Bioinformatics formulas

$P_{\text{error}}$ is the probability that the base call is wrong. Q = 20 means $P=10^{-2}$ (1 error in 100, 99% accuracy); Q = 30 means $P=10^{-3}$ (1 in 1000, 99.9%). In the common Phred+33 encoding, the quality character's ASCII code equals Q + 33.

$q_{ab}$ = observed frequency of residues a and b aligned in related sequences; $p_a$, $p_b$ = background frequencies of each residue; $\lambda$ = scaling constant (values are then rounded to integers). A positive score means the pair occurs more often than chance (a conservative or likely substitution), zero means as often as chance, and a negative score means less often than chance.

g = gap length; α = gap-opening penalty (the cost of the first gap position); β = gap-extension penalty (the cost of each further position), with α > β. Convention: some programs write the same idea as $W=o+eg$, where the opening cost is charged on top of a per-position cost for every position. Always check which form a question uses.

F(i,j) = best score for aligning the first i residues of X with the first j residues of Y; $s(x_i,y_j)$ = match/mismatch or matrix score; d = linear gap penalty, written as a positive number and subtracted. Initialisation: F(0,0) = 0, F(i,0) = −i·d, F(0,j) = −j·d. For sequences of lengths m and n this takes O(mn) time and memory.

The same as global alignment except for the 0 option and an initialisation of F(i,0) = F(0,j) = 0. The local alignment score is the largest F(i,j) anywhere in the matrix, and traceback stops at a cell with value 0.

E = the number of alignments with score ≥ S expected by chance alone in a search of this size; m = query length; n = total length of the database (in residues); S = the RAW alignment score; K and λ = statistical parameters of the scoring system. E rises in proportion to the search space (m·n) and falls exponentially as S rises. A smaller E-value means a more significant hit.

The bit score S′ normalises the raw score so that scores from different scoring systems can be compared. Each extra bit halves E. P is the probability of at least one chance hit scoring ≥ S; when E is small (below about 0.01), P ≈ E.

The distance from the merged cluster (i,j) to cluster k is the average of the two distances, weighted by the numbers of sequences $n_i$ and $n_j$ in each cluster. This equals the plain average over all member pairs. Using the simple average $(d_{i,k}+d_{j,k})/2$ without weights is a different method, WPGMA.

Valid for n ≥ 3 (unrooted) and n ≥ 2 (rooted); !! is the double factorial (the product of odd numbers). Check: n = 4 gives 3 unrooted and 15 rooted trees; n = 5 gives 15 unrooted and 105 rooted trees. The number of rooted trees for n taxa equals the number of unrooted trees for n + 1 taxa.

S = stoichiometric matrix (rows = metabolites, columns = reactions); v = vector of reaction fluxes. At steady state the net rate of change of every internal metabolite is zero, giving one linear equation per metabolite. The bounds encode irreversibility and uptake limits. c selects the objective (e.g. biomass or product flux). The problem is solved by linear programming, not by simulating dynamics. Before the bounds are applied, the steady-state solutions form a space of dimension (number of reactions) − rank(S).