Thermodynamic Uncertainty Relations in Stochastic Biochemical Networks

Abstract

The thermodynamic uncertainty relation (TUR) states that any current in a nonequilibrium steady state cannot be
made more precise without paying a corresponding entropic cost, but whether this bound is a tight, practically useful
constraint for real biochemical networks, as opposed to a loose asymptotic inequality, remains insufficiently
characterized. We evaluated the TUR bound numerically across 34 stochastic biochemical network topologies,
including linear and cyclic enzyme kinetics, kinetic proofreading cascades, and coupled receptor-ligand binding
schemes, simulated with the Gillespie algorithm over 2 x 10^4 trajectories per topology and reaction rates spanning
three orders of magnitude. Across the full parameter sweep of 1,220 individual rate combinations, the ratio Q of the
TUR-predicted minimum entropy production to the true simulated entropy production averaged 0.61 (SD 0.19),
indicating that the bound underestimates the actual dissipation by roughly a factor of 1.6 on average, with the
tightest agreement (Q = 0.86) found in simple unicyclic networks operating near equilibrium and the loosest (Q =
0.24) in multi-cycle kinetic proofreading schemes far from equilibrium. The bound's tightness correlated strongly
with network cyclomatic complexity (Spearman rho = -0.74, p < 0.001) and only weakly with the mean reaction flux
itself (rho = 0.18, p = 0.09), identifying topological branching rather than flux magnitude as the primary determinant
of TUR slack. In kinetic proofreading cascades specifically, designed to trade dissipation for discrimination fidelity,
the TUR-implied minimum cost of achieving a given error-correction fidelity underestimated the simulated cost by
58-71% across the four proofreading depths tested, an underestimate that grew with the number of proofreading
steps. A refined multidimensional TUR bound incorporating cross-correlations between multiple observable currents
recovered substantially more of the true entropy production, raising the average Q to 0.83 (SD 0.11) across the
same 1,220 parameter combinations, cutting the mean underestimation gap by more than half relative to the
single-current bound. These findings indicate that the standard scalar TUR, while qualitatively informative,
systematically underestimates dissipation in networks with substantial cyclic or branched topology, and that
multidimensional formulations incorporating multiple simultaneously observable currents are needed to extract
quantitatively reliable entropy-production estimates from single-molecule or single-cell trajectory data in biochemical
systems of realistic complexity.