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Parikh vectors
3ISD_1 9GDJ_1 1EGQ_1 Letter Amino acid
13 12 7 Q Glutamine
14 19 9 P Proline
36 3 8 K Lycine
7 8 5 M Methionine
19 21 22 T Threonine
21 20 13 D Aspartic acid
27 13 5 E Glutamic acid
9 12 4 H Histidine
26 24 14 L Leucine
14 15 6 F Phenylalanine
21 24 37 S Serine
1 4 2 W Tryptophan
18 16 19 V Valine
19 21 12 R Arginine
12 3 17 N Asparagine
22 23 33 G Glycine
23 8 11 I Isoleucine
12 11 17 Y Tyrosine
18 32 33 A Alanine
3 2 5 C Cysteine 

3ISD_1|Chain A|DNA polymerase beta|Homo sapiens (9606)
>9GDJ_1|Chains A, B, C|Methyltransferase|Streptomyces griseoviridis (45398)
>1EGQ_1|Chain A|PROTEINASE K|Engyodontium album (37998)
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 \(w=f(c)\)
3ISD , Knot 146 335 0.84 40 199 321 
MSKRKAPQETLNGGITDMLTELANFEKNVSQAIHKYNAYRKAASVIAKYPHKIKSGAEAKKLPGVGTKIAEKIDEFLATGKLRKLEKIRQDDTSSSINFLTRVSGIGPSAARKFVDEGIKTLEDLRKNEDKLNHHQRIGLKYFGDFEKRIPREEMLQMQDIVLNEVKKVDSEYIATVCGSFRRGAESSGDMDVLLTHPSFTSESTKQPKLLHQVVEQLQKVHFITDTLSKGETKFMGVCQLPSKNDEKEYPHRRIDIRLIPKDQYYCGVLYFTGSDIFNKNMRAHALEKGFTINEYTIRPLGVTGVAGEPLPVDSEKDIFDYIQWKYREPKDRSE
9GDJ , Knot 127 291 0.82 40 176 274 
MGSSHHHHHHSSGLVPRGSHMSSQTVTPDPYGNLAESYDRLAQWAIDQQQESPRDRVGDFLQTFWQSQDRPVRTVLEICCGTGLMLAELARRGYVVTGLDRSAAMLEQARARMGGKTTLIRAELPDIPAPAGEFDAVVSAAGGLNYLSESQISATFGAVARLLPAGGTFTFDVFGQGFYAKFFDPSAPRVMALELDDISYIWTFTKPAEAPFVDMSYTQFSPASRAVDGEPAFIRTRDLHRYYPLPHATVLRLAAEHGFTDARAHDNYSSDPSGPHTLYDTWTMVRTGSLE
1EGQ , Knot 119 279 0.80 40 169 265 
AAQTNAPWGLARISSTSPGTSTYYYDESAGQGSCVYVIDTGIEASHPEFEGRAQMVKTYYYSSRDGNGHGTHCAGTVGSRTYGVAKKTQLFGVKVLDDNGSGQYSTIIAGMDFVASDKNNRNCPKGVVASLSLGGGYSSSVNSAAARLQSSGVMVAVAAGNNNADARNYSPASEPSVCTVGASDRYDRRSSFSNYGSVLDIFGPGTSILSTWIGGSTRSISGTSMATPHVAGLAAYLMTLGKTTAASACRYIADTANKGDLSNIPFGTVNLLAYNNYQA

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 \(c\).

\(|P_{f(3ISD_1)}(2) \setminus P_{f(9GDJ_1)}(2)|=95\), \(|P_{f(9GDJ_1)}(2) \setminus P_{f(3ISD_1)}(2)|=72\). 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 1:10000110001011100110011010001001100001000110111001001001101001111100110010011101010010010000000010110010111101100110011001001000000100000111001101000110001101001110010010000110101010011000101011100101000000010110011001001011000100100011110011000000001000101011100000011101010011000101011001101000010111101111011110000011001010000100000
Pair \(Z_2\) Length of longest common subsequence
3ISD_1,9GDJ_1 167 3
3ISD_1,1EGQ_1 180 3
9GDJ_1,1EGQ_1 155 4

Newick tree

 
[
	3ISD_1:89.69,
	[
		9GDJ_1:77.5,1EGQ_1:77.5
	]:12.19
]

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 \((0.85)(0.8)(\frac{626 }{\log_{20} 626}-\frac{291}{\log_{20}291})=93.5\)
Status Protein1 Protein2 d d1/2
Query variables 3ISD_1 9GDJ_1 121 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]