Brownian Models of Chemical Reactions in Microdomains

  • Zeev Schuss
Part of the Applied Mathematical Sciences book series (AMS, volume 186)


Biological microstructures such as synapses, dendritic spines, subcellular domains, sensor cells, and many other structures are regulated by chemical reactions that involve only a small number of molecules, that is, between a few and up to thousands of molecules. Traditional chemical kinetics theory may provide an inadequate description of chemical reactions in such microdomains. Models with a small number of diffusers can be used to describe noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photoreceptors, and hair cells in the cochlea. A chemical reaction that involves only 10–100 proteins can cause a qualitative transition in the physiological behavior of a given part of a cell. Large fluctuations should be expected in a reaction if so few molecules are involved, both in transient and persistent binding and unbinding reactions. In the latter case, large fluctuations in the number of bound molecules can force the physiological state to change all the time, unless there is a specific mechanism that prevents the switch and stabilizes the physiological state. Therefore, a theory of chemical kinetics of such reactions is needed to predict the threshold at which switches occur and to explain how the physiological function is regulated in molecular terms at a subcellular level.


Dendritic Spine Calcium Dynamic Spine Head Myosin Molecule Dendritic Shaft 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Berne, B.J. and R. Pecora (1976), Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics. Wiley-Interscience NY.Google Scholar
  2. Blomberg, F., R.S. Cohen, and P. Siekevitz (1977), “The structure of postsynaptic densities isolated from dog cerebral cortex, II. Characterization and arrangement of some of the major protein within the structure,” J. Cell Biol., 74 (1), 204–225.Google Scholar
  3. Bonhoeffer, T. and R. Yuste (2002), “Spine motility: phenomenology, mechanisms, and function,” Neuron, 35 (6), 1019–1027.CrossRefGoogle Scholar
  4. Chandrasekhar, S. (1943), “Stochastic Problems In Physics and Astronomy,” Rev. Mod. Phys., 15, 2–89.CrossRefGoogle Scholar
  5. Crick, F. “Do dendritic spines twitch?” Trends Neurosci, 5, 44–46.Google Scholar
  6. Dunaevsky, A., A. Tashiro, A. Majewska, C. Mason, R. Yuste (1999), “Developmental regulation of spine motility in the mammalian central nervous system,” PNAS, 96 (23), 13438–13443.CrossRefGoogle Scholar
  7. Fischer, M., S. Kaech, D. Knutti, A. Matus (1998, “Rapid actin-based plasticity in dendritic spines,” Neuron, 20 (5), 847–854).Google Scholar
  8. Fischer, M., S. Kaech, U. Wagner, H. Brinkhaus, A. Matus (2000), “Glutamate receptors regulate actin-based plasticity in dendritic spines,” Nat. Neurosci., 3 (9), 887–894.CrossRefGoogle Scholar
  9. Hänggi, P., P. Talkner, and M. Borkovec (1990), “50 years after Kramers,” Rev. Mod. Phys., 62, 251–341.CrossRefGoogle Scholar
  10. Haynes, L.W., A.R. Kay, K.W. Yau (1986), “Single cyclic GMP-activated channel activity in excised patches of rod outer segment membrane,” Nature, 321 (6065), 66–70.CrossRefGoogle Scholar
  11. Holcman, D., Z. Schuss, and E. Korkotian (2004), “Calcium dynamics in dendritic spines and spine motility,” Biophys J., 87, 81–91.CrossRefGoogle Scholar
  12. Kandel, E.R., J.H. Schwartz, T.M. Jessell (2000), Principles of Neural Science, McGraw-Hill, New York, 4th edition.Google Scholar
  13. Koch, C. (1999), Biophysics of Computation, Oxford University Press, NY.Google Scholar
  14. Koch, C. and A. Zador (1993), “The function of dendritic spines: Devices subserving biochemical rather than electrical compartmentalization,” J. Neurosci., 13, 413–422.Google Scholar
  15. Koch, C. and I. Segev (editors) (2001), Methods in Neuronal Modeling (3rd printing), MIT Press, Cambridge, MA.Google Scholar
  16. Korkotian, E. and M. Segal (2001), “Spike-associated fast contraction of dendritic spines in cultured hippocampal neurons,” Neuron, 30 (3), 751–758.CrossRefGoogle Scholar
  17. Kramers, H.A. (1940), “Brownian motion in field of force and diffusion model of chemical reaction,” Physica, 7, 284–304.MathSciNetMATHCrossRefGoogle Scholar
  18. Landau, L.D. and E.M. Lifshitz (1975), Fluid Mechanics, Pergamon Press, Elmsford, NY.Google Scholar
  19. Lisman, J. (1994), “The CAM kinase II hypothesis for the storage of synaptic memory,” Trends Neurosci., 10, 406–412.CrossRefGoogle Scholar
  20. Lisman, J. (2003), “Long-term potentiation: outstanding questions and attempted synthesis,” Philos. Trans. R. Soc. Lond. B Biol. Sci., 29 (358(1432)), 829–842.Google Scholar
  21. Majewska, A., A. Tashiro, and R. Yuste (2000a), “Regulation of spine calcium dynamics by rapid spine motility,” J. Neurosci., 20 (22), 8262–8268.Google Scholar
  22. Majewska, A., E. Brown, J. Ross, R. Yuste (2000b), “Mechanisms of calcium decay kinetics in hippocampal spines: role of spine calcium pumps and calcium diffusion through the spine neck in biochemical compartmentalization,” J. Neurosci., 20 (5), 1722–1734.Google Scholar
  23. Malenka, R.C., J.A. Kauer, D.J. Perkel, and R.A. Nicoll (1989), “The impact of postsynaptic calcium on synaptic transmission—its role in long-term potentiation,” Trends Neurosci., 12 (11), 444–450.CrossRefGoogle Scholar
  24. Matkowsky, B.J. and Z. Schuss (1977), “The exit problem for randomly perturbed dynamical systems,” SIAM J. Appl. Math., 33, 365–382.MathSciNetMATHCrossRefGoogle Scholar
  25. Morales, M., E. Fifkova (1989), “Distribution of MAP2 in dendritic spines and its colocalization with actin. An immunogold electron-microscope study,” Cell Tissue Res., 256 (3), 447–456.Google Scholar
  26. Nadler, B., T. Naeh, and Z. Schuss (2002), “The stationary arrival process of diffusing particles from a continuum to an absorbing boundary is Poissonian,” SIAM J. Appl. Math., 62 (2), 433–447.MathSciNetCrossRefGoogle Scholar
  27. Nimchinsky, E.A., B.L. Sabatini, K. Svoboda (2002), “Structure and function of dendritic spines,” Annu. Rev. Physiol., 64, 313–335.CrossRefGoogle Scholar
  28. Picones, A. and J.I. Korenbrot (1994), “Analysis of fluctuations in the CGMP-dependent currents of cone photoreceptor outer segments,” Biophys. J. 66, (2, Part 1), 360–365.Google Scholar
  29. Ramón y Cajal, S. (1909), “Les nouvelles idées sur la structure du système nerveux chez l’homme et chez les vertébrés,” Transl. L. Azouly, Malaine, Paris, France. “New ideas on the structure of the nervous system of man and vertebrates,” Transl. N. & N.L. Swanson, MIT Press, Cambridge, MA 1991.Google Scholar
  30. Rieke, F. and D.A. Baylor (1996), “Molecular origin of continuous dark noise in rod photoreceptors,” Biophys J, 71, 2553–2572.CrossRefGoogle Scholar
  31. Sabatini, B.L., M. Maravall, and K. Svoboda (2001), “Ca2 +  signalling in dendritic spines,” Curr. Opin. Neurobiol., 11 (3), 349–356.CrossRefGoogle Scholar
  32. Schuss, Z. (2010b), Theory and Applications of Stochastic Processes, and Analytical Approach, Springer series on Applied Mathematical Sciences 170, NY.Google Scholar
  33. Segev, I. and W. Rall (1988), “Computational study of an excitable dendritic spine,” J. Neurophysiology, 60 (6), 499–523.Google Scholar
  34. Shepherd, G.M. (1996), “The dendritic spine: a multi-functional integrative unit,” J. Neurophysiology, 75 (6), 2197–2210.Google Scholar
  35. Volfovsky, N., H. Parnas, M. Segal, and E. Korkotian (1999), “Geometry of dendritic spines affects calcium dynamics in hippocampal neurons: theory and experiments,” J. Neurophysiol., 82, 450–454.Google Scholar
  36. Yuste, R. and W. Denk (1995), “Dendritic spines as basic functional units of neuronal integration,” Nature, 375 (6533), 682–684.CrossRefGoogle Scholar
  37. Zador, A., C. Koch, and T.H. Brown (1990), “Biophysical model of a Hebbian synapse,” PNAS, 87, 6718–6722.CrossRefGoogle Scholar
  38. Zucker, R.S. and W.G. Regehr (2002), “Short-term synaptic plasticity,” Ann. Rev. Physiol., 64, 355–405.CrossRefGoogle Scholar

Copyright information

© Author 2013

Authors and Affiliations

  • Zeev Schuss
    • 1
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations