PDE/ODE modeling and simulation to determine the role of diffusion in long-term and -range cellular signaling

BMC Biophysics

Elfriede Friedmann¹ 

[1]Department of Applied Mathematics, Im Neuenheimer Feld 294, Heidelberg, Germany

关键词: Cellular signaling;    Numerical simulation;    Reaction-diffusion systems;    Systems of coupled differential equations;   

Others  :  1230571
DOI  :  10.1186/s13628-015-0024-8

 received in 2014-11-05, accepted in 2015-09-21,  发布年份 2015

【 摘 要 】

Background

We study the relevance of diffusion for the dynamics of signaling pathways. Mathematical modeling of cellular diffusion leads to a coupled system of differential equations with Robin boundary conditions which requires a substantial knowledge in mathematical theory. Using our new developed analytical and numerical techniques together with modern experiments, we analyze and quantify various types of diffusive effects in intra- and inter-cellular signaling. The complexity of these models necessitates suitable numerical methods to perform the simulations precisely and within an acceptable period of time.

Methods

The numerical methods comprise a Galerkin finite element space discretization, an adaptive time stepping scheme and either an iterative operator splitting method or fully coupled multilevel algorithm as solver.

Results

The simulation outcome allows us to analyze different biological aspects. On the scale of a single cell, we showed the high cytoplasmic concentration gradients in irregular geometries. We found an 11 % maximum relative total STAT5-concentration variation in a fibroblast and a 70 % maximum relative pSTAT5-concentration variation in a fibroblast with an irregular cell shape. For pSMAD2 the maximum relative variation was 18 % in a hepatocyte with a box shape and 70 % in an irregular geometry. This result can be also obtained in a cell with a box shape if the molecules diffuse slowly (with D=1 μm ² /s instead of D=15 μm ² /s). On a scale of cell system in the lymph node, our simulations showed an inhomogeneous IL-2 pattern with an amount over three orders of magnitude (10 ⁻³ −1 pM) and high gradients in face of its fast diffusivity. We observed that 20 out of 125 cells were activated after 9 h and 33 in the steady state. Our in-silico experiments showed that the insertion of 31 regulatory T cells in our cell system can completely downregulate the signal.

Conclusions

We quantify the concentration gradients evolving from the diffusion of the molecules in several signaling pathways. For intracellular signaling pathways with nuclear accumulation the size of cytoplasmic gradients does not indicate the change in gene expression which has to be analyzed separately in future. For intercellular signaling the high cytokine concentration gradients play an essential role in the regulation of the molecular mechanism of the immune response. Furthermore, our simulation results can give the information on which signaling pathway diffusion may play a role. We conclude that a PDE model has to be considered for cells with an irregular shape or for slow diffusing molecules. Also the high gradients inside a cell or in a cell system can play an essential role in the regulation of the molecular mechanisms.

【 授权许可】

   
2015 Friedmann.

【 图 表 】

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【 参考文献 】

• [1]Kestler HA, Wawra C, Kracher B, Kühl M. Network modeling of signal transduction: establishing the global view. BioEssays. 2008; 30:1110-25.
• [2]Turing MA. The chemical basis of morphogenesis. Phil Trans R Soc B. 1952; 237:37-72.
• [3]Amonlirdviman K, Khare NA, Tree DRP, Chen WS, Axelrod JD, Tomlin CJ. Mathematical modeling of planar cell polarity to understand domineering nonautonomy. Science. 2005; 307:423-6.
• [4]Chaplain MAJ, Gerisch A. Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. J Theo Bio. 2008; 250(4):684-704.
• [5]Grimbs S, Arnold A, Koseska A, Kurths J, Selbig J, Nikoloski Z. Spatiotemporal dynamics of the calvin cycle: multistationarity and symmetry breaking instabilities. Biosystems. 2011; 103:212-23.
• [6]Brown GC, Kholodenko BN. Spatial gradients of cellular phosphoproteins. FEBS Lett. 1999; 457:452-4.
• [7]Caudron M, Bunt G, Bastiaens P, Karsenti E. Spatial coordination of spindle assembly by chromosome-mediated signaling gradients. Science. 2005; 309(5739):1373-6.
• [8]Crank J. Mathematics of Diffusion. Oxford & Clarendon Press, London; 1975.
• [9]Friedmann E, Pfeifer AC, Neumann R, Klingmüller U, Rannacher R. Interaction between experiment, modeling and simulation of spatial aspects in the jak2/stat5 signaling pathway. Model Based Parameter Estimation - Theorie and Application. Bock H, Carraro T, Jäger W, Körkel S, Rannacher R, Schlöder JP, editors. Springer, Berlin Heidelberg; 2013.
• [10]Klingmüller U, Bauer A, Bohl S, Nickel PJ, Breitkopf K, Dooley S et al.. Primary mouse hepatocytes for systems biology approaches: a standardized in vitro system for modelling of signal transduction pathways. Syst Biol. 2006; 153:433-7.
• [11]Friedmann E, Neumann R, Rannacher R. Well posedness of a linear spatio temporal model of the jak2/stat5 signaling pathway. Comm Math Anal. 2013; 5(2):76-102.
• [12]Claus J, Friedmann E, Klingmüller U, Szekeres T. Spatial aspects in the SMAD signaling pathway. J Math Biol. 2013; 67:1171-97.
• [13]Thurley K, Gerecht D, Friedmann E, Höfer T. Three-dimensional gradients of cytokine signaling between t cells. PLOS Comput Biol. 2015; doi:1-22.
• [14]Busse D, de la Rosa M, Hobiger K, Thurley K, Flossdorf M, Scheffold A, Höfer T. Competing feedback loops shape IL-2 signaling between helper and regulatory T lymphocytes in cellular microenvironments. Proc Nat Acad Sci USA. 2010; 107:3058-63.
• [15]de la Rosa M, Rutz S, Dorninger H, Scheffold A. Interleukin-2 is essential for CD4+CD25+regulatory T cell function. Eur J Immunol. 2004; 34:2480-8.
• [16]Scheffold A, Murphy KM, Höfer T. Competition for cytokines: T(reg) cells take all. Nat Immunol. 2007; 8:1285-7.
• [17]Thurley K. Numerische und analytische Lösungen eines räumzeitlichen Modells der Interleukin-2 Expression in T-Lymphozyten. PhD thesis, Humboldt-Universität zu Berlin; 2007.
• [18]GASCOIGNE. High Performance Adaptive Finite Element Toolkit. http://www.gascoigne.uni-hd.de. Accessed 2010.
• [19]Bangerth W, Hartmann R, Kanschat G. Deal.II - a genera purpose object oriented finite element library. ACM Trans Math Softw. 2007; 33:24-12427.
• [20]Bangerth W, Rannacher R. Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel; 2003.
• [21]Becker R, Rannacher R. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica. 2001; 10:1-102.
• [22]Kolb L. Visualizing High-Resolution Numerical Data with Isosurfaces using Topological Methods. PhD thesis, University Heidelberg, IWR, Numerical Geometry; 2013.
• [23]Deheuninck J, Luo K. Ski and SnoN, potent negative regulators of TGF-beta signaling. Cell Res. 2009; 19(1):47-57.
• [24]Wilkinson DS, Ogden SK, Stratton SA, L PJ, Nguyen TT, Smulian GA et al.. A direct intersection between p53 and transforming growth factor beta pathways targets chromatin modification and transcription repression of the alpha-fetoprotein gene. Mol Cell Biol. 2005; 25(3):1200-12.
• [25]Huse M, Lillemeier BF, Kuhns MS, Chen DS, Davis MM. T cells use two directionally distinct pathways for cytokine secretion. Nat Immunol. 2006; 7:247-55.
• [26]Perona-Wright G, Mohrs M. Sustained signaling by canonical helper T cell cytokines throughout the reactive lymph node. Nat Immunol. 2010; 11:520-6.
• [27]Feinerman O, Jentsch G, Tkach K, Coward J, Hathorn M, Sneddon MW et al.. Single-cell quantification of il-2 response by effector and regulatory t cells reveals critical plasticity in immune response. Mol Syst Biol. 2010; 6:437.
• [28]Sabatos C, Doh J, Chakravarti S, Friedman R, Pandurangi P, Tooley AJ et al.. A synaptic basis for paracrine interleukin-2 signaling during homotypic t cell interaction. Immunity. 2008; 29:6136-43.