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Colored extrinsic fluctuations and stochastic gene expression
Vahid Shahrezaei, Julien F Ollivier, Peter S Swain
TL;DR
Cellular variation is largely shaped by colored, nonspecific extrinsic fluctuations whose effects are not captured by intrinsic-noise models alone. The paper extends the standard stochastic simulation algorithm to include such fluctuations and shows that they alter protein numbers, noise, response times, and network-level stochasticity. Correlations and timescales determine whether these effects amplify or attenuate biochemical-network outputs.
Problem
Extrinsic fluctuations dominate cellular variation but their sources, timescales, and effects on biochemical-network stochasticity are incompletely understood.
Method
The paper extends the standard stochastic simulation algorithm to simulate time-varying extrinsic fluctuations with configurable properties and correlated or uncorrelated parameter changes.
Results
Extrinsic fluctuations alter mean protein numbers, intrinsic and total noise, response times, and output distributions, with correlated effects either amplifying or nearly canceling depending on network architecture.
Takeaways & Limitations
Both the timescales and nonspecificity of extrinsic fluctuations substantially affect biochemical-network function and performance.
Abstract
from arXiv · showhide
Stochasticity is both exploited and controlled by cells. Although the intrinsic stochasticity inherent in biochemistry is relatively well understood, cellular variation, or 'noise', is predominantly generated by interactions of the system of interest with other stochastic systems in the cell or its environment. Such extrinsic fluctuations are nonspecific, affecting many system components, and have a substantial lifetime, comparable to the cell cycle (they are 'colored'). Here, we extend the standard stochastic simulation algorithm to include extrinsic fluctuations. We show that these fluctuations affect mean protein numbers and intrinsic noise, can speed up typical network response times, and can explain trends in high-throughput measurements of variation. If extrinsic fluctuations in two components of the network are correlated, they may combine constructively (amplifying each other) or destructively (attenuating each other). Consequently, we predict that incoherent feedforward loops attenuate stochasticity, while coherent feedforwards amplify it. Our results demonstrate that both the timescales of extrinsic fluctuations and their nonspecificity substantially affect the function and performance of biochemical networks.
Subject category
The paper is categorized under computational methods for metabolic and regulatory networks.
- The paper concerns computational methods applied to metabolic and regulatory networks.
Introduction
Biochemical networks experience intrinsic fluctuations from reaction dynamics and extrinsic fluctuations from interactions with other stochastic systems. Extrinsic fluctuations dominate cellular variation, are colored on cell-cycle timescales, and alter network parameters and stochastic behavior.
- Biochemical noise arises intrinsically from reaction dynamics and extrinsically from interactions with other stochastic systems.Intrinsic fluctuation magnitude increases at low molecule copy numbers.
- Extrinsic fluctuations dominate cellular variation in both prokaryotes and eukaryotes and typically persist for lifetimes comparable to the cell cycle.
- Noise is commonly quantified using the coefficient of variation, while paired network measurements separate intrinsic differences from extrinsic correlation.Intrinsic fluctuations are uncorrelated between paired copies; extrinsic fluctuations affect both copies and correlate them.
- Extrinsic fluctuations vary network parameters, such as translation rate through changes in free-ribosome number, and their timescale can alter protein fluctuation lifetimes and mean numbers.Their nonspecific action on multiple parameters can combine constructively or destructively.
- The authors extend the standard stochastic simulation algorithm to model discontinuous, time-varying parameters and extrinsic fluctuations with desired properties.
Results
Extrinsic fluctuations reshape protein distributions, noise, response times, and regulatory effects. Their impact depends on magnitude, lifetime, parameter correlations, and network architecture, producing amplification or attenuation in feedforward motifs.
- Extrinsic fluctuations alter mean protein numbers and intrinsic noise: Extrinsic fluctuations can lower the mode and mean protein numbers, increase variance, lengthen high-number tails, and correlate paired protein outputs.Without extrinsic fluctuations, paired outputs are independent; with them, the joint distribution spreads along I1 = I2.
- Extrinsic fluctuations alter mean protein numbers and intrinsic noise: Increasing either a parameter’s coefficient of variation or extrinsic-fluctuation lifetime increases extrinsic noise in protein numbers.
- Extrinsic fluctuations alter mean protein numbers and intrinsic noise: Translation-rate fluctuations can produce over a twofold increase in intrinsic noise despite little change in mean protein number.Changes in extrinsic variables reshape the joint distribution and its projection onto measured protein variables.
- Extrinsic fluctuations can affect the performance of genetic networks: Parameter fluctuations enable stochastic sensitivity analysis: sensitive parameters generate high intrinsic, extrinsic, and total noise in the investigated network property.
- Extrinsic fluctuations can affect the performance of genetic networks: Extrinsic noise contributes to inverse-square-root scaling of total protein noise with mean protein number and correlates protein-fluctuation timescales with total noise.
- Extrinsic fluctuations can affect the performance of genetic networks: Negative feedback reduces extrinsic noise but also lowers mean protein numbers, so total noise may either increase or decrease depending on whether intrinsic or extrinsic noise dominates.
- Extrinsic fluctuations can affect the performance of genetic networks: Negative auto-regulation reduces the mean time to half steady-state protein number by at least a factor of two, while extrinsic noise increases response-time asymmetry and mean response time.
- Extrinsic fluctuations can combine destructively and constructively: Uncorrelated parameter fluctuations combine approximately additively, whereas correlated fluctuations amplify outputs when parameter effects align and attenuate them when effects oppose.
Discussion
The paper extends stochastic simulation to model extrinsic fluctuations and shows that their timescales, correlations, and nonspecific effects can reshape biochemical-network noise and function.
- Discussion: The extended simulation approach models extrinsic fluctuations with arbitrary properties, including correlated or uncorrelated variation across multiple parameters.Repeating simulations with the same extrinsic trajectory also allows intrinsic fluctuations to be averaged separately.
- Discussion: Both fluctuation magnitude and timescale are needed to predict how interactions between stochastic systems affect protein distributions and intrinsic noise.Timescale mixing can shift means, create asymmetry, and sometimes decrease intrinsic noise.
- Discussion: Transcription and translation rates are predicted to be the most significant sources of extrinsic fluctuations in the modeled gene-expression system.The proposed biological sources are fluctuations in ribosome and RNA-polymerase numbers, but the prediction is model-specific.
- Discussion: Correlated parameter fluctuations can either amplify protein-output noise far beyond independent fluctuations or nearly cancel through network architecture.The outcome depends on whether the network combines the fluctuations constructively or destructively.
- Discussion: Extrinsic fluctuation timescales and nonspecificity are important intracellular influences that networks may control or exploit.The simulation algorithm and mathematical analysis are intended to support quantitative understanding of endogenous and synthetic biochemical networks.
Materials and methods
The method extends first-reaction stochastic simulation to discontinuous, time-dependent propensities and generates positive, colored extrinsic fluctuations for network parameters.
- Materials and methods: The first-reaction algorithm calculates a putative time for every potential reaction and implements whichever reaction time occurs first.Each putative time is determined from the reaction propensity, the probability per unit time multiplied by the available reactant selections.
- Materials and methods: Time-dependent propensities are approximated with stepwise or piecewise-linear functions across sampled ∆t intervals.When a predicted reaction crosses an interval boundary, the simulation updates the propensity, advances to the boundary, and recalculates all putative reaction times.
- Materials and methods: For a linear propensity within an interval, Eq. 1 determines the reaction time when it lies within 0 ≤ τ ≤ ∆t; otherwise the reaction cannot occur there.A pre-generated extrinsic-noise time series changes reaction rates during simulation.
- Materials and methods: An Ornstein–Uhlenbeck process is exponentiated and normalized to generate positive extrinsic parameter fluctuations with fixed mean and finite autocorrelation time.This log-normal process avoids the negative parameter values that can arise when a normally distributed process is added directly to a rate.
- Materials and methods: Simulations use the Gibson–Bruck Gillespie method with the Facile network compiler and stochastic simulator.For Fig. 3, model parameters are sampled from log-normal distributions, with specified variance choices for most parameters and promoter activity.
- Materials and methods: Intrinsic and extrinsic noise are quantified using protein counts from two identical network copies and their time-averaged fluctuations.The copies share the same mean protein number, allowing the two-copy measurements to separate noise components.