Cooperativity folding units

Cooperative folding units are defined from a functional rather than a purely structural point of view, even though in many cases a one-to-one correspondence can be found. Cooperative folding units can be as large as the entire protein (e.g., small globular proteins that exhibit two-state behavior effectively behave as cooperative folding units), entire protein domains, subdomains, and structural motifs, or as small as single a helices. The hierarchical level at which the description is made depends on the desired level of detail and is, in principle, only constrained by the availability of structural information. 

According to the hierarchical approach, the number of states that need to be considered in the partition function is 2"cu, where ncu is the number of cooperative folding units. In order to develop a complete description of a system composed of ncu interacting cooperative folding units it is necessary to evaluate the intrinsic energetics of each... 

Fig. 6. Representation of states for a hypothetical protein composed of two cooperative folding units. Free energy differences designated in uppercase letters represent the intrinsic stabilities of the cooperative units and have a contribution to the statistical weights designated by k. Free energy differences designated in lowercase letters represent the interaction between cooperative units and have a contribution to the statistical weights designated by . [Reprinted from Freire and Murphy (1991).]...

An a-helix bundle may become a second-order cooperative folding unit if the interaction energy terms are such that the intermediate terms in the partition function become negligibly small [Eq. (14)] and the entire partition function reduces to a two-state partition function (i.e., a partition function of the form 1 + e G/RT). If such is the case, the a-helix bundle will be either completely folded or unfolded. Higher order cooperative folding units can be constructed from lower order ones following the same rules. The most immediate application of this approach is to proteins exhibiting pure a-helical structural motifs. 

Wallqvist A, Smythers GW, Covel DG (1997) Identification of cooperative folding units in a set of native proteins, Protein Sci, 6 1627—42... 

Murphy KP, Bhakuni V, Xie D, Freire E (1992) Molecular basis of cooperativity in protein folding. III. Structural identification of cooperative folding units and folding intermediates, J Mol Biol, 227 293-306... 

These considerations were used in the analysis of the kinetics of crystallization of three and of one dimensional systems (22, 68, 87). The main difference between the previous work and the analysis given here, is that in crystallization one must consider two different processes nucleation and crystal growth, even when the system is cooperative, whereas in the folding of proteins, the small size of the folded unit essentially reduces the process to nucleation alone, at least in a cooperative system. 

All-or-none transitions occur if the chain length is relatively short (n < 15 tripeptide units) and if the cooperativity is high (a < 1) since in this special case, the concentration of intermediates is negligibly low. Besides, in the case of short chains we may conclude that back folding and oligomerization are negligibly small because of the shortness of the chain ends beyond the helical part. A further simplification is the assumption that only one helical sequence exists, which excludes the formation of loops within a helical part, because of reasons of stability. Under these circumstances, only two different products exist in a measurable concentration at equilibrium. 

The lamellar thickening proceeds through many metastable states, each metastable state corresponding to a particular number of folds per chain, as illustrated in Fig. 8. In the original simulations of [22], Kg was monitored. Rg is actually very close to the lamellar thickness due to the asymmetric shape of the lamella. The number of folds indicated in Fig. 8 were identified by inspection of the coordinates of the united atoms. This quantization of the number of folds has been observed in experiments [50], as already mentioned. The process by which a state with p folds changes into a state with p - 1 folds is highly cooperative. The precursor lives in a quiescent state for a substantial time and suddenly it converts into the next state. By a succession of such processes, crystals thicken. If the simulation is run for a reasonably long time, the lamella settles down to the equilibrium number of folds per chain. 

Even more efficient bimetallic cooperativity was achieved by the dinuclear complex 36 [53]. It was demonstrated to cleave 2, 3 -cAMP (298 K) and ApA (323 K) with high efficiency at pH 6, which results in 300-500-fold rate increase compared to the mononuclear complex Cu(II)-[9]aneN at pH 7.3. The pH-metric study showed two overlapped deprotonations of the metal-bound water molecules near pH 6. The observed bell-shaped pH-rate profiles indicate that the monohydroxy form is the active species. The proposed mechanism for both 2, 3 -cAMP and ApA hydrolysis consists of a double Lewis-acid activation of the substrates, while the metal-bound hydroxide acts as general base for activating the nucleophilic 2 -OH group in the case of ApA (36a). Based on the 1000-fold higher activity of the dinuclear complex toward 2, 3 -cAMP, the authors suggest nucleophilic catalysis of the Cu(II)-OH unit in 36b. The latter mechanism is comparable to those of protein phosphatase 1 and fructose 1,6-diphosphatase. 

Flg.1. In the amino acid sequence of KO-42 is encoded its fold and its function as it controls the formation of a hairpin helix-loop-helix motif that dimerizes to form a four-helix bundle. On the surface of the folded motif a reactive site is formed that catalyzes hydrolysis, transesterification and amidation reactions of reactive esters, whereas unfolded peptides are incapable of cooperative catalysis. In addition the values, and thus the reactivities, of the histidine residues are controlled by the fold. The pK of each His residue of KO-42 is shown in the figure and deviate by as much as 1.2 units from that of random coil peptides which is 6.4... 

Applying the heUx coil theory to computational studies of the mPE backbone suggests that above a critical chain length of seven or eight repeat units the backbone can adopt a helical structure. The attachment of additional monomer units would further stabihze the helical structure and increase the cooperativity of the folding reaction [23]. 

The d(CGCGAATTCGCG)2 Duplex. When studied at comparable salt (NaCl) and duplex concentrations, the duplex of d(CGCGAATTCGCG) was found to have about the same Tm by DSC measurements (344 K) and by the temperature-dependent changes in the NMR chemical shifts of nonexchangeable GC and AT protons (345 K). At 0.01 M NaCl, the calorimetric enthalpy was 376 kJmol-1 of duplex, while the van t Hoff enthalpy evaluated from equation (16.28) was found to be 393 kJ-mol-1 of duplex. At a 10-fold higher salt concentration, the calorimetric enthalpy was 427 kJ-mol-1 of duplex, while the van t Hoff enthalpy was 310 kJ-mol-1 of duplex. From the ratio of the van t Hoff enthalpy to the calorimetric enthalpy at the higher salt concentration, it was estimated that 9 out of 12 base-pairs melt together in the cooperative unit, while all melt simultaneously at the low salt concentrations. 

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