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Parikh vectors
9JAP_1 8HWL_1 3LDG_1 Letter Amino acid
25 46 14 Q Glutamine
28 38 3 H Histidine
41 33 10 Y Tyrosine
59 61 25 D Aspartic acid
5 13 4 C Cysteine
37 52 23 F Phenylalanine
75 55 14 S Serine
41 100 26 V Valine
62 79 27 E Glutamic acid
86 56 24 I Isoleucine
5 3 6 W Tryptophan
90 105 34 L Leucine
17 35 8 M Methionine
56 38 20 N Asparagine
33 94 27 G Glycine
68 56 34 K Lycine
18 67 18 P Proline
39 59 16 T Threonine
45 116 29 A Alanine
39 72 22 R Arginine 

9JAP_1|Chains A, B, C, D, E, F, G, H|SIR2-like domain-containing protein|Escherichia fergusonii (564)
>8HWL_1|Chains A, B, C, D[auth E]|Pyruvate carboxylase, mitochondrial|Homo sapiens (9606)
>3LDG_1|Chain A|Putative uncharacterized protein SMU.472|Streptococcus mutans (210007)
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 \(w=f(c)\)
9JAP , Knot 321 869 0.83 40 295 779 
MGSSHHHHHHSSGLVPRGSHMDAMLNSLKLANSANESISDETYERLKRLISTGNAIVFVGAGFSKESINIIGSTPPLAKDLALQISNKSANYLKEVGADSHYIEEIKQCDDLMVASDFFLNNIPQKDELLQLLKDNYTIKDVTQEQIDIFSMKWRRIYTTNYDNAIELSLIKSGKSVTPLTLEDAPNQYKSAEDICLHINGRIERSKESDLDSAIKLTTSSYLSPEQFLTSSWYRQFKADIDNASAIVFLGYSMYDIDIQKIFFNDSSIKSKTFFITREGTTKFQNYKLAMFGEVINIGVNAFSHIAAKCIEESHQDKEVGLINSLELYTPGEEHDEIRDNDIANFMIFGKVSDRYIDEVTLNDNMHDKIILREEVSKIIEHIETDNDILIASDLGNGKSIMTRMLMSKLSRKGYLCFYYLYNEFSFSKDIERLSKLGQKIVIFIDDYSNCIDDTRYAIENRKDNIQLVLTTRHFGYENTKQHLLTMDMSSFKTHSVDYLSDSEVDNFVHIVDHLGAWGEKAGLSRHEKLSELDENARNQLSFLLLSILKSEAIQSRIREISNLALNDKEYKETVFAILLLDVIGLPLVRSLISDVAVNEKIYSAEFTENEGVKNLFIISNGMVKTKSSTLSRFLIANIFEHKYVVNQLLKVIEHLYVINKDAKDHRLQTLITSLLRFSIIEKLLPQRRVEINYFYEKVKHIIPNLINDPHFWVQYAMSMIPFKDYPSADRYLATAYSLAARKDNYHTKNIDTQRARLHLLVSLTKTGNEAYLEFEAGDNLIRIIPNDIYKYRQVLRYRDIYEKVYPTFNAKQKVFYEHAIKRIIKESESPELVEDLTYKIGVNWLDKLRGNLKLIVENIQENRPKGKK
8HWL , Knot 422 1178 0.84 40 319 1027 
MLKFRTVHGGLRLLGIRRTSTAPAASPNVRRLEYKPIKKVMVANRGEIAIRVFRACTELGIRTVAIYSEQDTGQMHRQKADEAYLIGRGLAPVQAYLHIPDIIKVAKENNVDAVHPGYGFLSERADFAQACQDAGVRFIGPSPEVVRKMGDKVEARAIAIAAGVPVVPGTDAPITSLHEAHEFSNTYGFPIIFKAAYGGGGRGMRVVHSYEELEENYTRAYSEALAAFGNGALFVEKFIEKPRHIEVQILGDQYGNILHLYERDCSIQRRHQKVVEIAPAAHLDPQLRTRLTSDSVKLAKQVGYENAGTVEFLVDRHGKHYFIEVNSRLQVEHTVTEEITDVDLVHAQIHVAEGRSLPDLGLRQENIRINGCAIQCRVTTEDPARSFQPDTGRIEVFRSGEGMGIRLDNASAFQGAVISPHYDSLLVKVIAHGKDHPTAATKMSRALAEFRVRGVKTNIAFLQNVLNNQQFLAGTVDTQFIDENPELFQLRPAQNRAQKLLHYLGHVMVNGPTTPIPVKASPSPTDPVVPAVPIGPPPAGFRDILLREGPEGFARAVRNHPGLLLMDTTFRDAHQSLLATRVRTHDLKKIAPYVAHNFSKLFSMENWGGATFDVAMRFLYECPWRRLQELRELIPNIPFQMLLRGANAVGYTNYPDNVVFKFCEVAKENGMDVFRVFDSLNYLPNMLLGMEAAGSAGGVVEAAISYTGDVADPSRTKYSLQYYMGLAEELVRAGTHILCIKDMAGLLKPTACTMLVSSLRDRFPDLPLHIHTHDTSGAGVAAMLACAQAGADVVDVAADSMSGMTSQPSMGALVACTRGTPLDTEVPMERVFDYSEYWEGARGLYAAFDCTATMKSGNSDVYENEIPGGQYTNLHFQAHSMGLGSKFKEVKKAYVEANQMLGDLIKVTPSSKIVGDLAQFMVQNGLSRAEAEAQAEELSFPRSVVEFLQGYIGVPHGGFPEPFRSKVLKDLPRVEGRPGASLPPLDLQALEKELVDRHGEEVTPEDVLSAAMYPDVFAHFKDFTATFGPLDSLNTRLFLQGPKIAEEFEVELERGKTLHIKALAVSDLNRAGQRQVFFELNGQLRSILVKDTQAMKEMHFHPKALKDVKGQIGAPMPGKVIDIKVVAGAKVAKGQPLCVLSAMKMETVVTSPMEGTVRKVHVTKDMTLEGDDLILEIE
3LDG , Knot 162 384 0.83 40 222 373 
MKKTFNLVATAAAGIEAVVGKELRNLGLDCQVENGRVLFKGNIETIAKSNLWLRSADRIKIVVGEFPARTFEELFQGVYALDWENYLPLGCQFPVAKAKSVKSKLHNEPSIQGITKKAIVKKLQHYFHRPDSVPLPENGPEFKIEISLLKDQARVMIDTTGPSLFKRGYRTEKGGAPIKENMAAAIILLSNWFPDKPFVDPTCGSGTFCIEAAMIGMNIAPGFNRDFAFEEWPWVDEALVTRVRNEADEQADYDIQLDISGFDFDGRMVEIARKNAREVGLEDVVKLKQMRLQDFKTNKINGVLISNPPYGERLLDDKAVDILYNEMGETFAPLKTWSQFILTNDTDFEQKFGRKADKKRKLYNGSLKVDLYQFYGQRVKRVLR

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 \(c\).

\(|P_{f(9JAP_1)}(2) \setminus P_{f(8HWL_1)}(2)|=35\), \(|P_{f(8HWL_1)}(2) \setminus P_{f(9JAP_1)}(2)|=59\). 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 1:11000000000011110100101110010110010001000000010011001011111111100001011100111100111010000100100111000010010000011110011100110000110110000010010000101101010010000000110101100100101101001100000100101010101000000010011010000010100110001000101010010111111001001010011100001000011100010001000011111011011101100111001000000001111001010011000001000011011111010000100101000100011100010011001000001111001101001100111001000101010010001010001001001100111110000001000001100000010111000011000000011010100100001001000010011011001111100111000001001000100010111101100011000100100111000000001111111011111110011001110001001010000110011110011100000010011110110000110011011001011000100001001100110101100111000101001000100111011001011100110111100010100011010011100000000010000101011101000100101010110011011100100000110000100010101010001100011001100000101100100011101100101010111001000010100
Pair \(Z_2\) Length of longest common subsequence
9JAP_1,8HWL_1 94 5
9JAP_1,3LDG_1 161 4
8HWL_1,3LDG_1 149 4

Newick tree

 
[
	3LDG_1:85.34,
	[
		9JAP_1:47,8HWL_1:47
	]:38.34
]

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 \((0.85)(0.8)(\frac{2047 }{\log_{20} 2047}-\frac{869}{\log_{20}869})=285.\)
Status Protein1 Protein2 d d1/2
Query variables 9JAP_1 8HWL_1 367 316.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]