About this model

This is a bond-graph model of cardiac contraction in the human cardiomyocyte.

  • Intracellular calcium transient [Ca]i
    • Changing sarcomere length
  • Tension generated by sarcomere
  • Re.TRPN_Ca: Binding of Ca to troponin (TRPN)
  • Re.BU: Spontaneous switching of crossbridge from B (blocked) to U (unbound) states
  • Re.UW: Spontaneous switching of crossbridge from U to W (pre-powerstroke) states
  • Re.WS: Spontaneous switching of crossbridge from W to S (powerstroke) states
  • Re.SU: Spontaneous switching of crossbridge from S to U states

Model status

The current CellML implementation runs in OpenCOR.

Model overview

This model is based on existing kinetic models, where the mathematics are translated into the bond-graph formalism. This describes the model in energetic terms and forces adherence to the laws of thermodynamics.

For the following bond-graphs, all enzymes are shown in maroon. A '0' node refers to a junction where all chemical potentials are the same. A '1' node refers to all fluxes being the same going in and out of the junction.

List of chemical species
Abbreviation Name
Cai Calcium, intracellular
TRPN troponin C
CaTRPN Complex of troponin C with attached Cai ion
B Crossbridge in blocked state
U Crossbridge in unbound state
W Crossbridge in pre-powerstroke state
S Crossbridge in powerstroke state
λ Extension ratio of sarcomere
Cd Sarcomere strain of passive component
ζx Distortion in state x
T Tension
Ta Active tension
Ftotal Passive tension
Tref Maximal active tension at resting length
Ttotal Total tension

Biochemical reactions

For reaction TRPN_Ca, the rate of Ca unbinding from TRPN is tension independent. The product CaTRPN is involved in changing the crossbridge from the bound state B to the unbound state U.


Fig. 1. Bond-graph formulation of calcium binding troponin, with the resulting complex participating in the crossbridge cycle with states B (blocked), U (unbound), W( pre-powerstroke), S (powerstroke).

Tension generation

For differential terms involving tension and displacement, a mass-spring-damper system was used to approximate the model in order to enable translation into bond-graph mathematics. Length terms were dimensionalised through the resting sarcomere length SL_0, and quantities of crossbridge units were dimensionalised through the estimated number of myosin binding sites per sarcomere.

Total tension is the sum of passive and active tension components. The total tension is then fed as an input into Re:TRPN_Ca.

Active tension

BG dampers

Fig. 2. Bond-graph formulations for distortion states, leading to calculation of active tension (Ta). Each of the three equations are modelled as a 1 node, with sum of potentials of the bonds equalling zero.

Passive tension

BG passive

Fig. 3. Bond-graph formulations of strain and extension contributing to passive tension (Ftotal). Each of the three equations are modelled as a 1 node, with sum of potentials of the bonds equalling zero.

Total tension

BG total

Fig. 4. Sum of active and passive tensions form total tension.

Parameter finding

A description of the process to find bond-graph parameters is shown in the folder parameter_finder, which relies on the:

  1. stoichiometry of system
  2. kinetic constants for forward/reverse reactions
  • If not already, all reactions are made reversible by assigning a small value to the reverse direction.
  1. linear algebra script.

Here, this solve process is performed in Python.

Derived from workspace BG_crossbridge_TRPN at changeset 959277c48267.
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