Bayesian Inference for Top-Down Protein Markov Models and the Ubiquity of Michaelis-Menten Kinetics
by Malte Norman Schäffner
Date of Examination:2023-11-06
Date of issue:2024-10-04
Advisor:Prof. Dr. Helmut Grubmüller
Referee:Prof. Dr. Helmut Grubmüller
Referee:Prof. Dr. Stefan Klumpp
Referee:Dr. Johannes Soeding
Referee:Prof. Dr. Timo Betz
Referee:Prof. Dr. Andreas Janshoff
Referee:Prof. Dr. Marcus Müller
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Abstract
English
The current understanding of protein function predominantly relies on the concept of describing proteins as comprising discrete conformational and chemical states including transitions between these states. The theory of Markov processes provides the mathematical foundation for such a description in the form of Markov models. Two distinct methodologies exist for the construction of such models. In a bottom-up approach, Markov model parameters are directly determined based on molecular dynamics simulations, which typically constrains the Markov models to short timescale dynamics. In contrast, in a top-down approach, the parameters are indirectly determined by fitting observables derived from the Markov model to experimental data. This second distinctive methodology permits the integration of long timescale dynamics. However, due to insufficient experimental data, many top-down Markov models may exist that fit the experimental data equally well. In the first thesis part, I employed Bayesian inference to address this underdetermination through a rigorous ranking of shared characteristics, such as molecular mechanisms, of well-fitting top-down Markov models in terms of posterior probability. I apply this approach to twin-ATPase ABCE1, which exhibits an unexpected kinetic asymmetry between its two ATP-binding sites that was observed in mutants. Thus far, the prevailing hypothesis to explain this asymmetry is a direct allosteric communication between the two binding sites. However, I challenge this hypothesis with successfully fitting top-down Markov models that are explicitly defined to exclude such direct allosteric communication. Instead, I demonstrate that a shift in steady-state population between reaction pathways, caused by avoiding kinetic trap states, provides an alternative explanation for the asymmetry. During this first part, the dependence of ATP turnover rate on a logarithmic substrate concentration approximated---at least piecewise---a sigmoid curve for the majority of ABCE1 Markov models. This dependence is also observed in one of the simplest enzyme kinetics models, namely the Michaelis-Menten kinetics. Despite the perceived ubiquity of the Michaelis-Menten kinetics, the prevalence of this dependence was an unexpected outcome, given that the Markov models of ABCE1 are considerably more intricate than the Markov model that is necessary to derive the Michaelis-Menten kinetics. In the second thesis part, I investigated this discrepancy in expectation by quantifying the occurrence of Michaelis-Menten kinetics in sparsely connected Markov models. Strikingly, I showed that even for large models with up to 200 Markov states, the turnover curves are almost exclusively (80%) comprised of such Michaelis-Menten-like pieces, as long as a fraction of concentration-dependent rates is less than 10%. As this low fraction covers the biologically relevant range, this result contributes to an explanation of the ubiquity of Michaelis-Menten kinetics. I explain this behavior based on the distance between possible Michaelis-Menten-like pieces and the combinatorial problem described by the hypergeometric distribution, which determines this distance. Both approaches---the employment of top-down Markov models to investigate allostery and the comprehensive analysis of the intrinsic characteristics of Markov models to elucidate common motifs in proteins---have the potential for broader applications and, thus, will contribute to a more profound understanding of proteins.
Keywords: ABC proteins; chemical reaction networks; allostery; enzyme kinetics; graph theory