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Feng J, Zhang X, Tian T. Mathematical Modeling and Inference of Epidermal Growth Factor-Induced Mitogen-Activated Protein Kinase Cell Signaling Pathways. Int J Mol Sci 2024; 25:10204. [PMID: 39337687 PMCID: PMC11432143 DOI: 10.3390/ijms251810204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2024] [Revised: 09/18/2024] [Accepted: 09/21/2024] [Indexed: 09/30/2024] Open
Abstract
The mitogen-activated protein kinase (MAPK) pathway is an important intracellular signaling cascade that plays a key role in various cellular processes. Understanding the regulatory mechanisms of this pathway is essential for developing effective interventions and targeted therapies for related diseases. Recent advances in single-cell proteomic technologies have provided unprecedented opportunities to investigate the heterogeneity and noise within complex, multi-signaling networks across diverse cells and cell types. Mathematical modeling has become a powerful interdisciplinary tool that bridges mathematics and experimental biology, providing valuable insights into these intricate cellular processes. In addition, statistical methods have been developed to infer pathway topologies and estimate unknown parameters within dynamic models. This review presents a comprehensive analysis of how mathematical modeling of the MAPK pathway deepens our understanding of its regulatory mechanisms, enhances the prediction of system behavior, and informs experimental research, with a particular focus on recent advances in modeling and inference using single-cell proteomic data.
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Affiliation(s)
- Jinping Feng
- School of Mathematics and Statistics, Henan University, Kaifeng 475001, China
| | - Xinan Zhang
- School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
| | - Tianhai Tian
- School of Mathematics, Monash University, Melbourne 3800, Australia
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2
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Turpin B, Bijman EY, Kaltenbach HM, Stelling J. Efficient design of synthetic gene circuits under cell-to-cell variability. BMC Bioinformatics 2023; 24:460. [PMID: 38062373 PMCID: PMC10701960 DOI: 10.1186/s12859-023-05538-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2022] [Accepted: 10/18/2023] [Indexed: 12/18/2023] Open
Abstract
BACKGROUND Synthetic biologists use and combine diverse biological parts to build systems such as genetic circuits that perform desirable functions in, for example, biomedical or industrial applications. Computer-aided design methods have been developed to help choose appropriate network structures and biological parts for a given design objective. However, they almost always model the behavior of the network in an average cell, despite pervasive cell-to-cell variability. RESULTS Here, we present a computational framework and an efficient algorithm to guide the design of synthetic biological circuits while accounting for cell-to-cell variability explicitly. Our design method integrates a Non-linear Mixed-Effects (NLME) framework into a Markov Chain Monte-Carlo (MCMC) algorithm for design based on ordinary differential equation (ODE) models. The analysis of a recently developed transcriptional controller demonstrates first insights into design guidelines when trying to achieve reliable performance under cell-to-cell variability. CONCLUSION We anticipate that our method not only facilitates the rational design of synthetic networks under cell-to-cell variability, but also enables novel applications by supporting design objectives that specify the desired behavior of cell populations.
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Affiliation(s)
- Baptiste Turpin
- Department of Biosystems Science and Engineering (D-BSSE) and SIB Swiss Institute of Bioinformatics, ETH Zurich, 4058, Basel, Switzerland
| | - Eline Y Bijman
- Department of Biosystems Science and Engineering (D-BSSE) and SIB Swiss Institute of Bioinformatics, ETH Zurich, 4058, Basel, Switzerland
| | - Hans-Michael Kaltenbach
- Department of Biosystems Science and Engineering (D-BSSE) and SIB Swiss Institute of Bioinformatics, ETH Zurich, 4058, Basel, Switzerland
| | - Jörg Stelling
- Department of Biosystems Science and Engineering (D-BSSE) and SIB Swiss Institute of Bioinformatics, ETH Zurich, 4058, Basel, Switzerland.
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Sarma U, Ripka L, Anyaegbunam UA, Legewie S. Modeling Cellular Signaling Variability Based on Single-Cell Data: The TGFβ-SMAD Signaling Pathway. Methods Mol Biol 2023; 2634:215-251. [PMID: 37074581 DOI: 10.1007/978-1-0716-3008-2_10] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/20/2023]
Abstract
Nongenetic heterogeneity is key to cellular decisions, as even genetically identical cells respond in very different ways to the same external stimulus, e.g., during cell differentiation or therapeutic treatment of disease. Strong heterogeneity is typically already observed at the level of signaling pathways that are the first sensors of external inputs and transmit information to the nucleus where decisions are made. Since heterogeneity arises from random fluctuations of cellular components, mathematical models are required to fully describe the phenomenon and to understand the dynamics of heterogeneous cell populations. Here, we review the experimental and theoretical literature on cellular signaling heterogeneity, with special focus on the TGFβ/SMAD signaling pathway.
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Affiliation(s)
- Uddipan Sarma
- Institute of Molecular Biology (IMB), Mainz, Germany
| | - Lorenz Ripka
- Institute of Molecular Biology (IMB), Mainz, Germany
- Department of Systems Biology, Institute for Biomedical Genetics, University of Stuttgart, Stuttgart, Germany
| | - Uchenna Alex Anyaegbunam
- Institute of Molecular Biology (IMB), Mainz, Germany
- Department of Systems Biology, Institute for Biomedical Genetics, University of Stuttgart, Stuttgart, Germany
| | - Stefan Legewie
- Institute of Molecular Biology (IMB), Mainz, Germany.
- Department of Systems Biology, Institute for Biomedical Genetics, University of Stuttgart, Stuttgart, Germany.
- Stuttgart Research Center for Systems Biology, University of Stuttgart, Stuttgart, Germany.
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4
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Browning AP, Drovandi C, Turner IW, Jenner AL, Simpson MJ. Efficient inference and identifiability analysis for differential equation models with random parameters. PLoS Comput Biol 2022; 18:e1010734. [PMID: 36441811 PMCID: PMC9731444 DOI: 10.1371/journal.pcbi.1010734] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Revised: 12/08/2022] [Accepted: 11/14/2022] [Indexed: 11/29/2022] Open
Abstract
Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.
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Affiliation(s)
- Alexander P. Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Christopher Drovandi
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
| | - Ian W. Turner
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Adrianne L. Jenner
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
| | - Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
- * E-mail:
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5
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Qureshi MH, Ozlu N, Bayraktar H. Adaptive tracking algorithm for trajectory analysis of cells and layer-by-layer assessment of motility dynamics. Comput Biol Med 2022; 150:106193. [PMID: 37859286 DOI: 10.1016/j.compbiomed.2022.106193] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Revised: 09/26/2022] [Accepted: 10/08/2022] [Indexed: 11/03/2022]
Abstract
Tracking biological objects such as cells or subcellular components imaged with time-lapse microscopy enables us to understand the molecular principles about the dynamics of cell behaviors. However, automatic object detection, segmentation and extracting trajectories remain as a rate-limiting step due to intrinsic challenges of video processing. This paper presents an adaptive tracking algorithm (Adtari) that automatically finds the optimum search radius and cell linkages to determine trajectories in consecutive frames. A critical assumption in most tracking studies is that displacement remains unchanged throughout the movie and cells in a few frames are usually analyzed to determine its magnitude. Tracking errors and inaccurate association of cells may occur if the user does not correctly evaluate the value or prior knowledge is not present on cell movement. The key novelty of our method is that minimum intercellular distance and maximum displacement of cells between frames are dynamically computed and used to determine the threshold distance. Since the space between cells is highly variable in a given frame, our software recursively alters the magnitude to determine all plausible matches in the trajectory analysis. Our method therefore eliminates a major preprocessing step where a constant distance was used to determine the neighbor cells in tracking methods. Cells having multiple overlaps and splitting events were further evaluated by using the shape attributes including perimeter, area, ellipticity and distance. The features were applied to determine the closest matches by minimizing the difference in their magnitudes. Finally, reporting section of our software were used to generate instant maps by overlaying cell features and trajectories. Adtari was validated by using videos with variable signal-to-noise, contrast ratio and cell density. We compared the adaptive tracking with constant distance and other methods to evaluate performance and its efficiency. Our algorithm yields reduced mismatch ratio, increased ratio of whole cell track, higher frame tracking efficiency and allows layer-by-layer assessment of motility to characterize single-cells. Adaptive tracking provides a reliable, accurate, time efficient and user-friendly open source software that is well suited for analysis of 2D fluorescence microscopy video datasets.
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Affiliation(s)
- Mohammad Haroon Qureshi
- Department of Molecular Biology and Genetics, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey; Center for Translational Research, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey
| | - Nurhan Ozlu
- Department of Molecular Biology and Genetics, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey
| | - Halil Bayraktar
- Department of Molecular Biology and Genetics, Istanbul Technical University, Maslak, Sariyer, 34467, Istanbul, Turkey.
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Kolbe N, Hexemer L, Bammert LM, Loewer A, Lukáčová-Medvid’ová M, Legewie S. Data-based stochastic modeling reveals sources of activity bursts in single-cell TGF-β signaling. PLoS Comput Biol 2022; 18:e1010266. [PMID: 35759468 PMCID: PMC9269928 DOI: 10.1371/journal.pcbi.1010266] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2021] [Revised: 07/08/2022] [Accepted: 05/30/2022] [Indexed: 11/30/2022] Open
Abstract
Cells sense their surrounding by employing intracellular signaling pathways that transmit hormonal signals from the cell membrane to the nucleus. TGF-β/SMAD signaling encodes various cell fates, controls tissue homeostasis and is deregulated in diseases such as cancer. The pathway shows strong heterogeneity at the single-cell level, but quantitative insights into mechanisms underlying fluctuations at various time scales are still missing, partly due to inefficiency in the calibration of stochastic models that mechanistically describe signaling processes. In this work we analyze single-cell TGF-β/SMAD signaling and show that it exhibits temporal stochastic bursts which are dose-dependent and whose number and magnitude correlate with cell migration. We propose a stochastic modeling approach to mechanistically describe these pathway fluctuations with high computational efficiency. Employing high-order numerical integration and fitting to burst statistics we enable efficient quantitative parameter estimation and discriminate models that assume noise in different reactions at the receptor level. This modeling approach suggests that stochasticity in the internalization of TGF-β receptors into endosomes plays a key role in the observed temporal bursting. Further, the model predicts the single-cell dynamics of TGF-β/SMAD signaling in untested conditions, e.g., successfully reflects memory effects of signaling noise and cellular sensitivity towards repeated stimulation. Taken together, our computational framework based on burst analysis, noise modeling and path computation scheme is a suitable tool for the data-based modeling of complex signaling pathways, capable of identifying the source of temporal noise.
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Affiliation(s)
- Niklas Kolbe
- Institute of Geometry and Practical Mathematics, RWTH Aachen University, Aachen, Germany
| | - Lorenz Hexemer
- Institute of Molecular Biology (IMB), Mainz, Germany
- Department of Systems Biology, Institute for Biomedical Genetics (IBMG), University of Stuttgart, Stuttgart, Germany
| | | | - Alexander Loewer
- Systems Biology of the Stress Response, Department of Biology, Technical University of Darmstadt, Darmstadt, Germany
| | | | - Stefan Legewie
- Department of Systems Biology, Institute for Biomedical Genetics (IBMG), University of Stuttgart, Stuttgart, Germany
- Stuttgart Research Center for Systems Biology (SRCSB), University of Stuttgart, Stuttgart, Germany
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Lambert B, Gavaghan DJ, Tavener SJ. A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data. J Theor Biol 2020; 511:110541. [PMID: 33271182 DOI: 10.1016/j.jtbi.2020.110541] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Revised: 07/16/2020] [Accepted: 11/10/2020] [Indexed: 11/26/2022]
Abstract
Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of "snapshots" representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named "Contour Monte Carlo" (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.
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Affiliation(s)
- Ben Lambert
- Department of Zoology, University of Oxford, Oxford, Oxfordshire, UK; MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London W2 1PG, UK.
| | - David J Gavaghan
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Simon J Tavener
- Department of Mathematics, Colorado State University, Fort Collins, Colorado, USA
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8
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Persson S, Welkenhuysen N, Shashkova S, Cvijovic M. Fine-Tuning of Energy Levels Regulates SUC2 via a SNF1-Dependent Feedback Loop. Front Physiol 2020; 11:954. [PMID: 32922308 PMCID: PMC7456839 DOI: 10.3389/fphys.2020.00954] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2020] [Accepted: 07/15/2020] [Indexed: 11/22/2022] Open
Abstract
Nutrient sensing pathways are playing an important role in cellular response to different energy levels. In budding yeast, Saccharomyces cerevisiae, the sucrose non-fermenting protein kinase complex SNF1 is a master regulator of energy homeostasis. It is affected by multiple inputs, among which energy levels is the most prominent. Cells which are exposed to a switch in carbon source availability display a change in the gene expression machinery. It has been shown that the magnitude of the change varies from cell to cell. In a glucose rich environment Snf1/Mig1 pathway represses the expression of its downstream target, such as SUC2. However, upon glucose depletion SNF1 is activated which leads to an increase in SUC2 expression. Our single cell experiments indicate that upon starvation, gene expression pattern of SUC2 shows rapid increase followed by a decrease to initial state with high cell-to-cell variability. The mechanism behind this behavior is currently unknown. In this work we study the long-term behavior of the Snf1/Mig1 pathway upon glucose starvation with a microfluidics and non-linear mixed effect modeling approach. We show a negative feedback mechanism, involving Snf1 and Reg1, which reduces SUC2 expression after the initial strong activation. Snf1 kinase activity plays a key role in this feedback mechanism. Our systems biology approach proposes a negative feedback mechanism that works through the SNF1 complex and is controlled by energy levels. We further show that Reg1 likely is involved in the negative feedback mechanism.
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Affiliation(s)
- Sebastian Persson
- Department of Mathematical Sciences, University of Gothenburg, Gothenburg, Sweden.,Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden
| | - Niek Welkenhuysen
- Department of Mathematical Sciences, University of Gothenburg, Gothenburg, Sweden.,Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden
| | - Sviatlana Shashkova
- Department of Microbiology and Immunology, Institute of Biomedicine, Sahlgrenska Academy, University of Gothenburg, Gothenburg, Sweden
| | - Marija Cvijovic
- Department of Mathematical Sciences, University of Gothenburg, Gothenburg, Sweden.,Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden
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Marguet A, Lavielle M, Cinquemani E. Inheritance and variability of kinetic gene expression parameters in microbial cells: modeling and inference from lineage tree data. Bioinformatics 2020; 35:i586-i595. [PMID: 31510690 PMCID: PMC6612834 DOI: 10.1093/bioinformatics/btz378] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
Motivation Modern experimental technologies enable monitoring of gene expression dynamics in individual cells and quantification of its variability in isogenic microbial populations. Among the sources of this variability is the randomness that affects inheritance of gene expression factors at cell division. Known parental relationships among individually observed cells provide invaluable information for the characterization of this extrinsic source of gene expression noise. Despite this fact, most existing methods to infer stochastic gene expression models from single-cell data dedicate little attention to the reconstruction of mother–daughter inheritance dynamics. Results Starting from a transcription and translation model of gene expression, we propose a stochastic model for the evolution of gene expression dynamics in a population of dividing cells. Based on this model, we develop a method for the direct quantification of inheritance and variability of kinetic gene expression parameters from single-cell gene expression and lineage data. We demonstrate that our approach provides unbiased estimates of mother–daughter inheritance parameters, whereas indirect approaches using lineage information only in the post-processing of individual-cell parameters underestimate inheritance. Finally, we show on yeast osmotic shock response data that daughter cell parameters are largely determined by the mother, thus confirming the relevance of our method for the correct assessment of the onset of gene expression variability and the study of the transmission of regulatory factors. Availability and implementation Software code is available at https://github.com/almarguet/IdentificationWithARME. Lineage tree data is available upon request. Supplementary information Supplementary material is available at Bioinformatics online.
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Affiliation(s)
| | - Marc Lavielle
- Inria Saclay & Ecole Polytechnique, Palaiseau, France
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Barbier I, Perez‐Carrasco R, Schaerli Y. Controlling spatiotemporal pattern formation in a concentration gradient with a synthetic toggle switch. Mol Syst Biol 2020; 16:e9361. [PMID: 32529808 PMCID: PMC7290156 DOI: 10.15252/msb.20199361] [Citation(s) in RCA: 25] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2019] [Revised: 04/29/2020] [Accepted: 05/08/2020] [Indexed: 11/20/2022] Open
Abstract
The formation of spatiotemporal patterns of gene expression is frequently guided by gradients of diffusible signaling molecules. The toggle switch subnetwork, composed of two cross-repressing transcription factors, is a common component of gene regulatory networks in charge of patterning, converting the continuous information provided by the gradient into discrete abutting stripes of gene expression. We present a synthetic biology framework to understand and characterize the spatiotemporal patterning properties of the toggle switch. To this end, we built a synthetic toggle switch controllable by diffusible molecules in Escherichia coli. We analyzed the patterning capabilities of the circuit by combining quantitative measurements with a mathematical reconstruction of the underlying dynamical system. The toggle switch can produce robust patterns with sharp boundaries, governed by bistability and hysteresis. We further demonstrate how the hysteresis, position, timing, and precision of the boundary can be controlled, highlighting the dynamical flexibility of the circuit.
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Affiliation(s)
- Içvara Barbier
- Department of Fundamental MicrobiologyUniversity of LausanneLausanneSwitzerland
| | - Rubén Perez‐Carrasco
- Department of Life SciencesImperial College LondonSouth Kensington CampusLondonUK
- Department of MathematicsUniversity College LondonLondonUK
| | - Yolanda Schaerli
- Department of Fundamental MicrobiologyUniversity of LausanneLausanneSwitzerland
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