Universal bounds on entropy production inferred from observed statistics

Nonequilibrium processes break time-reversal symmetry and generate entropy. Living systems are driven out-of-equilibrium at the microscopic level of molecular motors that exploit chemical potential gradients to transduce free energy to mechanical work, while dissipating energy. The amount of energy dissipation, or the entropy production rate (EPR), sets thermodynamic constraints on cellular processes. Practically, calculating the total EPR in experimental systems is challenging due to the limited spatiotemporal resolution and the lack of complete information on every degree of freedom. Here, we propose an inference approach for a tight lower bound on the total EPR given partial information, based on an optimization scheme that uses the observed transitions and waiting times statistics. We introduce hierarchical bounds relying on the first- and second-order transitions, and the moments of the observed waiting time distributions, and apply our approach to two generic systems of a hidden network and a molecular motor, with lumped states. Finally, we show that a lower bound on the total EPR can be obtained even when assuming a simpler network topology of the full system.

Figure 1

Coarse graining. (a) The full Markovian system (left) and the coarse-grained system (right). (b) An example of a full trajectory (left) containing the actual states and the corresponding coarse-grained trajectory (right) containing only the observed states.

Figure 2

Four-state system. (a) Illustration of the full four-state system topology, including the coarse graining of states 3 and 4 to state $H$. (b) Total EPR ${\sigma }_{\text{tot}}$ (solid black line), our bound ${\sigma }_{\text{opt}}^{\left(1\right)}$ (brown cross), KLD estimator ${\sigma }_{\text{KLD}}$ (dotted blue line), affinity estimator ${\sigma }_{\text{aff}}$ (dashed green line), two-step estimator ${\sigma }_2$ (yellow Asterisk), and the passive partial entropy production ${\sigma }_{\text{pp}}$ (dashed-dotted orange line). The rates used are $w_{12}=3{\text{s}}^{-1}, w_{13}=0{\text{s}}^{-1}, w_{14}=8{\text{s}}^{-1}, w_{21}=2{\text{s}}^{-1}, w_{23}=50{\text{s}}^{-1}, w_{24}=0.2{\text{s}}^{-1}, w_{31}=0{\text{s}}^{-1}, w_{32}=2{\text{s}}^{-1}, w_{34}=75{\text{s}}^{-1}, w_{41}=1{\text{s}}^{-1}, w_{42}=35{\text{s}}^{-1}, w_{43}=0.7{\text{s}}^{-1}$.

Figure 3

Molecular motor. (a) Illustration of the full molecular motor system, including the coarse graining of the active (red boxed square) and passive (ellipse) states. (b) Total EPR ${\sigma }_{\text{tot}}$ (solid black line), our bound ${\sigma }_{\text{opt}}^{\left(1\right)}$ (brown cross), KLD estimator ${\sigma }_{\text{KLD}}$ (dotted blue line), the affinity estimator ${\sigma }_{\text{aff}}$ (dashed green line), and the two-step estimator ${\sigma }_2$ (yellow asterisk). The rates used are $w_r=w_l=w_{u2}=w_{d2}=1{\text{s}}^{-1}, w_{u1}=w_{d1}=0.01{\text{s}}^{-1}$.

Figure 4

Importance of data accuracy. (a) The error of some statistics of the four-state system for different values of the trajectory length $N$. The absolute and relative errors are on the left and right axes, respectively. (b) The error of some statistics of the molecular motor system for different values of the trajectory length $N$. The absolute and relative errors are on the left and right axes, respectively. (c) The error of ${\sigma }_{\text{opt}}^{\left(1\right)}$ results for the four-state system for different constraint tolerance values, using the analytical statistics values. (d) The error of ${\sigma }_{\text{opt}}^{\left(1\right)}$ results for the molecular motor system for different constraint tolerance values, using the analytical statistics values. Error bars stand for the standard deviation of 10 different realizations.

Figure 5

Optimizing using a simple model. (a) Illustration of solving the optimization problem for a simple model with two hidden states (right), whereas the real system has more hidden states (left). (b) The results of ${\sigma }_{\text{opt}}^{\left(1\right)}$ assuming the simple four-state model (two hidden states), when the real system has two (red cross), three (green triangle), or four (blue circle) hidden states. (c) The results of ${\sigma }_{\text{opt}}^{\left(1\right)}$ assuming the simple molecular motor model (two hidden states), when the real system has two (red cross), three (green triangles), or four (blue circle) hidden states. For both systems, the results are presented for randomly generated transition rates (for each case) with statistics calculated from trajectories of length $N={10}^8$ using a constraint tolerance of ${10}^{-5}$.

Figure 6

Five-state system with four observed states. (a) Illustration of the full system topology, including the coarse graining of states 4 and 5 to state $H$. (b) Total EPR ${\sigma }_{\text{tot}}$ (black line), our bound ${\sigma }_{\text{opt}}^{\left(1\right)}$ (red cross), KLD estimator ${\sigma }_{\text{KLD}}$ (blue downward-pointing triangle), affinity estimator ${\sigma }_{\text{aff}}$ (green upward-pointing triangle), and two-step estimator ${\sigma }_2$ (yellow asterisk). The rates used are $w_{12}=11{\text{s}}^{-1}, w_{13}=0{\text{s}}^{-1}, w_{14}=71{\text{s}}^{-1}, w_{15}=81{\text{s}}^{-1}, w_{21}=31{\text{s}}^{-1}, w_{23}=0{\text{s}}^{-1}, w_{24}=12{\text{s}}^{-1}, w_{21}=96{\text{s}}^{-1}, w_{31}=0{\text{s}}^{-1}, w_{32}=0{\text{s}}^{-1}, w_{34}=92{\text{s}}^{-1}, w_{35}=12{\text{s}}^{-1}, w_{41}=69{\text{s}}^{-1}, w_{42}=15{\text{s}}^{-1}, w_{43}=14{\text{s}}^{-1}, w_{45}=91{\text{s}}^{-1}, w_{51}=100{\text{s}}^{-1}, w_{52}=71{\text{s}}^{-1}, w_{53}=29{\text{s}}^{-1}, w_{54}=30{\text{s}}^{-1}$.

Figure 7

Comparison to other bounds when assuming a simple topology. (a) Illustration of the full five-state system topology, including the coarse graining of states 3, 4, and 5 to state $H$ (left) and the full-system topology we assume (right). (b) Total EPR ${\sigma }_{\text{tot}}$ (solid black line), our bound ${\sigma }_{\text{opt}}^{\left(1\right)}$ (red cross), KLD estimator ${\sigma }_{\text{KLD}}$ (green upward-pointing triangle), and two-step estimator ${\sigma }_2$ (blue circle). The results are presented for randomly generated transition rates with statistics calculated from trajectories of length $N={10}^8$. Values of the estimators with the same ${\sigma }_{\text{tot}}$ correspond to the same system, showing that ${\sigma }_{\text{opt}}^{\left(1\right)}$ outperforms both ${\sigma }_{\text{KLD}}$ and ${\sigma }_2$.