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Parikh vectors
5JTY_1 6NKC_1 2XVW_1 Letter Amino acid
15 17 17 N Asparagine
16 9 36 D Aspartic acid
15 3 31 F Phenylalanine
7 0 35 C Cysteine
7 11 16 H Histidine
22 11 59 K Lycine
10 3 6 M Methionine
4 1 1 W Tryptophan
19 15 61 L Leucine
12 3 24 P Proline
10 9 18 Y Tyrosine
13 13 62 A Alanine
15 3 23 R Arginine
19 2 63 E Glutamic acid
19 18 12 G Glycine
17 14 8 I Isoleucine
20 15 41 V Valine
12 5 20 Q Glutamine
20 19 24 S Serine
20 10 28 T Threonine 

5JTY_1|Chain A|Glutamate receptor ionotropic, NMDA 1,Glutamate receptor ionotropic, NMDA 1|Rattus norvegicus (10116)
>6NKC_1|Chains A, B|Lipase, Lip_vut1|metagenome (256318)
>2XVW_1|Chain A|SERUM ALBUMIN|HOMO SAPIENS (9606)
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 \(w=f(c)\)
5JTY , Knot 135 292 0.87 40 207 287 
GMSTRLKIVTIHQEPFVYVKPTMSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMNFTYEVHLVADGKFGTQERVNNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKGTRITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQNVSLSILKSHENGFMEDLDKTWVRYQECDS
6NKC , Knot 83 181 0.79 38 117 165 
MASHNPVVMVHGIGGADYNFIGIKSYLQSQGWTSSELYAINFIDKTGNNINNAPRLSEYIKRVLNQTGASKVDIVAHSMGGANTLYYIKNLDGADKVGHVVTLGGANRLVTNTAPQNDKISYTSIYSTSDYIVLNSLSKLDGANNVQISGVSHVGLLFSSKVNALIKDGLTASGKHHHHHH
2XVW , Knot 229 585 0.83 40 237 530 
DAHKSEVAHRFKDLGEENFKALVLIAFAQYLQQCPFEDHVKLVNEVTEFAKTCVADESAENCDKSLHTLFGDKLCTVATLRETYGEMADCCAKQEPERNECFLQHKDDNPNLPRLVRPEVDVMCTAFHDNEETFLKKYLYEIARRHPYFYAPELLFFAKRYKAAFTECCQAADKAACLLPKLDELRDEGKASSAKQRLKCASLQKFGERAFKAWAVARLSQRFPKAEFAEVSKLVTDLTKVHTECCHGDLLECADDRADLAKYICENQDSISSKLKECCEKPLLEKSHCIAEVENDEMPADLPSLAADFVESKDVCKNYAEAKDVFLGMFLYEYARRHPDYSVVLLLRLAKTYETTLEKCCAAADPHECYAKVFDEFKPLVEEPQNLIKQNCELFEQLGEYKFQNALLVRYTKKVPQVSTPTLVEVSRNLGKVGSKCCKHPEAKRMPCAEDYLSVVLNQLCVLHEKTPVSDRVTKCCTESLVNRRPCFSALEVDETYVPKEFNAETFTFHADICTLSEKEEQIKKQTALVELVKHKPKATKEQLKAVMDDFAAFVEKCCKADDKETCFAEEGKKLVAASQAALGL

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 \(c\).

\(|P_{f(5JTY_1)}(2) \setminus P_{f(6NKC_1)}(2)|=138\), \(|P_{f(6NKC_1)}(2) \setminus P_{f(5JTY_1)}(2)|=48\). 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 1:1100010110100011101010100100000101010110011001100001101000110000110101110110010100010111010110000100000001011110110101011111101000010010100110001101110010010110010100100011010100001010100010100100010000000110110110000101111001110101000001100101110011111100001100010101100000111001000110000000
Pair \(Z_2\) Length of longest common subsequence
5JTY_1,6NKC_1 186 3
5JTY_1,2XVW_1 172 3
6NKC_1,2XVW_1 198 4

Newick tree

 
[
	6NKC_1:99.16,
	[
		5JTY_1:86,2XVW_1:86
	]:13.16
]

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 \((0.85)(0.8)(\frac{473 }{\log_{20} 473}-\frac{181}{\log_{20}181})=85.5\)
Status Protein1 Protein2 d d1/2
Query variables 5JTY_1 6NKC_1 115 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]