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Parikh vectors
6ORT_1 8ILT_1 1UDL_1 Letter Amino acid
18 14 2 H Histidine
2 20 5 K Lycine
6 4 3 W Tryptophan
9 13 3 Y Tyrosine
26 14 6 A Alanine
6 2 1 C Cysteine
26 16 6 V Valine
2 15 3 I Isoleucine
16 13 7 P Proline
21 13 6 D Aspartic acid
10 10 4 Q Glutamine
22 17 11 G Glycine
2 9 3 M Methionine
7 10 4 F Phenylalanine
11 10 4 T Threonine
32 7 1 R Arginine
3 6 6 N Asparagine
18 11 14 S Serine
22 28 4 E Glutamic acid
27 23 5 L Leucine 

6ORT_1|Chain A|Matrix remodeling-associated protein 8|Bos taurus (9913)
>8ILT_1|Chains A, B, C, D, E, F, G, H, I|Carboxylesterase|Geobacillus stearothermophilus (1422)
>1UDL_1|Chain A|intersectin 2|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 \(w=f(c)\)
6ORT , Knot 114 286 0.75 40 150 247 
SGPSGPATSDRSVVSESTVSWAAGARAVLRCQSPRMVWTQDRLHDRQRVVHWDLSGSKAGGPARRLVDMYSTGEQRVGEQRLGEQRVGEQRVYEPRDRGRLLLPPSAFHDGNFSLFIRAVEETDEGLYTCNLHHHYCHLYESLAVSLEVTDDPRAAGAHWDGEKEVLAVERGAPALLTCVNRAHVWTDRHLEEAQQVVHWDRQPPGVPHDRADRLLDLYASGERRAYGPPFLRDRVAVEADAFARGDFSLLIDPVEPADEGTYSCHLHHHYCGLHERRVFHLRVTE
8ILT , Knot 112 255 0.81 40 166 237 
MMKIVPPKPFFFEAGERAVLLLHGFTGNSADVRMLGRFLESKGYTCHAPIYKGHGVPPEELVHTGPDDWWQDVMNGYEFLKNKGYEKIAVAGLSLGGVFSLKLGYTVPIEGIVTMCAPMYIKSEETSYELVLEYAREYKKREGKSEEQIEQEMEKFKQTPMKTLKALQELKADVRDHLDLIYAPTFVVQARHDEMINPDSANIIYNEIESPVKQIKWYEQSGHVITLDQEKDQLHEDIYAFLESLDWLEHHHHHH
1UDL , Knot 50 98 0.78 40 80 90 
GSSGSSGQKGWFPASHVKLLGPSSERATPAFHPVCQVIAMYDYAANNEDELSFSKGQLINVMNKDDPDWWQGEINGVTGLFPSNYVKMTTDSSGPSSG

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 \(c\).

\(|P_{f(6ORT_1)}(2) \setminus P_{f(8ILT_1)}(2)|=78\), \(|P_{f(8ILT_1)}(2) \setminus P_{f(6ORT_1)}(2)|=94\). 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 1:0110111000001100001011111011100001011100001000001101010100111110011010001000110001100011000100100010111110110010101110110000011000010000001000111010100010111101010001111001111110010010110000100100110100011111000100110101010001011111000111010111010101110110110010000010000011000011010100
Pair \(Z_2\) Length of longest common subsequence
6ORT_1,8ILT_1 172 4
6ORT_1,1UDL_1 160 4
8ILT_1,1UDL_1 154 4

Newick tree

 
[
	6ORT_1:84.97,
	[
		1UDL_1:77,8ILT_1:77
	]:7.97
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{255}{\log_{20}255})=81.3\)
Status Protein1 Protein2 d d1/2
Query variables 6ORT_1 8ILT_1 96 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]