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Parikh vectors
8XMM_1 6IER_1 1ZYP_1 Letter Amino acid
65 11 3 Y Tyrosine
110 60 15 A Alanine
92 39 8 R Arginine
110 42 14 G Glycine
71 5 5 M Methionine
138 8 13 K Lycine
138 22 9 F Phenylalanine
172 20 11 S Serine
93 24 16 T Threonine
98 11 9 N Asparagine
105 27 11 D Aspartic acid
38 1 2 C Cysteine
26 17 4 H Histidine
30 9 1 W Tryptophan
133 25 17 E Glutamic acid
150 25 10 I Isoleucine
201 32 23 L Leucine
80 23 10 P Proline
55 8 8 Q Glutamine
126 37 13 V Valine 

8XMM_1|Chain A|Sodium channel protein type 9 subunit alpha|Homo sapiens (9606)
>6IER_1|Chain A|beta-glucosidase 1317|uncultured bacterium (77133)
>1ZYP_1|Chains A, B|Alkyl hydroperoxide reductase subunit F|Salmonella typhimurium (90371)
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 \(w=f(c)\)
8XMM , Knot 677 2031 0.84 40 369 1611 
MASWSHPQFEKGGGARGGSGGGSWSHPQFEKGFDYKDDDDKGTMAMLPPPGPQSFVHFTKQSLALIEQRIAERKSKEPKEEKKDDDEEAPKPSSDLEAGKQLPFIYGDIPPGMVSEPLEDLDPYYADKKTFIVLNKGKTIFRFNATPALYMLSPFSPLRRISIKILVHSLFSMLIMCTILTNCIFMTMNNPPDWTKNVEYTFTGIYTFESLVKILARGFCVGEFTFLRDPWNWLDFVVIVFAYLTEFVNLGNVSALRTFRVLRALKTISVIPGLKTIVGALIQSVKKLSDVMILTVFCLSVFALIGLQLFMGNLKHKCFRNSLENNETLESIMNTLESEEDFRKYFYYLEGSKDALLCGFSTDSGQCPEGYTCVKIGRNPDYGYTSFDTFSWAFLALFRLMTQDYWENLYQQTLRAAGKTYMIFFVVVIFLGSFYLINLILAVVAMAYEEQNQANIEEAKQKELEFQQMLDRLKKEQEEAEAIAAAAAEYTSIRRSRIMGLSESSSETSKLSSKSAKERRNRRKKKNQKKLSSGEEKGDAEKLSKSESEDSIRRKSFHLGVEGHRRAHEKRLSTPNQSPLSIRGSLFSARRSSRTSLFSFKGRGRDIGSETEFADDEHSIFGDNESRRGSLFVPHRPQERRSSNISQASRSPPMLPVNGKMHSAVDCNGVVSLVDGRSALMLPNGQLLPEVIIDKATSDDSGTTNQIHKKRRCSSYLLSEDMLNDPNLRQRAMSRASILTNTVEELEESRQKCPPWWYRFAHKFLIWNCSPYWIKFKKCIYFIVMDPFVDLAITICIVLNTLFMAMEHHPMTEEFKNVLAIGNLVFTGIFAAEMVLKLIAMDPYEYFQVGWNIFDSLIVTLSLVELFLADVEGLSVLRSFRLLRVFKLAKSWPTLNMLIKIIGNSVGAFGNLMLVLFIIVFIFAVVGMQLFGKSYKECVCKINDDCTLPRWHMNDFFHSFLIVFRVLCGEWIETMWDCMEVAGQAMCLIFYMMVFFIGNLVVLNLFLALLLSSFSSDNLTAIEEDPDANNLQIAVTRIKKGINYVKQTLREFILKAFSKKPKISREIRQAEDLNTKKENYISNHTLAEMSKGHNFLKEKDKISGFGSSVDKHLMEDSDGQSFIHNPSLTVTVPIAPGESDLENMNAEELSSDSDSEYSKVRLNRSSSSECSTVDNPLPGEGEEAEAEPMNSDEPEACFTDGCVWRFSCCQVNIESGKGKIWWNIRKTCYKIVEHSWFESFIVLMILLSSGALAFEDIYIERKKTIKIILEYADKIFTYIFILEMLLKWIAYGYKTYFTNAWCWLDFLIVDVSLVTLVANTLGYSDLGPIKSLRTLRALRPLRALSRFEGMRVVVNALIGAIPSIMNVLLVCLIFWLIFSIMGVNLFAGKFYECINTTDGSRFPASQVPNRSECFALMNVSQNVRWKNLKVNFDNVGLGYLSLLQVATFKGWTIIMYAAVDSVNVDKQPKYEYSLYMYIYFIFFIIFGSFFTLNLFICVIIDNFNQQKKKLGGQDIFMTEEQKKYYNAMKKLGSKKPQKPIPRPGNKIQGCIFDLVTNQAFDISIMVLICLNMVTMMVEKEGQSQHMTEVLYWINVVFIILFTGECVLKLISLRHYYFTVGWNIFDFVVVIISIVGMFLADLIETYFVSPTLFRVIRLARIGRILRLVKGAKGIRTLLFALMMSLPALFNIGLLLFLVMFIYAIFGMSNFAYVKKEDGINDMFNFETFGNSMICLFQITTSAGWDGLLAPILNSKPPDCDPKKVHPGSSVEGDCGNPSVGIFYFVSYIIISFLVVVNMYIAVILENFSVATEESTEPLSEDDFEMFYEVWEKFDPDATQFIEFSKLSDFAAALDPPLLIAKPNKVQLIAMDLPMVSGDRIHCLDILFAFTKRVLGESGEMDSLRSQMEERFMSANPSKVSYEPITTTLKRKQEDVSATVIQRAYRRYRLRQNVKNISSIYIKDGDRDDDLLNKKDMAFDNVNENSSPEKTDATSSTTSPPSYDSVTKPDKEKYEQDRTEKEDKGKDSKESKK
6IER , Knot 177 446 0.80 40 211 407 
MTRVSRRGLLAVGAATGGAVVATGTARADITARSGFLWGSAGAAYQIEGGNVASDLWVVEHVQPTIFREASGDAVDAYHRVFDDIALAASLGFNAHRFSIEWSRIEPEKGQISLAAIAYYRRVLEAIRSHGMTPVVTLHHFTSPRWFAAAGGFETRDGIEPFVRYAEIVSRHLGDLFGVVATFNEPNLGGLMSWGSLSKQIRPIVQASRASAARAVNSDKFAPLVLGDFRIQTPIIIEAHERAYDVIRRETGGRTPVGLTIAVNDERAGTPDAGLDAKLEDAVLPWVRARGDFIGVQNYTYALVGKDADLPNPEGVELTQMNYPFAPEALEGAIRLVARHTDKPIYVTENGVATEDDARRVAFIDRAVPAVFACMRDGIDVRGYIHWSFLDNWEWFAGFGPKFGLVAVDRTTFERTPKPSAAHLGRLARAGLPGDLRPLEHHHHHH
1ZYP , Knot 96 202 0.84 40 148 198 
MLDTNMKTQLRAYLEKLTKPVELIATLDDSAKSAEIKELLAEIAELSDKVTFKEDNTLPVRKPSFLITNPGSQQGPRFAGSPLGHEFTSLVLALLWTGGHPSKEAQSLLEQIRDIDGDFEFETYYSLSCHNCPDVVQALNLMAVLNPRIKHTAIDGGTFQNEITERNVMGVPAVFVNGKEFGQGRMTLTEIVAKVDTGAEKR

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 \(c\).

\(|P_{f(8XMM_1)}(2) \setminus P_{f(6IER_1)}(2)|=167\), \(|P_{f(6IER_1)}(2) \setminus P_{f(8XMM_1)}(2)|=9\). 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 1:110100101001111011011101001010011000000001011111111100110100001111000110000001000000000011010001011001111010111111001100101001000011110010011010101110110110110010101110011011110011000111010011010001000101100100110111011011010110011011011111110100110110101100101101100101111100111111001001001111011010111111101111010000100010000010011001000001000100101000111011000010010100010110010010001001011111110110000100100001011100011111111111010110111111111000000101001000010100110010000001011111110000100001111000000000100001000000000000001001000101001000000001000010111010001000010010001101010110100000001101010100110000110000011100000010111100100000001001000111111010100110001110110100111110101110111001000001000010000000001100011001010001100101100010010000000111100110011110001011010001011110111011101011100111110001100010011111011101111101110111101000101110110011101011011110101101100101101101100110101110111001111101111111111111111101110000001001000001101010011001111101101011001100101110110111011111110111101111111001000010110001010010111001001100100010011101100010100010010010000000100001101001001100000101110010001100001001100101010111111000100101001000000000010100000000001001111010010101100001010100101101000010100101011101000000110001100111111110011111001010000010111001001100111101110111010000100110110111101011011100110001111001001011011011001011011101111111011011110111111101111011110100010000100111001100000111101000101001010100111101011011010110111011100101000100000101010111111110110101110111001000000111001110000000001100110001001110110010101101100011010111110101101110001000010011011011111110100110110100001011101101111110111111101100011010110110110110110110110110011111110111110111111111110111110011010000110011010011001101101000111011111110001100010010110010100101011110110011101111101011111001011000000110000101100110010101001101001001111101111110100101111011110100100101111100011100101001000100011010100100011000100000010101100100000100010010010100100000110000111001000001000010000001100001001000000000000000100000000
Pair \(Z_2\) Length of longest common subsequence
8XMM_1,6IER_1 176 4
8XMM_1,1ZYP_1 229 4
6IER_1,1ZYP_1 149 5

Newick tree

 
[
	8XMM_1:10.78,
	[
		6IER_1:74.5,1ZYP_1:74.5
	]:35.28
]

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 \((0.85)(0.8)(\frac{2477 }{\log_{20} 2477}-\frac{446}{\log_{20}446})=496.\)
Status Protein1 Protein2 d d1/2
Query variables 8XMM_1 6IER_1 634 383
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]