Variation in per capita interaction strength: Thresholds due to nonlinear dynamics and nonequilibrium conditions

I measured the strength of interaction between a marine herbivore and its growing resource over a realistic range of absolute and relative abundances. The herbivores (hermit crabs: Pagurus spp.) have slow and/or weak functional and numerical responses to epiphytic diatoms (Isthmia nervosa), which show logistic growth in the absence of consumers. By isolating this interaction in containers in the field, I mimicked many of the physical and biological variables characteristic of the intertidal while controlling the densities of focal species. The per capita effects of consumers on the population dynamics of their resource (i.e., interaction strength) were defined by using the relationship between hermit crab density and proportional change in the resource. When this relationship is fit by a Weibull function, a single parameter distinguishes constant interaction strength from one that varies as a function of density. Constant interaction strength causes the proportion of diatoms to fall linearly or proportionally as hermit crab density increases whereas per capita effects that increase with density cause an accelerating decline. Although many mathematical models of species interactions assume linear dynamics and invariant parameters, at least near equilibrium, the per capita effects of hermit crabs on diatoms varied substantially, apparently crossing a threshold from weak to strong when consumption exceeded resource production. This threshold separates a domain of coexistence from one of local extinction of the resource. Such thresholds may help explain trophic cascades, resource compensation, and context-dependent interaction strengths, while indicating a way to predict trophic effects, despite nonlinearities, as a function of vital rates.

Nonequilibrium mechanism of transcription termination from observations of single RNA polymerase molecules

Cessation of transcription at specific terminator DNA sequences is used by viruses, bacteria, and eukaryotes to regulate the expression of downstream genes, but the mechanisms of transcription termination are poorly characterized. To elucidate the kinetic mechanism of termination at the intrinsic terminators of enteric bacteria, we observed, by using single-molecule light microscopy techniques, the behavior of surface-immobilized Escherichia coli RNA polymerase (RNAP) molecules in vitro. An RNAP molecule remains at a canonical intrinsic terminator for ≈64 s before releasing DNA, implying the formation of an elongation-incompetent (paused) intermediate by transcription complexes that terminate but not by those that read through the terminator. Analysis of pause lifetimes establishes a complete minimal mechanism of termination in which paused intermediate formation is both necessary and sufficient to induce release of RNAP at the terminator. The data suggest that intrinsic terminators function by a nonequilibrium process in which terminator effectiveness is determined by the relative rates of nucleotide addition and paused state entry by the transcription complex.

Free energy reconstruction from nonequilibrium single-molecule pulling experiments

Laser tweezers and atomic force microscopes are increasingly used to probe the interactions and mechanical properties of individual molecules. Unfortunately, using such time-dependent perturbations to force rare molecular events also drives the system away from equilibrium. Nevertheless, we show how equilibrium free energy profiles can be extracted rigorously from repeated nonequilibrium force measurements on the basis of an extension of Jarzynski's remarkable identity between free energies and the irreversible work.

Definition of amide protection factors for early kinetic intermediates in protein folding

Hydrogen–deuterium exchange experiments have been used previously to investigate the structures of well defined states of a given protein. These include the native state, the unfolded state, and any intermediates that can be stably populated at equilibrium. More recently, the hydrogen–deuterium exchange technique has been applied in kinetic labeling experiments to probe the structures of transiently formed intermediates on the kinetic folding pathway of a given protein. From these equilibrium and nonequilibrium studies, protection factors are usually obtained. These protection factors are defined as the ratio of the rate of exchange of a given backbone amide when it is in a fully solvent-exposed state (usually obtained from model peptides) to the rate of exchange of that amide in some state of the protein or in some intermediate on the folding pathway of the protein. This definition is straightforward for the case of equilibrium studies; however, it is less clear-cut for the case of transient kinetic intermediates. To clarify the concept for the case of burst-phase intermediates, we have introduced and mathematically defined two different types of protection factors: one is Pstruc, which is more related to the structure of the intermediate, and the other is Papp, which is more related to the stability of the intermediate. Kinetic hydrogen–deuterium exchange data from disulfide-intact ribonuclease A and from cytochrome c are discussed to explain the use and implications of these two definitions.

Spectral diffusion and the energy landscape of a protein

We present a novel type of spectral diffusion experiment in the millikelvin range to characterize the energy landscape of a protein as compared with that of a glass. We measure the time evolution of spectral holes for more than 300 hr after well-defined initial nonequilibrium conditions. We show that the model of noninteracting two-level systems can describe spectral diffusion in the glass, but fails for the protein. Our results further demonstrate that randomness in the energy landscape of a protein shows features of organization. There are “deep minimum” states separated by barriers, the heights of which we are able to estimate. The energy landscape of a glass is featureless by comparison.

Implications of macromolecular crowding for signal transduction and metabolite channeling

The effect of different total enzyme concentrations on the flux through the bacterial phosphoenolpyruvate:carbohydrate phosphotransferase system (PTS) in vitro was determined by measuring PTS-mediated carbohydrate phosphorylation at different dilutions of cell-free extract of Escherichia coli. The dependence of the flux on the protein concentration was more than linear but less than quadratic. The combined flux–response coefficient of the four enzymes constituting the glucose PTS decreased slightly from values of ≈1.8 with increasing protein concentrations in the assay. Addition of the macromolecular crowding agents polyethylene glycol (PEG) 6000 and PEG 35000 led to a sharper decrease in the combined flux–response coefficient, in one case to values of ≈1. PEG 6000 stimulated the PTS flux at lower protein concentrations and inhibited the flux at higher protein concentrations, with the transition depending on the PEG 6000 concentration. This suggests that macromolecular crowding decreases the dissociation rate constants of enzyme complexes. High concentrations of the microsolute glycerol did not affect the combined flux–response coefficient. The data could be explained with a kinetic model of macromolecular crowding in a two-enzyme group-transfer pathway. Our results suggest that, because of the crowded environment in the cell, the different PTS enzymes form complexes that live long on the time-scale of their turnover. The implications for the metabolic behavior and control properties of the PTS, and for the effect of macromolecular crowding on nonequilibrium processes, are discussed.

Directionality principles in thermodynamics and evolution

Directionality in populations of replicating organisms can be parametrized in terms of a statistical concept: evolutionary entropy. This parameter, a measure of the variability in the age of reproducing individuals in a population, is isometric with the macroscopic variable body size. Evolutionary trends in entropy due to mutation and natural selection fall into patterns modulated by ecological and demographic constraints, which are delineated as follows: (i) density-dependent conditions (a unidirectional increase in evolutionary entropy), and (ii) density-independent conditions, (a) slow exponential growth (an increase in entropy); (b) rapid exponential growth, low degree of iteroparity (a decrease in entropy); and (c) rapid exponential growth, high degree of iteroparity (random, nondirectional change in entropy). Directionality in aggregates of inanimate matter can be parametrized in terms of the statistical concept, thermodynamic entropy, a measure of disorder. Directional trends in entropy in aggregates of matter fall into patterns determined by the nature of the adiabatic constraints, which are characterized as follows: (i) irreversible processes (an increase in thermodynamic entropy) and (ii) reversible processes (a constant value for entropy). This article analyzes the relation between the concepts that underlie the directionality principles in evolutionary biology and physical systems. For models of cellular populations, an analytic relation is derived between generation time, the average length of the cell cycle, and temperature. This correspondence between generation time, an evolutionary parameter, and temperature, a thermodynamic variable, is exploited to show that the increase in evolutionary entropy that characterizes population processes under density-dependent conditions represents a nonequilibrium analogue of the second law of thermodynamics.

Ultrafast diffraction and structural dynamics: The nature of complex molecules far from equilibrium

Studies of molecular structures at or near their equilibrium configurations have long provided information on their geometry in terms of bond distances and angles. Far-from-equilibrium structures are relatively unknown—especially for complex systems—and generally, neither their dynamics nor their average geometries can be extrapolated from equilibrium values. For such nonequilibrium structures, vibrational amplitudes and bond distances play a central role in phenomena such as energy redistribution and chemical reactivity. Ultrafast electron diffraction, which was developed to study transient molecular structures, provides a direct method for probing the nature of complex molecules far from equilibrium. Here we present our ultrafast electron diffraction observations of transient structures for two cyclic hydrocarbons. At high internal energies of ≈4 eV, these molecules display markedly different behavior. For 1,3,5-cycloheptatriene, excitation results in the formation of hot ground-state structures with bond distances similar to those of the initial structure, but with nearly three times the average vibrational amplitude. Energy is redistributed within 5 ps, but with a negative temperature characterizing the nonequilibrium population. In contrast, the ring-opening reaction of 1,3-cyclohexadiene is shown to result in hot structures with a C—C bond distance of over 1.7 Å, which is 0.2 Å away from any expected equilibrium value. Even up to 400 ps, energy remains trapped in large-amplitude motions comprised of torsion and asymmetric stretching. These studies promise a new direction for studying structural dynamics in nonequilibrium complex systems.

Kinetics of self-assembling microtubules: an "inverse problem" in biochemistry.

Experimental time series for a nonequilibrium reaction may in some cases contain sufficient data to determine a unique kinetic model for the reaction by a systematic mathematical analysis. As an example, a kinetic model for the self-assembly of microtubules is derived here from turbidity time series for solutions in which microtubules assemble. The model may be seen as a generalization of Oosawa's classical nucleation-polymerization model. It reproduces the experimental data with a four-stage nucleation process and a critical nucleus of 15 monomers.