Analysis
What is the shortest peptide length that ensures sampling the entire (self) proteome?
In principle all human peptides can be recognized from sequences of length 5-6.
The selection condition is equivalent to the choice of the Extreme Value:
Probability distribution of the extremum of M objects:
Characteristics of the Extreme Value Distribution .gif)
:
Mean value: 
~ .gif)
Standard deviation: 
~ .gif)
Scaling in the large N limit: 
Due to the shaprpness of the distibution in the large N limit, the seletion condition can be written as:
where .gif)
and .gif)
are the mean and variance of interactions of the candidateTCR sequence.
The above condition is reminiscent of the micro-canonical constraints in Statistical Physics,
and it can be proved rigorously that in the large N limit, the probability to select a sequence is
where 
and 
have to be obtained self-consistently from
and 
Graphical solution:
0.gif)
How well does this work for finite N? (N=5 and M=10.000)
