Colored extrinsic fluctuations and stochastic gene expression
Vahid Shahrezaei1, Julien F Ollivier1 & Peter S Swain1
- Department of Physiology, Centre for Non-linear Dynamics, McGill University, Montreal, Quebec, Canada
Correspondence to: Peter S Swain1 Department of Physiology, Centre for Non-linear Dynamics, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec, Canada H3G 1Y6. Tel.: +1 514 398 4360; Fax: +1 514 398 7452; Email: swain@cnd.mcgill.ca
Received 3 January 2008; Accepted 3 April 2008; Published online 6 May 2008
Article highlights
- Biochemical networks experience both intrinsic fluctuations, generated by the stochasticity inherent to biochemical reactions within the network, and extrinsic fluctuations arising from the interactions of the network with other stochastic systems in the cell or its environment. Here we extend the standard stochastic simulation algorithm to include extrinsic fluctuations with any desired properties.
- We show that extrinsic fluctuations that are 'colored', or have a significant lifetime, can affect both measurements of mean protein numbers and the intrinsic noise and that these effects can explain trends in high throughput measurements of stochasticity.
- Extrinsic fluctuations can affect the performance of simple, genetic networks. For a negatively autoregulated network, we demonstrate that extrinsic fluctuations can enhance or degrade the ability of the network to attenuate stochasticity and can speed up the network's typical response time.
- Extrinsic fluctuations are non-specific, and extrinsic fluctuations in different components of a network can combine destructively to negate each other or constructively to amplify fluctuations in the network's output. These interference effects are likely to be exploited by cells in feedforward loops: incoherent feedforwards can attenuate stochasticity, while coherent feedforwards can amplify it.
Synopsis
Biochemical networks experience both intrinsic fluctuations, generated by the stochasticity inherent to biochemical reactions within the network, and extrinsic fluctuations arising from the interactions of the network with other stochastic systems in the cell or its environment. Both types of fluctuations can be controlled or exploited by cells.
Intrinsic fluctuations are relatively well understood. They arise from random collisions of reacting molecules with other molecules. Such collisions stochastically alter the time for two reactants to meet and their time to cross a reaction's energy barrier. Intrinsic fluctuations can be described mathematically by a master equation or simulated using a stochastic simulation algorithm, such as the Gillespie algorithm.
Despite usually being the dominant fluctuations in experimental measurements of in vivo protein levels, extrinsic fluctuations have been much less studied. They have been measured to have a lifetime comparable to the cell cycle, but their source is largely speculative. Here, we extend the Gillespie algorithm to include extrinsic fluctuations with any properties and so enable their systematic study, at least by simulation.
Extrinsic fluctuations have a significant lifetime, and, because their lifetime is greater than zero, they are called 'colored'. We show that their color causes extrinsic fluctuations to have a nonlinear effect on the dynamics of a network. They can shift the mean number of proteins and can alter the intrinsic noise found in protein levels. Intrinsic noise is measured by the variation in the difference between two distinguishable reporters, each expressed by a separate copy of the network under study. We describe how these effects, arising from the lifetime of extrinsic fluctuations, can help explain trends seen in high-throughput measurements of stochasticity in gene expression.
Extrinsic fluctuations are also nonspecific and can potentially simultaneously affect many cellular components. We show that extrinsic fluctuations in different components of a network can combine destructively to negate each other or constructively to amplify fluctuations in the network's output. These interference effects are likely to be exploited by cells in feedforward loops: some types of feedforward loops can attenuate stochasticity because they channel extrinsic fluctuations in the levels of the network's master transcription factor to combine destructively with those of the other regulating transcription factor and so decrease fluctuations in the network's output. Others can cause constructive extrinsic fluctuations in the network's transcription factors and so can amplify fluctuations in the network's output.
Our results show that the lifetime of extrinsic fluctuations increases their co-dependence with intrinsic fluctuations. More generally, they address how to model one stochastic system embedded in and interacting with a larger stochastic system. Mathematically, these interactions, or extrinsic fluctuations, cause the parameters of an already stochastic system to become stochastic themselves. Measurements of the stochasticity in the output of the system will have an intrinsic component, arising from the dynamics of the system itself, and an extrinsic component, arising from fluctuations in the system's parameters. For example, in gene expression, the rate of initiation of translation is stochastic because it is a function of the number of free ribosomes which is itself stochastic. Although the empirical definitions of extrinsic and intrinsic noise do not change, we show that the colored character of extrinsic fluctuations does alter their theoretical justification.
Our simulation algorithm and the concepts we propose should help to quantitatively understand endogenous networks and to design effective synthetic ones.
Acknowledgements
We thank M Chacron, M Elowitz, D Gillespie, and N Rosenfeld for useful conversations. In particular, we thank TJ Perkins for showing us the derivation of equation (1). PSS holds a Tier II Canada Research Chair. VS, JFO and PSS are supported by NSERC and the MITACS National Centre of Excellence.
