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Parikh vectors
5OXF_1 8OZT_1 4XQJ_1 Letter Amino acid
17 32 2 G Glycine
54 11 10 S Serine
19 25 2 T Threonine
3 10 0 W Tryptophan
15 22 5 Y Tyrosine
24 31 8 V Valine
13 17 8 R Arginine
60 18 8 N Asparagine
6 3 2 C Cysteine
8 21 5 H Histidine
115 36 6 L Leucine
45 16 7 F Phenylalanine
49 30 6 D Aspartic acid
70 20 8 E Glutamic acid
96 17 8 K Lycine
3 20 0 P Proline
32 36 2 A Alanine
34 16 3 Q Glutamine
60 26 8 I Isoleucine
9 7 1 M Methionine 

5OXF_1|Chains A, B|GTP-binding protein|Campylobacter jejuni (197)
>8OZT_1|Chains A, B, C, D, E, F|Alpha-L-fucosidase|Lactobacillus (1578)
>4XQJ_1|Chains A, D|Accessory gene regulator A|Staphylococcus aureus (strain COL) (93062)
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 \(w=f(c)\)
5OXF , Knot 259 732 0.77 40 213 575 
MKELFQKIWQNELQFLNFDAKFQDKSKLDTAECAIILSVNKDNYERYFLLKEFQELCKKIDLRVDIFSMQNAQICILNLFKSGFISKQDLLKALKILEKISKNTEIFDFILQEKVQSIDQKALFQNDFKELNTINLELQKLSFDENLKSRLQKTLEKFQNLEFNIAITGVMNAGKSSLLNALLKEDFLGVSNIPETANLTVLSYGKSEEAKIYFWDKKEWQNILESSHFNADLKEFIDKLDKSVNIEDFIKDKPLIQNIALCELKNFSSAKNKISALIKKIEIKSHLEFLKNNISIVDTPGLDDVVVQREIVTNEYLRESDFLIHLMNASQSLTQKDADFLVHCLLNSRLSKFLIVLTKADLLSKKDLEEVIVYTKESLKSRLVDLDENLVEKIDFLCVSAKMASDFYKGLASKESLQKSGMQEFENYLFNELYAGEKSKIALRAYKKELHLELKNILSEYEMQNRLIKENKQGVSEENQKLLLELQKQNTLLKEAQDEISNSIAKLKNIDSGIDNLVLLLAKKLKERLIDEFKYLKNNAQKLNLSRILNIVDITTKDGINDILREIKFENIKKIEELKTNLSLKYDFLKDDFDNGFEGFKDGISKNIDSIFQSEKFALLRLKIEKLSNLKSDLYELETNLDTVIFDTFKEFKMSEILNSLNINGAFFEFLNDKLKHYEKNQKSKLESLEKVLQSLKNQDANILNSFEENLEKIEKLKQLEMGLLNADKLHH
8OZT , Knot 175 414 0.85 40 221 399 
MTEPLPRIQHYEDLGLGLFIHWGLYSQMAVGEWTELIHHRNQHDYEQLIKTFTAAQFDAKKIAHAAKAVGAKYIVLTTKHHEGFFLYDTKGLSDFDVMHAPARRDLIAEFVAACREEDLLPFFYMATYDWHTPLYDDDFPAYLTYLQKSVEVLCRNYGPVGGFWFDGNWNKKDADWHLPELYGMIRHYQPNAIIVNNTGLKNRGQVSDPEIDVVTYQRRTPDEIYHGAPNEKYVAGEISITLNQHWGIAANDLNYKSPAEVIETVAHARHIGANILVNIGLTGTGAIPAAAQTYMHLLGRWTAMAAPVLYKGRPVPVTSAHGTRDFVLHTSKHDFLCILDLQVVGNDNVVLGGEGVNPRSFVGIGQPIQRIHWLDNDEVLSFTQDLDKKVLTVDATGYPYGSDWVVRIAQIDYE
4XQJ , Knot 53 99 0.82 36 83 97 
VETIELKRGSNSVYVQYDDIMFFESSTKSHRLIAHLDNRQIEFYGNLKELSQLDDRFFRCHNSFVVNRHNIESIDSKERIVYFKNKEHCYASVRNVKKI

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 \(c\).

\(|P_{f(5OXF_1)}(2) \setminus P_{f(8OZT_1)}(2)|=78\), \(|P_{f(8OZT_1)}(2) \setminus P_{f(5OXF_1)}(2)|=86\). 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 1:100110011000101101010100000100100111101000000001110010010001010101101001010110110011100001101101100100000110111000100100011100010010010101001010001000100010010010101110111011000110111000111100110010101100100001010110000100110000101010011001000101001100011100111001001001000101110010100010110001011001110011100011000010000111011010001000010111001100010011111001011000010011100000100011010001100101101010110010011100001000110010001100101100001110100001010100110000100011000001100000011101000001100100010001101001001100111111001000110010010001001010011011010000110011001010010010010001010001100010011011001100010011000011110101001001000100100010011100100101001100101011110110001000000000010010011001000010110010001001001001011110100100
Pair \(Z_2\) Length of longest common subsequence
5OXF_1,8OZT_1 164 4
5OXF_1,4XQJ_1 182 3
8OZT_1,4XQJ_1 208 3

Newick tree

 
[
	4XQJ_1:10.42,
	[
		5OXF_1:82,8OZT_1:82
	]:20.42
]

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 \((0.85)(0.8)(\frac{1146 }{\log_{20} 1146}-\frac{414}{\log_{20}414})=191.\)
Status Protein1 Protein2 d d1/2
Query variables 5OXF_1 8OZT_1 227 184.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]