MATLAB Stochastic System Identification Toolkit (SSIT) for modeling single-cell fluorescence microscopy data

Authors: Huy Vo, Joshua Cook, Eric Ron, Brian Munsky

The SSIT allows users to specifiy and solve the chemical master equation for discreate stochastic models, especially those used for the analysis of single-cell gene regulaton.

To learn more about the FSP theory that underlies the SSIT, please see the slides from our Nov. 3, 2022 BPPB Seminar

The SSIT includes command line tools and a graphical user interface to:

• Build, save, and load models
  • Create from propensity functions and stoichiometries
  • Load from / export to SimBiology
  • Load from / export to SBML
• Solve models
  • Solve using ODE analyses and basic moment closure analyses
  • Generate synthetic data from models using Stochastic Simulations (parallel)
  • Solve CME directly using the Finite State Projection algorithm, included automated FSP state set selection/expansion
  • All methods support non-linear, time varying propensity functions, including logical statements
• Compute sensitivity of solutions to parameter variations
• Compute first passage or escape time distributions for complex trajectories
• Load experimental single-cell data (e.g., from processes smFISH images)
• Fit Models to Experimental data
  • Compute the likelihood of data given model
  • Maximize likelihood using gradient and non-gradient based searches
  • Run efficient Metropolis Hastings algorithm (with custom proposal distribution or proposal distribution based on FIM) to estimate parameter uncertainties given single-cell data
  • Include custom priors on parameter distributions for Bayesian analysis
• Model and reject measurement noise
  • Calibrate empirical probability distortion operator (PDO) to quantify effects of image distortion
  • Include image distortions in parameter estimation
• Improve Experiment Designs
  • Compute the Fisher Information Matrix for CME models
  • Sample over model uncertainty (e.g., model prior or posterior from previous experiment round) for iterative experiment design.
  • Search experiment design space to find optimally informative experiments
  • Automatically adjust designs to account for image distortion effects
• Explore effects of Extrinsic Noise in parameters
• Form reduce order models
  • Run hybrid models with deterministic and stochastic species
  • Reduce models using Quasi-Steady Approximations on Fast Species
  • Reduce models using Eigenvalue decomposition
  • Reduce models using coarse meshes
  • Reduce models using Principle Orthogonal Decomposition
• Compare multiple models to different data sets with shared parameter sets
  • Identify parameters that change with genetic/environmental/experimental conditions
• Many Examples

Dependencies

For all basic functionalities:

• MATLAB R2023a
• Note that most, but not all, functionalities are also working in R2023b; we therefore encourage using R2023a for the time being.
• Symbolic Computing Toolbox.
• Image Processing Toolbox
• Global Optimization Toolbox (for model fitting only)
• Parallel Computing Toolbox (optional).
• SimBiology Toolbox (for loading/saving SBML models only)

Installation

Clone this package to a local folder on your computer. Then add the path to that folder (with subfolders) into MATLAB's search path. You can then call all functions from MATLAB.

Testing

To test your installation, navigate to the folder SSIT/tests and run the following test routines.

• PoissonTest % Tests various solution schemes, data generation/loading and model parameterization for a 1-species model
• Poisson2DTest % Tests various solutions for a 2-species model
• PoissonTVTest % Tests various solutions for a 1-species Time-varying model
• multiModelTests % Tests various solutions for a combinations of multiple models with different data sets.
• modelReductionTests % Tests various model reduction schemes for more efficient solutions of the Chemical Master Equation
• miscelaneousTests % Tests other aspects, including GUI functionality, loading/saving to SBME and SimBiology

Getting Started

The SSIT provides two basic interaction options: (1) command line tools and (2) a graphical user interface.

GUI Version

A GUI version of the SSIT has much of the functionality, and is a great way to familiarize yourself with the approach. However, for intenssive research tasks, we strongly recommend using the command line tools. To get started with the GUI, compile and launch to tool kit with the following commands:

src2app;

A = SSIT_GUI;

You should then see the model loading and building page of the graphical interface, and you are off to the races...

Command Line Version

To get started with the Command line Tools, navigate to the directory "Examples" and open one of the tutorial scripts "example_XXX.m". We recommend getting started using the following examples:

Examples/example_SSITBasics.m This example will get you started on creating and solving SSIT models.

Examples/example_EscapeTimes.m This example will get you started on using the SSIT to calculate escape time distributions.

Examples/example_FIMCalculation.m This example demonstrates the use of the SSIT to calculate Sensitivity of the CME solution and the Fisher Information Matrix.

Examples/example_FittingAndDesigningExperiments.m This example demonstrates the use of the SSIT and Fisher Information Matrix to design optimally informative experiments.

Examples/example_HybridModel.m This example demonstrates the use of the SSIT for creating hybrid models where some (upstream) species evolve determistically while other (downstream) species are stochastic.

Examples/example_ModelReductionTool.m This example demonstrates the use of the SSIT for applying linear transformations to reduce the size of the CME solution scheme (e.g., eigvenvalue decomposition, principle orthoginal decomposition, quasi-steady state assumptions, or coarse mesh solutions).

Examples/example_MultiModelTool.m This example demonstrates the use of the SSIT for fitting multiple different models with shared or related pameters, but fit to different data sets.

Or you can start creating and solving models as follows.

Example for generating an FSI model and fitting it to smFISH data for Dusp1 activation following glucocorticoid stimulation:

Define SSIT Model

Model = SSIT;

Model.species = {'OnGene';'rna'};

Model.initialCondition = [0; 0];

Model.propensityFunctions = {'kon * IGR * (2-OnGene)'; 'koff * OnGene'; 'kr * OnGene'; 'gr * rna'};

Model.stoichiometry = [1,-1,0,0; 0,0,1,-1];

Model.inputExpressions = {'IGR','a0 + a1 * exp(-r1 * t) * (1-exp(-r2 * t)) * (t>0)'};

Model.parameters = ({'koff',0.14; 'kon',0.14; 'kr',25; 'gr',0.01; 'a0',0.006; 'a1',0.4; 'r1',0.04; 'r2',0.1});

Model.fspOptions.initApproxSS = true; % Model is assumed to start at steady state at t=0;

Load and Fit smFISH Data

Model = Model.loadData('../ExampleData/DUSP1_Dex_100nM_Rep1_Rep2.csv',{'rna','RNA_nuc'});

Model.tSpan = unique([Model.initialTime,Model.dataSet.times]);

fitOptions = optimset('Display','iter','MaxIter',100);

[pars,likelihood] = Model.maximizeLikelihood([],fitOptions);

Update Model and Make Plots of Results

Model.parameters(:,2) = num2cell(pars);

Model.makeFitPlot

You should arrive at a fit of the model to the experimentally measured Dusp1 mRNA distributions looking something like this:

Acknowledgements

The SSIT tools in this repository make use of sparse tensors using the Tensor Toolbox for MATLAB (version 3.2.1) provided by Brett W. Bader, Tamara G. Kolda and others at www.tensortoolbox.org under Users of this software should cite the TTM creators at:

• B. W. Bader and T. G. Kolda, Efficient MATLAB Computations with Sparse and Factored Tensors, SIAM J. Scientific Computing, 30(1):205-231, 2007, http://dx.doi.org/10.1137/060676489.

The provided SSIT tools also make use a modified version of Expokit for the solution of time-invariant master equations (although these codes are not used for the current publication). Users of this software should cite the creators at:

• Sidje, R. B., Expokit Software Package for Computing Matrix Exponentials, ACM Trans. Math. Softw., 24:1, 1998.

How to Cite This Repository

The SSIT is free to use or copy with citation of the authors and the above referenced packages (Expokit from Sidje) and (TensorToolbox from Bader et al). If you use the SSIT for your research, we would appreciate it if you could cite us.

• H. D. Vo, J. Cook, E. Ron, B. Munsky, 2023, Stochastic System Identification Toolbox, v1.0.0, https://doi.org/10.5281/zenodo.10023437

To learn more about the history and use of the FSP, please explore the following papers.

For use of FSP, please read/cite one or more of the following:

• B. Munsky, M. Khammash, "The finite state projection algorithm for the solution of the chemical master equation," J. Chemical Physics, 124:4, 2006. https://doi.org/10.1063/1.2145882
• B Munsky, M Khammash, "The finite state projection approach for the analysis of stochastic noise in gene networks," IEEE Transactions on Automatic Control 53 (Special Issue), 201-214, 2008. https://ieeexplore.ieee.org/abstract/document/4439819
• Z Fox, G Neuert, B Munsky, "Finite state projection based bounds to compare chemical master equation models using single-cell data," The Journal of chemical physics, 145:7, 2016. https://pubs.aip.org/aip/jcp/article/145/7/074101/810475

For use of FSP tools with model reductions, please read/cite one or more of the following:

• S Peleš, B Munsky, M Khammash, "Reduction and solution of the chemical master equation using time scale separation and finite state projection," The Journal of chemical physics, 125:20, 2006. https://doi.org/10.1063/1.2397685
• JJ Tapia, JR Faeder, B Munsky, "Adaptive coarse-graining for transient and quasi-equilibrium analyses of stochastic gene regulation," 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 5361-5366, 2012. https://ieeexplore.ieee.org/abstract/document/6425828
• HD Vo, Z Fox, A Baetica, B Munsky, "Bayesian estimation for stochastic gene expression using multifidelity models," The Journal of Physical Chemistry B, 123:10, 2217-2234, 2019. https://pubs.acs.org/doi/full/10.1021/acs.jpcb.8b10946

For use of FSP for likelihood calculation and model parameter estimation, please read/cite one or more of the following:

• G. Neuert, B. Munsky, et al., "Systematic identification of signal-activated stochastic gene regulation", Science, 339:6119, 584-587, 2013. https://www.science.org/doi/full/10.1126/science.1231456
• B Munsky, Z Fox, G Neuert, "Integrating single-molecule experiments and discrete stochastic models to understand heterogeneous gene transcription dynamics," Methods 85, 12-21, 2015. https://www.sciencedirect.com/science/article/pii/S1046202315002510
• B Munsky, G Li, ZR Fox, DP Shepherd, G Neuert, "Distribution shapes govern the discovery of predictive models for gene regulation," Proceedings of the National Academy of Sciences, 115:29, 7533-7538, 2018. https://www.pnas.org/doi/abs/10.1073/pnas.1804060115
• D Kalb, HD Vo, S Adikari, E Hong-Geller, B Munsky, J Werner, Visualization and modeling of inhibition of IL-1β and TNF-α mRNA transcription at the single-cell level, Scientific Reports 11:1, 13692, 2021. https://www.nature.com/articles/s41598-021-92846-0
• C Lou, B Stanton, YJ Chen, B Munsky, CA Voigt, Ribozyme-based insulator parts buffer synthetic circuits from genetic context, Nature biotechnology 30:11, 1137-1142, 2012. https://www.nature.com/articles/nbt.2401
• DP Shepherd, N Li, et al, "Counting small RNA in pathogenic bacteria," Analytical Chemistry 85:10, 4938-4943, 2013. https://pubs.acs.org/doi/full/10.1021/ac303792p

For use of the FSP to calculate escape and first passage times, please read/cite the following:

• B Munsky, M Khammash, "Transient analysis of stochastic switches and trajectories with applications to gene regulatory networks," IET systems biology, 2:5, 323-333, 2008. https://digital-library.theiet.org/content/journals/10.1049/iet-syb_20070082

For use of FIM for experiment design, please read/cite one of the following:

• ZR Fox, B Munsky, "The finite state projection based Fisher information matrix approach to estimate information and optimize single-cell experiments," PLoS computational biology 15:1, e1006365, 2019. https://doi.org/10.1371/journal.pcbi.1006365
• ZR Fox, G Neuert, B Munsky, "Optimal design of single-cell experiments within temporally fluctuating environments," Complexity 2020, 1-15, 2020. https://doi.org/10.1155/2020/8536365
• HD Vo, LS Forero-Quintero, LU Aguilera, B Munsky, "Analysis and design of single-cell experiments to harvest fluctuation information while rejecting measurement noise," Frontiers in Cell and Developmental Biology 11, 1133994, 2023. https://doi.org/10.3389/fcell.2023.1133994

For more general insight on the use of quantitative models in biology, please read the q-bio Community written textbook:

• B Munsky, WS Hlavacek, LS Tsimring (Editors), "Quantitative biology: theory, computational methods, and models, MIT Press, 2018.