Lecture 2.1

Signals in Sequences

Learning objectives

After this lecture, you should be able to do the following.

Account for why a consensus spelling cannot locate a binding site in a genome, and propose what has to be added to find one.

  1. Given \(4^n\) supplied as a power of ten, determine roughly how many times an \(n\)-base consensus turns up by chance in a genome of stated length.
  2. Attribute the symptoms of a failed promoter scan to the right limit.
  3. Defend the claim that a validated model of a single site still cannot find that site in a genome, and propose what has to be added.

Reason about the detection metrics from the cases each one counts, and read an ROC curve against its landmarks.

  1. Interpret an ROC curve against its landmarks and explain why moving the cutoff only trades one error for the other.
  2. Sort a set of calls into TP, FP, FN, and TN, determine the sensitivity, precision, and false positive rate from those counts, and state which cases each denominator counts.

Read a position weight matrix built from aligned sites, judge which windows it favors, and account for what per-column independence costs.

  1. Define a motif, its consensus, and degeneracy, and state the evidence against a single spelling.
  2. Build an aligned stack of sites and explain why stacking the top-scoring words instead scrambles the columns.
  3. Determine the corrected frequency a pseudo-count gives a column holding a zero, \(p_{a,j} = (n_{a,j} + \alpha)/(n + 4\alpha)\), and predict how raising \(\alpha\) changes which windows score positively.
  4. Read a matrix entry \(M_{a,j} = \log(p_{a,j} / q_a)\) as a comparison of an observed column frequency against the background, determine the score of a window from supplied matrix entries, one per position, and explain why the logarithm lets those entries add rather than multiply.
  5. Analyze why the matrix ranks AAAAAAAA above TATATATA when the assay does the opposite, and attribute the inversion to per-column independence.

State the log observed-over-expected form shared by the scores in this lecture, and what changes from one model to the next.

  1. Define the enrichment score as a rank-based measure bounded by \(-0.5\) and \(+0.5\), and state what values near \(+0.5\) and near 0 mean.
  2. State the form shared by every score in this lecture, \(\log(\text{observed} / \text{expected})\), and identify what changes from one model to the next.

Interpret the composition statistics of a sequence from the form of their definitions, and explain what a single-letter count cannot see.

  1. Explain why single-letter statistics cannot tell ATATATAT from AAAATTTT.
  2. Determine the expected pair count \((L-1) f_a f_b\) from supplied frequencies and the observed-over-expected ratio, and interpret ratios near 1, near 2, and near 0.5.
  3. Given the base counts of a short sequence, determine the base frequencies \(f_a = n_a/L\) and the AT content, and check that the four frequencies add to 1.
  4. Determine the number of windows a scan of width \(w\) produces over a sequence of length \(L\), and predict how widening \(w\) trades resolution for stability.
  5. Differentiate a trend in group medians from a per-sequence predictor.
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