Our model of a fluctuating environment is based on an overdamped particle in a potential. The position

*x* is some generalized coordinate that describes the dominant genotype in the population, and hence a change in

*x* is a fixation event. This genotype changes according to an equivalent Langevin dynamics given by

where

*U* is the potential,

*γ* is the “friction”-like scale factor,

*η* is a white Gaussian noise with variance 2

*D*, where

*D* is the intrinsic diffusivity, presumably related in the population biology context to the population size [

8]. Notice that in the usual physics language, the process will minimize the potential so that

*U* is the negative fitness. Motion to minimize

*U* represents fixation of beneficial mutations, while Langevin noise allows low probability fixation of neutral and deleterious mutations. The first phenomenon is called natural selection/drift in evolutionary/physics languages, and the second is unfortunately referred to as drift/diffusion, respectively. To avoid confusion, in the remainder of the article we use the physics terminology.

We write the potential as

and we focus on the following range of parameters:

This models the emergence of novel functions in a population. Namely, the fitness is largely independent of time, as described by

*U*_{0}. However, small temporal changes in fitness are allowed. For example, acquiring a new enzyme is generically advantageous if its substrate is present, but deleterious if it is absent due to generic costs associated with protein overproduction [

16,

17]. We model this by adding a small fitness component Φ

*(x)* that fluctuates as

*S(t)*, representing, for example, changes in the availability of the metabolite due to seasonal or geological variations. Finally, we choose to separate the global, almost non-epistatic, fitness from the local, possibly highly-epistatic (but small) effects by making the gradient of

*U*_{0} smaller than that of Φ, even though the scale of Φ itself is smaller than that of

*U*_{0}. For example, this agrees with the observation that the ability of proteins to bind to DNA or to metabolic substrates is highly sensitive to the details of the protein sequence [

2–

5].

With the conditions above, we can redefine

*U*_{0}, Φ, and

*S* without much loss of generality, so that

*S*_{t} = 0. We then consider the simplest form of Φ

*(x)* and

*S(t)* that satisfies these conditions, and we will discuss how our results generalize to some other forms of the functions in Sect. 5. Namely, we choose Φ to be a zero-mean periodic saw-tooth potential, and

*S* to be a zero-mean periodic telegraph signal. These considerations allow us to write near a particular point

*x* in the genotype space

where

*v* is the intrinsic drift (in physics terms) or bias, defined as positive for the drift to the right, see . We always assume that

*h*/

*L* >

*v*, so that the fluctuating component of the potential can actually create local maxima and minima on top of the global landscape

*U*_{0}*(x)*. In what follows, we denote by

*T* the time between subsequent potential flips (the half-period of the fluctuations), and

*L* is half of the spatial period.

This model is similar to various stochastic ratchets considered in the literature [

18–

21]. Thus, the question of whether the fitness fluctuations can speed up the evolutionary search is a question similar to whether a rectified or a high-variance motion can appear due to ratcheting. We know from prior analysis [

22] that any unbiased spatially variable but temporally constant potential cannot give rise to rectified motion, and it will always slow down diffusion. Hence temporal fluctuations are an essential component of the model.

2.1 Rescaling of the Equation of Motion

Using the choices above, we can rewrite the equation of motion,

(1)as

where

*η* is a Gaussian white noise of unit variance. In

(8), the dynamics explicitly depends on five different parameters

*L, T, v, h*, and

*D*. Nevertheless, by rescaling the time, the space, and the potential as

*x/L* →

*x, t/T* →

*t*,

,

, we can reduce the number of parameters to only three: the ratio of the typical diffusion time over half the spatial period to half of the temporal period,

, the height of the fluctuating barriers in diffusivity (temperature) units,

, and the ratio between the slope of the average, large scale potential to the slope of the fluctuating perturbation,

*v*. In physical terms, if

*ω* is large, the particle has time to explore the entire valley of

*ϕ* before the potential flips. Further,

*β* measures the difficulty of crossing the peaks by diffusion. Finally, the condition that the perturbation induces local optima is

*v* < 1.

Using the rescaled variables, the dynamics becomes

We will use these rescaled variables in the rest of the article, unless noted otherwise. From this equation, it is easy to recover the dynamics in the original, non-scaled units by simple multiplications. In what follows we present simulation results obtained using first order Euler integration scheme of the dynamics defined in rescaled variables,

(9).