__Journal of Theoretical and
Applied Mechanics__

__ ____Sofia____, 2007, vol. 37, No. 4, pp
79-92__

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**GENERAL
THERMODYNAMIC THEORY FOR ACTIVE BIO-CONTINUUM**

*Institute** of **Mechanics**, **Bulgarian** **Academy** of Sciences,*

*Acad.
G. Bontchev Str., Bl. 4, 1113 Sofia, Bulgaria*

e-mail: petrov333@gmail.com

**Abstract.** Molecular motors are nanometric protein-based
devices, essential for the movement in living organisms. The aim of the present
study is to offer a general thermodynamic theory for the active bio-continua
considered as composition including continually distributed assemble of
molecular motors. The restrictions following from Clausius–Duhem inequality
formulation of Second principle of Thermodynamics on the constitutive equations
are under consideration. The capability of the new theory to model active
bio-mechanical processes is demonstrated for muscle mechanics by unifying the
well known classical heuristic models. Also the theory offers possibility for active extensions of the passive classical diffusion model, which is one of the candidates as a model of the canalicular-lacunar nutrient transport in cortical bone.

**1. Introduction**

The human body is composed of about 60 trillion cells. A close galaxy
order (_{}) is valid for all mammalians.
To be in stable condition such enormous system needs an extremely high
organization, specialization and communication. An important role for the shape
of the biological body, considered as bio-continuum, play the so called molecular
motors. They represent protein based nanometric devices transforming the free
energy, released by the hydrolysis of ATP (Adenosine TriPhospate), in
mechanical work [1]. The molecular motors are responsible for the muscle force,
for the shape and structural reconstruction of the cells at mitosis, adaptive reconstruction
of the connective tissues, organs and for the transportation of nutrients,
waste products and other important fractions. Their activity is controlled by
the central and peripheral nerve system with nerve signals, hormones and other
mediators. Some substances as alkaloids, alcohols, drugs are also able to affect
the molecular motors.

Our attempts to construct general thermodynamic theory for active bio-continua started 1978 with a model of the skeletal muscle tissue as actin-myosin composition [2]. In 1982 we, for first time in literature [3], offered a model for externally controlled mono continuum. The rapid progress in the field of nanoscience, for the last decade, makes the idea for modeling the activity of the bio-continua on the base of molecular motors composition actual and very attractive.

The principle aim of the present study is extension of the limit of the classical field mixture theory [3, 4, 5] in a way to be able to model the biomechanical activity on tissue level.

**2. Basic principles**

The present study is based on the following principles:

1. The active bio-continuum is characterized by multiple reference state.

2. The moment reference state is dependant on the molecular motor’s activity.

3. The control on the activity of the molecular motors is subjected to Le Chatelie–Brown principle: “Under the action of external factors the system reacts in a way to reduce the effect of the external action”. So the inputted by the activation signals entropy should be non positive in order to reduce the chaotic increase caused by the external factors.

4. The activation signals are not a principal source of internal energy, but of encoded bio-information, informational entropy and consequently, informational free energy. The inputted by the activation signal internal energy is negligibly small in respect to the inputted “informational” free energy.

5. The dissipation of the energy inputted by the activation signal is a negligibly small value in respect to the dissipation of the energy associated with the mechanical work done by the molecular motors.

6. The molecular motor at each moment could be in one of the two possible discreet states – active or inactive (in excited or in reference state).

7. Each closed volume of the active bio-continuum is an open multiphase thermodynamic system exchanging with surroundings metabolic products and activating informational signals.

**3. Kinematics**

The motion of each phase is described by the equation:

(1) ,

where _{} are
Cartesian coordinates at moment _{} of the _{} phase
particles, occupying at the initial moment a position in the material
coordinate system with Cartesian coordinates_{}.

The total mass density is sum of the partial densities:

(2) .

The barycentric velocity is expressed by the individual phase velocities:

(3) , .

The relative constituent’s velocities and fluxes are:

(4) _{} .

We denote with
_{} the dominating
constituent of the tissue, which is composed by the protein tissue skeleton, the
cell fluid, the bounded to the cytoskeleton water and other immovable in
respect to the skeleton fluids. Taking into account that because of the drag the
diffusion fluxes and the microcirculatory rates should be small values, we
reformulated the relative rates and the fluxes as:

(5) _{} .

**4. Conservation principles**

The conservation principles, for the continua under consideration, are presented by the equations:

**Mass balance -**

(6)

(7)

(8)

(9) ,

where and
_{} are constituent
concentration and the respective rate of the partial mass production;

M**omentum
balance -**

(10) ;

M**oment of momentum balance -**

(11) _{} ;

**Energy balance -**

(12) ,

where _{},
_{}, , _{} and
_{} are the Cauchy stress
tensor, the heat flux, the energy density, the rate of the internal
heat supply density, and the rate of the bio-informational energy density contributed
by the control signal.

**5. Second Law of Thermodynamics**

The Clausius-Duhem formulation of the Second thermodynamic law, for
mixture of _{} constituents, is:

(13) ,

where _{},
_{}, _{} and _{} are
respectively the entropy density, the chemical potential of _{}^{th}
constituent, the absolute temperature and the rate of density of the entropy inputted
by the activation signal. With the help of Legendre transformation we introduce
the free enthalpy:

(14) ,

where _{},
_{} and _{} are
the initial mass density, the second stress tensor of Piola and the Lagrangian
strain tensor. Substituting (6, 7, 9, 10, 12, 14) in (13) we reformulate the Second
law of thermodynamics for the mixture under consideration in the form:

(15) ,

where _{}

(16)

is the rate of the density of the free energy inputted by the activation signal.

**6. Kinetic equations**

The following form for the kinetics of the molecular motors and of the biochemical reactions rate is supposed:

(17) _{}

(18) _{}

_{} ,

where _{} is
a nonnegative parameter representing the activating signal and _{} is
the concentration of the active motors (**a.m.**). The values _{} and
_{} are the first terms of
the respective Taylor
series of _{}in terms of powers of_{}.

**7. Restriction following from the Second thermodynamic law on
the constitutive equations**

As thermodynamic state parameters we assume Piola’s stress tensor - _{},
the constituent’s concentrations - _{}, the gradients of concentrations
_{}, the density of the
active molecular motors - _{} and the temperature - _{}.
So for the constitutive equations we have:

(19)

(20)

(21)

(22)

(23)

.

Taking into account (19 - 23) in (15) we obtain:

(24)

_{}
.

The necessary and sufficient conditions for the validity of the inequality (24), under the assumptions done, are:

(25)

(26)

(27) ,
_{}

(28) (by definition) .

Next, for
simplicity, we will consider the partial mass density of the tissue skeleton as
constant - _{} (there is no growth) and
the biological continuum as isothermal medium. So we obtain from (24) and
(25-28) the following dissipation inequality:

(29) .

By substituting (16 - 18) in (28, 29) we receive:

(30) .

As this
inequality is necessary to be satisfied for all admissible values of _{},
it splits to the following two independent inequalities:

(31)

(32) .

According to the basic principles (3, 4)

(33)

(34) .

Consequently,

(35) .

To illustrate the capability of the theory to model active bioprocesses in bio-continua we consider the following two cases:

__A. Application of the theory to skeletal muscle mechanics__

The contraction of muscles is realized by muscle myofibrils. Each of them
is with diameter of 1_{} and contains 2500
actin-myosin filaments, which represent nanometric molecular motor. The pathway
for the filament activation is:

a)
in the
moment at which the electric nerve signal reaches the myofibril, specific meditative
substances diffuse through the myofibril membrane. The respective penetration time
is about 1 _{};

b) the diffused mediators generate the so called action membrane potential;

c)
the action
potential is the reason for release of the bounded within the sarco- plasmatic
reticulum Ca_{} ions, which activate the
actin-myosin protein motors to contract.

Following the theoretical model under consideration and the above pathway process consequence, we consider a simple linear model represented by the free enthalpy potential:

(36) ,

where _{} is
the concentration of the Ca_{} ions and *D, S, G, R, M* are
functions of the homeostatic state variables _{}, satisfying the
stability condition * D + S + G - (2R+2M) > or = 0* . The first term in (36)
represents the internal energy due to the elastic deformation, the second the
internal energy contributed by the active molecular motors, the third the
contribution due to the Ca_{} ions and the fourth and
the fifth terms – internal energy of the couple interaction between the stress,
molecular motors activity and Ca_{} ions. Following the relations
(26-36) we have:

(37)

(38)

(39)

(40)

(41) ,
_{}

(42) _{}

(43) ,

where _{},
_{}, _{}, _{} are
respectively the one dimensional stress and strain, the concentration of the active
actin-miosin motors and the concentration of the released calcium ions. Basing
on the Onsager’s theory [3] and on the idea of the linear dependence between
the generalized thermodynamic rates and the generalized thermodynamic forces we
have:

(44)

(45)

(46) ,

(47) , .

By substitution of (44 – 46) in (42, 43) we obtain:

(48)

(49)

(50)

(51)

(52) ,

where

(53) ,

are the relaxation times responsible for transition of the molecular motors in active state and for release of the calcium ions from sarco-plasmatic reticulum. One can see that these two phenomena are conjugated as it is predicted from cell physiology and experimental data [6, 7].

In the experimental and theoretical studies, rectangular electric potential impulse is usually used as an activating signal:

(54) .

One can see two intriguing peculiarities of the model from equations (37, 51, 52): the first one is the dependence of the calcium release relaxation time on the activation signal and the second one is the presence of the stress as effective parallel activation signal. The last is experimentally existing fact and it is called myogenic effect. One can see also that the present theory is able to model different kinds of experimental features including the influence of the activation on the relaxation times and the accumulative effect of stimulation as well. The present illustrative application to muscle mechanics unifies the classical heuristic models [8-14]. By the use of this illustrative example we have the ambition just to demonstrate the ability of the theory as an appropriate analytical tool.

__B. Application to the nutrient’s transport within the lacunar-canalicular
system of cortical bone__

The passive diffusion of glucose in the lacunar-canalicular system of cortical bone was studied by Petrov and Pollack in [15]. It was demonstrated analytically that few orders are lacking in such passive transport mechanism to sustain the vitality of bone cells (osteocytes). On the basis of this analytical result and on the fact that the cell processes within the canalicular canal are rich of actin filaments, the authors offered the hypothesis for the presence in active canalicular transport. One year later, in an experimental study Takai at al. [16] concluded that the theoretical findings of Petrov and Pollack are consistent with their experimental results. Next we will demonstrate the ability of the theory to model qualitatively the active diffusion problems in one dimensional microcanal (model of bone canaliculae).

As state
variables we consider the concentration of the diffusing substance _{}and
the concentration of the active molecular motors _{}. So, following
the general theory, for the present case in it’s linear simplification we have:

(55)

(56)

(57) ,

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68) ,

where
is non negatively defined matrix and _{} is non negative
constant.

We have
physical reasons to believe that the principle part of the active diffusive
flux is depending on the active molecular motors concentration, but not on the
concentration of the diffusing constituent. That is why we substitute in the
above relations _{}equal to zero. As a
result we obtain the following equation describing the active diffusion problem
in the active bio-continuum under consideration:

(69)

(70) .

In the case of steady state active diffusion we have:

(71) .

In Cartesian coordinate system with first axis parallel to the canal axis we obtain:

(72) , .

In non activated state the value of the diffusion constant is minimal –

(73) ,

and increases with the increase of the activation signal amplitude. Other peculiarity could be seen if we consider the response of the diffusion flux in respect to variation of the activation signal as step function:

(74) , .

Solving (74) in respect to concentration of the active motors and substituting the result in (69) we obtain:

(75) .

The first term of this equation could be considered as passive diffusion part and the second - as active diffusion term. When the activation is applied as a rectangular signal, then the relaxation time, according to (74) decreases, and, in such way, assists for the quick start of the molecular motors activity; when the signal is released the relaxation time increases and supports the slow turn out of the molecular motors. It is an attractive result following from the theory. Similar result is valid for the activation and deactivation of muscle contraction.

**8****. ****Conclusive remarks **

The key moment in the present theory is the generalized formulation of the Clausius-Duhem inequality, in which the entropy carried by the activation signals is taken into account.

The capability of the new theory to model active bio-mechanical processes is demonstrated for muscle mechanics by unifying the well known classical heuristic models [6-14]. Moreover, it is demonstrated that the theory offers possibilities for active extensions of the passive classical diffusion models with application to the canalicular diffusion transport in cortical bone.

Our efforts to
find some bibliographic information about other studies concerning active
bio-continuum

model based on the assumption for molecular motors activity and activation
signal entropy influx were not successful. It seems that the offered in the
present study thermodynamic model would be the first steps in this direction.

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