High-performance reconstruction of microscopic force fields from Brownian trajectories

Solving non-Hermitian physics for optical manipulation on a quantum computer

Intense laser light, with its ability to trap small particles, is providing us unprecedented access to the microscopic world. Nevertheless, owing to its open nature, optical force is nonconservative and can only be described by a non-Hermitian theory. This non-Hermiticity sets such system apart from conventional systems and has offered rich physics, such as the possession of the exceptional points. Consequently, analyzing and demonstrating the dynamics of large optically-bound clusters becomes an intricate challenge. Here, we developed a scalable quantum approach that allows us to predict the trajectories of optically trapped particles and tackle the associated non-Hermitian physics. This approach is based on the linear combination of unitary operations. With this, we experimentally revealed the non-Hermiticity and exceptional point for a single or multiple particles trapped by optical force fields, using a nuclear magnetic resonance quantum processor. Our method's scalability and stability have offering a promising path for large-scale optical manipulation with non-Hermitian dynamics.

Photon-efficient optical tweezers via wavefront shaping

Optical tweezers enable noncontact trapping of microscale objects using light. It is not known how tightly it is possible to three-dimensionally (3D) trap microparticles with a given photon budget. Reaching this elusive limit would enable maximally stiff particle trapping for precision measurements on the nanoscale and photon-efficient tweezing of light-sensitive objects. Here, we customize the shape of light fields to suit specific particles, with the aim of optimizing trapping stiffness in 3D. We show, theoretically, that the confinement volume of microspheres held in sculpted optical traps can be reduced by one to two orders of magnitude. Experimentally, we use a wavefront shaping-inspired strategy to passively suppress the Brownian fluctuations of microspheres in every direction concurrently, demonstrating order-of-magnitude reductions in their confinement volumes. Our work paves the way toward the fundamental limits of optical control over the mesoscopic realm.

Deep learning for optical tweezers

Optical tweezers exploit light-matter interactions to trap particles ranging from single atoms to micrometer-sized eukaryotic cells. For this reason, optical tweezers are a ubiquitous tool in physics, biology, and nanotechnology. Recently, the use of deep learning has started to enhance optical tweezers by improving their design, calibration, and real-time control as well as the tracking and analysis of the trapped objects, often outperforming classical methods thanks to the higher computational speed and versatility of deep learning. In this perspective, we show how cutting-edge deep learning approaches can remarkably improve optical tweezers, and explore the exciting, new future possibilities enabled by this dynamic synergy. Furthermore, we offer guidelines on integrating deep learning with optical trapping and optical manipulation in a reliable and trustworthy way.

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Shining Light in Mechanobiology: Optical Tweezers, Scissors, and Beyond

Mechanobiology helps us to decipher cell and tissue functions by looking at changes in their mechanical properties that contribute to development, cell differentiation, physiology, and disease. Mechanobiology sits at the interface of biology, physics and engineering. One of the key technologies that enables characterization of properties of cells and tissue is microscopy. Combining microscopy with other quantitative measurement techniques such as optical tweezers and scissors, gives a very powerful tool for unraveling the intricacies of mechanobiology enabling measurement of forces, torques and displacements at play. We review the field of some light based studies of mechanobiology and optical detection of signal transduction ranging from optical micromanipulation-optical tweezers and scissors, advanced fluorescence techniques and optogenentics. In the current perspective paper, we concentrate our efforts on elucidating interesting measurements of forces, torques, positions, viscoelastic properties, and optogenetics inside and outside a cell attained when using structured light in combination with optical tweezers and scissors. We give perspective on the field concentrating on the use of structured light in imaging in combination with tweezers and scissors pointing out how novel developments in quantum imaging in combination with tweezers and scissors can bring to this fast growing field.

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Optimal calibration of optical tweezers with arbitrary integration time and sampling frequencies: a general framework [Invited]

Optical tweezers (OT) have become an essential technique in several fields of physics, chemistry, and biology as precise micromanipulation tools and microscopic force transducers. Quantitative measurements require the accurate calibration of the trap stiffness of the optical trap and the diffusion constant of the optically trapped particle. This is typically done by statistical estimators constructed from the position signal of the particle, which is recorded by a digital camera or a quadrant photodiode. The finite integration time and sampling frequency of the detector need to be properly taken into account. Here, we present a general approach based on the joint probability density function of the sampled trajectory that corrects exactly the biases due to the detector's finite integration time and limited sampling frequency, providing theoretical formulas for the most widely employed calibration methods: equipartition, mean squared displacement, autocorrelation, power spectral density, and force reconstruction via maximum-likelihood-estimator analysis (FORMA). Our results, tested with experiments and Monte Carlo simulations, will permit users of OT to confidently estimate the trap stiffness and diffusion constant, extending their use to a broader set of experimental conditions.

Large deviations of the stochastic area for linear diffusions

The area enclosed by the two-dimensional Brownian motion in the plane was studied by Lévy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterizing the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.

Sparse inference and active learning of stochastic differential equations from data

Automatic machine learning of empirical models from experimental data has recently become possible as a result of increased availability of computational power and dedicated algorithms. Despite the successes of non-parametric inference and neural-network-based inference for empirical modelling, a physical interpretation of the results often remains challenging. Here, we focus on direct inference of governing differential equations from data, which can be formulated as a linear inverse problem. A Bayesian framework with a Laplacian prior distribution is employed for finding sparse solutions efficiently. The superior accuracy and robustness of the method is demonstrated for various cases, including ordinary, partial, and stochastic differential equations. Furthermore, we develop an active learning procedure for the automated discovery of stochastic differential equations. In this procedure, learning of the unknown dynamical equations is coupled to the application of perturbations to the measured system in a feedback loop. We show that active learning can significantly improve the inference of global models for systems with multiple energetic minima.

Efficient reconstruction of directed networks from noisy dynamics using stochastic force inference

We consider coupled network dynamics under uncorrelated noises that fluctuate about the noise-free long-time asymptotic state. Our goal is to reconstruct the directed network only from the time-series data of the dynamics of the nodes. By using the stochastic force inference method with a simple natural choice of linear polynomial basis, we derive a reconstruction scheme of the connection weights and the noise strength of each node. Explicit simulations for directed and undirected random networks with various node dynamics are carried out to demonstrate the good accuracy and high efficiency of the reconstruction scheme. We further consider the case when only a subset of the network and its node dynamics can be observed, and it is demonstrated that the directed weighted connections among the observed nodes can be easily and faithfully reconstructed. In addition, we propose a scheme to infer the number of hidden nodes and their effects on each observed node. The accuracy of these results is illustrated by simulations.

Information-driven transitions in projections of underdamped dynamics

Low-dimensional representations of underdamped systems often provide useful insights and analytical tractability. Here, we build such representations via information projections, obtaining an optimal model that captures the most information on observed spatial trajectories. We show that, in paradigmatic systems, the minimization of the information loss drives the appearance of a discontinuous transition in the optimal model parameters. Our results raise serious warnings for general inference approaches, and they unravel fundamental properties of effective dynamical representations impacting several fields, from biophysics to dimensionality reduction.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

While particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.

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A feedback trap was used to generate noisy Langevin microbead trajectories

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The potential energy surface is recovered using a Bayesian formulation

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The formulation uses a structured-kernel-interpolation Gaussian process (SKI-GP) to tractably approximate Gaussian process regression for larger datasets

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Thanks to our adaptation of SKI-GP, we have broadened the use of Gaussian processes for natural science applications

Learning non-stationary Langevin dynamics from stochastic observations of latent trajectories

Many complex systems operating far from the equilibrium exhibit stochastic dynamics that can be described by a Langevin equation. Inferring Langevin equations from data can reveal how transient dynamics of such systems give rise to their function. However, dynamics are often inaccessible directly and can be only gleaned through a stochastic observation process, which makes the inference challenging. Here we present a non-parametric framework for inferring the Langevin equation, which explicitly models the stochastic observation process and non-stationary latent dynamics. The framework accounts for the non-equilibrium initial and final states of the observed system and for the possibility that the system's dynamics define the duration of observations. Omitting any of these non-stationary components results in incorrect inference, in which erroneous features arise in the dynamics due to non-stationary data distribution. We illustrate the framework using models of neural dynamics underlying decision making in the brain.

Temperature Effects on Optical Trapping Stability

In recent years, optically trapped luminescent particles have emerged as a reliable probe for contactless thermal sensing because of the dependence of their luminescence on environmental conditions. Although the temperature effect in the optical trapping stability has not always been the object of study, the optical trapping of micro/nanoparticles above room temperature is hindered by disturbances caused by temperature increments of even a few degrees in the Brownian motion that may lead to the release of the particle from the trap. In this report, we summarize recent experimental results on thermal sensing experiments in which micro/nanoparticles are used as probes with the aim of providing the contemporary state of the art about temperature effects in the stability of potential trapping processes.

Optical manipulation: advances for biophotonics in the 21st century

SIGNIFICANCE

Optical trapping is a technique capable of applying minute forces that has been applied to studies spanning single molecules up to microorganisms.

AIM

The goal of this perspective is to highlight some of the main advances in the last decade in this field that are pertinent for a biomedical audience.

APPROACH

First, the direct determination of forces in optical tweezers and the combination of optical and acoustic traps, which allows studies across different length scales, are discussed. Then, a review of the progress made in the direct trapping of both single-molecules, and even single-viruses, and single cells with optical forces is outlined. Lastly, future directions for this methodology in biophotonics are discussed.

RESULTS

In the 21st century, optical manipulation has expanded its unique capabilities, enabling not only a more detailed study of single molecules and single cells but also of more complex living systems, giving us further insights into important biological activities.

CONCLUSIONS

Optical forces have played a large role in the biomedical landscape leading to exceptional new biological breakthroughs. The continuous advances in the world of optical trapping will certainly lead to further exploitation, including exciting in-vivo experiments.

Enhanced Signal-to-Noise and Fast Calibration of Optical Tweezers Using Single Trapping Events

The trap stiffness us the key property in using optical tweezers as a force transducer. Force reconstruction via maximum-likelihood-estimator analysis (FORMA) determines the optical trap stiffness based on estimation of the particle velocity from statistical trajectories. Using a modification of this technique, we determine the trap stiffness for a two micron particle within 2 ms to a precision of ∼10% using camera measurements at 10 kfps with the contribution of pixel noise to the signal being larger the level Brownian motion. This is done by observing a particle fall into an optical trap once at a high stiffness. This type of calibration is attractive, as it avoids the use of a nanopositioning stage, which makes it ideal for systems of large numbers of particles, e.g., micro-fluidics or active matter systems.

Large-scale, high-contrast glare suppression with low-transmittance eigenchannels of aperture-target transmission matrices

Glare suppression with wavefront shaping is a significant technique in terms of actively controlling the speckle light field. A novel glare suppression method based on transmission matrix (TM) measurement is demonstrated in this Letter. An aperture-target TM model is proposed, and its low-transmittance eigenchannel is utilized to minimize the speckle intensity inside a given target area. We verified the availability of this method by experimentally realizing high-contrast glare suppression in areas of various sizes and shapes. For a large-scale area containing 100 speckle grains, the average intensity was suppressed to 6.3% of the background intensity. We believe our method provides an ideal method for glare suppression, and it holds interesting prospects for areas such as speckle optical tweezers and imaging under scattering conditions.

Non-equilibrium properties of an active nanoparticle in a harmonic potential

Active particles break out of thermodynamic equilibrium thanks to their directed motion, which leads to complex and interesting behaviors in the presence of confining potentials. When dealing with active nanoparticles, however, the overwhelming presence of rotational diffusion hinders directed motion, leading to an increase of their effective temperature, but otherwise masking the effects of self-propulsion. Here, we demonstrate an experimental system where an active nanoparticle immersed in a critical solution and held in an optical harmonic potential features far-from-equilibrium behavior beyond an increase of its effective temperature. When increasing the laser power, we observe a cross-over from a Boltzmann distribution to a non-equilibrium state, where the particle performs fast orbital rotations about the beam axis. These findings are rationalized by solving the Fokker-Planck equation for the particle's position and orientation in terms of a moment expansion. The proposed self-propulsion mechanism results from the particle's non-sphericity and the lower critical point of the solution.

Single-trajectory characterization of active swimmers in a flow

We develop a maximum likelihood method to infer relevant physical properties of elongated active particles. Using individual trajectories of advected swimmers as input, we are able to accurately determine their rotational diffusion coefficients and an effective measure of their aspect ratio, also providing reliable estimators for the uncertainties of such quantities. We validate our theoretical construction using numerically generated active trajectories upon no flow, simple shear, and Poiseuille flow, with excellent results. Being designed to rely on single-particle data, our method eases applications in experimental conditions where swimmers exhibit a strong morphological diversity. We briefly discuss some of such ongoing experimental applications, specifically, in the characterization of swimming E. coli in a flow.

Fluid Viscoelasticity Triggers Fast Transitions of a Brownian Particle in a Double Well Optical Potential

Thermally activated transitions are ubiquitous in nature, occurring in complex environments which are typically conceived as ideal viscous fluids. We report the first direct observations of a Brownian bead transiting between the wells of a bistable optical potential in a viscoelastic fluid with a single long relaxation time. We precisely characterize both the potential and the fluid, thus enabling a neat comparison between our experimental results and a theoretical model based on the generalized Langevin equation. Our findings reveal a drastic amplification of the transition rates compared to those in a Newtonian fluid, stemming from the relaxation of the fluid during the particle crossing events.

Inverse design of nonequilibrium steady states: A large-deviation approach

The design of small scale nonequilibrium steady states (NESS) is a challenging, open ended question. While similar equilibrium problems are tractable using standard thermodynamics, a generalized description for nonequilibrium systems is lacking, making the design problem particularly difficult. Here we show we can exploit the large-deviation behavior of a Brownian particle and design a variety of geometrically complex steady-state density distributions and flux field flows. We achieve this design target from direct knowledge of the joint large-deviation functional for the empirical density and flow, and a "relaxation" algorithm on the desired target states via adjustable force field parameters. We validate the method by replicating analytical results, and demonstrate its capacity to yield complex prescribed targets, such as rose-curve or polygonal shapes on the plane. We consider this dynamical fluctuation approach a first step towards the design of more complex NESS where general frameworks are otherwise still lacking.

Predicting critical transitions in multiscale dynamical systems using reservoir computing

We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.

Optical Tweezers Exploring Neuroscience

Over the past decade, optical tweezers (OT) have been increasingly used in neuroscience for studies of molecules and neuronal dynamics, as well as for the study of model organisms as a whole. Compared to other areas of biology, it has taken much longer for OT to become an established tool in neuroscience. This is, in part, due to the complexity of the brain and the inherent difficulties in trapping individual molecules or manipulating cells located deep within biological tissue. Recent advances in OT, as well as parallel developments in imaging and adaptive optics, have significantly extended the capabilities of OT. In this review, we describe how OT became an established tool in neuroscience and we elaborate on possible future directions for the field. Rather than covering all applications of OT to neurons or related proteins and molecules, we focus our discussions on studies that provide crucial information to neuroscience, such as neuron dynamics, growth, and communication, as these studies have revealed meaningful information and provide direction for the field into the future.

Using the transient trajectories of an optically levitated nanoparticle to characterize a stochastic Duffing oscillator

We propose a novel methodology to estimate parameters characterizing a weakly nonlinear Duffing oscillator represented by an optically levitating nanoparticle. The method is based on averaging recorded trajectories with defined initial positions in the phase space of nanoparticle position and momentum and allows us to study the transient dynamics of the nonlinear system. This technique provides us with the parameters of a levitated nanoparticle such as eigenfrequency, damping, coefficient of nonlinearity and effective temperature directly from the recorded transient particle motion without any need for external driving or modification of an experimental system. Comparison of this innovative approach with a commonly used method based on fitting the power spectrum density profile shows that the proposed complementary method is applicable even at lower pressures where the nonlinearity starts to play a significant role and thus the power spectrum density method predicts steady state parameters. The technique is applicable also at low temperatures and extendable to recent quantum experiments. The proposed method is applied on experimental data and its validity for one-dimensional and three-dimensional motion of a levitated nanoparticle is verified by extensive numerical simulations.

Inferring the Dynamics of Underdamped Stochastic Systems

Many complex systems, ranging from migrating cells to animal groups, exhibit stochastic dynamics described by the underdamped Langevin equation. Inferring such an equation of motion from experimental data can provide profound insight into the physical laws governing the system. Here, we derive a principled framework to infer the dynamics of underdamped stochastic systems from realistic experimental trajectories, sampled at discrete times and subject to measurement errors. This framework yields an operational method, Underdamped Langevin Inference, which performs well on experimental trajectories of single migrating cells and in complex high-dimensional systems, including flocks with Viscek-like alignment interactions. Our method is robust to experimental measurement errors, and includes a self-consistent estimate of the inference error.

Ordering of binary colloidal crystals by random potentials

Structural defects are ubiquitous in condensed matter, and not always a nuisance. For example, they underlie phenomena such as Anderson localization and hyperuniformity, and they are now being exploited to engineer novel materials. Here, we show experimentally that the density of structural defects in a 2D binary colloidal crystal can be engineered with a random potential. We generate the random potential using an optical speckle pattern, whose induced forces act strongly on one species of particles (strong particles) and weakly on the other (weak particles). Thus, the strong particles are more attracted to the randomly distributed local minima of the optical potential, leaving a trail of defects in the crystalline structure of the colloidal crystal. While, as expected, the crystalline ordering initially decreases with an increasing fraction of strong particles, the crystalline order is surprisingly recovered for sufficiently large fractions. We confirm our experimental results with particle-based simulations, which permit us to elucidate how this non-monotonic behavior results from the competition between the particle-potential and particle-particle interactions.

Experimental optical trapping with frozen waves

We report, to the best of our knowledge, the first optical trapping experimental demonstration of microparticles with frozen waves. Frozen waves are an efficient method to model longitudinally the intensity of nondiffracting beams obtained by superposing copropagating Bessel beams with the same frequency and order. Based on this, we investigate the optical force distribution acting on microparticles of two types of frozen waves. The experimental setup of holographic optical tweezers using a spatial light modulator has been assembled and optimized. The results show that it is possible to obtain greater stability for optical trapping using frozen waves. The significant enhancement in trapping geometry from this approach shows promising applications for optical tweezers micromanipulations over a broad range.

Video microscopy-based accurate optical force measurement by exploring a frequency-changing sinusoidal stimulus

Optical tweezers are constantly evolving micromanipulation tools that can provide piconewton force measurement accuracy and greatly promote the development of bioscience at the single-molecule scale. Consequently, there is an urgent need to characterize the force field generated by optical tweezers in an accurate, cost-effective, and rapid manner. Thus, in this study, we conducted a deep survey of optically trapped particle dynamics and found that merely quantifying the response amplitude and phase delay of particle displacement under a sine input stimulus can yield sufficiently accurate force measurements. In addition, Nyquist-Shannon sampling theorem suggests that the entire recovery of the accessible particle sinusoidal response is possible, provided that the sampling theorem is satisfied, thereby eliminating the requirement for high-bandwidth (typically greater than 10 kHz) detectors. Based on this principle, we designed optical trapping experiments by loading a sinusoidal signal into the optical tweezers system and recording the trapped particle responses with 45 frames per second (fps) charge-coupled device (CCD) and 163 fps complementary metal-oxide-semiconductor (CMOS) cameras for video microscopy imaging. The experimental results demonstrate that the use of low-bandwidth detectors is suitable for highly accurate force quantification, thereby greatly reducing the complexity of constructing optical tweezers. The trap stiffness increases significantly as the frequency increases, and the experimental results demonstrate that the trapped particles shifting along the optical axis boost the transversal optical force.