  While modern physics and biology satisfactorily explain the passage from the
Big Bang to the formation of Earth and the first cells to present-day life,
respectively, the origins of biochemical life still remain an open question.
Since life, as we know it, requires extremely long genetic polymers, any answer
to the question must explain how an evolving system of polymers of
ever-increasing length could come about on a planet that otherwise consisted
only of small molecular building blocks. In this work, we show that, under
realistic constraints, an abstract polymer model can exhibit dynamics such that
attractors in the polymer population space with a higher average polymer length
are also more probable. We generalize from the model and formalize the notions
of complexity and evolution for chemical reaction networks with multiple
attractors. The complexity of a species is defined as the minimum number of
reactions needed to produce it from a set of building blocks, which in turn is
used to define a measure of complexity for an attractor. A transition between
attractors is considered to be a progressive evolution if the attractor with
the higher probability also has a higher complexity. In an environment where
only monomers are readily available, the attractor with a higher average
polymer length is more complex. Thus, our abstract polymer model can exhibit
progressive evolution for a range of thermodynamically plausible rate
constants. We also formalize criteria for open-ended and
historically-contingent evolution and explain the role of autocatalysis in
obtaining them. Our work provides a basis for searching for prebiotically
plausible scenarios in which long polymers can emerge and yield populations
with even longer polymers.
