Proteomics

Definition (Amino acid). An amino acid is a molecule with an amino group ($-\text{NH}_2$), a carboxyl group ($-\text{COOH}$), and a variable side chain (R group) attached to a central $\alpha$-carbon. There are 20 canonical amino acids, grouped by side chain chemistry: nonpolar/aliphatic (Gly, Ala, Val, Leu, Ile, Pro, Met), aromatic (Phe, Trp, Tyr), polar uncharged (Ser, Thr, Cys, Asn, Gln), positively charged (Lys, Arg, His), negatively charged (Asp, Glu). Each has a three-letter code (e.g., Ala) and a one-letter code (e.g., A).

Definition (Polypeptide, protein). A peptide bond is a covalent bond formed between the carboxyl group of one amino acid and the amino group of the next, with loss of a water molecule. A polypeptide is a linear chain of amino acids joined by peptide bonds. A protein is a polypeptide (or assembly of polypeptides) folded into a specific 3D conformation to carry out a biological function. The primary structure is the amino acid sequence, written from N-terminus (free amino group) to C-terminus (free carboxyl group).

Definition (Genetic code). The genetic code maps each of the 64 possible codons (RNA triplets) to one of 20 amino acids or a stop signal. Three codons are stop codons (UAA, UAG, UGA). The code is degenerate: most amino acids are encoded by 2-6 synonymous codons. Degeneracy is concentrated at the third codon position (wobble position): two codons differing only at position 3 often encode the same amino acid, because a tRNA anticodon can tolerate non-Watson-Crick pairing at that position. The code is nearly universal across all life.

Definition (Open reading frame). An open reading frame (ORF) is a contiguous sequence of codons beginning with a start codon (AUG, encoding Met) and ending with a stop codon. In a double-stranded DNA sequence of length $n$, there are six reading frames: three on each strand, shifted by 0, 1, or 2 nucleotides. ORF prediction uses codon usage bias, ORF length, and conservation to distinguish true protein-coding genes from random ORFs, which occur frequently by chance in long sequences.

Definition (Local alignment -- Smith-Waterman). Given two sequences $a = a_1 \cdots a_m$ and $b = b_1 \cdots b_n$ over an amino acid alphabet, a local alignment pairs a contiguous substring of $a$ with a contiguous substring of $b$ to maximize a score. The Smith-Waterman algorithm finds the optimal local alignment in $O(mn)$ time via the recurrence: for $1 \leq i \leq m$, $1 \leq j \leq n$, $$H(i,j) = \max\bigl(0,\; H(i-1,j-1)+s(a_i,b_j),\; H(i-1,j)-g,\; H(i,j-1)-g\bigr)$$ with $H(i,0) = H(0,j) = 0$, where $s(a_i,b_j)$ is the substitution score (from BLOSUM62 for proteins) and $g > 0$ is the linear gap penalty. The optimal score is $\max_{i,j} H(i,j)$, and the alignment is recovered by traceback from that cell.

Definition (BLAST). BLAST (Basic Local Alignment Search Tool) is a heuristic for fast database search that avoids full Smith-Waterman against every sequence. For protein search (BLASTP): (1) Seed: enumerate all words of length $w = 3$ in the query; for each query word, find all words in the amino acid alphabet scoring $\geq T$ against it under BLOSUM62 -- these are the neighborhood words. (2) Scan: scan the database for exact matches to neighborhood words. (3) Extend: extend each hit in both directions (without gaps first, then with gaps); report alignments of score $\geq S$.

Definition (E-value). The E-value of an alignment with score $S$ is the expected number of alignments of equal or better score that would be found by chance in a database of the same size. Under the Karlin-Altschul model: $$E = K \cdot m \cdot n \cdot e^{-\lambda S}$$ where $m$ is the query length, $n$ is the total database size in residues, and $K$, $\lambda$ are parameters determined by the scoring matrix and gap costs. A small E-value (e.g. $E < 10^{-5}$) indicates a statistically significant hit unlikely to arise by chance.

Definition (Peptide mass). The monoisotopic mass of a peptide of $n$ residues $r_1, \ldots, r_n$ is: $$M = \sum_{i=1}^{n} m_{r_i} + 18.010565 \text{ Da}$$ where $m_{r_i}$ is the monoisotopic residue mass of amino acid $r_i$ and $18.010565$ Da accounts for the water molecule completing the termini. The instrument measures the $m/z$ ratio of the protonated ion: $m/z = (M + z \cdot 1.007276) / z$, where $z$ is the charge state.

Definition (b-ions and y-ions). Fragmentation of a peptide $r_1 r_2 \cdots r_n$ at the peptide bond between positions $k$ and $k+1$ produces two ion series. b-ions retain the N-terminus: $$b_k = \sum_{i=1}^{k} m_{r_i} + 1.007276 \text{ Da}$$ y-ions retain the C-terminus (indexing from the C-terminal end): $$y_k = \sum_{i=n-k+1}^{n} m_{r_i} + 18.010565 + 1.007276 \text{ Da}$$ Together, the $b_1, \ldots, b_{n-1}$ and $y_1, \ldots, y_{n-1}$ series form a mass ladder that can be read to reconstruct the peptide sequence.

Definition (Peptide-spectrum match and FDR). A peptide-spectrum match (PSM) is a candidate assignment of a peptide sequence to an observed MS2 spectrum. A score quantifies agreement between the theoretical and observed spectra; a simple choice is the number of matching fragment ion masses within a tolerance $\delta$ (typically $\delta = 0.02$ Da). To estimate the false discovery rate (FDR), the target-decoy approach concatenates the real database with a decoy database of reversed sequences. At score threshold $\tau$: $$\widehat{\text{FDR}}(\tau) = \frac{\#\text{decoy PSMs} \geq \tau}{\#\text{target PSMs} \geq \tau}$$ The threshold is tuned so that $\widehat{\text{FDR}} \leq 0.01$ (1% FDR is standard).

Definition (Structural hierarchy). Protein structure is described at four levels. Primary structure: the amino acid sequence. Secondary structure: local regular structures stabilized by backbone hydrogen bonds -- chiefly $\alpha$-helices and $\beta$-sheets. Tertiary structure: the complete 3D fold of a single polypeptide, stabilized by hydrophobic packing, disulfide bonds, salt bridges, and hydrogen bonds. Quaternary structure: the arrangement of multiple polypeptide chains (subunits) into a functional complex.

Definition (Ramachandran plot). The backbone conformation of each residue is described by two dihedral angles: $\phi$ (rotation around the N-$C_\alpha$ bond) and $\psi$ (rotation around the $C_\alpha$-C bond), both in $(-180°, 180°]$. The Ramachandran plot is a scatter plot of $(\phi, \psi)$ values across all residues of a structure. Most residues cluster in sterically allowed regions: $\alpha$-helices ($\phi \approx -57°$, $\psi \approx -47°$) and $\beta$-sheets ($\phi \approx -120°$, $\psi \approx +130°$). Residues in disallowed regions indicate errors in the model.

Definition (Homology modeling). Homology modeling builds a 3D model of a target protein by aligning it onto a template protein of known structure. It is justified by the observation that proteins sharing more than ~30% sequence identity over their full length nearly always adopt the same overall fold. The pipeline: (1) search PDB for homologous templates using BLAST or PSI-BLAST; (2) align target and template sequences; (3) copy backbone coordinates from the template; (4) model loop regions and side chains; (5) energy minimize. Below 30% identity, fold recognition (threading) methods are needed.

Definition (RMSD). Given two sets of $n$ corresponding atom positions $r_1, \ldots, r_n$ and $r_1', \ldots, r_n'$ in $\mathbb{R}^3$, the root mean square deviation is: $$\text{RMSD} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \|r_i - r_i'\|^2}$$ RMSD measures the average distance between corresponding atoms after optimal superposition. It is typically computed over $C_\alpha$ atoms only. RMSD is sensitive to a few large deviations (flexible loops or termini) and depends on the choice of rotation and translation; the Kabsch algorithm finds the rotation that minimizes it.

Definition (TM-score). The TM-score is a length-normalized structural similarity measure less sensitive to local errors than RMSD: $$\text{TM-score} = \frac{1}{L_{\text{ref}}} \sum_{i=1}^{L_{\text{ali}}} \frac{1}{1 + (d_i / d_0)^2}$$ where $L_{\text{ref}}$ is the length of the reference protein, $L_{\text{ali}}$ is the number of aligned residue pairs, $d_i$ is the distance between the $i$-th aligned pair after superposition, and $d_0 = 1.24 (L_{\text{ref}} - 15)^{1/3} - 1.8$ Å normalizes by protein size. TM-score lies in $(0, 1]$. A TM-score $> 0.5$ generally indicates the same fold; $< 0.17$ is comparable to a random pair.

Definition (PPI network). A protein-protein interaction network (interactome) is a graph $G = (V, E)$ where each vertex $v \in V$ is a protein and each edge $\{u,v\} \in E$ represents a detected or predicted physical interaction. Edges may carry a confidence weight $w : E \to [0,1]$. The degree $\deg(v)$ is the number of interaction partners of $v$. A hub protein has anomalously high degree. PPI networks are approximately scale-free: the degree distribution follows a power law $P(\deg = k) \propto k^{-\gamma}$, with a small number of highly connected hubs and many low-degree proteins.

Definition (Centrality measures). The degree centrality of $v$ in a network of $n$ proteins is $C_d(v) = \deg(v)/(n-1)$. The betweenness centrality is: $$C_b(v) = \sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$$ where $\sigma_{st}$ is the total number of shortest paths from $s$ to $t$ and $\sigma_{st}(v)$ is the number passing through $v$. High betweenness identifies bridge proteins between functional modules. The clustering coefficient of $v$ is: $$C(v) = \frac{2 \,|\{\{i,j\} \in E : i,j \in N(v)\}|}{\deg(v)\,(\deg(v)-1)}$$ measuring how interconnected $v$'s neighbors are.

Definition (Protein domain and Pfam). A protein domain is a conserved sequence and structural unit that folds independently and carries a specific function. Most proteins are modular: they consist of one or more domains connected by flexible linkers. Pfam is a database of protein families and domains, each represented by a profile hidden Markov model (profile HMM) built from a curated multiple sequence alignment. Scanning a query sequence against Pfam identifies which known domains it contains, providing functional hypotheses without requiring experimental characterization.

Definition (Profile HMM). A profile HMM of length $L$ has three states per column: match ($M_k$), insert ($I_k$), and delete ($D_k$) for $k = 1, \ldots, L$. Match states have emission distributions over the 20 amino acids encoding the conservation at that alignment column; insert states model insertions; delete states are silent. Transition probabilities connect consecutive states. The probability of a sequence $x = x_1 \cdots x_n$ is computed by the forward algorithm in $O(nL)$ time; the most likely alignment is found by Viterbi decoding in the same complexity.

Definition (Gene Ontology). The Gene Ontology (GO) is a controlled vocabulary for annotating gene products across three independent ontologies: Biological Process (BP): the biological program in which the protein participates (e.g., "DNA repair", "glycolysis"); Molecular Function (MF): the biochemical activity at the molecular level (e.g., "kinase activity", "DNA binding"); Cellular Component (CC): where the protein exerts its function (e.g., "nucleus", "ribosome"). GO terms form a directed acyclic graph: if a protein is annotated with a term, it is implicitly annotated with all its ancestors (true path rule).