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The answer lies in considering the entropy change for the entire molecule We can explain this best by way of an example Suppose we have a simple, homogenous, double-helical DNA molecule 1000 base pairs long And let s say this DNA molecule has enough energy to melt 300 base pairs The question is which contributes more to a favorable free-energy change, melting 300 bases in a row, melting 2 segments of 150 bases each, 3 segments of 100 bases, etc up to 300 different segments of 1 base pair each We know that from an enthalpy point of view, one long segment of 300 base pairs is favored, because each new segment requires a small amount of additional enthalpy to break the extra stacking interaction required to start a new segment And we know that considering the entropy contribution of each individual base, one long segment is also favored, because again starting a new segment means melting a base with a smaller favorable entropy change compared with melting a base next to an already melted segment But what about the entropy change for the overall molecule To keep the discussion simple, for now, we ignore the small differences in the enthalpy change that result from having multiple melted segments, as compared with melting a single segment This difference in H is small when we are considering only a few segments, since it is just the addition of one more base-stacking interaction per segment, relative to total of at least 300 stacking interactions Even if we consider H resulting from melting 300 separate onebase-pair segments, 300 additional base-stacking interactions is relatively small compared to the large difference in S So for now we will assume that no matter how we arrange 300 unwound bases along a 1000-base-pair molecule the enthalpy level of the molecule is the same (only the entropy is different) With this assumption, we can say that the different ways of arranging 300 unwound bases along a 1000-base-pair molecule represent different ways of achieving the same energy level, and therefore are directly related to the entropy of the molecule The total number of ways to arrange 300 unwound bases and 700 helical bases along a 1000-base-pair molecule is given by W= 1000! (300!) (700!) (10-1)
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This is approximately equal to 5428 3 10263, a rather large number of ways to achieve the same state of 300 unwound base pairs But this large number includes all possible distributions of the 300 unwound base pairs, from those with a single 300-base-pair unwound segment, through those with 300 separate unwound segments In practice not all of these distributions of unwound bases
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are likely to occur Rather there will be some average cooperative unit length somewhere between a single base pair (not cooperative at all) and 300 base pairs Or, in terms of the number of melted segments, somewhere between 300 melted segments (of one base pair each) and a single segment (of 300 base pairs) In order to consider the entropy differences for different numbers of segments, we need to calculate the number of ways of arranging the unwound bases separately for each possible distribution of unwound segments The mathematics gets a little beyond our scope, so we ll just walk through the first few cases to give an idea where it s leading Take the case of a single segment of 300 unwound base pairs How many ways are there to arrange a single segment of 300 unwound base pairs in a 1000-basepair molecule The answer is illustrated in Fig 10-14 There are 701 possible ways to arrange 300 contiguous unwound bases within a 1000-base-pair DNA Therefore the case of a single 300 base pair melted segment represents only 701 of the 5428 3 10263 total number of ways to arrange 300 melted base pairs with any number of segments Now consider the case of two segments of unwound bases, each 150 base pairs in length This is illustrated in Fig 10-15 Let s count the number of ways to arrange two
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FIgure 10-14 Illustration of a 1000-base-pair DNA with 300 contiguously unwound bases This leaves 700 base pairs still in their helical state The 300-base-pair segment can be at the end of the molecule or anywhere in the middle with various numbers of the 700 helical base pairs on either side The illustration shows 14 of the 701 possible ways to have a single unwound segment of 300 base pairs
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