{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:42:03Z","timestamp":1777459323976,"version":"3.51.4"},"reference-count":44,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T00:00:00Z","timestamp":1615939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R15128162-01A1"],"award-info":[{"award-number":["R15128162-01A1"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Learning the underlying details of a gene network with feedback is critical in designing new synthetic circuits. Yet, quantitative characterization of these circuits remains limited. This is due to the fact that experiments can only measure partial information from which the details of the circuit must be inferred. One potentially useful avenue is to harness hidden information from single-cell stochastic gene expression time trajectories measured for long periods of time\u2014recorded at frequent intervals\u2014over multiple cells. This raises the feasibility vs. accuracy dilemma while deciding between different models of mining these stochastic trajectories. We demonstrate that inference based on the Maximum Caliber (MaxCal) principle is the method of choice by critically evaluating its computational efficiency and accuracy against two other typical modeling approaches: (i) a detailed model (DM) with explicit consideration of multiple molecules including protein-promoter interaction, and (ii) a coarse-grain model (CGM) using Hill type functions to model feedback. MaxCal provides a reasonably accurate model while being significantly more computationally efficient than DM and CGM. Furthermore, MaxCal requires minimal assumptions since it is a top-down approach and allows systematic model improvement by including constraints of higher order, in contrast to traditional bottom-up approaches that require more parameters or ad hoc assumptions. Thus, based on efficiency, accuracy, and ability to build minimal models, we propose MaxCal as a superior alternative to traditional approaches (DM, CGM) when inferring underlying details of gene circuits with feedback from limited data.<\/jats:p>","DOI":"10.3390\/e23030357","type":"journal-article","created":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T11:48:22Z","timestamp":1615981702000},"page":"357","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Critical Comparison of MaxCal and Other Stochastic Modeling Approaches in Analysis of Gene Networks"],"prefix":"10.3390","volume":"23","author":[{"given":"Taylor","family":"Firman","sequence":"first","affiliation":[{"name":"Molecular and Cellular Biophysics, University of Denver, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3120-2823","authenticated-orcid":false,"given":"Jonathan","family":"Huihui","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1470-5610","authenticated-orcid":false,"given":"Austin R.","family":"Clark","sequence":"additional","affiliation":[{"name":"Molecular and Cellular Biophysics, University of Denver, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kingshuk","family":"Ghosh","sequence":"additional","affiliation":[{"name":"Molecular and Cellular Biophysics, University of Denver, Denver, CO 80208, USA"},{"name":"Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1038\/35002131","article-title":"Construction of a Genetic Toggle Switch in Escherichia coli","volume":"403","author":"Gardner","year":"2000","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1038\/35002125","article-title":"A Synthetic Oscillatory Network of Transcriptional Regulators","volume":"403","author":"Elowitz","year":"2000","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1038\/nrg2102","article-title":"Network Motifs: Theory and Experimental Approaches","volume":"8","author":"Alon","year":"2007","journal-title":"Nat. Rev. Genet."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1126\/science.1156951","article-title":"Robust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback Loops","volume":"321","author":"Tsai","year":"2008","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Nevozhay, D., Adams, R., Itallie, E.V., Bennett, M., and Bal\u00e1zsi, G. (2012). Mapping the Environmental Fitness Landscape of a Synthetic Gene Circuit. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002480"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5123","DOI":"10.1073\/pnas.0809901106","article-title":"Negative auto regulation linearizes the dose\u2013response and suppresses the heterogeneity of gene expression","volume":"106","author":"Nevozhay","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lyons, S., Xu, W., Medford, J., and Prasad, A. (2014). Loads Bias Genetic and Signaling Switches in Synthetic and Natural Systems. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003533"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1039\/C5IB00252D","article-title":"Build to Understand: Synthetic Approaches to Biology","volume":"8","author":"Wang","year":"2016","journal-title":"Integr. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1038\/nrg2697","article-title":"Synthetic Biology: Understanding Biological Design from Synthetic Circuits","volume":"10","author":"Mukherji","year":"2009","journal-title":"Nat. Rev. Genet."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"244","DOI":"10.3184\/003685015X14368807556441","article-title":"Applications of Synthetic Gene Networks","volume":"98","author":"Wu","year":"2015","journal-title":"Sci. Prog."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"11304","DOI":"10.1038\/ncomms11304","article-title":"Real-time quantification of protein expression at the single-cell level via dynamic protein synthesis translocation reporters","volume":"7","author":"Aymoz","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3544","DOI":"10.1038\/s41467-019-11531-z","article-title":"Ultra-sensitive digital quantification of proteins and mRNA in single cells","volume":"10","author":"Lin","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"107499","DOI":"10.1016\/j.celrep.2020.03.063","article-title":"A Targeted Multi-omic Analysis Approach Measures Protein Expression and Low-Abundance Transcripts on the Single-Cell Level","volume":"31","author":"Mair","year":"2020","journal-title":"Cell Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1038\/msb.2009.75","article-title":"Listening to the Noise: Random Fluctuations Reveal Gene Network Parameters","volume":"5","author":"Munsky","year":"2009","journal-title":"Mol. Syst. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lillacci, G., and Khammash, M. (2010). Parameter Estimation and Model Selection in Computational Biology. PLoS Comput. Biol., 6.","DOI":"10.1371\/journal.pcbi.1000696"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8340","DOI":"10.1073\/pnas.1200161109","article-title":"Moment-Based Inference Predicts Bimodality in Transient Gene Expression","volume":"109","author":"Zechner","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1002\/rnc.2794","article-title":"A Distribution-Matching Method for Parameter Estimation and Model Selection in Computational Biology","volume":"22","author":"Lillacci","year":"2012","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"20130588","DOI":"10.1098\/rsif.2013.0588","article-title":"Designing Experiments to Understand the Variability in Biochemical Reaction Networks","volume":"10","author":"Ruess","year":"2013","journal-title":"J. R. Soc. Interface"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2311","DOI":"10.1093\/bioinformatics\/btt380","article-title":"The Signal within the Noise: Efficient Inference of Stochastic Gene Regulation Models Using Fluorescence Histograms and Stochastic Simulations","volume":"29","author":"Lillacci","year":"2013","journal-title":"Bioinformatics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3116","DOI":"10.1016\/S0006-3495(01)75949-8","article-title":"Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations","volume":"81","author":"Kepler","year":"2001","journal-title":"Biophys. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"188101","DOI":"10.1103\/PhysRevLett.96.188101","article-title":"Genetic Toggle Switch without Cooperative Binding","volume":"96","author":"Lipshtat","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","article-title":"Exact Stochastic Simulation of Coupled Chemical Reactions","volume":"81","author":"Gillespie","year":"1977","journal-title":"J. Phys. Chem."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1146\/annurev.physchem.58.032806.104637","article-title":"Stochastic simulation of chemical kinetic","volume":"58","author":"Gillespie","year":"2007","journal-title":"Annu. Rev. Phys. Chem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1089\/10665270252833208","article-title":"Modeling and simulation of genetic regulatory systems: A literature review","volume":"9","author":"Jong","year":"2004","journal-title":"J. Comput. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"044104","DOI":"10.1063\/1.2145882","article-title":"The Finite State Projection Algorithm for the Solution of the Chemical Master Equation","volume":"124","author":"Munsky","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1016\/j.jcp.2007.05.016","article-title":"A multiple time interval finite state projection algorithm for the solution to the chemical master equation","volume":"226","author":"Munsky","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Phillips, R., Kondev, J., Theriot, J., and Garcia, H.G. (2013). Physical Biology of the Cell, Garland Science.","DOI":"10.1201\/9781134111589"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1006\/jtbi.2000.1068","article-title":"How to make a biological switch","volume":"203","author":"Cherry","year":"2000","journal-title":"J. Theor. Biol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.cplett.2006.05.024","article-title":"Transient stochastic bistable kinetics of gene transcription during the cellular growth","volume":"424","author":"Zhdanov","year":"2006","journal-title":"Chem. Phys. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3776","DOI":"10.1016\/j.febslet.2008.10.005","article-title":"Robustness analysis of cellular memory in an autoactivating positive feedback system","volume":"582","author":"Cheng","year":"2008","journal-title":"FEBS Lett."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Frigola, D., Casanellas, L., Sancho, J., and Ibanes, M. (2012). Asymmetric Stochastic Switching Driven by Intrinsic Molecular Noise. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0031407"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1146\/annurev-physchem-071119-040206","article-title":"The Maximum Caliber Variational Principle for Nonequilibria","volume":"71","author":"Ghosh","year":"2020","journal-title":"Annu. Rev. Phys. Chem."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6202","DOI":"10.1021\/jp111112s","article-title":"Modeling Stochastic Dynamics in Biochemical Systems with Feedback Using Maximum Caliber","volume":"115","author":"Ghosh","year":"2011","journal-title":"J. Phys. Chem. B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1016\/j.bpj.2017.08.057","article-title":"Building Predictive Models of Genetic Circuits Using the Principle of Maximum Caliber","volume":"113","author":"Firman","year":"2017","journal-title":"Biophys. J."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Firman, T., Wedekind, S., McMorrow, T., and Ghosh, K. (2018). Maximum Caliber Can Characterize Genetic Switches with Multiple Hidden Species. J. Phys. Chem. B.","DOI":"10.1021\/acs.jpcb.7b12251"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1021\/acs.jpcb.8b07465","article-title":"Maximum Caliber can build and infer models of oscillation in three-gene feedback network","volume":"123","author":"Firman","year":"2019","journal-title":"J. Phys. Chem. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1103\/RevModPhys.85.1115","article-title":"Principle of Maximum Entropy and Maximum Caliber in Statistical Physics","volume":"85","author":"Ghosh","year":"2013","journal-title":"Rev. Mod. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"010901","DOI":"10.1063\/1.5012990","article-title":"Perspective: Maximum Caliber is a General Variational Principle for Dynamical Systems","volume":"148","author":"Dixit","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1006\/jtbi.1995.0014","article-title":"Model Genetic Circuits Encoding Autoregulatory Transcription Factors","volume":"172","author":"Keller","year":"1995","journal-title":"J. Theor. Biol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"C531","DOI":"10.1152\/ajpcell.1998.274.2.C531","article-title":"Frequency, Selectivity, Multistability, and Oscillations Emerge from Models of Genetic Regulatory Systems","volume":"274","author":"Smolen","year":"1998","journal-title":"Am. J. Physiol."},{"key":"ref_41","first-page":"2528","article-title":"Positive Feedback in Eukaryotic Gene Networks: Cell Differentiation by Graded to Binary Response Conversion","volume":"15","author":"Becksei","year":"2001","journal-title":"EMBO J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/S0955-0674(03)00017-6","article-title":"Sniffers, Buzzers, Toggles and Blinkers: Dynamics of Regulatory and Signaling Pathways in the Cell","volume":"15","author":"Tyson","year":"2003","journal-title":"Curr. Opin. Cell Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.bpj.2009.09.055","article-title":"Stochastic Bistability and Bifurcation in a Mesoscopic Signaling System with Autocatalytic Kinase","volume":"98","author":"Bishop","year":"2010","journal-title":"Biophys. J."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Faucon, P., Pardee, K., Kumar, R., Li, H., Loh, Y.-H., and Wang, X. (2014). Gene Networks of Fully Connected Triads with Complete Auto-Activation Enable Multistability and Stepwise Stochastic Transitions. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0102873"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/357\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:37:07Z","timestamp":1760161027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/357"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,17]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030357"],"URL":"https:\/\/doi.org\/10.3390\/e23030357","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,17]]}}}