Association kinetics with coupled diffusion. An extension to coiled-chain macromolecules applied to the lac repressor-operator system.

The association of a molecule onto a specified binding site on a large chain-like macromolecule is described in the "sliding" model, where the molecule is allowed to move along the chain in a one-dimensional diffusion which is coupled to the three-dimensional diffusion in solution. The present work extends a previous one by treating the chains more generally as coiled instead of straight. The model is applied to the lac repressor-operator association. A general expression for the rate of unspecific attachment to a chain-like macromolecule is also derived.     

Association kinetics with coupled diffusional flows: Special application to the lac repressor-operator system

The time development of the association of the lac repressor to the operator is considered in a model where the repressor is allowed to bind unspecifically to DNA and move along the DNA chain in a one-dimensional diffusion. The coupling to the three-dimensional diffusion outside the chain is introduced by letting the repressor associate and dissociate from the chain until it is finally bound to the operator. All distance correlations along the chain are included. The mean time of association is calculated and through a comparison with experimental data the molecular parameters are determined. The one-dimensional diffusion constant is found to lie of the order of 10^?9cm² s^?1. The model is sufficiently general to tie applicable to other similar systems.     

Rate enhancement by guided diffusion. Chain length dependence of repressor-operator association rates

Association rates are calculated for cases where one reaction partner belongs to a chain that has an unspecific affinity to the other. Provided that the unspecific attachment does not completely suppress diffusion along the chain, this channeling may considerably speed up the association. Explicit formulae are derived to show how this effect depends on the chain length and other parameters. The influence of electrostatic forces and reaction barriers is discussed. Time dependent solutions of the diffusion equations are analyzed in order to test the usual steady state assumptions. Experiments on the repressor-operator system seem to be in good agreement with our theory.     

Equilibria and kinetics of lac repressor-operator interactions by polyacrylamide gel electrophoresis.

We describe the use of gel electrophoresis in studies of equilibrium binding, site distribution, and kinetics of protein-DNA interactions. The method, which we call protein distribution analysis, is simple, sensitive and yields thermodynamically rigorous results. It is particularly well suited to studies of simultaneous binding of several proteins to a single nucleic acid. In studies of the lac repressor-operator interaction, we found that binding to the so-called third operator site (03) is 15-18 fold weaker than operator binding, and that the binding reactions with the first and third operators are uncoupled, implying that there is no communication between the sites. Pseudo-first order dissociation kinetics of the repressor-203 bp operator complex were found to be temperature sensitive, with delta E of 80 kcal mol-1 above 29 degrees C and 26 kcal mol-1 below. The half life of the complex (5 min at 21 degrees C) is shorter than that reported for very high molecular weight operator-containing DNAs, but longer than values reported for much shorter fragments. The binding of lac repressor core to DNA could not be detected by this technique: the maximum binding constant consistent with this finding is 10(5) M-1.     

Interaction of effecting ligands with lac repressor and repressor-operator complex.

The equilibrium association constants for the binding of a wide variety of effecting ligands of the lac repressor were measured by equilibrium dialysis. Also, detailed investigations of the apparent rate of dissociation of repressor-operator comples as a function of ligand concentration were carried out for several inducers and anti-inducers. The affinity of repressor-ligand comples for operator DNA was evaluated from the specific rate constants at saturating concentrations of effecting ligand. By fitting the experimental data depicting the functional dependence of the rate of dissociation upon ligand concentrations to calculated curves, assuming simple models of the induction mechanism, the equilibrium association constant for the binding of effecting ligand to repressor-operator comples was determined. Inducers reduce the affinity of lac repressor for operator DNA by a factor of approximately 1000 under standard conditions; the extent of destabilization depends on Mg2+ ion concentration. Anti-inducers increase the affinity of repressor for operator at most a factor of five. Only one neutral ligand, which binds to repressor without altering the stability of repressor-operator comples, was found. No homotropic or heterotropic interactions in the binding of effecting ligands either to repressor or to repressor-operator complex are evident.     

lac repressor-operator interaction. Analysis of the X86 repressor mutant

In vitro measurements show that the X86 repressor, which has an increased affinity for the lac operator as compared to wild-type repressor, also has an increased affinity for non-operator sites on Escherichia coli DNA. The rate constant of association of repressor and operator is decreased by E. coli DNA fivefold more for X86 repressor than for wild-type repressor. Low inducer concentrations increase the rate of association of X86 repressor and operator in the presence of E. coli DNA. In a partial equilibrium situation where part of the X86 repressor is bound to the operator, and part to either non-operator sites on E. coli DNA or to an O^c operator, the formation of complexes between X86 repressor and wild-type operator is favored by low inducer concentrations. Repression of the lac enzymes increases drastically in the X86 mutant in the absence of DNA synthesis in vivo. A new explanation for the in vivo characteristics of the X86 mutant is suggested.     

Analysis of ion concentration effects on the kinetics of protein-nucleic acid interactions: Application to lac repressor-operator interactions

The effects of monovalent and divalent cations on the bimolecular rate constant of the reaction of a positively charged ligand with a nucleic acid polyanion are analyzed for two possible reaction mechanisms. One mechanism postulates that the association reaction occurs without intermediates, and that ion effects on the rate constant result entirely from the screening of the charged reactants by ionic atmospheres of low molecular weight ions (a screening-controlled mechanism). This mechanism is analyzed by analogy with the Bronsted-Bjerrum theory for the kinetics of interaction of low molecular weight ions. The second mechanism to be considered here postulates the existence of a ligand-DNA intermediate which is in rapid equilibrium with the reactants (pre-equilibrium mechanism). Ion concentration effects on the association rate constants for the pre-equBibrium mechanism result mainly from the release of counterions from the DNA upon formation of the intermediate. Both of the above mechanisms predict that the logarithm of the association rate constant,_a, will be a linear function of the logarithm of the monovalent cation concentration, [M+] (in the absence of competition by divalent cations or anions). Knowledge of the salt dependences of K_a and of the observed equilibrium constant k_obs of the ligand-nucleic acid interaction should usually be sufficient to determine whether a screening-controlled mechanism or a pre-equilibrium mechanism is suitable to describe the process. If the association reaction can be described by a pre-equilibrium mechanism, the number of ionic interactions involved in the ligand-nucleic acid intermediate can be estimated. This analysis, extended to include the effects of divalent cations on screening or on the pre-equilibrium step, is applied to literature data on the salt dependence of the kinetics of the interaction of lac represser with lac operator DNA. When the operator is present on bacteriophage λ DNA, the observed reaction kinetics are consistent with the formation of an intermediate repressor-DNA complex in a pre-equilibrium step. On the other hand, the kinetics of association of toe represser with synthetic foe operator fragments may be an example of a screening-controlled reaction.     

Lac repressor-operator interaction. 8. Lactose is an anti-inducer of the lac operon

Lactose is shown to be an effective anti-inducer of the lac operon both in vivo and in vitro. When lactose is used as a carbon source, the synthesis of β-galaetosidase in Escheriahia coli is not fully induced. Moreover, lactose is able to partially inhibit induction by isopropyl-(β-d-thiogalactoside in strains synthesizing inactive as well as active β-galactosidase. These effects in vivo are not due to catabolite repression by the glucose derived from lactose. These in vivo results suggest that lactose is acting as an anti-inducer. This is confirmed in vitro by showing that lactose binds to the lac represser and stabilizes the represser-operator complex.     

Radiation disrupts protein-DNA complexes through damage to the protein. The lac repressor-operator system.

Eon, S., Culard, F., Sy, D., Charlier, M. and Spotheim-Maurizot, M. Radiation Disrupts Protein-DNA Complexes through Damage to the Protein. The lac Repressor-Operator System. Radiat. Res. 156, 110-117 (2001).Binding of a protein to its cognate DNA sequence is a key step in the regulation of gene expression. If radiation damage interferes with protein-DNA recognition, the entire regulation process may be perturbed. We have studied the effect of gamma rays on a model regulatory system, the E. coli lactose repressor-operator complex. We have observed the disruption of the complex upon irradiation in aerated solution. The complex is completely restored by the addition of nonirradiated repressor, but not by the addition of nonirradiated DNA. Thus radiation disrupts the DNA-protein complex by affecting the binding ability of the protein. This interpretation is supported by the dramatic loss of binding ability of a free irradiated repressor toward nonirradiated DNA. Interestingly, the dose necessary for the disruption of the irradiated complex is higher than that for inducing the complete loss of the binding ability of the free irradiated repressor. This may be due to the protection of key amino acids by the bound DNA. As seen from calculations of the accessibility of amino acids to radiolytic OH(.), the protection is due to both masking and conformational effects.     

Perturbation analysis of coupled diffusion and reaction with high-order kinetics

Asymptotic solutions for the effectiveness factor and the concentration profile are obtained for mth-order chemical reactions inside a slab catalyst pellet with Robin boundary condition at the pellet's outer surface. Using perturbation analysis in the limit of large reaction order m, the effectiveness factor and the concentration profile are explicitly determined up to O(1/m). Higher-order solutions can be obtained in a systematic way if desired.     

Ionic-strength dependence of the conformational change in the unphosphorylated sodium pump

The conformational change in the unphosphorylated sodium pump was studied as a function of ionic strength to learn whether the rate of the reaction is affected. The results corroborate our proposal [Smirnova, I. N., Lin, S.-H., and Faller, L. D. (1995) Biochemistry 34, 8657-8667] that competitive binding of the transported ions to two (or more) equivalent sites regulates a concerted change in protein conformation. An approximately 10-fold increase in ionic strength decreased the intrinsic affinity of the Na+ conformation of the enzyme for both Na+ and K+ roughly 3-fold, decreased the rate of the change from Na+ to K+ conformation by more than half, and increased the rate of the reverse reaction by about an order of magnitude. The logarithm of the first-order rate constant for the change from Na+ to K+ conformation depended inversely upon the square root of the ionic strength with the extrapolated value at zero ionic strength expressed as a second-order rate constant (1.1 x 10(9) M(-1) sec(-1)) approaching the limit for a diffusion-controlled reaction. The first-order rate constant for the change from K+ to Na+ conformation depended directly upon ionic strength and extrapolated to a zero-ionic-strength value (0.002 s(-1)) far below the diffusion limit. The results are compatible with shielding of oppositely charged domains that move through the solvent when the pump cycles between conformations. Electrostatic interactions between domains evidently contribute to the driving force for the change from Na+ to K+ conformation and to the stability of the K+ conformation.     

Wrapping of flanking non-operator DNA in lac repressor-operator complexes: implications for DNA looping

In our studies of lac repressor tetramer (T)-lac operator (O) interactions, we observed that the presence of extended regions of non-operator DNA flanking a single lac operator sequence embedded in plasmid DNA produced large and unusual cooperative and anticooperative effects on binding constants (Kobs) and their salt concentration dependences for the formation of 1:1 (TO) and especially 1:2 (TO2) complexes. To explore the origin of this striking behavior we report and analyze binding data on 1:1 (TO) and 1:2 (TO2) complexes between repressor and a single O(sym) operator embedded in 40 bp, 101 bp, and 2514 bp DNA, over very wide ranges of [salt]. We find large interrelated effects of flanking DNA length and [salt] on binding constants (K(TO)obs, K(TO2)obs) and on their [salt]-derivatives, and quantify these effects in terms of the free energy contributions of two wrapping modes, designated local and global. Both local and global wrapping of flanking DNA occur to an increasing extent as [salt] decreases. Global wrapping of plasmid-length DNA is extraordinarily dependent on [salt]. We propose that global wrapping is driven at low salt concentration by the polyelectrolyte effect, and involves a very large number (>/similar 20) of coulombic interactions between DNA phosphates and positively charged groups on lac repressor. Coulombic interactions in the global wrap must involve both the core and the second DNA-binding domain of lac repressor, and result in a complex which is looped by DNA wrapping. The non-coulombic contribution to the free energy of global wrapping is highly unfavorable ( approximately +30-50 kcal mol(-1)), which presumably results from a significant extent of DNA distortion and/or entropic constraints. We propose a structural model for global wrapping, and consider its implications for looping of intervening non-operator DNA in forming a complex between a tetrameric repressor (LacI) and one multi-operator DNA molecule in vivo and in vitro. The existence of DNA wrapping in LacI-DNA interactions motivates the proposal that most if not all DNA binding proteins may have evolved the capability to wrap and thereby organize flanking regions of DNA.     

A model of the lac repressor-operator complex based on physical and genetic data

Computer graphics were used to build a molecular model of the complex of Lac repressor and lac operator. The model is based (a) on the NMR data of the Kaptein group [Boelens, R., Lamerichs, R. M. J. N., Rullmann, J. A. C., van Boom, J. H. & Kaptein, R. (1988) Protein Sequence Data Anal. 1, 487-498] and (b) on our genetic and biochemical data including specificity changes [Lehming, N., Sartorius, J., Kisters-Woike, B., von Wilcken-Bergmann, B. & Müller-Hill, B. (1990) EMBO J. 9, 615-621]. Effects of amino acid exchanges in the recognition helix could be predicted by the model and were subsequently tested and confirmed by genetic experiments. Comparison of the modelled lac complex with the known crystallographic structures of several helix-turn-helix DNA complexes reveals striking similarities and suggests rules which govern the recognition between particular amino acid side chains and particular base pairs in these systems.     

Modified nucleotides reveal the indirect role of the central base pairs in stabilizing the lac repressor-operator complex.

Guanine residues in the lac operator were replaced by 2-aminopurine or purine analogues, pairing the modified nucleotides with C. The observed equilibrium dissociation constants for lac repressor binding to substituted operators were measured in 10 mM Tris, 150 mM KCl, 0.1 mM EDTA, 0.1 mM DTE, pH 7.6 at 25 degrees C. These measurements revealed five positions that destabilized the complex when substituted with either analogue. Two positions, which are related by a 2-fold symmetry, are in the major groove of the operator thought to directly interact with the protein. Three sites were in the central region of the operator. A purine analogue at a sixth site perturbed the local DNA structure and destabilized the complex. Alkylation interference experiments of the 2-aminopurine substituted operators demonstrated that, of the five affected, two substitutions displayed altered phosphate interference patterns at the phosphate adjacent to the substituted base. For these operators, complex formation was measured in different concentrations of KCl to assess the contribution of counterion release to the bimolecular process. The results indicated that both complexes were similar to wild-type, although minor changes were observed. The Kobs of the complex was then measured when 2-aminopurine or purine analogues were paired with uracil nucleotide, a base pair that serves to stabilize the DNA. The introduction of the new base pairs revealed two effects on the bimolecular interaction. For those operator sites that are thought to perturb the interaction directly, the affinity of the complex was weakened to levels observed for the singly-substituted operators. In contrast, the nucleotides of 2-aminopurine paired with uracil positioned in the central region of the operator served to enhance the stability of the complex. The purine-uracil base pair substitution on the other hand had a significant destabilizing effect on the interaction. We propose that the central base pairs modulate binding of the complex by altering the intrinsic properties of the DNA. Two specific attributes are required to achieve the lowest free energy of interaction. The DNA must have two interstrand hydrogen bonds to stabilize the duplex and it must have properties associated with directional bending or unwinding. This analysis does not rule out contributions by direct interactions between the protein and the central region of the operator but underscores how indirect effects play a major role in complex formation in this system.     

Thermodynamic origins of specificity in the lac repressor-operator interaction. Adaptability in the recognition of mutant operator sites

A system has been developed for facile generation and characterization of mutant lac operator sites, free of competing pseudo operator sequences. The interaction of lac repressor with these sites has been investigated by the nitrocellulose filter binding assay. The equilibrium binding affinity for each of three single-site changes was reduced by more than three orders of magnitude relative to the wild-type operator under standard assay conditions. The free-energy changes associated with single base-pair substitutions are not additive. We propose that adaptations in the recognition surface of the repressor involving significant trade-offs between electrostatic versus non-electrostatic interactions and between enthalpic versus entropic contributions to the binding free energy occur, in order to achieve the most stable complex with a given DNA sequence.     

Brownian dynamics study of the influences of electrostatic interaction and diffusion on protein-protein association kinetics.

A unified model is presented for protein-protein association processes that are under the influences of electrostatic interaction and diffusion (e.g., protein oligomerization, enzyme catalysis, electron and energy transfer). The proteins are modeled as spheres that bear point charges and undergo translational and rotational Brownian motion. Before association can occur the two spheres have to be aligned properly to form a reaction complex via diffusion. The reaction complex can either go on to form the product or it can dissociate into the separate reactants through diffusion. The electrostatic interaction, like diffusion, influences every step except the one that brings the reaction complex into the product. The interaction potential is obtained by extending the Kirkwood-Tanford protein model (Tanford, C., and J. G. Kirkwood. 1957. J. Am. Chem. Soc. 79:5333-5339) to two charge-embedded spheres and solving the consequent equations under a particular basis set. The time-dependent association rate coefficient is then obtained through Brownian dynamics simulations according an algorithm developed earlier (Zhou, H.-X. 1990. J. Phys. Chem. 94:8794-8800). This method is applied to a model system of the cytochrome c and cytochrome c peroxidase association process and the results confirm the experimental dependence of the association rate constant on the solution ionic strength. An important conclusion drawn from this study is that when the product is formed by very specific alignment of the reactants, as is often the case, the effect of the interaction potential is simply to scale the association rate constant by a Boltzmann factor. This explains why mutations in the interface of the reaction complex have strong influences on the association rate constant whereas those away from the interface have minimal effects. It comes about because the former mutations change the interaction potential of the reaction complex significantly and the latter ones do not.