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 :

Binding energies of a particular TCR sequencs:  

Extremal mean value:      ~     ,

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

Extremal standard deviation:        ~   

Note 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:

[ and  are the mean and variance of interactions of the candidate TCR 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)

A. Kosmrlj, A.K. Chakraborty, M.K., & E. Shakhnovich, PRL 103, 068103 (2009)

More elaborate variants of the above model can be studies (Hanrong Chen), such as

How does non-uniform usage of TCR sites, modify selection bias at different sites?

[B.D. Stadinski, ..., A.K. Chakraborty & E. Huseby, Nature Immunology (2006)]

Different sites could be in contact with the same amino-acid peptide. What are subsequent correlations between amino-acids at such sites?