Heterogeneous Michaelis-Menten model

The Michaelis-Menten model requires its reaction velocities to be measured from a preparation of homogeneous enzymes, with identical or near-identical catalytic activities. HetMM is a kinetic model that relaxes this requirement, by assuming there are an unknown number of enzyme species drawn from an unknown probability distribution. This model features one additional parameter over the Michaelis-Menten model, describing the standard deviation of this distribution.

HetMM is a program for estimating parameters under the heterogeneous and homogeneous Michaelis-Menten models using Markov chain Monte Carlo (MCMC). The two models are compared and the best model, plus its posterior probability, is reported. The Hill coefficient can also be estimated, allowing for positive or negative cooperativity between binding sites.

HetMM is implemented as a package for BEAST 2, which is traditionally used for Bayesian phylogenetics. However, BEAST 2 comes with a wide range of tools that that help with Bayesian inference, and also has great support and online documentation.

Dependencies

Required

• BEAST 2 is required to run HetMM | https://www.beast2.org/
• R will be used for preparing the data and generating figures | https://www.r-project.org/

Recommended

• Tracer is useful for analysing MCMC chains | https://www.beast2.org/tracer-2/

Installing HetMM

1. Open BEAUti, which is installed with BEAST 2. If you are using the command line, this can be done using

/path/to/beast/bin/beauti

2. Go to File -> Manage Packages

3. Install the HetMM package. Since HetMM is currently in pre-release, you may need to add an extra package repository as follows:

• click the packager repositories button. A dialog pops up.
• click the Add URL button. A dialog is shown where you can enter https://raw.githubusercontent.com/CompEvol/CBAN/master/packages-extra-2.7.xml
• click the OK button. There should be an extra entry in the list.
• click Done
• After a short delay, the HetMM package should appear in the list of packages.

4. Close BEAUti

Using HetMM

In this tutorial we will consider the hydrolysis of sucrose by Michaelis and Menten 1913.

1. Download the example file here.

2. Download the XML file template here. This file contains the full Bayesian model configuration, including parameters, prior distributions, and operators.

3. Use the csv2json.R script to convert the csv file into a json file so that it can be read in by BEAST 2:

Rscript csv2json.R MM.tsv

This will generate a file called MM.json.

Note that the first time you use R, you may need to select a mirror to install packages from. This can be done by:

# Open R
R

# Request to install a package, then follow the instructions to select a mirror
install.packages("jsonlite")

# Confirm the package was successfully installed
library(jsonlite)

# Quit R
q() 

4. Run BEAST 2 on the XML file from the command line using: /path/to/beast/bin/beast -df MM.json HetHill.xml

5. Now, BEAST should start running. On the sucrose dataset, it should take around 20 minutes to complete but you can proceed to the next step while you wait.

6. To diagnose the MCMC chain and understand its parameters, open Tracer and open HetHill.log. Ensure that the effective sample size (ESS) of all parameters exceeds 200, and if not, the MCMC chain should be run for longer. Please see the Tracer tutorial for further details: https://beast.community/analysing_beast_output

7. The probability that the dataset is heterogeneous is the mean value of the parameter called ModelIndicator. 0 is homogeneous, 1 is heterogeneous.

8. The distribution of Hill coefficient directions is under the parameter HillIndicator. -1 is negative cooperativity (h<1), 0 is neutral (h=1), and +1 is positive cooperativity (h>1).

9. To visualise the model and summarise the posterior distribution, use the plotMM.R script. From the command line this can be done using:

   Rscript plotMM.R HetHill.log

   A file named HetHill.log.png should have been generated. This image will contain 1 line per sample from the MCMC chain, coloured yellow for homogeneous and blue for heterogeneous. In this case, there are a mixture of blue and yellow lines. A larger dataset typically has just 1 colour, reflecting a higher level of confidence in the correct model. A summary of the inferred model parameters is also printed.

   

Summarising posterior distribution:
median (2.5 percentile, 97.5 percentile)
        Vmax = 3.92 (3.6, 4.26) 
          Km = 0.0182 (0.0144, 0.0414) 
           ε = 0.033 (0.0125, 0.0884) 
   p(hetero) = 0.102 
     p(Hill) = 0.0574 
  Best model = Homogeneous 

Preparing a data file

First, prepare a .csv (comma separated variable) file with two columns. The first column should contain substrate concentrations, and the second with reaction velocities. Columns should be separated by commas.

Then, use the csv2json.R script to convert the csv file into a json file

Rscript csv2json.R DATA.tsv

Lastly, run BEAST 2 on the newly generated DATA.json file /path/to/beast/bin/beast -df DATA.json HetHill.xml

Increasing the MCMC chain length

Open the XML file in a text editor, find the following line, and edit the value of chainLength accordingly.

<run id="mcmc" spec="MCMC" chainLength="10000000">

Model averaging

The following model indicators

1. ModelIndicator is 0 when the homogeneous model is being used, and 1 for heterogeneous
2. HillIndicator is -1 when the Hill coefficient is <1, 1 when Hill is 1, and +1 when Hill is greater than 1

These two indicators are estimated during MCMC. They can also be fixed.

In order to impose symmetry on the Hill coefficient, the hill0 parameter is given a Beta distribution

<prior id="hill.prior" name="distribution" x="@hill0">
    <Beta alpha="2" beta="4" name="distr" />
</prior>

When HillIndicator=-1, the coefficient is hill0. When HillIndicator=1, the Hill coefficient is set to 1. And, when HillIndicator=1, the Hill coefficient is set to the reciprocal of hill0. The Beta prior above places most of the probability mass away from h=1, so that the three models can be distinguished.

Parameters

This model contains the following parameters:

1. Vmax - the maximum reaction rate
2. Km - the substrate concentration that gives half of Vmax
3. errorSD ε - the standard deviation of error, in log-space. Error between expected and observed velocities are assumed to follow a log-normal(0, ε) distribution
4. kmSD - the standard deviation of Km across enzymes in the mixture. This parameter is only used in the heterogeneous model
5. Hill - the Hill coefficient. This parameter is only used in the Hill model

Changing prior distributions

Edit the XML file to change the prior distributions. For example, the default prior for vmax is a log-normal(mu=1, sigma=2) distribution.

<prior id="vmax.prior" name="distribution" x="@vmax">
    <LogNormal M="1" S="2" meanInRealSpace="false" name="distr" />
</prior> 

if you wanted to change this to a Gamma(alpha=2, beta=2) distribution, change it to:

<prior id="vmax.prior" name="distribution" x="@vmax">
    <Gamma alpha="2" beta="2" name="distr" />
</prior> 

or for an Exponential(mean=1000) distribution, change it to:

<prior id="vmax.prior" name="distribution" x="@vmax">
    <Exponential mean="1000" name="distr" />
</prior> 

The priors should reflect one's expectation about the parameters prior to observing the data. Adjusting the prior distributions from the default is especially important for Km and Vmax, as these are sensitive to the units. If Km is expressed in mM or M , their values will be 1000 times different.

Sampling from the prior

To sample from the prior distribution (and ignore the data), use the following command:

/path/to/beast/bin/beast -sampleFromPrior -df DATA.json HetHill.xml

Fixing a parameter or model indicator

Any parameter or model indicator is easily fixed, by editing the XML file. First, find the parameter in the state and set its value to the desired value. For example, to set the initial Vmax, edit the 1.0 below:

<parameter id="vmax" spec="parameter.RealParameter" lower="0.0" name="stateNode">1.0</parameter>

or to use the heterogeneous model, set the false to a true below:

<parameter id="modelIndicator" spec="parameter.BooleanParameter" name="stateNode">false</parameter>

Then, to ensure the value does not change during MCMC, delete all operators that change the value. For example, to fix Vmax, delete the following two lines (or comment then out using )

<operator id="vmax.scale" spec="kernel.BactrianScaleOperator" parameter="@vmax" scaleFactor="0.1" weight="2.0"/> 

and

<f idref="vmax"/>

or, to fix the model indicator, delete the following operator

<operator id="indicatorBitFlip" spec="BitFlipOperator" parameter="@modelIndicator" weight="1.0"/>

Contact

For any queries, please contact jordan.douglas@auckland.ac.nz, leave an issue on GitHub, or make a post on the BEAST user group https://groups.google.com/g/beast-users

References

Douglas, Jordan, Charles W. Carter Jr, and Peter R. Wills. "HetMM: A Michaelis-Menten model for non-homogeneous enzyme mixtures." bioRxiv (2023): 2023-10.