Left-to-right readers classify faster past events with motor responses on the left side of space and future events with responses on the right side. This suggests a left-to-right spatial organization in the mental representation of time. Here, we show that the significance and reliability of this representation are linked to the joint use of temporal and spatial codes in the task at hand. In a first unimanual Go/No-Go Implicit Association Test (IAT), attending selectively to “past” or to “future” words did not activate corresponding “left” or “right” spatial concepts and vice versa. In a second IAT, attending to both temporal  words and spatial targets  pointing arrows produced faster responses for congruent rather than incongruent combinations of temporal and spatial concepts in task instructions . This effect increased markedly in a STEARC task where spatial codes defined the selection between “left-side” and “right-side” button presses that were associated with “past” and “future” words. Two control experiments showed only partial or unreliable space–time congruency effects when (a) participants attended to superordinate semantic codes that included both spatial “left”/”right” or temporal “past/future” subordinate codes; (b) a primary speeded response was assigned to one dimension  and a nonspeeded one to the other dimension . These results help to define the conditions that trigger a stable and reliable spatial representation of time-related concepts.The online version contains supplementary material available at 10.3758/s13414-023-02746-w. People use space to think and talk about the flow of time , to represent the flow of time  and corresponding temporal concepts , then in an IAT, the mere and selective activation of one spatial concept  should consistently and reliably facilitate the activation and classification of the associated “past” temporal concept. Alternatively, if the positioning of “past” and “future” temporal concepts on the left and right side of an MTL depends on the simultaneous presence of contrasting “left vs. right” response codes in the task at hand, then a significant and reliable STA should be found only in (a) IATs that require discriminating, in different trials of the same task, both the “left vs. right” direction of arrow targets and classifying “past vs. future” word targets; (b) conventional STEARC tasks where contrasting “left vs. right” spatial response codes are associated to contrasting “past vs. future” temporal stimulus codes.f(U) = 0.4388 derived from the previous study of Santiago et al. . To avoid “carryover” effects that can take place in within-participants designs, the order of tasks administration followed that of the increasing complexity of task instructions. As an example, in the first experiment (see below), in different blocks of trials, unimanual responses were regulated by the consideration of a “single” time or space code . In the second experiment, unimanual responses were regulated by the “joint” association of a space code with a time code . Finally, in the third experiment, the “joint” association of a space with a time code determined the selection of a left vs. right side motor response . It is clear, for example, that performing the second “joint” experiment first would activate the explicit association between a space and a time code and that this association can be “carried over” to the performance of the first “single” code experiment, that is aimed at testing whether the activation of a single space or time codes produces the implicit activation of a corresponding and congruent time or space code, respectively. Vice versa, performing first the single code experiment does not activate the conceptual space–time association and does not produce corresponding carryover effects on the performance of the second “joint” code experiment. Following the same line of reasoning, we point out that the STEARC effect that in the present study was tested in the third final experiment is a very consolidated and replicated finding and that, therefore, the replication of this effect can hardly be attributed to the previous performance of the second unimanual experiment with the joint use of space and time codes.Based on this preliminary analysis, we tested 28 healthy adult participants. Each participant performed three different experiments included in the study  in the first three experiments.https://osdoc.cogsci.nl/3.3/; Mathôt al., https://www.jatos.org/). Participants accessed the experiment using a General Multiple Worker link. Participants were instructed to run the experiment in a quiet and isolated room and wear in-ear plug headphones to reduce environmental noise sources. They were also asked to keep their head positioned at a viewing distance of 60 cm from the screen. All participants had a normal or corrected-to-normal vision and were naïve to the aim of the study. Instructions for all experiments were provided during individual audio/video calls with one of the experimenters. A training block that included 24 trials was administered before the experimental session and always corresponded to a shortened version of the first experimental block.Due to COVID pandemic restrictions, experiments were administered through the open-source software OpenSesame , participants responded to targets according to instructions that required the discrimination of the left/right direction of arrows without requiring the discrimination of the past/future temporal targets or vice versa, the discrimination of the past/future temporal targets without requiring the discrimination of the left/right direction of arrows. The aim of the experiment was to test whether the selective activation of one  out of two contrasting spatial codes  determined the activation of the activation of a corresponding temporal code  and vice versa.https://www.trainingcognitivo.it/GC/nonparole/) and were equivalent in length and in the number of syllables to target word stimuli . In addition, past and future words have the same frequency of use in Italian  and an equivalent length . Similarly, in the time-to-space condition, in different trials, arrow targets were alternated with white circle no-go nontargets to promote the full processing of go left- and right-pointing arrows. Each block included 200 trials, with 40 repetitions for each stimulus category.In four different blocks of trials, participants had to (a) go only to left-pointing arrows and to words; (b) go only to right-pointing arrows and to words; (c) go only to past words and to arrows; (d) go only to future words and to arrows. Conditions “a” and “b” qualify the influence of space coding on time coding . Vice versa, conditions “c” and “d” qualify the influence of time coding on space coding . In the space-to-time condition, go temporal targets were intermixed with nonwords no-go nontargets to avoid participants responding to “all” word stimuli without full processing. Nonwords were generated using an online nonword generator  and trials in which responses were above and below two standard deviations from the mean of each experimental condition were excluded from the analyses. Following these criteria, 3.56% of trials were excluded from analyses.The space-to-time congruency effect was analyzed by entering RTs in congruent trials (past words of condition “a” and future words of condition “b”) and RTs in incongruent trials (past words of condition “b” and future words of condition “a”) in a 2 × 2 repeated-measures ANOVA, with time (past vs. future) and congruency (congruent vs. incongruent) as within-subjects factors.The time-to-space congruency effects were analyzed by entering RTs in congruent trials (left arrow of condition “c” and right arrow of condition “d”) and RTs in incongruent trials (right arrow of condition “d” and left arrow of condition “c”) in 2 × 2 repeated-measures ANOVA, with arrow direction (left vs. right) and congruency (congruent vs. incongruent) as within-subjects factors.r1, 2). Besides, we used the corrected Spearman–Brown correlation between the RTs advantages in the odd-numbered and even-numbered halves of trials as the reliability index (coefficient: rtt). To further assess the reliability of Congruency effects, the split-half test was also applied to 10,000 permutations of RTs .The reliability of congruency effects was examined using the split-half method. We first divided each experimental condition data into odd and even responses. Then we calculated the RTs advantages produced in the congruent respect to the incongruent condition (dRTs = RTs in the incongruent condition minus RTs in the congruent condition) for the odd-numbered and the even-numbered items. We then evaluated the correlation between the RTs advantages for odd and even-numbered items  = 0.336, p = .566, ηp2 = 0.012, or Time × Congruency interaction, F = 0.144, p = .70, ηp2 = 0.005; see Fig. The ANOVA highlighted no significant main effect of congruency (congruent vs. incongruent), F1, 7 = 0.336F = 5.071, p = .032, ηp2 = 0.158 . The same comparison with Left-pointing arrows was not statistically significant (p > .562). The reliability test showed inconsistent time-to-space congruency effect . This result was confirmed when 10,000 random halves were computed through a permutation method (see Supplementary File).The ANOVA revealed a significant Congruency × Arrow Direction interaction, F1, 7 = 5.071Based on the null results of the SC task in Experiment In four different conditions/blocks of trials, participants had to respond (a) only to left-pointing arrows and past words; (b) only to right-pointing arrows and future words; (c) only to left-pointing arrows and future words; (d) only to right-pointing arrows and past words. In conditions “a” and “b” spatial and temporal codes are congruent, while in conditions “c” and “d” they are Incongruent. RTs to temporal-target trials qualified the influence of spatial coding on temporal coding (space-to-time). Vice versa, RTs to arrow-direction trials qualified the influence of temporal coding on spatial coding (time-to-space). Each block consisted of 160 trials, 80 with temporal targets and 80 with arrow targets . Following the same exclusion criteria adopted in Experiment The space-to-time congruency effects were analyzed by entering RTs of congruent temporal-target trials (past words in condition “a” and future words in condition “b”) and incongruent temporal-target trials (past words in condition “c” and future words in condition “d”) in a 2 × 2 repeated-measures ANOVA, with Time (past vs. future) × Congruency (congruent vs. incongruent) as within-subjects factors.The time-to-space congruency effects were analyzed by entering RTs of congruent arrow direction-target trials (left arrows in condition “a” and right arrows in condition “b”) and incongruent arrow direction-target trials (left arrows in condition “c” and right arrows in condition “d”) in a 2 × 2 repeated-measures ANOVA, with arrow direction (left vs. right) and congruency (congruent vs. incongruent) as within-subjects factors for the investigation of the time-to-space congruency effect.The reliability of congruency effects was examined using the same procedure as in Experiment F = 7.403, p = .011, ηp2 = .215, with faster RTs in the congruent (745 ms) than in the incongruent (772 ms) condition. There was also a significant Congruency × Time interaction, F = 17.284, p < .001, ηp2 = .390. Bonferroni planned comparisons showed faster responses to congruent future targets (729 ms) versus incongruent future targets . The reliability test showed that the space-to-time congruency effect was consistent  = 7.899, p = .009, ηp2 = .226, with RTs being faster in the congruent (472 ms) than in the incongruent condition (493 ms). Reliability test showed that the time-to-space effect was consistent  in the “congruent” condition, they were asked to respond to past words with the left hand/button and future words with the right hand/button; (b) in the “incongruent” condition, vice versa, they were asked to respond to past words with the right hand/button and future words with the left hand/button. Each block consisted of 80 trials, 40 for each past/future condition. The order of blocks was counterbalanced among participants. In this experiment, 4.38% of trials were excluded from analyses due to the exclusion criteria adopted in Experiments Individual RTs of the congruent (past-words/left hand-button and future-words/right-hand-button) and incongruent (past-words/right-hand-button and future-words/left-hand-button) conditions were entered in a Time (past vs. future) × Congruency (congruent vs. incongruent) repeated-measures ANOVA.F = 50.560, p < .001, ηp2 = .651, with faster RTs to congruent (759 ms) than to incongruent targets (873 ms). The reliability analysis confirmed the consistency of the congruency effect  measured in the three tasks. Preliminary analyses were run to test the presence of multivariate normality in the data set  = 31.060, p < .001, ηp2 = .534. Bonferroni planned comparison showed a significant difference between Experiment ps < .001; see Fig. The ANOVA pointed out the main effect of task, F, 54 = 31p < .001). Other correlations were not significant .Pearson’s correlation  were asked to perform a modified version of the single code experiment. In each condition of the original single code experiment, participants were asked to put in contrast one conceptual dichotomy , while the remaining dichotomy was not activated by task instructions . In contrast, in this new version of the task, one conceptual code dichotomy was fully activated by task instructions, while the remaining dichotomy was only implicitly activated through its corresponding superordinate semantic code. For example, in the case of spatial codes, “horizontal” is superordinate to “left” and “right.” In the case of temporal codes, “not present-tense” is superordinate to “past” and” future” . Therefore, in different blocks of trials, participants had to respond (a) only to left-pointing arrows and not present-tense words (where past and future words targets were intermixed with present-tense nontargets words); (b) only to right-pointing arrows and not present-tense words; (c) only to past words and horizontal arrows ; (d) only to future words and horizontal arrows. Conditions “a” and “b” qualify the investigations of space-to-time congruency effects  triggers the activation of a corresponding temporal code . Vice versa, conditions “c” and “d” qualify the investigation of time-to-space congruency effects  that is not explicitly activated by the task instructions. The procedure of the superordinate category task was similar to that of the single code experiment except that, in this case, vertical up or down pointing arrows for conditions “a” and “b” and present-tense words were also for condition “c” and “d” presented as no-go nontargets. Each block included 200 trials, with 40 repetitions for each category of stimuli. Participants were tested in a quiet and isolated room. They wore in-ear plug headphones to reduce environmental noise. The head position was restrained with a chin rest at a viewing distance of 57.7 cm from the screen. All participants had a normal or corrected-to-normal vision and were naive to the aim of the study.Thirty-two right-handed Italian native speakers participants  and RTs in incongruent temporal trials (past words of condition “b” and future words of condition “a”) in a 2 × 2 repeated-measures ANOVA, with time (past vs. future) and congruency (congruent vs. incongruent) as within-subjects factors.The time-to-space congruency effects were analyzed by entering RTs in congruent spatial trials (left arrow of condition “c” and right arrow of condition “d”) and RTs in Incongruent spatial trials (right arrow of condition “d” and left arrow of condition “c”) in 2 × 2 repeated-measures ANOVA, with arrow direction (left vs. right) and congruency (congruent vs. incongruent) as within-subjects factors.The reliability of congruency effects was examined using the same procedure as in Experiment F = 12.939, p = 0.001, ηp2 = 0.294, with faster RTs to congruent trials (654 ms) than to incongruent ones (664 ms). Nonetheless, the reliability test showed this effect was inconsistent [split-half reliability: r1,2 = −.2563, rtt = −.407, p = .156; see Fig. The ANOVA pointed out a significant main effect of congruency (congruent vs. incongruent), t trials 54 ms thaF = 29.175, p < .001, ηp2 = 0. 484. Also, in this case, RTs were faster in the congruent (488 ms) than in the incongruent condition (514 ms). The reliability test showed that this effect was consistent , The single code task of Experiment SD = 2.55. In different experimental conditions, participants were asked to (a) “go” as fast as possible only to left-pointing arrows and to all words and then indicate, without time pressure, whether the word was a “past” or a “future” one by pressing the “green” or the “orange” button, respectively; (b) “go” as fast as possible only to right-pointing arrows and to all words, and then indicate, without time pressure, whether the word was a “past” or a “future” one by pressing the “green” or the “orange” button, respectively; (c) “go” as fast as possible only to past words and to all arrows and then indicate, without time pressure, whether the arrow was a left- or a right-pointing one by pressing by pressing the “green” or the “orange” button, respectively; (c) “go” as fast as possible only to future words and to all arrows and then indicate, without time pressure, whether the arrow was a left- or a right-pointing one by pressing the “green” or the “orange” button, respectively. Like in the other experiments, the central space bar was used for the primary speeded task. Buttons “B” and “Y” were used for the secondary nonspeeded task: These buttons are vertically arranged one above the other on the computer keyboard and were chosen not to include or suggest the implicit use of left versus right spatial response codes. Half of the participants had the “B” button covered with a green plastic button and the “Y” button with an orange plastic button, while for the other half of participants the position of green and orange buttons was reversed. Conditions “a” and “b” qualify space-to-time congruency effects, while conditions “c” and “d” qualify time-to-space congruency effects. The procedure and stimuli were as in Experiment In this second control experiment we tested 28 right-handed Italian native speakers  and RTs in Incongruent temporal trials (past words of condition “b” and future words of condition “a”) in a 2 × 2 repeated-measures ANOVA, with time (past vs. future) and congruency (congruent vs. incongruent) as within-subjects factors.The time-to-space congruency effects were analyzed by entering RTs in congruent spatial trials (left arrow of condition “c” and right arrow of condition “d”) and RTs in Incongruent spatial trials (right arrow of condition “d” and left arrow of condition “c”) in 2 × 2 repeated-measures ANOVA, with arrow direction (left vs. right) and congruency (congruent vs. incongruent) as within-subjects factors.The reliability of congruency effects was examined using the same procedure as in Experiment F = 0.057, p = .813, ηp2 = 0.002 , F = 10.477, p = 0.003, ηp2 = 0. 279. RTs were faster in the congruent (558 ms) than in the incongruent condition (573 ms). Nonetheless, the reliability test showed that this effect was inconsistent , gruent 55 ms than The main results of our study are as follows. First, in a single code task  shows that the STEARC arises at the response-related stage  gestures can be spontaneously adopted during the interpersonal communication of temporal information  while the contrast in the remaining dimension was only implicitly activated through a superordinate semantic code . In this case, we found significant congruency effects in the Space-to-Time and Time-to-Space directions .Current research mainly focuses on qualifying the types and the set of sensorimotor and cultural experiences that shape the spatial representation of abstract concepts like time and number. For example, Pitt and Casasanto  have sugIn the domain of space-number association, recent investigations have suggested that interindividual variations in scanning habits, finger counting style and spatial imagery can significantly affect the ability to generate and use mental spatial representations of numerical magnitudes Below is the link to the electronic supplementary material.


Nature Communications 10.1038/s41467-023-38850-6, published online 12 June 2023Correction to: M’ was incorrectly presented as mutation probability ‘zM’. These have been corrected in both the HTML and PDF versions of the Article.In the original version of this Article, in the third paragraph of the Discussion the equation ‘


However, it is often the case in practice that the input keys have intrinsic relationships that we can exploit to lower the bit complexity of f. For example, consider a string and the set of all its distinct k-mers as input keys: since two consecutive k-mers share an overlap of f to map consecutive k-mers to consecutive addresses, as to also preserve as much as possible their relationship in the codomain. This is a useful feature in practice as it guarantees a certain degree of locality of reference for f, resulting in a better evaluation time when querying consecutive k-mers.Minimal perfect hashing is the problem of mapping a static set of k-mers extracted consecutively from a collection of strings. We design a construction whose space usage decreases for growing k and discuss experiments with a practical implementation of the method: in practice, the functions built with our method can be several times smaller and even faster to query than the most efficient MPHFs in the literature.Motivated by these premises, we initiate the study of a new type of locality-preserving MPHF designed for  U, a function f maps each key of S into a distinct integer in [n]. The function is allowed to return any value in [n] for a key Given a universe set or |U|≫n . Minimalor |U|≫n  to databor |U|≫n , as wellor |U|≫n , compileor |U|≫n ,b.f act less randomly on S. This, in turn, would permit to achieve a lower space complexity for f.In its simplicity and versatility, the minimal perfect hashing problem does not take into account specific types of inputs, nor the intrinsic relationships between the input keys. Each key S are all the distinct sub-strings of length k, for some S are called k-mers. The crucial point is that any two consecutive k-mers in a string of k-mers as to reduce the randomness of f, thus lowering its bit complexity and evaluation time.We therefore consider in this article the following special setting of the minimal perfect hashing problem: the elements of Q, if k-mer f to hash k-mer following x in Q , or contig identifiers (e.g. unitigs) in a de Bruijn graph. Consecutive k-mers tend to have very similar—if not identical—satellite values, hence hashing consecutive k-mers to consecutive identifiers induce a natural clustering of the associated satellite values which is amenable to effective compression. The second important reason is, clearly, faster evaluation time when querying for consecutive k-mers in a sequence. This streaming query modality is the query modality employed by k-mer-based applications  which allows a more parsimonious memory layout. The space of the proposed solution decreases for growing k and the data structure is built in linear time in the size of the input (number of distinct k-mers). In Section 4 we present experiments across a breadth of datasets to show that the construction is practical too: the functions can be several times smaller and even faster to query than the most efficient, albeit “general-purpose”, MPHFs. We conclude in Section 5 where we also sketch some promising future directions. Our C++ implementation of the method is publicly available at https://github.com/jermp/lphash.We formalize the notion of locality-preserving MPHF along with other preliminary definitions in Section 2. We show how to obtain a locality-preserving MPHF in very compact space in Section 3. To achieve this result, we make use of two algorithmic tools: random minimizers  and a nok of a string k-mer of S.Definition 1The k-mer spectrum of is the set of all distinct k-mers of the strings in . Formally: (Spectrum). Definition 2A spectrum-preserving string set (or SPSS) of is a set of strings such that (i) each string of has length at least k, and (ii) (Spectrum-Preserving String Set). Let k-mers in a SPSS, we are interested in a SPSS k-mer is seen only once, i.e. for each k-mer x appears once. We assume that no k-mer appearing at the end of a string shares an overlap of k-mer of another string, otherwise we could reduce the number of strings in Since our goal is to build a MPHF for the see e.g. .k-mers. Without loss of generality, we index k-mers based on their positions in i-th k-mer in The input for our problem is therefore a SPSS n distinct k-mers in k-mers from a string of f as well as its evaluation time when querying k-mers in sequence.We want to build a MPHF Definition 3Let be a MPHF forand A be the set . The function f is -locality-preserving for if (LP-MPHF). We define a locality-preserving MPHF, or LP-MPHF, for Intuitively, the “best” LP-MPHF for k-mers is a very desirable property as motivated in Section 1. A different definition of locality-preservation could instead be given if we were considering generic input keys. Second, we did not use the term “order-preserving” to stress the distinction from classic order-preserving functions in the literature . Two more considerations are in order. First, it should be clear that the way we define locality-preservation in Definition 3 is only pertinent to SPSSs where having consecutive hash codes for consecutive terature  that makk-mers in k-mers do not share any overlap .From Definitions 3 and 4, it is easy to see that k-mers that will be one of the main ingredients used in Section 3.Definition 5k-mer). Given a k-mer x and a random hash function h, the minimizer of x is any m-mer such that for any other m-mer y of x, for some  Since h is a random hash function . Given a string S, a super-k-mer g is a maximal sub-string of S where each k-mer has the same minimizer and appears only once in g.. Let p be the start position of x. We can use this positional information p to compute Now, suppose we are given a query if x cannot possibly be in g and, hence, indexed by f.otherwise .Lemma 1.The strategy in for any Our strategy is to compute Proof. For k-mer of g, k-mers inside a super-k-mer.To sum up, the position of the minimizer in the first k-mers of g into k-mer g. We proceed as follows.Algorithm 1.f, given the k-mer x. The helper function x and the starting position p of x.Evaluation algorithm for function f(x):1: 2:  3:  return4:  From Layout. Let k-mer g is the only super-k-mer having minimizer k-mers have the same minimizer in Section 3.3.) We allocate an array L, that is, k-mers before those in g .Definition 7Let g be a super-k-mer. The first/last (FL) rule is as follows:(FL rule). Depending on the starting position of the minimizer in the first and last if and , then g is left-right-max; elseif and , then g is left-max; elseif and , then g is right-max; elseif and , then g is non-max.See Layout. Based on the FL-rule above, we derive a partitioned layout as follows. We store the type of each super-k-mer in an array k-mers to improve the space bound of Theorem 1 because:k-mers, we do not store L nor P;for all left-right-max super-k-mers, we only store L—precisely, two arrays k-mers, respectively;for all left/right-max super-k-mers, i.e. non-max, we store both L and P as explained before—let us indicate them with for all the other super-R: the result of this query is the number of super-k-mers that have type t in the prefix R[1.i]. If k-mer associated to t, we either do not perform any array access or access the j-th position of either Addressing the arrays d Pn see .R that also supports R in A succinct representation of let tree . In our Lookup.f. The complexity of the algorithm is still R) and a rank query as explained above but, on the other hand, it permits to save 2 array accesses in the left-right-max case or 1 in the left/right-max case compared to L and one to P). Hence, the overall number of array accesses performed by Algorithm 2.f. The quantities k-mers of Evaluation algorithm for a partitioned representation of function f(x):1: 2:  3:  4:  5:  6:  switch(t):7:  case left-right-max:8:    9:      break10:    case left-max:11:   12:    break13:    case right-max:14:   15:    break16:    case non-max:17:   18:    break19:    return20:  Compression. Intuitively, if the fraction of left-right-max super-k-mers and that of left/right-max super-k-mers is sufficiently high, we can save significant space compared to the data structure in Section 3.1 that stores both L and P for all minimizers. We therefore need to compute the proportions of the different types of super-k-mers as given by the FL rule. For ease of notation, let k-mer g.Remark 1.The FL rule is a partitioning rule, i.e. for any super-k-mer.k (k-mer length) and m (minimizer length). To achieve this goal we propose a simple model based on a (discrete-time) Markov chain.Our objective is to derive the expression for the probabilities k-mer. The corresponding Markov chain is illustrated in X, i.e. with each value in k-mer if, from state w we transition to state w which corresponds to the event that the right-most m-mer (that coming next to the right) is the new minimizer. If the chain reaches state 1, instead, we know that we are always going to see a new minimizer next. If h used by u Lemma 2.and We give the following lemma to prove Theorem 2. , the fastest MPHF in the literature, and the popular BBHash , 64 GB of RAM, and running the Linux 5.13.0 operating system. The whole code (LPHash and competitors) was compiled with gcc 11.2.0, using the flags -O3 and -march=native.Datasets. We use datasets of increasing size in terms of number of distinct k-mers; namely, the whole-genomes of: Saccharomyces cerevisiae , Caenorhabditis elegans , Gadus morhua , Falco tinnunculus , and Homo sapiens . For each dataset, we obtain the corresponding SPSS by first building the compacted de Bruijn graph using BCALM2 . A suitable value of m should clearly be not too small , nor too large  and Theorem 3 (partitioned) for b parameter models the number of bits per key spent by a MPHF of choice for the representation of the minimizer MPHF and the fallback MPHF. , LPHash still retains 2.3–3.7× better space.To make a concrete example, partitioned LPHash for We remark that, however, PTHash and BBHash are “general-purpose” MPHFs that can work with arbitrary keys, whereas the applicability of LPHash is restricted to spectrum-preserving string sets.k-mers read from the Human chromosome 13, for a total of Q, we query for each k-mer read “consecutively” from Q, that is, for Q[1.k], k-mers very cheap, as consecutive k-mers are likely to be part of the same super-k-mer.We distinguish between streaming and random queries (lookups) for LPHash. Given a query string k: while the streaming one slightly decreases for the better locality, the random one increases instead, for the more expensive hash calculations.Considering the result in LPHash is as fast as PTHash-v1 (fastest configuration) for streaming queries on the smaller Yeast dataset, but actually up to 1.4–2× faster on the larger datasets Elegans, Cod, Kestrel, and Human. Instead, it is up to k-mers . However, we do not regard this as a serious limitation since, as we already motivated, the streaming query modality is the one used in Bioinformatics tasks involving k-mers . We alsoWe now consider building time which is reported in k-mers since it builds two smaller PTHash functions  3 bits/k-mer and 100–200 nanoseconds per lookup. This work shows that it is possible to do significantly better than this when the k-mers come from a spectrum-preserving string set: for example, less than 0.6–0.9 bits/k-mer and 30–60 nanoseconds per lookup. Our code is open-source.We show that a concrete implementation of the method is practical as well. Before this paper, one used to build a BBHash function over the Miniception (As future work, we plan to further engineer the current implementation to accelerate construction and streaming queries. Other strategies for sampling the strings could be used other than random minimizers ; for exaiception  achievinbtad219_Supplementary_DataClick here for additional data file.


CRISPR/Cas9  is a popular and effective two-component technology used for targeted genetic manipulation. It is currently the most versatile and accurate method of gene and genome editing, which benefits from a large variety of practical applications. For example, in biomedicine, it has been used in research related to cancer, virus infections, pathogen detection, and genetic diseases. Current CRISPR/Cas9 research is based on data-driven models for on- and off-target prediction as a cleavage may occur at non-target sequence locations. Nowadays, conventional machine learning and deep learning methods are applied on a regular basis to accurately predict on-target knockout efficacy and off-target profile of given single-guide RNAs (sgRNAs). In this paper, we present an overview and a comparative analysis of traditional machine learning and deep learning models used in CRISPR/Cas9. We highlight the key research challenges and directions associated with target activity prediction. We discuss recent advances in the sgRNA–DNA sequence encoding used in state-of-the-art on- and off-target prediction models. Furthermore, we present the most popular deep learning neural network architectures used in CRISPR/Cas9 prediction models. Finally, we summarize the existing challenges and discuss possible future investigations in the field of on- and off-target prediction. Our paper provides valuable support for academic and industrial researchers interested in the application of machine learning methods in the field of CRISPR/Cas9 genome editing. Advances in the area of genome editing  in the 2010s revolutionized molecular biology, genetics, and biomedicine. Genome editing techniques allow precise manipulation, deletion, and insertion of sequence fragments within the DNA of living organisms. In recent years, three effective types of genome editing toolsets, called Zinc Finger Nucleases (ZFNs) .Highly effective CRISPR/Cas9 gene editing system, co-invented in 2012 by Emmanuelle Charpentier and Jennifer Doudna , has beeThe CRISPR/Cas9 genetic engineering system is an adapted version of the bacterial CRISPR-Cas9 antiviral defense system. CRISPR is a family of DNA sequences present in prokaryotic genomes that stems from DNA fragments of bacteriophages, which had infected prokaryotic genomes in the past. These DNAs are used as antiviral defense elements to recognize and eliminate DNA from similar bacteriophages during eventual infections . Cas9 isin vivo [In the CRISPR/Cas9 editing system, the Cas9 nuclease combined with a guide RNA (gRNA) is delivered into a cell, allowing the cell’s genome to be cut in a specific location, some targeted genes to be removed from it, and some other added to it, in vivo . Guide Rin vivo .The CRISPR/Cas9 system is nonetheless prone to unintended off-targets: a cleavage may occur at non-target locations . Thus, tOver the last few years, data-driven machine learning methods emerged as a new modeling approach which outperforms the common scoring prediction methods, such as MIT CRISPR Design Tool2 , CCTop aet al. [et al. [et al., only one employs deep learning. Chen et al. [et al. [in silico off-target prediction tools in order to evaluate their genome-wide CRISPR performances, and introduced an integrated Genome-Wide Off-target cleavage Search (iGWOS) platform designed for optimal genome-wide off-target predictions. The main goal of the study by Yaish et al. [et al. introduced predictive off-target in cellula models based on gradient boosting (i.e. the XGBoost decision tree-based ensemble learning framework was implemented by these authors) and compared them with state-of-the-art off-target prediction methods. The paper of O’Brien et al. [in silico optimization can successfully replace in vitro CRISPR-Cas9 designs. Thus, algorithmic solutions can be used for maximizing gRNA editing efficiency and minimizing gRNA specificity. Finally, a recent review of Konstantakos et al. [et al. assessed the performances of 10 deep learning and one conventional machine learning methods on six benchmark data sets, and provided recommendations for their use. The authors pointed out that the existing on-target prediction approaches still have some flaws, including their sensitivity to data heterogeneity, unclear decision making mechanism, and inability to produce general gRNA design rules.Among recent reviews and comparative studies discussing the use of computational methods for evaluating gRNA efficiency and predicting gRNA specificity, we need to mention the following works. Wang et al.  highligh [et al.  providedn et al.  first br [et al.  comparedh et al.  was to sn et al.  presentss et al.  addresseIn this review paper, we provide a summary of studies that have examined the effectiveness of artificial intelligence (AI) methods for on- and off-target activity prediction related to CRISPR/Cas9. In contrast to some previous reviews , 37–42, Our main contributions are as follows:To the best of our knowledge, this is the first comprehensive review of both traditional machine learning and deep learning methods, used for both on-target (guide efficiency) and off-target (guide specificity) outcome prediction in CRISPR/Cas9 genome editing;A description of the benchmark data sets used for on- and off-target prediction in CRISPR/Cas9 is provided. Most of the discussed data sets were curated and made available for researchers on our GitHub repository;The main sequence encoding techniques used in CRISPR/Cas9, applying to both traditional machine learning and deep learning methods, are discussed along with their intrinsic properties;The main deep learning models used for on- and off-target prediction in CRISPR/Cas9 are presented and their advantages and limitations are highlighted;Research challenges and avenues of future investigation regarding the application of traditional machine learning and deep learning methods in the field of CRISPR/Cas9 genome editing are discussed.In this section, we present the most popular CRISPR/Cas9 benchmark data sets used in the literature for on- and off-target prediction. These data sets can be divided into three categories: data sets including off-targets only, on-targets only, and both off- and on-targets.et al. [in vitro Reporting of CLeavage Effects by sequencing) screening strategy introduced by Tsai et al. [et al. [et al. [et al. [in vitro Cas9 genome-wide nuclease activity data set. CHANGE-seq was carried out to analyze 110 sgRNA targets across 13 therapeutically relevant loci in human primary T cells. A total of 201 934 off-target sites were identified with variable numbers of off-target sites, ranging from 19 to 61 415, for an individual sgRNA.The first group of benchmark data sets consists of off-targets. GUIDE-seq was one of the first off-target data repositories, based on the results of the GUIDE-seq technique developed by Tsai et al. . It can i et al.  was used [et al.  proposed [et al.  collecte [et al. , 47,  [et al. data set [et al. data set (4239 guides targeting 829 genes in Human Hct116 cells) [et al. data set (1234 guides targeting Human 293T cells) [et al. data set (50 guides targeting different genes in Caenorhabditis elegans genome) [et al. [crispor.org) that is intended to design, evaluate and clone guide sequences for the CRISPR/Cas9 system. This web tool incorporates several on- and off-target scoring algorithms. It also displays pre-calculated results for all human exons from the UCSC Genome Browser tracks. On the first page of crispor.org, the user enters three pieces of information: (i) a single genomic sequence, typically an exon under 2300 bp; (ii) a genome ; (iii) a PAM motif. The main output of CRISPOR is a web page that shows the annotated input sequence and the list of possible guides in this sequence. Furthermore, CRISPOR also generates a list of primers related to a selected guide. The related GitHub data repository organized by Haeussler comprises direct links to 22 experimental data sets along with some necessary data conversion scripts written in Python. Munoz et al. [et al. [The third group of benchmark data sets considered in the literature includes data containing both off- and on-target information. First, we need to mention the RESistance assays (RES) data set, i.e. Doench V2, made available by Doench et al. . It cons [et al.  aggregat0 cells) , 61, Koidata set , Doench data set , 59, Har6 cells) , Z_fish 6 cells) , Z_fish 6 cells) , Z_fish 6 cells) , Drosoph6 cells) , Chari eT cells) , Ciona dT cells) , and Far genome) . Moreovez et al.  designed [et al.  includeshttps://github.com/dagrate/public_data_crisprCas9. Moreover, several data sets presented here have been one-hot encoded and prepared for use in machine learning and deep learning experiments. The related Python scripts have been also made available to the scientific community.In Table In conclusion, we think that using the latest benchmark data containing large amounts of samples and features is likely to facilitate future work in CRISPR/Cas9, since such data can provide wide and complete coverage of intrinsic properties of both off- and on-targets under study, and thus be effectively exploited by state-of-the-art machine learning and deep learning methods.Before building AI models intended for on- and off-target prediction, the sgRNA–DNA sequence data must be pre-processed to be used as input. Data pre-processing, or data encoding, allows converting the sgRNA–DNA sequences of letters into sequences of numbers that AI models can read and interpret to build their predictions. Data pre-processing is an important milestone when trying to boost the predictive performance of AI models. The two most popular encoding techniques used in CRISPR-Cas9 are: (a) One-hot encoding and (b) Word embedding. Figure Word2Vec . This naet al., Wang et al., and Charlier et al. [It is worth noting that most of the sequence encoding schemes discussed in this section apply to both traditional machine learning and deep learning models. For example, the recent works of Lin r et al. , 73–75 uet al. [et al. is of size et al. [et al. [et al. combined a et al. [et al. [et al. the created seven-bit-long features allow one to take into account all possible correspondences existing between the original sgRNA–DNA pairs of features taking the values  technique, called word embedding. The idea of applying word embedding on sgRNAs is that off-targets are encoded closer to each other in the vector space than are on-targets. Liu et al.  combined [et al.  proposed [et al. , designe [et al.  proposed [et al. . The encet al. [et al. [et al. [et al.[et al. [et al. [et al. [et al. [et al., which are also known as thermodynamic properties, included Minimum Free Energy, the most favorable thermodynamic RNA–RNA interaction energy, local pair probabilities and specific sgRNA heat parameters. In their CRISPRO experiments, Schoonenberg et al. [et al. include the PROVEAN  deletion score of the targeted amino acids 1 and 2, position in the gene, predicted disorder score of amino acids 1 and 2, GC content of the 20-mer guide, length of the targeted exon, and off-target score of the gRNA. Chen et al. [et al. [et al. [et al. [et al. [One-hot encoding and word embedding are not the only ways to represent a sequence in machine learning models. The majority of conventional machine learning (ML) models used in CRISPR-Cas9 consider additional sequence features such as position-specific features, different counts, and structural/thermodynamic characteristics to best capture sequence information . Positi [et al. . Rahman  [et al.  considerg et al.  considern et al.  used two [et al.  proposed [et al.  explored [et al.  used seqe et al.  demonstrIn conclusion, we need to point out that feature comparison between different experiential CRISPR-Cas9 data sets uncover substantial discordance and that further research is warranted to identify the most significant generic predictors (i.e. explanatory features) in case of both guide efficiency (i.e. on-target activity) and guide specificity (i.e. off-target effects).In this section, we present different conventional machine learning models for on- and off-target prediction found in the genome editing literature related to CRISPR/Cas9. The presentation is organized based on the target categories: (1) off-target prediction only, (2) on-target prediction only, and finally (3) both on- and off-target prediction. Within each category, the works follow chronological order.et al. [et al. [et al. opened new directions for data aggregation in the field of genome editing. Chen et al. [et al. [et al. demonstrated that the ensemble-based AdaBoost algorithm was able to outperform the other predictive algorithms in terms of the area under the precision recall curve (AUPRC) and the area under the receiver operating characteristic curve (AUROC) metrics. Lazzarotto et al. [in vitro activity. Their CHANGE-Seq model was designed to better understand the specificity of genome editors. In their experiments, Lazzarotto et al. used the encoded one-dimensional vectors to train a gradient tree boosting model to predict off-target activities. The authors highlighted the importance of the protospacer and the PAM position to ensure accurate off-target predictions. Moreover, they showed that CHANGE-Seq generally outperforms the well-known GUIDE-seq [First, we discuss papers dealing with off-target activity prediction. Abadi et al.  proposed [et al.  were amo [et al. , GUIDE-s [et al. , HTGTS [ [et al. , CIRCLE- [et al. , and mul [et al. . The autn et al.  generaten et al. , and rann et al. , as well [et al.  explored [et al. , MIT Web [et al. , CFD [54 [et al. , and Cro [et al. , as well [et al. , random  [et al. , SVM, an [et al. . In theio et al.  proposedUIDE-seq  and CIRCUIDE-seq  models.et al. [et al. [et al. [et al. [et al. [in silico predictions of sgRNA on-target activity. CRISPRpred is capable to extract relevant features in order to use them in an SVM-based machine learning framework. The work of Rahman et al. emphasizes the importance of feature engineering in boosting the predictive performance of sgRNA on-target prediction models. Furthermore, Rafid et al. [et al. [in silico design of sgRNAs. Konstantakos et al. [et al. evaluated the performance of CRISPRedict, comparing it with state-of-the-art gRNAs design tools, including some deep learning models, and concluded that despite its simplicity, CRISPRedict provides interpretable efficiency predictions with comparable performance. Zarate et al. [Second, we present a summary of recent papers addressing the problem of machine learning prediction of on-target activities. Wang et al.  were amo [et al.  trained  [et al.  applied  [et al.  investig [et al.  introducd et al.  demonstrd et al.  based on [et al.  proposeds et al.  introduce et al.  developeet al. [et al. [Finally, we discuss papers addressing the problem of both on- and off-target activity prediction by means of conventional machine learning models. Doench et al.  designed [et al.  proposedAlthough the use of traditional machine learning algorithms, whose main advantages are their relative simplicity and fast training, led to some impressive on- and off-target activity prediction results, recent studies conducted using deep learning methods (see the next section) often demonstrated a superior performance. We are convinced that, in general, deep learning models are better suited for both on- and off-target activity prediction than conventional machine learning models, since modern CRISPR/Cas9 data sets contain hundreds of thousands, and sometimes millions, of samples, and state-of-the-art deep learning algorithms can be effectively used on such huge volumes of data encompassing complex non-linear patterns. However, for benchmark purpose, the results provided by deep learning algorithms should be always compared with those yielded by some well-performing traditional machine learning methods such as SVM, random forest and XGBoost, as well as their ensemble frameworks, which are capable of increasing the accuracy of individual methods.Table Deep learning applications across all research fields have recently gained popularity due to easier access to data, boosted computing power, and recent theoretical progress in supervised learning. DNNs are at the core of deep learning. They are capable of learning complex patterns from the data using multiple layers of interconnected neurons. Nonetheless, their training and optimization are still very challenging problems. This section is divided into two parts. First, we present the main properties of existing deep learning network architectures. Second, we discuss their applications in CRISPR/Cas9.In this section, we describe succinctly the three main types of DNNs used for on- and off-target activity prediction in CRISPR/Cas9. They are FNNs, CNNs, and RNNs. Figure FNNs are one of the most popular DNN models . In a stet al. [CNNs, introduced by Lecun et al. , rely, aet al. ).RNNs are another type of neural networks that can be used effectively for on- and off-target prediction in CRISPR/Cas9. In contrast to FNNs and CNNs, the information in RNNs does not always move forward; it can also move backward. The aim of RNNs is to replicate a memory process through a recurrent learning mechanism. RNNs aggregate information of the past inputs and that of the current input in order to produce the current output. An example of an RNN architecture used for off-target prediction in CRISPR/Cas9 is presented in Figure  follows : where For all deep learning network architectures presented here, the neurons rely on an activation function that is used to determine whether a given neuron should be activated or not. Thus, the activation function being used determines whether the neuron’s contribution to the network is important or not in the prediction process. An activation function allows the model to transform the weighted sum of the neuron’s input signals into an output signal. Table DNNs, by their nature, have a large number of parameters to fine-tune during their training. Special attention must be given to the problem of overfitting in the model’s training. Overfitting occurs when the model learns patterns only present in the training set and cannot generalize its predictive performance on the test set. Different techniques exist to limit the overfitting while training DNNs. The most important of them are early-stopping, network-reduction, regularization, and dropout , 128. WeThis section includes four thematic subsections. First, we discuss the studies emphasizing the use of novel sequence encoding strategies along with deep learning models. Second, we present works using different feature engineering approaches. Third, we highlight the papers applying class rebalancing techniques prior to carrying out deep learning algorithms. Forth, we describe some recent works relying on the use of attention mechanism. Within each category, the works follow the off-target, on-target and both on- and off-target prediction order.et al. [et al. [et al. [et al. [et al. [et al. compared the prediction performance of different FNN, CNN, and RNN network architectures  auto-encoder to learn the underlying representation of sgRNAs. The unsupervised deep representation learning approach was then used to transfer the encoded data to a hybrid DNN. The proposed network included a softmax activation function and an identity function for the classification and regression tasks, respectively. DeepCRISPR provided good prediction results in terms of both AUPRC and AUCROC, compared with the CFD prediction method [Concerning the prediction of both on- and off-targets, we need to mention the work of Chuai et al. , who wern method .Most of the aforementioned studies demonstrated impressive prediction results using novel sgRNA–DNA one-hot sequence encoding schemes combined with deep learning models. We think, however, that further prediction improvements will not anymore result from the use of sophisticated data encoding schemes, nor from the models complexity, but rather from an effective use of additional biological or physical features as well as from the application of the feature engineering and class rebalancing techniques.In addition to conventional nucleotide sequence data, some plausible biological and physical features, such as gene melting temperature, molecular weight or microhomology features, can be also used as input of CRISPR/Cas9 predictive models. Moreover, some new informative features can be created, or reengineered, from the existing ones. Feature engineering has proven to be an important milestone when developing and optimizing predictive models .et al. [et al. [et al. demonstrated the importance of both the sequence context and the chromatin accessibility for effective cleavage prediction. Niu et. al. [et al. [As regards the off-target prediction, Liu et al.  introducet al.  to extra [et al.  introduc [et al. . In thei et. al.  proposednd ReLUs  in the c [et al.  introducet al. [et al. [et al. demonstrated that the RNN that received as input the sequence data with added biological features was able to outperform the other competing models. Their second best-performing model was the RNN that received as input sequence data only. Furthermore, Wang et al. developed a DeepHF website to ease the access to WT-SpCas9, eSpCas9(1.1), and SpCas9-HF1 indel data. The authors also applied SHAP with XGBoost and RNNs to assess the feature importance of the sequence input. Moreover, they used the Deep SHAP algorithm [et al., we can conclude that additional biological information and plausible reengineered features increase the predictive performance of deep learning models, leaving opportunities for future model enhancement. Elkayam and Orenstein [et al. [Concerning the on-target prediction, Shrawgi et al.  introduc [et al. , who com [et al.  that com [et al.  with thelgorithm  to estimrenstein  proposedrenstein . It is b [et al. .et al. [Finally, the CRISPRon and CRISPRoff deep learning models, based on CNN and gradient boosting regression trees, and the related interactive webserver were developed by Xiang et al. . For theFeature engineering has proven to be essential in many research fields involving machine learning predictions. We think that this aspect has not yet been fully explored in CRISPR/Cas9 and that future investigations should contribute to higher classification performance of both traditional machine learning and deep learning models applied to predict on- and off-target activities.Training deep learning models with real-world CRISPR-Cas9 data is challenging because of a large natural imbalance existing between positive and negative samples. This leads to an imbalanced data classification problem with a much larger majority class and a much smaller minority class. Thus, the predictive models observe and learn more from samples of the majority class and, as a consequence, can fail to identify accurately samples from the minority class. This can negatively impact their overall predictive performance. In CRISPR/Cas9, the problem of data imbalance mainly applies to the task of off-target prediction.et al. [in vitro and cell-based experiments) to increase the size of the positive class samples (i.e. off-targets), and thus improve the model’s competency. Precisely, Zhang et al. proposed to increase synthetically the number of positive samples by rotating the sgRNA–DNA encoded images by 90, 180, and 270 degrees, respectively. Hence, the number of positive samples in the data set was quadrupled. The data were then passed to a four-layer CNN to perform the off-target prediction. The main finding of this work was that data augmentation was a critical step for improving the predictive performance of DL-CRISPR. Class rebalancing and data augmentation are still fast evolving domains. Alleviating data imbalance should boost the predictive performance of off-target prediction models as class imbalance remains one of the intrinsic properties of CRISPR-Cas9 data.In this context, we need to mention the work of Zhang et al.  and theiRecent progress in the development of deep learning models using attention mechanism  has triget al. [et al. [et al. [et al. introduced a third model, called seqCrispr, which harbors an LSTM component and a CNN component in parallel to provide accurate on-target efficiency predictions. Finally, Zhang et al. [Recently, Zhang et al.  have imp [et al.  designed [et al.  analyzedg et al.  proposedWe are confident that future promising methods involving attention mechanism and deep learning should soon emerge in the field.Table AI methods have emerged as state-of-the-art approach in the field of genome editing. We reviewed recent applications of traditional machine learning and deep learning algorithms for prediction of on- and off-target activity in CRISPR/Cas9. We believe that our review paper can serve as a guideline for CRISPR/Cas9 practitioners willing to apply AI methods in genome editing.The main conclusions of our study are as follows:First, we highlighted the importance of sequence encoding for sgRNA–DNA on- and off-target prediction. Initial models implied straightforward one-hot sequence encoding of the sgRNA–DNA sequence pairs . SubsequSecond, some recent works have demonstrated that the ensemble ML methods have generally outperformed non-ensemble ML methods , 95. ForThird, recent studies have highlighted the importance of feature selection and feature engineering for accurate activity prediction in CRISPR/Cas9. New methodologies have been introduced to incorporate sequence information, such as gene melting temperature, molecular weight, or microhomology features, into the model's input . Some woFourth, we observed that most of publicly available data sets have incomparable numbers of positive and negative samples, thus leading to a class imbalance problem that has a negative impact on the performance of both traditional ML and DL models, especially when predicting off-targets. Recent papers propose to use data augmentation to increase the number of samples of the minority class  or to apFifth, for sufficiently large data sets, DNNs have demonstrated their superior predictive performance in comparison with both scoring methods and conventional ML algorithms such as SVM, random forest, and XGBoost , 73–75. Sixth, attention-based DL models have been extensively used in some recent works in the field , 138. ThResearch gaps and future research directions related to the application of AI methods in genome editing include:et al. and Charlier et al. [Powerful deep learning models have a huge number of parameters that need to be tuned in the training process. Thus, to be effective, these models require large amounts of input data. Transfer learning started to be used in the field to leverage the problem of deep learning training on data sets with insufficient amount of training samples. For instance, Lin r et al. , 75 haveDNNs usually stack several layers of different types, each of which often containing dozens of neurons. Thus, designing efficient deep learning network architectures and finding optimal sets of hyperparameters remain extremely important and challenging tasks , 161. VaExplainability and interpretability of DNNs have been a topic of interest for the past few years , 162. ReAs we pointed out, some recent works in the field have focused on the use of features engineering to boost the predictive performance of machine learning models , 84. Infet al. [Uncertainty quantification is a key technique to improve the trustworthiness of predictions made by a trained network. This technique has become popular for evaluating uncertainty in various research fields . There aet al.  have recActive learning is a semi-supervised technique in which a learning algorithm is used to label unlabeled data. An active learning algorithm uses an initial subset of labeled data for training. The algorithm then predicts the most appropriate labels for unlabeled data. This technique is of particular interest in biology because obtaining labeled data is often costly and time-consuming . Active We reviewed current knowledge regarding the use of supervised machine learning methods for on- and off-target prediction in CRISPR/Cas9.We highlighted the importance of the data pre-processing step including encoding of the sgRNA–DNA sequence pairs without any information loss, embedding supplementary data with different channels reflecting insertions, deletions and mismatches, and considering some additional sequence information such as gene melting temperature, molecular weight, or microhomology features.Most of CRISPR/Cas9 data sets have incomparable numbers of positive and negative samples, thus leading to a class imbalance situation that should be mitigated using either data augmentation or data re-sampling techniques, especially in the case of off-target prediction.When training data sets were large enough, DNNs have demonstrated their superior predictive performance in comparison with scoring methods and traditional machine learning algorithms. However, for benchmark purpose, the results obtained using state-of-the-art deep learning methods should be compared with those provided by some effective conventional machine learning algorithms, such as SVM, random forest, and XGBoost.We emphasized the importance of feature selection for accurate on- and off-target prediction in CRISPR/Cas9. Thus, the automated feature learning and automated feature engineering techniques should be used to boost the performance of deep learning models.


Feedback oscillators, consisting of an amplifier whose output is partially fed back to its input, provide stable references for standardization and synchronization. Notably, the laser is such an oscillator whose performance can be limited by quantum fluctuations. The resulting frequency instability, quantified by the Schawlow-Townes formula, sets a limit to laser linewidth. Here, we show that the Schawlow-Townes formula applies universally to feedback oscillators beyond lasers. This is because it arises from quantum noise added by the amplifier and out-coupler in the feedback loop. Tracing the precise origin of quantum noise in an oscillator informs techniques to systematically evade it: we show how squeezing and entanglement can enable sub-Schawlow-Townes linewidth feedback oscillators. Our analysis clarifies the quantum limits to the stability of feedback oscillators in general, derives a standard quantum limit (SQL) for all such devices, and quantifies the efficacy of quantum strategies in realizing sub-SQL oscillators. Feedback oscillators are a fundamental tool in science and engineering. Here, Loughlin and Sudhir provide a generalized Schawlow-Townes-like formula for quantum-limited feedback oscillators, thus giving a general model to study the fundamental output noise of these devices and techniques to reduce their noise further. Frequency fluctuations of oscillators limit measurements of distance—using a radar6, lidar7, or for gravitational-wave detection10—and of time using optical atomic clocks11. The performance of information processors—classical14 or quantum16—and communication systems19 are also limited by the stability of their clocks. So is our sensitivity to new physics—at low energy20 or using high-energy colliders21; and of our planet23, or the universe25. Even modern economic practice is beholden to the ticks of a clock26.Feedback oscillators28. Its frequency noise is given by the (modified) Schawlow-Townes (ST) formula33,0. Here, P0 = ℏΩ0∣α∣2 is the oscillator’s mean output power and α is the amplitude of the mean photon flux. In a laser, the feedback element is a cavity, whose round-trip time is τ, average thermal occupation is η, equivalent to the cavity linewidth 34, nth = 0 .A paradigmatic example of a feedback oscillator whose stability is well understood is the laser37 to oscillators, and establish a bridge between the classical electronic theory of feedback oscillators and the quantum electronics of a laser. Based on these insights, we then consider methods of evading the ST bound by manipulating the oscillators’ quantum states. We find that squeezing, entanglement, and phase-sensitive amplification in feedback oscillators allow us to achieve stability beyond the Schawlow-Townes limit, providing a new avenue toward realizing ultra-stable oscillators.In this work, we show that the Schawlow-Townes limit applies to a much larger class of feedback oscillators: those that are constructed by positive feedback of the output of a broadband amplifier. In doing so, we precisely identify the origin of the ST limit, extend the well-known theory of quantum noise in amplifiersG embedded in a feedback loop of time delay τ, whose output, because of the no-cloning theorem38, is extracted from the loop using a beam-splitter. Clearly, the mode We consider, in Fig. 37 is that the classical input-output relation i, j ∈ {in, out})39.In the absence of the feedback loop, the observation of Haus-Caves41. We will now resolve these issues and show that a proper account of the saturation behavior leads to the threshold condition for oscillation: “gain = loss”, while a proper imposition of the commutation relation gives the correct quantum noise of the oscillator.These observations do not apply when the feedback loop is closed. First, positive feedback will lead to a large mean field at the input of the amplifier, which will saturate its output due to the intrinsic nonlinearity of the amplifier. Note that nonlinearity in the response is a fundamental requirement of any physical amplifier since its gain arises from an external source whose energy density has to be finite; it is precisely this nonlinearity that will determine the amplitude of the oscillating output field (as discussed below). Second, the commutation relations in eq.  only appi ∈ {in, out}) is the mean amplitude, and G0, such that this gain is larger than the loss via the out-coupler, parametrized by x for which The nonlinear input-output behavior of the (memoryless) amplifier can be cast as42. Small signal analysis shows that this oscillatory state is stable if the in-loop field amplitude, αss, satisfiesThe output of the amplifier ns, eqs.  and 6),,\documenG is the linear gain of the amplifier, the ancillary mode Around the steady state, the (Heisenberg-picture) operators representing the quantum fluctuations satisfy the set of linear equations:η < 1) need not coincide with those in open-loop operation (η = 0)43. In the closed-loop configuration, since the ancillary mode is not freely propagating, it need not satisfy the commutation relations π/τ (for some integer n) at which these spectra are singular are the poles of the transfer functions H0, HG, and are therefore the frequencies of steady-state oscillations of the closed loop. Physically, they correspond to constructive interference after subsequent traversals of the feedback loop. In a practical oscillator, the gain G[Ω] is typically engineered to sustain only one steady state at, say, frequency Ω0 as in Fig. ω from the carrier, i.e., Ω = Ω0 + ω where ωτ ≪ 1, the phase quadrature spectrum in eq. . Thus the frequency spectrum of the output around the carrier is η ≈ 1, in which case eqs.  ≤ ∣z∣ for any complex number z. In the first line of eq. . Gaussian state techniques allow the resulting output spectra to be derived  amplifiers and linear out-couplers but with the injection of squeezed or entangled states into the feedback loop. Recent workAn alternative method of reducing the oscillator’s output amplitude and phase spectra is to modify the feedback loop itself by replacing the phase-insensitive amplifier with a phase-sensitive amplifier as depicted in Fig. G = 1 in eq. (In the extreme case where the in-loop amplifier is purely phase-sensitive (1 in eq. ), i.e., 52 and pump noise in the squeezer will contaminate its output, precluding the zero phase noise around the carrier predicted by eq.  We define the cross-correlation between the two general (not necessarily hermitian and not necessarily commuting) operators ion [eq. ] of the For weak-stationary operators, i.e., those that pair-wise satisfy The spectrum of a weak-stationary (but not necessarily hermitian) operator is positive; i.e.,pi ≥ 0, ∑ipi = 1, and 〈ψi∣ψj〉 = δij, the expectation value may be written as,Since ary, eq. , togetheary, eq. , impliesc,(Uncertainty principle for spectra) For weak-stationary hermitian operators αj, it requires that the eigenvalues of the matrix with elements Let used eq. . SplittiConsider now the case where Equation  then reqives eq. . □αi} are classical field amplitudes and α0(t) sources the classical field that circulates in the loop. In reality, the input represented by α0 is pure noise, in the ideal case, just vacuum noise.Here, we analyze the classical steady-state behavior of the saturating feedback oscillator shown in Fig. α0 represents infinitesimally small fluctuations. In order to understand how the loop starts, consider that α0(0) is a small random value and zero for t > 0. Let this produce a small amplitude α0(0) produces a steady state amplitude, it must satisfyαss of eq.  is satisfied. We haveSimilarly, we can show that the operating point where all amplitudes are zero is unstable. That is, any initial fluctuation will drive the loop to its steady state eq. . Now, leαss and a linearized gain ofIn sum, the loop will reach a steady state with a large circulating classical field with amplitude αss circulating in the loop. This oscillating field is the fundamental feature of a positive feedback oscillator. Second, at this equilibrium point, the gain medium is linear for small perturbations around the large field, with a gain given by the requirement for the system to conserve round-trip power in steady-state, i.e., We emphasize two key points of this classical analysis of saturating feedback oscillators. First, any small initial fluctuation will be amplified, eventually reaching a stable equilibrium with a large-amplitude output field G is coupled back into its input after attenuation by a factor τ. The remaining fraction of the signal is coupled out of the loop to derive the out-of-loop field aout.Consider the positive feedback amplifier configuration shown in Fig. 37.The equations of motion for the system are obtained by going around the loop in Fig. H0,G are the corresponding linear response transfer functions. In steady state, η and the delay τ; soEquation  can be sIn this section, we show that a quantum-limited phase-sensitive amplifier can always be decomposed into an ideal phase-insensitive amplifier followed by an ideal single-mode squeezer.Consider a phase-sensitive amplifier that amplifies the input see Fig. A. We can prove this by explicit computation. In the case of the phase-sensitive amplifier, the output mode The primary benefit of this decomposition is that it clarifies the extent to which a phase-sensitive amplifier must add noise: only the phase-insensitive component in its decomposition adds noise, while the squeezer is noiseless.Supplementary InformationPeer Review File


Nature Communications 10.1038/s41467-022-32838-4, published online 19 September 2022Correction to: The original version of this article contained an error in Fig. 2E, in which the labels on the x-axis of “Leukocyte proliferation regulation” and “T cell proliferation regulation” were inadvertently swapped. This has been corrected in both the PDF and HTML versions of the article.The original version of the Updated Supplementary Information


Nature Communications 10.1038/s41467-023-37648-w, published online 18 April 2023Correction to: The original version of this article contained an error in the spelling of the author ‘Joseph N. Paulson’, which was incorrectly given as ‘Joe Paulson’.y axis was labeled ‘Cum Progression-free Survival’ rather than ‘Cum Survival’.The original version of this article contained an error in Figure 1d, in which the These errors have now been corrected in both the PDF and HTML versions of the article.


In this article, we propose adjusting this minimum Bayes factor with the information to approximate an exact Bayes factor, not only when p is a p-value but also when p is a pseudo-p-value. Additionally, we develop a version of the adjustment for linear models using the recent refinement of the Prior-Based BIC.Minimum Bayes factors are commonly used to transform two-sided  So p-values, and particularly the threshold of p-value. It is known that the evidence of a p-value against a point null hypothesis depends on the sample size. In [p-values in linear models and propose new monotonic minimum Bayes factors that depend on the sample size and converge to p-value increases with decreasing sample size. There are several proposals in the literature, and most do not depend on the sample size, while those that do continue to be Robust Lower Bounds; however, neither behaves like a real Bayes factor. In this article, we propose to adjust the Robust Lower Bound p-values are only pseudo-p-values [p-values. The effect of adjusting this minimum Bayes factor with the sample size is shown in a simulation in By now, it is well known by practitioners that p·log(p)  and (ii)α*=0.005 . These ssize. In , they cop-values  (p. 397)p-values using the p-value definition of [p-values in The outline of the article is as follows: In ition of  p. 397)ition of  and the ition of ; in bothition of p-values are well known to have Uniform; in [Under the null hypotheses, , 1); in  (p. 397)Definition 1. A p-value is a statistic satisfying xfor every sample point . Small values of give evidence that is true, where is some subset of the parameter space and is its complement. A p-value is valid if, for every and every p-values that are Uniformly Distributed in , that is,Based on this definition, we can say that there are valid Remark 1. We consider any valid p-value complying with  a pseudoThe “Robust Lower Bound”  has been appropriately chosen so that large values of Note that this calibration has already been proposed in . Another=1−p see .p-value under p-value.In contrast with Remark 1, if we consider lly α  α in (7)uated in  or (7), p in . Therefore, the approximate Bayes factor for this test calculated with 11) to prSince for any Bayes factor Equation , BFG theEquation , and BFLEquation .Consider comparing two normal means via the testSo, for Assuming priors:The Bayes factor ist-statistic with degrees of freedom On the other hand, considering +n2; see .r data points from each of the two normal distributions, K times. For all K simulations, K replicates, we test the hypotheses p-values lie between p-values would be considered sufficient to reject p-values obtained from samples where We now present a simulation that shows how our adjustment, or calibration, to p-values coming from samples, where p-values behaved K replicates. quations ), and whquations ) is usedp-value is a pseudo-p-value (see the example 8.3.30 in [This is an example where the .3.30 in ). Let S1p be the common value of p-value isUnder Remark 2. It does not seem to be simple to estimate the appropriate value in , in FiguIt is important to note that in Bayesian tests with a point null hypothesis, it is not possible to use continuous prior densities, because these distributions (as well as posterior distributions) will grant zero probability to So,Now, if we take p-value and not a pseudo-p-value.Consider comparing two nested linear models http://academic.uprm.edu/eacuna/datos.html (accessed on 13 January 2022). We want to predict the average mileage per gallon (denoted by mpg) of a set of As an example, we analyze a data set taken from , which cp-value of the F test is Through the Bayes factors Consider the following simple linear model Yi=1i·θ+mony. In , the incn grows and n is fixed and Here, we show through the posterior probabilities of the null hypothesis that the Bayes factor 1.p-value, which it should, but it is static, not a function of the sample size n. This limitation makes the bound of little use for moderate to large sample sizes, where it is arguably the correction to p-values more needed.Lower bounds have been an important development to give practitioners alternatives to classical testing with fixed 2.p-values, and sample size has a distinct advantage over other approximations, such as BIC, in that it is a valid approximation for any sample size.The approximation develops here as a function of 3.p-value Bayes factors of hypothesis tests. In this article, we extended the validity of the approximation for “pseudo-p-values,” which are ubiquitous in statistical practice. We hope that this development will give tools to the practice of statistics to make the posterior probability of hypotheses closer to everyday statistical practice, on which p-values  are calculated routinely. This allows an immediate and useful comparison between raw-p-values and (approximate) posterior odds.The (approximate) Bayes factors  and 11)11) are s