Moth1 Comput.

ModcUing, Vol.I I,

pp. 778-782,1988

0895-7177:X8 $3.00+000

Pergamon Press plc

Printed inGreatBritain PHYSIOLOGICAL

MODELS

II

ACTIVATION DYNAMICS FOR A DISTRIBUTION-MOMENT OF SKELETAL MUSCLE

HODEL

Shiping Ma and George I. Zahalak Department of Mechanical Engineering, Washington University, St. Louis, Missouri 63130, USA

Abstract. A mathematical representation is developed of the which activate the contractile calcium-mediated processes machinery of skeletal muscle. Keywords. Skeletal muscle; calcium factor; distribution-moment model.

activation;

pulse, X(t) is a dimensionless function discribing the time course of is the free Ca release rate, ical the concentration in calcium sarcoplasm, and [Cal* is a reference If [Cal+< is taken as concentration. then the calcium release infinite pulses are invariant, and do not decrease with increasing sarcoplasmic calcium concentration.

A simple mathematical model of calcium in skeletal muscle is activation proposed. This model is based on current biophysical understanding of the underlying molecular mechanisms, and is linked to an existing model of contration dynamics - the Distribution-Moment Model proposed by Zahalak (1981, 1986). The result is a nonlinear state variable representation of a single muscle fiber or a homogeous ensemble of such fiber as a contractile system driven by electrical stimulation and muscle length changes.

Uptake of Calcium by the Sarcoplasmic Reticulum Relaxation of muscle occurs when calcium is removed from myofibrillar space. The rate of calcium uptake by the sarcoplasmic reticulum is assumed linearly dependent on the concentration of free Ca ions in the sarcoplasm and the avaibable binding sites for Ca ions on the sarcoplasmic reticulum membrane (Ebashi and Endo, 1968). This results in a saturable, ATP-consuming calcium-uptake model which follows simple Michaelis-Menten kinetics

This complete stimulation-activationcontraction DE1 model includes the following submodels: (1) a model for from the the release of calcium sarcoplasmic reticulum; (2) a model for absorption of calcium by the sarcoplasmic reticulum; (3) a model for the mass balance of calcium ions in the myofibrillar space; (4) a model for calcium-troponinlactin-myosin interaction dynamics; and (5) a DM model for contraction dynamics based on a "self-consistent" molecular model (Hill et al, 1975, Ma and Zahalak, 1987). The first two of these submodels are based partially on a more complex calcium-dynamics model proposed by Lehman (1982).

[~alup = V,,, [Cal/([Cal+K,)

change

It is assumed that each muscle action potential causes a pulse of calcium release from sarcoplasmic the reticulum and, further, that the diffusion of calcium into the myofibrillar space is not a ratelimiting pcocess. The rate of calcium release, [Cal. , for a single pulse of electrical stigulus is assumed to be P,

x(t)

(

[Cal 1 - -

[Cal"

,

(2)

where the dot denotes differentiation with respect to time, [Cal of calcium uptake ?tkz ,',"z rate sarcoplasm, Vmax is the maximum uptake free-calcium the rate, Km is concentration at which half-maximal uptake rate occurs. The net rate of

Calcium Release and Diffusion

[