Model & Analysis

Focus on the energetics of binding of TCR to peptide:

model

Assume a pairwise interaction between the peptide amino-acids and corresponding ones on the TCR, resulting in binding energy

                     (Miazawa-Jernigan matrix J)

A. Kosmrlj, A. Jha, E.S. Huseby, M. Kardar, & A.K. Chakraborty, PNAS 105, 16671 (2008) (offlline)


Thymic selection of TCRs is modeled using the interaction E as follows:

Candidate TCR sequences are randomly generated. (This ignores potential V(D)J generation bias.)

The binding energy of any TCR is calculated against a repertoire of M self-peptides.

(Experimental justification)


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

     

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)