Approximation for Cooperative Interactions of a Spatially-Detailed Cardiac Sarcomere Model

Cellular and Molecular Bioengineering, Mar 2012

We developed a novel ordinary differential equation (ODE) model, which produced results that correlated well with the Monte Carlo (MC) simulation when applied to a spatially-detailed model of the cardiac sarcomere. Configuration of the novel ODE model was based on the Ising model of myofilaments, with the “co-operative activation” effect introduced to incorporate nearest-neighbor interactions. First, a set of parameters was estimated using arbitrary Ca transient data to reproduce the combinational probability for the states of three consecutive regulatory units, using single unit probabilities for central and neighboring units in the MC simulation. The parameter set thus obtained enabled the calculation of the state transition of each unit using the ODE model with reference to the neighboring states. The present ODE model not only provided good agreement with the MC simulation results but was also capable of reproducing a wide range of experimental results under both steady-state and dynamic conditions including shortening twitch. The simulation results suggested that the nearest-neighbor interaction is a reasonable approximation of the cooperativity based on end-to-end interactions. Utilizing the modified ODE model resulted in a reduction in computational costs but maintained spatial integrity and co-operative effects, making it a powerful tool in cardiac modeling.

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Approximation for Cooperative Interactions of a Spatially-Detailed Cardiac Sarcomere Model

0 Graduate School of Frontier Sciences, University of Tokyo , 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 1 of Frontier Sciences, University of Tokyo , 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-0882, Japan . Electronic mail: 2 Graduate School of Frontier Sciences, University of Tokyo , 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-0882, Japan We developed a novel ordinary differential equation (ODE) model, which produced results that correlated well with the Monte Carlo (MC) simulation when applied to a spatially-detailed model of the cardiac sarcomere. Configuration of the novel ODE model was based on the Ising model of myofilaments, with the ''co-operative activation'' effect introduced to incorporate nearest-neighbor interactions. First, a set of parameters was estimated using arbitrary Ca transient data to reproduce the combinational probability for the states of three consecutive regulatory units, using single unit probabilities for central and neighboring units in the MC simulation. The parameter set thus obtained enabled the calculation of the state transition of each unit using the ODE model with reference to the neighboring states. The present ODE model not only provided good agreement with the MC simulation results but was also capable of reproducing a wide range of experimental results under both steady-state and dynamic conditions including shortening twitch. The simulation results suggested that the nearestneighbor interaction is a reasonable approximation of the cooperativity based on end-to-end interactions. Utilizing the modified ODE model resulted in a reduction in computational costs but maintained spatial integrity and co-operative effects, making it a powerful tool in cardiac modeling. - Mathematical modeling is an indispensable tool in defining the mechanisms of activation and force generation of the cardiac sarcomere. Various mathematical models have been designed to replicate and characterize the cellular processes and activities of the sarcomere and, recently, detailed structure and filament properties have also been taken into account.2,4,10,16 However, current models have yet to replicate the anomalously high sensitivity of developed force to changes in the free cytosolic calcium (Ca) concentration, observed under both steady-state and dynamic conditions. This aberrant effect is suggested to be brought about by the co-operative interactions among intracellular molecules within the sarcomere. One postulated mechanism of cooperativity suggests that the strongly-bound cross-bridge releases the steric hindrance of tropomyosin to facilitate the attachment of nearby cross-bridges. A further potential mechanism underlying the co-operative interactions is the end-to-end interactions of regulatory troponin/tropomyosin (T/T) units along the thin filament. In either case, the physical arrangement of each molecular component is suggested to be a critical factor. To reproduce the co-operative effects that occur within the sarcomere, most current models utilize the phenomenological parameter tuning strategy to normalize the behavior of cross-bridges and to avoid the necessity of determining the state of each regulatory unit and the interactions among them (mean-field approximation). Although this approach enables the use of ordinary differential equations (ODE), has a lower computational cost, and has been reported to provide a fairly good representation of experimental data,1,9,13,20,21 the models lack a representation of spatial activity within the cell. This limits the predictive ability of the models and hampers the potential for direct comparisons with experimentally obtained data.18 Spatially-distributed models have been proposed that are capable of mimicking the physical arrangement of each functional unit within a cell, including the cross-bridges in the thick and thin filaments of the sarcomere.8,10,19,22 In these models, the transition rates of each unit are dependent on the states of neighboring units and/or the cross-bridge strain to reveal any potential co-operative mechanisms that occur. Moreover, the models have been found to have excellent reproducibility. However, the inherent and inevitable problem with this type of model is the necessity of using the computationally expensive Monte Carlo (MC) simulation. Although Rice et al.19 reported an analytical solution to their Ising model of myofilaments without MC simulation, its application is limited to the static state with a simple periodic boundary condition. Very recently, Campbell et al.3 proposed a Markov model approach to represent the states of regulatory units, but the computational costs again limited the number of units studied in the model. Here we propose a novel method for describing the behavior of a spatially detailed co-operative model using ODE in which the regulatory units are distributed along the sarcomere filament. Through modifications to the Ising model produced by Rice et al.,19 we produced an ODE model th (...truncated)


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Takumi Washio, Jun-ichi Okada, Seiryo Sugiura, Toshiaki Hisada. Approximation for Cooperative Interactions of a Spatially-Detailed Cardiac Sarcomere Model, Cellular and Molecular Bioengineering, 2012, pp. 113-126, Volume 5, Issue 1, DOI: 10.1007/s12195-011-0219-2