el rol del azar en el proceso de secreción celular isaac meilijson with ilan hammel and eyal...
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El rol del azar en el proceso de secreción celular
Isaac Meilijson with Ilan Hammel and Eyal Nitzany
Tel Aviv University
School of Mathematical Sciences and Faculty of Medicine
Escuela de Invierno Luis A. Santaló UBA, 23-27 de Julio, 2012
I. HAMMEL, I. MEILIJSONFUNCTION SUGGESTS NANO-STRUCTURE: ELECTROPHYSIOLOGY SUPPORTS THAT GRANULE MEMBRANES PLAY DICEJOURNAL OF THE ROYAL SOCIETY INTERFACE 2012
I. 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)
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
Two fundamental reasons for keeping granule inventory:
1. Uncertainty of demand: Granule stock is maintained as a buffer to meet uncertainty in demand by the extracellular environment.
2. Lead time for production: A source of supply during the lead time to produce granules of adaptive content.
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Most researchers hypothesize that granules stay as created and exit many-at-a-timeBernard Katz (Nobel Prize 1970) group in England
Cope & Williams 1992 review
Is Quantal content the number of homogeneous
vesicles released in a single response of cell activation?
Is Quantal size the synaptic content of a single vesicle out of a heterogeneous pool?
Gn = nG1
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Granule Quantal Growthaddition of unit granule
All agree on Granule-Membrane fusion
But not on Granule
+
- Granule fusion
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
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
Unit Addition Gn + G1 Gn+1
Poisson – like
Fre
quen
cy
G1 G2 G3 G4 G5 G6 G7 G8 G9
Gn/Gn-1= mean/(n-1)
Random Fusion Gn + Gm Gn+m
Geometric-like DistributionF
requ
ency
G1 G2 G3 G4 G5 G6 G7 G8 G9
Gn/Gn-1= 1-1/mean
G10 G11
Markov Stationary ModelMarkov process: The distribution of Future given History
is summarized by Present state. Granule Growth & Elimination : “state” is
granule size
Granule steady-state model Stationary distribution ≠ Exit distribution
Ryan et al. (1997) used optical microscopy and fluorescent dyes to track the spontaneous and evoked vesicle release.
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Three-parameter model μ/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)
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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)
The rudder: The SNARE SYSTEM
Rationale for rate shape SNARE: μ*nγ and λ*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*nγ or λ*nβ
Similar to Hua & Scheller PNAS 2001
Steady-state condition: flow into n+1 equals flow out of n+1
STAT(n)* λ*nβ = STAT(n+1)*(μ*(n+1)γ + λ*(n+1)β)
EXIT distribution
EXIT(n+) = GROW(1)* GROW(2)*...*GROW(n-1)
andGROW(m)=λ*mβ/(μ*mγ+λ*mβ)
EXIT(n) = EXIT(n+) - EXIT((n+1)+)
The role of γ ¨ STAT(n)/EXIT(n) = exp{-γ*log(n)-b(γ)}¨ STAT is exponential-type family with
EXIT as case γ=0
¨ Perhaps evoked case needs one hook-ring pair!
¨ Reminder: γ = -(2/3)(K-1)
Statistical tools
¨ EM algorithm for Gaussian mixture with equally spaced means and variance
¨ MLE of μ/λ , γ , β based on evoked and spontaneous data
¨ Omnibus program for five parameters¨ CUSUM detection of change-point
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.
CUSUM of Page (1953)
¨ Declare change when log likelihood ¨ Sn =Σ log g(Xi)/f(Xi) exhibits draw-up
¨ Sn - minm≤n Sm ¨ 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)
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].
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.
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)
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.