Affinity Maturation of B cells

Following infection, B cell receptors are released as Antibodies targeting specific pathogens (Antigens)

Antibodies mature the in Germinal Centers in lymph nodes:

Computational Model:

 

The processes of (hyper)mutation/competition/selection lead to rapid Population Evolution:

Fitness f  governs reproduction rates according to                 

Population mean  is defined by                                          

Mean fitness evolves as                                                     

green bullet Roland Fisher's Fundamental Theorem of Natural Selection

Affinity ω  is a trait correlated with fitness, and (including possibility of mutations) evolves as

green bullet George Price's Equation:                                                

To use the  equations in Affinity Maturation, we need to describe the dependence of fitness on affinity.

Vaccination with synthetic nano-particles coated with target antigens:

  -

Is there an optimal density n for coated target antigens?

Fitness of the maturing B cells depends on receptor affinity ω, and spike density n, say as:

Approximating the affinity distribution by a Gaussian of mean  and variance , Price's equation leads to

With these approximations, the equations are easily integrated to yield:

where         ,    and      .

In a variety of models, optimal affinity is achieved at an intermediate target density:

   

Optimal affinity is achieved at target (spike) density of less than ~1 per area spanned by the BC:

too few targets at low density to be productive, too many targets at high density to be competitive.

The low spike density of HIV may have evolved because of the effects of T helper cell depletion on affinity maturation

A. Amitai, A.K Chakraborty, and M. Kardar, PLOS Computational Biology (2018) (offline)

Broadly neutralizing Antibodies.