El rol del azar en el proceso de secreción celular

1. El rol del azar en el proceso de secrecióncelular Isaac Meilijson with Ilan Hammel and Eyal Nitzany Tel Aviv University School of Mathematical Sciences and Faculty of Medicine Escuela de Invierno Luis A. SantalóUBA, 23-27 de Julio, 2012

2. E. Nitzany, I. Hammel, I. Meilijson Quantal basis of vesicle growth and information content, a unified approach. Journal of Theoretical Biology, 266, 202–209, 2010 I. Hammel, I. MeilijsonFunction suggests nano-structure: Electrophysiology supports that granule membranes play diceJournal of the royal society interface 2012I. Hammel, M. KREPELOVA, I. MEILIJSONStatistical analysis of the quantal basis of secretory granule formation (in preparation)I. HAMMEL, I. MEILIJSONGranule size distribution suggests mechanism: the case for granule growth and elimination as a fusion nano-machine (Review, to appear)

3. Pancreatic Acinar Cell

4. Two fundamental reasons for keeping granule inventory: • Uncertainty of demand: Granule stock is maintained as a buffer to meet uncertainty in demand by the extracellular environment. • Lead time for production: A source of supply during the lead time to produce granules of adaptive content. 

5. Quantal nature of granular volume

6. Most researchers hypothesize that granules stay as created and exit many-at-a-time Bernard Katz (Nobel Prize 1970) group in England Cope & Williams 1992 review

7. Is Quantal sizethe synaptic content of a single vesicle out of a heterogeneous pool? Gn = nG1 Is Quantal content the number of homogeneous vesicles released in a singleresponse of cell activation?

8. But not on Granule - Granule fusion + Granule Quantal Growthaddition of unit granuleAll agree on Granule-Membrane fusion

9. Random cut area measurements, 1000 observations

10. Intra-cell monitoring by cuts or projectionsGaussian with empirical fluctuations? Lindau 1995, Baumeister-Lučić 2010, others Converting data from diameter or equivalent area (measured indirectly by electrophysiology techniques) to volume may disclose equally-spaced modes

11. Baumeister et all 2010 diameter data

12. Histogram of volume data

13. MODELING AND SIMULATIONSTheoretical Considerations on the Formation of Secretory Granules in the Rat Pancreas. Hammel I, Lagunoff D, Wysolmerski R. Exp cell Res 204:1-5 (1993). Unit Addition Gn + G1 Gn+1 Random Fusion Gn + Gm Gn+m

14. Unit Addition Gn + G1 Gn+1 Poisson – like Frequency Gn/Gn-1= mean/(n-1) G1 G2 G3 G4 G5 G6 G7 G8 G9

15. Random Fusion Gn + Gm Gn+m Geometric-like Distribution Gn/Gn-1= 1-1/mean Frequency G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11

16. Better fit to unit addition mechanism

17. Markov Stationary ModelMarkov process: The distribution of Future given History is summarized by Present state. Granule Growth & Elimination : “state” is granule size

18. Granule steady-state modelStationary distribution ≠ Exit distribution Ryan et al. (1997) used optical microscopy and fluorescent dyes to track the spontaneous and evoked vesicle release.

19. Three-parameter modelμ/λ , γ , β • Exit rate from n: μ*nγ • Transition rate from n to n+1: λ*nβ • Markov: independent exponentials, earliest wins • Mean sojourn in n: 1/(μ*nγ+λ*nβ) • Probability of exit μ*nγ /(μ*nγ+λ*nβ)

20. SNARE, ROSETTE, POROSOME Atomic force microscopy Porosome - Jena BP, 1997 20

21. The rudder: The SNARE SYSTEM Human subtlety will never devise an invention more beautiful, more simple or more direct than does nature because in her inventions nothing is lacking, and nothing is superfluous. —Leonardo da Vinci (http://www.brainyquote.com/quotes/authors/l/leonardo_da_vinci.html)

22. Rationale for rate shape SNARE: μ*nγandλ*nβ N “rings” diffuse randomly on surface (area n2/3) of granule, of which K must be close to each other and to K “hooks” in unit granule or membrane. Probability is of order (const/area)K-1 = const*n-(2/3)(K-1)= μ*nγor λ*nβ Similar to Hua & Scheller PNAS 2001

23. Steady-state condition: flow into n+1 equals flow out of n+1STAT(n)* λ*nβ = STAT(n+1)*(μ*(n+1)γ+λ*(n+1)β)

24. EXIT distributionEXIT(n+) = GROW(1)* GROW(2)*...*GROW(n-1) andGROW(m)=λ*mβ/(μ*mγ+λ*mβ)EXIT(n) = EXIT(n+) - EXIT((n+1)+)

25. The role of γ • STAT(n)/EXIT(n) = exp{-γ*log(n)-b(γ)} • STAT is exponential-type family with EXIT as caseγ=0 • Perhaps evoked case needs one hook-ring pair! • Reminder: γ = -(2/3)(K-1)

26. Statistical tools • EM algorithm for Gaussian mixture with equally spaced means and variance • MLE of μ/λ ,γ , βbased on evoked and spontaneous data • Omnibus program for five parameters • CUSUM detection of change-point

28. Detecting that spontaneous secretion turned evoked secretion • CUSUM - Statistical tool for detecting change-point in distribution • Performance measured by Kullback-Leibler divergence KLD • CUSUM calculations may use common action potential monitoring.

29. CUSUM of Page (1953) • Declare change when log likelihood • Sn=Σ log g(Xi)/f(Xi) exhibits draw-up • Sn - minm≤nSm • of at least a pre-assigned size. • Eg[log g(X)/f(X)]>0 is KLD(g,f) • Ef[log g(X)/f(X)]<0 is KLD(f,g)

30. Mean time to correct detection • Sn RW with incr Xi,S0=0. TH threshold. • TVTH = first n with Sn ≥ TH. • TDTH = first n with Sn - minm≤n Sm ≥ TH. • Clearly, E[TVTH] ≥E[TDTH] • Wald: E[TVTH]=E[STVTH]/E[X]≥TH/E[X] • So E[TDTH] approx (little smaller than something little bigger than) TH/E[X].

31. Mean time between false alarms • BM Bt with drift ν<0 and std σ. • Mean time to draw-up TH is • [exp(ρTH)-1-ρTH]/ρ • (Taylor 1975, Meilijson 2003) • Where coefficient of adjustment (Asmussen 2000) • ρ=-2ν/σ2 is such that E[exp(ρB1)]=1. • For random walks, this result holds approximately.

32. SUMMARY, FORMULA • For log likelihoods ρ=1. • Putting all together, Mean Granules to Detection is • ln[MGFA]+ln[KLD(E,S)+η2/η1-1-ln(η2/η1)] • ----------------------------------------------------- • KLD(S,E)+η1/η2-1-ln(η1/η2)

33. One word = 1 bit = 8-10 vesicles

34. Cellular communication: do we really need granule growth? If the rate of secretion is increased • by a small factor (≈2), granule size distribution plays a critical role • by significant bursts (rate of increase = η2 /η1 >10) the role of granule size distribution is minor. The G&E model suggests that granule polymerization has an advantage for the information gain it achieves under limited exocytosis of a small number of granules.

35. GRACIAS!