# ðŸ“’ Mullins 2013

A Mathematical Model of the Mouse Ventricular Myocyte Contraction

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## Introduction

- Changes in one of the subsystems can lead to abnormal behavior in others
- Dysfunction of the L-type Ca2+ channel => affects Ca2+ handling in cardiac cells [1], [2] resulting in cardiac arrhythmias

- Heterogeneities in cellular electrical activities in the heart, dysfunction of K+ channels, or acidosis can also produce pro-arrhythmic behavior
- Instability of Ca2+ dynamics (alternans) can lead to the action potential alternans [5] and alternans in mechanical contraction
- Force generation involves conformational changes in thick (myosin) and thin (actin, tropomyosin, and troponin) filaments. most mathematical models use a significantly simplified description of this process.

###### Schematic diagram of the mouse model cell and Markov model for force generation.

- New model by Land et al. (2012) adjusted the cellular contraction model of Rice et al. (2010)
- body temperature (310Â°K, or +37Â°C)
- absolute values are larger in the model
- did not study contraction force-frequency relationships in the cellular level

- Here:
- new electromechanical model for mouse ventricular myocyte contraction at room temperature (298Â°K, or +25Â°C)
- incorporated a myocyte contraction model from Rice et al
- the importance of using
**variable sarcomere lengths**

## Methods

Rice model adjusted model parameters to fit experimental data on myocyte contraction obtained for room temperatures

links Ca2+ dynamics and myocyte contraction

two nonpermissive tropomyosin states (N0 and N1) and four permissive tropomyosin states (P0, P1, P2, and P3)

All transition rates in the model are Ca2+-independent, except for kNP, which depends on the concentration of troponin with Ca2+ bound to a low-affinity binding site

twitch contraction, where Fcontrn is time-dependent: Hookeâ€™s law (SL0 = initial value of SL)

variable cell length: (L0â€Š=â€Š100 Âµm)

stimulated with different frequencies using a stimulus current (Istimâ€Š=â€Š80 pA/pF, Ï„stimâ€Š=â€Š0.5 ms) for at least 200,000 ms to reach a quasi-steady state

51 ordinary differential equations

## Results

### Steady-state Force-calcium Relationships

- an increasing sigmoid function of calcium concentration
- able to reproduce a shift in Ca2+ sensitivity for steady-state force-calcium relationships shown for three sarcomere lengths

### Dynamic Behavior of Contraction Force

- pacing @ 0.5 Hz
- endocardial cells show larger [Ca2+]i transients than epicardial cells
- significant differences in the experimental data obtained from different experimental groups on the time behavior of force, both in peak values and residual forces
- But clear similarity in the time-to-peak values and relaxation of the simulated forces

- The model includes
**changes in sarcomere length**during myocyte contraction

### Force-frequency Relationships

- the simulated amplitudes of [Ca2+]i transients for epicardial and endocardial cells are verified by the experimental data obtained by Dilly et al.
- able to reproduce peak contraction force-frequency relationships for mouse ventricular myocytes in the frequency range from 0.5 to 2.0 Hz

###### Simulated time courses for contraction forces, sarcomere lengths, and sarcomere shortenings for three different resting sarcomere lengths (1.9, 2.1, and 2.3 Âµm) for epicardial and endocardial cells

- increase in the resting sarcomere length increases twitch force and relative sarcomere shortening

### Constant versus Variable Sarcomere Length

- a variable SL when calculating the transition rate from non-permissive to permissive states, as well as in the detachment rates in permissive states in this model
- constant SL replaced the variable SL in the calculation of the normalized sarcomere length:
- The peak force when using a constant SL is clearly higher, while the residual force appears to be about the same
- Frequency dependence is similar

### Frequency Dependence of dL/dt and dF/dt

- dL/dt and dF/dt also show frequency dependence
- The epicardial cell demonstrates a monotonic increase in the magnitudes of peak values for dL/dt and dF/dt in the frequency range from 0.25 to 4 Hz
- the endocardial cell shows a biphasic behavior in the peak magnitudes of the derivatives: an increase when the stimulation frequency changes from 0.25 to 2 Hz, and a decrease in the frequency range from 2 to 4.0 Hz
- the model showed somewhat slower relaxation, thus lower values of +dL/dtmax, than experimental data (solid symbols)
- Simulated data for time-to-peak force shows good agreement with the experimental data
- while time-to-50% relaxation are somewhat longer in the simulated data

## Discussion

Mice’s contraction rate is about 10 Hz, faster than the rabbit (4 Hz) and human (1 Hz). APD in mouse ventricular myocytes is also considerably shorter (APD50 âˆ¼ 4.5 ms in mice versus âˆ¼200 ms in rabbits and âˆ¼300â€“400 ms in humans.

In mice, the peak value of Ca2+ transient occurs after almost complete repolarization of action potential

In larger species, such as rabbit, time scaling of the action potential, [Ca2+]i and contraction force transients is different

Mouse ventricular myocytes, unlike other species, demonstrate biphasic frequency dependence of intracellular [Ca2+]i transient and peak force

The model for an epicardial cell was also able to reproduce this physiological phenomenon.

However, our model for the endocardial cell does not show biphasic behavior in the frequency-dependence of both peak [Ca2+]i transients and peak contraction force, but at least qualitatively, reproduced saturation and even decrease in sarcomere shortening and contraction force amplitude at 4-Hz stimulation

the endocardial cells demonstrate significantly larger [Ca2+]i transients, and our modeling predicts larger contraction force and shortening in these ventricular myocytes.

Simulations with variable sarcomere length produce significantly smaller contraction force than the simulations with constant sarcomere length despite the same time course and amplitude of [Ca2+]i transient during twitch

The only difference between Rice Models 4 and 5 is the modulation of the kâˆ’ ltrpn rate by generated force. Because Model 4 and Model 5 yielded an approximately equal description of myocyte contraction, we implemented Model 4 in our electrophysiological model, as Model 5 led to unstable solutions.

### Limitations

- the model uses a simplified description of the relationships between contraction force and cellular shortening in the form of Hookâ€™s law, while the real dependence is more complicated
- It does not describe the effects of cellular shortening on Ca2+ transients, as does the model of Rice et al
- did not take into account intracellular spatial inhomogeneities of Ca2+ concentration and crossbridge binding sites (requires PDE)

## Reference

Mullins PD, Bondarenko VE. A mathematical model of the mouse ventricular myocyte contraction. PLoS ONE 2013;8(5):e63141. doi:10.1371journal.pone.0063141. PMC3650013 ↩︎