Statistical ensemble analysis for simulating extrinsic noise-driven response in NF-κB signaling networks

Joo et al. BMC Systems Biology 2013, 7:45 http://www.biomedcentral.com/1752-0509/7/45 RESEARCH ARTICLE Open Access Statistical ensemble analysis for simulating extrinsic noise-driven response in NF-κB signaling networks Jaewook Joo1*, Steven J Plimpton2 and Jean-Loup Faulon3 Abstract Background: Gene expression profiles and protein dynamics in single cells have a large cell-to-cell variability due to intracellular noise. Intracellular fluctuations originate from two sources: intrinsic noise due to the probabilistic nature of biochemical reactions and extrinsic noise due to randomized interactions of the cell with other cellular systems or its environment. Presently, there is no systematic parameterization and modeling scheme to simulate cellular response at the single cell level in the presence of extrinsic noise. Results: In this paper, we propose a novel statistical ensemble method to simulate the distribution of heterogeneous cellular responses in single cells. We capture the effects of extrinsic noise by randomizing values of the model parameters. In this context, a statistical ensemble is a large number of system replicates, each with randomly sampled model parameters from biologically feasible intervals. We apply this statistical ensemble approach to the well-studied NF-κB signaling system. We predict several characteristic dynamic features of NF-κB response distributions; one of them is the dosage-dependent distribution of the first translocation time of NF-κB. Conclusion: The distributions of heterogeneous cellular responses that our statistical ensemble formulation generates reveal the effect of different cellular conditions, e.g., effects due to wild type versus mutant cells or between different dosages of external stimulants. Distributions generated in the presence of extrinsic noise yield valuable insight into underlying regulatory mechanisms, which are sometimes otherwise hidden. Keywords: Statistical ensemble, Extrinsic noise, Cell to cell variability, NF-κB signal transduction network Background Single cell imaging generated a surge of interest in the intracellular dynamics of biochemical species, uncovering significant cell-to-cell variations in gene expression [1-8] and protein dynamics [9,10]. This variability originates from intrinsic [1-8] and extrinsic noise [3,6,10] and critically affects cellular decision-making processes [9-13]. Moreover, cellular response averaged over a population of cells is oftentimes noticeably different from the responses of single cells. The variability in the latter contains rich information regarding the regulatory mechanisms in operation. Here, we present a novel computational method to predict the distribution of extrinsic noise-driven heterogeneous cellular * Correspondence: 1 Department of Physics and Astronomy, University of Tennessee, Knoxville 37996, USA Full list of author information is available at the end of the article responses and to unravel discrepancies between single-cell versus population-averaged responses. Both intrinsic and extrinsic noise are the source of the large cell-to-cell variability in cellular responses [14]. Intrinsic noise refers to the pure probabilistic nature of individual biochemical reactions occurring within a cell. When the number of intracellular constituents is large, the cell’s behavior is well approximated by its expectation value according to the law of large numbers. But at the single-cell level, the number of molecules of certain species critical to a particular biochemical pathway can be small, and the range of statistical variation in the system needs to be considered [1-8]. Extrinsic noise refers to random interactions of the cell with other cells or its environment. Extrinsic fluctuations can originate from cells undergoing different stages of their cell cycle [15], fluctuations in the number of transcriptional regulators upstream of the signaling pathway of interest [3,6,9,10], © 2013 Joo et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Joo et al. BMC Systems Biology 2013, 7:45 http://www.biomedcentral.com/1752-0509/7/45 Page 2 of 18 and cell-to-cell variability in the copy number of proteins inherited from parent cells during cell division [10]. Extrinsic noise can affect the dynamics of cellular constituents locally in a specific signaling pathway or globally over the entire cell. In Figure 1, we summarize the effects of intrinsic and extrinsic fluctuations in the NF-κB signaling networks. The full effect of extrinsic noise should include “all” external stochastic effects that influence the cell, particularly the temporal fluctuations in the cellular kinetic conditions. However, in Ref. [10], Spencer et al. identified the most important source of extrinsic noise as the protein copy number inherited from the parent cell during cell division. Large cell-to-cell variations in the copy number of enzyme and regulatory protein could randomize the likelihood and the speed of any intracellular biochemical reaction. This means we can effectively “lump” all the effects of protein copy number variations into variations in kinetic rate constants. This is an attractive approach, because rate constants are an input into a variety of biochemical pathway modeling techniques. A pathway modeling framework that uses deterministic or stochastic differential equation models requires a priori knowledge of the structure of the biochemical reaction network, mathematical functional forms for the biochemical reactions, and associated reaction rate constants. Since limited or incomplete information is often all that is available to modelers, a computational model is often parameterized by using a nonlinear fitting algorithm. A conventional parameterization scheme identifies a single set of kinetic parameter values by minimizing the χ2 distance between experimental data and a prediction made by the model. Sloppy Cell and other similar parameterization algorithms include experimental errors in the parameterization by fitting to a rather large experimental error bar [16]. But both Intrinsic noisy systems External Signal IKK I B I B ,I B NF- B A20 conventional and Sloppy Cell parameterization schemes assume a deterministic and homogeneous biological response to a stimulus and aren’t designed to handle the heterogeneous, stochastic behavior of single cells and its dependence on extrinsic noise. In order to capture extrinsic noise and its effect on intracellular response, we propose a novel parameterization method, the “statistical ensemble” (SE) scheme, named after a key concept in statistical physics [17]. A cell is regarded as a complex system comprising a large number of components and elementary interactions among them (...truncated)