Noise-driven causal inference in biomolecular networks

Abstract

Single-cell RNA and protein concentrations dynamically fluctuate because of stochastic ("noisy") regulation. Consequently, biological signaling and genetic networks not only translate stimuli with functional response but also random fluctuations. Intuitively, this feature manifests as the accumulation of fluctuations from the network source to the target. Taking advantage of the fact that noise propagates directionally, we developed a method for causation prediction that does not require time-lagged observations and therefore can be applied to data generated by destructive assays such as immunohistochemistry. Our method for causation prediction, "Inference of Network Directionality Using Covariance Elements (INDUCE)," exploits the theoretical relationship between a change in the strength of a causal interaction and the associated changes in the single cell measured entries of the covariance matrix of protein concentrations. We validated our method for causation prediction in two experimental systems where causation is well established: in an E. coli synthetic gene network, and in MEK to ERK signaling in mammalian cells. We report the first analysis of covariance elements documenting noise propagation from a kinase to a phosphorylated substrate in an endogenous mammalian signaling network.

Fig 1. Biological noise is well approximated by the lognormal distribution.

(A) Single-cell quantification of fluorescent CFP in E. coli treated with 2 mM IPTG. (B) Flow cytometry measurements of ERK1/2 phosphorylated at (T202, Y204) in mouse T cells treated with 50nM PMA.

Fig 2. INDUCE (Inference of Network Directionality Using Covariance Elements) analysis for hypothetical model networks.

(A-D) INDUCE analysis of an isolated network connection. From left to right, (A) the Hill equation model of log concentration transfer function is color registered to y ₂. (B) The derivative of the transfer function is the network connection strength a _2,1, which peaks at half-maximal activation. (C-D) Variance versus covariance plots, the solution to Eq 8 applied at closely-spaced fixed points along the transfer function shown in A. (E-H) INDUCE analysis of a convergent (additive) regulation of node 2. The transfer function y ₂ versus y ₃ shown in black. (F) The connection strength a _2,3 shown in black. (H) Var(y2) versus Cov(y1, y2) has a loop, a hallmark of differential sensitivities to a stimulus. (I-L) INDUCE analysis of a linear cascade. (K, L) Both variance versus covariance plots exhibit loops, a hallmark of differential sensitivities to a stimulus. In all examples, biochemical parameters were chosen to illustrate qualitative differences in the variance versus covariance plots for different connectivities (see Section D in S1 text for parameter values).

Fig 3. E. coli synthetic transcriptional network.

(A) IPTG dose-response of G1. (B) IPTG dose-response of G2. (A-B) Averages of the log of the normalized G1 (A) and G2 (B) concentrations ±1 s.d. (C) Transfer function of G1 versus G2 (log scale). The negative slope is indicative of the repression of G2 by G1. (D) INDUCE analysis showing the fit of a two-node network model (solid curves). Error bars are 1 s.d. in computed in 1000 bootstraps of the data.

Fig 4. MAP kinase signaling.

(A) PMA dose-response of pMEK. (B) PMA dose-response of ppERK. (A-B) Averages of the log of the normalized protein concentrations ±1 s.d. (C) Transfer function of pMEK versus ppERK. The positive slope is indicative of ERK activation by MEK. (D) INDUCE analysis showing the fit of two-node network model (solid curves). Error bars show 1 s.d. in 1000 bootstraps of the data.