Bulletin of Mathematical Biology

, Volume 70, Issue 1, pp 297–321 | Cite as

Markov Random Field Modeling of the Spatial Distribution of Proteins on Cell Membranes

  • Jun Zhang
  • Stanly L. Steinberg
  • Bridget S. Wilson
  • Janet M. Oliver
  • Lance R. WilliamsEmail author
Original Article


Cell membranes display a range of receptors that bind ligands and activate signaling pathways. Signaling is characterized by dramatic changes in membrane molecular topography, including the co-clustering of receptors with signaling molecules and the segregation of other signaling molecules away from receptors. Electron microscopy of immunogold-labeled membranes is a critical technique to generate topographical information at the 5–10 nm resolution needed to understand how signaling complexes assemble and function. However, due to experimental limitations, only two molecular species can usually be labeled at a time. A formidable challenge is to integrate experimental data across multiple experiments where there are from 10 to 100 different proteins and lipids of interest and only the positions of two species can be observed simultaneously. As a solution, we propose the use of Markov random field (MRF) modeling to reconstruct the distribution of multiple cell membrane constituents from pair-wise data sets. MRFs are a powerful mathematical formalism for modeling correlations between states associated with neighboring sites in spatial lattices. The presence or absence of a protein of a specific type at a point on the cell membrane is a state. Since only two protein types can be observed, i.e., those bound to particles, and the rest cannot be observed, the problem is one of deducing the conditional distribution of a MRF with unobservable (hidden) states. Here, we develop a multiscale MRF model and use mathematical programming techniques to infer the conditional distribution of a MRF for proteins of three types from observations showing the spatial relationships between only two types. Application to synthesized data shows that the spatial distributions of three proteins can be reliably estimated. Application to experimental data provides the first maps of the spatial relationship between groups of three different signaling molecules. The work is an important step toward a more complete understanding of membrane spatial organization and dynamics during signaling.


Markov random field Spatial distribution Cell membrane Gold label Parameter estimation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrews, N.L. et al., 2007. Dynamics, topography, and microdomains in FcεRI signaling. Biophys. J., submitted. Google Scholar
  2. Barisas, B.G. et al., 2007. Compartmentalization of the Type I Fc epsilon receptor and MAFA on mast cell membranes. Biophys. Chem. 126, 209–217. CrossRefGoogle Scholar
  3. Berlin, R.D. et al., 1974. Control of cell surface topography. Nature 247, 45–46. CrossRefGoogle Scholar
  4. Besag, J., 1974. Spatial interaction and the statistical analysis of lattice systems. J. Roy. Stat. Soc. Ser. B 36(2), 192–236. zbMATHMathSciNetGoogle Scholar
  5. Besag, J., 1986. On the statistical analysis of dirty pictures. J. Roy. Stat. Soc. Ser. B 48(3), 259–302. zbMATHMathSciNetGoogle Scholar
  6. Bouman, C.A., Shapiro, M., 1994. A multiscale random field model for Bayesian image segmentation. IEEE Trans. Image Process. 3(2), 162–177. CrossRefGoogle Scholar
  7. Celeux, G., Forbes, F., Peyrard, N., 2003. EM procedures using mean field-like approximations for Markov model-based image segmentation. Pattern Recognit. 36(1), 131–144. zbMATHCrossRefGoogle Scholar
  8. Chalmond, B., 1989. An iterative Gibbsian technique for reconstruction of M-ary images. Pattern Recognit. 22(6), 747–762. CrossRefGoogle Scholar
  9. Cross, G.R., Jain, A.K., 1983. Markov random field texture models. IEEE Trans. Pattern Anal. Mach. Intell. 5(1), 25–39. Google Scholar
  10. Dempster, A.P., M. Laird, N., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. Ser. B 39(1), 1–38. zbMATHGoogle Scholar
  11. Derin, H., Elliott, H., 1987. Modeling and segmentation of noisy and textured images using Gibbs random fields. IEEE Trans. Pattern Anal. Mach. Intell. 9(1), 39–55. Google Scholar
  12. Efros, A.A., Leung, T.K., 1999. Texture synthesis by non-parametric sampling. In: ICCV ’99: Proceedings of the International Conference on Computer Vision, vol. 2, p. 1033, Washington, DC, USA. IEEE Computer Society. Google Scholar
  13. Geman, S., Geman, D., 1984. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6(6), 721–741. zbMATHGoogle Scholar
  14. Gill, P.E., Murray, W., Saunders, M.A., 2002. SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4), 979–1006. zbMATHCrossRefMathSciNetGoogle Scholar
  15. Gurelli, M.I., Onural, L., 1994. On a parameter estimation method for Gibbs–Markov random fields. IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 424–430. CrossRefGoogle Scholar
  16. Hancock, J.F., Prior, I.A., 2005. Electron microscopic imaging of ras signaling domains. Methods 37(2), 165–712. CrossRefGoogle Scholar
  17. Hernandez-Sanchez, B.A. et al., 2006. Synthesizing biofunctionalized nanoparticles to image cell signaling pathways. IEEE Trans. NanoBiosc. 5, 222–230. CrossRefGoogle Scholar
  18. Kato, Z., Berthod, M., Zerubia, J., 1996. A hierarchical Markov random field model and multitemperature annealing for parallel image classification. CVGIP: Graph. Model Image Process. 58(1), 18–37. CrossRefGoogle Scholar
  19. Kato, Z., Zerubia, J., Berthod, M., 1999. Unsupervised parallel image classification using Markovian models. Pattern Recognit. 32(4), 591–604. CrossRefGoogle Scholar
  20. Kim, J.H. et al., 2005. Independent trafficking of Ig-α/Ig-β and μ-heavy chain is facilitated by dissociation of the B cell antigen receptor complex. J. Immunol. 175, 147–154. Google Scholar
  21. Laferté, J.M., Pérez, P., Heitz, F., 2000. Discrete Markov image modeling and inference on the quadtree. IEEE Trans. Image Process. 9(3), 390–404. zbMATHCrossRefMathSciNetGoogle Scholar
  22. Li, S.Z., 1995. Markov Random Field Modeling in Computer Vision. Springer, London. Google Scholar
  23. Liang, K.H., Tjahjadi, T., 2006. Adaptive scale fixing for multiscale texture segmentation. IEEE Trans. Image Process. 15(1), 249–256. CrossRefGoogle Scholar
  24. Lidke, K.A. et al., 2007. Direct observation of membrane proteins confined by actin corrals. J. Cell Biol., submitted. Google Scholar
  25. Mignotte, M. et al., 2000. Sonar image segmentation using an unsupervised hierarchical MRF model. IEEE Trans. Image Process. 9(7), 1216–1231. CrossRefGoogle Scholar
  26. Nicolau, D.V. et al., 2006. Identifying optimal lipid raft characteristics required to promote nanoscale protein-protein interactions on the plasma membrane. Mol. Cell Biol. 26, 313–323. CrossRefGoogle Scholar
  27. Oliver, J.M. et al., 2004. Membrane receptor mapping: the membrane topography of FcεRI signaling. In: P. Quinn (Ed.), Membrane Dynamics and Domains, Subcellular Biochemistry, vol. 37. Kluwer Academic/Plenum, Dordecht/New York, pp. 3–34. Google Scholar
  28. Paget, R., Longstaff, I.D., 1998. Texture synthesis via a noncausal nonparametric multiscale Markov random field. IEEE Trans. Image Process. 7(6), 925–931. CrossRefGoogle Scholar
  29. Plowman, S.J. et al., 2005. H-ras, k-ras and inner plasma membrane raft proteins operate in nanoclusters with differential dependence on the actin cytoskeleton. Proc. Natl. Acad. Sci. 102, 15500–15505. CrossRefGoogle Scholar
  30. Prior, I.A. et al., 2003. Direct visualization of ras proteins in spatially distinct cell surface microdomains. J. Cell Biol. 160(2), 165–170. CrossRefGoogle Scholar
  31. Rabiner, L.R., 1989. A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257–286. CrossRefGoogle Scholar
  32. Rahman, N.A. et al., 1992. Rotational dynamics of type I Fc epsilon receptors on individually-selected rat mast cells studied by polarized fluorescence depletion. Biophys. J. 61, 334–346. Google Scholar
  33. Seagrave, J.C. et al., 1991. Relationship of IgE receptor topography to secretion in RBL-2H3 mast cells. J. Cell Physiol. 148, 139–151. CrossRefGoogle Scholar
  34. Thomas, J.L., Feder, T.J., Webb, W.W., 1992. Effects of protein concentration on IgE receptor mobility in rat basophilic leukemia cell plasma membranes. Biophys. J. 61, 1402–1412. CrossRefGoogle Scholar
  35. Tjelmeland, H., Besag, J., 1998. Markov random fields with higher order interactions. Scand. J. Stat. 25, 415–433. zbMATHCrossRefMathSciNetGoogle Scholar
  36. Tonazzini, A., Bedini, L., Salerno, E., 2006. A Markov model for blind image separation by a mean-field EM algorithm. IEEE Trans. Image Process. 15(2), 473–482. CrossRefMathSciNetGoogle Scholar
  37. Volna, P. et al., 2004. Negative regulation of mast cell signaling and function by the adaptor lab/ntal. J. Exp. Med. 200(8), 1001–1013. CrossRefGoogle Scholar
  38. Wilson, R., Li, C.T., 2003. A class of discrete multiresolution random fields and its application to image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 25(1), 42–56. CrossRefGoogle Scholar
  39. Wilson, B.S., Pfeiffer, J.R., Oliver, J.M., 2000. Observing FcεRI signaling from the inside of the mast cell membrane. J. Cell Biol. 149(5), 1131–1142. CrossRefGoogle Scholar
  40. Wilson, B.S. et al., 2001. High resolution mapping of mast cell membranes reveals primary and secondary domains of FcεRI and LAT. J. Cell Biol. 154(3), 645–658. CrossRefGoogle Scholar
  41. Wilson, B.S., Pfeiffer, J.R., Oliver, J.M., 2002. FcεRI signaling observed from the inside of the mast cell membrane. Mol. Immun. 38, 1259–1268. CrossRefGoogle Scholar
  42. Wilson, B.S. et al., 2004. Markers for detergent-resistant lipid rafts occupy distinct and dynamic domains in native membranes. Mol. Biol. Cell 15(6), 2580–2592. CrossRefGoogle Scholar
  43. Xue, M. et al., 2007. FPR and FcεRI occupy common signaling domains for localized crosstalk. Mol. Biol. Cell 18, 1410–1420. CrossRefGoogle Scholar
  44. Yang, S. et al., 2007. Mapping ErbB receptors on breast cancer cell membranes during signal transduction. J. Cell Sci. 120, 2763–2773. CrossRefGoogle Scholar
  45. Zhang, J., 1992. The mean field theory in EM procedures for Markov random fields. IEEE Trans. Image Process. 40(10), 2570–2583. zbMATHGoogle Scholar
  46. Zhang, J., Modestino, J.W., Langan, D.A., 1994. Maximum-likelihood parameter estimation for unsupervised stochastic model-based image segmentation. IEEE Trans. Image Process. 3(4), 404–420. CrossRefGoogle Scholar
  47. Zhang, J. et al., 2006. Characterizing the topography of membrane receptors and signaling molecules from spatial patterns obtained using nanometer-scale electron-dense probes and electron microscopy. Micron 37(1), 14–34. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • Jun Zhang
    • 1
  • Stanly L. Steinberg
    • 2
  • Bridget S. Wilson
    • 3
  • Janet M. Oliver
    • 3
  • Lance R. Williams
    • 1
    Email author
  1. 1.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA
  2. 2.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA
  3. 3.Department of PathologyUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations