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Parikh vectors
3TGW_1 1MAA_1 7GWF_1 Letter Amino acid
25 45 3 P Proline
20 31 9 S Serine
1 13 0 W Tryptophan
28 38 9 R Arginine
16 17 7 N Asparagine
8 10 3 H Histidine
21 8 4 K Lycine
27 29 8 D Aspartic acid
17 22 4 Q Glutamine
27 28 7 T Threonine
40 59 15 L Leucine
14 27 7 F Phenylalanine
33 45 10 V Valine
1 6 5 C Cysteine
31 27 6 E Glutamic acid
45 51 5 G Glycine
36 15 9 I Isoleucine
41 49 8 A Alanine
12 8 6 M Methionine
17 19 3 Y Tyrosine 

3TGW_1|Chains A, B|V-type ATP synthase beta chain|Methanosarcina mazei (192952)
>1MAA_1|Chains A, B, C, D|ACETYLCHOLINESTERASE|Mus musculus (10090)
>7GWF_1|Chain A|B-cell lymphoma 6 protein|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 \(w=f(c)\)
3TGW , Knot 187 460 0.83 40 221 429 
MVKEYKTITQIAGPLIFVEKTEPVGYNEIVNIKMGDGTVRRGQVLDSSADIVVVQVFEGTGGLDKDCGVIFTGETLKLPASVDLLGRILSGSGEPRDGGPRIVPDQLLDINGAAMNPYARLPPKDFIQTGISTIDGTNTLVRGQKLPIFSASGLPANEIALQIARQASVPGSESAFAVVFAAMGITNEEAQYFMSDFEKTGALERAVVFLNLADDPAVERIVTPRMALTAAEYLAYEHGMHVLVILTDITNYAEALRQMGAARNEVPGRRGYPGYMYTDLATLYERAGIVKGAKGSVTQIPILSMPGDDITHPIPDLSGYITEGQIVVARELHRKGIYPPINVLPSLSRLMNSGIGAGKTREDHKAVSDQMYAGYAEGRDLRGLVAIVGKEALSERDTKFLEFADLFEDKFVRQGRNENRTIEDTLEIGWQILTHLPENQLGRIDNKYIQKYHPAHRKAK
1MAA , Knot 218 547 0.83 40 252 510 
EGREDPQLLVRVRGGQLRGIRLKAPGGPVSAFLGIPFAEPPVGSRRFMPPEPKRPWSGVLDATTFQNVCYQYVDTLYPGFEGTEMWNPNRELSEDCLYLNVWTPYPRPASPTPVLIWIYGGGFYSGAASLDVYDGRFLAQVEGAVLVSMNYRVGTFGFLALPGSREAPGNVGLLDQRLALQWVQENIAAFGGDPMSVTLFGESAGAASVGMHILSLPSRSLFHRAVLQSGTPNGPWATVSAGEARRRATLLARLVGCPPGGAGGNDTELIACLRTRPAQDLVDHEWHVLPQESIFRFSFVPVVDGDFLSDTPEALINTGDFQDLQVLVGVVKDEGSYFLVYGVPGFSKDNESLISRAQFLAGVRIGVPQASDLAAEAVVLHYTDWLHPEDPTHLRDAMSAVVGDHNVVCPVAQLAGRLAAQGARVYAYIFEHRASTLTWPLWMGVPHGYEIEFIFGLPLDPSLNYTTEERIFAQRLMKYWTNFARTGDPNDPRDRKSPQWPPYTTAAQQYVSLNLKPLEVRRGLRAQTCAFWNRFLPKLLSATDTLD
7GWF , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE

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 \(c\).

\(|P_{f(3TGW_1)}(2) \setminus P_{f(1MAA_1)}(2)|=53\), \(|P_{f(1MAA_1)}(2) \setminus P_{f(3TGW_1)}(2)|=84\). 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 1:1100000100111111110000111000110101101010010110001011110110101110000111101001011101011101101010100111011100110101111010101110011001100101000110100111101011110011101100101110001111111111100001001100100011100111110110011100110101110110011000110111110010001011001111000111001011010001101000111101101010011110111001001110101010010111100100011011101110100110011111000000011000101101010010111111100110000001101101100011001000000100010111011001100011010000100001100010
Pair \(Z_2\) Length of longest common subsequence
3TGW_1,1MAA_1 137 4
3TGW_1,7GWF_1 173 4
1MAA_1,7GWF_1 212 3

Newick tree

 
[
	7GWF_1:10.47,
	[
		3TGW_1:68.5,1MAA_1:68.5
	]:35.97
]

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 \((0.85)(0.8)(\frac{1007 }{\log_{20} 1007}-\frac{460}{\log_{20}460})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 3TGW_1 1MAA_1 183 165.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]