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## Coarse grained population dynamics

We have seen so far that the presence of different time-scales is a typical situation in out of equilibrium systems. We will show how this feature will turn out to be very useful to extend our notion of temperature in the context of population dynamics far from equilibrium.

Let us now study the following situation. Consider a population in which the possible mutations can be grouped into two subsets, say A and B, each one containing several lineages. Mutations between lineages will happen with a certain average rate, 1/τ, and every time an individual will mutate, it will do to a lineage belonging to its current subset with probability p, and to one of the other subset with probability 1 − p. For simplicity, let us assume that the fitness is totally epistatic, and mutations can connect all lineages indifferently. Let us indicate lineages of group A with the label ai, and those of group B with bi. In a given state ai we will have a number mai of individuals, with fitness λai . The total number of individual is M, divided in the two subsets containing MA and MB = M −MA individuals respectively.

Such a situation can be thought of as referring to a situation in which a bi-allelic gene is responsible for choosing between two metabolic pathways, A and B, and each one of these can manifests itself in several different ways (the ai and bj). Or, and this is the situation in which we will concentrate on, we will consider this separation as originated by time-scale separation, with states in A being temporally far from state in B. Or one can think as A and B being two ecological niches, and individuals in one can mutate or migrate into a different niches, where their fitnesses change due to the different environments. This situation is probably the easier one to understand, and for a moment we will refer to A and B as niches, and ai and bj as species. Suppose that the mutation rate U = 1/τ is small, so that eq. 1.34 is satisfied for every difference in fitness between any species. Then the dynamics will satisfy detailed balance as described before in subsection 2.1.1, and the population will cluster in one or the other lineage, changing from time to time the niche, all this accordingly to the relative fitnesses of the species. But let us now consider a much higher mutation rate, or smaller τ , with p ∼ 1. Inside the two niches, we will have that the successive mutations regime does not hold any more, and clonal interference and polymorphism will be present.

However, mutations/migrations between the two niches will keep being rare, with a small rate of mutation U = (1 − p)/τ. Then what if we coarse grain our system, and we treat the two niches as two species, with their fitnesses being given by some average on their internal organizations? Let us show that, with proper definitions and caveats, one can get again detailed balance for these coarse grained competing mutants.

### Boolean satisfiability

A boolean variable x can assume only two values, TRUE or FALSE (resp. 1 or 0). We have three main binary operations, and(∧), or (∨) or xor (⊻). They return a TRUE result, for two variables x and y, only if: (∧) both x and y are TRUE; (∨) at least one is TRUE; (⊻) one and only one variable is TRUE. There is also the not(¬) operator, that invert the logical value of a variable. We will refer to ¬x as the conjugate of x. A combination of variables and their conjugates by means of these operators is what is called a boolean expression. The problem of finding a given set of variables xi so that a given expression is solvable, i.e. its final value is TRUE, is referred to as satisfiability problem. In the case in which it is possible to find such a set of variables, the expression is defined to be SAT (satisfiable), otherwise it is classified as UNSAT.

We will define clause of size k an expression that contains k variables, related by means of one kind of operator only. Let us concentrate on disjunctive clauses, that is clauses containing only ∨. More general expression can be obtained by relating this disjunctive clauses by means of conjunctive operator, namely ∧.What we have is referred to as the conjunctive normal form, CNF, which takes the general form: ^N i=0 _ki j zji.

#### Test of detailed balance

Crooks relation We have seen that our system decorrelates in time of the order t. We can interpret this time as the typical time it takes to the population to escape a metastable state of relative high fitness. Near the glass transition, when t ≫ τ , we can easily distinguish a fast evolution, which corresponds to the competition of simultaneously present lineages, happening close by in the fitness space , and a slow evolution, corresponding to the slow motion of the entire population in its different current lineages, all across the fitness landscape. In such a two-fold time-scale evolution, we can apply the coarse-graining mechanism on fast evolution that we have described in subsection 2.1.3, get rid of the fast dynamics, and show that the slow remaining degrees of freedom are in detailed balance. In particular, we will show that there are cases in which detailed balance does not hold between populations separated in time, but holds when the average value of these is considered.

**Phenomenology of population thermodynamics**

In this final part of the chapter, we will exploit the thermal correspondence to give new insights on the evolution of populations. Indeed, now it should be clear, for example, that bottlenecks in a population will give the same consequences as temperature spikes in glasses. Moreover, we can suggest that memory effects can arise, when a population is restrained to a size at which it was constrained for a long period before. Or we can figure out how to engineer sets of chemostats in order to speed up the evolution of bacterial colonies in an experiment. And we can understand how a fluctuating environment can limit the increase of fitness to a given threshold.

**Rejuvenation in fluctuating environment**

The last aspect we will investigate in this context of thermal populations is the property of rejuvenation. We have seen that dynamics in glasses is dominated by extremely slow relaxations, due to the difficulty of optimizing in an extremely rugged landscape, where the huge number of metastable states gives rise to entropic barriers difficult to overcome. In general, the optimization problem is extremely hard to solve, and equilibration takes place in extremely long, practically infinite, time-scales.

But what will happen if the time available for the system to relax is finite? Indeed, let us consider the case where the energy landscape is not fixed, but it is instead slowly changing. After some time, we can expect that the landscape will be totally different from the original one, and so the configuration acquired by the system when relaxing in the old landscape will result to be totally unoptimized in the new setting [77]. In other words, relaxation has to start again. It is not surprising to see that the system behaves as if it did not age at all during the previous time. This property is called rejuvenation. If there is a continuous change in the landscape, we expect the t necessary for relaxation to be bounded. Indeed, if the landscape decorrelates on a time tenv, the configurations of lower energy will typically be decorrelated in that time as well, and thus the system, by trying to optimize and reach the ground states, will decorrelate on time of the order of tenv. And since we know that the age of the system tw is of the order of the relaxation time, but relaxation is essentially reset every tenv, we see that a glass will not be able to be older than tw ∼ tenv. Let us stress that even if several different glasses will show the same age, their properties will be different. Indeed, a glass that would have aged more in a static environment will have slower dynamics, and its relaxation will be less efficient in reducing the energy. We will come back to this issue in a moment. We will investigate rejuvenation in our XORSAT population dynamics context. The change of the fitness landscape can be seen as a fluctuating environment, where different external parameters concur in changing the fitness of an individual (climate changes, food resources, incoming diseases, and so forth). We will implement the change in our simulations by modifying the XORSAT statement that provides (the logarithm of) the fitness function for an individual, eq. 2.17. Every time interval t, a given number of variables n in the clauses are changed randomly, by inverting their logic value. On average, the fitness landscape will then decorrelate on times of the order of tenv ∼ 3αNt/n. As a measure of the decorrelation time, we can use the t, computed by looking a the fluctuations of the correlation, χ4, as already described in subsection 2.2.2.

From what we said before, we expect t ∝ tenv, which is exactly what fig. 2.13 shows. A population in a rapidly fluctuating environment will decorrelate faster than a population in a quasi-static one.

**Table of contents :**

Definitions

**I Well mixed **

**1 Adaptation **

1.1 Stochastic evolutionary models

1.1.1 Moran model

Genetic drift

Constant selection

1.1.2 Wright-Fisher model

1.1.3 Galton-Watson model

1.2 Time-scales

1.2.1 Fixation times

Fixation time in presence of selection

Fixation time in presence of genetic drift

1.2.2 Comparisons

A first comparison: selection vs drift

A second comparison: mutation vs selection

1.3 Speed of adaptation

The Gaussian shape of the travelling wave

1.3.1 Successional mutations

1.3.2 Clonal interference

1.3.3 Concurrent mutations

More involved calculations

1.3.4 Infinite population size

**2 Population Temperature **

2.1 Population as a thermal system

2.1.1 Detailed Balance

2.1.2 Glasses

Real glasses

Correlation

Trap model and ergodicity breaking

Canyon model

2.1.3 Coarse grained population dynamics

2.2 Population glass

2.2.1 Boolean satisfiability

Physical interpretation

2.2.2 M population, 1/M temperature

Correlations and the glass-phase transition

2.2.3 Test of detailed balance

Crooks relation

A test of the Crooks relation

2.3 Phenomenology of population thermodynamics

2.3.1 Kovacs effect

Memory effects

2.3.2 Parallel tempering

2.3.3 Rejuvenation in fluctuating environment

**3 Coupled models **

3.1 Fitness REM

3.1.1 REM

3.1.2 A single model

3.2 Interacting models

3.2.1 Two systems

A time-scale separation mechanism

Limited jumps

3.2.2 More systems

**II Space and Mutation **

**4 Space as a limiting resource **

4.1 Constant population size in space

The wandering island of individuals

4.2 Freely growing population

4.2.1 Drift in space

Linear front

Circular front

4.2.2 Deterministic growth

Linear growth

Circular growth

**5 The selective effects of mutation **

5.1 Mutational Eden Model

5.1.1 Active and inactive phases

5.1.2 Front properties

5.2 Selective advantage of stability in well-mixed

5.3 Selective advantage of stability in space

5.3.1 Speed of growth

Saturation effects

5.3.2 Mutation rate evolution

**Conclusions **

**A Computations of the entropy **