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Robust Reconstruction of the Rate Constant Distribution Using the Phase Function Method

Yajun Zhou ^* and Xiaowei Zhuang ^*  

^* Department of Chemistry and Chemical Biology, Department of Physics, Howard Hughes Medical Institute, Harvard University, Massachusetts

Correspondence: Address reprint requests to X. Zhuang, Tel.: 617-496-9558; E-mail: zhuang{at}chemistry.harvard.edu.

Many biological processes exhibit complex kinetic behavior that involves a nontrivial distribution of rate constants. Characterization of the rate constant distribution is often critical for mechanistic understandings of these processes. However, it is difficult to extract a rate constant distribution from data measured in the time domain. This is due to the numerical instability of the inverse Laplace transform, a long-standing mathematical challenge that has hampered data analysis in many disciplines. Here, we present a method that allows us to reconstruct the probability distribution of rate constants from decay data in the time domain, without fitting to specific trial functions or requiring any prior knowledge of the rate distribution. The robustness (numerical stability) of this reconstruction method is numerically illustrated by analyzing data with realistic noise and theoretically proved by the continuity of the transformations connecting the relevant function spaces. This development enhances our ability to characterize kinetics and dynamics of biological processes. We expect this method to be useful in a broad range of disciplines considering the prevalence of complex exponential decays in many experimental systems.