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Where does the 299 come from This is the number of ways to split 300 unwound base pairs into two segments The smallest number of base pairs that can be in a segment is one so we start counting with a segment of 1 and a segment of 299 From there we go to 2 and 298, followed by 3 and 297, and so on, until we get to 299 and 1 But isn t 1 and 299 the same as 299 and 1 The answer is yes and no yes they are the same if order does not matter No, they are not the same, if order does matter in a homogenous sequence dNA order does not matter so for the case of homogenous sequence dNA we are double counting But most DNA sequences are heterogenous Certainly natural DNA are, so for most DNA we are not double counting The important point to realize here is that splitting the unwound base pairs up into multiple segments so greatly increases the number of ways of arranging them along the molecule, that it doesn t make a significant difference whether we are double counting (the additional factor of how to split up the base pairs among the segments) or not
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It s easy to see from this, that unwinding more, smaller segments results in a much larger positive entropy change than unwinding just a few or one contiguous segment So when we look at the entropy change for the whole molecule, having more segments is more favorable to the Gibbs energy change than having just a single segment However, entropy considerations at the individual base level have the opposite effect These two opposing effects balance each other out The end result is that DNA unwinds in multiple segments, of some average cooperative unit length To summarize, there are two competing entropy effects They are (1) the entropy of coil state base pairs related to their freedom of movement favoring longer contiguous unwound segments and (2) the entropy due to the number of ways of arranging unwound bases along the molecule This latter entropy favors more, shorter, unwound segments It is possible to calculate the balance between these two effects, after making some assumptions about the enthalpy change, to arrive at a theoretical cooperative unit length which can be compared to experimental observations
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The Assumption of the Same Energy State
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Now let s deal with the assumption that the energy of the molecule is the same regardless of how we arrange the unwound bases One of the ways we can deal with this assumption is by melting fewer base pairs when we need extra enthalpy to break the extra stacking interactions If we do this, we are not so much removing the assumption but actually strengthening the basis for it The technique is this: We hold the energy of the molecule constant by melting fewer base pairs in the case of those arrangements that would otherwise require more energy to break the additional stacking interactions By melting fewer base pairs we can keep the energy constant In this way we can assume that the energy of the molecule is constant for all of the different arrangements, with no doubts about that assumption If so, then all of the various arrangements (for a given energy level) truly represent different ways of achieving the same energy state of the molecule, and therefore truly represent the entropy state of the molecule So, for example, if the molecule has enough enthalpy to overcome 300 stacking interactions, we can unwind 300 contiguous bases at one end of the molecule If, as another arrangement, we unwind ten separate segments (instead of one), then we have ten additional stacking interactions to overcome Given the same amount of enthalpy, we can only unwind 290 base pairs So we would unwind 10 segments of 29 base pairs each (instead of 10 segments of 30 base pairs each) The same argument applies as we increase the number of segments In the extreme case we can melt a total 150 base pairs in 150 separate segments
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