We describe here the model of an adaptive syser [1], which can
modify its behavior in accordance with external environmental
change. We consider also the minisyser and compare selective
abilities of both sysers.
The minisyser (Fig. 1a) is a very simple syser, it includes only
such macromolecules, which are necessary and sufficient for
selfreproduction, namely, the matrix I , the
replication enzyme E_{1} , and the
translation enzyme E_{2} .
The adaptive syser (Fig. 1b) includes two additional enzymes:
the regulatory enzyme E_{3} and the adapting
enzyme (adapter) E_{4} . The regulatory enzyme E_{3}
recognizes the environment state and "turns on" or
"turns off" the synthesis of the adapter in accordance
with the environment changes.
Fig 1. The schemes of minisyser (a) and
adaptive syser (b). I is the polynucleotide matrix, E_{1}
, E_{2} , E_{3} , and E_{4}
are replication, translation, regulatory, and adapting enzymes,
respectively.
We suppose that there are two types of external
environment, A and B. The environment A is
a usual one, in which both sysers are able to reproduce
themselves. The environment B is unusual, in which the
macromolecular synthesis takes place only in the adaptive syser.
The regulatory enzyme E_{3} in the adaptive
syser is synthesized with a small rate in both A and B
environments, it recognizes the environment state and
"turns on" the synthesis of the adapter E_{4}
in the environment B and "turns off" this
synthesis in the environment A. The adapter E_{4}
provides the macromolecular synthesis in the adaptive syser in
the environment B.
For example, we may imply that the environment A
corresponds to the usual syser's "food", usual powerful
chemical substrate S_{A} , and the
environment B corresponds to the usually
"uneatable" chemical substrate S_{B}
, which can be transformed into "eatable food" S_{A}
by means of the adapter E_{4}. "For the
sake of economy", it is natural to synthesize the adapter
only then, when it is really needed, i.e. only in the environment
B. To recognize the environment state, the regulatory enzyme
E_{3}, which is synthesized always, but
"economically" (with a small rate), is included in the
syser structure. The scheme of this molecular control is similar
to the classical model by F. Jacob and J. Monod [2].
To describe quantitatively the syser's features, we use the
following assumptions: 1) the different sysers are placed into separate coacervates; 2) any coacervate volume is
proportional to the number of macromolecules inside it. From
these assumptions we obtain the following equations:
dN_{i }/dt
= Vf_{i} , 
V= c^{1}S_{i} N_{i}
, 
x_{i} = N_{i}
/V , 
(1) 
where N_{i} and
x_{i }are the number of molecules and
the concentration of ith macromolecules in a given
coacervate, respectively; V is coacervate volume; f_{i}
is the synthesis rate ith macromolecules; c is
the constant total concentration of macromolecules in a
coacervate (c = S_{i} x_{i}
= const); here the index i = 0 refers to matrix I,
and other i (= 1,2,3,4) refer ith enzymes (E_{1}
, E_{2} , E_{3} ,
E_{4}), respectively. We define the synthesis
rates as follows.
For the minisyser we set:
f_{0} = a_{0
}x_{0 }x_{1} , 
f_{i }= a_{i
}x_{0 }x_{2
}, 
i = 1,2, 
in environment A , 
(2a) 
f_{0} = f_{1}=
f_{2} = 0 , 
; 
; 
in environment B . 
(2b) 
For the adaptive syser we set:
f_{0} = a_{0
}x_{0 }x_{1} , 
f_{i }= a_{i}
x_{0 }x_{2 }, 
i = 1,2,3, 
in environment A , 
(3a) 
f_{0} = b_{0
}x_{0 }x_{1} , 
f_{i }= b_{i
}x_{0 }x_{2
}, 
i = 1,2,3,4, 
in environment B . 
(3b) 
Here a_{i }and b_{i
}are synthesis rate parameters. Eqs. (2), (3) state
that the matrix/enzyme synthesis rate is proportional to the matrix
and replication/translation enzyme concentrations.
From (1) we obtain:
dx_{i} /dt
= f_{i}  x_{i}
c^{1 }S_{j}
f_{j} , 
(4) 
dV/dt = V c^{1} S_{j}
f_{j }. 
(5) 
According to (2)  (4), the concentration
dynamics in a particular syser is described by nonlinear ordinary
differential equations, which were analyzed [1] by usual
qualitative methods. The analysis showed that the macromolecules
concentrations x_{i} converge to the
equilibrium stable state x^{0} = {x^{0}_{i}
}. In the environment A the values x^{0}_{i}
are expressed as (for both sysers):
x^{0}_{0} =
c a_{0 }a_{1 }D
, 
x^{0}_{i}
= c a_{i }a_{2 }D
, 
D = [a_{0}a_{1
}+ a_{2} (a_{1}+_{
}... + a_{n})]^{1}, 
i = 1, ..., n, 
(6) 
where n is a number of enzymes in a considered
syser (here we set a_{4} = 0) . For the adaptive
syser in the environment B, the equilibrium
concentrations x^{0}_{i} are
also determined by Eq. (6) after substitution b_{i}
instead a_{i }.
According to equation (5), the coacervate volume
rates are proportional to selective values:
W = c^{1} S_{j}
f_{j} . 
(7) 
Analogously to hypercycles,
we can consider the competition of minisyser and adaptive syser
explicitly, supposing that the total coacervates volume for both
types of sysers should be constant. During competition, a syser, having the maximal selective value W
, is selected. Assuming small convergence time to the equilibrium
state x^{0} and substituting values
x^{0}_{i} , determined by formulas
(6), into (7) , we obtain the selective values of the considered
sysers. For the minisyser we have
W_{Mini_A}
= c a_{0 }a_{1 }a_{2}
[a_{0}a_{1 }+
a_{2} (a_{1}+_{
}a_{2} )]^{1} , 
W_{Mini_B}
= 0, 
(8) 
in the environments A and B,
respectively. For the adaptive syser the corresponding
selective values are expressed as follows:
W_{Adaptive_A}
= c a_{0} a_{1} a_{2
}[a_{0} a_{1}+
a_{2} (a_{1}+ a_{2}
+ a_{3})]^{1} , W_{Adaptive_B}
= c b_{0 }b_{1
}b_{2} [b_{0}
b_{1 }+ b_{2} (b_{1}+
b_{2} + b_{3} + b_{4})]^{1}
.

(9) 
These expressions show, that in the environment A
, the minisyser has a selective advantage with respect to
the adaptive one, because W_{Mini_A} is
always greater, than W_{Adaptive_A }.
Such a disadvantage of the adaptive syser is due to the necessity
to synthesize always the additional regulatory enzyme E_{3}
. The disadvantage is small, if regulatory enzyme synthesis rate
is small (a_{3} << a_{1},
a_{2}). Obviously, the adaptive syser is preferable
in the environment B.
If the environment states (A and B)
are intermittent, we can introduce the effective selective values
of the considered sysers:
W_{Mini} = (1  P_{B})
W_{Mini_A} , 
(10) 
W_{Adaptive} = (1 
P_{B}) W_{Adaptive_A}
+ P_{B} W_{Adaptive_B}
, 
(11) 
where P_{B} is the
probability of the environment B . The adaptive syser
has a selective advantage with respect to the minisyser, if W_{Adaptive}
> W_{Mini} . From expressions (8)(11) we can see
that the adaptive syser is significantly preferable, if the
regulatory enzyme synthesis rate is small (a_{3}
<< a_{1}, a_{2}) and the
macromolecular synthesis rate in the environment B as
well as the probability of this environment are sufficiently
large (b_{i }~ a_{i}
and P_{B} ~ 1).
Thus, the adaptive syser does have a selective
advantage with respect to the minisyser, however not always, but
only if the "expenses", needed to support the molecular
control system operation, are sufficiently small.
Conclusion. The control system
of the adaptive syser could be the first control system, which
was "invented" by biological evolution. The adaptive
syser model demonstrates quantitatively, that the new
evolutionary invention has selective advantages, if
"invention profits" are greater than "invention
expenses".
References:
1. V.G.Red'ko. Biofizika. 1990. Vol. 35.
N.6. P. 1007 (In Russian).
2. F.Jacob and J.Monod. J. Mol. Biol.
1961. Vol. 3. P. 318.
Copyright© 1998 Principia Cybernetica 
Referencing this page