Decomposing fluctuations in the output of a biochemical system. In this example, we consider how fluctuations in three variables, denoted Y_i, affect fluctuations in Z, the system’s output. Y₁ is a biochemical species within the system being studied; Y₂ and Y₃ are variables in other stochastic systems that interact with the system of interest, but whose dynamics are principally generated independently of that system. We show Y₂ and Y₃ in the same system, but they need not be.

Designs of conjugate reporters to measure the effects of different cellular subsystems on variation in output. (A) To distinguish transcriptional from translational effects, three reporters are needed including a bicistronic mRNA with two independent ribosome binding sites. (B) Simulated results for the reporters in A assuming that extrinsic fluctuations affect only the rate of transcription, which fluctuates between three different levels (reactions and parameter values are given in SI Text). Blue dots show Z plotted against Z^′: The average spread along the Z = Z^′ diagonal equals the sum of V[Z] and the extrinsic variance; the average spread perpendicular to the diagonal equals the sum of the transcriptional and translational variation (SI Text). Red dots show Z plotted against Z^′′: the average spread along the diagonal equals the sum of V[Z], extrinsic, and transcriptional variation; the average spread perpendicular to the diagonal equals translational variation. For the parameters chosen (SI Text), the translational noise (coefficient of variation) is 0.12; the transcriptional noise is 0.39; and the extrinsic noise is 0.41. These numbers agree with Eqs. 9 through 11 to two decimal places. (C) Four reporters are needed to distinguish transductional variation from variation generated by gene expression. Here, a signaling network activates a transcription factor, T, in response to extracellular inputs. To measure variation in the output Z arising from gene expression, we require two conjugate reporters, Z and Z^′, whose expression is controlled by this transcription factor. To find a bound on transductional variation, we use two further conjugate and constitutively expressed reporters, Z_c and .

Determining informational variation for osmosensing in budding yeast allows us to predict the probability of the different osmotic conditions experienced by yeast. (A) Hyperosmotic stress is sensed by two pathways in budding yeast, which activate the MAP kinase kinase kinases Ste11 and Ssk2/22 (22). Both these kinases activate the MAP kinase kinase Pbs2, which in turn activates the MAP kinase Hog1. Activated Hog1 translocates from the cytosol to the nucleus and initiates new gene expression. (B) Histograms of fluorescence data from a YFP reporter expressed from the promoter for STL1 and measured by Pelet et al. (19). Fluorescence levels typically increase with increasing extracellular salt: Blue corresponds to zero extracellular salt; dark green to 0.05 M salt; red to 0.1 M; cyan to 0.15 M; magenta to 0.2 M; and brown to 0.4 M. Approximately 1,000 data points were measured for each concentration (19) and are shown using 20 bins for the fluorescence level (calculated in log-space). The left inset shows the same histograms but weighted by the probability of the different salt concentrations for an input distribution that has a low informational fraction of output variance; the right inset is analogous but for an input distribution that has a high informational fraction of output variance. (C) The five probability distributions for extracellular salt that give the five highest informational fractions (each approximately equal to 0.8 because of the high degree of overlap of the fluorescence distributions for zero and 0.05 M salt). Each distribution is read horizontally. We calculated the informational fraction for all possible probability distributions of the six concentrations of extracellular salt that were chosen experimentally. The informational fraction decreases continuously from around 0.8 to zero. A uniform probability distribution of salt gives an informational fraction of approximately 0.6.

A simple example: quantifying transcriptional and translational variation. By conditioning on the history of mRNA levels, , and of all processes extrinsic to gene expression,, we can decompose the variance in the level of the protein, Z, into three components. For example, translational variation is the extra variation generated on average given the joint history of mRNA and Y_e. This decomposition implies that we need a reporter for protein levels (Z), a reporter conjugate given the history of Y_e (Z′), and a reporter conjugate given the joint history of M and Y_e (Z′′). Scatter plots (here using simulated data) give a visual representation of the magnitude of the variance components. Each dot represents measurements of two reporters in an individual cell. For Z versus Z′, scatter perpendicular to the diagonal Z = Z′ gives the sum of transcriptional and translational variation and scatter parallel to the diagonal gives the sum of the variance of Z and extrinsic variation. For Z versus Z′′, perpendicular scatter is given by translational variation and parallel scatter is given by the variance of Z and both transcriptional and extrinsic variation. The extrinsic variation is equal to the covariance of Z and Z′.