(Super)systems and selection dynamics Eörs Szathmáry & Mauro Santos Collegium BudapestEötvös University Budapest.
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(Super)systems and selection dynamics
Eörs Szathmáry & Mauro Santos
Collegium Budapest Eötvös University Budapest
Haldane’s intellectual son: John Maynard Smith (1920-2004)
Units of evolution
hereditary traits affecting survival and/or reproduction
1. multiplication
2. heredity
3. variation
Parabolic replicators: survival of everybody
Szathmáry & Gladkih (1987)
Even a Lyapunov function could be proven:Varga & Szathmáry (1996) Bull. Math. Biol.
Growth laws and selection consequences
Szathmáry (1989) Trend Ecol. Evol.
• Parabolic: p < 1 survival of everybody
• Exponential: p = 1 survival of the fittest
• Hyperbolic: p > 1 survival of the common
Why would one do such a model?
A crucial insight: Eigen’s paradox (1971)
• Early replication must have been error-prone
• Error threshold sets the limit of maximal genome size to <100 nucleotides
• Not enough for several genes• Unlinked genes will compete• Genome collapses• Resolution???
Simplified error threshold
x + y = 1
Molecular hypercycle (Eigen, 1971)
autocatalysis
heterocatalytic aid
Parasites in the hypercycle (JMS)
parasite
short circuit
The Lotka-Volterra equation
The replicator equation
Game dynamics
Permanence
“Hypercyles spring to life”…
• Cellular automaton simulation on a 2D surface
• Reaction-diffusion
• Emergence of mesoscopic structure
• Conducive to resistance against parasites
• Good-bye to the well-stirred flow reactor
Mineral surfaces are a poor man’s form of compartmentation (?)
• A passive form of localisation (limited diffusion in 2D)
• Thermodynamic effect (when leaving group also leaves the surface)
• Kinetic effects: surface catalysis (cf. enzymes)• How general and diverse are these effects?• Good for polymerisation, not good for metabolism
(Orgel)• What about catalysis by the inner surface of the
bilayer (composomes)?
Surface metabolism catalysed by replicators (Czárán & Szathmáry, 2000)
I1-I3: metabolic replicators(template and enzyme)
M: metabolism (not detailed)
P: parasite (only template)
M
I1
I2
I3
P
Elements of the model• A cellular automaton model
simulating replication and dispersal in 2D
• ALL genes must be present in a limited METABOLIC neighbourhood for replication to occur
• Replication needs a template next door
• Replication probability proportional to rate constant (allowing for replication)
• Diffusion
X
i - 2 i - 1 i i + 1 i + 2
j -2
j - 1
j
j + 1
j + 2
S
Robust conclusions• Protected polymorphism of competitive
replicators (cost of commonness and advantage of rarity)
• This does NOT depend on mesoscopic structures (such as spirals, etc.)
• Parasites cannot drive the system to extinction• Unless the neighbourhood is too large (approaches
a well-stirred system)• Parasites can evolve into metabolic replicators• System survives perturbation (e.g. when death
rates are different in adjacent cells), exactly because no mesocopic structure is needed.
An interesting twist
• This system survives with arbitrary diffusion rates
• But metabolic neighbourhood size must remain small
• Why does excessive dispersal not ruin the system?
• Because it convergences to a trait-group model!
The trait group model (Wilson, 1980)
Random dispersal
Harvest
Applied to early coexistence: Szathmáry (1992)
Mixed global pool
Mixed global pool
Why does the trait group work?• It works only for cases when the “red hair
theorem” applies
• People with red hair overestimate the frequency of people with red hair, essentially because they know this about themselves
• “average subjective frequency”
• In short, molecules must be able to scratch their own back!
Error rates and the origin of replicators
Nature 420, 360-363 (2002).
Replicase RNA
Other RNA
Increase in efficiency• Target efficiency:
the acceptance of help
• Replicase efficiency: how much help it gives
• Copying fidelity
• Trade-off among all three traits: worst case
The dynamics becomes interesting on the rocks!
Evolving population
• Molecules interact with their neighbours• Have limited diffusion on the surface
Error rate Replicase activity
The stochastic corrector model for compartmentalized genomes
Szathmáry, E. & Demeter L. (1987) Group selection of early replicators and the origin of life. J. theor Biol. 128, 463-486.
Grey, D., Hutson, V. & Szathmáry, E. (1995) A re-examination of the stochastic corrector model. Proc. R. Soc. Lond. B 262, 29-35.
The stochastic corrector model (1986, ’87, ’95, 2002)
metabolic gene
replicase
membrane
The mathematical model• Inside compartments, there are numbers rather
than concentrations• Stochastic kinetics was applied:• Master equations instead of rate equations: P’(n, t)
= ……. Probabilities• Coupling of two timescales: replicator dynamics
and compartment fission• A quasipecies at the compartment level appears• Characterized by gene composition rather than
sequence
Dynamics of the SC model• Independently reassorting genes (ribozymes
in compartments)• Selection for optimal gene composition
between compartments• Competition among genes within the same
compartment• Stochasticity in replication and fission
generates variation on which natural selection acts
• A stationary compartment population emerges
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