(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