The model of evolution, discussed in Quasispecies, implies a strong assumption:
the selective value is determined by the Hamming distance between the
particular and unique master sequences. Only one maximum of the
selective value exists. Using the physical spinglass concept, we
can construct a similar model for a very large number of the local
maxima of a selective value.
D. Sherrington and S. Kirkpatrick proposed a
simple spinglass model to interpret the physical properties of
the systems, consisting of randomly interacting spins [1]. This
wellknown model can be described shortly as follows:
; 
1) There is a system S of
spins S_{i} , i =
1,..., N (the number of spins N is
supposed to be large, N >> 1). Spins take
the values: S_{i} = 1,
1 . 
; 
2) The exchange interactions between spins are
random. The energy of the spin system is defined as: 

E (S)
=  S_{i<j} J_{ij}
S_{i }S_{j },

(1) 

where J_{ij} are the
exchange interactions matrix elements. J_{ij}
are normally distributed random values: 

Probability_density{J_{ij}
} = (2p)^{1/2}
(N1)^{1/2} exp [ J_{ij}^{2}
(N1)2^{1}] . 
(2) 
From (1), (2) one can obtain, that the mean
spinglass energy is zero (<E > = 0) ,
and the mean square root value of the energy variation at
onespin reversal is equal to 2:
MSQR {dE
(S_{i} >
 S_{i}) } = 2 . 
(3) 
The model (1), (2) was intensively investigated.
For further consideration the following spinglass features are
essential :
 the number of local energy minima M is
very large: M ~ exp(aN)
, a = 0,2 ;
(a local energy minimum is defined as a state S_{L}
, at which any onespin reversal increases the energy E );
 the global energy minimum E_{0}
equals approximately: E_{0} =  0,8 N .
Let's construct the spinglass model of evolution
[2]. We suppose, that an informational sequence (a genome of a
model "organism") can be represented by a spinglass
system vector S . The evolved
population is a set {S_{k}}
of n sequences, k = 1,..., n. The
selective values of model "organisms" S_{k
}are defined as:
f(S_{k})
= exp[ b E(S_{k})]
, 
(4) 
where b is the selection
intensity parameter.
The definition (4) implies that the model genome S_{k}
consists of different elements S_{ki} ,
which pairwise interact in accordance with the random interaction
matrix J_{ij} . In order to maximize
the "organism" selective value (that is to minimize the
energy E(S)), it is necessary
to find a such combination of elements S_{i}
, that provides maximally cooperative interactions for given
matrix J_{ij} .
As in Quasispecies , we suppose,
that 1) the evolution process consists of consequent generations,
2) new generations are obtained by selection (in accordance with
selective values (4)) and mutations (sign reversals of sequence
symbols, S_{ki} >  S_{ki}
, with the probability P for any symbol) of sequences S_{k
}. The initial population is supposed to be random.
The described spinglass model of evolution was analyzed by
means of computer simulations and rough estimations [2]. The main
evolution features are illustrated by Fig.1. Here n(E)
is the number of sequences S , such
that E(S) = E in a
considered population; t is the generation number.
Fig. 1. The sequence distribution n(E)
at different generations t ; t_{3} >
t_{2} > t_{1} ; E_{0}
and E_{L} are global and local energy
minima, respectively; the global energy minimum E_{0}
equals approximately: E_{0} =  0,8 N.
Schematically, according to the computer simulations [2].
The spinglasstype evolution is analogous to the Hamming
distance case (see Quasispecies, Estimation of the evolution rate). But
unlike the Hammingdistance model, the evolution converges to one
of the local energy minima E_{L} ,
which can be different for different evolution realizations.
Because one mutation gives the energy change dE ~ 2 (see (3)), and the
typical time for one mutation per sequence dt
is of the order (PN)^{1}, the total number
of evolution generations T (at sufficiently large
selection intensity b) can be
estimated by the value T ~ (E_{0}/dE)xdt
~ (0.8N /2)x(PN)^{1}. This value
is close to the estimation of value T in the
Hammingdistance case. So, the estimations of the evolution rate
are roughly the same for both models, and we can use formulas
(1)(3) in Quasispecies to characterize
the spinglasstype evolution as well.
Analogously to the Hammingdistance case, we can consider the
sequential method of energy minimization, that is the consequent
changes of symbols (S_{i} >  S_{i})
of one sequence and fixation only successful reversals. The
sequential search needs smaller participant number than the
evolution search. Nevertheless, the evolution search provides in
average a more deep local energy minimum E_{L}
[2], because different valleys in energy landscape are looked
through in evolution process simultaneously with descending to
energy minima.
Thus, in the spinglass case, the evolutionary search has a certain
advantage with respect to the sequential search: it provides in
average the greater selective value.
Conclusion. The spinglass model of evolution
refers to the "organisms", which have many randomly
interacting genome elements. Evolution can be considered as a
search of such genome elements, which are able to cooperate in
the most successful manner.
References:
1. D.Sherrington , S.Kirkpatrick. //
Physical Review Letters. 1975. V.35. N.26. P.1792. S.Kirkpatrick,
D.Sherrington. // Physical Review B. 1978. V.17. N.11.
P.4384.
2.V.G.Red'ko. Biofizika. 1990. Vol. 35.
N.5. P. 831 (In Russian).
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