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Parikh vectors
6DKN_1 1FYA_1 2VFJ_1 Letter Amino acid
18 19 16 A Alanine
12 8 16 Q Glutamine
26 19 15 G Glycine
15 15 22 I Isoleucine
14 7 16 P Proline
5 0 8 W Tryptophan
29 11 24 R Arginine
24 10 20 D Aspartic acid
10 0 7 C Cysteine
28 20 30 E Glutamic acid
19 14 25 K Lycine
5 4 11 M Methionine
14 6 16 S Serine
13 3 12 Y Tyrosine
24 17 20 V Valine
14 3 19 N Asparagine
17 4 15 F Phenylalanine
12 0 13 H Histidine
34 20 41 L Leucine
16 13 20 T Threonine 

6DKN_1|Chains A, B|Decapping nuclease DXO homolog, chloroplastic|Arabidopsis thaliana (3702)
>1FYA_1|Chain A|60 KD CHAPERONIN|Escherichia coli (562)
>2VFJ_1|Chains A, B, C, D|TUMOR NECROSIS FACTOR|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 \(w=f(c)\)
6DKN , Knot 152 349 0.85 40 220 335 
RFVSELKLSKSKETLARKQTNFQEPCELTCYSRVEGGEVIYDDQGLRLFKRHIGEEIGADLNQGYDTFIEKKDLGSEGFGDLLGSIRAKNISLDNIHFVTFRNNLNKILGAAYNRHEPWEMGVHKRNGTIYLDVHKLPERPQSDLDRRRCYWGYCFESLATEDPGRAYGEEIHHVDANVEFCSVVRTKLGAHRVMMGAEMDCCDVSDKGKRFYVELKTTRELDDRTVDRFEREKLLKFWIQSFVAGVPYIVVGFRDDGGRLVRTERLTTRDIAHRARLKNYWQGGVCLAFADEVLCWLYGTVKENEDYILQFVHPFMRLELLQAQSCPDAITNHVHLLQNPASPPPPPQ
1FYA , Knot 87 193 0.79 34 127 186 
GLVPRGSEGMQFDRGYLSPYFINKPETGEVELESPFILLTDKKISNIRELLPVLEAVAKAGKPLLIIAEDVEGEALATLVVNTMRGIVKVAAVKAPGFGDRRKAMLQDIATLTGGTVISEELGMKLEKATLEDLGQAKRVVITKDTTTIIDGVGEEAAIQGRVAQIRQQIEEATSDYDREKLQERVAKLAGGV
2VFJ , Knot 159 366 0.85 40 221 347 
MAEQVLPQALYLSNMRKAVKIRERTPEDIFKPTNGIIHHFKTMHRYTLEMFRTCQFCPQFREIIHKALIDRNIQATLESQKKLNWCREVRKLVALKTNGDGNCLMHATSQYMWGVQDTDLVLRKALFSTLKETDTRNFKFRWQLESLKSQEFVETGLCYDTRNWNDEWDNLIKMASTDTPMARSGLQYNSLEEIHIFVLCNILRRPIIVISDKMLRSLESGSNFAPLKVGGIYLPLHWPAQECYRYPIVLGYDSHHFVPLVTLKDSGPEIRAVPLVNRDRGRFEDLKVHFLTDPENEMKEKLLKEYLMVIEIPVQGWDHGTTHLINAAKLDEANLPKEINLVDDYFELVQHEYKKWQENSEQGRRE

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 \(c\).

\(|P_{f(6DKN_1)}(2) \setminus P_{f(1FYA_1)}(2)|=128\), \(|P_{f(1FYA_1)}(2) \setminus P_{f(6DKN_1)}(2)|=35\). 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 1:0110010100000011000001001001000001011011000011011000110011101001000110000110011101110101001010010110100010011111000001101110000101010100110010001000000110010011000110101001001010101001100011100111110100001000100101010000010000100100001101110011111101111100011011000010000110010100010111011110011011010100000011011011101011010001011000101100110111110
Pair \(Z_2\) Length of longest common subsequence
6DKN_1,1FYA_1 163 4
6DKN_1,2VFJ_1 175 5
1FYA_1,2VFJ_1 160 4

Newick tree

 
[
	6DKN_1:86.01,
	[
		1FYA_1:80,2VFJ_1:80
	]:6.01
]

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 \((0.85)(0.8)(\frac{542 }{\log_{20} 542}-\frac{193}{\log_{20}193})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 6DKN_1 1FYA_1 127 96.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]