Local Self-Assembly of Dissipative Structures Sustained by Substrate Diffusion.

The coupling between energy-consuming molecular processes and the macroscopic dimension plays an important role in nature and in the development of active matter. Here, we study the temporal evolution of a macroscopic system upon the local activation of a dissipative self-assembly process. Injection of surfactant molecules in a substrate-containing hydrogel results in the local substrate-templated formation of assemblies, which are catalysts for the conversion of substrate into waste. We show that the system develops into a macroscopic (pseudo-)non-equilibrium steady state (NESS) characterized by the local presence of energy-dissipating assemblies and persistent substrate and waste concentration gradients. For elevated substrate concentrations, this state can be maintained for more than 4 days. The studies reveal an interdependence between the dissipative assemblies and the concentration gradients: catalytic activity by the assemblies results in sustained concentration gradients and, vice versa, continuous diffusion of substrate to the assemblies stabilizes their size. The possibility to activate dissipative processes with spatial control and create long lasting non-equilibrium steady states enables dissipative structures to be studied in the space-time domain, which is of relevance for understanding biological systems and for the development of active matter.

A Minimalistic Covalent Bond-Forming Chemical Reaction Cycle that Consumes Adenosine Diphosphate.

The development of synthetic active matter requires the ability to design materials capable of harnessing energy from a source to carry out work. Nature achieves this using chemical reaction cycles in which energy released from an exergonic chemical reaction is used to drive biochemical processes. Although many chemically fuelled synthetic reaction cycles that control transient responses, such as self-assembly, have been reported, the generally high complexity of the reported systems hampers a full understanding of how the available chemical energy is actually exploited by these systems. This lack of understanding is a limiting factor in the design of chemically fuelled active matter. Here, we report a minimalistic synthetic responsive reaction cycle in which adenosine diphosphate (ADP) triggers the formation of a catalyst for its own hydrolysis. This establishes an interdependence between the concentrations of the network components resulting in the transient formation of the catalyst. The network is sufficiently simple that all kinetic and thermodynamic parameters governing its behaviour can be characterised, allowing kinetic models to be built that simulate the progress of reactions within the network. While the current network does not enable the ADP-hydrolysis reaction to populate a non-equilibrium composition, these models provide insight into the way the network dissipates energy. Furthermore, essential design principles are revealed for constructing driven systems, in which the network composition is driven away from equilibrium through the consumption of chemical energy.

Erratum: Dissipation-recurrence inequalities at the steady state [Phys. Rev. E 103, 032112 (2021)].

This corrects the article DOI: 10.1103/PhysRevE.103.032112.

ATP Drives the Formation of a Catalytic Hydrazone through an Energy Ratchet Mechanism.

An energy ratchet mechanism is exploited for the synthesis of a molecule. In the presence of adenosine triphosphate (ATP), hydrazone-bond formation between an aldehyde and hydrazide is accelerated and the composition at thermodynamic equilibrium is shifted towards the hydrazone. Enzymatic hydrolysis of ATP installs a kinetically stable state, at which hydrazone is present at a higher concentration compared to the composition at thermodynamic equilibrium in the presence of the degradation products of ATP. It is shown that the kinetic state has an enhanced catalytic activity in the hydrolysis of an RNA-model compound.

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 ß (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→ß. 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.

Probability inequalities for direct and inverse dynamical outputs in driven fluctuating systems.

When a fluctuating system is subjected to a time-dependent drive or nonconservative forces, the direct-inverse symmetry of the dynamics can be broken so inducing an average bias. Here we start from the fluctuation theorem, a cornerstone of stochastic thermodynamics, for inspecting the unbalancing between direct and inverse dynamical outputs, here called "events," in a bidirectional forward-backward setup. The occurrence of an event might correspond to the realization of a quantitative output, or to the realization of a sequence of acts that compose a complex "narrative." The focus is on mutual bounds between the probabilities of occurrence of direct and inverse events in the forward and backward mode. The inspection is made for systems in contact with a thermal bath, and by assuming Markov dynamics on the uncontrolled degrees of freedom. The approach comprises both the case of systems under a time-dependent drive and time-independent external forces. The general formulation is then used to derive (or re-derive) specialized results valid for finite-time processes, and for systems taken into steady conditions (either periodic steady states or steady states) starting from equilibrium. Among the results, we find already known forms of "generalized" thermodynamic uncertainty relations, and derive useful constraints concerning the work distribution function for systems in steady conditions.

Persistent ATP-Concentration Gradients in a Hydrogel Sustained by Chemical Fuel Consumption.

We show the formation of macroscopic ATP-concentrations in an agarose gel and demonstrate that these gradients can be sustained in time at the expense of the consumption of a chemical fuel. The approach relies on the spatially controlled activation of ATP-producing and ATP-consuming reactions through the local injection of enzymes in the matrix. The reaction-diffusion system is maintained in a stationary non-equilibrium state as long as chemical fuel, phosphocreatine, is present. The reaction-diffusion system is coupled to a supramolecular system composed of monolayer protected gold nanoparticles and a fluorescent probe. As a result of this coupling, fluorescence signals emerge spontaneously in response to the ATP-concentration gradients. We show that the approach permits the rational formation of complex fluorescence patterns that change over time as a function of the evolution of the ATP-concentrations present in the system.

Shape dependence of the release rate of chemicals from plastic microparticles.

The release of chemical additives from plastic microparticles in the aqueous phase represents a potential indirect threat for environment and biota. The estimate of the release timescale is demanded for drawing sensible conclusions on quantitative grounds. While the microparticles are generally taken to be spherical for ease of modelling, in reality the variety of shapes is large. Here, we face the problem of working out an empirical simple expression for estimating the release times for arbitrary shapes, assuming that the plastic material is in the rubbery state, that the dynamics inside the particle is a diffusion process, and that the release is irreversible. Our inspection is based on numerical simulations of the release process for randomly generated instances of regular and irregular geometries. The expression that we obtain allows one to estimate the release time in terms of the corresponding time (easy to compute) for the equal-volume spherical particle taken as reference, and of the ratio between the surface areas of particle and equivalent sphere.

Sample size dependence of tagged molecule dynamics in steady-state networks with bimolecular reactions: Cycle times of a light-driven pump.

Here, steady-state reaction networks are inspected from the viewpoint of individual tagged molecules jumping among their chemical states upon the occurrence of reactive events. Such an agent-based viewpoint is useful for selectively characterizing the behavior of functional molecules, especially in the presence of bimolecular processes. We present the tools for simulating the jump dynamics both in the macroscopic limit and in the small-volume sample where the numbers of reactive molecules are of the order of few units with an inherently stochastic kinetics. The focus is on how an ideal spatial "compartmentalization" may affect the dynamical features of the tagged molecule. Our general approach is applied to a synthetic light-driven supramolecular pump composed of ring-like and axle-like molecules that dynamically assemble and disassemble, originating an average ring-through-axle directed motion under constant irradiation. In such an example, the dynamical feature of interest is the completion time of direct/inverse cycles of tagged rings and axles. We find a surprisingly strong robustness of the average cycle times with respect to the system's size. This is explained in the presence of rate-determining unimolecular processes, which may, therefore, play a crucial role in stabilizing the behavior of small chemical systems against strong fluctuations in the number of molecules.

Diffusive model to assess the release of chemicals from a material under intermittent release conditions.

We consider the archetype situation of a chemical species that diffuses in a material and irreversibly escapes through the interface. In our setup, the interface switches between two states corresponding to 'release phase' (absorbing boundary) during which the species is released to the exterior, and 'pause phase' (reflecting boundary) during which the species is not released and its concentration profile inside the material partially relaxes back to uniformity. By combining numerical solution of the diffusion equation and statistical analysis of the outcomes, we derive upper and lower bounds and an empirical approximation for the amount of species released up to a certain time, in which the only information about the release-pause alternation schedule is the number of release phases and the average duration of a release phase. The methodology is developed thinking especially to dermal exposure assessment in the case of a slab-like homogeneous material irreversibly releasing chemicals during a number of contacts. However, upon proper extensions, this approach might be useful for inspecting other situations that are encountered, for instance, when dealing with leakage of chemicals in environmental contexts and regulatory toxicology.

Dissipation-recurrence inequalities at the steady state.

For Markov jump processes in out-of-equilibrium steady state, we present inequalities which link the average rate of entropy production with the timing of the site-to-site recurrences. Such inequalities are upper bounds on the average rate of entropy production. The combination with the finite-time thermodynamic uncertainty relation (a lower bound) yields inequalities of the pure kinetic kind for the relative precision of a dynamical output. After having derived the main relations for the discrete case, we sketch the possible extension to overdamped Markov dynamics on continuous degrees of freedom, treating explicitly the case of one-dimensional diffusion in tilted periodic potentials; an upper bound on the average velocity is derived, in terms of the average rate of entropy production and the microscopic diffusion coefficient, which corresponds to the finite-time thermodynamic uncertainty relation in the limit of vanishingly small observation time.

Sensitivity analysis of the reaction occurrence and recurrence times in steady-state biochemical networks.

Continuous-time stationary Markov jump processes among discrete sites are encountered in disparate biochemical ambits. Sites and connecting dynamical events form a 'network' in which the sites are the available system's states, and the events are site-to-site transitions, or even neutral processes in which the system does not change site but the event is however detectable. Examples include conformational transitions in single biomolecules, stochastic chemical kinetics in the space of the molecules copy numbers, and even macroscopic steady-state reactive mixtures if one adopts the viewpoint of a tagged molecule (or even of a molecular moiety) whose state may change when it is involved in a chemical reaction. Among the variety of dynamical descriptors, here we focus on the first occurrence times (starting from a given site) and on the recurrence times of an event of interest. We develop the sensitivity analysis for the lowest moments of the statistical distribution of such times with respect to the rate constants of the network. In particular, simple expressions and inequalities allow us to establish a direct relationship between selective variation of rate constants and effect on average times and variances. As illustrative cases we treat the substrate inhibition in enzymatic catalysis in which a tagged enzyme molecule jumps between three states, and the basic two-site model of stochastic gene expression in which the single gene switches between active and inactive forms.

Chromatographic NMR spectroscopy: the effect of hollow silica microspheres on magnetic field inhomogeneities and resonance lineshapes.

Micrometric hollow silica spheres can effectively reduce magnetic field inhomogeneities when employed as a stationary phase in the context of NMR chromatography. We here provide a description of the NMR line broadening phenomenon for physically representative collections of hollow spheres with different geometries and filling factors. Our results highlight how, within the explored conditions, a proper modelling of the line broadening phenomenon should consider the enhanced relaxation of the spins during their diffusion across the spherical shells, and possibly other slow motional effects.

Individual-Molecule Perspective Analysis of Chemical Reaction Networks: The Case of a Light-Driven Supramolecular Pump.

The first study in which stochastic simulations of a two-component molecular machine are performed in the mass-action regime is presented. This system is an autonomous molecular pump consisting of a photoactive axle that creates a directed flow of rings through it by exploiting light energy away from equilibrium. The investigation demonstrates that the pump can operate in two regimes, both experimentally accessible, in which light-driven steps can be rate-determining or not. The number of photons exploited by an individual molecular pump, as well as the precision of cycling and the overall efficiency, critically rely on the operating regime of the machine. This approach provides useful information not only to guide the chemical design of a self-assembling molecular device with desired features, but also to elucidate the effect of the environment on its performance, thus facilitating its experimental investigation.

Tagged-moiety viewpoint of chemical reaction networks.

In this work we consider mass action chemical reaction networks, either closed or open, and focus on the hopping path that a tagged moiety makes from molecule to molecule because of the occurrence of the reactions. We develop the tool for simulating the stochastic paths by means of a Gillespie-like algorithm and provide examples of the master equation counterpart for simple archetype problems of general interest. Both stationary and transient conditions are taken into account. An explanatory case is adopted to illustrate the approach.

Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium.

The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.

Ion Pair Formation between Tertiary Aliphatic Amines and Perchlorate in the Biphasic Water/Dichloromethane System.

The ability of aliphatic amines (AAs), namely, tripropylamine (TPrA), trisobutylamine (TisoBuA), and tributylamine (TBuA), to form ion pairs with perchlorate anion (ClO4-) in biphasic aqueous/dichloromethane (CH2Cl2) mixtures containing ClO4- 0.1 M has been demonstrated by GC with flame ionization (FID) and mass detectors (MS) and by NMR measurements. The extraction efficiency of the AAs to the organic phase was modeled by equations that were used to fit the experimental GC data, allowing us to determine values for KP (partition constant of the free AA), KIP (formation constant of the ion pair), and KPIP (partition constant of the ion pair) for TPrA, TisoBuA, and TBuA at 25 °C. Ion pairs were shown to form in CH2Cl2 also when ClO4- is replaced by other inorganic anions, like NO3-, ClO3-, Cl-, H2PO4-, and IO3-. No ion pairs formed when CH2Cl2 was replaced by n-hexane, suggesting that aliphatic amine ion pairs can form in polar organic solvents but not in nonpolar ones.

Dissipation, lag, and drift in driven fluctuating systems.

This work deals with thermostated fluctuating systems subjected to driven transformations of the internal energetics. The main focus is on generally multidimensional systems with continuous configurational degrees of freedom over which overdamped Markovian fluctuations take place (diffusive regime of the motion). Mutual bounds are established between the average energy dissipation, the deviation between nonequilibrium probability density and underlying equilibrium distribution due to the system's lag, and the statistical properties of the components of the directed flow induced by the transformation itself. The directed flow is here expressed in terms of time-dependent "drift velocity" associated with the probability current in a advection-like formulation of the nonstationary Fokker-Planck equation. Consideration of the drift makes that the bounds achieved here extend the inequality derived by Vaikuntanathan and Jarzynski [Europhys. Lett. 87, 60005 (2009)EULEEJ0295-507510.1209/0295-5075/87/60005] involving only dissipation and lag. The key relations are then specified for the so-called stochastic pumps, i.e., systems that reach a periodic steady state in response of cyclic transformations and that are prototypes of nonautonomous dissipative converters of input energy into directed motion; a one-dimensional case model is adopted to illustrate the main features. Complementary results concerning bounds between the evolution rates of dissipation and lag, valid for both overdamped and underdamped dynamics, are also presented.

Fluctuating systems under cyclic perturbations: Relation between energy dissipation and intrinsic relaxation processes.

This study focuses on fluctuating classical systems in contact with a thermal bath, and whose configurational energetics undergoes cyclic transformations due to interaction with external perturbing agents. Under the assumptions that the configurational dynamics is a stochastic Markov process in the overdamped regime and that the nonequilibrium configurational distribution remains close to the underlying equilibrium one, we derived an analytic approximation of the average dissipated energy per cycle in the asymptotic limit (i.e., after many cycles of perturbation). The energy dissipation is then readily translated into average entropy production, per cycle, in the environment. The accuracy of the approximation was tested by comparing the outcomes with the exact values obtained by stochastic simulations of a model case: a "particle on a ring" that fluctuates in a bistable potential perturbed in two different ways. As pointed out in previous studies on the stochastic resonance phenomenon, the dependence of the average dissipation on the perturbation period may unveil the inner spectrum of the system's fluctuation rates. In this respect, the analytical approximation presented here makes it possible to unveil the connection between average dissipation, intrinsic rates and modes of fluctuation of the system at the unperturbed equilibrium, and features of the perturbation itself (namely, the period of the cycle and the projections of the energy perturbation over the system's modes). The possibilities of employing the analytical results as a guide to devising and rationalizing a sort of "spectroscopic calorimetry" experiment, and of employing them in strategies aiming to optimize the system's features on the basis of a target average dissipation, are briefly discussed.