Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.     

Handling and Sensing of Single Enzyme Molecules: From Fluorescence Detection towards Nanoscale Electrical Measurements

Classical methods to study single enzyme molecules have provided valuable information about the distribution of conformational heterogeneities, reaction mechanisms, and transients in enzymatic reactions when individual molecules instead of an averaging ensemble are studied. Here, we highlight major advances in all‐electrical single enzyme studies with a focus on recent micro‐ and nanofluidic tools, which offer new ways of handling and studying small numbers of molecules or even single enzyme molecules. We particularly emphasize nanofluidic devices, which enable the integration of electrochemical transduction and detection.     

Organic Reactions in Microdroplets: Reaction Acceleration Revealed by Mass Spectrometry

The striking finding that reaction acceleration occurs in confined‐volume solutions sets up an apparent conundrum: Microdroplets formed by spray ionization can be used to monitor the course of bulk‐phase reactions and also to accelerate reactions between the reagents in such a reaction. This Minireview introduces droplet and thin‐film acceleration phenomena and summarizes recent methods applied to study accelerated reactions in confined‐volume, high‐surface‐area solutions. Conditions that dictate either simple monitoring or acceleration are reconciled in the occurrence of discontinuous and complete desolvation as the endpoint of droplet evolution. The contrasting features of microdroplet and bulk‐solution reactions are described together with possible mechanisms that drive reaction acceleration in microdroplets. Current applications of droplet microreactors are noted as is reaction acceleration in confined volumes and possible future scale‐up.     

Water molecules in a protein cavity detected by a statistical-mechanical theory

Four water molecules confined in a small cavity of hen egg-white lysozyme were detected by means of the three-dimensional (3D) RISM theory, a statistical-mechanical theory of molecular solutions. This is the first theoretical realization of confined molecules in a protein without making nonsense tricks, such as placing the molecules in the space a priori. Possible impacts which the result may have on biochemistry and biophysics, including the molecular recognition, enzymatic reactions, etc., are discussed.     

Activation of Small Molecules by Cluster Complexes

The results of theoretical, experimental investigations on activation of small molecules on their coordination to cluster complexes of heavy transition metals with weak- and strong-field ligands are presented. Homogeneous catalytic redox reactions of the CO, N₂, H₂O molecules and the N₃ ^- molecular anion in the presence of cluster complexes of low-valent molybdenum and rhenium are studies. The reaction mechanism is established. Three modifications of the homogeneous cluster catalysis of redox reactions of small molecules are described.     

Mechanisms and Beyond: Elucidation of Fluxional Dynamics by Exchange NMR Spectroscopy

Detailed mechanistic information is crucial to our understanding of reaction pathways and selectivity. Dynamic exchange NMR techniques, in particular 2D exchange spectroscopy (EXSY) and its modifications, provide indispensable intricate information on the mechanisms of organic and inorganic reactions and other phenomena, for example, the dynamics of interfacial processes. In this Review, key results from exchange NMR studies of small molecules over the last few decades are systemised and discussed. After a brief introduction to the theory, the key types of dynamic processes are identified and fundamental examples given of intra- and intermolecular reactions, which, in turn, could involve, or not, bond-making and bond-breaking events. Following that logic, internal molecular rotation, intramolecular stereomutation and molecular recognition will first be considered because they do not typically involve bond breaking. Then, rearrangements, substitution-type reactions, cyclisations, additions and other processes affecting chemical bonds will be discussed. Finally, interfacial molecular dynamics and unexpected combinations of different types of fluxional processes will also be highlighted. How exchange NMR spectroscopy helps to identify conformational changes, coordination and molecular recognition processes as well as quantify reaction energy barriers and extract detailed mechanistic information by using reaction rate theory in conjunction with computational techniques will be shown.     

Single-molecule stochastic times in a reversible bimolecular reaction

In this work, we consider the reversible reaction between reactants of species A and B to form the product C. We consider this reaction as a prototype of many pseudobiomolecular reactions in biology, such as for instance molecular motors. We derive the exact probability density for the stochastic waiting time that a molecule of species A needs until the reaction with a molecule of species B takes place. We perform this computation taking fully into account the stochastic fluctuations in the number of molecules of species B. We show that at low numbers of participating molecules, the exact probability density differs from the exponential density derived by assuming the law of mass action. Finally, we discuss the condition of detailed balance in the exact stochastic and in the approximate treatment.     

Chemically Programmable and Switchable CAR-T Therapy

Although macromolecules on cell surfaces are predominantly targeted and drugged with antibodies, they harbor pockets that are only accessible to small molecules and constitutes a rich subset of binding sites with immense potential diagnostic and therapeutic utility. Compared to antibodies, however, small molecules are disadvantaged by a less confined biodistribution, shorter circulatory half-life, and inability to communicate with the immune system. Presented herein is a method that endows small molecules with the ability to recruit and activate chimeric antigen receptor T cells (CAR-Ts). It is based on a CAR-T platform that uses a chemically programmed antibody fragment (cp-Fab) as on/off switch. In proof-of-concept studies, this cp-Fab/CAR-T system targeting folate binding proteins on the cell surface mediated potent and specific eradication of folate-receptor-expressing cancer cells in vitro and in vivo.     

Remarkable nanoconfinement effects on chemical equilibrium manifested in nucleotide dimerization and H-D exchange reactions

Nanoconfinement entropic effects on chemical equilibrium involving a small number of molecules, which we term NCECE, are revealed by two widely diverse types of reactions. Employing statistical-mechanical principles, we show how the NCECE effect stabilizes nucleotide dimerization observed within self-assembled molecular cages. Furthermore, the effect provides the basis for dimerization even under an aqueous environment inside the nanocage. Likewise, the NCECE effect is pertinent to a longstanding issue in astrochemistry, namely the extra deuteration commonly observed for molecules reacting on interstellar dust grain surfaces. The origin of the NCECE effect is elucidated by means of the probability distributions of the reaction extent and related variations in the reactant-product mixing entropy. Theoretical modelling beyond our previous preliminary work highlights the role of the nanospace size in addition to that of the nanosystem size, namely the limited amount of molecules in the reaction mixture. Furthermore, the NCECE effect can depend also on the reaction mechanism, and on deviations from stoichiometry. The NCECE effect, leading to enhanced, greatly variable equilibrium "constants", constitutes a unique physical-chemical phenomenon, distinguished from the usual thermodynamical properties of macroscopically large systems. Being significant particularly for weakly exothermic reactions, the effects should stabilize products in other closed nanoscale structures, and thus can have notable implications for the growing nanotechnological utilization of chemical syntheses conducted within confined nanoreactors.     

Stochastic approaches for modelling in vivo reactions

In recent years, stochastic modelling has emerged as a physically more realistic alternative for modelling in vivo reactions. There are numerous stochastic approaches available in the literature; most of these assume that observed random fluctuations are a consequence of the small number of reacting molecules. We review some important developments of the stochastic approach and consider its suitability for modelling intracellular reactions. We then describe recent efforts to include the fluctuation effects caused by the structural organisation of the cytoplasm and the limited diffusion of molecules due to macromolecular crowding.     

Confined space regulated polymerization

The confined space produced during the polymerization has access for all small organic molecules or oligomers with small size to enter this confined space; however, it can prevent the macromolecules with big size from entering. Therefore, the reaction between two branched macromolecules is excluded in A₂+B₃ polymerization system, resulting uncrosslinked branched polymers, and there was no gel point observed throughout the polymerization. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 1730–1737, 2008     

Shearing of nanoscopic bridges in two-component thin liquid layers between chemically patterned walls

Bridge phases associated with a phase transition between two liquid phases occur when a two-component liquid mixture is confined between chemically patterned walls. In the bulk the liquid mixture with components A, B undergoes phase separation into an A-rich phase and a B-rich phase. The walls bear stripes attractive to A. In the bridge phase A-rich and B-rich regions alternate. Grand canonical Monte Carlo studies are performed with the alignment between stripes on opposite walls varied. Misalignment of the stripes places the nanoscopic liquid bridges under shear strain. The bridges exert a Hookean restoring force on the walls for small displacements from equilibrium. As the strain increases there are deviations from Hooke's law. Eventually there is an abrupt yielding of the bridges. Molecular dynamics simulations show the bridges form or disintegrate on time scales which are fast compared to wall motion and transport of molecules into or from the confined space. Some interesting possible applications of the phenomena are discussed.     

Intermittent single-molecule interfacial electron transfer dynamics

We report on single-molecule studies of photosensitized interfacial electron transfer (ET) processes in Coumarin 343 (C343)-TiO(2) nanoparticles (NP) and Cresyl Violet (CV(+))-TiO(2) NP systems, using time-correlated single-photon counting coupled with scanning confocal fluorescence microscopy. Fluorescence intensity trajectories of individual dye molecules adsorbed on a semiconductor NP surface showed fluorescence fluctuations and blinking, with time constants distributed from milliseconds to seconds. The fluorescence fluctuation dynamics were found to be inhomogeneous from molecule to molecule and from time to time, showing significant static and dynamic disorders in the interfacial ET reaction dynamics. We attribute fluorescence fluctuations to the interfacial ET reaction rate fluctuations, associating redox reactivity intermittency with the fluctuations of molecule-TiO(2) electronic and Franck-Condon coupling. Intermittent interfacial ET dynamics of individual molecules could be characteristic of a surface chemical reaction strongly involved with and regulated by molecule-surface interactions. The intermittent interfacial reaction dynamics that likely occur among single molecules in other interfacial and surface chemical processes can typically be observed by single-molecule studies but not by conventional ensemble-averaged experiments.     

Measurement of forces and structure between fluid interfaces

The forces and structures that develop at and between fluid interfaces are responsible for the stability of foam, emulsion and wetting films. Although these forces are less studied than the interactions occurring between solid surfaces, recent quantitative studies of films created between fluid interfaces are providing new information that both complements previous findings obtained with solid surfaces, and reveals unique and important differences for films confined between fluid interfaces. Noteworthy is that fluid interfaces can be much more mobile, thus fluctuations and interfacial boundary conditions can produce a rich variety of phenomena at these interfaces.     

Formation of GeO2 nanosheets using water thin layers in lamellar phase as a confined reaction field--in situ measurement of SAXS by synchrotron radiation

Highly crystallized GeO2 nanosheets were synthesized by hydrolysis and condensation reactions of germanium alkoxide using a 2-dimensional flat thin lamellar phase water layer containing surfactant molecules at the liquid-liquid interface as a confined reaction field.     

Laser vaporization: A versatile method for studying metal clusters

Vaporizing solid samples of metals and semiconductors with a YAG Laser is a method well suited for producing molecules and clusters of those materials. The clusters are examined by either laser-induced fluorescence (LIF) or mass spectroscopic methods. The technique is valuable for both gas phase and matrix studies. The method is described and some applications, studying either the structure of small metal molecules or their reactions, are reviewed, with emphasis on our recent results from the LIF studies of LiBe, Al₂ and the reaction of Al with oxygen, yielding A1₂O. For larger clusters, Ion Cyclotron Resonance is an extremely valuable method, as we demonstrate by its application to the reactions of small charged silicon clusters with strong oxidising agents.     

Chemically Programmable and Switchable CAR‐T Therapy

Although macromolecules on cell surfaces are predominantly targeted and drugged with antibodies, they harbor pockets that are only accessible to small molecules and constitutes a rich subset of binding sites with immense potential diagnostic and therapeutic utility. Compared to antibodies, however, small molecules are disadvantaged by a less confined biodistribution, shorter circulatory half‐life, and inability to communicate with the immune system. Presented herein is a method that endows small molecules with the ability to recruit and activate chimeric antigen receptor T cells (CAR‐Ts). It is based on a CAR‐T platform that uses a chemically programmed antibody fragment (cp‐Fab) as on/off switch. In proof‐of‐concept studies, this cp‐Fab/CAR‐T system targeting folate binding proteins on the cell surface mediated potent and specific eradication of folate‐receptor‐expressing cancer cells in vitro and in vivo.     

The nature of permeability anisotropy and catalytic activity

The phenomena of permeability anisotropy and an increase in the rates of catalytic reactions in porous membranes modified with highly dispersed catalytic systems were analyzed. A model of stochastic gas motions was proposed; this model is based on the hypothesis of the specific interaction of molecules with the inner surface of pores resulting in a nonisotropic distribution of molecules over traveling directions. The effects of asymmetric gas transfer in porous and gradient-porous membranes were considered to explain differences in the rates of heterogeneous catalytic reactions in a nanoporous membrane reactor under changes in the direction of supplying a reaction mixture. From the model proposed, it follows that the transversal diffusion of gas molecules is most probable in the porous medium of a ceramic membrane with a pore-size distribution gradient from large to small pores along the flow direction. This diffusion results in an increase in the frequency of molecular collisions with the wall of a microchannel and, correspondingly, in an increase in the contact time. The model proposed explains the intensification of a number of heterogeneous catalytic reactions performed in the porous media of catalytic porous membranes.