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In silico stochastic simulation of Ca 2+ triggered synaptic release Andrea Bracciali Enrico Cataldo Pierpaolo Degano Marcello Brunelli Dipartimento di Informatica Dipartimento di Biologia Universit` a di Pisa Universit` a di Pisa {

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SLIDE 1

In silico stochastic simulation of Ca2+ triggered synaptic release

Andrea Bracciali Enrico Cataldo Pierpaolo Degano Marcello Brunelli

Dipartimento di Informatica Dipartimento di Biologia Universit` a di Pisa Universit` a di Pisa

{braccia,degano}@di.unipi.it {ecataldo,mbrunelli}@biologia.unipi.it

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.1/23

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Models of the neural function

The functional capabilities of the nervous system arise from the complex organization of the neural network Models are needed to understand the ways in which neural circuits generate behavior, the ways in which experience alters the functional properties of circuits and therefore their behaviour (plasticity/memory), ... (and many other issues). (some) Key elements are the intrinsic biophysical/biochemical properties of the individual neurons the pattern of the synaptic connections amongst neurons the physiological properties of synaptic connections

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.2/23

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Models of the neural function

Focus:

  • 1. A model of a pre-synaptic calcium triggered release

Synapses: points of functional contact between neurons Chemical synapses: presynaptic action potentials cause chemical intermediary (neurotransmitters) to influence postsynaptic terminal Chemical synapses are plastic: modified by prior activity

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.3/23

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SLIDE 4

Models of the neural function

Focus:

  • 1. A model of a pre-synaptic calcium triggered release

Synapses: points of functional contact between neurons Chemical synapses: presynaptic action potentials cause chemical intermediary (neurotransmitters) to influence postsynaptic terminal Chemical synapses are plastic: modified by prior activity

  • 2. A (first) stochastic model

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.4/23

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SLIDE 5

Models of the neural function

Focus:

  • 1. A model of a pre-synaptic calcium triggered release

Synapses: points of functional contact between neurons Chemical synapses: presynaptic action potentials cause chemical intermediary (neurotransmitters) to influence postsynaptic terminal Chemical synapses are plastic: modified by prior activity

  • 2. A (first) stochastic model
  • 3. A computational (process-algebra based) approach

formal executable compositional

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.5/23

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Presynaptic calcium triggered release

From: Sudhof TC. The synaptic vesicle cycle. Annu Rev Neurosci. 27:509-47, 2004.

  • 1. Calcium gradient
  • 2. Vescicle activation

(exocitosis)

  • 3. Neuro-transimitter

release

  • 4. Calcium extrusion
  • 5. Vescicle recharging
  • 6. . . .
  • 7. Neuro-transmitter

reception

  • 8. . . .

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.6/23

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SLIDE 7

Presynaptic calcium concentration profile

From: Zucker RS, Kullmann DM, Schwartz

  • TL. Release of Neurotransmitters.

In: From molecules to networks - An introduction to cellu- lar and molecular neuroscience. Elsevier pp 197- 244 2004.

  • Microdomains of Calcium

concentrations near open channels

  • trigger the exocytosis of synaptic

vescicles.

  • Calcium concentration during release

is not homogeneous

  • unless subsequently in not effective

concentrations.

  • Uncaging:

An experimental method capable to induce spatially homo- geneous Calcium elevation in the presynaptic terminal. Applied to the synapse Calyx of Held.

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.7/23

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Calyx of Held: deterministic model

From: www.cs.stir.ac.uk/ bpg/research/syntran.html

By means of the uncaging method, a 5-step model of release has been defi ned based on concentrations, [SN00N]: Ca2+

i

+ V

5kon

− − − → ← − − − −

koff b0

VCa2+

i

+ Ca2+

i

. . . V4Ca2+

i

+ Ca2+

i kon

− − → ← − − − − −

5koff b4

V5Ca2+

i

γ

− → T where kon = 9 × 107 M −1s−1, koff = 9500 s−1, γ = 6000 s−1 and b = 0.25 have been defi ned by experimental fitting (complex).

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.8/23

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Calyx of Held: stochastic model of calcium uncaging

Deterministic model: unsuitable for small concentrations and volumes, e.g. if [Ca2+] = 10 µM, in a volume of 60 nm3 there is a single free ion; the assumption that binding of Ca2+ to vescicle does not affect the [Ca2+] concentration not adequate (with vescicle diameter ∼ 17 − 22 nm, V = 60 nm3 few Ca2+ ions).

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.9/23

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Calyx of Held: stochastic model of calcium uncaging

Deterministic model: unsuitable for small concentrations and volumes, e.g. if [Ca2+] = 10 µM, in a volume of 60 nm3 there is a single free ion; the assumption that binding of Ca2+ to vescicle does not affect the [Ca2+] concentration not adequate (with vescicle diameter ∼ 17 − 22 nm, V = 60 nm3 few Ca2+ ions). Stochastic model: actual quantities and stochastic rate constants: c = k 1st order c = k/(NA × V) 2nd order

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.9/23

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SLIDE 11

Calyx of Held: stochastic model of calcium uncaging

Deterministic model: unsuitable for small concentrations and volumes, e.g. if [Ca2+] = 10 µM, in a volume of 60 nm3 there is a single free ion; the assumption that binding of Ca2+ to vescicle does not affect the [Ca2+] concentration not adequate (with vescicle diameter ∼ 17 − 22 nm, V = 60 nm3 few Ca2+ ions). Stochastic model: actual quantities and stochastic rate constants: c = k 1st order c = k/(NA × V) 2nd order Calyx [SF06CTR]: a vast “parallel” arrangement of active zones (3-700) each one with up to 10 vescicles clustered in groups of about 10 in a volume with a diameter of almost 1 µm. action potential activates all the active zones.

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.9/23

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SLIDE 12

Calyx of Held: stochastic model of calcium uncaging

Deterministic model: unsuitable for small concentrations and volumes, e.g. if [Ca2+] = 10 µM, in a volume of 60 nm3 there is a single free ion; the assumption that binding of Ca2+ to vescicle does not affect the [Ca2+] concentration not adequate (with vescicle diameter ∼ 17 − 22 nm, V = 60 nm3 few Ca2+ ions). Stochastic model: actual quantities and stochastic rate constants: c = k 1st order c = k/(NA × V) 2nd order Calyx [SF06CTR]: a vast “parallel” arrangement of active zones (3-700) each one with up to 10 vescicles clustered in groups of about 10 in a volume with a diameter of almost 1 µm. action potential activates all the active zones. A cluster of 10 active zone each one with 10 vescicles in V = 0.5 10−15 liter

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.9/23

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Calyx of Held: stochastic model of calcium uncaging

The obtained stochastic model: con = 9 × 107 / (6.02 × 1023 × 0.5 × 10−15) s−1 = 0.3 s−1, coff = 9500 s−1, γ = 6000 s−1 b = 0.25. Ca2+ ions: 300, 3000 and 6000, corresponding to molar concentrations [Ca2+] of 1, 10 and 20 µM. Ca2+

i

+ V

5con

− − − → ← − − − −

coffb0

VCa2+

i

+ Ca2+

i

. . . V4Ca2+

i

+ Ca2+

i con

− − → ← − − − − −

5coff b4

V5Ca2+

i

γ

− → T

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.10/23

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SLIDE 14

Results

1000 2000 3000 4000 5000 6000 7000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 10 20 30 40 50 60 70 80 90 100 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Vstar T

Step-like calcium uncaging, V = 100, Ca2+ = 6000. Results are coherent with literature, [SN00N], e.g. High sensitivity of vescicles to Ca2+ concentration Calyx of Held triggers vescicle release with concentrations lower than 100 µM (usual values for other synapses are 100 − 300µM). In the fi gure 6000 Ca2+ correspond to 20µM.

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.11/23

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Results

1000 2000 3000 4000 5000 6000 7000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 10 20 30 40 50 60 70 80 90 100 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Vstar T 1000 2000 3000 4000 5000 6000 7000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 10 20 30 40 50 60 70 80 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Vstar T

Variation of b = 0.4 (was b = 0.25): lower and more uniform release rate

Ca2+ i + V 5con − − − − → ← − − − − − − coff b0 V Ca2+ i + Ca2+ i . . . V 4Ca2+ i + Ca2+ i con − − − → ← − − − − − − − 5coff b4 . . .

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.12/23

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SLIDE 16

Results

1000 2000 3000 4000 5000 6000 7000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 10 20 30 40 50 60 70 80 90 100 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Vstar T 1000 2000 3000 4000 5000 6000 7000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T 20 40 60 80 100 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Vstar T 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T

Variation of con = 0.5 (was con = 0.3): increase of the release rate

Ca2+ i + V 5con − − − − → ← − − − − − − coff b0 V Ca2+ i + Ca2+ i . . . V 4Ca2+ i + Ca2+ i con − − − → ← − − − − − − − 5coff b4 . . .

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.13/23

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SLIDE 17

Cell as computation [RS02N]

A process-algebra approach [recall previous Degano’s talk] interaction as communication processes [molecules, ions, proteins, vescicles, ...] defi ned as sequential, parallel, choice composition of communications stochastic reaction rates - associated to each communication - determine the next most probable interaction ... – Gillespie algorithm: rate × # Processes ready to interact – ... and the system evolves. A dialect of Pi-calculus as modeling language the SPiM interpreter [PC2004BC] as programming language/execution environment

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.14/23

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SLIDE 18

Model implementation — overview

directive sample 0.005 1 val con5 = 1.5 val b = 0.25 val coff5 = 47500.0 * b * b * b * b new vca@con5:chan ca() = do ?vca;()

  • r ?v2ca;()

... v() = !vca; v_ca() v_ca() = do !bvca; v()

  • r !v2ca; v_2ca()

Dv_ca() = ?bvca; ( ca() | Dv_ca() ) run 6000 of ca() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... )

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.15/23

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SLIDE 19

Model implementation — overview

directive sample 0.005 1 val con5 = 1.5 val b = 0.25 val coff5 = 47500.0 * b * b * b * b new vca@con5:chan ca() = do ?vca;()

  • r ?v2ca;()

... v() = !vca; v_ca() v_ca() = do !bvca; v()

  • r !v2ca; v_2ca()

Dv_ca() = ?bvca; ( ca() | Dv_ca() ) run 6000 of ca() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... ) Initial set-up (creation of communication channels)

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.16/23

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SLIDE 20

Model implementation — overview

directive sample 0.005 1 val con5 = 1.5 val b = 0.25 val coff5 = 47500.0 * b * b * b * b new vca@con5:chan ca() = do ?vca;()

  • r ?v2ca;()

... v() = !vca; v_ca() v_ca() = do !bvca; v()

  • r !v2ca; v_2ca()

Dv_ca() = ?bvca; ( ca() | Dv_ca() ) run 6000 of ca() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... ) Second order reaction

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.17/23

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SLIDE 21

Model implementation — overview

directive sample 0.005 1 val con5 = 1.5 val b = 0.25 val coff5 = 47500.0 * b * b * b * b new vca@con5:chan ca() = do ?vca;()

  • r ?v2ca;()

... v() = !vca; v_ca() v_ca() = do !bvca; v()

  • r !v2ca; v_2ca()

Dv_ca() = ?bvca; ( ca() | Dv_ca() ) run 6000 of ca() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... ) First order reaction (via single, dummy processes)

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.18/23

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SLIDE 22

Model implementation — overview

directive sample 0.005 1 val con5 = 1.5 val b = 0.25 val coff5 = 47500.0 * b * b * b * b new vca@con5:chan ca() = do ?vca;()

  • r ?v2ca;()

... v() = !vca; v_ca() v_ca() = do !bvca; v()

  • r !v2ca; v_2ca()

Dv_ca() = ?bvca; ( ca() | Dv_ca() ) run 6000 of ca() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... ) Setting initial conditions (quantities)

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.19/23

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SLIDE 23

A (modular) extension: Wave-like uncaging

1000 2000 3000 4000 5000 6000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P CaP 1 10 100 1000 10000 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P CaP 1 2 3 4 5 6 7 8 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Vstar T

Ca2+

i

+ P

c1

− → ← −

c2

CaP

c3

− → Ca2+

  • ...

val c1 = 8.00 new cp@c1:chan ca() = do ?vca;()

  • r ?v2ca;()

...

  • r ?cp;()

p() = !cp; ca_p() ca_p() = do !cpout; p()

  • r !cpback; ( p() | ca() )

... w( cnt : int) = do delay@40000.0; if 0 <= cnt then ( 80 of ca() | 80 of w(cnt - 1)) else ()

  • r !void; ()

run 1 of w(1) run 1000 of p() run 100 of v() run 1 of (Dv_ca() | Dv_2ca()| ... )

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.20/23

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SLIDE 24

Further ongoing developments [CMBS07]

1000 2000 3000 4000 5000 6000 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P CaP 1 10 100 1000 10000 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 V Ca2 V1Ca V2Ca V3Ca V4Ca Vstar T P CaP 0.5 1 1.5 2 2.5 3 3.5 4 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Vstar T

addressing plasticity: 2nd wave sensitive to residual calcium Moreover, compartimentalisation of the pre-synaptic terminal, ...

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.21/23

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SLIDE 25

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 26

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model further studying compartimentalisation

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 27

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model further studying compartimentalisation further pursuing the modular construction of one (more) synapses - post-synaptic terminal, ...

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 28

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model further studying compartimentalisation further pursuing the modular construction of one (more) synapses - post-synaptic terminal, ... studying linguistic and semantic suitable constructs, e.g. expressing locality, dynamically varying rates, ...

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 29

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model further studying compartimentalisation further pursuing the modular construction of one (more) synapses - post-synaptic terminal, ... studying linguistic and semantic suitable constructs, e.g. expressing locality, dynamically varying rates, ... devising qualitative/predictive analysis techniques

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 30

Conclusions

A fi rst stochastic model for (pre-)synaptic terminal A formal, executable, modular, verifi able computational model further studying compartimentalisation further pursuing the modular construction of one (more) synapses - post-synaptic terminal, ... studying linguistic and semantic suitable constructs, e.g. expressing locality, dynamically varying rates, ... devising qualitative/predictive analysis techniques making available results in an accessible (graphical) form.

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.22/23

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SLIDE 31

End of the talk

References

[SN00N] Schneggenburger N. and Neher E. “Intracellular calcium dependence of transmitter release rates at fast central synapse”, Nature 46:889-893, 2000. [SF06CTR] Schneggenburger N. and Fortsythe I.D. “The Calyx of Held”, Cell Tissue Res 326:311-337, 2006. [RS02N] Regev A. and Shapiro E. “Cellular Abstractions: Cells as Computation”, Nature 491:343, 2002. [CMSB] Bracciali A., Brunelli M., Cataldo E. and Degano P . “Expressive Models for Synaptic Plasticity”, CMSB 2007. To appear. [PC2004BC] Philips A. and Cardelli L. “A Correct Abstract Machine for the Stochastic Pi-Calculus”, Bioconcurr 2004, ENTCS.

Andrea Bracciali Enrico Cataldo Pierpaolo Degano Marcello Brunelli

Dipartimento di Informatica Dipartimento di Biologia Università di Pisa Università di Pisa {braccia,degano}@di.unipi.it {ecataldo,mbrunelli}@biologia.unipi.it

NETTAB 2007 – Pisa, – June 12-15, 2007 – p.23/23