THEORY AND APPLICATIONS OF GEOMETRICAL

25 Geometrical measures of localized calcium release (LCR) events have been used 26 to understand their biophysical basis. We found power-law scaling between three such 27 metrics maximum amplitude (MA), mass above half maximum amplitude (MHM) and 28 area at half maximum amplitude (AHM). In an effort to understand this scaling a minimal 29 analytic model was employed to simulate LCR events recorded by confocal line scan. 30 The distribution of logMHM as a function of logAHM, pMHM(pAHM), was dependent 31 on model parameters such as channel open time, current size, line scan off set and 32 apparent diffusion coefficient. The distribution of log[MHM/AHM] as a function of 33 logMA, p[MHM/AHM](pMA), was invariant reflecting the gross geometry of the LCR 34 event. The findings of the model were applied to real LCR line scan data from rabbit 35 portal vein myocytes, rat cerebral artery myocytes and guinea-pig fundus knurled cells. 36 pMHM(pAHM) could be used to distinguish two populations of LCR events in portal 37 vein, even at the scale of "calcium noise", and to calculate the relative current of the two. 38 The relative current was 2. pMHM(pAHM) could also be used to study pharmacological 39 effects. The pMHM(pAHM) distribution of knurled cell LCR events was markedly 40 contracted by ryanodine, suggesting a reduction in channel open time. The 41 p[MHM/AHM](pMA) distributions were invariant across all cell types and were 42 consistent with the model, underlying the common physical basis of their geometry. The 43 geometrical scaling of LCR events demonstrated here may help with their mechanistic 44 characterization. 45 46 INTRODUCTION 47 48 49 Calcium ions are actively sequestered by the intracellular organelles of almost all 50 cells. The release of these ions into the cytoplasm through channels in the organelle 51 membrane can take on a variety of forms dependent on the channels involved, the 52 stimulus for release, buffering by the cytoplasm and organelle and many other factors. 53 Localized calcium release (LCR) remains restricted to within a few microns of the 54 channels and lasts for period of milliseconds to a few seconds. Since they were first 55 imaged by Cheng (8), such LCR events have been demonstrated in a large range of cell 56 types, have been shown to be a major physiological mechanism and have inspired a 57 whole vocabulary, including the most common patterns of LCR sparks and puffs (7). 58 LCR events are imaged with fluorescence microscopy by loading the cytoplasm 59 with a dye that fluoresces when it binds calcium. A major concern has been to 60 quantitatively relate this fluorescence signal to the underlying biophysical properties such 61 as channel conductance, channel numbers and diffusion parameters. Two commonly used 62 metrics for this purpose are the maximum amplitude (MA) and the signal mass (SM). 63 MA, as the name suggests is simply the peak fluorescence of the LCR event. SM is the 64 sum of LCR fluorescence over space (6, 14, 24, 30-32). 65 Allometry is the study of the relationship between two metrics across a population 66 of objects similar in nature but of differing size. Specifically if two metrics, a and b, have 67 the relationship a = mb (a power law), if n =1 the relationship is said to be isometric but 68 if n is not 1 the relationship is allometric. Commonly this relationship is visualized by a 69 double log plot where log(a) = log(m) + n.log(b). Both allometric and isometric 70 relationships are common across the biological sciences and usually reflect underlying 71 physical laws. It was of interest to us to see if such a relationship applied to LCR events 72 and would be valuable in understanding LCR. We measured LCR events from various 73 cell types with three metrics the mass above half maximum amplitude (MHM), area at 74 half maximum amplitude (AHM) and MA. Isometric scaling was apparent between these 75 metrics. By simulation of LCR events with an analytic model it was found that this 76 scaling could be used to estimate relative current and understand changes wrought by 77 pharmacological manipulation. 78 79 METHODS 80 81 82 Data Collection 83 LCR events were recorded from three types of isolated cell guinea pig fundus 84 knurled cells, rabbit portal vein myocytes and rat middle cerebral artery myocytes as 85 described previously (12, 13, 21). Gastric fundus muscle was dissected in situ from the 86 stomach of guinea-pigs after cervical dislocation and exsanguination. Portal vein 87 upstream of the anastomosis of its right and left branches was removed from New 88 Zealand rabbits after overdose with pentobarbitone. Right and left middle cerebral 89 arteries were removed from Wistar rats after cervical dislocation and exsanguination. All 90 procedures were in line with UK guidelines (where all experiments took place) and the 91 UK Animals (Scientific Procedures) Act 1986. After obtaining muscle, cells were 92 isolated by enzymatic treatment (collagenase, protease, BSA, trypsin inhibitor) followed 93 by mechanical trituration. Cells were loaded with either fluo-3AM or fluo-4AM and 94 LCR events were recorded in the line scan mode with a Zeiss LSM 510 laser scanning 95 confocal microscope (Jena, Germany). 96 97 Computation 98 All data analysis was carried out with programs written in Java as a plugin for the 99 ImageJ image analysis software (NIH, Bethesda). We utilised Michael Thomas 100 Flanagan's scientific library (10), Piotr Wendykier's JTransforms library (28) and code 101 from Burger and Burge (5). All LCR simulations were carried out with programs written 102 in C++. 103 104 Data Processing 105 Line scan images were first processed by three algorithms, in order, 106 1) Lee Filtering. The Lee filter is a local statistics filter designed to dampen Poisson noise 107 whilst conserving signal shape (16). We used a σ = 5 and a kernel size of 5x5 pixels. 108 2) Zero-frequency band removal. Bands of brightness along the time dimension are 109 common in line scans of calcium fluorophore fluorescence, reflecting loading of 110 organelles. Izu et al. removed these bands by zeroing the zero-frequency component of 111 each time line's Fourier transform (15). We implemented a modification of this, 112 calculating the FFT for the first and last half of each time line separately and then using 113 the zero-frequency component from each half as interpolation points on a line that was 114 then subtracted from each time line as a whole. This accommodated for bleaching in 115 some bands. 116 3) Maximum region normalization. Line scans of LCR events are typically normalized to 117 the basal cytosolic fluorescence (F/F0). We used a novel algorithm to do this. Briefly, a 118 time line is segmented at different thresholds (T), beginning at the minimum pixel value 119 and then increasing in steps of Δ. At each threshold the number of segmented regions is 120 counted. If the number of regions at a threshold (Tcurrent) is less than the number counted 121 at T = Tcurrent nΔ, the algorithm stops and F0 = Tcurrent nΔ. This process is repeated for 122 each time line and the whole image is divided through by the average of these F0 values. 123 We used Δ = 0.1 and n = 5. 124