We introduce two methods based on statistical inference to calibrate optical tweezers. Both outperform well-established methods and cover a broader application field, including non-conservative force fields and out of equilibrium systems

Arzola, AV

Castillo, IP

García, LP

Magazzú, A

Selin, M

Volpe, G

English

UCL Discovery

FORMA and BEFORE: expanding applicationsof optical tweezersLaura Pérez Garcı́a,a, * Martin Selin,a Alejandro V. Arzola,b Giorgio Volpe, cAlessandro Magazzù,d Isaac Pérez Castilloe and Giovanni VolpeaaDepartment of Physics, University of Gothenburg, Gothenburg, SwedenbInstituto de Fı́sica, Universidad Nacional Autónoma de México,Cd. de México, C.P. 04510, MéxicoocDepartment of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UKdCNR-IPCF, Istituto Processi Chimico-Fisici, V.le F. Stagno D’Alcontres 37, 98158 Messina, Italye Departamento de Fı́sica, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad deMéxico 09340, Mexico*laura.perez.garcia@physics.gu.seAbstract: We introduce two methods based on statistical inference to calibrate opticaltweezers. Both outperform well-established methods and cover a broader application field,including non-conservative force fields and out of equilibrium systems.© 2021 The Author(s)1. Force reconstruction via Maximum Likelihood estimatorWe recently introduced force reconstruction via maximum-likelihood-estimator analysis (FORMA) to calculatethe force field acting on a Brownian particle by analyzing its trajectory displacements. FORMA exploits the linearrelationship between the time series of the position of a Brownian particle xn and drag forces fn = γ ∆xn∆tn at instantn in the vicinity of an equilibrium point:1fn =−κxn +σwn (1)Where κ is the stiffness of the force field and σ =√2Dγ2∆tn, D= 6πηR, D= kBT/γ , R is the radius of the particle,η is the viscosity of the medium in which the particle is moving; and finally, wn is a Gaussian random number ofzero mean and unit variance. Using a Maximum likelihood estimator, we find the solution:κ∗ =−∑n xn fn∑n x2n(2)Using the calculated κ∗ one can estimate the diffusion as:D∗ =1N ∑n∆tn2γ2[ fn +κ∗xn] (3)FORMA has proven to be faster, more precise, more accurate, and ten-fold less data-intensive than autocor-relation function method (ACF), power spectrum density method (PSD), mean squared displacement method(MSD), and potential method. Furthermore, with FORMA, we have characterized the force field’s conservativeand non-conservative components, having relevant advantages over well-established techniques. In Fig. (1), wecan see the application of FORMA to characterize the force field generated by a speckle pattern that featuresnon-conservative components due to the presence of optical vortices and unstable equilibrium points. Another keyadvantage of FORMA is that its application does not depend on the fine-tuning of a fitting parameter, which is thecase of ACF, PSD, and MSD [1].2. FORMA, the case of a flat prior in Bayes approachIn the context of inference statistics, FORMA uses a linear non-Bayesian approach to estimate the properties of theforce field and the diffusive properties of the particle. Recently, we have developed a method that uses Bayesianinference to expand and generalize FORMA.1This equation is valid in the vicinity of an equilibrium point F0(x) = 0; however, it can be applied to any other point by adding thecorrespondent term of the force at that point F0(x) = F0.(a) (b) (c)(d)1 µmFig. 1. FORMA has been used to characterize non-conservative force fields as the ones generatedby a speckle pattern. A 1µm polystyrene particle was suspended in water and exposed to a specklepattern formed by a spatial light modulator. The markers in (a) represent equilibrium points i.e.,where F0(r)= 0. In (b), we see the characteristic force field of a stable equilibrium point, specificallyan attractor; in (b), we see an unstable equilibrium point as all force lines are repelled from insideout, and in (c), we see a saddle point which happens to be in the middle of the line that connects twoattractors.With Bayesian Force Reconstruction (BEFORE), we have derived formulas for the posterior distribution for aset of parameters {D,γ,κ} denoted by θ , also called a model. The posterior distribution P(θ |x, f) gives us theprobability that a model θ explains the data-set provided by the vector pair {x, f} .Assuming P0(θ) to be the prior distribution for the model, by Baye’s rule, the posterior distribution is given by:P(θ |x, f) = L(f|x,θ)P0(θ)∫dθ ′L(f|x,θ ′)P0(θ ′)(4)where L is the likelihood of observing a vector f given the vector of positions x and the set of parameters θ .FORMA actually can be derived from a flat prior distribution; thus, the posterior distribution is proportional to thelikelihood L given by:L(f|x,θ) =√∏n ∆tn(4πDσ2)Nexp[−∑n∆tn4Dσ2( fn +κxn)2](5)Using more informative priors, BEFORE can provide reasonable estimates of the parameters even when littledata is available; it can also be generalized to more than one dimension and be used for time series with any timeinterval ∆t.Both FORMA and BEFORE represent an approach for calibration of optical tweezers using statistical inferencetechniques. Their advantages become relevant whenever we have few data points, out of equilibrium systems,non-conservative forces, underdamped regime, or inhomogeneous time sampling. All these advantages will allowexpanding experiments towards more complex scenarios such as biological systems, time-varying conditions, andactive baths [2].References1. Pérez Garcı́a, Laura and Donlucas Pérez, Jaime and Volpe, Giorgio and V. Arzola, Alejandro and Volpe, Giovanni,”High-performance reconstruction of microscopic force fields from Brownian trajectories” in Nature Communications,Vol. 9, 5166, (2018).2. Jan Gieseler, Juan Ruben Gomez Solano, Alessandro Magazzu, Isaac Pérez Castillo, Laura Pérez Garcı́a, MartaGironella Torrent, Xavier Viader-Godoy, Felix Ritort, Giuseppe Pesce, Alejandro Arzola, Karen Volke-Sepulveda, andGiovanni Volpe, ”Optical Tweezers: A Comprehensive Tutorial from Calibration to Applications” in Advances in Opticsand Photonics, (2021)

FORMA and BEFORE: Expanding applications of optical tweezers

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FORMA and BEFORE: Expanding applications of optical tweezers

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