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# Mechanisms of antibody variability during B-cell development

## Problem

The heavy chain and light chain subunits of antibodies both contain three regions called complementarity determining regions (CDR1, CDR2, and CDR3). CDRs are responsible for the high specificity of the fragment antigen binding (Fab) of antibodies. CDRs are hypervariable among antibodies, allowing for an extensive range of antigens with which different antibodies can interact.
During B-Cell Maturation, the gene for CDR3 of the heavy chain is formed when a random start text, V, end text, start subscript, start text, H, end text, end subscript segment, a random start text, D, end text, start subscript, start text, H, end text, end subscript segment, and a random start text, J, end text, start subscript, start text, H, end text, end subscript segment are joined together (Figure 1, start text, a, end text). A similar process occurs with only start text, V, end text, start subscript, start text, L, end text, end subscript and start text, J, end text, start subscript, start text, L, end text, end subscript segments to form CDR3 of the light chain. To further randomize CDR3, DNA breaks are introduced at the junction sites, leading to nucleotide addition or subtraction before the breaks are sealed. Three different mechanisms cause random nucleotide alteration at the junction sites (Figure 1, start text, b, end text).
P-addition (palindromic-sequence addition) involves the creation of a DNA hairpin at the ends of gene segments, which causes unpairing of nucleotides to form a single strand. Repair enzymes then add complementary nucleotides, generating a new palindromic sequence. N-addition (“N” number of random nucleotides) entails the insertion of a random number of nucleotides (up to 15) in between two gene fragments. Lastly, in a phenomenon known as junctional flexibility, a random number of nucleotides may be lost on the adjoining end of either gene fragment. These random nucleotide alterations come with the risk of non-productive rearrangements, which occur when the altered nucleotide sequence disrupts the reading frame of the heavy chain or light chain gene, resulting in a non-functional antibody. Non-productive rearrangements cause the B-cell to initiate cell death.
Another mechanism for Fab variability is somatic hypermutation. Somatic hypermutation occurs after a B-cell encounters its specific antigen and involves a greatly increased chance of mutation in all CDRs. In a process known as affinity maturation, activated B-cells that develop a greater affinity for their antigen through somatic hypermutation are selected for based on their affinity for antigen presented on the surfaces of follicular dendritic cells. Those that have a weaker affinity for their antigen are signaled to apoptose.
Figure 1: The start text, V, end text, start subscript, start text, H, end text, end subscript, start text, D, end text, start subscript, start text, H, end text, end subscript, and start text, J, end text, start subscript, start text, H, end text, end subscript gene segments in the coding region for the Fab of the heavy chain are randomly selected and joined to create CDR3. P-addition and N-addition, followed by junctional flexibility further increase the hypervariability of the region.
If there were 44 start text, V, end text, start subscript, start text, H, end text, end subscript gene segments, 27 start text, D, end text, start subscript, start text, H, end text, end subscript gene segments, and 6 start text, J, end text, start subscript, start text, H, end text, end subscript gene segments, how many potential SEQUENCES (x) can the antibody heavy chain have?