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# Yaniv_4-State_2005

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.

Model number: 0158

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## 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:

$\large \frac{dA}{dt} \quad = \quad k_l \: Ca \: R \quad - \quad \left(f \: + \: k_{-l}\right) \: A \quad + \quad \left(g_0 \: + \: g_1 \: V \right) \: T$

$\large \frac{dT}{dt} \quad = \quad f \: A \quad - \quad \left(g_0 \: + \: g_1 \: V \: + \:k_{-l}\right) \: T \quad + \quad k_l \: Ca \: U \quad$

$\large \frac{dU}{dt} \quad = \quad k_{-l} \: T \quad - \quad \left(g_0 \: + \: g_1 \: V \: + \: \left(k_l \: Ca\right)\right) \: U$

$\large \frac{dSL}{dt} \quad = \quad -V \quad$
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 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:
$\large k_{-l} \quad = \quad \frac{k_l}{K_0 \: + \: K_1 \: \left(\frac{N_{\rm XB}^n}{K_{1/2}^n \: + \: N_{\rm XB}^n}\right)}$

$\large N_{\rm XB} \quad = \quad N_c \: L_s \: (T \: + \: U)$

$\large L_s \quad = \quad \frac{SL \: - \: L_0}{2}$
where K0 is the calcium affinity to troponin at rest, K0 + K1 is the maximum caldium affinity, K1/2 is related to the number of force generating crossbridges at 50% of the calcium affinity range, n is the cooperativity coefficient, NXB is the number of force generating crossbridges in the sarcomere, Nc is the number of crossbridges per unit length, Ls 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:
$\large F \quad = \quad \overline{F} \: N_c \: L_s \: \frac{T \: + \: U}{\rm Tro} \: \left(1 \: - \: \frac{V}{V_u}\right)$
where $\overline{F}$ is the average force per crossbridge and Vu is the maximal shortening velocity.

## References

    Yaniv Y, Sivan R and Landesberg A.;"Analysis of hystereses in force length and
force calcium relations", Am J Physiol Heart Circ Physiol 288:H389-H399, 2005

Yaniv Y, Sivan R and Landesberg A.; "Stability, controllability and observability
of the 'Four State' Model for the sarcomeric control of contraction"
Ann Biomed Eng 34:778-789, 2006

Landesberg A and Sideman S.;"Mechanical regulation of cardiac muscle by coupling
calcium kinetics with cross-bridge cycling: A dynamic model", Am J Physiol Heart
Circ Physiol 267:H779-H795, 1994.



## Key Terms

Crossbridge, Cardiac Muscle, Sarcomere, Excitation-Contraction Coupling, Cardiovascular System, Cardiac_contraction, Landesberg_4-State