We present a simple model for biological aging. We studied it through computer simulations and we have found this model to reflect some features of real populations. Keywords:Aging, Monte Carlo Simulationsto appear in Journal of Statistical Physics * Electronic address: gfitjpp@vmhpo.uff.br 1The problem of aging has been studied recently [1] also by computer simulations, useful in understanding how the survival rates for younger and older individuals evolve in time and affect the preservation of the species [2]- [4]. The lowering of survival rates as time goes by is called senescence. Mutations play an important role in senescence modifying the survival rates either increasing them (helpful mutations) or decreasing them (deleterious mutations). In this paper we introduce a simple model for aging using the so-called "bit string strategy". This approach has been applied to other biological systems [5]. Our model is particularly suited for implementation in computers, although analytical results may be obtained providing some approximations. Since it is based on Boolean variables, bit-handling techniques have been used [6]. A complete description of our code will be presented elsewhere [7].Let us consider a population of N(t) individuals, at the time t, each one characterized by a genome which contains the information when a mutation will occur in the lifetime of a given individual. We consider the time as a discrete variable (t = 1, 2, . . . , B) as suited for implementation on computers. Hötzel [8] simulated the largest number of ages simulated up to now (five ages). We denote each time step as one generation since births can occur at each time step. Here, we will treat only hereditary mutations, although somatic mutations can be incorporated without great additional effort. Hence, a genome is a string of B bits defined in the birth and kept unchanged during the individual lifetime. The genome is built as follows: if an individual has the i-th bit in genome set to one it will suffer a deleterious mutation at age i. The parameter T represents a threshold, i.e., the maximum number of deleterious mutations that an individual can suffer and stay alive. In order to include the effect of food and space restrictions and to keep the population within the computer memory limits, we imposed the age-independent Verhulst factor. Hence, an individual which has passed the threshold test only stays alive with a probability (1 − N(t)/N max ). The next step is the birth: an individual whose age is larger than the reproduction age R generates one baby. As far as we know, this is the first model for biological aging where the reproduction age appears as a parameter. Although sex is not a difficulty neither for us nor for our model, we chose to work with asexual populations, for the sake of simplicity -sexual reproduction can be introduced, for example, by mixing the bit strings of two individuals (crossing over). Thus, the baby's genome will be the same as the parent, except by a fraction M of 2 randomly changed bits (mutat...

We propose a new Monte Carlo technique in which the degeneracy of energy states is obtained with a Markovian process analogous to that of Metropolis used currently in canonical simulations. The obtained histograms are much broader than those of the canonical histogram technique studied by Ferrenberg and Swendsen. Thus we can reliably reconstruct thermodynamic functions over a much larger temperature scale also away from the critical point. We show for the two-dimensional Ising model how our new method reproduces exact results more accurately and using less computer time than the conventional histogram method. We also show data in three dimensions for the Ising ferromagnet and the Edwards Anderson spin glass.In the simulation of thermal systems one typically needs to calculate a variable < Q > T as a function of temperature T . Usually one would have to perform independent Monte Carlo simulations at different values of temperature. An appealing strategy to avoid such multiple simulations is the canonical histogram method [1,2]: The histogram P T (E) of the energies at one given temperature T 0 is measured and then the distribution at a different temperature T is obtained by reweighting, i.e. by multiplying with exp(E/T − E/T 0 ) and normalizing. In order to obtain the average < Q > T , one needs to accumulate also the histogram of Q as a function of energy E. The thermal average at temperature T is thenwhere < Q(E) > means the average value of Q obtained at fixed energy E. For simplicity we set k B = 1 in this article. The histogram method has been carefully checked on various models [3]. Its main disadvantage is that the canonical distribution P T (E) of the energy is rather narrowly peaked around the average value < E > T 0 (and the more so the larger the system) so that when T is not close to T 0 the reweighting factor is very small there and very large near < E > T . One also has strong statistical fluctuations stemming from the tails of the distributions. So, in order to get reliable results the tails must be sampled very well by making good statistics and the temperatures considered should not be far from T 0 .We want to introduce here a completely different technique based on the calculation of the degeneracy g(E) of energy states from histograms of adequate macroscopic quantities defined below [4]. Figure 1 compares the classical canonical histogram for an Ising model on 1

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