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Parikh vectors
8XDH_1 4BXK_1 2JHG_1 Letter Amino acid
14 41 21 E Glutamic acid
20 24 30 K Lycine
17 12 9 M Methionine
25 30 18 F Phenylalanine
14 31 4 Y Tyrosine
11 36 17 D Aspartic acid
22 22 8 N Asparagine
21 42 8 Q Glutamine
48 68 25 L Leucine
15 38 20 P Proline
23 31 39 V Valine
7 33 12 R Arginine
36 20 24 I Isoleucine
25 36 26 S Serine
15 21 2 W Tryptophan
33 34 38 G Glycine
11 7 14 C Cysteine
6 19 7 H Histidine
22 30 24 T Threonine
41 54 28 A Alanine 

8XDH_1|Chains A, B, C|Urea transporter|Danio rerio (7955)
>4BXK_1|Chains A, B|ANGIOTENSIN-CONVERTING ENZYME|HOMO SAPIENS (9606)
>2JHG_1|Chains A, B|ALCOHOL DEHYDROGENASE E CHAIN|EQUUS CABALLUS (9796)
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 \(w=f(c)\)
8XDH , Knot 179 426 0.84 40 233 402 
MANPVHKITDEKKQQGLEKINSGQRFKANLIKWVSFISGDMAAFGEWMKGQFILLQIMDWVLRGAAQVMFVNNPLSGLIIFAGLILQNRWWALNGFVGTVFATISALILCQNRGAIAAGLYGYNGILVGLLMAVFSNAGDWYWWLLLPNIFMSMACPIVSSALASINSRWDLPVFTLPFNILVCLHMVATGHYNQYFPQILIQPTTSMSNLTWSELDYAQLFRSIPVGIGQVYGCDNAWTGGIFMIALFISSPITFAHATIGSAVGMVSGLALAAPFKNIYMGLWGYNCVLACIAIGGMFYALTWQTHLLAVACAFFCAYLGSAIGNVMSNFGLPACTWPFCLSALTFLLITTETKFIHKLPLAKVAYPEQNLRYYWKMKKEEKTQKGIQAKDTESQLEKEQISISMEKKDLDICTVDVQQEENTL
4BXK , Knot 250 629 0.85 40 284 597 
LDPGLQPGQFSADEAGAQLFAQSYQSSAEQVLFQSVAASWAHDTNITAENARRQEEAALLSQEFAEAWGQKAKELYEPIWQQFTDPQLRRIIGAVRTLGSANLPLAKRQQYNALLSQMSRIYSTAKVCLPQKTATCWSLDPDLTNILASSRSYAMLLFAWEGWHNAAGIPLKPLYEDFTALSNEAYKQDGFTDTGAYWRSWYNSPTFEDDLEHLYQQLEPLYLNLHAFVRRALHRRYGDRYINLRGPIPAHLLGDMWAQSWENIYDMVVPFPDKPNLDVTSTMLQQGWQATHMFRVAEEFFTSLELSPMPPEFWEGSMLEKPADGREVVCHASAWDFYNRKDFRIKQCTRVTMDQLSTVHHEMGHIQYYLQYKDLPVSLRRGANPGFHEAIGDVLALSVSTPEHLHKIGLLDRVTNDTESDINYLLKMALEKIAFLPFGYLVDQWRWGVFSGRTPPSRYNFDWWYLRTKYQGICPPVTRNETHFDAGAKFHVPNVTPYIRYFVSFVLQFQFHEALCKEAGYEGPLHQCDIYRSTKAGAKLRKVLRAGSSRPWQEVLKDMVGLDALDAQPLLKYFQLVTQWLQEQNQQNGEVLGWPEYQWHPPLPDNYPEGIDLVTDEAEASKFVEEYDL
2JHG , Knot 158 374 0.83 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRSDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(8XDH_1)}(2) \setminus P_{f(4BXK_1)}(2)|=55\), \(|P_{f(4BXK_1)}(2) \setminus P_{f(8XDH_1)}(2)|=106\). 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 1:110110010000000110010010010101101101101011111011010111101101110111011110011011111111110001111011110111010111100001111111010011111111111001101011111101110110111001110100010111101110111010111010000011011101000100101001001011001111110101000110111111111100110110101101111101111111100101111100011101111111011010001111101110101101110110011111001110101101111000001100111101101000100010100000000110100000010000101010000101001010000001
Pair \(Z_2\) Length of longest common subsequence
8XDH_1,4BXK_1 161 5
8XDH_1,2JHG_1 177 5
4BXK_1,2JHG_1 176 4

Newick tree

 
[
	2JHG_1:90.68,
	[
		8XDH_1:80.5,4BXK_1:80.5
	]:10.18
]

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 \((0.85)(0.8)(\frac{1055 }{\log_{20} 1055}-\frac{426}{\log_{20}426})=165.\)
Status Protein1 Protein2 d d1/2
Query variables 8XDH_1 4BXK_1 215 178
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]