The work of adhesion between two cells is a measure of the strength of adhesion between cells. The maximum work of adhesion correlates with the minimum surface energy (or energy left over for adhesion). Suppose that there are two cell types, cell type a and cell type b. If we are given that the work of adhesion between two cells of type a is greater than the work of adhesion between two cells of type b (Wa ≥ Wb), then we would like to draw the following conclusion about what would probably happen if we were to mix the two cell types: (Wa + Wb)/2 ≥ Wab ≥Wb. To do this, let us first assume that the work of adhesion is proportional to the number of adhesive sites per unit area of the opposing cell surfaces. Then let the frequency of adhesive sites per unit area for cell of type a be fa and for b be fb. Thus, the probability of apposition of adhesive sites in cell pairs is: 

• fa X fa for cell pairs of type a-a • fb X fb for cell pairs of type b-b 

• fa X fb for cell pairs o type a-b Now let the work of adhesion be related to the frequency of adhesive sites by a constant k: 

• Wa = kfa2 (for a-a pairs) • Wb = kfb2 (for b-b pairs) • Wab = Kfafb

(a) Given that Wa ≥ Wb, what can be said about the frequency of adhesion of sites on a-type cells compared with the frequency of adhesion sites on b-type of cells from the preceding information? 

(b) Manipulate the inequality that you have come up in part (a) to create an expression describing the relationship among fa, fb, fa2 and fb2. 

(c) Manipulate the inequality from part (a) to form a relationship between fa, fb and fb2. 

(d) Combine the expressions formed in parts (b) and (c) and manipulate to achieve the expression describing the behavior of the two cell types when mixed, given that Wa ≥ Wb.