the birth of smooth biological codes in a rough evolutionary world shalev itzkovitz, guy shinar, uri...
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The Birth of Smooth Biological Codes in a Rough Evolutionary
WorldShalev Itzkovitz, Guy Shinar, Uri Alon
T T
o Biological codes are information
channels or maps with natural
‘fitness’ measure.
o Codes are evolved and selected
according to their fitness or
‘smoothness’.
o The emergence of a code is a phase
transition in an information channel.
o Topology of errors (noise) governs
the emergent code.
Biological codes are (often) maps
• Biological code is a mapping between two sets of
molecules:
– Transcription net: Proteins → DNA binding sites
– Protein-protein recognition: immune system…
– Protein synthesis: DNA → Proteins
DNA Proteins
The genetic
code
Information flows from DNA to RNA to proteins through the
genetic code
• The 20 letters are the amino acids.
• Proteins are amino acid polymers.
DNA ACGGAGGTACCC 4 letters
RNA ACGGAGGUACCC 4 letters
Protein 20 lettersThr Glu Val Pro
Each of the 20 amino acidshas specific chemistry
• Amino acid = backbone + specific side group.
• Some amino acids are hydrophilic, hydrophobic, basic, acidic…
• The diversity of amino acids allows proteins to perform a wide variety of functions efficiently.
Each of the 20 amino acids isencoded by a triplet of RNA letters
• Genetic Code = mapping triplets to amino acids.
• 64 = 43 triplet codons encode only 20 amino acids
(degeneracy)
• Only 48 discernable codons due to U-C “wobble” at 3rd base.
Thr
Glu
Val
Pro
ACG
GUAGAG
CCC
The genetic code is smooth, degenerate and compact
• Redundancy – only 20 of 48.
• Degeneracy – mostly in the 3rd base
• Close codons separated by a single
letter (Hamming Distance = 1)
• Smoothness – Close codons encode
chemically similar amino acids.
( Hydrophobic xUx, hydrophilic
xAx).
• Compactness – single contiguous
domain per each amino-acid.
• The code is highly nonrandom
• (“one in a million” [Haig & Hurst] ). Shades: lighter (darker) – low (high) polarity.Letters: black (white) – hydrophobic (hydrophilic) yellow – medium. [Knight, Freeland, Landweber]
Biological codes evolve(d) to cope with inherent noise
• Messages are written in molecular words that are
read and interpreted by other molecules, which
calculate the response etc…
• Typical energy scale ~ a few kBT.
• Thermal noise → errors.
• Information channels adapt to errors through
evolutionary of selection-mutation
• Some errors = mutations are essential to evolution …
The code is an information channel with an average
distortion
,
i j
encoding misreading decoding
distortion
HUV = ∑paths Pαijβ Dαβ = ∑α,I,j,β PαUαiWijVjβDαβ
• U and V are binary matrices that determine the code
• W is the misreading (noise) stochastic matrix
U VW
Fitter code is one with less distortion
• The ‘error-load’ H measures the difference
between desired and the reproduced amino-acids.
• H is a natural measure for the fitness of the code.
• For better codes the encoding U and the decoding
V are optimized with respect to the reading W.
• The decoded amino-acids must be diverse enough
to map diverse chemical properties.
• However, to minimize the impact of errors it is
preferable to decode fewer amino-acids.
Theories on the origin of the code: Frozen accident or optimization?
Frozen accident hypothesis:
Any change in the code affects
all the proteins in the cell and
therefore will be too harmful:
Life began with very few amino-
acids. New amino-acids were
added until eventually the code
became frozen in its present
form.
[Crick 1968]
Load minimization
hypothesis:
Darwinian dynamics optimize
the code to minimize errors in
information flow
(due to mutations,
misreading).
[Sonneborn,
Zuckerkandl & Pauling…
1965]
Variant codes - evidence for ongoing optimization of the
code
• Variants of the “universal”
genetic code in many
organisms [Osawa, Jukes
1992].
• All variants use the same
twenty amino-acids
(universal invariant?)
• Continuity - Most changes
are to a neighboring amino-
acid.
(‘hydrodynamic’ flow ?)
GUG Val GCG Ala GAG Glu GGG Gly
GUA Val GCA Ala GAA Glu GGA Gly
G GUC Val GCC Ala GAC Asp GGC Gly
GUU Val GCU Ala GAU Asp GGU Gly
AUG Met ACG Thr AAG Lys AGG Arg
AUA Ile ACA Thr AAA Lys AGA Arg
A AUC Ile ACC Thr AAC Asn AGC Ser
AUU Ile ACU Thr AAU Asn AGU Ser
CUG Leu CCG Pro CAG Gln CGG Arg
CUA Leu CCA Pro CAA Gln CGA Arg
C CUC Leu CCC Pro CAC His CGC Arg
CUU Leu CCU Pro CAU His CGU Arg
UUG Leu UCG Ser UAG TER UGG Trp
UUA Leu UCA Ser UAA TER UGA TER
U UUC Phe UCC Ser UAC Tyr UGC Cys
UUU Phe UCU Ser UAU Tyr UGU Cys
U C A G
o Biological codes are information
channels or maps with natural
‘fitness’ measure.
o Codes are evolved and selected
according to their fitness.
o The emergence of a code is a phase
transition in an information channel.
o Topology of errors (noise) governs
the emergent code.
Codes compete by their error-load
• One letter change in DNA can change one amino
acid in one protein. If the new amino acid is
similar to the original the upset is minimal.
• The organism with the smallest error-load takes
over the population.
• - relatively small population
- high noise levels in protein synthesis
weak selection forces « random drift
Code’s evolution reaches steady-state
• Small effective population and strong drift.
• Population is in detailed balance and therefore
P(fitness) ~ exp(fitness/T) [Lassig,Sella & Hirsh]
• Smaller population is hotter: T ~ 1/Neff.
• The Boltzmannian probability PUV ~ exp(-HUV/T)
minimizes a ‘free energy’
F= <H>-TS = ∑HUV PUV + ∑ PUV logPUV
• F is used to optimize information channels …
2
, , ,
.ij ij i ji j
F T w d u u
At high T no code is chosen
• At high T (small populations) Boltzmann implies
that all codes are equally probable: <Uαi> = 1/NC
• The natural order parameter is uαi= <Uαi>-1/NC
• At high T the state is random ‘non-coding’ uαi=0
• Stability of F is determined by
• w – the preference of the reading w = W − 1/NC
d – normalized chemical distance matrix
δF ~ ut(TIδ×Iw –
w2×d)u
o Biological codes are information
channels or maps with natural
‘fitness’ measure.
o Codes are evolved and selected
according to their fitness.
o The emergence of a code is a phase
transition in an information channel.
o Topology of errors (noise) governs
the emergent code.
Code emerges at a phase transition
• When T is decreased below Tc an inhomogeneous coding
state appears
δF ~ ut(TIδ×Iw – w2×d)u
• Critical temperature Tc = λw2 × λd
• The code is the mode uαi of F that corresponds to these
maximal eigenvalues.
• Tc increases with the accuracy of reading w .
• The phase transition is continuous (2nd order).
• Analogous phase transition in information channels
Why twenty amino-acids?
• Code is the mode uαi that minimizes the free energy.
• This mode corresponds to the maximal w - eigenvalue.
• Knowledge of w at the phase transition yields code.
• What can we say without such knowledge? (Why 20?)
• More amino-acids more sensitivity to errors.
• Fewer amino-acids reduce functionality of proteins.
• Historical mechanisms : Freezing, Biosynthetic etc..
• Twenty as a topological feature of generic
evolutionary phase transition?
o Biological codes are information
channels or maps with natural
‘fitness’ measure.
o Codes are evolved and selected
according to their fitness.
o The emergence of a code is a phase
transition in an information channel.
o Topology of errors (noise) governs
the emergent code.
AAA
ACA
AAU
CAA
AAGAGA
AUA
GAA
UAA
UUA CCA
CAG
GAU
GUA
UGA
AGG
ACU
The probable errors define the graph and the topology of the
genetic code• Graph = codon vertices +
one-letter difference edges ( Hamming =
1 ) U
A
G
C
U
A
G
C
UC
A G
X XK4 X K4 X
K3
UU
UC
UA
CU
CA
CC
AC
AA
AU
Topology and genus of a simpler code
UU AU CU
UA AA CA
UC AC CC
V = vertices, E = edges, F = faces
Euler’s characteristic χ = V – E + F
Euler Genus (# holes) γ = 1 - (1/2) χ
Doublet Code with 3 bases is imbedded on a torusEach codon has 4 neighbors
Faces are quadrilateral mutation cycles F=V (d/4)= 9 ; E=V (d/2)=18
A C
U
A C
U
X
The genetic code graph is holey
• The 48-codon graph
:
– Each codon has degree d =
3+3+2 = 8 therefore
• E = 48 (d/2) = 192 edges
• F = 48 (d/4) = 96 faces
• The Euler characteristic is
χ = V – E + F = -48 and
– Euler’s genus is γ = 1 -
(1/2) χ = 25 (24 holes +
Klein)
– Embedding by group
Automorphism analysis
• Can one hear the shape of
The code?
K4 X K4 X K3
K
The genetic code has a spectrum
• uαi is average preference of codon i to encode α.
• Every mode corresponds to an amino-acid
-> number of modes = number of amino-acids.
• Misreading w is actually the graph Laplacian
w = -(Δ-Δrandom) where Δij=-Wij Δii=Σj≠iWij
• Δ measures the difference between codons and their
neighbors, a natural measure for error load.
• Maximal mode of w is the 2nd eigenmode of Δ
• Courant’s theorem: uαi have a single maximum
-> single contiguous domain for each amino-acid.
• uαi have single compact domains with one
maximum and one minimum (Courant’s
theorem).
• Compact organization reduces impact of errors
• Single domain in any direction (linearity) Σnαuαi
Embedding in RN-1 is tight
→ The code graph contains complete graph KN
[Banchoff 1965, Colin de Verdiére’s 1987]
amino-acids # = N = chr(γ)
Topology optimizes amino-acid assignment is in compact
domains
Coloring number of graph code is an upper limit for the number of amino-
acids• What is the minimal number of colors required in a
map so that no two adjacent regions have the
same color?
• The coloring number is a topological invariant and
therefore a function of the genus solely.
• Heawood’s conjecture [Ringel & Youngs, Appel &
Haken]
48172
1)(chr 4 7 8 9 10 11 12 12 13 13
14 15 15 16 16 16 17 17 18 18
19 19 19 20 20 20 21 21 21 22
22 22 23 23 23 24 24 24 24 25
25 25 25 26 26 26 27 27 27 27
( ) max( )Nchr K
( )codeN chr
The genetic code coevolves with increasing accuracy of translation
• A path for evolution of codes:
from early codes with higher
codon degeneracy and fewer
amino acids to lower degeneracy
codes with more amino acids.
• Preliminary simulations
• Twenty amino acids is invariant
even in variant codes. 21st and
22nd amino acids are context
dependent.
1st 2nd 3rd chr #
1 4 1 0 4
2 4 1 1 6
4 4 1 5 11
4 4 2 13 16
4 4 3 25 20
4 4 4 41 25
K4 X K4
Summary
• The 64 3-letter triplet code is patterned and degenerate,
maps only 20 amino acids.
• The governing evolutionary dynamics is interplay between
protein diversity and error penalty described by stochastic
diffusion equation.
• The 1st excited state of this diffusive mapping dynamics on
the high-genus surface of the code yield a pattern of ordered
20 amino acids (20 = the coloring number of the graph).
• Topology + dynamics Coloring (?)
Transcription network is a code that relates DNA sites and binding
proteins• Reading DNA to synthesize proteins is controlled by a
system of protein-DNA interactions (transcription net).
• Presence/absence of transcription factor may
repress/enhance synthesis of protein from nearby
gene.
• The transcription network is actually a code that
relates proteins with their DNA targets.
• Like the genetic code, transcription is subject
to evolutionary forces and
adapts to minimize errors.
PolTF
DNA
Probable recognition errors define the binding sequence
space
sphere packing (Shannon) Overlap and continuity
• Typical binding site: 4 base pairs = 12
bit
•Hamming = 1 K46 -> 4096 ‘codons’
TF AA
Codon binding site
Probable recognition errors define the binding sequence
space
• Coloring number
estimate:
v = 4L (L=6)
e ~ 4L(3/2)L
f ~ 4L(3/4)L
-> γ ~ 4L(3/8)L
• The coloring #
chr(γ) ~ 300103 104
100
101
102
103
104
number of genes
n-domain C2H2
winged helix
????
• Why does the code exhaust the coloring limit?• Other population dynamics models (‘quasi-species’)• Glassy 'almost-frozen' dynamics? • The necessity of the wobble (64/48)? 25 acids?
• Generic phase transition scenario that does not depend finely on missing details of the evolutionary pathway.
• Although not much is known about the primordial environment, minimal assumptions about the topology of probable errors can yield characteristics of biological codes.
• Esp. the number of twenty amino-acids in the present picture is reminiscent of a 'shell magic number‘.
Shalev Itzkovitz
Guy Shinar
Uri Alon
Guy Sella
J. –P. Eckmann
Elisha Moses