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Parikh vectors
3LMT_1 1MKW_1 8JGJ_1 Letter Amino acid
20 4 35 K Lycine
3 2 35 P Proline
4 2 71 A Alanine
3 4 14 Q Glutamine
3 1 16 H Histidine
20 1 63 I Isoleucine
8 1 53 V Valine
5 2 38 R Arginine
2 1 18 C Cysteine
11 8 45 E Glutamic acid
3 0 19 W Tryptophan
6 1 30 Y Tyrosine
9 4 63 G Glycine
15 3 80 L Leucine
2 0 23 M Methionine
7 3 49 T Threonine
20 1 26 N Asparagine
13 3 39 D Aspartic acid
7 6 44 F Phenylalanine
3 2 57 S Serine 

3LMT_1|Chains A, B, C, D, E, F|D-tyrosyl-tRNA(Tyr) deacylase|Plasmodium falciparum (36329)
>1MKW_1|Chain A[auth L]|ALPHA-THROMBIN|Bos taurus (9913)
>8JGJ_1|Chains A, B|H(+)/Cl(-) exchange transporter 3|Mus musculus (10090)
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 \(w=f(c)\)
3LMT , Knot 79 164 0.82 40 119 161 
MRVVIQRVKGAILSVRKENIGENEKELEIISEIKNGLICFLGIHKNDTWEDALYIIRKCLNLRLWNNDNKTWDKNVKDLNYELLIVSQFTLFGNTKKGNKPDFHLAKEPNEALIFYNKIIDEFKKQYNDDKIKIGKFGNYMNIDVTNDGPVTIYIDTHDINLNK
1MKW , Knot 31 49 0.82 36 44 46 
TSEDHFQPFFNEKTFGAGEADCGLRPLFEKKQVQDQTEKELFESYIEGR
8JGJ , Knot 310 818 0.84 40 304 752 
MESEQLFHRGYYRNSYNSITSASSDEELLDGAGAIMDFQTSEDDNLLDGDTAAGTHYTMTNGGSINSSTHLLDLLDEPIPGVGTYDDFHTIDWVREKCKDRERHRRINSKKKESAWEMTKSLYDAWSGWLVVTLTGLASGALAGLIDIAADWMTDLKEGICLSALWYNHEQCCWGSNETTFEERDKCPQWKTWAELIIGQAEGPGSYIMNYIMYIFWALSFAFLAVSLVKVFAPYACGSGIPEIKTILSGFIIRGYLGKWTLMIKTITLVLAVASGLSLGKEGPLVHVACCCGNIFSYLFPKYSTNEAKKREVLSAASAAGVSVAFGAPIGGVLFSLEEVSYYFPLKTLWRSFFAALVAAFVLRSINPFGNSRLVLFYVEYHTPWYLFELFPFILLGVFGGLWGAFFIRANIAWCRRRKSTKFGKYPVLEVIIVAAITAVIAFPNPYTRLNTSELIKELFTDCGPLESSSLCDYRNDMNASKIVDDIPDRPAGVGVYSAIWQLCLALIFKIIMTVFTFGIKVPSGLFIPSMAIGAIAGRIVGIAVEQLAYYHHDWFIFKEWCEVGADCITPGLYAMVGAAACLGGVTRMTVSLVVIVFELTGGLEYIVPLMAAVMTSKWVGDAFGREGIYEAHIRLNGYPFLDAKEEFTHTTLAADVMRPRRSDPPLAVLTQDNMTVDDIENMINETSYNGFPVIMSKESQRLVGFALRRDLTIAIESARKKQEGIVGSSRVCFAQHTPSLPAESPRPLKLRSILDMSPFTVTDHTPMEIVVDIFRKLGLRQCLVTHNGRLLGIITKKDILRHMAQTANQDPASIMFN

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 \(c\).

\(|P_{f(3LMT_1)}(2) \setminus P_{f(1MKW_1)}(2)|=102\), \(|P_{f(1MKW_1)}(2) \setminus P_{f(3LMT_1)}(2)|=27\). 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 1:10111001011110100001100000101100100111011110000010011011000101011000000100010010001111001011100001001010110010011110001100100000000101101100101010001110101000010100
Pair \(Z_2\) Length of longest common subsequence
3LMT_1,1MKW_1 129 4
3LMT_1,8JGJ_1 217 4
1MKW_1,8JGJ_1 274 4

Newick tree

 
[
	8JGJ_1:13.74,
	[
		3LMT_1:64.5,1MKW_1:64.5
	]:73.24
]

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 \((0.85)(0.8)(\frac{213 }{\log_{20} 213}-\frac{49}{\log_{20}49})=55.2\)
Status Protein1 Protein2 d d1/2
Query variables 3LMT_1 1MKW_1 67 42.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]