Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise

Modelling and simulation of complex biochemical reaction networks form cornerstones of modern biophysics. Many of the approaches developed so far capture temporal fluctuations due to the inherent stochasticity of the biophysical processes, referred to as intrinsic noise. Stochastic fluctuations, however, predominantly stem from the interplay of the network with many other - and mostly unknown - fluctuating processes, as well as with various random signals arising from the extracellular world; these sources contribute extrinsic noise. Here we provide a computational simulation method to probe the stochastic dynamics of biochemical systems subject to both intrinsic and extrinsic noise. We develop an extrinsic chemical Langevin equation - a physically motivated extension of the chemical Langevin equation - to model intrinsically noisy reaction networks embedded in a stochastically fluctuating environment. The extrinsic CLE is a continuous approximation to the Chemical Master Equation (CME) with time-varying propensities. In our approach, noise is incorporated at the level of the CME, and can account for the full dynamics of the exogenous noise process, irrespective of timescales and their mismatches. We show that our method accurately captures the first two moments of the stationary probability density when compared with exact stochastic simulation methods, while reducing the computational runtime by several orders of magnitude. Our approach provides a method that is practical, computationally efficient and physically accurate to study systems that are simultaneously subject to a variety of noise sources.

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Stationary Equations for Non-Markovian Biochemical Systems

10.1101/360123 ◽

2018 ◽

Author(s):

Jiajun Zhang ◽

Tianshou Zhou

Keyword(s):

Master Equation ◽

Stochastic Analysis ◽

Waiting Times ◽

Reaction Network ◽

State Variables ◽

New Approach ◽

Reaction Systems ◽

Linear Noise Approximation ◽

Biochemical Systems ◽

Markovian System

AbstractWe develop a new approach for stochastic analysis of biochemical reaction systems with arbitrary distributions of waiting times between reaction events. Specifically, we derive a stationary generalized chemical master equation for a non-Markovian reaction network. Importantly, this equation allows to transform the original non-Markovian problem into a Markovian one by introducing a mean reaction propensity function for every reaction in the network. Furthermore, we derive a stationary generalized linear noise approximation for the non-Markovian system, which is convenient to the direct estimation of the stationary noise in state variables. These derived equations can have broad applications, and exemplars of two representative non-Markovian models provide evidence of their applicability.

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CHARACTERIZATION OF INTRINSIC AND EXTRINSIC NOISE EFFECTS IN POSITIVELY REGULATED GENES

Journal of Biological System ◽

10.1142/s0218339019500165 ◽

2019 ◽

Vol 27 (03) ◽

pp. 383-398

Author(s):

AARÓN VÁZQUEZ-JIMÉNEZ ◽

MOISÉS SANTILLÁN ◽

JESÚS RODRÍGUEZ-GONZÁLEZ

Keyword(s):

Environmental Changes ◽

Intrinsic Noise ◽

Common Source ◽

Extrinsic Noise ◽

Noise Effects ◽

Intensity Parameters ◽

Degradation Rates ◽

External Stimuli ◽

Robust To Noise

Gene regulation is fundamental for cell survival. This regulation must be both robust to noise and sensitive enough to external stimuli to elicit the proper responses. In this work, we study, through stochastic numerical simulations, how a gene regulatory network with a positive feedback loop responds to environmental changes in the presence of intrinsic and extrinsic noises. Noise effects were characterized by measuring the statistical differences between two protein time series resulting from identical systems subject to the same source of extrinsic noise. A robust analysis was implemented by modifying the kinetic system parameters. We found that the common source of time-varying extrinsic fluctuations leads to a correlation in the systems it affects. The correlation and the extrinsic and intrinsic noise components are modulated by the update period and noise intensity parameters. Our results suggest that noise perception is controlled through the parameters associated with the response time: degradation rates and promoter dissociation constant.

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Noise control for molecular computing

Journal of The Royal Society Interface ◽

10.1098/rsif.2018.0199 ◽

2018 ◽

Vol 15 (144) ◽

pp. 20180199 ◽

Cited By ~ 3

Author(s):

Tomislav Plesa ◽

Konstantinos C. Zygalakis ◽

David F. Anderson ◽

Radek Erban

Keyword(s):

Stochastic Dynamics ◽

Noise Control ◽

Reaction System ◽

Reaction Network ◽

Intrinsic Noise ◽

Biochemical Networks ◽

Mass Action ◽

Mass Action Kinetics ◽

Interdisciplinary Field ◽

Biochemical Systems

Synthetic biology is a growing interdisciplinary field, with far-reaching applications, which aims to design biochemical systems that behave in a desired manner. With the advancement in nucleic-acid-based technology in general, and strand-displacement DNA computing in particular, a large class of abstract biochemical networks may be physically realized using nucleic acids. Methods for systematic design of the abstract systems with prescribed behaviours have been predominantly developed at the (less-detailed) deterministic level. However, stochastic effects, neglected at the deterministic level, are increasingly found to play an important role in biochemistry. In such circumstances, methods for controlling the intrinsic noise in the system are necessary for a successful network design at the (more-detailed) stochastic level. To bridge the gap, the noise-control algorithm for designing biochemical networks is developed in this paper. The algorithm structurally modifies any given reaction network under mass-action kinetics, in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics, while (ii) the deterministic dynamics are preserved. The capabilities of the algorithm are demonstrated on a production–decay reaction system, and on an exotic system displaying bistability. For the production–decay system, it is shown that the algorithm may be used to redesign the network to achieve noise-induced multistability. For the exotic system, the algorithm is used to redesign the network to control the stochastic switching, and achieve noise-induced oscillations.

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Extrinsic noise acts to lower protein production at higher translation initiation rates

10.1101/2020.08.21.260976 ◽

2020 ◽

Author(s):

Rati Sharma

Keyword(s):

Translation Initiation ◽

Protein Production ◽

Translation Termination ◽

Mrna Translation ◽

Intrinsic Noise ◽

Stochastic Simulations ◽

Residence Times ◽

Extrinsic Noise ◽

Initiation Rate ◽

The Individual

Any cellular process at the microscopic level is governed by both extrinsic and intrinsic noise. In this article, we incorporate extrinsic noise in a model of mRNA translation and carry out stochastic simulations of the same. We then evaluate various statistics related to the residence time of the ribosome on the mRNA and subsequent protein production. We also study the effect of slow codons. From our simulations, we show that noise in the translation initiation rate rather than the translation termination rate acts to significantly broaden the distribution of mRNA residence times near the membrane. Further, the presence of slow codons acts to increase the mean residence times. However, this increase also depends on the number and position of the slow codons on the lattice. We also show that the the slow codons act to mask any effect from the extrinsic noise themselves. Our results have implications towards a better understanding of the role the individual components play during the translation process.

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Pathway dynamics can delineate the sources of transcriptional noise in gene expression

eLife ◽

10.7554/elife.69324 ◽

2021 ◽

Vol 10 ◽

Author(s):

Lucy Ham ◽

Marcel Jackson ◽

Michael Stumpf

Keyword(s):

Intrinsic Noise ◽

Stochastic Simulations ◽

Transcriptional Noise ◽

Extrinsic Noise ◽

Cellular Processes ◽

Dual Reporter ◽

Extensive Cell ◽

Noise In Gene Expression ◽

Cell Expression ◽

Cell To Cell Variability

Single-cell expression profiling opens up new vistas on cellular processes. Extensive cell-to-cell variability at the transcriptomic and proteomic level has been one of the stand-out observations. Because most experimental analyses are destructive we only have access to snapshot data of cellular states. This loss of temporal information presents significant challenges for inferring dynamics, as well as causes of cell-to-cell variability. In particular, we typically cannot separate dynamic variability from within cells ('intrinsic noise') from variability across the population ('extrinsic noise'). Here we make this non-identifiability mathematically precise, allowing us to identify new experimental set-ups that can assist in resolving this non-identifiability. We show that multiple generic reporters from the same biochemical pathways (e.g. mRNA and protein) can infer magnitudes of intrinsic and extrinsic transcriptional noise, identifying sources of heterogeneity. Stochastic simulations support our theory, and demonstrate that 'pathway-reporters' compare favourably to the well-known, but often difficult to implement, dual-reporter method.

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Limits of noise for autoregulated gene expression

10.1101/100115 ◽

2017 ◽

Author(s):

Peter Czuppon ◽

Peter Pfaffelhuber

Keyword(s):

Gene Expression ◽

Negative Feedback ◽

Positive Feedback ◽

Fano Factor ◽

Intrinsic Noise ◽

Stochastic Averaging ◽

Jump Processes ◽

Markov Jump ◽

Extrinsic Noise ◽

Cellular Processes

AbstractGene 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, Kurtz and Popovic, 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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Comparing the Effects of Intrinsic and Extrinsic Noise on the Vicsek Model in Three Dimensions

Volume 2: Mechatronics; Estimation and Identification; Uncertain Systems and Robustness; Path Planning and Motion Control; Tracking Control Systems; Multi-Agent and Networked Systems; Manufacturing; Intelligent Transportation and Vehicles; Sensors and Actuators; Diagnostics and Detection; Unmanned, Ground and Surface Robotics; Motion and Vibration Control Applications ◽

10.1115/dscc2017-5303 ◽

2017 ◽

Author(s):

Masoud Jahromi Shirazi ◽

Nicole Abaid

Keyword(s):

Collective Behavior ◽

Intrinsic Noise ◽

Three Dimensions ◽

Extrinsic Noise ◽

Vicsek Model ◽

Fish Schools ◽

Natural Systems ◽

Order And Disorder ◽

Noise Models ◽

Intrinsic And Extrinsic Noise

A group of simple individuals may show ordered, complex behavior through local interactions. This phenomenon is called collective behavior, which has been observed in a vast variety of natural systems such as fish schools or bird flocks. The Vicsek model is a well-established mathematical model to study collective behavior through interaction of individuals with their neighbors in the presence of noise. How noise is modeled can impact the collective behavior of the group. Extrinsic noise captures uncertainty imposed on individuals, such as noise in measurements, while intrinsic noise models uncertainty inherent to individuals, akin to free will. In this paper, the effects of intrinsic and extrinsic noise on characteristics of the transition between order and disorder in the Vicsek model in three dimensions are studied through numerical simulation.

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ON EXPLORING EFFECTS OF MOLECULAR NOISE IN A SIMPLE VIRAL INFECTION MODEL

International Journal of Biomathematics ◽

10.1142/s1793524510000891 ◽

2010 ◽

Vol 03 (01) ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

ZHI XIE ◽

DON KULASIRI

Keyword(s):

Viral Infection ◽

Regulatory Networks ◽

Virus Assembly ◽

Intrinsic Noise ◽

Genetic Regulatory Networks ◽

Infection Model ◽

Extrinsic Noise ◽

Living Organisms ◽

High Level ◽

And Function

Intrinsic and extrinsic noises are all believed to be important in the development and function of many living organisms. In this study, we investigate the sources of the intrinsic noise and the influence of the extrinsic noise on an intracellular viral infection system. The contribution of the intrinsic noise from each reaction is measured by means of a special form of stochastic differential equations (SDEs), chemical Langevin equation. The intrinsic noise of the system is a linear sum of the noise in each of the reactions. The intrinsic noise mainly arises from the degradation of mRNA and the transcription processes. We then study the effects of extrinsic noise by the means of a general form of SDE. It is found that the noise of the viral components grows logarithmically with the increasing noise intensities. The system is most susceptible to the noise in the virus assembly process. A high level of noise in this process can even inhibit the growth of the viruses. This study also demonstrates the utility of SDEs in analyzing genetic regulatory networks perturbed by either inherent or parametric stochasticity.

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Network design principle for robust oscillatory behaviors with respect to biological noise

10.1101/2021.12.22.473835 ◽

2021 ◽

Author(s):

Lingxia Qiao ◽

Zhi-Bo Zhang ◽

Wei Zhao ◽

Ping Wei ◽

Lei Zhang

Keyword(s):

Chemical Reactions ◽

Regulatory Networks ◽

Transcriptional Regulatory Network ◽

Intrinsic Noise ◽

Biological Noise ◽

Extrinsic Noise ◽

Transcriptional Regulatory Networks ◽

Transcriptional Regulatory ◽

Activator Inhibitor ◽

Oscillatory Behaviors

Oscillatory behaviors, which are ubiquitous in transcriptional regulatory networks, are often subject to inevitable biological noise. Thus a natural question is how transcriptional regulatory networks can robustly achieve accurate oscillation in the presence of biological noise. Here, we search all two- and three-node transcriptional regulatory network topologies for those robustly capable of accurate oscillation against the parameter variability (extrinsic noise) or stochasticity of chemical reactions (intrinsic noise). We find that, no matter what source of the noise is applied, the topologies containing the repressilator with positive auto-regulation show higher robustness of accurate oscillation than those containing the activator-inhibitor oscillator, and additional positive auto-regulation enhances the robustness against noise. Nevertheless, the attenuation of different sources of noise is governed by distinct mechanisms: the parameter variability is buffered by the long period, while the stochasticity of chemical reactions is filtered by the high amplitude. Furthermore, we analyze the noise of a synthetic human nuclear factor κB (NF-κB) signaling network by varying three different topologies, and verify that the addition of a repressilator to the activator-inhibitor oscillator, which leads to the emergence of high-robustness motif—the repressilator with positive auto-regulation, improves the oscillation accuracy in comparison to the topology with only an activator-inhibitor oscillator. These design principles may be applicable to other oscillatory circuits.

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