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Parikh vectors
7LCZ_1 9EIW_1 4NVH_1 Letter Amino acid
13 13 6 H Histidine
38 51 25 L Leucine
22 32 15 S Serine
16 29 10 R Arginine
20 27 20 N Asparagine
12 13 1 C Cysteine
17 55 26 G Glycine
11 23 13 T Threonine
3 16 6 W Tryptophan
21 33 14 V Valine
14 39 10 I Isoleucine
17 42 23 K Lycine
5 19 5 M Methionine
8 44 14 P Proline
9 18 9 Q Glutamine
11 28 18 F Phenylalanine
31 48 16 A Alanine
12 26 24 D Aspartic acid
24 52 20 E Glutamic acid
5 14 14 Y Tyrosine 

7LCZ_1|Chains A, B|Isoform B of NAD(+) hydrolase sarm1|Drosophila melanogaster (7227)
>9EIW_1|Chain A|Phosphoenolpyruvate carboxykinase, cytosolic [GTP]|Rattus norvegicus (10116)
>4NVH_1|Chain A[auth B]|Cytochrome c peroxidase|Saccharomyces cerevisiae (285006)
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 \(w=f(c)\)
7LCZ , Knot 136 309 0.84 40 195 295 
KSQYLEKINEVIRRAWAVPTHGHELGYSLCNSLRQSGGLDLLMKNCVKPDLQFSSAQLLEQCLTTENRKHVVDNGLDKVVNVACVCTKNSNMEHSRVGTGILEHLFKHSEGTCSDVIRLGGLDAVLFECRTSDLETLRHCASALANLSLYGGAENQEEMILRKVPMWLFPLAFHNDDNIKYYACLAIAVLVANKEIEAEVLKSGCLDLVEPFVTSHDPSAFARSNLAHAHGQSKHWLKRLVPVLSSNREEARNLAAFHFCMEAGIKREQGNTDIFREINAIEALKNVASCPNAIASKFAAQALRLIGET
9EIW , Knot 254 622 0.87 40 281 577 
MPPQLHNGLDFSAKVIQGSLDSLPQEVRKFVEGNAQLCQPEYIHICDGSEEEYGRLLAHMQEEGVIRKLKKYDNCWLALTDPRDVARIESKTVIITQEQRDTVPIPKSGQSQLGRWMSEEDFEKAFNARFPGCMKGRTMYVIPFSMGPLGSPLAKIGIELTDSPYVVASMRIMTRMGTSVLEALGDGEFIKCLHSVGCPLPLKKPLVNNWACNPELTLIAHLPDRREIISFGSGYGGNSLLGKKCFALRIASRLAKEEGWLAEHMLILGITNPEGKKKYLAAAFPSACGKTNLAMMNPTLPGWKVECVGDDIAWMKFDAQGNLRAINPENGFFGVAPGTSVKTNPNAIKTIQKNTIFTNVAETSDGGVYWEGIDEPLAPGVTITSWKNKEWRPQDEEPCAHPNSRFCTPASQCPIIDPAWESPEGVPIEGIIFGGRRPAGVPLVYEALSWQHGVFVGAAMRSEATAAAEHKGKVIMHDPFAMRPFFGYNFGKYLAHWLSMAHRPAAKLPKIFHVNWFRKDKNGKFLWPGFGENSRVLEWMFGRIEGEDSAKLTPIGYVPKEDALNLKGLGDVNVEELFGISKEFWEKEVEEIDKYLEDQVNADLPYEIERELRALKQRISQM
4NVH , Knot 131 289 0.85 40 191 283 
LVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGGGANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL

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 \(c\).

\(|P_{f(7LCZ_1)}(2) \setminus P_{f(9EIW_1)}(2)|=48\), \(|P_{f(9EIW_1)}(2) \setminus P_{f(7LCZ_1)}(2)|=134\). 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 1:000010010011001111100100110010001000111011100010101010010110001000000011001100110110100000010000110111001100001000011011110111100000010010001011101010111000001110011111111110000010001011111111000101011001010110111000010111000110101000011001111100000010011110101011100001000110010110110011001011100111011011100
Pair \(Z_2\) Length of longest common subsequence
7LCZ_1,9EIW_1 182 4
7LCZ_1,4NVH_1 182 4
9EIW_1,4NVH_1 174 3

Newick tree

 
[
	7LCZ_1:92.29,
	[
		9EIW_1:87,4NVH_1:87
	]:5.29
]

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 \((0.85)(0.8)(\frac{931 }{\log_{20} 931}-\frac{309}{\log_{20}309})=167.\)
Status Protein1 Protein2 d d1/2
Query variables 7LCZ_1 9EIW_1 213 156.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]