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Parikh vectors
1LTH_1 7XNH_1 8CJO_1 Letter Amino acid
35 16 20 A Alanine
12 24 14 R Arginine
11 16 11 Q Glutamine
16 15 27 K Lycine
18 22 21 S Serine
16 12 8 N Asparagine
19 22 24 E Glutamic acid
25 27 21 G Glycine
8 6 4 M Methionine
21 18 16 T Threonine
5 13 16 Y Tyrosine
19 10 16 D Aspartic acid
2 10 9 C Cysteine
26 23 10 I Isoleucine
25 29 31 L Leucine
7 16 24 F Phenylalanine
17 14 18 P Proline
2 3 2 W Tryptophan
8 5 17 H Histidine
27 31 17 V Valine 

1LTH_1|Chains A[auth R], B[auth T]|L-LACTATE DEHYDROGENASE (T- AND R- STATE TETRAMER COMPLEX)|Bifidobacterium longum subsp. longum (1679)
>7XNH_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Gap junction delta-2 protein|Homo sapiens (9606)
>8CJO_1|Chain A|Tryptophan 5-hydroxylase 1|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 \(w=f(c)\)
1LTH , Knot 137 319 0.82 40 184 308 
AETTVKPTKLAVIGAGAVGSTLAFAAAQRGIAREIVLEDIAKERVEAEVLDMQHGSSFYPTVSIDGSDDPEICRDADMVVITAGPRQKPGQSRLELVGATVNILKAIMPNLVKVAPNAIYMLITNPVDIATHVAQKLTGLPENQIFGSGTNLDSARLRFLIAQQTGVNVKNVHAYIAGEHGDSEVPLWESATIGGVPMSDWTPLPGHDPLDADKREEIHQEVKNAAYKIINGKGATNYAIGMSGVDIIEAVLHDTNRILPVSSMLKDFHGISDICMSVPTLLNRQGVNNTINTPVSDKELAALKRSAETLKETAAQFGF
7XNH , Knot 145 332 0.84 40 213 318 
MGEWTILERLLEAAVQQHSTMIGRILLTVVVIFRILIVAIVGETVYDDEQTMFVCNTLQPGCNQACYDRAFPISHIRYWVFQIIMVCTPSLCFITYSVHQSAKQRERRYSTVFLALDRDPPESIGGPGGTGGGGSGGGKREDKKLQNAIVNGVLQNTENTSKETEPDCLEVKELTPHPSGLRTASKSKLRRQEGISRFYIIQVVFRNALEIGFLVGQYFLYGFSVPGLYECNRYPCIKEVECYVSRPTEKTVFLVFMFAVSGICVVLNLAELNHLGWRKIKLAVRGAQAKRKSIYEIRNKDLPRVSVPNFGRTQSSDSAYVSRGDMLEVLFQ
8CJO , Knot 142 326 0.84 40 198 311 
MKHHHHHHHGAAGTSLYKKAGENLYFQGSVPWFPKKISDLDHCANRVLMYGSELDADHPGFKDNVYRKRRKYFADLAMNYKHGDPIPKVEFTEEEIKTWGTVFQELNKLYPTHACREYLKNLPLLSKYCGYREDNIPQLEDVSNFLKERTGFSIRPVAGYLSPRDFLSGLAFRVFHCTQYVRHSSDPFYTPEPDTCHELLGHVPLLAEPSFAQFSQEIGLASLGASEEAVQKLATCYFFTVEFGLCKQDGQLRVFGAGLLSSISELKHALSGHAKVKPFDPKITCKQECLITTFQDVYFVSESFEDAKEKMREFTKTIKRPFGVKY

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 \(c\).

\(|P_{f(1LTH_1)}(2) \setminus P_{f(7XNH_1)}(2)|=62\), \(|P_{f(7XNH_1)}(2) \setminus P_{f(1LTH_1)}(2)|=91\). 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 1:1000101001111111111001111110011100111001100010101101001001010101010001010001011110111000110001011110101101111011011101101110011011001100101110001110100100101011110001101001010111001000111100101111110010111100110100000100010011001101011000111101101101110000011110011001011001010110110001100010011000011110001001000110111
Pair \(Z_2\) Length of longest common subsequence
1LTH_1,7XNH_1 153 4
1LTH_1,8CJO_1 176 4
7XNH_1,8CJO_1 179 3

Newick tree

 
[
	8CJO_1:92.47,
	[
		1LTH_1:76.5,7XNH_1:76.5
	]:15.97
]

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 \((0.85)(0.8)(\frac{651 }{\log_{20} 651}-\frac{319}{\log_{20}319})=91.9\)
Status Protein1 Protein2 d d1/2
Query variables 1LTH_1 7XNH_1 118 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]