isotherm was constructed experimentally. The acrylamide concentrations measured at different water activities could be modeled on the basis of a reaction scheme including not only acrylamide formation and elimination reactions but also an alternative Maillard reaction between both reactants. The corresponding rate constants and activation energies were estimated using nonlinear regression analysis. Whereas the rate constant for acrylamide formation varied only slightly with the initial water activity of the model system, the elimination rate constant showed a clear minimum around a water activity of 0.82. The opposite trend, namely, a maximum at a water activity of 0.82, was found for the Maillard reaction rate constant as a function of water activity, which confirms data from literature. The activation energies for the different reactions changed in a comparable way as the corresponding rate constant with water activity.     

Kinetics of phloretin binding to phosphatidylcholine vesicle membranes

The submillisecond kinetics for phloretin binding to unilamellar phosphatidylcholine (PC) vesicles was investigated using the temperature-jump technique. Spectrophotometric studies of the equilibrium binding performed at 328 nm demonstrated that phloretin binds to a single set of independent, equivalent sites on the vesicle with a dissociation constant of 8.0 microM and a lipid/site ratio of 4.0. The temperature of the phloretin-vesicle solution was jumped by 4 degrees C within 4 microseconds producing a monoexponential, concentration-dependent relaxation process with time constants in the 30--200-microseconds time range. An analysis of the concentration dependence of relaxation time constants at pH 7.30 and 24 degrees C yielded a binding rate constant of 2.7 X 10(8) M-1 s-1 and an unbinding constant of 2,900 s-1; approximately 66 percent of total binding sites are exposed at the outer vesicle surface. The value of the binding rate constant and three additional observations suggest that the binding kinetics are diffusion limited. The phloretin analogue, naringenin, which has a diffusion coefficient similar to phloretin yet a dissociation constant equal to 24 microM, bound to PC vesicle with the same rate constant as phloretin did. In addition, the phloretin-PC system was studied in buffers made one to six times more viscous than water by addition of sucrose or glycerol to the differ. The equilibrium affinity for phloretin binding to PC vesicles is independent of viscosity, yet the binding rate constant decreases with the expected dependence (kappa binding alpha 1/viscosity) for diffusion-limited processes. Thus, the binding rate constant is not altered by differences in binding affinity, yet depends upon the diffusion coefficient in buffer. Finally, studies of the pH dependence of the binding rate constant showed a dependence (kappa binding alpha [1 + 10pH-pK]) consistent with the diffusion-limited binding of a weak acid.