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Parikh vectors
1ARH_1 8FHI_1 8SWM_1 Letter Amino acid
5 15 4 C Cysteine
16 38 8 Q Glutamine
26 47 20 V Valine
10 24 10 Y Tyrosine
20 31 16 D Aspartic acid
27 46 24 E Glutamic acid
30 45 32 G Glycine
18 23 23 K Lycine
15 32 9 P Proline
5 13 1 W Tryptophan
17 39 33 I Isoleucine
20 35 13 F Phenylalanine
21 41 16 S Serine
22 42 16 T Threonine
47 56 35 A Alanine
22 43 11 R Arginine
23 26 16 N Asparagine
6 20 8 H Histidine
38 99 22 L Leucine
8 24 13 M Methionine 

1ARH_1|Chains A, B|ASPARTATE AMINOTRANSFERASE|Escherichia coli (562)
>8FHI_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 5|Oryctolagus cuniculus (9986)
>8SWM_1|Chain A|Ketol-acid reductoisomerase (NADP(+))|Campylobacter jejuni subsp. jejuni (32022)
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 \(w=f(c)\)
1ARH , Knot 169 396 0.85 40 218 370 
MFENITAAPADPILGLADLFRADERPGKINLGIGVYKDETGKTPVLTSVKKAEQYLLENETTKNYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGTGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFDFARQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIRANYSNPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGAVNVAGMTPDNMAPLCEAIVAVL
8FHI , Knot 281 739 0.83 40 305 676 
MGACPPKAKGPWAQLQKLLISWPVGEQDWEQYRDRVNMLQQERIRDSPLLQAAKENDLRLLKILLLNQSCDFQQRGAVGETALHVAALYDNLEAATLLMEAAPELAKEPALCEPFVGQTALHIAVMNQNLNLVRALLARGASVSARATGAAFRRSPHNLIYYGEHPLSFAACVGSEEIVRLLIEHGADIRAQDSLGNTVLHILILQPNKTFACQMYNLLLSYDEHSDHLQSLELVPNHQGLTPFKLAGVEGNTVMFQHLMQKRKHVQWTCGPLTSTLYDLTEIDSWGEELSFLELVVSSKKREARQILEQTPVKELVSFKWKKYGRPYFCVLASLYILYMICFTTCCIYRPLKLRDDNRTDPRDITILQQKLLQEAYVTHQDNIRLVGELVTVTGAVIILLLEIPDIFRVGASRYFGQTILGGPFHVIIITYASLVLLTMVMRLTNMNGEVVPLSFALVLGWCSVMYFARGFQMLGPFTIMIQKMIFGDLMRFCWLMAVVILGFASAFHITFQTEDPNNLGEFSDYPTALFSTFELFLTIIDGPANYSVDLPFMYCITYAAFAIIATLLMLNLFIAMMGDTHWRVAQERDELWRAQVVATTVMLERKMPRFLWPRSGICGYEYGLGDRWFLRVENHHDQNPLRVLRYVEAFKCSDKEDGQEQLSEKRPSTVESGMLSRASVAFQTPSLSRTTSQSSNSHRGWEILRRNTLGHLNLGLDLGEGDGEEVYHFTETSQVAPA
8SWM , Knot 143 330 0.83 40 192 313 
MAITVYYDKDCDLNLIKSKKVAIIGFGSQGHAHAMNLRDNGVNVTIGLREGSVSAVKAKNAGFEVMSVSEASKIADVIMILAPDEIQADIFNVEIKPNLSEGKAIAFAHGFNIHYGQIVVPKGVDVIMIAPKAPGHTVRNEFTLGGGTPCLIAIHQDESKNAKNLALSYASAIGGGRTGIIETTFKAETETDLFGEQAVLCGGLSALIQAGFETLVEAGYEPEMAYFECLHEMKLIVDLIYQGGIADMRYSISNTAEYGDYITGPKIITEETKKAMKGVLKDIQNGVFAKDFILERRAGFARMHAERKNMNDSLIEKTGRNLRAMMPWIS

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 \(c\).

\(|P_{f(1ARH_1)}(2) \setminus P_{f(8FHI_1)}(2)|=40\), \(|P_{f(8FHI_1)}(2) \setminus P_{f(1ARH_1)}(2)|=127\). 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 1:110010111101111110110100011010111110000010011100100100011000000001110111011000001111010111000010010011101110111011100001001110010110000110011101000100010000101011100100101101111010000101101010010011010100111111011001110110001011011111000111100000011100001110011110000100110010111010000111011011101100011011100010010001001001110010001100010111000111010110000110100011101110111011110100111100111111
Pair \(Z_2\) Length of longest common subsequence
1ARH_1,8FHI_1 167 4
1ARH_1,8SWM_1 150 4
8FHI_1,8SWM_1 187 4

Newick tree

 
[
	8FHI_1:92.74,
	[
		1ARH_1:75,8SWM_1:75
	]:17.74
]

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 \((0.85)(0.8)(\frac{1135 }{\log_{20} 1135}-\frac{396}{\log_{20}396})=193.\)
Status Protein1 Protein2 d d1/2
Query variables 1ARH_1 8FHI_1 246 186.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]