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Institute for
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Julian Martin Fernandez
Modeling of Biological
Evolution
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For my doctoral thesis, I studied the
problem of biological evolution. I proposed a model (Fernandez, Plastino & Diambra, 1995) involving genotype-phenotype interactions that
explained punctuated equilibrium and power law behavior as found in the
fossil record based on the internal dynamics of the ecological system, as
opposed to invoking recurrent external perturbations (like catastrophic
events). Details are given below. It is conjectured that biological
evolution takes place in terms of intermittent bursts of activity separating
relatively long periods of quiescence, rather than in a gradual manner. This
behavior was termed “punctuated equilibrium”. The intermittent
pattern has been observed in the fossil records. It was also found that the
distributions of both extinctions’ size and species’ lifetime
follow a power law. There are two main lines of thinking for
explaining this behavior. The first suggests that extinction events are
caused by external forces, such as changing sea levels, worldwide climate
pulses, or meteorites. A second line suggests that the ecology has evolved to
a self-organized critical state, a “poised” state far out of
equilibrium with propagating avalanches of activity of all sizes. The first
line is not very attractive from a dynamical point of view, and mathematical
models following the second line were in general too simple to allow for
reasonable conjectures concerning terrestrial biology. The second line also
required some degree of external perturbation to the system, although not as
dramatic as in the first line. Thus, we proposed a more complex model
of biological evolution. The model introduced a mapping between genotype and
phenotype (a novel feature in modeling biological evolution). The degree to
which a given species was adapted to the ecosystem was represented by a
quantity called its “fitness” (F). The F for a given species was
computed using a mathematical expression involving the interaction between
the phenotypic features of this species with those of the other species. Also
interactions between the phenotypic features of the given species with the
external environment were considered. The nonlinear mapping between genotype
and phenotype was demonstrated to be an essential feature for generating the
model’s dynamics, as it provided some degree of correlation between
phenotypic features. Without them, a given species might (eventually) attain,
after a series of appropriate mutations, any phenotypic feature whatsoever
(this does not happen in nature). Genetic changes, mimicked by
modifications in the genotype, drove the evolutionary process. The system
evolved in the following fashion: we started with an arbitrary initial
configuration and, in each of a series of time steps, mutation effects were
mimicked by slightly modifying (randomly) the genome of one of the species
chosen at random. A particular mutation was "accepted" if it
increased the corresponding F. The change in the genome was retained in that
case. Otherwise it was discarded and the genes ended up with their previous,
old values. Extensive numerical and analytical
analysis of the model provided many interesting results and explanations.
Briefly, some of them are: 1) The model shows punctuated
equilibrium and power-law behavior as expected (Fernandez, Plastino & Diambra, 1995). This behavior had its origin in the internal
dynamics (in particular, in the differences of structural stability of the
different genotypes (Fernandez
& Plastino, 2000), with
no need of external perturbations of any size. Self-organized criticality was
not present in the system (Fernandez & Plastino,
1997). 2) The modeled ecology presented
oscillatory modes that introduced some periodicity in the system, similar to
those claimed by some studies based on the fossil records (Fernandez & Plastino, 1999). 3) The model explained the mathematical
reasons underlying the selectivity of extinction events (Fernandez, Vaveliuk, Pennini & Kowalski, 1998). This fact had not previously received numerical
support. 4) It is the first model giving
numerical support to Kimura's neutral theory of molecular evolution (Fernandez, Vaveliuk, Pennini & Kowalski, 1998). 5) The effect of the external
environment introduced deformations in the power laws, similar to those
observed in the fossil records (giving to the power laws a rather concave
form) (Fernandez,
Plastino and Diambra, 1999).
The model also generated a smooth integration (a non trivial fact) at all
scales of both inter-species interactions and ambient (external) influences,
giving rise to a coherent dynamical picture 'a la Maturana. 6) When the effect of symbiosis was
introduced, the power laws were also deformed in the right way. Symbiosis
effects also gave numerical support to earlier speculations concerning an
"adaptive grid lock" mechanism (Fernandez, Plastino and Diambra, 1999). In addition, we studied the
“evolution vs. coevolution” issue. Our simulations in two model
scenarios—our model and the NKC family of models—indicate that,
contrary to previous claims, coevolution
does not constitute the crucial dynamical factor that accelerates
evolution but rather that the ecosystem evolves notwithstanding the fact that
coevolution may actually
‘‘retard’’ things (Fernandez,
Plastino, Diambra, Mostaccio, 1998). In a coevolutive system, organisms can keep evolving forever since
the fitness peaks may disappear because of the variations of the other
species. Thus, a species can keep climbing (and in the process may become
more complex) without necessarily becoming more fit. Indeed, the mechanism
has been called the red queen effect, referring to the red queen and Alice
who kept running without getting anywhere. |