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Parikh vectors
1MJK_1 8HIB_1 6YCJ_1 Letter Amino acid
1 8 3 C Cysteine
9 10 6 K Lycine
6 11 32 P Proline
6 17 22 T Threonine
8 7 42 A Alanine
5 13 26 D Aspartic acid
2 12 9 Q Glutamine
9 24 35 L Leucine
2 11 8 M Methionine
6 12 20 S Serine
4 7 30 V Valine
7 17 27 R Arginine
4 8 11 N Asparagine
4 11 31 G Glycine
3 8 13 H Histidine
2 5 10 W Tryptophan
3 9 13 Y Tyrosine
14 14 37 E Glutamic acid
7 12 20 I Isoleucine
2 14 14 F Phenylalanine 

1MJK_1|Chains A, B|METHIONINE REPRESSOR|Escherichia coli (562)
>8HIB_1|Chain A[auth V]|LIM domain-binding protein 1|Homo sapiens (9606)
>6YCJ_1|Chain A|Aromatic O-demethylase, cytochrome P450 subunit|Amycolatopsis sp. (strain ATCC 39116 / 75iv2) (385957)
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 \(w=f(c)\)
1MJK , Knot 54 104 0.80 40 84 102 
AEWSGEYISPYAEHGKKSEQVKKITVSIPLKVLKILTDERTRRKVNNLRHATNSELLCEAFLHAFTGQPLPDDADLRKERSDEIPEAAKEIMREMGINPETWEY
8HIB , Knot 108 230 0.85 40 167 223 
GIGRHTPYGNQTDYRIFELNKRLQNWTEECDNLWWDAFTTEFFEDDAMLTITFCLEDGPKRYTIGRTLIPRYFRSIFEGGATELYYVLKHPKEAFHSNFVSLDCDQGSMVTQHGKPMFTQVCVEGRLYLEFMFDDMMRIKTWHFSIRQHRELIPRSILAMHAQDPQMLDQLSKNITRCGLSNSTLNYLRLCVILEPMQELMSRHKTYSLSPRDCLKTCLFQKWQRMVAPP
6YCJ , Knot 168 409 0.82 40 219 390 
GPMTTTERPDLAWLDEVTMTQLERNPYEVYERLRAEAPLAFVPVLGSYVASTAEVCREVATSPDFEAVITPAGGRTFGHPAIIGVNGDIHADLRSMVEPALQPAEVDRWIDDLVRPIARRYLERFENDGHAELVAQYCEPVSVRSLGDLLGLQEVDSDKLREWFAKLNRSFTNAAVDENGEFANPEGFAEGDQAKAEIRAVVDPLIDKWIEHPDDSAISHWLHDGMPPGQTRDREYIYPTIYVYLLGAMQEPGHGMASTLVGLFSRPEQLEEVVDDPTLIPRAIAEGLRWTSPIWSASARISTKPVTIAGVDLPAGTPVMLSYGSANHDTGKYEAPSQYDLHRPPLPHLAFGAGNHACAGIYFANHVMRIALEELFEAIPNLERDTREGVEFWGWGFRGPTSLHVTWEV

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 \(c\).

\(|P_{f(1MJK_1)}(2) \setminus P_{f(8HIB_1)}(2)|=49\), \(|P_{f(8HIB_1)}(2) \setminus P_{f(1MJK_1)}(2)|=132\). 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 1:10101001010100100000100101011101101100000000100100100001100111011010111001010000000110110011001110100100
Pair \(Z_2\) Length of longest common subsequence
1MJK_1,8HIB_1 181 3
1MJK_1,6YCJ_1 191 4
8HIB_1,6YCJ_1 200 5

Newick tree

 
[
	6YCJ_1:10.08,
	[
		1MJK_1:90.5,8HIB_1:90.5
	]:9.58
]

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 \((0.85)(0.8)(\frac{334 }{\log_{20} 334}-\frac{104}{\log_{20}104})=71.4\)
Status Protein1 Protein2 d d1/2
Query variables 1MJK_1 8HIB_1 94 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]