**Analysis**

The selection condition is equivalent to the choice of the **Extreme Value**:

Probability distribution of the extremum of

Mobjects:

Characteristics of the Extreme Value Distribution :

Starting energyes:

Mean value: ~

Standard deviation: ~

Scaling in the large

Nlimit; (proteome)Due to the shaprpness of the distibution in the large

Nlimit, 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

Namino-acids in the sequenceThe "interaction" depends only on the sum of variances for the individual amino-acids

As such, in the large

Nlimit, 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)