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Parikh vectors
1SBF_1 4BKO_1 3UCE_1 Letter Amino acid
18 25 18 T Threonine
16 7 5 N Asparagine
0 3 0 C Cysteine
24 36 16 L Leucine
1 7 7 M Methionine
30 18 14 S Serine
21 56 30 A Alanine
17 31 12 D Aspartic acid
12 17 15 K Lycine
16 19 4 P Proline
6 5 1 W Tryptophan
5 11 11 Y Tyrosine
6 29 6 R Arginine
6 11 7 Q Glutamine
9 19 10 E Glutamic acid
13 40 22 G Glycine
6 16 6 H Histidine
15 22 13 I Isoleucine
14 12 5 F Phenylalanine
18 33 21 V Valine 

1SBF_1|Chain A|SOYBEAN AGGLUTININ|Glycine max (3847)
>4BKO_1|Chain A|PUTATIVE REDUCTASE BURPS305_1051|BURKHOLDERIA PSEUDOMALLEI (28450)
>3UCE_1|Chains A, B, C, D|Dehydrogenase|Vibrio vulnificus (196600)
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 \(w=f(c)\)
1SBF , Knot 114 253 0.83 38 164 243 
AETVSFSWNKFVPKQPNMILQGDAIVTSSGKLQLNKVDENGTPKPSSLGRALYSTPIHIWDKETGSVASFAASFNFTFYAPDTKRLADGLAFFLAPIDTKPQTHAGYLGLFNENESGDQVVAVEFDTFRNSWDPPNPHIGINVNSIRSIKTTSWDLANNKVAKVLITYDASTSLLVASLVYPSQRTSNILSDVVDLKTSLPEWVRIGFSAATGLDIPGESHDVLSWSFASNLPHASSNIDPLDLTSFVLHEAI
4BKO , Knot 165 417 0.79 40 202 390 
MGSSHHHHHHSSGLVPRGSHMIIKPRVRGFICVTTHPAGCAASVREQIAYVARRGPIERGPKKVLVIGASTGYGLAARIAAAFGAGAATLGVFFERAPADAKPGTAGWYNSAAFHDEAAARGLQATSVNGDAFSDEIKHKTIDAIRRDLGQVDLVVYSVAAPRRTHPKTGVTHQSTLKPIGHAVRLRGIDTDNEAIKETLLQPATPDEIADTVAVMGGEDWRMWIDALDAAGVLADGAKTTAFTYLGEQVTHDIYWNGSIGEAKKDLDRTVLALRGKLAARGGDARVSVLKAVVTQASSAIPMMPLYLSLLFKVMKARGTHEGCIEQVDGLLRDSLYGAQPHVDAEGRLRADRLELDPAVQARVLELWDQVTDDNLYTLTDFAGYKAEFLRLFGFGIDGVDYDAPVEPNVRIPNLIE
3UCE , Knot 100 223 0.80 38 150 212 
MMGSDKTVYVVLGGTSGIGAELAKQLESEHTIVHVASRQTGLDISDEKSVYHYFETIGAFDHLIVTAGSYAPAGKVVDVEVTQAKYAFDTKFWGAVLAAKHGARYLKQGGSITLTSGMLSRKVVANTYVKAAINAAIEATTKVLAKELAPIRVNAISPGLTKTEAYKGMNADDRDAMYQRTQSHLPVGKVGEASDIAMAYLFAIQNSYMTGTVIDVDGGALLG

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 \(c\).

\(|P_{f(1SBF_1)}(2) \setminus P_{f(4BKO_1)}(2)|=50\), \(|P_{f(4BKO_1)}(2) \setminus P_{f(1SBF_1)}(2)|=88\). 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 1:1001010100111001011101011100010101001000101010011011000110110000101101110101010110000110111111111000100011011110000010011110100100010110101110100100100001011000110111000100011110110100000011001101000110110111011011011100001101011001101000101101001110011
Pair \(Z_2\) Length of longest common subsequence
1SBF_1,4BKO_1 138 4
1SBF_1,3UCE_1 152 5
4BKO_1,3UCE_1 166 4

Newick tree

 
[
	3UCE_1:82.80,
	[
		1SBF_1:69,4BKO_1:69
	]:13.80
]

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 \((0.85)(0.8)(\frac{670 }{\log_{20} 670}-\frac{253}{\log_{20}253})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 1SBF_1 4BKO_1 141 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]