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Parikh vectors
7FKL_1 6KQG_1 2PHG_1 Letter Amino acid
14 15 12 D Aspartic acid
10 20 10 P Proline
21 17 12 T Threonine
5 1 0 W Tryptophan
7 4 5 Y Tyrosine
11 20 16 R Arginine
15 39 9 E Glutamic acid
23 17 17 I Isoleucine
18 13 13 S Serine
1 0 6 C Cysteine
5 5 1 H Histidine
10 10 9 F Phenylalanine
14 30 12 V Valine
19 18 15 K Lycine
8 3 6 M Methionine
18 22 24 A Alanine
14 10 7 N Asparagine
9 7 9 Q Glutamine
7 20 7 G Glycine
29 44 16 L Leucine 

7FKL_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>6KQG_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Thermus thermophilus (strain HB8 / ATCC 27634 / DSM 579) (300852)
>2PHG_1|Chain A|Transcription initiation factor IIB|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 \(w=f(c)\)
7FKL , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
6KQG , Knot 129 315 0.78 38 162 293 
MLDSKLKAPVFTVRTQGREYGEFVLEPLERGFGVTLGNPLRRILLSSIPGTAVTSVYIEDVLHEFSTIPGVKEDVVEIILNLKELVVRFLNPSLQTVTLLLKAEGPKEVKARDFLPVADVEIMNPDLHIATLEEGGRLNMEVRVDRGVGYVPAEKHGIKDRINAIPVDAVFSPVRRVAFQVEDTRLGQRTDLDKLTLRIWTDGSVTPLEALNQAVEILREHLTYFSNPQAAAVAAPEEAKEPEAPPEQEEELDLPLEELGLSTRVLHSLKEEGIESVRALLALNLKDLKNIPGIGERSLEEIKEALEKKGFTLKE
2PHG , Knot 91 206 0.78 38 138 195 
SRAMMNAFKEITTMADRINLPRNIVDRTNNLFKQVYEQKSLKGRANDAIASACLYIACRQEGVPRTFKEICAVSRISKKEIGRCFKLILKALETSVDLITTGDFMSRFCSNLCLPKQVQMAATHIARKAVELDLVPGRSPISVAAAAIYMASQASAEKRTQKEIGDIAGVADVTIRQSYRLIYPRAPDLFPTDFKFDTPVDKLPQL

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 \(c\).

\(|P_{f(7FKL_1)}(2) \setminus P_{f(6KQG_1)}(2)|=87\), \(|P_{f(6KQG_1)}(2) \setminus P_{f(7FKL_1)}(2)|=74\). 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 1:111000001011000101110000100101000101011001101011010100101110110001111000100110100100101110011000010011100011101101011011011101001011101110100100111010010111101000110010000110010111011000000101111001010100001110100001101000100111000100001010110000100111100101
Pair \(Z_2\) Length of longest common subsequence
7FKL_1,6KQG_1 161 4
7FKL_1,2PHG_1 157 3
6KQG_1,2PHG_1 152 4

Newick tree

 
[
	7FKL_1:80.64,
	[
		2PHG_1:76,6KQG_1:76
	]:4.64
]

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 \((0.85)(0.8)(\frac{573 }{\log_{20} 573}-\frac{258}{\log_{20}258})=89.1\)
Status Protein1 Protein2 d d1/2
Query variables 7FKL_1 6KQG_1 107 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]