One third of dynamic protein expression profiles can be predicted by a simple rate equation

Abstract

Cells respond to environmental stimuli with expression changes at both the mRNA and protein level, and a plethora of known and unknown regulators affect synthesis and degradation rates of the resulting proteins. To investigate the major principles of gene expression regulation in dynamic systems, we estimated protein synthesis and degradation rates from parallel time series data of mRNA and protein expression and tested the degree to which expression changes can be modeled by a simple linear differential equation. Examining three published datasets for yeast responding to diamide, rapamycin, and sodium chloride treatment, we find that almost one-third of genes can be well-modeled, and the estimated rates assume realistic values. Prediction quality is linked to low measurement noise and the shape of the expression profile. Synthesis and degradation rates do not correlate within one treatment, consistent with their independent regulation. When performing robustness analyses of the rate estimates, we observed that most genes adhere to one of two major modes of regulation, which we term synthesis- and degradation-independent regulation. These two modes, in which only one of the rates has to be tightly set, while the other one can assume various values, offer an efficient way for the cell to respond to stimuli and re-establish proteostasis. We experimentally validate degradation-independent regulation under oxidative stress for the heatshock protein Ssa4.

Figure 1. Predictability of protein expression can have several reasons

The panels show representative RNA and protein time-series profiles of yeast under diamide stress. Panels A–C show one of the genes whose protein levels are predicted well (FDR<30%) for all three data sets. Our model can predict increasing (A), decreasing (B), and more complex profiles (C, F). High predictability is defined by a high Spearman correlation (and FDR<30%) between the observed protein abundance profile (black) and the protein abundance profile predicted with the non-negative linear square fit ODE model (blue). Panels D–E display profiles with a low predictability. In both cases, lack of predictability is likely due to a systematic measurement error in the protein data. Panels E and F show two genes whose biological functions are similar but whose profiles and predictabilities vary. Diamide stress data is presented in absolute abundance units (number of molecules per cell), and the rest of the data is presented in arbitrary units (a.u.) derived from relative abundances.

Figure 2. Measurement noise impacts prediction quality

Each point represents a gene in the Diamide (A, D), Sodium chloride (B, E), or Rapamycin (C, F) stress data set. The y-coordinate of each point represents the prediction quality (measured in Spearman correlation R_s ) of the corresponding gene product, and the x-coordinate represents the number of direction changes in either the protein profile (A–C) or the RNA profile (D–F). The number of direction changes is used as a proxy for noise measurement. Panels A–C show that there is a dependency between prediction quality and noise in the protein profiles (F-test, ** for p<0.01, *** for p<0.001), with more noise corresponding to lower prediction quality. D–F show that there is no such relationship between noise in the RNA profiles and prediction quality. The box in each box plot denotes the interquartile range.

Figure 3. Relationship between synthesis and degradation rates

Each point represents a gene with its rates of protein synthesis or degradation (ks and kd values, respectively). Panels (a)–(c) describe the rates predicted by the ODE model under diamide, sodium chloride, and rapamycin stress respectively, with only the well-predicted genes plotted (FDR=30%). Panel (d) describes measured rates^, . We conclude the correlation is spurious and discuss the reasons in the supplement. a.u. – arbitrary units, P – protein concentration, R – RNA concentration, r – Pearson correlation coefficient, t - time.

Figure 4. Parameter landscapes exploring uniqueness of rate prediction

Four examples of parameter landscape profiles under diamide stress. A–B show the two most common profiles. Each point represents a certain combination of synthesis rate k_s and degradation rate k_d, and the color represents the quality of the fit that our model produces given these parameters. The quality of fit is given in fraction of variance unexplained (fvu) with darkest blue corresponding to values close to 0 (best fit) and the darkest red representing all fits with fvu greater than or equal to 1.0. The rest of the values fall in between as shown on the color scale on the rightmost column of every heatmap. C presents an example of a gene for which the fit is of approximately the same quality for a wide range of k_s and k_d values, and D shows an example where the solution is rather unique with very few additional solutions. The black circle on each heatmap represents the parameters that optimize the prediction (reported k_s and k_d values), and the white circle marks the center of the heatmap.

Figure 5. Experimental validation of degradation-independent regulation

Panels A and B show the mRNA and protein expression profile with the predicted protein concentrations and the parameter landscape for the SSA4 gene, respectively. In panel C, the western blot shows the abundance change of the GFP-tagged heat shock protein Ssa4 in cells under diamide stress. After 90 min of 1.5 mM diamide stress, we observe a large increase in Ssa4 abundance, confirming the mass spectrometry results . Consistent with Ssa4’s degradation-independent mode of regulation, inhibition of protein synthesis with cycloheximide (CHX) substantially affects protein expression under stress, and inhibition of protein degradation with MG-132 has only a minor effect on Ssa4 protein expression levels. GAPDH was used as loading control.

Figure 6. Parameter landscape clusters correspond to profile clusters

Each heatmap shows a representative example for a cluster of parameter landscapes (see text). The k_s and k_d axes denote the synthesis and the degradation rates, respectively. The color represents the fraction of variance unexplained (fvu), with the bluer hue representing a lower value. The darkest red represents all values of fvu greater than or equal to 1.0. The line graph to the right of each heatmap represents the median profile of the profile cluster that has the largest intersection (by the number of genes in it) with the given cluster of parameter landscapes (see Table 2, Table S3). There are two major types of parameter landscape / profile cluster pairs: the “up-up” (left) and the “down-down” (right). These pairs occur in all three experiments (Diamide, Sodium chloride, Rapamycin), but contain different functional members (Table 2). P – protein concentration; R – RNA concentration, k_s – synthesis rate, k_d – degradation rate.

Figure 7. Predicted rates are not random

The histograms show log (base e)-ratios of predicted Diamide stress and observed steady-state synthesis (A) and degradation rates (B). The bars in represent the data points where the stress-induced k_s values are inferred from the well-modeled genes (at FDR 30%). The mean of both of the distributions is significantly different from 0 (one-sample t-test, p=1e-4 and p=3e-4 respectively). The log ratio of 0 is denoted by a black vertical line.