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Parikh vectors
4BAE_1 7DTE_1 4CGR_1 Letter Amino acid
17 61 9 T Threonine
7 26 3 M Methionine
6 53 14 S Serine
3 9 6 W Tryptophan
20 74 17 V Valine
14 83 20 L Leucine
8 51 3 K Lycine
0 29 1 C Cysteine
18 31 21 E Glutamic acid
8 33 3 I Isoleucine
8 57 5 F Phenylalanine
7 58 3 Y Tyrosine
14 64 46 A Alanine
4 56 2 N Asparagine
4 28 4 Q Glutamine
25 46 14 G Glycine
8 37 5 H Histidine
6 30 12 P Proline
14 43 25 R Arginine
14 75 3 D Aspartic acid 

4BAE_1|Chains A, B, C, D|DNA GYRASE SUBUNIT B|MYCOBACTERIUM SMEGMATIS (1772)
>7DTE_1|Chain A|RNA-directed RNA polymerase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4CGR_1|Chains A, B|PUTATIVE TETR-FAMILY TRANSCRIPTIONAL REGULATOR|STREPTOMYCES COELICOLOR (1902)
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 \(w=f(c)\)
4BAE , Knot 94 205 0.81 38 136 198 
MLEGLEAVRKRPGMYIGSTGERGLHHLIWEVVDNAVDEAMAGFATRVDVKIHADGSVEVRDDGRGIPVEMHATGMPTIDVVMTQLHAGGKFDGETYAVSGGLHGVGVSVVNALSTRLEATVLRDGYEWFQYYDRSVPGKLKQGGETKETGTTIRFWADPEIFETTDYNFETVARRLQEMAFLNKGLTIELTDERDGKHRVFHYPG
7DTE , Knot 355 944 0.85 40 326 864 
MGSADAQSFLNRVCGVSAARLTPCGTGTSTDVVYRAFDIYNDKVAGFAKFLKTNCCRFQEKDEDDNLIDSYFVVKRHTFSNYQHEETIYNLLKDCPAVAKHDFFKFRIDGDMVPHISRQRLTKYTMADLVYALRHFDEGNCDTLKEILVTYNCCDDDYFNKKDWYDFVENPDILRVYANLGERVRQALLKTVQFCDAMRNAGIVGVLTLDNQDLNGNWYDFGDFIQTTPGSGVPVVDSYYSLLMPILTLTRALTAESHVDTDLTKPYIKWDLLKYDFTEERLKLFDRYFKYWDQTYHPNCVNCLDDRCILHCANFNVLFSTVFPPTSFGPLVRKIFVDGVPFVVSTGYHFRELGVVHNQDVNLHSSRLSFKELLVYAADPAMHAASGNLLLDKRTTCFSVAALTNNVAFQTVKPGNFNKDFYDFAVSKGFFKEGSSVELKHFFFAQDGNAAISDYDYYRYNLPTMCDIRQLLFVVEVVDKYFDCYDGGCINANQVIVNNLDKSAGFPFNKWGKARLYYDSMSYEDQDALFAYTKRNVIPTITQMNLKYAISAKNRARTVAGVSICSTMTNRQFHQKLLKSIAATRGATVVIGTSKFYGGWHNMLKTVYSDVENPHLMGWDYPKCDRAMPNMLRIMASLVLARKHTTCCSLSHRFYRLANECAQVLSEMVMCGGSLYVKPGGTSSGDATTAYANSVFNICQAVTANVNALLSTDGNKIADKYVRNLQHRLYECLYRNRDVDTDFVNEFYAYLRKHFSMMILSDDAVVCFNSTYASQGLVASIKNFKSVLYYQNNVFMSEAKCWTETDLTKGPHEFCSQHTMLVKQGDDYVYLPYPDPSRILGAGCFVDDIVKTDGTLMIERFVSLAIDAYPLTKHPNQEYADVFHLYLQYIRKLHDELTGHMLDMYSVMLTNDNTSRYWEPEFYEAMYTPHTVLQHHHHHHHHHH
4CGR , Knot 94 216 0.78 40 117 197 
HMVSLTERRKAETRMEIARAAARLFVGQGLRATRAEDIARAAGVAPRTFYRYFATKEEAVAPLYALGAERWVRAVREAPAELSPPEALERAVRHTLTPGAGVSAPSWEWARTLIRLAESSPALRKVWAEVCHSTERGLVQALAARMSGGDDNVAVRLAASPRLHFAAAVAGASVRVAAEHWASSSPQGARSPLEQALLNLEVLRGFAWEAGPAEEG

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 \(c\).

\(|P_{f(4BAE_1)}(2) \setminus P_{f(7DTE_1)}(2)|=15\), \(|P_{f(7DTE_1)}(2) \setminus P_{f(4BAE_1)}(2)|=205\). 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 1:1101101100011101100100110011101100110011111100101010101010100010111101010111010111001011101010001101110111101101100010101100100110000001110100110000010010111010110000001001100100111100110101000001000110011
Pair \(Z_2\) Length of longest common subsequence
4BAE_1,7DTE_1 220 4
4BAE_1,4CGR_1 151 4
7DTE_1,4CGR_1 251 4

Newick tree

 
[
	7DTE_1:12.09,
	[
		4BAE_1:75.5,4CGR_1:75.5
	]:53.59
]

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 \((0.85)(0.8)(\frac{1149 }{\log_{20} 1149}-\frac{205}{\log_{20}205})=253.\)
Status Protein1 Protein2 d d1/2
Query variables 4BAE_1 7DTE_1 322 194
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]