1
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Heltberg MS, Jiang Y, Fan Y, Zhang Z, Nordentoft MS, Lin W, Qian L, Ouyang Q, Jensen MH, Wei P. Coupled oscillator cooperativity as a control mechanism in chronobiology. Cell Syst 2023; 14:382-391.e5. [PMID: 37201507 DOI: 10.1016/j.cels.2023.04.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Revised: 12/16/2022] [Accepted: 04/04/2023] [Indexed: 05/20/2023]
Abstract
Control of dynamical processes is vital for maintaining correct cell regulation and cell-fate decisions. Numerous regulatory networks show oscillatory behavior; however, our knowledge of how one oscillator behaves when stimulated by two or more external oscillatory signals is still missing. We explore this problem by constructing a synthetic oscillatory system in yeast and stimulate it with two external oscillatory signals. Letting model verification and prediction operate in a tight interplay with experimental observations, we find that stimulation with two external signals expands the plateau of entrainment and reduces the fluctuations of oscillations. Furthermore, by adjusting the phase differences of external signals, one can control the amplitude of oscillations, which is understood through the signal delay of the unperturbed oscillatory network. With this we reveal a direct amplitude dependency of downstream gene transcription. Taken together, these results suggest a new path to control oscillatory systems by coupled oscillator cooperativity.
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Affiliation(s)
- Mathias S Heltberg
- Center for Cell and Gene Circuit Design, CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China; Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Yuanxu Jiang
- Center for Cell and Gene Circuit Design, CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China; Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Yingying Fan
- Center for Cell and Gene Circuit Design, CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China; Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Zhibo Zhang
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | | | - Wei Lin
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Long Qian
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Qi Ouyang
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Mogens H Jensen
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark.
| | - Ping Wei
- Center for Cell and Gene Circuit Design, CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China; Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China; Faculty of Synthetic Biology, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China.
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2
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Öcal K, Sanguinetti G, Grima R. Model reduction for the Chemical Master Equation: An information-theoretic approach. J Chem Phys 2023; 158:114113. [PMID: 36948813 DOI: 10.1063/5.0131445] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/24/2023] Open
Abstract
The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, Linear Mapping Approximation, and state-space lumping. Despite the success of these techniques, they appear to be rather disparate, and at present, no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper, we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimizing a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimization approaches. In addition, we derive general expressions for propensities of a reduced system that generalize those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system, and a genetic oscillator.
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Affiliation(s)
- Kaan Öcal
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
| | - Guido Sanguinetti
- Scuola Internazionale Superiore di Studi Avanzati, 34136 Trieste, Italy
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, United Kingdom
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3
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Heltberg MS, Lucchetti A, Hsieh FS, Minh Nguyen DP, Chen SH, Jensen MH. Enhanced DNA repair through droplet formation and p53 oscillations. Cell 2022; 185:4394-4408.e10. [PMID: 36368307 DOI: 10.1016/j.cell.2022.10.004] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2022] [Revised: 08/23/2022] [Accepted: 10/05/2022] [Indexed: 11/11/2022]
Abstract
Living organisms are constantly exposed to DNA damage, and optimal repair is therefore crucial. A characteristic hallmark of the response is the formation of sub-compartments around the site of damage, known as foci. Following multiple DNA breaks, the transcription factor p53 exhibits oscillations in its nuclear concentration, but how this dynamics can affect the repair remains unknown. Here, we formulate a theory for foci formation through droplet condensation and discover how oscillations in p53, with its specific periodicity and amplitude, optimize the repair process by preventing Ostwald ripening and distributing protein material in space and time. Based on the theory predictions, we reveal experimentally that the oscillatory dynamics of p53 does enhance the repair efficiency. These results connect the dynamical signaling of p53 with the microscopic repair process and create a new paradigm for the interplay of complex dynamics and phase transitions in biology.
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Affiliation(s)
- Mathias S Heltberg
- Niels Bohr Institute, University of Copenhagen, Copenhagen, 2100, Denmark.
| | | | - Feng-Shu Hsieh
- Lab for Cell Dynamics, Institute of Molecular Biology, Academia Sinica, Taipei, 115, Taiwan
| | - Duy Pham Minh Nguyen
- Lab for Cell Dynamics, Institute of Molecular Biology, Academia Sinica, Taipei, 115, Taiwan
| | - Sheng-Hong Chen
- Lab for Cell Dynamics, Institute of Molecular Biology, Academia Sinica, Taipei, 115, Taiwan; National Center for Theoretical Sciences, Physics Division, Complex Systems, Taipei, 10617, Taiwan
| | - Mogens H Jensen
- Niels Bohr Institute, University of Copenhagen, Copenhagen, 2100, Denmark.
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4
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Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation à la Fenichel. J Math Biol 2022; 85:3. [PMID: 35776210 DOI: 10.1007/s00285-022-01768-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2021] [Revised: 04/17/2022] [Accepted: 06/02/2022] [Indexed: 10/17/2022]
Abstract
The linear noise approximation models the random fluctuations from the mean-field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order [Formula: see text], where [Formula: see text] is the size of the chemical system (usually the volume). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the slow scale linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis -Menten reaction mechanism.
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5
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Nordick B, Yu PY, Liao G, Hong T. Nonmodular oscillator and switch based on RNA decay drive regeneration of multimodal gene expression. Nucleic Acids Res 2022; 50:3693-3708. [PMID: 35380686 PMCID: PMC9023291 DOI: 10.1093/nar/gkac217] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Revised: 03/13/2022] [Accepted: 03/21/2022] [Indexed: 12/15/2022] Open
Abstract
Periodic gene expression dynamics are key to cell and organism physiology. Studies of oscillatory expression have focused on networks with intuitive regulatory negative feedback loops, leaving unknown whether other common biochemical reactions can produce oscillations. Oscillation and noise have been proposed to support mammalian progenitor cells’ capacity to restore heterogenous, multimodal expression from extreme subpopulations, but underlying networks and specific roles of noise remained elusive. We use mass-action-based models to show that regulated RNA degradation involving as few as two RNA species—applicable to nearly half of human protein-coding genes—can generate sustained oscillations without explicit feedback. Diverging oscillation periods synergize with noise to robustly restore cell populations’ bimodal expression on timescales of days. The global bifurcation organizing this divergence relies on an oscillator and bistable switch which cannot be decomposed into two structural modules. Our work reveals surprisingly rich dynamics of post-transcriptional reactions and a potentially widespread mechanism underlying development, tissue regeneration, and cancer cell heterogeneity.
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Affiliation(s)
- Benjamin Nordick
- School of Genome Science and Technology, The University of Tennessee, Knoxville, Tennessee 37916, USA
| | - Polly Y Yu
- NSF-Simons Center for Mathematical and Statistical Analysis of Biology, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Guangyuan Liao
- Department of Biochemistry & Cellular and Molecular Biology, The University of Tennessee, Knoxville, Tennessee 37916, USA
| | - Tian Hong
- Department of Biochemistry & Cellular and Molecular Biology, The University of Tennessee, Knoxville, Tennessee 37916, USA.,National Institute for Mathematical and Biological Synthesis, Knoxville, Tennessee 37916, USA
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6
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Eilertsen J, Schnell S. On the Validity of the Stochastic Quasi-Steady-State Approximation in Open Enzyme Catalyzed Reactions: Timescale Separation or Singular Perturbation? Bull Math Biol 2021; 84:7. [PMID: 34825985 PMCID: PMC8768927 DOI: 10.1007/s11538-021-00966-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2021] [Accepted: 10/26/2021] [Indexed: 10/19/2022]
Abstract
The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction in the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis-Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.
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Affiliation(s)
- Justin Eilertsen
- Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA
- Mathematical Reviews, American Mathematical Society, 416 4th Street, Ann Arbor, MI, 48103, USA
| | - Santiago Schnell
- Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.
- Department of Computational Medicine & Bioinformatics, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.
- Department of Biological Sciences, and Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA.
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7
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Song YM, Hong H, Kim JK. Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities. PLoS Comput Biol 2021; 17:e1008952. [PMID: 34662330 PMCID: PMC8562860 DOI: 10.1371/journal.pcbi.1008952] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Revised: 11/02/2021] [Accepted: 10/04/2021] [Indexed: 11/19/2022] Open
Abstract
Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.
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Affiliation(s)
- Yun Min Song
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
| | - Hyukpyo Hong
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
| | - Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
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8
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Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells. PLoS Comput Biol 2021; 17:e1008803. [PMID: 34260581 PMCID: PMC8330952 DOI: 10.1371/journal.pcbi.1008803] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2021] [Revised: 08/03/2021] [Accepted: 06/08/2021] [Indexed: 01/21/2023] Open
Abstract
During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration. Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.
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9
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Puchhammer F, Ben Abdellah A, L'Ecuyer P. Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks. Bull Math Biol 2021; 83:91. [PMID: 34236503 DOI: 10.1007/s11538-021-00920-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2020] [Accepted: 06/09/2021] [Indexed: 10/20/2022]
Abstract
We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with [Formula: see text]-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time T by the sample average over n sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.
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Affiliation(s)
- Florian Puchhammer
- Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country, Spain.
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada.
| | - Amal Ben Abdellah
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
| | - Pierre L'Ecuyer
- DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
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10
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Lunz D. On rapid oscillations driving biological processes at disparate timescales. Phys Biol 2021; 18:036002. [PMID: 33418553 DOI: 10.1088/1478-3975/abd9db] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
We consider a generic biological process described by a dynamical system, subject to an input signal with a high-frequency periodic component. The rapid oscillations of the input signal induce inherently multiscale dynamics, motivating order-reduction techniques. It is intuitive that the system behaviour is well approximated by its response to the averaged input signal. However, changes to the high-frequency component that preserve the average signal are beyond the reach of such intuitive reasoning. In this study, we explore system response under the influence of such an input signal by exploiting the timescale separation between high-frequency input variations and system response time. Employing the asymptotic method of multiple scales, we establish that, in some circumstances, the intuitive approach is simply the leading-order asymptotic contribution. We focus on higher-order corrections that capture the response to the details of the high-frequency component beyond its average. This approach achieves a reduction in system complexity while providing valuable insight into the structure of the response to the oscillations. We develop the general theory for nonlinear systems, while highlighting the important case of systems affine in the state and the input signal, presenting examples of both discrete and continuum state spaces. Importantly, this class of systems encompasses biochemical reaction networks described by the chemical master equation and its continuum approximations. Finally, we apply the framework to a nonlinear system describing mRNA translation and protein expression previously studied in the literature. The analysis shines new light on several aspects of the system quantification and both extends and simplifies results previously obtained.
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Affiliation(s)
- Davin Lunz
- INRIA Saclay-Île de France, 91120 Palaiseau, France.,École Polytechnique, CMAP 91128 Palaiseau, France.,Institut Pasteur, 75015 Paris, France
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11
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Rombouts J, Gelens L. Dynamic bistable switches enhance robustness and accuracy of cell cycle transitions. PLoS Comput Biol 2021; 17:e1008231. [PMID: 33411761 PMCID: PMC7817062 DOI: 10.1371/journal.pcbi.1008231] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2020] [Revised: 01/20/2021] [Accepted: 11/18/2020] [Indexed: 02/06/2023] Open
Abstract
Bistability is a common mechanism to ensure robust and irreversible cell cycle transitions. Whenever biological parameters or external conditions change such that a threshold is crossed, the system abruptly switches between different cell cycle states. Experimental studies have uncovered mechanisms that can make the shape of the bistable response curve change dynamically in time. Here, we show how such a dynamically changing bistable switch can provide a cell with better control over the timing of cell cycle transitions. Moreover, cell cycle oscillations built on bistable switches are more robust when the bistability is modulated in time. Our results are not specific to cell cycle models and may apply to other bistable systems in which the bistable response curve is time-dependent. Many systems in nature show bistability, which means they can evolve to one of two stable steady states under exactly the same conditions. Which state they evolve to depends on where the system comes from. Such bistability underlies the switching behavior that is essential for cells to progress in the cell division cycle. A quick switch happens when the cell jumps from one steady state to another steady state. Typical of this switching behavior is its robustness and irreversibility. In this paper, we expand this viewpoint of the dynamics of the cell cycle by considering bistable switches which themselves are changing in time. This gives the cell an extra layer of control over transitions both in time and in space, and can make those transitions more robust. Such dynamically changing bistability can appear very naturally. We show this in a model of mitotic entry, in which we include a nuclear and cytoplasmic compartment. The activity of a crucial cell cycle protein follows a bistable switch in each compartment, but the shape of its response is changing in time as proteins are imported into and exported from the nucleus.
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Affiliation(s)
- Jan Rombouts
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), B-3000 Leuven, Belgium
- * E-mail: (J.R.); (L.G.)
| | - Lendert Gelens
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), B-3000 Leuven, Belgium
- * E-mail: (J.R.); (L.G.)
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12
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Holehouse J, Sukys A, Grima R. Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality. J Chem Phys 2020; 153:164113. [PMID: 33138415 DOI: 10.1063/5.0017573] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.
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Affiliation(s)
- James Holehouse
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Augustinas Sukys
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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13
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Kim JK, Tyson JJ. Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy. PLoS Comput Biol 2020; 16:e1008258. [PMID: 33090989 PMCID: PMC7581366 DOI: 10.1371/journal.pcbi.1008258] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.
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Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
| | - John J. Tyson
- Department of Biological Sciences, Virginia Tech, Blacksburg, Virginia, United States of America
- Division of Systems Biology, Virginia Tech, Blacksburg, Virginia, United States of America
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14
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Joanito I, Yan CCS, Chu JW, Wu SH, Hsu CP. Basal leakage in oscillation: Coupled transcriptional and translational control using feed-forward loops. PLoS Comput Biol 2020; 16:e1007740. [PMID: 32881861 PMCID: PMC7494099 DOI: 10.1371/journal.pcbi.1007740] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2020] [Revised: 09/16/2020] [Accepted: 06/26/2020] [Indexed: 11/19/2022] Open
Abstract
The circadian clock is a complex system that plays many important roles in most organisms. Previously, many mathematical models have been used to sharpen our understanding of the Arabidopsis clock, which brought to light the roles of each transcriptional and post-translational regulations. However, the presence of both regulations, instead of either transcription or post-translation, raised curiosity of whether the combination of these two regulations is important for the clock’s system. In this study, we built a series of simplified oscillators with different regulations to study the importance of post-translational regulation (specifically, 26S proteasome degradation) in the clock system. We found that a simple transcriptional-based oscillator can already generate sustained oscillation, but the oscillation can be easily destroyed in the presence of transcriptional leakage. Coupling post-translational control with transcriptional-based oscillator in a feed-forward loop will greatly improve the robustness of the oscillator in the presence of basal leakage. Using these general models, we were able to replicate the increased variability observed in the E3 ligase mutant for both plant and mammalian clocks. With this insight, we also predict a plausible regulator of several E3 ligase genes in the plant’s clock. Thus, our results provide insights into and the plausible importance in coupling transcription and post-translation controls in the clock system. For circadian clocks, several current models had successfully captured the essential dynamic behavior of the clock system mainly with transcriptional regulation. Previous studies have shown that the 26S proteasome degradation controls are important in maintaining the stability of circadian rhythms. However, how the loss-of-function or over-expression mutant of this targeted degradations lead to unstable oscillation is still unclear. In this work, we investigate the importance of coupled transcriptional and post-translational feedback loop in the circadian oscillator. With general models our study indicate that the unstable behavior of degradation mutants could be caused by the increase in the basal level of the clock genes. We found that coupling a non-linear degradation control into this transcriptional based oscillator using feed-forward loop improves the robustness of the oscillator. Using this finding, we further predict some plausible regulators of Arabidopsis’s E3 ligase protein such as COP1 and SINAT5. Hence, our results provide insights on the importance of coupling transcription and post-translation controls in the clock system.
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Affiliation(s)
- Ignasius Joanito
- Institute of Chemistry, Academia Sinica, Taipei, Taiwan
- Bioinformatics Program, Taiwan International Graduate Program, Academia Sinica, Taipei, Taiwan and Institute of Bioinformatics and System Biology, National Chiao Tung University, Hsinchu, Taiwan
| | | | - Jhih-Wei Chu
- Bioinformatics Program, Taiwan International Graduate Program, Academia Sinica, Taipei, Taiwan and Institute of Bioinformatics and System Biology, National Chiao Tung University, Hsinchu, Taiwan
- Department of Biological Science and Technology, National Chiao Tung University, Hsinchu, Taiwan
| | - Shu-Hsing Wu
- Institute of Plant and Microbial Biology, Academia Sinica, Taipei, Taiwan
- Genome and Systems Biology Degree Program, National Taiwan University, Taipei, Taiwan
| | - Chao-Ping Hsu
- Institute of Chemistry, Academia Sinica, Taipei, Taiwan
- Genome and Systems Biology Degree Program, National Taiwan University, Taipei, Taiwan
- * E-mail:
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15
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van Soest I, del Olmo M, Schmal C, Herzel H. Nonlinear phenomena in models of the circadian clock. J R Soc Interface 2020; 17:20200556. [PMID: 32993432 PMCID: PMC7536064 DOI: 10.1098/rsif.2020.0556] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2020] [Accepted: 08/27/2020] [Indexed: 12/14/2022] Open
Abstract
The mammalian circadian clock is well-known to be important for our sleep-wake cycles, as well as other daily rhythms such as temperature regulation, hormone release or feeding-fasting cycles. Under normal conditions, these daily cyclic events follow 24 h limit cycle oscillations, but under some circumstances, more complex nonlinear phenomena, such as the emergence of chaos, or the splitting of physiological dynamics into oscillations with two different periods, can be observed. These nonlinear events have been described at the organismic and tissue level, but whether they occur at the cellular level is still unknown. Our results show that period-doubling, chaos and splitting appear in different models of the mammalian circadian clock with interlocked feedback loops and in the absence of external forcing. We find that changes in the degradation of clock genes and proteins greatly alter the dynamics of the system and can induce complex nonlinear events. Our findings highlight the role of degradation rates in determining the oscillatory behaviour of clock components, and can contribute to the understanding of molecular mechanisms of circadian dysregulation.
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Affiliation(s)
- Inge van Soest
- Institute for Theoretical Biology, Charité and Humboldt Universität zu Berlin, 10115 Berlin, Germany
- Master Program Neuroscience and Cognition, Utrecht University, Utrecht, The Netherlands
| | - Marta del Olmo
- Institute for Theoretical Biology, Charité and Humboldt Universität zu Berlin, 10115 Berlin, Germany
| | - Christoph Schmal
- Institute for Theoretical Biology, Charité and Humboldt Universität zu Berlin, 10115 Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Charité and Humboldt Universität zu Berlin, 10115 Berlin, Germany
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16
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Holehouse J, Cao Z, Grima R. Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study. Biophys J 2020; 118:1517-1525. [PMID: 32155410 PMCID: PMC7136347 DOI: 10.1016/j.bpj.2020.02.016] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2019] [Revised: 01/27/2020] [Accepted: 02/11/2020] [Indexed: 02/08/2023] Open
Abstract
Autoregulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to 1) which subcellular processes are explicitly modeled, 2) the modeling methodology employed (discrete, continuous, or hybrid), and 3) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them, and highlight some of the insights gained through modeling.
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Affiliation(s)
- James Holehouse
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Zhixing Cao
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom; The Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai, People's Republic of China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom.
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17
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Holehouse J, Grima R. Revisiting the Reduction of Stochastic Models of Genetic Feedback Loops with Fast Promoter Switching. Biophys J 2019; 117:1311-1330. [PMID: 31540707 PMCID: PMC6818172 DOI: 10.1016/j.bpj.2019.08.021] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Revised: 07/29/2019] [Accepted: 08/20/2019] [Indexed: 12/19/2022] Open
Abstract
Propensity functions of the Hill type are commonly used to model transcriptional regulation in stochastic models of gene expression. This leads to an effective reduced master equation for the mRNA and protein dynamics only. Based on deterministic considerations, it is often stated or tacitly assumed that such models are valid in the limit of rapid promoter switching. Here, starting from the chemical master equation describing promoter-protein interactions, mRNA transcription, protein translation, and decay, we prove that in the limit of fast promoter switching, the distribution of protein numbers is different than that given by standard stochastic models with Hill-type propensities. We show the differences are pronounced whenever the protein-DNA binding rate is much larger than the unbinding rate, a special case of fast promoter switching. Furthermore, we show using both theory and simulations that use of the standard stochastic models leads to drastically incorrect predictions for the switching properties of positive feedback loops and that these differences decrease with increasing mean protein burst size. Our results confirm that commonly used stochastic models of gene regulatory networks are only accurate in a subset of the parameter space consistent with rapid promoter switching.
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Affiliation(s)
- James Holehouse
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom.
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18
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Patsatzis DG, Goussis DA. A new Michaelis-Menten equation valid everywhere multi-scale dynamics prevails. Math Biosci 2019; 315:108220. [PMID: 31255632 DOI: 10.1016/j.mbs.2019.108220] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2018] [Revised: 06/25/2019] [Accepted: 06/25/2019] [Indexed: 12/17/2022]
Abstract
The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.
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Affiliation(s)
- Dimitris G Patsatzis
- School of Mathematical and Physical Sciences, National Technical University of Athens, Athens, 15780, Greece
| | - Dimitris A Goussis
- Department of Mechanical Engineering, Khalifa University, Abu Dhabi,127788, United Arab Emirates.
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19
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Del Olmo M, Kramer A, Herzel H. A Robust Model for Circadian Redox Oscillations. Int J Mol Sci 2019; 20:E2368. [PMID: 31086108 PMCID: PMC6539027 DOI: 10.3390/ijms20092368] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Revised: 05/08/2019] [Accepted: 05/08/2019] [Indexed: 12/12/2022] Open
Abstract
The circadian clock is an endogenous oscillator that controls daily rhythms in metabolism, physiology, and behavior. Although the timekeeping components differ among species, a common design principle is a transcription-translation negative feedback loop. However, it is becoming clear that other mechanisms can contribute to the generation of 24 h rhythms. Peroxiredoxins (Prxs) exhibit 24 h rhythms in their redox state in all kingdoms of life. In mammalian adrenal gland, heart and brown adipose tissue, such rhythms are generated as a result of an inactivating hyperoxidation reaction that is reduced by coordinated import of sulfiredoxin (Srx) into the mitochondria. However, a quantitative description of the Prx/Srx oscillating system is still missing. We investigate the basic principles that generate mitochondrial Prx/Srx rhythms using computational modeling. We observe that the previously described delay in mitochondrial Srx import, in combination with an appropriate separation of fast and slow reactions, is sufficient to generate robust self-sustained relaxation-like oscillations. We find that our conceptual model can be regarded as a series of three consecutive phases and two temporal switches, highlighting the importance of delayed negative feedback and switches in the generation of oscillations.
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Affiliation(s)
- Marta Del Olmo
- Institute for Theoretical Biology, Charité and Humboldt-Universität zu Berlin, 10115 Berlin, Germany.
| | - Achim Kramer
- Laboratory of Chronobiology, Charité-Universitätsmedizin Berlin, 10117 Berlin, Germany.
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Charité and Humboldt-Universität zu Berlin, 10115 Berlin, Germany.
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20
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Quasi-Steady-State Approximations Derived from the Stochastic Model of Enzyme Kinetics. Bull Math Biol 2019; 81:1303-1336. [DOI: 10.1007/s11538-019-00574-4] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2017] [Accepted: 01/29/2019] [Indexed: 10/27/2022]
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21
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Eilertsen J, Schnell S. A Kinetic Analysis of Coupled (or Auxiliary) Enzyme Reactions. Bull Math Biol 2018; 80:3154-3183. [DOI: 10.1007/s11538-018-0513-4] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2018] [Accepted: 09/20/2018] [Indexed: 10/28/2022]
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22
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Abstract
Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721-759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.
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23
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Caranica C, Al-Omari A, Deng Z, Griffith J, Nilsen R, Mao L, Arnold J, Schüttler HB. Ensemble methods for stochastic networks with special reference to the biological clock of Neurospora crassa. PLoS One 2018; 13:e0196435. [PMID: 29768444 PMCID: PMC5955539 DOI: 10.1371/journal.pone.0196435] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2018] [Accepted: 04/12/2018] [Indexed: 11/18/2022] Open
Abstract
A major challenge in systems biology is to infer the parameters of regulatory networks that operate in a noisy environment, such as in a single cell. In a stochastic regime it is hard to distinguish noise from the real signal and to infer the noise contribution to the dynamical behavior. When the genetic network displays oscillatory dynamics, it is even harder to infer the parameters that produce the oscillations. To address this issue we introduce a new estimation method built on a combination of stochastic simulations, mass action kinetics and ensemble network simulations in which we match the average periodogram and phase of the model to that of the data. The method is relatively fast (compared to Metropolis-Hastings Monte Carlo Methods), easy to parallelize, applicable to large oscillatory networks and large (~2000 cells) single cell expression data sets, and it quantifies the noise impact on the observed dynamics. Standard errors of estimated rate coefficients are typically two orders of magnitude smaller than the mean from single cell experiments with on the order of ~1000 cells. We also provide a method to assess the goodness of fit of the stochastic network using the Hilbert phase of single cells. An analysis of phase departures from the null model with no communication between cells is consistent with a hypothesis of Stochastic Resonance describing single cell oscillators. Stochastic Resonance provides a physical mechanism whereby intracellular noise plays a positive role in establishing oscillatory behavior, but may require model parameters, such as rate coefficients, that differ substantially from those extracted at the macroscopic level from measurements on populations of millions of communicating, synchronized cells.
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Affiliation(s)
- C. Caranica
- Department of Statistics, University of Georgia, Athens, Georgia
| | - A. Al-Omari
- Department of Biomedical Systems and Informatics Engineering, Yarmouk University, Irbid, Jordan
| | - Z. Deng
- School of Electrical and Computer Engineering, College of Engineering, University of Georgia, Athens, Georgia
| | - J. Griffith
- Genetics Department, University of Georgia, Athens, Georgia
- College of Agricultural and Environmental Sciences, University of Georgia, Athens, Georgia
| | - R. Nilsen
- Genetics Department, University of Georgia, Athens, Georgia
| | - L. Mao
- School of Electrical and Computer Engineering, College of Engineering, University of Georgia, Athens, Georgia
| | - J. Arnold
- Genetics Department, University of Georgia, Athens, Georgia
- * E-mail:
| | - H.-B. Schüttler
- Department of Physics and Astronomy, University of Georgia, Athens, Georgia
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24
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Herath N, Del Vecchio D. Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+. J Chem Phys 2018. [DOI: 10.1063/1.5012752] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Affiliation(s)
- Narmada Herath
- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - Domitilla Del Vecchio
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
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25
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Choi B, Rempala GA, Kim JK. Beyond the Michaelis-Menten equation: Accurate and efficient estimation of enzyme kinetic parameters. Sci Rep 2017; 7:17018. [PMID: 29208922 PMCID: PMC5717222 DOI: 10.1038/s41598-017-17072-z] [Citation(s) in RCA: 63] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2017] [Accepted: 11/22/2017] [Indexed: 11/09/2022] Open
Abstract
Examining enzyme kinetics is critical for understanding cellular systems and for using enzymes in industry. The Michaelis-Menten equation has been widely used for over a century to estimate the enzyme kinetic parameters from reaction progress curves of substrates, which is known as the progress curve assay. However, this canonical approach works in limited conditions, such as when there is a large excess of substrate over enzyme. Even when this condition is satisfied, the identifiability of parameters is not always guaranteed, and often not verifiable in practice. To overcome such limitations of the canonical approach for the progress curve assay, here we propose a Bayesian approach based on an equation derived with the total quasi-steady-state approximation. In contrast to the canonical approach, estimates obtained with this proposed approach exhibit little bias for any combination of enzyme and substrate concentrations. Importantly, unlike the canonical approach, an optimal experiment to identify parameters with certainty can be easily designed without any prior information. Indeed, with this proposed design, the kinetic parameters of diverse enzymes with disparate catalytic efficiencies, such as chymotrypsin, fumarase, and urease, can be accurately and precisely estimated from a minimal amount of timecourse data. A publicly accessible computational package performing such accurate and efficient Bayesian inference for enzyme kinetics is provided.
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Affiliation(s)
- Boseung Choi
- Korea University Sejong campus, Division of Economics and Statistics, Department of National Statistics, Sejong, 30019, Korea
| | - Grzegorz A Rempala
- The Ohio State University, Division of Biostatistics and Mathematical Biosciences Institute, Columbus, OH, 43210, USA
| | - Jae Kyoung Kim
- Korea Advanced Institute of Science and Technology, Department of Mathematical Sciences, Daejeon, 34141, Korea.
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26
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D'Alessandro M, Beesley S, Kim JK, Jones Z, Chen R, Wi J, Kyle K, Vera D, Pagano M, Nowakowski R, Lee C. Stability of Wake-Sleep Cycles Requires Robust Degradation of the PERIOD Protein. Curr Biol 2017; 27:3454-3467.e8. [PMID: 29103939 DOI: 10.1016/j.cub.2017.10.014] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2017] [Revised: 09/14/2017] [Accepted: 10/04/2017] [Indexed: 10/18/2022]
Abstract
Robustness in biology is the stability of phenotype under diverse genetic and/or environmental perturbations. The circadian clock has remarkable stability of period and phase that-unlike other biological oscillators-is maintained over a wide range of conditions. Here, we show that the high fidelity of the circadian system stems from robust degradation of the clock protein PERIOD. We show that PERIOD degradation is regulated by a balance between ubiquitination and deubiquitination, and that disruption of this balance can destabilize the clock. In mice with a loss-of-function mutation of the E3 ligase gene β-Trcp2, the balance of PERIOD degradation is perturbed and the clock becomes dramatically unstable, presenting a unique behavioral phenotype unlike other circadian mutant animal models. We believe that our data provide a molecular explanation for how circadian phases, such as wake-sleep onset times, can become unstable in humans, and we present a unique mouse model to study human circadian disorders with unstable circadian rhythm phases.
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Affiliation(s)
- Matthew D'Alessandro
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Stephen Beesley
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, South Korea
| | - Zachary Jones
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Rongmin Chen
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Julie Wi
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Kathleen Kyle
- Center for Genomics and Personalized Medicine, Florida State University, 319 Stadium Drive, Tallahassee, FL 32306, USA
| | - Daniel Vera
- Center for Genomics and Personalized Medicine, Florida State University, 319 Stadium Drive, Tallahassee, FL 32306, USA
| | - Michele Pagano
- Howard Hughes Medical Institute, Department of Pathology, New York University School of Medicine, 550 First Avenue, MSB 599, New York, NY 10016, USA
| | - Richard Nowakowski
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA
| | - Choogon Lee
- Department of Biomedical Sciences, Program in Neuroscience, College of Medicine, Florida State University, 1115 West Call Street, Tallahassee, FL 32306, USA.
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27
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Kim JK, Sontag ED. Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation. PLoS Comput Biol 2017; 13:e1005571. [PMID: 28582397 PMCID: PMC5481150 DOI: 10.1371/journal.pcbi.1005571] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2017] [Revised: 06/22/2017] [Accepted: 05/12/2017] [Indexed: 12/14/2022] Open
Abstract
Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.
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Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea
- * E-mail: (JKK); , (EDS)
| | - Eduardo D. Sontag
- Department of Mathematics and Center for Quantitative Biology, Rutgers University, New Brunswick, New Jersey, United States of America
- * E-mail: (JKK); , (EDS)
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28
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Huang B, Lu M, Jia D, Ben-Jacob E, Levine H, Onuchic JN. Interrogating the topological robustness of gene regulatory circuits by randomization. PLoS Comput Biol 2017; 13:e1005456. [PMID: 28362798 PMCID: PMC5391964 DOI: 10.1371/journal.pcbi.1005456] [Citation(s) in RCA: 103] [Impact Index Per Article: 14.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2016] [Revised: 04/14/2017] [Accepted: 03/15/2017] [Indexed: 01/06/2023] Open
Abstract
One of the most important roles of cells is performing their cellular tasks properly for survival. Cells usually achieve robust functionality, for example, cell-fate decision-making and signal transduction, through multiple layers of regulation involving many genes. Despite the combinatorial complexity of gene regulation, its quantitative behavior has been typically studied on the basis of experimentally verified core gene regulatory circuitry, composed of a small set of important elements. It is still unclear how such a core circuit operates in the presence of many other regulatory molecules and in a crowded and noisy cellular environment. Here we report a new computational method, named random circuit perturbation (RACIPE), for interrogating the robust dynamical behavior of a gene regulatory circuit even without accurate measurements of circuit kinetic parameters. RACIPE generates an ensemble of random kinetic models corresponding to a fixed circuit topology, and utilizes statistical tools to identify generic properties of the circuit. By applying RACIPE to simple toggle-switch-like motifs, we observed that the stable states of all models converge to experimentally observed gene state clusters even when the parameters are strongly perturbed. RACIPE was further applied to a proposed 22-gene network of the Epithelial-to-Mesenchymal Transition (EMT), from which we identified four experimentally observed gene states, including the states that are associated with two different types of hybrid Epithelial/Mesenchymal phenotypes. Our results suggest that dynamics of a gene circuit is mainly determined by its topology, not by detailed circuit parameters. Our work provides a theoretical foundation for circuit-based systems biology modeling. We anticipate RACIPE to be a powerful tool to predict and decode circuit design principles in an unbiased manner, and to quantitatively evaluate the robustness and heterogeneity of gene expression. Cells are able to robustly carry out their essential biological functions, possibly because of multiple layers of tight regulation via complex, yet well-designed, gene regulatory networks involving a substantial number of genes. State-of-the-art genomics technology has enabled the mapping of these large gene networks, yet it remains a tremendous challenge to elucidate their design principles and the regulatory mechanisms underlying their biological functions such as signal processing and decision-making. One of the key barriers is the absence of accurate kinetics for the regulatory interactions, especially from in vivo experiments. To this end, we have developed a new computational modeling method, Random Circuit Perturbation (RACIPE), to explore the dynamic behaviors of gene regulatory circuits without the requirement of detailed kinetic parameters. RACIPE takes a network topology as the input, and generates an unbiased ensemble of models with varying kinetic parameters. Each model is subjected to simulation, followed by statistical analysis for the ensemble. We tested RACIPE on several gene circuits, and found that the predicted gene expression patterns from all of the models converge to experimentally observed gene state clusters. We expect RACIPE to be a powerful method to identify the role of network topology in determining network operating principles.
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Affiliation(s)
- Bin Huang
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- Department of Chemistry, Rice University, Houston, TX, United States of America
| | - Mingyang Lu
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- The Jackson Laboratory, Bar Harbor, ME, United States of America
| | - Dongya Jia
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- Program in Systems, Synthetic and Physical Biology, Rice University, Houston, TX, United States of America
| | - Eshel Ben-Jacob
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- School of Physics and Astronomy, and The Sagol School of Neuroscience, Tel-Aviv University, Tel-Aviv, Israel
| | - Herbert Levine
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- Department of Bioengineering, Rice University, Houston, TX, United States of America
- Department of Biosciences, Rice University, Houston, TX, United States of America
- Department of Physics and Astronomy, Rice University, Houston, TX, United States of America
- * E-mail: (HL); (JNO)
| | - Jose N. Onuchic
- Center for Theoretical Biological Physics, Rice University, Houston, TX, United States of America
- Department of Chemistry, Rice University, Houston, TX, United States of America
- Department of Biosciences, Rice University, Houston, TX, United States of America
- Department of Physics and Astronomy, Rice University, Houston, TX, United States of America
- * E-mail: (HL); (JNO)
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Paijmans J, Lubensky DK, ten Wolde PR. A thermodynamically consistent model of the post-translational Kai circadian clock. PLoS Comput Biol 2017; 13:e1005415. [PMID: 28296888 PMCID: PMC5371392 DOI: 10.1371/journal.pcbi.1005415] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2016] [Revised: 03/29/2017] [Accepted: 02/15/2017] [Indexed: 11/18/2022] Open
Abstract
The principal pacemaker of the circadian clock of the cyanobacterium S. elongatus is a protein phosphorylation cycle consisting of three proteins, KaiA, KaiB and KaiC. KaiC forms a homohexamer, with each monomer consisting of two domains, CI and CII. Both domains can bind and hydrolyze ATP, but only the CII domain can be phosphorylated, at two residues, in a well-defined sequence. While this system has been studied extensively, how the clock is driven thermodynamically has remained elusive. Inspired by recent experimental observations and building on ideas from previous mathematical models, we present a new, thermodynamically consistent, statistical-mechanical model of the clock. At its heart are two main ideas: i) ATP hydrolysis in the CI domain provides the thermodynamic driving force for the clock, switching KaiC between an active conformational state in which its phosphorylation level tends to rise and an inactive one in which it tends to fall; ii) phosphorylation of the CII domain provides the timer for the hydrolysis in the CI domain. The model also naturally explains how KaiA, by acting as a nucleotide exchange factor, can stimulate phosphorylation of KaiC, and how the differential affinity of KaiA for the different KaiC phosphoforms generates the characteristic temporal order of KaiC phosphorylation. As the phosphorylation level in the CII domain rises, the release of ADP from CI slows down, making the inactive conformational state of KaiC more stable. In the inactive state, KaiC binds KaiB, which not only stabilizes this state further, but also leads to the sequestration of KaiA, and hence to KaiC dephosphorylation. Using a dedicated kinetic Monte Carlo algorithm, which makes it possible to efficiently simulate this system consisting of more than a billion reactions, we show that the model can describe a wealth of experimental data. Circadian clocks are biological timekeeping devices with a rhythm of 24 hours in living cells pertaining to all kingdoms of life. They help organisms to coordinate their behavior with the day-night cycle. The circadian clock of the cyanobacterium Synechococcus elongatus is one of the simplest and best characterized clocks in biology. The central clock component is the protein KaiC, which is phosphorylated and dephosphorylated in a cyclical manner with a 24 hr period. While we know from elementary thermodynamics that oscillations require a net turnover of fuel molecules, in this case ATP, how ATP hydrolysis drives the clock has remained elusive. Based on recent experimental observations and building on ideas from existing models, we construct the most detailed mathematical model of this system to date. KaiC consists of two domains, CI and CII, which each can bind ATP, yet only CII can be phosphorylated. Moreover, KaiC can exist in two conformational states, an active one in which the phosphorylation level tends to rise, and an inactive one in which it tends to fall. Our model predicts that ATP hydrolysis in the CI domain is the principal energetic driver of the clock, driving the switching between the two conformational states, while phosphorylation in the CII domain provides the timer for the conformational switch. The coupling between ATP hydrolysis in the CI domain and phosphorylation in the CII domain leads to novel testable predictions.
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Affiliation(s)
| | - David K. Lubensky
- Department of Physics, University of Michigan, Ann Arbor, Michigan, United States of America
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