Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.Comment: 13 pages, 8 figures. pdfLaTeX with RevTeX4 macro

Estimating topological entropy from the motion of stirring rods

Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch' on the rods. But how good is this lower bound? We present examples from numerical simulations and speculate on what affects the 'gap' between the lower bound and the measured topological entropy. The key is the sign of the rod motion's action on first homology of the orientation double cover of the punctured disk.Comment: 10 pages, 20 figures. IUTAM Procedia style (included). Submitted to volume "Topological Fluid Dynamics II.

Loop-closure principles in protein folding

Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.Comment: 27 pages, 5 figures; to appear in Archives of Biochemistry and Biophysic

Macromolecular crowding modulates folding mechanism of alpha/beta protein apoflavodoxin

Protein dynamics in cells may be different from that in dilute solutions in vitro since the environment in cells is highly concentrated with other macromolecules. This volume exclusion due to macromolecular crowding is predicted to affect both equilibrium and kinetic processes involving protein conformational changes. To quantify macromolecular crowding effects on protein folding mechanisms, here we have investigated the folding energy landscape of an alpha/beta protein, apoflavodoxin, in the presence of inert macromolecular crowding agents using in silico and in vitro approaches. By coarse-grained molecular simulations and topology-based potential interactions, we probed the effects of increased volume fraction of crowding agents (phi_c) as well as of crowding agent geometry (sphere or spherocylinder) at high phi_c. Parallel kinetic folding experiments with purified Desulfovibro desulfuricans apoflavodoxin in vitro were performed in the presence of Ficoll (sphere) and Dextran (spherocylinder) synthetic crowding agents. In conclusion, we have identified in silico crowding conditions that best enhance protein stability and discovered that upon manipulation of the crowding conditions, folding routes experiencing topological frustrations can be either enhanced or relieved. The test-tube experiments confirmed that apoflavodoxin's time-resolved folding path is modulated by crowding agent geometry. We propose that macromolecular crowding effects may be a tool for manipulation of protein folding and function in living cells.Comment: to appear in Biophysical Journal (2009). to appear in Biophysical Journal (2009

A simple measure of native-state topology and chain connectivity predicts the folding rates of two-state proteins with and without crosslinks

The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO or 'localness' of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the the native contacts. These changes in localness can be taken into account by the graph-theoretical concept of effective contact order (ECO). The relative ECO, however, the natural extension of the relative CO for proteins with crosslinks, overestimates the changes in the folding rates caused by crosslinks. We suggest here a novel measure of native-state topology, the relative logCO, and its natural extension, the relative logECO. The relative logCO is the average value for the logarithm of the CO of all contacts, divided by the logarithm of the chain length. The relative log(E)CO reproduces the folding rates of a set of 26 two-state proteins without crosslinks with essentially the same high correlation coefficient as the relative CO. In addition, it also captures the folding rates of 8 two-state proteins with crosslinks.Comment: 13 pages, 2 tables, and 2 figure

Protein folding in high-dimensional spaces:hypergutters and the role of non-native interactions

We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential barriers will commonly be found, even in "two-state"proteins; (4) very early times will show charactreristic departures from single-exponential kinetics; (5) contributions of non-native interactions to phi-values are calculable, and may be significant. The example of a three-helix bundle is treated in more detail as an illustration. The model also shows that high-dimensional structures provide conceptual relations between the "folding funnel", "diffusion-collision", "nucleation-condensation" and "topomer search" models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in non-native, rather than native interactions. The predictions are related to very recent findings in experiment and simulation.Comment: Submitted to Biophys.

How native state topology affects the folding of Dihydrofolate Reductase and Interleukin-1beta

The overall structure of the transition state and intermediate ensembles experimentally observed for Dihydrofolate Reductase and Interleukin-1beta can be obtained utilizing simplified models which have almost no energetic frustration. The predictive power of these models suggest that, even for these very large proteins with completely different folding mechanisms and functions, real protein sequences are sufficiently well designed and much of the structural heterogeneity observed in the intermediates and the transition state ensembles is determined by topological effects.Comment: Proc. Natl. Acad. Sci. USA, in press (11 pages, 4 color PS figures) Higher resolution PS files can be found at http://www-physics.ucsd.edu/~cecilia/pub_list.htm