3D trajectories adopted by coding and regulatory DNA elements: first-passage times for genomic interactions.

During B lymphocyte development, immunoglobulin heavy-chain variable (VH), diversity (DH), and joining (JH) segments assemble to generate a diverse antigen receptor repertoire. Here, we have marked the distal VH and DH-JH-Eµ regions with Tet-operator binding sites and traced their 3D trajectories in pro-B cells transduced with a retrovirus encoding Tet-repressor-EGFP. We found that these elements displayed fractional Langevin motion (fLm) due to the viscoelastic hindrance from the surrounding network of proteins and chromatin fibers. Using fractional Langevin dynamics modeling, we found that, with high probability, DHJH elements reach a VH element within minutes. Spatial confinement emerged as the dominant parameter that determined the frequency of such encounters. We propose that the viscoelastic nature of the nuclear environment causes coding elements and regulatory elements to bounce back and forth in a spring-like fashion until specific genomic interactions are established and that spatial confinement of topological domains largely controls first-passage times for genomic interactions.

Decoding the mechanical fingerprints of biomolecules.

The capacity of biological macromolecules to act as exceedingly sophisticated and highly efficient cellular machines - switches, assembly factors, pumps, or motors - is realized through their conformational transitions, that is, their folding into distinct shapes and selective binding to other molecules. Conformational transitions can be induced, monitored, and manipulated by pulling individual macromolecules apart with an applied force. Pulling experiments reveal, for a given biomolecule, the relationship between applied force and molecular extension. Distinct signatures in the force-extension relationship identify a given biomolecule and thus serve as the molecule's 'mechanical fingerprints'. But, how can these fingerprints be decoded to uncover the energy barriers crossed by the molecule in the course of its conformational transition, as well as the associated timescales? This review summarizes a powerful class of approaches to interpreting single-molecule force spectroscopy measurements - namely, analytically tractable approaches. On the fundamental side, analytical theories have the power to reveal the unifying principles underneath the bewildering diversity of biomolecules and their behaviors. On the practical side, analytical expressions that result from these theories are particularly well suited for a direct fit to experimental data, yielding the important parameters that govern biological processes at the molecular level.

Distinguishing Signatures of Multipathway Conformational Transitions.

The folding and binding of biomolecules into functional conformations are thought to be commonly mediated by multiple pathways rather than a unique route. Yet even in experiments where one can "see" individual conformational transitions, their stochastic nature generally precludes one from determining whether the transitions occurred through one or multiple pathways. We establish model-free, observable signatures in the response of macromolecules to force that unambiguously identify multiple pathways-even when the pathways themselves cannot be resolved. The unified analytical description reveals that, through multiple pathways, the response of molecules to external forces can be shaped in diverse ways, resulting in a rich design space for a tailored biological function already at the single-molecule level.

A transformation for the mechanical fingerprints of complex biomolecular interactions.

Biological processes are carried out through molecular conformational transitions, ranging from the structural changes within biomolecules to the formation of macromolecular complexes and the associations between the complexes themselves. These transitions cover a vast range of timescales and are governed by a tangled network of molecular interactions. The resulting hierarchy of interactions, in turn, becomes encoded in the experimentally measurable "mechanical fingerprints" of the biomolecules, their force-extension curves. However, how can we decode these fingerprints so that they reveal the kinetic barriers and the associated timescales of a biological process? Here, we show that this can be accomplished with a simple, model-free transformation that is general enough to be applicable to molecular interactions involving an arbitrarily large number of kinetic barriers. Specifically, the transformation converts the mechanical fingerprints of the system directly into a map of force-dependent rate constants. This map reveals the kinetics of the multitude of rate processes in the system beyond what is typically accessible to direct measurements. With the contributions from individual barriers to the interaction network now "untangled", the map is straightforward to analyze in terms of the prominent barriers and timescales. Practical implementation of the transformation is illustrated with simulated biomolecular interactions that comprise different patterns of complexity--from a cascade of activation barriers to competing dissociation pathways.

Statistical mechanics of viral entry.

Viruses that have lipid-membrane envelopes infect cells by fusing with the cell membrane to release viral genes. Membrane fusion is known to be hindered by high kinetic barriers associated with drastic structural rearrangements-yet viral infection, which occurs by fusion, proceeds on remarkably short time scales. Here, we present a quantitative framework that captures the principles behind the invasion strategy shared by all enveloped viruses. The key to this strategy-ligand-triggered conformational changes in the viral proteins that pull the membranes together-is treated as a set of concurrent, bias field-induced activated rate processes. The framework results in analytical solutions for experimentally measurable characteristics of virus-cell fusion and enables us to express the efficiency of the viral strategy in quantitative terms. The predictive value of the theory is validated through simulations and illustrated through recent experimental data on influenza virus infection.

Conformational dynamics through an intermediate.

The self-assembly of biological and synthetic nanostructures commonly proceeds via intermediate states. In living systems in particular, the intermediates have the capacity to tilt the balance between functional and potentially fatal behavior. This work develops a statistical mechanical treatment of conformational dynamics through an intermediate under a variable force. An analytical solution is derived for the key experimentally measurable quantity-the distribution of forces at which a conformational transition occurs. The solution reveals rich kinetics over a broad range of parameters and enables one to locate the intermediate and extract the activation barriers and rate constants.

Kinetics and energetics of biomolecular folding and binding.

The ability of biomolecules to fold and to bind to other molecules is fundamental to virtually every living process. Advanced experimental techniques can now reveal how single biomolecules fold or bind against mechanical force, with the force serving as both the regulator and the probe of folding and binding transitions. Here, we present analytical expressions suitable for fitting the major experimental outputs from such experiments to enable their analysis and interpretation. The fit yields the key determinants of the folding and binding processes: the intrinsic on-rate and the location and height of the activation barrier.

Single molecules in an extension clamp: extracting rates and activation barriers.

When a macromolecule, held at a fixed end-to-end separation, undergoes conformational rearrangements, the fluctuating mechanical force generated by the molecule can be used as a reporter of the molecular conformational dynamics. We present an analytical framework for extracting the intrinsic rates of conformational transitions and the locations and heights of the rate-limiting barriers from such extension clamp measurements. The unique nature of the bias imposed by the extension clamp on the activation barriers allows access to biomolecular transitions currently not accessible in pulling experiments. A mapping rule is established between the outputs of different types of experiments, providing information about poorly accessible regions of the molecular landscape.

Locating the barrier for folding of single molecules under an external force.

Single-molecule pulling experiments on the folding of biomolecules are usually interpreted with one-dimensional models in which the dynamics occurs on the "pulling coordinate." Paradoxically, the free-energy profile along this coordinate may lack a refolding barrier, yet a barrier is known to exist for folding; thus, it has been argued that pulling experiments do not probe folding. Here, we show that transitions monitored in pulling experiments probe the true folding barrier but that the barrier may be hidden in the projection onto the pulling coordinate. However, one-dimensional theory using the pulling coordinate still yields physically meaningful energy landscape parameters.

Biomolecules under mechanical stress: a simple mechanism of complex behavior.

The unfolding of a biomolecule by stretching force is commonly treated theoretically as one-dimensional dynamics along the reaction coordinate coincident with the direction of pulling. Here we explore a situation, particularly relevant to complex biomolecules, when the pulling direction alone is not an adequate reaction coordinate for the unfolding or rupture process. We show that in this case the system can respond to pulling force in unusual ways. Our theory points out a remarkably simple, but largely overlooked, mechanism of the complex responses of biomolecules to force. The mechanism originates from the basic property of the transition state to change its structure under applied force. A relationship is established between a key experimental observable--force-dependent lifetime--and the microscopic properties of the biomolecule in the form of an analytical solution to the problem of a force-induced molecular transition in two dimensions. The theory is applicable to biological contexts ranging from protein folding to ligand-receptor interactions.

Theory, analysis, and interpretation of single-molecule force spectroscopy experiments.

Dynamic force spectroscopy probes the kinetic and thermodynamic properties of single molecules and molecular assemblies. Here, we propose a simple procedure to extract kinetic information from such experiments. The cornerstone of our method is a transformation of the rupture-force histograms obtained at different force-loading rates into the force-dependent lifetimes measurable in constant-force experiments. To interpret the force-dependent lifetimes, we derive a generalization of Bell's formula that is formally exact within the framework of Kramers theory. This result complements the analytical expression for the lifetime that we derived previously for a class of model potentials. We illustrate our procedure by analyzing the nanopore unzipping of DNA hairpins and the unfolding of a protein attached by flexible linkers to an atomic force microscope. Our procedure to transform rupture-force histograms into the force-dependent lifetimes remains valid even when the molecular extension is a poor reaction coordinate and higher-dimensional free-energy surfaces must be considered. In this case the microscopic interpretation of the lifetimes becomes more challenging because the lifetimes can reveal richer, and even nonmonotonic, dependence on the force.

A theory of synaptic transmission.

Rapid and precise neuronal communication is enabled through a highly synchronous release of signaling molecules neurotransmitters within just milliseconds of the action potential. Yet neurotransmitter release lacks a theoretical framework that is both phenomenologically accurate and mechanistically realistic. Here, we present an analytic theory of the action-potential-triggered neurotransmitter release at the chemical synapse. The theory is demonstrated to be in detailed quantitative agreement with existing data on a wide variety of synapses from electrophysiological recordings in vivo and fluorescence experiments in vitro. Despite up to ten orders of magnitude of variation in the release rates among the synapses, the theory reveals that synaptic transmission obeys a simple, universal scaling law, which we confirm through a collapse of the data from strikingly diverse synapses onto a single master curve. This universality is complemented by the capacity of the theory to readily extract, through a fit to the data, the kinetic and energetic parameters that uniquely identify each synapse. The theory provides a means to detect cooperativity among the SNARE complexes that mediate vesicle fusion and reveals such cooperativity in several existing data sets. The theory is further applied to establish connections between molecular constituents of synapses and synaptic function. The theory allows competing hypotheses of short-term plasticity to be tested and identifies the regimes where particular mechanisms of synaptic facilitation dominate or, conversely, fail to account for the existing data for the paired-pulse ratio. The derived trade-off relation between the transmission rate and fidelity shows how transmission failure can be controlled by changing the microscopic properties of the vesicle pool and SNARE complexes. The established condition for the maximal synaptic efficacy reveals that no fine tuning is needed for certain synapses to maintain near-optimal transmission. We discuss the limitations of the theory and propose possible routes to extend it. These results provide a quantitative basis for the notion that the molecular-level properties of synapses are crucial determinants of the computational and information-processing functions in synaptic transmission.

Single-molecule rupture dynamics on multidimensional landscapes.

We explore emergent effects of multidimensionality of the free energy landscape on single-molecule kinetics under constant force. The proposed minimal model reveals the existence of a spectrum of unusual scenarios for the force-dependent lifetime, all of which are shown to occur on a free energy landscape with a single transition state. We present an analytical solution that governs single-molecule responses to a constant force and relates them to microscopic parameters of the system.

Chromosome dynamics near the sol-gel phase transition dictate the timing of remote genomic interactions.

Diverse antibody repertoires are generated through remote genomic interactions involving immunoglobulin variable (VH), diversity (DH) and joining (JH) gene segments. How such interactions are orchestrated remains unknown. Here we develop a strategy to track VH-DHJH motion in B-lymphocytes. We find that VH and DHJH segments are trapped in configurations that allow only local motion, such that spatially proximal segments remain in proximity, while spatially remote segments remain remote. Within a subset of cells, however, abrupt changes in VH-DHJH motion are observed, plausibly caused by temporal alterations in chromatin configurations. Comparison of experimental and simulated data suggests that constrained motion is imposed by a network of cross-linked chromatin chains characteristic of a gel phase, yet poised near the sol phase, a solution of independent chromatin chains. These results suggest that chromosome organization near the sol-gel phase transition dictates the timing of genomic interactions to orchestrate gene expression and somatic recombination.

Pulling direction as a reaction coordinate for the mechanical unfolding of single molecules.

The folding and unfolding kinetics of single molecules, such as proteins or nucleic acids, can be explored by mechanical pulling experiments. Determining intrinsic kinetic information, at zero stretching force, usually requires an extrapolation by fitting a theoretical model. Here, we apply a recent theoretical approach describing molecular rupture in the presence of force to unfolding kinetic data obtained from coarse-grained simulations of ubiquitin. Unfolding rates calculated from simulations over a broad range of stretching forces, for different pulling directions, reveal a remarkable "turnover" from a force-independent process at low force to a force-dependent process at high force, akin to the "roll-over" in unfolding rates sometimes seen in studies using chemical denaturant. While such a turnover in rates is unexpected in one dimension, we demonstrate that it can occur for dynamics in just two dimensions. We relate the turnover to the quality of the pulling direction as a reaction coordinate for the intrinsic folding mechanism. A novel pulling direction, designed to be the most relevant to the intrinsic folding pathway, results in the smallest turnover. Our results are in accord with protein engineering experiments and simulations which indicate that the unfolding mechanism at high force can differ from the intrinsic mechanism. The apparent similarity between extrapolated and intrinsic rates in experiments, unexpected for different unfolding barriers, can be explained if the turnover occurs at low forces.

Mean time-of-flight of photons in transillumination measurements of optically anisotropic tissue with an inclusion.

We study the effect of optical anisotropy on the mean time-of-flight of photons in a slab of turbid medium containing an inclusion whose optical properties differ from those of the bulk. For this analysis the difference in the mean time for a photon introduced into the slab to reach a specified target point with and without the inclusion is calculated. This difference is defined to be a measure of the contrast. The theoretical model is based on a continuous-time random walk on a lattice, which can be solved exactly and furnishes an exact expression for the contrast. Qualitative and quantitative characteristics of the contrast are analysed as functions of the geometric configuration of the system components (locations of the source, the inclusion and the detector), parameters that specify the optical anisotropy of the medium, and either the scattering properties of the inclusion or the lifetime of the small fluorophore in the case of the time-resolved fluorescence experimental configuration.

First-Passage Processes in the Genome.

Many essential processes in biology share a common fundamental step-establishing physical contact between distant segments of DNA. How fast this step is accomplished sets the "speed limit" for the larger-scale processes it enables, whether the process is antibody production by the immune system or tissue differentiation in a developing embryo. This naturally leads us to ask, How long does it take for DNA segments that are strung out over millions of base pairs along the chromatin fiber to find each other in the crowded cell? This question, fundamental to biology, can be recognized as the physics problem of the first-passage time, or the waiting time for the first encounter. Here, we review a number of approaches to revealing the physical principles by which cells solve, with astonishing efficiency, the first-passage problem for remote genomic interactions.

Time-dependent rate coefficients for diffusion-influenced reactions with centrosymmetric potentials.

Simple closed-form expressions are presented for the time-dependent rate coefficients of diffusion-influenced reactions in the presence of spherically symmetric potentials. For diffusion-controlled contact reactions, our expression reproduces the first two terms in both the short- and long-time expansions of the rate coefficient. At intermediate times, agreement with numerical results for the Debye-Hückel potential is found to be within a few percent for a wide range of parameters. For diffusion-influenced contact reactions (described by the radiation boundary condition), the agreement is even better. When the reactivity depends on the distance between the reactants (e.g., exponentially), our analytic result is less accurate, because it reproduces the two terms in the long-time expansion only to the linear order of the reciprocal of the diffusion coefficient. Our results should prove useful in the analysis of experimental data for diffusion-influenced reactions with centrosymmetric interaction potentials.

Time-dependent diffusion coefficients in periodic porous materials.

We derive an approximate solution for the Laplace transform of the time-dependent diffusion coefficient, D(t), of a molecule diffusing in a periodic porous material. In our model, the material is represented by a simple cubic lattice of identical cubic cavities filled with a solvent and connected by small circular apertures in otherwise reflecting cavity walls, the thickness of which can be neglected. The solution describes the decrease of D(t) from its initial value, D(0) = D, where D is the diffusion constant in the free solvent, to its asymptotic value, D(infinity) = D(eff), which is much smaller than D. A simple heuristic formula for the mean-squared displacement of the diffusing molecule is suggested. The theoretically predicted results are in good agreement with the data obtained from Brownian dynamics simulations.