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1.  Multiple regimes of operation in bimodal AFM: understanding the energy of cantilever eigenmodes 
One of the key goals in atomic force microscopy (AFM) imaging is to enhance material property contrast with high resolution. Bimodal AFM, where two eigenmodes are simultaneously excited, confers significant advantages over conventional single-frequency tapping mode AFM due to its ability to provide contrast between regions with different material properties under gentle imaging conditions. Bimodal AFM traditionally uses the first two eigenmodes of the AFM cantilever. In this work, the authors explore the use of higher eigenmodes in bimodal AFM (e.g., exciting the first and fourth eigenmodes). It is found that such operation leads to interesting contrast reversals compared to traditional bimodal AFM. A series of experiments and numerical simulations shows that the primary cause of the contrast reversals is not the choice of eigenmode itself (e.g., second versus fourth), but rather the relative kinetic energy between the higher eigenmode and the first eigenmode. This leads to the identification of three distinct imaging regimes in bimodal AFM. This result, which is applicable even to traditional bimodal AFM, should allow researchers to choose cantilever and operating parameters in a more rational manner in order to optimize resolution and contrast during nanoscale imaging of materials.
PMCID: PMC3701429  PMID: 23844344
atomic force microscopy; bimodal AFM; cantilever eigenmodes; polymer characterization
2.  Trade-offs in sensitivity and sampling depth in bimodal atomic force microscopy and comparison to the trimodal case 
This paper presents experiments on Nafion® proton exchange membranes and numerical simulations illustrating the trade-offs between the optimization of compositional contrast and the modulation of tip indentation depth in bimodal atomic force microscopy (AFM). We focus on the original bimodal AFM method, which uses amplitude modulation to acquire the topography through the first cantilever eigenmode, and drives a higher eigenmode in open-loop to perform compositional mapping. This method is attractive due to its relative simplicity, robustness and commercial availability. We show that this technique offers the capability to modulate tip indentation depth, in addition to providing sample topography and material property contrast, although there are important competing effects between the optimization of sensitivity and the control of indentation depth, both of which strongly influence the contrast quality. Furthermore, we demonstrate that the two eigenmodes can be highly coupled in practice, especially when highly repulsive imaging conditions are used. Finally, we also offer a comparison with a previously reported trimodal AFM method, where the above competing effects are minimized.
PMCID: PMC4142983  PMID: 25161847
amplitude modulation; bimodal; multifrequency atomic force microscopy; indentation depth modulation; Nafion; open loop; proton exchange membranes; trimodal
3.  Multi-frequency tapping-mode atomic force microscopy beyond three eigenmodes in ambient air 
We present an exploratory study of multimodal tapping-mode atomic force microscopy driving more than three cantilever eigenmodes. We present tetramodal (4-eigenmode) imaging experiments conducted on a thin polytetrafluoroethylene (PTFE) film and computational simulations of pentamodal (5-eigenmode) cantilever dynamics and spectroscopy, focusing on the case of large amplitude ratios between the fundamental eigenmode and the higher eigenmodes. We discuss the dynamic complexities of the tip response in time and frequency space, as well as the average amplitude and phase response. We also illustrate typical images and spectroscopy curves and provide a very brief description of the observed contrast. Overall, our findings are promising in that they help to open the door to increasing sophistication and greater versatility in multi-frequency AFM through the incorporation of a larger number of driven eigenmodes, and in highlighting specific future research opportunities.
PMCID: PMC4222484  PMID: 25383276
amplitude-modulation; bimodal; frequency-modulation; multi-frequency atomic force microscopy; multimodal; open loop; trimodal
4.  Bimodal atomic force microscopy driving the higher eigenmode in frequency-modulation mode: Implementation, advantages, disadvantages and comparison to the open-loop case 
We present an overview of the bimodal amplitude–frequency-modulation (AM-FM) imaging mode of atomic force microscopy (AFM), whereby the fundamental eigenmode is driven by using the amplitude-modulation technique (AM-AFM) while a higher eigenmode is driven by using either the constant-excitation or the constant-amplitude variant of the frequency-modulation (FM-AFM) technique. We also offer a comparison to the original bimodal AFM method, in which the higher eigenmode is driven with constant frequency and constant excitation amplitude. General as well as particular characteristics of the different driving schemes are highlighted from theoretical and experimental points of view, revealing the advantages and disadvantages of each. This study provides information and guidelines that can be useful in selecting the most appropriate operation mode to characterize different samples in the most efficient and reliable way.
PMCID: PMC3628543  PMID: 23616939
amplitude-modulation; atomic force microscopy; frequency-modulation; phase-locked loop; spectroscopy
5.  Theoretical study of the frequency shift in bimodal FM-AFM by fractional calculus 
Bimodal atomic force microscopy is a force-microscopy method that requires the simultaneous excitation of two eigenmodes of the cantilever. This method enables the simultaneous recording of several material properties and, at the same time, it also increases the sensitivity of the microscope. Here we apply fractional calculus to express the frequency shift of the second eigenmode in terms of the fractional derivative of the interaction force. We show that this approximation is valid for situations in which the amplitude of the first mode is larger than the length of scale of the force, corresponding to the most common experimental case. We also show that this approximation is valid for very different types of tip–surface forces such as the Lennard-Jones and Derjaguin–Muller–Toporov forces.
PMCID: PMC3323908  PMID: 22496992
AFM; atomic force microscopy; bimodal AFM; frequency shift; integral calculus applications
6.  Challenges and complexities of multifrequency atomic force microscopy in liquid environments 
This paper illustrates through numerical simulation the complexities encountered in high-damping AFM imaging, as in liquid enviroments, within the specific context of multifrequency atomic force microscopy (AFM). The focus is primarily on (i) the amplitude and phase relaxation of driven higher eigenmodes between successive tip–sample impacts, (ii) the momentary excitation of non-driven higher eigenmodes and (iii) base excitation artifacts. The results and discussion are mostly applicable to the cases where higher eigenmodes are driven in open loop and frequency modulation within bimodal schemes, but some concepts are also applicable to other types of multifrequency operations and to single-eigenmode amplitude and frequency modulation methods.
PMCID: PMC3999742  PMID: 24778952
amplitude-modulation; bimodal; frequency-modulation; liquids; multifrequency atomic force microscopy
7.  Interpreting motion and force for narrow-band intermodulation atomic force microscopy 
Intermodulation atomic force microscopy (ImAFM) is a mode of dynamic atomic force microscopy that probes the nonlinear tip–surface force by measurement of the mixing of multiple modes in a frequency comb. A high-quality factor cantilever resonance and a suitable drive comb will result in tip motion described by a narrow-band frequency comb. We show, by a separation of time scales, that such motion is equivalent to rapid oscillations at the cantilever resonance with a slow amplitude and phase or frequency modulation. With this time-domain perspective, we analyze single oscillation cycles in ImAFM to extract the Fourier components of the tip–surface force that are in-phase with the tip motion (F I) and quadrature to the motion (F Q). Traditionally, these force components have been considered as a function of the static-probe height only. Here we show that F I and F Q actually depend on both static-probe height and oscillation amplitude. We demonstrate on simulated data how to reconstruct the amplitude dependence of F I and F Q from a single ImAFM measurement. Furthermore, we introduce ImAFM approach measurements with which we reconstruct the full amplitude and probe-height dependence of the force components F I and F Q, providing deeper insight into the tip–surface interaction. We demonstrate the capabilities of ImAFM approach measurements on a polystyrene polymer surface.
PMCID: PMC3566785  PMID: 23400552
atomic force microscopy; AFM; frequency combs; force spectroscopy; high-quality-factor resonators; intermodulation; multifrequency
8.  Dynamic calibration of higher eigenmode parameters of a cantilever in atomic force microscopy by using tip–surface interactions 
We present a theoretical framework for the dynamic calibration of the higher eigenmode parameters (stiffness and optical lever inverse responsivity) of a cantilever. The method is based on the tip–surface force reconstruction technique and does not require any prior knowledge of the eigenmode shape or the particular form of the tip–surface interaction. The calibration method proposed requires a single-point force measurement by using a multimodal drive and its accuracy is independent of the unknown physical amplitude of a higher eigenmode.
PMCID: PMC4222386  PMID: 25383301
atomic force microscopy; calibration; multimodal AFM; multifrequency AFM
9.  Frequency, amplitude, and phase measurements in contact resonance atomic force microscopies 
The resonance frequency, amplitude, and phase response of the first two eigenmodes of two contact-resonance atomic force microscopy (CR-AFM) configurations, which differ in the method used to excite the system (cantilever base vs sample excitation), are analyzed in this work. Similarities and differences in the observables of the cantilever dynamics, as well as the different effect of the tip–sample contact properties on those observables in each configuration are discussed. Finally, the expected accuracy of CR-AFM using phase-locked loop detection is investigated and quantification of the typical errors incurred during measurements is provided.
PMCID: PMC3999761  PMID: 24778949
contact-resonance AFM; dynamic AFM; frequency modulation; phase-locked loop; viscoelasticity
10.  Measurement of Mechanical Properties of Cantilever Shaped Materials 
Sensors (Basel, Switzerland)  2008;8(5):3497-3541.
Microcantilevers were first introduced as imaging probes in Atomic Force Microscopy (AFM) due to their extremely high sensitivity in measuring surface forces. The versatility of these probes, however, allows the sensing and measurement of a host of mechanical properties of various materials. Sensor parameters such as resonance frequency, quality factor, amplitude of vibration and bending due to a differential stress can all be simultaneously determined for a cantilever. When measuring the mechanical properties of materials, identifying and discerning the most influential parameters responsible for the observed changes in the cantilever response are important. We will, therefore, discuss the effects of various force fields such as those induced by mass loading, residual stress, internal friction of the material, and other changes in the mechanical properties of the microcantilevers. Methods to measure variations in temperature, pressure, or molecular adsorption of water molecules are also discussed. Often these effects occur simultaneously, increasing the number of parameters that need to be concurrently measured to ensure the reliability of the sensors. We therefore systematically investigate the geometric and environmental effects on cantilever measurements including the chemical nature of the underlying interactions. To address the geometric effects we have considered cantilevers with a rectangular or circular cross section. The chemical nature is addressed by using cantilevers fabricated with metals and/or dielectrics. Selective chemical etching, swelling or changes in Young's modulus of the surface were investigated by means of polymeric and inorganic coatings. Finally to address the effect of the environment in which the cantilever operates, the Knudsen number was determined to characterize the molecule-cantilever collisions. Also bimaterial cantilevers with high thermal sensitivity were used to discern the effect of temperature variations. When appropriate, we use continuum mechanics, which is justified according to the ratio between the cantilever thickness and the grain size of the materials. We will also address other potential applications such as the ageing process of nuclear materials, building materials, and optical fibers, which can be investigated by monitoring their mechanical changes with time. In summary, by virtue of the dynamic response of a miniaturized cantilever shaped material, we present useful measurements of the associated elastic properties.
PMCID: PMC3675557
Microcantilever; mechanics; ageing; environment; stress; gas; materials; sensor; pressure; temperature
11.  Atomic Force Microscopy and Wettability Study of Oxidized Patterns at the Surface of Polystyrene 
The surface properties of patterned surfaces made by a combination of photolithography and oxygen plasma treatment of polystyrene (PS) were investigated. PS and plasma-treated PS (PSox) were first characterized using X-ray photoelectron spectroscopy and the study of wetting dynamics (Wilhelmy plate method) in water and in solutions of different pH. The results indicated that the PSox surface may be viewed as covered with a polyelectrolyte-like gel, which swells depending on pH. It was then shown, using atomic force microscopy (AFM), that the adhesion force measured on PS with a silicon tip in water was higher compared with that measured on PSox. This feature allowed imaging of the oxidation patterns using the adhesion mapping mode. The origin of the pulloff force contrast, which could not be explained by combining Johnson–Kendall–Roberts theory and thermodynamic considerations, was attributed to repulsion between the tip and hydrated polymer chains present on the oxidized surface. Imaging was also performed in the lateral force mode, a higher friction being recorded On PS than On PSOX.
PMCID: PMC3261235  PMID: 10550254
atomic force microscopy; plasma treatment; polystyrene; Wilhelmy plate method; Johnson–Kendall–Roberts model; surface forces
12.  Spectroscopic Investigation of Local Mechanical Impedance of Living Cells 
PLoS ONE  2014;9(7):e101687.
We studied nanoscale mechanical properties of PC12 living cells with a Force Feedback Microscope using two experimental approaches. The first one consists in measuring the local mechanical impedance of the cell membrane while simultaneously mapping the cell morphology at constant force. As the interaction force is increased, we observe the appearance of the sub-membrane cytoskeleton. We compare our findings with the outcome of other techniques. The second experimental approach consists in a spectroscopic investigation of the cell while varying the tip indentation into the membrane and consequently the applied force. At variance with conventional dynamic Atomic Force Microscopy techniques, here it is not mandatory to work at the first oscillation eigenmode of the cantilever: the excitation frequency of the tip can be chosen arbitrary leading then to new spectroscopic AFM techniques. We found in this way that the mechanical response of the PC12 cell membrane is found to be frequency dependent in the 1 kHz - 10 kHz range. In particular, we observe that the damping coefficient consistently decreases when the excitation frequency is increased.
PMCID: PMC4084948  PMID: 24999625
13.  Acute Degradation of Surface-Bound Unsaturated Polyolefins in Common Solvents under Ambient Conditions 
We describe the detachment of covalently grafted polybutadiene and polynorbornenechains – which were prepared by surface-initiated ring-opening metathesis polymerization (SiROMP) – from Si/SiO2 substrates upon brief exposure to common solvents in air. Degradation and disappearance of grafted polybutadiene films after successive rinses with dichloromethane was monitored by ellipsometry. Changes in surface topography were analyzed by atomic force microscopy. The rapid auto-oxidation of allylic carbon-hydrogen bonds renders these thin films extremely susceptible to degradation under ambient conditions. Polymers in the tethered state suffer more acute degradation (on the time scale of seconds) compared to those dissolved in solution (not detectable after days). To prevent degradation, unsaturated polymersprepared by SiROMP and the subsequent conversion (of unsaturated groups) need to be carriedout under inert atmosphere. For example, smooth polybutadiene thin films of ~ 100 Å thick were covalently attached to silicon substrates via SiROMP of cyclooctadiene in the vapor phase. Solvent rinsing to remove unreacted monomers and free oligomers/polymers was carried out prior to the conversion of double bonds to epoxide groups. When these steps were carried out under nitrogen, negligible film loss was observed and surface topography of the thin film was preserved. Once the unsaturation was removed from polybutadiene, the epoxidized and hydroxylated polybutadiene films were stable toward solvent exposure in air.
PMCID: PMC2756074  PMID: 19788207
14.  The nanoscale phase distinguishing of PCL-PB-PCL blended in epoxy resin by tapping mode atomic force microscopy 
Nanoscale Research Letters  2012;7(1):153.
In this work, we investigated the bulk phase distinguishing of the poly(ε-caprolactone)-polybutadiene-poly(ε-caprolactone) (PCL-PB-PCL) triblock copolymer blended in epoxy resin by tapping mode atomic force microscopy (TM-AFM). We found that at a set-point amplitude ratio (rsp) less than or equal to 0.85, a clear phase contrast could be obtained using a probe with a force constant of 40 N/m. When rsp was decreased to 0.1 or less, the measured size of the PB-rich domain relatively shrank; however, the height images of the PB-rich domain would take reverse (translating from the original light to dark) at rsp = 0.85. Force-probe measurements were carried out on the phase-separated regions by TM-AFM. According to the phase shift angle vs. rsp curve, it could be concluded that the different force exerting on the epoxy matrix or on the PB-rich domain might result in the height and phase image reversion. Furthermore, the indentation depth vs. rsp plot showed that with large tapping force (lower rsp), the indentation depth for the PB-rich domain was nearly identical for the epoxy resin matrix.
PMCID: PMC3311068  PMID: 22360980
tapping mode AFM; PCL-PB-PCL; phase image; force-probe
15.  Tuning Core vs. Shell Dimensions to Adjust the Performance of Nanoscopic Containers for the Loading and Release of Doxorubicin 
Detailed studies were performed to probe the effects of the core and shell dimensions of amphiphilic, shell crosslinked, knedel-like polymer nanoparticles (SCKs) on the loading and release of doxorubicin (DOX), a widely-used chemotherapy agent, in aqueous buffer, as a function of the solution pH. Effects of the nanoparticle composition were held constant, by employing SCKs constructed from a single type of amphiphilic diblock copolymer, poly(acrylic acid)-b-polystyrene (PAA-b-PS). A series of four SCK nanoparticle samples, ranging in number-average hydrodynamic diameter from 14–30 nm, was prepared from four block copolymers having different relative block lengths and absolute degrees of polymerization. The ratios of acrylic acid to styrene block lengths ranged from 0.65 to 3.0, giving SCKs with ratios of shell to core volumes ranging from 0.44 to 2.1. Although the shell thicknesses were calculated to be similar (1.5–3.1 nm by transmission electron microscopy (TEM) calculations and 3.5–4.9 nm by small angle neutron scattering (SANS) analyses), two of the SCK nanoparticles had relatively large core diameters (19 ± 2 and 20 ± 2 nm by TEM; 17.4 and 15.3 nm by SANS), while two had similar, smaller core diameters (11 ± 2 and 13 ± 2 nm by TEM; 9.0 and 8.9 nm by SANS). The SCKs were capable of being loaded with 1500–9700 DOX molecules per each particle, with larger numbers of DOX molecules packaged within the larger core SCKs. Their shell-to-core volume ratio showed impact on the rates and extents of release of DOX, with the volume occupied by the poly(acrylic acid) shell relative to the volume occupied by the polystyrene core correlating inversely with the diffusion-based release of DOX. Given that the same amount of polymer was used to construct each SCK sample, SCKs having smaller cores and higher acrylic acid vs. styrene volume ratios were present at higher concentrations than were the larger core SCKs, and gave lower final extents of release., Higher final extents of release and faster rates of release were observed for all DOX-loaded particle samples at pH 5.0 vs. pH 7.4, respectively, ca. 60% vs. 40% at 60 h, suggesting promise for enhanced delivery within tumors and cells. By fitting the data to the Higuchi model, quantitative determination of the kinetics of release was made, giving rate constants ranging from 0.0431 to 0.0540 h−½ at pH 7.4 and 0.106 to 0.136 h−½ at pH 5.0. In comparison, the non-crosslinked polymer micelle analogs exhibited rate constants for release of DOX of 0.245 and 0.278 h−½ at pH 7.4 and 5.0, respectively. These studies point to future directions to craft sophisticated devices for controlled drug release.
PMCID: PMC3119510  PMID: 21241750
core-shell nanoparticles; block copolymers; doxorubicin; drug delivery; nanoparticle dimensions; release kinetics
16.  Structural Evolution of Low-Molecular-Weight Poly(ethylene oxide)-block-polystyrene Diblock Copolymer Thin Film 
The Scientific World Journal  2013;2013:539457.
The structural evolution of low-molecular-weight poly(ethylene oxide)-block-polystyrene (PEO-b-PS) diblock copolymer thin film with various initial film thicknesses on silicon substrate under thermal annealing was investigated by atomic force microscopy, optical microscopy, and contact angle measurement. At film thickness below half of the interlamellar spacing of the diblock copolymer (6.2 nm), the entire silicon is covered by a polymer brush with PEO blocks anchored on the Si substrate due to the substrate-induced effect. When the film is thicker than 6.2 nm, a dense polymer brush which is equal to half of an interlamellar layer was formed on the silicon, while the excess material dewet this layer to form droplets. The droplet surface was rich with PS block and the PEO block crystallized inside the bigger droplet to form spherulite.
PMCID: PMC3835911  PMID: 24302862
17.  Modeling viscoelasticity through spring–dashpot models in intermittent-contact atomic force microscopy 
We examine different approaches to model viscoelasticity within atomic force microscopy (AFM) simulation. Our study ranges from very simple linear spring–dashpot models to more sophisticated nonlinear systems that are able to reproduce fundamental properties of viscoelastic surfaces, including creep, stress relaxation and the presence of multiple relaxation times. Some of the models examined have been previously used in AFM simulation, but their applicability to different situations has not yet been examined in detail. The behavior of each model is analyzed here in terms of force–distance curves, dissipated energy and any inherent unphysical artifacts. We focus in this paper on single-eigenmode tip–sample impacts, but the models and results can also be useful in the context of multifrequency AFM, in which the tip trajectories are very complex and there is a wider range of sample deformation frequencies (descriptions of tip–sample model behaviors in the context of multifrequency AFM require detailed studies and are beyond the scope of this work).
PMCID: PMC4273292  PMID: 25551043
atomic force microscopy; creep; dissipated energy; multifrequency; stress relaxation; tapping mode; viscoelasticity
18.  Bimodal Analysis of Mammary Epithelial Cell Migration in Two Dimensions 
Annals of biomedical engineering  2008;37(1):230-245.
Cell migration paths of mammary epithelial cells (expressing different versions of the promigratory tyrosine kinase receptor Her2/Neu) were analyzed within a bimodal framework that is a generalization of the run-and-tumble description applicable to bacterial migration. The mammalian cell trajectories were segregated into two types of alternating modes, namely, the “directional-mode” (mode I, the more persistent mode, analogous to the bacterial run phase) and the “re-orientation-mode” (mode II, the less persistent mode, analogous to the bacterial tumble phase). Higher resolution (more pixel information, relative to cell size) and smaller sampling intervals (time between images) were found to give a better estimate of the deduced single cell dynamics (such as directional-mode time and turn angle distribution) of the various cell types from the bimodal analysis. The bimodal analysis tool permits the deduction of short-time dynamics of cell motion such as the turn angle distributions and turn frequencies during the course of cell migration compared to standard methods of cell migration analysis. We find that the two-hour mammalian cell tracking data do not fall into the diffusive regime implying that the often-used random motility expressions for mammalian cell motion (based on assuming diffusive motion) are invalid over the time steps (fraction of minute) typically used in modeling mammalian cell migration.
PMCID: PMC3586332  PMID: 18982450
Bimodal framework; mammalian cell; turn angle distribution; random motility coefficient
19.  Optimization of Functionalized Polymer Layers for Specific Targeting of Mobile Receptors on Cell Surfaces 
The reversible binding between a planar polymer layer functionalized by targeting groups and a planar cell surface containing different densities of mobile receptors has been studied by Monte Carlo simulations. Using the acceptance-ratio method the distance-dependent profiles for the average number of ligands bound to receptors, the total free energy for the polymer layer-cell surface interaction and the interaction force were obtained. Four main design parameters for the polymer layer were considered: the degree of functionalization, chain degree of polymerization, polymer density of grafting and the binding energy for the targeting group-receptor interaction. We found that an increase in the degree of functionalization or in the absolute energy of ligand-receptor binding results in a larger number of ligands bound to the receptors, lower free energy and stronger attractive force. Polymer layers composed of shorter chains were found to exhibit a deeper and narrower free energy profile and a larger attractive force, while longer tethers can interact with the cell surface at a larger and broader range of separation distances, in agreement with experimental observations. Our simulation results show that the increase in polymer grafting density from the mushroom to brush regime enhances the ligand availability and results in a stronger attractive force, increases the maximum binding distance, but exhibits a shallower free energy minimum due to the smaller tolerance to compression for polymer layers with high grafting density. We used two measures of the polymer layer binding affinity to the cell surface: the free energy minimum, related to the equilibrium binding constant and the fraction of bound ligands. We found that the polymer layers with a smaller chain length and grafting density, larger degree of functionalization and larger absolute binding energy exhibit both a larger equilibrium binding constant to the cell surface and a larger average number of bound ligands, except for high binding energies when the maximum level of binding is reached independently of polymer length and grafting density. We showed that high binding specificity can be achieved by the polymer layers with intermediate ligand-receptor binding energies or an intermediate number of ligands, as a larger binding energy or number of ligands ensures a high binding affinity but lacks specificity while a smaller binding energy or number of ligands provides inadequate affinity. We found that the results for polymer layers with different properties follow a similar pattern when both high binding affinity to cells with high receptor density and high binding specificity are considered. As a result, the optimal design of the polymer layers can be achieved by using several different strategies, which are discussed.
PMCID: PMC2731659  PMID: 18834163
20.  On nodes and modes in resting state fMRI 
Neuroimage  2014;99(100):533-547.
This paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fMRI timeseries — and simulations of neuronal fluctuations based upon the connectome. Its particular focus is on patterns or modes of distributed activity that underlie functional connectivity. We first demonstrate that the eigenmodes of functional connectivity – or covariance among regions or nodes – are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections. This symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling. Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory. Technically, these modes have small negative Lyapunov exponents that approach zero from below. Interestingly, the superposition of modes – whose exponents are sampled from a power law distribution – produces classical 1/f (scale free) spectra. We conjecture that the emergence of dynamical instability – that underlies intrinsic brain networks – is inevitable in any system that is separated from external states by a Markov blanket. This conjecture appeals to a free energy formulation of nonequilibrium steady-state dynamics. The common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes. We use this as the basis of a dynamic causal model (DCM) of resting state fluctuations — as measured in terms of their complex cross spectra. In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents — that can also be interpreted as locations in a multidimensional scaling space. Model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space. Here, we focus on conceptual issues with simulated fMRI data and provide an illustrative application using an empirical multi-region timeseries.
•This paper presents a formal characterisation of [eigen] modes in resting state fMRI.•It relates these to dynamical instabilities in neuronal (inferential) processing.•It uses the equivalence between eigenmodes of functional and effective connectivity for dynamical causal modelling.
PMCID: PMC4121089  PMID: 24862075
Dynamic causal modelling; Effective connectivity; Functional connectivity; Resting state; fMRI; Proximity graph; Free energy; Criticality; Self-organisation; Lyapunov exponents; Bayesian
21.  Probing viscoelastic surfaces with bimodal tapping-mode atomic force microscopy: Underlying physics and observables for a standard linear solid model 
This paper presents computational simulations of single-mode and bimodal atomic force microscopy (AFM) with particular focus on the viscoelastic interactions occurring during tip–sample impact. The surface is modeled by using a standard linear solid model, which is the simplest system that can reproduce creep compliance and stress relaxation, which are fundamental behaviors exhibited by viscoelastic surfaces. The relaxation of the surface in combination with the complexities of bimodal tip–sample impacts gives rise to unique dynamic behaviors that have important consequences with regards to the acquisition of quantitative relationships between the sample properties and the AFM observables. The physics of the tip–sample interactions and its effect on the observables are illustrated and discussed, and a brief research outlook on viscoelasticity measurement with intermittent-contact AFM is provided.
PMCID: PMC4222404  PMID: 25383277
amplitude-modulation; bimodal; dissipation; frequency modulation; multi-frequency atomic force microscopy; viscoelasticity; standard linear solid
22.  Drive-amplitude-modulation atomic force microscopy: From vacuum to liquids 
We introduce drive-amplitude-modulation atomic force microscopy as a dynamic mode with outstanding performance in all environments from vacuum to liquids. As with frequency modulation, the new mode follows a feedback scheme with two nested loops: The first keeps the cantilever oscillation amplitude constant by regulating the driving force, and the second uses the driving force as the feedback variable for topography. Additionally, a phase-locked loop can be used as a parallel feedback allowing separation of the conservative and nonconservative interactions. We describe the basis of this mode and present some examples of its performance in three different environments. Drive-amplutide modulation is a very stable, intuitive and easy to use mode that is free of the feedback instability associated with the noncontact-to-contact transition that occurs in the frequency-modulation mode.
PMCID: PMC3343270  PMID: 22563531
atomic force microscopy; control systems; dissipation; frequency modulation; noncontact
23.  Multimode circuit optomechanics near the quantum limit 
Nature Communications  2012;3:987-.
The coupling of distinct systems underlies nearly all physical phenomena. A basic instance is that of interacting harmonic oscillators, giving rise to, for example, the phonon eigenmodes in a lattice. Of particular importance are the interactions in hybrid quantum systems, which can combine the benefits of each part in quantum technologies. Here we investigate a hybrid optomechanical system having three degrees of freedom, consisting of a microwave cavity and two micromechanical beams with closely spaced frequencies around 32 MHz and no direct interaction. We record the first evidence of tripartite optomechanical mixing, implying that the eigenmodes are combinations of one photonic and two phononic modes. We identify an asymmetric dark mode having a long lifetime. Simultaneously, we operate the nearly macroscopic mechanical modes close to the motional quantum ground state, down to 1.8 thermal quanta, achieved by back-action cooling. These results constitute an important advance towards engineering of entangled motional states.
Optomechanical systems allow for the exploration of macroscopic behaviour at or near the quantum limit. Massel et al. use micromechanical resonators to study the hybridisation of one photonic and two phononic modes with phonon numbers down to 1.8, showing a coupling between all three degrees of freedom.
PMCID: PMC3432470  PMID: 22871806
24.  Detection and Localization of Single LysM-Peptidoglycan Interactions▿  
Journal of Bacteriology  2008;190(21):7079-7086.
The lysin motif (LysM) is a ubiquitous protein module that binds peptidoglycan and structurally related molecules. Here, we used single-molecule force spectroscopy (SMFS) to measure and localize individual LysM-peptidoglycan interactions on both model and cellular surfaces. LysM modules of the major autolysin AcmA of Lactococcus lactis were bound to gold-coated atomic force microscopy tips, while peptidoglycan was covalently attached onto model supports. Multiple force curves recorded between the LysM tips and peptidoglycan surfaces yielded a bimodal distribution of binding forces, presumably reflecting the occurrence of one and two LysM-peptidoglycan interactions, respectively. The specificity of the measured interaction was confirmed by performing blocking experiments with free peptidoglycan. Next, the LysM tips were used to map single LysM interactions on the surfaces of L. lactis cells. Strikingly, native cells showed very poor binding, suggesting that peptidoglycan was hindered by other cell wall constituents. Consistent with this notion, treatment of the cells with trichloroacetic acid, which removes peptidoglycan-associated polymers, resulted in substantial and homogeneous binding of the LysM tip. These results provide novel insight into the binding forces of bacterial LysMs and show that SMFS is a promising tool for studying the heterologous display of proteins or peptides on bacterial surfaces.
PMCID: PMC2580685  PMID: 18757536
25.  Atomic force microscopy with sub-picoNewton force stability for biological applications 
Methods (San Diego, Calif.)  2013;60(2):131-141.
Atomic force microscopy (AFM) is widely used in the biological sciences. Despite 25 years of technical developments, two popular modes of bioAFM, imaging and single molecule force spectroscopy, remain hindered by relatively poor force precision and stability. Recently, we achieved both sub-pN force precision and stability under biologically useful conditions (in liquid at room temperature). Importantly, this sub-pN level of performance is routinely accessible using a commercial cantilever on a commercial instrument. The two critical results are that (i) force precision and stability were limited by the gold coating on the cantilevers, and (ii) smaller yet stiffer cantilevers did not lead to better force precision on time scales longer than 25 ms. These new findings complement our previous work that addressed tip-sample stability. In this review, we detail the methods needed to achieve this sub-pN force stability and demonstrate improvements in force spectroscopy and imaging when using uncoated cantilevers. With this improved cantilever performance, the widespread use of nonspecific biomolecular attachments becomes a limiting factor in high-precision studies. Thus, we conclude by briefly reviewing site-specific covalent-immobilization protocols for linking a biomolecule to the substrate and to the AFM tip.
PMCID: PMC3669665  PMID: 23562681
Atomic force microscopy; scanning probe microscopy; single molecule force spectroscopy; drift; overstretching DNA; cantilever; imaging

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