Selection Limits to Adaptive Walks on Correlated Landscapes

FREQ-Seq2: a method for precise high-throughput combinatorial quantification of allele frequencies

The accurate determination of allele frequencies is crucially important across a wide range of problems in genetics, such as developing population genetic models, making inferences from genome-wide association studies, determining genetic risk for diseases, as well as other scientific and medical applications. Furthermore, understanding how allele frequencies change over time in populations is central to ascertaining their evolutionary dynamics. We present a precise, efficient, and economical method (FREQ-Seq2) for quantifying the relative frequencies of different alleles at loci of interest in mixed population samples. Through the creative use of paired barcode sequences, we exponentially increased the throughput of the original FREQ-Seq method from 48 to 2,304 samples. FREQ-Seq2 can be targeted to specific genomic regions of interest, which are amplified using universal barcoded adapters to generate Illumina sequencing libraries. Our enhanced method, available as a kit along with open-source software for analyzing sequenced libraries, enables the detection and removal of errors that are undetectable in the original FREQ-Seq method as well as other conventional methods for allele frequency quantification. Finally, we validated the performance of our sequencing-based approach with a highly multiplexed set of control samples as well as a competitive evolution experiment in Escherichia coli and compare the latter to estimates derived from manual colony counting. Our analyses demonstrate that FREQ-Seq2 is flexible, inexpensive, and produces large amounts of data with low error, low noise, and desirable statistical properties. In summary, FREQ-Seq2 is a powerful method for quantifying allele frequency that provides a versatile approach for profiling mixed populations.

Adaptation, fitness landscape learning and fast evolution

We consider evolution of a large population, where fitness of each organism is defined by many phenotypical traits. These traits result from expression of many genes. Under some assumptions on  fitness we prove that such model organisms  are capable, to some extent, to recognize the fitness landscape. That fitness landscape learning sharply reduces the number of mutations needed for adaptation. Moreover, this learning increases phenotype robustness with respect to mutations, i.e., canalizes the phenotype.  We show that learning and canalization work only when evolution is gradual. Organisms can be adapted to  many constraints associated with a hard environment, if that environment becomes harder step by step. Our results explain why evolution can involve genetic changes of a relatively large effect and why the total number of changes are surprisingly small.

Surfing on the seascape: Adaptation in a changing environment

The environment changes constantly at various time scales and, in order to survive, species need to keep adapting. Whether these species succeed in avoiding extinction is a major evolutionary question. Using a multilocus evolutionary model of a mutation-limited population adapting under strong selection, we investigate the effects of the frequency of environmental fluctuations on adaptation. Our results rely on an "adaptive-walk" approximation and use mathematical methods from evolutionary computation theory to investigate the interplay between fluctuation frequency, the similarity of environments, and the number of loci contributing to adaptation. First, we assume a linear additive fitness function, but later generalize our results to include several types of epistasis. We show that frequent environmental changes prevent populations from reaching a fitness peak, but they may also prevent the large fitness loss that occurs after a single environmental change. Thus, the population can survive, although not thrive, in a wide range of conditions. Furthermore, we show that in a frequently changing environment, the similarity of threats that a population faces affects the level of adaptation that it is able to achieve. We check and supplement our analytical results with simulations.

Adaptation, fitness landscape learning and fast evolution

We consider evolution of a large population, where fitness of each organism is defined by many phenotypical traits. These traits result from expression of many genes. Under some assumptions on  fitness we prove that such model organisms  are capable, to some extent, to recognize the fitness landscape. That fitness landscape learning sharply reduces the number of mutations needed for adaptation. Moreover, this learning increases phenotype robustness with respect to mutations, i.e., canalizes the phenotype.  We show that learning and canalization work only when evolution is gradual. Organisms can be adapted to  many constraints associated with a hard environment, if that environment becomes harder step by step. Our results explain why evolution can involve genetic changes of a relatively large effect and why the total number of changes are surprisingly small.

Computational Complexity as an Ultimate Constraint on Evolution

Experiments show that evolutionary fitness landscapes can have a rich combinatorial structure due to epistasis. For some landscapes, this structure can produce a computational constraint that prevents evolution from finding local fitness optima-thus overturning the traditional assumption that local fitness peaks can always be reached quickly if no other evolutionary forces challenge natural selection. Here, I introduce a distinction between easy landscapes of traditional theory where local fitness peaks can be found in a moderate number of steps, and hard landscapes where finding local optima requires an infeasible amount of time. Hard examples exist even among landscapes with no reciprocal sign epistasis; on these semismooth fitness landscapes, strong selection weak mutation dynamics cannot find the unique peak in polynomial time. More generally, on hard rugged fitness landscapes that include reciprocal sign epistasis, no evolutionary dynamics-even ones that do not follow adaptive paths-can find a local fitness optimum quickly. Moreover, on hard landscapes, the fitness advantage of nearby mutants cannot drop off exponentially fast but has to follow a power-law that long-term evolution experiments have associated with unbounded growth in fitness. Thus, the constraint of computational complexity enables open-ended evolution on finite landscapes. Knowing this constraint allows us to use the tools of theoretical computer science and combinatorial optimization to characterize the fitness landscapes that we expect to see in nature. I present candidates for hard landscapes at scales from single genes, to microbes, to complex organisms with costly learning (Baldwin effect) or maintained cooperation (Hankshaw effect). Just how ubiquitous hard landscapes (and the corresponding ultimate constraint on evolution) are in nature becomes an open empirical question.

Modelling Evolutionary Algorithms with Stochastic Differential Equations

There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.

How to Escape Local Optima in Black Box Optimisation: When Non-elitism Outperforms Elitism

Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in fitness. For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The ( 1 + 1 ) EA is a simple and well-studied evolutionary algorithm that has to jump across the valley to a point of higher fitness because it does not accept worsening moves (elitism). In contrast, the Metropolis algorithm and the Strong Selection Weak Mutation (SSWM) algorithm, a famous process in population genetics, are both able to cross the fitness valley by accepting worsening moves. We show that the runtime of the ( 1 + 1 ) EA depends critically on the length of the valley while the runtimes of the non-elitist algorithms depend crucially on the depth of the valley. Moreover, we show that both SSWM and Metropolis can also efficiently optimise a rugged function consisting of consecutive valleys.