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Parikh vectors
7TYL_1 7TMM_1 8IMQ_1 Letter Amino acid
27 50 14 A Alanine
28 27 4 R Arginine
31 46 14 D Aspartic acid
26 41 1 K Lycine
18 53 2 S Serine
17 39 4 T Threonine
4 7 4 W Tryptophan
8 6 1 C Cysteine
21 18 9 Q Glutamine
32 47 24 E Glutamic acid
30 46 8 L Leucine
20 24 6 F Phenylalanine
14 27 3 Y Tyrosine
26 42 4 V Valine
7 11 1 H Histidine
23 41 5 I Isoleucine
14 30 4 P Proline
24 17 4 N Asparagine
17 52 8 G Glycine
7 15 4 M Methionine 

7TYL_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|Homo sapiens (9606)
>7TMM_1|Chains A, C, E|H(+)-transporting two-sector ATPase|Saccharomyces cerevisiae (4932)
>8IMQ_1|Chain A|Acidic protein MsyB|Escherichia coli (562)
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 \(w=f(c)\)
7TYL , Knot 169 394 0.85 40 227 378 
MGCLGNSKTEDQRNEEKAQREANKKIEKQLQKDKQVYRATHRLLLLGAGESGKNTIVKQMRILHVNGFNGEGGEEDPQAARSNSDGEKATKVQDIKNNLKEAIETIVAAMSNLVPPVELANPENQFRVDYILSVMNVPDFDFPPEFYEHAKALWEDEGVRACYERSNEYQLIDCAQYFLDKIDVIKQADYVPSDQDLLRCRVLTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVASSSYNMVIREDNQTNRLQAALKLFDSIWNNKWLRDTSVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENIRRVFNDCRDIIQRMHLRQYELL
7TMM , Knot 251 639 0.84 40 270 587 
MAGAIENARKEIKRISLEDHAESEYGAIYSVSGPVVIAENMIGCAMYELVKVGHDNLVGEVIRIDGDKATIQVYEETAGLTVGDPVLRTGKPLSVELGPGLMETIYDGIQRPLKAIKEESQSIYIPRGIDTPALDRTIKWQFTPGKFQVGDHISGGDIYGSVFENSLISSHKILLPPRSRGTITWIAPAGEYTLDEKILEVEFDGKKSDFTLYHTWPVRVPRPVTEKLSADYPLLTGQRVLDALFPCVQGGTTCIPGAFGCGKTVISQSLSKYSNSDAIIYVGCGERGNEMAEVLMEFPELYTEMSGTKEPIMKRTTLVANTSNMPVAAREASIYTGITLAEYFRDQGKNVSMIADSSSRWAEALREISGRLGEMPADQGFPAYLGAKLASFYERAGKAVALGSPDRTGSVSIVAAVSPAGGDFSDPVTTATLGITQVFWGLDKKLAQRKHFPSINTSVSYSKYTNVLNKFYDSNYPEFPVLRDRMKEILSNAEELEQVVQLVGKSALSDSDKITLDVATLIKEDFLQQNGYSTYDAFCPIWKTFDMMRAFISYHDEAQKAVANGANWSKLADSTGDVKHAVSSSKFFEPSRGEKEVHGEFEKLLSTMQERFAESTDDYKDHDGDYKDHDIDYKDDDDK
8IMQ , Knot 56 124 0.72 40 87 113 
MTMYATLEEAIDAAREEFLADNPGIDAEDANVQQFNAQKYVLQDGDIMWQVEFFADEGEEGECLPMLSGEAAQSVFDGDYDEIEIRQEWQEENTLHEWDEGEFQLEPPLDTEEGRAAADEWDER

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 \(c\).

\(|P_{f(7TYL_1)}(2) \setminus P_{f(7TMM_1)}(2)|=61\), \(|P_{f(7TMM_1)}(2) \setminus P_{f(7TYL_1)}(2)|=104\). 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 1:1101100000000000010001000100010000010010001111111001000110010110101101011000101100000100100100100010011001111100111110110100010100110110110101110100010111000110100000000011001001100101100100110000110001100111000101001010110111000000011001001011111110000011100000000101110110011000110000111110000111001111000100011011000010010101100101001001100011010010101000001010001000010011000001100101000011
Pair \(Z_2\) Length of longest common subsequence
7TYL_1,7TMM_1 165 4
7TYL_1,8IMQ_1 188 3
7TMM_1,8IMQ_1 215 4

Newick tree

 
[
	8IMQ_1:10.42,
	[
		7TYL_1:82.5,7TMM_1:82.5
	]:23.92
]

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 \((0.85)(0.8)(\frac{1033 }{\log_{20} 1033}-\frac{394}{\log_{20}394})=168.\)
Status Protein1 Protein2 d d1/2
Query variables 7TYL_1 7TMM_1 214 173
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]