One area of research has been the development of simulation software which can be easily used to model Ca2+ influx through channels and the subsequent imaging via a widefield microscope. This software has been used to model Caffeine activated plasma membrane channels as well as the relationship between STOCS (spontaneous transient outward currents) and sparks.

Discrete localized fluorescence transients due to openings of a single plasma membrane Ca2+ permeable cation channel were recorded using wide-field digital imaging microscopy with fluo-3 as the Ca2+ indicator. These transients were obtained while simultaneously recording the unitary channel currents using the whole-cell current-recording configuration of the patch-clamp technique. This cation channel in smooth muscle cells is opened by caffeine (Guerrero, A., F.S. Fay, and J.J. Singer. 1994. J. Gen. Physiol. 104:375–394). The localized fluorescence transients appeared to occur at random locations on the cell membrane, with the duration of the rising phase matching the duration of the channel opening. Moreover, these transients were only observed in the presence of sufficient extracellular Ca2+, suggesting that they are due to Ca2+ influx from the bathing solution. The fluorescence transient is characterized by an initial fast rising phase when the channel opens, followed by a slower rising phase during prolonged openings. When the channel closes there is an immediate fast falling phase followed by a slower falling phase. Computer simulations of the underlying events were used to interpret the time course of the transients. The rapid phases are mainly due to the establishment or removal of Ca2+ and Ca2+-bound fluo-3 gradients near the channel when the channel opens or closes, while the slow phases are due to the diffusion of Ca2+ and Ca2+-bound fluo-3 into the cytoplasm. Transients due to short channel openings have a "Ca2+ spark-like" appearance, suggesting that the rising and early falling components of sparks (due to openings of ryanodine receptors) reflect the fast phases of the fluorescence change. The results presented here suggest methods to determine the relationship between the fluorescence transient and the underlying Ca2+ current, to study intracellular localized Ca2+ handling as might occur from single Ca2+ channel openings, and to localize Ca2+ permeable ion channels on the plasma membrane.

(to be filled in shortly I hope)

Numerical simulations of Caffeine activated channel (CAC) activity is useful for many reasons. Obtaining a model which fits the data demonstrates the feasibility of the proposed physiological processes related to handling Ca²⁺ influx. By modifying the parameters of a model one can determine the sensitivity of the results to parameter variability. Simulation also provides access to information which is not available experimentally. Behavior close to a channel (< 100nm) can be analyzed. In addition, a simulation provides not only [CaFluo3] (which yields the fluorescence measurements) but also the [Ca²⁺] (which is not in equilibrium with CaFluo3 near the channel).

Several groups have been active in simulating reaction-diffusion equations inside the cell. The most relevant simulations (to our work) typically involve examination of sparks. Two main factors motivate simulations of sparks. First, almost all spark images are acquired via a confocal linescan microscope. The resulting data is fluorescence(x,t). Most sparks are not precisely in focus and most are not centered exactly on the line being scanned. Simulations are therefore used to understand the influence of these uncontrollable experimental factors on the resulting images, and on the derived data (e.g., histograms of spark amplitudes). The second reason is to determine whether sparks are comprised of a single channel opening or multiple channels. Simulations help in two ways. They can be used to determine total Ca²⁺ influx. Using estimates of single channel current (e.g., from lipid bilayers), total current then can be converted into an estimate of channel number. Simulations can also be used to indicate the spatial extent of the Ca²⁺ release region. Several studies have shown that a spatially extended source (e.g., several channels near each other) produce a simulated profile which matches the data better than a single source.

Another research area using simulations of cellular Ca²⁺ handling processes does so to understand neurotransmitter release. The primary motivation for simulations in this area is to provide information about [Ca²⁺] very close (<100nm) to the membrane. This is a region which is currently impossible to resolve via fluorescence microscopy. In addition, since near the channel Ca²⁺ is not in equilibrium with a fluorescent indicator it is unreliable to directly interpret any fluorescent images as indicative of [Ca²⁺]. In fact, these nonequilibrium domains may be an important control mechanism in neurotransmitter release [Neher, "Vesicle Pools and Ca2+ Microdomains: New Tools for Understanding Their Roles in Neurotransmitter Release, Neuron, vol. 20, 389-399, 1998].

Most simulations of intracellular Ca²⁺ handling model Ca²⁺ release as occurring at the center of a spherically symmetric cell. Since most sparks are inside the cell, away from the plasma membrane, this approximation is often reasonable. Spherical symmetry permits a 3D volume to be simulated in the time it takes to simulate a 1D "volume" (i.e., a much smaller volume). (Technically, going from one dimension to higher dimensions also introduces additional issues related to the control of stability and convergence of the algorithm). Spherical simulations are, of course, unable to include any structure or process which does not have this symmetry (except a ½ ball, e.g., a spherical cell cut by a flat membrane).

Most simulations of intracellular Ca²⁺ handling near a channel are based upon solving a set of partial differential equations (PDEs), the reaction-diffusion equations. Each buffer or ion in the model typically adds two equations to the system. The system of equations is nonlinear and an iterative method is typically used to determine how the model evolves in time. Solution of the equations can be quite time consuming. Some factors influencing the time required are the numerical technique used to approximate the derivatives, the number of elements into which the volume is subdivided, the size of the temporal step each iteration of the algorithm can take (frequently restricted due to stability problems in the numerical solution) and the duration of the opening simulated. Derivatives are typically approximated by either finite differences or finite elements. The number of elements involved in a subdivision depends upon the total volume being simulated and the size of each element. Adaptive subdivision methods use small elements in regions of rapid change (e.g., near the channel) and large elements in regions of little change, thereby using fewer elements than a fixed size element approach.

Currently simulations can be performed in cylindrical coordinates, 2D Cartesian coordinates, or 3D Cartesian coordinates. Cylindrical coordinates only require two spatial parameters, yet simulate a (cylindrically symmetric) 3D geometry. This two dimensional formulation can be simulated much quicker than a fully three dimensional model. This model also matches our application better than a spherical one. Our data is acquired from smooth muscle cells. These cells are, approximately, shaped like a cylinder. This is important since the widefield microscope used to acquire the images has a much larger depth of field than a confocal. Therefore the shape of the volume being imaged can affect the fluorescence image. CACs also typically permit current to flow for a much longer time than sparks. This potentially can produce a spatially larger fluorescence spot (depending upon buffers and current levels), making cell geometry important. In addition, since CACs are located on the plasma membrane, it is not as valid to assume that the Ca²⁺ is really coming in at the center of a sphere (although locally for short time periods the membrane might considered a flat plane so the cell could be modeled as a ½ ball). The membrane restricts diffusion (it also possesses Ca²⁺ pumps but we’ve seen no evidence that this is a significant factor) thereby changing the CaFluo3 distribution from what it would otherwise be.

However, the cylindrical symmetry of this simulation restricts the geometry of the cells which can be modeled; a single channel cannot be placed along the (curved) side of a cylindrical cell. Therefore cylindrical simulations have either placed the channel in the center (r=0, x = L/2) of a cell (radius = 6m m, length = 30m m) or on the end (r=0, x = 0) of a cell (radius = 6m m, length = 12m m). The latter configuration can model a single channel in the middle of a (flat) membrane (PM pumps are present on the ends of the cylinder as well as the sides).

The simulation currently permits an arbitrary number of buffers with arbitrary diffusion coefficients (e.g., mobile fluo-3 and an immobile endogenous buffer), as well as Ca²⁺. The SR can be located at any specific location (within the constraints of cylindrical symmetry). We typically locate this directly under the plasma membrane (as we are simulating smooth muscle cells). This differs from most other simulations which model SR as being everywhere present. In addition, since the SR is an explicit organelle it can have a different diffusion rate within it. The SR can contains ATPase Ca²⁺ pumps and can exhibit calcium induced calcium release (CICR).

Ca²⁺ current during a simulation can be controlled by a current trace obtained during an experiment. This allows the simulation to briefly close the channel in exactly the same manner as sometimes seen during an otherwise long open time. By incorporating short closures precisely into the simulation we are able to compare experimental data, which otherwise would have been compromised by the closures, to simulated results.

An explicit time, finite difference approximation is used to solve the partial differential equations (PDEs) comprising the reaction-diffusion equations. The finite difference equations are solved with a sub-millisecond time step (to ensure stability), with images of intracellular concentrations typically saved every millisecond of simulation time. Simulations are usually performed at 100nm spatial resolution. Concentration images are then converted to three dimensional Cartesian coordinates. These images are then "oriented" relative to an optical axis, and numerically blurred (convolved) with the theoretical point spread function (PSF) of a widefield microscope. We then typically examine the longer cylinder with its axis perpendicular to the optical axis, and the shorter cylinder with its axis either perpendicular to or aligned with the optical axis. These orientations of the shorter cylinder provided, respectively, a view of the transient with the channel on the side or on the top/bottom of the cell.

The simulation has been parallelized to run on a Beowulf system (we run it on Linux PCs with Alpha CPUs, connected via Myrinet). Variable sized voxels can be specified, thus enabling a locally high spatial resolution (e.g., 25 nm near a channel opening) without requiring similar resolution (and its concomitant increased simulation time) elsewhere (e.g. 100 nm voxels elsewhere).

We have found initial rates of fluorescent rise proportional to current and signal mass units (SMU, total integrated light) proportional to time and invariant to focus. Unlike calculations of SMU from confocal linescan data, we do not have to weight values by the cube of the radius (since we have 2D images and the large depth of field from the widefield microscope). We are therefore not bothered by noise magnification.

Several aspects of our experimental system cause acquired images to be influenced by events at a larger spatial scale than confocal linescan images of cardiac sparks. A widefield microscope has a much larger depth of field than a confocal. In addition, total light collected is independent of focus (e.g., the SMUs of each z-plane acquired from a point source is constant). The CAC open times are typically long (100s of ms to seconds). Experiments are often performed with SR uptake blocked. This larger scale requires that simulations take into account more global aspects of cell shape and size (at least in some circumstances).

F₀ (initial fluorescent intensity) can be significantly affected by the size of a cell. An unrestored widefield image of a cell filled with a uniform [CaFluo3] will have an F₀ approximately proportional to cell depth (at any given xy position). Therefore, identical D F (e.g., due to current influx) will produce different of D F/F₀ in cells of different depth. Of course restoration (which we plan on applying to fast 3D images of CACs) eliminates this dependency of F₀ on cell depth.

Widefield has better collection efficiency and longer dwell time than confocal. Unlike confocal linescan images it can directly determine the 2D spatial profile of Ca²⁺ entering a channel. This should make it possible to directly recognize any spatial inhomogeneities affecting Ca²⁺ diffusion into the cell (and then incorporate them into a simulation). In addition, confocal linescan images (and hence some parameters like peak F/F₀) are significantly affected by the xy position of the linescan relative to the spark. Widefield imaging does not have these difficulties.