194

14

The Nature of Living Things

plexes that are also recognized by T-cells, a process that leads to the destruction of

(or the attempt to destroy) the foreign objects (since they “present” antigens to the

T-cells, they are known as antigen-presenting cells, APC).

The number upper NN of foreign antigens that must be recognized by an organism is

very large, perhaps greater than10 Superscript 1610¹⁶, and at the same time there is a smaller number,

upper N prime tilde 10 Superscript 6N ^' ∼10⁶, of self-antigens that must not be recognized. Yet, according to Tonegawa’s

theory, the immunoglobulin and T-cell receptors may only containn tilde 10 Superscript 7n ∼10⁷ different

motifs. Recognition is presumed to be accomplished by a generalized lock-and-key

mechanism involving complementary amino acid sequences. How large should the

complementary region be, supposing that the system has evolved to optimize the task?

Ifupper P Subscript upper SPS is the probability that a random receptor recognizes a random antigen, the value

of its complementupper P Subscript upper F Baseline equals 1 minus upper P Subscript upper SPF = 1 −PS maximizing the product of the probabilities that each

antigen is recognized by at least one receptor and that none of the self-antigens is

recognized (i.e., left parenthesis 1 minus upper P Subscript upper F Superscript n Baseline right parenthesis Superscript upper N Baseline upper P Subscript upper F Superscript n upper N prime(1 −Pⁿ

F^{)N PnN '}

F

) is ⁴⁰

upper P Subscript upper F Baseline equals left parenthesis 1 plus StartFraction upper N Over upper N Superscript prime Baseline EndFraction right parenthesis Superscript negative 1 divided by n Baseline periodPF =

(

1 + ^N

N ^'

)−1/n

.

(14.2)

Using the estimated values fornn,upper NN, andupper N primeN ^', one computesupper P Subscript upper S Baseline almost equals 2 times 10 Superscript negative 6PS ≈2 × 10⁻⁶. Suppose

that the complementary sequence is composed ofmm classes of amino acids and that at

leastcc complementary pairs on a sequence ofss amino acids are required for recogni-

tion. Since the probability of a long match is very small, to a good approximation the

individual contributions to the match can be regarded as being independent. A pair

is thus matched with probability 1 divided by m1/m and mismatched with probability 1 minus 1 divided by m1 −1/m.

Starting at one end of the sequence, runs ofcc matches occur as with probabilitym Superscript negative cm^−c,

and elsewhere they are preceded by a mismatch and can start ats minus cs −c possible sites.

Hence

upper P Subscript upper S Baseline equals left bracket left parenthesis s minus c right parenthesis left parenthesis m minus 1 right parenthesis divided by m plus 1 right bracket divided by m Superscript c Baseline periodPS = [(s −c)(m −1)/m + 1]/m^c .

(14.3)

If s much greater than c greater than 1s >> c > 1, one obtains

c equals log Subscript m Baseline left bracket s left parenthesis m minus 1 right parenthesis divided by m right bracket minus log Subscript m Baseline upper P Subscript upper S Baseline periodc = logm[s(m −1)/m] −logm PS .

(14.4)

Problem. Estimate cc, supposing ss to be a few tens, m equals 3m = 3 (positive, negative, and

neutral residues), and using the numbers given above (because they all enter as

logarithms the exact values are not critical). How does it compare with observation?

Another challenge for the immune system is to be able to detect small increments

of an antigen concentration in the presence of a large background noise. The challenge

can be met by a multidentate strategy, in which a minimum threshold number of

antigens needs to bind in order to trigger a response from the antibody. The molecular

detection efficiency increases with increasing threshold number. ⁴¹

40 Percus et al. (1993).

41 Manghani and Ramsden (2003).