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A Model of Nucleic Acid Elongation and Replication
using Stochastic Cellular Automata
Chrisantha Fernando1 2 3, Johan Elf 4, Jon Rowe3, Eörs Szathmáry2, 5, 6
1
Center for Computational Neuroscience and Robotics, Dept. of Informatics, University
of Sussex, Falmer, Brighton BN1 9RH, UK
2
Dept of Plant Taxonomy and Ecology, Eötvös University, and Collegium Budapest
(Institute for Advanced Study), Szentháromság u. 2, H-1014 Budapest, Hungary
3
Systems Biology Centre & School of Computer Science, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK.
4
Dept. of Cell & Molecular Biology, Molecular Biology Programme, Biomedical
Center, Box 596, Husarg. 3, SE-751 24 Uppsala, Sweden
5
Dept of Plant Taxonomy and Ecology, Institute of Biology, Eötvös University
6
Parmenides Center for the Study of Thinking, Kardinal-Faulhaber-Str, 14a, D-80333
Munich, Germany
The origin of template replication is a major unsolved problem in science.
Although in vitro selection of ribozymes1 lends credence to the idea of an RNA
world2, we still have no replicase ribozyme3. Non-enzymatic synthesis of templates
up to 55 nucleotides has been achieved on mineral surfaces4, but there is no
replication30, because templates do not recycle by unzipping5. Short oligonucleotide
analogues can self-replicate6,7, but longer ones cannot because self-inhibition by
strand association becomes prohibitive9, and ideas based on oscillating monomer
concentrations26 or salinity27 failed in simulations28. Three main obstacles to long
template replication observed in simulation were: (i) competition by successfully
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replicating short templates; (ii) creation of new short templates by premature
detachment of incomplete copies from longer strands; and ultimately (iii)
elongation side-reactions occurring at staggered ends. The spontaneous formation
of libraries of diverse sequences was observed, in agreement with experiments29,
and is a pre-requisite for the natural selection of long catalytic sequences.
A model of nucleic acid oligomer dynamics was designed to elucidate the reactions
occurring in non-enzymatic systems consisting of activated dinucleotide building
blocks8. Figure 1 shows an outline of the algorithm. Each oligomer was represented on
its own grid as an arrangement of contiguous discrete (phosphoamidate) p-bonds and
(Watson-Crick base pairs) h-bonds. Cellular automata neighbourhood rules determined
the propensity for bond breakage and formation24,23. Using these propensities, reaction
events were executed stochastically using a variant of the Gillespie algorithm10,18,22. The
rules implemented nearest neighbour16,17 and more distant19,20 stacking dependent hbond breakage, catalysed and zipper h-bond formation11, and p-bond breakage and
formation. Polymers were selected for collision and possible association by h-bonds or
‘stacking bonds’ (s-bonds), assuming a well mixed reactor of known volume (see
supplementary material).
Major obstacles to circumvent in the simulations were: i) the vast number of distinct
configurations that nucleic acid polymers could adopt transiently, resulting in many
possible types of interaction, ii) the order-of magnitude difference in characteristic time
scales between h-bond and p-bond formation and breakage; and iii) the high
interdependency of reaction rates, due to stacking and inter-polymer reactions. These
problems were solved28 respectively by i) limiting the secondary structures to those not
including hairpins, and simulating only very small volumes containing few oligomers ii)
using a relaxation method to allow h-bond dynamics to run approximately to
equilibrium before each p-bond event, and iii) using an efficient stochastic algorithm.
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Realistic reaction kinetics and free energies were obtained from empirical
studies12,13,14,15.
Figure 1. The reactor was initialized with dinucleotides. In the first phase, the fast h-bond events were
simulated until steady state, whereupon N random microstates were sampled, i.e. a snapshot was taken of
the instantaneous state of templates in the reactor at regular intervals, revealing a variety of configurations
(some unlikely) in which p-bonds could be formed or broken. A p-bond event was chosen based on the
propensity of p-bond formation and breakage configurations observed in the microstates sampled. The
event was then percolated to and executed upon a specific bond in a microstate by hierarchical roulette
wheel selection. This process was iterated. See supplementary information for a full description of the
algorithm and its validation.
The behaviour of a reactor initialized with CG dimers is shown in figure 2, in
comparison to reactors initialized with TA, TG and CA dimers. The reaction
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mechanisms could be directly observed by visualization of individual oligomers.
Elongation occurred by two processes: spontaneous ligation between staking stabilized
oligomers; and templated ligation at the staggered ends of duplexes. Although
replication of short oligomers (< 6-mers) was observed, premature detachment of
oligomers from templates resulted in considerable partial copying (see supplementary
material). As dimers were consumed, a combinatorial explosion of elongation reactions
tapped the weakly coupled autocatalytic replication cycles resulting in an `elongation
catastrophe’ (see supplementary material).
Figure 2. Left: Elongation reactions occurring in a reactor initialized with 500 dinucleotides at
concentration 20mM at 30oC. CG elongates the fastest. At this temperature, template directed p-bond
formation is highly unlikely between dimers. Spontaneous ligation is permitted only between strands less
than 10 nucleotides in length, and accounts for the initial p-bond formation events. Right: Details of
further elongation in the CG case to 4-mers, 6-mers and 8-mers, longer strands not shown. Bottom: A
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sample of the configurations in which p-bonds formed in simulation. On the left are 3 bimolecular
complexes stabilized by s-bonds at which spontaneous p-bond formation occurs (s). On the right are 8
configurations that show template directed p-bond formation (a-h). Reactions c., d., f., and g. are
elongating. Reactions a., and b. make perfect complementary templates. Reactions e. and h. make partial
copies.
The rate of elongation was greatest with CG dimers because their high stability
provided many staggered-end duplexes for ligation reactions, see figure 3. The other
self-complementary dimer, TA, also elongated effectively due to its capacity to form
staggered-end duplexes, however the initial phase of elongation was critically
dependent upon spontaneous p-bond formation of 6-mers that were then capable of
significant duplex formation and subsequent templated ligation at 2oC. Bimolecular
mixtures of CG and, TG, or CA, produced a reduced rate of elongation since both TG
and CA are not self-complementary and so prevented extension at their staggered ends.
1:1:1:1 mixtures of dimers showed less elongation than CG or TA dimers alone, but
greater elongation than the 1:1 mixtures of CG + CA, and CG + TG. CG + TA showed
greater elongation than the other 1:1 cases since staggered ends formed by TA are selfcomplementary; in agreement with experimental results29. In all mixed reactors, the full
range of dimer types were observed in the elongated strands.
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Figure 3. Left: The oligomer length distribution after 3 days is shown for 1:1:1:1 (top), 1:1 mixtures of
CG + either CA, TG or TA, and CG alone. Total initial dimer concentration was initialized at 20mM in
all cases, at 2oC. CG alone shows greatest elongation. Elongation in the 1:1:1:1 case is greater than in the
1:1 cases CG + TG and CG + CA. However, elongation with CG + TA is also effective. Right: Examples
of oligomer complexes showing staggered ends. Note that CG + TG and CG + CA form ‘runs’ of dimers
on elongating strands since TG and CA dimers spend the initial phase as single strands, elongating by
spontaneous ligation, only later becoming incorporated into elongating strands consisting of CG runs.
Complexes also often consist of three or more coupled double strands.
These experiments suggest a hypothesis for the function of a minimal replicase
ribozyme. When ligation can occur by non-enzymatic means, elongating side-reactions
prevent replication. In such a reactor, the minimal replicase ribozyme must be capable
of self-cleavage and, consecutively, of the next round of replication.
A ‘self-cleaving
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Supplementary Information accompanies the paper on www.nature.com/nature
Acknowledgements: Acknowledgements: Guenter Von Kiedrowski, Karin Achilles, Simon McGregor,
Mons Ehrenburg. Cost D27 STSM grants for travel to Budapest, Uppsala, Crete (thanks to Denis
Neibecker). European 6th Framework Grant ESIGNET for the study of evolution in cell signalling
networks.
Correspondence and requests for materials should be addressed to Chrisantha Fernando. (e-mail:
C.T.Fernando@cs.bham.ac.uk).