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Parikh vectors
6KNM_1 6GKK_1 3BVM_1 Letter Amino acid
23 6 18 Y Tyrosine
14 2 8 M Methionine
28 11 14 T Threonine
7 6 5 Q Glutamine
21 11 12 G Glycine
45 17 15 L Leucine
13 15 30 K Lycine
28 8 10 F Phenylalanine
36 11 16 V Valine
17 12 3 R Arginine
26 15 21 D Aspartic acid
18 11 9 I Isoleucine
17 6 7 P Proline
13 11 23 N Asparagine
16 4 2 C Cysteine
12 5 6 H Histidine
24 11 16 S Serine
9 8 3 W Tryptophan
28 10 6 A Alanine
12 10 13 E Glutamic acid 

6KNM_1|Chain A[auth B]|Apelin receptor,Rubredoxin,Apelin receptor|Homo sapiens (9606)
>6GKK_1|Chains A, B, C, D|Eukaryotic translation initiation factor 4E|Mus musculus (10090)
>3BVM_1|Chain A|Enterotoxin type C-3|Staphylococcus aureus (1280)
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 \(w=f(c)\)
6KNM , Knot 169 407 0.83 40 215 384 
MKTIIALSYIFCLVFADYKDDDDKFDNYYGADNQSECEYTDWKSSGALIPAIYMLVFLLGTTGNGLVLWTVFRSSREKRRSADIFIASLAVADLTFVVTLPLWATYTYRDYDWPFGTFFCKLSSYLIFVNMYASAFCLTGLSFDRYLAIVRPVANARLRLRVSGAVATAVLWVLAALLAMPVMVLRTTGDLENTNKVQCYMDYSMVATVSSEWAWEVGLGVSSTTVGFVVPFTIMLTCYFFIAQTIAMKKYTCTVCGYIYNPEDGDPDNGVNPGTDFKDIPDDWVCPLCGVGKDQFEEVEERRRLLSIIVVLVVTFALCKMPYHLVKTLYMLGSLLHWPCDFDLFLMNIFPYCTCISYVNSCLNPFLYAFFDPRFRQACTSMLLMGQSRLEVLFQGPHHHHHHHHHH
6GKK , Knot 94 190 0.86 40 152 186 
VANPEHYIKHPLQNRWALWFFKNDKSKTWQANLRLISKFDTVEDFWALYNHIQLSSNLMPGCDYSLFKDGIEPMWEDEKNKRGGRWLITLNKQQRRSDLDRFWLETLLCLIGESFDDYSDDVCGAVVNVRAKGDKIAIWTTECENRDAVTHIGRVYKERLGLPPKIVIGYQSHADTATKSGSTTKNRFVV
3BVM , Knot 107 237 0.82 40 159 227 
ESQPDPMPDDLHKSSEFTGTMGNMKYLYDDHYVSATKVKSVDKFLAHDLIYNISDKKLKNYDKVKTELLNEDLAKKYKDEVVDVYGSNYYVNCYFSSKDNKWWPGKTCMYGGITKHEGNHFDNGNLQNVLVRVYENKRNTISFEVQTDKKSVTAQELDIKARNFLINKKNLYEFNSSPYETGYIKFIENNGNTFWYDMMPAPGDKFDQSKYLMMYNDNKTVDSKSVKIEVHLTTKNG

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 \(c\).

\(|P_{f(6KNM_1)}(2) \setminus P_{f(6GKK_1)}(2)|=126\), \(|P_{f(6GKK_1)}(2) \setminus P_{f(6KNM_1)}(2)|=63\). 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 1:10011110011011110000000010000110000000000100011111110111111100101111101100000000010111101111010111011111000000001111011001000111101010110101101000111101110101010101111011111111111111111000101000001000100011101000111011111000011111110111000111100111000000101010010010100110110010011001101101110001001000001101111111011100110011001011101101100101111011100001001000101110111010100100011111000101110110000000000
Pair \(Z_2\) Length of longest common subsequence
6KNM_1,6GKK_1 189 3
6KNM_1,3BVM_1 186 4
6GKK_1,3BVM_1 177 4

Newick tree

 
[
	6KNM_1:95.43,
	[
		3BVM_1:88.5,6GKK_1:88.5
	]:6.93
]

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 \((0.85)(0.8)(\frac{597 }{\log_{20} 597}-\frac{190}{\log_{20}190})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6KNM_1 6GKK_1 146 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]