When you read a theoretical paper, it is easy to nod and agree with statements that really depend upon some unwritten mathematical steps. It�s good to get in the habit of trying to fill in these steps, and asking for help when you need it. Here I�ll try to guide you with some questions based on Hopfield�s classic 1974 paper on kinetic proofreading.

Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity. JJ Hopfield, Proc Nat Acad Sci (USA) 71, 4135–4139 (1974).

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1. Although we covered this in class, be sure you understand (using the notation in the paper) how to get to Eq [3], and the sentence after this equation about the minimum error fraction.

2. Go through the algebra to derive Eq [5] from the kinetic scheme in Eq [4].

3. Explain the meaning and origin of the �equilibrium constraint� in Eq [6].

4. After Eq [6] there is the statement that, with the equilibrium constraint, the error fraction in Eq [5] never falls below the minimum error fraction from the single step kinetic scheme. Can you show that this is true, mathematically? That is, can you show that the error fraction in Eq [5], with parameters constrained as in Eq [6], has a minimum value? Intuitively, why should this be true?

5. The text immediately following Eq [9] is the critical piece, where �proofreading� is established and the error probability from the single step scheme gets squared. Write out all the equations that go with these words!