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Parikh vectors
4OKU_1 5QGV_1 6TRV_1 Letter Amino acid
17 14 13 P Proline
24 18 23 V Valine
15 10 14 R Arginine
13 7 12 Q Glutamine
28 11 43 G Glycine
13 12 9 K Lycine
6 3 1 M Methionine
27 4 23 S Serine
5 8 16 Y Tyrosine
25 14 33 A Alanine
7 6 14 N Asparagine
23 13 12 D Aspartic acid
16 4 1 C Cysteine
16 11 10 E Glutamic acid
20 11 16 T Threonine
4 0 14 W Tryptophan
12 9 3 H Histidine
16 9 19 I Isoleucine
22 23 14 L Leucine
6 9 7 F Phenylalanine 

4OKU_1|Chains A, B|Micronemal protein MIC2|Toxoplasma gondii (5811)
>5QGV_1|Chain A|Peroxisomal coenzyme A diphosphatase NUDT7|Homo sapiens (9606)
>6TRV_1|Chains A[auth AAA], B[auth BBB]|Uncharacterized protein|Scedosporium apiospermum (563466)
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 \(w=f(c)\)
4OKU , Knot 136 315 0.82 40 194 303 
SIVDALRKRYDTSRGGDANGVDTSGVEDVIQSDSAIGAAEGCTNQLDICFLIDSSGSIGIQNFRLVKQFLHTFLMVLPIGPEEVNNAVVTYSTDVHLQWDLQSPNAVDKQLAAHAVLEMPYKKGSTNTADGLKACKQILFTGSRPGREHVPKLVIGMTDGESDSDFRTVRAAKEIRELGGIVTVLAVGHYVKHSECRSMCGCSGTSDDDSPCPLYLRADWGQLATAIKPMLKEVCKTLPQDAICSDWSAWSPCSVSCGDGSQIRTRTEVSAPQPGTPTCPDCPAPMGRTCVEQGGLEEIRECSAGVCAHHHHHHA
5QGV , Knot 92 196 0.82 38 147 187 
SMLDDAKARLRKYDIGGKYSHLPYNKYSVLLPLVAKEGKLHLLFTVRSEKLRRAPGEVCFPGGKRDPTDMDDAATALREAQEEVGLRPHQVEVVCCLVPCLIDTDTLITPFVGLIDHNFQAQPNPAEVKDVFLVPLAYFLHPQVHDQHYVTRLGHRFINHIFEYTNPEDGVTYQIKGMTANLAVLVAFIILEKKPT
6TRV , Knot 121 297 0.77 40 163 263 
GHMSGVLQISFPAGIAAIRNNSSLRVYEAALDGGVREAQYEGRWAGGKPDNVIATGKIGTPIAATSVGFQYIRVYYVGADNKAREACWDGKGWYTGAFVKDVAPYSSIGAVFLGKNIVVRVYTQNHDNTIQEWVWDSPSTGWTAGANFGAALPGTAIAATSWGAGPYHIRVYFQDTNRNVIESGWDGSGWYTGGLKISNQSPRASLGATSWGESGSSLGIRLYYATQDNLIKEKAWDGGGGWYDGGFQQRSIPGSRVAAIPLPVLRVYLQNGTEVSGITEYAWNSGWVVGQAVLPPA

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 \(c\).

\(|P_{f(4OKU_1)}(2) \setminus P_{f(5QGV_1)}(2)|=100\), \(|P_{f(5QGV_1)}(2) \setminus P_{f(4OKU_1)}(2)|=53\). 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 1:011011000000001101011000110011000011111010000101011100010111001011001100111111111001001110000010101010010110001110111011000100001011010001110100110001101111100100000100101100100111110111110010000000101001000000101101010110110110111001000110011000101101001001010010000010110110100100111110001001110010000111010000001
Pair \(Z_2\) Length of longest common subsequence
4OKU_1,5QGV_1 153 3
4OKU_1,6TRV_1 171 4
5QGV_1,6TRV_1 164 4

Newick tree

 
[
	6TRV_1:86.05,
	[
		4OKU_1:76.5,5QGV_1:76.5
	]:9.55
]

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 \((0.85)(0.8)(\frac{511 }{\log_{20} 511}-\frac{196}{\log_{20}196})=91.2\)
Status Protein1 Protein2 d d1/2
Query variables 4OKU_1 5QGV_1 114 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]