  Accurate and robust spatial orders are ubiquitous in living systems. In 1952,
Alan Turing proposed an elegant mechanism for pattern formation based on
spontaneous breaking of the spatial translational symmetry in the underlying
reaction-diffusion system. Much is understood about dynamics and structure of
Turing patterns. However, little is known about the energetic cost of Turing
pattern. Here, we study nonequilibrium thermodynamics of a small spatially
extended biochemical reaction-diffusion system by using analytical and
numerical methods. We find that the onset of Turing pattern requires a minimum
energy dissipation to drive the nonequilibrium chemical reactions. Above onset,
only a small fraction of the total energy expenditure is used to overcome
diffusion for maintaining the spatial pattern. We show that the positioning
error decreases as energy dissipation increases following the same tradeoff
relationship between timing error and energy cost in biochemical oscillatory
systems. In a finite system, we find that a specific Turing pattern exists only
within a finite range of total molecule number, and energy dissipation broadens
the range, which enhances the robustness of the Turing pattern against molecule
number fluctuations in living cells. These results are verified in a realistic
model of the Muk system underlying DNA segregation in E. coli, and testable
predictions are made for the dependence of the accuracy and robustness of the
spatial pattern on the ATP/ADP ratio. In general, the theoretical framework
developed here can be applied to study nonequilibrium thermodynamics of
spatially extended biochemical systems.
