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Parikh vectors
7UIW_1 5XTI_1 5OGC_1 Letter Amino acid
64 32 20 E Glutamic acid
21 10 17 H Histidine
46 26 18 I Isoleucine
78 32 33 L Leucine
32 12 27 N Asparagine
44 21 19 D Aspartic acid
3 12 5 C Cysteine
36 10 15 Q Glutamine
11 10 12 Y Tyrosine
50 29 16 R Arginine
14 8 9 M Methionine
27 17 12 F Phenylalanine
39 22 28 T Threonine
69 33 23 A Alanine
44 25 31 K Lycine
25 22 9 P Proline
60 28 33 V Valine
50 52 20 G Glycine
42 21 30 S Serine
3 9 0 W Tryptophan 

7UIW_1|Chains A, B, C, D, E, F|ATP-dependent Clp protease ATP-binding subunit ClpA|Escherichia coli (562)
>5XTI_1|Chains A, CC[auth BA]|NADH dehydrogenase [ubiquinone] flavoprotein 1, mitochondrial|Homo sapiens (9606)
>5OGC_1|Chain A[auth K]|Kinesin-like protein KIF18A|Homo sapiens (9606)
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 \(w=f(c)\)
7UIW , Knot 286 758 0.83 40 270 687 
MLNQELELSLNMAFARAREHRHEFMTVEHLLLALLSNPSAREALEACSVDLVALRQELEAFIEQTTPVLPASEEERDTQPTLSFQRVLQRAVFHVQSSGRNEVTGANVLVAIFSEQESQAAYLLRKHEVSRLDVVNFISHGTRKDEPTQSSDPGSQPNSEEQAGGEERTENFTTNLNQLARVGGIDPLIGREKELERAIQVLCRRRKNNPLLVGESGVGKTAIAEGLAWRIVQGDVPEVMADCTIYSLDIGSLLAGTKYRGDFEKRFKALLKQLEQDTNSILFIDEIHTIIGAGAASGGQVDAANLIKPLLSSGKIRVIGSTTYQEFSNIFEKDRALARRFQKIDITEPSIEETVQIINGLKPKYEAHHDVRYTAKAVRAAVELAVKYINDRHLPDKAIDVIDEAGARARLMPVSKRKKTVNVADIESVVARIARIPEKSVSQSDRDTLKNLGDRLKMLVFGQDKAIEALTEAIKMARAGLGHEHKPVGSFLFAGPTGVGKTEVTVQLSKALGIELLRFDMSEYMERHTVSRLIGAPPGYVGFDQGGLLTDAVIKHPHAVLLLDEIEKAHPDVFNILLQVMDNGTLTDNNGRKADFRNVVLVMTTNAGVRETERKSIGLIHQDNSTDAMEEIKKIFTPEFRNRLDNIIWFDHLSTDVIHQVVDKFIVELQVQLDQKGVSLEVSQEARNWLAEKGYDRAMGARPMARVIQDNLKKPLANELLFGSLVDGGQVTVALDKEKNELTYGFQSAQKHKAEAAH
5XTI , Knot 176 431 0.82 40 231 405 
PKKTSFGSLKDEDRIFTNLYGRHDWRLKGSLSRGDWYKTKEILLKGPDWILGEIKTSGLRGRGGAGFPTGLKWSFMNKPSDGRPKYLVVNADEGEPGTCKDREILRHDPHKLLEGCLVGGRAMGARAAYIYIRGEFYNEASNLQVAIREAYEAGLIGKNACGSGYDFDVFVVRGAGAYICGEETALIESIEGKQGKPRLKPPFPADVGVFGCPTTVANVETVAVSPTICRRGGTWFAGFGRERNSGTKLFNISGHVNHPCTVEEEMSVPLKELIEKHAGGVTGGWDNLLAVIPGGSSTPLIPKSVCETVLMDFDALVQAQTGLGTAAVIVMDRSTDIVKAIARLIEFYKHESCGQCTPCREGVDWMNKVMARFVRGDARPAEIDSLWEISKQIEGHTICALGDGAAWPVQGLIRHFRPELEERMQRFAQQH
5OGC , Knot 163 377 0.85 38 216 362 
GSHMSVTEEDLCHHMKVVVRVRPENTKEKAAGFHKVVHVVDKHILVFDPKQEEVSFFHGKKTTNQNVIKKQNKDLKFVFDAVFDETSTQSEVFEHTTKPILRSFLNGYNCTVLAYGATGAGKTHTMLGSADEPGVMYLTMLHLYKCMDEIKEEKICSTAVSYLEVYNEQIRDLLVNSGPLAVREDTQKGVVVHGLTLHQPKSSEEILHLLDNGNKNRTQHPTDMNATSSRSHAVFQIYLRQQDKTASINQNVRIAKMSLIDLAGSERASTSGAKGTRFVEGTNINRSLLALGNVINALADSKRKNQHIPYRNSKLTRLLKDSLGGNCQTIMIAAVSPSSVFYDDTYNTLKYANRAKDIKSSLKSNVLNVNNHITQYV

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 \(c\).

\(|P_{f(7UIW_1)}(2) \setminus P_{f(5XTI_1)}(2)|=88\), \(|P_{f(5XTI_1)}(2) \setminus P_{f(7UIW_1)}(2)|=49\). 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 1:11000101010111101000000110100111111001010011010010111100010111000011111000000001010100110011101000100010110111111000000110110000100101101100100000100000110010000011100000010001001101111011110000100110110000000111110011100111011110110101101110001001011011110000101000101110010000001111001001111111011010110110111001010111000000100110000111001001010010100010110110100010001000101101110111001000011001101100111010111100000010110100111011011000100000001001100101111100011011001101101111000011101111110111000101010011110110101000100001001111111011100111100111001011111001001010110111011001010000100101001111100011100000001111000000011001001101010001001111001000110011001110101010001101010001001110010001111011101100010011100111101101101011100000010011001000010110
Pair \(Z_2\) Length of longest common subsequence
7UIW_1,5XTI_1 137 6
7UIW_1,5OGC_1 144 4
5XTI_1,5OGC_1 175 4

Newick tree

 
[
	5OGC_1:83.64,
	[
		7UIW_1:68.5,5XTI_1:68.5
	]:15.14
]

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 \((0.85)(0.8)(\frac{1189 }{\log_{20} 1189}-\frac{431}{\log_{20}431})=197.\)
Status Protein1 Protein2 d d1/2
Query variables 7UIW_1 5XTI_1 243 189.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]