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Parikh vectors
7RYK_1 5YOG_1 2IBG_1 Letter Amino acid
8 5 0 C Cysteine
8 12 16 L Leucine
3 6 9 F Phenylalanine
6 9 11 V Valine
3 3 7 Q Glutamine
7 7 15 T Threonine
6 9 8 I Isoleucine
2 2 16 P Proline
10 8 21 S Serine
12 19 19 A Alanine
11 7 11 R Arginine
14 1 10 N Asparagine
12 18 9 G Glycine
1 1 4 H Histidine
3 4 13 Y Tyrosine
7 17 8 D Aspartic acid
2 13 14 E Glutamic acid
6 4 14 K Lycine
2 0 5 M Methionine
6 3 4 W Tryptophan 

7RYK_1|Chains A[auth AAA], B[auth BBB]|Lysozyme C|Gallus gallus (9031)
>5YOG_1|Chain A|Flavodoxin|Desulfovibrio vulgaris (strain Hildenborough / ATCC 29579 / DSM 644 / NCIMB 8303) (882)
>2IBG_1|Chains A, B, C, D|CG9211-PA|Drosophila melanogaster (7227)
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 \(w=f(c)\)
7RYK , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
5YOG , Knot 71 148 0.80 38 109 146 
AAKALIVYGSTTGCTEYTAETIARELADAGYEVDSRDAASVEAGGLFEGFDLVLLGCSTWGDDSIELQDDFIPLFDSLEETGAQGRKVACFGCGDSSWEYFCGAVDAIEEKLKNLGAEIVQDGLRIDGDPRAARDDIVGWAHDVRGAI
2IBG , Knot 103 214 0.86 38 159 209 
STYPPTPPNVTRLSDESVMLRWMVPRNDGLPIVIFKVQYRMVGKRKNWQTTNDNIPYGKPKWNSELGKSFTASVTDLKPQHTYRFRILAVYSNNDNKESNTSAKFYLQPGAALDPMPVPELLEIEEYSETAVVLHWSLASDADEHLITGYYAYYRPSSSAGEYFKATIEGAHARSFKIAPLETATMYEFKLQSFSAASASEFSALKQGRTQRPK

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 \(c\).

\(|P_{f(7RYK_1)}(2) \setminus P_{f(5YOG_1)}(2)|=70\), \(|P_{f(5YOG_1)}(2) \setminus P_{f(7RYK_1)}(2)|=75\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
7RYK_1,5YOG_1 145 3
7RYK_1,2IBG_1 159 4
5YOG_1,2IBG_1 150 3

Newick tree

 
[
	2IBG_1:78.81,
	[
		7RYK_1:72.5,5YOG_1:72.5
	]:6.31
]

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 \((0.85)(0.8)(\frac{277 }{\log_{20} 277}-\frac{129}{\log_{20}129})=46.2\)
Status Protein1 Protein2 d d1/2
Query variables 7RYK_1 5YOG_1 57 55
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]