Advertisement

Simulation Strategies for Calcium Microdomains and Calcium Noise

  • Nicolas Wieder
  • Rainer H. A. Fink
  • Frederic von WegnerEmail author
Chapter
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 1131)

Abstract

In this article, we present an overview of simulation strategies in the context of subcellular domains where calcium-dependent signaling plays an important role. The presentation follows the spatial and temporal scales involved and represented by each algorithm. As an exemplary cell type, we will mainly cite work done on striated muscle cells, i.e. skeletal and cardiac muscle. For these cells, a wealth of ultrastructural, biophysical and electrophysiological data is at hand. Moreover, these cells also express ubiquitous signaling pathways as they are found in many other cell types and thus, the generalization of the methods and results presented here is straightforward.

The models considered comprise the basic calcium signaling machinery as found in most excitable cell types including Ca2+ ions, diffusible and stationary buffer systems, and calcium regulated calcium release channels. Simulation strategies can be differentiated in stochastic and deterministic algorithms. Historically, deterministic approaches based on the macroscopic reaction rate equations were the first models considered. As experimental methods elucidated highly localized Ca2+ signaling events occurring in femtoliter volumes, stochastic methods were increasingly considered. However, detailed simulations of single molecule trajectories are rarely performed as the computational cost implied is too large. On the mesoscopic level, Gillespie’s algorithm is extensively used in the systems biology community and with increasing frequency also in models of microdomain calcium signaling. To increase computational speed, fast approximations were derived from Gillespie’s exact algorithm, most notably the chemical Langevin equation and the τ-leap algorithm. Finally, in order to integrate deterministic and stochastic effects in multiscale simulations, hybrid algorithms are increasingly used. These include stochastic models of ion channels combined with deterministic descriptions of the calcium buffering and diffusion system on the one hand, and algorithms that switch between deterministic and stochastic simulation steps in a context-dependent manner on the other. The basic assumptions of the listed methods as well as implementation schemes are given in the text. We conclude with a perspective on possible future developments of the field.

Keywords

Calcium Calcium signaling Microdomains Stochastic modeling Calcium noise Colored noise Chemical master equation Gillespie’s algorithm Langevin equation IP3R 

References

  1. 1.
    Clapham DE (2007) Calcium Signaling. Cell 131:1047–1058CrossRefGoogle Scholar
  2. 2.
    Berridge MJ (2006) Calcium microdomains: organization and function. Cell Calcium 40:405–412.  https://doi.org/10.1016/j.ceca.2006.09.002 CrossRefPubMedGoogle Scholar
  3. 3.
    Rizzuto R, Pozzan T (2006) Microdomains of intracellular ca 2+ : molecular determinants and functional consequences. Physiol Rev 86:369–408.  https://doi.org/10.1152/physrev.00004.2005 CrossRefGoogle Scholar
  4. 4.
    Martín F, Soria B (1996) Glucose-induced [Ca2+]i oscillations in single human pancreatic islets. Cell Calcium 20:409–414.  https://doi.org/10.1016/S0143-4160(96)90003-2 CrossRefPubMedGoogle Scholar
  5. 5.
    Jaffe LF (1999) Organization of early development by calcium patterns. BioEssays 21:657–667CrossRefGoogle Scholar
  6. 6.
    Giorgi C, Missiroli S, Patergnani S et al (2015) Mitochondria-associated membranes: composition, molecular mechanisms, and physiopathological implications. Antioxid Redox Signal 22(12):995–1019.  https://doi.org/10.1089/ars.2014.6223 CrossRefPubMedGoogle Scholar
  7. 7.
    Rieusset J, Fauconnier J, Paillard M et al (2016) Disruption of calcium transfer from ER to mitochondria links alterations of mitochondria-associated ER membrane integrity to hepatic insulin resistance. Diabetologia 59:614–623.  https://doi.org/10.1007/s00125-015-3829-8 CrossRefPubMedPubMedCentralGoogle Scholar
  8. 8.
    Orrenius S, Zhivotovsky B, Nicotera P (2003) Regulation of cell death: the calcium-apoptosis link. Nat Rev Mol Cell Biol 4:552–565CrossRefGoogle Scholar
  9. 9.
    Myoga MH, Regehr WG (2011) Calcium microdomains near R-type calcium channels control the induction of presynaptic long-term potentiation at parallel Fiber to Purkinje cell synapses. J Neurosci 31:5235–5243.  https://doi.org/10.1523/JNEUROSCI.5252-10.2011 CrossRefPubMedPubMedCentralGoogle Scholar
  10. 10.
    von Wegner F, Wieder N, Fink RHA (2014) Microdomain calcium fluctuations as a colored noise process. Front Genet 5:376.  https://doi.org/10.3389/fgene.2014.00376 CrossRefGoogle Scholar
  11. 11.
    Keller DX, Franks KM, Bartol TM, Sejnowski TJ (2008) Calmodulin activation by calcium transients in the postsynaptic density of dendritic spines. PLoS One 3:e2045.  https://doi.org/10.1371/journal.pone.0002045 CrossRefPubMedPubMedCentralGoogle Scholar
  12. 12.
    Zeng S, Holmes WR (2010) The effect of noise on CaMKII activation in a dendritic spine during LTP induction. J Neurophysiol 103:1798–1808.  https://doi.org/10.1152/jn.91235.2008 CrossRefPubMedPubMedCentralGoogle Scholar
  13. 13.
    Berridge MJ (2009) Inositol trisphosphate and calcium signalling mechanisms. Biochim Biophys Acta, Mol Cell Res 1793:933–940CrossRefGoogle Scholar
  14. 14.
    Gardiner CW (1996) Handbook of stochastic methods: for physics, chemistry and the natural sciences, Springer series in synergetics. Springer, BerlinGoogle Scholar
  15. 15.
    Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35–55.  https://doi.org/10.1146/annurev.physchem.58.032806.104637 CrossRefPubMedGoogle Scholar
  16. 16.
    Higham DJ (2008) Modeling and simulating chemical reactions. SIAM Rev 50:347–368.  https://doi.org/10.1137/060666457 CrossRefGoogle Scholar
  17. 17.
    Higham DJ, Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43:525–546.  https://doi.org/10.1137/S0036144500378302 CrossRefGoogle Scholar
  18. 18.
    Bezprozvanny I, Watras J, Ehrlich BE (1991) Bell-shaped calcium-response curves of lns(l,4,5)P3- and calcium-gated channels from endoplasmic reticulum of cerebellum. Nature 351:751–754.  https://doi.org/10.1038/351751a0 CrossRefPubMedGoogle Scholar
  19. 19.
    Mak D-OD, McBride S, Foskett JK (1998) Inositol 1,4,5-tris-phosphate activation of inositol tris-phosphate receptor Ca2+ channel by ligand tuning of Ca2+ inhibition. Proc Natl Acad Sci 95:15821–15825.  https://doi.org/10.1073/pnas.95.26.15821 CrossRefPubMedGoogle Scholar
  20. 20.
    Andrews SS, Bray D (2004) Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol 1:137–151.  https://doi.org/10.1088/1478-3967/1/3/001 CrossRefPubMedGoogle Scholar
  21. 21.
    Franks KM, Bartol TM, Sejnowski TJ (2002) A Monte Carlo model reveals independent signaling at central glutamatergic synapses. Biophys J 83:2333–2348.  https://doi.org/10.1016/S0006-3495(02)75248-X CrossRefPubMedPubMedCentralGoogle Scholar
  22. 22.
    Shahrezaei V, Delaney KR (2004) Consequences of molecular-level Ca2+ channel and synaptic vesicle colocalization for the Ca2+ microdomain and neurotransmitter exocytosis: a Monte Carlo study. Biophys J 87:2352–2364.  https://doi.org/10.1529/biophysj.104.043380 CrossRefPubMedPubMedCentralGoogle Scholar
  23. 23.
    Tanskanen AJ, Greenstein JL, O’Rourke B, Winslow RL (2005) The role of stochastic and modal gating of cardiac L-type Ca2+ channels on early after-depolarizations. Biophys J 88:85–95.  https://doi.org/10.1529/biophysj.104.051508 CrossRefPubMedGoogle Scholar
  24. 24.
    Hake J, Lines GT (2008) Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion. Biophys J 94:4184–4201.  https://doi.org/10.1529/biophysj.106.103523 CrossRefPubMedPubMedCentralGoogle Scholar
  25. 25.
    Flegg MB, Rüdiger S, Erban R (2013) Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release. J Chem Phys 138:154103.  https://doi.org/10.1063/1.4796417 CrossRefPubMedGoogle Scholar
  26. 26.
    Dobramysl U, Rüdiger S, Erban R (2016) Particle-based multiscale modeling of calcium puff dynamics. Multiscale Model Simul 14:997–1016.  https://doi.org/10.1137/15M1015030 CrossRefGoogle Scholar
  27. 27.
    Nguyen V, Mathias R, Smith GD (2005) A stochastic automata network descriptor for Markov chain models of instantaneously coupled intracellular Ca2+channels. Bull Math Biol 67:393–432.  https://doi.org/10.1016/j.bulm.2004.08.010 CrossRefPubMedGoogle Scholar
  28. 28.
    Munsky B, Khammash M (2006) The finite state projection algorithm for the solution of the chemical master equation. J Chem Phys 124:044104.  https://doi.org/10.1063/1.2145882 CrossRefPubMedGoogle Scholar
  29. 29.
    Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361.  https://doi.org/10.1021/j100540a008 CrossRefGoogle Scholar
  30. 30.
    Gibson MA, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem A 104:1876–1889.  https://doi.org/10.1021/jp993732q CrossRefGoogle Scholar
  31. 31.
    Weinberg SH (2016) Microdomain [ca 2+ ] fluctuations Alter temporal dynamics in models of ca 2+ −dependent signaling cascades and synaptic vesicle release. Neural Comput 28:493–524.  https://doi.org/10.1162/NECO_a_00811 CrossRefPubMedGoogle Scholar
  32. 32.
    Li H, Hou Z, Xin H (2005) Internal noise stochastic resonance for intracellular calcium oscillations in a cell system. Phys Rev E Stat Nonlin Soft Matter Phys 71:061916.  https://doi.org/10.1103/PhysRevE.71.061916 CrossRefPubMedGoogle Scholar
  33. 33.
    Kummer U, Krajnc B, Pahle J et al (2005) Transition from stochastic to deterministic behavior in calcium oscillations. Biophys J 89:1603–1611.  https://doi.org/10.1529/biophysj.104.057216 CrossRefPubMedPubMedCentralGoogle Scholar
  34. 34.
    Von Wegner F, Fink RHA (2010) Stochastic simulation of calcium microdomains in the vicinity of an L-type calcium channel. Eur Biophys J 39:1079–1088CrossRefGoogle Scholar
  35. 35.
    Gillespie DT (2000) Chemical Langevin equation. J Chem Phys 113:297–306.  https://doi.org/10.1063/1.481811 CrossRefGoogle Scholar
  36. 36.
    Cao Y, Gillespie DT, Petzold LR (2006) Efficient step size selection for the tau-leaping simulation method. J Chem Phys 124:044109.  https://doi.org/10.1063/1.2159468 CrossRefPubMedGoogle Scholar
  37. 37.
    Tian T, Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys 121:10356–10364.  https://doi.org/10.1063/1.1810475 CrossRefPubMedGoogle Scholar
  38. 38.
    Wieder N, Fink R, Von Wegner F (2015) Exact stochastic simulation of a calcium microdomain reveals the impact of Ca2+ fluctuations on IP3R gating. Biophys J 108:557–567.  https://doi.org/10.1016/j.bpj.2014.11.3458 CrossRefPubMedPubMedCentralGoogle Scholar
  39. 39.
    Goldwyn JH, Imennov NS, Famulare M, Shea-Brown E (2011) Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons. Phys Rev E Stat Nonlin Soft Matter Phys 83:041908.  https://doi.org/10.1103/PhysRevE.83.041908 CrossRefPubMedPubMedCentralGoogle Scholar
  40. 40.
    Dangerfield CE, Kay D, Burrage K (2012) Modeling ion channel dynamics through reflected stochastic differential equations. Phys Rev E Stat Nonlin Soft Matter Phys 85:051907.  https://doi.org/10.1103/PhysRevE.85.051907 CrossRefPubMedGoogle Scholar
  41. 41.
    Alfonsi A (2005) On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl 11:355–384.  https://doi.org/10.1163/156939605777438569 CrossRefGoogle Scholar
  42. 42.
    Manninen T, Linne M-L, Ruohonen K (2006) Developing Itô stochastic differential equation models for neuronal signal transduction pathways. Comput Biol Chem 30:280–291.  https://doi.org/10.1016/j.compbiolchem.2006.04.002 CrossRefPubMedGoogle Scholar
  43. 43.
    Zhang J, Hou Z, Xin H (2004) System-size biresonance for intracellular calcium signaling. ChemPhysChem 5:1041–1045.  https://doi.org/10.1002/cphc.200400089 CrossRefPubMedGoogle Scholar
  44. 44.
    lian ZC, Jia Y, Liu Q et al (2007) A mesoscopic stochastic mechanism of cytosolic calcium oscillations. Biophys Chem 125:201–212.  https://doi.org/10.1016/j.bpc.2006.08.001 CrossRefGoogle Scholar
  45. 45.
    Wang X, Hao Y, Weinberg SH, Smith GD (2015) Ca2+−activation kinetics modulate successive puff/spark amplitude, duration and inter-event-interval correlations in a Langevin model of stochastic Ca2+release. Math Biosci 264:101–107.  https://doi.org/10.1016/j.mbs.2015.03.012 CrossRefPubMedGoogle Scholar
  46. 46.
    Winslow RL, Tanskanen A, Chen M, Greenstein JL (2006) Multiscale modeling of calcium signaling in the cardiac dyad. Ann NY Acad Sci 1080:362–375CrossRefGoogle Scholar
  47. 47.
    Soeller C, Cannell MB (1997) Numerical simulation of local calcium movements during L-type calcium channel gating in the cardiac diad. Biophys J 73:97–111.  https://doi.org/10.1016/S0006-3495(97)78051-2 CrossRefPubMedPubMedCentralGoogle Scholar
  48. 48.
    Cannell MB, Soeller C (1997) Numerical analysis of ryanodine receptor activation by L-type channel activity in the cardiac muscle diad. Biophys J. doi: S0006-3495(97)78052-4 [pii]\r10.1016/S0006-3495(97)78052-4, vol 73, pp 112–122Google Scholar
  49. 49.
    Smith GD, Keizer JE, Stern MD et al (1998) A simple numerical model of calcium spark formation and detection in cardiac myocytes. Biophys J 75:15–32.  https://doi.org/10.1016/S0006-3495(98)77491-0 CrossRefPubMedPubMedCentralGoogle Scholar
  50. 50.
    Jiang YH, Klein MG, Schneider MF (1999) Numerical simulation of Ca2+ “sparks” in skeletal muscle. Biophys J 92:308–332.  https://doi.org/10.1016/j.pbiomolbio.2005.05.016 CrossRefGoogle Scholar
  51. 51.
    Baylor SM, Hollingworth S (2007) Simulation of ca 2+ movements within the sarcomere of fast-twitch mouse fibers stimulated by action potentials. J Gen Physiol 130:283–302.  https://doi.org/10.1085/jgp.200709827 CrossRefPubMedPubMedCentralGoogle Scholar
  52. 52.
    Stern MD, Pizarro G, Ríos E (1997) Local control model of excitation-contraction coupling in skeletal muscle. J Gen Physiol 110:415–440.  https://doi.org/10.1085/jgp.110.4.415 CrossRefPubMedPubMedCentralGoogle Scholar
  53. 53.
    Greenstein JL, Winslow RL (2002) An integrative model of the cardiac ventricular myocyte incorporating local control of Ca2+ release. Biophys J 83:2918–2945.  https://doi.org/10.1016/S0006-3495(02)75301-0 CrossRefPubMedPubMedCentralGoogle Scholar
  54. 54.
    Rüdiger S, Shuai JW, Huisinga W et al (2007) Hybrid stochastic and deterministic simulations of calcium blips. Biophys J 93:1847–1857.  https://doi.org/10.1529/biophysj.106.099879 CrossRefPubMedPubMedCentralGoogle Scholar
  55. 55.
    Kalantzis G (2009) Hybrid stochastic simulations of intracellular reaction-diffusion systems. Comput Biol Chem 33:205–215.  https://doi.org/10.1016/j.compbiolchem.2009.03.002 CrossRefPubMedPubMedCentralGoogle Scholar
  56. 56.
    Choi T, Maurya MR, Tartakovsky DM, Subramaniam S (2010) Stochastic hybrid modeling of intracellular calcium dynamics. J Chem Phys 133:165101.  https://doi.org/10.1063/1.3496996 CrossRefPubMedPubMedCentralGoogle Scholar
  57. 57.
    Krishnamurthy V, Chung SH (2007) Large-scale dynamical models and estimation for permeation in biological membrane ion channels. Proc IEEE 95:853–880.  https://doi.org/10.1109/JPROC.2007.893246 CrossRefGoogle Scholar
  58. 58.
    Elf J, Doncic A, Ehrenberg M (2003) Mesoscopic reaction-diffusion in intracellular signaling. In: Bezrukov S, Frauenfelder H, Moss F (eds) Fluctuations and noise in biological, biophysical, and biomedical systems. Proceedings of the SPIE, pp 114–124Google Scholar
  59. 59.
    Weinberg SH, Smith GD (2014) The influence of Ca2+ buffers on free [Ca2+] fluctuations and the effective volume of Ca2+ microdomains. Biophys J 106:2693–2709.  https://doi.org/10.1016/j.bpj.2014.04.045 CrossRefPubMedPubMedCentralGoogle Scholar
  60. 60.
    Sherman A, Smith GD, Dai L, Miura RM (2001) Asymptotic analysis of buffered calcium diffusion near a point source. SIAM J Appl Math 61:1816–1838.  https://doi.org/10.1137/S0036139900368996 CrossRefGoogle Scholar
  61. 61.
    Li QS, Wang P (2004) Internal signal stochastic resonance induced by colored noise in an intracellular calcium oscillations model. Chem Phys Lett 387:383–387.  https://doi.org/10.1016/j.cplett.2004.02.042 CrossRefGoogle Scholar
  62. 62.
    Blomberg C (2006) Fluctuations for good and bad: the role of noise in living systems. Phys Life Rev 3:133–161CrossRefGoogle Scholar
  63. 63.
    Faisal AA, Selen LPJ, Wolpert DM (2008) Noise in the nervous system. Nat Rev Neurosci 9:292–303CrossRefGoogle Scholar
  64. 64.
    Zhong S, Qi F, Xin H (2001) Internal stochastic resonance in a model system for intracellular calcium oscillations. Chem Phys Lett 342:583–586.  https://doi.org/10.1016/S0009-2614(01)00625-X CrossRefGoogle Scholar
  65. 65.
    Thul R, Falcke M (2004) Release currents of IP3 Receptor Channel clusters and concentration profiles. Biophys J 86:2660–2673.  https://doi.org/10.1016/S0006-3495(04)74322-2 CrossRefPubMedPubMedCentralGoogle Scholar
  66. 66.
    Skupin A, Falcke M (2009) From puffs to global Ca2+ signals: how molecular properties shape global signals. Chaos 19:037111.  https://doi.org/10.1063/1.3184537 CrossRefPubMedGoogle Scholar
  67. 67.
    Marchant JS, Parker I (2001) Role of elementary Ca2+ puffs in generating repetitive Ca2+ oscillations. EMBO J 20:65–76.  https://doi.org/10.1093/emboj/20.1.65 CrossRefPubMedPubMedCentralGoogle Scholar
  68. 68.
    De Young GW, Keizer J (1992) A single-pool inositol 1,4,5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration. Proc Natl Acad Sci 89:9895–9899.  https://doi.org/10.1073/pnas.89.20.9895 CrossRefPubMedGoogle Scholar
  69. 69.
    Ullah G, Daniel Mak D-O, Pearson JE (2012) A data-driven model of a modal gated ion channel: the inositol 1,4,5-trisphosphate receptor in insect Sf9 cells. J Gen Physiol 140:159–173.  https://doi.org/10.1085/jgp.201110753 CrossRefPubMedPubMedCentralGoogle Scholar
  70. 70.
    Siekmann I, Wagner LE, Yule D et al (2012) A kinetic model for type i and II IP3R accounting for mode changes. Biophys J 103:658–668.  https://doi.org/10.1016/j.bpj.2012.07.016 CrossRefPubMedPubMedCentralGoogle Scholar
  71. 71.
    Xu T, Yu X, Perlik AJ et al (2009) Rapid formation and selective stabilization of synapses for enduring motor memories. Nature 462:915–919.  https://doi.org/10.1038/nature08389 CrossRefPubMedPubMedCentralGoogle Scholar
  72. 72.
    Bers DM (2002) Cardiac excitation contraction coupling. Nature 415:198–215.  https://doi.org/10.1016/B978-0-12-378630-2.00221-8 CrossRefPubMedPubMedCentralGoogle Scholar
  73. 73.
    Bers DM, Despa S (2013) Cardiac excitation-contraction coupling. In: Lennarz WJ, Lane MD (eds) The Encyclopedia of biological chemistry, 2nd edn. Academic Press, Waltham, MA, pp 379–383CrossRefGoogle Scholar
  74. 74.
    Artyomov MN, Das J, Kardar M, Chakraborty AK (2007) Purely stochastic binary decisions in cell signaling models without underlying deterministic bistabilities. Proc Natl Acad Sci U S A 104:18958–18963.  https://doi.org/10.1073/pnas.0706110104 CrossRefPubMedPubMedCentralGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Nicolas Wieder
    • 1
    • 2
    • 3
  • Rainer H. A. Fink
    • 3
  • Frederic von Wegner
    • 4
    • 3
    Email author
  1. 1.Broad Institute of MIT and HarvardCambridgeUSA
  2. 2.Department of MedicineBrigham and Women’s Hospital and Harvard Medical SchoolBostonUSA
  3. 3.Medical Biophysics Group, Institute of Physiology and PathophysiologyUniversity of HeidelbergHeidelbergGermany
  4. 4.Department of Neurology and Brain Imaging CenterGoethe University FrankfurtFrankfurt am MainGermany

Personalised recommendations