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Parikh vectors
4RDO_1 8ZGH_1 2WZN_1 Letter Amino acid
6 34 11 T Threonine
12 50 39 V Valine
7 38 29 A Alanine
3 10 1 C Cysteine
14 29 23 K Lycine
4 15 13 M Methionine
6 38 10 F Phenylalanine
17 39 12 S Serine
5 30 10 Q Glutamine
8 50 29 E Glutamic acid
4 17 2 W Tryptophan
8 36 9 Y Tyrosine
9 54 18 R Arginine
11 27 10 N Asparagine
8 60 26 D Aspartic acid
9 24 26 I Isoleucine
8 85 24 L Leucine
4 50 21 P Proline
11 59 31 G Glycine
13 33 10 H Histidine 

4RDO_1|Chains A, B, C, D, E, F|YTH domain-containing family protein 2|Homo sapiens (9606)
>8ZGH_1|Chains A, B, C, D|Multifunctional procollagen lysine hydroxylase and glycosyltransferase LH3|Homo sapiens (9606)
>2WZN_1|Chain A|354AA LONG HYPOTHETICAL OPERON PROTEIN FRV|PYROCOCCUS HORIKOSHII (53953)
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 \(w=f(c)\)
4RDO , Knot 82 167 0.83 40 135 162 
MGSSYHHHHHHSSGENLYFQHMKHGRVFIIKSYSEDDIHRSIKYNIWCSTEHGNKRLDAAYRSMNGKGPVYLLFSVNGSGHFCGVAEMKSAVDYNTCAGVWSQDKWKGRFDVRWIFVKDVPNSQLRHIRLENNENKPVTNSRDTQEVPLEKAKQVLKIIASYKHTTS
8ZGH , Knot 297 778 0.84 40 306 704 
MTSSGPGPRFLLLLPLLLPPAASASDRPRGRDPVNPEKLLVITVATAETEGYLRFLRSAEFFNYTVRTLGLGEEWRGGDVARTVGGGQKVRWLKKEMEKYADREDMIIMFVDSYDVILAGSPTELLKKFVQSGSRLLFSAESFCWPEWGLAEQYPEVGTGKRFLNSGGFIGFATTIHQIVRQWKYKDDDDDQLFYTRLYLDPGLREKLSLNLDHKSRIFQNLNGALDEVVLKFDRNRVRIRNVAYDTLPIVVHGNGPTKLQLNYLGNYVPNGWTPEGGCGFCNQDRRTLPGGQPPPRVFLAVFVEQPTPFLPRFLQRLLLLDYPPDRVTLFLHNNEVFHEPHIADSWPQLQDHFSAVKLVGPEEALSPGEARDMAMDLCRQDPECEFYFSLDADAVLTNLQTLRILIEENRKVIAPMLSRHGKLWSNFWGALSPDEYYARSEDYVELVQRKRVGVWNVPYISQAYVIRGDTLRMELPQRDVFSGSDTDPDMAFCKSFRDKGIFLHLSNQHEFGRLLATSRYDTEHLHPDLWQIFDNPVDWKEQYIHENYSRALEGEGIVEQPCPDVYWFPLLSEQMCDELVAEMEHYGQWSGGRHEDSRLAGGYENVPTVDIHMKQVGYEDQWLQLLRTYVGPMTESLFPGYHTKARAVMNFVVRYRPDEQPSLRPHHDSSTFTLNVALNHKGLDYEGGGCRFLRYDCVISSPRKGWALLHPGRLTHYHEGLPTTWGTRYIMVSFVDPAAAENLYFQGDYKDHDGDYKDHDIDYKDDDDKHHHHHHHH
2WZN , Knot 147 354 0.81 40 199 334 
MDLKGGESMVDWKLMQEIIEAPGVSGYEHLGIRDIVVDVLKEVADEVKVDKLGNVIAHFKGSSPRIMVAAHMDKIGVMVNHIDKDGYLHIVPIGGVLPETLVAQRIRFFTEKGERYGVVGVLPPHLRRGQEDKGSKIDWDQIVVDVGASSKEEAEEMGFRVGTVGEFAPNFTRLNEHRFATPYLDDRICLYAMIEAARQLGDHEADIYIVGSVQEEVGLRGARVASYAINPEVGIAMDVTFAKQPHDKGKIVPELGKGPVMDVGPNINPKLRAFADEVAKKYEIPLQVEPSPRPTGTDANVMQINKEGVATAVLSIPIRYMHSQVELADARDVDNTIKLAKALLEELKPMDFTP

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 \(c\).

\(|P_{f(4RDO_1)}(2) \setminus P_{f(8ZGH_1)}(2)|=24\), \(|P_{f(8ZGH_1)}(2) \setminus P_{f(4RDO_1)}(2)|=195\). 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 1:11000000000001001010010010111100000001000100011000001000101100010101110111010101010111010011000001111000010101010111100110001001010000001100000000111001001101110000000
Pair \(Z_2\) Length of longest common subsequence
4RDO_1,8ZGH_1 219 6
4RDO_1,2WZN_1 184 5
8ZGH_1,2WZN_1 175 4

Newick tree

 
[
	4RDO_1:10.28,
	[
		2WZN_1:87.5,8ZGH_1:87.5
	]:17.78
]

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 \((0.85)(0.8)(\frac{945 }{\log_{20} 945}-\frac{167}{\log_{20}167})=214.\)
Status Protein1 Protein2 d d1/2
Query variables 4RDO_1 8ZGH_1 271 162.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]