In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene transcription takes place out of equilibrium. In this paper we discuss a simple mapping between dynamic models of gene expression and stochastic systems driven out of equilibrium. Using this mapping, results of nonequilibrium statistical mechanics such as the Jarzynski equality and the fluctuation theorem are demonstrated for gene expression dynamics. Applications of this approach include the determination of regulatory interactions between genes from experimental gene expression data.
Deep Dive into Non-equilibrium dynamics of gene expression and the Jarzynski equality.
In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene transcription takes place out of equilibrium. In this paper we discuss a simple mapping between dynamic models of gene expression and stochastic systems driven out of equilibrium. Using this mapping, results of nonequilibrium statistical mechanics such as the Jarzynski equality and the fluctuation theorem are demonstrated for gene expression dynamics. Applications of this approach include the determination of regulatory interactions between genes from experimental gene expression data.
Cellular dynamics is based on the expression of specific genes at specific times. The control over gene expression is a crucial feature of nearly all forms of life, as it allows an organism to respond to changing external and internal conditions. With perfect regulatory control, only the DNA of those genes whose products are required at a given instant would be transcribed to m(essenger)RNA molecules. These mRNA molecules are in turn translated to proteins. For example, enzymes to break down nutrients are produced only when nutrients are present, or repair proteins are assembled to respond to DNA damage.
To initiate the transcription of a gene, specific molecules, called transcription factors, locate and bind to DNA near the starting site of a gene. These molecules attract and activate an enzyme which reads off DNA, producing an RNA chain molecule according to the DNA template. Transcription factor molecules are themselves proteins and thus subject to regulatory control, through other transcription factors, or through themselves. As a result, mRNA and protein concentrations of different genes may have highly non-trivial interdependencies. A prominent example is the spatial-temporal evolution of protein concentrations in the early stages of embryonic development, leading to the formation of the body plan of an organism [1].
Despite the need for stringent control, gene regulation is an inherently noisy process [2]. At the level of single cells, only few molecules are involved, with single events potentially having a large impact [3].
In this paper, the dynamics of mRNA concentrations in synchronized cell populations is studied. The simplest model for the concentration x(t) of a given mRNA is [4,5,6]
where η is the decay constant of the mRNA molecule and f is the average rate at which new molecules are produced by transcription of the corresponding gene. Transcription of a gene leads to the production of mRNA molecules at some rate f . mRNA molecules decay at a rate η per molecule. b) The resulting dynamics of mRNA concentration x can be mapped onto an harmonic oscillator subject to a restoring force -ηx and an external force f driving the system out of equilibrium.
The term ξ(t) describes all other processes, including changes in the transcription rate due to changing transcription factor concentrations. Their influence has been modeled by a random uncorrelated variable with mean zero and covariance ξ(t)ξ(t ′ ) = δ(t -t ′ ) [5,6]. Equation (1) is well-known as the Langevin-equation of an Ornstein-Uhlenbeck process describing the motion of an overdamped particle with position x in a quadratic potential V (x) = (ηx -f ) 2 /(2η) [7]. A thermal bath with inverse temperature β = 2/D given by the Einstein relation exerts a random force leading to an equilibrium solution P eq (x) ∼ exp{-βV (x)}, see Fig. 1.
We probe this equilibrium scenario using experimental measurements [8] of expression levels of all yeast genes taken at discrete intervals ∆ t [31]. In order to allow comparison across genes, we rescale the expression levels x of each gene using q = 2/(Dη)(ηx -f ) so the distribution of q in equilibrium is P (q) ∼ exp{-q 2 /2}. The parameters η, f, D for each gene were determined by maximizing the likelihood P η,f,D (x) of the expression levels x ≡ {x(t)} with respect to the free parameters. The
is given in terms of the short-term propagator of the Langevin equation (1). Drift and diffusion under this propagator can be compared in detail with the experimentally measured expression levels [9].
Figure 2 shows the distribution of rescaled expression levels q across all genes and times. While the observed distribution P (q) is roughly compatible with the equilibrium Gaussian distribution, the statistics of expression levels is not stationary. As an example, we consider the set of target genes of a transcription factor called Swi4 [32]. The average value q(t) Swi4 of the target genes at different times varies over the experimental time course, and these average values are correlated with the expression levels of the transcription factor Swi4, see inset of Fig. 2.
The set of (rescaled) expression levels of all ast genes at different times along the cell cycle has a distribution roughly compatible with the equilibrium distribution of the Langevin equation (1) (solid red line). Deviations at high and low expression levels might in principle be due to non-linearities of DNA hybridisation to probes. Inset: However, the distribution of expression levels is not stationary, but changes with the expression level of transcription factors. Here the mean expression levels q(t) Swi4 of Swi4 target genes at a given time t are plotted against the expression level y(t -∆t) of their transcription factor Swi4 at the preceding measurement. The mean expression level of target genes is positively correlated with the expression level of the transcription factor, which changes continuously over the cell cycle.
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