(653d) Regularization of Ill-Conditioned Inverse Problems Resulting From Determining Transcription Factor Profiles From Fluorescent Reporter Images

Signal transduction pathways can be viewed as biochemical reaction networks that contain a large number of proteins and even larger number of reactions involving these proteins. It is one of the defining features of signal transduction networks that significant interactions exist among the proteins, which further complicates the task of identifying the role of specific proteins within these networks (Huang et al 2010).  One of the aims of Systems Biology is to extract information about proteins from experimental data in order to improve our understanding and derive models of signal transduction pathways. However, this task can be challenging as the amount of data available from experiments is limited, the data is often qualitative and contains a significant amount of noise. For instance, images obtained from Green Fluorescent Protein (GFP) Reporters have been widely used as an indicator of gene expression and transcription factor activity. However, determining the dynamics and concentrations of transcription factor from the observed fluorescence is not straightforward as GFP requires time for protein folding and fluorophore formation after transcription (van Roessel 2002). Furthermore, it is non-trivial to determine the amount of fluorescence seen in fluorescence microscopy images due to experimental as well as measurement noise. The purpose of this study is to develop a mathematical formulation for solving inverse problems for determining dynamic profiles of transcription factors from a series of fluorescent images available from GFP reporter systems.    

The ODE model originally proposed by Huang et al. (2008) is used to link the transcription factor dynamics to the observed fluorescence intensity. However, unlike the work presented by Huang et al. (2008) where only specific profiles of the transcription factor dynamics could be computed, we used a more general approach for our work. The transcription factor profile was discretized at several points in time and an inverse problem was formulated that computes the transcription factor profile at these discrete time points from the available data which allowed the overall profiles to take any shape.  Due to the significant amount of noise present in the fluorescence intensity profiles, it is required to include a regularization procedure in the inverse problem formulation.   Two regularization methods - Truncated Singular Value Decomposition (TSVD) and Tikhonov Regularization – have been implemented and their results for the solution of this inverse problem are discussed in detail.

The regularization techniques have been evaluated using both simulated data and experimental data available for the transcription factor STAT3 in hepatocytes which are continuously stimulated by IL-6. For each of the cases, the two regularization techniques have been applied for several different choices of the regularization parameters. Both methods performed equally well if little noise was present. However, it was found that the results from the Tikhonov regularization provide a better fit for the simulated data if the noise levels are large. The reason for this is that the tuning parameters of the Tikhonov regularization are continuous and thus can be freely chosen, while the TSVD approach is more restrictive because of discrete values of the tuning parameters.  For the case study involving real data, the dynamic profiles of the transcription factor STAT3 obtained after solving the inverse problem were in close agreement with results reported in the literature. The results showed a high initial response in STAT3 concentrations as compared to the long term response which was also observed by Singh et al (2006) from their mathematical model of the IL-6 –STAT3 signal transduction pathway. Further, the second peak in the STAT3 concentration was observed at around 5 hours, previously also reported by Fischer et al (2004). Both the Tikhonov regularization and the TSVD approach are able to predict the time and intensity of the initial peak. However, similar to the investigation based upon simulated data, the results returned by the Tikhonov regularization seem to provide a better fit for the experimental data as compared to TSVD. 

References

Abhay Singh, Arul Jayaraman, Juergen Hahn. "Modeling Regulatory Mechanisms in Il-6 Signal Transduction in Hepatocytes." Biotechnology and Bioengineering 95, no. 5 (2006): 850- 862.

Patrick Fischer, Ute Lehmann, Radoslaw M. Sobota, Jochen Schmitz, Claudia Niemand, Sonja Linnemann, Serge Haan, Iris  Behrmann, Akihiko Yoshimura, James A. Johnston, Gerhard Muller-Newen, Peter C. Heinrich and Fred Schaper. "The Role of the Inhibitors of Interleukin-6 Signal Transduction Shp2 and Socs3 for Desensitization of Interleukin-6 Signalling." Biochemistry Journal 378 (2004): 449 - 460.

Peter van Roessel, Andrea H. Brand. "Imaging into the Future: Visualizing Gene Expression and Protein Interactions with Fluorescent Proteins." Nature Cell Biology 4 (2002).

Zuyi (Jacky) Huang, Yunfei Chu, Juergen Hahn. "Model Simplification Procedure for Signal Transduction Pathway Models: An Application to Il-6 Signaling." Chemical Engineering Science 65 (2010).

Zuyi Huang, Fatih Senochak, Arul Jayaraman and Juergen Hahn. "Integrated Modeling and Experimental Approach for Determining Transcription Factor Profiles from Fluorescent Reporter Data." BMC Systems Biology 2 (2008): 11.