We use the mesoscopic nonequilibrium thermodynamics theory to derive the general kinetic equation of a system in the presence of potential barriers. The result is applied to a description of the evolution of systems whose dynamics is influenced by entropic barriers. We analyze in detail the case of diffusion in a domain of irregular geometry in which the presence of the boundaries induces an entropy barrier when approaching the exact dynamics by a coarsening of the description. The corresponding kinetic equation, named the Fick-Jacobs equation, is obtained, and its validity is generalized through the formulation of a scaling law for the diffusion coefficient which depends on the shape of the boundaries. The method we propose can be useful to analyze the dynamics of systems at the nanoscale where the presence of entropy barriers is a common feature.

We show that transport in the presence of entropic barriers exhibits peculiar characteristics which makes it distinctly different from that occurring through energy barriers. The constrained dynamics yields a scaling regime for the particle current and the diffusion coefficient in terms of the ratio between the work done to the particles and available thermal energy. This interesting property, genuine to the entropic nature of the barriers, can be utilized to effectively control transport through quasi-one-dimensional structures in which irregularities or tortuosity of the boundaries cause entropic effects. The accuracy of the kinetic description has been corroborated by simulations. Applications to different dynamic situations involving entropic barriers are outlined. Transport through quasi-one-dimensional structures as pores, ion channels, and zeolites is ubiquitous in biological and physicochemical systems and constitute a basic mechanism in processes as catalysis, osmosis, and particle separation [1][2][3][4][5][6]. A common characteristic of these systems is the confinement arising from the presence of boundaries which very often exhibit an irregular geometry. Variations of the shape of the structure along the propagation direction implies changes in the number of accessible states of the particles. Consequently, entropy is spatially varying, and the system evolves through entropic barriers, which controls the transport, promoting or hampering the transfer of mass and energy to certain regions. Motion in the system can be induced by the presence of external driving forces supplying the particles with the energy necessary to proceed. The study of the kinetics of the entropic transport, the properties of transport coefficients in far from equilibrium situations, and the possibility for transport control mechanisms are objectives of major importance in the dynamical characterization of those systems.Our purpose in this Letter is to demonstrate that entropic transport exhibits striking features, sometimes counterintuitive, which are different from those observed in the more familiar case with energy barriers [7]. We propose a general scenario describing the dynamics through entropic barriers and show the existence of a scaling regime for the current of particles and the effective diffusion coefficient. The presence of this regime might have important implications in the control of transport.Entropic transport.-The origin of the entropic barriers can be inherent to the intimate nature of the system or may emerge as a consequence of a coarsening of the description employed. A typical example presents the motion of a Brownian particle in an enclosure of varying cross section. This basic situation constitutes the starting point in the study of transport processes in the type of confined systems that are very often encountered at subcellular level, nanoporous materials, and in microfluidic applications. As shown in Ref. [8], the complicated boundary conditions of the diffusion equation in irregular channels can be ...

With few exceptions, the shells (capsids) of sphere-like viruses have the symmetry of an icosahedron and are composed of coat proteins (subunits) assembled in special motifs, the T-number structures. Although the synthesis of artificial protein cages is a rapidly developing area of materials science, the design criteria for self-assembled shells that can reproduce the remarkable properties of viral capsids are only beginning to be understood. We present here a minimal model for equilibrium capsid structure, introducing an explicit interaction between protein multimers (capsomers). Using Monte Carlo simulation we show that the model reproduces the main structures of viruses in vivo (T-number icosahedra) and important nonicosahedral structures (with octahedral and cubic symmetry) observed in vitro. Our model can also predict capsid strength and shed light on genome release mechanisms. Icosahedral symmetry is ubiquitous among spherical viruses (1). A classic example is the cowpea chlorotic mottle virus (CCMV), a well studied RNA virus with a shell composed of exactly 180 identical proteins (subunits) (2, 3). Fig. 1a is a cryo-transmission electron microscopy reconstruction showing 5-and 6-fold morphological units (capsomers), and Fig. 1b shows the arrangement of the individual subunits within the capsomers. The capsid has 6 5-fold rotation axes, 10 3-fold axes, and 15 2-fold axes, the symmetry elements of an icosahedron. The subunits can be divided into 12 capsomers that contain five subunits (pentamers) and 20 capsomers that contain six subunits (hexamers).The current understanding of sphere-like virus structures, like that adopted by CCMV, is based on the Caspar and Klug (CK) ''quasi-equivalence'' principle (4), which provides the foundation of modern structural virology. CK showed that closed icosahedral shells can be constructed from pentamers and hexamers by minimizing the number T of nonequivalent locations that subunits occupy, with the T-number adopting the particular integer values 1, 3, 4, 7, 12, 13, . . . (T ϭ h 2 ϩ k 2 ϩ hk, with h, k equal to nonnegative integers). These CK shells always contain 12 pentamers plus 10 (T-1) hexamers, and these structures have indeed been found to characterize a predominantly large fraction of sphere-like viruses (the CCMV capsid, for example, being a T ϭ 3 structure). Nevertheless, exceptions among WT capsids have been documented (5, 6), including the retroviruses like HIV (7); assembly of subunits with point mutations can also produce breakdown of CK-type capsid icosahedral symmetry (8).It is significant that the icosahedral point group generates the maximum enclosed volume for shells comprised of a given size subunit (4). But the fact that many viruses, including CCMV, self-assemble spontaneously from their molecular components under in vitro conditions (9) indicates that both icosahedral symmetry and the CK construction should be generic features of the free energy minima of aggregates of viral capsid proteins. Because of current computational limitations, direct eval...

Concepts of everyday use such as energy, heat, and temperature have acquired a precise meaning after the development of thermodynamics. Thermodynamics provides the basis for understanding how heat and work are related and the general rules that the macroscopic properties of systems at equilibrium follow. Outside equilibrium and away from macroscopic regimes, most of those rules cannot be applied directly. Here we present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how the probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain Fokker−Planck equations for the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as nonlinear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, such as translocation and stretching.

We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation, which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one-dimensional diffusion. The validity of this approximation, based on the assumption of an instantaneous equilibration of the particle distribution in the cross section of the structure, is analyzed by comparing the different time scales that characterize the problem. A validity criterion is established in terms of the shape of the structure and of the applied force. It is analytically corroborated and verified by numerical simulations that the critical value of the force up to which this description holds true scales as the square of the periodicity of the structure. The criterion can be visualized by means of a diagram representing the regions where the Fick-Jacobs description becomes inaccurate in terms of the scaled force versus the periodicity of the structure.