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Parikh vectors
4OWP_1 5VRV_1 8ITL_1 Letter Amino acid
11 13 21 I Isoleucine
13 21 22 V Valine
6 22 37 R Arginine
11 8 10 N Asparagine
18 20 24 E Glutamic acid
3 1 10 C Cysteine
21 30 58 L Leucine
5 22 17 F Phenylalanine
7 12 18 Q Glutamine
3 13 4 M Methionine
11 11 17 T Threonine
6 7 8 H Histidine
15 15 6 K Lycine
6 11 11 P Proline
9 21 17 S Serine
2 5 11 W Tryptophan
12 40 28 A Alanine
12 24 6 D Aspartic acid
10 28 24 G Glycine
5 6 15 Y Tyrosine 

4OWP_1|Chain A|26S proteasome regulatory subunit RPN8|Saccharomyces cerevisiae (559292)
>5VRV_1|Chains A, B, C, D, E, F|Protein regulated by acid pH|Agrobacterium fabrum (strain C58 / ATCC 33970) (176299)
>8ITL_1|Chain A[auth R]|Gastric inhibitory polypeptide receptor|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 \(w=f(c)\)
4OWP , Knot 87 186 0.81 40 138 180 
MSLQHEKVTIAPLVLLSALDHYERTQTKENKRCVGVILGDANSSTIRVTNSFALPFEEDEKNSDVWFLDHNYIENMNEMCKKINAKEKLIGWYHSGPKLRASDLKINELFKKYTQNNPLLLIVDVKQQGVGLPTDAYVAIEQVKDDGTSTEKTFLHLPCTIEAEEAEEIGVEHLLRDVRDQAAGGL
5VRV , Knot 140 330 0.82 40 193 311 
SNAEDVERATDIVMRQDSADANLVRMGDKHVMFSESGNAFIMYGIQGRSWIAFADPVGDEEDFPDLVWQFVEAARGAGARAAFYQISPFLLSHCADAGLRAFKLGELALVDLTAFELKGGKLATLRQSLSRGARDGLTFEVVEQSQVPDIMDELQQVSDGWLAHHNTREKRFSLGAFEPDYILSQPVAVLRKDGKITAFANLMVTETKKEATIDLMRFSADAPRGSMDFLFVSIMQHLREAGYESFNLGMAPMSGMSKRDAAPVWDRIGSTLFEHGERFYNFKGLRAFKAKFHPKWEPRYLAVQNGADAALALMDATVLISGGVRGVIGK
8ITL , Knot 158 364 0.85 40 204 346 
RAETGSKGQTAGELYQRWERYRRECQETLAAAEPPSVAAGFVLRQCGSDGQWGLWRDHTQCENPEKNEAFLDQRLILERLQVMYTVGYSLSLATLLLALLILSLFRRLHCTRNYIHINLFTSFMLRAAAILSRDRLLPRPGPYLGDQALALWNQALAACRTAQIVTQYCVGANYTWLLVEGVYLHSLLVLVGGSEEGHFRYYLLLGWGAPALFVIPWVIVRYLYENTQCWERNEVKAIWWIIRTPILMTILINFLIFIRILGILLSKLRTRQMRCRDYRLRLARSTLFLVPLLGVHEVVFAPVTEEQARGALRFAKLGFEIFLSSFQGFLVSVLYCFINKEVQSEIRRGWHHCRLRRSLGEEQR

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 \(c\).

\(|P_{f(4OWP_1)}(2) \setminus P_{f(5VRV_1)}(2)|=55\), \(|P_{f(5VRV_1)}(2) \setminus P_{f(4OWP_1)}(2)|=110\). 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 1:101000010111111101100000000000000111111010000101000111110000000011110000100100100010100011110001101010010100110000000111111010001111100101110010001000000110110010100100111001100100011111
Pair \(Z_2\) Length of longest common subsequence
4OWP_1,5VRV_1 165 3
4OWP_1,8ITL_1 176 3
5VRV_1,8ITL_1 183 4

Newick tree

 
[
	8ITL_1:92.06,
	[
		4OWP_1:82.5,5VRV_1:82.5
	]:9.56
]

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 \((0.85)(0.8)(\frac{516 }{\log_{20} 516}-\frac{186}{\log_{20}186})=95.7\)
Status Protein1 Protein2 d d1/2
Query variables 4OWP_1 5VRV_1 122 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]