The Annals of Applied Statistics

Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach

Ye Zhang, Zhigang Yao, Patrik Forssén, and Torgny Fornstedt

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The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reactants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hormone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

Article information

Ann. Appl. Stat., Volume 13, Number 4 (2019), 2011-2042.

Received: September 2018
Revised: April 2019
First available in Project Euclid: 28 November 2019

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Rate constant biosensor Bayesian variational method integral equation adaptive discretization algorithm


Zhang, Ye; Yao, Zhigang; Forssén, Patrik; Fornstedt, Torgny. Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach. Ann. Appl. Stat. 13 (2019), no. 4, 2011--2042. doi:10.1214/19-AOAS1263.

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Supplemental materials

  • Supplement to “Estimating the rate constant from biosensor data via an adaptive variational Bayesian approach”. We provide additional material of the proof of Theorem 1, finite element approximation of integral equations, as well as a demonstration of our main algorithm.