Bayesian modeling reveals metabolite‐dependent ultrasensitivity in the cyanobacterial circadian clock
2020; Springer Nature; Volume: 16; Issue: 6 Linguagem: Inglês
10.15252/msb.20199355
ISSN1744-4292
AutoresLu Hong, Danylo Lavrentovich, Archana G. Chavan, Eugene Leypunskiy, Eileen Li, Charles H. Matthews, Andy LiWang, Michael J. Rust, Aaron R. Dinner,
Tópico(s)Photosynthetic Processes and Mechanisms
ResumoArticle4 June 2020Open Access Source DataTransparent process Bayesian modeling reveals metabolite-dependent ultrasensitivity in the cyanobacterial circadian clock Lu Hong Lu Hong orcid.org/0000-0001-7812-0617 Graduate Program in Biophysical Sciences, University of Chicago, Chicago, IL, USA Search for more papers by this author Danylo O Lavrentovich Danylo O Lavrentovich orcid.org/0000-0002-8432-9596 Department of Chemistry, University of Chicago, Chicago, IL, USA Search for more papers by this author Archana Chavan Archana Chavan School of Natural Sciences, University of California, Merced, CA, USA Search for more papers by this author Eugene Leypunskiy Eugene Leypunskiy orcid.org/0000-0002-3335-1099 Graduate Program in Biophysical Sciences, University of Chicago, Chicago, IL, USA Search for more papers by this author Eileen Li Eileen Li Department of Statistics, University of Chicago, Chicago, IL, USA Search for more papers by this author Charles Matthews Charles Matthews Department of Statistics, University of Chicago, Chicago, IL, USA Search for more papers by this author Andy LiWang Andy LiWang orcid.org/0000-0003-4741-6946 School of Natural Sciences, University of California, Merced, CA, USA Quantitative and Systems Biology, University of California, Merced, CA, USA Center for Circadian Biology, University of California, San Diego, La Jolla, CA, USA Chemistry and Chemical Biology, University of California, Merced, CA, USA Health Sciences Research Institute, University of California, Merced, CA, USA Center for Cellular and Biomolecular Machines, University of California, Merced, CA, USA Search for more papers by this author Michael J Rust Corresponding Author Michael J Rust [email protected] Department of Molecular Genetics and Cell Biology, University of Chicago, Chicago, IL, USA Institute for Biophysical Dynamics, University of Chicago, Chicago, IL, USA Institute for Genomics and Systems Biology, University of Chicago, Chicago, IL, USA Search for more papers by this author Aaron R Dinner Corresponding Author Aaron R Dinner [email protected] orcid.org/0000-0001-8328-6427 Department of Chemistry, University of Chicago, Chicago, IL, USA Institute for Biophysical Dynamics, University of Chicago, Chicago, IL, USA James Franck Institute, University of Chicago, Chicago, IL, USA Search for more papers by this author Lu Hong Lu Hong orcid.org/0000-0001-7812-0617 Graduate Program in Biophysical Sciences, University of Chicago, Chicago, IL, USA Search for more papers by this author Danylo O Lavrentovich Danylo O Lavrentovich orcid.org/0000-0002-8432-9596 Department of Chemistry, University of Chicago, Chicago, IL, USA Search for more papers by this author Archana Chavan Archana Chavan School of Natural Sciences, University of California, Merced, CA, USA Search for more papers by this author Eugene Leypunskiy Eugene Leypunskiy orcid.org/0000-0002-3335-1099 Graduate Program in Biophysical Sciences, University of Chicago, Chicago, IL, USA Search for more papers by this author Eileen Li Eileen Li Department of Statistics, University of Chicago, Chicago, IL, USA Search for more papers by this author Charles Matthews Charles Matthews Department of Statistics, University of Chicago, Chicago, IL, USA Search for more papers by this author Andy LiWang Andy LiWang orcid.org/0000-0003-4741-6946 School of Natural Sciences, University of California, Merced, CA, USA Quantitative and Systems Biology, University of California, Merced, CA, USA Center for Circadian Biology, University of California, San Diego, La Jolla, CA, USA Chemistry and Chemical Biology, University of California, Merced, CA, USA Health Sciences Research Institute, University of California, Merced, CA, USA Center for Cellular and Biomolecular Machines, University of California, Merced, CA, USA Search for more papers by this author Michael J Rust Corresponding Author Michael J Rust [email protected] Department of Molecular Genetics and Cell Biology, University of Chicago, Chicago, IL, USA Institute for Biophysical Dynamics, University of Chicago, Chicago, IL, USA Institute for Genomics and Systems Biology, University of Chicago, Chicago, IL, USA Search for more papers by this author Aaron R Dinner Corresponding Author Aaron R Dinner [email protected] orcid.org/0000-0001-8328-6427 Department of Chemistry, University of Chicago, Chicago, IL, USA Institute for Biophysical Dynamics, University of Chicago, Chicago, IL, USA James Franck Institute, University of Chicago, Chicago, IL, USA Search for more papers by this author Author Information Lu Hong1, Danylo O Lavrentovich2,14, Archana Chavan3, Eugene Leypunskiy1, Eileen Li4, Charles Matthews4,15, Andy LiWang3,5,6,7,8,9, Michael J Rust *,10,11,12 and Aaron R Dinner *,2,11,13 1Graduate Program in Biophysical Sciences, University of Chicago, Chicago, IL, USA 2Department of Chemistry, University of Chicago, Chicago, IL, USA 3School of Natural Sciences, University of California, Merced, CA, USA 4Department of Statistics, University of Chicago, Chicago, IL, USA 5Quantitative and Systems Biology, University of California, Merced, CA, USA 6Center for Circadian Biology, University of California, San Diego, La Jolla, CA, USA 7Chemistry and Chemical Biology, University of California, Merced, CA, USA 8Health Sciences Research Institute, University of California, Merced, CA, USA 9Center for Cellular and Biomolecular Machines, University of California, Merced, CA, USA 10Department of Molecular Genetics and Cell Biology, University of Chicago, Chicago, IL, USA 11Institute for Biophysical Dynamics, University of Chicago, Chicago, IL, USA 12Institute for Genomics and Systems Biology, University of Chicago, Chicago, IL, USA 13James Franck Institute, University of Chicago, Chicago, IL, USA 14Present address: Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA 15Present address: School of Mathematics, University of Edinburgh, Edinburgh, UK *Corresponding author. Tel: +1 773 834 1463; E-mail: [email protected] *Corresponding author. Tel: +1 773 702 2330; E-mail: [email protected] Molecular Systems Biology (2020)16:e9355https://doi.org/10.15252/msb.20199355 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Abstract Mathematical models can enable a predictive understanding of mechanism in cell biology by quantitatively describing complex networks of interactions, but such models are often poorly constrained by available data. Owing to its relative biochemical simplicity, the core circadian oscillator in Synechococcus elongatus has become a prototypical system for studying how collective dynamics emerge from molecular interactions. The oscillator consists of only three proteins, KaiA, KaiB, and KaiC, and near-24-h cycles of KaiC phosphorylation can be reconstituted in vitro. Here, we formulate a molecularly detailed but mechanistically naive model of the KaiA—KaiC subsystem and fit it directly to experimental data within a Bayesian parameter estimation framework. Analysis of the fits consistently reveals an ultrasensitive response for KaiC phosphorylation as a function of KaiA concentration, which we confirm experimentally. This ultrasensitivity primarily results from the differential affinity of KaiA for competing nucleotide-bound states of KaiC. We argue that the ultrasensitive stimulus–response relation likely plays an important role in metabolic compensation by suppressing premature phosphorylation at nighttime. Synopsis This study takes a data-driven kinetic modeling approach to characterize the interaction between KaiA and KaiC in the cyanobacterial circadian clock to understand how the oscillator responds to changes in cellular metabolic conditions. An extensive dataset of KaiC autophosphorylation measurements is generated and used to constrain a detailed yet mechanistically naive kinetic model within a Bayesian parameter estimation framework. KaiA concentration tunes the sensitivity of KaiC autophosphorylation and the period of the full oscillator to %ATP. The model reveals an ultrasensitive dependence of KaiC phosphorylation on KaiA concentration as a result of differential KaiA binding affinity to ADP- vs. ATP-bound KaiC. Ultrasensitivity in KaiC phosphorylation likely contributes to metabolic compensation by suppressing premature phosphorylation at nighttime. Introduction Achieving a predictive understanding of biological systems and chemical reaction networks is challenging because complex behavior can emerge from even a small number of interacting components. Classic examples include the propagation of action potentials in neurobiology and chemical oscillators such as the Belousov–Zhabotinsky reaction. The collective dynamics in such systems cannot be easily intuited through qualitative reasoning alone, and thus, mathematical modeling plays an important role in summarizing and interpreting existing observations and formulating testable, quantitative hypotheses. However, it can be difficult to rationalize how collective dynamics emerge from specific molecular features. One approach to addressing this issue is to compare mathematical representations of competing molecular mechanisms based on their abilities to fit experimental data. The circadian clock from the cyanobacterium Synechococcus elongatus (Johnson et al, 2011) represents a unique opportunity to use model fitting to learn biochemical mechanisms. This is because the core oscillator can be reconstituted in a test tube from a small number of components. The simplicity of the system makes it possible to both cleanly model the basic biochemical events in the circadian cycle and to collect quantitative data under well-controlled conditions. The core oscillator of S. elongatus consists of three proteins, KaiA, KaiB, and KaiC, which self-organize to generate a near-24-h rhythm in KaiC phosphorylation. The basic biochemical events are well-established (Swan et al, 2018). KaiC is an ATPase (Terauchi et al, 2007) that phosphorylates and dephosphorylates itself by transfer of phosphoryl groups from and to bound nucleotides (Egli et al, 2012; Nishiwaki & Kondo, 2012); KaiA-dependent nucleotide exchange reactions drive the phosphorylation phase of the cycle (Nishiwaki-Ohkawa et al, 2014), and KaiB-mediated sequestration of KaiA leads to dephosphorylation. The reconstituted oscillator retains many of the hallmarks of circadian rhythms in living organisms (Nakajima et al, 2005; Yoshida et al, 2009; Rust et al, 2011; Leypunskiy et al, 2017). Yet, questions remain about how the clock couples to environmental conditions while maintaining a robust ~ 24-h rhythm (Yoshida et al, 2009; Phong et al, 2013; Leypunskiy et al, 2017; Murayama et al, 2017). The Kai oscillator senses changes in the relative concentrations of ATP to ADP in solution, which allows entrainment to metabolic rhythms (Rust et al, 2011; Phong et al, 2013; Leypunskiy et al, 2017). KaiA modulates these dynamics via its function as a nucleotide exchange factor, but how the system adapts and responds to changes in metabolic conditions is not clear. Models that account for the possible protein complexes, including the interplay of nucleotide-bound and phosphorylation states, can be dauntingly complex (e.g., Lin et al, 2014; Paijmans et al, 2017b). How to fit them to data and interpret the results is an area of active research. Here, we use a data-driven Bayesian approach to estimate the parameters of a molecularly detailed kinetic model of the KaiA–KaiC subsystem, with the goal of learning the features required to capture the behavior of the system during the phosphorylation phase of the clock cycle quantitatively (Fig 1A). The model describes the coupling between KaiA, nucleotides (ATP and ADP) in solution, KaiC phosphorylation, and KaiC nucleotide-bound states. To provide training data for the model, we collected kinetic time series measuring KaiC phosphorylation kinetics over a wide range of KaiA concentrations ([KaiA]) and %ATP (defined as 100%[ATP]/([ATP] + [ADP])). Although such data do not give us direct access to all relevant states of the KaiA–KaiC subsystem, they place constraints on the underlying molecular interactions. Bayesian statistics (Wasserman, 2000; MacKay & Kay, 2003) have found diverse applications in systems biology (Flaherty et al, 2008; Klinke, 2009; Toni et al, 2009; Xu et al, 2010; Schmidl et al, 2012; Eydgahi et al, 2013; Pullen & Morris, 2014; Mello et al, 2018), including circadian biology (Higham & Husmeier, 2013; Trejo Banos et al, 2015; Martins et al, 2018). Here, they provide a unified framework for estimating parameter values, quantifying the importance of specific model elements, and making mechanistic predictions from the model. Figure 1. Phosphorylation data are fit by a mechanistically naive kinetic model An outline of the data-driven Bayesian model fitting approach employed in this work. To constrain the model, measurements of KaiC phosphorylation kinetics were collected at six [KaiA] and three %ATP conditions. The curves represent the best fit model prediction. A schematic of the mass-action kinetics model. The model elaborates on the autophosphorylation reactions of KaiC by explicitly keeping track of the time evolution of the KaiC phosphoforms and nucleotide-bound states; conversions among these states are mediated by phosphotransfer, nucleotide exchange, ATP hydrolysis, and KaiA (un)binding. Note that the KaiA binding reactions are second-order, but KaiA concentration ([A]) is written as part of the effective first-order rate constant. See the main text for a discussion of the state and rate constant nomenclature and Fig EV1A for a schematic of the full model. The posterior distributions for log KaiA dissociation constants (base 10). The horizontal axis represents the affinity for ADP-bound KaiC, and the vertical axis represents the affinity for ATP-bound KaiC; X ∈ {U, T, S, D} and corresponds to the four colors of the KaiC phosphoforms, as in panel C. The asterisks represent the best fit, and the contour lines represent the 95% and 68% highest posterior density regions (HDR). The dashed line represents the line, so that densities above the line indicate higher affinity for the ADP-bound states and densities below the line indicate higher affinity for the ATP-bound states. Source data are available online for this figure. Source Data for Figure 1 [msb199355-sup-0003-SDataFig1B.csv] Download figure Download PowerPoint By Markov chain Monte Carlo (MCMC) sampling, we obtain an ensemble of parameter sets that fit the data. Even with extensive training data, many microscopic parameters in the model are not tightly constrained. Despite this, we show that this ensemble of fits robustly makes predictions that are borne out in experimental tests (Brown & Sethna, 2003; Gutenkunst et al, 2007). In particular, the model reveals a previously unappreciated ultrasensitive dependence of phosphorylation on the concentration of KaiA, with strong nonlinearity at low [KaiA], conditions that likely apply near the nighttime to daytime transition point, when a large fraction of KaiA molecules are inhibited. Importantly, we find that the threshold KaiA concentration varies with the %ATP in solution. This ultrasensitive response primarily arises from a differential affinity of KaiA for different nucleotide-bound states of KaiC. This mechanism is analogous to substrate competition (Ferrell & Ha, 2014b), where kinetic competition of multiple enzyme substrates leads to ultrasensitivity. Lastly, we consider the implications of these results for the full oscillator, in which KaiC rhythmically switches between phosphorylation and dephosphorylation. A well-known mechanism for the inhibition of KaiA is its sequestration into KaiBC complexes (Chang et al, 2012), which form when KaiC is sufficiently phosphorylated (Rust et al, 2007); this mechanism serves as a delayed negative feedback loop in the oscillator. The KaiB-independent thresholding phenomenon we describe here is strongest when KaiC phosphorylation is low and when KaiC is ADP-bound. This suggests there are at least two mechanisms that work together during the cycle to prevent KaiA from acting at inappropriate times, and that the relative strength of the two mechanisms varies with the nucleotide pool. Incorporation of the ultrasensitive response to KaiA into a mathematical model of the full oscillator suggests that this effect both stabilizes the period against changes in the nucleotide pool and allows oscillations to persist even when KaiB binds KaiA relatively weakly. Consistent with this prediction, we find that a substantial amount of KaiA is not bound by KaiB even when the clock is dephosphorylating. These results shed new light on metabolic compensation, a property that allows robust 24-h oscillation in spite of changes in %ATP conditions (Johnson & Egli, 2014). Taken together, our results show how the Bayesian framework combined with extensive training data can be used to discover unanticipated mechanisms and direct experimental investigations. Results A molecular model of KaiA—KaiC dynamics To probe the response of KaiC phosphorylation to a wide range of metabolic conditions, we made kinetic measurements of KaiC phosphorylation at three %ATP conditions and six [KaiA] conditions while holding the KaiC concentration constant (Fig 1B). KaiC is a homohexamer and each subunit has two duplicated ATPase domains, termed CI and CII (Hayashi et al, 2003; Pattanayek et al, 2004; Terauchi et al, 2007). The CII domain can autophosphorylate via a bidirectional phosphotransferase mechanism (Egli et al, 2012; Nishiwaki & Kondo, 2012) with two phosphorylation sites. Each KaiC subunit thus has four phosphoforms: the unphosphorylated (U), phosphoserine-431 (S), phosphothreonine-432 (T), and doubly phosphorylated (D) states, each of which peaks at a distinct time in the oscillation (Xu et al, 2004; Nishiwaki et al, 2007; Rust et al, 2007). Our strategy is to fit these data with a model of the KaiC catalytic cycle in CII with a minimum of simplifying assumptions. To this end, we formulate a model based on mass-action kinetics. We explicitly keep track of three properties of the CII domain of each KaiC subunit: its phosphorylation status (right superscripts in Fig 1C), nucleotide-bound state (right subscript), and whether or not KaiA is bound (left superscript). We do not consider CI or the hexameric nature of KaiC explicitly (see Appendix for further discussion). There are thus 16 possible KaiC states, eight of which are shown in Fig 1C, along with the phosphotransfer, nucleotide exchange, KaiA (un)binding, and hydrolysis reactions that connect the states (see Fig EV1A for the full model structure). We also considered the possibility that nucleotides might interact directly with KaiA, which could allow KaiA's activity to directly depend on nucleotides in solution. However, we did not detect any direct interaction between KaiA and ATP or ADP using NMR spectroscopy (Appendix Fig S1), so we do not allow for this scenario in the model. Click here to expand this figure. Figure EV1. Overview of the model A schematic of the full mass-action kinetic model. Here, each arrow represents a reaction, and the associated rate constant is represented using the notation introduced in the main text. The thickness of the arrows is proportional to the best fit rate on a log scale (base 10) at 100% ATP and 1.5 μM KaiA. The posterior distributions for all rate constants, initial conditions, and the global error hyperparameter. The three distributions represent the results from three independent runs; the log-posterior values for the best fits from the three runs are listed. The red lines represent the best fit from the best run (i.e., the blue distributions). See Materials and Methods and Appendix Fig S5 for further details on the model parameterization method. Download figure Download PowerPoint In the rest of this section, we further elaborate our model by stepping through the four classes of reactions that we include; additional details can be found in Materials and Methods. In the next section, we describe in qualitative terms the fitting procedure and analyze the extent to which the model was constrained by the training dataset. For readers who are primarily interested in the biochemical conclusions of the model, these results can be skipped without loss of continuity. Phosphotransfer The KaiC CII domain has reversible phosphotransferase activity (Egli et al, 2012; Nishiwaki & Kondo, 2012); it can transfer a γ-phosphate group from a bound ATP to a phosphorylation site, but unlike a typical phosphatase, it regenerates ATP from ADP during dephosphorylation, i.e., (1)where (X,Y)∈{(U,T),(U,S),(T,D),(S,D)}. This mechanism implies that the nucleotide-bound state of KaiC has a significant impact on the net direction of its phosphotransferase activity: An ATP-bound KaiC presumably cannot dephosphorylate, and an ADP-bound KaiC cannot phosphorylate. Nucleotide exchange KaiA binding to the CII domain (Kim et al, 2008; Pattanayek & Egli, 2015) stimulates KaiC autophosphorylation (Iwasaki et al, 2002; Williams et al, 2002; Kageyama et al, 2006). Recent work has shown that KaiA can bind to KaiC and act as a nucleotide exchange factor (Nishiwaki-Ohkawa et al, 2014) by facilitating conformational changes at the subunit interface that promote solvent exposure of the nucleotide-binding pocket (Hong et al, 2018). It is currently unclear whether this nucleotide exchange activity is responsible for all of KaiA's effect on KaiC or whether it alters the KaiC catalytic cycle in other ways (see Appendix for further analysis of this issue). The reversible binding of KaiA (2)contributes two classes of rate constants, ka and kb. Because the CII domain of KaiC releases its bound nucleotides very slowly in the absence of KaiA (Nishiwaki-Ohkawa et al, 2014), we ignore the possibility of KaiA-independent nucleotide exchange in the model. Under the assumptions that (i) the apo state is in a quasi-steady state, (ii) the ADP and ATP on rates are identical, and (iii) ATP release is slow, nucleotide exchange can be modeled as a one-step reaction: (3)where (4)and is the ADP dissociation rate constant. Nucleotide exchange thus contributes one class of rate constant, . See Materials and Methods for the derivation of 4. ATP hydrolysis Finally, we allow for irreversible ATP hydrolysis in the CII domain (5)which contributes one class of rate constants, kh. This assumption is important because this pathway is required in the model for sustained dephosphorylation by allowing ADP-bound KaiC to accumulate in the absence of KaiA. Because each KaiC molecule consumes relatively little ATP on the timescale of the simulations (Terauchi et al, 2007), we assume the solution ATP and ADP concentrations are constant. State-dependent rates Given the six classes of rate constants, kp, kd, ka, kb, , and kh, we make the model maximally general, or mechanistically naive, by allowing each rate constant to potentially depend on the specific molecular state involved in the reaction. For example, the KaiA dissociation rate constant is allowed to vary depending on the nucleotide-bound state and phosphoform background of KaiC, and thus, the dissociation rate constants for the ADP-bound U () and ATP-bound T phosphoforms () are two independent model parameters. In this way, the parameter fitting and model comparison procedures automatically test specific biochemical hypotheses about the functions of KaiA and KaiC. For example, allowing the KaiA off rates to depend on the nucleotide-bound states is equivalent to the hypothesis that KaiA has different dwell times for ATP- vs. ADP-bound states of KaiC. In fact, because each reaction has an independent rate constant, except for thermodynamic constraints of detailed balance, the fitting procedure effectively allows for simultaneous testing of all possible two-way interactions of the three categories of KaiC properties, without a priori preference for any particular mechanism. The data constrain the parameters to widely varying degrees We estimate the model parameters through a Bayesian framework. In this framework, we maximize the posterior probability, which is proportional to the product of the prior distribution and the likelihood function. Here, we interpret the prior as representing subjective beliefs on the model parameters before experimental inputs, while the likelihood function quantifies the goodness of fit. Bayesian parameter estimation reduces to least-squares fitting under the assumption of normally distributed residuals and uniform priors. In practice, we find that direct numerical optimization of the posterior usually results in fits that are trapped in low probability local maxima (Appendix Fig S2B). Thus, we instead draw parameters from the prior distribution and then use a heuristic combination of MCMC sampling and optimization (Powell's algorithm) to explore the parameter space. The MCMC method that we use (Goodman & Weare, 2010; Foreman-Mackey et al, 2013) efficiently searches the parameter space by simulating an ensemble of parameter sets in parallel; the spread of the ensemble reflects the geometry of the posterior distribution and is used to guide the directions of Monte Carlo moves. See Materials and Methods for a more mathematical treatment of the fitting procedure and comparison of different numerical optimization and sampling methods. We use this approach to fit the phosphorylation data (Fig 1B) together with previously published data on dephosphorylation (Rust et al, 2011), ATP hydrolysis rate (Terauchi et al, 2007), and the KaiA dwell time for each KaiC phosphoform (Kageyama et al, 2006; Mori et al, 2018) (see Materials and Methods). Overall, the model achieves excellent agreement with the training data (Figs 1B and EV2A–C). In the following analyses, we refer to model predictions using the best fit parameter values and quantify the uncertainties using the posterior distribution (see Appendix for further discussion of the convergence of the simulations). Click here to expand this figure. Figure EV2. Behavior of the model Model fit to the dephosphorylation dataset (Rust et al, 2007). The best fit KaiA dwell time as a function of KaiC phosphoform and nucleotide-bound state. The error bars represent the 95% posterior interval, computed from the final 30,000 MCMC steps of the 224 walkers of the ensemble of the inference run, shown in blue in Fig EV1B. The dashed lines represent the experimental measurements (Kageyama et al, 2006; Mori et al, 2018), which did not resolve the nucleotide-bound states. Inorganic phosphate production per KaiC monomer over the course of a phosphorylation reaction. The gray region represents the experimental bounds on the KaiC hydrolysis rate with 1.2 μM KaiA and no KaiA (Terauchi et al, 2007). The kinetics of the dephosphorylation reaction in the absence of KaiA, broken down into the eight individual KaiC states. The gray region represents the 95% posterior interval. Refer to Fig EV1A for the KaiC state names. The predicted phosphorylation kinetics at 7 and 1.75 μM KaiC, both at 100% ATP and 1.5 μM KaiA, compared to experimental measurements. Note that these two time series are not part of the training set. The posterior distributions for and , the dissociation rates for ATP and ADP, respectively, in an early iteration of the model. The horizontal axis is on a base 10 log scale. The long tail to the left of the posterior distribution for suggests that the model can be simplified by setting the rate to zero. Download figure Download PowerPoint We find that certain parameters, such as the hydrolysis rates in the U and T phosphoforms and the KaiA off rates from the U phosphoform, are tightly constrained, while many others, mainly involving S and D phosphoforms, are less constrained, in the sense that their posterior distributions span multiple orders of magnitude, exhibit multimodality, or cannot be reproduced over multiple independent runs (Fig EV1B). Some parameters are highly correlated, and certain combinations of the parameters are much better constrained than the individual parameters. For example, the posterior distributions for the KaiA binding affinities (Fig 1D) appear better constrained than the on/off rates (Appendix Fig S3B). Taken together, these results are consistent with the notion that collective fits of multiparameter models are generally "sloppy", meaning that the sensitivities of different combinations of parameters can range over orders of magnitude with no obvious gaps in the spectrum (Brown & Sethna, 2003; Gutenkunst et al, 2007). As we will see, we can nonetheless make useful predictions using the ensemble of model parameters, because the model behavior is constrained along the stiffest directions of the posterior distribution. By contrast, direct parameter measurements need to be both complete and precise to achieve similar predictive validity (Gutenkunst et al, 2007). We further characterize the structure of the parameter space in Appendix and Appendix Fig S3. KaiC (de)phosphorylation goes through transient kinetic intermediates In the model, we can break down the kinetics of KaiC phosphorylation reactions and interpret the underlying molecular events. Here, we consider the phosphorylation kinetics at a standa
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