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Parikh vectors
5OTT_1 6MOZ_1 7MBY_1 Letter Amino acid
12 36 28 S Serine
5 39 30 A Alanine
12 18 9 E Glutamic acid
1 19 8 H Histidine
8 54 30 L Leucine
11 34 5 P Proline
10 21 22 R Arginine
0 21 18 I Isoleucine
3 22 10 K Lycine
4 19 8 Y Tyrosine
6 30 17 V Valine
6 26 30 D Aspartic acid
6 7 14 C Cysteine
6 20 12 Q Glutamine
4 26 11 F Phenylalanine
6 31 28 T Threonine
3 19 17 N Asparagine
6 34 27 G Glycine
0 9 8 M Methionine
7 12 8 W Tryptophan 

5OTT_1|Chain A|Glucagon-like peptide 1 receptor|Homo sapiens (9606)
>6MOZ_1|Chains A, B|Glucosylceramidase|Homo sapiens (9606)
>7MBY_1|Chain A[auth B]|Guanine nucleotide-binding protein G(I)/G(S)/G(T) subunit beta-1|Rattus norvegicus (10116)
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 \(w=f(c)\)
5OTT , Knot 59 116 0.80 36 92 112 
RPQGATVSLWETVQKWREYRRQCQRSLTEDPPPATDLFCNRTFDEYACWPDGEPGSFVNVSCPWYLPWASSVPQGHVYRFCTAEGLWLQKDNSSLPWRDLSECEESKRGERSSPEE
6MOZ , Knot 208 497 0.86 40 260 481 
ARPCIPKSFGYSSVVCVCNATYCDSFDPPTFPALGTFSRYESTRSGRRMELSMGPIQANHTGTGLLLTLQPEQKFQKVKGFGGAMTDAAALNILALSPPAQNLLLKSYFSEEGIGYNIIRVPMASCDFSIRTYTYADTPDDFQLHNFSLPEEDTKLKIPLIHRALQLAQRPVSLLASPWTSPTWLKTNGAVNGKGSLKGQPGDIYHQTWARYFVKFLDAYAEHKLQFWAVTAENEPSAGLLSGYPFQCLGFTPEHQRDFIARDLGPTLANSTHHNVRLLMLDDQRLLLPHWAKVVLTDPEAAKYVHGIAVHWYLDFLAPAKATLGETHRLFPNTMLFASEACVGSKFWEQSVRLGSWDRGMQYSHSIITNLLYHVVGWTDWNLALNPEGGPNWVRNFVDSPIIVDITKDTFYKQPMFYHLGHFSKFIPEGSQRVGLVASQKNDLDAVALMHPDGSAVVVVLNRSSKDVPLTIKDPAVGFLETISPGYSIHTYLWHRQ
7MBY , Knot 146 340 0.83 40 192 317 
MSELDQLRQEAEQLKNQIRDARKACADATLSQITNNIDPVGRIQMRTRRTLRGHLAKIYAMHWGTDSRLLVSASQDGKLIIWDSYTTNKVHAIPLRSSWVMTCAYAPSGNYVACGGLDNICSIYNLKTREGNVRVSRELAGHTGYLSCCRFLDDNQIVTSSGDTTCALWDIETGQQTTTFTGHTGDVMSLSLAPDTRLFVSGACDASAKLWDVREGMCRQTFTGHESDINAICFFPNGNAFATGSDDATCRLFDLRADQELMTYSHDNIICGITSVSFSKSGRLLLAGYDDFNCNVWDALKADRAGVLAGHDNRVSCLGVTDDGMAVATGSWDSFLKIWN

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 \(c\).

\(|P_{f(5OTT_1)}(2) \setminus P_{f(6MOZ_1)}(2)|=24\), \(|P_{f(6MOZ_1)}(2) \setminus P_{f(5OTT_1)}(2)|=192\). 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 1:01011010110010010000000000100011110011000010001011010110110100110111100110101001001011110000001110010000000010000100
Pair \(Z_2\) Length of longest common subsequence
5OTT_1,6MOZ_1 216 3
5OTT_1,7MBY_1 196 3
6MOZ_1,7MBY_1 160 4

Newick tree

 
[
	5OTT_1:10.75,
	[
		7MBY_1:80,6MOZ_1:80
	]:29.75
]

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 \((0.85)(0.8)(\frac{613 }{\log_{20} 613}-\frac{116}{\log_{20}116})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 5OTT_1 6MOZ_1 186 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]