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Parikh vectors
7CZO_1 5DOU_1 1EAF_1 Letter Amino acid
8 101 13 S Serine
5 42 4 Y Tyrosine
0 20 2 C Cysteine
5 98 15 I Isoleucine
13 128 32 L Leucine
10 59 9 F Phenylalanine
8 109 15 G Glycine
3 41 5 M Methionine
1 87 10 T Threonine
4 119 18 V Valine
6 49 9 Q Glutamine
4 100 16 K Lycine
8 98 11 E Glutamic acid
5 34 5 H Histidine
6 72 16 P Proline
1 14 2 W Tryptophan
3 114 33 A Alanine
6 53 11 R Arginine
4 73 6 N Asparagine
6 78 11 D Aspartic acid 

7CZO_1|Chain A|Uncharacterized protein HI_0666|Haemophilus influenzae (strain ATCC 51907 / DSM 11121 / KW20 / Rd) (71421)
>5DOU_1|Chains A, B, C, D|Carbamoyl-phosphate synthase [ammonia], mitochondrial|Homo sapiens (9606)
>1EAF_1|Chain A|DIHYDROLIPOYL-TRANSACETYLASE|Azotobacter vinelandii (354)
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 \(w=f(c)\)
7CZO , Knot 56 106 0.82 38 96 104 
MRDLVRSGKVFLYGEFIGLLREDHRGFHFSYNPDYQGIPLSLSFPIEQSPFHSDTLFPYFASLVPEGWLKHKYALHQRIDESDMFRFLLNNGENMLGAVQIQEEKQ
5DOU , Knot 517 1489 0.84 40 345 1274 
MSYYHHHHHHDYDIPTTENLYFQGAMDPSVKAQTAHIVLEDGTKMKGYSFGHPSSVAGEVVFNTGLGGYPEAITDPAYKGQILTMANPIIGNGGAPDTTALDELGLSKYLESNGIKVSGLLVLDYSKDYNHWLATKSLGQWLQEEKVPAIYGVDTRMLTKIIRDKGTMLGKIEFEGQPVDFVDPNKQNLIAEVSTKDVKVYGKGNPTKVVAVDCGIKNNVIRLLVKRGAEVHLVPWNHDFTKMEYDGILIAGGPGNPALAEPLIQNVRKILESDRKEPLFGISTGNLITGLAAGAKTYKMSMANRGQNQPVLNITNKQAFITAQNHGYALDNTLPAGWKPLFVNVNDQTNEGIMHESKPFFAVQFHPEVTPGPIDTEYLFDSFFSLIKKGKATTITSVLPKPALVASRVEVSKVLILGSGGLSIGQAGEFDYSGSQAVKAMKEENVKTVLMNPNIASVQTNEVGLKQADTVYFLPITPQFVTEVIKAEQPDGLILGMGGQTALNCGVELFKRGVLKEYGVKVLGTSVESIMATEDRQLFSDKLNEINEKIAPSFAVESIEDALKAADTIGYPVMIRSAYALGGLGSGICPNRETLMDLSTKAFAMTNQILVEKSVTGWKEIEYEVVRDADDNCVTVCNMENVDAMGVHTGDSVVVAPAQTLSNAEFQMLRRTSINVVRHLGIVGECNIQFALHPTSMEYCIIEVNARLSRSSALASKATGYPLAFIAAKIALGIPLPEIKNVVSGKTSACFEPSLDYMVTKIPRWDLDRFHGTSSRIGSSMKSVGEVMAIGRTFEESFQKALRMCHPSIEGFTPRLPMNKEWPSNLDLRKELSEPSSTRIYAIAKAIDDNMSLDEIEKLTYIDKWFLYKMRDILNMEKTLKGLNSESMTEETLKRAKEIGFSDKQISKCLGLTEAQTRELRLKKNIHPWVKQIDTLAAEYPSVTNYLYVTYNGQEHDVNFDDHGMMVLGCGPYHIGSSVEFDWCAVSSIRTLRQLGKKTVVVNCNPETVSTDFDECDKLYFEELSLERILDIYHQEACGGCIISVGGQIPNNLAVPLYKNGVKIMGTSPLQIDRAEDRSIFSAVLDELKVAQAPWKAVNTLNEALEFAKSVDYPCLLRPSYVLSGSAMNVVFSEDEMKKFLEEATRVSQEHPVVLTKFVEGAREVEMDAVGKDGRVISHAISEHVEDAGVHSGDATLMLPTQTISQGAIEKVKDATRKIAKAFAISGPFNVQFLVKGNDVLVIECNLRASRSFPFVSKTLGVDFIDVATKVMIGENVDEKHLPTLDHPIIPADYVAIKAPMFSWPRLRDADPILRCEMASTGEVACFGEGIHTAFLKAMLSTGFKIPQKGILIGIQQSFRPRFLGVAEQLHNEGFKLFATEATSDWLNANNVPATPVAWPSQEGQNPSLSSIRKLIRDGSIDLVINLPNNNTKFVHDNYVIRRTAVDSGIPLLTNFQVTKLFAEAVQKSRKVDSKSLFHYRQYSAGKAA
1EAF , Knot 108 243 0.81 40 150 231 
IPPIPPVDFAKYGEIEEVPMTRLMQIGATNLHRSWLNVPHVTQFESADITELEAFRVAQKAVAKKAGVKLTVLPLLLKACAYLLKELPDFNSSLAPSGQALIRKKYVHIGFAVDTPDGLLVPVIRNVDQKSLLQLAAEAAELAEKARSKKLGADAMQGACFTISSLGHIGGTAFTPIVNAPEVAILGVSKASMQPVWDGKAFQPRLMLPLSLSYDHRVINGAAAARFTKRLGDLLADIRAILL

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 \(c\).

\(|P_{f(7CZO_1)}(2) \setminus P_{f(5DOU_1)}(2)|=6\), \(|P_{f(5DOU_1)}(2) \setminus P_{f(7CZO_1)}(2)|=255\). 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 1:1001100101110101111100000110100010001111010111000110000111011011101110000110001000011011100100111110100000
Pair \(Z_2\) Length of longest common subsequence
7CZO_1,5DOU_1 261 4
7CZO_1,1EAF_1 158 5
5DOU_1,1EAF_1 207 4

Newick tree

 
[
	5DOU_1:12.11,
	[
		7CZO_1:79,1EAF_1:79
	]:49.11
]

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 \((0.85)(0.8)(\frac{1595 }{\log_{20} 1595}-\frac{106}{\log_{20}106})=394.\)
Status Protein1 Protein2 d d1/2
Query variables 7CZO_1 5DOU_1 500 267
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]