Measuring Flux Distributions for Diffusion in the Small-Numbers Limit

For the classical diffusion of independent particles, Fick's law gives a well-known relationship between the average flux and the average concentration gradient. What has not yet been explored experimentally, however, is the dynamical distribution of diffusion rates in the limit of small particle numbers. Here, we measure the distribution of diffusional fluxes using a microfluidics device filled with a colloidal suspension of a small number of microspheres. Our experiments show that (1) the flux distribution is accurately described by a Gaussian function; (2) Fick's law, that the average flux is proportional to the particle gradient, holds even for particle gradients down to a single particle difference; (3) the variance in the flux is proportional to the sum of the particle numbers; and (4) there are backward flows, where particles flow up a concentration gradient, rather than down it. In addition, in recent years, two key theorems about nonequilibrium systems have been introduced:  Evans' fluctuation theorem for the distribution of entropies and Jarzynski's work theorem. Here, we introduce a new fluctuation theorem, for the fluxes, and we find that it is confirmed quantitatively by our experiments

RNA–protein binding kinetics in an automated microfluidic reactor

Microfluidic chips can automate biochemical assays on the nanoliter scale, which is of considerable utility for RNA–protein binding reactions that would otherwise require large quantities of proteins. Unfortunately, complex reactions involving multiple reactants cannot be prepared in current microfluidic mixer designs, nor is investigation of long-time scale reactions possible. Here, a microfluidic ‘Riboreactor’ has been designed and constructed to facilitate the study of kinetics of RNA–protein complex formation over long time scales. With computer automation, the reactor can prepare binding reactions from any combination of eight reagents, and is optimized to monitor long reaction times. By integrating a two-photon microscope into the microfluidic platform, 5-nl reactions can be observed for longer than 1000 s with single-molecule sensitivity and negligible photobleaching. Using the Riboreactor, RNA–protein binding reactions with a fragment of the bacterial 30S ribosome were prepared in a fully automated fashion and binding rates were consistent with rates obtained from conventional assays. The microfluidic chip successfully combines automation, low sample consumption, ultra-sensitive fluorescence detection and a high degree of reproducibility. The chip should be able to probe complex reaction networks describing the assembly of large multicomponent RNPs such as the ribosome

Non-Equilibrium Dynamics: Diffusion in Small Numbers and Ribosomal Self-Assembly

Biological systems are encountered in states that are far from equilibrium. A change in the cell's condition triggers the flow of energy and matter that causes the cell's transition from that non-equilibrium state to a different state. Our interest is on non-equilibrium systems and the way these relate to the cell's "small numbers" limit as well as to the mechanisms of self-assembly. Cells contain proteins and nucleotides in numbers smaller than Avogadro's. In addition, advances in single-molecule experiments, which are, by definition, a case of the "small numbers" problem, have emphasized the importance of fluctuations. Does the result we get from a single-molecule measurement agree with what we would get from a bulk measurement? Is it a fluctuation from the mean? It is, thus, of biological interest to see the behavior of non-equilibrium systems at the "small numbers" limit where fluctuations become important. Using microfluidics, we concentrate on the diffusion of a small number of submicron particles in a system that is away from equilibrium. Therefore, we study the "small numbers" limit of Fick's Law, with special reference to the fluctuations that attend diffusive dynamics in order to experimentally test the theoretical predictions obtained via the use of E. T. Jaynes' "principle of maximum caliber." The process of macromolecular self-assembly is also highly dynamical. The system's components come together, defeating in this way entropic effects, to form the system. In the case of the ribosome, whose importance lies in its ability to synthesize proteins, understanding the mechanism of the highly dimensional process of self-assembly becomes relevant when designing, for example, new antibiotics. The second part of this thesis concentrates on the RNA-protein interactions which, in the case of the ribosome, determine the mechanism of self-assembly. With the use of microfluidic technology and a fluorescence assay we determine the thermodynamics and kinetics of RNA folding and RNA-protein binding for a fragment of the bacterial 30S ribosomal subunit, paving the way for the study of the complete assembly of the 30S subunit.</p

Simple Brownian diffusion: an introduction to the standard theoretical models

Brownian diffusion, the motion of large molecules in a sea of very many much smaller molecules, is topical because it is one of the ways in which biologically important molecules move about inside living cells. This book presents the mathematical physics that underlies the four simplest models of Brownian diffusion

Validity conditions for stochastic chemical kinetics in diffusion-limited systems.

The chemical master equation (CME) and the mathematically equivalent stochastic simulation algorithm (SSA) assume that the reactant molecules in a chemically reacting system are "dilute" and "well-mixed" throughout the containing volume. Here we clarify what those two conditions mean, and we show why their satisfaction is necessary in order for bimolecular reactions to physically occur in the manner assumed by the CME and the SSA. We prove that these conditions are closely connected, in that a system will stay well-mixed if and only if it is dilute. We explore the implications of these validity conditions for the reaction-diffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily well-mixed, but can be partitioned into cubical subvolumes (voxels) that are. We show that the validity conditions, together with an additional condition that is needed to ensure the physical validity of the diffusion-induced jump probability rates of molecules between voxels, require the voxel edge length to have a strictly positive lower bound. We prove that if the voxel edge length is steadily decreased in a way that respects that lower bound, the average rate at which bimolecular reactions occur in the reaction-diffusion CME and SSA will remain constant, while the average rate of diffusive transfer reactions will increase as the inverse square of the voxel edge length. We conclude that even though the reaction-diffusion CME and SSA are inherently approximate, and cannot be made exact by shrinking the voxel size to zero, they should nevertheless be useful in many practical situations

Measuring flux distributions for diffusion in the small-numbers limit

For the classical diffusion of independent particles, Fick’s law gives a well-known relationship between the average flux and the average concentration gradient. What has not yet been explored experimentally, however, is the dynamical distribution of diffusion rates in the limit of small particle numbers. Here, we measure the distribution of diffusional fluxes using a microfluidics device filled with a colloidal suspension of a small number of microspheres. Our experiments show that (1) the flux distribution is accurately described by a Gaussian function; (2) Fick’s law, that the average flux is proportional to the particle gradient, holds even for particle gradients down to a single particle difference; (3) the variance in the flux is proportional to the sum of the particle numbers; and (4) there are backward flows, where particles flow up a concentration gradient, rather than down it. In addition, in recent years, two key theorems about nonequilibrium systems have been introduced: Evans ’ fluctuation theorem for the distribution of entropies and Jarzynski’s work theorem. Here, we introduce a new fluctuation theorem, for the fluxes, and we find that it is confirmed quantitatively by our experiments