Exact Expectations of Minimal Spanning Trees for Graphs With Random Edge Weights

Two methods are used to compute the expected value of the length of the minimal spanning tree (MST) of a graph whose edges are assigned lengths which are independent and uniformly distributed. The first method yields an exact formula in terms of the Tutte polynomial. As an illustration, the expected length of the MST of the Petersen graph is found to be 34877/12012 = 2.9035 .... A second, more elementary, method for computing the expected length of the MST is then derived by conditioning on the length of the shortest edge. Both methods in principle apply to any finite graph. To illustrate the method we compute the expected lengths of the MSTs for complete graphs

Calcium Regulation of Single Ryanodine Receptor Channel Gating Analyzed Using HMM/MCMC Statistical Methods

Type-II ryanodine receptor channels (RYRs) play a fundamental role in intracellular Ca2+ dynamics in heart. The processes of activation, inactivation, and regulation of these channels have been the subject of intensive research and the focus of recent debates. Typically, approaches to understand these processes involve statistical analysis of single RYRs, involving signal restoration, model estimation, and selection. These tasks are usually performed by following rather phenomenological criteria that turn models into self-fulfilling prophecies. Here, a thorough statistical treatment is applied by modeling single RYRs using aggregated hidden Markov models. Inferences are made using Bayesian statistics and stochastic search methods known as Markov chain Monte Carlo. These methods allow extension of the temporal resolution of the analysis far beyond the limits of previous approaches and provide a direct measure of the uncertainties associated with every estimation step, together with a direct assessment of why and where a particular model fails. Analyses of single RYRs at several Ca2+ concentrations are made by considering 16 models, some of them previously reported in the literature. Results clearly show that single RYRs have Ca2+-dependent gating modes. Moreover, our results demonstrate that single RYRs responding to a sudden change in Ca2+ display adaptation kinetics. Interestingly, best ranked models predict microscopic reversibility when monovalent cations are used as the main permeating species. Finally, the extended bandwidth revealed the existence of novel fast buzz-mode at low Ca2+ concentrations

Human beings as islands of stability: Monitoring body states using breath profiles

By checking the reproducibility of conventional mid-infrared Fourier spectroscopy of human breath in a small test study (15 individuals), we found that a set of volatile organic compounds (VOC) of the individual breath samples remains reproducible at least for 18 months. This set forms a unique individual's "island of stability" (IOS) in a multidimensional VOC concentration space. The IOS stability can simultaneously be affected by various life effects as well as the onset of a disease. Reflecting the body state, they both should have different characteristics. Namely, they could be distinguished by different temporal profiles: In the case of life effects (beverage intake, physical or mental exercises, smoking etc.), there is a non-monotonic shift of the IOS position with the return to the steady state, whereas a progressing disease corresponds to a monotonic IOS shift. As a first step of proving these dependencies, we studied various life effects with the focus on the strength and characteristic time of the IOS shift. In general, our results support homeostasis on a long time scale of months, allostasis on scales of hours to weeks or until smoke quitting for smokers, as well as resilience in the case of recovery from a disease

Unitary Ca2+ Current through Mammalian Cardiac and Amphibian Skeletal Muscle Ryanodine Receptor Channels under Near-physiological Ionic Conditions

Ryanodine receptor (RyR) channels from mammalian cardiac and amphibian skeletal muscle were incorporated into planar lipid bilayers. Unitary Ca2+ currents in the SR lumen-to-cytosol direction were recorded at 0 mV in the presence of caffeine (to minimize gating fluctuations). Currents measured with 20 mM lumenal Ca2+ as exclusive charge carrier were 4.00 and 4.07 pA, respectively, and not significantly different. Currents recorded at 1–30 mM lumenal Ca2+ concentrations were attenuated by physiological [K+] (150 mM) and [Mg2+] (1 mM), in the same proportion (∼55%) in mammalian and amphibian channels. Two amplitudes, differing by ∼35%, were found in amphibian channel studies, probably corresponding to α and β RyR isoforms. In physiological [Mg2+], [K+], and lumenal [Ca2+] (1 mM), the Ca2+ current was just less than 0.5 pA. Comparison of this value with the Ca2+ flux underlying Ca2+ sparks suggests that sparks in mammalian cardiac and amphibian skeletal muscles are generated by opening of multiple RyR channels. Further, symmetric high concentrations of Mg2+ substantially reduced the current carried by 10 mM Ca2+ (∼40% at 10 mM Mg2+), suggesting that high Mg2+ may make sparks smaller by both inhibiting RyR gating and reducing unitary current

Single Channel Function of Inositol 1,4,5-trisphosphate Receptor Type-1 and -2 Isoform Domain-Swap Chimeras

The InsP3R proteins have three recognized domains, the InsP3-binding, regulatory/coupling, and channel domains (Mignery, G.A., and T.C. Südhof. 1990. EMBO J. 9:3893–3898). The InsP3 binding domain and the channel-forming domain are at opposite ends of the protein. Ligand regulation of the channel must involve communication between these different regions of the protein. This communication likely involves the interceding sequence (i.e., the regulatory/coupling domain). The single channel functional attributes of the full-length recombinant type-1, -2, and -3 InsP3R channels have been defined. Here, two type-1/type-2 InsP3R regulatory/coupling domain chimeras were created and their single channel function defined. One chimera (1-2-1) contained the type-2 regulatory/coupling domain in a type-1 backbone. The other chimera (2-1-2) contained the type-1 regulatory/coupling domain in a type-2 backbone. These chimeric proteins were expressed in COS cells, isolated, and then reconstituted in proteoliposomes. The proteoliposomes were incorporated into artificial planar lipid bilayers and the single-channel function of the chimeras defined. The chimeras had permeation properties like that of wild-type channels. The ligand regulatory properties of the chimeras were altered. The InsP3 and Ca2+ regulation had some unique features but also had features in common with wild-type channels. These results suggest that different independent structural determinants govern InsP3R permeation and ligand regulation. It also suggests that ligand regulation is a multideterminant process that involves several different regions of the protein. This study also demonstrates that a chimera approach can be applied to define InsP3R structure-function

Sarcoplasmic Reticulum K+ (TRIC) Channel Does Not Carry Essential Countercurrent during Ca2+ Release

AbstractThe charge translocation associated with sarcoplasmic reticulum (SR) Ca2+ efflux is compensated for by a simultaneous SR K+ influx. This influx is essential because, with no countercurrent, the SR membrane potential (Vm) would quickly (<1 ms) reach the Ca2+ equilibrium potential and SR Ca2+ release would cease. The SR K+ trimeric intracellular cation (TRIC) channel has been proposed to carry the essential countercurrent. However, the ryanodine receptor (RyR) itself also carries a substantial K+ countercurrent during release. To better define the physiological role of the SR K+ channel, we compared SR Ca2+ transport in saponin-permeabilized cardiomyocytes before and after limiting SR K+ channel function. Specifically, we reduced SR K+ channel conduction 35 and 88% by replacing cytosolic K+ for Na+ or Cs+ (respectively), changes that have little effect on RyR function. Calcium sparks, SR Ca2+ reloading, and caffeine-evoked Ca2+ release amplitude (and rate) were unaffected by these ionic changes. Our results show that countercurrent carried by SR K+ (TRIC) channels is not required to support SR Ca2+ release (or uptake). Because K+ enters the SR through RyRs during release, the SR K+ (TRIC) channel most likely is needed to restore trans-SR K+ balance after RyRs close, assuring SR Vm stays near 0 mV

Luminal Ca2+ Regulation of Single Cardiac Ryanodine Receptors: Insights Provided by Calsequestrin and its Mutants

The luminal Ca2+ regulation of cardiac ryanodine receptor (RyR2) was explored at the single channel level. The luminal Ca2+ and Mg2+ sensitivity of single CSQ2-stripped and CSQ2-associated RyR2 channels was defined. Action of wild-type CSQ2 and of two mutant CSQ2s (R33Q and L167H) was also compared. Two luminal Ca2+ regulatory mechanism(s) were identified. One is a RyR2-resident mechanism that is CSQ2 independent and does not distinguish between luminal Ca2+ and Mg2+. This mechanism modulates the maximal efficacy of cytosolic Ca2+ activation. The second luminal Ca2+ regulatory mechanism is CSQ2 dependent and distinguishes between luminal Ca2+ and Mg2+. It does not depend on CSQ2 oligomerization or CSQ2 monomer Ca2+ binding affinity. The key Ca2+-sensitive step in this mechanism may be the Ca2+-dependent CSQ2 interaction with triadin. The CSQ2-dependent mechanism alters the cytosolic Ca2+ sensitivity of the channel. The R33Q CSQ2 mutant can participate in luminal RyR2 Ca2+ regulation but less effectively than wild-type (WT) CSQ2. CSQ2-L167H does not participate in luminal RyR2 Ca2+ regulation. The disparate actions of these two catecholaminergic polymorphic ventricular tachycardia (CPVT)–linked mutants implies that either alteration or elimination of CSQ2-dependent luminal RyR2 regulation can generate the CPVT phenotype. We propose that the RyR2-resident, CSQ2-independent luminal Ca2+ mechanism may assure that all channels respond robustly to large (>5 μM) local cytosolic Ca2+ stimuli, whereas the CSQ2-dependent mechanism may help close RyR2 channels after luminal Ca2+ falls below ∼0.5 mM