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# 4-State_Sarcomere_Energetics

The 4-State model of sarcomeric contraction created by Landesberg and Sideman is investigated in terms of the energy liberated by the system as a function of the rate of shortening.

Model number: 0070

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## Description

This model characterizes the energy liberation rate during steady sarcomeric contraction as developed by Tchaicheeyan and Landesberg (Am J Physiol 289:H2176-H2182, 2005). Three hypotheses are applied to the basic Landesberg 4-State model of sarcomeric function in an effort to explain the biphasic energy liberation rate - velocity relationship. They are:

1) Crossbridge turnover rate from non-force generating (weak) to force generating (strong) conformation decreases as velocity increases.

2) Crossbridge kinetics is determined by the number of strong crossbridges

3) The affinity of troponin for calcium is modulated by the number of strong crossbridges.

All three hypotheses yeild biphasic energy liberation rate - velocity relationships. The equations given below show how these three hypotheses are integrated into the basic Landesberg 4-State model.

## Equations

The basic 4-State sarcomere model relates force generation to the two strong states of troponin. The set of ODEs governing the transition of the troponin regulatory unit through its four states is given by:
$\large \frac{dA}{dt} \quad = \quad k_l \: Ca \: R \quad - \quad \left(f \: + \: k_{-l}\right) \: A \quad + \quad \left(g_0 \: + \: g_1 \: V \right) \: T$
$\large \frac{dT}{dt} \quad = \quad f \: A \quad - \quad \left(g_0 \: + \: g_1 \: V \: + \:k_{-l}\right) \: T \quad + \quad k_l \: Ca \: U \quad$
$\large \frac{dU}{dt} \quad = \quad k_{-l} \: T \quad - \quad \left(g_0 \: + \: g_1 \: V \: + \: \left(k_l \: Ca\right)\right) \: U$
$\large \frac{dSL}{dt} \quad = \quad -V \quad$
and
$\large \mathrm{Tro} \quad = \quad R \quad + \quad A \quad + \quad T \quad + \quad U$
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 three variants of the model affect the kinetics of the basic 4-State model in the following ways: Variant 1: Velocity affecting weak to strong transition rate
$\large f \quad = \quad f_0 \: + \: f_1 \: V$
Variant 2: Crossbridge - crossbridge cooperativity
$\large f \quad = \quad f_0 \: + \: f_1 \: \frac{N_{XB}^n}{f_m^n \: + \: N_{XB}^n}$

Variant 3: Crossbridge - calcium cooperativity
$\large k_{-l} \quad = \quad k_0 \: - \: k_1 \: \frac{N_{XB}^n}{k_m^n \: + \: N_{XB}^n}$
where f is the weak to strong transition rate of troponin, k-l is the calcium bound to unbound transition rate, NXB is the number of strong crossbridges, and fm and km represent the number of strong crossbridges which represents an f or k-1, respectively, at the midpoint of its range.

## References

Tchaicheeyan O and Landesberg A.
Regulation of energy liberation during steady sarcomere shortening.
American Journal of Physiology, Heart and Circulatory Physiology 289:H2176-H2182, 2005.

Landesberg A and Sideman S.
Force-velocity relationship and biochemical-to-mechanical energy conversion by the sarcomere.
American Journal of Physiology, Heart and Circulatory Physiology 278:H1274-H1284, 2000.

## Key Terms

Sarcomere, Toponin, Energy liberation, Crossbridge model, Actin, Myosin, Cardiovascular_System, Cardiac_contraction, Crossbridge, 4-State_Model_Energetics