Sarcomere length dependence and Heart Beat simulation

Here you will include influence of the sarcomere length in your model, and then you will perform the heart beat simulation. The heart beat simulation test is introduced in Heart Beat.

Sarcomere Length (SL) dependence

Deformation of the continuum affects the sarcomere length. The sarcomere length SL of the deformed continuum is given multiplying the natural sarcomere length SL0 to the stretch ratio λ along the filament direction. The degree of filament overlap depends on the sarcomere length (SL) as shown below. When the sarcomere is overstretched (A), there are some myosin heads near the center without the actin filament over them. When the sarcomere is over-shrunk (B), the actin filament coming from the other side may inhibit binding of the myosin head to the actin filament in the same side. To bring these effects into your model, you can use the function xi_overlap() provided by XB Modeler. xi_overlap() returns values between 0 and 1 for each myosin head according to the overlap condition of the actin filament above it as depicted below. See [ Washio et al., 2012] for details for this function. The degree of overlap is determined from the sarcomere geometry, and you can check the default setting and define your own geometry by clicking Edit Parameters and then clicking on the Sarcomere tab.

Relation between the sarcomere length and the stretch of continuum

[Washio et al., 2012] T. Washio, J. Okada, S. Sugiura and T. Hisada. Approximation for Cooperative Interactions of a Spatially-Detailed Cardiac Sarcomere Model. Cell Mol Bioeng 5, 113-126 (2012)

The figure below shows a typical tension – sarcomere length relationship for a fixed calcium ion concentration observed in real muscle. The slope in the shorter SL range is steeper than one in the longer SL range. This phenomenon can not be explained only by the overlapping degree of actin filaments. Thus you may adjust some transition rates according to the sarcomere length SL. XB Modeler provides a function sarcomere_length() which returns the current sarcomere length.

A typical tension – sarcomere length relationship for a fixed calcium ion concentration observed in real muscle

Copying your model

Go to the Cell Models page in your dashboard and click on the Copy button on the row of the 4 state model (model name: tutorial Ca-F regulatory) tested in Running the Isometric Contraction simulation. Then a copy of the model appears on the top row. Click the Edit button.

Giving sarcomere length dependent factors

To bring the sarcomere length effect into your model. You will modify the transition from N_XB to P_XB. Click on the arrow from N_XB to P_XB. Click on the Define a rate button. Replace the definition of "r" in the Transition dialog.

Input the following codes in the Transition dialog. Here, first, you will multiply the function xi_overlap() to the the previous rate to take the overlapping degree into account. Second, to reproduce the steep slope in the shorter sarcomere length range of the SL-tension curve, you will multiply a factor "SL_factor" to r.

r =xi_overlap() * get(TT, knp) * GAMMA**(get(MH, ng, -1) + get(MH, ng, 1))
SL = sarcomere_length()
SL_Q0 = 2.2
if (SL >= SL_Q0) then
  SL_factor = 1.0
 SL_factor = max( 1.0-0.2 * (SL_Q0 - SL), 0.0)
end if
r =SL_factor * r

Click on the Close button.

Saving your model

After modifying the transition rate of your model, you will save the model. Click Save in the menu bar. Enter a model name and note (optional) in the text boxes of the Save dialog, and click on the Save button.

Running a Heart Beat simulation

Submitting your job

Go back to the Dashboard page, and then click on the Run Simulation button on the row corresponding to the model that you have saved.

Then click on the Heart Beat radio button from the Observations in the Simulation Setup dialog. Click on the Submit button.

When the simulation job is successfully submitted by XB Modeler, the notification dialog appears on your screen. Then click on the Close button.

Looking your simulation results

When the simulation is finished, you receive an email notification as follows.

Click on the Simulation Results button in your dashboard page, and then Click on the Plot button in the top row that corresponds to the latest simulation result.

Then time courses of the cavity pressure and volume appear on the left and the work rates of the blood ejection and the ATP consumption appear on the right for three heart beats of the rotationally symmetric left ventricle model. We intended to reproduce standard heart beats where the cavity volume at the end-diastole is around 120 ml and 60% of the volume is ejected in the systolic phase. However, the end-diastolic volume of the current model is less than 90ml. What was wrong?

Results of the Heart Beat test

Checking what was wrong

To see insights of this problem, adjust the sliders below the pressure and volume figure from 2 sec to 3 sec. And then click on the Plot state ratio button. Then time courses of state ratios appear. Though it is small, there is still a small amount of population of XB_PostR. This prevents relaxation of the heart wall thus filling of the blood for the next cycle.

State ratio results of the Heart Beat test

Modifying your model

Click on the Cell Models button in your dashboard page, and then click on the Edit button in the top row that corresponds to the previous model. Click Edit Parameters in the menu bar on your model editor page. Click on the Your Custom Parameter tab. Then change the value of the parameter Q from 3.6 to 3.2, the parameter h0 from 200 to 400, and the parameter g" from 100 to 200. The parameter Q determines the balance of transition rates between the no binding state N_XB and the other states. By increasing the parameter h0 you will accelerate the transitions between XB_PreR and XB_PostR. By increasing the parameter g", you will accelerate the cycling from P_XBXB_PreRXB_PostRP_XB , thus the cycle is more activated. However, it increases the ATP consumption, since you assumed that one ATP molecule is consumed at the transition from XB_PostR to P_XB. Then click on the Save button and go back to your dashboard page to run again the Heart Beat test.

Looking the results of modified model

The end-diastolic volume is improved compared to the previous model. There may be many ways to construct a better model. Please try to construct your own model to reproduce a beating heart and then see also how the results of other tests agree with the experimental data.

Results of the Heart Beat test with the modified model