- Methodology Article
- Open Access
Towards in vivo estimation of reaction kinetics using high-throughput metabolomics data: a maximum likelihood approach
- Weiruo Zhang^{1},
- Ritesh Kolte^{1} and
- David L Dill^{2}Email author
Received: 17 March 2015
Accepted: 15 September 2015
Published: 5 October 2015
Abstract
Background
High-throughput assays such as mass spectrometry have opened up the possibility for large-scale in vivo measurements of the metabolome. This data could potentially be used to estimate kinetic parameters for many metabolic reactions. However, high-throughput in vivo measurements have special properties that are not taken into account in existing methods for estimating kinetic parameters, including significant relative errors in measurements of metabolite concentrations and reaction rates, and reactions with multiple substrates and products, which are sometimes reversible. A new method is needed to estimate kinetic parameters taking into account these factors.
Results
A new method, InVEst (In Vivo Estimation), is described for estimating reaction kinetic parameters, which addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on a model where all measurements have relative errors. Simulations show that InVEst produces accurate estimates for a reversible enzymatic reaction with multiple reactants and products, that estimated parameters can be used to predict the effects of genetic variants, and that InVEst is more accurate than general least squares and graphic methods on data with relative errors. InVEst uses the bootstrap method to evaluate the accuracy of its estimates.
Conclusions
InVEst addresses several challenges of in vivo data, which are not taken into account by existing methods. When data have relative errors, InVEst produces more accurate and robust estimates. InVEst also provides useful information about estimation accuracy using bootstrapping. It has potential applications of quantifying the effects of genetic variants, inference of the target of a mutation or drug treatment and improving flux estimation.
Keywords
- Relative error
- Enzymatic reaction
- Parameter estimation
- Maximum likelihood
- Error-in-all-measurements
- In vivo data
Background
High-throughput assays such as mass spectrometry are improving rapidly, which creates an opportunity for large scale in vivo measurements of the metabolome. Those in vivo data could enable estimation of kinetic parameters of metabolic reactions which are hard to estimate using in vitro data.
Metabolic reactions are normally enzyme-catalyzed reactions, and quantitative estimates of their kinetic parameters could be very useful. Knowledge of kinetic parameters allows estimation of reaction rates directly from concentration measurements. Comparing the estimated kinetic parameters of a reaction in the wild type and mutant cells permits quantification of the effects of genetic variants, which may change the abundance or activity of a metabolic enzyme. Similarly, the effect of a drug that targets a particular enzyme could be estimated. If parameters can be estimated for many reactions in a pathway, it would enable inference of the target of a mutation or drug treatment – if the estimates show that one enzyme is particularly strongly affected, that enzyme is probably the target. Finally, estimated parameters also allow estimation of maximum reaction rates, which can then be used as constraints to improve flux balance analysis [1].
We explore the central problem of how to estimate the kinetic parameters of individual reactions using in vivo high-throughput measurements of metabolite concentrations and reaction rates at steady state, obtained by mass spectrometry or by nuclear magnetic resonance. The method requires metabolite concentration and reaction rate data in multiple experiments under varying conditions. For example, data could consist of several experiments obtained by perturbing the system through changes in nutrient media, drug treatment, or genetic alterations. From such data, the kinetics of many individual reactions can potentially be estimated.
Enzyme kinetic parameters have been measured for at least a century [2]. The basic method involves mixing a measured amount of substrate and enzyme, and measuring the concentration of product at various points in time, creating a progress curve [3]. In this setting, the experimenter has control over the initial concentrations of enzyme and substrate and thus can obtain relatively accurate measurements for concentrations. Although the experimental conditions are not at steady state, the mathematical formula for the kinetics can be simplified to the familiar Michaelis Menten kinetics by assuming that some elementary reactions are in near-equilibrium (this is called the quasi-steady-state assumption).
In contrast with an in vitro experiment, one major challenge with in vivo measurements of concentrations and reaction rates is the presence of significant error. Except for very low abundance metabolites, the errors are normally relative, meaning that they are proportional to the metabolite concentrations, instead of additive. (Relative error is shown in available experimental data in Additional file 1: Figure S1.) To quantify measurement precision with relative errors, experimentalists often use the coefficient of variation (CV), which is calculated by dividing the standard deviation of peak area/height by the mean peak area/height [4–6]. Methods such as least squares, which assume additive errors, are often not going to produce accurate estimates of parameters with relative errors. Because of such significant relative errors, it might not be reasonable to assume that errors are only in reaction rates as most of the in vitro enzyme kinetics methods assume. Relative errors in both concentrations and reaction rates need to be considered. Furthermore, many in vivo experiments are not time courses, so the data are assumed to be at steady-state. Another challenge with in vivo measurements is the difficulty of measuring enzyme and intermediate enzyme complex concentrations [7, 8], so these are typically unknown. Finally, control over metabolite concentrations in the cell is limited, so the range of experimental data points may be suboptimally distributed for accurate estimation of all parameters, making it difficult to estimate some parameters of a reaction.
A new estimation method, InVEst, standing for In Vivo Estimation, is described for estimating reaction parameters that addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation, based on a model where all measurements have relative errors. As described, InVEst uses a family of reversible reaction mechanisms with multiple reactants and products with a single displacement mechanism. It is not always possible to obtain data from the entire range of metabolite concentrations and reaction rates, so some parameters may not be identifiable. InVEst estimates the standard deviations of parameter estimates using bootstrapping (a method of estimating variation in statistics by random subsampling of a data set), so that the user can understand the range of errors for the estimates.
Features of different enzyme kinetic parameter estimation methods. “WLS” stands for the weighted least squares method. “TLS” stands for the total least squares method. “Raaijmakers” is the maximum likelihood method of Raaijmakers
Multiple substrates/ | Reversible | Relative | Error in all | |
---|---|---|---|---|
products | reaction | error | variables | |
Double reciprocal | ✗ | ✗ | ✗ | ✗ |
Direct linear | ✗ | ✗ | ✓ | ✓ |
WLS | ✓ | ✓ | ✗ | ✗ |
TLS | ✓ | ✓ | ✗ | ✓ |
Raaijmakers | ✗ | ✗ | ✓ | ✗ |
InVEst | ✓ | ✓ | ✓ | ✓ |
In this paper, our goal is to focus on the specific problem of estimating kinetic parameters as accurately as possible, given realistic assumptions about data errors. We discuss the formulation of InVEst, and evaluate the method on simulated data. We show that InVEst works well on data with relative errors in all measurements. We also demonstrate the application of InVEst and discuss the parameter identifiability issue.
Methods
Like most methods of kinetic parameter estimation, we assume that temperature and pressure are constant, so rate constants in mass action kinetic equations are constant, and the Gibbs Free Energy of Formation is constant. We also assume that the measured system is at steady state, meaning that the time derivatives of metabolite concentrations and reaction rates are zero.
Also, we assume that there are measurements of stable reactants and products of enzyme reactions, but not substrate-enzyme complexes, product-enzyme complexes and free enzyme concentrations, as they are generally difficult to measure experimentally. It is assumed that metabolite concentrations are obtained by high-throughput methods, such as chromatography, mass spectroscopy, or nuclear magnetic resonance spectroscopy [15]. For example, reasonably accurate concentration data can be obtained by mass spectroscopy with internal standards. Normally, average value of coefficient of variation for mass spectrometry below 0.2 is considered as good measurements [16–18], and thus it is not unreasonable to expect such data to have a constant coefficient of variation (i.e., normally distributed relative error) of 20 %.
We also assume that it is possible to obtain measurements of reaction rates. For steady state reaction rate measurement, one widely used method is C ^{13} labeling, which uses a cell culture at steady state in a medium with labeled-carbon substrates. Reaction rates can be determined by analyzing the labeling pattern of targeted metabolites from mass spectrometry [19]. In addition, we assume that the Gibb’s Free Energies of Formation of metabolites are known, since these are used to compute the equilibrium constants (K _{ eq }) for enzymatic reactions.
Single substrate and product reversible reactions
We use a standard simple but general reaction mechanism to represent most metabolic reversible reactions [20]. This subsection considers single reactant/product case. The more general case consisting of multiple reactants and multiple products will be discussed later. The reaction is a three step process, namely binding, conversion and release:
where a is the reactant, b is the product, E is the free enzyme, aE and bE are the intermediate complexes, and k _{ i } and k _{−i } are reaction rate constants for i∈{1,2,3}.
where \(K_{\textit {eq}} = \frac {k_{1}k_{2}k_{3}}{k_{-1}k_{-2}k_{-3}}\) is an equilibrium constant, obtained from the Standard Gibbs Free Energy of Formation of the reactants and products.
If K _{ eq } is very large and the reversible reactions’ rate constants (k _{−2} and k _{−3}) are small, c _{3} can be neglected and the rate Eq. 2 can be reduced to standard irreversible Michaelis Menten equation.
This rate equation can be derived from the ordinary differential equations for mass action kinetics of a reaction (1), by setting the derivatives of the concentrations of all chemical species to zero (since the system is assumed to be at steady state) and solving for [ E _{ tot }]. The detailed derivation and calculation for the steady state equation and equilibrium constant are presented in Additional files 2 and 3.
Parameter estimation by maximum likelihood for single substrate/product reversible reaction
The InVEst method estimates the parameters of kinetic rate Eq. (2) using maximum likelihood, assuming relative error in all measurements. Parameters are estimated from a set of n experiments, each with data values for a _{ i } (substrate), b _{ i } (product), v _{ i } (reaction rate), for experiment i.
Each data value has some known relative error. Specifically, we have a _{ i }=a _{ i0} ε _{ a }, b _{ i }=b _{ i0} ε _{ b } and v _{ i }=v _{ i0} ε _{ v }, where a _{ i0}, b _{ i0}, and v _{ i0} are latent variables representing the data values without measurement error, multiplied by a normally distributed error with mean 1 and standard deviation σ: \(\epsilon _{x} \sim N \left (1,{\sigma _{x}^{2}}\right)\) (where x is a, b, or v).
In the implementation, this is simplified to an unconstrained optimization problem by substituting the right-hand side of Eq. 2 for v _{ i0}.
Generalization to multiple substrates and products
where c _{1}, c _{2}, c _{3}, K _{ eq }, and E _{ tot } are as before.
In the implementation, this can also be simplified to an unconstrained optimization problem by substituting the right-hand side of Eq. 3 for v _{ i0}.
Parameter identifiability
So changes in c _{3} will have little effect on v. More importantly, changes in data values resulting from erroneous estimates of c _{3} will be small relative to the noise in the data, so estimates of c _{3} tend to have large errors. Similarly, estimates of c _{1} tend to have large errors when c _{2} a+c _{3} b≫c _{1} and estimates of c _{2} have large errors when c _{1}+c _{3} b≫c _{2} a.
Estimates of the accuracy of parameter estimates must be obtained using the available data. InVEst uses bootstrapping to estimate the variance of the parameter estimates.
Bootstrap estimation of standard error
The c parameter estimates can vary widely in accuracy, depending on the experimental data. Bootstrapping [23] is used to estimate the relative standard errors and bias of the parameter estimates, so users can tell whether the parameter estimation is good or not. Let \(\hat {c}\) be the estimate from the data, and \(\hat {c_{i}}^{*}\) be the estimate from a bootstrap sample. A typical recommendation is to use N=n ^{2} bootstrap samples for n experimental measurements [24]. The bootstrap estimation of standard errors is calculated from \(SE_{B}(\hat {c}) = \left [\frac {1}{N}\sum (\hat {c_{i}}^{*}-\hat {c})^{2}\right ]^{\frac {1}{2}}\) and bias estimation is calculated by \(\mathit {Bias} = \frac {1}{N}\sum \hat {c_{i}}^{*}-\hat {c}\)[25]. As the c parameters have a large range of possible values, it is more appropriate to use relative errors and relative bias to describe the estimate. The relative standard error is calculated by \(SE_{B}/\hat {c}\) and the relative bias is calculated by \(\mathit {Bias}/\hat {c}\).
Estimation of total enzyme change
Estimating kinetic parameters can be useful for identifying the effects of genetic changes or drug treatments that target metabolic enzymes. The total concentration of the enzyme in the cell may change because of changes in gene expression or loss of function in one or more copies of the gene coding for the enzyme, or the activity may change because of changes in the protein sequence or post translational modifications. Estimating these changes for specific enzymes in each sample can help identify the target of a mutation or drug (it’s the enzyme whose activity changes the most), and may be useful for estimating the impact of such a change on flux through a network.
Results
Evaluate InVEst using simulated data
We evaluate the parameter estimation method on simulated data. For MATLAB code for reproducing the results of this work, please refer to [26]. The simulations were carried out in MATLAB on a laptop computer with an Intel Core i5-4200u 2.3 GHz processor and 8 GB installed memory.
Many reactions in metabolic pathways have multiple substrates and products and are reversible reactions. The simulation is based on the reaction acetylornithine aminotransferase from Saccharomyces Cerevisiae Arginine biosynthesis pathway with Arg8 [27]. Kinetic parameters and the total enzyme concentration are not available, and thus we use some heuristic numbers for them. The experimental data are chosen to be well-distributed, since poorly distributed data would guarantee inaccurate parameter estimates even for the best possible estimation method.
The standard Gibbs Free Energy of Formation for the metabolites are taken from MetaCyc database [29], and are provided in the Additional file 4. The standard Gibbs Free Energy of Formation can be used to compute K _{ eq }=1.7281, and, assuming E _{ tot }=1 M, the c parameters are c _{1}=2.5783, c _{2}=3.7327 and c _{3}=3.5238.
To characterize the amount of data for effective use of InVEst, we evaluated the accuracy of parameter estimates for varying numbers of simulated experiments. Data sets of 12, 24 and 30 experiments were generated by choosing values for substrate and product concentrations and computing v exactly for each choice based on Eq. 3. Relative errors were introduced by multiplying a random value from the normal distribution of N(1,σ ^{2}). A value of 0.2 was used for σ for metabolites, and σ _{ v } of 0.2 was used for reaction rates.
Average c parameter estimates, relative standard errors and relative bias as a function of number of experiments for acetylornithine aminotransferase when σ _{ v }=0.2. Results are based on 1,000 simulated data sets. “n” is the number of experiments. “Avg Est” is the average value of the estimates. “Rel SE” is the relative standard error, and “Rel bias” is the relative bias
n | Run time | True | Avg Est | Rel SE | Rel bias |
---|---|---|---|---|---|
12 | 1.74sec/simulation | c _{1}:2.578 | 2.315 | 0.188 | 0.102 |
c _{2}:3.733 | 3.68 | 0.108 | 0.014 | ||
c _{3}:3.524 | 3.54 | 0.10 | 0.007 | ||
24 | 7.98sec/simulation | c _{1}:2.578 | 2.567 | 0.143 | 0.004 |
c _{2}:3.733 | 3.755 | 0.081 | 0.006 | ||
c _{3}:3.524 | 3.544 | 0.087 | 0.006 | ||
30 | 20.04sec/simulation | c _{1}:2.578 | 2.573 | 0.129 | 0.002 |
c _{2}:3.733 | 3.742 | 0.062 | 0.002 | ||
c _{3}:3.524 | 3.517 | 0.073 | 0.002 |
c parameter estimates for acetylornithine aminotransferase when σ _{ v }=0.5. Results are based on 1000 simulated data sets of 30 experiments, each
True | Avg Est | Rel SE | Rel bias | |
---|---|---|---|---|
c _{1} | 2.578 | 2.555 | 0.189 | 0.009 |
c _{2} | 3.733 | 3.806 | 0.127 | 0.020 |
c _{3} | 3.524 | 3.652 | 0.140 | 0.036 |
c parameter estimates for acetylornithine aminotransferase when σ _{ v }=0.2. Estimates are from a single simulated data set of 30 experiments. The bootstrap method was used to estimate relative standard error (“Rel SE”) and relative bias (“Rel bias”)
True | Est | Rel SE | Rel bias | |
---|---|---|---|---|
c _{1} | 2.578 | 2.750 | 0.111 | 0.008 |
c _{2} | 3.733 | 3.902 | 0.066 | 0.005 |
c _{3} | 3.524 | 3.552 | 0.094 | 0.016 |
c parameter estimates for acetylornithine aminotransferase when σ _{ v }=0.5. Estimates are from a single simulated data set of 30 experiments. The bootstrap method was used to estimate relative standard error (“Rel SE”) and relative bias (“Rel bias”)
True | Est | Rel SE | Rel bias | |
---|---|---|---|---|
c _{1} | 2.578 | 2.933 | 0.152 | 0.021 |
c _{2} | 3.733 | 4.081 | 0.184 | 0.052 |
c _{3} | 3.524 | 3.343 | 0.243 | 0.059 |
Comparison of InVEst with prior methods
Most current methods produce optimal estimates only when errors are additive and when errors occur only in reaction rate measurements. These assumptions are generally not true with in vivo data. In this subsection, we compare InVEst to some existing methods and show that InVEst produces better estimates when data have relative errors in all measurements.
Comparison of the accuracy of prior methods: total least square (TLS), ordinary least square (OLS), direct linear plot (DLP), double reciprocal plot(DRP) and InVEst. True c _{1}=1.5, True c _{2}=0.8. Data have relative errors in all variables. Results are based on 1,000 simulated data sets of 30 experiments, each. “Avg Est” is the average value of the estimates. “Rel SE” is the relative standard error
Avg Est c _{1} | Avg Est c _{2} | Rel SE c _{1} | Rel SE c _{2} | |
---|---|---|---|---|
TLS | 0.840 | 0.940 | 0.389 | 0.143 |
OLS | 1.036 | 0.921 | 0.413 | 0.147 |
DLP | 1.396 | 0.883 | 0.429 | 0.262 |
DRP | 1.859 | 0.498 | 0.307 | 1.124 |
InVEst | 1.518 | 0.766 | 0.128 | 0.112 |
Predicting total enzyme concentration change
c parameter estimates for acetylornithine aminotransferase from mutant/drug treated sample. Results are based on 1,000 simulated data sets
True | Avg Est | Rel SE | Rel bias | |
---|---|---|---|---|
c _{1} | 25.783 | 24.784 | 0.119 | 0.039 |
c _{2} | 37.327 | 37.480 | 0.061 | 0.004 |
c _{3} | 35.238 | 35.518 | 0.065 | 0.008 |
E _{ tot } change prediction based on 1,000 simulated data sets
True | Avg Est | Rel SE | Rel bias | |
---|---|---|---|---|
\(\frac {E_{\textit {tot}}^{wt}}{E_{\textit {tot}}^{mt}} = \frac {c_{1}^{mt}}{c_{1}^{wt}}\) | 10 | 10.214 | 0.091 | 0.021 |
\(\frac {E_{\textit {tot}}^{wt}}{E_{\textit {tot}}^{mt}} = \frac {c_{2}^{mt}}{c_{2}^{wt}}\) | 10 | 9.957 | 0.022 | 0.004 |
\(\frac {E_{\textit {tot}}^{wt}}{E_{\textit {tot}}^{mt}} = \frac {c_{3}^{mt}}{c_{3}^{wt}}\) | 10 | 10.115 | 0.049 | 0.012 |
Since any of the c _{ i } parameters can be used to estimate the change in E _{ tot }, the one that gives minimum standard error, c _{2}, was chosen. This also demonstrates that even though sometimes identifiability issues can occur and some parameters cannot be estimated, our method could still be very useful if one parameter can be estimated accurately.
Discussion
This work is intended to be a first step towards estimating parameters for reactions in large metabolic networks in vivo. In vivo estimation will need to be based on data that have relatively large relative errors in all measured parameters, and will have to deal with a variety of reaction kinetics, including reactions that are reversible and have multiple substrates and/or products. Although measurement and estimation of enzyme kinetics has been studied for many decades, there is no single existing estimation method that addresses all of these issues. We have proposed a maximum likelihood approach to estimate kinetic parameters using nonlinear optimization, with estimates on the standard error and bias of the results using the bootstrap.
Simulations show that InVEst produces accurate estimates for realistic high-throughput metabolomics data. For example, with 20–30 samples with coefficients of 20 % in metabolite concentrations and 50 % in reaction rate estimates, estimates have a relative standard error of less than 20 %. Collecting data of this quality would be technically difficult, but is within the current state of the art.
An advantage of the method is that it estimates each set of reaction parameters independently. If measurements are not available for some metabolites, it can still estimate parameters for those reactions for which the data include all substrates and products.
Solving the problem of in vivo parameter estimation in its full generality will require meeting a number of additional challenges. Some reactions have more complex kinetics than those we consider, especially various kinds of inhibition. When the inhibiting metabolite and mechanism of inhibition are known, the approach described here can probably be generalized to accommodate the inhibition mechanism in our future work. Otherwise, a process of model selection may be necessary, where competing models are estimated and the quality of the results compared, with appropriate adjustments for model complexity. In addition, it will be necessary to deal with the kinetics of transport reactions, and to take account of different compartments in the cell.
Parameter identifiability is a difficult issue in in vivo estimation. We have shown that accurate estimates of all parameters require data that is well-distributed over the kinetics curve, but such data will not often be obtainable for several reasons. Experimental data must be obtained by perturbing metabolites and fluxes, for example, by adjusting nutrient media, testing mutants, and targeting reactions with drugs. First, accurate estimation may require non-physiological concentrations of metabolites – estimating c _{3} for a reaction that is nearly irreversible being an example. More generally, there is usually inadequate controllability of metabolite concentrations and reaction fluxes to obtain the experimental values needed for accurate estimation, for many reasons including concentrations are toxic or inadequate to sustain life, and rate-limiting reactions that make high fluxes in other reactions impossible to obtain. Since we can’t estimate everything accurately, it is important to produce estimates of the standard errors of parameter estimates, so we can tell which ones are meaningful. Also, as we note above, if some but not all parameters of a reaction can be estimated accurately, the results still may be useful. For example, it is possible to estimate the total concentration or relative activity of an enzyme in wild-type vs. mutant cells when only one of the kinetic parameters is accurately estimated.
Conclusion
In conclusion, a new method, InVEst, is developed for estimating reaction kinetic parameters in metabolic networks that addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on models where all measurements have potentially relative errors. It can be applied to multiple substrate/product reversible enzymatic reactions with a generalized single displacement mechanism. Because it is not always possible to obtain good data covering full range of possible metabolite concentrations and reaction rates, certain parameters may be non-identifiable. InVEst uses bootstrap to estimate the standard errors of parameter estimations that can tell which estimates are reliable.
InVEst enables the estimation of reaction rates directly from concentration measurements. Also, comparing the estimated kinetic parameters of a reaction in the wild type and mutant cells can quantify enzyme abundance or activity change due to genetic variants. The same method can also be used to measure the effect of a drug that targets a particular enzyme. Moreover, estimated parameters can be used to estimate maximum reaction rates, which could be used as constraints to improve flux-balance analysis.
Declarations
Acknowledgements
D.L.D. and W.Z. were supported by a King Abdullah University of Science and Technology (KAUST) research grant under the KAUST Stanford Academic Excellence Alliance program. R. K. was supported by Stanford Graduate Fellowship.
We thank Prof. Chaitan Khosla, Chemical Engineering, Stanford University and Prof. Douglas Brutlag, Biochemistry, Stanford University, for their valuable advice and comments on our work. We also thank Prof. Chao Du, Statistics, University of Virginia, for his suggestions on the bootstrap.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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