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Parikh vectors
3WXF_1 1FXJ_1 4CTR_1 Letter Amino acid
16 31 25 G Glycine
17 27 24 V Valine
13 8 11 M Methionine
13 19 25 T Threonine
14 9 16 Y Tyrosine
20 23 26 D Aspartic acid
8 3 11 C Cysteine
8 17 18 N Asparagine
20 20 25 E Glutamic acid
16 23 23 I Isoleucine
36 37 36 L Leucine
21 14 27 K Lycine
19 4 19 F Phenylalanine
14 32 20 A Alanine
14 18 22 R Arginine
2 2 13 W Tryptophan
15 9 23 P Proline
23 8 26 S Serine
15 17 18 Q Glutamine
8 10 14 H Histidine 

3WXF_1|Chains A, C|Uncharacterized protein|Danio rerio (7955)
>1FXJ_1|Chains A, B|UDP-N-ACETYLGLUCOSAMINE PYROPHOSPHORYLASE|Escherichia coli (562)
>4CTR_1|Chains A, B|NITRIC OXIDE SYNTHASE, BRAIN|RATTUS NORVEGICUS (10116)
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 \(w=f(c)\)
3WXF , Knot 140 312 0.86 40 209 305 
SHMGLEVMVGKKKGIQGHYNSSYLDSTLFCLFSFSSVLDTVLLRPRSKTDVEYYKETQELLRTEIVNPLRIHGYVCATKIMKLRRILEKVEAASGFTSEQKDPEEFLNILFHHILRVDPLLRLRSAGQKVQDCYFYQIFMDKKDKVMVPTSQQLLEWSFINSDLKFAEAPSCLIIQMPRFGKDFKMFNKIFPSLELDITDLLDDTPLEGGPHQQMELFAVLCIETSHYVAFVKYGSADSAWLFFDSMADRDGGQNGFNIPQVSRCPEVGEYLKMTPEELHALDPKNIQGYARRLLCDAYMCMYQSPTMSLYK
1FXJ , Knot 141 331 0.82 40 183 320 
MLNNAMSVVILAAGKGTRMYSDLPKVLHTLAGKAMVQHVIDAANELGAAHVHLVYGHGGDLLKQALKDDNLNWVLQAEQLGTGHAMQQAAPFFADDEDILMLYGDVPLISVETLQRLRDAKPQGGIGLLTVKLDDPTGYGRITRENGKVTGIVEHKDATDEQRQIQEINTGILIANGADMKRWLAKLTNNNAQGEYYITDIIALAYQEGREIVAVHPQRLSEVEGVNNRLQLSRLERVYQSEQAEKLLLAGVMLRDPARFDLRGTLTHGRDVEIDTNVIIEGNVTLGHRVKIGTGCVIKNSVIGDDCEISPYTVVEDANLAAACTIGPFAR
4CTR , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG

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 \(c\).

\(|P_{f(3WXF_1)}(2) \setminus P_{f(1FXJ_1)}(2)|=107\), \(|P_{f(1FXJ_1)}(2) \setminus P_{f(3WXF_1)}(2)|=81\). 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 1:001110111100011010000001000110110100110011101000001000000001100011011010101010011010011001011011000000100110111001101011101001100100001001110000011110000110101100010110110011101101100101100111010101001100011011100010111110100000111100101001111100110001100110110100010110010101001011010010101001100101010001010100
Pair \(Z_2\) Length of longest common subsequence
3WXF_1,1FXJ_1 188 3
3WXF_1,4CTR_1 181 4
1FXJ_1,4CTR_1 177 3

Newick tree

 
[
	3WXF_1:93.48,
	[
		4CTR_1:88.5,1FXJ_1:88.5
	]:4.98
]

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 \((0.85)(0.8)(\frac{643 }{\log_{20} 643}-\frac{312}{\log_{20}312})=91.9\)
Status Protein1 Protein2 d d1/2
Query variables 3WXF_1 1FXJ_1 116 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]