  To celebrate Hans Frauenfelder's achievements, we examine energy(-like)
"landscapes" for complex living systems. Energy landscapes summarize all
possible dynamics of some physical systems. Energy(-like) landscapes can
explain some biomolecular processes, including gene expression and, as
Frauenfelder showed, protein folding. But energy-like landscapes and existing
frameworks like statistical mechanics seem impractical for describing many
living systems. Difficulties stem from living systems being high dimensional,
nonlinear, and governed by many, tightly coupled constituents that are noisy.
The predominant modeling approach is devising differential equations that are
tailored to each living system. This ad hoc approach faces the notorious
"parameter problem": models have numerous nonlinear, mathematical functions
with unknown parameter values, even for describing just a few intracellular
processes. One cannot measure many intracellular parameters or can only measure
them as snapshots in time. Another modeling approach uses cellular automata to
represent living systems as discrete dynamical systems with binary variables.
Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g.,
Cellular Potts Model). But numerous biological features, in current practice,
are qualitatively described rather than quantitatively (e.g., gene is (highly)
expressed or not (highly) expressed). Cellular automata governed by verbal
rules are useful representations for living systems and can mitigate the
parameter problem. However, they can yield complex dynamics that are difficult
to understand because much of the existing mathematical tools and theorems
apply to continuous but not discrete dynamical systems. Recent studies found
ways to overcome this challenge by discovering a predictive "landscape" that
yield low-dimensional representations of cellular automata dynamics. We review
these studies.
