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Parikh vectors
1UBT_1 1IQF_1 6NCX_1 Letter Amino acid
7 15 33 R Arginine
12 17 24 K Lycine
16 16 36 V Valine
31 13 31 A Alanine
23 22 44 G Glycine
11 11 31 D Aspartic acid
15 19 51 E Glutamic acid
14 13 36 I Isoleucine
19 13 43 L Leucine
6 5 6 M Methionine
22 6 28 P Proline
10 11 42 S Serine
10 8 32 Y Tyrosine
16 7 22 N Asparagine
7 9 19 Q Glutamine
8 5 23 H Histidine
11 12 23 F Phenylalanine
15 20 21 T Threonine
3 4 13 W Tryptophan
11 9 16 C Cysteine 

1UBT_1|Chain A[auth S]|Periplasmic [NiFe] hydrogenase Small subunit|Desulfovibrio vulgaris str. 'Miyazaki F' (883)
>1IQF_1|Chain A|coagulation Factor Xa|Homo sapiens (9606)
>6NCX_1|Chains A, B, C, D|Beta-galacturonidase|Eisenbergiella tayi (1432052)
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 \(w=f(c)\)
1UBT , Knot 120 267 0.83 40 183 259 
LMGPRRPSVVYLHNAECTGCSESVLRAFEPYIDTLILDTLSLDYHETIMAAAGDAAEAALEQAVNSPHGFIAVVEGGIPTAANGIYGKVANHTMLDICSRILPKAQAVIAYGTCATFGGVQAAKPNPTGAKGVNDALKHLGVKAINIAGCPPNPYNLVGTIVYYLKNKAAPELDSLNRPTMFFGQTVHEQCPRLPHFDAGEFAPSFESEEARKGWCLYELGCKGPVTMNNCPKIKFNQTNWPVDAGHPCIGCSEPDFWDAMTPFYQN
1IQF , Knot 109 235 0.84 40 171 227 
IVGGQECKDGECPWQALLINEENEGFCGGTILSEFYILTAAHCLYQAKRFKVRVGDRNTEQEEGGEAVHEVEVVIKHNRFTKETYDFDIAVLRLKTPITFRMNVAPACLPERDWAESTLMTQKTGIVSGFGRTHEKGRQSTRLKMLEVPYVDRNSCKLSSSFIITQNMFCAGYDTKQEDACQGDSGGPHVTRFKDTYFVTGIVSWGEGCARKGKYGIYTKVTAFLKWIDRSMKTR
6NCX , Knot 230 574 0.84 40 267 537 
MIRTFETHKIRKTAELSSALWNFHTIGTQGEEAVIQAPVPGCWENYPDTVSYRGQASYSREFEAKGNIRLEFKGVSHTASVLVDGKPVGSHYNAYTPFDVVLKDIRPGIHQLEVIADNSFGPDSALHVPNDYQSYGGISRGVVLEELGEAYLSWIHFTPFLRKDGWYGKAEICVRNLSSGRLDGSVEVEIGKNSFAVLPIVLEGEEEKSFSTEELPCPWAECWSPESPVLYLITAVLRTADGAADDIIDRVGFREIRTEGKDILLNGRKLRIKGFCRHEDHPQFGCALPFSAMQHDLMLIKDLGANSIRTVHYPNDELFLDLCDEQGILVWEENHARGLSEENMRNPHFKQQCGDCIREMITAHYNHPSIYIWGILNECASDTEYGRECYSEQYELIKSLDPYRPRSSASCRFKTDICLGYPEVVSYNIYPKWYHDVPVEDYLDELYQWIQNESEGTGKPFLITEIGAGAIYGYRTPAHVKWSEEYQVQALKEQLQAVFSREGCSGVYIWQFCDVRVCDSWFGSRPRTMNNKGIVDEYRRPKLAYEVVKDSYRSLGNYFENLYFQSGSHHHHHH

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 \(c\).

\(|P_{f(1UBT_1)}(2) \setminus P_{f(1IQF_1)}(2)|=90\), \(|P_{f(1IQF_1)}(2) \setminus P_{f(1UBT_1)}(2)|=78\). 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 1:111100101101001000100001101101010011100101000001111110110111001100101111110111101101101011000110100011101011110100101111011010101101100110011101101110110100111011001000111010010010111100100001011010110111010000100110100110011101000101010000111011010110001011011011000
Pair \(Z_2\) Length of longest common subsequence
1UBT_1,1IQF_1 168 4
1UBT_1,6NCX_1 174 4
1IQF_1,6NCX_1 176 4

Newick tree

 
[
	6NCX_1:88.63,
	[
		1UBT_1:84,1IQF_1:84
	]:4.63
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{235}{\log_{20}235})=76.7\)
Status Protein1 Protein2 d d1/2
Query variables 1UBT_1 1IQF_1 97 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]