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Parikh vectors
3EDK_1 8CSY_1 8ZLH_1 Letter Amino acid
24 41 10 Q Glutamine
59 62 13 G Glycine
24 24 4 M Methionine
26 58 6 S Serine
27 23 1 Y Tyrosine
0 5 2 C Cysteine
26 43 3 T Threonine
25 55 6 F Phenylalanine
41 41 8 D Aspartic acid
49 136 16 L Leucine
15 11 3 W Tryptophan
34 72 9 V Valine
46 45 4 R Arginine
31 15 7 N Asparagine
30 65 15 E Glutamic acid
17 18 2 H Histidine
22 49 11 I Isoleucine
19 29 8 K Lycine
37 56 10 P Proline
49 63 4 A Alanine 

3EDK_1|Chains A, B|Cyclomaltodextrinase|Flavobacterium sp. 92 (197856)
>8CSY_1|Chains A[auth C], B[auth E]|Band 3 anion transport protein|Homo sapiens (9606)
>8ZLH_1|Chains A, B|All1292 protein|Nostoc sp. PCC 7120 = FACHB-418 (103690)
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 \(w=f(c)\)
3EDK , Knot 242 601 0.86 38 279 560 
AAPTAIEHMEPPFWWAGMQHKGLQLMVHGRDIGRMEAALDYPGVRLVSPTRVPNANYLFVDLEIGPEAQPGSFDIVFKGDGRSERYRYRLLAREQGSAQRQGFGPGDAIYQIMPDRFANGDPSNDNVAGMREQADRRHGGGRHGGDIRGTIDHLDYIAGLGFTQLWPTPLVENDAAAYSYHGYAATDHYRIDPRYGSNEDFVRLSTEARKRGMGLIQDVVLSHIGKHHWWMKDLPTPDWINYGGKFVPTQHHRVAVQDPYAAQADSENFTKGWFVEGMPDLNQTNPLVANYLIQNNIWWIEYAGLSGLRIDTYGYSDGAFLTEYTRRLMAEYPRLNMVGQEWSTRVPVVARWQRGKANFDGYTSHLPSLMDFPLVDAMRNALSKTGEENGLNEVYETLSLDYLYPEPQNLVLFGGNHDMARMFSAAGEDFDRWRMNLVFLMTMPRIPQFYSGDEILMTSTVKGRDDASYRRDFPGGWAGDKANAFSGAGLTSQQRAAQDLVRKLANWRKNQPVIHNGRLMHFGPEENTWVYFRYNKDKRIMVAMNNNDKPMTLPTARFQEMLKGAPSGVDFLSGKTVGLGRELRLAPKSVVVIELPGLPEA
8CSY , Knot 330 911 0.82 40 289 781 
MEELQDDYEDMMEENLEQEEYEDPDIPESQMEEPAAHDTEATATDYHTTSHPGTHKVYVELQELVMDEKNQELRWMEAARWVQLEENLGENGAWGRPHLSHLTFWSLLELRRVFTKGTVLLDLQETSLAGVANQLLDRFIFEDQIRPQDREELLRALLLKHSHAGELEALGGVKPAVLTRSGDPSQPLLPQHSSLETQLFCEQGDGGTEGHSPSGILEKIPPDSEATLVLVGRADFLEQPVLGFVRLQEAAELEAVELPVPIRFLFVLLGPEAPHIDYTQLGRAAATLMSERVFRIDAYMAQSRGELLHSLEGFLDCSLVLPPTDAPSEQALLSLVPVQRELLRRRYQSSPAKPDSSFYKGLDLNGGPDDPLQQTGQLFGGLVRDIRRRYPYYLSDITDAFSPQVLAAVIFIYFAALSPAITFGGLLGEKTRNQMGVSELLISTAVQGILFALLGAQPLLVVGFSGPLLVFEEAFFSFCETNGLEYIVGRVWIGFWLILLVVLVVAFEGSFLVRFISRYTQEIFSFLISLIFIYETFSKLIKIFQDHPLQKTYNYNVLMVPKPQGPLPNTALLSLVLMAGTFFFAMMLRKFKNSSYFPGKLRRVIGDFGVPISILIMVLVDFFIQDTYTQKLSVPDGFKVSNSSARGWVIHPLGLRSEFPIWMMFASALPALLVFILIFLESQITTLIVSKPERKMVKGSGFHLDLLLVVGMGGVAALFGMPWLSATTVRSVTHANALTVMGKASTPGAAAQIQEVKEQRISGLLVAVLVGLSILMEPILSRIPLAVLFGIFLYMGVTSLSGIQLFDRILLLFKPPKYHPDVPYVKRVKTWRMHLFTGIQIICLAVLWVVKSTPASLALPFVLILTVPLRRVLLPLIFRNVELQCLDADDAKATFDEEEGRDEYDEVAMPV
8ZLH , Knot 70 142 0.81 40 105 134 
MMFGSTKPESGDSKWRSQLDRFVKENQQDLAALFWGLWLENGDSQGTIGIDLQPTPHFVYCPKDAVEKLNNNVENRLQELLGIIEHNQPEIEVLMIGIGKGEIKLIQFAPEPPPPVCFEQVGKDIDGLLELLEQRMSGEIVV

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 \(c\).

\(|P_{f(3EDK_1)}(2) \setminus P_{f(8CSY_1)}(2)|=53\), \(|P_{f(8CSY_1)}(2) \setminus P_{f(3EDK_1)}(2)|=63\). 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 1:1110110010111111110001101110100110101110011101101001101001110101110101101011101010000000011100010100011111011001110011010100001111000100001110011010101001001111110011101110001110000101100000101001000011010001000111110011100110001110011010110011011100000111001011010000100111101110100001111001100011110011101101000100011110000001110010101110010001111101001010101000011011011110110011000100011001000101001010100111111000110110111001001010111110110110100100111000101000100000111111100101101111000001100110011010000111001011011100001101000000011111000001101101010011011101101101001111001011100111101111101
Pair \(Z_2\) Length of longest common subsequence
3EDK_1,8CSY_1 116 4
3EDK_1,8ZLH_1 214 3
8CSY_1,8ZLH_1 214 4

Newick tree

 
[
	8ZLH_1:11.92,
	[
		3EDK_1:58,8CSY_1:58
	]:60.92
]

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 \((0.85)(0.8)(\frac{1512 }{\log_{20} 1512}-\frac{601}{\log_{20}601})=229.\)
Status Protein1 Protein2 d d1/2
Query variables 3EDK_1 8CSY_1 288 238.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]