▪ Abstract The energy landscape theory of protein folding is a statistical description of a protein's potential surface. It assumes that folding occurs through organizing an ensemble of structures rather than through only a few uniquely defined structural intermediates. It suggests that the most realistic model of a protein is a minimally frustrated heteropolymer with a rugged funnel-like landscape biased toward the native structure. This statistical description has been developed using tools from the statistical mechanics of disordered systems, polymers, and phase transitions of finite systems. We review here its analytical background and contrast the phenomena in homopolymers, random heteropolymers, and protein-like heteropolymers that are kinetically and thermodynamically capable of folding. The connection between these statistical concepts and the results of minimalist models used in computer simulations is discussed. The review concludes with a brief discussion of how the theory helps in the interpre...

THEORY OF PROTEIN FOLDING: The Energy Landscape Perspective

A large source of uncertainty in present understanding of the global carbon cycle is the distribution and dynamics of the soil organic carbon reservoir. Most of the organic carbon in soils is degraded to inorganic forms slowly, on timescales from centuries to millennia1. Soil minerals are known to play a stabilizing role, but how spatial and temporal variation in soil mineralogy controls the quantity and turnover of long-residence-time organic carbon is not well known2. Here we use radiocarbon analyses to explore interactions between soil mineralogy and soil organic carbon along two natural gradients—of soil-age and of climate—in volcanic soil environments. During the first ∼150,000 years of soil development, the volcanic parent material weathered to metastable, non-crystalline minerals. Thereafter, the amount of non-crystalline minerals declined, and more stable crystalline minerals accumulated. Soil organic carbon content followed a similar trend, accumulating to a maximum after 150,000 years, and then decreasing by 50% over the next four million years. A positive relationship between non-crystalline minerals and organic carbon was also observed in soils through the climate gradient, indicating that the accumulation and subsequent loss of organic matter were largely driven by changes in the millennial scale cycling of mineral-stabilized carbon, rather than by changes in the amount of fast-cycling organic matter or in net primary productivity. Soil mineralogy is therefore important in determining the quantity of organic carbon stored in soil, its turnover time, and atmosphere–ecosystem carbon fluxes during long-term soil development; this conclusion should be generalizable at least to other humid environments.

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Mineral control of soil organic carbon storage and turnover

The cytoplasm of vertebrate cells contains three distinct filamentous biopolymers, the microtubules, microfilaments, and intermediate filaments. The basic structural elements of these three filaments are linear polymers of the proteins tubulin, actin, and vimentin or another related intermediate filament protein, respectively. The viscoelastic properties of cytoplasmic filaments are likely to be relevant to their biologic function, because their extreme length and rodlike structure dominate the rheologic behavior of cytoplasm, and changes in their structure may cause gel-sol transitions observed when cells are activated or begin to move. This paper describes parallel measurements of the viscoelasticity of tubulin, actin, and vimentin polymers. The rheologic differences among the three types of cytoplasmic polymers suggest possible specialized roles for the different classes of filaments in vivo. Actin forms networks of highest rigidity that fluidize at high strains, consistent with a role in cell motility in which stable protrusions can deform rapidly in response to controlled filament rupture. Vimentin networks, which have not previously been studied by rheologic methods, exhibit some unusual viscoelastic properties not shared by actin or tubulin. They are less rigid (have lower shear moduli) at low strain but harden at high strains and resist breakage, suggesting they maintain cell integrity. The differences between F-actin and vimentin are optimal for the formation of a composite material with a range of properties that cannot be achieved by either polymer alone. Microtubules are unlikely to contribute significantly to interphase cell rheology alone, but may help stabilize the other networks.

https://rupress.org/jcb/article-pdf/113/1/155/1061095/155.pdf

Viscoelastic properties of vimentin compared with other filamentous biopolymer networks.

Publisher Summary The chapter describes the understanding of the physics and physical chemistry of sickle cell hemoglobin polymerization in solutions and in red cells. The polymerization of sickle cell hemoglobin has probably become the best understood of all protein self-assembly systems. The structure of the hemoglobin S molecule, the structure of the various aggregated forms of hemoglobin S, and the structural analysis of the polymers are discussed in the chapter. The chapter discusses the thermodynamics of hemoglobin S polymerization, and includes a description of the nonideal behavior of concentrated hemoglobin S solutions and the effects of physiologically relevant variables, especially oxygen, and the presence of non-S hemoglobins on the polymerization process. Understanding the polymerization process is not only important for understanding the pathophysiology of sickle cell disease, but is critical to the major problem of developing a specific therapy that could be used in the treatment of patients. The kinetic and thermodynamic studies have played a major role by providing relevant and sensitive assays for potential therapeutic agents. The results of the thermodynamic and kinetic studies of solutions are used to explain various properties of cells, including morphological and rheological properties.

Sickle cell hemoglobin polymerization.

Long-chain polymer molecules are approximated by a model consisting of isodimensional segments which tend to arrange themselves in co-linear succession. A fraction f of the bonds is assumed to be ‘bent’ out of the co-linear direction of preceding segments. The fact that the free energy of solution derived using the lattice model separates into (a) a mixing term dependent only on the concentration, and (b) a disorientation term depending on f but not on the concentration, leads to the concept of an equilibrium ‘flexibility’ f which is characteristic of the polymer chain at a given temperature. This f must exceed a specified critical value, which decreases with dilution, if the disordered phase usually considered to occur is to be more stable than an ordered one in which the chains assume their preferred rod-like form and lie parallel to one another. Transition between the two states should be co-operative, and should involve a latent heat (owing to the change in intramolecular configurational energy) even in the absence of a change in the intermolecular energy. The concept of a phase transition due solely to intramolecular forces thus arises. Configurational dimensions of various polymers are such as to suggest that inflexibility may be a dominant factor in causing them to crystallize. This is certainly true for cellulose derivatives, probably also for polytetrafluoroethylene, and possible for polymethylene as well. Intramolecular forces favouring the rod-like form are, doubtless of major importance also in bringing about crystallization of proteins in the α-form.

Statistical thermodynamics of semi-flexible chain molecules

A partition function for a system of rigid rod-like particles with partial orientation about an axis is derived through the use of a modified lattice model. In the limit of perfect orientation the partition function reduces to the ideal mixing law ; for complete disorientation it corresponds to the polymer mixing law for rigid chains. A general expression is given for the free energy of mixing as a function of the mole numbers, the axis ratio of the solute particles, and a disorientation parameter. This function passes through a minimum followed by a maximum with increase in the disorientation parameter, provided the latter exceeds a critical value which is 2e for the pure solute and which increases with dilution. Assigning this parameter the value which minimizes the free energy, the chemical potentials display discontinuities a t the concentration a t which the minimum first appears. Separation into an isotropic phase and a some what more concentrated anisotropic phase arises because of the discontinuity, in confirmation of the theories of Onsager and Isihara, which treat only the second virial coefficient. Phase separation thus arises as a consequence of particle asymmetry, unassisted by an energy term . Whereas for a large-particle asymmetry both phases in equilibrium are predicted to be fairly dilute when mixing is athermal, a comparatively small positive energy of interaction causes the concentration in the anisotropic phase to increase sharply, while the concentration in the isotropic phase becomes vanishingly small. The theory offers a statistical mechanical basis for interpreting precipitation of rod-like colloidal particles with the formation of fibrillar structures such as are prominent in the fibrous proteins. The asymmetry of tobacco mosaic virus particles (with or without inclusion of their electric double layers) is insufficient alone to explain the well-known phase separation which occurs from their dilute solutions at very low ionic strengths. Higher-order interaction between electric double layers appears to be a major factor in bringing about dilute phase separation for these and other asymmetric colloidal particles bearing large charges, as was pointed out previously by Oster.

P. J. Flory

Papers

Phase equilibria in solutions of rod-like particles