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Parikh vectors
4ZLH_1 4GEY_1 1EWV_1 Letter Amino acid
28 19 32 E Glutamic acid
18 47 39 G Glycine
18 22 25 K Lycine
9 24 19 F Phenylalanine
18 19 16 T Threonine
12 19 17 Y Tyrosine
27 18 31 R Arginine
6 2 10 C Cysteine
23 23 19 Q Glutamine
15 4 13 M Methionine
4 14 21 P Proline
4 14 7 W Tryptophan
36 33 29 A Alanine
23 31 33 D Aspartic acid
5 28 20 N Asparagine
39 32 44 L Leucine
15 25 35 S Serine
17 34 35 V Valine
8 16 14 H Histidine
14 12 31 I Isoleucine 

4ZLH_1|Chains A, B|Lipopolysaccharide assembly protein B|Escherichia coli O157:H7 (83334)
>4GEY_1|Chain A|Porin B|Pseudomonas putida (931281)
>1EWV_1|Chains A, B|METABOTROPIC GLUTAMATE RECEPTOR SUBTYPE 1|Rattus norvegicus (10116)
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 \(w=f(c)\)
4ZLH , Knot 148 339 0.84 40 202 323 
QDKAVDLFLDMLKEDTGTVEAHLTLGNLFRSRGEVDRAIRIHQTLMESASLTYEQRLLAIQQLGRDYMAAGLYDRAEDMFNQLTDETDFRIGALQQLLQIYQATSEWQKAIDVAERLVKLGKDKQRVEIAHFYCELALQHMASDDLDRAMTLLKKGAAADKNSARVSIMMGRVFMAKGEYAKAVESLQRVISQDRELVSETLEMLQTCYQQLGKTAEWAEFLQRAVEENTGADAELMLADIIEARDGSEAAQVYITRQLQRHPTMRVFHKLMDYHLNEAEEGRAKESLMVLRDMVGEKVRSKPRYRCQKCGFTAYTLYWHCPSCRAWSTIKPIRGLDGL
4GEY , Knot 183 436 0.85 40 230 410 
ANVRLQHHHHHHHLEAEAFSSESKWMTGDWGGTRTELLDKGYDFTLDYVGEVAGNLHGGYNDDKTARYSDQFALGAHLDLQKILGWHDAEFKLAITERSGRNLSNDRISDPRAGQFSSVQEVWGRGQTWRLTQMWIKQKYFDGALDVKFGRFGEGEDFNSFPCDFQNLAFCGSQVGNWVGGIWYNWPVSQWALRVKYNITPAFFVQVGAFEQNPSNLETGNGFKLSGSGTKGAIMPMEAVWSPKVNGLPGEYRLGYYYSTAKADDVYDDVNGNPQALTGEAFKSHSSKHGWWVVAQQQVTAHGGDVNRGLSLFANFTVHDKATNVVDNYQQVGLVYKGAFDARPKDDIGFGVARIHVNDDVKKRAELLNAQSGINDYDNPGFVPLQRTEYNAELYYGFHVTNWLTVRPNLQYIKSPGGVDEVDNALVAGLKIQSSF
1EWV , Knot 201 490 0.84 40 256 458 
SSQRSVARMDGDVIIGALFSVHHQPPAEKVPERKCGEIREQYGIQRVEAMFHTLDKINADPVLLPNITLGSEIRDSCWHSSVALEQSIEFIRDSLISIRDEKDGLNRCLPDGQTLPPGRTKKPIAGVIGPGSSSVAIQVQNLLQLFDIPQIAYSATSIDLSDKTLYKYFLRVVPSDTLQARAMLDIVKRYNWTYVSAVHTEGNYGESGMDAFKELAAQEGLCIAHSDKIYSNAGEKSFDRLLRKLRERLPKARVVVCFCEGMTVRGLLSAMRRLGVVGEFSLIGSDGWADRDEVIEGYEVEANGGITIKLQSPEVRSFDDYFLKLRLDTNTRNPWFPEFWQHRFQCRLPGHLLENPNFKKVCTGNESLEENYVQDSKMGFVINAIYAMAHGLQNMHHALCPGHVGLCDAMKPIDGRKLLDFLIKSSFVGVSGEEVWFDEKGDAPGRYDIMNLQYTEANRYDYVHVGTWHEGVLNIDDYKIQMNKSGMVRS

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 \(c\).

\(|P_{f(4ZLH_1)}(2) \setminus P_{f(4GEY_1)}(2)|=83\), \(|P_{f(4GEY_1)}(2) \setminus P_{f(4ZLH_1)}(2)|=111\). 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 1:000110111011000010101010110110001010011010001100101000001111001100011111000100110010000010111100110100100010011011001101100000101101000111001100010011011001111000010101111011110100101100100110000011000101100000011001011011001100001101011110110100100110101000100010101100110001001001010001111001110010001000000011010010100100011001011011011
Pair \(Z_2\) Length of longest common subsequence
4ZLH_1,4GEY_1 194 3
4ZLH_1,1EWV_1 166 4
4GEY_1,1EWV_1 152 4

Newick tree

 
[
	4ZLH_1:94.55,
	[
		1EWV_1:76,4GEY_1:76
	]:18.55
]

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 \((0.85)(0.8)(\frac{775 }{\log_{20} 775}-\frac{339}{\log_{20}339})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 4ZLH_1 4GEY_1 152 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]