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Wegner recognized that: “The equilibrium constant (K) for association of a monomer with a polymer is the same at both ends, since the addition of a monomer to each end leads to the same polymer.”; a simple reversible polymer can’t treadmill; ATP hydrolysis is required. GTP is hydrolyzed for microtubule treadmilling.
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain, with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base.
One can obtain an addition chain for from an addition chain for by including one additional sum = +, from which follows the inequality () + on the lengths of the chains for and . However, this is not always an equality, as in some cases 2 n {\displaystyle 2n} may have a shorter chain than the one obtained in this way.
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An addition reaction is the reverse of an elimination reaction, in which one molecule divides into two or more molecules. For instance, the hydration of an alkene to an alcohol is reversed by dehydration. There are two main types of polar addition reactions: electrophilic addition and nucleophilic addition.
In a free-radical addition, there are two chain propagation steps. In one, the adding radical attaches to a multiply-bonded precursor to give a radical with lesser bond order. In the other, the newly-formed radical product abstracts another substituent from the adding reagent to regenerate the adding radical. [3]: 743–744