Aspects of protein-DNA interactions: a review of quantitative thermodynamic theory for modelling synthetic circuits utilising LacI and CI repressors, IPTG and the reporter gene lacZ.

Protein-DNA interactions are central to the control of gene expression across all forms of life. The development of approaches to rigorously model such interactions has often been hindered both by a lack of quantitative binding data and by the difficulty in accounting for parameters relevant to the intracellular situation, such as DNA looping and thermodynamic non-ideality. Here, we review these considerations by developing a thermodynamically based mathematical model that attempts to simulate the functioning of an Escherichia coli expression system incorporating two of the best characterised prokaryotic DNA binding proteins, Lac repressor and lambda CI repressor. The key aim was to reproduce experimentally observed reporter gene activities arising from the expression of either wild-type CI repressor or one of three positive-control CI mutants. The model considers the role of several potentially important, but sometimes neglected, biochemical features, including DNA looping, macromolecular crowding and non-specific binding, and allowed us to obtain association constants for the binding of CI and its variants to a specific operator sequence.

Kinetic trapping of a key hemoglobin intermediate.

The complete binding cascade of human hemoglobin consists of a series of partially ligated intermediates. The individual intermediate binding constants cannot be distinguished in O(2) binding curves, however, each constant can be determined from the O(2)-induced change in assembly constant for the α(2)ß(2) tetramer from its constituent αß dimers. The characterization of these O(2) binding constants has shown the Hb cascade to be asymmetric in nature, with binding dependent upon the specific distribution of O(2) among the four hemesites. A stopped-flow approach to measuring the dissociation constant of a key doubly ligated intermediate, that in which one dimer is oxygenated and the other is not, is described. The intermediate is transiently formed in the absence of O(2) and then allowed to dissociate in the presence of O(2). The free dimers thus released are trapped by the plasma protein haptoglobin, the rate limiting step being that of tetramer dissociation. The kinetic constant observed for the dissociation of this intermediate confirms the value for its equilibrium O(2) binding constant, previously determined under equilibrium conditions by subzero isoelectric focusing.

The Hill coefficient: inadequate resolution of cooperativity in human hemoglobin.

The Hill coefficient nH is a dimensionless parameter that has long been used as a measure of the extent of cooperativity. Originally derived from the oxygen-binding curve of human hemoglobin (Hb) by A. V. Hill in 1910, and reinvented by J. Wyman several decades later, nH is indexed to the stoichiometry of ligation and is indirectly related to the overall cooperative free energy for binding all four oxygen ligands. However, the overall cooperative free energy of Hb ligation can be measured directly by experimental methods. The microscopic cooperative free energies that relate to energetic coupling between specific subunit pairs can also be experimentally determined, while the Hill coefficient is, by its nature, a macroscopic parameter that cannot detect differences among specific subunit-subunit couplings. Its continued use in studies of the mechanism of cooperativity in Hb is therefore of increasingly limited value.

Asymmetric distribution of cooperativity in the binding cascade of normal human hemoglobin. 2. Stepwise cooperative free energy.

Stepwise cooperative free energies and intermediate Hill coefficients are used to assess the presence of noncooperative sequences in the database of binding free energies previously obtained for the eight partially ligated intermediates of human hemoglobin, encompassing a variety of hemesite analog substitutions. This analysis is prompted by the observed noncooperative binding of two ligands to hemoglobins that are partially substituted with Zn2+-heme, an analog of deoxy Fe2+-heme (Holt et al. (2005) Biochemistry 44, XXXXX). The results show that noncooperative binding sequences are observed in all hemesite analog studied to date. The noncooperative binding observed in (alpha2Znbeta2FeO2) and (alpha2FeO2beta2Zn) is therefore not a Zn-specific substitution artifact. One of several binding sequences from singly to triply ligated hemoglobin is also observed to occur with little or no positive cooperativity. These results demonstrate the variability possible among different ligation pathways in a highly cooperative multi-subunit system such as hemoglobin. As a direct consequence of this variability, differences among ligation pathways are not always detectable using cooperativity functions based on statistical distributions, such as the Hill coefficient n(H). The limitations of Hill coefficient analysis in evaluating cooperativity in intermediates of complex systems is contrasted with the utility of the stepwise binding parameters.

Asymmetric distribution of cooperativity in the binding cascade of normal human hemoglobin. 1. Cooperative and noncooperative oxygen binding in Zn-substituted hemoglobin.

The complete binding cascade of human hemoglobin consists of eight partially ligated intermediates and 16 binding constants. Each intermediate binding constant can be evaluated via dimer-tetramer assembly when ligand configurations within the tetramer are fixed through the use of hemesite analogs. The Zn/Fe analog, in which the nonbinding Zn2+ heme substitutes for deoxy Fe2+ heme, also permits direct measurement of O2 binding to the remaining Fe2+ hemesites within the symmetrically ligated Hb tetramers. Measurement of O2 binding over a range of Zn/Fe Hb concentrations to both alpha-subunits (species 23) or to both beta-subunits (species 24) shows noncooperative binding and incomplete saturation of the available Fe2+ hemesites. In contrast, the asymmetrically ligated Zn/FeO2 species 21, in which both oxygens are bound to one of the dimers within the tetramer, exhibits positive cooperativity and >90% ligation under atmospheric conditions. These properties are confirmed in the present study by measurement of the rate constant for tetramer dissociation to free dimer. The binding constants thus derived for these partially ligated intermediates are consistent with the stoichiometric constants measured for native hemoglobin by standard O2 binding techniques, providing additional evidence that Zn2+-heme substitution provides an excellent deoxy hemoglobin analog. There is no evidence that Zn-substitution stabilizes a low-affinity form of the tetramer, as previously suggested. These characterizations demonstrate distinct, nonadditive physical properties of the doubly ligated tetrameric species, yielding an asymmetric distribution of cooperativity within the cascade of O2 binding by human hemoglobin.

The molecular code for hemoglobin allostery revealed by linking the thermodynamics and kinetics of quaternary structural change. 1. Microstate linear free energy relations.

A novel model linking the thermodynamics and kinetics of hemoglobin's allosteric (R --> T) and ligand binding reactions is applied to photolysis data for human HbCO. To describe hemoglobin's kinetics at the microscopic level of structural transitions and ligand-binding events for individual [ij]-ligation microstates ((ij)R --> (ij)T, (ij)R + CO --> ((i)(+1))(k)R, and (ij)T + CO --> ((i)(+1))(k)T), the model calculates activation energies, (ij)DeltaG(++), from previously measured cooperative free energies of the equilibrium microstates (Huang, Y., and Ackers, G. K. (1996) Biochemistry 35, 704-718) by using linear free energy relations ((ij)DeltaG(++) - (01)DeltaG(++) = alpha[(ij)DeltaG - (01)DeltaG], where the parameter alpha, describing the variation of activation energy with reaction energy perturbation, can depend on the natures of both the reaction and the perturbation). The alpha value measured here for the allosteric dynamics, 0.21 +/- 0.03, corresponds closely to values observed previously, strongly suggesting that the thermodynamic microstate energies directly underlie the allosteric kinetics (as opposed to the alpha((ij)DeltaG(RT)) serving merely as arbitrary fitting parameters). Besides systematizing the study of hemoglobin kinetics, the utility of the microstate linear free energy model lies in the ability to test microscopic aspects of allosteric dynamics such as the "symmetry rule" for quaternary change deduced previously from thermodynamic evidence (Ackers, G. K., et al. (1992) Science 255, 54-63). Reflecting a remarkably detailed correspondence between thermodynamics and kinetics, we find that a kinetic model that includes the large free energy splitting between doubly ligated T microstates implied by the symmetry rule fits the data significantly better than one that does not.

The molecular code for hemoglobin allostery revealed by linking the thermodynamics and kinetics of quaternary structural change. 2. Cooperative free energies of (alphaFeCObetaFe)2 and (alphaFebetaFeCO)2 T-state tetramers.

Ligand photodissociation experiments are used to measure the prephotolysis equilibria between doubly liganded R and T quaternary conformers of the symmetric Fe-Co HbCO hybrids, (alpha(FeCO)beta(Co))(2) and (alpha(Co)beta(FeCO))(2). The free energies obtained from these data are used to calculate the cooperative free energies of the (alpha(FeCO)beta(Fe))(2) and (alpha(Fe)beta(FeCO))(2) intermediate CO-ligation states of normal hemoglobin in the T conformation, quantities important to the evaluation of current models of cooperativity. The symmetry rule model, incorporating sequential cooperativity of T-state ligand binding within an alphabeta dimer in addition to the traditional two-state cooperativity of the tetramer, predicts a larger free energy penalty for disturbing both dimers in a doubly liganded T tetramer than would be expected in the two-state model as currently formulated. (Cooperative energy penalties are simply proportional to the number of tetramer-bound ligands in the traditional two-state model.) The value found here for the energies of doubly liganded T microstates in which both dimers are perturbed, 7.9 +/- 0.3 kcal/mol, is consistent with the symmetry rule model but significantly higher than that expected (5-6 kcal/mol) in the two-state model of cooperativity.

Single residue modification of only one dimer within the hemoglobin tetramer reveals autonomous dimer function.

The mechanism of cooperativity in the human hemoglobin tetramer (a dimer of alpha beta dimers) has historically been modeled as a simple two-state system in which a low-affinity structural form (T) switches, on ligation, to a high-affinity form (R), yielding a net loss of hydrogen bonds and salt bridges in the dimer-dimer interface. Modifications that weaken these cross-dimer contacts destabilize the quaternary T tetramer, leading to decreased cooperativity and enhanced ligand affinity, as demonstrated in many studies on symmetric double modifications, i.e., a residue site modified in both alpha- or both beta-subunits. In this work, hybrid tetramers have been prepared with only one modified residue, yielding molecules composed of a wild-type dimer and a modified dimer. It is observed that the cooperative free energy of ligation to the modified dimer is perturbed to the same extent whether in the hybrid tetramer or in the doubly modified tetramer. The cooperative free energy of ligation to the wild-type dimer is unperturbed, even in the hybrid tetramer, and despite the overall destabilization of the T tetramer by the modification. This asymmetric response by the two dimers within the same tetramer shows that loss of dimer-dimer contacts is not communicated across the dimer-dimer interface, but is transmitted through the dimer that bears the modified residue. These observations are interpreted in terms of a previously proposed dimer-based model of cooperativity with an additional quaternary (T/R) component.