Our planet came into existence 4.6 billion years ago. There is clear chemical evidence for life on earth 3.5 billion years ago [1], [2]. The evolutionary process generated procaria, eucaria and complex multi-cellular organisms. Throughout the history of life, evolution had to discover sequences of biological polymers that perform specific, complicated functions. The average length of bacterial genes is about 1000 nucleotides, that of human genes about 3000 nucleotides. The longest known bacterial gene contains more than nucleotides, the longest human gene more than . A basic question is what is the time scale required by evolution to discover the sequences that perform desired functions. While many results exist for the fixation time of individual mutants [3]–[15], here we ask how the time scale of evolution depends on the length of the sequence that needs to be adapted. We consider the crucial distinction of polynomial versus exponential time [16]–[18]. A time scale that grows exponentially in is infeasible for long sequences.

Evolutionary dynamics operates in sequence space, which can be imagined as a discrete multi-dimensional lattice that arises when all sequences of a given length are arranged such that nearest neighbors differ by one point mutation [19]. For constant selection, each point in sequence space is associated with a non-negative fitness value (reproductive rate). The resulting fitness landscape is a high dimensional mountain range. Populations explore fitness landscapes searching for elevated regions, ridges, and peaks [20]–[27].

A question that has been extensively studied is how long does it take for existing biological functions to improve under natural selection. This problem leads to the study of adaptive walks on fitness landscapes [15], [20], [21], [28], [29]. In this paper we ask a different question: how long does it take for evolution to discover a new function? More specifically, our aim is to estimate the expected discovery time of new biological functions: how long does it take for a population of reproducing organisms to discover a biological function that is not present at the beginning of the search. We will discuss two approximations for rugged fitness landscapes. We also discuss the significance of clustered peaks.

We consider an alphabet of size four, as is the case for DNA and RNA, and a nucleotide sequence of length . We consider a population of size , which reproduces asexually. The mutation rate, , is small: individual mutations are introduced and evaluated by natural selection and random drift one at a time. The probability that the evolutionary process moves from a sequence to a sequence , which is at Hamming distance one from , is given by , where is the fixation probability of sequence in a population consisting of sequence . In the special case of a flat fitness landscape, we have , and . Thus we have an evolutionary random walk, where each step is a jump to a neighboring sequence of Hamming distance one.

Consider a high-dimensional sequence space. A particular biological function can be instantiated by some of the sequences. Each sequence has a fitness value , which measures the ability of the sequence to encode the desired function. Biological fitness landscapes are typically expected to have many peaks [29]–[31]. They can be highly rugged due to epistatic effects of mutations [32]–[34]. They can also contain large regions or networks of neutrality [20], [21]. Empirical studies of short RNA sequences have revealed that the underlying fitness landscape has low peak density [35]: around peaks in sequences.

For the purpose of estimating the expected discovery time we can approximate the fitness landscape with a binary step function over the sequence space. We discuss two different approximations (Figure 1). For the first approximation, we consider the scenario where fitness values below some threshold, , have negligible contribution; those sequences do not instantiate the desired function (either not at all or only below the minimum level that could be detected by natural selection). We approximate the rugged fitness landscape as follows: if then ; if then . The set of sequences with constitutes the target set, and the remaining fitness landscape is neutral.

The second approximation works as follows. Consider the evolutionary process exploring a rugged fitness landscape where the goal is to attain a fitness level . Local maxima below slow down the evolutionary process to attain , because the evolutionary walk might get stuck in those local maxima. In order to derive lower bounds for the expected discovery time, the rugged fitness landscape can be approximated as follows. Let be the fitness value of the highest local maximum below . Then for every sequence in a mountain range with a local maximum below we assign the fitness value . The mountain ranges with local maxima above are the target sequences. Note that the target set includes sequences that start at the upslope of mountain ranges with peaks above . Thus, again we obtain a fitness landscape with clustered targets and neutral region, where the neutral region consists of all sequences whose fitness values have been assigned to . The two approximations are illustrated in Figure 1. For the second approximation generates larger target areas than the first approximation and is therefore more lenient.

Our key results for estimating the discovery time can now be formulated for binary fitness landscapes, but they apply to any type of rugged landscape using one of the two approximations. We note that our methods can also be applied for certain non-binary fitness landscapes, and an example of a fitness landscape with a large gradient arising from multiplicative fitness effects is discussed in Sections 6 and 7 of Text S1.

We now present our main results in the following order. We first estimate the discovery time of a single search aiming to find a single broad peak. Then we study multiple simultaneous searches for a single broad peak. Finally, we consider multiple broad peaks that are uniformly randomly distributed in sequence space.

We first study a broad peak of target sequences described as follows: consider a specific sequence; any sequence within a certain Hamming distance of that sequence belongs to the target set. Specifically, we consider that the evolutionary process has succeeded, if the population discovers a sequence that differs from the specific sequence in no more than a fraction of positions. We refer to the specific sequence as the target center and as the width (or radius) of the peak. For example, if and , then the target center is surrounded by a cloud of approximately sequences. For a single broad peak with width , the target set contains at least sequences, which is an exponential function of . The fitness landscape outside the broad peak is flat. We refer this binary fitness landscape as a broad peak landscape. The population needs to discover any one of the target sequences in the broad peak, starting from some sequence that is not in the broad peak. We establish the following result.

The regeneration process formalizes the role of several existing ideas. First, it ties in with the proposal that gene duplications and genome rearrangements are major events leading to the emergence of new genes [43]. Second, evolution can be seen as a tinkerer playing around with small modifications of existing sequences rather than creating entirely new ones [44]. Third, the process is related to Gillespie's suggestion [29] that the starting sequence for an evolutionary search must have high fitness. In our theory, proximity in fitness value is replaced by proximity in sequence space. However, our results show that proximity alone is insufficient to break the exponential barrier, and only when combined with the process of regeneration it yields polynomial discovery time with high probability. Our process can also explain the emergence of orphan genes arising from non-coding regions [45]. Section 12 of the Text S1 discusses the connection of our approach to existing results.

There is one other scenario that must be mentioned. It is possible that certain biological functions are hyper-abundant in sequence space [21] and that a process generating a large number of random sequences will find the function with high probability. For example, Bartel & Szostak [46] isolated a new ribozyme from a pool of about random sequences of length . While such a process is conceivable for small effective sequence length, it cannot represent a general solution for large .

Our theory has clear empirical implications. The regeneration process can be tested in systems of in vitro evolution [47]. A starting sequence can be generated by introducing point mutations in a known protein encoding sequence of length . If these point mutations destroy the function of the protein, then the expected discovery time of any one attempt to find the original sequence should be exponential in . But only polynomially many searches in are required to find the target with high probability in polynomially many steps. The same setup can be used to explore whether the biological function can be found elsewhere in sequence space: the evolutionary trajectory beginning with the starting sequence could discover new solutions. Our theory also highlights how important it is to explore the distribution of biological functions in sequence space both for RNA [20], [21], [35], [46] and in the protein universe [48].

In summary, we have developed a theory that allows us to estimate time scales of evolutionary trajectories. We have shown that various natural processes of evolution take exponential time as function of the sequence length, . In some cases we have established strong dichotomy results for precise boundary conditions. We have proposed a mechanism that allows evolution in polynomial time scales. Some interesting directions of future work are as follows: (1) Consider various forms of rugged fitness landscapes and study more refined approximations as compared to the ones we consider; and then estimate the expected discovery time for the refined approximations. (2) While in this paper we characterize the difference between exponential and polynomial for the expected discovery time, more refined analysis (such as efficiency for polynomial time, like cubic vs quadratic time) for specific fitness landscapes using mechanisms like recombination is another interesting problem.

Our results are based on a mathematical analysis of the underlying stochastic processes. For Markov chains on the one-dimensional grid, we describe recurrence relations for the expected hitting time and present lower and upper bounds on the expected hitting time using combinatorial analysis (see Text S1 for details). We now present the basic intuitive arguments of the main results.

For a single broad peak, due to symmetry we can interpret the evolutionary random walk as a Markov chain on the one-dimensional grid. A sequence of type is steps away from the target, where is the Hamming distance between this sequence and the target. The probability that a type sequence mutates to a type sequence is given by . The stochastic process of the evolutionary random walk is a Markov chain on the one-dimensional grid .

Consider a Markov chain on the one-dimensional grid, and let denote the expected hitting time from to . The general recurrence relation for the expected hitting time is as follows:(1)for , with boundary condition . The interpretation is as follows. Given the current state , if , at least one transition will be made to a neighboring state , with probability , from which the hitting time is .

Theorem 1 is derived by obtaining precise bounds for the recurrence relation of the hitting time (Equation 1). Consider that for all (i.e., progress towards state is always possible), as otherwise is never reached from . We show (see Lemma 2 in the Text S1) that we can write as a sum, , where is the sequence defined as:(2)

The basic intuition obtained from Equation 2 is as follows: (i) If , for some constant , then the sequence grows at least as fast as a geometric series with factor . (ii) On the other hand, if and for some constant , then the sequence grows at most as fast as an arithmetic series with difference . From the above case analysis the result for Theorem 1 is obtained as follows: If , then for all , we have for some , and hence the sequence grows geometrically for a linear length in . Then, for all states (i.e., for all sequences outside of the target set). This corresponds to case 1 of Theorem 1. On the other hand, if , then it is , and case 2 of Theorem 1 is derived (for details see Corollary 2 in Text S1).

The basic intuition for the result is as follows: consider a single search for which the expected hitting time is exponential. Then for the single search the probability to succeed in polynomially many steps is negligible (as otherwise the expectation would not have been exponential). In case of independent searches, the independence ensures that the probability that all searches fail is the product of the probabilities that every single search fails. Using the above arguments we establish Theorem 2 (for details see Section 8 in Text S1).

For this result, it is first convenient to view the evolutionary walk taking place in the sequence space of all sequences of length , under no selection. Each sequence has neighbors, and considering that a point mutation happens, the transition probability to each of them is . The underlying Markov chain due to symmetry has fast mixing time, i.e., the number of steps to converge to the stationary distribution (the mixing time) is . Again by symmetry the stationary distribution is the uniform distribution. If , then from Theorem 1 we obtain that the expected time to reach a single broad peak is exponential. By union bound, if , the probability to reach any of the broad peaks within steps is negligible. Since after the first steps the Markov chain converges to the stationary distribution, then each step of the process can be interpreted as selection of sequences uniformly at random among all sequences. Using Hoeffding's inequality, we show that with high probability, in expectation such steps are required before a sequence is found that belongs to the target set. Thus we obtain the result of Theorem 3 (for details see Section 9 in Text S1).

An important aspect of our work is that we establish our results using elementary techniques for analysis of Markov chains. The use of more advanced mathematical machinery, such as martingales [49] or drift analysis [50], [51], can possibly be used to derive more refined results. While in this work our goal is to distinguish between exponential and polynomial time, whether the techniques from [49]–[51] can lead to a more refined characterization within polynomial time is an interesting direction for future work.