Cooperativity partition function

In this book, we define cooperativity in probabilistic terms. This is not the most common or popular definition, yet it conveys the spirit and essence of what researchers mean when they use this term. Since the partition function embodies the probabilities of the occupancy events, the definition of cooperativity can... 

The possibility to calculate the anomalous properties of water quantitatively with a simple two step function of the orientations partition function of water molecules gives the possibility to estimate the size of the network of H-bonded molecules in liquid water too. We have to take into account the cooperative mechanism of H-bonds and the preference of C2v symmetry of intermolecular arrangements in... 

To generate sharp transitions the interaction term T in Eq. (10.2) must be large. An alternative approach has been used by Sorai and Seki34) following the theory of heterophase fluctuations. They assume cooperative domains of LS and HS molecules, resp., with a strong interaction within the domains, so that the complexes change spin simultaneously. For n atomes per domain the partition function becomes... 

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... 

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. 

However, using good quality ab initio calculations (extended basis set, electron correlation included, some relativistic effects accounted for) appropriate partition functions can be generated consequently, all the thermodynamic functions are evaluable (for an isolated pair of LS and HS molecules). Unfortunately, cooperativity effects in the solid (condensed) phase contribute to the thermodynamic functions and a proper estimation of such contributions is of extreme difficulty. 

The above-mentioned results may be alternatively described in terms of the one dimensioned Ising model in order to extract the feature of the interactions. There are several theories that treat DNA-ligand interactions, but one of the simplest was conveniently chosen to express cooperative nature of the phenomena. The phosphate groups on DNA u e viewed as an array of binding sites along the poly(nucleotide) helices, each site being occupied or vacant. The partition function, z, for such system is written as. 

C)ur aim is to compute the partition function, based on the combinatorics for counting the numbers of arrangements of H and C units in the chain. We will consider three different models for the partition function. To show the nature of cooperativity, it is useful to start with a model that has no cooperativity. 

Suppose the helix-coil system has only two possible states CCCC...C or HHH. ..H. Assume that all other states have zero probability. Maximum cooperativity means that if one monomer is H, all are H if one monomer is C, all are C. Now the partition function for each of the two states is q - or q, so the partition function for the system is... 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

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