Mapping the energy landscape of biomolecules using single molecule force correlation spectroscopy: theory and applications

The Effect of Pulling and Twisting Forces on Chameleon Sequence Peptides

Chameleon sequences are amino acid sequences found in several distinct configurations in experiment. They challenge our understanding of the link between sequence and structure, and provide insight into structural competition in proteins. Here, we study the energy landscapes for three such sequences, and interrogate how pulling and twisting forces impact the available structural ensembles. Chameleon sequences do not necessarily exhibit multiple structural ensembles on a multifunnel energy landscape when we consider them in isolation. The application of even small forces leads to drastic changes in the energy landscapes. For pulling forces, we observe transitions from helical to extended structures in a very small span of forces. For twisting forces, the picture is much more complex, and highly dependent on the magnitude and handedness of the applied force as well as the reference angle for the twist. Depending on these parameters, more complex and more simplistic energy landscapes are observed alongside more and less diverse structural ensembles. The impact of even small forces is significant, confirming their likely role in folding events. In addition, small forces exerted by the remaining scaffold of a protein may be sufficient to lead to the adoption of a specific structural ensemble by a chameleon sequence.

Statistical Learning from Single-Molecule Experiments: Support Vector Machines and Expectation-Maximization Approaches to Understanding Protein Unfolding Data

Single-molecule force spectroscopy has become a powerful tool for the exploration of dynamic processes that involve proteins; yet, meaningful interpretation of the experimental data remains challenging. Owing to low signal-to-noise ratio, experimental force-extension spectra contain force signals due to nonspecific interactions, tip or substrate detachment, and protein desorption. Unravelling of complex protein structures results in the unfolding transitions of different types. Here, we test the performance of Support Vector Machines (SVM) and Expectation Maximization (EM) approaches in statistical learning from dynamic force experiments. When the output from molecular modeling in silico (or other studies) is used as a training set, SVM and EM can be applied to understand the unfolding force data. The maximal margin or maximum likelihood classifier can be used to separate experimental test observations into the unfolding transitions of different types, and EM optimization can then be utilized to resolve the statistics of unfolding forces: weights, average forces, and standard deviations. We designed an EM-based approach, which can be directly applied to the experimental data without data classification and division into training and test observations. This approach performs well even when the sample size is small and when the unfolding transitions are characterized by overlapping force ranges.

Does protein unfolding play a functional role in vivo?

Unfolding and refolding of multidomain proteins under force have yet to be recognized as a major mechanism of function for proteins in vivo. In this review, we discuss the inherent properties of multidomain proteins under a force vector from a structural and functional perspective. We then characterize three main systems where multidomain proteins could play major roles through mechanical unfolding: muscular contraction, cellular mechanotransduction, and bacterial adhesion. We analyze how key multidomain proteins for each system can produce a gain-of-function from the perspective of a fine-tuned quantized response, a molecular battery, delivery of mechanical work through refolding, elasticity tuning, protection and exposure of cryptic sites, and binding-induced mechanical changes. Understanding how mechanical unfolding and refolding affect function will have important implications in designing mechano-active drugs against conditions such as muscular dystrophy, cancer, or novel antibiotics.

Blue light-induced low mechanical stability of ruthenium-based coordination bonds: an AFM-based single-molecule force spectroscopy study

A HA–Ru^II complex was conjugated to a hyaluronan polymer through amide bonds. In AFM experiments using the “multi-fishhook” approach, the cantilever tip made contact with the polymeric molecule, resulting in stretching, indicated by sawtooth-like force-extension curves.

TensorCalculator: exploring the evolution of mechanical stress in the CCMV capsid

A new computational methodology for the accurate numerical calculation of the Cauchy stress tensor, stress invariants, principal stress components, von Mises and Tresca tensors is developed. The methodology is based on the atomic stress approach which permits the calculation of stress tensors, widely used in continuum mechanics modeling of materials properties, using the output from the MD simulations of discrete atomic and [Formula: see text]-based coarse-grained structural models of biological particles. The methodology mapped into the software package TensorCalculator was successfully applied to the empty cowpea chlorotic mottle virus (CCMV) shell to explore the evolution of mechanical stress in this mechanically-tested specific example of a soft virus capsid. We found an inhomogeneous stress distribution in various portions of the CCMV structure and stress transfer from one portion of the virus structure to another, which also points to the importance of entropic effects, often ignored in finite element analysis and elastic network modeling. We formulate a criterion for elastic deformation using the first principal stress components. Furthermore, we show that von Mises and Tresca stress tensors can be used to predict the onset of a viral capsid's mechanical failure, which leads to total structural collapse. TensorCalculator can be used to study stress evolution and dynamics of defects in viral capsids and other large-size protein assemblies.

Dynamic Transition from α-Helices to β-Sheets in Polypeptide Coiled-Coil Motifs

We carried out dynamic force manipulations in silico on a variety of coiled-coil protein fragments from myosin, chemotaxis receptor, vimentin, fibrin, and phenylalanine zippers that vary in size and topology of their α-helical packing. When stretched along the superhelical axis, all superhelices show elastic, plastic, and inelastic elongation regimes and undergo a dynamic transition from the α-helices to the β-sheets, which marks the onset of plastic deformation. Using the Abeyaratne-Knowles formulation of phase transitions, we developed a new theoretical methodology to model mechanical and kinetic properties of protein coiled-coils under mechanical nonequilibrium conditions and to map out their energy landscapes. The theory was successfully validated by comparing the simulated and theoretical force-strain spectra. We derived the scaling laws for the elastic force and the force for α-to-β transition, which can be used to understand natural proteins' properties as well as to rationally design novel biomaterials of required mechanical strength with desired balance between stiffness and plasticity.

Mechanistic Basis for the Binding of RGD- and AGDV-Peptides to the Platelet Integrin αIIbβ3

Binding of soluble fibrinogen to the activated conformation of the integrin αIIbβ3 is required for platelet aggregation and is mediated exclusively by the C-terminal AGDV-containing dodecapeptide (γC-12) sequence of the fibrinogen γ chain. However, peptides containing the Arg-Gly-Asp (RGD) sequences located in two places in the fibrinogen Aα chain inhibit soluble fibrinogen binding to αIIbβ3 and make substantial contributions to αIIbβ3 binding when fibrinogen is immobilized and when it is converted to fibrin. Here, we employed optical trap-based nanomechanical measurements and computational molecular modeling to determine the kinetics, energetics, and structural details of cyclic RGDFK (cRGDFK) and γC-12 binding to αIIbβ3. Docking analysis revealed that NMR-determined solution structures of cRGDFK and γC-12 bind to both the open and closed αIIbβ3 conformers at the interface between the αIIb β-propeller domain and the β3 βI domain. The nanomechanical measurements revealed that cRGDFK binds to αIIbβ3 at least as tightly as γC-12. A subsequent analysis of molecular force profiles and the number of peptide-αIIbβ3 binding contacts revealed that both peptides form stable bimolecular complexes with αIIbβ3 that dissociate in the 60-120 pN range. The Gibbs free energy profiles of the αIIbβ3-peptide complexes revealed that the overall stability of the αIIbβ3-cRGDFK complex was comparable with that of the αIIbβ3-γC-12 complex. Thus, these results provide a mechanistic explanation for previous observations that RGD- and AGDV-containing peptides are both potent inhibitors of the αIIbβ3-fibrinogen interactions and are consistent with the observation that RGD motifs, in addition to AGDV, support interaction of αIIbβ3 with immobilized fibrinogen and fibrin.

The physics of pulling polyproteins: a review of single molecule force spectroscopy using the AFM to study protein unfolding

One of the most exciting developments in the field of biological physics in recent years is the ability to manipulate single molecules and probe their properties and function. Since its emergence over two decades ago, single molecule force spectroscopy has become a powerful tool to explore the response of biological molecules, including proteins, DNA, RNA and their complexes, to the application of an applied force. The force versus extension response of molecules can provide valuable insight into its mechanical stability, as well as details of the underlying energy landscape. In this review we will introduce the technique of single molecule force spectroscopy using the atomic force microscope (AFM), with particular focus on its application to study proteins. We will review the models which have been developed and employed to extract information from single molecule force spectroscopy experiments. Finally, we will end with a discussion of future directions in this field.

Structural transitions and energy landscape for Cowpea Chlorotic Mottle Virus capsid mechanics from nanomanipulation in vitro and in silico

Physical properties of capsids of plant and animal viruses are important factors in capsid self-assembly, survival of viruses in the extracellular environment, and their cell infectivity. Combined AFM experiments and computational modeling on subsecond timescales of the indentation nanomechanics of Cowpea Chlorotic Mottle Virus capsid show that the capsid's physical properties are dynamic and local characteristics of the structure, which change with the depth of indentation and depend on the magnitude and geometry of mechanical input. Under large deformations, the Cowpea Chlorotic Mottle Virus capsid transitions to the collapsed state without substantial local structural alterations. The enthalpy change in this deformation state ΔHind = 11.5-12.8 MJ/mol is mostly due to large-amplitude out-of-plane excitations, which contribute to the capsid bending; the entropy change TΔSind = 5.1-5.8 MJ/mol is due to coherent in-plane rearrangements of protein chains, which mediate the capsid stiffening. Direct coupling of these modes defines the extent of (ir)reversibility of capsid indentation dynamics correlated with its (in)elastic mechanical response to the compressive force. This emerging picture illuminates how unique physico-chemical properties of protein nanoshells help define their structure and morphology, and determine their viruses' biological function.

Resolving two-dimensional kinetics of the integrin αIIbβ3-fibrinogen interactions using binding-unbinding correlation spectroscopy

Using a combined experimental and theoretical approach named binding-unbinding correlation spectroscopy (BUCS), we describe the two-dimensional kinetics of interactions between fibrinogen and the integrin αIIbβ3, the ligand-receptor pair essential for platelet function during hemostasis and thrombosis. The methodology uses the optical trap to probe force-free association of individual surface-attached fibrinogen and αIIbβ3 molecules and forced dissociation of an αIIbβ3-fibrinogen complex. This novel approach combines force clamp measurements of bond lifetimes with the binding mode to quantify the dependence of the binding probability on the interaction time. We found that fibrinogen-reactive αIIbβ3 pre-exists in at least two states that differ in their zero force on-rates (k(on1) = 1.4 × 10(-4) and k(on2) = 2.3 × 10(-4) μm(2)/s), off-rates (k(off1) = 2.42 and k(off2) = 0.60 s(-1)), and dissociation constants (K(d)(1) = 1.7 × 10(4) and K(d)(2) = 2.6 × 10(3) μm(-2)). The integrin activator Mn(2+) changed the on-rates and affinities (K(d)(1) = 5 × 10(4) and K(d)(2) = 0.3 × 10(3) μm(-2)) but did not affect the off-rates. The strength of αIIbβ3-fibrinogen interactions was time-dependent due to a progressive increase in the fraction of the high affinity state of the αIIbβ3-fibrinogen complex characterized by a faster on-rate. Upon Mn(2+)-induced integrin activation, the force-dependent off-rates decrease while the complex undergoes a conformational transition from a lower to higher affinity state. The results obtained provide quantitative estimates of the two-dimensional kinetic rates for the low and high affinity αIIbβ3 and fibrinogen interactions at the single molecule level and offer direct evidence for the time- and force-dependent changes in αIIbβ3 conformation and ligand binding activity, underlying the dynamics of fibrinogen-mediated platelet adhesion and aggregation.

Mechanism of fibrin(ogen) forced unfolding

Fibrinogen, upon enzymatic conversion to monomeric fibrin, provides the building blocks for fibrin polymer, the scaffold of blood clots and thrombi. Little has been known about the force-induced unfolding of fibrin(ogen), even though it is the foundation for the mechanical and rheological properties of fibrin, which are essential for hemostasis. We determined mechanisms and mapped the free energy landscape of the elongation of fibrin(ogen) monomers and oligomers through combined experimental and theoretical studies of the nanomechanical properties of fibrin(ogen), using atomic force microscopy-based single-molecule unfolding and simulations in the experimentally relevant timescale. We have found that mechanical unraveling of fibrin(ogen) is determined by the combined molecular transitions that couple stepwise unfolding of the γ chain nodules and reversible extension-contraction of the α-helical coiled-coil connectors. These findings provide important characteristics of the fibrin(ogen) nanomechanics necessary to understand the molecular origins of fibrin viscoelasticity at the fiber and whole clot levels.

Dielectrophoretic tweezers as a platform for molecular force spectroscopy in a highly parallel format

We demonstrated the application of a simple electrode geometry for dielectrophoresis (DEP) on colloidal probes as a form of molecular force spectroscopy in a highly parallel format. The electric field between parallel plates is perturbed with dielectric microstructures, generating uniform DEP forces on colloidal probes in the range of several hundred piconewtons across a macroscopic sample area. We determined the approximate crossover frequency between negative and positive DEP using electrodes without dielectric microstructures-a simplification over standard experimental methods involving quadrupoles or optical trapping. 2D and 3D simulations of the electric field distributions validated the experimental behavior of several of our DEP tweezers geometries and provided insight into potential improvements. We applied the DEP tweezers to the stretching of a short DNA oligomer and detected its extension using total-internal reflection fluorescence microscopy. The combination of a simple cell fabrication, a uniform distribution of high axial forces, and a facile optical detection of our DEP tweezers makes this form of molecular force spectroscopy ideal for highly parallel detection of stretching or unbinding kinetics of biomolecules.

Sop-GPU: accelerating biomolecular simulations in the centisecond timescale using graphics processors

Theoretical exploration of fundamental biological processes involving the forced unraveling of multimeric proteins, the sliding motion in protein fibers and the mechanical deformation of biomolecular assemblies under physiological force loads is challenging even for distributed computing systems. Using a C(α)-based coarse-grained self organized polymer (SOP) model, we implemented the Langevin simulations of proteins on graphics processing units (SOP-GPU program). We assessed the computational performance of an end-to-end application of the program, where all the steps of the algorithm are running on a GPU, by profiling the simulation time and memory usage for a number of test systems. The ∼90-fold computational speedup on a GPU, compared with an optimized central processing unit program, enabled us to follow the dynamics in the centisecond timescale, and to obtain the force-extension profiles using experimental pulling speeds (v(f) = 1-10 μm/s) employed in atomic force microscopy and in optical tweezers-based dynamic force spectroscopy. We found that the mechanical molecular response critically depends on the conditions of force application and that the kinetics and pathways for unfolding change drastically even upon a modest 10-fold increase in v(f). This implies that, to resolve accurately the free energy landscape and to relate the results of single-molecule experiments in vitro and in silico, molecular simulations should be carried out under the experimentally relevant force loads. This can be accomplished in reasonable wall-clock time for biomolecules of size as large as 10(5) residues using the SOP-GPU package.

Generation of random numbers on graphics processors: forced indentation in silico of the bacteriophage HK97

The use of graphics processing units (GPUs) in simulation applications offers a significant speed gain as compared to computations on central processing units (CPUs). Many simulation methods require a large number of independent random variables generated at each step. We present two approaches for implementation of random number generators (RNGs) on a GPU. In the one-RNG-per-thread approach, one RNG produces a stream of random numbers in each thread of execution, whereas the one-RNG-for-all-threads method builds on the ability of different threads to communicate, thus, sharing random seeds across an entire GPU device. We used these approaches to implement Ran2, Hybrid Taus, and Lagged Fibonacci algorithms on a GPU. We profiled the performance of these generators in terms of the computational time, memory usage, and the speedup factor (CPU time/GPU time). These generators have been incorporated into the program for Langevin simulations of biomolecules fully implemented on the GPU. The ∼250-fold computational speedup on the GPU allowed us to carry out single-molecule dynamic force measurements in silico to explore the mechanical properties of the bacteriophage HK97 in the experimental subsecond time scale. We found that the nanomechanical response of HK97 depends on the conditions of force application, including the rate of change and geometry of the mechanical perturbation. Hence, using the GPU-based implementation of RNGs, presented here, in conjunction with Langevin simulations, makes it possible to directly compare the results of dynamic force measurements in vitro and in silico.

Direct observation of an ensemble of stable collapsed states in the mechanical folding of ubiquitin

Statistical theories of protein folding have long predicted plausible mechanisms for reducing the vast conformational space through distinct ensembles of structures. However, these predictions have remained untested by bulk techniques, because the conformational diversity of folding molecules has been experimentally unapproachable. Owing to recent advances in single molecule force-clamp spectroscopy, we are now able to probe the structure and dynamics of the small protein ubiquitin by measuring its length and mechanical stability during each stage of folding. Here, we discover that upon hydrophobic collapse, the protein rapidly selects a subset of minimum energy structures that are mechanically weak and essential precursors of the native fold. From this much reduced ensemble, the native state is acquired through a barrier-limited transition. Our results support the validity of statistical mechanics models in describing the folding of a small protein on biological timescales.

Estimating kinetic and thermodynamic parameters from single molecule enzyme-inhibitor interactions

We report the application of recently developed microscopic models to estimate the apparent kinetic and thermodynamic parameters in a single molecule force spectroscopy study of the carbonic anhydrase enzyme and a complementary sulfonamide inhibitor. The most probable rupture force for the enzyme-inhibitor interaction shows a nonlinear dependency on the log-loading rate. Estimates for the kinetic and thermodynamic parameters were obtained by fitting the nonlinear dependency to linear cubic potential and cusp potential models and compared to the standard Bell-Evans model. The reliability of the estimated parameters was verified by modeling the experimental rupture force distributions by the theoretically predicted distributions at rupture. We also report that linkers that are attached to the enzyme and inhibitor show appreciable effects on the apparent kinetic and thermodynamic parameters.

Single-molecule biophysics: at the interface of biology, physics and chemistry

Single-molecule methods have matured into powerful and popular tools to probe the complex behaviour of biological molecules, due to their unique abilities to probe molecular structure, dynamics and function, unhindered by the averaging inherent in ensemble experiments. This review presents an overview of the burgeoning field of single-molecule biophysics, discussing key highlights and selected examples from its genesis to our projections for its future. Following brief introductions to a few popular single-molecule fluorescence and manipulation methods, we discuss novel insights gained from single-molecule studies in key biological areas ranging from biological folding to experiments performed in vivo.