  Intracellular protein patterns are described by (nearly) mass-conserving
reaction-diffusion systems. While these patterns initially form out of a
homogeneous steady state due to the well-understood Turing instability, no
general theory exists for the dynamics of fully nonlinear patterns. We develop
a unifying theory for wavelength-selection dynamics in (nearly) mass-conserving
two-component reaction-diffusion systems independent of the specific
mathematical model chosen. This encompasses both the dynamics of the mesa- and
peak-shaped patterns found in these systems. Our analysis uncovers a diffusion-
and a reaction-limited regime of the dynamics, which provides a systematic link
between the dynamics of mass-conserving reaction-diffusion systems and the
Cahn-Hilliard as well as conserved Allen-Cahn equations, respectively. A
stability threshold in the family of stationary patterns with different
wavelengths predicts the wavelength selected for the final stationary pattern.
At short wavelengths, self-amplifying mass transport between single pattern
domains drives coarsening while at large wavelengths weak source terms that
break strict mass conservation lead to an arrest of the coarsening process. The
rate of mass competition between pattern domains is calculated analytically
using singular perturbation theory, and rationalized in terms of the underlying
physical processes. The resulting closed-form analytical expressions enable us
to quantitatively predict the coarsening dynamics and the final pattern
wavelength. We find excellent agreement of these expressions with numerical
results. The systematic understanding of the length-scale dynamics of fully
nonlinear patterns in two-component systems provided here builds the basis to
reveal the mechanisms underlying wavelength selection in multi-component
systems with potentially several conservation laws.
