J. theor. Biol. (1992) 157, 83-94
A Stochastic Evolutionary Model of Molecular Sequences LIAOFU Luo~" AND L. E. H. TRAINOR
Department of Physics, University of Toronto, Toronto M5S 1A7, Canada (Received on 7 May 1991, Accepted on 14 December 1991) A stochastic evolutionary model of molecular sequences is proposed. The basic forces in evolution are supposed to be mutation and selection. The concept is somewhat similar to Kauffman-Levin's concept of adaptive walks and corresponding analytical expressions have been developed. The selective force is divided into two parts: a slowly-varying part and a rapidly-changing fluctuation. The latter influences the distribution of sequences and results in an equation of motion along the flow line. The former plays a more important role in the emergence of evolutionary order. It is demonstrated that the asymmetry of selective forces would lead to a definite order of the system.
1. lntroduction--A Stochastic Evolutionary Model The evolution of nucleotide sequences is one of the most fascinating problems in theoretical biology. The reason is two-fold. First, nucleotide sequences are prototypes o f life which contain, in principle, full information about the evolution of species. Second, they are one-dimensional informational systems with a four-letter alphabet, and consequently it is comparatively easy to deal with them. One key point is to explain how order emerges from an evolutionary system. Recently, several authors (e.g. Kauffman & Levin, 1987) have pointed out that the order arises as a result o f self-organization of complex systems and the selective forces play a role in further molding. They suggested that a new theory of evolution should encompass the marriage o f selection and self-organization. Evidently, this viewpoint is very important. However, to understand the problem more deeply one should investigate a concrete system. In this paper we will study a simplified model of molecular sequences which comprise an ensemble where each sequence is composed of just two symbols, A and B. It is reasonable to conjecture that the model belongs to the same "universality class" with realistic nucleotide sequences (Stein & Anderson, 1985), that is, they share many common features of evolution. The basic assumption of our model is (Luo et al., 1990) (a) Mutation via A,-~-B takes place with probability ~ in one time step; (b) Selection. Each site in a sequence with length N accepts K inputs, that is, the fitness contribution of each site depends on itself and K other sites among the N. Furthermore, the selective force, as a whole, is stochastic. The K + l sites compose a selection unit which we shall refer to as a codon. For simplicity we
, Permanent address : Physics Department, Inner Mongolia University, Huhehote, China. 83 0022-5193/92/130083 + 12 $03.00/0
© 1992 Academic Press Limited
84
L.
LUO
AND
L.
E.
H.
TRAINOR
suppose that the sequence can be divided into L = N / ( K + 1) non-overlapping selection units (codons). The total number ofcodons is 9' = 2K÷ I. For example, for K = 0 there are two independent codons, namely A and B; for K= 1 there are four independent codons, namely AA, AB, BA and BB; for K = 2 there are eight codons, namely AAA, A A B . . . BBB. To express the stochastic property of selection we introduce a stochastic fitness function S = S({ni} ) in which ne denotes the number of ith type codons in a sequence (i= 1, 2 . . . . . 9'). We assume that if the mutation in one step leads to 6 S > T+
(la)
for a given sequence then it reproduces a new identical one with a probability/3 and the number of sequences in the ensemble increases by 1. If the mutation in one step leads to SS 0 such that I¢(p, t)l _
A Stochastic Evolutionary Model of Molecular Sequences LIAOFU Luo~" AND L. E. H. TRAINOR
Department of Physics, University of Toronto, Toronto M5S 1A7, Canada (Received on 7 May 1991, Accepted on 14 December 1991) A stochastic evolutionary model of molecular sequences is proposed. The basic forces in evolution are supposed to be mutation and selection. The concept is somewhat similar to Kauffman-Levin's concept of adaptive walks and corresponding analytical expressions have been developed. The selective force is divided into two parts: a slowly-varying part and a rapidly-changing fluctuation. The latter influences the distribution of sequences and results in an equation of motion along the flow line. The former plays a more important role in the emergence of evolutionary order. It is demonstrated that the asymmetry of selective forces would lead to a definite order of the system.
1. lntroduction--A Stochastic Evolutionary Model The evolution of nucleotide sequences is one of the most fascinating problems in theoretical biology. The reason is two-fold. First, nucleotide sequences are prototypes o f life which contain, in principle, full information about the evolution of species. Second, they are one-dimensional informational systems with a four-letter alphabet, and consequently it is comparatively easy to deal with them. One key point is to explain how order emerges from an evolutionary system. Recently, several authors (e.g. Kauffman & Levin, 1987) have pointed out that the order arises as a result o f self-organization of complex systems and the selective forces play a role in further molding. They suggested that a new theory of evolution should encompass the marriage o f selection and self-organization. Evidently, this viewpoint is very important. However, to understand the problem more deeply one should investigate a concrete system. In this paper we will study a simplified model of molecular sequences which comprise an ensemble where each sequence is composed of just two symbols, A and B. It is reasonable to conjecture that the model belongs to the same "universality class" with realistic nucleotide sequences (Stein & Anderson, 1985), that is, they share many common features of evolution. The basic assumption of our model is (Luo et al., 1990) (a) Mutation via A,-~-B takes place with probability ~ in one time step; (b) Selection. Each site in a sequence with length N accepts K inputs, that is, the fitness contribution of each site depends on itself and K other sites among the N. Furthermore, the selective force, as a whole, is stochastic. The K + l sites compose a selection unit which we shall refer to as a codon. For simplicity we
, Permanent address : Physics Department, Inner Mongolia University, Huhehote, China. 83 0022-5193/92/130083 + 12 $03.00/0
© 1992 Academic Press Limited
84
L.
LUO
AND
L.
E.
H.
TRAINOR
suppose that the sequence can be divided into L = N / ( K + 1) non-overlapping selection units (codons). The total number ofcodons is 9' = 2K÷ I. For example, for K = 0 there are two independent codons, namely A and B; for K= 1 there are four independent codons, namely AA, AB, BA and BB; for K = 2 there are eight codons, namely AAA, A A B . . . BBB. To express the stochastic property of selection we introduce a stochastic fitness function S = S({ni} ) in which ne denotes the number of ith type codons in a sequence (i= 1, 2 . . . . . 9'). We assume that if the mutation in one step leads to 6 S > T+
(la)
for a given sequence then it reproduces a new identical one with a probability/3 and the number of sequences in the ensemble increases by 1. If the mutation in one step leads to SS 0 such that I¢(p, t)l _
