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Parikh vectors
5LDW_1 1MIF_1 5DLH_1 Letter Amino acid
6 4 2 M Methionine
9 4 10 F Phenylalanine
3 5 28 Y Tyrosine
0 2 8 H Histidine
6 6 18 I Isoleucine
26 11 16 L Leucine
11 11 28 A Alanine
5 7 20 N Asparagine
2 4 24 D Aspartic acid
6 11 26 S Serine
10 4 25 T Threonine
4 1 3 W Tryptophan
1 5 10 R Arginine
6 3 7 E Glutamic acid
3 9 36 G Glycine
6 8 8 P Proline
4 8 22 V Valine
1 3 0 C Cysteine
3 6 14 Q Glutamine
3 3 11 K Lycine 

5LDW_1|Chain A|NADH-ubiquinone oxidoreductase chain 3|Bos taurus (9913)
>1MIF_1|Chains A, B, C|MACROPHAGE MIGRATION INHIBITORY FACTOR|Homo sapiens (9606)
>5DLH_1|Chain A|Thermolysin|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
5LDW , Knot 57 115 0.78 38 84 111 
MNLMLALLTNFTLATLLVIIAFWLPQLNVYSEKTSPYECGFDPMGSARLPFSMKFFLVAITFLLFDLEIALLLPLPWASQTANLNTMLTMALFLIILLAVSLAYEWTQKGLEWTE
1MIF , Knot 59 115 0.81 40 98 111 
MPMFIVNTNVPRASVPDGFLSELTQQLAQATGKPPQYIAVHVVPDQLMAFGGSSEPCALCSLHSIGKIGGAQNRSYSKLLCGLLAERLRISPDRVYINYYDMNAASVGWNNSTFA
5DLH , Knot 132 316 0.80 38 179 302 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGNGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSQMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK

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 \(c\).

\(|P_{f(5LDW_1)}(2) \setminus P_{f(1MIF_1)}(2)|=53\), \(|P_{f(1MIF_1)}(2) \setminus P_{f(5LDW_1)}(2)|=67\). 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 1:1011111100101101111111111010100000010001101110101110101111110111101011111111110001010011011111111111011001000110100
Pair \(Z_2\) Length of longest common subsequence
5LDW_1,1MIF_1 120 3
5LDW_1,5DLH_1 171 3
1MIF_1,5DLH_1 165 3

Newick tree

 
[
	5DLH_1:90.61,
	[
		5LDW_1:60,1MIF_1:60
	]:30.61
]

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 \((0.85)(0.8)(\frac{230 }{\log_{20} 230}-\frac{115}{\log_{20}115})=36.7\)
Status Protein1 Protein2 d d1/2
Query variables 5LDW_1 1MIF_1 46 45.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]