Abstract
Biological function often springs from the intricate synchronization of individual proteins, rather than from bulk interactions. High-throughput single-molecule techniques now allow us to move beyond bulk rates to record distributions of reaction times. Such distributions can greatly help mechanistic modeling efforts, as they often contain signatures of the underlying reaction path. With a tentative model at hand, correctly judging its predictive power is predicated on correctly estimating its parameters from the available data. For complex models, such parameter estimation can be far from trivial, and the choice of method can significantly influence the result. We here provide a self-contained introduction to maximum-likelihood estimation aimed at single-molecule experimenters. By considering relevant examples, we explain how to use maximum-likelihood estimation and we compare its performance to that of popular least-squares methods. Considering single-molecule data, we argue that maximum-likelihood estimation is generally the superior choice and conclude with a discussion of how to estimate the spread in parameter estimates through bootstrapping.
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Notes
- 1.
Note that we do not need to know the actual constant value of \( \sigma_{b} \), as it will not affect the position of the minimum of \( R^{\text{uwLS}} \).
- 2.
A more intuitive way of writing this might be in the form \( p\left( {{\text{model}}|{\text{data}}} \right) = \frac{{p\left( {\text{model}} \right)}}{{p\left( {\text{data}} \right)}}p\left( {{\text{data}}|{\text{model}}} \right). \)
- 3.
There are subtleties here relating to variable changes [18], but these lie outside our present scope.
- 4.
It should be noted that as the logarithm takes a unit-less argument, while the PDF has units (inverse time in case of the unbinding experiments). Strictly, we therefore need to multiply the PDF with some constant that renders the argument of the logarithm unit less in the definition of \( L^{\text{ML}} \left( {\left\{ \tau \right\}_{M} } \right) \). As the value of this constant does not affect the position of the minimum, we drop it for notational convenience.
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Acknowledgements
We thank Tao Ju (Thijs) Cui, Misha Klein, and Olivera Rakic for careful reading of the manuscript and thoughtful feedback. B. Eslami-Mosallam acknowledges financial support through the research program Crowd management: the physics of genome processing in complex environments, which is financed by the Netherlands Organisation for Scientific Research. I. Katechis acknowledges financial support from the Netherlands Organisation for Scientific Research, as part of the Frontiers in Nanoscience program.
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Eslami-Mosallam, B., Katechis, I., Depken, M. (2019). Fitting in the Age of Single-Molecule Experiments: A Guide to Maximum-Likelihood Estimation and Its Advantages. In: Joo, C., Rueda, D. (eds) Biophysics of RNA-Protein Interactions. Biological and Medical Physics, Biomedical Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9726-8_5
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