Data-driven approach to decomposing complex enzyme kinetics with surrogate models

Quantifying transient 3D dynamical phenomena of single mRNA particles in live yeast cell measurements

Single-particle tracking (SPT) has been extensively used to obtain information about diffusion and directed motion in a wide range of biological applications. Recently, new methods have appeared for obtaining precise (10s of nm) spatial information in three dimensions (3D) with high temporal resolution (measurements obtained every 4 ms), which promise to more accurately sense the true dynamical behavior in the natural 3D cellular environment. Despite the quantitative 3D tracking information, the range of mathematical methods for extracting information about the underlying system has been limited mostly to mean-squared displacement analysis and other techniques not accounting for complex 3D kinetic interactions. There is a great need for new analysis tools aiming to more fully extract the biological information content from in vivo SPT measurements. High-resolution SPT experimental data has enormous potential to objectively scrutinize various proposed mechanistic schemes arising from theoretical biophysics and cell biology. At the same time, methods for rigorously checking the statistical consistency of both model assumptions and estimated parameters against observed experimental data (i.e., goodness-of-fit tests) have not received great attention. We demonstrate methods enabling (1) estimation of the parameters of 3D stochastic differential equation (SDE) models of the underlying dynamics given only one trajectory; and (2) construction of hypothesis tests checking the consistency of the fitted model with the observed trajectory so that extracted parameters are not overinterpreted (the tools are applicable to linear or nonlinear SDEs calibrated from nonstationary time series data). The approach is demonstrated on high-resolution 3D trajectories of single ARG3 mRNA particles in yeast cells in order to show the power of the methods in detecting signatures of transient directed transport. The methods presented are generally relevant to a wide variety of 2D and 3D SPT tracking applications.

P-SPLINES USING DERIVATIVE INFORMATION

Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biomolecular systems. We demonstrate how a collection of Penalized-splines (P-splines) can be useful in quantitatively summarizing such data. In this work, functions estimated using P-splines are associated with stochastic differential equations (SDEs). It is shown how quantities estimated in a single SDE summarize fast-scale phenomena, whereas variation between curves associated with different SDEs partially reflects noise induced by motion evolving on a slower time scale. P-splines assist in "semiparametrically" estimating nonlinear SDEs in situations where a time-dependent external force is applied to a single-molecule system. The P-splines introduced simultaneously use function and derivative scatterplot information to refine curve estimates. We refer to the approach as the PuDI (P-splines using Derivative Information) method. It is shown how generalized least squares ideas fit seamlessly into the PuDI method. Applications demonstrating how utilizing uncertainty information/approximations along with generalized least squares techniques improve PuDI fits are presented. Although the primary application here is in estimating nonlinear SDEs, the PuDI method is applicable to situations where both unbiased function and derivative estimates are available.