Simulation of Calsequestrin Mutations on Excitation-Contraction Coupling with a GPU based Computational Model By: Tingjun ’ Tim ’ Bi ABSTRACT We have implemented a genotype-phenotype (cGP) model that simulates how the Calsequestrin (Casq) gene affects excitation-contraction (EC) coupling. Catecholaminergic Polymorphic Ventricular Tachycardia (CPVT2) is a lethal cardiac disease for which the treatment is currently not well understood. Computational models that have attempted to contribute to the understanding often lacked the processing power to simulate the spontaneously triggered heart beats at the tissue level. Additionally, the cGP model was implemented on a Graphical Processing Unit (GPU) and achieved a 20x speed up when compared to the CPU implementation. We have utilized the message passing interface to distribute the cGP simulations across multiple GPUs on a cluster of servers as a step towards simulating multi-cell calcium dynamics that produces Sudden Cardiac Deaths caused by CPVTs on a tissue level. TABLE OF CONTENTS INTRODUCTION LITERATURE REVIEW Cardiac Electrophyical Signaling Sudden Cardiac Deaths CICR related arrhythmias Calsequestrin Treatments for Ventricular Tachycardia induced SCD Personalized treatment of VT patients MODEL DESCRIPTION CICR model Tania model for Calsequstrin Simulation Difference between the Tania model and the GPU Implementation PROGRAM IMPLEMENTATION RESULTS DISCUSSION REFERENCES APPENDIX INTRODUCTION Computational modeling can help elucidate how the Casq mutations affect both the normal CICR process and of the abnormal Sudden Cardiac Death scenario caused by irregular calcium signaling. The calsequestrin protein has a simple buffering function that can be quantified. Variations to the calsequestrin gene translate to a quantifiable change in the protein's calcium binding capacity. The effect of the alteration impacts the storage capacity of calcium in the SR and hence affects the calcium transients of the CICR compartments. The key opportunity that calsequstrin presents with computational modeling is that the Casq gene and protein a simple and clear 'input' to change in vivo, that can translate to consistent 'outputs' of experimental data. Both with single cell experiments and mice studies. Since the Null Casq mutation is nonlethal, experiments are thus enabled a definite range from cells where Casq is nonexistent, to cells where Casq is malfunctioning, and to cells where Casq is normally functioning. The in vivo data of the different Casq mutations can provide a concrete link between mathematical models and the real world data. In addition, the function of Casq as a 'shock-absorber' buffering protein that protects cells from scenarios of cateconic stress has a direct impact on the key scenario that causes Sudden Cardiac Death. Since the exact effects of anti- arrthymic drugs are uncertain and large amounts of resources are being spent by private drug companies and publicly funded organizations to clarify each drug's impact, the 'input-output' dynamics of the Casq mutation could provide a quantifiable means of drug validation. An invivo-insillco experiment could be done for example on the effects of the Casq mutations on mice in advanced stages of coronary artery disease, with the addition of how an AA drug such as Flacinde influences the treatment of each differing mice genotype. This study examines how the calsequestrin mutations impact the CICR process, particularly how the heterozygote calsequestrin genotype affects the frequencies of the spontaneously generated calcium sparks during CICR that can potentially then trigger a self sustaining calcium wave. LITERATURE REVIEW CARDIAC ELECTROPHYICAL SIGNALING In myocardial cells, muscle contraction is facilitated by a process known as calcium induced calcium release (CICR). After receiving the triggering electrical signal from the SA node, muscle cells from the atrium and the ventricular compartments contract sequentially to pump blood throughout the body. In a single cardiomyocyte, the cell starts a depolarization process by propagating electrical wave along the cell wall that consists of sodium (Na+) and potassium (K) ions. During the initial phase, sodium ions first enter the cell. It's entry across the sacrolemma generates a signal through the T-tubule that activates the voltage-gated DHPR channels to release the initial triggering calcium of CICR. The ryanodine receptors (RyRs) on the SR near the open DHPR channel release larger amount calcium to the cytosol in response to the triggering calcium. During the plateau phase, the slow calcium current and the subsequent influx of calcium into the cell trigger myosin and tradin contractions. The increased calcium binds to the myofilament protein troponin C which then induces crossbridge cycling leading to muscle contraction. During the relaxation phase, calcium is removed from the cytosol via the SR calcium atpase (SERCA), that pumps calcium back to the SR and by the Na/Ca exchanger (NCX). The NCX exports one Ca+ ion (two positive charges) in exchange for three Na+ ions (three positive charges) that enters the cell, which exchanges the efflux of one Ca+ ion. The activation of the NCX generates an inward current. The repolarization is conducted via the activation of the outward potassium current. After the Ca+ influx occurs, the potassium current continues to flow inward completing the restitution period. SUDDEN CARDIAC DEATHS If an irregularity occurs during the intricately regulated CICR process, spontaneous intracellular calcium disruptions may be triggered. If the disruptions are triggered in multiple adjacent cells, they can lead to uncoordinated electrical signaling (defined as arrhythmias) and irregular contractions of the heart. When the uncoordinated muscular contractions are sufficiently severe, the heart cannot pump blood throughout the body, and can result in sudden cardiac death (SCD). Sudden cardiac death is the unexpected death of an individual due to the loss of heart function. It is the largest cause of natural death in the United States, causing over 325,000 deaths each year. Deaths from SCD each year out numbers the deaths from all cancers in the United States. The prevalence ratio is 50 to 100 sudden cardiac deaths per 100,000 individuals occur every year in the US, and rises to 1 per 1000 for adults older than 35 years. More than 80% of all SCDs are caused by arrhythmias. Which subsequently can be further categorized to either ventricular tachycardia (VT) or ventricular fibrillation (VF). VT and VF cause the deaths of 100,000 individuals a year. Along with atrial fibrillation (AF), they compose the most common forms of arrhythmias. The conduction disturbances that are caused by arrthymias are most likely the result of cell damage from an underlying structural heart disease. Coronary artery disease is present in 80-85% SCDs that undergo ventricular arrhythmia; the most frequent cause of the coronary disease is atherosclerosis. CICR RELATED ARRHYTHMIAS This section describes the steps in which arrthymias are triggered from CICR. It presents 6 steps to approach the question of why life threatening calcium dependent SCD 'heart attacks' occur. 1-Longterm and short term triggering conditions for calcium leaks 2-Calcium leaks from cell happen and grow past threshold to a calcium spark 3-Sparks grow into calcium waves from cell 4-Waves grow past threshold, depolarizes the cell spontaneously and prematurely 5-Multiple cells from tissue region spontaneously depolarizes as a group past source-sink threshold 6-Ongoing disruption from tissue region permanently disrupts heart rhythm Stage 1 The triggering conditions start with the propensity of damaged cells to leak calcium from the ryanodine receptor. When the cell is stressed via internal factors such as cell damage or external factors such as exposure to caffeine, it is often the case that a number of the CICR components slightly malfunction and additional calcium is released into the cell. Since CICR requires the intricate regulation of intracellular calcium levels, the additional amount of calcium caused by the malfunction of the components is often enough to upset the equilibrium and cause the triggering of spontaneous calcium sparks. Damaged myocytes with faulty cellular components are the primary internal cause of the calcium leak, usually comes in the form of acute illnesses that triggers metabolic abnormalities. An example could be seen from coronary artery disease via atherosclerosis where large tissue regions are constantly corroded by extracellular calcium. For the external triggering conditions, the release of catecholamine during substance intake, certain medications, caffeine intake, beta adrenergic stimulation, exercise, and emotional stress can also result in the increase of the myocyte's SR calcium load. The increased intracellular calcium and SR calcium content with this scenario are explained by the increased heart rate, the increased Ca+ channel current, leading to the increased SR uptake. In the presence of high levels of SR Ca+ concentration, the regions related to the RyRs can spontaneously leak calcium. Regulation of the RyR channel permeability is complex, both cytostoic and SR luminal free Ca+ concentration affects the open probability of the channel. The cell damage of the RyRs by free radicals or coronary artery disease increases RyR leakage. SR calcium leak rate is proportional to the luminal SR calcium concentration, the more the SRs are overloaded with calcium, the more likely the RyR regions become prone to calcium leaks. Stage 2 to 4 During this stage, the initial Ca+ leaks diffuse to neighboring RyRs and triggers more calcium release. The spontaneous larger calcium release from the initial calcium leak is termed a calcium spark. If the calcium sparks from the leaks occur from multiple regions of the cell, the process can become self sustaining and generates a wave of calcium release that propagates along the cell. Calcium released during a wave is removed from the cell by NCX. Waves of sufficient speed and amplitude activate the NCX and cause an inward current that spontaneously depolarizes the cell membrane. Stage 5 When only one cardiac myocyte which reaches the sodium channel activation threshold and depolarizes from the calcium wave, the other surrounding cells act as an electrical sink to suppress the irregularity from the single cell. Life threatening arrhythmic conditions can only be initiated when a number of cells from the same cellular region uniformly depolarize at an intensity sufficient to overcome the resistance caused by the multi celled sink property. Uniform depolorizations from a significant tissue region can trigger premature action potentials that are termed 'delayed afterdepolerizations' and 'earlyafter depolerizations'. Stage 6 Multiple EADs and DADs can trigger the faster heart beats, which in mild cases result episodes of syncope (consisting of 1 to 2 minutes) that eventually stabilizes. In the more difficult cases, initial heartbeat disruptions develop to indefinite VTs and other variations of ventricular arrhythmias that requires hospitalization of the patient and may escalate to sudden cardiac death. Sudden cardiac death is triggered when the depolerizations push the heart rhythm to an irrecoverable dissonance that stops the patient's blood flow permanently. CALSEQUESTRIN Calsequestrin (Casq) is a high capacity and low affinity calcium buffering protein that is localized to the SR lumen in the myocyte. Modest disruptions in calsequestrin function increases arrhythmia susceptibility. Similarly, modest disruptions of Casq by drugs can increase the risk of sudden cardiac death. Calsequestrin serves two functions within the myocyte cell. During a high stress catecholamic event, calsequestrin binds calcium by absorbing the extra calcium from the SR, calsequestrin acts as shock absorbers to absorb the extra calcium generated via the cellular anomaly. It is not only able to buffer the calcium for the SR but also releases calcium into the cytosol upon the opening of the RyR2 channels. The stored calcium from Casq is released in an orderly fashion to maintain the normal CICR process. Experimental studies have documented the ability of calsequestrin to modulate high calcium load that occurs during adrenergic stimulation and also proved that the protein prevents the occurrence spontaneous sparks. Calsequstrin is composed of three domains each with negatively charged folds with four a-helix surrounding a b-sheet core. To stabilize the molecular core, a divalent ion such as calcium is required, inducing at the same time confirmation changes that further increase its calcium binding capability. The binding occurs through the N-terminal arms of Casq, at higher Ca+ concentrations, back to back binding of 3 Ca+ ions occurs through the c-terminal tail. Deletion of the N or C terminal of monomers results in the incapability of molecule to form linear polymers. The second function of the Calsequstrin is to contribute to the direct regulation of the calcium gating process at the junctional SR. Casq is connected to the junctional SR localized at the terminal cisternae and binds to the RyR via the two proteins of triadin and junctin. The resulting protein complex interacts with the ryanodine receptor and affects the luminal sensing capabilities and modulates RyR function. The calsequstrin gene is relatively short (24k bp) compared to the longer gene for the ryanodine receptor (120k bp). One of the main opportunities in studying Casq related calcium dynamics is conducted with the homozygous knockout Casq mice. Experimental results from the homozygous knockout mice with both alleles set to recessive results in an absence of the Casq binding proteins. The Casq null mice survive past infancy and experience normal live functioning until the unset of a catecholamic stress event. Without the shock absorbing attributes of Casq, the null mice experience increased spontaneous SR calcium release during stress and a higher incidence of SCDs from arrthymias. The SR load threshold at which spontaneous calcium release is lowered. The main phenotypical change for the null mice is mainly the increased SR volume to compensate for the removal of the shock absorbing aspects of the protein. The overall intercellular calcium content for the null mice remains the same, and calcium induced SR releases are also the same. Heterozygous knockout mice with only one copy of Casq allele exhibit fewer phenotypical changes. The physiology of the heterozygote mice with the wild type mice remains the same, and only exhibits a slightly higher SR calcium leak during catecholamic events at the same free intra SR calcium concentrations. CPVT2 INDUCED VENTRICULAR TACHYCARDIA Catecholaminergic polymorphic ventricular tachycardia is a genetic disorder that affects the Ryanodine and Calsequestrin proteins that are integral to the CICR process. CPVT causes 15% of all SCDs in patients who are under the age of 30. CPVT1 is designated for the mutations with the ryanodine two receptor, and CPVT2 for the Casq mutations. Of the two variations, CPVT2 is rarer but it is also more lethal. The patients with CPVT2 are diagnosed the during their infancy, and few reach their 30th year. The cardiac ECG for the CPVT2 patient at rest is normal. During an exercise however, the pace of the cardiac beat changes first starts with ventricular premature ectopic beats (VPEs). That often escalates into spontaneous polymorphic ventricular tachycardia (PVT) or ventricular fibulation (VF). PVT and VF consists of a period of elevated heart rate functioning that can range both from beats of 100-333 bpm, where ventricular tachycardia are indicated by a rapid heart rhythm of a rate greater than 150 beats per minute. The most common naturally occurring mutations for the calsequestrin protein are the R33Q and DNN3 mutations. The mutations that consists of Casq dysfunction contributes to an increased arrhythmia risk, by either decreasing the threshold for a spontaneous spark due to the increased SR luminal calcium sensitivity. While it has been proven that the mutations alter the protein's structure, it is unclear what exact changes to the calsequestrin protein from the mutations cause the development of spontaneous spark releases and the subsequent development of VTs. TREATMENT OPTIONS FOR CPVT Anti-arrhythmic (AA) drugs influence the cardiac conduction properties and works to revert an abnormal heart rhythm to a normal rhythm. Most anti- arthymic drugs are modulators of ion channels, and center on the prevention and inhibition of the precursor condition at the cellular level. A key question being asked during the initial phase of new anti-arrhythmic drug development is: which ion channels are the best candidates? Blockers and stabilizers of the ion channels related to CICR are constantly being considered as drug targets aimed at preventing calcium dependent arrhymias. The prescribed treatment for CPVT consists of a combination of different AA drugs, in conjunction of the cardioverter defibrillator (ICD) implantation if needed. A large amount of AA drugs either inhibit the NCX or inhibit the Na+ channels, a wider categorization could be relegated to Class 2 drugs (beta blockers) and Class 4 (calcium channel antagonists). The drugs are utilized together prevent the stress induced increase in SR calcium content. The effects of the drugs consist of the reducing the heart rate, the calcium influx into the cell, and the calcium uptake into the SR. Class 4 AA agents decrease the rate of calcium influx, and acts to reduce the SR content via modulating the triggering conditions of the CPVT. The implantation of an ICD has been considered the most reliable treatment option for patients with CPVT. However, the surgical implantation is dangerous and costly and the treatment results from the ICD do not protect all patients. The paper will focus on four different drugs as treatment options: beta blockers, verapamil, diltiazem, and flacinde. B-adrenerigic blockers constitute mainstream therapy for the treatment of CPVT symptoms. Beta blockers aim to prevent or slowdown SCD symptoms. After treatment with the beta blockers, 30% of patients still experience cardiac arrhythmias. Verapamil and diltiazem are two of the calcium channel blockers that are categorized to the class 4 category of antiarryhmic medication. The main effect of the drugs is to slow the SA node automaticity and prolong the AV refractoriness. The role of the two drugs are limited to the rate-slowing non- dihydropyridine calcium channel blockers by slowing the rate for atrial fibrillation or VT. Verapamil can reduce the amplitude of DAD below the threshold for required for triggering VT. Experimental results involving transgenic mice resulted in an 80% prevention of the onset of CPVT symptoms with the verapamil drug(class 4), and yet the combination of the two drugs only prevented a total of 5 out of 11 of the CPVT patients. The contrast between the 100% and 45% rate relates to one of the key questions in CPVT therapy. The drug of Flacinde is a sodium channel blocker. Flecainide acts on two levels of the CICR cascade first by reducing the RyR open probability, and secondly by inhibiting the sodium current. Which can reduce the effects of the NCX from propagating the current. Since the cystoic inward calcium current is a direct function of the activity level of the NCX exchanger. Flecainide decreases both the spark frequency and amplitude. Increases the Ca+ spark frequency and reduces the spark mass, but has no effect on the Ca+ content, there is evidence that it significantly increases the threshold required for triggering of an AP. Flecainide is effective in the reduction of the symptoms on the CPVT knockout mice. A single injection of flecainide instituted the protective effects for upwards of 6 hours after exposure to isoproterenol, there is evidence to conclude that it reduces the opening probability of RyR2 channels. There is evidence that Flacinde directly targets the molecular defect in CPVT and in addition to the regular Class 1c action of blocking the sodium channels. Flacinde has proven effective with patients for whom the combination of Class 2 and 4 drugs has failed (achieving an 86% success rate). Several experiments have been conducted with Flacainide on the genetic variants of Casq. The effectiveness of the Flacidine on the Class 2 and 4 ineffective patients may contribute to the understanding of the ineffectiveness and variability of the effect AA drugs on CPVT patients in general. Research involving the understanding and treatment of CPVT will directly impact our ability to save the lives of the CPVT patients. At a broader level, the research will also indirectly impact the treatment options for the 60,000 patients that are annually diagnosed with VT each year. PERSONALIZED TREATMENT OF VT PATIENTS The treatment of ventricular arrhythmia has consistently been problematic. Treatment of VT symptoms via AA drugs only have a 30% success rate. Patients with VF or sustained VT symptoms for whom the AA drug treatment prove ineffective, necessitate more dangerous treatment with defibrillation or electrical cardioversion. In addition, there is a chance of AA drug poisoning. The toxicity associated with AA drugs can aggravate arrhythmias and can be life-threatening. Poisoning due to high doses is common. The ongoing challenge for personalized medicine is utilizing an individual's biological data to predict the most likely drug response. Since an individual's genetic variation provides a foundation for the different possible drug reactions, current efforts focus on sequencing an individual's genome as a foundational touchstone that other personalized information (such as BMI and blood pressure) can build upon. Determining the mutation variation can have an impact of the differing treatment responses. An example in the treatment of CPVT consists of a scenario where a heterozygote mutation (R2401H) on the exon 47 of the RyR2 protein is relegated to the FKBP12.6 binding region. Experimental results have shown that beta blockers are ineffective for treating this mutation. If the information about R2401H is immediately available to the cardiologist, he or she can rule out the beta blocker option as ineffective and proceed to other viable options. Another area where the research of the CPVT mutations can have an impact is with drug trials on mice with the altered Casq gene. Since the calsequstrin mutations are not initially lethal and have a spectrum in which the "shock absorbing" abilities are lightly, moderately, or severely impacted. In-vivo experiments can be conducted in the lab can update in-silico simulated models to investigate the effectiveness of AA drugs. Thusly possibly prevent the chance of AA drug poisoning while increasing drug potency. The mice studies will be particularly helpful during the troublesome scenario of administering multiple drugs. An attempt could be made to fine tune the different dosage levels of drug combinations with CPVT mice with different mutations to the Casq protein to improve the treatment for CPVT patients. The optimized AA dosage can also save lives. For both the CPVT and VT cases, successful treatment via the AA drugs will prevent more extreme solutions like the surgical implantation of an ICD. MODEL DESCRIPTION To understand the cardiac CICR process, both in-vivo experiments and in-silico simulations are needed. Within a single myocyte cell there is a large multitude of biochemical reactions, with each reaction necessitating the representation of numerous biochemical molecular parameters. Many of these processes are stochastic in nature. In- vivo studies are only able to provide an incomplete picture as we have a limited capacity to measure the parameters. Computational models enable us to extrapolate the underlining regulatory mechanisms of CICR from the data provided by in-vivo experiments. The Markov Chain Monte Carlo method model stochastic processes and hence is a powerful approach to modeling the CICR process. They provide an alternative to modeling the open fraction of a population with ordinary differential equations (ODEs) that is commonly used. However, these deterministic model fail to capture some essential features of cardiac calcium dynamics. Hence, the stochastic methods are needed. While the CICR models based on the Monte Carlo method are able to provide accurate and detailed description of calcium cycling, they are also very computationally intensive. A Monte Carlo simulation such as the whole cell CICR model that incorporates millions bio- molecular components will require a large amount of computational resources. We have implemented the simplified cGP CICR model on a Graphical Processing Unit (GPU). While the design of the Central Processing Unit is oriented around maximizing the speed of each individual processor core, the design of the GPU is oriented towards aggregating as many cores as possible and running all of the cores in parallel. Due to the refined architecture, the GPU is optimized to perform scientific modeling computations that require simultaneous analysis of a multitude of biological parameters. The model will attempt to incorporate the calcium concentrations of the various CICR compartments and analyze the impact of genetic variation on calsequstrin's capacity to modulate high calcium load and prevent spontaneous sparks. The paper will develop and compare simplified models for the regular homozygote and the clinically altered heterozygote whole cell calcium transients for a single cardiac myocyte. The Monte Carlo CICR model that was adapted for the GPU program is a based on a two state Markov chain that provides a minimal simulation of EC coupling. A large part of CICR occurs within the space that is between the SR and the cell wall. Within this sub- membrane space there are dyadic junctions where the voltage-gated DHPR channels (1 to 5 per junction) pumps extracellular calcium into the intra-cellular sub-membrane space. The RyR channels (50 to 200 per junction) then mediates the further release of calcium within the myocyte. We assume the dyadic junctions (also called calcium release units, CaRUs, dyadic clefts, and dyadic subspaces) are spread evenly across the cell wall. The circular junction space is assumed to have a 160 nm radius that is evenly spaced across the cell every 1.2 uM apart from each other, this dynamic is described in Figure 1. If we assume a cell membrane area of 15,000 uM, we can then derive a total of 20,000 dyadic subspaces. (Shannon, 2004) In our model we ignore the effects of the voltage-gated DHPR channels and focus only on the steady state RyR gating. It has been shown that the each cluster of RyRs within a single dyadic subspace membrane operates in an all-or-none release pattern. As a result, we can construct a two state Markov chain (equation 1) related to the open and close states of the RyR cluster (or RyR "mega-channel"). Where k(RyR+) transition is a function of the Ca concentrations from both the junctional SR (JSR) and the dyadic subspace (DS). EQUATIONS (1-2): k(RyR+) [closed] -- [open] k(RyR-) k(RyR+) = K(open max rate) * c(ds,n)^4 / [ K(max binding constant) - a(RyR)*c(JSR,n) ] ^4 + c(ds,n)^4 The transfer of in calcium between compartments during each time step (alternatively know as calcium flux) is provided by the equations 3 through 6. EQUATIONS (3-6): J(RyR,n) = Y(RyR,n) * v(RyR,T)/N * ( c(JSR,n) - c(ds,n) ) J(refill,n) = v(refill,T)/N * ( c(nsr) - c(JSR,n) ) J(efflux,T) = sum(1,N) v(efflux,T)/ N * c(ds,n) - c(myo) 0 = b(ds)/L(ds) * ( J(RyR,n) - J(efflux,n) ) Y(RyR,n) from equation 3 is the number of open RyR at each release site, the elements of the matrix takes on the value of 1 or 0 depending on RyR open/closed. This illustrates that the JSR calcium level and the RyR open probability are mutually dependent. Since the activation is a decreasing function of the JSR and JSR depletion will make it harder to activate the CaRU after release terminates. J(RyR,n) is related to the Ca flux through RyR channels. J(efflux,n) is related to the diffusion from the CaRU's DS to the bulk myoplasm and J(refill,n) is related to the diffusion from the CaRU's NSR to JSR. By obtaining the calcium flux, we are able to calculate the change in the calcium concentrations within the DS and JSR compartments. EQUATIONS (7-8): dca(ds,n) = 1/L(ds) * ( J(dhpr,n) + J(RyR,n) - J(efflux,n) ) dca(JSR,n)/dt = 1/L(JSR) * J(refill,n) - J(RyR,n) The effective volume ratios of L(ds) and L(JSR) are defined with by the physical volume and the buffering capacity for the myoplasm. The volume ratios of each subspace are defined with relation to the physical volume. Equations 9,10,11,12 Q = x(oc)*k- + x(co)*C*K+ / Q = S_1^T * K- + (S_2*C)^T * K+ x(oc) = exp( -a(x) * [N(c,Ecc) - (N(o) - 1)(Eoo) ] ) x(co) = exp( -a(x) * [N(o,Eoo) - (N(c) - 1)(Ecc) ] ) C = eM * c When this Markov chain is extended to include 20 RyRs, it can be represented by a 21 state Markov chain. The Markov chain is represented by transition rate matrix Q. Q takes as input the parameters of K-, K+. The x(co) and x(oc) parameters from equations 10 and 11 are coupling factors, and are derived from a(x) which is the average connectivity and E(ij) which is the change in energy by the channel in state i when connected to channel in state j. EQUATIONS (13-16) dt = -0.1 / Q(min) P = S^T + Q*dt A(i,j) = 1 if X(i,j) < sum of (N,k=1) * P(i,k) 0 otherwise S(j,i) = (A(i,j+1) - A(i,j) )^T After calculating Q, the adaptive time step is calculated by retrieving the smallest diagonal entry in Q. We can then utilize Q and dt to calculate the new transition probability matrix P. Each element of P is compared to a random number between 0 and 1 to determine A that describes the open/closed state of the current gating channels. To calculate the first component of the new channel open/closed states for the next time step in matrix S, we subtract the current element of A with the previous vertical element and take the transpose of the result. EQUATIONS (17-18) Ca(ds) = Ca(ds)(current) + dt*dca(ds,n); Ca(JSR) = Ca(JSR)(current) + dt*dca(JSR,n); Euler's method is utilized to calculate the ODEs of Ca(ds) and Ca(JSR) for the next time step. The output of the simulation consists of a dataset that simulates the change in calcium concentrations within the JSR and DS compartments for each of the corresponding dyadic clefts. As a final step, we obtain an average of the local cleft activity with the overall cellular transient. The locally controlled Ca cycling is different from common-pool models (which simulate a single common dyadic junction) in that it provides a simulation of the differing concentration levels and duration of Ca transients from each individual cleft that leads to more accurate depiction of the cellular Ca cycling. The simplified model is insufficient when compared to real life whole cell CICR. It does not simulate the effects of the DHPR voltage clamp's role in the first stage of CICR. The model simulates the DS and JSR compartments, while ignoring the important NSR and bulk myoplasm compartments of the cell. The model also assumes spatial homogeneity in the distribution of the dyadic junctions, which compared to the heterogeneity of the real life SR can cause further deviations from the experimental results (Figures 1 and 2). The paper utilizes the calsequestrin model developed by Tania and Keener. The Tania model utilizes the model for EC Coupling developed by Shannon et. al as the base model. Tania et al. adjusted the diffusion rate for the NSR and JSR compartments by adding the rapid buffering approximation to account for the differing quantities of Casq in the NSR. Additionally, a six state Markov model is introduced to account for the Casq-RyR interactions that affect the RyR opening probabilities from the gating level. TANIA MODEL FOR CALSEQUSTRIN SIMULATION The Tania model adds two additional parts to model the effect of casquestrin. The first adjustment made is for the Casq buffering properties in the luminal SR. Casq is the primary Ca buffer molecule within the luminal SR, each Casq molecule in the SR has capacity to bind 40-50 Ca ions, each Ca molecule has m Ca binding sites, all-or-nothing reaction of 40-50 binding sites in single molecule. Based on the binding capacity and the total number of Casq molecules within the SR, the equilibrium binding constant / Ca half saturation constant of casquestrin is 0.63 mM. When the constant is lower, it denotes an overall decreased binding capacity of the total Casq in the SR, when it is higher it denotes an increased binding capacity. The most important variable is the total Casq concentration within the junctional subspace is 14mM. The time frame of Casq bind/unbind to RyR is 1/10th of the opening closing constant. The first addition to the Tania model that adjusts for the calcium concentration in the Junctional SR by altering the rapid buffering approximation constant for the B(JSR). The new approximation constant takes into account the two variables of total Casq buffer concentration and the casquestrin-calcium disassociation constant. The constant is the defined by the kinetic equilibrium constant of when calsequestrin binding to calcium, free SR calcium during re-uptake sink for SR calcium that can modulate restitution after each cycle.(Tania, 2010) The JSR refilling rate to the original levels can range from 10 ms to 100 ms. The rate is affected by the inter-spark intervals for B (q) via luminal sensing. The scaling factor related to the B (JSR) provides the luminal sensing variable of A. When A is set equal to 1, there is no effect, and when A greater than 1 indicates that there is an effect. (Tania, 2010) mCaCSQ/dt = K(on)CaCSQ - K(off)M*Ca*Casq K(on) = association K(off) = dissociation rate constant steady state due to timeframe difference K(on)CaCSQ - K(off)M*Ca*Casq = 0 Casq - total concentration in JSR Casq(tot) = Casq + mCA*Casq Casq = Casq(tot)*K(constant) / K(constant) + Ca(JSR) where K(constant) = 400-650uM or 0.65mM Casq varies inversely with C(JSR)-- C(JSR) low -> Casq high m = 3 -- coefficient of Casq buffering of 2.95 B(JSR) = 1/ [ 1 + m * Ca(JSR) * Casq(tot) * K(m,csq) / (k(csq) + Ca(JSR) ) ^2 ] The final equation to calculate the JSR refilling B(JSR) rate first multiplies coefficient of Casq buffering (m) with the total calsequstrin and the calcium disassociation constant. The result of total Casq binding capacity within the junctional subspace is matched against the with the calcium with in the junctional subspace Ca(JSR). The result number is divided by the calcium disassociation constant plus the calcium with in the junctional subspace squared. The inverse is taken for the final JSR refilling rate. To account for how Casq modulates the calcium release via interaction with the RyR and the two additional proteins of tridin and junctin, Tania implements the RyR Casq interactions with a kinetic model of the RyR channel. The second addition considers the calcium flux changes when calcium binding to Casq/JSR that causes conformational change of Casq. The conformational change affects RyR activity, where the previous RyR open when bind. The precondition for the change is low SR-Ca. When Casq unbinds from the RyR, the RyR open probability increases. Tania assumes that q is the free Casq concentration that can bind or unbind to a RyR channel and changing the opening probabilities of the channel. The state transition simulates the RyR gating property of the calsequestrin molecule. A six state Markov model is introduced to account for the Casq-RyR interactions that affect the RyR opening probabilities from the gating level. When calsequestrin binds with the RyR, the open probability decreases, during the unbinding, the open probability increases. There are no refractory and inactive states for the Tania model due to the fact that it has been shown that cystic calcium does not play a significant role in the CICR process. (Tania, 2010) DIFFERENCE BETWEEN THE TANIA MODEL AND THE GPU IMPLEMENTATION There are a number of areas where the GPU model tries to improve upon the Tania model. Firstly the GPU model is geared towards genetic simulation with the key variable being the total calquestrin available within the junctional subspace with the intention of the Casq buffering with the luminal SR as accurate as possible. Since the Casq-RyR interaction is hard to quantify, the GPU model does not implement it. Where Tania's model is based on two compartments, the GPU model incorporates 4 compartments. As a result, the shape is not a smooth curve but a junctured curve that provides a plot result that is of a more realistic shape, and corresponds to the initial spike from the action potential and the subsequent tapering off. The code was implemented on a GPU, which not only helped with the speed up issues, but also enables the extension of the code to multi cell model for tissue simulation. The code was also implemented on python, which we are able to extend the code to gui and web based interface. Although the current model only encompasses only one whole cell, the eventual goal is to extend the model to the multiple cells at the tissue level to get a better idea of how changes in the calsequestrin gene affects cardiac arrhythmias. PROGRAM IMPLEMENTATION As a first step, we have adapted a simplified steady state matlab code into a CPU version on python. (Williams, 2007) The CPU python code is implemented primarily with the matrix operations of the Numpy Python module. We then adapted the CPU version of the python steady state code to be run on the GPU with PyCUDA. PyCUDA is a python library designed to facilitate GPU programming with the Python language. It accesses CUDA's parallel computation API and provides a layer of abstraction over the CUDA C interface. As a result, it retains the efficiencies of C while being easy to understand and implement. PyCUDA implements a gpuarray subroutine that declares an array data structure on the GPU, the subroutine also includes a number of matrix operations such as addition and dot products. We implemented two versions of the GPU code to conduct the Monte Carlo CICR simulations. In version one, we utilized the default matrix calculation modules that was included with the PyCUDA package. For the second version, we wrote C kernels to supplement the default modules to improve the program performance. PyCUDA's SourceKernel module is able to incorporate C level CUDA code directly into the CICR program. The benefit of utilizing the SourceKernel module is that it accesses low level GPU memory and thus is able to directly allocate the computational resources on GPU. A number of different operations were utilized in the process of implementing the SourceKernel code for the second CICR GPU program. A number of simple operations like addition and multiplication are implemented similarly on the GPU as they are on the CPU. We worked extensively to optimize the complex matrix operations such as conjugate transpose, dot, and cumulative sum to obtain the maximum possible GPU performance. As another alternative for speeding up the code, we explored the possibility running the code on multiple GPUs. We have developed a program that utilizes the message passing interface (MPI) to coordinate its execution across different GPUs on multiple server nodes. Python MPI implements a partitioned local data access to initiate multiple processes on all the nodes of the cluster. Utilizing the MPI interface we were able to start a GPU thread on each declared MPI instance and enabled each of the MPI instances to communicate with each other. We have adopted the GPU code with MPI to conduct a preliminary simula