Stereochemical studies on cyclic peptides: Detailed energy minimization studies on hydrogen bonded all-trans cyclic pentapeptide backbones

Conformational studies have been carried out on hydrogenbonded all-trans cyclic pentapeptide backbone. Application of a combination of grid search and energy minimization on this system has resulted in obtaining 23 minimum energy conformations, which are characterized by unique patterns of hydrogen bonding comprising of β- and γ-turns. A study of the minimum energy conformationsvis-a-vis non-planar deviation of the peptide units reveals that non-planarity is an inherent feature in many cases. A study on conformational clustering of minimum energy conformations shows that the minimum energy conformations fall into 6 distinct conformational families. Preliminary comparison with available X-ray structures of cyclic pentapeptide indicates that only some of the minimum energy conformations have formed crystal structures. The set of minimum energy conformations worked out in the present study can form a consolidated database of prototypes for hydrogen bonded backbone and be useful for modelling cyclic pentapeptides both synthetic and bioactive in nature. This is part XV of the series. Part XIV in this series is Ramakrishnanet al 1987.     

Mimicry of beta II'-turns of proteins in cyclic pentapeptides with one and without D-amino acids.

The solution structure of eight cyclic pentapeptides has been determined by two-dimensional 1H-NMR spectroscopy combined with spectra simulations and restrained molecular dynamic simulations. Six of the cyclic pentapeptides were derived from the C-terminal cholecystokinin fragment CCK-4 enlarged with Asp1 resulting in the sequence (Asp-Trp-Met-Asp-Phe), one L-amino acid after the other was substituted by its D-analog. In addition, two peptides, including an all-L-amino-acid-containing cyclic pentapeptide, cyclo(Asp-Phe-Lys-Ala-Thr) and cyclo(Asp-Phe-Lys-Ala-D-Thr) were investigated. All D-amino-acid-containing peptides show beta II'-turn conformations with the D-amino acid in the i + 1 position, excepting the D-aspartic-acid-containing peptides. These two peptides are characterized by the lack of beta-turns at pH values less than 4, suggesting that D-aspartic acid in the full-protonized state avoids the formation of beta-turns in these compounds. At pH values greater than 5, a conformational change into the beta II'-turn conformation was also observed for these peptides. Conformations without beta-turns are expected for cyclic all-L pentapeptides, but both cyclo(Asp-Phe-Lys-Ala-Thr) and the D-Thr analog cyclo(Asp-Phe-Lys-Ala-D-Thr) exhibit beta II'-turn conformations around Thr-Asp and D-Thr-Asp. Thus cyclic all-L pentapeptides and those with one D-amino acid are able to form similar structures preferably with a beta II'-turn. The beta-turn formation in cyclic pentapeptides containing a D-aspartic acid is dependent on the ionization state. The relevance of the work to the design of beta'-turn mimetics is discussed.     

Conformations of an ion-binding cyclic peptide analogue of valinomycin, cyclo(L-Val-Gly-Gly-L-Pro)3.

A 270-MHz 1H nuclear magnetic resonance investigation of an ion-binding cyclic peptide analogue of valinomycin, cyclo(L-Val-Gly-Gly-L-Pro)3, and its cation complexes is reported. In CD2Cl2 and CDCl3, the peptide is proposed to occur in a C3-symmetric conformer with the N--H's of all six glycine residues intramolecularly hydrogen bonded. This conformation is different from the familiar valinomycin bracelet structure and lacks any "cavity". Cations do not bind, or bind only weakly, to the peptide in these solvents. Uncomplexed cyclo(L-Val-Gly-Gly-L-Pro)3 in acetonitrile appears to be averaging among several conformations with no evidence found for any preferred intramolecular hydrogen bonds. The strong 1:1 complexes of cyclo(L-Val-Gly-Gly-L-Pro)3 with K+ ANd Ba2+ in acetonitrile are structurally analogous to the bracelet conformation of valinomycin and involve the N--H's of the Val residues and of the Gly's preceding Pro in intramolecular hydrogen bonding. Tl+ was also found to form strong 1:1 complexes with the dodecapeptide.     

Cyclic peptides — Small and big and their conformational aspects

Cyclic peptides form an interesting class of compounds for study by conformational analysis, by virtue of their unique conformational features and biological properties. The small cyclic peptides having 3-6 peptide units in their ring, show a variety of conformational characteristics such as occurrence ofcis peptide units, flexibility of peptide dimension and variety in hydrogen bonding. The different possible conformations of cyclic tri- and hexa-peptides are given and certain specific conformational features are discussed for cyclic tetra and pentapeptides. For higher cyclic peptides, the hydrogen bonding requirement for stability of the backbone of the ring, is seen to be kept to a minimum. These various features and their significance are examined and discussed in the light of energy minimization studies and analysis of available experimental data.     

Backbone modifications in cyclic peptides. Conformational analysis of a cyclic pseudopentapeptide containing a thiomethylene ether amide bond replacement

NMR and X-ray crystallographic studies have shown that cyclic pentapeptides of the general structure cyclo(D-Xxx-Pro-Gly-Pro-Gly) possess beta- and gamma-turn intramolecular hydrogen bonds. As part of our continuing series surveying the compatibility of various amide bond replacements on peptide structure, we have synthesized cyclo(D-Phe-Pro psi[CH2S]Gly-Pro-Gly). The pseudopeptide was prepared by solid phase methods and cleaved from the resin by a new procedure involving phase transfer catalysis using K2CO3 and tetrabutylammonium hydrogen sulfate. Cyclization was carried out with the use of DPPA, HOBt, and DMAP to afford the product in 69% yield. The conformational behavior of the pseudopeptide was analyzed by 1H and 13C (1D and 2D) NMR techniques. The backbone modification replaced the amide bond that is involved in a gamma-turn intramolecular hydrogen bond in the all-amide structure. In CDCl3, the pseudopeptide adopted the same all-trans conformation as its parent, although the remaining beta-turn hydrogen bond was weaker according to delta delta/delta TNH measurements. In DMSO-d6, the all-trans conformer and a second conformer were observed in a ratio of 55:45. These conformers, which slowly interconverted on the NMR time scale, could be separately assigned; peaks due to chemical exchange were readily distinguishable by the ROESY technique as reported earlier by others. 13C and ROESY experiments suggested the minor conformer contained one cis amide bond at the Gly1-Pro2 position. Thus, both the location and type of amide surrogate are important determinants affecting the compatibility of the replacement with a particular conformational feature.     

Combined use of molecular dynamics simulations and NMR to explore peptide bond isomerization and multiple intramolecular hydrogen-bonding possibilities in a cyclic pentapeptide, cyclo(Gly-Pro-D-Phe-Gly-Val).

The conformational behavior of a model cyclic pentapeptide--cyclo(Gly-L-Pro-D-Phe-Gly-L-Val)--has been explored through the combined use of in vacuo molecular dynamics simulations and a range of nmr experiments (preceding paper). The molecular dynamics analysis suggests that, despite the conformational constraints imposed by formation of the pentapeptide cycle, this pentapeptide undergoes conformational transitions between various hydrogen-bonded conformations, characterized by low energy barriers. An inverse gamma turn with Pro in position i + 1 and a gamma turn with D-Phe in position i + 1 are two alternatives occurring frequently. Like other DLDDL cyclic pentapeptides, cyclo(Gly-Pro-D-Phe-Gly-Val) is also stabilized by an inverse gamma-turn structure with the beta-branched Val residue in position i + 1, and this hydrogen bond is retained in the different conformational families. The gamma-turn around D-Phe3 and the inverse gamma turn around Val5 are consistent with the nmr observations. 3JNH-CH alpha coupling constants of the all-trans forms were calculated from one of the molecular dynamics trajectories and are comparable to nmr experimental data, suggesting that the conformational states visited during the simulation are representative of the conformational distribution in solution. In addition to the equilibrium among various hydrogen-bonded all-trans conformers, the observation in nmr spectra of two sets of resonances for all peptide protons indicated a slow conformational interconversion of the Gly-Pro peptide bond between trans and cis isomers. The activation energy between these two conformers was determined experimentally by magnetization transfer and was calculated by high temperature constrained molecular dynamics simulation. Both methods yield a free energy of activation of ca. 20 kcal/mol. Furthermore, the free energy of activation is dependent on the direction of rotation of the Gly-Pro peptide bond.     

Nucleo amino acids as arginine mimetics in cyclic peptides

Summary A general strategy for the synthesis of Fmoc protected nucleobase modifed amino acids is presented. Fmoc protected nucleo amino acids bearing a natural purine (guanine) as well as an artificial purine (isoadenine) in the side chain have been synthesized and incorporated into cyclic pentapeptides. The structure of the cyclic peptides is based on the well known RGD peptides, which act as selective integrin antagonists. The nucleo amino acids serve as conformationally constrained arginine mimetics with a reduced basicity of the guanidino moiety.     

Cyclic pentapeptides as models for reverse turns: determination of the equilibrium distribution between type I and type II conformations of Pro-Asn and Pro-Ala beta-turns

Cyclic pentapeptides are excellent models for reverse turns and have been used extensively in our laboratory to explore the influence of different amino acid sequences on turn preference. This paper is divided into two parts: In the first, we review our previous studies of cyclic pentapeptides. We summarize work that demonstrates the range of conformations possible within the cyclic pentapeptide backbone, the importance of sequence chirality in determining the backbone fold, and the utility of these cyclic pentapeptides as models for various turns. In the second, we present new results on two cyclic pentapeptides that contain beta-turns with Pro-Ala or Pro-Asn sequences in the i + 1 and i + 2 positions. By stereochemical criteria, a type I beta-turn is expected to be preferred by such L-L sequences. On the other hand, in proteins Asn occurs frequently in the i + 2 position of type II turns. We asked whether the same propensity would be manifest in an isolated model peptide, and if so, what the interactions were that influenced the relative stability of the type I and type II turns. To address these questions we have compared the conformational behavior of two peptides: cyclo(Gly-Pro-Ala-D-Phe-Pro) and cyclo(D-Ala-Pro-Asn-Gly-Pro). From previous studies, we anticipated that both peptides would contain an inverse gamma-turn and a beta-turn which consisted of either Gly-Pro-Ala-D-Phe or D-Ala-Pro-Asn-Gly in positions i to i + 3, respectively. Nuclear magnetic resonance analysis confirms this overall backbone conformation. Furthermore, quantitative nuclear Overhauser effect measurements in combination with molecular dynamics simulations and torsionally-forced energy minimizations have enabled us to determine that both type I and type II beta-turns are present in equilibrium in these peptides. The introduction of Asn in position i + 2 shifts this equilibrium significantly towards type II. We have done preliminary assessment of the possible side-chain/backbone conformations that contribute to the shift in populations.     

Stereochemical studies on cyclic peptides. IX. Conformational studies on cyclic tetrapeptides containing alternating cis and trans peptide units

Conformational analyses of cyclic tetrapeptides consisting of alternating cis and trans peptide units have been made using contact criteria and energy calculations. This study has been restricted to those structures having a symmetry element in the backbone ring, such as a twofold axis (d) or a center of inversion (i). There are five main results. (1) There are two distinct types of conformations, which are stereochemically favorable corresponding to each of twofold and inversion-symmetrical structures, designated as d₁, d₂ (for twofold symmetrical) and i₁, i₂ (for inversion-symmetrical). Among these, the i₁ type has the lowest energy when glycyl residues occur at all four α-carbon atoms. (2) With the glycyl residue at all four α-carbon atoms, methyl substitution at the cis peptide nitrogen atoms is possible in all the four types, whereas the substitution at trans peptide nitrogen atoms is possible only for the i₁ type. Thus only in the i₁ type can all the nitrogen atoms be methylated simultaneously. The conformation of the molecule in the crystal structure of cyclotetrasarcosyl belongs to the i₁ type. (3) When alanyl residues occur at all four α-carbon atoms, the possible symmetrical type is dependent on the enantiomorphic form and the actual sequence of the alanyl residues. (4) The methyl substitution at peptide nitrogen atoms for cyclic tetrapeptides having alanyl residues causes more stereochemical restriction in the allowed conformations than with glycyl residues. (5) The prolyl residue can be incorporated favorably at the cis-trans junction of both d and i types of structures. The results of the present study are compared with the data on cyclic tetrapeptides available from the crystal structure and nmr studies. The results show an overall agreement both regarding the type of symmetry and the conformational parameters.     

Studies on the synthesis of cyclic pentapeptides as LHRH antagonists and the factors that influence cyclization yield.

Six cyclic pentapeptides containing two or three non-protein amino acids have been synthesized by cyclization of linear precursors in dilute solution and characterized by TLC. HPLC, NMR, melting point. specific rotation etc. A total of 72 cyclization reactions were carried out to study the factors that influence head-to-tail cyclization: linear precursor sequence, coupling reagent, residue configuration, the proportion of DMAP additive, concentration, reaction temperature and reaction time. The cyclic pentapeptides will be modified by active moieties and evaluated as LHRH antagonists.     

Energetics of ribonuclease A catalysis. 2. Nonenzymatic hydrolysis of cytidine cyclic 2',3'-phosphate

Various kinetic aspects of the nonenzymatic hydrolysis of cytidine cyclic 2',3'-phosphate and uridine cyclic 2',3'-phosphate have been studied in order to provide a basis for comparison with the ribonuclease A catalyzed hydrolysis reaction. Studies of the pH dependence of the nonenzymatic reaction reveal mechanisms that are first order in hydroxide concentration and second order in hydrogen ion concentration, in addition to a "water" reaction. The rate constant for the water reaction was found to be very small, approximately equal to 2.5 X 10(-6) min-1. General base catalyzed hydrolysis reactions were also studied with imidazole as the catalyst. At pH values in which both the protonated and neutral forms of imidazole are present, a kinetic mechanism was observed that appears to be second order in total imidazole concentration, thus suggesting that bifunctional catalysis occurs. The activation enthalpy for the hydroxide, hydrogen ion, water, and imidazole catalyzed reactions was determined.     

Opioid receptor affinity and selectivity effects of second residue and carboxy terminal residue variation in a cyclic disulfide-containing opioid tetrapeptide.

The previously described cyclic, delta opioid receptor-selective tetrapeptide H-Tyr-D-Cys-Phe-D-Pen-OH, where Pen, penicillamine, is beta-beta-dimethylcysteine, was modified at residues 2 and 4 by varying combinations of D- and L-Cys and D- and L-Pen, and effects on mu and delta opioid receptor binding affinities and on potency in the mouse vas deferens (MVD) smooth muscle assay were evaluated. A comparison was drawn between consequences of alterations in this series of analogs and those of analogous modifications in the related cyclic pentapeptide series which includes the highly delta receptor-selective [D-Pen2,D-Pen5]enkephalin, DPDPE. Unlike effects observed in the cyclic pentapeptide series, the mu receptor binding affinities of the cyclic tetrapeptides are not dramatically influenced by substitution of Pen for Cys at residue 2. Conversely, while binding of the pentapeptides is only slightly affected by alteration of the chirality of the carboxy-terminal residue, modification of stereochemistry at the carboxy terminus in the tetrapeptides critically alters binding behavior at both mu and delta sites. In contrast with the pentapeptide series, the tetrapeptides appear to be highly dependent upon primary sequence for binding and activity, as only the lead compound binds with high affinity to the delta site. Results suggest that the less flexible cyclic tetrapeptides, lacking the Gly3 residue, display more stringent structural requirements for binding and activity than do the corresponding cyclic pentapeptides.     

Design of high affinity cyclic pentapeptide ligands for kappa-opioid receptors.

Using results from our previously reported cyclic opioid peptide series and reliable models for mu-, delta-, and kappa-opioid receptors (MOR, DOR, and KOR, respectively) and their complexes with peptide ligands, we have designed and synthesized a series of cyclic pentapeptides of structure Tyr-C[D-Cys-Phe-Phe-X]-NH2, cyclized via disulfide, methylene, or ethylene dithioethers, and where X = D- or L-Cys; or D- or L-penicillamine (Pen; beta,beta-dimethylcysteine). Determination of binding affinities to MOR, DOR, and KOR revealed that members of this series with X = D- or L-Cys display KOR affinities in the low nanomolar range, demonstrating that a 'DPDPE-like' tetrapeptide scaffold is suitable not only for DOR and MOR ligands, but also for KOR ligands. The cyclic pentapeptides reported here are not, however, selective for KOR, rather they display significant selectivity and high affinity for MOR. Indeed, peptide 8, Tyr-C[D-Cys-Phe-Phe-Cys]-NH2-cyclized via a methylene dithioether, shows picomolar binding affinity for MOR ( = 16 pm) with more than 100-fold selectivity for MOR vs. DOR or KOR, and may be of interest as a high affinity, high selectivity MOR ligand. Nonetheless, the high affinity KOR peptides in this series represent excellent leads for the development of structurally related, selective KOR ligands designed to exploit structurally specific features of KOR, MOR, and DOR.     

Unique molecular conformation of aureobasidin A, a highly amide N-methylated cyclic depsipeptide with potent antifungal activity: X-ray crystal structure and molecular modeling studies.

A structural feature of aureobasidins, cyclic depsipeptide antibiotics produced by Aureobasidium pullulans R106, is the N-methylation of four out of seven amide bonds. In order to investigate possible relationship between the molecular conformation and the amide N-methylation, aureobasidin A (AbA), which exhibits the potent antifungal activity, was subjected to X-ray crystal analysis. The crystal, recrystallized from ether (orthorhombic, space group P2(1)2(1)2(1), a = 21.643 (3) A, b = 49.865(10) A, c = 12.427 (1) A, z= 8), contained two independent conformers per asymmetric unit and they took on a similar arrowhead-like conformation. The conformation consisted of three secondary structures of antiparallel beta-sheet, and beta- and gamma-turns, and was stabilized by three intramolecular and transannular N-H O=C hydrogen bonds. The beta-hydroxy-N-methyl-l-valine residue, which is indispensable for its bioactivity, was located at the tip of the corner. Since a nearly identical conformation has been observed for aureobasidin E, a related cyclic depsipeptide, this arrowhead-like conformation may be energetically stable and important for biological activity. The contribution of the amide N-methylation to the conformation was investigated by model building and energy calculations. The energy-minimizations of AbA analogs, in which some (one to four) of four N-methylated amide bonds were replaced with usual amide bond, led to some conformers which are fairly different from the arrowhead form of AbA, although they are stabilized by three intramolecular N-H...O=C hydrogen bonds. This result explains the reason why four out of the seven amide bonds have to be methylated to manifest biological activity, i.e. the high N-methylation of aureobasidin is necessary to form only one well-defined conformation.     

Solvent accessibility studies of β-bends and application to cyclic hexapeptides

The solvent-accessible surface areas (ASAs), of the atoms in tripeptides around the minimum-energy conformations of the β-bend types I, I′, II, and II′ have been computed as a first step in the systematic solvent accessibiity study of secondary structures. The side chains chosen at the two middle positions of the bend are L -Ala, D -Ala, and Gly. The ASAs of the hydrogen atoms are reported here and are found useful in determining the type of β-bends in six examples of cyclic hexapeptides whose crystal structures are known. Comparison with observation showed that all the β-bends in these cyclic hexapeptides were correctly identified by the present method. This points to a possible use of the method in identifying β-bend types in solution.     

Computational design and evaluation of β-sheet breaker peptides for destabilizing Alzheimer's amyloid-β42 protofibrils

The β-sheet breaker (BSB) peptides interfere with amyloid fibril assembly and used as therapeutic agents in the treatment of Alzheimer's disease (AD). In this regard, a simple yet effective in silico screening methodology was applied in the present study to evaluate a potential 867 pentapeptide library based on known BSB peptide, LPFFD, for destabilizing Aβ₄₂ protofibrils. The molecular docking based virtual screening was used to filter out pentapeptides having binding affinities stronger than LPFFD. In the next step, binding free energies of the top 10 pentapeptides were evaluated using the MM-PBSA method. The residue-wise binding free energy analysis reveals that two pentapeptides, PVFFE, and PPFYE, bind to the surface of Aβ₄₂ protofibril and another pentapeptide, PPFFE, bind in the core region of Aβ₄₂ protofibril. By employing molecular dynamics simulation as a post filter for the top-hit peptides from MM-PBSA, the pentapeptides, PPFFE, PVFFE, and PPFYE, have been identified as potential BSB peptides for destabilizing Aβ₄₂ protofibril structure. The conformational microstate analysis, a significant decrease in the β-sheet content of Aβ₄₂ protofibril, a loss in the total number of hydrogen bonds in Aβ₄₂ protofibril, Asp23-Lys28 salt bridge destabilization and analysis of the free energy surfaces highlight Aβ₄₂ protofibril structure destabilization in presence of pentapeptides. Among three top-hit pentapeptides, PPFFE displayed the most potent Aβ₄₂ protofibril destabilization effect that shifted the energy minima toward lowest value of β-sheet content as well as lowest number of hydrogen bonds in Aβ₄₂ protofibril. The in silico screening workflow presented in the study highlight an alternative tool for designing novel peptides with enhanced BSB ability as potential therapeutic agents for AD.     

LH and FSH stimulation of cyclic AMP in specific cell types isolated from the testes.

The direct effect of LH and FSH on cyclic AMP levels in specific cell types, isolated from the rat testes, was investigated in vitro. LH significantly stimulated cyclic AMP production in isolated interstitial cells and had only a slight effect on the isolated germ cells. FSH significantly stimulated cyclic AMP production in isolated seminiferous tubules, organ cultures of testes explants, and isolated Sertoli cells, with only a small response elicited in the germ cells. FSH had no effect on the cyclic AMP levels in interstitial cells and either freshly isolated or cultured peritubular cells. These data indicate that the Sertoli cells and interstitial cells are the main cell types in the testes which respond to FSH and LH respectively with increased cyclic AMP production. A possible slight effect of either hormone on the cyclic AMP level in the germ cells has not be ruled out.     

NMR studies and semi-empirical energy calculations for cyclic ADP-ribose

A possible pH-dependent conformational switch was investigated for cyclic ADP-ribose. NMR signals for the exchangeable protons were observed in H2O at low temperature, but there was no direct evidence for the protonation of N-3 at neutral pH that has previously been postulated. MNDO calculations indicated that pH dependent 31P chemical shift changes are attributable to protonation of the phosphate adjacent to the N-1 of adenine, and not due to trans-annular hydrogen bonding with a protonated N-3.     

Structure,stability and water permeation of ([D-Leu-L-Lys-(D-Gln-L-Ala)3]) cyclic peptide nanotube: a molecular dynamics study

The structural stability of 8 × ([D-Leu-L-Lys-(D-Gln-L-Ala)₃]) cyclic peptide nanotube (CPN) in water and different phospholipid bilayers were explored by 100 ns independent molecular dynamics (MD) simulations. The role of non-bonded interaction energy between the side and main chains of cyclic peptide rings in different membrane environments assessed, wherein the repulsive electrostatic interaction energy between neighbouring cyclic peptide rings was found adequate to break hydrogen bond energy thereby to crumple CPN. Further, the water permeation across the CPN channel was studied in four types of phospholipid bilayers- DMPG (1,2-Dimyristoyl-sn-glycero-3-phosphorylglycerol), DMPS (1,2-Dimyristoyl-sn-glycero-3-phosphoserine), POPC (1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine) and POPE (1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine) from MD simulations. DMPS membrane shows higher non-bonded interaction energies (?1913.06 kJ/mol of electrostatic interaction energy and ?994.13 kJ/mol of van der Waals interaction energy) with CPN due to the presence of polar molecules in lipid structure. Thusly, the non-bonded interaction energies were essential towards the stability of CPN than hydrogen bonds between the nearby cyclic peptides. The result also reveals the role of side chains, hydrogen bonds and non-bonded interaction energies in an aqueous environment. The diffusion coefficient of water obtained from means square deviation calculation shows similar coefficients irrespective of the lipid surroundings. However, the permeation coefficients demonstrate water flow in the channel relies upon the environment.     

Interaction of cyclic adenosine 3':5'-monophosphate with protein kinase. Equilibrium binding models.

A number of potential models for the interaction of cyclic AMP with protein kinase (RC or R2C2) have been examined. These include: Model 1, the simultaneous binding of cyclic AMP and release of C (catalytic subunit) from an independent RC protomer; Model 2, dissociation of an independent RC protomer prior to cyclic AMP binding to R (regulatory subunit); Model 3, cyclic AMP binding to RC prior to the dissociation of C; Model 4, random binding of cyclic AMP and dissociation of C with an interaction factor alpha less than 1; Model 5, release of 2C concomitant with the binding of one cyclic AMP to R2C2 followed by binding of the second cyclic AMP to the vacant R subunit; and Model 6, the simultaneous binding of cyclic AMP and release of C from one RC protomer resulting in a greater "affinity" of the other RC protomer for cyclic AMP, i.e., a cooperative version of Model 1. All the above models yield [cyclic AMP]0.5 values that increase with increasing protein concentration and Hill plots with average slopes equal to or less than 1.0 in the usual experimental range (10 to 90% of saturation). The Hill plots can be nonlinear, but for each model the exact shape of the plot changes in a characteristic (diagnostic) manner with changing protein concentration. Skeletal muscle protein kinase yields relatively linear Hill plots with napp values greater than 1.0. Consequently, Models 1 to 6 are not likely candidates. However, Model 2 is an excellent alternative model for proteins that display "negative cooperativity" with respect to the binding of a ligand. The properties of several "linear", "tetrahedral", and "all-or-nothing" cooperative models have also been examined. These include Models 7, A, B, and C and 8, A, B, and C which are cooperative versions of Models 2 and 3, respectively, and Model 9, a cooperative version of random Model 4. Model 9 is the most general model from which all others can be derived. Models 9 and 7, A, B, and C in which the prior dissociation of C greatly enhances or is an absolute requirement for cyclic AMP binding to R, are likely candidates for skeletal muscle protein kinase. All four of these models are capable of yielding Hill plots with average slopes greater than 1, and napp values that decrease with increasing protein concentration (in agreement with published data). In addition, in all four models the tight binding of MgATP to R2C2 yields decreased napp values and increased [cyclic AMP]0.5 values (also consistent with published data).