Piconewton Scale Research Articles

Control of the Probe-Sample Interaction Force at the Piconewton Scale by a Magnetic Microprobe in Aqueous Solutions

This article describes the design and implementation of a control system that controls the probe-sample interaction force at the piconewton scale by manipulating a magnetic microprobe in aqueous solutions under a microscope. The control system has two functions, namely accurate force generation by magnetic flux control and precise control of the probe-sample interaction force. Magnetic flux control uses an improved six-input-six-output digital control law along with a disturbance estimator to achieve two control objectives. First, the magnetic flux at each pole tip of a previously developed hexapole electromagnetic actuator can be individually and precisely controlled at frequencies over 4 kHz, and the experimental results precisely followed the theoretical predictions based on the design. Second, together with the optimized flux allocation, the flux control system generates precise magnetic forces, making the six-input hexapole actuator behave like a decoupled three-axis force generator with a bandwidth of more than 4 kHz. Interaction force control compares the measured deformation of the sample with the expected deformation, updated in real time by a deformation predictor, to generate the magnetic force to control the interaction between the probe and the sample. Digital control laws, the estimator, and the predictor are all implemented using a high-speed Field Programmable Gate Array (FPGA) system. Experiments confirm that through mathematical modeling, digital control technology, high-speed electronics, and real-time computation, accurate three-dimensional force generation at the piconewton scale with high bandwidth and precise interaction force control with zero-mean random error, attributed to random thermal force and measurement noise, are achieved.

Improved Prediction of Molecular Response to Pulling by Combining Force Tempering with Replica Exchange Methods.

Small mechanical forces play important functional roles in many crucial cellular processes, including in the dynamic behavior of the cytoskeleton and in the regulation of osmotic pressure through membrane-bound proteins. Molecular simulations offer the promise of being able to design the behavior of proteins that sense and respond to these forces. However, it is difficult to predict and identify the effect of the relevant piconewton (pN) scale forces due to their small magnitude. Previously, we introduced the Infinite Switch Simulated Tempering in Force (FISST) method, which allows one to estimate the effect of a range of applied forces from a single molecular dynamics simulation, and also demonstrated that FISST additionally accelerates sampling of a molecule's conformational landscape. For some problems, we find that this acceleration is not sufficient to capture all relevant conformational fluctuations, and hence, here we demonstrate that FISST can be combined with either temperature replica exchange or solute tempering approaches to produce a hybrid method that enables more robust prediction of the effect of small forces on molecular systems.

Unravelling DNA Organization with Single-Molecule Force Spectroscopy Using Magnetic Tweezers.

Genomes carry the genetic blueprint of all living organisms. Their organization requires strong condensation as well as carefully regulated accessibility to specific genes for proper functioning of their hosts. The study of the structure and dynamics of the proteins that organize the genome has benefited tremendously from the development of single-molecule force spectroscopy techniques that allow for real-time, nanometer accuracy measurements of the compaction of DNA and manipulation with pico-Newton scale forces. Magnetic tweezers, in particular, have the unique ability to complement such force spectroscopy with the control over the linking number of the DNA molecule, which plays an important role when DNA-organizing proteins form or release wraps, loops, and bends in DNA. Here, we describe all the necessary steps to prepare DNA substrates for magnetic tweezers experiments, assemble flow cells, tether DNA to a magnetic bead inside a flow cell, and manipulate and record the extension of such DNA tethers. Furthermore, we explain how mechanical parameters of nucleoprotein filaments can be extracted from the data.

Force-Regulated Spontaneous Conformational Changes of Integrins α5β1 and αVβ3.

Integrins are cell surface nanosized receptors crucial for cell motility and mechanosensing of the extracellular environment, which are often targeted for the development of biomaterials and nanomedicines. As a key feature of integrins, their activity, structure and behavior are highly mechanosensitive, which are regulated by mechanical forces down to pico-Newton scale. Using single-molecule biomechanical approaches, we compared the force-modulated ectodomain bending/unbending conformational changes of two integrin species, α5β1 and αVβ3. It was found that the conformation of integrin α5β1 is determined by a threshold head-to-tail tension. By comparison, integrin αVβ3 exhibits bistability even without force and can spontaneously transition between the bent and extended conformations with an apparent transition time under a wide range of forces. Molecular dynamics simulations observed almost concurrent disruption of ∼2 hydrogen bonds during integrin α5β1 unbending, but consecutive disruption of ∼7 hydrogen bonds during integrin αVβ3 unbending. Accordingly, we constructed a canonical energy landscape for integrin α5β1 with a single energy well that traps the integrin in the bent state until sufficient force tilts the energy landscape to allow the conformational transition. In contrast, the energy landscape of integrin αVβ3 conformational changes was constructed with hexa-stable intermediate states and intermediate energy barriers that segregate the conformational change process into multiple small steps. Our study elucidates the different biomechanical inner workings of integrins α5β1 and αVβ3 at the submolecular level, helps understand their mechanosignaling processes and how their respective functions are facilitated by their distinctive mechanosensitivities, and provides useful design principles for the engineering of protein-based biomechanical nanomachines.

Single molecule magnetic tweezers for unfolding dynamics of Acyl-CoA binding protein

There are significant differences in mechanical stability and unfolding dynamics among proteins with different structural compositions. Compared with proteins with β-sheets and subjected to shearing forces, proteins that are composed entirely of α-helices often undergo rapid unfolding under low stretching forces, thus requiring quantitative studies by using experimental tools that can precisely control forces on a pico-Newton scale. Magnetic tweezers with intrinsic force-control capability and great stability for long-time continuous measurement are suitable to measure force-induced conformation transitions of protein subjected to low forces of several pico-Newton. Acyl-CoA binding protein (ACBP) is a model protein used to study the folding/unfolding kinetics of complete α-helices protein. It is composed of 86 amino acid residues, forming a helical bundle of four α-helices. When its N- and C-terminal are stretched, the first and last α-helix are subjected to shear force in parallel. Previous biochemical studies showed that ACBP folding and unfolding in a two-state manner. In this paper, we use magnetic tweezers to stretch ACBP from its N- and C-end and obtain the distribution of the unfolding force at different loading rates ranging from 0.25 pN/s to 4 pN/s. The most probable unfolding forces are all less than 10 pN, which confirms that ACBP is not mechanically stable. At a constant loading rate, the unfolding force distribution and the most probable unfolding force as a function of loading rate have well-defined analytical formulas based on Bell’s model. Therefore, the experimental results of unfolding force can be fitted directly to obtain the important kinetic parameter of unfolding distance which is defined as the difference in extension between the native state and the transition state. Data analysis shows that ACBP has an extraordinarily long unfolding distance of 7.8 nm. Steered molecular dynamics simulations of ACBP stretching gives the transition state with N-terminal α-helix fully unfolded and C-terminal α-helix partially unfolded, which is consistent with the long unfolding distance obtained in the experiment on magnetic tweezers. According to the simulation results, the unfolding of α-helices is less cooperative than that of β-sheet structures. This characteristic makes α-helix proteins sensitive to mechanical forces, rendering them suitable as force sensors in cells. This study shows that single-molecule stretching experiment combined with molecular dynamics simulations is a reliable method to reveal the molecular mechanism of protein conformationtransitions under stretching forces.

Measuring Cellular Traction Forces with Micropillar Arrays.

Accurately evaluating cellular forces is critical for studying mechanosensing and mechanotransduction processes, and it necessitates sensitive measurements on the piconewton scale. Here we describe a specialized method that employs elastic polydimethylsiloxane (PDMS) micropillar arrays, which cells can adhere to and bend. The flexibility of the pillars correlates with their heights; the longer they are, the easier they are to bend. Thus, an array of taller pillars mimics a relatively soft substrate that readily yields in response to cellular forces. Tracking cell movements and pillar displacements using live-cell microscopy enables the calculation of cellular forces and the tracking of their dynamic features throughout early and late stages of cell spreading on the pillars. This technique offers the advantage of high spatial and temporal resolution analyses and constitutes a method to investigate the effect of substrate rigidities on cellular functions.

On non-Kolmogorov turbulence in blood flow and its possible role in mechanobiological stimulation

The study of turbulence in physiologic blood flow is important due to its strong relevance to endothelial mechanobiology and vascular disease. Recently, Saqr et al. (Sci Rep 10, 15,492, 2020) discovered non-Kolmogorov turbulence in physiologic blood flow in vivo, traced its origins to the Navier–Stokes equation and demonstrated some of its properties using chaos and hydrodynamic-stability theories. The present work extends these findings and investigates some inherent characteristics of non-Kolmogorov turbulence in monoharmonic and multiharmonic pulsatile flow under ideal physiologic conditions. The purpose of this work is to propose a conjecture for the origins for picoNewton forces that are known to regulate endothelial cells’ functions. The new conjecture relates these forces to physiologic momentum-viscous interactions in the near-wall region of the flow. Here, we used high-resolution large eddy simulation (HRLES) to study pulsatile incompressible flow in a straight pipe of L/D=20. The simulations presented Newtonian and Carreau–Yasuda fluid flows, at R{e}_{m}approx 250, each represented by one, two and three boundary harmonics. Comparison was established based on maintaining constant time-averaged mass flow rate in all simulations. First, we report the effect of primary harmonics on the global power budget using primitive variables in phase space. Second, we describe the non-Kolmogorov turbulence in frequency domain. Third, we investigate the near-wall coherent structures in time and space domains. Finally, we propose a new conjecture for the role of turbulence in endothelial cells’ mechanobiology. The proposed conjecture correlates near-wall turbulence to a force field of picoNewton scale, suggesting possible relevance to endothelial cells mechanobiology.

Tension Gauge Tether Probes for Quantifying Growth Factor Mediated Integrin Mechanics and Adhesion.

Multicellular organisms rely on interactions between membrane receptors and cognate ligands in the surrounding extracellular matrix (ECM) to orchestrate multiple functions, including adhesion, proliferation, migration, and differentiation. Mechanical forces can be transmitted from the cell via the adhesion receptor integrin to ligands in the ECM. The amount and spatial organization of these cell-generated forces can be modulated by growth factor receptors, including epidermal growth factor receptor (EGFR). The tools currently available to quantify crosstalk-mediated changes in cell mechanics and relate them tofocal adhesions, cellular morphology, and signaling are limited. DNA-based molecular force sensors known as tension gauge tethers (TGTs) have been employed to quantify these changes. TGT probes are unique in their ability to both modulate the underlying force threshold and report piconewton scale receptor forces across the entire adherent cell surface at diffraction-limited spatial resolution. The TGT probes used here rely on the irreversible dissociation of a DNA duplex by receptor-ligand forces that generate a fluorescent signal. This allows quantification of the cumulative integrin tension (force history) of the cell. This article describes a protocol employing TGTs to study the impact of EGFR on integrin mechanics and adhesion formation. The assembly ofthe TGT mechanical sensing platform is systematically detailed and the procedure to image forces, focal adhesions, and cell spreading is outlined. Overall, the ability to modulate the underlying force threshold of the probe, the adhesion ligand, and the type and concentration of growth factor employed for stimulation make this a robust platform for studying the interplay of diverse membrane receptors in regulating integrin-mediated forces.

Tension Gauge Tether Probes for Quantifying Growth Factor Mediated Integrin Mechanics and Adhesion

Multicellular organisms rely on interactions between membrane receptors and cognate ligands in the surrounding extracellular matrix (ECM) to orchestrate multiple functions, including adhesion, proliferation, migration, and differentiation. Mechanical forces can be transmitted from the cell via the adhesion receptor integrin to ligands in the ECM. The amount and spatial organization of these cell-generated forces can be modulated by growth factor receptors, including epidermal growth factor receptor (EGFR). The tools currently available to quantify crosstalk-mediated changes in cell mechanics and relate them to focal adhesions, cellular morphology, and signaling are limited. DNA-based molecular force sensors known as tension gauge tethers (TGTs) have been employed to quantify these changes. TGT probes are unique in their ability to both modulate the underlying force threshold and report piconewton scale receptor forces across the entire adherent cell surface at diffraction-limited spatial resolution. The TGT probes used here rely on the irreversible dissociation of a DNA duplex by receptor-ligand forces that generate a fluorescent signal. This allows quantification of the cumulative integrin tension (force history) of the cell. This article describes a protocol employing TGTs to study the impact of EGFR on integrin mechanics and adhesion formation. The assembly of the TGT mechanical sensing platform is systematically detailed and the procedure to image forces, focal adhesions, and cell spreading is outlined. Overall, the ability to modulate the underlying force threshold of the probe, the adhesion ligand, and the type and concentration of growth factor employed for stimulation make this a robust platform for studying the interplay of diverse membrane receptors in regulating integrin-mediated forces.

DNA tension probes show that cardiomyocyte maturation is sensitive to piconewton scale traction forces

Cardiac muscle cells (CMCs) are the unit cells that comprise the heart. CMCs go through different stages of differentiation and maturation pathways to fully mature into beating cells. These cells can sense and respond to mechanical cues through receptors such as integrins which influence maturation pathways. For example, cell traction forces are important for the differentiation and development of functional CMCs, as CMCs cultured on varying substrate stiffness function differently. Most work in this area has focused on understanding the role of bulk ECM (extracellular matrix) stiffness in mediating the functional fate of CMC.

High-precision atomic force microscopy with atomically-characterized tips

Traditionally, atomic force microscopy (AFM) experiments are conducted at tip–sample distances where the tip strongly interacts with the surface. This increases the signal-to-noise ratio, but poses the problem of relaxations in both tip and sample that hamper the theoretical description of experimental data. Here, we employ AFM at relatively large tip–sample distances where forces are only on the piconewton and subpiconewton scale to prevent tip and sample distortions. Acquiring data relatively far from the surface requires low noise measurements. We probed the CaF2(111) surface with an atomically-characterized metal tip and show that the experimental data can be reproduced with an electrostatic model. By experimentally characterizing the second layer of tip atoms, we were able to reproduce the data with 99.5% accuracy. Our work links the capabilities of non-invasive imaging at large tip–sample distances and controlling the tip apex at the atomic scale.

Probing nanomechanical responses of cell membranes

Despite the importance in various cellular processes, the nanomechanical responses of the living cell membrane have been elusive due to complexities in the membrane associated with the hidden architecture of multiple molecular components, including the lipid bilayer. Here, combined experimental and theoretical frameworks that can probe and interpret nanomechanical responses of the cell membrane are demonstrated. A magnetic tweezer assay was introduced to apply pico-Newton scale forces to lipids and E-cadherin molecules at the living cell surface. Two unique classes of force-extension curves were identified: one with a deflection transition (Type I) and another with a discontinuous transition (Type II). The repeated observations of these responses, regardless of cell type and targeted cell surface molecule, suggest the Type I and II curves are the primary nanomechanical responses of cell membranes. To reproduce these responses in vitro, a model system using synthetic lipid vesicles was also developed. Together with a finite element model of lipid bilayers, the reproduced responses suggest that the confined fluidity and curvature constraints imposed on the lipid bilayer components of the cell membrane are the main parameters responsible for the generation of these responses. This work provides an insight into how forces on membrane molecules propagate to the lipid bilayer components to generate specific nanomechanical responses. In addition, the consistent results obtained using different methodologies demonstrate that the presented force-probing assays and the theoretical model can serve a combined testbed to investigate nanoscale mechanics of the living cell membrane.

The Detection of Long-Chain Bio-Markers Using Atomic Force Microscopy

The detection of long-chain biomolecules on mineral surfaces is presented using an atomic force microscope (AFM). This is achieved by using the AFM’s ability to manipulate molecules and measure forces at the pico-newton scale. We show that a highly characteristic force-distance signal is produced when the AFM tip is used to detach long-chain molecules from a surface. This AFM force spectroscopy method is demonstrated on bio-films, spores, fossils and mineral surfaces. The method works with AFM imaging and correlated tip enhanced infrared spectroscopy. The use of AFM force spectroscopy to detect this class of long chain bio-markers has applications in paleontology, life detection and planetary science.

Forces of Phagocytosis with Two-Channel Live Cell Bessel Light Sheet 3D Imaging

A critical function of the immune system is the search and elimination of foreign pathogens by macrophages, neutrophils and other cells. These cells eliminate large particles (>0.5μm) by consuming them through a process called phagocytosis. There are several receptor-ligand pairs and associated signaling pathways that initiate phagocytosis. The best understood is the Fcϒ receptor (FcϒR)-immunoglobulin G (IgG) pair. Recognition by FcϒR of an IgG coated particle triggers the cell's cytoskeleton to form a pseudopod that reaches around and engulfs the particle. Evaluating the mechanics of macrophage target engulfment requires careful monitoring of forces and high quality florescence imaging of the membrane and cytoskeleton. We combined an atomic force microscope (AFM) with a versatile optics system, a first of its kind, to monitor piconewton scale forces while imaging the phagocytosis engulfment from the side using Pathway Rotated Imaging for Sideways Microscopy with vertical light sheet (PRISM-LS). The macrophage produces a dynamic response a few nanonewtons during its envelopment of an IgG covered bead attached to an AFM cantilever. We observe the macrophages pull on the bead with downward forces in the range of 300pN-8nN. The macrophage exerts these downward forces before the pseudopod envelops past the midpoint of the bead. These downward forces can last from 5 seconds to 2 minutes. The phagocytic cup formation is associated with punctate brightening of actin underneath the bead. In addition, the versatile optics system can be used to record two-channel, three dimensional images of cells while engaging with the AFM. Bessel beam light sheet imaging with controlled force data will inform mechanical models of phagocytosis which will improve understanding of this important immunological process and inform mammalian disease progression.

Design, modeling, and control of a horizontal magnetic micromanipulator

Magnetic micromanipulators with a wide range of force generating capabilities are able to manipulate micron size particles for various applications and measurements. These magnetic particles can be coated with receptors to specifically bind to target biomolecules. In this work, a horizontal magnetic micromanipulator is designed, modeled and controlled for single micron size magnetic particle manipulations. A method is presented for dynamic modeling of magnetic micromanipulators. A feedback control method is designed that allows direct linearization of the system. It is shown that the proposed controller guarantees the stability of the closed-loop system, and yields zero steady-state error in a wide range of operation conditions. We show that the micromanipulator is able to generate a wide range of piconewton (pN) scale forces on a superparamagnetic particle for single molecule separation, and biosensor developments.

SensorFRET: A Standardless Approach to Measuring Pixel-based Spectral Bleed-through and FRET Efficiency using Spectral Imaging

Fluorescence microscopy of FRET-based biosensors allow nanoscale interactions to be probed in living cells. This paper describes a novel approach to spectrally resolved fluorescence microscopy, termed sensorFRET, that enables quantitative measurement of FRET efficiency. This approach is an improvement on existing methods (FLIM, sRET, luxFRET, pFRET), as it does not require single fluorophore standards to be measured with every experiment and the acquisition is intensity independent, allowing the laser power to be optimized for varying levels of fluorophore expression. Additionally, it was found that all spectral based methods, including sensorFRET, fail at specific fluorophore-excitation wavelength combinations. These combinations can be determined a priori using sensorFRET, whereas other methods would give no indication of inaccuracies. This method was thoroughly validated and compared to existing methods using simulated spectra, Fluorescein and TAMRA dye mixtures as a zero FRET control, and Cerulean-Venus FRET standards as positive FRET controls. Simulations also provided a means of quantifying the uncertainty in each measurement by relating the fit residual of noisy spectra to the standard deviation of the measured FRET efficiency. As an example application, Teal-Venus force sensitive biosensors integrated into E-cadherin were used to resolve piconewton scale forces along different parts of an individual cell junction.

Mechanics of Bacterial Cells and Initial Surface Colonisation.

The mechanical properties of bacterial cells play an important role in crucial bacterial processes such as cell growth, colonisation and biofilm formation. Recent developments in the field of nanotechnology and atomic force microscopy (AFM) have made it possible to observe, characterise and understand the nanomechanic behaviour of live bacterial cells as never before. Unlike traditional techniques, AFM makes it possible to employ living bacteria in their physiological environment with minimal or no sample preparation. The technique of AFM nanoindentation opens new possibilities to study bacterial cell wall stiffness under different mechanical and buffer conditions. Also, by attaching bacterial cells to functionalised AFM cantilevers, single-cell force spectroscopy (SCFS) can be used to measure the adhesion of bacteria to biological and non-biological substrates at the nano-newton and pico-newton scale, and provide specific information on receptor-ligand interactions. By studying the biophysics of the bacterial-surface interaction with the abovementioned techniques, it has been possible to gain new insight on the early stages of bacterial colonisation and biofilm formation.

Simple horizontal magnetic tweezers for micromanipulation of single DNA molecules and DNA-protein complexes.

We report the development of a simple-to-implement magnetic force transducer that can apply a wide range of piconewton (pN) scale forces on single DNA molecules and DNA-protein complexes in the horizontal plane. The resulting low-noise force-extension data enable very high-resolution detection of changes in the DNA tether's extension: ~0.05 pN in force and <10 nm change in extension. We have also verified that we can manipulate DNA in near equilibrium conditions through the wide range of forces by ramping the force from low to high and back again, and observing minimal hysteresis in the molecule's force response. Using a calibration technique based on Stokes' drag law, we have confirmed our force measurements from DNA force-extension experiments obtained using the fluctuation-dissipation theorem applied to transverse fluctuations of the magnetic microsphere. We present data on the force-distance characteristics of a DNA molecule complexed with histones. The results illustrate how the tweezers can be used to study DNA binding proteins at the single molecule level.

Lighting Up the Force: Investigating Mechanisms of Mechanotransduction Using Fluorescent Tension Probes.

The ability of cells to sense the physical nature of their surroundings is critical to the survival of multicellular organisms. Cellular response to physical cues from adjacent cells and the extracellular matrix leads to a dynamic cycle in which cells respond by remodeling their local microenvironment, fine-tuning cell stiffness, polarity, and shape. Mechanical regulation is important in cellular development, normal morphogenesis, and wound healing. The mechanisms by which these finely balanced mechanotransduction events occur, however, are not well understood. In large part, this is due to the limited availability of tools to study molecular mechanotransduction events in live cells. Several classes of molecular tension probes have been recently developed which are rapidly transforming the study of mechanotransduction. Molecular tension probes are primarily based on fluorescence resonance energy transfer (FRET) and report on piconewton scale tension events in live cells. In this minireview, we describe the two main classes of tension probes, genetically encoded tension sensors and immobilized tension sensors, and discuss the advantages and limitations of each type. We discuss future opportunities to address major biological questions and outline the challenges facing the next generation of molecular tension probes.

Calibration of optical tweezers with non-spherical probes via high-resolution detection of Brownian motion

Optical tweezers are commonly used and powerful tools to perform force measurements on the piconewton scale and to detect nanometer-scaled displacements. However, the precision of these instruments relies to a great extent on the accuracy of the calibration method. A well-known calibration procedure is to record the stochastic motion of the trapped particle and compare its statistical behavior with the theory of the Brownian motion in a harmonic potential. Here we present an interactive calibration software which allows for the simultaneous fitting of three different statistical observables (power spectral density, mean square displacement and velocity autocorrelation function) calculated from the trajectory of the probe to enhance fitting accuracy. The fitted theory involves the hydrodynamic interactions experimentally observable at high sampling rates. Furthermore, a qualitative extension is included in our model to handle the thermal fluctuations in the orientation of optically trapped asymmetric objects. The presented calibration methodology requires no prior knowledge of the bead size and can be applied to non-spherical probes as well. The software was validated on synthetic and experimental data. Program summaryProgram title: PFMCalCatalogue identifier: AEXH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXH_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 206,399No. of bytes in distributed program, including test data, etc.: 10,319,465Distribution format: tar.gzProgramming language: MatLab 2011a (MathWorks Inc.).Computer: General computer running MatLab (MathWorks Inc.), using Statistics Toolbox.Operating system: Any which supports Matlab using Statistics Toolbox.RAM: 10 MBClassification: 3, 4.9, 18, 23.Nature of problem: Calibration of optical tweezers by measuring the Brownian motion of the trapped object. The voltage-to-displacement ratio of the detection system (the inverse of the sensitivity), the stiffness of the trap and the size of the bead are obtained via the simultaneous fitting of the power spectral density (PSD), mean square displacement (MSD) and velocity autocorrelation (VAF) functions calculated from the trajectory. The calibration can be performed for non-spherical probes as well.Solution method: Initialization points for all parameters are inferred from characteristic features of the statistical observables (PSD, MSD and VAF) based on the method developed by Grimm et al. in [1]. Theoretical functions for the PSD, the MSD and the VAF are calculated from the model of Brownian motion confined by a harmonic potential taking hydrodynamic interactions into consideration [2–4]. This calibration methodology has been successfully used in actual experiments with micro-spheres [5, 6]. Calculated functions are fitted to the measurement data via the Levenberg–Marquardt least square fitting routine available in the MatLab Optimalization Toolbox, using the nlinfit function. If the error to the measured data has been estimated, the corresponding data values can be weighted by the inverse of the standard error squared in order to eliminate bias introduced by heteroscedasticity. In order to increase robustness and avoid convergence to local minima, minimum search from multiple initial values in the vicinity of the first guess is possible.Additional comments: Input of the program is the experimental PSDexp, MSDexp, and VAFexp data points calculated from the measured x,y and z projections of the trajectory of the particle. The data may best be blocked and may optionally contain an error for each data point for better results. The data should be formatted into three columns: 1. independent variables (time or frequency array), 2. function values, 3. optionally, error values (e.g. standard error calculated from the binning) to improve fitting efficiency. In case error values were not estimated, the third column should be filled with zeros. The first row is reserved for a header, data is read from the second row. Data size should not be larger than a few hundreds of rows, otherwise, using larger blocks for binning is advised. In order to observe the short time behavior of the Brownian motion, influenced by hydrodynamic effects, the sampling rate should be typically higher than 100 kHz. Total sampling time is typically tens of seconds, to achieve a good resolution in the frequency range. The optical axis of the laser, the microscope and the detector system should be co-aligned to exclude artificial crosstalk between the x,y and z channels of the position detector.Running time: Seconds