Simple Statistical Mechanical Model Research Articles

Accurate prediction of protein folding mechanisms by simple structure-based statistical mechanical models

Recent breakthroughs in highly accurate protein structure prediction using deep neural networks have made considerable progress in solving the structure prediction component of the ‘protein folding problem’. However, predicting detailed mechanisms of how proteins fold into specific native structures remains challenging, especially for multidomain proteins constituting most of the proteomes. Here, we develop a simple structure-based statistical mechanical model that introduces nonlocal interactions driving the folding of multidomain proteins. Our model successfully predicts protein folding processes consistent with experiments, without the limitations of protein size and shape. Furthermore, slight modifications of the model allow prediction of disulfide-oxidative and disulfide-intact protein folding. These predictions depict details of the folding processes beyond reproducing experimental results and provide a rationale for the folding mechanisms. Thus, our physics-based models enable accurate prediction of protein folding mechanisms with low computational complexity, paving the way for solving the folding process component of the ‘protein folding problem’.

Simple rose model of water in constant electric field.

A simple two-dimensional statistical mechanical water model, called the rose model, was used in this work. We studied how a homogeneous constant electric field affects the properties of water. The rose model is a very simple model that helps explain the anomalous properties of water. Rose water molecules are represented as two-dimensional Lennard-Jones disks with potentials for orientation-dependent pairwise interactions mimicking formations of hydrogen bonds. The original model is modified by addition of charges for interaction with the electric field. We studied what kind of influence the electric field strength has on the model's properties. To determine the structure and thermodynamics of the rose model under the influence of the electric field we used Monte Carlo simulations. Under the influence of a weak electric field the anomalous properties and phase transitions of the water do not change. On the other hand, the strong fields shift the phase transition points as well as the position of the density maximum.

De novo design of beta hairpin peptides with custom installed halogen-bonding groups.

Much of the success of computational protein design relies on the huge number of experimental crystal structures available for training and testing. Such information is simply not available for designing solution-stable hairpin peptides, much less ones that incorporate non-canonical amino acids. We tackle this problem using a two-step physics-based approach. First, we use a simple statistical mechanical model, trained on a number of designed hairpin sequences known to be stable in solution, to rank a number of potential designs with desired properties. Then, we perform all-atom folding simulations of the highest-ranked sequences using large-scale distributed computing. To validate our predictions, a selection of designed sequences predicted to have sensitive halogen bond donor-acceptor pairs are synthesized and experimentally characterized using solution-state NMR spectroscopy. Our results show the utility of simple statistical mechanical models, coupled with massively parallel simulations for de novo molecular design of non-natural foldamers. Such designed sequences will help develop model systems to understand the role of halogen bonding in peptide stability.

The electric field changes the anomalous properties of the Mercedes Benz water model.

The influence of a homogeneous constant electric field on water properties was assessed. We used a simple two-dimensional statistical mechanical model called the Mercedes-Benz (MB) model of water in the study. The MB water molecules are two-dimensional disks with Gaussian arms that mimic the formation of hydrogen bonds. The model is modified with added charges for interaction with the electric field. The influence of the strength of the electric field on the water's properties was studied using Monte Carlo simulations. The structure and thermodynamics of the water were determined as a function of the strength of the electric field. We observed that the properties and phase transitions of the water in the low strength electric field does not change. In contrast, the high strength electric field shifts boiling and melting points as well as the position of the density maxima. After further increasing the strength of the electric field the density anomaly disappears.

Thermodynamics and surface properties of Fe–V and Fe–Ti liquid alloys

Abstract A simple statistical mechanical model, based on a quasi-lattice approximation in which one assumes the formation of complexes, has been used to study bulk properties, such as free energy of mixing, the thermodynamic activity and enthalpy of mixing, in liquid Fe –Vand Fe –Ti alloys. The energetics and its effect on the alloying behavior of the liquid alloys has been investigated with the aim of correlating bulk phenomena with surface effects. The analysis shows that, assuming the formation of intermetallic complexes of the form Fe2V and FeTi in the liquid alloys, one can explain the energetics of the bulk alloys. Our results for the bulk calculations indicate that Fe –V and Fe –Ti both exhibit a significant tendency for compound formation. From a perusal of the diffusion coefficient D, we observe the same trend towards compound formation, as demonstrated by the chemical short-range order parameter (CSRO) close to the assumed stoichiometric composition. Furthermore, using the model calculations in the bulk, we study some surface properties. Our calculations indicate that Fe segregates to the surface at all bulk compositions in Fe –Vand Fe –Ti liquid, though the segregation effect is more pronounced in the former alloy. The reason for this is that Fe –Ti is a more ordered system than Fe –V and, thus, the driving force for surface segregation in these alloys is their energetics.

Small size effects in open and closed systems: What can we learn from ideal gases about systems with interacting particles?

Small systems have higher surface area-to-volume ratios than macroscopic systems. The thermodynamics of small systems therefore deviates from the description of classical thermodynamics. One consequence of this is that properties of small systems can be dependent on the system's ensemble. By comparing the properties in grand canonical (open) and canonical (closed) systems, we investigate how a small number of particles can induce an ensemble dependence. Emphasis is placed on the insight that can be gained by investigating ideal gases. The ensemble equivalence of small ideal gas systems is investigated by deriving the properties analytically, while the ensemble equivalence of small systems with particles interacting via the Lennard-Jones or the Weeks-Chandler-Andersen potential is investigated through Monte Carlo simulations. For all the investigated small systems, we find clear differences between the properties in open and closed systems. For systems with interacting particles, the difference between the pressure contribution to the internal energy, and the difference between the chemical potential contribution to the internal energy, are both increasing with the number density. The difference in chemical potential is, with the exception of the density dependence, qualitatively described by the analytic formula derived for an ideal gas system. The difference in pressure, however, is not captured by the ideal gas model. For the difference between the properties in the open and closed systems, the response of increasing the particles' excluded volume is similar to the response of increasing the repulsive forces on the system walls. This indicates that the magnitude of the difference between the properties in open and closed systems is related to the restricted movement of the particles in the system. The work presented in this paper gives insight into the mechanisms behind ensemble in-equivalence in small systems, and illustrates how a simple statistical mechanical model, such as the ideal gas, can be a useful tool in these investigations.

Dissecting Multiple Pathways in the Relaxation Dynamics of Helix Coil Transitions with Optimum Dimensionality Reduction.

Fast kinetic experiments with dramatically improved time resolution have contributed significantly to understanding the fundamental processes in protein folding pathways involving the formation of a-helices and b-hairpin, contact formation, and overall collapse of the peptide chain. Interpretation of experimental results through application of a simple statistical mechanical model was key to this understanding. Atomistic description of all events observed in the experimental findings was challenging. Recent advancements in theory, more sophisticated algorithms, and a true long-term trajectory made way for an atomically detailed description of kinetics, examining folding pathways, validating experimental results, and reporting new findings for a wide range of molecular processes in biophysical chemistry. This review describes how optimum dimensionality reduction theory can construct a simplified coarse-grained model with low dimensionality involving a kinetic matrix that captures novel insights into folding pathways. A set of metastable states derived from molecular dynamics analysis generate an optimally reduced dimensionality rate matrix following transition pathway analysis. Analysis of the actual long-term simulation trajectory extracts a relaxation time directly comparable to the experimental results and confirms the validity of the combined approach. The application of the theory is discussed and illustrated using several examples of helix <==> coil transition pathways. This paper focuses primarily on a combined approach of time-resolved experiments and long-term molecular dynamics simulation from our ongoing work.

The Complexity of Analytical Tasks: A Driving Force for Sensor Array Design

Introduction Analytical tasks, the detection goals and scoping of quantitative measurements, are a foundational consideration for the design of chemical sensor arrays. Often sensor arrays are expected to support a variety of possible tasks which are only vaguely defined by both the end user and array designer. Nonetheless, the expedient choices made by or forced upon array designers and fabricators can significantly impact the analytical tasks that could be supported by an array. Analytical tasks may be defined as boundaries in an abstract space defining chemically and practically relevant quantities that are to be measured; chemical sensor and instrument responses may be defined in related spaces specific to their physics. Intelligently finding a mapping between these two abstract spaces defines our problem and informs the impact of analytical tasks upon sensor array design while avoiding the “black box” problems of popular machine learning approaches. This presentation details recent work at NRL that probes the meaning and impact of analytical tasks on chemical sensor array design through general, device agnostic figures of merit developed via techniques originating in statistical mechanics, information theory, information retrieval, and theoretical computer science. Approach Multiple theoretical approaches are explored to determine and measure the impact of analytical task on chemical sensor array design. To begin, we use Ising model-like descriptions of binary sensor responses first described in a biological context by Murugan et. al. [1-3] and subsequently developed by us for chemical sensing with environmental interferents. Analytical tasks are then scoped as molecular and concentration identification against an unknown chemical background. On this framework, we explore the capabilities of a variety of approaches towards assessing the impact of analytical task on sensor array design.The first set of approaches originate in information theory and we consider the mutual information and Kullback-Leibler divergences between different sensor array configurations considered over probability distributions defining the likelihood of that sensor array needing to perform different analytical tasks. Platform-agnostic figures of merit are derived and used to show the impact of analytical tasks on sensor array design.Next, Van Riijsbergen and Melucci [4,5] have developed and elaborated upon an approach to representing and searching information in Hilbert spaces using operator theoretic methods reminiscent to those used in quantum mechanical calculations. By transferring our probabilistic sensor array models to a Hilbert space, we are able to perform similar searches against sensor arrays and by operator methods to ask logical questions of the sensor array using operators that encode analytical tasks and specific exigencies like comparisons to the effectiveness of the array for other analytical tasks.Finally, Almagor et. al. [6-9] have recently developed a finite state machine approach for formally describing the complexity of sensing tasks within formal languages. We attempt to map our binary chemical sensor array model to this methodology and to describe the formal complexity of this toy model for chemical sensing. Analytical tasks are input as restrictions on available formal words in the finite state language. Conclusions Using a simple statistical mechanical model for a chemical sensor array with background interferents, we have been able to probe multiple approaches to quantifying the impact of analytical tasks on the design and relevance of a sensor array for a set of chemical detection problems. Importantly, these approaches avoid drawbacks associated with more empirical, machine learning-based approaches, enabling a more clear understanding of the chemical landscape and how chemical sensor array design interacts with it. Our investigation has explored ways to construct figures of merit for specific families of analytical tasks using methodologies as diverse as information theory, Hilbert space search methods, and complexity techniques for automata-defined sensing.

Statistical Mechanical Model of Gas Adsorption in a Metal-Organic Framework Harboring a Rotaxane Molecular Shuttle.

Metal-organic frameworks (MOFs) are modular and tunable nanoporous materials with applications in gas storage, separations, and sensing. Integrating flexible/dynamic, gas-responsive components into MOFs can give them unique or enhanced adsorption properties. Here, we explore the adsorption properties that could be imparted to a MOF by a rotaxane molecular shuttle (RMS) in its pores. In the unit cell of an RMS-MOF, a macrocyclic wheel is mechanically interlocked with a strut of the MOF scaffold. The wheel shuttles between stations on the strut that are also gas adsorption sites. At a level of abstraction similar to the seminal Langmuir adsorption model, we pose and analyze a simple statistical mechanical model of gas adsorption in an RMS-MOF that accounts for (i) wheel/gas competition for sites on the strut and (ii) gas-induced changes in the configurational entropy of the shuttling wheel. We determine how the amount of gas adsorbed, the position of the wheel, and the differential energy of adsorption depend on temperature, pressure, and the interactions of the gas and wheel with the stations on the strut. Our model reveals that, compared to a rigid, Langmuir material, the chemistry of the RMS-MOF can be tuned to render gas adsorption more or less temperature sensitive and to release more or less heat upon adsorption. The model also uncovers that, if gas-wheel competition for a station is fierce, temperature influences the position of the wheel differently depending on the amount of gas adsorbed.

The Complexity of Analytical Tasks: A Driving Force for Sensor Array Design

Introduction Analytical tasks, the detection goals and scoping of quantitative measurements, are a foundational consideration for the design of chemical sensor arrays. Often sensor arrays are expected to support a variety of possible tasks which are only vaguely defined by both the end user and array designer. Nonetheless, the expedient choices made by or forced upon array designers and fabricators can significantly impact the analytical tasks that could be supported by an array. Analytical tasks may be defined as boundaries in an abstract space defining chemically and practically relevant quantities that are to be measured; chemical sensor and instrument responses may be defined in related spaces specific to their physics. Intelligently finding a mapping between these two abstract spaces defines our problem and informs the impact of analytical tasks upon sensor array design while avoiding the “black box” problems of popular machine learning approaches. This presentation details recent work at NRL that probes the meaning and impact of analytical tasks on chemical sensor array design through general, device agnostic figures of merit developed via techniques originating in statistical mechanics, information theory, information retrieval, and theoretical computer science. Approaches Multiple theoretical approaches are explored to determine and measure the impact of analytical task on chemical sensor array design. To begin, we use Ising model-like descriptions of binary sensor responses first described in a biological context by Murugan et. al. [1-3] and subsequently developed by us for chemical sensing with environmental interferents. Analytical tasks are then scoped as molecular and concentration identification against an unknown chemical background. On this framework, we explore the capabilities of a variety of approaches towards assessing the impact of analytical task on sensor array design.The first set of approaches originate in information theory and we consider the mutual information and Kullback-Leibler divergences between different sensor array configurations considered over probability distributions defining the likelihood of that sensor array needing to perform different analytical tasks. Platform-agnostic figures of merit are derived and used to show the impact of analytical tasks on sensor array design.Next, Van Riijsbergen and Melucci [4,5] have developed and elaborated upon an approach to representing and searching information in Hilbert spaces using operator theoretic methods reminiscent to those used in quantum mechanical calculations. By transferring our probabilistic sensor array models to a Hilbert space, we are able to perform similar searches against sensor arrays and by operator methods to ask logical questions of the sensor array using operators that encode analytical tasks and specific exigencies like comparisons to the effectiveness of the array for other analytical tasks.Finally, Almagor et. al. [6-9] have recently developed a finite state machine approach for formally describing the complexity of sensing tasks within formal languages. We attempt to map our binary chemical sensor array model to this methodology and to describe the formal complexity of this toy model for chemical sensing. Analytical tasks are input as restrictions on available formal words in the finite state language. Conclusions Using a simple statistical mechanical model for a chemical sensor array with background interferents, we have been able to probe multiple approaches to quantifying the impact of analytical tasks on the design and relevance of a sensor array for a set of chemical detection problems. Importantly, these approaches avoid drawbacks associated with more empirical, machine learning-based approaches, enabling a more clear understanding of the chemical landscape and how chemical sensor array design interacts with it. Our investigation has explored ways to construct figures of merit for specific families of analytical tasks using methodologies as diverse as information theory, Hilbert space search methods, and complexity techniques for automata-defined sensing.

Understanding the Binding of Aromatic Hydrocarbons on Rutile TiO2(110)

The adsorption of cyclohexane, benzene, and alkyl-substituted benzene derivatives is studied on rutile TiO2(110) by a combination of molecular beam dosing, temperature-programmed desorption, and density functional theory (DFT). An inversion analysis is used to extract the coverage-dependent desorption energies from TiO2(110). The values of the suitable prefactors are derived from simple statistical mechanical models assuming different limits in the adsorbate mobility on the surface. The prefactor values determined using the vibrational frequencies from DFT calculations corroborate this analysis and show that the adsorbates are mobile in one or two dimensions on a corrugated TiO2(110) surface. The adsorption of benzene derivatives is found to be dominated by the dative Lewis acid–base interactions of the π system with the surface Ti ions. While the desorption energy generally increases with increasing the length and the number of substituents, the difference between the desorption energies decreases as the...

Ferroic multipolar order and disorder in cyanoelpasolite molecular perovskites.

We use a combination of variable-temperature high-resolution synchrotron X-ray powder diffraction measurements and Monte Carlo simulations to characterize the evolution of two different types of ferroic multipolar order in a series of cyanoelpasolite molecular perovskites. We show that ferroquadrupolar order in [C3N2H5]2Rb[Co(CN)6] is a first-order process that is well described by a four-state Potts model on the simple cubic lattice. Likewise, ferrooctupolar order in [NMe4]2B[Co(CN)6] (B = K, Rb, Cs) also emerges via a first-order transition that now corresponds to a six-state Potts model. Hence, for these particular cases, the dominant symmetry breaking mechanisms are well understood in terms of simple statistical mechanical models. By varying composition, we find that the effective coupling between multipolar degrees of freedom-and hence the temperature at which ferromultipolar order emerges-can be tuned in a chemically sensible manner. This article is part of the theme issue 'Mineralomimesis: natural and synthetic frameworks in science and technology'.

Hybrid local-order mechanism for inversion symmetry breaking

Using classical Monte Carlo simulations, we study a simple statistical mechanical model of relevance to the emergence of polarisation from local displacements on the square and cubic lattices. Our model contains two key ingredients: a Kitaev-like orientation-dependent interaction between nearest neighbours, and a steric term that acts between next-nearest neighbours. Taken by themselves, each of these two ingredients is incapable of driving long-range symmetry breaking, despite the presence of a broad feature in the corresponding heat capacity functions. Instead each component results in a "hidden" transition on cooling to a manifold of degenerate states, the two manifolds are different in the sense that they reflect distinct types of local order. Remarkably, their intersection---\emph{i.e.} the ground state when both interaction terms are included in the Hamiltonian---supports a spontaneous polarisation. In this way, our study demonstrates how local ordering mechanisms might be combined to break global inversion symmetry in a manner conceptually similar to that operating in the "hybrid" improper ferroelectrics. We discuss the relevance of our analysis to the emergence of spontaneous polarisation in well-studied ferroelectrics such as BaTiO$_3$ and KNbO$_3$.

Talk about toy models

Scientific models are frequently discussed in philosophy of science. A great deal of the discussion is centred on approximation, idealisation, and on how these models achieve their representational function. Despite the importance, distinct nature, and high presence of toy models, they have received little attention from philosophers. This paper hopes to remedy this situation. It aims to elevate the status of toy models: by distinguishing them from approximations and idealisations, by highlighting and elaborating on several ways the Kac ring, a simple statistical mechanical model, is used as a toy model, and by explaining why toy models can be used to successfully carry out important work without performing a representational function.

Cooperativity and modularity in protein folding.

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

Characterization of protein folding by a Φ-value calculation with a statistical-mechanical model.

The Φ-value analysis approach provides information about transition-state structures along the folding pathway of a protein by measuring the effects of an amino acid mutation on folding kinetics. Here we compared the theoretically calculated Φ values of 27 proteins with their experimentally observed Φ values; the theoretical values were calculated using a simple statistical-mechanical model of protein folding. The theoretically calculated Φ values reflected the corresponding experimentally observed Φ values with reasonable accuracy for many of the proteins, but not for all. The correlation between the theoretically calculated and experimentally observed Φ values strongly depends on whether the protein-folding mechanism assumed in the model holds true in real proteins. In other words, the correlation coefficient can be expected to illuminate the folding mechanisms of proteins, providing the answer to the question of which model more accurately describes protein folding: the framework model or the nucleation-condensation model. In addition, we tried to characterize protein folding with respect to various properties of each protein apart from the size and fold class, such as the free-energy profile, contact-order profile, and sensitivity to the parameters used in the Φ-value calculation. The results showed that any one of these properties alone was not enough to explain protein folding, although each one played a significant role in it. We have confirmed the importance of characterizing protein folding from various perspectives. Our findings have also highlighted that protein folding is highly variable and unique across different proteins, and this should be considered while pursuing a unified theory of protein folding.

A simple theoretical model goes a long way in explaining complex behavior in protein folding.

Understanding how natural proteins fold spontaneously onto their specific, biologically functional 3D structures is both a fascinating fundamental problem in modern biochemistry and a necessary step toward developing technologies for protein engineering and designing protein-based nanodevices. One of the limitations that scientists working in this area have encountered in the past, however, has been the difficulty in connecting analytical theory to experimental results. For a long time experimentalists could not use theory to interpret their results. Theoretical predictions, moreover, were not amenable to experimental testing. Such limitations have been progressively eliminated by the combination of key theoretical concepts, improved simulations, and new experiments and their detailed quantitative analysis with simple statistical mechanical models. The work of Inanami et al. in PNAS (1) provides a remarkable example of how powerful these simple theoretical models can be in explaining the complexities and nuances of protein folding reactions.

Moral foundations in an interacting neural networks society: A statistical mechanics analysis

The moral foundations theory supports that people, across cultures, tend to consider a small number of dimensions when classifying issues on a moral basis. The data also show that the statistics of weights attributed to each moral dimension is related to self-declared political affiliation, which in turn has been connected to cognitive learning styles by the recent literature in neuroscience and psychology. Inspired by these data, we propose a simple statistical mechanics model with interacting neural networks classifying vectors and learning from members of their social neighbourhood about their average opinion on a large set of issues. The purpose of learning is to reduce dissension among agents when disagreeing. We consider a family of learning algorithms parametrized by δ, that represents the importance given to corroborating (same sign) opinions. We define an order parameter that quantifies the diversity of opinions in a group with homogeneous learning style. Using Monte Carlo simulations and a mean field approximation we find the relation between the order parameter and the learning parameter δ at a temperature we associate with the importance of social influence in a given group. In concordance with data, groups that rely more strongly on corroborating evidence sustain less opinion diversity. We discuss predictions of the model and propose possible experimental tests.

GRAVITATIONAL COLLAPSE AND BLACK HOLE THERMODYNAMICS IN BRANEWORLD SCENARIO

We examine the dynamics of the gravitational collapse in a 4-dim Lorentzian brane embedded in a 5-dim bulk with an extra timelike dimension. By considering the collapse of pure dust on the brane we derive a bouncing FLRW interior solution and match it with a corrected Schwarzschild exterior geometry. In the physical domain considered for the parameters of the solution, the analytical extension is built, exhibiting an exterior event horizon and a Cauchy horizon, analogous to the Reissner–Nordstrom solution. For such an exterior geometry we examine the effects of the bulk-brane corrections in the Hawking radiation. In this scenario the model extends Bekenstein's black hole geometrical thermodynamics for quasi-extremal configurations, with an extra work term in the laws associated with variations of the brane tension. We also propose a simple statistical mechanics model for the entropy of the bouncing collapsed matter by quantizing its fluctuations and constructing the associated partition function. This entropy differs from the geometrical entropy by an additive constant proportional to the area of the extremal black hole and satisfies an analogous first law of thermodynamics. A possible connection between both entropies is discussed.

Measurements of Force-Driven Changes in Bound Protein Numbers on a Single DNA

DNA compaction and chromosome organization involve the dynamic interaction of long DNA molecules and many copies of various proteins. Determining numbers of proteins bound to large DNAs is important to understand their chromosomal functions and protein numbers may be affected not only by chemical factors, but also by physical factors such as mechanical forces generated in DNA, e.g., by transcription or replication (1). We performed single-DNA stretching experiments with bacterial nucleoid proteins HU (2) and Fis, where we verified that the force-extension measurements were in thermodynamic (chemical-mechanical) equilibrium. Given thermal equilibrium of protein binding, we could use a thermodynamic Maxwell relation to deduce the change of protein number on a single stretched DNA due to varied applied force. For the binding of both HU and Fis under conditions where they compact DNA, the numbers of bound proteins decreased as force was increased from 0.03 to 12 pN. This effect saturated with force for HU, but did not for Fis, reflecting the tighter binding of the latter to DNA. The experimental results agree well with expectations of binding numbers based on electrophoretic mobility shift assay data, and the HU data agree well with results from a simple statistical-mechanical model of DNA-bending proteins. This thermodynamic approach may be applied to measure force-driven changes in numbers of a wide variety of molecules bound to DNA or to other polymers; in the case of proteins binding to DNA, force-dependent binding suggests mechano-chemical mechanisms for gene regulation. 1. C. Bustamante, Z. Bryant, S. B. Smith, Nature, (2003). 2. B. Xiao, R. C. Johnson, J. F. Marko, Nucleic Acids Res, (2010).