Thymic Selection
Immunological Tolerance of the T cell repertoire is shaped in the thymus gland:
where a diverse repertoire of thymocites (expressing distinct TCRs, produced by random recombination)
is culled by encounters with self-pMHC, via
Positive selection: TCR must bind sufficiently strongly to at least one self pMHC
(implicated in MHC restriction, and sensitivity).
Negative selection: TCR must not bind any self pMHC too strongly
(deleting autoimmune TCR).
Model for Thymic selection [Kosmrlj, Jha, Huseby, Kardar, & Chakraborty, PNAS 105, 16671 (2008) (offlline)]
Candidate TCR, sequences of N amino-acids, 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)
Binding energy E: Assumepairwise interactions between aligned amino-acids of peptide and TCR (Miazawa-Jernigan matrix J)
What are characteristics of amino-acids in the post-selection TCR repertoire?
Simulations were performed for TCRs of length N=5 and varying values of M
(strong binding) 
(weak binding)
(Typical number of peptide encounters in thymus is 1,000-10,000)
Specificity of TCR comes from cooperative binding of weakly interacting amino-acids (team effort).
Verified in transgenic mice [Huseby, et. al, Nat Immunol 7, 1191 (2006)] 
Created mice expressing only one self peptide (M=1) in their thymus!
Compared specificity of TCRs from notmal and transgenic mice to mutations of a recognized peptide.
Normal mice TCR typically do not recognize peptides after single site mutations,
while transgenic mice TCR are tolerant of most single site mutations, except at a few "hot spots".
Elite controllers of HIV appear to better tolerate mutations of the virus. Is a similar mechanism at play?
A. Košmrlj, ..., A.K. Chakraborty, Nature online (5 May 2010) WebMed, ...
Connection to Statistical Physics:
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: =E_.jpeg)
where .gif)
and .gif)
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:
The above selection condition is reminiscent of the micro-canonical rule 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
0.gif)
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)
Selective enrichment of amino-acids in TCR.
