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Parikh vectors
4GYP_1 2GPW_1 4AZI_1 Letter Amino acid
25 21 32 D Aspartic acid
28 35 30 E Glutamic acid
15 19 6 H Histidine
22 43 19 I Isoleucine
8 1 8 W Tryptophan
21 22 14 P Proline
15 18 29 N Asparagine
21 14 12 Q Glutamine
44 46 29 G Glycine
39 29 44 L Leucine
19 25 42 K Lycine
18 15 12 M Methionine
18 13 18 F Phenylalanine
10 15 28 Y Tyrosine
28 15 25 T Threonine
40 46 48 A Alanine
22 30 9 R Arginine
4 8 0 C Cysteine
19 23 21 S Serine
30 31 16 V Valine 

4GYP_1|Chains A, B|Glucarate dehydratase|Escherichia coli (83333)
>2GPW_1|Chains A, B, C, D|Biotin carboxylase|Escherichia coli (562)
>4AZI_1|Chains A, B|BETA-N-ACETYLHEXOSAMINIDASE|STREPTOCOCCUS PNEUMONIAE (170187)
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 \(w=f(c)\)
4GYP , Knot 184 446 0.84 40 252 429 
MSSQFTTPVVTEMQVIPVAGHDSMLMNLSGAHAPFFTRNIVIIKDNSGHTGVGEIPGGEKIRKTLEDAIPLVVGKTLGEYKNVLTLVRNTFADRDAGGRGLQTFDLRTTIHVVTGIEAAMLDLLGQHLGVNVASLLGDGQQRSEVEMLGYLFFVGNRKATPLPYQSQPDDSCDWYRLRHEEAMTPDAVVRLAEAAYEKYGFNDFKLKGGVLAGEEEAESIVALAQRFPQARITLDPNGAWSLNEAIKIGKYLKGSLAYAEDPCGAEQGFSGREVMAEFRRATGLPTATNMIATDWRQMGHTLSLQSVDIPLADPHFWTMQGSVRVAQMCHEFGLTWGSHSNNHFDISLAMFTHVAAAAPGKITAIDTHWIWQEGNQRLTKEPFEIKGGLVQVPEKPGLGVEIDMDQVMKAHELYQKHGLGARDDAMGMQYLIPGWTFDNKRPCMVR
2GPW , Knot 191 469 0.83 40 237 439 
MGSSHHHHHHSSGLVPRGSHMLDKIVIANRGEIALRILRACKELGIKTVAVHSSADRDLKHVLLADETVCIGPAPSVKSYLNIPAIISAAEITGAVAIHPGYGFLSENANFAEQVERSGFIFIGPKAETIRLMGDKVSAIAAMKKAGVPCVPGSDGPLGDDMDKNRAIAKRIGYPVIIKASGGGGGRGMRVVRGDAELAQSISMTRAEAKAAFSNDMVYMEKYLENPRHVEIQVLADGQGNAIYLAERDCSMQRRHQKVVEEAPAPGITPELRRYIGERCAKACVDIGYRGAGTFEFLFENGEFYFIEMNTRIQVEHPVTEMITGVDLIKEQLRIAAGQPLSIKQEEVHVRGHAVECRINAEDPNTFLPSPGKITRFHAPGGAGVRWESHIYAGYTVPPYYDSMIGKLICYGENRDVAIARMKNALQELIIDGIKTNVDLQIRIMNDENFQHGGTNIHYLEKKLGLQEK
4AZI , Knot 184 442 0.84 38 224 413 
GSHMEKLAKNKVISIDAGRKYFTLNQLKRIVDKASELGYSDVHLLLGNDGLRFLLDDMTITANGKTYASDDVKKAIIEGTKAYYDDPNGTALTQAEVTELIEYAKSKDIGLIPAINSPGHMDAMLVAMEKLGIKNPQAHFDKVSKTTMDLKNEEAMNFVKALIGKYMDFFAGKTKIFNFGTDEYANDATSAQGWYYLKWYQLYGKFAEYANTLAAMAKERGLQPMAFNDGFYYEDKDDVQFDKDVLISYWSKGWWGYNLASPQYLASKGYKFLNTNGDWYYILGQKPEDGWYFLKKAIENTGKTPFNQLASTKYPEVDLPTVGSMLSIWADRPSAEYKEEEIFELMTAFADHNKDYFRANYNALREELAKIPTNLEGYSKESLEALDAAKTALNYNLNRNKQAELDTLVANLKAALQGLKPAVTHSGSLDENEVAANVETRP

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 \(c\).

\(|P_{f(4GYP_1)}(2) \setminus P_{f(2GPW_1)}(2)|=93\), \(|P_{f(2GPW_1)}(2) \setminus P_{f(4GYP_1)}(2)|=78\). 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 1:10001001110010111111000111010110111100011110000100111011110010001001111111001100001101100011000111011001010001011011011110111001110110111010000010111011111000101110000100000100100001101011101101100001100101011111100010011111001101010101011101001101100101011010010110011010011101001011101001110010011001010010111101011010101011010001110110000001010111100111111101011000111001000100011010111101100111110101001101001000011110001111001111101000010110
Pair \(Z_2\) Length of longest common subsequence
4GYP_1,2GPW_1 171 5
4GYP_1,4AZI_1 144 4
2GPW_1,4AZI_1 159 4

Newick tree

 
[
	2GPW_1:85.78,
	[
		4GYP_1:72,4AZI_1:72
	]:13.78
]

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 \((0.85)(0.8)(\frac{915 }{\log_{20} 915}-\frac{446}{\log_{20}446})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 4GYP_1 2GPW_1 153 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]