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