What I do. (and what I want to do) Berton Earnshaw. March 1, Department of Mathematics, University of Utah Salt Lake City, Utah 84112

Transcription

1 What I do (and what I want to do) Berton Earnshaw Department of Mathematics, University of Utah Salt Lake City, Utah March 1, 2008

2 Current projects What I do AMPA receptor trafficking Synaptic plasticity

3 Current projects What I do AMPA receptor trafficking Synaptic plasticity What I want to do Just count things

4 Mathematical neuroscience at Utah Cognition Systems Cortical Areas Small Networks Neurons Dendrites Synapses

5 The synapse E.R. Kandel et al. Principles of Neural Science (2000) M.B. Kennedy Science (2000)

6 Synaptic transmission E.R. Kandel et al. Principles of Neural Science (2000)

7 Synaptic plasticity Collingridge et al., Nat. Rev. Neurosci. (2004)

8 AMPA receptor trafficking at a single spine Sheng & Kim, 2002 Surface AMPARs constitutively recycle with intracellular stores Laterally diffuse within postsynaptic membrane Crosslink to scaffolding proteins in PSD

9 Model of trafficking at a single spine β EXO END Q P h R Ω U DEL DEG PSD α(z-q) σ EXO k Spine Head AMPA receptor scaffolding protein δ C σ DEG Spine head: PSD unbound: PSD bound: Intracellular: dr dt = 1 (Ω[U R] kr h[r P]) A dp dt = h a [R P] α[z Q]P + βq + σexo C a dq = α[z Q]P βq dt dc dt = σexo C σ DEG C + kr + δ,

10 Trafficking during LTP/LTD number of receptors in PSD Total Bound GluR1/2 Free GluR1/2 Bound GluR2/3 Free GluR2/3 Scaffolding number of receptors in PSD t [min] t [min] O Connor et al. PNAS (2005) Dudek & Bear PNAS (1992)

11 Fast or slow recycling? Passafaro et al., 2001 Adesnik et al., 2005

12 AMPA receptor trafficking along a spiny dendrite 1. somatic exocytosis 2. lateral membrane diffusion 3. surface entry into spine 4. local exo/endocytosis Groc & Choquet, 2006 AMPARs trafficked in vesicles along microtubules? AMPARs diffuse from soma to synapse?

13 Model of trafficking along a spiny dendrite U t = D 2 U ρ(x)ω(x)[u(x,t) R(x,t)] x2 ρ(x) = spine density at x receptor scaffolding protein PSD EXO END J soma DEL DEG x = 0 spine x = L

14 Steady-state receptor concentrations number or concentration [µm -2 ] concentration in dendrite number in PSD number bound in PSD distance from soma [mm] Piccini & Malinow, 2002 GluR1/2 GluR2/3 1,000 identical spines distributed uniformly Two sources of AMPARs at soma local intracellular delivery

15 Recovery rate depends on distance from soma recovery [% baseline] µm 300 µm time [hr] Recovery exhibits many time-scales!

16 Now that we re done with that......let s count something!

17 A single strand of RNA Primary structure: sequence of bases (A,G,U,C) Secondary structure: pairing of bases Watson-Crick pairs: A-U, G-C (less often U-G) Tertiary structure: resulting 3D molecule Different tertiary structures different enzymatic properties

18 A single strand of RNA: An example Primary structure: AACCAUGUGGUACUUGAUGGCGAC

19 A single strand of RNA: An example Primary structure: Secondary structure: AACCAUGUGGUACUUGAUGGCGAC

20 A single strand of RNA: An example Primary structure: Secondary structure: AACCAUGUGGUACUUGAUGGCGAC Tertiary structure: extremely difficult to predict (probably NP-hard)

21 RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1,...,n} all vertices have degree 1 there do not exist k arches {i 1, j 1 },...,{i k, j k } such that i 1 < < i k < j 1 < < j k

22 RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1,...,n} all vertices have degree 1 there do not exist k arches {i 1, j 1 },...,{i k, j k } such that i 1 < < i k < j 1 < < j k

23 RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1,...,n} all vertices have degree 1 there do not exist k arches {i 1, j 1 },...,{i k, j k } such that i 1 < < i k < j 1 < < j k RNA secondary structure of n bases, pseudoknot type k 2 k-noncrossing (but not k 1) arch diagram of order n no 1-arches {i, i + 1} abstract secondary structure (no primary structure)

24 k-noncrossing arch diagrams and walks in Weyl chamber Walk in Z m of length n sequence of vectors x 0,x 1,...,x n Z m s.t. x i+1 x i = 0 or 1

25 k-noncrossing arch diagrams and walks in Weyl chamber Walk in Z m of length n sequence of vectors x 0,x 1,...,x n Z m s.t. x i+1 x i = 0 or 1 Weyl chamber subset of vectors x = (x 1,...,x m ) Z m s.t. x 1 > > x m > 0

26 k-noncrossing arch diagrams and walks in Weyl chamber Walk in Z m of length n sequence of vectors x 0,x 1,...,x n Z m s.t. x i+1 x i = 0 or 1 Weyl chamber subset of vectors x = (x 1,...,x m ) Z m s.t. x 1 > > x m > 0 Theorem (Chen et al. (2007) Trans. Am. Math. Soc. 359) There exists a bijection between k-noncrossing arch diagrams of order n and walks of length n in Z k 1 which start and end at a = (k 1,k 2,...,1) and remain in the Weyl chamber.

27 Idea of proof: The bijection (4,3,2,1),(5,3,2,1),(6,3,2,1),(6,4,2,1),(6,4,2,1),(6,4,2,1), (6,4,3,1),(6,4,3,1),(5,4,3,1),(5,4,3,2),(6,4,3,2),(6,4,3,1),.(6,4,3,1),(6,4,2,1),(6,4,2,1),(6,3,2,1),(5,3,2,1),(4,3,2,1)

28 Counting k-noncrossing RNA secondary structures Set A k (n,l) = # k-nc arch diagrams of order n, l isolated nodes B k (n,l) = # k-nc RNA structures of n bases, l isolated bases

29 Counting k-noncrossing RNA secondary structures Set A k (n,l) = # k-nc arch diagrams of order n, l isolated nodes B k (n,l) = # k-nc RNA structures of n bases, l isolated bases Theorem (Jin, Qin & Reidys, 2008) B k (n,l) = (n l)/2 b=0 ( 1) b ( n b b ) A k (n 2b,l) where A k (n,l) is given by the generating function n n=1 l=0 A k (n,l) xn n! = ex det[i i j (2x) I i+j (2x)] k 1 i,j=1 and I r (2x) = j=0 x2r+j /(j!(r + j)!) is hyperbolic Bessel function of 1st kind of order r.

30 The end Thank you!