The Pair Approach Applied to Kinetics in Restricted Geometries: Strengths and Weaknesses of the Method

Multiparticle diffusion limited [Formula: see text] reaction in small volumes

A multiparticle reaction model in which particles A annihilate in clusters of size k as [Formula: see text] is investigated analytically. The system is studied for arbitrary reaction order k > 2, dimension d, and size L. Particles diffuse with diffusion constant D, and annihilate with rate σ which depends on the positions of kA particles in the cluster prior to a reaction. The particles are assumed to be spatially extended objects with radius a. Exclusion effects are not taken into account since A particles are allowed to overlap. The master equation is rephrased in the language of a field theory which, in turn, is used to derive the equations of motion for many-point densities. An approximate form of the equations of motion was solved analytically in the diffusion-controlled limit (infinite reaction rate). An explicit expression for the effective reaction rate has been found in the form of the Laplace transform. It was shown that the number of particles saturates to a constant value for large times. The value is approached through an exponential decay. The exponential decay constant is the non-algebraic function of particle size a and system size L.

Exact equilibrium-state solution of an intracellular complex formation model: kA↔P reaction in a small volume

A generic model of complex formation in small volumes was studied under the assumption of perfect mixing. Particles A react in clusters, and each reaction converts k A particles into a P particle. The back reaction is also allowed. The equilibrium state of the model is solved exactly. Fluctuations in product particle number are reduced by increasing the degree of cooperativity k. Three qualitatively distinct reactant fluctuation characteristics emerge.

Liposomes: technologies and analytical applications

Liposomes are structurally and functionally some of the most versatile supramolecular assemblies in existence. Since the beginning of active research on lipid vesicles in 1965, the field has progressed enormously and applications are well established in several areas, such as drug and gene delivery. In the analytical sciences, liposomes serve a dual purpose: Either they are analytes, typically in quality-assessment procedures of liposome preparations, or they are functional components in a variety of new analytical systems. Liposome immunoassays, for example, benefit greatly from the amplification provided by encapsulated markers, and nanotube-interconnected liposome networks have emerged as ultrasmall-scale analytical devices. This review provides information about new developments in some of the most actively researched liposome-related topics.

Interplay between chemical reactions and transport in structured spaces

The main motivation behind this study is to understand the interplay between the reactions and transport in a geometries that are not compact. Typical examples of compact geometries are a box or a sphere. A network made of containers C(1) , C(2),..., C(N) and tubes is an example of the space that is structured and noncompact. In containers, particles react with the rate lambda. Tubes connecting containers allow for the exchange of chemicals with the transport rate D. A situation is considered where a number of reactants is small and kinetics is noise dominated. A method is presented that can be used to calculate the average and higher moments of the reaction time. A number of different chemical reactions are studied and their performance compared in various ways. Two schemes are discussed in general, the reaction on a fixed geometry ensemble (ROGE) and the geometry on a fixed reaction ensemble, examples are given in the ROGE case. The most important findings are as follows. (i) There is a large number of reactions that run faster in a networklike geometry. Such reactions contain antagonistic catalytic influences in the intermediate stages of a reaction scheme that are best dealt with in a networklike structure. (ii) Antagonistic catalytic influences are hard to identify since they are strongly connected to the pattern of injected molecules (inject pattern) and depend on the choice of molecules that have to be synthesized at the end (task pattern). (iii) The reaction time depends strongly on the details of the inject and task patterns.

Controlled Initiation of Enzymatic Reactions in Micrometer-Sized Biomimetic Compartments

We present a technique to initiate chemical reactions involving few reactants inside micrometer-scale biomimetic vesicles (10(-12) to 10(-15) L) integral to three-dimensional surfactant networks. The shape of these networks is under dynamic control, allowing for transfer and mixing of two or several reactants at will. Specifically, two nanotube-connected vesicles were filled with reactants (substrate and enzyme, respectively) by microinjection. Initially, the vesicles are far apart and any diffusive mixing (on relevant experimental time scales) between the contents of the separated vesicles is hindered because of the narrow diameter and long axial extension of the nanotube. To initiate a reaction, the vesicles were brought close together, the nanotube was consumed by the vesicles and at a critical distance, the nanotube-vesicle junctions were dilated leading to formation of one spherical reactor, and hence mixing of the contents. We demonstrate the concept using a model enzymatic reaction, which yields a fluorescent product (two-step hydrolysis of fluorescein diphosphate by alkaline phosphatase), where product formation was measured as a function of time using a FRAP fluorescence microscopy protocol. By comparing the enzymatic activity with bulk measurements, the enzyme concentration inside the vesicle could be determined. Reactions could be followed for systems having as few as approximately 15 enzyme molecules confined to a reactor vesicle. To describe the experiments we use a simple diffusion-controlled reaction model and solve it using a survival probability approach. The agreement with experiment is qualitative, but the model describes the trends well. It is shown that the model correctly predicts (i) single-exponential decay after a few seconds, and (ii) that the substrate decay constant depends on the number of enzymes and geometry of reaction container. The numerical correction factor Lambda is introduced in order to ensure semiquantitative agreement between experiment and theory. It was shown that this numerical factor depends weakly on vesicle radius and number of enzymes, thus it is sufficient to determine this factor only once in a single calibration measurement.

BIOMIMETIC NANOSCALE REACTORS AND NETWORKS

Methods based on self-assembly, self-organization, and forced shape transformations to form synthetic or semisynthetic enclosed lipid bilayer structures with several properties similar to biological nanocompartments are reviewed. The procedures offer unconventional micro- and nanofabrication routes to yield complex soft-matter devices for a variety of applications for example, in physical chemistry and nanotechnology. In particular, we describe novel micromanipulation methods for producing fluid-state lipid bilayer networks of nanotubes and surface-immobilized vesicles with controlled geometry, topology, membrane composition, and interior contents. Mass transport in nanotubes and materials exchange, for example, between conjugated containers, can be controlled by creating a surface tension gradient that gives rise to a moving boundary or by induced shape transformations. The network devices can operate with extremely small volume elements and low mass, to the limit of single molecules and particles at a length scale where a continuum mechanics approximation may break down. Thus, we also describe some concepts of anomalous fluctuation-dominated kinetics and anomalous diffusive behaviours, including hindered transport, as they might become important in studying chemistry and transport phenomena in these confined systems. The networks are suitable for initiating and controlling chemical reactions in confined biomimetic compartments for rationalizing, for example, enzyme behaviors, as well as for applications in nanofluidics, bioanalytical devices, and to construct computational and complex sensor systems with operations building on chemical kinetics, coupled reactions and controlled mass transport.