El rol del azar en el proceso de secreción celular

Isaac Meilijson with Ilan Hammel and Eyal Nitzany

Tel Aviv UniversitySchool 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. MeilijsonQuantal basis of vesicle growth and information content, a unified approach. Journal of Theoretical Biology, 266, 202–209, 2010

Pancreatic Acinar Cell

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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Quantal nature of granular volume

Most researchers hypothesize that granules stay as created and exit many-at-a-timeBernard Katz (Nobel Prize 1970) group in EnglandCope & 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

Random cut area measurements, 1000 observations

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

Baumeister et all 2010 diameter data

Histogram of volume data

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

Freq

uenc

y

G1 G2 G3 G4 G5 G6 G7 G8 G9

Gn/Gn-1= mean/(n-1)

Random Fusion Gn + Gm Gn+m

Geometric-like DistributionFr

eque

ncy

G1 G2 G3 G4 G5 G6 G7 G8 G9

Gn/Gn-1= 1-1/mean

G10 G11

Better fit to unit addition mechanism

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)

20Porosome - Jena BP, 1997

Atomic force microscopy

SNARE, ROSETTE, POROSOM

E

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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)

One word = 1 bit = 8-10 vesicles

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.

GRACIAS!