Universal Bound on the Fano Factor in Enzyme Kinetics

Thermodynamic cost for precision of general counting observables

We analytically derive universal bounds that describe the tradeoff between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the waiting times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting-time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

Time-Resolved Statistics of Snippets as General Framework for Model-Free Entropy Estimators

Irreversibility is commonly quantified by entropy production. An external observer can estimate it through measuring an observable that is antisymmetric under time reversal like a current. We introduce a general framework that allows us to infer a lower bound on entropy production through measuring the time-resolved statistics of events with any symmetry under time reversal, in particular, time-symmetric instantaneous events. We emphasize Markovianity as a property of certain events rather than of the full system and introduce an operationally accessible criterion for this weakened Markov property. Conceptually, the approach is based on snippets as particular sections of trajectories between two Markovian events, for which a generalized detailed balance relation is discussed.

Upper bounding the average residence times in partially observed steady-state Markov jump processes

Several types of stochastic dynamics can be modeled as a continuous-time Markov jump process among a finite number of sites. Within such framework, we face the problem of getting an upper bound on the average residence time of the system in a given site β (i.e., the average lifetime of the site) if what we can observe is only the permanence of the system in an adjacent site α and the occurrence of the transitions α→β. Supposing to have a long time record of this partial monitoring of the network under steady-state conditions, we show that an upper bound on the average time spent in the unobserved site can indeed be given. The bound is formally proved, tested by means of simulations, and illustrated for a multicyclic enzymatic reaction scheme.

Theoretical Tools to Quantify Stochastic Fluctuations in Single-Molecule Catalysis by Enzymes and Nanoparticles

Single-molecule microscopic techniques allow the counting of successive turnover events and the study of the time-dependent fluctuations of the catalytic activities of individual enzymes and different sites on a single heterogeneous nanocatalyst. It is important to establish theoretical methods to obtain the statistical measurements of such stochastic fluctuations that provide insight into the catalytic mechanism. In this review, we discuss a few theoretical frameworks for evaluating the first passage time distribution functions using a self-consistent pathway approach and chemical master equations, to establish a connection with experimental observables. The measurable probability distribution functions and their moments depend on the molecular details of the reaction and provide a way to quantify the molecular mechanisms of the reaction process. The statistical measurements of these fluctuations should provide insight into the enzymatic mechanism.

Tight uncertainty relations for cycle currents

Several recent inequalities bound the precision of a current, i.e., a counter of the net number of transitions in a system, by a thermodynamic measure of dissipation. However, while currents may be defined locally, dissipation is a global property. Inspired by the fact that, ever since Carnot, cycles are the unit elements of thermodynamic processes, we prove similar bounds tailored to cycle currents, counting net cycle completions, in terms of their conjugate affinities. We show that these inequalities are stricter than previous ones, even far from equilibrium, and that they allow us to tighten those on transition currents. We illustrate our results with a simple model and discuss some technical and conceptual issues related to shifting attention from transition to cycle observables.

Cycle counts and affinities in stochastic models of nonequilibrium systems

For nonequilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any given cycle is described by universal formulas which depend on the cycle affinity, but are otherwise independent of system details. We discuss the similarities and differences of this result to fluctuation theorems, and generalize the result to families of cycles, relevant under coarse graining. Finally, we describe the application of large deviation theory to this cycle-counting problem.

Understanding the Optimal Cooperativity of Human Glucokinase: Kinetic Resonance in Nonequilibrium Conformational Fluctuations

The cooperativity of a monomeric enzyme arises from dynamic correlation instead of spatial correlation and is a consequence of nonequilibrium conformation fluctuations. We investigate the conformation-modulated kinetics of human glucokinase, a monomeric enzyme with important physiological functions, using a five-state kinetic model. We derive the non-Michealis-Menten (MM) correction term of the activity (i.e., turnover rate), predict its relationship to cooperativity, and reveal the violation of conformational detailed balance. Most importantly, we reproduce and explain the observed resonance effect in human glucokinase (i.e., maximal cooperativity when the conformational fluctuation rate is comparable to the catalytic rate). With the realistic parameters, our theoretical results are in quantitative agreement with the reported measurement by Miller and co-workers. The analysis can be extended to a general chemical network beyond the five-state model, suggesting the generality of kinetic cooperativity and resonance.

Modeling work-speed-accuracy trade-offs in a stochastic rotary machine

Molecular machines are stochastic systems that catalyze the energetic processes keeping living cells alive and structured. Inspired by the examples of F_{1}-ATP synthase and the bacterial flagellum, we present a minimal model of an externally driven stochastic rotary machine. We explore the trade-offs of work, driving speed, and driving accuracy when changing driving strength, speed, and the underlying system dynamics. We find an upper bound on accuracy and work for a particular speed. Our results favor slow driving when tasked with minimizing the work-accuracy ratio and maximizing the rate of successful cycles. Finally, in the parameter regime mapping to the dynamics of F_{1}-ATP synthase, we find a significant decay of driving accuracy at physiological rotation rates, raising questions about how ATP synthase achieves reasonable or even remarkable efficiency in vivo.

Quantum Thermodynamic Uncertainty Relation for Continuous Measurement

We use quantum estimation theory to derive a thermodynamic uncertainty relation in Markovian open quantum systems, which bounds the fluctuation of continuous measurements. The derived quantum thermodynamic uncertainty relation holds for arbitrary continuous measurements satisfying a scaling condition. We derive two relations; the first relation bounds the fluctuation by the dynamical activity and the second one does so by the entropy production. We apply our bounds to a two-level atom driven by a laser field and a three-level quantum thermal machine with jump and diffusion measurements. Our result shows that there exists a universal bound upon the fluctuations, regardless of continuous measurements.

Stochastic thermodynamics of chemical reactions coupled to finite reservoirs: A case study for the Brusselator

Biomolecular processes are typically modeled using chemical reaction networks coupled to infinitely large chemical reservoirs. A difference in chemical potential between these reservoirs can drive the system into a non-equilibrium steady-state (NESS). In reality, these processes take place in finite systems containing a finite number of molecules. In such systems, a NESS can be reached with the help of an externally driven pump for which we introduce a simple model. The crucial parameters are the pumping rate and the finite size of the chemical reservoir. We apply this model to a simple biochemical oscillator, the Brusselator, and quantify the performance using the number of coherent oscillations. As a surprising result, we find that higher precision can be achieved with finite-size reservoirs even though the corresponding current fluctuations are larger than in the ideal infinite case.

Thermodynamic uncertainty relations and molecular-scale energy conversion

The thermodynamic uncertainty relation (TUR) is a universal constraint for nonequilibrium steady states that requires the entropy production rate to be greater than the relative magnitude of current fluctuations. It has potentially important implications for the thermodynamic efficiency of molecular-scale energy conversion in both biological and artificial systems. An alternative multidimensional thermodynamic uncertainty relation (MTUR) has also been proposed. In this paper we apply the TUR and the MTUR to a description of molecular-scale energy conversion that explicitly contains the degrees of freedom exchanging energy via a time-independent multidimensional periodic potential. The TUR and the MTUR are found to be universal lower bounds on the entropy generation rate and provide upper bounds on the thermodynamic efficiency. The TUR is found to provide only a weak bound while the MTUR provides a much tighter constraint by taking into account correlations between degrees of freedom. The MTUR is found to provide a tight bound in the near or far from equilibrium regimes but not in the intermediate force regime. Collectively, these results demonstrate that the MTUR is more appropriate than the TUR for energy conversion processes, but that both diverge from the actual entropy generation in certain regimes.

Inferring Entropy Production from Short Experiments

We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in nonequilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines.

Geometry and Flexibility of Optimal Catalysts in a Minimal Elastic Model

We have general knowledge of the principles by which catalysts accelerate the rate of chemical reactions but no precise understanding of the geometrical and physical constraints to which their design is subject. To analyze these constraints, we introduce a minimal model of catalysis based on elastic networks where the implications of the geometry and flexibility of a catalyst can be studied systematically. The model demonstrates the relevance and limitations of the principle of transition-state stabilization: optimal catalysts are found to have a geometry complementary to the transition state but a degree of flexibility that nontrivially depends on the parameters of the reaction as well as on external parameters such as the concentrations of reactants and products. The results illustrate how simple physical models can provide valuable insights into the design of catalysts.

Kinetic proofreading and the limits of thermodynamic uncertainty

To mitigate errors induced by the cell's heterogeneous noisy environment, its main information channels and production networks utilize the kinetic proofreading (KPR) mechanism. Here, we examine two extensively studied KPR circuits, DNA replication by the T7 DNA polymerase and translation by the E. coli ribosome. Using experimental data, we analyze the performance of these two vital systems in light of the fundamental bounds set by the recently discovered thermodynamic uncertainty relation (TUR), which places an inherent trade-off between the precision of a desirable output and the amount of energy dissipation required. We show that the DNA polymerase operates close to the TUR lower bound, while the ribosome operates ∼5 times farther from this bound. This difference originates from the enhanced binding discrimination of the polymerase which allows it to operate effectively as a reduced reaction cycle prioritizing correct product formation. We show that approaching this limit also decouples the thermodynamic uncertainty factor from speed and error, thereby relaxing the accuracy-speed trade-off of the system. Altogether, our results show that operating near this reduced cycle limit not only minimizes thermodynamic uncertainty, but also results in global performance enhancement of KPR circuits.

Uncertainty relations for underdamped Langevin dynamics

A trade-off between the precision of an arbitrary current and the dissipation, known as the thermodynamic uncertainty relation, has been investigated for various Markovian systems. Here, we study the thermodynamic uncertainty relation for underdamped Langevin dynamics. By employing information inequalities, we prove that for such systems, the relative fluctuation of a current at a steady state is constrained by both the entropy production and the average dynamical activity. We find that unlike what is the case for overdamped dynamics, the dynamical activity plays an important role in the bound. We illustrate our results with two systems, a single-well potential system and a periodically driven Brownian particle model, and numerically verify the inequalities.

Fluctuation Theorem Uncertainty Relation

The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty relation was discovered, which states that the fluctuation of observables is lower bounded by the entropy production. In the present Letter, we derive a thermodynamic uncertainty relation from the fluctuation theorem. We refer to the obtained relation as the fluctuation theorem uncertainty relation, and it is valid for arbitrary dynamics, stochastic as well as deterministic, and for arbitrary antisymmetric observables for which a fluctuation theorem holds. We apply the fluctuation theorem uncertainty relation to an overdamped Langevin dynamics for an antisymmetric observable. We demonstrate that the antisymmetric observable satisfies the fluctuation theorem uncertainty relation but does not satisfy the relation reported for current-type observables in continuous-time Markov chains. Moreover, we show that the fluctuation theorem uncertainty relation can handle systems controlled by time-symmetric external protocols, in which the lower bound is given by the work exerted on the systems.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Uncertainty relations for time-delayed Langevin systems

The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback-Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmetric under time reversal, the fluctuation is bounded from below by a function of a quantity that can be identified as a generalization of the total entropy production in Markovian systems. We also provide a lower bound for arbitrary observables that are odd under position reversal. The term in this bound reflects the extent to which the position symmetry has been broken in the system and can be positive even in equilibrium. Our results hold for finite observation times and a large class of time-delayed systems because detailed underlying dynamics are not required for the derivation. We numerically verify the derived uncertainty relations using two single time-delay systems and one distributed time-delay system.

Uncertainty relations in stochastic processes: An information inequality approach

The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound. Inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities, we apply the latter to systems described by Langevin equations, and we derive the bound for the fluctuation of thermodynamic quantities. When applying the Cramér-Rao inequality, the obtained inequality reduces to the fluctuation-response inequality. We find that the thermodynamic uncertainty relation is a particular case of the Cramér-Rao inequality, in which the Fisher information is the total entropy production. Using the equality condition of the Cramér-Rao inequality, we find that the stochastic total entropy production is the only quantity that can attain equality in the thermodynamic uncertainty relation. Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations.

Unified Thermodynamic Uncertainty Relations in Linear Response

Thermodynamic uncertainty relations (TURs) are recently established relations between the relative uncertainty of time-integrated currents and entropy production in nonequilibrium systems. For small perturbations away from equilibrium, linear response (LR) theory provides the natural framework to study generic nonequilibrium processes. Here, we use LR to derive TURs in a straightforward and unified way. Our approach allows us to generalize TURs to systems without local time-reversal symmetry, including, e.g., ballistic transport and periodically driven classical and quantum systems. We find that, for broken time reversal, the bounds on the relative uncertainty are controlled both by dissipation and by a parameter encoding the asymmetry of the Onsager matrix. We illustrate our results with an example from mesoscopic physics. We also extend our approach beyond linear response: for Markovian dynamics, it reveals a connection between the TUR and current fluctuation theorems.

Insights into noise modulation in oligomerization systems of increasing complexity

Understanding under which conditions the increase of systems complexity is evolutionarily advantageous, and how this trend is related to the modulation of the intrinsic noise, are fascinating issues of utmost importance for synthetic and systems biology. To get insights into these matters, we analyzed a series of chemical reaction networks with different topologies and complexity, described by mass-action kinetics. We showed, analytically and numerically, that the global level of fluctuations at the steady state, measured by the sum over all species of the Fano factors of the number of molecules, is directly related to the network's deficiency. For zero-deficiency systems, this sum is constant and equal to the rank of the network. For higher deficiencies, additional terms appear in the Fano factor sum, which are proportional to the net reaction fluxes between the molecular complexes. We showed that the system's global intrinsic noise is reduced when all fluxes flow from lower to higher degree oligomers, or equivalently, towards the species of higher complexity, whereas it is amplified when the fluxes are directed towards lower complexity species.

Phase transition in thermodynamically consistent biochemical oscillators

Biochemical oscillations are ubiquitous in living organisms. In an autonomous system, not influenced by an external signal, they can only occur out of equilibrium. We show that they emerge through a generic nonequilibrium phase transition, with a characteristic qualitative behavior at criticality. The control parameter is the thermodynamic force which must be above a certain threshold for the onset of biochemical oscillations. This critical behavior is characterized by the thermodynamic flux associated with the thermodynamic force, its diffusion coefficient, and the stationary distribution of the oscillating chemical species. We discuss metrics for the precision of biochemical oscillations by comparing two observables, the Fano factor associated with the thermodynamic flux and the number of coherent oscillations. Since the Fano factor can be small even when there are no biochemical oscillations, we argue that the number of coherent oscillations is more appropriate to quantify the precision of biochemical oscillations. Our results are obtained with three thermodynamically consistent versions of known models: the Brusselator, the activator-inhibitor model, and a model for KaiC oscillations.

Quantifying fluctuations in reversible enzymatic cycles and clocks

Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but it also allows fluctuation measurements that may reveal the structure and dynamics of the underlying biochemical network. Here, we study nonequilibrium reaction cycles, such as the mechanochemical cycle of molecular motors, the phosphorylation cycle of circadian clock proteins, or the transition state cycle of enzymes. Fluctuations in such cycles may be measured using either of two classical definitions of the randomness parameter, which we show to be equivalent in general microscopically reversible cycles. We define a stochastic period for reversible cycles and present analytical solutions for its moments. Furthermore, we associate the two forms of the randomness parameter with the thermodynamic uncertainty relation, which sets limits on the timing precision of the cycle in terms of thermodynamic quantities. Our results should prove useful also for the study of temporal fluctuations in more general networks.

Current fluctuations in quantum absorption refrigerators

Absorption refrigerators transfer thermal energy from a cold bath to a hot bath without input power by utilizing heat from an additional "work" reservoir. Particularly interesting is a three-level design for a quantum absorption refrigerator, which can be optimized to reach the maximal (Carnot) cooling efficiency. Previous studies of three-level chillers focused on the behavior of the averaged cooling current. Here, we go beyond that and study the full counting statistics of heat exchange in a three-level chiller model. We explain how to obtain the complete cumulant generating function of the refrigerator in a steady state, then derive a partial cumulant generating function, which yields closed-form expressions for both the averaged cooling current and its noise. Our analytical results and simulations are beneficial for the design of nanoscale engines and cooling systems far from equilibrium, with their performance optimized according to different criteria, efficiency, power, fluctuations, and dissipation.

Single-molecule theory of enzymatic inhibition

The classical theory of enzymatic inhibition takes a deterministic, bulk based approach to quantitatively describe how inhibitors affect the progression of enzymatic reactions. Catalysis at the single-enzyme level is, however, inherently stochastic which could lead to strong deviations from classical predictions. To explore this, we take the single-enzyme perspective and rebuild the theory of enzymatic inhibition from the bottom up. We find that accounting for multi-conformational enzyme structure and intrinsic randomness should strongly change our view on the uncompetitive and mixed modes of inhibition. There, stochastic fluctuations at the single-enzyme level could make inhibitors act as activators; and we state—in terms of experimentally measurable quantities—a mathematical condition for the emergence of this surprising phenomenon. Our findings could explain why certain molecules that inhibit enzymatic activity when substrate concentrations are high, elicit a non-monotonic dose response when substrate concentrations are low.

Single molecule approaches demonstrated that enzymatic catalysis is stochastic which could lead to deviations from classical predictions. Here authors rebuild the theory of enzymatic inhibition to show that stochastic fluctuations on the single enzyme level could make inhibitors act as activators.

First-passage times in renewal and nonrenewal systems

Fluctuations in stochastic systems are usually characterized by full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e., the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach, it is shown that when the subsequent first-passage times are uncorrelated, there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.

Frenetic Bounds on the Entropy Production

We give a systematic derivation of positive lower bounds for the expected entropy production (EP) rate in classical statistical mechanical systems obeying a dynamical large deviation principle. The logic is the same for the return to thermodynamic equilibrium as it is for steady nonequilibria working under the condition of local detailed balance. We recover there recently studied "uncertainty" relations for the EP, appearing in studies about the effectiveness of mesoscopic machines. In general our refinement of the positivity of the expected EP rate is obtained in terms of a positive and even function of the expected current(s) which measures the dynamical activity in the system, a time-symmetric estimate of the changes in the system's configuration. Also underdamped diffusions can be included in the analysis.

Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation

In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. On the basis of a discrete kinetic network model, we use an integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its general validity. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics can be achieved in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics.

Proof of the finite-time thermodynamic uncertainty relation for steady-state currents

The thermodynamic uncertainty relation offers a universal energetic constraint on the relative magnitude of current fluctuations in nonequilibrium steady states. However, it has only been derived for long observation times. Here, we prove a recently conjectured finite-time thermodynamic uncertainty relation for steady-state current fluctuations. Our proof is based on a quadratic bound to the large deviation rate function for currents in the limit of a large ensemble of many copies.

Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials

Using Brownian motion in periodic potentials V(x) tilted by a force f, we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation q, and in order to limit the output fluctuation within a relative uncertainty ε, at least 2k_{B}T/ε^{2} of heat must be dissipated. Our model shows that this bound is attained not only at near-equilibrium [f≪V^{'}(x)] but also at far-from-equilibrium [f≫V^{'}(x)], more generally when the dissipated heat is normally distributed. Furthermore, the energetic cost is maximized near the critical force when the barrier separating the potential wells is about to vanish and the fluctuation of Brownian particles is maximized. These findings indicate that the deviation of heat distribution from Gaussianity gives rise to the inequality of the uncertainty relation, further clarifying the meaning of the uncertainty relation. Our derivation of the uncertainty relation also recognizes a bound of nonequilibrium fluctuations that the variance of dissipated heat (σ_{q}^{2}) increases with its mean (μ_{q}), and it cannot be smaller than 2k_{B}Tμ_{q}.

Finite-time generalization of the thermodynamic uncertainty relation

For fluctuating currents in nonequilibrium steady states, the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving. We show that this relation holds not only for the long-time limit of fluctuations, as described by large deviation theory, but also for fluctuations on arbitrary finite time scales. This generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments, for which infinite time scales are not accessible. Importantly, often this finite-time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long-time limit. We illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle.

Coherence of biochemical oscillations is bounded by driving force and network topology

Biochemical oscillations are prevalent in living organisms. Systems with a small number of constituents cannot sustain coherent oscillations for an indefinite time because of fluctuations in the period of oscillation. We show that the number of coherent oscillations that quantifies the precision of the oscillator is universally bounded by the thermodynamic force that drives the system out of equilibrium and by the topology of the underlying biochemical network of states. Our results are valid for arbitrary Markov processes, which are commonly used to model biochemical reactions. We apply our results to a model for a single KaiC protein and to an activator-inhibitor model that consists of several molecules. From a mathematical perspective, based on strong numerical evidence, we conjecture a universal constraint relating the imaginary and real parts of the first nontrivial eigenvalue of a stochastic matrix.

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.

First Passage under Restart

First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart-but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a generic approach to first passage under restart. Key features of diffusion under restart-the ultimate poster boy for this wide and diverse class of problems-are then shown to be completely universal.

Tightening the uncertainty principle for stochastic currents

We connect two recent advances in the stochastic analysis of nonequilibrium systems: the (loose) uncertainty principle for the currents, which states that statistical errors are bounded by thermodynamic dissipation, and the analysis of thermodynamic consistency of the currents in the light of symmetries. Employing the large deviation techniques presented by Gingrich et al. [Phys. Rev. Lett. 116, 120601 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.120601] and Pietzonka, Barato, and Seifert [Phys. Rev. E 93, 052145 (2016)2470-004510.1103/PhysRevE.93.052145], we provide a short proof of the loose uncertainty principle, and prove a tighter uncertainty relation for a class of thermodynamically consistent currents J. Our bound involves a measure of partial entropy production, that we interpret as the least amount of entropy that a system sustaining current J can possibly produce, at a given steady state. We provide a complete mathematical discussion of quadratic bounds which allows one to determine which are optimal, and finally we argue that the relationship for the Fano factor of the entropy production rate varσ/meanσ≥2 is the most significant realization of the loose bound. We base our analysis both on the formalism of diffusions, and of Markov jump processes in the light of Schnakenberg's cycle analysis.

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.

Dissipation Bounds All Steady-State Current Fluctuations

Near equilibrium, small current fluctuations are described by a Gaussian distribution with a linear-response variance regulated by the dissipation. Here, we demonstrate that dissipation still plays a dominant role in structuring large fluctuations arbitrarily far from equilibrium. In particular, we prove a linear-response-like bound on the large deviation function for currents in Markov jump processes. We find that nonequilibrium current fluctuations are always more likely than what is expected from a linear-response analysis. As a small-fluctuations corollary, we derive a recently conjectured uncertainty bound on the variance of current fluctuations.

Constraints on Fluctuations in Sparsely Characterized Biological Systems

Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.

Shot-noise Fano factor

A variability measure of the times of uniform events based on a shot-noise process is proposed and studied. The measure is inspired by the Fano factor, which we generalize by considering the time-weighted influence of the events given by a shot-noise response function. The sequence of events is assumed to be an equilibrium renewal process, and based on this assumption we present formulas describing the behavior of the variability measure. The formulas are derived for a general response function, restricted only by some natural conditions, but the main focus is given to the shot noise with exponential decrease. The proposed measure is analyzed and compared with the Fano factor.

Skewness and Kurtosis in Statistical Kinetics

We obtain lower and upper bounds on the skewness and kurtosis associated with the cycle completion time of unicyclic enzymatic reaction schemes. Analogous to a well-known lower bound on the randomness parameter, the lower bounds on skewness and kurtosis are related to the number of intermediate states in the underlying chemical reaction network. Our results demonstrate that evaluating these higher order moments with single molecule data can lead to information about the enzymatic scheme that is not contained in the randomness parameter.

Dispersion for two classes of random variables: general theory and application to inference of an external ligand concentration by a cell

We derive expressions for the dispersion for two classes of random variables in Markov processes. Random variables such as current and activity pertain to the first class, which is composed of random variables that change whenever a jump in the stochastic trajectory occurs. The second class corresponds to the time the trajectory spends in a state (or cluster of states). While the expression for the first class follows straightforwardly from known results in the literature, we show that a similar formalism can be used to derive an expression for the second class. As an application, we use this formalism to analyze a cellular two-component network estimating an external ligand concentration. The uncertainty related to this external concentration is calculated by monitoring different random variables related to an internal protein. We show that, inter alia, monitoring the time spent in the phosphorylated state of the protein leads to a finite uncertainty only if there is dissipation, whereas the uncertainty obtained from the activity of the transitions of the internal protein can reach the Berg-Purcell limit even in equilibrium.