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Parikh vectors
4TRH_1 3IQZ_1 5MQN_1 Letter Amino acid
36 9 51 S Serine
26 6 66 T Threonine
24 6 40 Y Tyrosine
4 3 16 C Cysteine
26 3 42 Q Glutamine
40 13 59 I Isoleucine
55 18 76 K Lycine
25 14 78 G Glycine
15 2 20 H Histidine
3 17 26 M Methionine
2 0 29 W Tryptophan
26 35 93 A Alanine
27 8 52 N Asparagine
46 24 62 D Aspartic acid
44 37 86 E Glutamic acid
20 14 59 R Arginine
22 10 40 F Phenylalanine
49 22 92 L Leucine
23 16 62 P Proline
29 26 56 V Valine 

4TRH_1|Chains A, B|SidC|Legionella pneumophila (446)
>3IQZ_1|Chains A, B, C, D, E, F|F420-dependent methylenetetrahydromethanopterin dehydrogenase|Methanopyrus kandleri (2320)
>5MQN_1|Chain A|Glycosyl hydrolases family 2, sugar binding domain|Bacteroides thetaiotaomicron (818)
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 \(w=f(c)\)
4TRH , Knot 213 542 0.82 40 246 502 
MVINMVDVIKFKEPERCDYLYVDENNKVHILLPIVGGDEIGLDNTCQTAVELITFFYGSAHSGVTKYSAEHQLSEYKRQLEEDIKAINSQKKISPHAYDDLLKEKIERLQQIEKYIELIQVLKKQYDEQNDIRQLRTGGIPQLPSGVKEIIKSSENAFAVRLSPYDNDKFTRFDDPLFNVKRNISKYDTPSRQAPIPIYEGLGYRLRSTLFPEDKTPTPINKKSLRDKVKSTVLSHYKDEDRIDGEKKDEKLNELITNLQNELVKELVKSDPQYSKLSLSKDPRGKEINYDYLVNSLMLVDNDSEIGDWIDTILDATVDSTVWVAQASSPFYDGAKEISSDRDADKISIRVQYLLAEANIYCKTNKLSDANFGEFFDKEPHATEIAKRVKEGFTQGADIEPIIYDYINSNHAELGLKSPLTGKQQQEITDKFTKHYNTIKESPHFDEFFVADPDKKGNIFSHQGRISCHFLDFFTRQTKGKHPLGDLASHQEALQEGTSNRLHHKNEVVAQGYEKLDQFKKEVVKLLAENKPKELLDYLVAT
3IQZ , Knot 121 283 0.80 38 164 270 
MTVAKAIFIKCGNLGTSMMMDMLLDERADREDVEFRVVGTSVKMDPECVEAAVEMALDIAEDFEPDFIVYGGPNPAAPGPSKAREMLADSEYPAVIIGDAPGLKVKDEMEEQGLGYILVKPDAMLGARREFLDPVEMAIYNADLMKVLAATGVFRVVQEAFDELIEKAKEDEISENDLPKLVIDRNTLLEREEFENPYAMVKAMAALEIAENVADVSVEGCFVEQDKERYVPIVASAHEMMRKAAELADEARELEKSNDAVLRTPHAPDGKVLSKRKFMEDPE
5MQN , Knot 407 1105 0.86 40 341 998 
MKSRLKQQIFAISLLACTAISPANALQTHLLREQFQNPSDEAKPWTFWYWMFGAVSKEGITADLEAMKRAGLGGTYLMPIKGIKEGPQYNGKAQQLTPEWWEMVRFSMEEADRLGLKLGMHICDGFALAGGPWMTPKESMQKIVWSDTIVDGGKIKGLHLPQPEAYEGFYEDISLFALPVKEEAADVMPAQITCANIATGNHIDIKKTVNMDDAGVIRSSYPCYIQYEYEQPFTCRNIEIILSGNNYQAHRLKVMASDDGVNYRLVKQLVPARQGWQNTDENSTHAIPATTARYFRFYWTPEGSEPGSEDMDAAKWKPNLKIKELRLHREARLDQWEGKAGLVWRVASSTKKEEIGEQDCYALSQIINLTDPFKNSPSDNFKERTLTATLPKGKWKLLRMGHTATGHTNATAGGGKGLECDKFNPKAVRKQFDNWFAQAFVKTNPDVARRVLKYMHVDSWECGSQNWSDTFAAEFRKRRGYDLMPYLPLLAGIPMESAERSEKILRDVRTTIGELVVDVFYQVLADCAKEYDCQFSAECVAPTMVSDGLLHYQKVDLPMGEFWLNSPTHDKPNDMLDAISGAHIYGKNIIQAEGFTEVRGTWNEHPGILKALLDRNYALGINRLFFHVYVHNPWLDRKPGMTLDGIGLFFQRDQTWWNKGAKAFCEYITRCQSLLQYGHPVADIAVFTGEEMPRRSILPERLVPSLPGIFGAERVESERIRLANEGQPLRVRPVGVTHSANMSDPEKWVNPLRGYAYDSFNKDALLRLAKAENGRMTLPGGASYKVLVLPLPRPMNPDPAALSPEVKQKINELKEAGILIPSLPYKEDDFSSYGLERDLIVPENIAWTHRQGEQGDIYFIANQLEETRTFTASMRIDGRKPECWNPVTGEINADIPYEQKSHRTEITLTLAPNESVFIVYPAEEDDKETSEKERKEKKDSVKEASETGLEATEYTVTFTANGKTIQRQELFDWSKEEDEQIRYYSGTAVYKTTFRWKSKVKEDQQVYLNLGKVCDLATVRVNGIDCGTIWTAPYRADITAALKKGVNELEIEVTNTWANALKGADEGKAPFDGIWTNAKYRRAENTLLPAGLLGPLNFDVANKNK

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 \(c\).

\(|P_{f(4TRH_1)}(2) \setminus P_{f(3IQZ_1)}(2)|=126\), \(|P_{f(3IQZ_1)}(2) \setminus P_{f(4TRH_1)}(2)|=44\). 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 1:11101101101001000001010000010111111110011100000011011011010100110000100010000001000101100000101010001100010010010001011011000000000100100111101101100110000011110101000001001001110100010000010001111100111001000111000010110000100010001100000000101000000100110010001100110001000010100010100100001100111100000110110011010100011110100110011001000001001010100111010100000010010110110001010011001001100110101110001000010111001101000001000100000010001010011110100010110001010001101100000100111011000011001000010000011101000100100011011100010011001110
Pair \(Z_2\) Length of longest common subsequence
4TRH_1,3IQZ_1 170 4
4TRH_1,5MQN_1 133 4
3IQZ_1,5MQN_1 203 4

Newick tree

 
[
	3IQZ_1:10.04,
	[
		4TRH_1:66.5,5MQN_1:66.5
	]:34.54
]

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 \((0.85)(0.8)(\frac{825 }{\log_{20} 825}-\frac{283}{\log_{20}283})=148.\)
Status Protein1 Protein2 d d1/2
Query variables 4TRH_1 3IQZ_1 190 141
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]