PHYS3410 BIOPHYSICS II
Lecture Notes

Section I: Thermodynamics of Biological Systems

Lecture 6: Entropy and Life

Consider a cell: not a closed system

Important parameters:

(i) Entropy
(ii) Entropy production

Entropy change dS consists of entropy due to dissipative processes, d_iS (chemical reactions, contractile processes, conformal changes in proteins) and exchange processes with the environment, d_eS (heat exchange, heat flow, material flow, charge exchange, volume change).

The d_eS term can be negative or positive, but the d_iS term can be greater or equal to zero.

The entropy density

In non-equilibrium system, the local entropy production can be defined as entropy density s

   (6.1)

For multitude of chemical reactions:

   (6.2)

In eqn. (6.2) the entropy production is a sum over all the chemical reactions. Thus there may be some reactions, which produce negative entropy and these are “coupled” to ones with positive entropy production.

Diffusion

Living cells continually exchange materials with the environment or between various compartments within the cell. Therefore, diffusion processes must be included in derivation of entropy production.

Considering diffusion allows us to introduce “div” and “grad” formalism, which provides an elegant short-hand and also some useful general identities.  The change of concentration c of substance i in arbitrary volume V, can be expressed:

          (6.3)

If the substance i is involved in chemical reaction

     (6.4)

where n_i is the stoichiometric coefficient

A similar analysis may be applied to local changes in entropy. Define local entropy density s_v. Then change of total entropy with time:

      (6.5)

The flow of entropy J_s is related to the exchange of entropy with surroundings

          (6.6)

The total entropy production due to irreversible processes

           (6.7)

For a local change in entropy

           (6.8)

This expression makes possible to isolate s from the total change in entropy and evaluate the dependence of s on flows and forces.

The flow of energy, J_w, may consist from a flow of pure heat and the energy carried by substances diffusing in or out of the cell. Experimentally, these are difficult to distinguish.

  (6.9)

Thermodynamic forces

The scalar forces, such as chemical Affinity, need no further clarification, but spatial characteristics must be assigned to vectorial forces. Consider the chemical potential as a function of space coordinates, obtaining equipotential surfaces. The maximal change (vector perpendicular to the equipotential surfaces) is the gradient (grad).

A driving force F occurs, whenever a difference in potential exists and its direction is that of maximal increase in the potential.

F = -grad m

Total entropy production and the Dissipation Function

The entropy production in open system, such as a living cell, can be written:

            (6.10)

To include the flow of ions, which is very important for living systems, we have to extend the chemical potential into electrochemical potential:

                (6.11)

Ñ partial molar volume
m_o standard chemical potential dependent only on temperature
z valency (ions)
F Faraday constant (~96450 coulombs/mole)
y electrostatic potential

Eqn (6.10) gives the local rate of entropy density production for the case of heat, matter, ion and chemical reaction flows. The entropy production is given in the form of flows J_i and their conjugate forces F_i:

              (6.12)

It is sometime convenient to use another form of eqn.(6.10), which is multiplied by T. Lord Rayleigh called this form the Dissipation Function.  It has dimensions of free energy per unit time and is a measure of the rate of local dissipation of free energy by irreversible processes:

            (6.13)

Note that here instead of heat flow, we have entropy flow driven by grad (-T). The flows of matter, ions and chemical reaction remain unaltered, but their driving forces assume more familiar form.

Example

Stationary state

Biological systems are not in equilibrium, but often are in steady (stationary) state: independent of time. The time-independence of thermodynamic state variables yields, from eqns.(6.4), (6.9) and (6.8) the following conditions for stationary state:

                    (6.14)

The dividing line between living and inanimate systems is often difficult to define. In living systems, many reactions are integrated into harmonious whole. The thermodynamic analysis of irreversible processes can give us some useful insight into the overall behaviour of living systems.

The stationary state, of which equilibrium is but a special case, is the end point of any evolutionary physical process. The rate of entropy production assumes a minimum in a stationary state. However, there are forces, which do not vanish in stationary state: these can be regarded as constraints acting on the system. If the constraints are removed, the rate of entropy will decrease further: