A general kinetic analysis of transport. Tests of the carrier model based on predicted relations among experimental parameters.

The analysis of transport kinetics has lacked both a unified treatment in which general rate equations are written entirely in terms of experimental parameters, and a convention by which these parameters may be designated in a concise yet immediately recognizable manner. Such a treatment is presented here in an easily accessible form, and a simple system of nomenclature is proposed resembling that in use in enzyme kinetics. The treatment is independent of assumptions about rate-limiting steps in transport, and applies to both active and facilitated systems, including obligatory exchange. A single substrate is characterized by twelve different parameters, only five of which are required in theory to calculate the others. If a second substrate is present on the trans side of the membrane there are six more parameters. All eighteen parameters are linked by multiple relationships which provide a complete set of rejection criteria for the generalized form of the mobile carrier. Relationships among parameters are also defined that give information on the rate-limiting steps in transport. Equations governing any individual experiment, involving only experimental parameters, are easily written out from the general expressions, for example under conditions of zero trans and infinite trans flux, equilibrium exchange, or competitive inhibition.     

A simple experimental approach to the Determination of carrier transport parameters for unlabeled substrate analogs

A method is described by which affinities and transport rates for unlabeled substrate analogs are readily determined, and which is based on the effect of an unlabeled analog upon the rate of transport of a labeled substrate present at a low concentration on the trans side of the membrane. The procedure is widely applicable since it does not depend on assumptions about rate-limiting steps and holds for both active and non-active systems. Here it is applied in an experimental study of the facilitated diffusion system for choline in erythrocytes, and it is shown that the transport parameters for a test substrate obtained by this method are the same as those found when the transport of the substrate is followed directly.     

Implications of the Alternating Access Model for Organic Anion Transporter Kinetics

Many transport proteins, including the clinically important organic anion transporters (OATs), appear to function via an “alternating access” mechanism. In analyzing the kinetics of these transporters, the terms K _m and V _max are often treated in the field as denoting, respectively, the affinity of the substrate for the transporter and the turnover (conformational switch) rate of the substrate–transporter complex. In fact, the expressions for both these parameters have very complex forms comprising multiple rate constants from conformational switch as well as association/dissociation steps in the cycling of the transporter and, therefore, do not have straightforward physical meanings. However, if the rapid equilibrium assumption is made (namely, that the association/dissociation steps occur far more rapidly than the conformational switch steps), these expressions become greatly simplified and their physical meaning clear, though still distinct from the conventional interpretations. V _max will be a function of not just the rate of substrate–transporter complex turnover but also the rate of the “return” conformational switch and will vary largely with the slower of these two steps (the rate-limiting step). K _m will be seen to be related to substrate affinity by a term that varies inversely with the substrate–transporter complex turnover rate, essentially because the greater this rate, the greater the extent to which transporters will be distributed in a conformation inaccessible to substrate. Here, an intuitive approach is presented to demonstrate these conclusions. The phenomena of trans-stimulation and trans-inhibition are discussed in the context of this analysis.     

The hypoxanthine transporter of Novikoff rat hepatoma cells exhibits directional symmetry and equal mobility when empty or substrate-loaded

The kinetics of hypoxanthine transport were measured in hypoxanthine phosphoribosyltransferase-deficient Novikoff cells by rapid kinetic techniques applying both zero-trans and equilibrium exchange protocols. The data indicate operation of a simple carrier with directional symmetry and equal mobility when substrate loaded and empty. Zero-trans influx and efflux were about equivalent and so were zero-trans influx and equilibrium exchange flux. The apparent Michaelis-Menten constant and maximum velocity were about 500 μM and 100 pmol/s per μl cell H₂O, respectively. The time courses of accumulation of radioactively labeled hypoxanthine at a concentration above the Michaelis-Menten constant differed noticeably in zero-trans and equilibrium exchange mode, but computer simulations showed that the difference is predicted by the symmetrical carrier model and does not reflect trans-stimulation.     

Presteady-state kinetics and carrier-mediated transport: A theoretical analysis

Summary Kinetic studies of cotransport mechanisms have so far been limited to the conventional steady-state approach which does not allow in general to resolve either isomerization or ratelimiting steps and to determine the values of the individual rate constants for the elementary reactions involved along a given transport pathway. Such questions can only be answered using presteady-state or relaxation experiments which, for technical reasons, have not yet been introduced into the field of cotransport kinetics. However, since two recent reports seem compatible with the observation of such transient kinetics, it would appear that theoretical studies are needed to evaluate the validity of such claims and to critically evaluate the expectations from a presteady-state approach. We thus report such a study which was performed on a simple four-state mechanism of carrier-mediated transport. The time-dependent equation for zero-trans substrate uptake was thus derived and then extended to models withp intermediary steps. It is concluded that (p-1) exponential terms will describe the approach to the steady state but that such equations have low analytical value since the parameters of the flux equation cannot be expressed in terms of the individual rate constants of the elementary reactions for models withp>5. We thus propose realistic simplifications based on the time-scale separation hypothesis which allows replacement of the rate constants of the rapid steps by their equilibrium constants, thereby reducing the complexity of the kinetic system. Assuming that only one relaxation can be observed, this treatment generates approximate models for which analytical expressions can easily be derived and simulated through computer modeling. When performed on the four-state mechanism of carrier-mediated transport, the simulations demonstrate the validity of the approximate solutions derived according to this hypothesis. Moreover, our approach clearly shows that presteady-state kinetics, should they become applicable to (co)transport kinetics, could be invaluable in determining more precise transport mechanisms.     

The trans-to-cis proline isomerization in E. coli Trx folding is accelerated by trans prolines

The conserved fold of thioredoxin (Trx)-like thiol/disulfide oxidoreductases contains an invariant cis-proline residue (P76 in Escherichia coli Trx) that is essential for Trx function and that is responsible for the folding rate-limiting step. E. coli Trx contains four additional prolines, which are all in the trans conformation in the native state. Notably, a recent study revealed that replacement of all four trans prolines in Trx by alanines (Trx variant Trx1P) further slowed the rate-limiting step 25-fold, indicating that one or several of the four trans prolines accelerate the trans-to-cis transition of P76 in Trx wild-type (wt). Here, we characterized the folding kinetics of Trx variants containing cisP76 and one or several of the natural trans prolines of Trx wt with NMR spectroscopy. First, we demonstrate that the isomerization reaction in Trx1P is a pure two-state transition between two distinct tertiary structures, in which all observed NMR resonances changes follow the same first-order kinetics. Moreover, we show that trans-P68 is the critical residue responsible for the faster folding of wt Trx relative to the single-proline (P76) variant Trx1P, as the two-proline variant Trx2P(P76P68) already folds seven times faster than Trx1P. trans-P34 also accelerates trans-to-cis isomerization of P76, albeit to a smaller extent. Overall, the results demonstrate that trans prolines can significantly modulate the kinetics of rate-limiting trans-to-cis proline isomerization in protein folding. Finally, we discuss possible mechanisms of acceleration and the potential significance of a protein-internal folding acceleration mechanism for Trx in a living cell.     

Trans effects in cell membrane transport

The efflux of a substrate from preloaded cells may be decelerated by an inhibitor in the external medium or accelerated by a compatible substrate in the external medium. The derivations of rate equations for the initial velocity of the zero-trans reaction, trans efflux inhibition, and accelerated exchange diffusion are described for steady state conditions. The rate constants making up the Michaelis constant for the trans inhibition reaction are the same as the corresponding parameters in the zero-trans reaction. The rate constants making up the Michaelis constant for the accelerated exchange reaction, however, are different from the corresponding parameters in the zero-trans reaction. The rate equation for trans inhibition shows that the velocity constant for recovery of the unloaded carrier may be determined with steady state experimental data. It is suggested that the observed recovery constant is independent of the substrates and trans inhibitors chosen for an assay of a particular carrier system. An experiment is briefly described to show a determination of a tentative value for the recovery constant of the unloaded nucleoside carrier in yeast cells and the apparent inhibition constant for a trans inhibitor.     

L-Phenylalanine ammonia-lyase (maize, potato, and Rhodotorula glutinis) Explaining the kinetic effects of substrate modification by linear free-energy relationships

The action of phenylalanine ammonia-lyase [EC 4.3.1.5] on a series of para-substituted l-phenylalanines has been investigated. Multiple linear regression analysis has been used to relate the logs of k_cat, K_m, and $\text{k}{}_{\mathrm{<mtext>cat</mtext>}}\text{K}{}_{\mathrm{m}}$ to substituent parameters for electron withdrawal, hydrophobic bonding, and size. The inhibitory action of the enantiomeric d-phenylalanines was also investigated. The results indicate that the rate-limiting step is not subsequent to the release of cinnamate from the enzyme. Explanations for the observed regression constants are discussed in terms of the influence of the substituent parameters on the dissociation constant for the bound substrate, the rate-limiting step, and intermediate steps such as the elimination process. The discussion utilizes a new theoretical treatment of the application of linear free-energy relationships to steady-state enzyme kinetics. It is shown that, in order to interpret structure-activity correlations in terms of rate and equilibrium constants for an unbranched catalytic sequence, a restricted model must apply. The reaction must have a single rate-limiting step so that quasi-equilibrium conditions prevail. In the QE-DS (quasi-equilibrium dominant-state) model, a single state of the enzyme-substrate complex is assumed to predominate. In the QE-FR (quasi-equilibrium fixed-ratio) model, changes in the enzyme substituents are assumed not to alter the ratio between the different forms of the enzyme-substrate complex prior to the rate-limiting step.     

Bacterial Sec Protein Transport Is Rate-limited by Precursor Length: A Single Turnover Study

An in vitro real-time single turnover assay for the Escherichia coli Sec transport system was developed based on fluorescence dequenching. This assay corrects for the fluorescence quenching that occurs when fluorescent precursor proteins are transported into the lumen of inverted membrane vesicles. We found that 1) the kinetics were well fit by a single exponential, even when the ATP concentration was rate-limiting; 2) ATP hydrolysis occurred during most of the observable reaction period; and 3) longer precursor proteins transported more slowly than shorter precursor proteins. If protein transport through the SecYEG pore is the rate-limiting step of transport, which seems likely, these conclusions argue against a model in which precursor movement through the SecYEG translocon is mechanically driven by a series of rate-limiting, discrete translocation steps that result from conformational cycling of the SecA ATPase. Instead, we propose that precursor movement results predominantly from Brownian motion and that the SecA ATPase regulates pore accessibility.     

Kinetics of Mixed Microbial Assemblages Enhance Removal of Highly Dilute Organic Substrates

Our experiments with selected organic substrates reveal that the rate-limiting process governing microbial degradation rates changes with substrate concentration, S, in such a manner that substrate removal is enhanced at lower values of S. This enhancement is the result of the dominance of very efficient systems for substrate removal at low substrate concentrations. The variability of dominant kinetic parameters over a range of S causes the kinetics of complex assemblages to be profoundly dissimilar to those of systems possessing a single set of kinetic parameters; these findings necessitate taking a new approach to predicting substrate removal rates over wide ranges of S.     

Platelet Transport Rates and Binding Kinetics at High Shear over a?Thrombus

Thrombus formation over a ruptured atherosclerotic plaque cap can occlude an artery with fatal consequences. We describe a computational model of platelet transport and binding to interpret rate-limiting steps seen in experimental thrombus formation over a collagen-coated stenosis. The model is used to compute shear rates in stenoses with growing boundaries. In the model, moving erythrocytes influence platelet transport based on shear-dependent enhanced diffusivity and a nonuniform platelet distribution. Adhesion is modeled as platelet-platelet binding kinetics. The results indicate that observed thrombus growth rates are limited by platelet transport to the wall for shear rates up to 6000 s⁻¹. Above 7000 s⁻¹, the thrombus growth rate is likely limited by binding kinetics (10⁻⁴ m/s). Thrombus growth computed from these rate-limiting steps match the thrombus location and occlusion times for experimental conditions if a lag time for platelet activation is included. Using fitted parameters, the model is then used to predict thrombus size and shape at a higher Reynolds number flow consistent with coronary artery disease.     

Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

In Silico Kinetic Study of the Glucose Transporter

Glucose transport in plasma membranes is the prototypic example of facilitated diffusion through biological membranes, and transport in erythrocytes is the most widely studied. One of the oldest and simplest models describing the kinetics of the transport reaction is that of alternating conformers, schematized in a cycle of four partial reactions where glucose binds and dissociates at two opposite steps, and the transporter undergoes transconformations at the other two opposite steps. The transport kinetics is entirely defined by the forward and backward rate constants of the partial reactions and the glucose and transporter concentrations at each side of the membrane, related by the law of mass action. We studied, in silico, the effect of modifications of the variables on the transient kinetics of the transport reaction. The simulations took into account thermodynamic constraints and provided results regarding initial velocities of transport, maximal velocities in different conditions, apparent influx and efflux affinities, and the turnover number of the transporter. The results are in the range of those experimentally reported. Maximal initial velocities are obtained when the affinities of the ligand for the transporter are the same at the extra- and intracellular binding sites and when the equilibrium constants of the transconformation steps are equal among them and equal to 1, independently of the obvious effect of the increase of the rate constant values. The results are well adjusted to Michaelis–Menten kinetics. A larger initial velocity for efflux than for uptake described in human erythrocytes is demonstrated in a model with the same dissociation constants at the outer and inner sites of the membrane. The larger velocities observed for uptake and efflux when transport occurs towards a glucose-containing trans side can also be reproduced with the alternating conformer model, depending on how transport velocities are measured.     

Voltage-dependent processes in the electroneutral amino acid exchanger ASCT2

Neutral amino acid exchange by the alanine serine cysteine transporter (ASCT)2 was reported to be electroneutral and coupled to the cotransport of one Na⁺ ion. The cotransported sodium ion carries positive charge. Therefore, it is possible that amino acid exchange is voltage dependent. However, little information is available on the electrical properties of the ASCT2 amino acid transport process. Here, we have used a combination of experimental and computational approaches to determine the details of the amino acid exchange mechanism of ASCT2. The [Na⁺] dependence of ASCT2-associated currents indicates that the Na⁺/amino acid stoichiometry is at least 2:1, with at least one sodium ion binding to the amino acid–free apo form of the transporter. When the substrate and two Na⁺ ions are bound, the valence of the transport domain is +0.81. Consistently, voltage steps applied to ASCT2 in the fully loaded configuration elicit transient currents that decay on a millisecond time scale. Alanine concentration jumps at the extracellular side of the membrane are followed by inwardly directed transient currents, indicative of translocation of net positive charge during exchange. Molecular dynamics simulations are consistent with these results and point to a sequential binding process in which one or two modulatory Na⁺ ions bind with high affinity to the empty transporter, followed by binding of the amino acid substrate and the subsequent binding of a final Na⁺ ion. Overall, our results are consistent with voltage-dependent amino acid exchange occurring on a millisecond time scale, the kinetics of which we predict with simulations. Despite some differences, transport mechanism and interaction with Na⁺ appear to be highly conserved between ASCT2 and the other members of the solute carrier 1 family, which transport acidic amino acids.     

On the functional organization of electron transport from plastoquinone to Photosystem I

Signal I, the EPR signal of P-700, induced by long flashes as well as the rate of linear electron transport are investigated at partial inhibition of electron transport in chloroplasts. Inhibition of plastoquinol oxidation by dibromothymoquinone and bathophenanthroline, inhibition of plastocyanin by KCN and HgCl₂, and inhibition by 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide are used to study a possible electron exchange between electron-transport chains after plastoquinone. (1) At partial inhibition of plastocyanin the reduction kinetics of P-700⁺ show a fast component comparable to that in control chloroplasts and a new slow component. The slow component indicates P-700⁺ which is not accessible to residual active plastocyanin under these conditions. We conclude that P-700 is reduced via complexed plastocyanin. (2) The rate of linear electron transport at continuous illumination decreases immediately when increasing amounts of plastocyanin are inhibited by KCN incubation. This is not consistent with an oxidation of cytochrome f by a mobile pool of plastocyanin with respect to the reaction rates of plastocyanin being more than an order of magnitude faster than the rate-limiting step of linear electron transport. It is evidence for a complex between the cytochrome b₆ - f complex and plastocyanin. The number of these complexes with active plastocyanin is concluded to control the rate-limiting plastoquinol oxidation. (3) Partial inhibition of the electron transfer between plastoquinone and cytochrome f by dibromothymoquinone and bathophenanthroline causes decelerated monophasic reduction of total P-700⁺. The P-700 kinetics indicate an electron transfer from the cytochrome b₆ - f complex to more than ten Photosystem I reaction center complexes. This cooperation is concluded to occur by lateral diffusion of both complexes in the membrane. (4) The proposed functional organization of electron transport from plastoquinone to P-700 in situ is supported by further kinetic details and is discussed in terms of the spatial distribution of the electron carriers in the thylakoid membrane.     

Theoretical analysis of intrinsic reaction kinetics and the behavior of immobilized enzymes system for steady-state conditions

Mathematical modeling of immobilized enzymes under different kinetics mechanism viz. simple Michaelis–Menten, uncompetitive substrate inhibition, total competitive product inhibition, total non-competitive product inhibition and reversible Michaelis–Menten reaction are discussed. These five kinetic models are based on reaction diffusion equations containing non-linear terms related to Michaelis–Menten kinetics of the enzymatic reaction. Modified Adomian decomposition method is employed to derive the general analytical expressions of substrate and product concentration for all these five mechanisms for all possible values of the parameters Φ_S (Thiele modulus for substrate), Φ_P (Thiele modulus for product) and α (dimensionless inhibition degree). Also we have presented the general analytical expressions for the mean integrated effectiveness factor for all values of parameters. Analytical results are compared with the numerical results and also with the limiting case results, which are found to be good in agreement.     

Kinetics of glucose transport in human erythrocytes: zero-trans efflux and infinite-trans efflux at 0°C

The kinetic features of glucose transport in human erythrocytes have been the subject of many studies, but no model is consistent with both the kinetic observations and the characteristics of the purified transporter. In order to reevaluate some of the kinetic features, initial rate measurements were performed at 0°C. The following kinetic parameters were obtained for fresh blood: zero-trans efflux K_m = 3.4 mM, V_max = 5.5 mM/min; infinite-trans efflux K_m = 8.7 mM, V_max = 28 mM/min. For outdated blood, somewhat different parameters were obtained: zero-trans efflux K_m = 2.7 mM, V_max = 2.4 mM/min; infinite-trans efflux K_m = 19 mM, V_max = 23 mM/min. The K_m values for fresh blood differ from the previously reported values of 16 mM and 3.4 mM for zero-trans and infinite-trans efflux, respectively (Baker, G.F. and Naftalin, R.J. (1979) Biochim. Biophys. Acta 550, 474–484). The use of 50 mM galactose rather than 100 mM glucose as the infinite-trans sugar produced no change in the infinite-trans efflux K_m values but somewhat lower V_max values. Simulations indicate that initial rates were closely approximated by the experimental conditions. The observed time courses of efflux are inconsistent with a model involving rate-limiting dissociation of glucose from hemoglobin (Naftalin, R.J., Smith, P.M. and Roselaar, S.E. (1985) Biochim. Biophys. Acta 820, 235–249). The results presented here support the adequacy of the carrier model to account for the kinetics.     

Facilitated diffusion with consecutive reaction: optimal carrier affinity

The interplay between facilitated diffusion of a substrate through a membrane and a consecutive enzymic reaction, both of which follow Michaelis-Menten kinetics, has been theoretically investigated and the effect of the kinetic and transport parameters on the rate of substrate uptake is graphically illustrated. At steady state two characteristic features of the system have been identified. First, the substrate concentration at the internal enzymic side of the membrane cannot exceed a given value even at much higher external substrate concentrations. Second, the uptake rate is maximum at a given value of K_T, the kinetic parameter of the transport system that expresses the reciprocal carrier affinity of the substrate. The optimum value of K_T is approximately equal to the external substrate concentration. This particular dependence of the uptake rate on the carrier affinity is expected to play an important role in hormonal regulation.     

The transport of chloroquine across human erythrocyte membranes is mediated by a simple symmetric carrier

The kinetic properties of the mediated transport of chloroquine in human erythrocytes are investigated. The high rates of translocation across the cell membrane and high adsorbance properties to glass surfaces have led to the development of new techniques for measuring initial rates of transport. Three different methodological procedures are used to accomplish a complete kinetic characterization of the system. All measurements were done at 25°C. Under zero-trans conditions the system displays complete symmetry, the Michaelis constants being 39.2±2.4 μM for influx and 36.6±5.6 μM for efflux. The respective maximal velocities are 206.4±36.0 μM·min^?1 and 190.0±7.8 μM·min^?1. Under equilibrium-exchange conditions the Michaelis constant is 108.6±15.6 μM and the maximal velocity is 630.3±50.4 μM·min^?1. This 3-fold increase in both K and V over the zero-trans values indicates that the rate-limiting step in the transport of chloroquine is the movement of the unloaded carrier. The kinetic data are consistent with the prediction of a simple carrier model.     

The relationship between active transport and the exchange diffusion effect

Dependences of unidirectional ionic fluxes across biological membranes on the trans concentrations of the same ion, commonly described as exchange diffusion, and the association of this phenomenon with active transport, are noted. It is suggested that this effect could arise as a result of energetic coupling between the movement of ions conveyed in each direction by the pump if the latter operates near thermodynamic equilibrium and if the rate of the energizing reactions are restricted. This hypothesis is supported by an analysis in which the transport step and the energizing reactions are separated and described according to the laws of chemical kinetics. A likely cause for such restriction of the maximum rate of energy supply is shown to lie in evolutionary optimization of the efficiency of active transport if the energizing reaction is not perfectly coupled. Similar optimization will produce gross ionic fluxes large compared with the net flux, especially if the transport step approaches perfect coupling, when restriction of the rate of energy supply will cause a large exchange diffusion effect. The range of validity of the analysis is examined with particular reference to the ionic exchanges between osmoregulating animals and their surroundings.