# Comparative Statistical Mechanics of Muscle and Non-Muscle Contractile Systems: Stationary States of Near-Equilibrium Systems in A Linear Regime

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. General Mechanical Parameters

#### 2.2. Molecular Mechanics of Actin–Myosin Molecular Motors

^{−1}) value is 1/tc, where tc is the total duration of the actin–myosin CB cycle. Again, the highest value of kcat was observed in the heart tissues, which was about 7000 times higher than in the placental tissues (Figure 1C and Table 1).

#### 2.3. Statistical Mechanics

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Experimental Procedure

#### 4.2. Experimental Set-Up

_{2}–5% CO

_{2}and maintained at pH 7.4. Contractile samples were stimulated either electrically by means of two platinum electrodes or chemically by means of KCl (0.05 M). Electrical stimulus: 5 ms duration; stimulation frequency: 50–100 Hz; train duration: 250–5000 ms; train frequency: 0.17 Hz. For muscle strips, experiments were carried out at the resting length corresponding to the apex of the initial length-active tension curve (Lo). For non-muscle placental samples, the preload was the load that induced neither shortening nor lengthening. The cross-sectional area of the contractile sample (in mm

^{2}) was calculated from the weight/length ratio of the sample at Lo at the end of the experiment. The experimental protocol and the electromagnetic transducer have been described in an earlier study [40].

^{−1}) was assessed by means of the zero-load clamp technique. The peak isometric tension, i.e., the peak force normalized per cross-sectional area (total isometric tension, To, in mN·mm

^{−2}) was measured from the fully isometric contraction. The Hill hyperbolic tension–velocity relationship [15] was determined from the peak velocity (V) of 8 to 10 isotonic after loaded contractions, which were plotted against the isotonic tension (T), and by successive increments of load from zero-load up to the total isometric tension [22]. The T–V relationship was fitted according to Hill’s classic equation, (T + a) (V + b) = [To + a] b where –a and –b were the asymptotes of the hyperbola. For all muscle or non-muscle samples, the T–V relationship was accurately fitted by means of a hyperbola. The G curvature of the T–V relationship was To/a = Vmax/b [15,41].

#### 4.3. Huxley Formalism

_{Hux}) and the isotonic tension (P

_{Hux})—as a function of the shortening velocity (V) of the contractile structure—were obtained by the following equations:

_{Hux}= (Ne) (h/2 l) (f1/(f1 + g1)) [g1 + f1 (V/Φ) [(1 − exp (−Φ/V)]]

_{Hux}= N (sw/2 l) (f1/(f1 + g1)) [1 − (V/Φ) [(1 − exp (−Φ/V)) (1 + (1/2) ((f1 + g1)/g2)

^{2}(V/Φ)]

_{ATP}, is roughly −60 kJ/mol. The value used for e is 10

^{−19}J [27]. The tilt or swing of the myosin head relative to actin varies from 0 to the molecular step size (h); f1 and g1 correspond to a tilt from 0 to h; g2 corresponds to a tilt > h; Φ = (f1 + g1) h/2 = b; N is the number of cycling CBs per mm

^{2}at peak isometric tension. The molecular step size represents the translocation distance of the actin filament per ATP hydrolysis, produced by the tilt of the myosin head. Parameter l represents the distance between two successive actin sites with which any myosin site can combine. In agreement with the Huxley condition (𝓁 >> h), the h and 𝓁 values are h = 10 nm and 𝓁 = 28.6 nm (this is close to the semi-helicoidal turn of the actin filament) [42]. The value of h was confirmed by the three-dimensional head structure of the muscle myosin II [43,44,45]. Calculations of f1, g1, and g2 were obtained from the following equations [13]:

^{−1}) and the Avogadro number. The myosin ATPase activity is the product of the kcat and myosin content. The thermodynamic flow is v = kcat × CB mole number per liter (mol·L

^{−1}·s

^{−1}). The rate of mechanical work (W

_{M}) is equal to P

_{Hux}·V [22]. At any given load level, the efficiency of the contractile tissue is defined as the ratio of W

_{M}to E

_{Hux}. The peak efficiency is the peak value of efficiency.

#### 4.4. Determination of CB Probabilities of the Six States

**→**A2, and so on (Figure 2). These values can be calculated from Huxley’s equations [13]. The probability PD1 = tD1/tc = (1/g2)/tc = kcat/g2 was low, due to the fact that tD1 << tc and g2 >> 1/tc [20]. In addition, tD2 was approximated to 10 tD1 [41]. Consequently, PD2 was approximated to 10 × PD1. The probability PA1 = tA1/tc = (1/f1)/tc = kcat/f1 was also low, due to the fact that tA1 = 1/f1 was << tc. The probability PA2 = tA2/tc = (h/vo) × kcat was low, due to the fact that tA2 was << tc [19,20]. Thus, the most probable detached state was D3.

_{r}of the six states increased from E

_{0}to E

_{5}. By convention, the lowest level (E

_{0}) coincided with the ground state (gs), and was equal to zero (E

_{0}= E

_{gs}= 0) [8]. The highest level was E

_{5.}The probability P

_{r}of each state r diminished from P

_{0}(the most probable state, i.e., PD3) to P

_{5}(the least probable state). The distribution of energy was characterized by the average number N

_{r}of cycling CBs that occupied a given state of energy level E

_{r}. The average internal energy E is the sum of their individual energies, i.e., $\mathrm{E}={\displaystyle \sum _{\mathrm{r}}{\mathrm{N}}_{\mathrm{r}}{\mathrm{E}}_{\mathrm{r}}}$.

_{5}− E

_{0}= kT (ln P

_{0}/P

_{5}) = 10

^{−19}J. As 0.5 < PD3 < 1 and E

_{5}− E

_{0}= kT ln (P

_{0}/P

_{5}); this means that P

_{5}was << to PA1, PA2, PD1, and PD2. Consequently, the least probable state was A3, which implies that P

_{5}= PA3. The highest state level E

_{5}was E

_{A3}= 10

^{−19}J. In addition, PA3 + PD3 = 1 − (PA1 + PA2 + PD1 + PD2). We know the ratio PD3/PA3 and the sum PA3 + PD3. Thus, we can deduce PA3 and PD3.

#### 4.5. Statistical Mechanics

_{2}, and produce ATP that drives the chemo-mechanical processes. In CB myosin molecular motors, there is a one-to-one chemo-mechanical coupling. This means that only one ATP is consumed per CB cycle [7]. The number of independent and distinguishable cycling CBs is equal to the number of ATPs that are consumed during contractile processes. In statistical mechanics, the grand canonical ensemble can be applied to complex open systems such as muscle and non-muscle contractile tissues. In our study, the open contractile system (CS) was in a container (

**C**). The following molecules made up the system and the container: myosin CBs, actin, and small soluble molecules such as ATP, ADP and inorganic phosphate Pi. These molecules could be exchanged between C

**S**and

**C**. CBs were either attached to or detached from actin, and bound or not with ATP, ADP or Pi. Moreover, the number of cycling CBs could fluctuate slightly with the number of non-cycling CBs, which became cycling CBs and vice versa. C

**S**was composed of all the active cycling CBs which were each in one of the six states.

**C**was composed of all the non-cycling CBs, all the non-cycling actin molecules, and all the ATP, ADP and Pi that were not attached to the cycling CBs.

**S**was calculated from the Huxley equations [7] and was expressed in nM·L

^{−1}of contractile tissue. Let A be the chemical affinity of the CB cycle, S the statistical entropy, E the internal energy, and T (Kelvin) the temperature of

**S**. The grand potential is linked to E, S, A and T according to the classic relationship ψ = E − TS − A. Statistical entropy $\mathrm{S}=-\mathrm{R}{\displaystyle \sum _{\mathrm{r}}{\mathrm{P}}_{\mathrm{r}}\mathrm{ln}{\mathrm{P}}_{\mathrm{r}}}$ characterizes the dispersal of energy and makes it possible to calculate the degree of disorder in the system. The molecular partition function is z = 1/Pmax (where Pmax was the highest probability PD3). The Boltzmann distribution is given by the equation:

**S**tended towards zero. Under these conditions, A also tended towards zero. Thus, E − TS tended towards ψ. The extrapolation at v = 0 of the E − TS versus v relationship (i.e., the ordinate of this relationship) was equal to ψ. The affinity A was calculated by the equation:

_{i}S/dt) due to chemical reactions is the product of the thermodynamic force (A/T) and the thermodynamic flow:

## 5. Statistical Analysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Mechanical parameters of contractile tissues and crossbridge (CB) properties: (

**A**) maximum unloaded shortening velocity (Vmax); (

**B**) total isometric tension; (

**C**) kcat, the inverse of the total CB time cycle; (

**D**) unitary CB force (po); (

**E**) maximum efficiency; and (

**F**) myosin content.

**Figure 2.**The ATP-ADP-Pi actin–myosin CB cycle with six different conformational steps, i.e., three detached steps (D1, D2 and D3) and three attached steps (A1, A2 and A3). Transition A3

**→**D1 is the ATP binding step which induces the CB detachment after the ATP binding with the actin (Act)-myosin (M) complex (AM). The rate constant for detachment is g2: AM → Act + M. Transition D1 → D2 is the ATP hydrolysis: M-ATP

**→**M-ADP-Pi. Transition D2

**→**D3 is M-ADP-Pi

**→**M*-ADP-Pi. D3 is the step with the highest probability. Transition D3 → A1 is the attachment state: the myosin head (M*-ADP-Pi) binds with Act and the rate constant for attachment is f1: M-ADP-Pi + Act

**→**AM-ADP-Pi. Transition A1 → A2 is the power stroke which is triggered by the Pi release: AM-ADP-Pi

**→**AM-ADP + Pi. The power stroke is characterized by the generation of a unitary CB force ($\cong $picoN) and an elementary CB step ($\cong $nm). Transition A2

**→**A3 is the release of ADP: AM-ADP

**→**AM + ADP.

**Figure 3.**Probabilities of the six steps of the CB cycle (see paragraph 4.4). (

**A**) PD1 probability; (

**B**) PD2 probability; (

**C**) PD3 probability; (

**D**) PA1 probability; (

**E**) PA2 probability; (

**F**) PA3 probability.

**Figure 4.**Attachment (f1) and detachment (g1 and g2) constants of crossbridges: (

**A**) Attachment (f1) constant of crossbridges; (

**B**) detachment (g2) constant of crossbridges; and (

**C**) detachment (g1) constant of crossbridges.

**Figure 5.**Crossbridge properties and curvature (G) of the tension–velocity relationship: (

**A**) Frictional drag force (FDF) of CBs; (

**B**) coefficient of frictional drag force; (

**C**) G curvature of the tension-velocity relationship; (

**D**) mean velocity of the CB tilt.

**Figure 6.**Thermodynamic parameters: (

**A**) statistical entropy (S); (

**B**) internal energy (E); (

**C**) S’ = dS/dPD3: derivative of S according to PD3; (

**D**) grand potential.

**Figure 7.**Thermodynamic parameters: (

**A**) thermodynamic flow; (

**B**) thermodynamic force; (

**C**) entropy production rate; (

**D**) affinity.

**Figure 8.**Relationships between thermodynamics parameters and the unitary CB force: (

**A**) relationship between the statistical entropy and the microcanonical partition function; (

**B**) relationship between the statistical entropy and the unitary CB force; (

**C**): relationship between A–RT–GP (grand potential GP is ψ) and the derivative of S with respect to PD3, i.e., S’; (

**D**) relationship between affinity and the CB force.

**Table 1.**Mean values of contractile parameters, myosin CB properties, and statistical mechanics for the heart and the placental tissues. The last column presents the ratio of the mean in the heart tissues and the mean in the placental tissues.

Placenta (Mean) | Heart (Mean) | Heart/Placenta Ratio | |
---|---|---|---|

Vmax | 0.002 | 3.7 | 1850 |

Tension | 1.5 | 42 | 28 |

Kcat | 0.003 | 20.6 | 6866 |

Unitary CB force | 2.0 | 1.6 | 0.80 |

max Efficiency (%) | 36 | 28 | 0.78 |

Myosin content | 0.14 | 12.5 | 89 |

Frictional drag force (FDF) | 3.5 | 4.1 | 1.2 |

FDF coefficient | 3943 | 1 | 0.0003 |

Vo | 0.002 | 4.7 | 2350 |

G | 3.7 | 1.6 | 0.43 |

f1 | 0.075 | 306 | 4080 |

g1 | 0.028 | 194 | 6928 |

g2 | 0.341 | 731 | 2143 |

Statistical entropy | 5.6 | 9.3 | 1.7 |

Internal energy | 1097 | 1401 | 1.3 |

Affinity | 354 | 535 | 1.5 |

Thermodynamic force | 1.2 | 1.8 | 1.5 |

Thermodynamic flow | 4.8 × 10^{−6} | 2.8 | 0.58 × 10^{6} |

Entropy Production Rate | 8.2 × 10^{−13} | 6.5 × 10^{−7} | 0.8 × 10^{6} |

**Table 2.**There was a relationship of proportionality between the thermodynamic force and the thermodynamic flow. The second column represents the slope of this relationship.

Slope | R | P | |
---|---|---|---|

PLACENTA | 2.4 × 10^{13} | 0.85 | 0.001 |

UTERUS | 1.6 × 10^{10} | 0.61 | 0.04 |

TRACHEA | 4.9 × 10^{9} | 0.76 | 0.01 |

HEART | 5.2 × 10^{7} | 0.89 | 0.0001 |

DIAPHRAGM (tw) | 1.1 × 10^{9} | 0.91 | 0.0001 |

DIAPHRAGM (TET) | 4.2 × 10^{8} | 0.85 | 0.002 |

SOLEUS | 3.5 × 10^{8} | 0.60 | 0.0001 |

EDL | 7.9 × 10^{7} | 0.71 | 0.0001 |

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**MDPI and ACS Style**

Lecarpentier, Y.; Claes, V.; Krokidis, X.; Hébert, J.-L.; Timbely, O.; Blanc, F.-X.; Michel, F.; Vallée, A. Comparative Statistical Mechanics of Muscle and Non-Muscle Contractile Systems: Stationary States of Near-Equilibrium Systems in A Linear Regime. *Entropy* **2017**, *19*, 558.
https://doi.org/10.3390/e19100558

**AMA Style**

Lecarpentier Y, Claes V, Krokidis X, Hébert J-L, Timbely O, Blanc F-X, Michel F, Vallée A. Comparative Statistical Mechanics of Muscle and Non-Muscle Contractile Systems: Stationary States of Near-Equilibrium Systems in A Linear Regime. *Entropy*. 2017; 19(10):558.
https://doi.org/10.3390/e19100558

**Chicago/Turabian Style**

Lecarpentier, Yves, Victor Claes, Xénophon Krokidis, Jean-Louis Hébert, Oumar Timbely, François-Xavier Blanc, Francine Michel, and Alexandre Vallée. 2017. "Comparative Statistical Mechanics of Muscle and Non-Muscle Contractile Systems: Stationary States of Near-Equilibrium Systems in A Linear Regime" *Entropy* 19, no. 10: 558.
https://doi.org/10.3390/e19100558