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Parikh vectors
1FDO_1 4WMY_1 5NIR_1 Letter Amino acid
31 14 6 P Proline
45 21 4 T Threonine
35 15 1 N Asparagine
48 16 5 E Glutamic acid
15 8 0 H Histidine
20 17 1 F Phenylalanine
31 20 1 Y Tyrosine
27 13 3 Q Glutamine
39 13 3 K Lycine
18 4 1 M Methionine
32 26 4 S Serine
13 9 1 W Tryptophan
76 19 4 A Alanine
38 13 2 R Arginine
54 34 6 G Glycine
44 13 3 L Leucine
39 18 9 D Aspartic acid
15 10 10 C Cysteine
44 8 5 I Isoleucine
50 15 5 V Valine 

1FDO_1|Chain A|FORMATE DEHYDROGENASE H|Escherichia coli (562)
>4WMY_1|Chains A, B|Intelectin-1|Homo sapiens (9606)
>5NIR_1|Chains A, B|Collagen alpha-1(II) chain|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 \(w=f(c)\)
1FDO , Knot 282 715 0.86 42 300 659 
MKKVVTVCPYCASGCKINLVVDNGKIVRAEAAQGKTNQGTLCLKGYYGWDFINDTQILTPRLKTPMIRRQRGGKLEPVSWDEALNYVAERLSAIKEKYGPDAIQTTGSSRGTGNETNYVMQKFARAVIGTNNVDCCARVUHGPSVAGLHQSVGNGAMSNAINEIDNTDLVFVFGYNPADSHPIVANHVINAKRNGAKIIVCDPRKIETARIADMHIALKNGSNIALLNAMGHVIIEENLYDKAFVASRTEGFEEYRKIVEGYTPESVEDITGVSASEIRQAARMYAQAKSAAILWGMGVTQFYQGVETVRSLTSLAMLTGNLGKPHAGVNPVRGQNNVQGACDMGALPDTYPGYQYVKDPANREKFAKAWGVESLPAHTGYRISELPHRAAHGEVRAAYIMGEDPLQTDAELSAVRKAFEDLELVIVQDIFMTKTASAADVILPSTSWGEHEGVFTAADRGFQRFFKAVEPKWDLKTDWQIISEIATRMGYPMHYNNTQEIWDELRHLCPDFYGATYEKMGELGFIQWPCRDTSDADQGTSYLFKEKFDTPNGLAQFFTCDWVAPIDKLTDEYPMVLSTVREVGHYSCRSMTGNCAALAALADEPGYAQINTEDAKRLGIEDEALVWVHSRKGKIITRAQVSDRPNKGAIYMTYQWWIGACNELVTENLSPITKTPEYKYCAVRVEPIADQRAAEQYVIDEYNKLKTRLREAALA
4WMY , Knot 137 306 0.85 40 200 296 
TDWSHPQFEKSTDEANTYFKEWTCSSSPSLPRSCKEIKDECPSAFDGLYFLRTENGVIYQTFCDMTSGGGGWTLVASVHENDMRGKCTVGDRWSSQQGSKAVYPEGDGNWANYNTFGSAEAATSDDYKNPGYYDIQAKDLGIWHVPNKSPMQHWRNSSLLRYRTDTGFLQTLGHNLFGIYQKYPVKYGEGKCWTDNGPVIPVVYDFGDAQKTASYYSPYGQREFTAGFVQFRVFNNERAANALCAGMRVTGCNTEHHCIGGGGYFPEASPQQCGDFSGFDWSGYGTHVGYSSSREITEAAVLLFYR
5NIR , Knot 41 74 0.79 38 64 72 
GSMAQEAGSCVQDGQRYNDKDVWKPEPCRICVCDTGTVLCDDIICEDVKDCLSPEIPFGECCPICPTDLATASG

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 \(c\).

\(|P_{f(1FDO_1)}(2) \setminus P_{f(4WMY_1)}(2)|=145\), \(|P_{f(4WMY_1)}(2) \setminus P_{f(1FDO_1)}(2)|=45\). 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 1:1001101010010100101110010110101101000010101010011011000011010100111000011010110100110011001011000011011000100010100000110011011110001000101001101111000110111001100100001111110011000111100110100011011100100100101101011100100111101110111000100011110000110000011010010010010110100100110101010011111111100100110010010011110101101011101101000101100111110001100010011000011011110011100100100110011010101101110011000101011001100101111001110001011011110001100011101100110011011010101000101100110011011000000011001001010101100001101111011000000100100011000100101110110001111100100001111001001100000010100111111100110101000010011100011111000010110010100010011101000111110001100010110001000001101011100011000110000010001001111
Pair \(Z_2\) Length of longest common subsequence
1FDO_1,4WMY_1 190 4
1FDO_1,5NIR_1 260 3
4WMY_1,5NIR_1 190 3

Newick tree

 
[
	5NIR_1:11.47,
	[
		1FDO_1:95,4WMY_1:95
	]:24.47
]

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 \((0.85)(0.8)(\frac{1021 }{\log_{20} 1021}-\frac{306}{\log_{20}306})=191.\)
Status Protein1 Protein2 d d1/2
Query variables 1FDO_1 4WMY_1 249 175
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]