Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops.

A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.

The diagnostic vibrational signature of pentacoordination in heme carbonyls.

Heme-carbonyl complexes are widely exploited for the insight they provide into the structural basis of function in heme-based proteins, by revealing the nature of their bonded and nonbonded interactions with the protein. This report presents two novel results which clearly establish a FeCO vibrational signature for crystallographically verified pentacoordination. First, anisotropy in the NRVS density of states for ν(Fe-C) and δ(FeCO) in oriented single crystals of [Fe(OEP)(CO)] clearly reveals that the Fe-C stretch occurs at higher frequency than the FeCO bend and considerably higher than any previously reported heme carbonyl. Second, DFT calculations on a series of heme carbonyls reveal that the frequency crossover occurs near the weak trans O atom donor, furan. As ν(Fe-C) occurs at lower frequencies than δ(FeCO) in all heme protein carbonyls reported to date, the results reported herein suggest that they are all hexacoordinate.

Effects of imidazole deprotonation on vibrational spectra of high-spin iron(II) porphyrinates.

The effects of the deprotonation of coordinated imidazole on the vibrational dynamics of five-coordinate high-spin iron(II) porphyrinates have been investigated using nuclear resonance vibrational spectroscopy. Two complexes have been studied in detail with both powder and oriented single-crystal measurements. Changes in the vibrational spectra are clearly related to structural differences in the molecular structures that occur when imidazole is deprotonated. Most modes involving the simultaneous motion of iron and imidazolate are unresolved, but the one mode that is resolved is found at higher frequency in the imidazolates. These out-of-plane results are in accord with earlier resonance Raman studies of heme proteins. We also show the imidazole vs imidazolate differences in the in-plane vibrations that are not accessible to resonance Raman studies. The in-plane vibrations are at lower frequency in the imidazolate derivatives; the doming mode shifts are inconclusive. The stiffness, an experimentally determined force constant that averages the vibrational details to quantify the nearest-neighbor interactions, confirms that deprotonation inverts the relative strengths of axial and equatorial coordination.

Vibrational probes and determinants of the S = 0 ⇌ S = 2 spin crossover in five-coordinate [Fe(TPP)(CN)]-.

The low-frequency vibrational characterization of the spin-crossover complex, five-coordinate cyano(tetraphenylporphyrinato)iron(II), [Fe(TPP)(CN)](-), is reported. Nuclear resonance vibrational spectroscopy has been used to measure all low-frequency vibrations involving iron at several temperatures; this yields vibrational spectra of both the low- (S = 0) and high-spin (S = 2) states. Multitemperature oriented single-crystal measurements facilitate assignments of the vibrational character of all modes and are consistent with the DFT-predicted spectra. The availability of the entire iron vibrational spectrum allows for the complete correlation of the modes between the two spin states. These data demonstrate that not only do the frequencies of the vibrations shift to lower values for the high-spin species as would be expected owing to the weaker bonds in the high-spin state, but also the mixing of iron modes with ligand modes changes substantially. Diagrams illustrating the changing character of the modes and their correlation are given. The reduced iron-ligand frequencies are the primary factor in the entropic stabilization of the high-spin state responsible for the spin crossover.

Nuclear resonance vibrational spectra of five-coordinate imidazole-ligated iron(II) porphyrinates.

Nuclear resonance vibrational spectra have been obtained for six five-coordinate imidazole-ligated iron(II) porphyrinates, [Fe(Por)(L)] (Por = tetraphenylporphyrinate, octaethylporphyrinate, tetratolylporphyrinate, or protoporphyrinate IX and L = 2-methylimidazole or 1,2-dimethylimidazole). Measurements have been made on both powder and oriented crystal samples. The spectra are dominated by strong signals around 200-300 cm(-1). Although the in-plane and out-of-plane vibrations are seriously overlapped, oriented crystal spectra allow their deconvolution. Thus, oriented crystal experimental data, along with density functional theory (DFT) calculations, enable the assignment of key vibrations in the spectra. Molecular dynamics are also discussed. The nature of the Fe-N(Im) vibrations has been elaborated further than was possible from resonance Raman studies. Our study suggests that the Fe motions are coupled with the porphyrin core and peripheral groups motions. Both peripheral groups and their conformations have significant influence on the vibrational spectra (position and shape).

New perspectives on iron-ligand vibrations of oxyheme complexes.

We report our studies of the vibrational dynamics of iron for three imidazole-ligated oxyheme derivatives that mimic the active sites of histidine-ligated heme proteins complexed with dioxygen. The experimental vibrational data are obtained from nuclear resonance vibrational spectroscopy (NRVS) measurements conducted on both powder samples and oriented single crystals, and which includes several in-plane (ip) and out-of-plane (oop) measurements. Vibrational spectral assignments have been made through a combination of the oriented sample spectra and predictions based on density functional theory (DFT) calculations. The two Fe-O(2) modes that have been previously observed by resonance Raman spectroscopy in heme proteins are clearly shown to be very strongly mixed and are not simply either a bending or stretching mode. In addition, a third Fe-O(2) mode, not previously reported, has been identified. The long-sought Fe-Im stretch, not observed in resonance Raman spectra, has been identified and compared with the frequencies observed for the analogous CO and NO species. The studies also suggest that the in-plane iron motion is anisotropic and is controlled by the orientation of the Fe-O(2) group and not sensitive to the in-plane Fe-N(p) bonds and/or imidazole orientations.

Spectroscopic identification of reactive porphyrin motions.

Nuclear resonance vibrational spectroscopy (NRVS) reveals the vibrational dynamics of a Mössbauer probe nucleus. Here, (57)Fe NRVS measurements yield the complete spectrum of Fe vibrations in halide complexes of iron porphyrins. Iron porphine serves as a useful symmetric model for the more complex spectrum of asymmetric heme molecules that contribute to numerous essential biological processes. Quantitative comparison with the vibrational density of states (VDOS) predicted for the Fe atom by density functional theory calculations unambiguously identifies the correct sextet ground state in each case. These experimentally authenticated calculations then provide detailed normal mode descriptions for each observed vibration. All Fe-ligand vibrations are clearly identified despite the high symmetry of the Fe environment. Low frequency molecular distortions and acoustic lattice modes also contribute to the experimental signal. Correlation matrices compare vibrations between different molecules and yield a detailed picture of how heme vibrations evolve in response to (a) halide binding and (b) asymmetric placement of porphyrin side chains. The side chains strongly influence the energetics of heme doming motions that control Fe reactivity, which are easily observed in the experimental signal.

Electronic structure and dynamics of nitrosyl porphyrins.

Nitric oxide (NO) is a signaling molecule employed to regulate essential physiological processes. Thus, there is great interest in understanding the interaction of NO with heme, which is found at the active site of many proteins that recognize NO, as well as those involved in its creation and elimination. We summarize what we have learned from investigations of the structure, vibrational properties, and conformational dynamics of NO complexes with ferrous porphyrins, as well as computational investigations in support of these experimental studies. Multitemperature crystallographic data reveal variations in the orientational disorder of the nitrosyl ligand. In some cases, equilibria among NO orientations can be analyzed using the van't Hoff relationship and the free energy and enthalpy of the solid-state transitions evaluated experimentally. Density functional theory (DFT) calculations predict that intrinsic barriers to torsional rotation are smaller than thermal energies at physiological temperatures, and the coincidence of observed NO orientations with minima in molecular mechanics potentials indicates that nonbonded interactions with other chemical groups control the conformational freedom of the bound NO. In favorable cases, reduced disorder at low temperatures exposes subtle structural features including off-axis tilting of the Fe-NO bond and anisotropy of the equatorial Fe-N bonds. We also present the results of nuclear resonance vibrational spectroscopy measurements on oriented single crystals of [Fe(TPP)(NO)] and [Fe(TPP)(1-MeIm)(NO)]. These describe the anisotropic vibrational motion of iron in five- and six-coordinate heme-NO complexes and reveal vibrations of all Fe-ligand bonds as well as low-frequency molecular distortions associated with the doming of the heme upon ligand binding. A quantitative comparison with predicted frequencies, amplitudes, and directions facilitates identification of the vibrational modes but also suggests that commonly used DFT functionals are not fully successful at capturing the trans interaction between the axial NO and imidazole ligands. This supports previous conclusions that heme-NO complexes exhibit an unusual degree of variability with respect to the computational method, and we speculate that this variability hints at a genuine electronic instability that a protein can exploit to tune its reactivity. We anticipate that ongoing characterization of heme-NO complexes will deepen our understanding of their structure, dynamics, and reactivity.

Mapping NO movements in crystalline [Fe(Porph)(NO)(1-MeIm)].

Orientational disorder of the distal nitrosyl (NO) ligand in iron porphyrinates is a common phenomenon. We present an analysis of multitemperature crystallographic data for the order/disorder phenomenon. The observed temperature-dependent order/disorder and variable rotational orientations of nitrosyl ligands for six different six-coordinate iron porphyrinates have been examined in terms of the nonbonded contacts found in the solid state. Favorable orientations for NO can be identified either by calculation of the close nonbonded contacts or by evaluation of the geometry-dependent potential energy using semiempirical nonbonded potential functions. The nonbonded contacts display temperature-dependent differences consistent with observed structural differences. The motion of NO appears to be controlled by intermolecular interactions that allow a limited set of orientations, and under some conditions, only a single NO orientation is allowed. In some cases, the equilibria involving the orientations of NO can be analyzed using the van't Hoff relationship, and the free energy and enthalpy of the solid-state transitions can be evaluated. The intrinsic barriers to rotation of the NO were examined using a fine-meshed series of DFT calculations. The calculations also showed the detailed effects of the variation of the NO orientation on the equatorial bond distances.

Low-frequency mode activity of heme: femtosecond coherence spectroscopy of iron porphine halides and nitrophorin.

The low-frequency mode activity of metalloporphyrins has been studied for iron porphine-halides (Fe(P)(X), X = Cl, Br) and nitrophorin 4 (NP4) using femtosecond coherence spectroscopy (FCS) in combination with polarized resonance Raman spectroscopy and density functional theory (DFT). It is confirmed that the mode symmetry selection rules for FCS are the same as for Raman scattering and that both Franck-Condon and Jahn-Teller mode activities are observed for Fe(P)(X) under Soret resonance conditions. The DFT-calculated low-frequency (20-400 cm (-1)) modes, and their frequency shifts upon halide substitution, are in good agreement with experimental Raman and coherence data, so that mode assignments can be made. The doming mode is located at approximately 80 cm (-1) for Fe(P)(Cl) and at approximately 60 cm (-1) for Fe(P)(Br). NP4 is also studied with coherence techniques, and the NO-bound species of ferric and ferrous NP4 display a mode at approximately 30-40 cm (-1) that is associated with transient heme doming motion following NO photolysis. The coherence spectra of three ferric derivatives of NP4 with different degrees of heme ruffling distortion are also investigated. We find a mode at approximately 60 cm (-1) whose relative intensity in the coherence spectra depends quadratically on the magnitude of the ruffling distortion. To quantitatively account for this correlation, a new "distortion-induced" Raman enhancement mechanism is presented. This mechanism is unique to low-frequency "soft modes" of the molecular framework that can be distorted by environmental forces. These results demonstrate the potential of FCS as a sensitive probe of dynamic and functionally important nonplanar heme vibrational excitations that are induced by the protein environmental forces or by the chemical reactions in the aqueous phase.

Vibrational dynamics of oxygenated heme proteins.

Advanced spectroscopic techniques coupled with DFT calculations reveal the vibrational dynamics of the iron in stable dioxygen complexes with myoglobin and with a mutant engineered to model the catalytic site of heme-copper oxidases. The unprecedented level of detail will constrain computational modelling of reactions with oxygen.