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Parikh vectors
7TQY_1 4RHK_1 2FZC_1 Letter Amino acid
13 5 14 Q Glutamine
29 20 34 A Alanine
28 25 21 D Aspartic acid
28 17 15 K Lycine
9 14 8 M Methionine
13 12 12 F Phenylalanine
0 2 2 W Tryptophan
47 9 15 N Asparagine
31 17 15 I Isoleucine
43 32 38 L Leucine
44 31 20 S Serine
11 13 8 Y Tyrosine
35 20 15 R Arginine
26 31 15 G Glycine
15 15 11 H Histidine
17 24 12 P Proline
37 13 18 T Threonine
26 30 22 V Valine
9 4 1 C Cysteine
30 17 14 E Glutamic acid 

7TQY_1|Chain A[auth K]|Kinesin-like protein|Candida albicans (5476)
>4RHK_1|Chain A|Arginase|Trypanosoma brucei brucei (185431)
>2FZC_1|Chains A, C|Aspartate carbamoyltransferase catalytic chain|Escherichia coli (562)
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 \(w=f(c)\)
7TQY , Knot 196 491 0.82 38 236 454 
MASYPNSLGSPATVTSTSVPTAKQSSISVAVRVRPFTEAESNRLVKIDNDDVFLGDGCLTSDNNNNNNNSNSNGNGNGNGSSAANSSGASTSRRAIFNTLGGLRKIINVVDDRMLIFDPPETNPLTKMQRNAFPNSFKGSRIREHRFVFDRLFDEDCTQDQVYRNTTQPLLDSVLDGYNATVFAYGATGCGKTHTISGTPEDPGVIFLTMKELYNRIEELKDTKIIDISLSYLEIYNETIRDLLNPMTQCKNLVIREDANNKISVSNLSRHRPNSVEEVMQLILEGNKNRTCSPTEANATSSRSHAVLQINVIQKDRTGDITEEHTFATLSIIDLAGSERAAATKNRGARLNEGANINKSLLALGNCINALCDPRRRNHVPYRDSKLTRLLKFSLGGNCKTVMIVCVSPSSQHYDETLNTLKYADRAKEIKTKLIRNQHNLDRHVGSYLKMITEQKQEIEELRARESKMVESTINKRKDLESKLEHHHHHH
4RHK , Knot 149 351 0.83 40 208 336 
MGSSHHHHHHSSGLVPRGSHMFSKFLMNVKGVTPRGSDWANRLGPVALFGYGAGMPRRAPLLDFFLQSPRDCDHYAELTIHDKGPIECPPETVMFMPVLNCGQMLDEAAGTETPTSDEWYLGSLEASTELLEKGYVPVSVGGDGSATLSMVEAYKRLFPSDDIVIVHFSARPSVSDPRSPLRVLLDKGLLKGVVSVGNRQVSSEDRKVRKLHKMFYMDMHAIYSKGLFCIRDIRNDYPVFISIDASVLDPAFAPAVDSPVAGGLSTRDLLHIMNGIRGPKVVGIDVYGYNPDLDVYRKDNVGLTAIALSKIIKEGILKAYSISTHTEEEGMERVKMLQRQGTVSENPYPDH
2FZC , Knot 139 310 0.85 40 190 299 
ANPLYQKHIISINDLSRDDLNLVLATAAKLKANPQPELLKHKVIASCFFEASTRTRLSFETSMHRLGASVVGFSDSANTSLGKKGETLADTISVISTYVDAIVMRHPQEGAARLATEFSGNVPVLNAGDGSNQHPTQTLLDLFTIQETQGRLDNLHVAMVGDLKYGRTVHSLTQALAKFDGNRFYFIAPDALAMPQYILDMLDEKGIAWSLHSSIEEVMAEVDILYMTRVQKERLDPSEYANVKAQFVLRASDLHNAKANMKVLHPLPRVDEIATDVDKTPHAWYFQQAGNGIFARQALLALVLNRDLVL

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 \(c\).

\(|P_{f(7TQY_1)}(2) \setminus P_{f(4RHK_1)}(2)|=85\), \(|P_{f(4RHK_1)}(2) \setminus P_{f(7TQY_1)}(2)|=57\). 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 1:11001001101101000011010000101110101100100001101000011110101000000000000001010101001100011000001110011110011011000111101100011001000111001010010000111001100000000100000011100110100101110110101000010101001111110100100010010000110101001010000100110110000011100010001010010000100100110111010000000100101000000111010110000010100000110101101110001110000110100110100011111001011001000001100000100110101110000111101010000000010010010010010001100000100011001011000000100101000011000100000100010000000
Pair \(Z_2\) Length of longest common subsequence
7TQY_1,4RHK_1 142 6
7TQY_1,2FZC_1 162 4
4RHK_1,2FZC_1 156 4

Newick tree

 
[
	2FZC_1:82.15,
	[
		7TQY_1:71,4RHK_1:71
	]:11.15
]

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 \((0.85)(0.8)(\frac{842 }{\log_{20} 842}-\frac{351}{\log_{20}351})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 7TQY_1 4RHK_1 162 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]