Genomes are large systems
with small-system statistics:
Genome Growth by Duplication
National Tsinghua University
February 19, 2003
Institute of Physics, Academia Sinica
March 20, 2003
HC Lee
Dept Physics & Dept Life Sciences
National Central University
Plan of Presentation
• Introduction
• Frequency of words in genomes
• Large system & small-system statistics
• Model for genome growth & evolution
• Some results
• Discussion - The RNA world, spandrels,
codons, punctuated equilibrium, the
Universal Ancestor, etc.
• Outlook
The Book of Life
Many completed genomes
Many
completed
genomes
1995-2002 – Bacteria 細菌 (about 80 organisms);
0.5-5 Mb; hundreds to 2000 genes
1996 April – Yeast 酵母 (Saccharomyces cerevisiae)
12 Mb, 5,500 genes
1998 Dec. - Worm 線蟲 (Caenorhabditis elegans)
97 Mb, 19,000 genes
2000 March – Fly 果蠅 (Drosophila melanogaster)
137 Mb, 13,500 genes
2000 Dec. - Mustard 芥末子 (Arabidopsis thaliana)
125 Mb, 25,498 genes
2001 Feb. – Human 人類 (Homo sapiens)
3000 Mb, 35,000~40,000 genes
CBL@NCU
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Life Science Research
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CBL@NCU
Life Science in silico
CBL@NCU
= [biology]
+ [computer-science]
+ [math & physics]
+ [sequence data]
“It is much easier to teach biology to people
from a math, physics or computer-science
background than to teach a biologist how to
code well.”
- Nature, February 15, 2001, p963
Two approaches to Life Science
• Local - “Biology”
– Individual, specificity, uniqueness
• Global - “Physics”
– Class, generality, universality
Today’s talk:
Global treatment of microbial genome
Identify universality
Hypothesis for
early growth of universal ancestral genome
Structure of genome is complex
• Many levels – genes, intergenic region,
regulatory sections
• Gene – network of introns and exons
• Genome – network of genes
• Random mutation
• Genes are products of “blind
watchmaker”
• Once made, gene is repeatedly copied
– paralogues, orthologues and pseudogenes
• Genes are protected against rapid
mutation
Genome as text
• Genome is a text of four letters –
A,C,G,T
• Frequencies of k-mers characterize the
whole genome
– E.g. counting frequencies of 7-mers with a
“sliding window”
N(GTTACCC) = N(GTTACCC) +1
Textual statistics of genome almost
random but NOT TRIVIALLY so
• Looks like a random text to casual observer
• We know parts of it are coded
– Coded text also appears random but occupies
almost no volume in space of all tests
• Very hard to construct dictionary
• Distribution of frequencies of k-mers
– Characterizes whole genomes
– Similar in coding and non-coding regions
– For short oligos width of width of distribution
many times (up to 80) wider than normal
– Disparity greater for smaller k
• Similar for other kinds of distributions
21 century random text generator
- Courtesy PY Lai
Genomes violently disobey
rule of large systems
• Large systems have sharply defined averages
• Genomes are large texts with very fuzzy
averages
– There are 64 3-letter words (3-mers), each should
appear 15,625 +/- 125 times in a 1 Mb long
genome
– In random sequence, chances one 3-mer would
appear more (less) than 24,000 (8,000) times is
10-830 (10-980)
- In Treponema pallidum (syphilis;1 Mb long), 6 3mers (CGC, GCG, AAA, TTT, GCA, TGC) occur
more than 24,000 times and 2 (CTA, TAG) appear
less than 8,000 times
Bacterial genomes
are UNLIKE
random sequences
M. jannaschii, 70% A+T
B. subtilis, 57% A+T
E. coli, 50% A+T
If genome grows randomly by single
nucleotide then distribution is Poisson
Poisson P(f=k) = lk e-l /k!
<f> = l, D (stand. dev.) = <f>1/2
Gamma G(f) = fa-1 e-f/b /baG(a)
<f> = ab,
D = a1/2 b
Random single nucleotide; D = 15.5
E. coli, a=3.05, b = 80.0; D = 140
Non-uniform nucleotide composition breaks the
n-mer Poisson distribution into n+1 peaks
Number of 6-mers
62.0
26.6
Given [at]/[cg]=70/30. If mean frequency
is 244, then mean frequency of 6-mers with
144
K a or t’s and 6-k c or g’s is
fk = 244 (0.7)k (0.3)6-k/(.5)6
Random single nucleotide
k
fk
337
787
11.4
M. janaschii
________
0 11.4
1 26.6
2
62.0
3
144
4
337
5
787
6 1837
1837
Frequency of 6-mers
Similar discrepancy in other genomes
and for other word lengths
rms deviation of word count in genomes
Word length
(k)
Average word
count/1Mb
Genomic
deviation (error)
Poisson
deviation
2
3
4
5
6
7
8
9
10
62,500
15,625
3,906
977
244
61
15.3
3.81
0.95
10,580 (2,040)
4,080 (630)
1,490 (210)
469 (66)
141 (21)
41.9 (6.7)
12.4 (2.3)
3.84 (0.84)
1.33 (0.34)
250
125
62.5
31.2
15.6
7.8
3.9
1.9
0.98
Statistically genomes resemble random
sequences of much short lengths
Effective length:
Length of sequence
with Poisson
distribution having
same mean to s.d.
ratio as genome
sequence.
Recall for Poisson,
s.d.= sqrt(mean)
Leff = ((mean/s.d.)gen)2 4k
k
Mean
s.d.
Effective
genome
length (kb)
2
3
4
5
6
7
8
9
10
62,500
15,625
3,906
977
244
61
15.3
3.81
0.95
10,580
4,080
1,490
469
141
41.9
12.4
3.84
1.33
0.56
0.94
1.8
4.4
12
35
100
260
540
How does a genome evolve and grow?
•
Evolve by random mutation
–
•
Plus natural selection
–
–
•
Fitness acts only on phenotype, not directly on
genome
Selection is made on genome generated
randomly
Genome cannot grow through random
mutation alone
–
•
replacement, insertion, deletion
Otherwise Poisson distribution
Must grow to long length while retaining
statistical characteristics of SHORT genome
The genome is a self plagiarizer
• Genomes have many homologous genes
• 50%, probably much more, of human
genome composed of recent repeats
– Many traces of repeats obliterated by mutation
– Lower organisms may have longer genomes
• Five types of repeats
– transposable elements; processed pseudogenes;
simple k-mer repeats; segmental duplications (10-300
kb); (large) blocks of tandemly repeated sequences
A Hypothesis for Genome
Growth
• Random early growth
• Followed by
1. random duplication and
2. random mutation
Self copying – strategy for retaining and
multiple usage of hard-to-come-by
coded sequences (i.e. genes)
The Model
• The genome grows by random
single base addition from nothing to
an initial length much shorter than
final length
• Thereafter the genome evolves by
random mutation and random
duplication, with a fixed frequency
ratio
The Model (continued)
• Mutation is standard single-point
replacement (no insertion and deletion)
• Segmental duplication involves three
stochastic steps
– random selection of site of copied
segment
– weighed random selection of length of
copied segment
– random selection of insertion site of
copied segment
Stochastic selection of the length of
self-copied segments
• Use Erlang density distribution function for
segment length l
f(l) = 1/(s m!) (l/s)m exp(-l/s)
(gamma function when m is real)
• Mean < l > = (m+1) s
standard deviation = (m+1)½ s
 Nothing special about this particular function,
but mean and s.d. important
First generation result
LS Hsieh, LF Luo, FM Ji and HCL, PRL 90 (2003) 18101
• Distribution of 6-mer frequency
• Starting genome length 1000
• Final genome length 1 million
• Mutation to duplication event ratio
100 < h < 4000
• Length scale for copied segments
2500 < s < 100 K
• Compared with E. coli (4.5 Mbp), B.
subtilis (4.2 Mbp), M. jannaschii (1.7
Mbp) (all normalized to 1 Mbp)
Number of 6-mers
E. coli
[at]/[cg]=50/50
E. coli vs
mutation + repeat
Ratio 500:1
Sigma = 15k
D= 140, 144
Frequency of 6-mers
E. coli vs
random
D= 140, 15.5
Number of 6-mers
B. subtilis
[at]/[cg]=60/40
B. subtilis vs
mutation + repeat
Ratio 600:1
Sigma = 15k
D= 167, 169
Frequency of 6-mers
B. subtilis vs
random
D= 167, 79
Number of 6-mers
M. jannaschii
[at]/[cg]=70/30
M. jannaschii vs
mutation + repeat
Ratio 600:1
Sigma = 15k
D= 320, 321
Frequency of 6-mers
M. jannaschii vs
random
D= 320, 265
Gamma function reproduce higher moments
Organism
[at]/[gc]
a
b
D(2)
D(3)
D(4)
D(5)
E. coli
50/50
gamma distribution
3.05 80.0
radom w/o self-copy (Poisson)
w/ self-copy (h = 500 s = 15K)
140
140
15.6
144
147
146
3.6
148
213
208
20.7
212
252
243
10
247
B. subtilis
60/40
gamma distribution
2.12 115
radom w/o self-copy (Poisson/7)
w/ self-copy (h = 600 s = 15K)
168
168
79
169
223
186
68
194
316
261
109
266
400
310
117
311
M. jannaschii 70/30
gamma distribution
0.58 418
radom w/o self-copy (Poisson/7)
w/ self-copy (h = 600 s = 15K)
320
320
264
321
465
439
369
462
650
609
500
635
810
767
603
783
Gamma distribution:
D(n) = (<(x - <x>)n>)1/n;
D(x) = xa-1 b-a exp(-x/b)/G(a)
<x> = 244 = a b;
D(2) = a1/2 b
What about other k’s?
• Initial model good for k=6 but for other k’s
not so good. Over-compensation (too
broad) when k>6 and under-compensation
(too narrow) when k<6.
• Good result for k=6 (length = 1 Mb) requires
h ~ 0.04 s. In the limit of very small mutation
to duplication event ratio, or h ~1, s ~25 b.
• New model with short duplication length,
s ~ 25 b, and without mutation.
Density function for duplication
segment length
• Recall Erlang density distribution function
has mean and rms deviation
< l > = (m+1) s; Dl = sqrt(m+1) s
• For < l > = 25, have:
m
0
2
4
s
25
8
5
Dl
25
14
11  Good!
Comparison
of k-mer
distributions,
k=5-9, for
model
sequence D
and genome
Treponema
Length of
duplicated
segements:
25 +/- 12 bp
Model sequence almost reproduces
shape of genomic distributions
rms deviation of word count in genomes
Word
length
T. pallidum
Genomic
average (error)
Poisson
Present
model
2
3
4
5
6
7
8
9
10
8260
3870
1380
432
129
37.5
11.0
3.4
1.3
10,580 (2,040)
4,080 (630)
1,490 (210)
469 (66)
141 (21)
41.9 (6.7)
12.4 (2.3)
3.84 (0.84)
1.33 (0.34)
250
125
62.5
31.2
15.6
7.8
3.9
1.9
0.98
8207
3415
1202
402
134
45.3
15.9
5.9
2.3
Counts of dinucleotdies (k=2)
Random sequence
at 62500+/-250
Counts of trinucleotdies (k=3)
Random sequence
at 15625+/-125
Counts of tetranucleotdies (k=4)
Random sequence
at 3906+/-63
Methanoccocus jannaschii
70% A+T, 30% C+G
Model sequence generated
Exactly as before, except
70% A+T in initial random seq
Random sequence
Result sensitive to parameters
• Paremeter values for “good” model
sequence:
- Initial random sequence length L0 ~1 kb;
- Mean copied segment length <l> ~ 25 b
- rms Dl ~ 12 b
If L0 > 10 kb, no good results
If <l> = 15 b, sequence too random for k<5
If <l> = 40 b, sequence too choppy for k>6
If <l> = 25 b, Dl ~ 15 b; agreement worsens
Discussion: The RNA World
• RNA was discovered in early 80’s to
have enzymatic activity – ribozymes can
splice and replicate DNA sequences
(Cech et al. (1981), Guerrier-Takada et al. 1983)
• The RNA world conjecture – early had no
proteins, only RNAs, which played the
dual roles of genotype and phenotype
• Some present-day ribozymes are very
small; smallest hammerhead ribozyme
only 31 nucleotides; ribozymes in early life
need not be much larger
RNA World & size of early genome
• In our model the small initial size of the
genome necessarily implies an early RNA
world
• A genome ~ 1K nt long is long enough to
code the many small ribozymes (but not
proteins) needed to propagate life
• Origin of this initial genome not addressed in
the model. It (or its presursor) could have
arisen spontaneously - artificial ribozymes
have been succcessfully isolated from pools
of random RNA sequences (Ekland et al. 1995)
RNA World & length of duplicated
segments
• Recall that present-day ribozyme can be
as small as 31 nt
• The average duplicated segment length
of 25 nt in the model is very short
compared to present-day genes that
code for proteins, but likely represents a
good portion of the length of a typical
ribozyme encoded in the early universal
genome of the RNA world
Are codons “spandrels”?
• Spandrels
– In architecture - the roughly
triangular space between an
arch, a wall and the ceiling
– In evolution – major category of important
evolutionary features that were originally
side effects and did not arise as
adaptations (Gould and Lewontin 1979)
• Wide 3-mer/codon distribution or natural
selection, which came first?
Are codons “spandrels”? (cont’d)
• Frequency of 3-mer distribution in genomes is
about 40 x wider than Poisson. Was the widening
caused by
– Uneven codon usage + natural selection? Or,
– Genome growth by segmental duplication?
• In RNA world, codons came after RNA and
existence of replication machinery. Hence the
following scenario:
RNA + recombination > genome growth by stochastic
dupliction > extreme bias in 3-mer population > rise of
codon
• In our model, codons are most likely spandrels
More spandrels
•Same goes with other oligonucleotides
Many oligonucleotides that are grossly overor under-represented have biological
functions. Evolution being an opportunistic
process, these oligonucleotides could have
been drafted to serve special biological
purposes because they had already been
made very copious or very rare by stochastic
genome growth
Duplication continued and expanded
after the rise of proteins
• In bacterial genomes typically about 12% of
genes represent recent duplication events
– Average gene is about 1000 bases long. Suggest
about 12% of genome generated by
duplications of ~ 1000 b segments. Not yet
incorporated into the model.
• In higher organisms a large number of
repeat sequences with lengths ranging from
1 base to many kilobases are believed to
have resulted from at least five modes of
duplication
Grow by duplication (of gene-size
segments) may explain:
• How have genes been duplicated at the high
rate of about 1% per gene per million years?
(Lynch 2000)
• Why are there so many duplicate genes in all
life forms? (Maynard 1998, Otto & Yong 2001)
• Was duplicate genes selected because they
contribute to genetic robustness (by
protecting the genome against harmful
mutations)? (Gu et al. 2003)
– Likely not; Most likely high frequency of occurrence
duplicate genes is a spandrel
Classical Darwinian Gradualism
or Punctuated equilibrium?
• Great debated in palaeontology and
evolution - Dawkins & others vs. (the late)
Gould & Eldridge: evolution went
gradually and evenly vs. by stochastic
bursts with intervals of stasis
Our model provides genetic basis for both.
Mutation and small duplication induce gradual
change; occasional large duplication can induce
abrupt and seemingly discontinuous change
Discussion (cont’d)
• Phylogeny and the Universal Ancestor
– If extremely frequent and extremely rare oligos
(EFERO) are the remnants of much shorter early
sequence, then there should exist such a short
sequence during some stage of the genome growth.
– Then we may be able to use the set of EFEROs in
whole genomes to construct phylogenetic trees of
whole genomes.
– At each node of the tree would be an ancestor
sequence characterized by a set of EFEROs.
– The ancestor of Life would be characterized by the
minimum set of EFEROs.
Summary
• Distribution of frequency of k-mers in bacterial
genomes hugely wider than Poisson – larger for
smaller k
• Can be explained by simple two-phase genome
growth model:
– first grow to short (~1 kb) random sequence
– then grow by random duplications of segments of
length 25 +- 12 b long
• Reproduces genomic statistics for k=2-8
• Universal ancestral genome lived in an RNA world
– Replication carried by ribozymes ~ 30 nt;
– Codons and many signal sequences are spandreals
Outlook
• Need to understand distribution for ALL k’s
– There are repeated k-mers of k up to ~1000
• Other oddities
– E.g. Distribution of entropies of k-mers
• Empirical verification
– Can duplication growth be independently verified?
• Time scale
– When did growth happen? At what rate? How did
growth stabilize? Has it stabilized?
• Phylogeny
– Can we build a good tree based on model? Can
we learn anything about the Universal Ancestor ?
Is there a Universal Ancestor ?
CBL Lab @NCU
Phys. & Life Sci.
* L.C. Hsieh
# J.L. Lo
# T.Y. Chen
# J.P. Yiu
# Z.Y. Guo
# Z.R. Chiu
# H.Y.Bai
# C.H. Chang
# H.D. Chen
# W.L. Fan
#
Collaborators
J.Z. Horng
# F.M. Lin
Horng Lab, NCU, Comp. Sci.
*L.F. Luo
Univ. Outer Mongolia
*F.M. Ji
Beijing Jiaotong Univ.
Rosie Redfied
Zoology, UBC
*This work; # students
* # All simulation in this work done by L.C. Hsieh
Thank you for your
attention
Genomes are large systems
with small-system statistics:
Genome Growth by Duplication
Lecture II
Winter School on Modern Biophysics
National Taiwan University
December 16-18, 2002
HC Lee
Dept Physics & Dept Life Science
National Central University
Result sensitive to values of two
parameters
• Mutation to duplication event ratio h
– bacterial genomes, 200 < h ~ 0.04s < 800
– If h >> 800 (@ s ~ 15K)
• too many mutations
• gets long genome with Poisson distribution
– If h << 200 (@ s ~ 15K)
• too much duplication
• too few mutations
• gets multiple copies of random short (initial)
genome (distribution too wide)
Mutation to self-copy ratio is 500 +/- 100
Mutation/self-copy = h
Scale of repeat length = s = 15K
P(l)/P(l’) = exp{-(l-l’)/s}
[at]:[cg] = 70:30
h = 100
h = 250
h = 500
h = 2000
h = 4000
(genome-like)
Result sensitive to values of two
parameters (cont’d)
• Length scale s for copied segments
– s ~ 10 K to 25 K for bacterial genomes
– If s << 5 K (@ h ~ 600)
• genome grows too slowly
• too many mutations
• gets long genome with Poisson distribution
– If s >> 25 K (@ h ~ 600)
• genome grows too quickly
• too few mutations
• gets multiple copies of random short (initial)
genome (distribution too wide)
Scale of repeat length cannot be too short
Scale of repeat length = s
P(l)/P(l’) = exp{-(l-l’)/s}
Mutation/self-copy = h = 500
[at]:[cg] = 70:30
s= 0.5K
s =2.5K
s =15K
s =50K
s =1000K
(genome-like)
Number of oligos
Frequency distribution of 6-mers
Frequency of oligo