## Thermodynamics of living organisms

The processes of self-organisation and evolution of living organisms can be interpreted from the thermodynamics point of view. Namely, the problem was considered in the basic works of Ostwald (1931), Bauer (1935), von Bertallanffy (1942, 1952, 1956), Schrodinger (1944) and Prigogine and Wiame (1946). In this section, we shall follow the book of Ebeling et al. (1990).

To keep the organism at steady state, the energy balance must be maintained, dU/dt = 0. Nevertheless, it is not sufficient: the entropy balance must also be fulfilled, — deS/dt = dS/dt > 0. The function of metabolism in a living organism, from the thermodynamic point of view, is to "suck" the amount of entropy which is at least equal to entropy produced within the organism by means of the exchange of matter and heat with the environment.

Let d'Q be heat brought to the organism during time dt, and denk be a number of moles or molecules of kth component brought to the organism during the same time. Then d Q

T k where sk is a specific entropy of kth component. The required export of entropy is provided by the following processes: (1) heat transfer, (2) matter exchange, and (3) transformation of matter within the system. The first (heat conductivity and heat irradiation) plays the main role. For this, a certain difference of temperatures has to be maintained between the organism and its environment, Torg — Tenv = DT > 0. There are two groups of organisms with different types of thermal control: poikilotherms with DT < const and homeotherms with Torg < const. The latter can transfer a quantity of heat qorg in a unit time into the environment by means of heat conductivity. In addition, the organism can get the radiation energy qrad in the course of the same time.

If we assume radiation incoming to the organism to be radiation from a "black body" with temperature Trad, then it corresponds to the entropy inflow qSad = 4qrad/3Trad, where the factor (4/3) is Planck's form-factor, which takes into account some peculiarity of entropy transport by means of radiation (Landau and Lifshitz, 1995). Finally, also taking into account matter exchange, the entropy export is written as deS 4<?rad . / 1 1 \ V denk

A more exact balance has to take into account different types of radiation. For instance, if we consider the energy of solar radiation absorbed by plants during the process of photosynthesis, qphot, then for a more correct description we must assume the existence of different thermal components qphot with different radiative temperatures Tphot (see also Chapter 10). Summing all these contributions we get

— M < — V 4qphot . / — , 4girrad — 4gassim — y s denk dt ^ 3T' ' q°rgl t T I ' 3T 3T ^Sk dt i 3 T rad \ T env T org J 3 T org 3 T env k

Here, qirrad and qassim are the energies of radiation emitted and assimilated by the organism into and from the environment. A necessary condition of functioning and developing for the living organism is a positive balance of the entropy export (see Eq. (2.4))

where

W dAdiss _ y dt Torg^ dt dt diS 1 ( dAdiss dink 1 o

and the term dA'diss/dt is the power of dissipative forces.

The phenomenological theory of cell fission can be a good illustration of this feature (Volkenstein, 1988). The entropy balance in a cell considered as a sphere with radius R is described by the equation dS _ diS dt dt deS

We assume that the entropy production within the cell is proportional to its volume, and the entropy outflow is proportional to its surface (with a and ¡3 as the coefficients of proportionality). The cell grows until the stationary state at dS/dt = 0 with R* = 33/2a will be attained. If R > R*, then the internal entropy production does not compensate its outflow, and the cell has to die. However, if the cell is divided, then the volume is conserved but the surface increases. As a result, dS=dt again becomes negative. Since at this moment DiS = 2a(4p(R')3/3 = a4pR3/3, then the radius, R1, of two new cells will be equal to R' = R/^2. The entropy outflow for these cells is DeS = 2^4pR2 = 8)8pR2/v/4, and at R = R* = 33/a the negative increment of the total entropy is AS = 36p(33/a2) (1 - s/2) < -29.4(33/a2).

The picture will be more complex for organisms which are more complex than a cell. Theoretically, it should be expected that two processes—the total growth of biomass and cells differentiation—most strongly influence entropy change. In order to separate these effects, we consider the change of specific entropy, i.e. entropy per unit of biomass, s = S/B, where B is the total biomass:

The process of differentiation leads to a decrease in the specific entropy, since system order increases, while biomass growth corresponds to the positiveness of the derivative dB=dt. Therefore, the change of entropy of the organism is determined by a combination of negative (differentiation) and positive (growth) terms, and a monotonous dependence of the entropy change on time should not be expected.

Returning to Eq. (5.6), we can rewrite it in a more general form. For this, we write the entropy as S = psV where p is the biomass density and V is the organism volume, so that B = pV. We assume again that diS/dt = aV and deS/dt = bF, but now the values of a and b are not coefficients. Indeed, since (1/V)diS/dt = a is nothing else than the specific entropy production, in accordance with Prigogine, da/dt # 0, a(t)! a* when t As before, b is assumed constant. We also assume that volume Vand surface area F are connected by the relation F(F, V) = 0. For standard, simply-connected geometric figures with central symmetry, where volume and surface area are determined by a single characteristic size, F = fVy where y = 2/3. For instance, for a sphere with radius R we have V = (4/3)pR3 and F = 4pR2 then y = 2/3 and f = (4p)1/3(3)2/3 < 4.84 and for a cube with side l we have V = l3 and F = 6l2 then f = 6 and y = 2/3, too. By combining Eqs. (5.6) and (5.7), we get (p = const)

We assume that there are the limit values of volume, V *, specific entropy production, a* = limt!1 a(t), and specific entropy, s* = limt!1 s(t), so that (1) we can neglect by the term (V/s)ds/dt for large t, (2) we can express the value of V * as a function of parameters: (V*)1-y = bf/a*. By denoting a new variable, z = V/V*, we can write the asymptotic analogue of Eq. (5.8) as follows dz dt

It is easy to see that the derivative of z is negative within the interval (0,1); this implies that the equilibrium with z* = 1 is unstable! This means that, in the process of growing old, organisms do not reach their final stationary state, where the constant rate of entropy production would be observed. The latter means death for living organisms. So, we have proved (thermodynamically) the impossibility of infinite life for any organism. It is interesting that if we use the Kostitzin-von Bertallanffy (Kostitzin, 1937; von Bertallanffy, 1956) growth equation based on the conservation law of energy, we would get another result.

Let the energy contained in an organism be equal to E = peV where e is a specific energy per unit of biomass. The energy balance is dE

The main assumptions in the model are the following: (1) input of energy is proportional to surface area of the organism, Qem = epF, (2) metabolism M is proportional to its volume (or biomass), M = rpV, where r is a specific metabolism per unit of biomass. Then Eq. (5.10) can be represented as (p = const, e = const)

dt e

It is easy to see that if r(t)! r*, i.e. to the stationary level of basic metabolism, then Eq. (5.11) has the equilibrium V* = 1-ysf/r*, and it is stable! From this follows the so-called allometric principle:

where r * is, for instance, the intensity of oxygen uptake per unit mass, and W * is the total mass or weight of the organism. The principle is very popular in physiological ecology (see, for instance, Pianka, 1978).

Finally, we consider one curious example. Let there be two organisms which differ from each other only by the geometry of the surface across which energy transports into organisms: values of form-factors, f1 and f2, and such a geometric index as y: y1 and y2. Note that y = 2/3 for figures of simple shape. Let V* > V* then the following inequality must hold: f2(1 — y1) > f1(1 — y2). Assume that the organism strives to increase its size (mass, weight, volume) in the process of its evolution. As is seen from the equality, there are two strategies to realise the aim: (1) to increase f and (2) to increase y (of course, a mixed strategy is also possible). In order to realise the first strategy, the organism has to increase its form-factor, for instance to transform from cubic to spherical. Aristotle could be right when he spoke about pre-humans as spherical figures. This could be a reason that we do not observe living organisms of cubic shape.

The second strategy is a principal complication of surface. For instance, if the surface is a fractal with dimension greater than 2 (but less than 3), then y > 2/3. Let us look at lungs of superior animals: this is a typical fractal structure.