Kylie
Defeater of Illogic
What I came up with as a "Fundamental Theorem of Biology" is P = (n^a)*f, where:
P = probability of viable organisms
n = number of nested levels
a = acceleration factor
f = lowest probability of self-assembly of a fundamental element
Initially all I focused on was the 'f' factor.
Given a set of fundamental elements, S, what are the probabilities of self-assembly? For example, suppose S = <P,Q,R,S,T>. Further, suppose we define rules of the form, If C then PQ, meaning that if condition C is met, P and Q assemble (join together). If the rules can be expressed mathematically, the probability of any statement (e.g. If C then PQ) can be calculated.
Given the probabilities of all these statements, a Markov transition matrix can be created. A Markov transition matrix is just an eigenvalue problem, which means that for simple cases, we can derive closed form equations for the probabilities of self-assembly.
If we assume a simple rule such as for any list of fundamental elements, there are equal probabilities of assembling with the element preceding and following it in the list, the equation that results is: f = 1 / (2^(n-2) + sum{i=2,n}(2^(i-2))). Several cases like this can be solved. However, they quickly become so complex as to make closed form solutions impractical and numerical solutions become necessary.
One means of numerical solution is TAM (Tile Assembly Method). If one programs rules for Markov chains as described above, it can be quickly demonstrated that TAM correctly approximates the closed form cases.
In what units do you measure the acceleration factor, and how do you determine the probability of self assembly of a fundamental element? For that matter, how do you even define if something is a fundamental element or not?
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