Breaking bonds in the atomic force microscope: theory and analysis

Understanding the Mechanochemistry of Mechano-Radicals in Self-Growth Materials by Single-Molecule Force Spectroscopy

Recent research on mechano-radicals has provided valuable insights into self-growth and adaptive responsive materials. Typically, mechanophores must remain inert in the absence of force but respond quickly to external tension before other linkages within the polymer network. Azo compounds exhibit promising combinations of mechanical stability and force-triggered reactivity, making them widely used as mechano-radicals in force-responsive materials. However, the activation conditions and behavior of azo compounds have yet to be quantitatively explored. In this study, we investigated the mechanical strength of three azo compounds using single-molecule force spectroscopy. Our results revealed that these compounds exhibit rupture forces ranging from ~500 to 1000 pN, at a loading rate of 3×104 pN s-1. Importantly, these mechanophores demonstrate distinct kinetic properties. Their unique mechanical attributes enable azo bond scission and free radical generation before causing major polymer backbone damage of entire material during polymer network deformation. This fundamental understanding of mechanophores holds significant promise for the development of self-growth materials and their related applications.

Force Dependent Barriers from Analytic Potentials within Elastic Environments

Bond rupture under the action of external forces is usually induced by temperature fluctuations, where the key quantity is the force dependent barrier that needs to be overcome. Using analytic potentials we find that these barriers are fully determined by the dissociation energy and the maximal force the potential can withstand. The barrier shows a simple dependence on these two quantities that allows for a re-interpretation of the Eyring-Zhurkov-Bell length Δ x ‡ and the expressions in theories going beyond that. It is shown that solely elastic environments do not change this barrier in contrast to the predictions of constraint geometry simulate external force (COGEF) strategies. The findings are confirmed by explicit calculations of bond rupture in a polydimethylsiloxane model.

Polymers and biopolymers at interfaces

This review updates recent progress in the understanding of the behaviour of polymers at surfaces and interfaces, highlighting examples in the areas of wetting, dewetting, crystallization, and 'smart' materials. Recent developments in analysis tools have yielded a large increase in the study of biological systems, and some of these will also be discussed, focussing on areas where surfaces are important. These areas include molecular binding events and protein adsorption as well as the mapping of the surfaces of cells. Important techniques commonly used for the analysis of surfaces and interfaces are discussed separately to aid the understanding of their application.

Influence of ligand-receptor interactions on force-extension behavior within the freely jointed chain model

We study the influence of receptor-ligand interactions on the force response of single polymer chains theoretically. The extension of the chain is modeled in terms of freely jointed chain or elastic freely jointed chain (EFJC) models. The situation involving noninteracting bonds is solved exactly, while effects of interactions are treated within a mean-field approximation. The form with shorter bonds governs the low force situation, while the form with longer bonds is relevant in the high force regime. We further discuss the accuracy of approximate relations, which were used to describe the response of the EFJC model.

Size extensivity of elastic properties of alkane fragments

Using MP2, CCSD, and B3LYP methods of computational chemistry, we show length dependence in the intrinsic elastic properties of short alkane fragments. For isolated alkane fragments of finite length in the gas phase and zero temperature, the intrinsic elasticity constants are found to vary with the number of carbon atoms and its parity. From extrapolation of the elasticity constants calculations to infinite chain length, and by comparing with in-situ elasticity constant of single poly(ethylene) molecule obtained with atomic force microscopy, we estimate the softening effect of environment on the extension response of the polymer.

A Simply Synthesized, Tough Polyarylene with Transient Mechanochromic Response

A simple and high-yielding route to tough polyarylenes of the type poly(meta,meta,para-phenylene) (PmmpP) is developed. PmmpP is tough even in its as-synthesized state which has an intermediate molar mass of M_w ≈60 kg mol^-1 and exhibits outstanding mechanical properties at further optimized molecular weight of M_w =96 kg mol^-1 , E=0.9 GPa, ϵ=300 %. Statistical copolymers with para,para-spiropyran (SP) are mechanochromic, and the toughness allows mechanochromism to be investigated. Strained samples instantaneously lose color upon force release. DFT calculations show this phenomenon to be caused by the PmmpP matrix that allows build-up of sufficiently large forces to be transduced to SP, and the relatively unstable corresponding merocyanine (MC) form arising from the aromatic co-monomer. MC units covalently incorporated into PmmpP show a drastically reduced half life time of 3.1 s compared to 4.5 h obtained for SP derivatives with common 6-nitro substitution.

A density functional theory model of mechanically activated silyl ester hydrolysis

To elucidate the mechanism of the mechanically activated dissociation of chemical bonds between carboxymethylated amylose (CMA) and silane functionalized silicon dioxide, we have investigated the dissociation kinetics of the bonds connecting CMA to silicon oxide surfaces with density functional calculations including the effects of force, solvent polarizability, and pH. We have determined the activation energies, the pre-exponential factors, and the reaction rate constants of candidate reactions. The weakest bond was found to be the silyl ester bond between the silicon and the alkoxy oxygen atom. Under acidic conditions, spontaneous proton addition occurs close to the silyl ester such that neutral reactions become insignificant. Upon proton addition at the most favored position, the activation energy for bond hydrolysis becomes 31 kJ mol(-1), which agrees very well with experimental observation. Heterolytic bond scission in the protonated molecule has a much higher activation energy. The experimentally observed bi-exponential rupture kinetics can be explained by different side groups attached to the silicon atom of the silyl ester. The fact that different side groups lead to different dissociation kinetics provides an opportunity to deliberately modify and tune the kinetic parameters of mechanically activated bond dissociation of silyl esters.

Stochastic simulation of single-molecule pulling experiments

Single-molecule pulling experiments are widely used for studying the structure, dynamics, and function of single biological molecules via applying mechanical forces on them in a controlled way. Pulling at a constant speed or at a constant force builds up a mechanical force on a molecule or molecular complex leading to a molecular transition such as the dissociation of a molecular complex, unfolding of a protein, or unwrapping of a higher-order structure. We perform Brownian dynamics and Monte Carlo simulations of single-molecule pulling experiments. Through our simulations we demonstrate that the molecular transition rate based on the Kramers theory in the high-barrier limit becomes unsuitable as the applied force approaches the critical force at which the barrier disappears. We also demonstrate that use of molecular transition rate based on mean first passage time (MFPT) approach would be more relevant in describing molecular transition especially as the applied force approaches the critical force.

Force spectroscopy of polymer desorption: theory and molecular dynamics simulations

Forced detachment of a single polymer chain, strongly adsorbed on a solid substrate, is investigated by two complementary methods: a coarse-grained analytical dynamical model, based on the Onsager stochastic equation, and Molecular Dynamics (MD) simulations with a Langevin thermostat. The suggested approach makes it possible to go beyond the limitations of the conventional Bell-Evans model. We observe a series of characteristic force spikes when the pulling force is measured against the cantilever displacement during detachment at constant velocity vc (displacement control mode) and find that the average magnitude of this force increases as vc increases. The probability distributions of the pulling force and the end-monomer distance from the surface at the moment of the final detachment are investigated for different adsorption energies ε and pulling velocities vc. Our extensive MD simulations validate and support the main theoretical findings. Moreover, the simulations reveal a novel behavior: for a strong-friction and massive cantilever the force spike pattern is smeared out at large vc. As a challenging task for experimental bio-polymer sequencing in future we suggest the fabrication of a stiff, super-light, nanometer-sized AFM probe.

Mechanically induced silyl ester cleavage under acidic conditions investigated by AFM-based single-molecule force spectroscopy in the force-ramp mode

AFM-based dynamic single-molecule force spectroscopy was used to stretch carboxymethylated amylose (CMA) polymers, which have been covalently tethered between a silanized glass substrate and a silanized AFM tip via acid-catalyzed ester condensation at pH 2.0. Rupture forces were measured as a function of temperature and force loading rate in the force-ramp mode. The data exhibit significant statistical scattering, which is fitted with a maximum likelihood estimation (MLE) algorithm. Bond rupture is described with a Morse potential based Arrhenius kinetics model. The fit yields a bond dissociation energy D_e = 35 kJ mol⁻¹ and an Arrhenius pre-factor A = 6.6 × 10⁴ s⁻¹. The bond dissociation energy is consistent with previous experiments under identical conditions, where the force-clamp mode was employed. However, the bi-exponential decay kinetics, which the force-clamp results unambiguously revealed, are not evident in the force-ramp data. While it is possible to fit the force-ramp data with a bi-exponential model, the fit parameters differ from the force-clamp experiments. Overall, single-molecule force spectroscopy in the force-ramp mode yields data whose information content is more limited than force-clamp data. It may, however, still be necessary and advantageous to perform force-ramp experiments. The number of successful events is often higher in the force-ramp mode, and competing reaction pathways may make force-clamp experiments impossible.

Effect of viscoelasticity on the analysis of single-molecule force spectroscopy on live cells

Single-molecule force spectroscopy is used to probe the kinetics of receptor-ligand bonds by applying mechanical forces to an intermediate media on which the molecules reside. When this intermediate media is a live cell, the viscoelastic properties can affect the calculation of rate constants. We theoretically investigate the effect of media viscoelasticity on the common assumption that the bond force is equal to the instantaneous applied force. Dynamic force spectroscopy is simulated between two cells of varying micromechanical properties adhered by a single bond with a constant kinetic off-rate. We show that cell and microvilli deformation, and hydrodynamic drag contribute to bond forces that can be 28-90% lower than the applied force for loading rates of 10(3)-10(7) pN/s, resulting in longer bond lifetimes. These longer bond lifetimes are not caused by changes in bond kinetics; rather, they are due to the mechanical response of the intermediate media on which the bonds reside. Under the assumption that the instantaneous bond force is equal to the applied force--thereby ignoring viscoelasticity--leads to 14-39% error in the determination of off-rates. We present an approach that incorporates viscoelastic properties in calculating the instantaneous bond force and kinetic dissociation parameter of the intermolecular bond.

Anti-Arrhenius cleavage of covalent bonds in bottlebrush macromolecules on substrate

Spontaneous degradation of bottlebrush macromolecules on aqueous substrates was monitored by atomic force microscopy. Scission of C ─ C covalent bonds in the brush backbone occurred due to steric repulsion between the adsorbed side chains, which generated bond tension on the order of several nano-Newtons. Unlike conventional chemical reactions, the rate of bond scission was shown to decrease with temperature. This apparent anti-Arrhenius behavior was caused by a decrease in the surface energy of the underlying substrate upon heating, which results in a corresponding decrease of bond tension in the adsorbed macromolecules. Even though the tension dropped minimally from 2.16 to 1.89 nN, this was sufficient to overpower the increase in the thermal energy (k(B)T) in the Arrhenius equation. The rate constant of the bond-scission reaction was measured as a function of temperature and surface energy. Fitting the experimental data by a perturbed Morse potential V = V(0)(1 - e(-βx))(2) - fx, we determined the depth and width of the potential to be V(0) = 141 ± 19 kJ/mol and β(-1) = 0.18 ± 0.03 Å, respectively. Whereas the V(0) value is in reasonable agreement with the activation energy E(a) = 80-220 kJ/mol of mechanical and thermal degradation of organic polymers, it is significantly lower than the dissociation energy of a C ─ C bond D(e) = 350 kJ/mol. Moreover, the force constant K(x) = 2β(2)V(0) = 1.45 ± 0.36 kN/m of a strained bottlebrush along its backbone is markedly larger than the force constant of a C ─ C bond K(l) = 0.44 kN/m, which is attributed to additional stiffness due to deformation of the side chains.

Hidden multiple bond effects in dynamic force spectroscopy

In dynamic force spectroscopy, a (bio-)molecular complex is subjected to a steadily increasing force until the chemical bond breaks. Repeating the same experiment many times results in a broad distribution of rupture forces, whose quantitative interpretation represents a formidable theoretical challenge. In this study we address the situation that more than a single molecular bond is involved in one experimental run, giving rise to multiple rupture events that are even more difficult to analyze and thus are usually eliminated as far as possible from the further evaluation of the experimental data. We develop and numerically solve a detailed model of a complete dynamic force spectroscopy experiment including a possible clustering of molecules on the substrate surface, the formation of bonds, their dissociation under load, and the postprocessing of the force extension curves. We show that the data, remaining after elimination of obvious multiple rupture events, may still contain a considerable number of hidden multiple bonds, which are experimentally indistinguishable from true single bonds, but which have considerable effects on the resulting rupture force statistics and its consistent theoretical interpretation.

Single-molecule force-clamp experiments reveal kinetics of mechanically activated silyl ester hydrolysis

We have investigated the strength of silyl ester bonds formed between carboxymethylated amylose (CMA) molecules and silane-functionalized silicon oxide surfaces using AFM-based single-molecule force spectroscopy in the force-clamp mode. Single tethered CMA molecules were picked up, and bond lifetimes were determined at constant clamp forces of 0.8, 1.0, and 1.2 nN at seven temperatures between 295 and 320 K at pH 2.0. The results reveal biexponential rupture kinetics. To obtain the reaction rate constants for each force and temperature individually, the results were analyzed with a biexponential kinetic model using the maximum likelihood estimation (MLE) method. The force-independent kinetic and structural parameters of the underlying bond rupture mechanisms were extracted by fitting the entire data set with a parallel MLE fit procedure using the Zhurkov/Bell model and, alternatively, an Arrhenius kinetics model combined with a Morse potential as an analytic representation of the binding potential. With activation energies between 37 and 40 kJ mol(-1), and with Arrhenius prefactors between 5 × 10(4) and 2 × 10(6) s(-1), the results point to the hydrolysis of the silyl ester bond.

Model for stretching and unfolding the giant multidomain muscle protein using single-molecule force spectroscopy

Single-molecule manipulation has allowed the forced unfolding of multidomain proteins. Here we outline a theory that not only explains these experiments but also points out a number of difficulties in their interpretation and makes suggestions for further experiments. For titin we reproduce force-extension curves, the dependence of break force on pulling speed, and break-force distributions and also validate two common experimental views: Unfolding titin Ig domains can be explained as stepwise increases in contour length, and increasing force peaks in native Ig sequences represent a hierarchy of bond strengths. Our theory is valid for essentially any molecule that can be unfolded in atomic force microscopy; as a further example, we present force-extension curves for the unfolding of RNA hairpins.

Effects of multiple-bond ruptures on kinetic parameters extracted from force spectroscopy measurements: revisiting biotin-streptavidin interactions

Force spectroscopy measurements of the rupture of the molecular bond between biotin and streptavidin often results in a wide distribution of rupture forces. We attribute the long tail of high rupture forces to the nearly simultaneous rupture of more than one molecular bond. To decrease the number of possible bonds, we employed hydrophilic polymeric tethers to attach biotin molecules to the atomic force microscope probe. It is shown that the measured distributions of rupture forces still contain high forces that cannot be described by the forced dissociation from a deep potential well. We employed a recently developed analytical model of simultaneous rupture of two bonds connected by polymer tethers with uneven length to fit the measured distributions. The resulting kinetic parameters agree with the energy landscape predicted by molecular dynamics simulations. It is demonstrated that when more than one molecular bond might rupture during the pulling measurements there is a noise-limited range of probe velocities where the kinetic parameters measured by force spectroscopy correspond to the true energy landscape. Outside this range of velocities, the kinetic parameters extracted by using the standard most probable force approach might be interpreted as artificial energy barriers that are not present in the actual energy landscape. Factors that affect the range of useful velocities are discussed.

Microorganisms, mineral surfaces, and aquatic environments: learning from the past for future progress

The interactions between the geosphere and the biosphere are central questions in environmental and geological research. The relationship between bacteria and their environment is an important example of these interactions. By studying microbial communities in modern environments, it is possible to understand the underlying mechanisms that shape these environments and apply this knowledge to the rock record. Recently, new experimental and theoretical methods, ranging from nano- and biotechnology to mathematical and conceptual modelling, have come into play. Thus, new opportunities for interdisciplinary research in the field of geobiology have emerged. In this paper, we review aspects of state-of-the-art imaging and modelling techniques and propose a research concept linking the experimental and the theoretical approaches.

Asymptotic strength limit of hydrogen-bond assemblies in proteins at vanishing pulling rates

We develop a fracture-mechanics-based theoretical framework that considers the free energy competition between entropic elasticity of polypeptide chains and rupture of peptide hydrogen bonds, which we use here to provide an explanation for the intrinsic strength limit of protein domains at vanishing rates. Our analysis predicts that individual protein domains stabilized only by hydrogen bonds cannot exhibit rupture forces larger than approximately 200 pN in the asymptotic limit. This result explains earlier experimental and computational observations that indicate an asymptotical strength limit at vanishing pulling rates.

Dynamic strength of the silicon-carbon bond observed over three decades of force-loading rates

The mechanical strength of individual Si-C bonds was determined as a function of the applied force-loading rate by dynamic single-molecule force spectroscopy, using an atomic force microscope. The applied force-loading rates ranged from 0.5 to 267 nN/s, spanning 3 orders of magnitude. As predicted by Arrhenius kinetics models, a logarithmic increase of the bond rupture force with increasing force-loading rate was observed, with average rupture forces ranging from 1.1 nN for 0.5 nN/s to 1.8 nN for 267 nN/s. Three different theoretical models, all based on Arrhenius kinetics and analytic forms of the binding potential, were used to analyze the experimental data and to extract the parameters fmax and D(e) of the binding potential, together with the Arrhenius A-factor. All three models well reproduced the experimental data, including statistical scattering; nevertheless, the three free parameters allow so much flexibility that they cannot be extracted unambiguously from the experimental data. Successful fits with a Morse potential were achieved with fmax = 2.0-4.8 nN and D(e) = 76-87 kJ/mol, with the Arrhenius A-factor covering 2.45 x 10(-10)-3 x 10(-5) s(-1), respectively. The Morse potential parameters and A-factor taken from gas-phase density functional calculations, on the other hand, did not reproduce the experimental forces and force-loading rate dependence.

Analyzing single-bond experiments: influence of the shape of the energy landscape and universal law between the width, depth, and force spectrum of the bond

Experimentalists who measure the rupture force of a single molecular bond usually pull on that bond at a constant speed, keeping the loading rate r=df/dt constant. The challenge is to extract the energy landscape of the interaction between the two molecules involved from the experimental rupture force distribution under several loading rates. This analysis requires the use of a model for the shape of this energy landscape. Several barriers can compose the landscape, though molecular bonds with a single barrier are often observed. The Bell model is commonly used for the analysis of rupture force measurements with bonds displaying a single barrier. It provides an analytical expression of the most likely rupture force which makes it very simple to use. However, in principle, it can only be applied to landscapes with extrema whose positions do not vary under force. Here, we evaluate the general relevance of the Bell model by comparing it with another analytical model for which the landscape is harmonic in the vicinity of its extrema. Similar shapes of force distributions are obtained with both models, making it difficult to confirm the validity of the Bell model for a given set of experimental data. Nevertheless, we show that the analysis of rupture force experiments on such harmonic landscapes with the Bell model provides excellent results in most cases. However, numerical computation of the distributions of the rupture forces on piecewise-linear energy landscapes indicates that the blind use of any model such as the Bell model may be risky, since there often exist several landscapes compatible with a given set of experimental data. Finally, we derive a universal relation between the range and energy of the bond and the force spectrum. This relation does not depend on the shape of the energy landscape and can thus be used to characterize unambiguously any one-barrier landscape from experiments. All the results are illustrated with the streptavidin-biotin bond.

Optimal evaluation of single-molecule force spectroscopy experiments

The forced rupture of single chemical bonds under external load is addressed. A general framework is put forward to optimally utilize the experimentally observed rupture force data for estimating the parameters of a theoretical model. As an application, we explore to what extent a distinction between several recently proposed models is feasible on the basis of realistic experimental data sets.

Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of alpha-helical and beta-sheet protein domains

The fundamental fracture mechanisms of biological protein materials remain largely unknown, in part, because of a lack of understanding of how individual protein building blocks respond to mechanical load. For instance, it remains controversial whether the free energy landscape of the unfolding behavior of proteins consists of multiple, discrete transition states or the location of the transition state changes continuously with the pulling velocity. This lack in understanding has thus far prevented us from developing predictive strength models of protein materials. Here, we report direct atomistic simulation that over four orders of magnitude in time scales of the unfolding behavior of alpha-helical (AH) and beta-sheet (BS) domains, the key building blocks of hair, hoof, and wool as well as spider silk, amyloids, and titin. We find that two discrete transition states corresponding to two fracture mechanisms exist. Whereas the unfolding mechanism at fast pulling rates is sequential rupture of individual hydrogen bonds (HBs), unfolding at slow pulling rates proceeds by simultaneous rupture of several HBs. We derive the hierarchical Bell model, a theory that explicitly considers the hierarchical architecture of proteins, providing a rigorous structure-property relationship. We exemplify our model in a study of AHs, and show that 3-4 parallel HBs per turn are favorable in light of the protein's mechanical and thermodynamical stability, in agreement with experimental findings that AHs feature 3.6 HBs per turn. Our results provide evidence that the molecular structure of AHs maximizes its robustness at minimal use of building materials.

Rupture force analysis and the associated systematic errors in force spectroscopy by AFM

Force spectroscopy is a new and valuable tool in physical chemistry and biophysics. However, data analysis has yet to be standardized, hindering the advancement of the technique. In this article, treatment of the rupture forces is described in the framework of the Bell-Evans model, and the systematic errors associated with the tether effect for approaches that utilize the most probable, the median, and the mean rupture forces are compared. It is shown that significant systematic errors in the dissociation rate can result from nonlinear loading with polymeric tethers even if the apparent loading rate is used in the analysis. Analytical expressions for the systematic errors are provided for the most probable and median forces. The use of these expressions to correct the associated systematic errors is illustrated by the analysis of the measured rupture forces between single hexadecane molecules in water. It is noted that the measured distributions of rupture forces often contain high forces that are unaccounted for by theoretical models. Experimental data indicate that the most significant effect of the high forces "tail" is on the dissociation rate obtained from the median force analysis whereas the barrier width appears to be unaffected.