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Parikh vectors
4XRG_1 1GKA_1 4OIO_1 Letter Amino acid
42 12 22 A Alanine
9 13 18 K Lycine
27 9 17 T Threonine
39 12 30 V Valine
32 10 15 D Aspartic acid
8 2 5 H Histidine
25 14 13 S Serine
28 9 20 P Proline
15 14 4 Y Tyrosine
33 6 20 R Arginine
7 6 0 C Cysteine
29 7 39 E Glutamic acid
44 9 20 G Glycine
43 11 44 L Leucine
8 0 3 M Methionine
17 16 10 N Asparagine
15 7 7 Q Glutamine
21 9 17 I Isoleucine
19 12 10 F Phenylalanine
13 2 1 W Tryptophan 

4XRG_1|Chains A, B|Homospermidine synthase|Blastochloris viridis (1079)
>1GKA_1|Chain A|CRUSTACYANIN A1 SUBUNIT|HOMARUS GAMMARUS (6707)
>4OIO_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Thermus thermophilus (300852)
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 \(w=f(c)\)
4XRG , Knot 197 474 0.85 40 245 449 
DWPVYHRIDGPIVMIGFGSIGRGTLPLIERHFAFDRSKLVVIDPSDEARKLAEARGVRFIQQAVTRDNYRELLVPLLTAGPGQGFCVNLSVDTSSLDIMELARENGALYIDTVVEPWLGFYFDPDLKPEARSNYALRETVLAARRNKPGGTTAVSCCGANPGMVSWFVKQALVNLAADLGVTGEEPTTREEWARLAMDLGVKGIHIAERDTQRASFPKPFDVFVNTWSVEGFVSEGLQPAELGWGTFERWMPDNARGHDSGCGAGIYLLQPGANTRVRSWTPTAMAQYGFLVTSNESISIADFLTVRDAAGQAVYRPTCHYAYHPCNDAVLSLHEMFGSGKRQSDWRILDETEIVDGIDELGVLLYGHGKNAYWYGSQLSIEETRRIAPDQNATGLQVSSAVLAGMVWALENPNAGIVEADDLDFRRCLEVQTPYLGPVVGVYTDWTPLAGRPGLFPEDIDTSDPWQFRNVLVR
1GKA , Knot 88 180 0.84 38 138 176 
KIPNFVVPGKCASVDRNKLWAEQTPNRNSYAGVWYQFALTNNPYQLIEKCVRNEYSFDGKQFVIKSTGIAYDGNLLKRNGKLYPNPFGEPHLSIDYENSFAAPLVILETDYSNYACLYSCIDYNFGYHSDFSFIFSRSANLADQYVKKCEAAFKNINVDTTRFVKTVQGSSCPYDTQKTV
4OIO , Knot 129 315 0.78 38 162 293 
MLDSKLKAPVFTVRTQGREYGEFVLEPLERGFGVTLGNPLRRILLSSIPGTAVTSVYIEDVLHEFSTIPGVKEDVVEIILNLKELVVRFLNPSLQTVTLLLKAEGPKEVKARDFLPVADVEIMNPDLHIATLEEGGRLNMEVRVDRGVGYVPAEKHGIKDRINAIPVDAVFSPVRRVAFQVEDTRLGQRTDLDKLTLRIWTDGSVTPLEALNQAVEILREHLTYFSNPQAAAVAAPEEAKEPEAPPEQEEELDLPLEELGLSTRVLHSLKEEGIESVRALLALNLKDLKNIPGIGERSLEEIKEALEKKGFTLKE

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 \(c\).

\(|P_{f(4XRG_1)}(2) \setminus P_{f(1GKA_1)}(2)|=150\), \(|P_{f(1GKA_1)}(2) \setminus P_{f(4XRG_1)}(2)|=43\). 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 1:011100010111111111011010111100011100001111010001001101011011001100000001111110111101101010100001011011000111010011011111010101010100001100011110000111001100011011110111001110111011101001000001101110111011011000000101101101110010101110011011011110100111001010001011110110111000100101011100111100000101101101001110110010000100100011101001110100000101100001101100111110101001010100101000001110001011010011111111110010111101001010001010010111111100010111101111100100001101001110
Pair \(Z_2\) Length of longest common subsequence
4XRG_1,1GKA_1 193 4
4XRG_1,4OIO_1 165 3
1GKA_1,4OIO_1 166 3

Newick tree

 
[
	1GKA_1:92.36,
	[
		4XRG_1:82.5,4OIO_1:82.5
	]:9.86
]

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 \((0.85)(0.8)(\frac{654 }{\log_{20} 654}-\frac{180}{\log_{20}180})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 4XRG_1 1GKA_1 172 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]