Time dependent rate of diffusion-influenced ligand binding to receptors on cell surfaces.

The theory of the kinetics of binding of ligands to a sphere partially covered by receptors is extended to provide the full time dependence of the reactive flux. The ligands diffuse to the receptors; the receptors are either fully or partially absorbing. The total flux into the sphere with many receptors is expressed analytically in terms of the flux into a single isolated receptor on the sphere. At steady state, the Berg-Purcell formula is generalized to the case where the binding to a single receptor is only partially diffusion controlled. At short times, the receptors behave independently and the total flux is the sum of the fluxes to the isolated receptors.     

Propagators and time-dependent diffusion coefficients for anomalous diffusion

Complex diffusive dynamics are often observed when one is investigating the mobility of macromolecules in living cells and other complex environments, yet the underlying physical or chemical causes of anomalous diffusion are often not fully understood and are thus a topic of ongoing research interest. Theoretical models capturing anomalous dynamics are widely used to analyze mobility data from fluorescence correlation spectroscopy and other experimental measurements, yet there is significant confusion regarding these models because published versions are not entirely consistent and in some cases do not appear to satisfy the diffusion equation. Further confusion is introduced through variations in how fitting parameters are reported. A clear definition of fitting parameters and their physical significance is essential for accurate interpretation of experimental data and comparison of results from different studies acquired under varied experimental conditions. This article aims to clarify the physical meaning of the time-dependent diffusion coefficients associated with commonly used fitting models to facilitate their use for investigating the underlying causes of anomalous diffusion. We discuss a propagator for anomalous diffusion that captures the power law dependence of the mean-square displacement and can be shown to rigorously satisfy the extended diffusion equation provided one correctly defines the time-dependent diffusion coefficient. We also clarify explicitly the relation between the time-dependent diffusion coefficient and fitting parameters in fluorescence correlation spectroscopy.     

Macromolecular assemblages. Theory and simulation

A selection of World Wide Web sites relevant to papers published in this issue of Current Opinion in Structural Biology.     

The equilibrium assumption is valid for the kinetic treatment of most time-dependent protein-modification reactions.

To facilitate mechanistic interpretation of the kinetics of time-dependent inhibition of enzymes and of similar protein modification reactions, it is important to know when the equilibrium assumption may be applied to the model: formula: (see text). The conventional criterion of quasi-equilibrium, k + 2 less than k-1, is not always easy to assess, particularly when k + 2 cannot be separately determined. It is demonstrated that the condition k + 2 less than k-1 is necessarily true, however, when the value of the apparent second-order rate constant for the modification reaction is much smaller than the value of k + 1. Since k + 1 is commonly at least 10(7)M-1.S-1 for substrates, it is probable that the equilibrium assumption may be properly applied to most irreversible inhibitions and modification reactions.     

Analysis of proton exchange kinetics with time-dependent exchange rate

Mass spectrometry is used to probe the kinetics of hydrogen–deuterium exchange in lysozyme in pH 5, 6 and 7.4. An analysis based on a Verhulst growth model is proposed and effectively applied to the kinetics of the hydrogen exchange. The data are described by a power-like function which is based on a time-dependence of the exchange rate. Experimental data ranging over many time scales is considered and accurate fits of a power-like function are obtained. Results of fittings show correlation between faster hydrogen–deuterium exchange and increase of pH. Furthermore a model is presented that discriminates between easily exchangeable hydrogens (located in close proximity to the protein surface) and those protected from the exchange (located in the protein interior). A possible interpretation of the model and its biological significance are discussed.     

Integrated rate equations for enzyme-catalysed first-order and second-order reactions.

Generalized rate equations covering all mechanisms giving hyperbolic initial-rate kinetics with stoichiometry A in equilibrium P, A in equilibrium P + Q, A + B in equilibrium P and A + B in equilibrium P + Q were integrated. The results are regular and reasonably economical.     

Alternative nonlinear model for estimating second-order rate coefficients for biodegradation.

A modification of the second-order model for biodegradation was derived, applied to an example data set, and shown to be superior for describing the anaerobic biodegradation of p-cresol by an enriched bacterial consortium. The modified model circumvents the no-growth assumption implicit in the use of the second-order rate equation, but still requires the assumption of first-order kinetics over the course of substrate depletion. Violation of the no-growth assumption is particularly important since overestimates of the pseudo-first-order rate coefficient lead to underestimates of the time required for the removal of a xenobiotic chemical from a contaminated environment. Our calculations show that the errors introduced into the pseudo-first-order rate coefficient (and the resulting estimates of the second-order rate coefficient) approach 100% if one doubling occurs in activity over the course of substrate depletion. For an exemplary data set, use of a first-order model resulted in a 100% overestimate of the first-order decay coefficient, which would in turn lead to a corresponding overestimate of the second-order rate coefficient. The modified model we describe is a potential alternative to the pseudo-first-order model for the routine estimation of second-order rate coefficients.     

Using time-dependent models to investigate body condition and growth rate of the giant gartersnake

Identifying links between phenotypic attributes and fitness is a primary goal of reproductive ecology. Differences in within-year patterns of body condition between sexes of gartersnakes in relation to reproduction and growth are not fully understood. We conducted an 11-year field study of body condition and growth rate of the giant gartersnake Thamnophis gigas across 13 study areas in the Central Valley of California, USA. We developed a priori mixed effects models of body condition index (BCI), which included covariates of time, sex and snout–vent length and reported the best-approximating models using an information theoretic approach. Also, we developed models of growth rate index (GRI) using covariates of sex and periods based on reproductive behavior. The largest difference in BCI between sexes, as predicted by a non-linear (cubic) time model, occurred during the mating period when female body condition (0.014±0.001  se ) was substantially greater than males (−0.027±0.002  se ). Males likely allocated energy to search for mates, while females likely stored energy for embryonic development. We also provided evidence that males use more body energy reserves than females during hibernation, perhaps because of different body temperatures between sexes. We found GRI of male snakes was substantially lower during the mating period than during a non-mating period, which indicated that a trade-off existed between searching for mates and growth. These findings contribute to our understanding of snake ecology in a Mediterranean climate.     

On the rotational brownian motion of a bacterial idle motor. I. Theory of time-dependent fluorescence depolarization

A rotational diffusion equation and its Green's function for a spheroidal particle such as a bacterial body, to which an actively driving but idle motor is attached, are presented. As an application of the theory, general expressions for the time-dependent fluorescence depolarization caused by such a particle have been obtained. Measurement of such depolarizalion should provide a useful tool for determination of the rate of revolution of the rotating motor attached to cell bodies such as bacteria under various solution conditions, if a fluorescent (or phosphorescent) label is attached to the motor shaft.     

Determination the rate constants of some biexponential reactions

Reactions that are described by biexponential functions are typical for many biological processes. The kinetics of these reactions is described by transcendental irrational equations interconnecting the reagent concentrations, time and rate constants. Meantime, their graphical representation in the semi-logarithmic coordinates can be decomposed into two straight lines that intercept at some angle. New simple methods for asymptotic numerical solution of the equations describing these reactions are suggested. These methods permit determining the rate constants using the kinetic data of initial substance concentration, which transform into final product according to a two-component model, a sequential model or a competitive model.     

Effect of local molecular shape and anisotropic reactivity on the rate of diffusion-controlled reactions.

The role of distance-dependent anisotropic reactivity and molecular geometry in the vicinity of localized reaction centers in influencing the rate of bimolecular diffusion-controlled reactions is analyzed in detail, both analytically and numerically. The effect of local molecular shape is considered within the model of reflective hemispheres of small radius l(h) on the surfaces of otherwise spherical molecules of radius R (l(h) < R). The distance-dependent reactivity is modeled by reactive hemispheres of radius l(r) on top of the reflective hemispheres (l(r) < R). It is shown that the presence of the reflective hemispheres leads to a markedly large increase of the reaction rate. The maximum effect is ~R/l(h) > 1 times, as described by the ratio of local to average molecular curvature. It is observed for l(h) approximately R(l(r)/R)(1/2) > l(r). The effect of thickness of the reaction regions is described within the model of reactive cylinders of height l(r) and angular radius theta < 1. It is shown that the characteristic parameter in the expansion of the reaction rate as a function of l(r)/R is l(r)/(Rtheta(2)), and therefore, even for small relative thickness d = l(r)/theta, its effect on the rate is very strong, i.e., the conventional model of reactive patches, which assumes zero thickness of the reaction region, may considerably underestimate the reaction rate.     

Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations

This article describes the numerical solution of the time-dependent Smoluchowski equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.     

Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

Effect of the dilution rate on the biomass yield ofBacillus thuringiensis and determination of its rate coefficients under steady-state conditions

Depending on the biomass yield on glucose and the cell morphology ofBacillus thuringiensis, three different metabolic states were observed in continuous culture. At dilution rates between 0.18 h^–1 and 0.31 h^–1 vegetative cells, sporulating bacteria and spores coexisted, while glucose and amino acids were consumed. Only vegetative cells were observed at dilution rates between 0.42 h^–1 and 0.47 h^–1 and glucose was used as the main carbon and energy source. AtD = 0.50 h^–1 the biomass yield on glucose decreases sharply. To define better the specific growth rate range in which the microorganism uses mainly glucose, a dilution rate of 0.25–0.45 h^–1 was studied. The experimental data could be adjusted to a Monod model and the following rate coefficients and growth yields were determined: maximum specific growth rate 0.54 h^–1, saturation constant 0.56 mg glucose ml^–1, biomass growth yields 0.43 g cells (g glucose)^–1, and 0.76 g cells (g oxygen)^–1, and maintenance coefficients 0.065 g glucose (g cells)^–1 h^–1 and 0.039 g oxygen (g cells)^–1 h^–1.