This is a reproduction of the 4 state model of a cardiac sarcomere originally developed by Landesberg and Sideman. The particular version presented here was used in Yaniv, Sivan and Landesberg Am J Physiol 288:H389-H399, 2005.

Description

  This code represents the simplified Four State model of the sarcomere as 
  developed by Landsberg et al and used by Yaniv in the paper analyzing the
  sarcomeric control of contraction (AJP 288:H389-H399, 2005, Reference 1 
  below).  The model simplification over previous Landesberg models is that 
  the movement of troponin units between overlap regions is ignored and only 
  the troponin in the single overlap region is considered as the sarcomere 
  extends and contracts.  

  This model describes the regulation of crossbridge cycling and force production 
  present in a single cardiac sarcomere using a four state representation of the 
  troponin regulatory units. Troponin is either in a strong or a weak state and 
  either has calcium bound to it or is unbound. The force generated in the sarcomere 
  is determined by the number of troponin units in the single overlap region that 
  are in the strong bound and unbound states, T and U respectively. This model is 
  a simplification of previous models that considered the states of troponin in 
  the single double and no overlap region of the sarcomere.

  This model incorporates two feedback mechanisms. The first is a positive feedback 
  mechanism affecting the affinity of troponin for calcium. Calcium bound to troponin 
  increases the chance of further bonding of calcium to other available sites on the 
  troponin regulatory unit and is an example of cooperativity. The second is a negative 
  feedback mechanism and involves the influence of the filament sliding velocity on 
  crossbridge cycling. The parameters given in Yaniv et al. are not sufficient to yield 
  the results presented. Recent communication with the authors and a more complete set 
  of model parameters present in a paper in press allow us to estimate the values for 
  some of the parameters used here. See the model notes for more details.

Equations

The set of ODEs governing the transition of the troponin regulatory unit through its four states is given by:

and

where R, A, T and U are the unbound weak, bound weak, bound strong and unbound strong states of troponin, respectively. Tro is the total concentration of troponin regulatory units, SL is the sarcomere length and V is the sarcomere contraction velocity defined as positive in the case of sarcomere shortening. The negative mechanical feedback is evident in the alteration of the reaction rate from strong to weak states in the form of g₀ + g₁*V term. The cooperativity mechanism affects the reaction rate from bound to unbound states, k_-l, and is given by this set of equations:

where K₀ is the calcium affinity to troponin at rest, K₀ + K₁ is the maximum caldium affinity, K_1/2 is related to the number of force generating crossbridges at 50% of the calcium affinity range, n is the cooperativity coefficient, N_XB is the number of force generating crossbridges in the sarcomere, N_c is the number of crossbridges per unit length, L_s is the single overlap length and L₀ is a length constant based on the actin and myosin filament lengths.
The sarcomeric force is calculated from:

where  is the average force per crossbridge and V_u is the maximal shortening velocity.