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Parikh vectors
3KOF_1 1LGD_1 4MVL_1 Letter Amino acid
3 0 2 C Cysteine
26 9 9 I Isoleucine
34 26 15 L Leucine
22 24 15 K Lycine
21 19 10 D Aspartic acid
4 7 3 W Tryptophan
16 7 7 R Arginine
15 11 10 Q Glutamine
20 22 12 G Glycine
11 18 12 P Proline
21 19 14 S Serine
38 13 7 A Alanine
26 13 12 E Glutamic acid
9 12 4 H Histidine
6 2 2 M Methionine
6 12 12 F Phenylalanine
18 11 11 T Threonine
13 8 7 Y Tyrosine
17 17 14 V Valine
11 10 10 N Asparagine 

3KOF_1|Chains A, B|Transaldolase B|Escherichia coli K-12 (83333)
>1LGD_1|Chain A|Carbonic anhydrase II|Homo sapiens (9606)
>4MVL_1|Chains A, B, C, D|Neutrophil gelatinase-associated lipocalin|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 \(w=f(c)\)
3KOF , Knot 141 337 0.81 40 183 304 
MGSSHHHHHHSSGLVPRGSHMTDKLTSLRQYTTVVADTGDIAAMKLYQPQDATTNPSLILNAAQIPEYRKLIDDAVAWAKQQSNDRAQQIVDATDKLAVNIGLEILKLVPGRISTEVDARLSYDTEASIAKAKRLIKLYNDAGISNDRILIKLASTWQGIRAAEQLEKEGINCNLTLLFSFAQARACAEAGVFLISPYVGEILDWYKANTDKKEYAPAEDPGVVSVSEIYQYYKEHGYETVVMGASFRNIGEILELAGCDRLTIAPTLLKELAESEGAIERKLSYTGEVKARPARITESEFLWQHNQDPMAVDKLAEGIRKFAIDQEKLEKMIGDLL
1LGD , Knot 112 260 0.79 38 173 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLPTPPLLESVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
4MVL , Knot 94 188 0.87 40 142 183 
QDSTSDLIPAPPLSKVPLQQNFQDNQFHGKWYVVGAAGNVLLREDKDPLKMYATIYELKEDKSYNVTSVGFDDKKCLYKIRTFVPGSQPGEFTLGRIKSEPGGTSWLVRVVSTNYNQHAMVFFKEVAQNRETFNITLYGRTKELTSELKENFIRFSKSLGLPENHIVFPVPIDQCIDGSAWSHPQFEK

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 \(c\).

\(|P_{f(3KOF_1)}(2) \setminus P_{f(1LGD_1)}(2)|=91\), \(|P_{f(1LGD_1)}(2) \setminus P_{f(3KOF_1)}(2)|=81\). 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 1:1100000000001111010010001001000001110010111101001001000101110110110000110011111000000010011010001110111011011110100010101000001011010011010001110000111011001011011001000110001011101101010101111110101101101001000000011100111101001000000010001111101001101101110001011101100110001110001000101010110100001110000011110011011001110000100111011
Pair \(Z_2\) Length of longest common subsequence
3KOF_1,1LGD_1 172 4
3KOF_1,4MVL_1 173 3
1LGD_1,4MVL_1 163 3

Newick tree

 
[
	3KOF_1:87.77,
	[
		1LGD_1:81.5,4MVL_1:81.5
	]:6.27
]

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 \((0.85)(0.8)(\frac{597 }{\log_{20} 597}-\frac{260}{\log_{20}260})=95.0\)
Status Protein1 Protein2 d d1/2
Query variables 3KOF_1 1LGD_1 116 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]