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Parikh vectors
6EKM_1 5VXK_1 8BCX_1 Letter Amino acid
13 8 29 D Aspartic acid
13 5 22 I Isoleucine
6 3 13 M Methionine
1 2 8 W Tryptophan
8 5 14 N Asparagine
12 13 24 P Proline
10 14 22 S Serine
8 9 52 A Alanine
9 11 21 R Arginine
2 15 33 G Glycine
16 5 47 K Lycine
7 3 21 F Phenylalanine
11 11 26 T Threonine
5 5 18 Y Tyrosine
3 4 11 C Cysteine
11 12 20 Q Glutamine
15 7 30 E Glutamic acid
10 2 13 H Histidine
25 10 34 L Leucine
26 11 34 V Valine 

6EKM_1|Chain A|Mitotic spindle assembly checkpoint protein MAD2B|Mus musculus (10090)
>5VXK_1|Chain A[auth B]|single-domain antibody JMK-H2|Vicugna pacos (30538)
>8BCX_1|Chains A, B, C, D|Nitrite reductase (cytochrome; ammonia-forming)|Geobacter metallireducens GS-15 (269799)
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 \(w=f(c)\)
6EKM , Knot 97 211 0.82 40 149 203 
MTTLTRQDLNSAQVVADVLSEFLEVAVHLILYVREVYPVGIFQKRKKYNVPVQMSCHPELNQYIQDTLHCVKPLLEKNDVEKVVVVILDKEHRPVEKFVFEITQPPLLSINSDSLLSHVEQLLRAFILKISKVDKVLDHNPPGCTFTVLVHTREAATRNMEKIQVIKDFPWILADEQDVHMHDPRLIPLKTMTSDILKMQLYVEERAHKNS
5VXK , Knot 75 155 0.81 40 120 151 
GSTQLQLVESGGGLVQPGGSLRLSCAASGFTLDDQPIAWFRQAPGKEREGVSCISIDGNTQSYSDSVKGRFTISRDTANNRVHLQMNNLKPEDTAVYYCAADRYTSVRQMCTMIEGLHRVWGQGTQVTVSSEPKTPKPQPRQGAPVPYPDPLEPR
8BCX , Knot 198 492 0.83 40 247 453 
MKKTAIAIAVALAGFATVAQAASWSHPQFEKGAAQPVKTVQIPDGEVDPAVWGKAYPTEYEMWKKTEEPEPPGKSKYKRGFDADHVTYDKLSEFPYMALLFNGWGFGIAYNEPRGHANMVRDQLEIDSARLKSGGVCLTCKTPYAPKLEKEMGIDYFKTPFKDVLAKIPEKHKTLGVACIDCHDNKDMSLRISRGFTLGEALKKLGVDQAKLSRQEMRSLVCAQCHVTYNIPKDADKKSIGVYFPWQGSKMGNISVENIIKQIRSDASVGEWTQTVTGFKLGFIRHPEYELFSNNSVHWKAGAACTDCHMPYTRVGAFKVSDHRVMSPLKNDMKACIQCHTEKPEWLRDQVIAIQDRTVSLMLRSGYATATVAKLFEKAHAAQAQGKQIDKALYDRAKDLYEEAFYRCVFIGAENSVGFHNPTEAMRVLGDATAFATKAEALLRQALAKAGVDVPLTVNLELNKYLDQRGEKKLTFDPKVEIKDPYGVQVRF

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 \(c\).

\(|P_{f(6EKM_1)}(2) \setminus P_{f(5VXK_1)}(2)|=97\), \(|P_{f(5VXK_1)}(2) \setminus P_{f(6EKM_1)}(2)|=68\). 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 1:1001000010010111011001101110111010010111110000000111010001010001000100101110000100111111000001100111010011110100001100100110111101001001100011100101110000110001001011001111110000101001011110010001101010100010000
Pair \(Z_2\) Length of longest common subsequence
6EKM_1,5VXK_1 165 3
6EKM_1,8BCX_1 180 4
5VXK_1,8BCX_1 201 4

Newick tree

 
[
	8BCX_1:99.32,
	[
		6EKM_1:82.5,5VXK_1:82.5
	]:16.82
]

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 \((0.85)(0.8)(\frac{366 }{\log_{20} 366}-\frac{155}{\log_{20}155})=63.7\)
Status Protein1 Protein2 d d1/2
Query variables 6EKM_1 5VXK_1 83 72
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]