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Parikh vectors
3KCO_1 2HVA_1 2OBR_1 Letter Amino acid
28 16 12 E Glutamic acid
10 1 11 H Histidine
8 7 6 M Methionine
16 13 33 T Threonine
19 18 26 V Valine
37 9 14 D Aspartic acid
1 2 5 C Cysteine
11 7 19 Q Glutamine
11 11 18 S Serine
9 10 10 Y Tyrosine
47 13 18 A Alanine
10 5 21 N Asparagine
39 9 26 L Leucine
11 5 9 I Isoleucine
23 7 17 F Phenylalanine
19 13 28 P Proline
6 4 4 W Tryptophan
35 8 14 R Arginine
37 21 28 G Glycine
11 13 8 K Lycine 

3KCO_1|Chain A|Xylose isomerase|Streptomyces rubiginosus (36817)
>2HVA_1|Chain A|Heme-binding protein 1|Mus musculus (10090)
>2OBR_1|Chain A|Capsid protein|Norovirus (142786)
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 \(w=f(c)\)
3KCO , Knot 159 388 0.81 40 206 357 
MNYQPTPEDRFTFGLWTVGWQGRDPFGDATRRALDPVESVRRLAELGAHGVTFHDDDLIPFGSSDSEREEHVKRFRQALDDTGMKVPMATTNLFTHPVFKDGGFTANDRDVRRYALRKTIRNIDLAVELGAETYVAWGGREGAESGGAKDVRDALDRMKEAFDLLGEYVTSQGYDIRFAIEPKPNEPRGDILLPTVGHALAFIERLERPELYGVNPEVGHEQMAGLNFPHGIAQALWAGKLFHIDLNGQNGIKYDQDLRFGAGDLRAAFWLVDLLESAGYSGPRHFDFKPPRTEDFDGVWASAAGCMRNYLILKERAAAFRADPEVQEALRASRLDELARPTAADGLQALLDDRSAFEEFDVDAAAARGMAFERLDQLAMDHLLGARG
2HVA , Knot 91 192 0.83 40 138 188 
GSMLGMIRNSLFGSVETWPWQVLSTGGKEDVSYEERACEGGKFATVEVTDKPVDEALREAMPKIMKYVGGTNDKGVGMGMTVPVSFAVFPNEDGSLQKKLKVWFRIPNQFQGSPPAPSDESVKIEEREGITVYSTQFGGYAKEADYVAHATQLRTTLEGTPATYQGDVYYCAGYDPPMKPYGRRNEVWLVKA
2OBR , Knot 143 327 0.84 40 197 315 
GFLVPPTVESRTKPFTVPILTVEEMSNSRFPIPLEKLYTGPSSAFVVQPQNGRCTTDGVLLGTTQLSAVNICTFRGDVTHIAGSHDYIMNLASQNWNNYDPTEEIPAPLGTPDFVGKIQGMLTQTTREDGSTRAHKATVSTGSVHFTPKLGSVQYTTDTNNDLQTGQNTKFTPVGVIQDGNNHQNEPQQWVLPNYSGRTGHNVHLAPAVAPTFPGEQLLFFRSTMPGCSGYPNMNLDCLLPQEWVQHFYQEAAPAQSDVALLRFVNPDTGRVLFECKLHKSGYVTVAHTGPHDLVIPPNGYFRFDSWVNQFYTLAPMGNGAGRRRAL

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 \(c\).

\(|P_{f(3KCO_1)}(2) \setminus P_{f(2HVA_1)}(2)|=117\), \(|P_{f(2HVA_1)}(2) \setminus P_{f(3KCO_1)}(2)|=49\). 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 1:1000101000101111011101001110100011011001001101110110100001111100000000010010011000110111100011001110011101000010001100010010111011100011111001100111001001100100110111001000100101110101001010111101101111100100101011010110001111011011101111101101010100110000010111101011111101100110011001010110000101111011101000111000111101010100110100100110101101101110000110010101111011110010011100111101
Pair \(Z_2\) Length of longest common subsequence
3KCO_1,2HVA_1 166 4
3KCO_1,2OBR_1 169 4
2HVA_1,2OBR_1 173 4

Newick tree

 
[
	2OBR_1:86.32,
	[
		3KCO_1:83,2HVA_1:83
	]:3.32
]

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 \((0.85)(0.8)(\frac{580 }{\log_{20} 580}-\frac{192}{\log_{20}192})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 3KCO_1 2HVA_1 136 102
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]