
Compression of Molecular Sequences.
The complexity
(information content, entropy, compressibility, ... )
of DNA and other biological sequences is
of interest for a variety of reasons:
Repetitive subsequences (polyA, (AT)^{*}, etc.)
have low information content;
they may be of interest in their own right or may be given
a low weighting in sequence matches if they can be identified.
Informally, two extreme kinds of sequence are "boring",
those with near zero information content, e.g. AAAAAAA.... ,
and those with maximum information content, i.e. random sequences,
as "typed" by the proverbial monkey.
Generally, interesting sequences have some structure or pattern
and lie somewhere between these extremes.
The Alu sequences form a family of sequences, each about 300 bases
long, that occur thousands of time in human DNA.
The typical Alu is about 87% similar to the concensus Alu therefore
the information content of an Alu is much less than 600 bits.
Gene duplication, regulatory signals, codon bias, etc.
all reduce the information content below 2 bits/base
and may be of biological interest.
[Example (click)]
D.R. Powell, D.L. Dowe, L. Allison and T.I. Dix.
Discovering Simple DNA Sequences by Compression.
Pacific Symposium on Biocomputing 3 (PSB98), pp.595606, 1998.
L. Allison, T. Edgoose & T. I. Dix,
Compression of Strings with Approximate Repeats,
Intelligent Systems in Molecular Biology (ISMB'98),
pp.816, Montreal, 28 June  1 July 1998,
[HTML]
[pdf].
AED's Approximate Repeats Model (ARM) for DNA
sequences allows for approximate forward and reversecomplementary repeats
and achieves good sensitivity at pattern discovery.
L. Allison, L. Stern, T. Edgoose & T. I. Dix.
Sequence Complexity for Biological Sequence Analysis.
Computers and Chemistry 24(1), pp.4355, Jan' 2000.

L. Stern, L. Allison, R. L. Coppel, & T. I. Dix.
Discovering Patterns in Plasmodium falciparum Genomic DNA.
Molecular and Biochemical Parasitology, 118(2),
pp.175186, Dec' 2001.

Minh Duc Cao, Trevor I. Dix, Lloyd Allison & Chris Mears.
A Simple Statistical Algorithm for Biological Sequence Compression.
IEEE Data Compression Conference (DCC), 2007.
The expert model (XM) gives good compression of DNA
and also protein sequences, and it is fast.

See also the alignment of compressible sequences
[below], Comp. Jrnl (1999), etc..
One
of the products of this work is the alignment density plot
(right). This is a good way to visualize the collection of all
possible alignments of two strings and the greyscale level
indicates the degree of (un)certainty as to
which is the correct or "true" alignment.
The plot from the sequence alignment algorithm
shows the degree of certainty
and uncertainty in various sequence alignments, see:
L. Allison, C. S. Wallace & C. N. Yee.
When is a String Like a String?
Artificial Intelligence in Mathematics (AIM),
Ft. Lauderdale, Florida, January 1990
[HTML].
L.Allison, C.S.Wallace & C.N.Yee.
Inductive Inference over MacroMolecules,
Technical Report 90/148 [HTML]
or....
L.Allison, C.S.Wallace & C.N.Yee,
FiniteState Models in the Alignment of MacroMolecules,
J.Molec.Evol. 35(1) 7789, 1992.
C. N. Yee & L. Allison.
Reconstruction of Strings Past,
Comp. Appl. in BioSciences (CABIOS) 9(1) pp.17, 1993,
[HTML].
By summing the probability over all alignments
even very distant relationships can be detected.
By averaging parameter estimates over all alignments,
rather than just the optimal alignment,
unbiased estimates of parameter values are obtained.
This is demonstrated for the 1state (simple) and
3state (linear, affine gap costs) models.
The 1, 3 and 5state (etc.) models are models of alignment,
evolution or mutation having increasing complexity.
3state mutation machine (right) from the MML computational biology
sequence alignment program corresponds to affine
or linear gap costs as commonly used in aligning DNA sequences, see:
L.Allison,
Normalization of Affine Gap Costs Used in Optimal Sequence Alignment,
J.Theor.Biol. 161 263269, 1993
[www inc' pdf paper].
You can interpret your favourite integer penalties
or scores as probabilities or you can choose integer penalties (scores)
that approximate probabilities that you believe are appropriate.
Low (or moderate)
information content
(nonrandom, repetitive, compressible,...) sequences cause problems
in alignment algorithms and in database searching, causing falsepositive
matches, for example. The information content of the sequences
can be taken into account giving better results; see:
D. R. Powell, L. Allison, T. I. Dix and D. L. Dowe.
Alignment of low information sequences.
Proceedings of the Fourth Australasian Theory Symposium
[CATS98]
pp.215229, February 1998.
L. Allison.
InformationTheoretic Sequence Alignment (HTML),
TR98/14 School of Comp. Sci. & SWE, Monash University, June 1998,
 on the alignment of nonrandom (compressible, repetitive,
lowinformation content) sequences.
L. Allison, D. Powell & T. I. Dix.
Compression and Approximate Matching,
Computer Journal, 42(1), pp.110, 1999
[www inc' pdf paper].
L. Allison, D. Powell and T. I. Dix.
Modelling Is More Versatile Than Shuffling.
[TR 2000/98] (HTML),
2000.
Fewer false positives, fewer false negatives,
can (and should) change rankorder of alignments,
applicable to more general models than the "shuffling"
correction for nonuniform populations of sequences,
identifies correct population model(s).
[Example]
Seminars:
(DNA) Sequence Complexity and Alignment,
to the Walter and Eliza Hall (WEHI) Bioinformatics Group,
13 Feb' 2001,
on the relationship between sequence alignment,
compression & modelling of biological sequences, and
alignment of nonrandom sequences.
Modelling v. Shuffling,
to the Bioinformatics Group & Dept. Computer Science
[UWA],
11 Aug' 2004,
on PRSS, masking, shuffling, Malignment,
low to medium information content sequences, and alignment accuracy,
significance & ranking.

D. R. Powell, L. Allison, T. I. Dix.
ModellingAlignment for NonRandom Sequences.
17th ACS Australian Joint Conf. on Artificial Intelligence
(AI2004), pp.203214, 610 Dec 2004,
SpringerVerlag, LNCS/LNAI 3339, isbn:3540240594.
Comparison with PRSS from FASTA package.
Link leads to software and paper.

Variations on Sequence Alignment
(2007),
dynamic programming and related algorithms
for two or more sequences.
Minimum message length [MML]
is used to compare sequences objectively.
MML also gives a natural nulltheory and a significancetest for hypotheses.
The models are based on finitestate machines
(~ hidden Markov models).
The theory of multistate distributions is used to calculate
the accuracy of the parameter estimates.
L. Allison & T. I. Dix.
A bitstring longest common subsequence algorithm,
Inf. Proc. Lett. 23(6), pp.305310, 1986
[paper]
[code].
Uses bitvector operations to get a speedup
roughly equal to the wordlength of the computer.
L. Allison, Lazy dynamic programming can be eager.
Inf. Proc. Lett., 43(4), pp.207212, Sept' 1992
[paper]
When written in a
lazy functional programming language,
the simple, O(n^{2})time dynamic programming algorithm (DPA)
can be made to run in just O(n.d)time by adjusting
the central comparison test.
L. Allison.
Using Hirschberg's algorithm to generate random alignments.
Inf. Proc. Lett., 51 pp.251255, 1994.
Allows alignments of two strings to be sampled at random
from their posterior probability distribution.
This gives a MonteCarlo method for estimating evolutionary parameters and,
when cooled, gives a simulated annealing method for optimal alignment.
It has been applied to multiple alignment,
see Allison & Wallace Jrnl. Molec. Evol., 39, pp.418430, 1994
under multiple alignment.
D. R. Powell, L. Allison & T. I. Dix.
A versatile divide and conquer technique for optimal string alignment.
Inf. Proc Lett., 70(3), pp.127139, 1999
[www inc' pdf document]
Common string alignment algorithms such as the
basic dynamic programming algorithm (DPA) and the time efficient
Ukkonen algorithm use quadratic space to determine an alignment between
two strings. In this paper we present a technique that can be applied
to these algorithms to obtain an alignment using only linear space,
while having little or no effect on the time complexity.
This new technique has several advantages over previous methods
for determining alignments in linear space, such as: simplicity,
the ability to use essentially the same technique when using different
cost functions, and the practical advantage of easily being able to
trade available memory for running time.
L. Allison.
Longest biased interval and longest nonnegative sum interval.
Bioinformatics, 19(10), pp.12941295, 1 July 2003.
Finds the longest interval of a sequence that has
at least a specified minimum bias (e.g. 80%) towards certain specified
characters (bases, amino acids, e.g. AT). It can process very long
sequences quickly.
L. Allison.
Finding approximate palindromes quickly and simply.
TR 2004/162,
School of Computer Science & Software Eng., Monash U., 2004.
Fast 3way alignment procedure:
O(n.d^{2}) time at worst, O(n+d^{3}) on average,
where the strings' lengths are ~n and d is the 3way editdistance.
L. Allison.
A fast algorithm for the optimal alignment of three strings.
J. Theor. Biol. 164(2) pp.261269, 1993
[www inc' pdf paper].
Simple mutation costs;
for affine gap costs see Powell et al 2000
D. R. Powell, L. Allison & T. I. Dix.
Fast, optimal alignment of three sequences using linear gap costs.
J. Theor. Biol. 207(3) pp.325336 Dec 2000
[www inc' pdf paper].
Gap (indel) costs of the form a+b*k for a gap of length k,
where a and b are small integers. The algorithm runs in
O(n+d^{3}) time on average, where the string lengths are ~n
and d is the 3way edit distance.
i.e. It is fast when the strings are similar, and the edit distance,
not string length, is the main influence on running time.
Note that each internal node in an unrooted binary tree has three neighbours
so that a good heuristic for kway alignment, given an evolutionarytree,
is to do repeated 3way alignment, inferring the strings at internal
nodes in the process.
Multiple Alignment and Evolutionary Trees.
Personal
publications in multiple alignment (short).
Technical Report 91/155
Minimum Message Length Encoding,
Evolutionary Trees and Multiple Alignment.
L. Allison, C. S. Wallace and C. N. Yee.
Minimum message length encoding, evolutionary trees and multiple
alignment.
Hawaii Int. Conf. Sys. Sci., HICCS25, v1, pp.663674, Jan 1992
[paper].
L.Allison & C. S. Wallace.
The Posterior Probability Distribution of Alignments and its
Application to Estimation of Evolutionary Trees and
Optimisation of Multiple Alignments,
Technical Report 93/188,
July 1993.
L. Allison & C. S. Wallace.
An Information Measure for the String to String Correction
Problem with Applications,
17th Annual Annual Comp. Sci. Conf., ACSC17,
Christchurch, New Zealand, and
Australian Computer Science Communications 16(1C),
pp.659668, Jan' 1994
[paper (HTML)]
L. Allison and C. S. Wallace.
The Posterior Probability Distribution of Alignments and its Application to Estimation of Evolutionary Trees and Optimisation of Multiple Alignments.
Jrnl. Molec. Evol. 39 pp.418430, 1994.
Minimum message length [MML] is used to determine
the accuracy with which branchlengths in an evolutionary tree can
be estimated  it is rather low in general. Simulated annealing is
used as a way out of the localoptima problem for a practical
multiplealignment algorithm. Gibbssampling is used as a practical
way of estimating edge lengths, and standard deviations of estimates
give another indication of reliable accuracy.
On
Multiple Sequence Alignment
Source code
of a stochastic multiple alignment computer program [6/'95].
This does iterated, stochastic 3way alignment at each internal
node of the tree to sample ancestral strings. This strategy
should be straightforward to adapt to sophisticated mutation models.
There is also good potential for speed up on a parallel computer
or on a network of workstations.
Technical Report 95/225 postscript (50K)
Towards Modelling Evolution = Mutation modulo Selection
in Sequence Alignment.
Biological evolution takes place through the mutation of DNA modulated
by the pressure of selection which determines which changes are viable.
A proposal for modelling both of these factors
in the multiplealignment problem is presented.
It is argued that treebased alignment methods primarily model mutations
events, and that most nontree methods (e.g. profiles) model
selection pressures on a family of sequences.
A treebased method specifies mutation probabilities and,
if used in isolation, the hope must be that biases
due to selection are small or average out.
A nontree method can be used to model the family of sequences 
which sequences are fit and viable,
and which are nonviable, as members of the family.
It is suggested that the two approaches can be combined:
the latter can be used to modify the mutation probabilities
of the former  to model which mutations are viable and are observed.
See also Trevor Dix, Darren Platt and C. N. Yee.
Protein Secondary Structure Prediction.
Personal
publications in protein structure (short).
Technical Report 92/163
A Decision Graph Explanation of Protein Secondary Structure Prediction.
[HTML]
D. L. Dowe, J. Oliver, T. I. Dix, L. Allison & C. S. Wallace
A decision graph explanation of protein secondary structure prediction.
26th Hawaii Int. Conf. Sys. Sci. Vol.1 pp.669678 Jan' 1993
[HTML]
D. L. Dowe, L. Allison, T. I. Dix, L. Hunter,
C. S. Wallace & T. Edgoose
Circular clustering of protein dihedral angles by
minimum message length.
Pacific Symposium on Biocomputing '96 (PSB96),
World Scientific, pp.242255, Jan' 1996
[paper (ps)]
T. Edgoose, L. Allison & D. L. Dowe.
An MML classification of protein structure that knows about angles and
sequence.
Pacific Symposium on Biocomputing '98 (PSB98),
pp.585596, Jan' 1998
[paper (pdf)]
Rubik's snake (tm) (right) makes a good digital model
for protein folding and is also fun. It is made of 24 triangular
prisms which meet on their square faces. Two meeting
prisms can rotate relative to each other about the centre
of the meeting face so as to vary the "dihedral" angle.
The snake can be packed in a ball as a "globular" protein
and can form "helices" and "extended"
conformations.
Minimum Message Length Encoding.
Many of the above projects make use of a method of
inductive inference known as
Minimum Message Length [MML] encoding.
MML is a realisation of Occam's razor.
It is invaluable for dealing with error and uncertainty in data
and for comparing models that have different complexities.
For example, an alignmentmodel with gappenalties is more
complex (has more degrees of freedom) than one without gappenalties.
The former will in general fit the data strings better than the latter 
but is the improvement enough to "pay for" the extra complexity of the model?
MML can judge this tradeoff fairly.
(Note that maximumlikelihood ignores the complexity of the model
which can result in the well known phenomenon of overfitting.)
(Hidden) Markov Models
(HMM) are often called probabilistic finite state automata (PFSA),
or probabilistic finite state machines (PFSM),
in computing because of their close relationship
to finite state automata, as used in formal language theory.
Whatever they are called, they are useful
in pattern & structure discovery, compression, alignment of
pair of sequences, etc..
[more]
Misc'
Compression and Analysis of Biological Sequences. Why and How [2007],
Modelling Alignment,
Bioinformatics at CSSE
[9/2001],
Sequence Complexity.
Dr. Lloyd Allison
,
Dr. Trevor Dix,
Prof. C. S. Wallace.

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