2. LOS HUI: UNA MINORÍA A LA QUE PERTENECEN 20 MILLONES DE PERSONAS.
2.2. El problema de la desigualdad y la indiferencia.
For the sake of simplicity, we assume the following:
, , j j j j k k k j j k k yµ > y′ ′µ z µ < z′ ′µ
∑
∑
∑
∑
(9.9) . j j k k j k j k j k y µ + z µ = y′ ′µ + z′ ′µ∑
∑
∑
∑
(9.10)This means that the Gibbs free energies produced from (ii) supply are all used for (iii) reproduction. From this situation, large energy and negative entropy are produced and they become the energy and entropy for activation. That is, so-called negative entropy has to be interpreted as activation entropy. In order that smart give and receive of the Gibbs free energies can be performed, it is not possible if the activated complex of the
reacting components of the materials of (i), (ii) and (iii) takes common feature with large entropy, but it can be possible if it makes the organization with small entropy. Here is the role of proteins. Therefore, such materials like proteins thermodynamically collapse and they proceed the reactions of Equation (9.5) by give and take of the Gibbs free energy before their collapse, and constantly reproduce the materials with holding the materials that collapse are going to collapse. We have to interpret the concept of negative entropy in this meaning. The point of view of Schrödinger [147] and Brillouin
[148] has a danger to recognize the negative entropy as a permanent existence.
According to Equation (9.10), the consumption (or exhaustion) per second asso- ciated with qs is given by
(∑
ixisµ
i−∑
ixis′ ′µ
i)
qs. Therefore,. K is i is i s s i i G = x µ − x′ ′µq
∑ ∑
∑
(9.11)On the other hand, the reproduction per second is given by . ks k ks k s s k k M= z′ ′µ − z µ q
∑ ∑
∑
(9.12)Now when the system is in the stationary state, we have to hold the following relation:
.
K
M =G (9.13)
As is discussed in the section VII, Motoyosi Sugita applied the method of maximum principle of Equation (7.16) to the above problem. Assuming that M =cont. and
giving the variation of
δ
qs, we seek for the condition that GK becomes the maximum. Let us define by λ a Lagrange multiplier (i.e., unknown parameter of Lagrange). We obtain 0. is i is i s s is i is i s i i s i i ks k ks k s s ks k ks k s k k s k k x x q q x x z z q q z z µ µ δ δ µ µ λ µ µ δ λ δ µ µ − ′ ′ + − ′ ′ ′ ′ ′ ′ + − + − = ∑ ∑
∑
∑
∑
∑
∑ ∑
∑
∑
∑
∑
(9.14) Now the relationship between∑
ixisµi−∑
ixis′ ′µi and qs is given by Equation (9.6). We can assume that Equation (9.11) can be written in the quadratic form in the first approximation as was discussed before [Equation (7.41) in Section 7]. Within this assumption, for each s the following relation has to be satisfied0. is i is i ks k ks k i i k k x µ − x′ ′µ λ+ z′ ′µ − z µ =
∑
∑
∑
∑
(9.15)Here multiplying qs for both sides of Equation (9.15) and take the sum for s, we find 0. is i is i s ks k ks k s s i i s k k x µ x µ q λ z µ z µ q − ′ ′ + ′ ′− =
∑ ∑
∑
∑ ∑
∑
(9.16)Comparing Equation (9.16) with Equation (9.13), we obtain
λ
= −1. Hence, we, is i is i ks k ks k i i k k x µ − x′ ′µ = z′ ′µ − z µ
∑
∑
∑
∑
(9.17) or . is i is i s ks k ks k s i i k k x µ x µ q z µ z µ q − ′ ′ = ′ ′− ∑
∑
∑
∑
(9.18)This is the detailed balance equation in the process of the chemical reaction s. From Equation (9.18) together with the help of Equation (9.1'), if we can know the ratios of qs between different s, then we can know the relationship between
µ
i. Theseµ
i are the chemical potentials in the living state, and therefore, they usually cannot be measured biochemically; i.e., they are the quantities that dominate the living function biochemically.To understand this point, let us assume that there is a material K such that it is produced by the rate of z qks and it is consumed by the ratio of x qjkj. In the
stationary state, the balance of income and outgo of the material K is given by .
k k kj j
j
z q =
∑
x q (9.19)As is shown in Figure 8, suppose that z qkk occurs as a combination with x qkjk. If
Figure 8. Schematic diagram of combined chemical reactions in metabolism of life. Ai stand for the species of materials.
the difference of chemical potential in each reaction is written as ∆
µ
k , then corresponding to Equation (9.18) the balance of Gibbs free energy is given by. kj j k k k k j x
µ
q zµ
q ∆ = ∆ ∑
(9.20)Or if we use Equation (9.19) then
. kj j k k jk j j j x
µ
qµ
x q ∆ = ∆ ∑
∑
(9.21)Motoyosi Sugita points out that if we write as
kj k kj
x q =q (9.22)
then Equation (9.21) has the same form of the equation of simple reproduction in economics, where GK corresponds to the utility function in economics. And hence he suggests that once we can establish the analogy between biological phenomena and economics, there appears the way to use such mathematical analysis to apply to biology. Similarly to what was discussed on Equation (9.18), what we can say about Equation (9.21) is the following: If we are able to know the ratio between each reaction rate in the metabolism in the body with keeping the life body in the stationary state, then we are able to find the relation between the chemical potentials. From this we are able to measure the negentropy in the living state by the kinematical method. On the other hand, if we want to measure it by the usual chemical method then we have to make the living system in the equilibrium state and hence inevitably we have to kill the life. However, if we seek for the inter-relationship between the chemical rates with keeping the living chemical reactions, then we can measure the free energy of the system that is in the living state.
As a special example for the above theory of Motoyosi Sugita, he discussed the combined chemical reactions of ATP and proteins. Suppose that all protein reactions are averaged for all proteins conceptually such that we can assume that there is only one protein. Denote by x qPAP the speed of decomposition of ATP to produce the protein, where the Gibbs free energy of ATP decomposed for the production of protein is given by ∆
µ
A PA Px q . On the other hand, denote by x qAPA the speed of de- composition of protein to produce ATP, where the Gibbs free energy of the protein decomposed for the production of ATP is given by ∆µ
Px qAPA. Then in the stationary state we have,
AP A P PA P A
x q ∆
µ
=x q ∆µ
(9.23)where ∆
µ
P (∆µ
A) is the chemical potential difference between decomposition and reproduction of protein (ATP) (see Figure 9). Therefore, if we can know the value ofA
µ
∆ then we can know ∆
µ
P, vice versa. The free energy of proteins manufactures the complex organization of life and it makes the smart deliver and receive of the free energy possible. Therefore, once it is cut down to the outside, then the meaning ofP
µ
∆ is lost. Now, according to experiments the function of protein depends on the speeds of production and decomposition.
Figure 9. Schematic diagram of combined chemical reactions of Protein and ATP in metabolism of life.
Thus, when we think in this way, the negentropy (=negative entropy) is not permanently existing, but constantly produced by the reaction (iii), and constantly increasing by the reaction (i). If one doubts why entropy cannot increase in life body, then one forgets the combination of reactions (iii) and (i). Even though ni =0, as long
as qs≠0, the local production of entropy is carried out in life body.