Controlling chemistry by geometry in nanoscale systems.

Scientific literature dealing with the rates, mechanisms, and thermodynamic properties of chemical reactions in condensed media almost exclusively assumes that reactions take place in volumes that do not change over time. The reaction volumes are compact (such as a sphere, a cube, or a cylinder) and do not vary in shape. In this review article, we discuss two important systems at small length scales (approximately 10 nm to 5 microm), in which these basic assumptions are violated. The first system exists in cell biology and is represented by the tiniest functional components (i.e., single cells, organelles, and other physically delineated cellular microenvironments). The second system comprises nanofluidic devices, in particular devices made from soft-matter materials such as lipid nanotube-vesicle networks. In these two systems, transport, mixing, and shape changes can be achieved at or very close to thermal energy levels. In further contrast to macroscopic systems, mixing by diffusion is extremely efficient, and kinetics can be controlled by shape and volume changes.

Tunable filtering of chemical signals in a simple nanoscale reaction-diffusion network.

We study numerically the filtering capabilities of a nanoscale network of two micrometer-sized containers joined by a nanotube, one of which hosts an enzymatic chemical reaction. Spatiotemporal chemical signals of substrate molecules are injected into the network. The substrate propagates by diffusion and reacts with enzymes distributed in the network prior to the injections. The dimensions of the network are tailored in a way that the transport and enzymatic reaction rates are comparable in size, a situation in which the overall behavior is highly influenced by the geometry and topology of the network. This property is crucial for the functionality of the filter developed in here. It is demonstrated that input signals can be classified in a crude way using a simple setup (a two-container network) and that the classification can be tuned by changing the geometry of the network (the length of the tube connecting the two containers). The filter device we investigate can also be viewed as a primitive chemistry-based computational element in the sense that the information encoded in the signals is processed using chemical reactions. In particular, it is demonstrated that the two-container device may filter out signals based on the average injection frequency.

Diffusive transport in networks built of containers and tubes.

We have developed analytical and numerical methods to study the transport of noninteracting particles in large networks consisting of M d -dimensional containers C1,...,C(M) with radii R(i) linked together by tubes of length l(ij) and radii a(ij) where i,j = 1,2,...,M. Tubes may join directly with each other, forming junctions. It is possible that some links are absent. Instead of solving the diffusion equation for the full problem we formulated an approach that is computationally more efficient. We derived a set of rate equations that govern the time dependence of the number of particles in each container, N1(t), N2(t),...,N(M)(t). In such a way the complicated transport problem is reduced to a set of M first-order integro-differential equations in time, which can be solved efficiently by the algorithm presented here. The workings of the method have been demonstrated on a couple of examples: networks involving three, four, and seven containers and one network with a three-point junction. Already simple networks with relatively few containers exhibit interesting transport behavior. For example, we showed that it is possible to adjust the geometry of the networks so that the particle concentration varies in time in a wave-like manner. Such behavior deviates from simple exponential growth and decay occurring in the two-container system.

Embedding a native state into a random heteropolymer model: the dynamic approach.

We study a random heteropolymer model with Langevin dynamics, in the supersymmetric formulation. Employing a procedure similar to one that has been used in static calculations, we construct an ensemble in which the affinity of the system for a native state is controlled by a "selection temperature" T0. In the limit of high T0, the model reduces to a random heteropolymer, while for T0-->0 the system is forced into the native state. Within the Gaussian variational approach that we employed previously for the random heteropolymer, we explore the phases of the system for high and low T0. For high T0, the system exhibits a (dynamical) spin-glass phase, like that found for the random heteropolymer, below a temperature T(g). For low T0, we find an ordered phase, characterized by a nonzero overlap with the native state, below a temperature T(n) proportional to 1/T(0)>T(g). However, the random-globule phase remains locally stable below T(n), down to the dynamical glass transition at T(g). Thus, in this model, folding is rapid for temperatures between T(g) and T(n), but below T(g) the system can get trapped in conformations uncorrelated with the native state. At a lower temperature, the ordered phase can also undergo a dynamical glass transition, splitting into substates separated by large barriers.

Random heteropolymer dynamics.

We study the Langevin dynamics of the standard random heteropolymer model by mapping the problem to a supersymmetric field theory using the Martin-Siggia-Rose formalism. The resulting model is solved nonperturbatively employing a Gaussian variational approach. In constructing the solution, we assume that the chain is very long and impose the translational invariance which is expected to be present in the bulk of the globule by averaging over the center of mass coordinate. In this way we derive equations of motion for the correlation and response functions C(t,t') and R(t,t'). The order parameters are extracted from the asymptotic behavior of these functions. We find a dynamical phase diagram with frozen (glassy) and melted (ergodic) phases. In the glassy phase the system fails to reach equilibrium and exhibits aging of the type found in p-spin glasses. Within the approximations used in this study, the random heteropolymer model can be mapped to the problem of a manifold in a random potential with power law correlations.

Two-species reaction-diffusion system with equal diffusion constants: anomalous density decay at large times

We study a two-species reaction-diffusion model where A+A-->, A+B-->, and B+B-->, with annihilation rates lambda(0), delta(0)>lambda(0), and lambda(0), respectively. The initial particle configuration is taken to be randomly mixed with mean densities n(A)(0)>n(B)(0), and with the two species A and B diffusing with the same diffusion constant. A field-theoretic renormalization group analysis suggests that, contrary to expectation, the large-time density of the minority species decays at the same rate as the majority when d