Asymptotic analysis for vesicular release at neuronal synapses - PowerPoint PPT Presentation

Asymptotic analysis for vesicular release at neuronal synapses Claire Guerrier Applied Mathematics and Computational Biology, ´ Ecole Normale Sup´ erieure S´ eminaire Les probabilit´ es de demain May 17 th , 2016

Outline 1 Overview of synaptic transmission at chemical synapses 2 Asymptotic analysis of the narrow escape problem at a cusp 1 / 10

1 Overview of synaptic transmission at chemical synapses 2 Asymptotic analysis of the narrow escape problem at a cusp

Overview of synaptic transmission at chemical synapses Functional organization of chemical synapses 10 11 neurons in the human brain, each containing 10 3 synapses.         AP        Pre-synaptic terminal   Neurotransmitters        Calcium ions      Calcium channels � Synaptic cleft Neurotransmitters � receptors Post-synaptic terminal 2 / 10

Overview of synaptic transmission at chemical synapses Modeling SNARE complex activation by calcium ions Ca 2+ Synaptic vesicle Ca 2+ SNARE Cx Ca 2+ Ca 2+ Ca 2+ Ca 2+ Ca 2+ 3 / 10

Overview of synaptic transmission at chemical synapses Modeling SNARE complex activation by calcium ions Ca 2+ Synaptic vesicle Ca 2+ SNARE Cx Ca 2+ Ca 2+ Ca 2+ Ca 2+ Ca 2+ Calcium ions: Brownian particles. Docked vesicle: a sphere tangent to the surface of the Active Zone. Binding on the SNARE Complex: a particle reaches the red cylinder between the vesicle and the pre-synaptic membrane. 3 / 10

1 Overview of synaptic transmission at chemical synapses 2 Asymptotic analysis of the narrow escape problem at a cusp

Asymptotic analysis of the narrow escape problem at a cusp The narrow escape problem in a cusp A Brownian particle is described by the stochastic equation √ ˙ X = 2 D ˙ w . The first time to exit the domain ¯ Ω through the small hole ∂ ¯ Ω a , starting from x is ∈ ¯ Ω | X (0) = x ∈ ¯ τ ( x ) = inf { t > 0; X ( t ) / Ω } . The mean first passage time u ( x ) = E ( τ ( x )) is the solution of the mixed boundary value problem XXXXXXXXX e 3 ∆ D ∆ u ( x ) = − 1 for x ∈ ¯ Ω       ∂u ¯  ∂n ( x ) = 0 for x ∈ ∂ ¯ Ω \ ∂ ¯ Ω Ω a    ×   u ( x ) = 0 for x ∈ ∂ ¯  Ω a , where | ∂ ¯ Ω a | ≪ | ∂ ¯ Ω | . e 2 0 ∂ ¯ Ω a (Dynkin, 1961) e 1 4 / 10

Asymptotic analysis of the narrow escape problem at a cusp Reduction to a 2D problem Integrating over θ , we get e 3 ∆ ¯ Ω × e 2 0 ∂ ¯ Ω a e 1 The problem is independent of θ in cylindrical coordinates x = ( r, θ, z ) . It is equivalent to the following problem in Ω : 5 / 10

Asymptotic analysis of the narrow escape problem at a cusp Reduction to a 2D problem Integrating over θ , we get z e 3 ∆ ¯ R 2 Ω Ω | ∂ Ω a | = ε ≪ R 1 , × × � 2 R 1 R 2 r a = R 2 − R 1 ε (1 + o (1)) . R 1 × e 2 0 ∂ ¯ Ω a + r a r 0 ∂ Ω a e 1 . The problem is independent of θ in ∂ 2 u ∂r ( r, z )+ ∂ 2 u  ∂r 2 ( r, z ) + 1 ∂u cylindrical coordinates x = ( r, θ, z ) . ∂z 2 ( r, z )    r It is equivalent to the following problem     = − 1 in Ω :   D for ( r, z ) ∈ Ω  ∂u   ∂n ( r, z ) =0 for ( r, z ) ∈ ∂ Ω \ ∂ Ω a         u ( r, z ) =0 for ( r, z ) ∈ ∂ Ω a . 5 / 10

Asymptotic analysis of the narrow escape problem at a cusp Conformal mapping of domain Ω z t 1 s a = 0 × r a s 1 − 2R 2 1 R 2 f ( r + iz ) = z 1 ˜ r + iz Ω Ω z 2 + + × + r a 0 ∂ ˜ Ω a R 1 ∂ Ω a × 1 − 2R 1 r 0 ∂ Ω a 1 Boundary value problem for v ( s, t ) = u ( r, z ) , where f ( r + iz ) = r + iz = s + it : � ∂s ( s 2 + t 2 ) 2 ∆ v ( s, t ) + s 2 + t 2 �  ∂s ( s, t ) + ∂t ∂v ∂v = − 1 D for ( s, t ) ∈ ˜ ∂t ( s, t ) Ω   s ∂r ∂r     ∂v ∂n ( s, t ) =0 for ( s, t ) ∈ ∂ ˜ Ω \ ∂ ˜ Ω a      v ( s, t ) =0 for ( s, t ) ∈ ∂ ˜  Ω a . 6 / 10

Asymptotic analysis of the narrow escape problem at a cusp The inner solution near the absorbing boundary Scaling: √ √ � R 1 R 2 � ζ = s 2 Rε = s ε, ˜ R = R 2 − R 1 Y ( ζ, t ) = v ( s, t ) , and a regular expansion of Y in power of ˜ ε ε 2 Y 2 ( ζ, t ) + ... Y ( ζ, t ) = Y 0 ( ζ, t ) + ˜ εY 1 ( ζ, t ) + ˜ gives the expansion for the equation in the mapped domain: ζ 4 ∂ 2 Y 0 ζ 4 ∂ 2 Y 1 + ζ 4 ∂ 2 Y 0 ∂ζ 2 + 2 ζ 2 t 2 ∂ 2 Y 0 1 � � + 1 � − ζ 3 ∂Y 0 ∂ζ + 2 ζ 2 t∂Y 0 � = O (1) . ε 2 ˜ ∂t 2 ε ˜ ∂t 2 ∂t 2 ∂t 7 / 10

Asymptotic analysis of the narrow escape problem at a cusp The inner solution near the absorbing boundary Scaling: √ √ � R 1 R 2 � ζ = s 2 Rε = s ε, ˜ R = R 2 − R 1 Y ( ζ, t ) = v ( s, t ) , and a regular expansion of Y in power of ˜ ε ε 2 Y 2 ( ζ, t ) + ... Y ( ζ, t ) = Y 0 ( ζ, t ) + ˜ εY 1 ( ζ, t ) + ˜ gives the expansion for the equation in the mapped domain: ζ 4 ∂ 2 Y 0 ζ 4 ∂ 2 Y 1 + ζ 4 ∂ 2 Y 0 ∂ζ 2 + 2 ζ 2 t 2 ∂ 2 Y 0 1 � � + 1 � − ζ 3 ∂Y 0 ∂ζ + 2 ζ 2 t∂Y 0 � = O (1) . ε 2 ˜ ∂t 2 ε ˜ ∂t 2 ∂t 2 ∂t ε − 2 ) : Leading order term O (˜ Using the boundary conditions, ∂Y 0 ∂t ( ζ, t ) = 0 . 7 / 10

Asymptotic analysis of the narrow escape problem at a cusp The inner solution near the absorbing boundary Scaling: √ √ � R 1 R 2 � ζ = s 2 Rε = s ε, ˜ R = R 2 − R 1 Y ( ζ, t ) = v ( s, t ) , and a regular expansion of Y in power of ˜ ε ε 2 Y 2 ( ζ, t ) + ... Y ( ζ, t ) = Y 0 ( ζ, t ) + ˜ εY 1 ( ζ, t ) + ˜ gives the expansion for the equation in the mapped domain: ζ 4 ∂ 2 Y 0 ζ 4 ∂ 2 Y 1 + ζ 4 ∂ 2 Y 0 ∂ζ 2 + 2 ζ 2 t 2 ∂ 2 Y 0 1 � � + 1 � − ζ 3 ∂Y 0 ∂ζ + 2 ζ 2 t∂Y 0 � = O (1) . ε 2 ˜ ∂t 2 ε ˜ ∂t 2 ∂t 2 ∂t ε − 1 ) : Second order term O (˜ ζ 4 ∂ 2 Y 1 ( ζ, t ) + ζ 4 ∂ 2 Y 0 ( ζ ) − ζ 3 ∂Y 0 ( ζ ) ε − 2 ) : = 0 . Leading order term O (˜ ∂t 2 ∂ζ 2 ∂ζ Using the boundary conditions, Integrating over t and using the boundary ∂Y 0 ∂t ( ζ, t ) = 0 . conditions, we obtain 1 − ζ 2 � � Y 0 ( ζ ) = A , and 1 − s 2 ˜ � � v ( s, t ) = A ε . 7 / 10

Asymptotic analysis of the narrow escape problem at a cusp Computation of A using the divergence theorem v ( s, t ) = A � 1 − 2 Rεs 2 � . The constant A is determined from the divergence theorem � � ∂u ∆ u = ∂n ¯ ∂ ¯ Ω Ω � ε − | ¯ √ Ω | � � ∂u ∂u D = ∆ u = ∂n = 2 π 2 Rε ∂r dz = − 4 πAε. ¯ ∂ ¯ Ω Ω 0 Thus | ¯ Ω | A = 4 πDε . The leading order term of the mean first passage time outside of the boundary layer is obtained by setting s = 0 . It is independent of the initial position: | ¯ Ω | τ = 4 πDε . In the boundary layer, the leading order term of the mean first passage time is: | ¯ Ω | � 1 − 2 Rεs 2 � . v ( s, t ) = 4 πDε 8 / 10

Asymptotic analysis of the narrow escape problem at a cusp Summary Summary We computed the leading order term of the mean first passage time to a small ribbon located between two tangent spheres. The mean first passage time is constant outside of a boundar layer near the cusp, and is well approximated by a Poisson process. ( Schuss et al. , PNAS 2007) Next steps of the project: We built a model of the Active Zone, and investigated the influence of channels and vesicular organization on the release probability. We combined our previous result on the mean first passage time and the model of the Active Zone to build a model of the pre-synaptic terminal. This approach allows us to replace a model initially described using a system of PDE, with a system of ODE coupled to a Markov chain. We could the realize fast stochastic simulations using a Gillespie algorithm. 9 / 10

Asymptotic analysis of the narrow escape problem at a cusp Acknowledgements Acknowledgements IBENS David Holcman Jurgen Reingruber Jing Yang Assaf Amitai Nathanael Hoz´ e Khanh Dao Duc J´ erˆ ome Cartailler Ofir Shukron Pierre Parutto 10 / 10

Questions Comparison with MFPT in other geometries XXXXXXXXX XXXXXXXXX | ¯ Ω | τ = ¯ Ω 4 πDε √ × Surface of the hole: | ∂ ¯ 2 Rε 3 / 2 . Ω a | = 2 π ( Guerrier et al. , MMS , 2015) ∂ ¯ Ω a XXXXXXXXX XXXXXXXXX ∂ S a Small hole on a sphere: Small hole at the end of Σ a funnel-shaped cusp: τ = |S| √ S 4 Da . τ = | Σ | R Da 3 / 2 . × Surface of the hole: | ∂ S a | = πa 2 . Surface of the hole: | ∂ Σ a | = πa 2 . ( Singer et al. , J. Stat. ∂ Σ a Phys. , 2006) ( Holcman et al. , MMS , 2012)

Questions The Climbing Fiber to Purkinje cell synapses We observe several vesicles in the terminal. Some vesicles are docked to the pre-synaptic membrane. They are docked at the Active Zone, where calcium channels are also located. Serial Electron Microscopy section of Climbing Fiber (CF) synapses. Blue: CF pre-synaptic terminal Pink: Purkinje cell. Yellow: Astrocytes. ( Xu-Friedman et al. , J Neuroscience 2001.)

Questions Modeling the Active Zone Active Zone: a dense region apposed to the post-synaptic neuron where calcium channels and docked vesicles are located. A.Z.