Steady-state probabilities for Markov jump processes in terms of powers of the transition rate matrix.

Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N - 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.

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.

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.

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.

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.

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.

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.

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.

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.

Loop Electrostatics Asymmetry Modulates the Preexisting Conformational Equilibrium in Thrombin.

Thrombin exists as an ensemble of active (E) and inactive (E*) conformations that differ in their accessibility to the active site. Here we show that redistribution of the E*-E equilibrium can be achieved by perturbing the electrostatic properties of the enzyme. Removal of the negative charge of the catalytic Asp102 or Asp189 in the primary specificity site destabilizes the E form and causes a shift in the 215-217 segment that compromises substrate entrance. Solution studies and existing structures of D102N document stabilization of the E* form. A new high-resolution structure of D189A also reveals the mutant in the collapsed E* form. These findings establish a new paradigm for the control of the E*-E equilibrium in the trypsin fold.

Towards bulk thermodynamics via non-equilibrium methods: gaseous methane as a case study.

We illustrate how the Jarzynski equality (JE), which is the progenitor of non-equilibrium methods aimed at constructing free energy landscapes for molecular-sized fluctuating systems subjected to steered transformations, can be applied to derive equations of state for bulk systems. The key-step consists of physically framing the computational strategy of "total energy morphing", recently presented by us as an efficient implementation of the JE [M. Zerbetto, A. Piserchia, D. Frezzato, J. Comput. Chem., 2014, 35, 1865-1881], in terms of build-up of the real thermodynamic state of a bulk material from the corresponding ideal state, in which the particles are non-interacting. In this context, the JE machinery yields the excess free energy versus suitably chosen controlled state variables, whose thermodynamic derivatives eventually lead to the equation of state. As an explanatory case study, we apply the methodology to derive the equation of state of gaseous methane by constructing the Helmholtz free energy versus the particle density (at fixed temperature) and then evaluating the thermodynamic derivative with respect to the volume. In our intent, this "old-style" work on gaseous methane should open the way for the investigation of thermodynamics of extended systems via non-equilibrium methods.

Probing the conformational energetics of alkyl thiols on gold surfaces by means of a morphing/steering non-equilibrium tool.

In this work we show that a non-equilibrium statistical tool based on Jarzynski's equality (JE) can be applied to achieve a sufficiently accurate mapping of the torsion free energy, bond-by-bond, for an alkyl thiol ligand tethered to a gold surface and sensing the presence of the surrounding cluster of similar chains. The strength of our approach is the employment of a strategy to let grow the internal energetics of the whole system (namely, the "energy morphing" stage recently presented by us in J. Comput. Chem., 2014, 35, 1865-1881) before initiating the rotational steering, which yields accurate results in terms of statistical uncertainties and bias on the free energy profiles. The work is mainly methodological and illustrates the feasibility of this kind of inspection on nanoscale molecular clusters with conformational flexibility. The outcomes for the archetype of self-assembled-monolayers considered here, a regular pattern of 10-carbon alkyl thiols on an ideal gold surface, give information on the conformational mobility of the ligands. Notably, such information is unlikely to be obtained by means of standard equilibrium techniques or by conventional molecular dynamics simulations.

Features in chemical kinetics. III. Attracting subspaces in a hyper-spherical representation of the reactive system.

In this work, we deal with general reactive systems involving N species and M elementary reactions under applicability of the mass-action law. Starting from the dynamic variables introduced in two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138(23), 234101 (2013); 138(23), 234102 (2013)], we turn to a new representation in which the system state is specified in a (N × M)(2)-dimensional space by a point whose coordinates have physical dimension of inverse-of-time. By adopting hyper-spherical coordinates (a set of dimensionless "angular" variables and a single "radial" one with physical dimension of inverse-of-time) and by examining the properties of their evolution law both formally and numerically on model kinetic schemes, we show that the system evolves towards the equilibrium as being attracted by a sequence of fixed subspaces (one at a time) each associated with a compact domain of the concentration space. Thus, we point out that also for general non-linear kinetics there exist fixed "objects" on the global scale, although they are conceived in such an abstract and extended space. Moreover, we propose a link between the persistence of the belonging of a trajectory to such subspaces and the closeness to the slow manifold which would be perceived by looking at the bundling of the trajectories in the concentration space.

Looking for some free energy? Call JEFREE ( ).

In this communication, we present the Jarzynski's Equality FREe Energy (JEFREE) library, an efficient and easy-to-use C++ library targeted to the calculation of the free energy profile along a selected generalized coordinate of a system, within the framework of the nonequilibrium steered transformations as introduced by Jarzynski [Phys. Rev. E, 1997, 56, 5018]. JEFREE can be readily integrated into any code, since both C and FORTRAN wrappers have been developed, and easily customizable by a user thanks to the object-oriented programming paradigm offered by the C++ language. Also, JEFREE implements the novel idea of making a total "morphing" of the system energy landscape before initiating the proper steering stage. This proves to be an efficient mean to overtake the problematic sampling of the initial equilibrium state when the number of degrees of freedom is high and the landscape owns many local minima separated by large energy barriers. The calculation of the free energy profile for the rotation along torsion angles in alkyl chains is presented as an example of application of our tool.

Toward quantitative estimates of binding affinities for protein-ligand systems involving large inhibitor compounds: a steered molecular dynamics simulation route.

Understanding binding mechanisms between enzymes and potential inhibitors and quantifying protein-ligand affinities in terms of binding free energy is of primary importance in drug design studies. In this respect, several approaches based on molecular dynamics simulations, often combined with docking techniques, have been exploited to investigate the physicochemical properties of complexes of pharmaceutical interest. Even if the geometric properties of a modeled protein-ligand complex can be well predicted by computational methods, it is still challenging to rank with chemical accuracy a series of ligand analogues in a consistent way. In this article, we face this issue calculating relative binding free energies of a focal adhesion kinase, an important target for the development of anticancer drugs, with pyrrolopyrimidine-based ligands having different inhibitory power. To this aim, we employ steered molecular dynamics simulations combined with nonequilibrium work theorems for free energy calculations. This technique proves very powerful when a series of ligand analogues is considered, allowing one to tackle estimation of protein-ligand relative binding free energies in a reasonable time. In our cases, the calculated binding affinities are comparable with those recovered from experiments by exploiting the Michaelis-Menten mechanism with a competitive inhibitor.

Features in chemical kinetics. I. Signatures of self-emerging dimensional reduction from a general format of the evolution law.

Simplification of chemical kinetics description through dimensional reduction is particularly important to achieve an accurate numerical treatment of complex reacting systems, especially when stiff kinetics are considered and a comprehensive picture of the evolving system is required. To this aim several tools have been proposed in the past decades, such as sensitivity analysis, lumping approaches, and exploitation of time scales separation. In addition, there are methods based on the existence of the so-called slow manifolds, which are hyper-surfaces of lower dimension than the one of the whole phase-space and in whose neighborhood the slow evolution occurs after an initial fast transient. On the other hand, all tools contain to some extent a degree of subjectivity which seems to be irremovable. With reference to macroscopic and spatially homogeneous reacting systems under isothermal conditions, in this work we shall adopt a phenomenological approach to let self-emerge the dimensional reduction from the mathematical structure of the evolution law. By transforming the original system of polynomial differential equations, which describes the chemical evolution, into a universal quadratic format, and making a direct inspection of the high-order time-derivatives of the new dynamic variables, we then formulate a conjecture which leads to the concept of an "attractiveness" region in the phase-space where a well-defined state-dependent rate function ω has the simple evolution ω[over dot]=-ω(2) along any trajectory up to the stationary state. This constitutes, by itself, a drastic dimensional reduction from a system of N-dimensional equations (being N the number of chemical species) to a one-dimensional and universal evolution law for such a characteristic rate. Step-by-step numerical inspections on model kinetic schemes are presented. In the companion paper [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234102 (2013)] this outcome will be naturally related to the appearance (and hence, to the definition) of the slow manifolds.

Features in chemical kinetics. II. A self-emerging definition of slow manifolds.

In the preceding paper of this series (Part I [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013)]) we have unveiled some ubiquitous features encoded in the systems of polynomial differential equations normally applied in the description of homogeneous and isothermal chemical kinetics (mass-action law). Here we proceed by investigating a deeply related feature: the appearance of so-called slow manifolds (SMs) which are low-dimensional hyper-surfaces in the neighborhood of which the slow evolution of the reacting system occurs after an initial fast transient. Indeed a geometrical definition of SM, devoid of subjectivity, "naturally" follows in terms of a specific sub-dimensional domain embedded in the peculiar region of the concentrations phase-space that in Part I we termed as "attractiveness region." Numerical inspections on simple low-dimensional model cases are presented, including the benchmark case of Davis and Skodje [J. Chem. Phys. 111, 859 (1999)] and the preliminary analysis of a simplified model mechanism of hydrogen combustion.

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.

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.