#### Title

Using Stochastic Differential Equations to Model Gap-Junction Gating Dynamics in Cardiac Myocytes

#### Abstract

The cell-to-cell propagation of the cardiac action potential allows for the electro-mechanical coupling of cells, which promotes the coordinated contraction of cardiac tissue, often referred to as the heartbeat. The main structures that promote electrical coupling between adjacent cardiac cells are pore-like proteins called gap junctions that line the membranes of such cells, allowing a channel for electrically charged ions to travel between cells. It is known that the conformational, and hence conducting, properties of gap-junction channels change as a function of local gap-junctional voltage and local ionic concentrations and are stochastic in nature. Many previous models of gap junctions have made a constant-resistance approximation or used an ODE model relating gating state to a local voltage. In this thesis, we extend a previous ODE model of gap-junction gating state by Henriquez et al. and formulate it as a system of stochastic differential equations (SDEs) by deriving the expected change vector and covariance matrix of the model and integrating the covariance with respect to a stochastic process, the Wiener Process. In doing so, we construct the first SDE-based model of gap-junction gating dynamics. This SDE description of the electrical coupling between cardiac cells is integrated into a 1D cable model where intracellular current dynamics are described using the Luo-Rudy 1 formulation. Monte Carlo simulations are performed on the resulting model in order to gather data used to construct distributions of several model responses of interest, including conduction block, conduction velocity, gap-junction current and gap-junction conductance. We find a smoothing effect occurs as the number of gap junctions considered increases, but at small numbers of gap junctions, such as those observed in many diseased states, stochastic effects can be pronounced. In such decoupled regimes, stochastic effects are found to have a large effect on the occurrence of conduction block, the cessation of action potential propagation at some tissue location, and are found to increase the variance in conduction velocity from cell to cell. The waiting time between when two consecutive gap junctions reach their maximum current was found to conform to a gamma distribution, with shape and scale parameters a function of the number of gap junctions. As the number of gap junctions increases, the spread of the waiting time distributions decreases. Gap-junctional conductance was modeled as a time-dependent Gaussian distribution, with a temporal variance decreasing as a function of the elapsed time after depolarization. In the case of conduction block, we show that an emulator function can be constructed to estimate the probability of occurrence, thereby reducing the need for a large number of computationally intensive Monte Carlo simulations. Along with probabilistically describing the stochastic gap- junction model, these distributions can be leveraged in larger-scale tissue-level simulations to incorporate stochastic gap-junction gating at a reduced computational cost.

#### Library of Congress Subject Headings

Gap junctions (Cell biology)--Mathematical models; Heart conduction system--Mathematical models; Electrophysiology--Mathematical models; Stochastic differential equations

#### Publication Date

7-31-2015

#### Document Type

Thesis

#### Student Type

Graduate

#### Degree Name

Applied and Computational Mathematics (MS)

#### Department, Program, or Center

School of Mathematical Sciences (COS)

#### Advisor

Elizabeth Cherry

#### Advisor/Committee Member

Seshavadhani Kumar

#### Advisor/Committee Member

Laura Munoz

#### Recommended Citation

Consagra, Will, "Using Stochastic Differential Equations to Model Gap-Junction Gating Dynamics in Cardiac Myocytes" (2015). Thesis. Rochester Institute of Technology. Accessed from

https://scholarworks.rit.edu/theses/8793

#### Campus

RIT – Main Campus

#### Plan Codes

ACMTH-MS

## Comments

Physical copy available from RIT's Wallace Library at QH603.C4 C66 2015