Transition States in Protein Folding Kinetics: The Structural Interpretation of Phi-values

Phi-values are experimental measures of the effects of mutations on the folding kinetics of a protein. A central question is which structural information Phi-values contain about the transition state of folding. Traditionally, a Phi-value is interpreted as the 'nativeness' of a mutated residue in the transition state. However, this interpretation is often problematic because it assumes a linear relation between the nativeness of the residue and its free-energy contribution. We present here a better structural interpretation of Phi-values for mutations within a given helix. Our interpretation is based on a simple physical model that distinguishes between secondary and tertiary free-energy contributions of helical residues. From a linear fit of our model to the experimental data, we obtain two structural parameters: the extent of helix formation in the transition state, and the nativeness of tertiary interactions in the transition state. We apply our model to all proteins with well-characterized helices for which more than 10 Phi-values are available: protein A, CI2, and protein L. The model captures nonclassical Phi-values 1 in these helices, and explains how different mutations at a given site can lead to different Phi-values.Comment: 26 pages, 7 figures, 5 table

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.Comment: 9 pages, 6 figures, 1 tabl

Adhesion-induced phase separation of multiple species of membrane junctions

A theory is presented for the membrane junction separation induced by the adhesion between two biomimetic membranes that contain two different types of anchored junctions (receptor/ligand complexes). The analysis shows that several mechanisms contribute to the membrane junction separation. These mechanisms include (i) the height difference between type-1 and type-2 junctions is the main factor which drives the junction separation, (ii) when type-1 and type-2 junctions have different rigidities against stretch and compression, the ``softer'' junctions are the ``favored'' species, and the aggregation of the softer junction can occur, (iii) the elasticity of the membranes mediates a non-local interaction between the junctions, (iv) the thermally activated shape fluctuations of the membranes also contribute to the junction separation by inducing another non-local interaction between the junctions and renormalizing the binding energy of the junctions. The combined effect of these mechanisms is that when junction separation occurs, the system separates into two domains with different relative and total junction densities.Comment: 23 pages, 6 figure

Segregation of receptor-ligand complexes in cell adhesion zones: Phase diagrams and role of thermal membrane roughness

The adhesion zone of immune cells, the 'immunological synapse', exhibits characteristic domains of receptor-ligand complexes. The domain formation is likely caused by a length difference of the receptor-ligand complexes, and has been investigated in experiments in which T cells adhere to supported membranes with anchored ligands. For supported membranes with two types of anchored ligands, MHCp and ICAM1, that bind to the receptors TCR and LFA1 in the cell membrane, the coexistence of domains of TCR-MHCp and LFA1-ICAM1 complexes in the cell adhesion zone has been observed for a wide range of ligand concentrations and affinities. For supported membranes with long and short ligands that bind to the same cell receptor CD2, in contrast, domain coexistence has been observed for a rather narrow ratio of ligand concentrations. In this article, we determine detailed phase diagrams for cells adhering to supported membranes with a statistical-physical model of cell adhesion. We find a characteristic difference between the adhesion scenarios in which two types of ligands in a supported membrane bind (i) to the same cell receptor or (ii) to two different cell receptors, which helps to explain the experimental observations. Our phase diagrams fully include thermal shape fluctuations of the cell membranes on nanometer scales, which lead to a critical point for the domain formation and to a cooperative binding of the receptors and ligands.Comment: 23 pages, 6 figure

The linear tearing instability in three dimensional, toroidal gyrokinetic simulations

Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro kinetic turbulence code, GKW . The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate and frequency of the mode were investigated by varying the current profile, collisionality and the pressure gradients. Both collision-less and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absense of a pressure gradient is observed which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However the island rotation reverses direction at high collisionality. The growth rate is found to follow a $\eta^{1/7}$ scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability using resistive MHD theory, however a modification due to toroidal coupling and pressure effects is seen

Phi-values in protein folding kinetics have energetic and structural components

Phi-values are experimental measures of how the kinetics of protein folding is changed by single-site mutations. Phi-values measure energetic quantities, but are often interpreted in terms of the structures of the transition state ensemble. Here we describe a simple analytical model of the folding kinetics in terms of the formation of protein substructures. The model shows that Phi-values have both structural and energetic components. In addition, it provides a natural and general interpretation of "nonclassical" Phi-values (i.e., less than zero, or greater than one). The model reproduces the Phi-values for 20 single-residue mutations in the alpha-helix of the protein CI2, including several nonclassical Phi-values, in good agreement with experiments.Comment: 15 pages, 3 figures, 1 tabl

Fluctuation induced interactions between domains in membranes

We study a model lipid bilayer composed of a mixture of two incompatible lipid types which have a natural tendency to segregate in the absence of membrane fluctuations. The membrane is mechanically characterized by a local bending rigidity $\kappa(\phi)$ which varies with the average local lipid composition $\phi$. We show, in the case where $\kappa$ varies weakly with $\phi$, that the effective interaction between lipids of the same type can either be everywhere attractive or can have a repulsive component at intermediate distances greater than the typical lipid size. When this interaction has a repulsive component, it can prevent macro-phase separation and lead to separation in mesophases with a finite domain size. This effect could be relevant to certain experimental and numerical observations of mesoscopic domains in such systems.Comment: 9 pages RevTex, 1 eps figur

Dynamic phase separation of fluid membranes with rigid inclusions

Membrane shape fluctuations induce attractive interactions between rigid inclusions. Previous analytical studies showed that the fluctuation-induced pair interactions are rather small compared to thermal energies, but also that multi-body interactions cannot be neglected. In this article, it is shown numerically that shape fluctuations indeed lead to the dynamic separation of the membrane into phases with different inclusion concentrations. The tendency of lateral phase separation strongly increases with the inclusion size. Large inclusions aggregate at very small inclusion concentrations and for relatively small values of the inclusions' elastic modulus.Comment: 6 pages, 6 figure