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Parikh vectors
2YSO_1 8HAK_1 2VDD_1 Letter Amino acid
1 6 11 P Proline
0 4 23 D Aspartic acid
2 7 20 I Isoleucine
7 7 22 G Glycine
3 2 10 H Histidine
1 2 6 M Methionine
10 5 41 S Serine
2 6 24 V Valine
2 18 20 R Arginine
2 2 0 C Cysteine
0 10 31 T Threonine
0 0 1 W Tryptophan
4 7 21 E Glutamic acid
2 12 55 L Leucine
4 8 40 Q Glutamine
2 13 17 K Lycine
2 4 10 F Phenylalanine
1 3 19 Y Tyrosine
0 18 51 A Alanine
1 1 38 N Asparagine 

2YSO_1|Chain A|Zinc finger protein 95 homolog|Homo sapiens (9606)
>8HAK_1|Chains A, E|Histone H3.1|Homo sapiens (9606)
>2VDD_1|Chains A, B, C|OUTER MEMBRANE PROTEIN TOLC|ESCHERICHIA COLI (83333)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
2YSO , Knot 25 46 0.69 32 36 40 
GSSGSSGSREKSHQCRECGEIFFQYVSLIEHQVLHMGQKNSGPSSG
8HAK , Knot 65 135 0.78 38 104 127 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEACEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
2VDD , Knot 182 460 0.80 38 204 417 
MRKLLPILIGLSLSGFSSLSQAENLMQVYQQARLSNPELRKSAADRDAAFEKINEARSPLLPQLGLGADYTYSNGYRDANGINSNATSASLQLTQSIFDMSKWRALTLQEKAAGIQDVTYQTDQQTLILNTATAYFNVLNAIDVLSYTQAQKEAIYRQLDQTTQRFNVGLVAITDVQNARAQYDTVLANELTARNNLDNAVEQLRQITGNYYPELAALNVENFKTDKPQPVNALLKEAEKRNLSLLQARLSQDLAREQIRQAQDGHLPTLDLTASTGISDTSYSGSKTRGAAGTQYDDSNMGQNKVGLSFSLPIYQGGMVNSQVKQAQYNFVGASEQLESAHRSVVQTVRSSFNNINASISSINAYKQAVVSAQSSLDAMEAGFSVGTETIVDVLDATTTLYNAKQELANARYNYLINQLNIKSALGTLNEQDLLALNNALSKPVSTNPELEHHHHHHHH

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(2YSO_1)}(2) \setminus P_{f(8HAK_1)}(2)|=28\), \(|P_{f(8HAK_1)}(2) \setminus P_{f(2YSO_1)}(2)|=96\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1001001000000000001011100101100011011000011001
Pair \(Z_2\) Length of longest common subsequence
2YSO_1,8HAK_1 124 2
2YSO_1,2VDD_1 204 3
8HAK_1,2VDD_1 172 4

Newick tree

 
[
	2VDD_1:10.88,
	[
		2YSO_1:62,8HAK_1:62
	]:40.88
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{181 }{\log_{20} 181}-\frac{46}{\log_{20}46})=46.4\)
Status Protein1 Protein2 d d1/2
Query variables 2YSO_1 8HAK_1 58 37.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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