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Parikh vectors
2MHA_1 2XSX_1 2OMX_1 Letter Amino acid
13 17 18 P Proline
18 17 59 T Threonine
20 17 5 R Arginine
15 26 28 D Aspartic acid
4 9 2 M Methionine
6 16 9 F Phenylalanine
19 47 26 A Alanine
23 41 19 G Glycine
6 27 36 I Isoleucine
11 23 41 S Serine
8 3 3 W Tryptophan
17 11 8 Y Tyrosine
7 22 57 N Asparagine
5 6 0 C Cysteine
13 13 23 Q Glutamine
25 38 76 L Leucine
25 27 12 E Glutamic acid
8 7 1 H Histidine
11 37 20 K Lycine
16 31 19 V Valine 

2MHA_1|Chains A, C|CLASS I HISTOCOMPATIBILITY ANTIGEN (H-2KB) (ALPHA CHAIN)|Mus musculus (10090)
>2XSX_1|Chains A, B|BETA-ENOLASE|HOMO SAPIENS (9606)
>2OMX_1|Chain A|Internalin-A|Listeria monocytogenes (169963)
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 \(w=f(c)\)
2MHA , Knot 121 270 0.83 40 175 261 
GPHSLRYFVTAVSRPGLGEPRYMEVGYVDDTEFVRFDSDAENPRYEPRARWMEQEGPEYWERETQKAKGNEQSFRVDLRTLLGYYNQSKGGSHTIQVISGCEVGSDGRLLRGYQQYAYDGCDYIALNEDLKTWTAADMAALITKHKWEQAGEAERLRAYLEGTCVEWLRRYLKNGNATLLRTDSPKAHVTHHSRPEDKVTLRCWALGFYPADITLTWQLNGEELIQDMELVETRPAGDGTFQKWASVVVPLGKEQYYTCHVYHQGLPEPL
2XSX , Knot 180 435 0.83 40 220 405 
SMAMQKIFAREILDSRGNPTVEVDLHTAKGRFRAAVPSGASTGIYEALELRDGDKGRYLGKGVLKAVENINSTLGPALLQKKLSVADQEKVDKFMIELDGTENKSKFGANAILGVSLAVCKAGAAEKGVPLYRHIADLAGNPDLILPVPAFNVINGGSHAGNKLAMQEFMILPVGASSFKEAMRIGAEVYHHLKGVIKAKYGKDATNVGDEGGFAPNILENNEALELLKTAIQAAGYPDKVVIGMDVAASEFYRNGKYDLDFKSPDDPARHITGEKLGELYKSFIKNYPVVSIEDPFDQDDWATWTSFLSGVNIQIVGDDLTVTNPKRIAQAVEKKACNCLLLKVNQIGSVTESIQACKLAQSNGWGVMVSHRSGETEDTFIADLVVGLCTGQIKTGAPCRSERLAKYNQLMRIEEALGDKAIFAGRKFRNPKAK
2OMX , Knot 155 462 0.68 38 157 341 
ATITQDTPINQIFTDTALAEKMKTVLGKTNVTDTVSQTDLDQVTTLQADRLGIKSIDGVEYLNNLTQINFSNNQLTDITPLKNLTKLVDILMNNNQIADITPLANLTNLTGLTLFNNQITDIDPLKNLTNLNRLELSSNTISDISALSGLTSLQQLNFSSNQVTDLKPLANLTTLERLDISSNKVSDISVLAKLTNLESLIATNNQISDITPLGILTNLDELSLNGNQLKDIGTLASLTNLTDLDLANNQISNLAPLSGLTKLTELKLGANQISNISPLAGLTALTNLELNENQLEDISPISNLKNLTYLTLYFNNISDISPVSSLTKLQRLFFYNNKVSDVSSLANLTNINWLSAGHNQISDLTPLANLTRITQLGLNDQAWTNAPVNYKANVSIPNTVKNVTGALIAPATISDGGSYTEPDITWNLPSYTNEVSYTFSQPVTIGKGTTTFSGTVTQPLKA

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 \(c\).

\(|P_{f(2MHA_1)}(2) \setminus P_{f(2XSX_1)}(2)|=62\), \(|P_{f(2XSX_1)}(2) \setminus P_{f(2MHA_1)}(2)|=107\). 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 1:110010011011001111010010110100001101000100100010101100011001000000101000010101001110000001100010110100110010110100001001000111000100101101111100001001101001010101001011000100101011000010101000001000101001111101101010101010011001011000111010100110111111000000001000111011
Pair \(Z_2\) Length of longest common subsequence
2MHA_1,2XSX_1 169 4
2MHA_1,2OMX_1 176 4
2XSX_1,2OMX_1 159 3

Newick tree

 
[
	2MHA_1:88.40,
	[
		2XSX_1:79.5,2OMX_1:79.5
	]:8.90
]

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 \((0.85)(0.8)(\frac{705 }{\log_{20} 705}-\frac{270}{\log_{20}270})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 2MHA_1 2XSX_1 154 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]