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 :

Starting energyes:            

Mean value:                ~    

Standard deviation:                    ~   

Scaling in the large N limit;                       (proteome)

Due to the shaprpness of the distibution in the large N limit, the seletion condition can be written as:

where  and  are the mean and variance of interactions of the candidateTCR sequence.

The above selection condition is reminiscent of the micro-canonical constraints in Statistical Physics.

The "energy" involves interactions amongst the N amino-acids in the sequence

The "interaction" depends only on the sum of variances for the individual amino-acids

As such, in the large N limit, the probability to select a sequence can be written as a product


where  and  have to be obtained self-consistently from

         and         .

Graphical solution for

Because of the restriction to an energy interval, there is a range of parameters where = 0.

  How well does this work for finite N? (N=5 and M=10.000)

Kosmrlj, Chakraborty, MK, & Shakhnovich, PRL 103, 068103 (2009)