Andre Cardoso Barato

Thermodynamic uncertainty relation for biomolecular processes.

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions, can be described as Markov processes on a suitable network. We show quite generally that, in a steady state, the dispersion of observables, like the number of consumed or produced molecules or the number of steps of a motor, is constrained by the thermodynamic cost of generating it. An uncertainty ε requires at least a cost of 2k(B)T/ε2 independent of the time required to generate the output.

Unifying three perspectives on information processing in stochastic thermodynamics.

So far, feedback-driven systems have been discussed using (i) measurement and control, (ii) a tape interacting with a system, or (iii) by identifying an implicit Maxwell demon in steady-state transport. We derive the corresponding second laws from one master fluctuation theorem and discuss their relationship. In particular, we show that both the entropy production involving mutual information between system and controller and the one involving a Shannon entropy difference of an information reservoir like a tape carry an extra term different from the usual current times affinity. We, thus, generalize stochastic thermodynamics to the presence of an information reservoir.

Efficiency of cellular information processing

We show that a rate of conditional Shannon entropy reduction, characterizing the learning of an internal process about an external process, is bounded by the thermodynamic entropy production. This approach allows for the definition of an informational efficiency that can be used to study cellular information processing. We analyze three models of increasing complexity inspired by the E. coli sensory network, where the external process is an external ligand concentration jumping between two values. We start with a simple model for which ATP must be consumed so that a protein inside the cell can learn about the external concentration. With a second model for a single receptor we show that the rate at which the receptor learns about the external environment can be nonzero even without any dissipation inside the cell since chemical work done by the external process compensates for this learning rate. The third model is more complete, also containing adaptation. For this model we show inter alia that a bacterium in an environment that changes at a very slow time-scale is quite inefficient, dissipating much more than it learns. Using the concept of a coarse-grained learning rate, we show for the model with adaptation that while the activity learns about the external signal the option of changing the methylation level increases the concentration range for which the learning rate is substantial.

Stochastic thermodynamics of bipartite systems: transfer entropy inequalities and a Maxwell’s demon interpretation

We consider the stationary state of a Markov process on a bipartite system from the perspective of stochastic thermodynamics. One subsystem is used to extract work from a heat bath while being affected by the second subsystem. We show that the latter allows for a transparent and thermodynamically consistent interpretation of a Maxwells demon. Moreover, we obtain an integral fluctuation theorem involving the transfer entropy from one subsystem to the other. Comparing three different inequalities, we show that the entropy decrease of the first subsystem provides a tighter bound on the rate of extracted work than either the rate of transfer entropy from this subsystem to the demon or the heat dissipated through the dynamics of the demon. The latter two rates cannot be ordered by an inequality, as shown with the illustrative example of a four state system.

An autonomous and reversible Maxwell's demon

Building on a model introduced by Mandal and Jarzynski (Proc. Natl. Acad. Sci. U.S.A., 109 (2012) 11641), we present a simple version of an autonomous reversible Maxwells demon. By changing the entropy of a tape consisting of a sequence of bits passing through the demon, the demon can lift a mass using the coupling to a heat bath. Our model becomes reversible by allowing the tape to move in both directions. In this thermodynamically consistent model, total entropy production consists of three terms one of which recovers the irreversible limit studied by MJ. Our demon allows an interpretation in terms of an enzyme transporting and transforming molecules between compartments. Moreover, both genuine equilibrium and a linear response regime with corresponding Onsager coefficients are well defined. Efficiency and efficiency at maximum power are calculated. In linear response, the latter is shown to be bounded by 1/2, if the demon operates as a machine and by 1/3, if it is operated as an eraser.

Universal bounds on current fluctuations.

For current fluctuations in nonequilibrium steady states of Markovian processes, we derive four different universal bounds valid beyond the Gaussian regime. Different variants of these bounds apply to either the entropy change or any individual current, e.g., the rate of substrate consumption in a chemical reaction or the electron current in an electronic device. The bounds vary with respect to their degree of universality and tightness. A universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production. A second, stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states. These two bounds are conjectures based on extensive numerics. An exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved. This bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium. An asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived. This bound allows for the prediction of the asymptotic growth of the generating function. Even though our results are restricted to networks with a finite number of states, we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle. Our bounds provide a general class of constraints for nonequilibrium systems.

Stochastic thermodynamics with information reservoirs.

We generalize stochastic thermodynamics to include information reservoirs. Such information reservoirs, which can be modeled as a sequence of bits, modify the second law. For example, work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir. We obtain an inequality, and the corresponding fluctuation theorem, generalizing the standard entropy production of stochastic thermodynamics. From this inequality we can derive an information processing entropy production, which gives the second law in the presence of information reservoirs. We also develop a systematic linear response theory for information processing machines. For a unicyclic machine powered by an information reservoir, the efficiency at maximum power can deviate from the standard value of 1/2. For the case where energy is consumed to erase the tape, the efficiency at maximum erasure rate is found to be 1/2.

Information-theoretic versus thermodynamic entropy production in autonomous sensory networks.

For sensory networks, we determine the rate with which they acquire information about the changing external conditions. Comparing this rate with the thermodynamic entropy production that quantifies the cost of maintaining the network, we find that there is no universal bound restricting the rate of obtaining information to be less than this thermodynamic cost. These results are obtained within a general bipartite model consisting of a stochastically changing environment that affects the instantaneous transition rates within the system. Moreover, they are illustrated with a simple four-states model motivated by cellular sensing. On the technical level, we obtain an upper bound on the rate of mutual information analytically and calculate this rate with a numerical method that estimates the entropy of a time series generated with a simulation.

Sensory capacity: An information theoretical measure of the performance of a sensor

For a general sensory system following an external stochastic signal, we introduce the sensory capacity. This quantity characterizes the performance of a sensor: sensory capacity is maximal if the instantaneous state of the sensor has as much information about a signal as the whole time series of the sensor. We show that adding a memory to the sensor increases the sensory capacity. This increase quantifies the improvement of the sensor with the addition of the memory. Our results are obtained with the framework of stochastic thermodynamics of bipartite systems, which allows for the definition of an efficiency that relates the rate with which the sensor learns about the signal with the energy dissipated by the sensor, which is given by the thermodynamic entropy production. We demonstrate a general trade-off between sensory capacity and efficiency: if the sensory capacity is equal to its maximum 1, then the efficiency must be less than 1/2. As a physical realization of a sensor we consider a two-component cellular network estimating a fluctuating external ligand concentration as signal. This model leads to coupled linear Langevin equations that allow us to obtain explicit analytical results.

Nonequilibrium sensing and its analogy to kinetic proofreading

For a paradigmatic model of chemotaxis, we analyze the effect of how a nonzero affinity driving receptors out of equilibrium affects sensitivity. This affinity arises whenever changes in receptor activity involve adenosine triphosphate hydrolysis. The sensitivity integrated over a ligand concentration range is shown to be enhanced by the affinity, providing a measure of how much energy consumption improves sensing. With this integrated sensitivity we can establish an intriguing analogy between sensing with nonequilibrium receptors and kinetic proofreading: the increase in integrated sensitivity is equivalent to the decrease of the error in kinetic proofreading. The influence of the occupancy of the receptor on the phosphorylation and dephosphorylation reaction rates is shown to be crucial for the relation between integrated sensitivity and affinity. This influence can even lead to a regime where a nonzero affinity decreases the integrated sensitivity, which corresponds to anti-proofreading.