Evolutionary adaptation is usually described as the product of a directional process involving the selection of hereditary variations. The nature and origin of these hereditary variations, and their relationship to the environment in which the organism lives are obviously of fundamental importance for understanding this and other evolutionary processes. It has been customary to assume that the variations are DNA variations, and that their origin is random with respect to the selecting environment. Today both of these assumptions are being questioned. Some variations in DNA, such as phase variations in bacteria, are locus-specific and have very high transition rates between a limited number of locus-specific states (Robertson and Meyer 1992, Moxon et al. 1994). Moreover, developmentally and environmentally induced changes in DNA sequence are recognized in both unicellular and multicellular organisms (Volume 8, issue 12, of Trends in Genetics 1992 has been devoted to the description of some such systems; see also Schneeberger and Cullis 1991 ). Induced mutations and locus-specific mutation rates are, of course, not necessarily adaptive. The role of adaptive mutations in bacteria is at present a hotly debated subject in evolutionary genetics (for opposing views concerning this issue see Foster and Cairns 1992 and Lenski and Mittler 1993). However, the existence of non-classical DNA variations such as the high, locus-specific mutation rates found in many pathogenic microorganisms, and the developmentally regulated changes in DNA sequences as, for example, those studied in yeast, raises the problem of the evolution of such systems.
Some heritable variations may not involve a change in DNA sequence (Holliday 1987; Jablonka and Lamb 1989 ; Jablonka, Lachmann, and Lamb 1992 ). Such heritable variations are normal in complex multicellular organisms, where different cell types are generated from a fertilized egg. The various determined states are stable and are transmitted through many cell divisions, in the absence of the stimuli which originally induced the differences. Inheritance systems additional to the system of DNA replication must operate to maintain the stability of these determined states. Epigenetic inheritance systems (abbreviated EIS by Maynard Smith 1990 ) are responsible for the inheritance of the functional states of genes and cell structures in cell lineages. Several types of cellular inheritance systems are known: in the chromatin marking system chromatin marks such as DNA methylation patterns, or patterns of proteins associated with DNA are carried and transmitted on the chromosomes through cell divisions; in a steady-state system inheritance is based on the self-perpetuating properties of reactions involving positive transcriptional self-regulation; in the structural inheritance system a 3-dimensional structure is used as a template for the same structures in daughter cells (for a discussion of the different systems see Jablonka, Lachmann, and Lamb 1992 ). The epigenetic variations may be random with respect to the environment (these were termed epimutations by Holliday 1987 ), or they may be induced by the environment. With a chromatin marking EIS, the number of variant heritable states that a locus has is specific to the locus. In both cultured cell lines (Holliday 1987 ; Harris 1989; Meins 1985 ), and in unicellular organisms (Jollos 1921 ; Nanney 1960; Pillus and Rine 1989) hereditary variations which were initially assumed to be classical mutations, were later shown to be epigenetic variations. Inheritance of epigenetic information therefore does occur, and information can be acquired and transmitted in ways that do not involve changes in DNA base sequence.
In addition to genetic and epigenetic inheritance, information can be transmitted via behavioral channels. Social learning, which includes various learning mechanisms, allows transgenerational transmission of behavioral phenotypes in both birds and non-human mammals (Galef 1988). In humans, cultural evolution is driven by behavioral transmission mediated by language, and is a major factor in the evolution of individual and social behavior (see for example, Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985).
The existence of epigenetic variations, transmissible behaviors and non-classical DNA variations, raises the question of the evolution of the inheritance systems underlying them. Under what circumstances is there an advantage in transmitting information acquired by parents to progeny, and how is the fidelity of the transmission related to environmental conditions? Much of the information acquired as a response to transient environmental stimuli may be irrelevant or deleterious if transmitted to progeny, for it may be caused by factors such as accidents, injuries, and aging. Are there any circumstances in which it is advantageous to inherit a response rather than depending on a renewed response to a stimulus?
The inheritance of a phenotypic response will be advantageous in a changing environment when (a) the environmental conditions which favor this response last longer than the generation time of the organism; (b) there is a lag period before the adaptive response is manifest, and this lag causes a selective stress on the organism. (This could happen either because the stimulus for transition is rare or absent, or because the transition to an active state takes a long time). Phenotypic transmission can be advantageous under such circumstances, for example when the environment periodically fluctuates.
We shall define a periodically fluctuating environment with a cycle length that is longer than the generation time of the organism, but not so long as to allow adaptation through the fixation of classical mutations, as an Intermediate Length Cycle (ILC) environment. Typical periodic cycles are seasonal fluctuations, diurnal fluctuations, and fluctuations in population density, which may be due to intra-population dynamics, or to interactions between different species such as host-parasite interactions. Organisms that transmit their adaptive functional state to progeny in this type of fluctuating environment will have an advantage. For the progeny of such organisms, some of the cost of being transiently in a non-adaptive state is avoided. It is clear that the periodicity of the environmental fluctuation will determine the evolution of the transition rate from one phenotypic state to another.
In this paper we discuss the evolution of the rate of heritable variations in asexually reproducing organisms, in a periodically fluctuating environment, under non-inducing and under inducing conditions. Our basic model is similar to previous models describing the evolution of spontaneous mutation rates (e.g Kimura 1967 ; Levins 1967 ; Leigh 1973 ; Lieberman and Feldman 1986 ; Ishii et al. 1989 ), but we suggest a new interpretation of the model's results for ILC environments, and we describe a model for inducible heritable mutations. We base the interpretation of our models on the properties of transmissible behavior, EISs, and non-classical mutations. The properties of these systems permit the following assumptions: (i) there is a high probability that the states transmissible will be adaptive in specific environmental conditions; (ii) the rate of transition from one state to another may be either spontaneous or induced by specific stimuli; (iii) the range of rates of transition between variants is expected to be large, varying from a very high rate to a very low rate -- comparable to the rate of classical mutations; (iv) the rate of transition (for example, the change in the pattern of methylation of a locus) will be intrinsic to the locus or to the phenotypic state. This latter feature means that it is not necessary to assume the existence of a mutator (or modifier) locus which affects the rate of variation in the whole genome. Consequently, in our model, as in some previous models (for example that of Leigh 1973, Ishii et al. 1989, Moxon et al. 1994), a high transition rate in one locus or for one phenotype does not increase the genetic load in the population through the accumulation of deleterious variations elsewhere in the genome. The independence of the rate of transition in one locus from the transition rates in other loci means that in our model, an evolutionary stable transition rate for a locus is identical to its optimal transition rate. The models we describe are particularly applicable to the evolution of phase variations in bacteria, and to the inheritance of phenotypes through behavioral transmission.