Partition function cell cluster

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

Although cadherins do not exclusively cluster, the formation of adherens junctions is important for several tissue functions [18]. In epithelial tissues cadherins mediate zonula adherens junctions, which partition the cell membrane into apical and basolateral surfaces. In neuronal cells they mechanically stabilize synaptic junctions by mediating adhesion between the presynaptic and postsynaptic cells. In cardiac myocytes they establish intercalated discs, which both stabilize gap junctional coupling and transmit contractile forces. 

Let us first derive the expression for the cell-cluster partition function. Let us consider I particles (1, 2. .. Q sharing a cell cluster formed by cells a,. .. A. In this cell cluster the molecules move imder the influence of ... 

These partition functions depend only on the number I of cells in the cluster, and their mutual arrangement which will be indicated by the index A. To specify the arrangement k we shall use the notation Wik instead of Pa - A- The set of partition functions. .. Wit has as its first member the usual cell partition function (7.1.1) and as its last member the configurational partition function for the whole liquid (apart from a normalization factor). In effect. 

Formula (7.8.13) expresses the configurational partition function (7.8.6) as the sum of contributions from all possible subdivisions of the iV-ceUs into cell clusters. A similar expression is well known in the theory of imperfect gases where Q is expressed as a sum of contributions from all possible molecular clusters (cf. for example Mayer and Mayer [1940], p. 277). 

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