Colored extrinsic fluctuations and stochastic gene expression

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.     

Robust filtering circuit design for stochastic gene networks under intrinsic and extrinsic molecular noises

How to design a robust gene network to tolerate more intrinsic kinetic parameter variations and to attenuate more extrinsic environmental noises to achieve a desired filtering level will be an important topic for systems biology and synthetic biology. At present, there is no good systematic design method to achieve robust gene network design. In this study, a gene network suffering from intrinsic kinetic parameter fluctuations and extrinsic environmental noises is modeled as a Langevin equation with state-dependent stochastic noises. Based on the nonlinear stochastic filtering theory, a systematic gene circuit design method is proposed to make gene networks improve their robustness to tolerate more intrinsic noises and to attenuate extrinsic noises to a prescribed filtering level. The robust gene network design principles have not only yielded a comprehensive design theory of robust gene networks, but also gained valuable insights into the molecular noise filtering of gene networks from the systematic perspective.     

Adjusting phenotypes by noise control

Genetically identical cells can show phenotypic variability. This is often caused by stochastic events that originate from randomness in biochemical processes involving in gene expression and other extrinsic cellular processes. From an engineering perspective, there have been efforts focused on theory and experiments to control noise levels by perturbing and replacing gene network components. However, systematic methods for noise control are lacking mainly due to the intractable mathematical structure of noise propagation through reaction networks. Here, we provide a numerical analysis method by quantifying the parametric sensitivity of noise characteristics at the level of the linear noise approximation. Our analysis is readily applicable to various types of noise control and to different types of system; for example, we can orthogonally control the mean and noise levels and can control system dynamics such as noisy oscillations. As an illustration we applied our method to HIV and yeast gene expression systems and metabolic networks. The oscillatory signal control was applied to p53 oscillations from DNA damage. Furthermore, we showed that the efficiency of orthogonal control can be enhanced by applying extrinsic noise and feedback. Our noise control analysis can be applied to any stochastic model belonging to continuous time Markovian systems such as biological and chemical reaction systems, and even computer and social networks. We anticipate the proposed analysis to be a useful tool for designing and controlling synthetic gene networks.     

Noise-reduction through interaction in gene expression and biochemical reaction processes

We demonstrate that interaction in gene expression and biochemical reaction processes has a significant influence on reducing fluctuations. Especially, we have found that the interaction between synthesized proteins and background molecules can reduce the fluctuation level in gene expression, which is a counter example to the intuition that background factors disturb information processing in genetic networks by increasing the noise level. This fact also indicates that the macromolecular crowding observed in actual cells can contribute to reduce the noise level. In addition, the noise-reduction phenomenon is not limited to the interaction between the proteins and background molecules, but can be applied to other reactions such as a dimerization process and the coupling of reactions with large fluctuations by intrinsic noise. Finally, on the basis of these results, we propose a new and plausible method for reducing the fluctuations generated in synthesized genetic networks, and also discuss the applicability of this method to the stabilization of system dynamics by using a toggle switch model.     

Intrinsic and external noise in an auto-regulatory genetic network

A single gene auto-regulatory network is analysed. The main goal is to investigate the effects of the negative and positive feedbacks on the intrinsic and external noises. The central finding of this paper is that: for the intrinsic noise, both the negative and positive feedback regulations increase the fluctuation strength of mRNA levels (where the fluctuation strength is measured by the Fano factor for both the fluctuations of mRNAs and proteins), and the negative feedback decreases, but the positive feedback increases, the fluctuation strength of proteins; for the external noise, the negative feedback not only increase the fluctuation strength of mRNA levels but also the fluctuation strength of proteins, and though the effect of the positive feedback on the fluctuation strength of mRNA levels depends on the size of positive feedback parameter k, the positive feedback must decrease the fluctuation strength of proteins.     

Noise suppression in stochastic genetic circuits using PID controllers

Inside individual cells, protein population counts are subject to molecular noise due to low copy numbers and the inherent probabilistic nature of biochemical processes. We investigate the effectiveness of proportional, integral and derivative (PID) based feedback controllers to suppress protein count fluctuations originating from two noise sources: bursty expression of the protein, and external disturbance in protein synthesis. Designs of biochemical reactions that function as PID controllers are discussed, with particular focus on individual controllers separately, and the corresponding closed-loop system is analyzed for stochastic controller realizations. Our results show that proportional controllers are effective in buffering protein copy number fluctuations from both noise sources, but this noise suppression comes at the cost of reduced static sensitivity of the output to the input signal. In contrast, integral feedback has no effect on the protein noise level from stochastic expression, but significantly minimizes the impact of external disturbances, particularly when the disturbance comes at low frequencies. Counter-intuitively, integral feedback is found to amplify external disturbances at intermediate frequencies. Next, we discuss the design of a coupled feedforward-feedback biochemical circuit that approximately functions as a derivate controller. Analysis using both analytical methods and Monte Carlo simulations reveals that this derivative controller effectively buffers output fluctuations from bursty stochastic expression, while maintaining the static input-output sensitivity of the open-loop system. In summary, this study provides a systematic stochastic analysis of biochemical controllers, and paves the way for their synthetic design and implementation to minimize deleterious fluctuations in gene product levels.     

Effect of feedback regulation on stochastic gene expression

Stochastic noise in gene expression arises as a result of species in small copy number undergoing transitions between discrete chemical states. Here the noise in a single gene network is investigated using the Omega-expansion techniques. We show that the linear noise approximation implies an invariant relationship between the normalized variances and normalized covariance in steady-state statistics. This invariant relationship provides an exactly statistical interpretation for why the stochastic noise in gene expression should be measured by the normalized variance. The nature of the normalized variance reveals the basic relationship between the stochasticity and system size in gene expression. The linear noise approximation implies also that for both mRNA and protein, the total noise can be decomposed into two basic components, one concerns the contribution of average number of molecules, and other the contribution of interactions between mRNA and protein. For the situation with linear feedback, our results clearly show that for two genes with the same average number of protein molecules, the gene with negative feedback will have a small protein noise, i.e., the negative feedback will reduce the protein noise. For the effect of the burst size on the protein noise, we show also that the protein intrinsic noise will decrease with the increase of the burst size, but the protein extrinsic noise is independent of the burst size.