INTRODUCTION

The purpose of this paper is to study the problem of iso-kinetic shortening contraction of the sarcomer. For that reason we will base upon a recently developed theory (Tsili, 2013a). The main assumption of this work is that during contraction the change of cross-section area of the sarcomer is small and could be neglected.   

THE PHYSICAL APPROXIMATION OF THE PROBLEM

The basic equations of our theory are in our previous paper (Tsili, 2013a) and they will not be repeated here. We supposed that at t <0, the sarcomer was free of stress. In resting state it had a length Lo, a radius r_o and a temperature θ_ο, as indicated in Fig. 1. Consequently its cross -section area and initial volume of sarcomer were S_o and ξ_ο, respectively, where:

 S_o =πr_o²  and ξ_o = S_o.L_o  (2.1)   

Figure 1

In resting state the sarcomer had a length Lo and a radius So .

We assume that at t=0 the motor neuron sends a stimulative pulse to the motor unit, in order to start the process of muscle contraction. Due to the delay of the response, at t1 >0 a mechanical stress is produced. The last simultaneously displaces with constant speed upwards and downwards the material particles of the sarcomer that lie below and above the z line respectively. As a result of the above displacements, the sarcomer shortens ( see Fig. 2.). The process of shortening continues until a time t_s. At t >t_s the contraction stops and the inverse process of relaxation starts.

Figure 2

The sarcomer at time t &ge; t1, during shortening. Accordingly to all that we stated in previous paragraph, the displacements are: ur= u&theta; = 0 and uz = a&theta;&omicron;tz (2.2)1-2-3

Accordingly to all that we stated in previous paragraph, the displacements are:

 u_r= u_θ = 0  and u_z = aθ_οtz  (2.2)1-2-3

where a is an unkown constant. Therefore the velocity of the material particle of sarcomer is:

 v_r =du_r/dt=0 v_θ =du_θ/dt =0 and v_z=du_z /dt =aθ_οz (2.3)1-2-3

Τhen strain–displacement equations become:

E_rr = E_θθ = Ε_rθ = Ε_zθ = Ε_rz= 0 and E_zz= aθ_οt  (2.4)1-2-3-4-5-6

 Substituting the above into stress - strain equation (see Tsili, 2013a) it follows that all stresses vanish:

  T_rr= T_θθ = Τ_θz = T_rz= T_rθ = 0   (2.5)1-2-3-4-5

except from the axial stress:

  T_zz=c₃₃aθ_οt +a_z +a_zzaθ_ο   (2.6)

CALCULATION OF STRESS, FORCE, LENGTH AND VOLUME OF THE SARCOMER, USING “RAW” DATA

Spector et., al., (1980); Roy et., al., (1982); Powel et., al., (1984) calculated the maximal isometric stress P_o:

   P_o =225KN/m²  (3.1)

Accordingly to Gordon et., al., (1966) and to Pollack (1990), the initial length and diameter of sarcomer are :

L_o= 2.22 μm  and d_o= 2r_o=1μm  (3.2)_1-2  respectively (1μm=10^─6m). Therefore:

S_o =πr_o²= 0.7854 μm² and V_o=S_o.L_o =1.7436 μm³  (3.3)_1-2

The maximum isometric force F_o could be calculated by replacing (3.1) and (3.3)₂ into:

F_o = P_o.S_o =176.715x10^─9N (3.4)  

Burkholder and Lieber (2001, p.1531) superimposed data taken from 36 studies for bird, cat, rat, rabbit, mouse, frog, horse, human and derived a general normalized force-length relation for the sarcomer. Accordingly to their graph, the ascend limb of this relation is :

where f = F/F_o and k= L/L_o. It has been shown (Pollack 1990, p.3; Epstein and Herzog,1998, p.76) that during shortening the length of the sarcomer linearly decreases :

  L(t) = At + B  for t₁≤ t ≤t_s in units of μm  (3.6)

 Then the contraction speed is:

 v_z= du_z/dt= d(L_o ─ L(t))/dt = ─A (3.7)  

Faulkner et., al.,(1986) reported values for the contraction velocity 6 lengths/sec and 2 lengths/sec for fast and slow fibres respectively at temperature 37^οC. We assume that we have to do only with fast fibres. Then from (3.7), it results: 

   A = ─6μm/sec   (3.8)

In addition for fast- fibres t₁= 0.02sec (see for example Makrorpoulou, 2009, p.16). 

Figure 3

Electromyograph for muscle contraction. Taken from Makropoulou (2009)

Substituting the initial condition L(t₁=0.02) = L_o= 2.22μm into (3.6), it follows:

 L(t) = ─6t + 2.34 in units of μm (3.9)

Consequently by the help of (3.1), (3.2), (3.3), (3.4), (3.5) and (3.9) it is possible to calculate the normalized stresses, forces, lengths and volumes of the sarcomer, for various time moments. All results are concentrated in Table 1.Also Table 2. contains the results derived from linear regression analysis on data of Table 1. Particularly the forces and the lengths have been calculated by:

  f= F/F_o=(P.S_o)/(Po.S_o)= P/P_o =1.4─ 6.933t and  

  k= L/L_o = (V/S_o)/(V_o/S_o)=V/V_o = 1.0542─2.705t  (3.10)_1-2

THEORETICAL RESULTS FOR CONTRACTION SPEED AND FOR NORMALIZED STRESS, FORCE, LENGTH AND VOLUME OF THE SARCOMER

In (2.2)₃ we restrict z to a constant z₁ such that:

z₁= ─0.00954x10⁹  (4.1)   and we choose:

a=─17x10^─9 and  θ_ο=37^οC (4.2)_1-2   

Table 1

All results for stresses, forces, lengths and volumes of the sarcomer, for various time moments of isokinetic contraction. The initial data are Po=225KN/m2, Fo=176.715Nx10─9, So =0.7854 &mu;m2, Vo= 1.7436&mu;m3 ,Lo = 2.22 &mu;m and are picked from cites refered in the text.

Table 2

All results for stresses, forces, lengths and volumes of the sarcomer de-rived by linear regression analysis F/Fo= 1.4─6.933t and L/Lo= 1.0542─2.705t on data of Table 1.

 Replacing (4.1) and (4.2) into (2.2)₃ and (2.3)₃, it implies:

  u_z =6t in units of μm and v_z=6μm/sec  (4.3)_1-2

respectively. The theoretical result (4.3)₂ predicts the measurement of Faulkner et., al., (1986) for the contraction speed of fast fibres.

We normalize sarcomer’s length:

 k=L(t)/ L_o=L_o─u_z/ L_o=2.22─6t/ 2.22= (V/S_o) /(V_o/L_o)=V/V_o (4.4) 

where we have used (3.3)₂ and (4.3)₁. At continuity we divide (2.6) with the maximum isometric stress (3.1) and we account that the coefficient az coincides with P_o ( Tsili, 2013b. p.3). Then it implies:

  T_zz/P_o = P/ P_o = c₃₃aθ_οt/P_o + 1+a_zzaθ_ο/P_o  (4.5)  Hatta et., al. (1984) measured the elastic constant of muscle c33 in active state:

  c₃₃ =2,47x10⁹N/ m²  (4.6)

and concluded that muscle elastically lies near an unstable state. Therefore more implicit situations such viscoelasticity are unstable. The last means that it is rather impossible to compute a standard value for the viscoelastic coefficient a_zz.

We only could predict the order of the magnitude of the a-bove parameter. We choose:

   a_zz = ─0.143x10⁹N/m²  (4.7)

Then (4.4) because of (3.1), (3.2)₁, (4.2), (4.6) and (4.7) concludes to:

 P/P_o = 1.4─6.905t= (F/S_o)/ (F_o/S_o) = F/F_o (4.8)

Therefore using (4.4) and (4.8) it is possible to calculate all values for normalized stresses, forces, lengths and volumes for the sarcomer for the same time moments as we did earlier. All these results are found in Table 3. 

DISCUSSION

Our model for proper choice of the unknown parameters, i) it predicts the contraction speed for fast fibres measured by Faulkner et., al., (1986) and ii) it is very closed to results for normalized stresses, forces, lengths and volumes of the sarcomer derived from statistical analysis on “raw” data (Table 2.), picked by Spector et., al., (1980); Roy et., al.,(1982); (1982); Powel et., al., (1984); Gordon et., al., (1966); Pollack (1990), Burkholder and Lieber (2001); Faulkner et., al., (1986), Epstein and Herzog (1998); Hatta et., al.,(1984); Makropoulou (2009). Compare our theoretical results in Table 3. with those of Table 2. 

Table 3

The results for normalized stress ,force, length and volume of sarcomer derived from our model for a = ─17x.10─9 , &theta;&omicron;=37&omicron;C, c33= 2.47x109, azz=─0.143x109