Bioinformatics Advance Access originally published online on August 29, 2006
Bioinformatics 2006 22(21):2709-2710; doi:10.1093/bioinformatics/btl456
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Tropical—parameter estimation and simulation of reaction–diffusion models based on spatio-temporal microscopy images
1 Department of Theoretical Bioinformatics, German Cancer Research Center Im Neuenheimer Feld 580, 69120 Heidelberg, Germany
2 Institute for Pharmacy and Molecular Biotechnology, University of Heidelberg Im Neuenheimer Feld 364, 69120 Heidelberg, Germany
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 PROGRAM OVERVIEW3 SUMMARY AND FUTURE...REFERENCES |
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Summary: Tropical is a software for simulation and parameter estimation of reaction–diffusion models. Based on spatio-temporal microscopy images, Tropical estimates reaction and diffusion coefficients for user-defined models. Tropical allows the investigation of systems with an inhomogeneous distribution of molecules, making it well suited for quantitative analyses of microscopy experiments such as fluorescence recovery after photobleaching (FRAP).
Availability: Tropical is available free of charge for academic use at http://www.dkfz.de/tbi/projects/modellingAndSimulationOfCelluarSystems/tropical.jsp after signing a material transfer agreement.
Contact: r.eils{at}dkfz.de
Supplementary information: http://www.dkfz.de/tbi/projects/modellingAndSimulationOfCelluarSystems/tropical.jsp
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 PROGRAM OVERVIEW3 SUMMARY AND FUTURE...REFERENCES |
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Fluorescence microscopy techniques such as time-lapse microscopy and fluorescence recovery after photobleaching (FRAP) have emerged as standard tools for visualizing molecule dynamics in living cells. Quantitative characterization of the motility of molecules requires the fitting of reaction–diffusion models to spatio-temporal data extracted from images (Beaudouin et al., 2006; Delon et al., 2006; Sprague et al., 2004). Computational tools for such quantitative analysis are still missing. Our motivation was thus to develop a program to enable parameter estimation and simulation for complex reaction–diffusion models. Tropicals' main advantages are that (1) an inhomogeneous distribution of binding partners can be considered and (2) that it directly operates on microscopy images. So far only a few studies have incorporated the inhomogeneous distribution of binding sites (Beaudouin et al., 2006; Sprague et al., 2006).
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2 PROGRAM OVERVIEW |
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TOPABSTRACT1 INTRODUCTION2 PROGRAM OVERVIEW3 SUMMARY AND FUTURE...REFERENCES |
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An overview of Tropical is given in Figure 1. Tropical estimates the specified parameters and runs a simulation with the best estimation result based on the input of (1) spatio-temporal microscopy images, (2) initial images for all state variables and (3) a user-defined model. The model is composed of one ordinary differential equation describing the reaction of each molecule. The diffusion term is added automatically. We applied a normal diffusion term which can be modified as described by Siggia et al. (2000). Binary compartment images serve as masks for the simulation grid. They define the area where parameters are estimated and specify regions for computing a recovery curve. Additionally, an output window provides the intermediate results during runtime. A log file of the complete parameter estimation process and a result file with estimated parameters and numerical control values are also written. Results are also provided in tiff formatted images and a text file. The complete source code is written in C++ and is available for Linux and Windows. The graphical user interface was developed using QT, version 4.1.2 (www.trolltech.com).
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2.1 Algorithms and libraries
Tiff image I/O is handled by the open source library libtiff (v.3.7.2) (www.remotesensing.org/libtiff/). The model description is compiled from a text file by the open watcom C++ compiler (www.openwatcom.org). Spatial discretization of the partial differential diffusion equations is done by a finite differences approach with Neumann boundary conditions. The underlying grid for this discretization is represented by the pixels of the input images. An optional binning algorithm reduces the resolution and thus accelerates computation. The differential equations are solved by a Runge-Kutta fourth order algorithm with adaptive step size [adapted from Press et al. (2002)]. For parameter estimation the Levenberg–Marquardt algorithm, published by Press et al. (2002) or a modified version of it, which can lead to faster convergence (for details see handbook), may be used.
2.2 Tests and application
To verify Tropical's functionality, we simulated two scenarios with defined parameters using Berkley Madonna (www.berkeleymadonna.com), one diffusion and one reaction–diffusion system. The resulting matrices were transformed into image series using ImageJ (http://rsb.info.nih.gov/ij/) which were then used as input for parameter estimation.
First, we tested parameter estimation on a diffusion system with different initial parameters and different numerical parameters, such as initial step size or step size tolerance for the ODE solver and number of maximum iterations for the Levenberg–Marquardt algorithm. Second, we tested a 2D reaction–diffusion system similar to a FRAP experiment, with an interaction of one observable diffusing molecule on a non-diffusing invisible binding site. The image series for parameter estimation showed the effective sum of the diffusing and the bound observable molecule. This corresponds to a realistic setting, where free and bound forms of a protein cannot readily be distinguished.
In both test cases highly accurate diffusion and reaction coefficients were achieved. For pure diffusion, results differed by 1.7 ± 1.99% from the expected values. For the reaction–diffusion model, D differed by 1.86 ± 0.85% and koff by 0.14 ± 0.04% from the original D and koff, respectively. Deviations of the parameters were calculated by performing 10 parameter estimation runs using different initial parameters. The mean of the 10 estimated parameters was compared to the parameters used to generate the original data with Berkeley Madonna.
Adding noise up to 7% SD of the maximum intensity did not strongly affect the accuracy. The estimated parameters deviated 5–10% from the expected values showing that Tropical can cope with typical levels of noise in fluorescence microscopy images. To test Tropical on a realistic example, we estimated the dissociation and diffusion coefficients from a FRAP experiment on the nuclear protein nucleophosmin (B23). B23 diffuses in the nucleoplasm and reversibly binds to unspecified binding sites located inside the nucleoli (Fig. 1B) (Chen and Huang, 2001). A model of this binding-diffusion system was created which consisted of two state variables (free and bound pool) and two parameters (diffusion coefficient D of the free pool and dissociation rate koff of the binding reaction):
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3 SUMMARY AND FUTURE DEVELOPMENT |
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TOPABSTRACT1 INTRODUCTION2 PROGRAM OVERVIEW3 SUMMARY AND FUTURE...REFERENCES |
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Tropical is an easy to use software for parameter estimation and simulation of reaction–diffusion models using microscopy images and user-defined models. We presently apply this system to a number of proteins for estimating their reaction–diffusion parameters in vivo.
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Acknowledgments |
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The B23 construct was provided by Sui Huang Lab, Northwestern University Medical School. We also wish to thank D. Jackson for carefully editing this paper. We acknowledge financial support by the European Science Foundation and the DFG [EuroDyna EI358/3-1/2] and a project funded by the Human Frontiers in Science Program [RGP19/2003-C302]. Funding to pay the Open Access publication charges was provided by the German Cancer Research Center.
Conflict of Interest: none declared.
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FOOTNOTES |
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The authors wish it to be known that, in their opinion, the first two authors should be regarded as joint First Authors. 
Associate Editor: Alfonso Valencia
Received on June 11, 2006; revised on August 15, 2006; accepted on August 19, 2006
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REFERENCES |
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TOPABSTRACT1 INTRODUCTION2 PROGRAM OVERVIEW3 SUMMARY AND FUTURE...REFERENCES |
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Beaudouin, J., et al. (2006) Dissecting the contribution of diffusion and interactions to the mobility of nuclear proteins. Biophys. J, . 90, 1878–1894
Chen, D. and Huang, S. (2001) Nucleolar components involved in ribosome biogenesis cycle between the nucleolus and nucleoplasm in interphase cells. J. Cell Biol, . 153, 169–176
Delon, A., et al. (2006) Continuous photobleaching in vesicles and living cells: a measure of diffusion and compartmentation. Biophys. J, . 90, 2548–2562
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. Numerical Recipes in C++. The Art of Scentific Computing, (2002) , Cambridge, UK Cambridge University Press.
Siggia, E.D., et al. (2000) Diffusion in inhomogeneous media: theory and simulations applied to whole cell photobleach recovery. Biophys. J, . 79, 1761–1770
Sprague, B.L., et al. (2006) Analysis of binding at a single spatially localized cluster of binding sites by fluorescence recovery after photobleaching. Biophys. J, . 91, 1169–1191
Sprague, B.L., et al. (2004) Analysis of binding reactions by fluorescence recovery after photobleaching. Biophys. J, . 86, 3473–3495
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