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Parikh vectors
6PTO_1 8VOM_1 5VBU_1 Letter Amino acid
26 1 26 Q Glutamine
68 7 30 E Glutamic acid
90 8 70 L Leucine
19 2 13 M Methionine
85 12 28 S Serine
31 4 30 R Arginine
12 1 7 C Cysteine
42 1 16 F Phenylalanine
37 6 38 P Proline
52 17 21 T Threonine
32 1 11 Y Tyrosine
50 13 29 V Valine
56 9 30 A Alanine
67 4 20 D Aspartic acid
46 16 31 G Glycine
24 3 16 H Histidine
58 3 8 N Asparagine
57 5 22 I Isoleucine
64 5 22 K Lycine
11 0 8 W Tryptophan 

6PTO_1|Chains A[auth X], B[auth Y], C[auth Z]|DNA polymerase alpha-binding protein|Saccharomyces cerevisiae (4932)
>8VOM_1|Chains A, B, C, D, E, F|Spike protein|Bacteriophage P2 (2905681)
>5VBU_1|Chains A, B, C|Cytochrome P450 21-hydroxylase|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 \(w=f(c)\)
6PTO , Knot 352 927 0.86 40 313 842 
MVSVIDKLVFDFGGKTLVSLAPDNNTLCVANKNGLTKILKTNNPEEEPETLDSSKLVSSIKCYSNSHFLMTTMQGDALRYNIDSSQEELLARFALPLRDCCVIHSGKMAVFGGDDLELILLELDDETHKKHAIKIDEQVSQISYNSQMNILAVSMINGKVQIFSLTSTIPNKVHELNDYIVANSYDDTHRDKILSNMMDDIDKDNDNDLSETADPDENNVADPEFCAANRICTRVAWHPKGLHFALPCADDTVKIFSIKGYSLQKTLSTNLSSTKAHFIDLQFDPLRGTYIAAVDLNNKLTVWNWETSEIHYTREFKRKITNIAWKIQADSKTLDLVLGTWSGSIAIVQNLAESVVSNIPDQSVAESSTKHGLFVDSESDLENLEGNDDINKSDKLFSDITQEANAEDVFTQTHDGPSGLSEKRKYNFEDEEDFIDDDDGAGYISGKKPHNEHSYSRVHKTHSFPISLANTGKFRYMPFSPAGTPFGFTDRRYLTMNEVGYVSTVKNSEQYSITVSFFDVGRFREYHFEDLFGYDLCFLNEKGTLFGQSKTGQIQYRPHDSIHSNWTKIIPLQAGERITSVAATPVRVIVGTSLGYFRSFNQFGVPFAVEKTSPIVALTAQNYRVFSVHYSQFHGLSYSLSELGTSSKRYYKRECPLPMSLPNINSDMKKDANLDYYNFNPMGIKSLFFSSYGDPCIFGSDNTLLLLSKWRSPEESKWLPILDSNMEIWKMSGGKETTDIHVWPLALAYDTLNCILVKGKHIWPEFPLPLPSEMEIRMPVFVKSKLLEENKAILNKKNEIGADTEAEEGEEDKEIQIPVSMAAEEEYLRSKVLSELLTDTLENDGEMYGNENEVLAALNGAYDKALLRLFASACSDQNVEKALSLAHELKQDRALTAAVKISERAELPSLVKKINNIREARYEQQLK
8VOM , Knot 59 118 0.79 38 87 111 
GSGSDALHIRFPDGAVIEYEPETSALTVSGIKTASVTASGSVTATVPVVMVKASTRVTLDTPEVVCTNRLITGTLEVQKGGTMRGNIEHTGGELSSNGKVLHTAKAPGDSGGTTGSPL
5VBU , Knot 195 476 0.84 40 251 446 
MAKKTSSKGKLPPLAPGFLHLLQPDLPIYLLGLTQKFGPIYRLHLGLQDVVVLNSKRTIEEAMVKKWADFAGRPEPLTYKLVSRNYPDLSLGDYSLLWKAHKKLTRSALLLGIRDSMEPVVEQLTQEFCERMRAQPGTPVAIEEEFSLLTCSIICYLTFGDKIKDDNLMPAYYKCIQEVLKTWSHWSIQIVDVIPFLRFFPNPGLRRLKQAIEKRDHIVEMQLRQHKESLVAGQWRDMMDYMLQGVAQPSMEEGSGQLLEGHVHMAAVDLLIGGTETTANTLSWAVVFLLHHPEIQQRLQEELDHELGPGASSSRVPYKDRARLPLLNATIAEVLRLRPVVPLALPHRTTRPSSISGYDIPEGTVIIPNLQGAHLDETVWERPHEFWPDRFLEPGKNSRALAFGCGARVCLGEPLARLELFVVLTRLLQAFTLLPSGDALPSLQPLPHCSVILKMQPFQVRLQPRGMGAHSPGQSQ

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 \(c\).

\(|P_{f(6PTO_1)}(2) \setminus P_{f(8VOM_1)}(2)|=234\), \(|P_{f(8VOM_1)}(2) \setminus P_{f(6PTO_1)}(2)|=8\). 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 1:110110011101110011011100001011000110011000010001001000011001000000011100101011000100000011101111100001100101111110010111101000000001101000100100000101111011010101101000110010010001110000000000110011001000000010001010000110101011001000111010110111101000101101010010001000100001011010101101001111010001011010000100000100010011101010000101111010101111001100110011000110000001111000001001010001000001100100010100110000011011000000010000011000011101010010000000010000011101100101001110111011110000010100110100100000001010110110100001001110010110001011100001010001000100010011110110010011101101111001101001001111111000011111010000110100001011000100110000000000011110110100010001010000101111001110001010111000011110010010000111110001011010110000010111111100010011101001110111111001010111110001100001110000011100010010000010111011100001000110011000100010101000011111011000111011101000001001101100100001101110100010110110010010010000010
Pair \(Z_2\) Length of longest common subsequence
6PTO_1,8VOM_1 242 4
6PTO_1,5VBU_1 140 5
8VOM_1,5VBU_1 208 3

Newick tree

 
[
	8VOM_1:12.84,
	[
		6PTO_1:70,5VBU_1:70
	]:53.84
]

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 \((0.85)(0.8)(\frac{1045 }{\log_{20} 1045}-\frac{118}{\log_{20}118})=255.\)
Status Protein1 Protein2 d d1/2
Query variables 6PTO_1 8VOM_1 330 184
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]