1. Deterministic vs Stochastic simulation Stationary solutions: “dual-Padé” approximation Modeling localized Ca 2+ dynamics: interesting Math questions Victor Matveev Department of Mathematical Sciences New Jersey Institute of Technology B(r) r Ca(r) r 1 vs Supported in part by NSF grant DMS-1517085

2. Roles of Ca 2+ in cell physiology 1880s: Ca 2+ triggers muscle contraction (Sydney Ringer) 1950s: Ca 2+ spike in muscle cell (Fatt & Katz) 1960s: Ca 2+ triggers synaptic transmission and facilitation (Katz, Miledi, Rahamimoff) 1960s: Ca 2+ triggers hormone release (William Douglas) 1980s: Ca 2+ triggers T-lymphocyte cytotoxic activity 1980s: Ca 2+ triggers gene transcription (Michael Greenberg) Other functions: cell cycle, apoptosis, fertilization, LTP (learning and memory),…  Ca 2+ signals compartmentalized in time and space

3. Roles of Ca 2+ in cell physiology ▪ An ideal signal: [Ca 2+ ] out / [Ca 2+ ] in ~10 5 ▪ Multitude of sources and interaction partners ▪ Multiple Ca 2+ pathways  localization of Ca 2+ signals [Ca 2+ ] in =0.1 μM [Ca 2+ ] out =2-10 mM Ca 2+ Signal 1 Signal 2

4. Processes driven by local Ca 2+ elevation Synaptic transmission Endocrine hormone release Muscle cell contraction Graham Johnson Medical Media Ca 2+

5. Localized Ca 2+ signaling: exocytosis Action potential (depolarizing pulse) Ca 2+ ions Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm Ca 2+ buffers (calbindin, parvalbumin, calmodulin, …)

6. Localized Ca 2+ signaling: exocytosis Ca 2+ channels open Several Ca 2+ ions bound to release site Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm

7. Localized Ca 2+ signaling: exocytosis Exocytosis (fusion) Neurotransmitter receptors/channels Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm

8. Localized Ca 2+ signaling: exocytosis Neurotranmitter binds to postsynaptic receptors Neurotransmitter receptors/channels Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm

9. Localized Ca 2+ signaling: exocytosis Postsynaptic ion channels open Neurotransmitter receptors/channels Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm

10. Localized Ca 2+ signaling: exocytosis Endocrine cell T-lymphocyte Blood stream Alien cell Chemical synapseNeuromuscular junction Myocyte Neuron SR

11. Localized Ca 2+ signaling: exocytosis Ca 2+ “nanodomain” Area of elevated [Ca 2+ ] > 10 μM within ~30-100 nm of the Ca 2+ channel 1981: Ca 2+ microdomain hypothesis (Rodolfo Llinas)

12. Interest in modeling local Ca 2+ dynamics Single-channel Ca 2+ domains with sharp gradients physiologically relevant Ca 2+ -sensitive exocytosis steps still debated Complex dynamics: Ca 2+ is strongly buffered Limited resolution of Ca 2+ imaging Interpreting imaging requires modeling (Ca 2+ dyes are buffers) 1980s: First “Ca 2+ domain” simulations: Simon & Llinas, Chad & Eckert, Fogelson & Zucker, Neher

13. Deterministic vs Stochastic simulation Stationary solutions: beyond asymptotics Modeling cell Ca 2+ dynamics: challenges and open problems vs VCell, CalC, …Mcell, Smoldyn, …

14. Deterministic solver: CalC http://www.calciumcalculator.org Simple and efficient Allows parameter sensitivity analysis All standard geometries implemented Cartesian 1D, 2D, 3D Spherical 1D, 2D, 3D Cylindrical 2D, 3D Polar 1D, 2D Arbitrary number of buffers Flow control & MATLAB integration New features: 2-to-1 Ca 2+ buffers Non-linear boundary flux (pumps) Spherical obstacles

15. Deterministic approach: “mass-action” Ca 2+ ions Presynaptic cell (axon) Postsynaptic cell (neuron or muscle) ~40nm Ca 2+ buffers (calbindin, parvalbumin, calmodulin, …) V0V0

16.  Nanodomains form and collapse within 10  s  Stationary solution is a good approximation Deterministic simulation: Ca 2+ nanodomain Buffer: 100  M, D B =0.05  m 2 /ms, k - /k + =1  M Current: I Ca = 0.5 pA Ca 2+ Movie / Animation (will be shown at talk)

17. Deterministic vs Stochastic simulation Deterministic vs. Stochastic Simulation VCell, CalC, …Mcell, Smoldyn, … vs

18.  Number of Ca 2+ ions (1 ms pulse, I Ca =0.2 pA): N Ca ~600  Most Ca 2+ quickly buffered: only ~30 free ions  Ca 2+ binding reactions are non-linear  Two sources of stochasticity: 1. Stochastic channel gating: can be combined with deterministic Ca 2+ dynamics 2. Stochastic diffusion and reaction Significance of stochastic fluctuations: Stochastic Approach 1 pA  3121 ions per ms

19. Stochastic modeling of Ca dynamics Simulation using Smoldyn (www.smoldyn.org)  Buffer molecules  Ca 2+ ions  Bound buffer Release sensor 200 nm Channel location

20. Deterministic vs Stochastic simulation Modchang Nadkarni Bartol Triampo Sejnowski Levine Rappel (2010) “Comparison of deterministic & stochastic simulations…” Physical Biology Stochastic (MCell) vs. Deterministic: difference < 20% Greater discrepancy for small concentrations and release probability Rigorous analysis of this comparison is lacking Deterministic vs. Stochastic Simulation VCell, CalC,.. MCell, Smoldyn, … vs

21. Mass action works much better than expected! Monte-Carlo: 40,000 Smoldyn iterations (N Ca =312: 1ms 0.1pA pulse) Vs. Weinberg and Smith (2014) Biophys J: Mobile buffers increase fluctuations Ca 2+ sensor I Ca =0.1pA I Ca =0

22. Mass action kinetics: issues Mass action kinetics not exact: Issue 1: Continuous limit (  V  0) Issue 2: Correlations between N Ca and N B Issue 3: Variance in N Ca

23. Mass action issue #1: spatial scale Exact for  V  0 Fails for  V  0 Exact VV N Ca =4 N B =5

24. Mass action issue #1: spatial scale Fails for  V  0 VV N Ca =4 N B =5 Number of pairs: N pair = N Ca  N B =  V 2  Ca  B Concentration of pairs: N pair /  V =  V  Ca  B  0 Fail!  V V

25. Mass action issue #1: spatial scale K int (|  |) – interaction kernel |||| 1 nm Assumes B and Ca not correlated Mass action = coarse-grain over interaction radius

26. Mass action kinetics: issues Mass action kinetics not exact: Issue 1: Continuous limit (  V  0) Issue 2: Correlations between N Ca and N B Issue 3: Variance in N Ca

27. Mass action issue #2: correlation Ca   N Ca  : average number of Ca 2+ ions B   N B  : average number of B molecules Average number of Ca-B pairs:  N Ca  N B    N Ca    N B  !! VV N Ca N B Mass action = independence

28. Mass action kinetics: issues Mass action kinetics not exact: Issue 1: Continuous limit (  V  0) Issue 2: Correlations between N Ca and N B Issue 3: Variance in N Ca

29. Mass action issue #3: variance Ca 2+  N Ca   δ Stochastic  Set diffusion aside; consider local reactions: Incorrect logic…  N Ca  Deterministic 

30. Mass action issue #3: variance  N Ca   δ Stochastic  Consider argument #2:  N Ca  Deterministic  Ca 2+ Need rigorous analysis… Set diffusion aside; consider local reactions:

31. Deterministic vs Stochastic simulation Deterministic vs. Stochastic Simulation vs

32. Mass action vs Monte-Carlo: issues Mass action kinetics not exact: Issue 1: Continuous limit (  V  0): also an issue with Monte-Carlo Issue 2: Correlations between N Ca and N B : main source of discrepancy Issue 3: Variance in N Ca : not an issue!

33. Stochastic reaction-diffusion simulation algorithms Time-based particle Brownian reaction dynamics (MCell, Smoldyn) Unlimited spatial resolution (not coarse-grained) Inexact bi-molecular reaction Coarse-grained Gillespie solvers (STEPS, MesoRD, SmartCell) Exact local reaction (Gillespie in each sub-volume) Finite spatial resolution; accurate but inexact reaction-diffusion integration Event-based particle Brownian dynamics and reaction (CDS, GFRD) Unlimited spatial resolution (not coarse-grained) More accurate bi-molecular reaction, excluded volume effects Computationally expensive (D  t) 1/2

34. Mass action issues: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv Erban & Chapman (2009) Phys Biol “Stochastic modeling of reaction-diffusion processes”

35. Deterministic vs Stochastic: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv Erban & Chapman (2009) Phys Biol “Stochastic modeling of reaction-diffusion processes”

36. Deterministic vs Stochastic: rigorous analysis But... such reaction are physically impossible! Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv

37. Mass action issues: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv

38. Mass action issues: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv

39. Mass action issues: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv

40. Mass action issues: rigorous analysis Erban, Chapman, Maini (2007) “Practical guide to stochastic simulations of reaction-diffusion processes” arXiv

41. Mass action works much better than expected! Monte-Carlo: 40,000 Smoldyn iterations (N Ca =300: 1ms 0.1pA pulse) Vs. Hypothesis: correlations very small unless N ca  5 Ca 2+ sensor I Ca =0.1pA I Ca =0

42. Deterministic vs stochastic simulation Conclusion: Agreement better than expected Monte-Carlo and Mass-action approaches share common problems Monte-Carlo: reaction and diffusion cannot be combined “exactly” Homo-species reactions: deterministic approach suffers because of variance Hetero-species reactions: deterministic approach suffers because of correlations Need to examine size of correlations between Ca 2+ and Buffer Deterministic vs. Stochastic Simulation vs

43. Deterministic vs stochastic simulation Stationary solutions: beyond asymptotics Modeling cell Ca 2+ dynamics: challenges and open problems B(r) r Ca(r) r BTBT vs

44. Ca 2+ Modeling cell Ca 2+ dynamics Good modeling approach: Simulate stochastic channel opening/closing Simulate Ca 2+ dynamics deterministically

45. Modeling cell Ca 2+ dynamics Good modeling approach: Simulate stochastic channel opening/closing Simulate Ca 2+ dynamics deterministically

46. Modeling cell Ca 2+ dynamics Good modeling approach: Simulate stochastic channel opening/closing Use steady-state Ca 2+ solution near each open channel Nguyen Mathias Smith (2005) Bull Math Biol, 67(3):393-432.

47. Stationary solutions Ca 2+ B(r) r Ca(r) r BTBT

48. Non-dimensionalized equations for equilibrium Smith, Wagner, Keizer Biophys J (1996); Smith Dai Miura Sherman SIAM (2001) B(r) r Ca(r) r BTBT b(  )  c(  )  1

49. Stationary solution: asymptotic approximations Neher (1998); Smith, Wagner, Keizer Biophys J (1996); Smith Dai Miura Sherman SIAM (2001) b(  )  c(  )  1

50. Stationary solutions: alternative approach Problems with asymptotic approximations: Poor performance unless <<1 and/or  <<1 Hard to extend to complex buffers (2Ca 2+ + B  B ** ) Alternative approach: “Dual-Padé Approximation”

51. Alternative approach: “Dual-Padé” approximation b(  )  1 b o = ? b(x) x 1 b o = ? Large  behavior: x=1/ 

52. Stationary approximation: alternative approach b(  )  c(  )  1 This problem is hard: Nonlinear Singular (regular sing’ty at 0) Not autonomous BVP, not IVP “Weak” BC at  =0 Additional constraint: 0  b  1 ?   Channel location

53. Stationary approximation: alternative approach b(  )  1 b o = ? Finite radius of convergence for any b o

54. Stationary approximation: alternative approach b(  )  1 b o >0 Large  behavior: x=1/  b(x) x 1 b o = ? Finite radius of convergence for any b o

55. Stationary approximation: alternative approach b(  )  1 b o = ? Large  behavior: x=1/  b(x) x 1 b o = ? Asymptotic series exists Finite radius of convergence for any b o

56. Stationary solution: rational approximation b(  )  1 b o = ? Large  behavior: x=1/  b(x) x 1 b o = ?

57. Rational approximation: lowest order b(  )  1 b o = ? Large  behavior: x=1/  b(x) x 1 b o = ?

58. Rational approximation: lowest order

59. RBA: µ=1,  =0.1

60. Rational approximation: lowest order EBA: µ=0.1,  =1 RBA: µ=1,  =0.1

61. Rational approximation: lowest order RBA: µ=1,  =0.1 EBA: µ=0.1,  =1 Padé: µ=1,  =1

62. Rational approximation: second order RBA: µ=1,  =0.1 EBA: µ=0.1,  =1 Padé: µ=1,  =1

63. Rational approximation: third order, =  =1 Note: Rapid Buffer Approximation still superior for <<1,  =O(1): allows arbitrary set of channel and buffers All coefficients A k, B k obtained in closed form (muPad in MATLAB)

64. Stationary solutions: “Dual Padé” approximation Simplify expressions for coefficients for  2 nd order Analysis of singularities in complex plane More accurate Ansatz? Extension to “complex” (2-to-1) buffers (2Ca+B  B ** ) Stationary Ca 2+ nanodomain approximations