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Parikh vectors
5GNE_1 4LBM_1 8VDK_1 Letter Amino acid
25 6 30 S Serine
4 1 1 W Tryptophan
21 9 35 G Glycine
26 10 25 I Isoleucine
15 10 17 P Proline
4 1 0 C Cysteine
24 8 34 K Lycine
16 8 11 F Phenylalanine
20 4 14 Q Glutamine
20 6 19 E Glutamic acid
5 2 6 M Methionine
18 3 29 Y Tyrosine
24 13 44 V Valine
26 6 18 A Alanine
12 9 3 R Arginine
22 14 50 N Asparagine
26 5 45 T Threonine
27 7 37 D Aspartic acid
9 4 3 H Histidine
31 13 23 L Leucine 

5GNE_1|Chains A, B|Leucine aminopeptidase|Legionella pneumophila (272624)
>4LBM_1|Chain A|Galectin-3|Homo sapiens (9606)
>8VDK_1|Chain A|Serine-aspartate repeat-containing protein D|Staphylococcus aureus subsp. aureus JH1 (359787)
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 \(w=f(c)\)
5GNE , Knot 162 375 0.85 40 220 355 
GSQSPEEDRIIVSNCLYEQLKDKARLIAQSDHFSFIAIPIDENITNTLQKMRPVCGNYLNAEPYIQQTSNRFLDSKKLLDKLTSYHSISTANPYPINHEAQVHSLFEQIDPAKIWQTNQHLTSYINRSAKSRTGVEAAQWFKQQFDTLAQDYGRKDVESYFVKTGNKFIQPSVVTVIGKDKPGEAIVIGAHIDTLDGNMPGADDDSSGISVELEMARVVFSSNFELNRPIYFIAYAAEERGLIGSGYVVQDFLQKKIPVKAVMQLDQAGYRANAKDQTIWLLKDYVDKGLTEFTAELLTRYVKTPVGYTKCGYACSDHVNWTNEGFKTTYPSATTLDDDNPYVHTSNDTLDILNLEHMVNFTKLGLAFIVELGLN
4LBM , Knot 69 139 0.81 40 113 137 
GPLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI
8VDK , Knot 173 444 0.79 38 191 409 
SKNVNDLITSNTTLTVVDADNSKTIVPAQDYLSLKSQITVDDKVKSGDYFTIKYSDTVQVYGLNPEDIKNIGDIKDPNNGETIATAKHDTANNLITYTFTDYVDRFNSVKMGINYSIYMDADTIPVDKKDVPFSVTIGNQITTTTADITYPAYKEADNNSIGSAFTETVSHVGNVEDPGYYNQVVYVNPMDKDLKGAKLKVEAYHPKYPTNIGQINQNVTNIKIYRVPEGYTLNKGYDVNTNDLVDVTDEFKNKMTYGSNQSVNLDFGDITSAYVVMVNTKFQYTNSESPTLVQMATLSSTGNKSVSTGNALGFTNNQSGGAGQEVYKIGNYVWEDTNKNGVQELGEKGVGNVTVTVFDNNTNTKVGEAVTKEDGSYLIPNLPNGDYRVEFSNLPKGYEVTPSKQGNNEELDSNGLSSVITVNGKDNLSADLGIYKPKYNLGDY

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 \(c\).

\(|P_{f(5GNE_1)}(2) \setminus P_{f(4LBM_1)}(2)|=145\), \(|P_{f(4LBM_1)}(2) \setminus P_{f(5GNE_1)}(2)|=38\). 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 1:100010000111000100010001011100001011111100010001001011010010101010000001100001100100000100101011000101001100101101100000100010001000011011011000100110001000100011001001101011011100011011111101001010111100000110101011011100010100110111011000111101011001100011101110100110010100001111000100110010101100010011100001010000101000110000101001000010100000010110100110100111111101110
Pair \(Z_2\) Length of longest common subsequence
5GNE_1,4LBM_1 183 3
5GNE_1,8VDK_1 137 4
4LBM_1,8VDK_1 166 3

Newick tree

 
[
	4LBM_1:92.79,
	[
		5GNE_1:68.5,8VDK_1:68.5
	]:24.29
]

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 \((0.85)(0.8)(\frac{514 }{\log_{20} 514}-\frac{139}{\log_{20}139})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 5GNE_1 4LBM_1 140 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]