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Parikh vectors
4XBF_1 2BGS_1 3SXG_1 Letter Amino acid
34 16 16 T Threonine
30 23 14 D Aspartic acid
9 5 4 C Cysteine
11 10 9 M Methionine
27 8 18 F Phenylalanine
39 16 11 S Serine
9 8 6 W Tryptophan
53 25 31 V Valine
37 7 15 Q Glutamine
43 28 17 E Glutamic acid
15 24 10 H Histidine
73 26 29 L Leucine
41 31 13 K Lycine
37 15 17 P Proline
18 7 13 Y Tyrosine
55 26 16 A Alanine
32 14 22 R Arginine
27 12 7 N Asparagine
45 26 16 G Glycine
31 17 9 I Isoleucine 

4XBF_1|Chain A|Lysine-specific histone demethylase 1A|Homo sapiens (9606)
>2BGS_1|Chain A|ALDOSE REDUCTASE|HORDEUM VULGARE (4513)
>3SXG_1|Chain A|Histo-blood group ABO system transferase|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 \(w=f(c)\)
4XBF , Knot 262 666 0.85 40 278 616 
PSGVEGAAFQSRLPHDRMTSQEAACFPDIISGPQQTQKVFLFIRNRTLQLWLDNPKIQLTFEATLQQLEAPYNSDTVLVHRVHSYLERHGLINFGIYKRIKPLPTKKTGKVIIIGSGVSGLAAARQLQSFGMDVTLLEARDRVGGRVATFRKGNYVADLGAMVVTGLGGNPMAVVSKQVNMELAKIKQKCPLYEANGQAVPKEKDEMVEQEFNRLLEATSYLSHQLDFNVLNNKPVSLGQALEVVIQLQEKHVKDEQIEHWKKIVKTQEELKELLNKMVNLKEKIKELHQQYKEASEVKPPRDITAEFLVKSKHRDLTALCKEYDELAETQGKLEEKLQELEANPPSDVYLSSRDRQILDWHFANLEFANATPLSTLSLKHWDQDDDFEFTGSHLTVRNGYSCVPVALAEGLDIKLNTAVRQVRYTASGCEVIAVNTRSTSQTFIYKCDAVLCTLPLGVLKQQPPAVQFVPPLPEWKTSAVQRMGFGNLNKVVLCFDRVFWDPSVNLFGHVGSTTASRGELFLFWNLYKAPILLALVAGEAAGIMENISDDVIVGRCLAILKGIFGSSAVPQPKETVVSRWRADPWARGSYSYVAAGSSGNDYDLMAQPITPGPSIPGAPQPIPRLFFAGEHTIRNYPATVHGALLSGLREAGRIADQFLGAMYTL
2BGS , Knot 145 344 0.82 40 204 317 
MGHHHHHHHHHSSGHIDDDDKHMLEASAKATMGQGEQDHFVLKSGHAMPAVGLGTWRAGSDTAHSVRTAITEAGYRHVDTAAEYGVEKEVGKGLKAAMEAGIDRKDLFVTSKIWCTNLAPERVRPALENTLKDLQLDYIDLYHIHWPFRLKDGAHMPPEAGEVLEFDMEGVWKEMENLVKDGLVKDIGVCNYTVTKLNRLLRSAKIPPAVCQMEMHPGWKNDKIFEACKKHGIHITAYSPLGSSEKNLAHDPVVEKVANKLNKTPGQVLIKWALQRGTSVIPKSSKDERIKENIQVFGWEIPEEDFKVLCSIKDEKRVLTGEELFVNKTHGPYRSARDVWDHEN
3SXG , Knot 132 293 0.85 40 195 285 
FMVSLPRMVYPQPKVLTPCRKDVLVVTPWLAPIVWEGTFNIDILNEQFRLQNTTIGLTVFAIKKYVAFLKLFLETAEKHFMVGHRVHYYVFTDQPAAVPRVTLGTGRQLSVLEVRAYKRWQDVSMRRMEMISDFCERRFLSEVDYLVCVDVDMEFRDHVGVEILTPLFGTLHPGFYGSSREAFTYERRPQSQAYIPKDEGDFYYLGGFFGGSVQEVQRLTRACHQAMMVDQANGIEAVWHDESHLNKYLLRHKPTKVLSPEYLWDQQLLGWPAVLRKLRFTAVPKNHQAVRNP

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 \(c\).

\(|P_{f(4XBF_1)}(2) \setminus P_{f(2BGS_1)}(2)|=121\), \(|P_{f(2BGS_1)}(2) \setminus P_{f(4XBF_1)}(2)|=47\). 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 1:101101111000110001000011011011011000001111100001011100101010101010010110000011100100010001110111000101110000101111101101111100100111010110100011101101001001101111110111101111100010101101000011001010111000001100010011010001000101011000110110110111010000100001001001100000100110011010001001000000100101100101011100000010110000001100010100010010101100101000000110101101011010110010100100000101010010100100011111101101010011001000101001111000000001100001110011111100011110111111010001100111101001110100111010101110110001001011111010011111111110111110010001111001111011110011101000110010101110100001111001000011101101110111110111011111000100011010111101100110110011111001
Pair \(Z_2\) Length of longest common subsequence
4XBF_1,2BGS_1 168 4
4XBF_1,3SXG_1 167 4
2BGS_1,3SXG_1 185 3

Newick tree

 
[
	2BGS_1:89.91,
	[
		4XBF_1:83.5,3SXG_1:83.5
	]:6.41
]

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 \((0.85)(0.8)(\frac{1010 }{\log_{20} 1010}-\frac{344}{\log_{20}344})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 4XBF_1 2BGS_1 222 166.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]