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Sun Y, Zhang F, Ouyang Q, Luo C. The dynamic-process characterization and prediction of synthetic gene circuits by dynamic delay model. iScience 2024; 27:109142. [PMID: 38384832 PMCID: PMC10879701 DOI: 10.1016/j.isci.2024.109142] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2023] [Revised: 01/17/2024] [Accepted: 02/01/2024] [Indexed: 02/23/2024] Open
Abstract
Differential equation models are widely used to describe genetic regulations, predict multicomponent regulatory circuits, and provide quantitative insights. However, it is still challenging to quantitatively link the dynamic behaviors with measured parameters in synthetic circuits. Here, we propose a dynamic delay model (DDM) which includes two simple parts: the dynamic determining part and the doses-related steady-state-determining part. The dynamic determining part is usually supposed as the delay time but without a clear formula. For the first time, we give the detail formula of the dynamic determining function and provide a method for measuring all parameters of synthetic elements (include 8 activators and 5 repressors) by microfluidic system. Three synthetic circuits were built to show that the DDM can notably improve the prediction accuracy and can be used in various synthetic biology applications.
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Affiliation(s)
- Yanhong Sun
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Fengyu Zhang
- Wenzhou Institute University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
| | - Qi Ouyang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
| | - Chunxiong Luo
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
- Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
- Wenzhou Institute University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China
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Islam MS, Faruque IA. Insect visuomotor delay adjustments in group flight support swarm cohesion. Sci Rep 2023; 13:6407. [PMID: 37076527 PMCID: PMC10115836 DOI: 10.1038/s41598-023-32675-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Accepted: 03/31/2023] [Indexed: 04/21/2023] Open
Abstract
Flying insects routinely demonstrate coordinated flight in crowded assemblies despite strict communication and processing constraints. This study experimentally records multiple flying insects tracking a moving visual stimulus. System identification techniques are used to robustly identify the tracking dynamics, including a visuomotor delay. The population delay distributions are quantified for solo and group behaviors. An interconnected visual swarm model incorporating heterogeneous delays is developed, and bifurcation analysis and swarm simulation are applied to assess swarm stability under the delays. The experiment recorded 450 insect trajectories and quantified visual tracking delay variation. Solitary tasks showed a 30ms average delay and standard deviation of 50ms, while group behaviors show a 15ms average and 8ms standard deviation. Analysis and simulation indicate that the delay adjustments during group flight support swarm formation and center stability, and are robust to measurement noise. These results quantify the role of visuomotor delay heterogeneity in flying insects and their role in supporting swarm cohesion through implicit communication.
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Sun Y, Zhang F, Li L, Chen K, Wang S, Ouyang Q, Luo C. Two-Layered Microfluidic Devices for High-Throughput Dynamic Analysis of Synthetic Gene Circuits in E. coli. ACS Synth Biol 2022; 11:3954-3965. [PMID: 36283074 DOI: 10.1021/acssynbio.2c00307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
Abstract
Escherichia coli is a common chassis for synthetic gene circuit studies. In addition to the dose-response of synthetic gene circuits, the analysis of dynamic responses is also an important part of the future design of more complicated synthetic systems. Recently, microfluidic-based methods have been widely used for the analysis of gene expression dynamics. Here, we established a two-layered microfluidic platform for the systematic characterization of synthetic gene circuits (eight strains in eight different culture environments could be observed simultaneously with a 5 min time resolution). With this platform, both dose responses and dynamic responses with a high temporal resolution could be easily derived for further analysis. A controlled environment ensures the stability of the bacterial growth rate, excluding changes in gene expression dynamics caused by changes of the growth dilution rate. The precise environmental switch and automatic micrograph shooting ensured that there was nearly no time lag between the inducer addition and the data recording. We studied four four-node incoherent-feedforward-loop (IFFL) networks with different operators using this device. The experimental results showed that as the effect of inhibition increased, two of the IFFL networks generated pulselike dynamic gene expressions in the range of the inducer concentrations, which was different from the dynamics of the two other circuits with only a simple pattern of rising to the platform. Through fitting the dose-response curves and the dynamic response curves, corresponding parameters were derived and introduced to a simple model that could qualitatively explain the generation of pulse dynamics.
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Affiliation(s)
- Yanhong Sun
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics School of Physics, Peking University, Beijing100871, China
| | - Fengyu Zhang
- School of Life Sciences and Peking-Tsinghua Center for Life Sciences, Peking University, Beijing100871, China
| | - Lusi Li
- Academy of Advanced Interdisciplinary Studies, Peking University, Beijing100871, China
| | - Kaiyue Chen
- Wenzhou Institute University of Chinese Academy of Sciences, Wenzhou, Zhejiang325001, China
| | - Shujing Wang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics School of Physics, Peking University, Beijing100871, China
| | - Qi Ouyang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics School of Physics, Peking University, Beijing100871, China.,Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing100871, China
| | - Chunxiong Luo
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics School of Physics, Peking University, Beijing100871, China.,Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing100871, China.,Wenzhou Institute University of Chinese Academy of Sciences, Wenzhou, Zhejiang325001, China
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Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics. MATHEMATICS 2021. [DOI: 10.3390/math9060680] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich the model introduced in (Gaurav et al., 2019) with the mechanism of self-information that accounts for information about the product performance basing on consumers’ experience on the recent past. We obtain conditions for which the model exhibits a forward or a backward phenomenology and evaluate the impact of self-information on both these scenarios. Our analysis suggests that, even if hysteretic dynamics in referral campaigns is intimately linked to the mechanism of referrals, an adequate level of self-information and a fairly high level of customer-satisfaction can act as strategic tools to manage hysteresis and allow the campaign to spread in more controllable conditions.
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5
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Nonlinear delay differential equations and their application to modeling biological network motifs. Nat Commun 2021; 12:1788. [PMID: 33741909 PMCID: PMC7979834 DOI: 10.1038/s41467-021-21700-8] [Citation(s) in RCA: 22] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2020] [Accepted: 02/01/2021] [Indexed: 12/24/2022] Open
Abstract
Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs. Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.
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Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach. J Math Biol 2020; 81:343-367. [PMID: 32583030 PMCID: PMC7363733 DOI: 10.1007/s00285-020-01512-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2019] [Revised: 04/11/2020] [Indexed: 12/21/2022]
Abstract
Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.
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Kyrychko YN, Schwartz IB. Enhancing noise-induced switching times in systems with distributed delays. CHAOS (WOODBURY, N.Y.) 2018; 28:063106. [PMID: 29960399 DOI: 10.1063/1.5034106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution leads to an increase in the switching times even for longer values of mean time delays for both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution. Analytical predictions are compared to the direct numerical simulations and show excellent agreement between theory and numerical experiment.
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Affiliation(s)
- Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - I B Schwartz
- US Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section, Plasma Physics Division, Washington, DC 20375, USA
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Rombouts J, Vandervelde A, Gelens L. Delay models for the early embryonic cell cycle oscillator. PLoS One 2018; 13:e0194769. [PMID: 29579091 PMCID: PMC5868829 DOI: 10.1371/journal.pone.0194769] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2017] [Accepted: 03/09/2018] [Indexed: 11/19/2022] Open
Abstract
Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.
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Affiliation(s)
- Jan Rombouts
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven, 3000 Leuven, Belgium
| | - Alexandra Vandervelde
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven, 3000 Leuven, Belgium
| | - Lendert Gelens
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven, 3000 Leuven, Belgium
- * E-mail:
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Ingalls B, Mincheva M, Roussel MR. Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation. Bull Math Biol 2017; 79:1539-1563. [PMID: 28608044 DOI: 10.1007/s11538-017-0298-x] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2016] [Accepted: 05/17/2017] [Indexed: 11/25/2022]
Abstract
A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.
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Affiliation(s)
- Brian Ingalls
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada.
| | - Maya Mincheva
- Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115, USA
| | - Marc R Roussel
- Alberta RNA Research and Training Institute, Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada
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