BENG 276 HHMI Interfaces Lab 2: Numerical Analysis for Multi-Scale Biology Intracellular Transport Modeling Andrew McCulloch Department of Bioengineering

Example: Ca 2+ Signaling, Buffering and Diffusion In Atrial Myocytes Calcium is an important second messenger that regulates many dynamic physiological processes in the cell. Confocal laser scanning microscopy in combination with fluorescent indicators enables subcellular processes such as intracellular Ca 2+ dynamics to be visualized. However, these methods are still limited. For example the indicator itself acts as a calcium buffer and this influences the signal. Images are two-dimensional. Thus models of calcium diffusion in the cell are valuable.

Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 1999, 76:A459 A: Fluo-3 fluorescence (yellow line). B: Ca 2+ signal recorded in the cell center (blue) and periphery (red). C: Spatially averaged Ca 2+ signal across the entire cell. D: L-type Ca 2+ current and voltage-clamp protocol. E: A line-scan image recorded from the yellow line in (A). F: Spatial profile of Ca 2+ during the rising phase of Ca 2+. G: Ca 2+ signal as a surface plot, computed from the line-scan image in (E). Experimental Recordings

Cylindrical cell geometry is assumed. Model cell has diameter and volume corresponding to the real atrial myocyte. The cell has two spaces - restricted subsarcolemmal (RSP) and myofibrillar (MYOF). Ca 2+ and mobile buffers fluo-3 and calmodulin diffuse throughout the cell. SR Ca 2+ release and uptake are not included Ca 2+ enters via L-type Ca 2+ current. The activity of the Na + / Ca 2+ exchanger at rest is compensated by a Ca 2+ -leak. Ca 2+ binds to fluo-3, calmodulin, troponin-C and phospholipids without cooperativity. Initial total concentrations of the mobile buffers (fluo-3 or calmodulin) are spatially uniform. Diffusion constants for Ca 2+ bound to fluo-3 or calmodulin are approximately equal to the diffusion constants for fluo-3 or calmodulin. Model Formulation

Governing Equations where: [C] intracellular species concentration; D C diffusion constant for [C]; J i source – source i for [C]; J j sink –sink j for [C]; k m +, k m - - kinetic rates

Model Formulation

Example: Intracellular Ca 2+ buffering by Troponin C where: [Ca 2+ ] i free intracellular Ca 2+ ; Tn stationary buffer troponin C; D Ca diffusion constant for free Ca 2+ ; k Tn +, k Tn - kinetic rate constants; J LCC L-type Ca 2+ current; J NCX Na/Ca exchange current; J leak Ca 2+ leak. Sources and sinks for Ca 2+ were modeled as in Michailova et al., Biophys J 2002.

Finite Difference Method for 1-D Heat Equation

Implicit Scheme

Time Discretization: The  -Rule

Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 1999, 76:A459 A: Model cell has a diameter (15.6 μm) and capacitance (41 pF) corresponding to the real cell. B: Ca 2+ signal calculated for the cell center (blue), periphery (red) and fuzzy space (green). C: Time-course of spatially averaged Ca 2+ signal. D: Simulated Ca 2+ current. E: Calculated diffusion of Ca 2+ in space and time as a line-scan image. F: Spatial profile of Ca 2+ during the rising phase of Ca 2+ (at 100 ms). G: Ca 2+ signal as a surface plot. Model Results

Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 1999, 76:A459 Ca 2+ Signaling, Buffering and Diffusion in Atrial Myocytes

Effects of Fluo-3 concentration: Surface plots A-C were reconstructed from simulations with 0 μM, 30 μM and 400 μM Fluo-3. Panel D shows Ca 2+ signal in the cell center for Fluo μM. Panel E shows Ca 2+ concentration at 100 ms for various Fluo-3 concentrations for the restricted space (red), the periphery (green), the average (black) and the cell center (blue). Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 2002

Virtual Cell Fokker–Planck equation for Brownian motion of a particle in a fluid drift D 1 (x,t) and diffusion D 2 (x,t)

Virtual Cell Experiment and simulation of calcium dynamics following BK stimulation of a neuroblastoma cell. A 250 nM solution of BK was applied at time 0, and the [Ca 2+ ] cyt is monitored with fura-2 to produce the experimental record (left) obtained at 15 frames/sec. Representative frames are shown, and the change in calcium in the neurite (green box) and soma (yellow box) are plotted in the inset. The Virtual Cell simulation shown in the next column provides a good match to the experiment. The third and fourth columns display the simulation results for [InsP3] cyt and Po, the open probability of the InsP3-sensitive calcium channel in the ER membrane (Slepchenko BM et al. Annu Rev Biophys Biomol Struct 2002;31:423-41)

Temporal and spatial control of signaling Steinberg SF, Brunton LL. Annu Rev Pharmacol Toxicol. 2001;41: Need experimental techniques with high spatiotemporal resolution Temporal control differential response rates positive and negative feedback loops Spatial control membrane microdomains protein complexes restricted diffusion

Imaging PKA activity in live neonatal cardiac myocytes with AKAR Zhang J, Ma Y, Taylor SS, Tsien RY. Proc Natl Acad Sci U S A Dec 18;98(26): membrane-targeted pmAKAR2 25  m AKAR2

Imaging PKA-mediated phosphorylation gradients with local cAMP uncaging before after global UV local UV

cAMP network model Modeling PKA-mediated phosphorylation gradients from local cAMP uncaging 50  m Cardiac myocyte geometry cAMP AKAR 2d model of local cAMP uncaging 50  m Local UV spot

Imaging and modeling PKA-mediated phosphorylation gradients with local cAMP uncaging UV 0 µm 65 µm

PKA-mediated phosphorylation gradients 0 µm 65 µm experimentmodel global UV local UV baseline ΔR P / ΔR D Δt MAX (s)

Local cAMP uncaging in model D cAMP (μm 2 /s) Δt MAX ΔR P / ΔR D Dotted lines: experimental mean Sensitivity to changing D cAMP Red arrow: 270 µm 2 /s (simple cell) D cAMP = 100 μm 2 /s D cAMP = 10 4 μm 2 /s inhibit PDE inhibit PDE + AC ΔR P / ΔR D Gradient magnitude w/ perturbations no cAMP buffering by PKA